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2024arXiv240911012S

Supernova remnants SNRs have long been suspected to be the primary sources of Galactic cosmic rays. Over the past decades great strides have been made in the modelling of particle acceleration magnetic field amplification and escape from SNRs. Yet while many SNRs have been observed in nonthermal emission in radio Xrays and gammarays there is no evidence for any individual object contributing to the locally observed flux. Here we propose a particular spectral signature from individual remnants that is due to the energydependent escape from SNRs. For young and nearby sources we predict fluxes enhanced by tens of percent in narrow rigidity intervals given the percentlevel flux uncertainties of contemporary cosmicray data such features should be readily detectable. We model the spatial and temporal distribution of sources and the resulting distribution of fluxes with a Monte Carlo approach. The decision tree that we have trained on simulated data is able to discriminate with very high significance between the null hypothesis of a smooth distribution of sources and the scenario with a stochastic distribution of individual sources. We suggest that this cosmicray energydependent injection time CREDIT scenario be considered in experimental searches to identify individual SNRs as cosmicray sources.

2024-09-01T00:00:00Z

['2024arXiv240911012S', '10.48550/arXiv.2409.11012', 'arXiv:2409.11012']

['Astrophysics - High Energy Astrophysical Phenomena', 'Astrophysics - Astrophysics of Galaxies']

Investigating the CREDIT history of supernova remnants as cosmicray sources

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.11012.pdf

{'Investigating the CREDIT history of supernova remnants as cosmic-ray sources': 'Anton Stall, ∗ Chun Khai Loo, † and Philipp Mertsch ‡ Institute for Theoretical Particle Physics and Cosmology (TTK), RWTH Aachen University, 52056 Aachen, Germany (Dated: September 18, 2024) \nSupernova remnants (SNRs) have long been suspected to be the primary sources of Galactic cosmic rays. Over the past decades, great strides have been made in the modelling of particle acceleration, magnetic field amplification, and escape from SNRs. Yet, while many SNRs have been observed in non-thermal emission in radio, X-rays, and gamma-rays, there is no evidence for any individual object contributing to the locally observed flux. Here, we propose a particular spectral signature from individual remnants that is due to the energy-dependent escape from SNRs. For young and nearby sources, we predict fluxes enhanced by tens of percent in narrow rigidity intervals; given the percent-level flux uncertainties of contemporary cosmic-ray data, such features should be readily detectable. We model the spatial and temporal distribution of sources and the resulting distribution of fluxes with a Monte Carlo approach. The decision tree that we have trained on simulated data is able to discriminate with very high significance between the null hypothesis of a smooth distribution of sources and the scenario with a stochastic distribution of individual sources. We suggest that this cosmic-ray energy-dependent injection time (CREDIT) scenario be considered in experimental searches to identify individual SNRs as cosmic-ray sources.', 'INTRODUCTION': "Cosmic rays with energies up to at least E knee ≃ 3 PeV are usually assumed to be of Galactic origin [1-3]. Supernova remnants (SNRs) [4] have long been considered the prime candidate for the origin of these Galactic cosmic rays (GCRs) for a number of reasons: First, observations of SNRs oftentimes show power-law spectra, e.g. in radio [5], X-rays [6] or gamma-rays [7]. Second, diffusive shock acceleration (e.g. [8]) can operate at the SNRs' blast waves, thus providing a likely acceleration mechanism. Finally, the locally observed energy density of cosmic rays can be explained if on average ∼ 10 % of the shock kinetic energy is converted to GCRs [9]. Yet, observational evidence for SNRs to accelerate cosmic rays up to E knee is scarce [2]. \nFrom a modelling perspective, it is not clear how SNRs can accelerate particles to a few PeV either. Adopting shock speeds of O (10 4 ) km s -1 and a (turbulent) magnetic field strength typical of the interstellar medium, i.e. ∼ 1 µ G, it can be shown [10] that the maximum particle energy is of the order of 10 TeV which falls short of E knee by some three orders of magnitude. Attaining energies of E knee instead would require significant amplification of the turbulent magnetic field. This amplification is believed to be achieved thanks to the non-resonant hybrid instability (also known as Bell instability) [11], which is driven by the macroscopic current of escaping cosmic rays. \nIn the following, we lay out the concordance view of time-dependent acceleration at and escape from SNR shocks, e.g. [12-15]. The amplification of magnetic fields is time-dependent due to its dependence on environmental parameters, most importantly the shock speed. The non-resonant hybrid instability operates most efficiently at high shock speeds, which can be achieved in the ejecta- \ndominated phase of the SNR evolution. Consequently, the maximum energy, or better rigidity R max 1 , is expected to be attained at t Sed , the start of the SedovTaylor phase, when the amount of matter accumulated by the shock equals the ejecta mass. For t > t Sed , the shock speed decreases and so does the maximum rigidity R max ( t ). Only particles of rigidities R < R max ( t ) can be confined by the magnetised turbulence across the shock, and so particles of rigidity R max ( t ) escape at time t . The trend continues down to a rigidity R b (e.g. 10 TV), below which particles can be confined without magnetic field amplification. All particles with R < R b escape at the same time t life when the supernova remnant shock dissipates. One could parametrise this time-dependence of cosmic-ray escape with a source term, that is the number of GCRs released from a source per unit time and rigidity, of the form Q ( R , t ) ≡ δ ( t -t esc ( R )) Q R ( R ) where t esc ( R ) is the inverse function of R max ( t ) and Q R ( R ) is an arbitrary function of rigidity, but likely close to powerlaw form. This concordance scenario has been widely adopted in the literature and is also supported by numerical simulations [16, 17]. We will refer to this as the Cosmic Ray Energy-Dependent Injection Time (CREDIT) scenario. \nThe details of the cosmic-ray escape are usually not considered when trying to explain the locally observed GCR measurements. Instead, the rate of injection of GCRs into the interstellar medium is modelled as the product of a smooth source density n src ( ⃗r ) and a steady rate per unit rigidity ¯ Q ( R ). This rate can be considered the temporal average of the time-dependent source term \n¯ Q ( R ) ≡ ν ∫ ∞ 0 d t δ ( t -t esc ( R )) Q R ( R ) = νQ R ( R ) . (1) \nHere, ν is the Galactic rate of supernovae. \nIt is hence natural to ask if there are observable consequences of the CREDIT scenario for locally measured cosmic rays. If one adopts a distribution of sources that is spatially smooth and steady, as is usually assumed, e.g. in numerical finite difference codes (see, however, [18]) then there are none, that is observationally the timedependent escape scenario cannot be distinguished from a time-independent scenario. \nIn reality, SNRs are individual objects with spatial and temporal extents much smaller than the distances and times over which cosmic rays propagate. The smooth and steady assumption can be justified in situations where a large number of sources contribute to the flux of GCRs. In such a situation, the measured flux for a particular distribution of sources will differ little from the mean of an ensemble of such source distributions. This mean is identical to the prediction from the smooth source distribution due to the linearity of GCR transport. For instance, it is easy to estimate that for standard parameters of GCRs, the number of sources contributing to the flux at a 10 GV is about 100 times larger than at 100 TV, where only some tens of sources cause the bulk of the flux. This justifies the use of the smooth scenario at GV energies. \nThere are other situations, however, when the assumption is not justified. Examples include the transport of GCR electrons and positrons above a few hundred GeV [19, 20] or of GCRs below a GV [21]; in both cases, the ranges are limited by energy losses. Consequently, the spectrum becomes very sensitive to the actual distribution of sources. For certain realisations of the source distribution, this can lead to deviations from the prediction of a smooth model and might be observable. Given our limited knowledge of the true source distances and ages, we have to resort to Monte Carlo simulations in which the total flux is simulated for different realisations of sources drawn from a smooth distribution [16, 22-24]. We refer to models of this sort as stochastic models. \nIn the following, we argue that the CREDIT scenario put into the context of a stochastic model can give large and, more importantly, observable spectral features at TV rigidities. We illustrate this by computing the fluxes of GCR protons for a large ensemble of source distributions. We characterise the spectral features and employ machine learning to reliably discriminate the spectral fluctuations in a CREDIT scenario from the fluctuations due to statistical errors in a smooth model.", 'METHODOLOGY': "To study the influence of source stochasticity on the locally observed GCR spectrum, we consider a simplified model of the Galaxy. We assume that all sources lie in the Galactic disk ( z = 0) and cosmic rays diffuse through the Galactic halo that extends to z = ± H . We draw the source ages from a uniform distribution adopting the canonical supernova rate ν of 0 . 03 yr -1 [25, 26] and the source positions from an axisymmetric distribution just depending on galactocentric distance [27], assuming that the Sun is located at radius R ⊙ . \nWe will only consider GCR protons with rigidities above some GV such that inelastic collision and advection can be ignored. We also assume an isotropic diffusion coefficient that follows a power-law in rigidity κ ( R ) = κ 0 β ( R ) R δ 2 . So, the transport equation is \n∂ψ R ( t, x , R ) ∂t -κ ( R ) ∇ 2 ψ R ( t, x , R ) = Q ( R , x , t ) , (2) \nwhere ψ R ( t, x , R ) = dn/d R denotes the isotropic CR density ( n is the number density). It is related to the differential flux Φ R = ( d 4 n ) / ( d R dAdtd Ω) = v/ (4 π ) ψ R and to the phase-space density f = f ( x , p, t ) through ψ R = (4 πp 2 Ze ) /c f . \nWe assume that a single population of SNRs inject all GCR protons up to the knee rigidity R knee . In the CREDIT scenario, the particles' escape times are rigidity-dependent. This escape time t esc ( R ) = t 0 + ∆ t esc ( R ) depends on the time of the supernova explosion t 0 and the time it takes a proton of a certain rigidity to escape the source ∆ t esc ( R ). We assume that the latter follows a broken power-law (as indicated by [17]): \n∆ t esc ( R ) = t Sed ( R R knee ) a , a = ln ( t life /t Sed ) ln ( R b / R knee ) , (3) \nfor R > R b and t esc ( R ) = t life otherwise. Choosing R b = O (10 TV) is motivated by the maximum rigidity that can be confined in SNR environments according to [10]. Setting R b → ∞ lets all protons escape at the same time. We will refer to this scenario, which is usually considered as the standard injection in stochastic modelling [22-24], as burst-like injection. \nThe source injection term on the RHS of equation (2) for a single source at ( t i , x i ) is given by \nQ i ( R , x , t ) = Q R ( R ) δ (3) ( x -x i ) δ ( t -t esc ,i ( R )) , (4) \nwhere Q R ( R ) is the time-integrated source spectrum as described in equation (1) and is the same for all sources. Motivated by diffusive shock acceleration [8], it has a \nTABLE I. Fiducial values for the GCR proton spectrum \npower-law dependence on rigidity given by Q R ( R ) ∝ R -2 . 2 . Note that the deviation from the canonical R -2 spectrum can be explained as due to the relative speed of the scattering centres [28, 29] or shock obliquity [30]. As the source spectrum normalisation only enters as an overall factor and we will only be interested in relative deviations of stochastic proton spectra from the ensemble mean, we do not need to fix it in this study. \nWe solve the transport equation (2) under the assumption of two free escape boundary conditions, i.e. ψ ( z = ± H ) = 0, using the method of mirror charges. We do not consider a radial boundary condition, as we assume the observer to be sufficiently far away from the edge of the Galaxy such that the escape is dominantly in the z-direction. The Green's function of equation (2), that is the solution for a single source, is given by \nG ( t, x ; t i , x i ) = Q R ( R ) (2 πσ 2 ) 3 / 2 e -( x i -x ) 2 2 σ 2 · ϑ ( z, σ 2 , z max ) (5) \nwhere σ 2 ( R , t ; t i ) = 2 κ ( R ) ( t -t esc ,i ( R )). The Green's function contains the power-law source spectrum, a Gaussian, and a correction function ϑ , which accounts for the free escape boundary condition. This function is an infinite sum related to the Jacobi theta function [22]. It has values in [0 , 1] and gets much smaller than 1 if z ≈ ± H or if the diffusion length ( ≈ σ ) is comparable with H . We exclude source injections that lie outside the observer's past light-cone to remedy potential non-causal solutions of the diffusion equation [23]. \nPredictions for the local spectrum of GCR protons can be obtained by summing the Green's functions of tens of millions of sources, whose positions and ages are sampled from the respective distributions described above. This embarrassingly parallel problem can be efficiently solved on graphical processing units (GPUs). To this end, we have used the python package jax [31]. Pronounced jump-like features originating in the time-dependent escape can be seen in sizeable parts of the realisations. Locally measured GCR fluxes should thus reveal information about the escape of cosmic rays from sources.", 'RESULTS': "In the left panel of Fig. 1, we show a number of example spectra for a CREDIT scenario where we adopted the fiducial parameters presented in Tab. I. To better illustrate the fluctuations, we present the spectra for ten random realisations divided by the ensemble mean for R b = 1TV. One can see fluctuations around the average spectrum for all rigidities, but their character changes markedly at R b : For R < R b , the spectral features are very smooth and typically only extend a few percent above or below the average value. For R > R b instead, the spectral features are much more peaked, and the maxima of individual features can be as large as a few times the average spectrum. \nOf course, given the limited event statistics, experimental bins are typically wider than these spectral features. In the middle panel of Fig. 1, we, therefore, present the same example spectra as in the left panel but sampled only at the central rigidities of bins employed by experimental collaborations. Specifically, for rigidities below and above ∼ 2 TV, we have adopted the published rigidity bins of the AMS-02 and DAMPE collaborations, respectively. While the features in the DAMPE range are not as high and as nicely resolved as before, the most dramatic spectral features are still easy to make out by eye. We checked that integrating a finely resolved spectrum over rigidity bins gives similar statistics of the fluxes. \nStatistical errors also arise due to the finite number of registered events. For a large enough number of events per bin, the distribution of the measured fluxes will be Gaussian and the measured fluxes in different bins can be considered as independent. We use reported statistical errors of the calorimetric cosmic-ray experiments AMS-02 [32] and DAMPE [33] as standard deviations in the individual rigidity bins. We found that the relation σ stat [%] = 0 . 42 · ( R [TV]) 0 . 6 approximates the statistical errors of both experiments well. In the right panel of Fig. 1, we show the spectrum from the smooth model divided by the average, a flat line, plus ten random draws from a distribution of statistical errors only. Generally, the statistical fluctuations are much smaller than the expected features under the CREDIT scenario, rendering their detection possible. \nHowever, it is conceivable that statistical fluctuations can produce a flux similar to a realisation with less dramatic features. We trained a decision tree (DT) to discriminate between the null hypothesis of a smooth source distribution (plus statistical errors) and the alternative hypotheses of a CREDIT ( R b ∈ [1 , 23] TV) or a burstlike ( R b → ∞ ) scenario 3 . The DT aims to split the \nFIG. 1. GCR proton fluxes (normalised to the ensemble mean) predicted for the CREDIT scenario with R b = 1TV. The alternating white and grey vertical lines indicate the rigidity bins. Left panel: The coloured lines show 10 random realisations, adopting a resolution in rigidity much higher than currently realised by experiments. Middle panel: Same random realisation, but sampled at the centres of the rigidity bins employed by AMS-02 and DAMPE. Right panel: Fluxes in a smooth scenario plus statistical errors due to finite event numbers of the sizes published by AMS-02 and DAMPE. \n<!-- image --> \nFIG. 2. Confusion matrices (normalised to rows/ true labels for the percentages) with 20 distinct R b ∈ [1 , 23] TV for the CREDIT model \n<!-- image --> \ndata, the flux value in each rigidity bin, scattered in 81dimensional space, into hypercuboids that contain realisations of the same label [34]. This is done by continuously splitting one hypercuboid (and the dataset) into two smaller hypercuboids. This can be thought of as an if-statement, where if the value of a realisation at the bin is smaller than a fixed value, it is in one hypercube and otherwise in the other. Making a prediction for a new data point is just travelling through a number of ifstatements and obtaining the label at the end. We used \nthe sklearn package [35] to train and validate our DT. No tuning of hyperparameters was required to improve the results. We estimate the significance of our hypothesis test and its power through the so-called confusion matrix. The confusion matrix shows how often an element of the validation set with a true label is classified with a predicted label. A diagonal matrix corresponds to a perfect classification. Using 10-fold cross-validation on our 4 · 10 6 realisations for each scenario, we find an almost diagonal confusion matrix, Fig. 2, which shows the strength of the DT to reliably distinguish the three models. \nWe checked for the effect of the uncorrelated part of the systematic error by adding in quadrature an uncorrelated, relative systematic error of 1% to the statistical errors [36, 37]; the classifier's performance did not change significantly. We also considered a relaxation of the peaked injection of each rigidity at a single point in time and found that smoothing the injection by a Gaussian kernel with a conservative width of 1 kyr does not alter the results for R b ≳ 1 TV.", 'SUMMARY AND CONCLUSION': "We considered a stochastic source model for simulating the GCR proton spectrum. At rigidities above some TV, the spectrum is determined by relatively few sources. We discussed that rigidity-dependent escape times of protons from their sources (CREDIT scenario) enhance the observability of single source contributions in the spectrum. Such an escape model, motivated by the same physical mechanisms that are usually invoked to explain parti- \ncle acceleration in Galactic SNRs beyond O (10 TV), predicts a high likelihood for source configurations that produce pronounced peaked features in the proton spectrum. The emergence of such peaks is characteristic of CREDIT models (see Fig. 1). Here, we used a decision tree to discriminate spectra produced by a CREDIT or burst-like source injection from a power-law spectrum and a smooth source distribution. The classifier works extremely well, reliably deciding whether a spectrum is compatible with a smooth source injection or a CREDIT/burst-like model. \nWe suggest applying our classifier to experimental data in the future. If the data get classified as CREDIT-like, one can determine the time of the supernova event t i for an observed spectral feature, if other model parameters are assumed, i.e. diffusion coefficient κ ( R ), break rigidity R b , Sedov time t Sed , and lifetime of sources t life . Alternatively, for a candidate supernova time t i , one can constrain a combination of κ ( R ), R b , t Sed , and t life that fit a detected spectral feature. Instead, if the data get classified as burst-like, this corresponds to a lower bound on R b , which informs models of acceleration. Finally, if the classifier finds a smooth model, this should trigger a reckoning with the SNR paradigm for GCRs. \nFor this, however, the cosmic-ray model parameters have to be chosen carefully such that no correlated spectral features attributed to the transport model emerge in the normalised spectra (like Fig. 1). This can be done using extensive fits like e.g. [38]. Deviations from the fiducial values given in Tab. I will change the spectral features, enhancing them whenever the effective number of contributing sources decreases (e.g. lower source rate, smaller halo height, or enhanced diffusion). \nProducing extensive Monte Carlo simulation samples using state-of-the-art programming paradigms opens up the possibility to train classifiers to reliably distinguish features of different models in data. We expect this to be a powerful tool to constrain GCR models. The possibilities of this will be explored in future work, but are already demonstrated by the classification power to discriminate CREDIT against burst-like models. \nAs the number of contributing sources falls with rigidity, extensions of high precision measurements beyond ≈ 100 TV can enhance the predictive power of the classifiers. One proposed space based direct detection experiment is AMS-100 [39] which proposedly will extend high precision proton measurements beyond the cosmicray knee. The leading high-precision direct detection experiment DAMPE is currently limited by statistics and suffers from increasingly limited power to discriminate proton and helium at rigidities beyond ≈ 100 TV [40]. The inclusion of helium spectra in our model and the use of the p+He spectra provided by DAMPE can also increase the predictive power of our model. In addition, an unbinned analysis would further improve the sensitivity to the spectral features. We postpone such a study to future work. \n- ∗ stall@physik.rwth-aachen.de\n- † khai.loo@rwth-aachen.de\n- ‡ pmertsch@physik.rwth-aachen.de\n- [1] S. Navas et al. (Particle Data Group), Phys. Rev. D 110 , 030001 (2024).\n- [2] S. Gabici, C. Evoli, D. Gaggero, P. Lipari, P. Mertsch, E. Orlando, A. Strong, and A. Vittino, Int. J. Mod. Phys. D 28 , 1930022 (2019), arXiv:1903.11584 [astro-ph.HE].\n- [3] M. Kachelriess and D. V. Semikoz, Prog. Part. Nucl. Phys. 109 , 103710 (2019), arXiv:1904.08160 [astroph.HE].\n- [4] S. P. 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2024arXiv240907639K

We establish the spacetime Penrose inequality in spherical symmetry in spacetime dimensions n1geq3 with charge and cosmological constant from the initial data perspective. We also show that this result extends to the GaussBonnet theory of gravity.

2024-09-01T00:00:00Z

['arXiv:2409.07639', '2024arXiv240907639K', '10.48550/arXiv.2409.07639']

['General Relativity and Quantum Cosmology', 'Mathematical Physics']

The Penrose inequality in spherical symmetry with charge and in GaussBonnet gravity

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.07639.pdf

{'THE PENROSE INEQUALITY IN SPHERICAL SYMMETRY WITH CHARGE AND IN GAUSS-BONNET GRAVITY': 'HARI K. KUNDURI, JUAN MARGALEF-BENTABOL, AND SARAH MUTH \nAbstract. We establish the spacetime Penrose inequality in spherical symmetry in spacetime dimensions n +1 ≥ 3 with charge and cosmological constant from the initial data perspective. We also show that this result extends to the Gauss-Bonnet theory of gravity.', '1. Introduction': 'The Penrose inequality [28] is a conjectured inequality which asserts that the mass of an asymptotically flat black hole spacetime is bounded below by a function of the area of a spatial cross-section of its event horizon (see the comprehensive review [27]). The bound is saturated if and only if the spacetime is isometric to the Schwarzschild solution. The conjecture is based upon heuristic arguments arising from our physical model of gravitational collapse (the weak cosmic censorship conjecture and final state conjecture [20]). A counterexample would cast doubt on at least one of these two central conjectures. Generalizations of the inequality that incorporate charge and angular momentum are expected to hold [9]. \nThe conjecture can be stated precisely in terms of initial data set ( M,g,k ) where ( M,g ) is a Riemannian manifold of dimension n and k is a symmetric rank-2 tensor. The triplet satisfies the appropriate constraint equations (see Equations (2.4) and (3.4) below) for some energy-momentum density µ, J , which encode the matter fields present in the spacetime. They are assumed to satisfy the dominant energy condition µ ≥ | J | g .', '1.1. Asymptotically flat case.': "The initial data set is asymptotically flat if there exists a compact set K ⊂ M and a diffeomorphism Φ : M \\ K → R n \\ B where B is a closed coordinate ball such that in the coordinates { x i } associated to Φ, \n(1.1) Φ ∗ g = δ + O 2 ( r -κ ) , Φ ∗ k = O 1 ( r -κ -1 ) , Φ ∗ µ, Φ ∗ J = O ( r -2 κ ) , \nfor some κ > ( n -2) / 2. We have the well-defined geometric invariants known as ADM energy and linear momenta which are defined by \nP i = 1 ( n -1) ω n -1 lim r →∞ ∫ S n -1 r ( k ij -(Tr g k ) g ij ) ν j vol( S n -1 r ) , (1.2b) \nE ADM = 1 2( n -1) ω n -1 lim r →∞ ∫ S n -1 r ( ∂ i g ij -∂ j g ii ) ν j vol( S n -1 r ) (1.2a) \nwhere ω n denotes the volume of the unit sphere S n , ν j denotes the unit-normal vector field to the n -sphere S n -1 r of radius r and vol( S n -1 r ) = r n -1 vol( S n -1 ). The ADM mass m is then defined as m = √ E 2 -| P | 2 where P is the momentum n -vector associated with the initial data set. By the \npositive mass theorem, m ≥ 0 with equality if and only if the initial data can be embedded into Minkowski spacetime [29, 30, 32]. A quasi-local-in-time proxy for the horizon of a black hole is a marginally outer trapped surface (MOTS). This is a hypersurface Σ ⊂ M (co-dimension 2 in spacetime) which has the property that the outward null expansion θ + vanishes, where the outward and inward null expansions θ ± are given by \n(1.3) θ ± = H Σ ± Tr Σ k, \nwhere H Σ denotes the mean curvature of Σ within M . The notion of a MOTS captures the intuitive idea that outward-directed light rays cannot escape Σ. The outermost MOTS (often referred to as an apparent horizon) is defined to be a MOTS that is not enclosed within any other MOTS. The Penrose conjecture can then be stated as \n(1.4) m ≥ 1 2 ( A ω n -1 ) n -2 n -1 . \nHere A is understood to be the smallest area required to enclose the apparent horizon. In the present work, as we will explain in detail later, A may be identified with the area of the apparent horizon itself. Here equality is to be achieved if and only if the initial data is isometric to a spatial hypersurface in the Schwarzschild spacetime in dimension n +1. \n/negationslash \nThe conjecture has been rigorously established in n = 3 for time-symmetric ( k = 0) initial data sets (this is called the 'Riemannian-Penrose inequality') in the seminal work of Huisken and Ilmanen [16] and Bray [4] using inverse mean curvature flow and conformal flow respectively. These results were subsequently extended to n ≤ 7 by Bray-Lee [3]. For k = 0, the problem is much more difficult. Progress has been made in the spherically symmetric setting (for a clear exposition, see [22, Theorem 7.46]) and more recently for cohomogeneity-one initial data [18, Theorem 1.1] using the generalized Jang equation approach developed by Bray-Khuri [2]. In all these cases, there is an associated rigidity statement, namely that the equality is satisfied if and only if the initial data set is isometric to Schwarzschild initial data.", '1.2. Asymptotically AdS case.': 'In the presence of a negative cosmological constant, it is natural to consider spacetimes that are asymptotically Anti-de Sitter (AdS n +1 ). The initial data ( M,g,k ) associated with spacelike hypersurfaces of asymptotically AdS spacetimes are asymptotically hyperbolic. More precisely, let ( H n , b ) denote R n with the hyperbolic metric \n(1.5) b = d r 2 V hyp + r 2 g S . \nwhere V hyp ( r ) := 1 + r 2 . The metric, scaled so that Ric( b ) = -( n -1) b , models constant time spacelike hypersurfaces in asymptoticallyAdS n +1 with spacetime metric g = -V hyp d t 2 + b and negative cosmological constant Λ = -n ( n -1) / 2 (equivalent to fixing the AdS length scale to /lscript = 1). \nAn initial data set ( M,g,k ) is asymptotically hyperbolic if there is a compact set K ⊂ M and a diffeomorphism Φ : M n \\ K → H n \\ B where B is a closed coordinate ball such that in the asymptotic chart \n(1.6) ˆ g = Φ ∗ g -b = O 2 ( r -κ ) , Φ ∗ k = O 1 ( r -κ ) , Φ ∗ µ, Φ ∗ J = O ( r -2 κ ) , \nTHE PENROSE INEQUALITY IN SPHERICAL SYMMETRY WITH CHARGE AND IN GAUSS-BONNET GRAVITY 3 \nfor κ > n/ 2. A well-defined notion of energy valid in this setting is the Chrusciel-Nagy energy [7], which relies on the so-called static potential (or static lapse function) W := √ V hyp : \n(1.7) E hyp = 1 2( n -1) ω n -1 lim r →∞ ∫ S n -1 r [ W div b ˆ g -W d(Tr b ˆ g ) + (Tr b ˆ g )d W -ˆ g ( D W, )]( V )vol( S n -1 r ) , \nwhere V i = W∂ i r is the unitary outward b -normal vector to the coordinate spheres { r = r 0 } and D the b -Levi-Civita connection. \nAn extension of (1.4) valid in the AdS setting is expected to hold, which is of particular interest from the holographic perspective [12]. The inequality has been established in spherical symmetry ([11, Theorem 6], [14, Theorem 1], [17]) with additional various hypotheses. We will recover these results from the initial data perspective along the lines of [22]. More generally, the Penrose inequality in the asymptotically hyperbolic setting has been established for small time symmetric perturbations of Schwarzschild-AdS data [1] and for graphs [8]. The spacetime Penrose inequality for the wider class of cohomogeneity-one initial data has also been established using the Jang equation approach [18, Theorem 1.2]. \nIn this note, we will focus on establishing two extensions of the spacetime Penrose inequality in spherical symmetry: the Einstein-Maxwell theory and the Gauss-Bonnet theory.', '1.3. Einstein Maxwell initial data.': 'Consider charged initial data sets ( M,g,k,E,B ) in n dimensions, which satisfy the constraints arising from Einstein-Maxwell theory with additional (uncharged) matter sources. Here ( E,B ) denote the electric field one-form and magnetic field n -2 form (see Section 2 for further details). Under appropriate fall-off conditions on the electric field, we define the electric charge (both in the asymptotically flat and asymptotically AdS) as \n(1.8) q = ( ( n -2)( n -1) 2 ) -1 / 2 1 ω n -1 lim r →∞ ∫ S n -1 r E ( ν ) vol( S n -1 r ) . \n/negationslash \nConsider the asymptotically flat setting in dimension n = 3. The charged Riemannian-Penrose inequality ( k = 0 but without any symmetry assumptions, and for multiple black holes) has been rigorously established [19]. The spacetime with k = 0 case in spherical symmetry was established in [10] using a Jang equation approach. In higher dimensions n > 3, the charged RiemannianPenrose inequality was proved for Einstein-Maxwell initial data sets, which could be embedded as a hypersurface in Euclidean space R n +1 [23]. We prove \nTheorem 1.1. Consider a spherically symmetric charged asymptotically flat/hyperbolic initial data set ( M,g,k,E,B ) of the Einstein-Maxwell equations with uncharged matter sources (with a negative cosmological constant in the latter case) satisfying the constraint equations (2.4) and the dominant energy condition. Then \n(1.9) E ≥ 1 2 [ ( A ω n -1 ) n -2 n -1 + q 2 ( ω n -1 A ) n -2 n -1 + 1 /lscript 2 ( A ω n -1 ) n n -1 ] , \nwhere E is the ADM mass (asymptotically flat case, /lscript → ∞ ) or the Chrusciel-Nagy energy E hyp (asymptotically hyperbolic case, /lscript = 1 ) respectively, q the charge (1.8) , and A the area of the apparent horizon. Moreover, equality holds if and only if the initial data can be isometrically embedded as a spatial hypersurface in the Reissner-Nordstrom(-AdS n +1 ) spacetime of mass E and charge q . \nHere, by spherically symmetric, we mean that M is diffeomorphic to [0 , ∞ ) × S n -1 with boundary ∂M that is an outermost MOTS (see subsection 2.3 below). The asymptotically flat case of this result was proved in [15] under the assumption that the initial data set is maximal (Tr k = 0).', '1.4. Gauss-Bonnet gravity.': "Our second result concerns the spacetime Penrose inequality in spherical symmetry within the setting of Einstein Gauss-Bonnet gravity (the reader is referred to the recent review [13] and references therein). There is a vast variety of proposed modifications and generalizations of general relativity, typically including additional fundamental fields and coupling parameters. A particular mathematically natural generalization is Einstein-Gauss-Bonnet theory, first proposed by Lanczos [21] and later generalized by Lovelock [24, 25]. The latter's family of theories does not introduce any new fields and is unique in preserving the property that the field equations only contain up to second derivatives of the metric tensor. In particular, Lovelock showed that in spacetime dimension 4, the Einstein-Hilbert functional is the unique choice which gives rise to a left-hand side of the field equations (i.e. not involving matter fields) that is symmetric, divergenceless, and involves no more than second derivatives of the metric. In dimensions 5 and 6, one may add an additional term L GB to the Einstein-Hilbert action consisting of a certain combination of scalar invariants while preserving these properties: \n(1.10) L GB := R 2 -4 R ab R ab + R abcd R abcd , \nwhere R , R ab , R abcd denote respectively the scalar curvature, Ricci tensor, and Riemann tensor of the spacetime metric g . (1.10) is referred to as the 'Gauss-Bonnet term'. It is a topological invariant in four spacetime dimensions and does not contribute to the equations of motion (its variation is a pure divergence). In dimensions greater than 6, Lovelock's theorem allows for additional possibilities beyond the Gauss-Bonnet term, but we will restrict attention to (1.10) in the present work. \nAn obvious question is whether the chain of arguments which led to the Penrose conjecture continues to hold within Gauss-Bonnet gravity. If so, one would expect an analogous geometric inequality to hold. We answer this in the affirmative within spherical symmetry: \nTheorem 1.2. Consider a spherically symmetric asymptotically flat or asymptotically hyperbolic initial data set ( M,g,k ) of the Einstein-Gauss-Bonnet equations (with a negative cosmological constant in the latter case) satisfying (3.4) with matter sources satisfying the dominant energy condition. Let ˜ α = ( n -2)( n -3) α where α is the Gauss-Bonnet coupling constant. Then \n(1.11) E ≥ 1 2 [ ( A ω n -1 ) n -2 n -1 + 1 /lscript 2 ( A ω n -1 ) n n -1 + ˜ α ( A ω n -1 ) n -4 n -1 ] , \nwhere E is the ADM mass in the asymptotically flat case ( /lscript → ∞ ) and the Chrusciel-Nagy energy E hyp in the asymptotically hyperbolic case (with length scale /lscript = (1 -˜ α ) -1 / 2 with ˜ α ∈ (0 , 1 / 2) ) respectively, and A is the area of the apparent horizon. Moreover, equality holds if and only if the initial data can be isometrically embedded as a spatial hypersurface in Schwarzschild(AdS n +1 ) Gauss-Bonnet spacetime of mass E (3.11) .", '2.1. Equations of motion.': 'Consider a spacetime ( M , g ) of dimension d = n +1 and the Einstein-Maxwell action \nTHE PENROSE INEQUALITY IN SPHERICAL SYMMETRY WITH CHARGE AND IN GAUSS-BONNET GRAVITY 5 \n(2.1) S ( g , A, φ ) = 1 16 π ∫ M ( R ( g ) + n ( n -1) /lscript 2 -2 | F | 2 g ) vol( g ) + ∫ M ˆ L m ( g , φ ) , \nwhere R ( g ) is the g -Ricci scalar, F := d A , | F | 2 g := 1 2! F ab F ab , φ denotes the collection of non-charged fields, and ˆ L m their Lagrangian. The equations of motion are \nG ab = 2 ( F ac F c b -1 2 | F | 2 g g ab ) + n ( n -1) 2 /lscript 2 g ab +8 π ˆ T ab , (2.2a) \nd /star g F = 0 , (2.2b) \nwhere G ab is the g -Einstein tensor, the term in parentheses of (2.2a) is the electromagnetic energymomentum tensor T ab and ˆ T ab is the energy-momentum tensor associated with the non-charged fields. Notice that we have also the trivial condition d F = 0 that follows from the definition of F .', '2.2. Initial Value Problem.': 'We now work out the constraint equations induced on a spacelike hypersurface M , relevant for the initial value formulation. We use { i, j, k . . . } abstract indices on M and denote ı : M ↪ → M the inclusion map (hence ı i a denotes the pullback/pushforward) which induces the metric g := ι ∗ g and its Levi-Civita covariant derivative D . We also have the future-pointing unit g -normal vector field n a over M . The orientation of M is given by vol( g ) := ı ∗ ι /vectorn vol( g ) (equivalently, with a slight abuse of notation, vol( g ) = -n ∧ vol( g )). It will be useful to introduce the following fields: \nContracting (2.2a) with n a n b and ı a i n b , and contracting (2.2b) with n a , leads to the constraints \n16 π ˆ µ = R ( g ) + (tr g k ) 2 -| k | 2 g -2 | E | 2 g -2 | B | 2 g + n ( n -1) /lscript 2 (2.4a) \n8 π ˆ J = div g ( k -(tr g k ) g ) + 2 /star g ( E ∧ B ) (2.4b) \ndiv g E = 0 ≡ D i E i = 0 , (2.4c) \nwhere | ω | 2 := ω a 1 ...a p ω a 1 ...a p /p ! denotes the norm of a p -form ω . Finally, from d F = 0, we obtain the last constraint \ndiv g B = 0 ≡ D i B ij 2 ··· j n -3 = 0 . (2.4d) \nThe Riemannian manifold ( M,g,k,E,B ) forms an Einstein-Maxwell initial data set if (2.4a)(2.4d) are satisfied for some (ˆ µ, ˆ J ).', '2.3. Spherically symmetric initial data set.': "We consider a spherically symmetric initial data set ( M,g,k,E,B ) satisfying (2.4). This symmetric condition forces several restrictions on all the elements of the initial data set. First, notice that M is a warped product I × ρ S n -1 for some interval I (we assume the initial data set has an apparent horizon boundary, which is invariant under the isometries of the ambient geometry, thus I = [ r + , ∞ ) for some r + > 0) and some positive function ρ : I → R + . That means that if we introduce a radial coordinate s , the metric can be written as \n(2.5) g = d s 2 + ρ ( s ) 2 g S . \nNotice that the pullback of g S to S n -1 (with abstract indices { A,B,C ... } ) is the standard round metric dΩ 2 . It is worth mentioning that the vector field ( ∂ s ) i = g ij (d s ) j is tangent to the unit radial geodesics. \nThe symmetric condition on the extrinsic curvature implies that there exist functions ˜ k I , ˜ k S : I → R such that \nFinally, the symmetric condition implies also that the electric 1-form field and the magnetic ( n -2)form field must be purely radial, so that E is given by \n(2.6) k = ˜ k I ( s )d s 2 + ˜ k S ( s ) n -1 ρ ( s ) 2 g S . \n(2.7) E = E I ( s )d s \nfor some function E I : I → R . Meanwhile, the magnetic field must vanish (unless n -2 = 1, in which case it is a 1-form, which also has to be radial), as we now argue. We first demonstrate the following lemma. \nLemma 2.1. There are no SO ( n ) -invariant ( n -2) -forms on S n -1 . \nProof. Suppose there was a non-vanishing SO ( n )-invariant ( n -2)-form α A 1 ...A n -2 . Then \nis a traceless symmetric SO ( n )-invariant (0 , 2) tensor on S n -1 (dΩ 2 is used to raise/lower the indices and compute the norm). Then, γ t := dΩ 2 + tβ is a 1-parameter family of SO ( n )-invariant metrics on S n -1 for sufficiently small | t | . However, up to scaling, there is a unique such metric, namely dΩ 2 . Thus, β (which is traceless) must be proportional to dΩ 2 (which is not traceless), and hence it must be zero, leading to \nβ AB = α AC 2 ··· C n -2 α C 2 ··· C n -2 B -| α | 2 S n -1 (dΩ 2 ) AB \n(2.8) α AC 2 ...C n -2 α BC 2 ...C n -2 = | α | 2 S n -1 δ B A . \n/negationslash \nSince α is non-vanishing, being a ( n -2)-form ('codimension 1'), it must have a nontrivial kernel. Thus, there exists V A = 0 such that α AC 2 ...C n -2 V A = 0. Plugging this into (2.8), shows that V A = 0, leading to a contradiction. /square \nLet X s : S n -1 ↪ → M be the embedding mapping of the unit sphere into the sphere of radius s in M . Define a := ι ∂ s B and b := B -d s ∧ a . We may then write B = d s ∧ a + b . Note that B and d s are both rotation-invariant, and hence so are a and b . Consider the pullback ˜ b = ( X s ) ∗ b ∈ Ω n -2 ( S n -1 ). By the above Lemma, ( X s ) ∗ b = 0 for all s and hence b = 0 since b has no d s term. Now consider µ := /star g B = /star g (d s ∧ a ) ∈ Ω 2 ( M ). It is clear that µ has no d s term, hence ι ∂ s µ = 0. Since d /star g B = 0, µ must be independent of s , so µ can be understood (with a slight abuse of notation) as a 2-form on \nthe sphere µ ∈ Ω 2 ( S n -1 ). This means that, in particular, µ is closed (and in fact exact for n > 3). Since µ can be locally expressed at a point p ∈ S n -1 as a square antisymmetric matrix, its rank must be even. Hence if n -1 is odd, µ p must have a non-trivial kernel, and a minor modification of the argument used in Lemma 2.1 implies µ p = 0. By rotational invariance it must vanish everywhere on the sphere. If n -1 > 2 is even, µ cannot have maximal rank either, for otherwise it would define a symplectic form (a closed, non-degenerate two form), which does not exist on even-dimensional spheres S n -1 with n > 3. Hence µ = 0 by Lemma 2.1. The only other case is n = 3, in which case B = B ( s )d s . Then E ∧ B = 0. By electromagnetic duality, we may consider a transformation into a duality frame in which B = 0. In summary, we will focus on the case with B = 0 in the following. \nRecall that the null expansions are given by θ ± = k S ± H , where S is a level set of r , k S coincides with the trace of k with respect to the induced metric ρ ( s ) 2 dΩ 2 on S (see (2.6)), and H is the mean curvature of the level set. The requirement that the level set be a MOTS is θ + = 0. The assumption that the apparent horizon boundary is an outermost MOTS is equivalent to imposing ρ ' ( s ) > 0. Furthermore, the outermost property implies that the outermost MOTS is also outer minimizing [22, Theorem 7.46]. We may thus use r := ρ ( s ) as a coordinate. Denoting ˜ V ( s ) := ρ ' ( s ) 2 and V ( r ) := V ( ρ -1 ( r )), the metric (2.5) and extrinsic curvature (2.6) read \nwhere k I ( r ) = ˜ k I ( ρ -1 ( r )) and k S ( r ) := ˜ k S ( ρ -1 ( r )). Observe that the outward unit normal to a surface of constant r is given by ν i = √ V ∂ i r , which implies that ν i = (d r ) i / √ V . In particular, its mean curvature is given by \n˜ (2.9) g = d r 2 V ( r ) + r 2 g S k = k I ( r ) d r 2 V ( r ) + k S ( r ) n -1 r 2 g S , \n(2.10) H = ( n -1) √ V ( r ) r .", '2.4. Definition of charged mass and its properties.': 'We generalize the elementary proof of the spacetime Penrose inequality in spherical symmetry outlined by [22]. The approach is to define a quasi-local mass (the Hawking mass) associated with spheres (level sets of an appropriate radial function defined on the initial data set) and then show that this quantity is non-decreasing along a radial flow and approaches the total energy in the designated asymptotic region. Motivated by Disconzi-Khuri [10], we modify the Hawking mass by adding a term involving a cosmological constant and the total charge of the system (see Equation (1.8)).', 'Definition 2.2 (Charged Hawking (Misner-Sharp) mass) .': "(2.11) m H ( r ) := r n -2 2 [ 1 + r 2 ( n -1) 2 ( k 2 S -H 2 ) ] m q ( r ) := q 2 2 r n -2 m /lscript ( r ) := r n 2 /lscript 2 -→ m CH ( r ) := m H ( r ) + m /lscript ( r ) + m q ( r ) \nObserve that k 2 S -H 2 = θ + θ -, the product of the outward and inward expansions. For a MOTS, this term vanishes. Hence, the quasi-local mass can be rewritten in a standard form as an integral over a surface S . \nAs mentioned before, once we have defined the Charged Hawking mass, we must prove two things. First, that it is increasing, and second, that it tends to the total energy of the space-time (ADM energy or AdS energy) in the asymptotic limit r →∞ . \n2.4.1. Monotonicity. \nProposition 2.3. If the constraint equations (2.4) are satisfied and the system is spherically symmetric (2.9) , the charged Hawking mass is non-decreasing as a function of r . \nProof. First, notice that using (2.10), we can rewrite the Hawking mass as \n(2.12) m H ( r ) = r n -2 2 [ 1 -V ( r ) + r 2 ( k S n -1 ) 2 ] . \nIt is easy to check that the scalar curvature of the spherically symmetric metric (2.9) is \n(2.13) R ( g ) = n -1 r n -1 d d r [ r n -2 (1 -V ( r )) ] . Thus, \nFrom the first to the second line, we have used that in spherical symmetry, Tr g k = k I + k S and | k | 2 g = k 2 I + k 2 S / ( n -1). Meanwhile, the last line is obtained by contracting the RHS of (2.4b) (recall that we are assuming /star g ( E ∧ B ) = 0) with the normal ν i = (d r ) i / √ V . Assuming the constraints (2.4), we have \n(2.14) m H ' ( r ) + m ' /lscript ( r ) = 1 2 r n -1 n -1 R ( g ) + r n -1 2 ( n ( k S n -1 ) 2 +2 r k S n -1 k ' S n -1 ) + n r n -1 2 /lscript 2 = r n -1 2( n -1) [ R ( g ) -n n -1 k 2 S + n ( n -1) /lscript 2 ] + r n -1 k S n -1 ( n k S n -1 + r k ' S n -1 ) = r n -1 2( n -1) { R ( g ) + (Tr g k ) 2 -| k | 2 g + n ( n -1) /lscript 2 -2 k S ( Tr g k -nk S n -1 -r k ' S n -1 )} = r n -1 2( n -1) { R ( g ) + (Tr g k ) 2 -| k | 2 g + n ( n -1) /lscript 2 -2 k S H ( ν · div g ( k -(tr g k ) g )) } . \n(2.15) m ' CH ( r ) = m ' H ( r ) + m ' /lscript ( r ) + m ' q ( r ) = 8 πr n -1 n -1 [ ˆ µ -k S H ( ˆ J · ν ) ] + r n -1 n -1 | E | 2 g -n -2 2 r n -1 q 2 . \nm ' q ( r ) = -n -2 2 r n -1 q 2 (1.8) = -n -2 2 r n -1 2 ( n -1)( n -2) 1 ω 2 n -1 ( lim R →∞ ∫ S R E ( ν ) vol( S n -1 R ) ) 2 = -1 ( n -1) r n -1 ω 2 n -1 lim R →∞ (∫ V rR div g E vol( V rR ) -∫ S r E ( ν ) vol( S n -1 r ) ) 2 ≥ ≥ -r n -1 ( n -1)( r n -1 ω n -1 ) 2 lim R →∞ (∫ S r E ( ν ) 2 vol( S n -1 r ) )(∫ S r 1 2 vol( S n -1 r ) ) = -r n -1 ( n -1) r n -1 ω n -1 ∫ S r | E | 2 g vol( S n -1 r ) = -r n -1 | E | 2 g n -1 , \nWe now focus on the last term: \nwhere V rR denotes the space in between the spheres S r and S R . In the second line, we have used the divergence theorem; in the third, the Cauchy-Schwarz inequality for integrals together with the constraint div g E = 0, and in the last two equalities, we have used that E is purely radial. \nGathering the previous two equations, we have \n(2.16) m ' CH ( r ) ≥ 8 πr n -1 n -1 [ ˆ µ -k S H ( ˆ J · ν ) ] ≥ 0 . \nThe last inequality follows from the fact that, as argued in [22], the outermost condition implies H > k S in the interior of M = [ r + , ∞ ) × S n -1 (otherwise, at some point H = k S by continuity, implying the existence of another MOTS outside the boundary). Then the dominant energy condition ˆ µ ≥ | ˆ J | ≥ | ( ˆ J · ν ) | implies m ' CH ( r ) ≥ 0, with equality if and only if ˆ µ = | ˆ J | = 0 and | E | = 0. /square", '2.4.2. Asymptotic limit.': 'We wish to show the charged Hawking mass coincides with the total energy in the limit r → ∞ , i.e., with E ADM (1.2a) (notice that a spherically symmetric initial data set can be shown to have vanishing linear momenta; thus the ADM energy coincides with the ADM mass) or E hyp (1.7). This result is quite standard, but we give the argument for convenience.', 'Lemma 2.4.': '(2.17) lim r →∞ m CH ( r ) = { E ADM asymptotically flat ( /lscript = ∞ ) E hyp asymptotically hyperbolic ( /lscript = 1) \nProof. Since the charged term m q decreases to zero when r → ∞ , we can ignore that term in the following. \nConsider first the asymptotic flatness case, which implies m /lscript = 0 and Tr g k = O 1 ( r -κ ) for some κ > ( n -2) / 2. The latter implies k S = O 1 ( r -κ -2 ), so r n k 2 S = O 1 ( r n -2 κ -4 ) = O 1 ( r -2 -/epsilon1 ) for some /epsilon1 > 0. We then have that the limit of (2.12) is \n(2.18) lim r →∞ m H ( r ) = lim r →∞ r n -2 2 (1 -V ( r )) . \nFor the spherically symmetric metric (2.9), the fall-off relative to the Euclidean metric is \nwhere { x i } is a standard Cartesian chart with r = √ x i x j δ ij . Substituting this into the ADM energy formula, one finds \n(2.19) g -δ = ( 1 V -1 ) d r 2 = 1 -V V x i x j r 2 d x i d x j , \n(2.20) E ADM = lim r →∞ r n -1 2 1 -V ( r ) rV ( r ) (2.18) = lim r →∞ m H ( r ) , \nwhere in the last equality, we also used V ( r ) → 1 when r →∞ . \n(2.21) lim r →∞ m CH ( r ) = lim r →∞ r n -2 2 ( 1 + r 2 /lscript 2 -V ( r ) ) . \nConsider now the asymptotically hyperbolic case (we fix /lscript = 1 without loss of generality). From equation (1.6), observe that V hyp ( r ) /V ( r ) → 1 and that the term r n k 2 S = O 1 ( r n -2 κ -4 ) = O 1 ( r -2 -/epsilon1 ) can be neglected in the limit r →∞ . Thus, \nFor a spherically symmetric metric, the deviation tensor (1.6) is given by \n(2.22) ˆ g := g -b = V hyp -V V hyp V d r 2 = V hyp -V V V ⊗ V , \n(2.23) E hyp = lim r →∞ V hyp ( r ) V ( r ) r n -2 2 ( V hyp ( r ) -V ( r )) = lim r →∞ m CH ( r ) , \nwhere V i = (d r ) i / √ V hyp . A computation shows \nwhere we are using the asymptotic behaviour of V ( r ) as r →∞ .', '2.5. Penrose inequality.': "In this section, we prove Theorem 1.1. \nProof. Let r + correspond to the location of the apparent horizon. Then \n(2.24) E lemma = 2 . 4 lim r →∞ m CH ( r ) lemma ≥ 2 . 3 m CH ( r + ) = r n -2 + 2 + r n + 2 /lscript 2 + q 2 2 r n -2 + , \nwhere in the last equality we have used the definition of m CH and the equality k 2 S -H 2 = θ + θ -, which vanishes for a MOTS. The Penrose inequality (1.9) follows from the area being given by A = r n -1 + ω n -1 . \nNow it remains to prove the rigidity statement. Namely, if the equality saturates, the initial data set must arise from a spatial hypersurface in the static Reissner-Nordstrom(-AdS n +1 ) spacetime. Let us recall the definition and the required properties of this space-time. Its metric can be written as \n(2.25) g RN = -V RN ( r )d t 2 + d r 2 V RN ( r ) + r 2 g S , V RN ( r ) = 1 -2 M r n -2 + Q 2 r 2 n -4 + r 2 /lscript 2 ; \nwhere r ranges in ( r 0 , ∞ ), the unbounded interval where V RN ( r ) > 0. \nThe initial value problem to be considered is given on a spherically symmetric hypersurface, i.e. given as the level set M ( f ) = { t = f ( r ) } for some f satisfying | V RN f ' | < 1 ( M ( f ) must be space-like). In the following, we denote s the sign of f ' . \nLemma 2.5. The initial data on M ( f ) induced by (2.25) satisfying the constraint equations (2.4) is: \n(2.26) g ( f ) = d r 2 V f + r 2 g S ˆ µ ( f ) = 0 E ( f ) = E ( f ) I d r k ( f ) = k ( f ) I d r 2 V f + k ( f ) S n -1 r 2 g S ˆ J ( f ) = 0 B ( f ) = 0 , \nwhere \n(2.27) V f := V RN 1 -( V RN f ' ) 2 k ( f ) I = d d r ( r k ( f ) S n -1 ) k ( f ) S = -s n -1 r √ V f -V RN E ( f ) I = √ ( n -1)( n -2) 2 V f Q r n -1 . \nMoreover, the parameters ( M,Q ) of V RN correspond to the ADM/hyperbolic mass and the charge. \nProof. We introduce the new time coordinate τ = t -f ( r ). It is then a short computation to check \nThus, the induced metric on M ( f ) = { τ = 0 } is simply \n(2.28) g RN = -V f d τ 2 + 1 V f ( d r -V f V RN f ' d τ ) 2 + r 2 g S . \n(2.29) g ( f ) = d r 2 V f + r 2 g S , \nwith lapse and shift \n(2.30) N = √ V f , N i = -V RN f ' (d r ) i . \nTHE PENROSE INEQUALITY IN SPHERICAL SYMMETRY WITH CHARGE AND IN GAUSS-BONNET GRAVITY11 \nThe extrinsic curvature can be obtained from the lapse and the shift through the formula \n(2.31) k ( f ) ij = 1 2 N [ D i N j + D j N i ] , \nwhere D denotes the g ( f ) -Levi-Civita connection. It is not hard to check that the non-zero components are \n(2.32) k ( f ) S n -1 = -s r √ V f -V RN k ( f ) I := ν i ν j k ( f ) ij = d d r ( r k ( f ) S n -1 ) , \n(2.33) V RN f ' = s √ 1 -V RN V f . \nwhere ν i = √ V f ∂ i r . For that computation, it is useful to realize \nSince the Reissner-Nordstrom solution has no uncharged matter, ˆ µ ( f ) = 0 and ˆ J ( f ) = 0. Recall, because of the spherical symmetry, B ( f ) = 0 and the electric field is radial E = E I d r . The term E I can be easily obtained using the constraint equation (2.4a) \n(2.34) 2 V f ( E ( f ) I ) 2 = 2 | E ( f ) | 2 g ( f ) = ( n -1)( n -2) ( Q r n -1 ) 2 , \nwhere we have used Tr g k ( f ) = k ( f ) I + k ( f ) S , | k ( f ) | 2 g = ( k ( f ) I ) 2 + ( k ( f ) S ) 2 / ( n -1) and Equations (2.32), Equation (2.13) with V f instead of V , and the definition of V RN given in (2.25). Therefore, we indeed recover the initial data given in Equation (2.26). \nIt only remains to prove M is the ADM/hyperbolic mass and Q is the charge. The former is immediate using lemma 2.21 and the limits given in (2.18) or (2.4). For the charge we use the definition given in (1.8), the electric field we computed in (2.34), and the fact E ( ν ) = E I √ V f : \n(2.35) q = 1 ω n -1 lim r →∞ ∫ S r Q r n -1 vol( S n -1 r ) = Q. /square \nThe goal now is to show, for a given initial data set ( g, k, E, B ) that saturates the charged spherically symmetric Penrose inequality, there is some function f ( r ) such that g = g ( f ) and k = k ( f ) . \nRecall that ( g, k ) are spherically symmetric, hence they are given by Equation (2.9) for some V , k I and k S . It is clear that in order to have g = g ( f ) , we need to have V = V f for some f . \nSuppose now equality holds in the Penrose inequality. This implies \n(2.36) E = m CH ( r ) = Q 2 2 r n -2 + r n -2 2 [ 1 -V ( r ) + r 2 /lscript 2 + ( r k S n -1 ) 2 ] \nfor every r ≥ r + (of course, this means the RHS is constant). It is immediate to solve for V and find \n(2.37) V ( r ) = V RN ( r ) + ( r k S n -1 ) 2 + 2 r n -2 ( M -E ) . \nNotice that the mass M of the RN slice M ( f ) must be equal to the mass E of the original slice M . Moreover, we have seen that V = V f . Plugging these expressions back in (2.37) and solving for f ' \nleads to \nwhich satisfies the condition | f ' ( r ) V RN | < 1, since V RN is positive in the interval ( r 0 , ∞ ). For this choice of f (defined up to an overall constant and up to a sign on each interval of the support of f ), we obtain from lemma 2.5 the value of the spherical part of the extrinsic curvature: \n(2.38) f ' ( r ) 2 = 1 V 2 RN ( r k S n -1 ) 2 ( r k S n -1 ) 2 + V RN , \n(2.39) k ( f ) S = -s n -1 r √ V f -V RN (2.37) = -s | k S | . \n(2.40) H ( ˆ J · ν ) = k I -k S n -1 -r k ' S n -1 (2.39) -→ k I = d d r ( r k ( f ) S n -1 ) + H ( ˆ J · ν ) (2.27) = k ( f ) I + H ( ˆ J · ν ) . \nWe can choose the sign of f to get k ( f ) S = k S . To obtain the radial part of the extrinsic curvature, recall that contracting the RHS of (2.4b) with ν i = (d r ) i / √ V (as we used to derive Equation (2.14)) leads to \nLet us now prove ( ˆ J, ˆ µ ) = ( ˆ J ( f ) , ˆ µ ( f ) ) = 0, which incidentally would also prove k I = k ( f ) I . For that, we use m ' CH = 0. From (2.15) we get \n(2.41) | E | 2 g = ( n -1)( n -2) 2 q 2 r 2( n -1) , ˆ µ = k S H ( ˆ J · ν ) . \nThe first condition implies, following the results of lemma 2.5, the electric field E = ( E · ν ) ν of the initial data set is that of a Reissner-Nordstrom-AdS black hole with charge q = Q . The second equation implies that both ˆ µ and ˆ J · ν vanish (and hence so does ˆ J , since it must be radial). Indeed, we have seen already that H > k S for all r > r + , but the dominant energy condition forces ˆ µ ≥ | ˆ J | . \nTherefore, we have shown that the inequality is saturated if and only if the initial data set can be isometrically embedded as a spatial hypersurface in the Reissner-Nordstrom(-AdS) spacetime. /square", '3. The Gauss-Bonnet gravity theory': 'In this section, we prove Theorem 1.2, the spacetime Penrose inequality in spherical symmetry within the setting of Einstein-Gauss-Bonnet gravity (see the review [13]).', '3.1. Equations of motion.': "Consider the action functional in spacetime dimension n +1: \nwhere α > 0 is a constant coupling parameter, ψ denotes the collection of additional fields (charged and non-charged), and L their Lagrangian. The equations of motion are \n(3.1) S ( g , A, ψ ) = ∫ ( R + n ( n -1) /lscript 2 + αL GB ) vol( g ) + ∫ M ˜ L ( g , ψ ) , \nwhere ˜ T ab is the energy-momentum tensor associated with the non-metric fields, and we have the geometric 'source' term \n˜ (3.2) G ab + α H ab = n ( n -1) 2 /lscript 2 g ab +8 π ˜ T ab , \n(3.3) H ab := 2 [ RR ab -2 R ac R c b -2 R cd R acbd + R cde a R bcde ] -1 2 g ab L GB . \nTHE PENROSE INEQUALITY IN SPHERICAL SYMMETRY WITH CHARGE AND IN GAUSS-BONNET GRAVITY13", '3.2. Initial Value Problem.': 'Consider an initial data set ( M,g,k ) with the conventions and definitions of section 2.2. Contracting the field equation with n a n b and ı a i n b (and after a long computation), we obtain the constraints [31]: \n16 π µ = M + n ( n -1) /lscript 2 + α ( M 2 -4 M ij M ij + M ijkl M ijkl ) (3.4a) \n˜ where M ijkl := ( ı ∗ R ) ijkl = ı a i ı b j ı c k ı d l R abcd is the pullback of the g -Riemann tensor (as usual, we define M jl := g ik M ijkl and M = g jl M jl ), and N ijk := ı a i ı b j ı c k n d R abcd (we also define N j := g ik N ijk ). In the previous computation, we have used the Gauss-Codazzi and the Codazzi-Mainardi equations: \n˜ 8 π J j = N j +2 α ( MN j -2 M i j N i +2 M ik N jik -M lik j N ikl ) , (3.4b) \nM ijkl = R ijkl + k ik k jl -k il k jk (3.5a) \nN ijk = D i k jk -D j k ik , (3.5b) \nwhere R i jkl is the g -Riemann tensor. Notice \nM = R ( g ) + (Tr g k ) 2 -k ij k ij (3.6a) \nN j = D i k ij -D j (Tr g k ) , (3.6b) \nso when α = 0, (3.4) reduces to the usual constraints of general relativity. \nIt is a straightforward exercise to include a Maxwell field using the results from the previous section. We will restrict attention to the Gauss-Bonnet term here. Choquet-Bruhat has addressed the well-posedness of the Cauchy problem associated with these constraints [6].', '3.3. Spherically symmetric initial data set.': 'We consider spherically symmetric data ( M,g,k ) with metric g and extrinsic curvature k as in (2.9).', '3.4. Definition of Gauss-Bonnet quasi-local mass and its properties.': "Let's first introduce a quasi-local mass analogous to the charged Hawking mass, adapted for such data satisfying the constraint equations (3.4a) and (3.4b): \nDefinition 3.1 (Generalized Gauss-Bonnet Hawking mass) . \n(3.7) m GB ( r ) := m H ( r ) + m /lscript ( r ) + 2 α m H ( r ) 2 r n \n˜ We recall that ˜ α := ( n -2)( n -3) α , which vanishes for n = 3 ( d = 4), in which case m GB equals the Hawking mass. This is consistent with the GB term being topological. A related definition of a quasi-local Misner-Sharp mass valid for spherically symmetric spacetimes was presented in [26].", '3.4.1. Monotonicity.': "Proposition 3.2. For spherically symmetric initial data satisfying the Gauss-Bonnet constraints, the generalized Gauss-Bonnet quasi-local mass is a monotonically non-decreasing function of r . \nProof. We start by computing the α factors of (3.4), which we denote: \n˜ µ ( α ) := M 2 -4 M ij M ij + M ijkl M ijkl (3.8a) ˜ J ( α ) i := MN j -2 M i j N i +2 M ik N jik -M lik j N ikl , (3.8b) \nin a spherical symmetric setting (2.9). For that, we break the indices into the normal and tangent parts using the decomposition g ij = ν i ν j + X i A X j B (dΩ 2 ) AB /r 2 where ν i = √ V ∂ i r and X i A denotes the pullback/pushforward of the embedding X : S n -1 ↪ → M . We compute the following terms: \nM ABCD = 2 m H ( r ) r n r 4 (dΩ 2 AB dΩ 2 BD -dΩ 2 AD dΩ 2 CD ) (3.9b) \nM 1 B 1 D := ν i X j A ν k X l B M ijkl = ( -r '' r + k S n -1 k I ) r 2 dΩ 2 BD (3.9a) \n( \nM \n11 \n= ( \nn \n- \n1) \n- \n+ \nn \n1 \nk \nI \n) \n(3.9c) \nM = 2( n -1) ( -r '' r + k S n -1 k I ) +( n -1)( n -2) 2 m H ( r ) r n (3.9e) \n(3.9f) \nN \n1 \n= \nH \nr \nk \nS \nr -M AB = [ -r '' r + k S n -1 k I +( n -2) 2 m H ( r ) r n ] r 2 dΩ 2 AB (3.9d) \n[ \nk \nI \n- \n- \nN A 1 B = N 1 n -1 r 2 dΩ 2 AB , (3.9h) \nk \nS \nn -1 N 1 AB = -N 1 n -1 r 2 dΩ 2 AB (3.9g) \nwhere we recall m H is given by equation (2.12). After a long computation and using the previous expressions, one can check that (3.8) can be rewritten as \n˜ µ ( α ) = ( n -1)( n -2)( n -3) 8 m H ( r ) r n [ m ' H ( r ) r n -1 -n m H ( r ) 2 r n + k S n -1 N 1 H ] (3.10a) \n( ν · ˜ J ( α ) ) = 2 N 1 ( n -2)( n -3) m H ( r ) r n . (3.10b) Finally, we compute the derivative of the GB-mass: \n' 2 \nm GB ' ( r ) = m ' H ( r ) + m ' /lscript ( r ) + 4 ˜ α m H ( r ) m H ( r ) r n -2 n ˜ α m H ( r ) r n +1 (2.14) = (3.6) = r n -1 2( n -1) { M + n ( n -1) /lscript 2 -2 k S H ( ν · N ) } + 2 ˜ αm H ( r ) r n ( 2 m ' H ( r ) -n m H ( r ) r ) (3.4) = = r n -1 2( n -1) { 16 π ˜ µ -α ˜ µ ( α ) -2 k S H (8 πν · ˜ J -2 αν · ˜ J ( α ) ) } + 2 ˜ αm H ( r ) r n ( 2 m ' H ( r ) -n m H ( r ) r ) (3.10) = = 8 πr n -1 n -1 { ˜ µ -k S H ( ν · ˜ J ) } ≥ 0 . For the final inequality, we have used the same reasoning as for equation (2.16). /square", '3.4.2. Assymptotic limit.': "Since m H ( r ) 2 r n → 0 as r → ∞ , it is clear the same arguments as in the previous section show, assuming the ambient spaces have the appropriate asymptotic behaviour, m GB ( r ) approaches the ADM/hyperbolic energies in the limit r →∞ . \nk \n' \nS \n- \n] \nr \n'' \nn \n1", '3.5. Penrose inequality.': "The proof of the Penrose inequality (1.11) is almost identical to the charged Penrose inequality. The rigidity statement of Theorem 1.2 is similar but requires a bit more work. First, notice that the Einstein-Gauss-Bonnet equations admit static, spherically symmetric solutions [5] \n(3.11) d s 2 = -V GB ( r )d t 2 + d r 2 V GB ( r ) + r 2 g S , V GB ( r ) = 1 + r 2 2 α ( 1 ∓ [ 1 + 8 M ˜ α r n -4 ˜ α /lscript 2 ] 1 / 2 ) . \n˜ The strategy now, as for the charged Einstein-Maxwell case, is to show that if equality is achieved, the resulting data set can be realized as a spatial hypersurface defined by the level set t = f ( r ) in the spherically symmetric Gauss-Bonnet black hole spacetime (3.11). Following the method discussed in the Reissner-Nordstrom case, one finds this is possible provided f ( r ) is chosen to satisfy (2.38), with the simple replacement V RN → V GB . The requirement m ' GB = 0 implies both µ and J vanish identically. This demonstrates that the inequality is saturated if and only if the initial data set can be isometrically embedded as a spatial hypersurface in the Gauss-Bonnet vacuum spacetime defined by (3.11). \n˜ We must choose the upper ( -) sign so the solutions reduce to Schwarzschild-AdS in the limit α → 0. A t -constant spatial hypersurface furnishes a time-symmetric initial data set satisfying (3.4a) and (3.4b). In the case of vanishing cosmological constant ( /lscript → ∞ ), the above spacetime metric is asymptotically flat, and one finds the parameter M is equal to the ADM mass defined by (1.2a). For finite /lscript , the spacetime is asymptotically AdS n +1 , but with a rescaled length scale /lscript 2 eff = /lscript 2 (1 + √ 1 -4 ˜ α//lscript 2 ) / 2. In our conventions, if we fix /lscript = (1 -˜ α ) -1 / 2 with ˜ α ∈ (0 , 1 / 2) (this ensures V GB ( r ) ∼ 1 + r 2 as r →∞ ), then the hyperbolic energy (1.7) of (3.11) E hyp = M/ (1 -2 α ).", 'References': "- [1] L. C. Ambrozio, On Perturbations of the Schwarzschild Anti-De Sitter Spaces of Positive Mass , Commun. Math. Phys. 337 , no.2, 767-783 (2015) [arXiv:1402.4317 [math.DG]].\n- [2] H. L. Bray and M. A. Khuri, A Jang Equation Approach to the Penrose Inequality , Discrete Contin. Dyn. Syst. A 27 , 741-766 (2010) [arXiv:0910.4785 [math.DG]].\n- [3] H. L. Bray and D. A. Lee, On the Riemannian-Penrose inequality in dimensions less than 8 , Duke Math. J. 148 (2009), 81-106 [arXiv:0705.1128 [math.DG]].\n- [4] H. L. Bray, Proof of the Riemannian-Penrose inequality using the positive mass theorem, J. Differential Geom. 59 (2001), no.2, 177-267.\n- [5] R. G. Cai, 'Gauss-Bonnet black holes in AdS spaces,' Phys. Rev. D 65 , 084014 (2002) [arXiv:hep-th/0109133 [hep-th]].\n- [6] Y. Choquet-Bruhat, 'The Cauchy Problem for Stringy Gravity,' J. Math. Phys. 29 , 1891-1895 (1988).\n- [7] P. T. Chrusciel and G. Nagy, The Mass of space - like hypersurfaces in asymptotically anti-de Sitter space-times , Adv. Theor. 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Yamada, Proof of the Riemannian-Penrose Inequality with Charge for Multiple Black Holes , J. Diff. Geom. 106 , no.3, 451-498 (2017) doi:10.4310/jdg/1500084023 [arXiv:1409.3271 [gr-qc]].\n- [20] Sergui Klainerman, Mathematical challenges of General Relativity , Rendiconti di Matematica, Serie VII 27 , Roma (2007), 105-122.\n- [21] C. Lanczos, A Remarkable property of the Riemann-Christoffel tensor in four dimensions , Annals Math. 39 , 842-850 (1938).\n- [22] D. Lee, Geometric relativity, Grad. Stud. Math., 201, American Mathematical Society, Providence, RI, 2019. xii+361 pp. ISBN:978-1-4704-5081-6\n- [23] L. Lopes de Lima, F. Gir˜ao, W. Loz'orio and J. Silva, Penrose inequalities and a positive mass theorem for charged black holes in higher dimensions , Class. Quant. Grav. 33 , no.3, 035008 (2016) [arXiv:1401.0945 [math.DG]].\n- [24] D. Lovelock, Divergence-free tensorial concomitants , Aequat. Math. 4 , no.1, 127-138 (1970)\n- [25] D. Lovelock, The Einstein tensor and its generalizations , J. Math. Phys. 12 , 498-501 (1971).\n- [26] H. Maeda and M. Nozawa, 'Generalized Misner-Sharp quasi-local mass in Einstein-Gauss-Bonnet gravity,' Phys. Rev. D 77 , 064031 (2008) [arXiv:0709.1199 [hep-th]].\n- [27] M. Mars, Present status of the Penrose inequality , Class. Quant. Grav. 26 , 193001 (2009) [arXiv:0906.5566 [gr-qc]].\n- [28] R. Penrose, Gravitational collapse: the role of general relativity , Nuovo Cimento, Rivista Serie., 1 (1969), 252-276.\n- [29] R. Schoen and S.T. Yau On the proof of the positive mass conjecture in general relativity , Commun. Math. Phys. 65 (1979), no.1, 45-76.\n- [30] R. Schoen and S.T. Yau Proof of the positive mass theorem. II , Commun. Math. Phys 79 (1981), no.2, 231-260.\n- [31] T. Torii and H. a. Shinkai, 'N+1 formalism in Einstein-Gauss-Bonnet gravity,' Phys. Rev. D 78 , 084037 (2008) [arXiv:0810.1790 [gr-qc]].\n- [32] E. Witten, A Simple Proof of the Positive Energy Theorem , Commun. Math. Phys. 80 , 381 (1981) \nDepartment of Mathematics and Statistics and Department of Physics and Astronomy, McMaster University, Hamilton, ON Canada \nEmail address : kundurih@mcmaster.ca \nDepartamento de Matem'aticas, Universidad Carlos III de Madrid, Legan'es, Spain, Grupo de Teor'ıas de Campos y F'ısica Estad'ıstica, Instituto Gregorio Mill'an, Unidad Asociada al Instituto de Estructura de la Materia CSIC, Madrid, Spain, \nDepartment of Mathematics and Statistics, Memorial University, St John's, NL Canada Email address : juan.margalef@uc3m.es \nDepartment of Mathematics and Statistics, Memorial University, St John's, NL Canada Email address : smmmuth@mun.ca"}

2024arXiv240704743A

Cosmic strings that are attached to rapidly spinning black holes can extract significant amounts of rotational energy and angular momentum. Here we study the effect on primordial black holes which are expected to form with one or more cosmic strings attached. Although large primordial black holes are predicted to rapidly spin up due to accretion soon after forming we argue that cosmic strings will spin them down again. We show that if there are cosmic strings with tension greater than 1020 the spins of large primordial black holes of mass greater than 30 Modot should consequently be observed to be near zero. We also investigate the effect on a supermassive black hole of capturing a cosmic string and the possibility of observing the subsequent spin down by its effect on a pulsar orbiting the black hole.

2024-07-01T00:00:00Z

['10.48550/arXiv.2407.04743', '2024arXiv240704743A', 'arXiv:2407.04743']

['Astrophysics - Cosmology and Nongalactic Astrophysics', 'Astrophysics - High Energy Astrophysical Phenomena', 'General Relativity and Quantum Cosmology', 'High Energy Physics - Theory']

Searching for cosmic strings via black hole spindown

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2407.04743.pdf

{'Sameer Ahmed, a Michael Kavic, a Steven L. Liebling, b Matthew Lippert, a Mohammad Mian, a and John Simonetti c': 'a \nDepartment of Chemistry and Physics, \nSUNY Old Westbury, \nOld Westbury, NY, United States \n- b Department of Physics, \nLong Island University, \nBrookville, NY, United States \n- c Department of Physics, Virginia Tech, Blacksburg, VA, United States \nE-mail: \nkavicm@oldwestbury.edu , Steve.Liebling@liu.edu , \nlippertm@oldwestbury.edu , jhs@vt.edu \nAbstract: Cosmic strings that are attached to rapidly spinning black holes can extract significant amounts of rotational energy and angular momentum. Here we study the effect on primordial black holes, which are expected to form with one or more cosmic strings attached. Although large primordial black holes are predicted to rapidly spin up due to accretion soon after forming, we argue that cosmic strings will spin them down again. We show that if there are cosmic strings with tension greater than 10 -20 , the spins of large primordial black holes of mass greater than 30 M ⊙ should consequently be observed to be near zero. We also investigate the effect on a supermassive black hole of capturing a cosmic string and the possibility of observing the subsequent spin down by its effect on a pulsar orbiting the black hole.', '1 Introduction': "Cosmic strings are conjectural remnants of phase transitions in the early universe and represent an intriguing astronomical manifestation of high-energy physics. They are one-dimensional topological defects naturally appearing in many particle physics models, including many GUT theories (see [1-3] for reviews). In the context of string theory, fundamental strings and D-strings can also manifest as cosmic superstrings [4]. \nEfforts to detect cosmic strings have exploited several possible phenomena. Oscillating strings produce a stochastic gravitational wave background [5]. Cosmic strings can also develop cusps and kinks which emit strong bursts of gravitational radiation [6-8]. The amplitude of this gravitational wave signal depends primarily on the string's mass density, or tension, µ . 1 \nPerhaps most exciting are the recent results from the North American Nanohertz Observatory for Gravitational Waves (NANOGrav), looking for ultra-low-frequency gravitational waves by observing millisecond pulsars. NANOGrav has found evidence of a stochastic gravitational wave background compatible with a network of cosmic strings in the range 10 -12 < µ < 10 -10 [9, 10]. Although not a firm detection, the observations offer tantalizing evidence for the existence of cosmic strings in the early universe. \nDirect searches by the LIGO/Virgo/KAGRA (LVK) gravitational wave (GW) observatory network for burst signals have set an upper bound of µ < 3 × 10 -7 , and analysis of the stochastic GW background yields a tighter constraint of µ < 10 -10 [11]. A stochastic background would also induce characteristic noise in the timing of pulsar signals. Results from the Parkes Pulsar Timing Array currently give the strongest bound: µ < 1 . 5 × 10 -11 [12, 13]. Microlensing of extra-galactic sources by strings could probe down to tensions as small as µ = 10 -13 [14]. Future GW detectors, such as LISA, could potentially be sensitive to µ = 5 . 8 × 10 -18 [13]. \nHere, we investigate the interaction between a cosmic string (CS) and a black hole (BH), with a view toward observable signatures. A CS colliding with a BH will be gravitationally captured, with \nthe black hole ending up stuck like a bead on a string [15-17]. A short CS loop might be expected to quickly fall in to the BH, but a very long CS, in particular one extending beyond the cosmological horizon, could persist attached to the BH for a long time. \nThe behavior a CS attached to a rotating black hole was studied in [18]. Interestingly, the cosmic string can extract rotational energy and angular momentum [19, 20] from the BH by a generalization of the Penrose Process [21] using one-dimensional, relativistic strings rather than particles. The wellknown Blanford-Znajek mechanism [22], in which a force-free electromagnetic field extracts rotational energy from an accreting BH, is also an example of this stringy Penrose Process but with magnetic flux lines acting as the strings [23]. Unlike the Blanford-Znajek mechanism, in which the rotational energy powers highly visible jets emanating from accreting black holes, the energy extracted by cosmic strings is not easily observable. However, the spin down and energy loss of the BH is potentially detectable. \nAn important question to consider is how likely is it that a given BH will, in fact, collide with and capture a CS. In [24], the rate for a black hole to capture cosmic strings was estimated. A stellar-mass BH presents a small target, and the chance of capturing a CS is extremely low; the expected time to capture a single cosmic string would be far longer than the age of the universe. However, for a supermassive black hole, such as Sagittarius A* (Sgr A*), and for very low cosmic string tension, µ ≲ 10 -18 , there is an order-one probability for one or more collisions during the black hole's lifetime. \nObserving the spin down of a supermassive back hole due to cosmic strings, although challenging, is possible, at least in principle. We propose a technique by which the energy lost by the BH could be detected by its effect on a pulsar orbiting the BH. However, in addition to finding an appropriate pulsar sufficiently close to the BH, the precision of pulsar timing measurements would need to improve significantly in order to detect the mass loss due to phenomenologically viable cosmic strings with µ < 10 -11 . \nAparticularly interesting alternative scenario is the interaction of a cosmic string with a primordial black hole (PBH) [25]. PBHs may have been created by large density fluctuations in the early universe with a very wide range of possible masses, in principle down to the Planck scale (see [26-29] for recent reviews). The possibility that large black holes recently observed by LVK may have primordial origin [30] has spurred renewed interest. \nIn the early universe, at the time of PBH formation, the cosmic string network would be sufficiently dense that there will be, on average, one or more cosmic strings in a typical over-dense region which collapses to form a PBH. The PBH is then created with one or more cosmic strings already attached [25]. \nA typical PBH formed by over-densities during the radiation-dominated era is not expected to form with significant spin [31, 32]. However, there are mechanisms by which PBHs form with large spins, e.g. [33, 34]. Even if the PBH is formed with low spin, subsequent accretion of gas and dark matter can spin it up. In particular, it was shown in [35] that this accretion process is mostly completed early in the history of the universe and that large PBHs are expected to have higher accretion rates, resulting in higher spins. As a result, a PBH with mass greater than about 30 M ⊙ is predicted to have near-extremal spin when later observed. \nHowever, if the PBH formed with cosmic strings attached, then after being spun up by accretion, the PBH would subsequently be spun down by those strings. As a result, later observations would find high-mass PBHs with very small angular momentum, in contrast to the prediction of [35]. \nAccurate measurements of BH spins have become increasingly possible as techniques have improved from gravitational wave observations of binary BH mergers, and from X-ray emission spectroscopy of BH accretion disks (for a review, see [36]). With a sufficiently large data set of black hole spin observations, it will become possible to distinguish, at least statistically, those PBHs formed \nin the early universe from other BHs formed later, for example, via stellar collapse. Observations of highly spinning PBHs would then rule out, or at least highly constrain, the existence of cosmic strings at the time of PBH formation. \nThe outline of the remainder of this paper is as follows: In section 2, we review the mechanism by which cosmic strings remove energy from black holes. Then, in section 3, we argue that cosmic strings would have effectively spun down PBHs being observed in the present epoch. In section 4, we discuss the possibility of observing the cosmic string-induced spin down of a supermassive black hole via pulsar timing measurements. Finally, we discuss open questions and future directions in section 5.", '2 Energy extraction by cosmic strings': 'The Penrose Process [21] is a mechanism in classical general relativity by which energy can be removed from a rotating black hole. An object that enters within the ergosphere of the black hole will necessarily co-rotate with the black hole. The object is then split into two parts, and one part launched in the backward (contra-rotating) direction. This piece will still be co-rotating, albeit slower, and can have negative energy and angular momentum as measured by an asymptotic observer. When this negative energy piece ultimately falls through the event horizon, the energy and angular momentum of the black hole are reduced. The remaining part of the object receives a kick and can escape the black hole with greater energy and angular momentum than it had originally. \nThe available energy is limited by the spin of the black hole. A Kerr black hole 2 with angular momentum J has a mass M given by \nM = √ M 2 irr + J 2 4 M 2 irr , (2.1) \nwhere the irreducible mass M irr is given by the horizon area: M 2 irr = A 16 π . The Penrose process can continue extracting energy until J = 0 and the black hole is completely spun down, while the area theorem ensures that M irr does not decrease. At this point all of the rotational energy E rot = M -M irr has been removed, leaving a non-rotating BH with M = M irr . As a result, at most 29% of the original BH energy may be extracted. \nThe Penrose process is actually a quite general description of classical energy extraction from a rotating black hole, with a wide range of physical scenarios. Superradiance (for a review, see [37]) is one such example. Another such process involves an externally sourced magnetic field threading the black hole, extracting rotational energy via the Blandford-Znajek (BZ) process [22]. The equivalence of the BZ process and energy extraction by a protruding CS is demonstrated in [20, 23]. \nIn particular, a long, relativistic CS with one end entering the ergosphere is dragged around by the black hole. If the CS is rigidly rotating with an angular velocity ω that is less than the angular velocity Ω h of the BH, i.e. if ω < Ω h , then there is an outward radial flow of angular momentum and energy along the string [19]. In the limit of a slowly rotating black hole, J < M 2 , and averaging over the angle at which the string emerges from the ergosphere, the outward angular momentum flux q is \nq = 2 M 2 (Ω h -ω ) . (2.2) \nFor a CS with tension µ , the energy extraction rate is dE/dt = µqω . Restoring the dimensionful constants, this can be written as [19]: \ndE dt = ( 4 . 5 × 10 58 erg s ) µ ( q/a 1 / 2 )( ω/ Ω h 1 / 2 ) u ( a ) , (2.3) \nwhere the dimensionless function u ( a ) = ( 1 -√ 1 -a 2 /M 2 ) . When a = 0 the BH is no longer spinning and u (0) = 0, which shuts off the energy extraction. \nAs the black hole loses rotational energy to the string, its total mass decreases. For typical values of a , q , and ω , the final three factors of (2.3) are order one. The rate at which the BH is losing mass, assuming a > 0 and ω < Ω h is then approximately \ndM dt ≈ -( 10 34 kg s ) µ ≈ -( 10 4 M ⊙ s ) µ (2.4) \nwhich agrees with the results of [24]. \nIf the CS is very long, energy will continually be drawn away from the BH and dissipated down the length of the string. If the other end is sufficiently far away, in particular if the CS extends beyond the cosmological horizon, outward-moving fluctuations will never be reflected and return to the BH. As the string extracts angular momentum from the BH, its angular velocity will not increase significantly because that angular momentum propagates outward. The BH will eventually spin down completely. \nIn the process of spinning down, the black hole can lose at most 29% of its original mass. For a typical, rapidly rotating, but non-extemal BH, the mass loss will be of order 10%. Taking ∆ M ≈ M/ 10 and ignoring the dependence of (2.3) on the BH spin, the timescale t SD over which the black hole spins down is given by: \nt SD ≈ 10 6 s ( M M ⊙ )( 10 -11 µ ) . (2.5) \nThe fiducial tension µ = 10 -11 is roughly the maximum value allowed by the observational bounds, implying the last factor will be greater than one. Because the actual mass-loss rate goes to zero as a → 0, the spin-down time given by (2.5) is an overestimate. \nFor a stellar mass black hole with, say, M BH = 30 M ⊙ , the approximate spin-down time (2.5) gives \nt SD ≈ 1 yr ( 10 -11 µ ) . (2.6) \nThis implies that, unless µ is very small, a stellar mass black hole will spin down rapidly unless it is being actively being spun up by, for example, accretion. In particular, a typical observation of a stellar BH will be extremely unlikely to see the spin down in progress. \nIn contrast, for a supermassive mass black hole, M BH = 10 7 M ⊙ , (2.5) gives \nt SD ≈ 10 6 yr ( µ 10 -11 ) -1 . (2.7) \nThe spin down of a supermassive BH is a much slower process, especially for small µ . Consequently, the chance of observing such a BH in the process of spinning down will be much higher.', '3.1 Formation and spin up': 'Black holes may have been created in the early universe by a variety of mechanisms. The collapse of large density fluctuations during the radiation dominated era [38, 39] has received the most attention 3 , but other scenarios have been considered such as the collapse of vacuum bubbles or domain walls [4148] and of cosmic string loops or cusps [41, 49, 50]. These primordial black holes (PBHs) can, in principle, be created at any mass within a very wide range. Although the idea of PBHs is not new, the possibility that large black holes recently observed by LVK may have primordial origin [30] has spurred renewed interest. In addition, recent work has argued that NANOgrav may have observed stochastic GW signals from PBH formation in the early universe, suggesting that PBHs comprise a significant fraction of dark matter [51-53]. \nThe abundance of PBHs is constrained by a variety of phenomenological bounds [26-29]. Extremely light PBHs, with M PBH < 10 -19 M ⊙ , would have completely evaporated by now due to Hawking radiation and would no longer be present. Somewhat more massive PBHs with M PBH < 10 -16 M ⊙ would not have evaporated yet, but they would be emitting sufficient Hawking radiation that their abundance must be quite limited to have avoided observation. The abundance of PBHs in the planetary to sub-solar-mass range, 10 -11 M ⊙ < M PBH < 10 -1 M ⊙ , is modestly constrained by the frequency of stellar microlensing events, though for solar-mass and larger PBHs, M PBH > M ⊙ , dynamical effects, limits on radiation from accreted gas, and GW signals impose more stringent limits. The sub-planetary mass range, 10 -16 M ⊙ < M PBH < 10 -11 M ⊙ , however, is still quite unconstrained. \nThe mass at which a PBH forms is sensitively related to the time at which it forms. A region will collapse into a PBH of mass M when its Schwarzschild radius is the size of the Hubble radius. During the radiation-dominated era, the average mass within a Hubble radius is M H = M ⊙ ( t 10 -5 s ) As a result, a PBH of mass M will form at a time [25, 28] \nt PBH ∼ 10 -5 s ( M M ⊙ ) . (3.1) \nUnlike black holes formed by stellar collapse, PBHs formed by collapse of over-densities typically have very low spins. Estimates in [31, 32] predict an initial spin parameter of order a ∼ 10 -2 . Larger spins have been suggested, however, as a possibility for PBHs formed via other mechanisms [33, 34]. \nOver the long history of the universe, both the mass and spin of a PBH can evolve significantly. While many phenomena can affect the PBH during its lifetime, accretion of the surrounding interstellar gas has been shown to be a significant driver of mass and spin evolution [35]. In particular, the accretion effects grow with BH mass and become significant for M ≳ 10. 4 At early times, z > 10, the accretion rate for these large BHs can exceed the Eddington rate of 1 . 44 × 10 14 kg / s ( M/M ⊙ ). For super-Eddington accretion rates, the accreting matter will form a thin disk [54] which leads to efficient spin-up of the PBH. \nThis rapid accretion process is expected to decrease with the onset of structure formation. Due to the attraction of large-scale structures, a typical PBH will speed up relative to the ambient gas, and the resulting large proper velocities strongly suppress accretion [54, 55]. Although uncertainties \nremain regarding the dynamics of PBHs and structure formation, significant accretion can be expected to end around z ∼ 10. \nThe evolution of PBH mass and spin assuming a sharp accretion cut-off at z = 10, was computed in [35] (labeled as Model 1). A small PBH with initial mass M ≲ 30 M ⊙ and low spin χ ∼ 0 experiences negligible accretion and remains small and slowly spinning. However, larger PBHs with M > 30 M ⊙ rapidly accrete as z → 10, spinning up to near extremality χ ≲ 1. In both cases, after accretion ends at z = 10, PBHs are then assumed to continue with constant mass and spin until the present (see Fig. 4 of [35]). 5 \nThe results of [35] yield a prediction for the late-time spins of PBHs. In the absence of further dynamical evolution, a large PBH M > 30 M ⊙ observed any time after z = 10 would be expected to have near-extremal spin. A small PBH, by contrast, should be observed with nearly zero spin.', '3.2 Spin down by cosmic strings': 'The presence of cosmic strings changes the predicted distribution of PBH spins significantly. Cosmic strings form early in the history of the universe, generally before the formation of PBHs. In order to be relevant to PBHs, however, they must have formed after inflation; otherwise, the string network at the time of PBH formation would be much too sparse. Focusing on cosmic strings created during symmetry-breaking phase transitions, the energy scale η of the phase transition, which is related to the string tension µ = ( m p η ) 2 , determines the time of string formation [25]: \nt s ∼ ( m p η ) 2 t p ∼ t p µ (3.2) \nwhere t p is the Planck time. Compared with the PBH formation time (3.1), cosmic string form much earlier than PBHs, as long as the CS tension µ is not extremely small; for example, for a 30 M ⊙ PBHs any tension µ > 10 -38 will yield t s ≪ t PBH . \nAfter the strings are created, the network begins to evolve. Long strings spanning across Hubble volumes stretch as the universe expands. Through self-intersections and intersections with other strings they can reconnect and break off smaller loops. Loops oscillate, lose energy by emitting gravitational waves, and shrink. Numerical simulations predict a scale-invariant string network, in which a given Hubble patch contains several long strings and a larger number of loops ranging over all sizes [56-58]. \nLater, after this CS network has emerged, PBHs begin to form. Overdense regions begin to collapse when their apparent horizon comes within the cosmological horizon (3.1). This roughly cosmologicalhorizon-sized region, which includes O (10) long strings crossing through it, then collapses, forming a new PBH. For every long CS originally in the collapsing region, the new PBH will have a pair of long strings sticking out of the horizon [25]. 6 \nThe subsequent evolution of the coupled PBH-CS system was studied in detail in [25]. A long string attached to a black hole horizon exerts a pull on the black hole, but can not, by itself, become unattached from the black hole. If, for example, a string connects two black holes, they can be pulled toward each other, eventually merging into a single larger black hole with no cosmic string. However, if the typical distance between PBHs is large, the string pulling the PBHs together will have to compete with the cosmological expansion. \nA pair of strings emerging from a BH horizon can intersect, potentially reconnecting to form a half loop, whose ends both enter the horizon, and a long string which is disconnected from the PBH. The half loop will eventually fall into the horizon and disappear, leaving the PBH with two fewer strings. Initially, with O (10) strings attached to the PBH, the probability for any two to intersect is high, but, as the number of strings falls, intersections become increasingly unlikely. Once the number of strings drops to 2, the chance that they then intersect can become very low [25]. \nThe details are complicated and depend on the properties of the cosmic strings, the density of the CS network, and the distance between PBHs. The likelihood that a PBH will eventually lose all its strings, and the timescale for this to occur, are open questions and will likely require numerical simulations to address. However, it seems reasonable to assume at least one pair of long strings remains attached. \nAs described above in Sec. 3.1, a typical PBH forms with very low spin, but large PBHs will subsequently be spun up by accretion. However, assuming the large PBH has one or more cosmic strings attached during the accretion period, the angular momentum will then be extracted by the cosmic strings, spinning the PBH back down. \nThe spin-down rate for a 30 M ⊙ mass PBH (2.6) is quite rapid, as long as the cosmic string tension is not extremely small. Even for µ = 10 -20 , orders of magnitude below current observational bounds, the spin-down time is 10 9 years; by the present day, the PBH will have long since lost its angular momentum. For larger string tensions, closer to the observational bound µ ∼ 10 -11 , the spin-down rate is faster than the accretion rate [35]. In this case, the PBH never even gets spun up, with the angular momentum extracted as fast as it falls in. \nThis allows us to establish a clear prediction and a definitive test for the existence of cosmic strings in an unprobed range of string tensions. If there are cosmic strings with µ > 10 -20 , PBHs with mass larger than 30 M ⊙ will be observed to have near-zero spin. Otherwise, according to [35], these high mass PBHs will instead have near-extremal spin.', '3.3 Observations of black hole spin': "Determining the spin of a BH presents a significant observational challenge. Currently, spin measurements are possible for two types of BH systems: Spins of accreting BHs can be measured through electromagnetic radiation, mostly through X-ray reflection spectroscopy, and thermal continuum fitting, while spins of merging black holes are constrained through GW observation (see [36] for a review). \nFor accreting BHs in the regime between geometrically thin and optically thick accretion disks, Xray reflection spectroscopy is the primary means for determining the spin. X-ray reflection spectroscopy refers to the soft X-ray lines emitted by the photo-ionized outer region of the optically thick accretion disk. The applicable regime corresponds to accretion rates between about 0.01 and 0.3 of the Eddington accretion rate and, in some circumstances, is applicable up to about the Eddington limit. This technique is applicable to classical Seyfert galaxies, moderately luminous quasars, and black hole X-ray binaries in their luminous hard X-ray state. \nThermal continuum fitting utilizes the fact that the spin of an accreting BH influences the location of the innermost stable circular orbit (ISCO) in the accretion disk, thus determining the temperature of the blackbody radiation from that region. For prograde accretion, the higher the spin rate of the BH, the smaller the ISCO and the higher the temperature of the inner disk. This technique has been applied more to stellar-mass BH X-ray binaries than to accreting supermassive BHs. \nUsing these methods, many accreting supermassive BHs have been found to be rapidly spinning. However, there is a population of more slowly spinning BHs with masses greater than 3 × 10 7 M ⊙ , consistent with what is expected from structure formation models. Accreting stellar-mass BHs in \nX-ray binary systems have also been observed to be rapidly spinning, but, in many cases, their spins must have been rapid at birth instead of having been produced by spin-up accretion. Spin-up from near zero spin to maximal spin requires that a significant fraction of the BH mass be provided by accretion. In the case of high-mass X-ray binaries, the accretion process is time-limited due to the rapid evolution of the massive companion star. Even Eddington-limited cases would not have sufficient time to spin up the BH by accretion. For low-mass X-ray binaries, however, spin up by accretion is possible. \nLikely more relevant to the PBHs being discussed here are measurements of the spins of merging binary black holes (BBH) using the GW signal. These observations are cleaner because details of the accretion do not come into play. However, the effects of spin in the GW signal are subtle, and the sensitivity of LVK is such that, for the cases observed, the results presented in [36] interestingly favor low spins for the merging objects. \nBeyond measuring a black hole's spin, there remains the task of identifying whether it is primordial in origin. BBH merger events observed in the LVK third GW Transient Catalog (GWTC-3) [59] have widely been considered as a mixed population of astrophysical black holes (ABHs) and PBHs [60-62]. There are aspects of these observations that suggest the existence of a sub-population of PBHs. This includes the merger event rate of high mass BBH systems, including events that reside in the upper mass gap that applies to ABHs [63]. A recent Bayesian analysis estimated that 1 / 4 of the events in the GWTC-3 are due to PBHs [62]. However, because the mass spectrum of PBHs is expected to peak at high mass (30 M ⊙ ) [61], observations of high-mass BBH merger events are significantly more likely to be PBH systems. These systems are the ideal candidates for the observation described here because of the spin-up process described above. Repeated observations of such high-mass merger events will allow for further delineation of sub-populations of BBH mergers and would allow for a definitive test for the existence of cosmic strings with a string tension µ > 10 -20 .", '4 Spin-down of Supermassive Black Holes': "As discussed in Sec. 3, primordial black holes form with cosmic strings pre-attached. However, laterforming, non-primordial black holes can only become attached to a CS if the two collide. A stellar mass BH is an extremely small target, and the probability for a passing cosmic string to hit it is vanishingly low. A supermassive BH, on the other hand, is large enough that collisions with cosmic strings, although still rare, can be expected to occur over the lifetime of the BH in certain circumstances. \nThe rate at which strings collide with and become captured by a supermassive BH depends on size of the BH and the density of the CS network. Cosmic strings tend to cluster in galactic halos [64] where the BHs are located, improving the chances for a collision. The expected time for a CS collision was estimated in [24] as \nt c = 3 × 10 9 yr ( µ 10 -18 ) 2 ( M M SgrA ∗ ) -2 . 5 ( η (p)) -1 (4.1) \nwhere the mass of Sgr A*, M SgrA ∗ = 4 × 10 6 M ⊙ , is taken as fiducial mass and η ( p ) is an order-one function of the CS reconnection probability p . In the particular case of Sgr A*, if the cosmic strings are rather light, µ < 10 -18 , we would expect at least one CS to have been captured since the BH's formation, order 10 10 years ago. 7 \nIf a supermassive BH were to capture a CS, there would be observational consequences. For example, gravitational lensing could be used to detect the acceleration of a BH caused by the net force exerted by attached cosmic strings [65]. Oscillating cosmic strings, extracting rotational energy from the BH, could emit observable gravitational waves, although the particular signatures from such a process are still to be determinded [24]. We propose an alternative method to observe the CS spinning down the BH. \nSupermassive BHs are expected to be rapidly spinning, and those that have been observed all have large spins [36]. Lower mass supermassive BHs, those with M < 3 × 10 7 M ⊙ , all have near-extremal spins, χ > 0 . 9. The heaviest supermassive BHs, on the other hand, tend to have more modest spins. \nAs shown in Sec. 2, once the CS has been captured by the BH, it will begin extracting angular momentum and, along with it, energy. This spin-down process takes a long time (2.7) because there is a lot of angular momentum to extract. In the case of Sgr A*, for µ < 10 14 , the spin-down time would be t SD > 10 9 yr, implying that, however long ago the string was captured, the spin down would be currently ongoing. \nIn principle, the decreasing BH spin could be measured using the methods discussed in Sec. 3.3, for example X-ray reflection spectroscopy for an accreting supermassive BH. However, the spin is only decreasing at a rate of ˙ a/a ≈ 10 -9 yr -1 and the accuracy of these techniques is not nearly good enough to measure the spin to one part in 10 9 [36]. \nAlthough the BH's spin is challenging to measure, its mass M can be readily determined from observations of orbiting objects. In particular, a pulsar orbiting the BH would allow a very accurate measurement of the mass. Pulsars are the most accurate clocks in nature and have long played an important role in high-precision astrophysical measurements. Famously, the gradually decreasing period of the Hulse-Taylor binary pulsar was measured to one part in 10 15 , providing the first observational evidence for gravitational waves [66, 67]. \nSupermassive BHs are found in dense galactic centers, which are also expected to be the home of large numbers of compact objects. In our own galaxy, around 100 pulsars are predicted to be orbiting very close to Sgr A*, with orbital periods P ≲ 10 yr [68-70]. The Square Kilometer Array (SKA) is expected to probe the galactic center and observe these pulsars. \nAssuming such a pulsar is in fact found close to Sgr A*, timing observations will be able to very accurately measure P . Ongoing observations over an entire period will then determine the rate at which the period is changing. If Sgr A* is losing mass due to extraction by cosmic strings, its gravitational attraction on the pulsar will decrease, causing the pulsar to spiral outward and the period to increase 8 [71-73] \n˙ P P = -2 ˙ M SgrA ∗ M SgrA ∗ , (4.2) \nwhere the mass loss rate ˙ M is given by (2.4). For a pulsar with period P = 10yr, the period would be increasing at a rate of ˙ P ∼ 10 6 µ . Only sufficiently light strings, with µ < 10 -18 , would have a significant chance of being captured by Sgr A*, implying the effect on the period would be quite small: \n˙ P ≲ 10 -12 . (4.3) \nWould such a small effect be observable? With current observational technology, the orbital period of an isolated pulsar orbiting near Sgr A* could, in principle, be measurable to an accuracy of about \n∆ P ∼ 10 -11 yr [74-76]. Using two measurements spaced apart by one period, the rate of change of the period could then be measured to an accuracy of ˙ P ∼ ∆ P/P ∼ 10 -12 , which is within the order of magnitude of the size of the expected effect (4.3). However, the neighborhood around Sgr A* is rather crowded, and a pulsar in this region would likely not be sufficiently isolated to achieve this precision. Although the mass loss is, in principle, observable via pulsar measurements, significant observational improvements would be required to make it viable in practice. \nIn addition, other factors complicate this proposed observational approach. Sgr A* is continually accreting matter from its surroundings, and the resulting increase to the BH mass would be both large and uneven. However, a pulsar orbiting well outside any material surrounding the BH would measure via Kepler's 3rd law the mass of both the BH and the accreting matter. The gravitational effect on the pulsar of matter in an accretion disk wouldn't change once it fell into the BH. In contrast, a long CS extracts the mass far beyond the pulsar orbit and so reduces the gravitational pull on the pulsar and increases the orbital period. \nAs in the PBH case discussed in Sec. 3.2, long, horizon-crossing strings are required to effectively carry away the extracted energy and angular momentum and spin down the BH. A small CS loop captured by the BH would not spin it down, as the energy could just travel all the way around the loop and fall back in. Furthermore, the loop itself would eventually fall into the BH.", '5 Discussion and open questions': "Many physical scenarios for the early universe predict cosmic strings, and, indeed, F- or D-strings of string theory could manifest as cosmic strings. Despite their ubiquity in such theories, scant evidence of their actual existence has been found, but a very intriguing statistical GW background signal was found recently by NANOGrav consistent with cosmic strings. However this GW background turns out, alternative methods of testing for the existence of cosmic strings are quite valuable. Here, we have considered the resulting spin-down of a black hole due to attached cosmic strings, for various black hole sizes, finding something of an inverse 'goldilocks' problem in which the extremes may be preferred avenues for observation. \nThe prospects of observing the spin-down of stellar-mass black holes (with masses up to hundreds of solar masses) are dim for two reasons. First, the likelihood of such a black hole encountering a cosmic string is quite small, and, second, the spin down of the black hole would be quite fast and unlikely to be observed. \nA supermassive black hole (of order a million or more solar masses), on the other hand, suffers from neither of these problems. Its size makes for a significant probability of encountering a cosmic string and for a long spin-down time. Because the spin down is very gradual, observing it requires extremely sensitive measurements of the black hole spin. We suggest observing a pulsar orbiting close to Sgr A* and looking for out-spiraling due to energy being extracted from the black hole by the cosmic string. However, the signal is likely too small even for high-precision pulsar observations, at least with current technology. \nPrimordial black holes, though small, are likely to form with one or more cosmic strings already attached. Although they typically don't form with significant spin, PBHs soon get spun up by accretion, and, in the absence of cosmic strings to spin them down, large PBHs with masses over 30 M ⊙ are predicted to reach near-extremal spins. This allows a sharp test for the existence of cosmic strings: Large PBHs should be observed with almost no spin if cosmic strings exist and with very high spins if they do not. \nSeveral factors complicate the interpretation offered above which we have largely omitted from our analysis but which could play important roles. \nWe have presented a simplified description of the dynamics of the black hole-cosmic string system. For the purposes of estimating the energy extraction rate, the strings have been treated as rigidly rotating. Multiple long strings attached to a BH can interact with each other, for example, by intersecting and reconnecting, resulting in a short segment with both ends attached and which falls into the BH and another segment no longer attached to the BH. Other than the energy and spin extraction, other effects of the string on the black hole have been neglected. For example, an attached string would exert a force on the black hole, causing it to move. A more thorough discussion of these dynamics can be found elsewhere [24, 25]. \nWe have also neglected other aspects of black hole evolution, in particular, merger events. Typically, BH mergers result in highly spinning remnants, affecting, for example, the distribution of masses and spins of the BH population. As a particularly interesting example, two PBHs connected by a cosmic string would get pulled toward each other and eventually merge. If other strings remained attached, the resulting remnant could then be spun down. Such a scenario merits future investigation. More discussion of PBH evolution can be found in [55] and [35] 9 . \nFinally, differentiating PBHs from conventional stellar-sized BHs presents an interesting statistical challenge (see Ref. [35]), especially given that future generations of GW detectors will greatly expand the volume range of BH observations and that merger remnant BHs will be highly spinning.", 'Acknowledgments': 'We would like to thank Edward Smakov, Michael Ramsey, and Timmy Dhakaia who were involved in early stages of this project. The work was supported by the National Science Foundation grants PHY1820733, PHY-2000398, PHY-2310608 (MSL); AST-2011731 (MK); PHY-2011383, PHY-2308861 (SL); and AST-2011757 (JS). ML would also like to thank both the APCTP and the BNL EIC Theory Institute for warm hospitality while this work was being completed.', 'References': "- [1] A. Vilenkin and E. P. S. Shellard, Cosmic Strings and Other Topological Defects . 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2024arXiv240912898A

Treelevel gravitational amplitudes satisfy an infinite hierarchy of soft factorization theorems. The existence of these theorems has been recently linked with the existence of an infinite tower of asymptotic symmetries. In this paper we analyze the relevance of the soft graviton theorems beyond subleading order in the context of classical gravitational scattering in four dimensions. More in detail we show that the infinite impact parameter limit of the latetime gravitational field emitted during a classical scattering can be derived using these factorization theorems. The classical field obtained in this infinite impact parameter regime has an expansion in the frequency of the detector where the modes scale as omeganlogomega with a vanishing memory.

2024-09-01T00:00:00Z

['arXiv:2409.12898', '10.48550/arXiv.2409.12898', '2024arXiv240912898A']

['High Energy Physics - Theory', 'General Relativity and Quantum Cosmology']

On the classical limit of the subnleading soft graviton theorems in D 4 without deflection

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.12898.pdf

{'Samim Akhtar a,b': '- a The Institute of Mathematical Sciences \nIV Cross Road, C.I.T. Campus, Taramani, Chennai 600 113, India \n- b Homi Bhabha National Institute \nTraining School Complex, Anushakti Nagar, Mumbai 400 094, India \nE-mail: samimakhtar@imsc.res.in \nAbstract: Tree-level gravitational amplitudes satisfy an infinite hierarchy of soft factorization theorems. The existence of these theorems has been recently linked with the existence of an infinite tower of asymptotic symmetries. In this paper, we analyze the relevance of the soft graviton theorems beyond sub-leading order in the context of classical gravitational scattering in four dimensions. More in detail, we show that the infinite impact parameter limit of the late-time gravitational field emitted during a classical scattering can be derived using these factorization theorems. The classical field obtained in this (infinite impact parameter) regime has an expansion in the frequency of the detector where the modes scale as ω n log ω with a vanishing memory.', '1 Introduction': "The two most prominent perturbative techniques to analyze classical gravitational scattering of two objects whose Schwarzschild radii are parametrically small compared to the impact parameter of the scattering are the post-Minkowskian (PM) and post-Newtonian (PN) schemes [1-49]. In the former, one computes relevant classical observables such as scattering angle or the gravitational waveform as a perturbation series in the gravitational constant G . On the other hand, PN expansion can be used when the incoming velocities of the scattering particles or objects in a bound orbit are non-relativistic. The two perturbative schemes are intricately tied to each other in the case of large impact parameter as beautifully explained in [50, 51] and are the most potent tools in analyzing the relationship between quantum amplitudes and classical scattering. However, a complementary perturbative expansion leads to different insights about gravitational radiation emitted in a scattering process. This expansion is not in terms of parameters intrinsic to the scattering process but the characteristic frequency of the detector placed at null infinity. It is known as the soft expansion of gravitational radiation. At any given order in soft expansion, the radiative field is exact to all orders in PM and PN expansion. It is hence a non-perturbative \nprobe to gravitational scattering and offers remarkable insights into universal modes of gravitational radiation in classical scattering [52-66]. \nIt is by now well known that for a generic gravitational scattering, the radiative field has the following form under soft expansion, \nh µν ( ω,r, ˆ n ) = 1 r e iωr ∞ /summation.disp N =-1 ω N h N µν ( ˆ n ) + ∞ /summation.disp m = 0 ω m ( log ω ) m + 1 h log m µν ( ˆ n ) + /summation.disp N,M /divides.alt0 M -N > 1 ω M ( log ω ) N h log ( N,M ) µν ( ˆ n ) + O /parenleft.alt4 1 r 2 /parenright.alt4 , (1.1) \nwhere ˆ n is the unit vector on the celestial sphere and all the terms subleading in 1 r do not contribute to the radiative flux at null infinity [67-69]. \nThe classification of soft terms into three distinct families in equation 1.1 is as follows. The first one is a Laurent expansion in the detector frequency. The second set of terms that scale as ω N ln ω N + 1 capture the (infinite) set of leading logarithmic terms at all orders in the soft expansion and were discovered in a series of papers [55, 57, 58, 60-62]. Finally, the third family encapsulates the logarithmic terms which (for a given N ) are sub-leading relative to the corresponding leading logarithmic soft factors. \nThe reason for such a classification is intricately tied to the idea of universality inherent in soft expansion. Given a set of incoming and outgoing momenta and spins of the scattering objects, if a specific mode in the soft expansion does not depend on the details of the underlying equations of motion and the details of the scattering, then we refer to such a mode as a universal term in the soft expansion. It has been known since the early 60s that in a generic classical gravitational scattering in which massive objects with arbitrary multipole moments emit gravitational radiation, then the leading soft factor h -1 µν ( ˆ n ) is universal and only depends on the incoming and outgoing momenta of the scattering particles 1 . In last ten years, B. Sahoo, A. Sen, and their collaborators have shown that universality of gravitational radiation in the soft expansion is not only tied to the leading soft factor. It has been now rigorously shown that the leading logs N ≤ 2 are universal and only depend on the momenta of the incoming and outgoing massive objects [52, 55, 57, 58, 60-62, 76]. It has been conjectured that the universality extends to all N and a specific formula for the coefficient of ω N ln ω N + 1 has been put forward by Heissenberg et al in the case of 2 -2 scattering [76]. \nAn interesting aspect of the leading log soft factor h log µν ( ˆ n ) is that, unlike h -1 µν , it is sourced even in the limit when n particles which undergo scattering are mutually so far apart that each of them experiences a vanishing deflection! The source of such a radiative mode then is the asymptotic interaction between the incoming or outgoing states, leading to the emission of gravitational radiation only from t → ±∞ . In this paper, we analyze the classical gravitational radiation in D = 4 dimensions where two massive objects with momenta p 1 , p 2 scatter at finite impact parameter. We then analyze the soft expansion of gravitational radiation in /divides.alt0 b /divides.alt0 → ∞ limit such that ω /divides.alt0 b /divides.alt0 is fixed. As we show below, \nin this regime, all the terms of the form ω m log ω /divides.alt0 m ≥ 1 survive and can be completely determined by the so-called (sub) n -leading soft graviton theorems for tree-level gravitational amplitudes. These theorems reveal the extent to which a gravitational amplitude factorizes when one of the gravitons becomes soft as compared to other external momenta. This is discussed in more detail in Section 3. For any n ≥ 1 , (sub) n -leading soft expansion of the tree-level scattering amplitude for a generic theory of gravity with arbitrary matter coupling has the following form: \nlim ω → 0 ∂ n ω ω A 5 ( ˜ p 1 , ˜ p 2 → p 1 , p 2 , k ) = ˆ S n A 4 +B n ( p 1 , p 2 , p ' 1 , p ' 2 , ˆ k ) A 4 +R n ( p 1 , p 2 , p ' 1 , p ' 2 , ˆ k ) , (1.2) \nwhere B n is the non-universal part of the factorization formula that depends on the irrelevant three-point couplings in the theory [53, 77] and R n ≠ 0 ∀ n > 2 . Hence the soft expansion of the tree-level amplitude does not factorize. However, the tensorial structure of the remainder term R n , which is linear in the polarization of the soft graviton leads to the following 'factorization formula' for tree-level amplitude at all orders in the soft expansion [78]: \nlim ω → 0 ∂ n ω ω Π -ˆ n A 5 ( ˜ p 1 , ˜ p 2 → p 1 , p 2 , k ) = Π -ˆ n ˆ S n A 4 ( ˜ p 1 , ˜ p 2 → p 1 , p 2 ) , (1.3) \nwhere in the present paper we consider scattering of two scalar particles minimally coupled to gravity, hence B n = 0 ∀ n . Π -ˆ n ∶= D n + 1 z ( 1 + /divides.alt0 z /divides.alt0 2 ) -1 is the projection operator, where ˆ k = ( 1 , ˆ n ( z, ¯ z )) and z, ¯ z are the stereographic coordinates. \nWe thus see that higher-order soft theorems appear to be rather limited in their ability to constrain the gravitational scattering as they are only a component of the radiative field which is 'orthogonal' to the remainder term. In this paper, we prove that despite their limitations, the (sub) n -leading soft theorems can capture all the logarithmic soft factors in the limit of vanishing deflection. As these higher-order tree-level soft theorems are intricately tied to the discovery of the w 1 +∞ asymptotic symmetry algebra [79-82], our analysis can potentially reveal the link between higher spin asymptotic symmetries and a subset of logarithmic terms in the soft expansion of gravitational radiation. \nThis paper is organized as follows. In section 2, we review the KMOC formalism used to compute classical observables from scattering amplitudes. Then we review the infinite hierarchy of soft graviton theorems for tree-level amplitudes in section 3. Next, we briefly review the classical log soft theorems in section 3.1. In section 4, we compute the radiative field for the scattering of scalar particles via gravitational interaction using the KMOC formalism and take the soft expansion. In section 5, we compare the soft radiation obtained via the quantum soft theorems with the soft expansion of the radiation kernel to (sub) n -leading order, extract out the logarithmic contributions, and identify the remainder terms. We then summarise the important results for n = 3 i.e. the (sub) 3 -leading order in the soft expansion in section 6. We conclude by discussing some open questions in section 7. In the appendices we state our conventions, review the calculation of obtaining the logarithmic contributions to the soft radiation at (sub) 2 -leading order in frequency, and evaluate some integrals that are used in the main text.", '2 KMOC formalism in a nutshell': "The KMOC formalism [83, 84] is a framework used to compute classical observables, such as linear impulse or radiation emitted from on-shell scattering amplitudes for large impact parameter scattering 2 . The basic idea is to take wave packets for incoming (classical) particles, evolve them using a quantum S-matrix operator, and then compute the expectation value of an observable in the final state. An appropriate classical limit is then applied to obtain the classical result. The primary characteristic of the formalism is that the classical limit is taken before evaluating the full amplitude which makes the computation substantially simpler. Additionally, radiation reaction effects are inherently encoded within the framework. For a short sample of the results obtained with the formalism, we refer the reader to [6, 85-88]. \nThe initial state is described as: \nwhere \n/divides.alt0 Ψ /uni27E9 = /integral.disp 2 /product.disp i = 1 d Φ ( p i ) e ip 2 · b /slash.left /uni0335 h ϕ i ( p i ) /divides.alt0 /uni20D7 p 1 ; /uni20D7 p 2 /uni27E9 , (2.1) \nd Φ ( p ) = d 4 p ( 2 π ) 4 ˆ δ ( p 2 -m 2 ) Θ ( p 0 ) , /integral.disp d Φ ( p ) /divides.alt0 ϕ ( p )/divides.alt0 2 = 1 , (2.2) \nϕ i ( p i ) s are the minimum uncertainty wave packets for the particles. The wavepacket of the second particle is translated, with respect to the first particle's wavepacket, by a distance of b - the impact parameter. As the initial particles are described by coherent states we have \n/uni27E8 P µ i /uni27E9 = m i u µ i +O( /uni0335 h ) , σ 2 i m 2 i = (/uni27E8 P 2 i /uni27E9 -/uni27E8 P i /uni27E9 2 ) m 2 i /uni0335 h → 0 /leftrightline/leftrightline→ 0 , (2.3) \nwhere σ 2 i is the variance and m i s are the masses of the particles. Here the expectation value of the momentum operator is with respect to the initial state in equation (2.1). For massive spinning particles and more detailed construction of these wavefunctions, we refer the reader to [89]. \nThe computation of classical observables involves the change in the expectation value of a quantum mechanical operator: \n/uni27E8 ∆ O A /uni27E9 = /uni27E8 Ψ /divides.alt0 S † O A S /divides.alt0 Ψ /uni27E9 -/uni27E8 Ψ /divides.alt0 O A /divides.alt0 Ψ /uni27E9 . (2.4) \nwhere S = I + iT is the S-matrix. For the linear impulse, O A = P µ , the momentum operator, and O A = h µν ( x ) , the graviton field operator, from which we read off the radiation kernel. The expression in equation (2.4) can be simplified using the on-shell completeness relation and unitarity of the S-matrix [83]. To compute radiation, an important intermediate quantity introduced by KMOC is the so-called radiation kernel R µν ( k ) . Radiation kernel is simply the field radiated at momentum k µ = ω ( 1 , ˆ n ) and is a result of in-elastic scattering. \nThe radiation kernel has the following compact expression [84] \n/uni27E8R µν ( k )/uni27E9 = /uni0335 h 3 /slash.left 2 /integral.disp 2 /product.disp i = 1 ˆ d 4 q i ˆ δ ( 2 p i · q i + q 2 i ) e -iq 2 · b /slash.left /uni0335 h ˆ δ ( 4 ) ( q 1 + q 2 -k ) A µν 5 ( p 1 + q 1 , p 2 + q 2 → p 1 , p 2 , k ) + O( T † T ) , (2.5) \nwhere A µν 5 is the tree-level five-point amplitude. The classical limit is taken at the level of the integrand, by expressing massless momenta in terms of their wave numbers (e.g. q i = ¯ q i /uni0335 h ), appropriately rescaling the dimensionful couplings and keeping leading order terms in /uni0335 h . In gravity, the dimensionless coupling is obtained by κ → κ /slash.left √ /uni0335 h , where κ = √ 32 πG . Here G is the gravitational constant. We can now write down the classical limit of the radiation kernel, at leading order in the coupling as follows \nR µν ( ¯ k ) = /uni27EA.alt3 /integral.disp 2 /product.disp i = 1 ˆ d 4 ¯ q i ˆ δ ( 2 p i · ¯ q i ) e -i ¯ q 2 · b ˆ δ ( 4 ) ( ¯ q 1 + ¯ q 2 -¯ k ) /parenleft.alt1 /uni0335 h 2 A µν 5 ( p 1 + /uni0335 h ¯ q 1 , p 2 + /uni0335 h ¯ q 2 → p 1 , p 2 , /uni0335 h ¯ k )/parenright.alt1/uni27EB.alt3 . (2.6) \nHere /uni27EA.alt3 f ( p 1 , p 2 , q . . . )/uni27EB.alt3 denotes the integration over the minimum uncertainty wave packets which localizes the momenta onto their classical values. Note that in the above expression, we have stripped the /uni0335 h -scaling of the coupling constant. The classical limit of the radiation kernel is the radiative gravitational field. To obtain the soft radiative field, we then carry out a soft expansion of the radiation kernel in the frequency of the emitted graviton. This will give us the classical radiation to (sub) n -leading order in frequency. The KMOC formalism has been generalized to describe many types of scattering. For example, in [84] the formalism has been extended to include incoming waves in the initial state. It has also been extended to include additional internal degrees of freedom, such as color charges in [90]. It has also been generalized to describe scattering in curved backgrounds [91, 92].", '3 A brief review of soft graviton theorems for tree-level amplitude': "In this section, we review the infinite hierarchy of soft graviton theorems for tree-level amplitudes. We consider massive scalars coupled to gravity and analyze five-point scattering amplitude in which two scalar particles with momenta ˜ p 1 , ˜ p 2 scatter into p 1 , p 2 and a graviton. Our kinematics satisfy the following on-shell conditions \np 2 i = ˜ p 2 i = m 2 i . (3.1) \nFor a generic theory of gravity with arbitrary matter coupling, the (sub) n /divides.alt0 n ≥ 1 soft graviton theorem can be written as follows [93, 94]. \nlim ω → 0 ∂ n ω ω A 5 ( ˜ p 1 , ˜ p 2 → p 1 , p 2 , k ) = ˆ S n A 4 +B n ( p 1 , p 2 , p ' 1 , p ' 2 , ˆ k ) A 4 +R n ( p 1 , p 2 , p ' 1 , p ' 2 , ˆ k ) (3.2) where the (sub) n -leading soft factor is defined as \nˆ S n = /uni23A7 /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23A8 /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23A9 i κ 2 ∑ i = 1 , 2 ϵ µν /bracketleft.alt3 1 p i · k p ( µ i ˆ J ν ) ρ i k ρ -1 ˜ p i · k ˜ p ( µ i ˆ ˜ J ν ) ρ i k ρ /bracketright.alt3 , if n = 1 κ 2 ∑ i = 1 , 2 ϵ µν /bracketleft.alt4 ˆ J µρ i k ρ ˆ J νσ i k σ p i · k /parenleft.alt2 k · ∂ ∂p i /parenright.alt2 n -2 + ˆ ˜ J µρ i k ρ ˆ ˜ J νσ i k σ ˜ p i · k /parenleft.alt2 k · ∂ ∂ ˜ p i /parenright.alt2 n -2 /bracketright.alt4 if n ≥ 2 . (3.3) \nB n is the non-universal part of the factorization formula which depends on the irrelevant three-point couplings in the theory [53, 77]. \nR n = ϵ µν ˆ k α 1 ˆ k α 2 /uni22EF ˆ k α n -1 A µνα 1 α 2 /uni22EF α n -1 (3.4) \nis the so called Remainder term which vanishes for n < 3 and spoils factorisation for n ≥ 3 . A is antisymmetric in any two indices among µ and α 's. As was shown in [78], however, the Remainder term always satisfies the following constraint simply due to the tensor structure of the amplitude. \nD n + 1 z ( 1 +/divides.alt0 z /divides.alt0 2 ) -1 R n = 0 , (3.5) \nwhere ˆ k = ( 1 , ˆ n ( z, ¯ z )) and z, ¯ z are the stereographic coordinates. In the present paper, we consider the gravitational scattering of two minimally coupled scalars ϕ 1 , ϕ 2 with masses m 1 , m 2 respectively, and as a result \nB n = 0 ∀ n (3.6) \nHowever, as discussed above, the remainder term R n continue to persist ∀ n ≥ 3 . \nThe leading order contribution to the gravitational radiation arises from the tree-level five-point amplitude which in this case is a sum over seven Feynman diagrams shown in Figure 1. \nFigure 1 : Tree-level five-point amplitudes for gravitational scattering of massive particles \n<!-- image --> \nVII \nThe unstripped five-point amplitude is given by \n¯ M µν 5 [ p 1 + /uni0335 h ¯ q 1 , p 2 + /uni0335 h ¯ q 2 → p 1 , p 2 , , /uni0335 h ¯ k ] = ˆ δ ( 4 ) ( ¯ q 1 + ¯ q 2 -¯ k ) ¯ A µν 5 [ p 1 + /uni0335 h ¯ q 1 , p 2 + /uni0335 h ¯ q 2 → p 1 , p 2 , /uni0335 h ¯ k ] . (3.7) \nHere the stripped amplitude ¯ A µν 5 is given by [95, 96]: \n¯ A µν 5 [ p 1 + /uni0335 h ¯ q 1 , p 2 + /uni0335 h ¯ q 2 → p 1 , p 2 , , /uni0335 h ¯ k ] = -κ 3 m 2 1 m 2 2 /uni0335 h 2 /bracketleft.alt3 4 P µ P ν ¯ q 2 1 ¯ q 2 2 + 2 γ ¯ q 2 1 ¯ q 2 2 /parenleft.alt3 Q µ P ν + Q ν P µ /parenright.alt3 +/parenleft.alt3 γ 2 -1 2 /parenright.alt3/parenleft.alt3 Q µ Q ν ¯ q 2 1 ¯ q 2 2 -P µ P ν ω 2 1 ω 2 2 /parenright.alt3/bracketright.alt3 , (3.8) \nwhere κ = √ 32 πG and \nP µ = -ω 1 u µ 2 + ω 2 u µ 1 Q µ = ( ¯ q 1 -¯ q 2 ) µ + ¯ q 2 1 ω 1 u µ 1 -¯ q 2 2 ω 2 u µ 2 , ω 1 = -¯ k · u 1 , ω 2 = -¯ k · u 2 . (3.9) \nAs k µ ¯ A µν 5 = k ν ¯ A µν 5 = 0 , the amplitude is gauge-invariant. The (sub) n -leading soft graviton theorem for the tree-level amplitude can be restated as follows \nˆ δ ( 4 ) ( q 1 + q 2 -k )A µν 5 [ p 1 + q 1 , p 2 + q 2 → p 1 , p 2 , k ] = n /summation.disp r = 0 (-1 ) n -r ( n -r ) ! S ( r ) ,µν ( k · ∂ ) n -r ( ˆ δ ( 4 ) ( q 1 + q 2 ))A 4 [ p 1 + q 1 , p 2 + q 2 → p 1 , p 2 ] + X µν , (3.10) \nwhere the soft graviton factors are given by \nS ( 0 ) ,µν = κ /summation.disp i = 1 , 2 /bracketleft.alt4 1 p i · k p ( µ i p ν ) i -1 ˜ p i · k ˜ p ( µ i ˜ p ν ) i /bracketright.alt4 S ( 1 ) ,µν = i κ 2 /summation.disp i = 1 , 2 /bracketleft.alt4 1 p i · k p ( µ i ˆ J ν ) ρ i k ρ -1 ˜ p i · k ˜ p ( µ i ˆ ˜ J ν ) ρ i k ρ /bracketright.alt4 S ( n ) ,µν = κ 2 /summation.disp i = 1 , 2 /uni23A1 /uni23A2 /uni23A2 /uni23A2 /uni23A2 /uni23A2 /uni23A3 ˆ J µρ i k ρ ˆ J νσ i k σ p i · k /parenleft.alt4 k · ∂ ∂p i /parenright.alt4 n -2 + ˆ ˜ J µρ i k ρ ˆ ˜ J νσ i k σ ˜ p i · k /parenleft.alt4 k · ∂ ∂ ˜ p i /parenright.alt4 n -2 /uni23A4 /uni23A5 /uni23A5 /uni23A5 /uni23A5 /uni23A5 /uni23A6 , n ≥ 2 . (3.11) \nR n = ϵ µν X µν is the remainder term that spoils the factorization beyond the sub-subleading order in the soft expansion. For the amplitude (3.8), it is given by the following expression \nX µν = κ 3 m 2 1 m 2 2 4 n /summation.disp r = 3 (-1 ) n -r ( n -r ) ! ( k · ∂ ) n -r ( ˆ δ ( 4 ) ( q 1 + q 2 )) Λ µν r -1 +( 1 ↔ 2 ) , (3.12) \nwhere the polynomial Λ µν n is defined as \nΛ µν n ≥ 2 = H µν 2 2 n -2 ( ¯ q · ¯ k ) n -2 ( ¯ q 2 ) n -2 . (3.13) \nHere \nwhere \nH µν 2 = -4 ( ¯ q 2 ) 2 /parenleft.alt3 ω 2 2 u µ 1 u ν 1 -ω 1 ω 2 2 ( u µ 2 u ν 1 + u ν 2 u µ 1 )/parenright.alt3 . (3.14) \nAs can be explicitly checked R n satisfies the constraint equation (3.5) and the resulting A is antisymmetric in the indices as stated in equation (3.4). The factorized terms that were obtained via the soft graviton factors on the four-point amplitude in equation (3.10) can be written as follows \nwhere the polynomials K µν n are defined as \nκ 3 m 2 1 m 2 2 4 n /summation.disp r = 0 (-1 ) n -r ( n -r ) ! ( k · ∂ ) n -r ( ˆ δ ( 4 ) ( q 1 + q 2 )) K µν r -1 +( 1 ↔ 2 ) , (3.15) \nK \nK µν -1 = -1 ¯ q 2 /parenleft.alt3 γ 2 -1 2 /parenright.alt3/braceleft.alt3 -1 ω 1 ¯ q ( µ u ν ) 1 -1 ω 2 1 ( ¯ k · ¯ q ) u µ 1 u ν 1 /braceright.alt3 K µν 0 = -2 γ ¯ q 2 /parenleft.alt3 -u ( µ 1 u ν ) 2 + 2 ω 2 ω 1 u µ 1 u ν 1 /parenright.alt3 + H µν 0 K µν + 1 = H µν 1 + H µν 0 2 ( ¯ q · ¯ k ) ¯ q 2 µν n 2 = H µν 1 2 n -1 ( ¯ q · ¯ k ) n -1 ( ¯ q 2 ) n 1 + H µν 0 2 n ( ¯ q · ¯ k ) n ( ¯ q 2 ) n , (3.16) \n≥ - \nH µν 1 = 4 γ ¯ q 2 ω 2 ¯ q 2 ¯ q ( µ u ν ) 1 and H µν 0 = -2 ¯ q 2 /parenleft.alt3 γ 2 -1 2 /parenright.alt3 ¯ q µ ¯ q ν ¯ q 2 (3.17)", '3.1 A brief review of classical log soft theorems': "Classical soft theorems govern the non-analytic behavior of electromagnetic and gravitational waveforms in the low-frequency domain of the detector. The most well-known example of the classical soft graviton theorem is the gravitational memory effect [70-73] which predicts a permanent change in the asymptotic metric fluctuation caused by the passage of a gravitational wave. Memory effect is an observable of classical scattering and is simply the coefficient of the leading term in the soft expansion of the radiative field [97-99]. As reviewed in the introduction, in recent years, a hierarchy of universal soft theorems in four dimensions have been discovered [55, 57, 58, 60-62]. \nThe table in [62] summarizes the different orders of the low-frequency gravitational waveform in the ω → 0 limit and their relations to PM expansion. It also illustrates the late and early time behavior of the gravitational waveform at large retarded time u . In this paper, we focus on tree-level scattering. The loop-level corrections to the leading logs will be pursued elsewhere. \nAs stated before, soft graviton theorems are exact statements describing the gravitational radiation in soft frequency expansion. We consider a 2 → 2 scattering process with a large impact parameter. These processes can then be studied within perturbation theory. If p ' a is the final momentum of a particle and the initial momentum is p a , then we have \np ' µ a = p µ a + ∞ /summation.disp n = 1 κ 2 n ∆ p ( n ) µ a , (3.18) \nwhere ∆ p ( 1 ) µ a is the LO linear impulse and κ 2 n -th term is the N n -1 LO impulse. Consistency requires that the radiation at any perturbative order matches the classical soft factor. Plugging the equation (3.18) in the radiative field (1.1), it is evident that both log ω and ω log ω survive even at leading order in the coupling. This then connects the appearance of ω n log ω from tree-level amplitudes which we will compute in the subsequent sections.", '4 Gravitational Scattering of massive spinless particles': "We consider gravitational scattering of two scalars ϕ 1 , ϕ 2 with masses m 1 , m 2 respectively. KMOC formalism provides us with a formula to compute the gravitational radiative field. In a scattering process where two scalar particles with initial momenta p 1 , p 2 gravitationally scatter at impact parameter distance b , the radiative field is given by the following equation \nh µν ( ω, ˆ n ) = lim /uni0335 h → 0 /uni0335 h 3 /slash.left 2 /integral.disp 2 /product.disp i = 1 ˆ d 4 ¯ q i ˆ δ ( 2 p i · ¯ q i ) e i ¯ q 1 · b ˆ δ ( 4 ) ( ¯ q 1 + ¯ q 2 -¯ k ) /parenleft.alt1A µν 5 ( p 1 + /uni0335 h ¯ q 1 , p 2 + /uni0335 h ¯ q 2 → p 1 , p 2 , /uni0335 h ¯ k /parenright.alt1 +/uni22EF + O( κ 5 ) (4.1) \nwhere A µν 5 is the five point amplitude. The final momenta are integrated over and this integration is written in terms of the momentum exchange q i = ˜ p i -p i . The leading order contribution to the gravitational radiation arises from the tree-level amplitude (3.8) and it is given by \nR µν ( ω, ˆ n ) = -κ 3 m 1 m 2 4 /integral.disp ˆ d 4 ¯ q 1 ˆ d 4 ¯ q 2 ˆ δ ( u 1 · ¯ q 1 ) ˆ δ ( u 2 · ¯ q 2 ) ˆ δ ( 4 ) ( ¯ k -¯ q 1 -¯ q 2 ) e ib · ¯ q 1 ×/bracketleft.alt3 4 ¯ q 2 1 ¯ q 2 2 /parenleft.alt3 ω 2 1 u µ 2 u ν 2 + ω 2 2 u µ 1 u ν 1 -ω 1 ω 2 ( u µ 2 u ν 1 + u ν 2 u µ 1 )/parenright.alt3 + 2 γ ¯ q 2 1 ¯ q 2 2 /parenleft.alt3 -ω 1 ( ¯ q 1 -¯ q 2 ) ( µ u ν ) 2 + ω 2 ( ¯ q 1 -¯ q 2 ) ( µ u ν ) 1 -¯ q 2 1 u ( µ 1 u ν ) 2 -¯ q 2 2 u ( µ 2 u ν ) 1 + 2 ω 2 ω 1 ¯ q 2 1 u µ 1 u ν 1 + 2 ω 1 ω 2 ¯ q 2 2 u µ 2 u ν 2 /parenright.alt3 +/parenleft.alt3 γ 2 -1 2 /parenright.alt3/braceleft.alt3 1 ¯ q 2 1 ¯ q 2 2 /parenleft.alt3( ¯ q 1 -¯ q 2 ) µ ( ¯ q 1 -¯ q 2 ) ν + ¯ q 2 1 ω 1 ( ¯ q 1 -¯ q 2 ) ( µ u ν ) 1 -¯ q 2 2 ω 2 ( ¯ q 1 -¯ q 2 ) ( µ u ν ) 2 /parenright.alt3 + 1 ω 2 1 /parenleft.alt3 -2 ¯ k · ¯ q 2 ¯ q 2 2 /parenright.alt3 u µ 1 u ν 1 + 1 ω 2 2 /parenleft.alt3 -2 ¯ k · ¯ q 1 ¯ q 2 1 /parenright.alt3 u µ 2 u ν 2 /braceright.alt3/bracketright.alt3 . (4.2) \nWe can rewrite the expression as follows \nR µν ( ω, ˆ n ) = -κ 3 m 1 m 2 4 /integral.disp ˆ d 4 ¯ q ˆ δ ( u 1 · ¯ k -u 1 · ¯ q ) ˆ δ ( u 2 · ¯ q ) e ib ·( ¯ k -¯ q ) ×/bracketleft.alt3 4 ( ¯ k -¯ q ) 2 ¯ q 2 /parenleft.alt3 ω 2 2 u µ 1 u ν 1 -ω 1 ω 2 2 ( u µ 2 u ν 1 + u ν 2 u µ 1 )/parenright.alt3 + 2 γ ¯ q 2 /parenleft.alt3 -2 ω 2 ( ¯ k -¯ q ) 2 ¯ q ( µ u ν ) 1 -u ( µ 1 u ν ) 2 + 2 ω 2 ω 1 u µ 1 u ν 1 /parenright.alt3 + 2 ¯ q 2 /parenleft.alt3 γ 2 -1 2 /parenright.alt3/braceleft.alt3 1 ( ¯ k -¯ q ) 2 ¯ q µ ¯ q ν -1 ω 1 ¯ q ( µ u ν ) 1 -1 ω 2 1 ( ¯ k · ¯ q ) u µ 1 u ν 1 /braceright.alt3/bracketright.alt3 -κ 3 m 1 m 2 4 /integral.disp ˆ d 4 ¯ q ˆ δ ( u 1 · ¯ q ) ˆ δ ( u 2 · ¯ k -u 2 · ¯ q ) e ib · ¯ q ×/bracketleft.alt3 4 ( ¯ k -¯ q ) 2 ¯ q 2 /parenleft.alt3 ω 2 1 u µ 2 u ν 2 -ω 1 ω 2 2 ( u µ 2 u ν 1 + u ν 2 u µ 1 )/parenright.alt3 + 2 γ ¯ q 2 /parenleft.alt3 -2 ω 1 ( ¯ k -¯ q ) 2 ¯ q ( µ u ν ) 2 -u ( µ 2 u ν ) 1 + 2 ω 1 ω 2 u µ 2 u ν 2 /parenright.alt3 + 2 ¯ q 2 /parenleft.alt3 γ 2 -1 2 /parenright.alt3/braceleft.alt3 1 ( ¯ k -¯ q ) 2 ¯ q µ ¯ q ν -1 ω 2 ¯ q ( µ u ν ) 2 -1 ω 2 2 ( ¯ k · ¯ q ) u µ 2 u ν 2 /braceright.alt3/bracketright.alt3 . (4.3) \nSoft expansion of the radiation kernel is obtained via the following two expansions: \nˆ δ ( u 1 · ¯ k -u 1 · ¯ q ) = ˆ δ ( u 1 · ¯ q ) - ( u 1 · ¯ k ) ˆ δ ' ( u 1 · ¯ q ) + ( u 1 · ¯ k ) 2 2! ˆ δ '' ( u 1 · ¯ q ) +/uni22EF + (1 ) n ( u 1 · ¯ k ) n n ! ˆ δ ( n ) ( u 1 · ¯ q ) (4.4) \nand \n1 ( ¯ k -¯ q ) 2 = 1 ¯ q 2 -2 ( ¯ q · ¯ k ) = 1 ¯ q 2 /parenleft.alt3 1 + 2 ( ¯ q · ¯ k ) ¯ q 2 + 4 ( ¯ q · ¯ k ) 2 ( ¯ q 2 ) 2 + 8 ( ¯ q · ¯ k ) 3 ( ¯ q 2 ) 3 +/uni22EF/parenright.alt3 (4.5) \nNote that the range of integration over exchange momentum is ω < /divides.alt0 q ⊥ /divides.alt0 < b -1 , where ω = /divides.alt0 k /divides.alt0 and hence the integral of the terms from the expansion in equation (4.5) is IR finite. \nThe leading order soft radiation can be written as follows \nR ( 0 ) ,µν = κ 3 m 1 m 2 4 /integral.disp ˆ d 4 ¯ q /braceleft.alt3 e -ib · ¯ q ˆ δ ( u 1 · ¯ q ) ˆ δ ( u 2 · ¯ q )/parenleft.alt3 K µν -1 /parenright.alt3 + e ib · ¯ q /parenleft.alt3 1 ↔ 2 /parenright.alt3/braceright.alt3 = κ 3 m 1 m 2 4 /parenleft.alt3 γ 2 -1 2 /parenright.alt3 /integral.disp ˆ d 4 ¯ q ¯ q 2 e -ib · ¯ q ˆ δ ( u 1 · ¯ q ) ˆ δ ( u 2 · ¯ q )/parenleft.alt3 1 ω 1 ¯ q ( µ u ν ) 1 + ( ¯ k · ¯ q ) ω 2 1 u µ 1 u ν 1 /parenright.alt3 +( 1 ↔ 2 ) = -κ 3 m 1 m 2 8 πγβb 2 /parenleft.alt3 γ 2 -1 2 /parenright.alt3/parenleft.alt3 1 ω 1 b ( µ u ν ) 1 + ( ¯ k · b ) ω 2 1 u µ 1 u ν 1 /parenright.alt3 +( 1 ↔ 2 ) . (4.6) \nNote that in the /divides.alt0 b /divides.alt0 → ∞ limit, this term doesn't contribute to the kernel. This is expected as the particles are so far apart in this limit that they experience no deflection, and hence the memory effect vanishes! However, we show below that, unlike the memory effect, the leading log soft factor survives in this vanishing deflection limit. \nUsing the expansion of the delta functions and the five-point amplitude, the sub-leading order soft radiation can be expressed as follows \nR ( 1 ) ,µν = κ 3 m 1 m 2 4 /integral.disp ˆ d 4 ¯ q /bracketleft.alt3/braceleft.alt3 e -ib · ¯ q ˆ δ ( u 1 · ¯ q ) ˆ δ ( u 2 · ¯ q )/parenleft.alt3 K µν 0 /parenright.alt3 + e ib · ¯ q /parenleft.alt3 1 ↔ 2 /parenright.alt3/braceright.alt3 -/braceleft.alt3 e -ib · ¯ q ˆ δ ' ( u 1 · ¯ q ) ˆ δ ( u 2 · ¯ q )( u 1 · k )/parenleft.alt3 K µν -1 /parenright.alt3 + e ib · ¯ q /parenleft.alt3 1 ↔ 2 /parenright.alt3/braceright.alt3 + e -ib · ¯ q ˆ δ ( u 1 · ¯ q ) ˆ δ ( u 2 · ¯ q )( ib · ¯ k ) K µν -1 /bracketright.alt3 . (4.7) \nR ( 1 ) ,µν = κ 3 m 1 m 2 4 /parenleft.alt1 I µν 1 + I µν 2 + I µν 3 /parenright.alt1 (4.8) \nHere, the superscript in R ( 1 ) ,µν denotes the truncation of the radiation kernel to O( ω 0 ) . We can re-write R ( 1 ) ,µν as \nAll three terms are evaluated in Appendix C and the final result is summarised below. \nI µν 1 = /integral.disp ˆ d 4 ¯ qe -ib · ¯ q ˆ δ ( u 1 · ¯ q ) ˆ δ ( u 2 · ¯ q )/bracketleft.alt3 2 γ ¯ q 2 /parenleft.alt3 -u ( µ 1 u ν ) 2 + 2 ω 2 ω 1 u µ 1 u ν 1 /parenright.alt3 + 2 ¯ q 2 /parenleft.alt3 γ 2 -1 2 /parenright.alt3 ¯ q µ ¯ q ν ¯ q 2 /bracketright.alt3 = γ πγβ log ( ωb )/parenleft.alt3 u ( µ 1 u ν ) 2 -2 ( u 2 · ¯ k ) ( u 1 · ¯ k ) u µ 1 u ν 1 /parenright.alt3 + O ( ω 0 ) (4.9) \nI µν 2 = -/integral.disp ˆ d 4 ¯ q ¯ q 2 e -ib · ¯ q ˆ δ ' ( u 1 · ¯ q ) ˆ δ ( u 2 · ¯ q )/parenleft.alt3 γ 2 -1 2 /parenright.alt3( u 1 · ¯ k )/parenleft.alt3 1 ω 1 ¯ q ( µ u ν ) 1 + ( ¯ k · ¯ q ) ω 2 1 u µ 1 u ν 1 /parenright.alt3 = 1 2 πγ 3 β 3 /parenleft.alt3 γ 2 -1 2 /parenright.alt3 log ( ωb )/braceleft.alt3 -( γu 2 -u 1 ) ( µ u ν ) 1 +/parenleft.alt1 γ ( u 2 · ¯ k ) u 1 · ¯ k -1 /parenright.alt1 u µ 1 u ν 1 /braceright.alt3 , (4.10) \nand finally, \nI µν 3 = /integral.disp ˆ d 4 ¯ q ¯ q 2 e -ib · ¯ q ˆ δ ( u 1 · ¯ q ) ˆ δ ( u 2 · ¯ q )( ib · ¯ k )/parenleft.alt3 γ 2 -1 2 /parenright.alt3/parenleft.alt3 1 ω 1 ¯ q ( µ u ν ) 1 + ( ¯ k · ¯ q ) ω 2 1 u µ 1 u ν 1 /parenright.alt3 = -( b · k ) 2 πγβb 2 /parenleft.alt3 γ 2 -1 2 /parenright.alt3/parenleft.alt3 1 ω 1 b ( µ u ν ) 1 + ( ¯ k · b ) ω 2 1 u µ 1 u ν 1 /parenright.alt3 , (4.11) \nWe note that I µν 3 has no logarithmic terms unlike I µν 1 and I µν 2 . Therefore the logarithmic contribution of the first particle with initial momentum p 1 to the gravitational radiation is given by \nR log ω,µν 1 = κ 3 m 1 m 2 4 πγ 3 β 3 log ( ωb ) γ ( 2 γ 2 -3 )/parenleft.alt3 u ( µ 1 u ν ) 2 -( u 2 · ¯ k ) ( u 1 · ¯ k ) u ( µ 1 u ν ) 1 /parenright.alt3 . (4.12) \nThis expression matches with the classical result for sub-leading log soft graviton factor [55, 57] and with the result obtained using quantum soft theorems [100]. As stated before, in the deflection less limit ( /divides.alt0 b /divides.alt0 → ∞ ) such that ωb is fixed, the leading logarithmic contribution survives! In fact, all the log terms of the form ω m log ω /divides.alt0 m ≥ 1 survive and can be completely determined by the (sub) n -leading soft graviton theorems for tree-level gravitational amplitudes. \nWe can similarly compute the radiation kernel at (sub) 2 -leading order in the soft expansion at leading order in the coupling given in Appendix B and the result matches with existing results in the literature. \nSimilarly, using the expansion of the delta functions and the five-point amplitude given in equations (4.4) and (4.5) respectively, the (sub) n -leading order soft radiation is given by \nR ( n ) ,µν ( ¯ k ) = κ 3 m 1 m 2 4 /integral.disp ˆ d 4 ¯ q /bracketleft.alt3 n /summation.disp r = 0 1 ( n -r ) ! e -ib · ¯ q ˆ δ ( u 1 · ¯ q ) ˆ δ ( u 2 · ¯ q )( ib · ¯ k ) n -r K µν r -1 + n -1 /summation.disp r = 0 (-1 ) n -r ( n -r ) ! /braceleft.alt3 e -ib · ¯ q ˆ δ ( n -r ) ( u 1 · ¯ q ) ˆ δ ( u 2 · ¯ q )( u 1 · ¯ k ) n -r /parenleft.alt3 K µν r -1 /parenright.alt3 + e ib · ¯ q /parenleft.alt3 1 ↔ 2 /parenright.alt3/braceright.alt3 + n -2 /summation.disp r = 0 /summation.disp t,s ≥ 1 ∋( t + s )= n -r (-1 ) s t ! s ! e -ib · ¯ q ( ib · ¯ k ) t ( u 1 · ¯ k ) s ˆ δ ( s ) ( u 1 · ¯ q ) ˆ δ ( u 2 · ¯ q ) K µν r -1 /bracketright.alt3 , (4.13) \nUsing the integrals evaluated in Appendix C, we will isolate the logarithmic contributions ( ω n -1 log ( ωb ) ) that come only from the following two types of integrals: \nwhere the polynomial K µν n is defined in Section 3. Here, the superscript in R ( n ) ,µν denotes the truncation of the radiation kernel to O( ω n -1 ) . \nI µν 1 = /integral.disp ˆ d 4 ¯ q /bracketleft.alt3 1 ( n -1 ) ! e -ib · ¯ q ˆ δ ( u 1 · ¯ q ) ˆ δ ( u 2 · ¯ q )( ib · ¯ k ) n -1 K µν 0 = -1 ( n -1 ) ! /integral.disp ˆ d 4 ¯ qe -ib · ¯ q ˆ δ ( u 1 · ¯ q ) ˆ δ ( u 2 · ¯ q )( ib · ¯ k ) n -1 /bracketleft.alt3 2 γ ¯ q 2 /parenleft.alt3 -u ( µ 1 u ν ) 2 + 2 ω 2 ω 1 u µ 1 u ν 1 /parenright.alt3 + 2 ¯ q 2 /parenleft.alt3 γ 2 -1 2 /parenright.alt3 ¯ q µ ¯ q ν ¯ q 2 /bracketright.alt3 = i n -1 γ ( n -1 ) ! πγβ ( ωb ) n -1 log ( ωb )/parenleft.alt3 u ( µ 1 u ν ) 2 -( u 2 · ¯ k ) ( u 1 · ¯ k ) u ( µ 1 u ν ) 1 /parenright.alt3 +O( ω n -1 ) , (4.14) \nwhere we have used the integral result of equation (C.2) and \nI µν 2 = -1 ( n -1 ) ! e -ib · ¯ q ( ib · ¯ k ) n -1 ( u 1 · ¯ k ) ˆ δ ' ( u 1 · ¯ q ) ˆ δ ( u 2 · ¯ q ) K µν -1 = 1 ( n -1 ) ! /integral.disp ˆ d 4 ¯ qe -ib · ¯ q ˆ δ ' ( u 1 · ¯ q ) ˆ δ ( u 2 · ¯ q )( u 1 · ¯ k )( ib · ¯ k ) n -1 1 ¯ q 2 /parenleft.alt3 γ 2 -1 2 /parenright.alt3/braceleft.alt3 -1 ω 1 ¯ q ( µ u ν ) 1 -1 ω 2 1 ( ¯ k · ¯ q ) u µ 1 u ν 1 /braceright.alt3 = -i n -1 2 ( n -1 ) ! πγ 3 β 3 ( ωb ) n -1 log ( ωb )/parenleft.alt3 γ 2 -1 2 /parenright.alt3/braceleft.alt3( γu 2 -u 1 ) ( µ u ν ) 1 -1 ( u 1 · ¯ k ) ( ¯ k · ( γu 2 -u 1 )) u µ 1 u ν 1 /braceright.alt3 , (4.15) \nwhere we have used the integral result of equation (C.10). \nTherefore upon simplifying, the log term in (sub) n -leading radiation kernel corresponding to particle 1 is given by \nR ( ωb ) n -1 log ( ωb ) ,µν 1 = i n -1 m 1 m 2 κ 3 4 π ( n -1 ) ! γ 3 β 3 γ ( 2 γ 2 -3 )( ωb ) n -1 log ( ωb )/parenleft.alt3 u ( µ 1 u ν ) 2 -( u 2 · ¯ k ) ( u 1 · ¯ k ) u ( µ 1 u ν ) 1 /parenright.alt3 . (4.16) \nNote that in the deflection less limit ( /divides.alt0 b /divides.alt0 → ∞ ) such that ωb is fixed, the logarithmic contributions survive. The rest of the terms constitute integrals similar to the ones described in section 5.1 and they do not give any logarithmic contributions. Also, the terms obtained using quantum soft theorems and then taking the classical limit, match with the counterparts in the soft expansion of the radiation kernel in equation (4.13).", '5 Radiation kernel to (sub) n -leading order in frequency': 'In this section, we prove our main result. Given a (sub) n -leading soft graviton theorem of a tree-level gravitational amplitude, the so-called remainder terms never contribute to the leading log contribution that arises in the classical limit.', '5.1 (sub) n -leading order soft radiation from quantum soft theorems': "We will now compute the soft radiation by applying (sub) n -leading soft graviton operator on the quantum four-point amplitude and then take the classical limit. We will see that not all the terms in the soft expansion of the radiation kernel can be recovered by applying the soft theorems and the remainder terms do not correspond to any logarithmic contribution. As ω n -1 log ω is more dominant than the ω n -1 terms, the low-frequency classical radiation during a scattering process is simply obtained from the soft theorems. One can discard the remainder terms then. \nFrom quantum soft theorems, the (sub) n -leading radiation kernel is given by \nR µν ω ( n -1 ) = 1 4 m 1 m 2 /integral.disp ˆ d 4 q 1 ˆ d 4 q 2 e iq 1 · b ˆ δ ( u 1 · q 1 ) ˆ δ ( u 2 · q 2 ) × κ 2 /summation.disp i = 1 , 2 /bracketleft.alt3 J µρ i k ρ J νσ i k σ p i · k /parenleft.alt3 k · ∂ ∂p i /parenright.alt3 n -2 + ˜ J µρ i k ρ ˜ J νσ i k σ ˜ p i · k /parenleft.alt3 k · ∂ ∂ ˜ p i /parenright.alt3 n -2 /bracketright.alt3 /parenleft.alt3 ˆ δ ( 4 ) ( q 1 + q 2 )A 4 /parenright.alt3 , (5.1) \nwhere \nA 4 [ p 1 , ˜ p 1 , p 2 , ˜ p 2 ] = κ 2 2 q 2 2 /bracketleft.alt3( p 2 · ˜ p 2 )( m 2 1 -p 1 · ˜ p 1 ) + m 2 2 ( p 1 · ˜ p 1 -2 m 2 1 ) +( p 1 · ˜ p 2 )( p 2 · ˜ p 1 ) + ( p 1 · p 2 )( ˜ p 1 · ˜ p 2 )/bracketright.alt3 . (5.2) \nFirst, let us evaluate the soft operators' action on A 4 . We consider the contribution from particle 1 for now. The action of the soft operators on the numerator of the amplitudes is given by \nκ J µρ 1 k ρ J νσ 1 k σ p 1 · k /parenleft.alt3 k · ∂ ∂p i /parenright.alt3 2 [A 4 ] N = κ 3 k ρ k σ 2 q 2 2 ( p 1 · k ) /parenleft.alt3 p 1 ∧ ∂ ∂p 1 /parenright.alt3 µρ /parenleft.alt3 p 1 ∧ ∂ ∂p 1 /parenright.alt3 νσ /parenleft.alt3 k · ∂ ∂p 1 /parenright.alt3 /bracketleft.alt3( k · ˜ p 1 )( m 2 2 -p 2 · ˜ p 2 ) + ( k · ˜ p 2 )( p 2 · ˜ p 1 ) + ( k · p 2 )( ˜ p 1 · ˜ p 2 )/bracketright.alt3 = 0 . (5.3) \nand \nκ ˜ J µρ 1 k ρ ˜ J νσ 1 k σ ˜ p 1 · k /parenleft.alt3 k · ∂ ∂ ˜ p 1 /parenright.alt3 2 [A 4 ] N = κ 3 k ρ k σ 2 q 2 2 ( ˜ p 1 · k ) /parenleft.alt3 ˜ p 1 ∧ ∂ ∂ ˜ p 1 /parenright.alt3 µρ /parenleft.alt3 ˜ p 1 ∧ ∂ ∂ ˜ p 1 /parenright.alt3 νσ /parenleft.alt3 k · ∂ ∂ ˜ p 1 /parenright.alt3 /bracketleft.alt3( k · p 1 )( m 2 2 -p 2 · ˜ p 2 ) + ( k · ˜ p 2 )( p 1 · p 2 ) + ( k · p 2 )( p 1 · ˜ p 2 )/bracketright.alt3 = 0 . (5.4) \nThe classical contribution comes from the action of the soft operators on the denominator of the amplitude and it is given by \nκ 2 J µρ 1 k ρ J νσ 1 k σ p 1 · k /parenleft.alt3 k · ∂ ∂p 1 /parenright.alt3 n -2 A 4 + κ 2 ˜ J µρ 1 k ρ ˜ J νσ 1 k σ ˜ p 1 · k /parenleft.alt3 k · ∂ ∂p 1 /parenright.alt3 n -2 A 4 = (-1 ) n + 1 κ 3 2 n -3 ( ¯ q 2 ) n + 1 ( ¯ q · ¯ k ) n -1 /parenleft.alt4 ¯ q µ ¯ q ν (( p 1 · p 2 ) 2 -1 2 m 2 1 m 2 2 ) + ¯ q 2 ( p 2 · ¯ k ) ( ¯ q · ¯ k ) ¯ q ( µ p ν ) 1 /parenright.alt4 . (5.5) \nTherefore, the classical contribution to soft radiation from the action of (sub) n -leading soft operator ( S ( n ) ,µν ) on the four-point amplitude alone is given by \nR µν ω ( n -1 ) ,A = (-1 ) n + 1 κ 3 m 1 m 2 4 /integral.disp ˆ d 4 ¯ qe -i ¯ q · b ˆ δ ( u 1 · ¯ q ) ˆ δ ( u 2 · ¯ q ) 2 n -1 ( ¯ q 2 ) n + 1 ( ¯ q · ¯ k ) n -1 ×/parenleft.alt4 ¯ q µ ¯ q ν /parenleft.alt3 γ 2 -1 2 /parenright.alt3 + ¯ q 2 ( u 2 · ¯ k ) ( ¯ q · ¯ k ) ¯ q ( µ u ν ) 1 /parenright.alt4 . (5.6) \nThis term doesn't give any logarithmic contributions using the integral results of Appendix C. Let us evaluate the soft operators' action on the delta function now. Again we restrict to the contribution from particle 1 where we use the following distributional identity: \nS ( n ) ,µν ˆ δ ( 4 ) ( q 1 + q 2 ) = ˆ δ ( 4 ) ( q 1 + q 2 ) S ( n ) ,µν -( k · ∂ ) ˆ δ ( 4 ) ( q 1 + q 2 ) S ( n -1 ) ,µν + 1 2 ( k · ∂ ) 2 ˆ δ ( 4 ) ( q 1 + q 2 ) S ( n -2 ) ,µν +/uni22EF + (-1 ) n n ! ( k · ∂ ) n ˆ δ ( 4 ) ( q 1 + q 2 ) S ( 0 ) ,µν . (5.7) \nWe have, \nR µν ω ( n -1 ) ,D = 1 m 1 m 2 /integral.disp ˆ d 4 q 1 ˆ d 4 q 2 e iq 1 · b ˆ δ ( u 1 · q 1 ) ˆ δ ( u 2 · q 2 )/bracketleft.alt3 -( k · ∂ ) ˆ δ ( 4 ) ( q 1 + q 2 ) S ( n -1 ) ,µν + 1 2 ( k · ∂ ) 2 ˆ δ ( 4 ) ( q 1 + q 2 ) S ( n -2 ) ,µν +/uni22EF + (-1 ) n -2 ( n -2 ) ! ( k · ∂ ) n -2 ˆ δ ( 4 ) ( q 1 + q 2 ) S ( 2 ) ,µν + (-1 ) n -1 ( n -1 ) ! ( k · ∂ ) n -1 ˆ δ ( 4 ) ( q 1 + q 2 ) S ( 1 ) ,µν + (-1 ) n n ! ( k · ∂ ) n ˆ δ ( 4 ) ( q 1 + q 2 ) S ( 0 ) ,µν /bracketright.alt3A 4 , (5.8) \nas the action of S ( n ) ,µν soft operator on the amplitude is given in equation (5.5). We have the following terms that contribute to the classical soft radiation: \nR µν ω ( n -1 ) , 1 = 1 m 1 m 2 /integral.disp ˆ d 4 q 1 ˆ d 4 q 2 e iq 1 · b ˆ δ ( u 1 · q 1 ) ˆ δ ( u 2 · q 2 )/braceleft.alt3 -( k · ∂ ) ˆ δ ( 4 ) ( q 1 + q 2 ) S ( n -1 ) ,µν A 4 /braceright.alt3 = 1 m 1 m 2 /integral.disp ˆ d 4 q 1 ˆ d 4 q 2 e iq 1 · b ˆ δ ( u 1 · q 1 ) ˆ δ ( u 2 · q 2 )/braceleft.alt3 ˆ δ ( 4 ) ( q 1 + q 2 -k ) -ˆ δ ( 4 ) ( q 1 + q 2 )/braceright.alt3 S ( n -1 ) ,µν A 4 . (5.9) \nIntegrating q 1 and relabelling q 2 → q and keeping only O( ω n -1 ) terms, we have, \nR µν ω ( n -1 ) , 1 = 1 m 1 m 2 /integral.disp ˆ d 4 qe -iq · b ˆ δ ( u 2 · q )/braceleft.alt3( ik · b ) ˆ δ ( u 1 · q ) - ( u 1 · k ) ˆ δ ' ( u 1 · q )/braceright.alt3 S ( n -1 ) ,µν A 4 = 1 m 1 m 2 /integral.disp ˆ d 4 ¯ qe -i ¯ q · b ˆ δ ( u 2 · ¯ q )/braceleft.alt3( i ¯ k · b ) ˆ δ ( u 1 · ¯ q ) - ( u 1 · ¯ k ) ˆ δ ' ( u 1 · ¯ q )/braceright.alt3 ×(-1 ) n κ 3 2 n -1 ( ¯ q 2 ) n ( ¯ q · ¯ k ) n -2 /parenleft.alt4 ¯ q µ ¯ q ν (( p 1 · p 2 ) 2 -1 2 m 2 1 m 2 2 ) + ¯ q 2 ( p 2 · ¯ k ) ( ¯ q · ¯ k ) ¯ q ( µ p ν ) 1 /parenright.alt4 . (5.10) \nas the classical contribution of S ( n -1 ) ,µν on the amplitude is given in equation (5.5). This doesn't give any logarithmic contributions using the integral results of Appendix C. Therefore, we compute the other terms. \nR µν ω ( n -1 ) , 2 = 1 m 1 m 2 /integral.disp ˆ d 4 q 1 ˆ d 4 q 2 e iq 1 · b ˆ δ ( u 1 · q 1 ) ˆ δ ( u 2 · q 2 ) 1 2 ( k · ∂ ) 2 ˆ δ ( 4 ) ( q 1 + q 2 ) S ( n -2 ) ,µν A 4 = 1 m 1 m 2 /integral.disp ˆ d 4 q 1 ˆ d 4 q 2 e iq 1 · b ˆ δ ( u 1 · q 1 ) ˆ δ ( u 2 · q 2 )/braceleft.alt3 ˆ δ ( 4 ) ( q 1 + q 2 -k ) -ˆ δ ( 4 ) ( q 1 + q 2 ) +( k · ∂ ) ˆ δ ( 4 ) ( q 1 + q 2 )/braceright.alt3 S ( n -2 ) ,µν A 4 . (5.11) \nIntegrating q 1 and relabelling q 2 → q and keeping only O( ω n -1 ) terms, we have, \nR µν ω ( n -1 ) , 2 = 1 m 1 m 2 /integral.disp ˆ d 4 ¯ qe -i ¯ q · b ˆ δ ( u 2 · ¯ q )/braceleft.alt3 ( i ¯ k · b ) 2 2 ˆ δ ( u 1 · ¯ q ) - ( ib · ¯ k )( u 1 · ¯ k ) ˆ δ ' ( u 1 · ¯ q ) + 1 2 ( u 1 · ¯ k ) 2 ˆ δ '' ( u 1 · ¯ q )/braceright.alt3 S ( n -2 ) ,µν A 4 . (5.12) \nThis too doesn't give any logarithmic contributions. Similarly, all the other terms till the action of S ( 2 ) ,µν do not lead to any log terms using the integral results of Appendix C. For \ninstance \nR µν ω ( n -1 ) , 3 = 1 m 1 m 2 /integral.disp ˆ d 4 q 1 ˆ d 4 q 2 e iq 1 · b ˆ δ ( u 1 · q 1 ) ˆ δ ( u 2 · q 2 ) (-1 ) n -2 ( k · ∂ ) n -2 ( n -2 ) ! ˆ δ ( 4 ) ( q 1 + q 2 ) S ( 2 ) ,µν A 4 = 1 m 1 m 2 /integral.disp ˆ d 4 q 1 ˆ d 4 q 2 e iq 1 · b ˆ δ ( u 1 · q 1 ) ˆ δ ( u 2 · q 2 )/braceleft.alt3 ˆ δ ( 4 ) ( q 1 + q 2 -k ) -ˆ δ ( 4 ) ( q 1 + q 2 ) +( k · ∂ ) ˆ δ ( 4 ) ( q 1 + q 2 ) + /uni22EF -(-1 ) n -3 ( n -3 ) ! ( k · ∂ ) n -3 ˆ δ ( 4 ) ( q 1 + q 2 )/braceright.alt3 S ( 2 ) ,µν A 4 \n(5.13) \nIntegrating q 1 and relabelling q 2 → q and keeping only O( ω n -1 ) terms, we have, \nR µν ω ( n -1 ) , 3 = 1 m 1 m 2 /integral.disp ˆ d 4 qe -iq · b ˆ δ ( u 2 · q )/braceleft.alt3 ( ik · b ) n -2 ( n -2 ) ! ˆ δ ( u 1 · q ) + /summation.disp r,s ≥ 1 ∋( r + s )= n -2 (-1 ) s r ! s ! ( ib · k ) r ( u 1 · k ) s ˆ δ ( s ) ( u 1 · q ) + (-1 ) n -2 ( n -2 ) ! ( u 1 · k ) n -2 ˆ δ ( n -2 ) ( u 1 · q )/braceright.alt3 S ( 2 ) ,µν A 4 = -1 m 1 m 2 κ 3 /integral.disp ˆ d 4 ¯ qe -i ¯ q · b ˆ δ ( u 2 · ¯ q )/braceleft.alt3 ( i ¯ k · b ) n -2 ( n -2 ) ! ˆ δ ( u 1 · ¯ q ) + /summation.disp r,s ≥ 1 ∋( r + s )= n -2 (-1 ) s r ! s ! ( ib · ¯ k ) r ( u 1 · ¯ k ) s ˆ δ ( s ) ( u 1 · ¯ q ) + (-1 ) n -2 ( n -2 ) ! ( u 1 · ¯ k ) n -2 ˆ δ ( n -2 ) ( u 1 · ¯ q )/braceright.alt3 1 ( ¯ q 2 ) 3 ( ¯ q · ¯ k ) ×/parenleft.alt4 ¯ q µ ¯ q ν (( p 1 · p 2 ) 2 -1 2 m 2 1 m 2 2 ) + ¯ q 2 ( p 2 · ¯ k ) ( ¯ q · ¯ k ) ¯ q ( µ p ν ) 1 /parenright.alt4 , (5.14) \nas the the classical contribution of S ( 2 ) ,µν on the amplitude is given in equation (B.5). Therefore, we compute the remaining two terms which should give logarithmic contributions. \nR µν ω ( n -1 ) , 4 = 1 m 1 m 2 /integral.disp ˆ d 4 q 1 ˆ d 4 q 2 e iq 1 · b ˆ δ ( u 1 · q 1 ) ˆ δ ( u 2 · q 2 ) (-1 ) n -1 ( k · ∂ ) n -1 ( n -1 ) ! ˆ δ ( 4 ) ( q 1 + q 2 ) S ( 1 ) ,µν A 4 = 1 m 1 m 2 /integral.disp ˆ d 4 q 1 ˆ d 4 q 2 e iq 1 · b ˆ δ ( u 1 · q 1 ) ˆ δ ( u 2 · q 2 )/braceleft.alt3 ˆ δ ( 4 ) ( q 1 + q 2 -k ) -ˆ δ ( 4 ) ( q 1 + q 2 ) +( k · ∂ ) ˆ δ ( 4 ) ( q 1 + q 2 ) + /uni22EF -(-1 ) n -2 ( n -2 ) ! ( k · ∂ ) n -2 ˆ δ ( 4 ) ( q 1 + q 2 )/braceright.alt3 S ( 1 ) ,µν A 4 . \n(5.15) \nIntegrating q 1 and relabelling q 2 → q and keeping only O( ω n -1 ) terms, we have, \nR µν ω ( n -1 ) , 4 = 1 m 1 m 2 /integral.disp ˆ d 4 qe -iq · b ˆ δ ( u 2 · q )/braceleft.alt3 ( ik · b ) n -1 ( n -1 ) ! ˆ δ ( u 1 · q ) + /summation.disp r,s ≥ 1 ∋( r + s )= n -1 (-1 ) s r ! s ! ( ib · k ) r ( u 1 · k ) s ˆ δ ( s ) ( u 1 · q ) + (-1 ) n -1 ( n -1 ) ! ( u 1 · k ) n -1 ˆ δ ( n -1 ) ( u 1 · q )/braceright.alt3 S ( 1 ) ,µν A 4 . (5.16) \nFrom equation (B.17), the classical contribution of S ( 1 ) ,µν on the amplitude is given by \nS ( 1 ) ,µν A 4 = -2 κ 3 ( ¯ k · p 1 ) /bracketleft.alt3 p ( µ 1 p ν ) 2 ( ¯ k · p 1 )( p 1 · p 2 ) -p ( µ 1 p ν ) 1 ( ¯ k · p 2 )( p 1 · p 2 ) + 1 2¯ q 2 ¯ q µ ¯ q ν ( ¯ k · p 1 )/bracketright.alt3 . (5.17) \nTherefore, by substituting the above expression we get \nR µν ω ( n -1 ) , 4 = 1 m 1 m 2 /integral.disp ˆ d 4 ¯ qe -i ¯ q · b ˆ δ ( u 2 · ¯ q )/braceleft.alt3 ( i ¯ k · b ) n -1 ( n -1 ) ! ˆ δ ( u 1 · ¯ q ) + /summation.disp r,s ≥ 1 ∋( r + s )= n -1 (-1 ) s r ! s ! ( ib · ¯ k ) r ( u 1 · ¯ k ) s ˆ δ ( s ) ( u 1 · ¯ q ) + (-1 ) n -1 ( n -1 ) ! ( u 1 · ¯ k ) n -1 ˆ δ ( n -1 ) ( u 1 · ¯ q )/braceright.alt3 ×-2 κ 3 ( ¯ k · p 1 ) ¯ q 2 p ( µ 1 /bracketleft.alt3 p ν ) 2 ( ¯ k · p 1 )( p 1 · p 2 ) -p ν ) 1 ( ¯ k · p 2 )( p 1 · p 2 )/bracketright.alt3 +O( ω n -1 ) = i n -1 m 1 m 2 κ 3 γ ( n -1 ) ! πγβ ( ωb ) n -1 log ( ωb ) u ( µ 1 /parenleft.alt3 u ν ) 2 -u ν ) 1 ( ¯ k · u 2 ) ( ¯ k · u 1 ) /parenright.alt3 +O( ω n -1 ) , (5.18) \nwhere we have used the integral result of equation (C.2). The second integral and the third one do not give log terms following the result of equation (C.13). We are now left with computing one last term. \nR µν ω ( n -1 ) , 5 = 1 m 1 m 2 /integral.disp ˆ d 4 q 1 ˆ d 4 q 2 e iq 1 · b ˆ δ ( u 1 · q 1 ) ˆ δ ( u 2 · q 2 ) (-1 ) n ( k · ∂ ) n n ! ˆ δ ( 4 ) ( q 1 + q 2 ) S ( 0 ) ,µν A 4 = 1 m 1 m 2 /integral.disp ˆ d 4 q 1 ˆ d 4 q 2 e iq 1 · b ˆ δ ( u 1 · q 1 ) ˆ δ ( u 2 · q 2 )/braceleft.alt3 ˆ δ ( 4 ) ( q 1 + q 2 -k ) -ˆ δ ( 4 ) ( q 1 + q 2 ) +( k · ∂ ) ˆ δ ( 4 ) ( q 1 + q 2 ) + /uni22EF -(-1 ) n -1 ( n -1 ) ! ( k · ∂ ) n -1 ˆ δ ( 4 ) ( q 1 + q 2 )/braceright.alt3 S ( 0 ) ,µν A 4 \n(5.19) \nIntegrating q 1 and relabelling q 2 → q and keeping only O( ω n -1 ) terms, we have, \nR µν ω ( n -1 ) , 5 = 1 m 1 m 2 /integral.disp ˆ d 4 qe -iq · b ˆ δ ( u 2 · q )/braceleft.alt3 ( ik · b ) n n ! ˆ δ ( u 1 · q ) + /summation.disp r,s ∋( r + s )= n (-1 ) s r ! s ! ( ib · k ) r ( u 1 · k ) s ˆ δ ( s ) ( u 1 · q ) + (-1 ) n n ! ( u 1 · k ) n ˆ δ ( n ) ( u 1 · q )/braceright.alt3 S ( 0 ) ,µν A 4 . \n(5.20) \nWe have, for particle 1 \nS ( 0 ) ,µν = 1 p 1 · k p ( µ 1 p ν ) 1 -1 ˜ p 1 · k ˜ p ( µ 1 ˜ p ν ) 1 = -¯ q ( µ p ν ) 1 p 1 · ¯ k + ( ¯ q · ¯ k ) p ( µ 1 p ν ) 1 ( p 1 · ¯ k ) 2 . (5.21) \nWe have the following integrals \nI µν 1 = -κ 3 ( 2 ( p 1 · p 2 ) 2 -m 2 1 m 2 2 ) ( i ¯ k · b ) n 2 n ! /integral.disp ˆ d 4 ¯ q ¯ q 2 e -i ¯ q · b ˆ δ ( u 1 · ¯ q ) ˆ δ ( u 2 · ¯ q )/parenleft.alt3 ¯ q ( µ p ν ) 1 p 1 · ¯ k -( ¯ q · ¯ k ) p ( µ 1 p ν ) 1 ( p 1 · ¯ k ) 2 /parenright.alt3 = -iκ 3 ( 2 ( p 1 · p 2 ) 2 -m 2 1 m 2 2 ) ( i ¯ k · b ) n 4 n ! πγβ /parenleft.alt3 b ( µ p ν ) 1 p 1 · ¯ k -( b · ¯ k ) p ( µ 1 p ν ) 1 ( p 1 · ¯ k ) 2 /parenright.alt3 , (5.22) \nusing the integral result of equation (C.3). Next, we have the integral \nI µν 2 = -κ 3 ( 2 ( p 1 · p 2 ) 2 -m 2 1 m 2 2 ) /summation.disp r,s r s n (-1 ) s 2 r ! s ! ( ib · ¯ k ) r ( u 1 · ¯ k ) s /integral.disp ˆ d 4 ¯ q ¯ q 2 e -i ¯ q · b ˆ δ ( s ) ( u 1 · ¯ q ) ˆ δ ( u 2 · ¯ q \n∋( + )= ) ×/parenleft.alt3 ¯ q ( µ p ν ) 1 p 1 · ¯ k -( ¯ q · ¯ k ) p ( µ 1 p ν ) 1 ( p 1 · ¯ k ) 2 /parenright.alt3 = κ 3 ( 2 ( p 1 · p 2 ) 2 -m 2 1 m 2 2 ) /summation.disp r,s ∋( r + s )= n (-1 ) s 4 r ! γβ ( ib · ¯ k ) r ( u 1 · ¯ k ) s -1 /integral.disp ˆ d 2 ¯ q ⊥ e -i ¯ q ⊥ · b 1 ¯ q 2 ⊥ /bracketleft.alt3 1 ( ¯ q 2 ⊥ γ 2 β 2 ) s /slash.left 2 +/parenleft.alt3 -1 /radical.alt1 ¯ q 2 ⊥ γ 2 β 2 /parenright.alt3 s /bracketright.alt3/parenleft.alt3 ¯ q ( µ ⊥ u ν ) 1 -( ¯ q ⊥ · ¯ k ) u ( µ 1 u ν ) 1 ( u 1 · ¯ k ) /parenright.alt3 -κ 3 ( 2 ( p 1 · p 2 ) 2 -m 2 1 m 2 2 ) 1 2 ( n -1 ) ! γβ ( ib · ¯ k ) n -1 /integral.disp ˆ d 2 ¯ q ⊥ e -i ¯ q ⊥ · b 1 ¯ q 2 ⊥ ∂ ∂ ( u 1 · ¯ q ) /parenleft.alt3 ¯ q ( µ u ν ) 1 -( ¯ q · ¯ k ) u ( µ 1 u ν ) 1 ( u 1 · ¯ k ) /parenright.alt3 = -κ 3 ( 2 ( p 1 · p 2 ) 2 -m 2 1 m 2 2 ) i n -1 4 π ( n -1 ) ! γ 3 β 3 ( ωb ) n -1 log ( ωb ) ×/parenleft.alt3( γu 2 -u 1 ) ( µ u ν ) 1 -(( γu 2 -u 1 ) · ¯ k ) u ( µ 1 u ν ) 1 ( u 1 · ¯ k ) /parenright.alt3 +O( ω n -1 ) , (5.23) \nwhere we have used the integral result of equation (C.10), and lastly \nI µν 3 = -κ 3 ( 2 ( p 1 · p 2 ) 2 -m 2 1 m 2 2 ) (-1 ) n ( u 1 · ¯ k ) n 2 n ! /integral.disp ˆ d 4 ¯ q ¯ q 2 e -i ¯ q · b ˆ δ ( n ) ( u 1 · ¯ q ) ˆ δ ( u 2 · ¯ q ) /parenleft.alt3 ¯ q ( µ p ν ) 1 p 1 · ¯ k -( ¯ q · ¯ k ) p ( µ 1 p ν ) 1 ( p 1 · ¯ k ) 2 /parenright.alt3 = κ 3 ( 2 ( p 1 · p 2 ) 2 -m 2 1 m 2 2 ) (-1 ) n ( u 1 · ¯ k ) n 2 n ! γβ /integral.disp ˆ d 2 ¯ q ⊥ e -i ¯ q ⊥ · b ∂ n ∂ ( u 1 · ¯ q ) n 1 ¯ q 2 /parenleft.alt3 ¯ q ( µ p ν ) 1 p 1 · ¯ k -( ¯ q · ¯ k ) p ( µ 1 p ν ) 1 ( p 1 · ¯ k ) 2 /parenright.alt3 . (5.24) \nUsing the integral result of equation (C.10), we get \nI µν 3 = /uni23A7 /uni23AA /uni23AA /uni23AA /uni23AA /uni23A8 /uni23AA /uni23AA /uni23AA /uni23AA /uni23A9 O( ω n -1 ) , if n ≥ 2 . -κ 3 ( 2 ( p 1 · p 2 ) 2 -m 2 1 m 2 2 ) 1 4 πγ 3 β 3 log ( ωb )/parenleft.alt3( γu 2 -u 1 ) ( µ u ν ) 1 -(( γu 2 -u 1 )· ¯ k ) u ( µ 1 u ν ) 1 ( u 1 · ¯ k ) /parenright.alt3 , if n = 1 . (5.25) \nHere the log ( ωb ) contribution comes only from n = 1 . \nTherefore we collect the log terms and upon simplifying the ω n -1 log ω terms of radiation kernel w.r.t particle 1 from the quantum soft theorems is given by \nR µν ω n -1 log ω = i n -1 m 1 m 2 κ 3 4 π ( n -1 ) ! γ 3 β 3 γ ( 2 γ 2 -3 )( ωb ) n -1 log ( ωb )/parenleft.alt3 u ( µ 1 u ν ) 2 -( u 2 · ¯ k ) ( u 1 · ¯ k ) u ( µ 1 u ν ) 1 /parenright.alt3 , (5.26) \nwhich matches with the log terms of (sub) n -leading order soft expansion of the radiation kernel in equation (4.16). One can also see that the rest of the terms obtained using quantum soft theorems also match with the counterparts in the soft expansion of the radiation kernel in equation (4.13).", '5.2 Remainder terms in (sub) n -leading order soft radiation': "Comparing the soft expansion of the radiation kernel and the radiation kernel obtained using quantum soft theorems to (sub) n -leading order in frequency, we see that not all the terms in the soft expansion of the radiation kernel in equation (4.13) are recovered by applying the soft theorems. Such terms in the (unstripped) five-point amplitude do not factorize as soft factors times the four-point amplitude. These are known as the 'Remainder terms.' We have identified such terms at (sub) n -leading order for n ≥ 3 in the soft radiation kernel given by \nX µν R ,ω ( n -1 ) = κ 3 m 1 m 2 4 /integral.disp ˆ d 4 ¯ q /bracketleft.alt3 n /summation.disp r = 3 1 ( n -r ) ! e -ib · ¯ q ˆ δ ( u 1 · ¯ q ) ˆ δ ( u 2 · ¯ q )( ib · ¯ k ) n -r Λ µν r -1 + n -1 /summation.disp r = 3 (-1 ) n -r ( n -r ) ! /braceleft.alt3 e -ib · ¯ q ˆ δ ( n -r ) ( u 1 · ¯ q ) ˆ δ ( u 2 · ¯ q )( u 1 · ¯ k ) n -r /parenleft.alt3 Λ µν r -1 /parenright.alt3 + e ib · ¯ q /parenleft.alt3 1 ↔ 2 /parenright.alt3/braceright.alt3 + n -2 /summation.disp r = 3 /summation.disp t,s ≥ 1 ∋( t + s )= n -r (-1 ) s t ! s ! e -ib · ¯ q ( ib · ¯ k ) t ( u 1 · ¯ k ) s ˆ δ ( s ) ( u 1 · ¯ q ) ˆ δ ( u 2 · ¯ q ) Λ µν r -1 /bracketright.alt3 , (5.27) \nwhere the polynomial Λ µν n is defined in Section 3. Using the integral results of Appendix C, it is evident that these remainder terms do not give any logarithmic contributions. As ω n -1 log ω is more dominant than the ω n -1 terms, one can simply discard the remainder terms in computing the low-frequency classical radiation during a scattering process.", '6 Radiation kernel to (sub) 3 -leading order in frequency': 'In this section, we will compute the soft radiative gravitational field to (sub) 3 -leading order in frequency. One can simply substitute n = 3 in the previous section for the analysis. We will only summarise the important results here. \n- · The leading logarithmic contribution to the radiation kernel in this order is given by \nR µν ω 2 log ω = -κ 3 m 1 m 2 8 πγ 3 β 3 ( ωb ) 2 log ( ωb ) γ ( 2 γ 2 -3 )/parenleft.alt3 u ( µ 1 u ν ) 2 -( u 2 · ¯ k ) ( u 1 · ¯ k ) u ( µ 1 u ν ) 1 /parenright.alt3 +( 1 ↔ 2 ) . (6.1) \nAs stated before, in the deflection less limit ( /divides.alt0 b /divides.alt0 → ∞ ) such that ωb is fixed, the logarithmic contribution survives. \n- · The factorized terms in the radiation kernel that are obtained via the quantum soft graviton theorems match with the counterparts obtained from the soft expansion of the classical radiation kernel at (sub) 3 -leading order. However, at this order ( n ≥ 3 ) , the radiation kernel is infected with the presence of non-factorizing remainder terms.\n- · We have identified such remainder terms at (sub) 3 -leading order in the soft radiation kernel given by \nwhere \nX µν R ,ω 2 = κ 3 m 1 m 2 4 /integral.disp ˆ d 4 ¯ q /braceleft.alt3 e -ib · ¯ q ˆ δ ( u 1 · ¯ q ) ˆ δ ( u 2 · ¯ q ) H µν 2 + e ib · ¯ q /parenleft.alt3 1 ↔ 2 /parenright.alt3/braceright.alt3 , (6.2) \nH µν 2 = -4 ( ¯ q 2 ) 2 /parenleft.alt3 ω 2 2 u µ 1 u ν 1 -ω 1 ω 2 2 ( u µ 2 u ν 1 + u ν 2 u µ 1 )/parenright.alt3 . (6.3) \nAs expected, the remainder term does not give any logarithmic contributions. As ω 2 log ω is more dominant than the ω 2 terms, one can simply discard the remainder term in computing the low-frequency classical radiation during a scattering process.', '7 Discussion': "In this work, we have shown that the tree-level (sub) n -leading soft graviton theorems for two massive scalar fields minimally coupled to gravity generate all the logarithmic terms in the soft expansion in the limit of vanishing deflection. It would be interesting to explore the effect of the non-universal terms in the soft factors which are generated by irrelevant terms in the Lagrangian. Already at the (sub) 2 -leading order in the soft expansion, where the remainder terms are zero, the corresponding soft factor is modified by the presence of \na finite set of higher derivative terms in the Lagrangian [77]. Higher derivative terms will certainly change the higher-order tree-level soft factors, but we believe that they will not alter the leading logs in the deflection-less limit. However, this needs to be investigated further. \nIt would be interesting to compute the soft gravitational radiation for D > 4 and analyze the soft spectra. In contrast to the case in D = 4 , the sub-leading contribution in higher dimensions arises from the integration region where /divides.alt0 l /divides.alt0 ∼ b -1 . This aligns with the classical soft theorem in higher dimensions, as discussed in [59], where it was shown that during scattering, the 'outer' space-time region with size ≥ b contributes to the subleading radiation. There is a reversal of order in the behavior of soft emission between D = 4 and D > 4 . In dimensions higher than four, the integral yields ω 0 terms from the 'UV region' where /divides.alt0 l /divides.alt0 ∼ b -1 , and terms proportional to ω D -4 from the 'IR region' where /divides.alt0 l /divides.alt0 ≥ ω . In D = 4 , however, the integral produces logarithmic terms log ω from the IR region and ω 0 terms from the UV region. It would be worth examining whether any logarithmic contributions appear at (sub) n -leading order in D = 5 , as this represents the first non-trivial case, and also reviewing the remainder terms. This would prove to be highly useful to have an interpretation of the classical soft theorems in D > 4 spacetime dimensions.", 'Acknowledgments': 'I am grateful to Alok Laddha for suggesting the problem, numerous insightful discussions, and constant encouragement. I thank him for going through the draft and providing valuable suggestions and comments on it. I thank Sujay K. Ashok for constant encouragement and valuable comments on the draft. I would also like to thank Arkajyoti Manna and Akavoor Manu for useful discussions.', 'A Conventions': 'Throughout the paper, we will use the metric signature as (+ , -, -, -) , unless otherwise stated. So, the on-shell condition is p 2 = m 2 . Since the impact parameter is spacelike we have -b 2 > 0 . The rescaled delta functions appearing in the main text are defined as \nˆ δ ( p · q ) ∶= 2 πδ ( p · q ) , ˆ δ ( 4 ) ( p + q ) ∶= ( 2 π ) 4 δ ( 4 ) ( p + q ) . (A.1) \nwhere p µ and q µ are generic four vectors. We also absorb the 2 π factor in the measure d 4 q and define the rescaled measure as \nˆ d 4 q ∶= d 4 q ( 2 π ) 4 . (A.2)', 'B (sub) 2 -leading order soft radiation from quantum soft theorems': "In this appendix, we will review the computation of the soft radiation by applying (sub) 2 -leading soft graviton operator on the quantum four-point amplitude and then take the \nclassical limit. \nFrom quantum soft theorems, the (sub) 2 -leading radiation kernel is given by \nR µν ω = 1 4 m 1 m 2 /integral.disp ˆ d 4 q 1 ˆ d 4 q 2 e iq 1 · b /slash.left /uni0335 h ˆ δ ( u 1 · q 1 ) ˆ δ ( u 2 · q 2 ) κ /summation.disp i = 1 , 2 /bracketleft.alt3 J µρ i k ρ J νσ i k σ p i · k + ˜ J µρ i k ρ ˜ J νσ i k σ ˜ p i · k /bracketright.alt3 /parenleft.alt3 ˆ δ ( 4 ) ( q 1 + q 2 )A 4 /parenright.alt3 , (B.1) \nwhere \nA 4 [ p 1 , ˜ p 1 , p 2 , ˜ p 2 ] = κ 2 2 q 2 2 /bracketleft.alt3( p 2 · ˜ p 2 )( m 2 1 -p 1 · ˜ p 1 ) + m 2 2 ( p 1 · ˜ p 1 -2 m 2 1 ) +( p 1 · ˜ p 2 )( p 2 · ˜ p 1 ) + ( p 1 · p 2 )( ˜ p 1 · ˜ p 2 )/bracketright.alt3 . (B.2) \nFirst, let us evaluate the soft operators' action on A 4 . We consider the contribution from particle 1 for now. The action of the soft operators on the numerator of the amplitudes is given by \nκ J µρ 1 k ρ J νσ 1 k σ p 1 · k [A 4 ] N = -κ 3 k ρ k σ 2 q 2 2 ( p 1 · k ) /parenleft.alt3 p 1 ∧ ∂ ∂p 1 /parenright.alt3 µρ /parenleft.alt3 p 1 ∧ ∂ ∂p 1 /parenright.alt3 νσ /bracketleft.alt3( p 2 · ˜ p 2 )( m 2 1 -p 1 · ˜ p 1 ) + m 2 2 ( p 1 · ˜ p 1 -2 m 2 1 ) + ( p 1 · ˜ p 2 )( p 2 · ˜ p 1 ) + ( p 1 · p 2 )( ˜ p 1 · ˜ p 2 )/bracketright.alt3 \n= 0 . (B.3) \nand \nκ ˜ J µρ 1 k ρ ˜ J νσ 1 k σ ˜ p 1 · k [A 4 ] N = -κ 3 k ρ k σ 2 q 2 2 ( ˜ p 1 · k ) /parenleft.alt3 ˜ p 1 ∧ ∂ ∂ ˜ p 1 /parenright.alt3 µρ /parenleft.alt3 ˜ p 1 ∧ ∂ ∂ ˜ p 1 /parenright.alt3 νσ /bracketleft.alt3( p 2 · ˜ p 2 )( m 2 1 -p 1 · ˜ p 1 ) + m 2 2 ( p 1 · ˜ p 1 -2 m 2 1 ) + ( p 1 · ˜ p 2 )( p 2 · ˜ p 1 ) + ( p 1 · p 2 )( ˜ p 1 · ˜ p 2 )/bracketright.alt3 \n= 0 . \n(B.4) \nThe classical contribution from the action of the soft operators on the denominator of the amplitude is given by \n-κ 3 2 ( ¯ q 2 ) 3 ( ¯ q · ¯ k ) /parenleft.alt4 ¯ q µ ¯ q ν (( p 1 · p 2 ) 2 -1 2 m 2 1 m 2 2 ) + ¯ q 2 ( p 2 · ¯ k ) ( ¯ q · ¯ k ) ¯ q ( µ p ν ) 1 /parenright.alt4 . (B.5) \nTherefore the classical contribution to soft radiation from the action of the (sub) 2 -leading soft operator ( S ( 2 ) ,µν ) on the four-point amplitude alone is given by \nR µν ω,A = = -κ 3 m 1 m 2 4 /integral.disp ˆ d 4 ¯ qe -i ¯ q · b ˆ δ ( u 1 · ¯ q ) ˆ δ ( u 2 · ¯ q ) 1 2 ( ¯ q 2 ) 3 ( ¯ q · ¯ k ) ×/parenleft.alt4 ¯ q µ ¯ q ν /parenleft.alt3 γ 2 -1 2 /parenright.alt3 + ¯ q 2 ( u 2 · ¯ k ) ( ¯ q · ¯ k ) ¯ q ( µ u ν ) 1 /parenright.alt4 . (B.6) \nLet us evaluate the soft operators' action on the delta function now. Again we restrict to the contribution from particle 1. We use the distributional identity: \nS ( 2 ) ,µν ˆ δ ( 4 ) ( q 1 + q 2 ) = ˆ δ ( 4 ) ( q 1 + q 2 ) S ( 2 ) ,µν -( k · ∂ ) ˆ δ ( 4 ) ( q 1 + q 2 ) S ( 1 ) ,µν + 1 2 ( k · ∂ ) 2 ˆ δ ( 4 ) ( q 1 + q 2 ) S ( 0 ) ,µν . (B.7) \nHere, S ( 0 ) ,µν , S ( 1 ) ,µν , S ( 2 ) ,µν are the leading, sub-leading and (sub) 2 -leading soft operators respectively. We have, \nR µν ω,D = 1 m 1 m 2 /integral.disp ˆ d 4 q 1 ˆ d 4 q 2 e iq 1 · b /slash.left /uni0335 h ˆ δ ( u 1 · q 1 ) ˆ δ ( u 2 · q 2 )/bracketleft.alt3 ˆ δ ( 4 ) ( q 1 + q 2 ) S ( 2 ) ,µν -( k · ∂ ) ˆ δ ( 4 ) ( q 1 + q 2 ) S ( 1 ) ,µν + 1 2 ( k · ∂ ) 2 ˆ δ ( 4 ) ( q 1 + q 2 ) S ( 0 ) ,µν /bracketright.alt3A 4 . (B.8) \nFrom equation (B.5), we have the classical contribution of S ( 2 ) ,µν on the amplitude. Therefore, we compute the remaining two terms. We have \nR µν ω, 1 = -1 m 1 m 2 /integral.disp ˆ d 4 q 1 ˆ d 4 q 2 e iq 1 · b /slash.left /uni0335 h ˆ δ ( u 1 · q 1 ) ˆ δ ( u 2 · q 2 )( k · ∂ ) ˆ δ ( 4 ) ( q 1 + q 2 ) S ( 1 ) ,µν A 4 = 1 m 1 m 2 /integral.disp ˆ d 4 q 1 ˆ d 4 q 2 e iq 1 · b /slash.left /uni0335 h ˆ δ ( u 1 · q 1 ) ˆ δ ( u 2 · q 2 )/braceleft.alt3 ˆ δ ( 4 ) ( q 1 + q 2 -k ) -ˆ δ 4 ( q 1 + q 2 )/braceright.alt3 S 1 ,µν A 4 . \n( ) ( ) (B.9) \nIntegrating q 1 and relabelling q 2 → q , we have \nR µν ω, 1 = /integral.disp ˆ d 4 qe -iq · b /slash.left /uni0335 h ˆ δ ( u 2 · q )/braceleft.alt3 ˆ δ ( u 1 · ( k -q )) e ik · b /slash.left /uni0335 h -ˆ δ ( u 1 · q )/braceright.alt3 S ( 1 ) ,µν A 4 . (B.10) \nWriting only the O( ω ) term, \nR µν ω, 1 = /integral.disp ˆ d 4 qe -iq · b /slash.left /uni0335 h ˆ δ ( u 2 · q )/braceleft.alt3 ˆ δ ( u 1 · q )( ik · b /uni0335 h ) - ( u 1 · k ) ˆ δ ' ( u 1 · q )/braceright.alt3 S ( 1 ) ,µν A 4 . (B.11) \nThe sub-leading soft graviton operator is given by \nS ( 1 ) ,µν A 4 = κ 2 /bracketleft.alt3 p ( µ 1 J ν ) ρ 1 k ρ ( p 1 · k ) -˜ p ( µ 1 ˜ J ν ) ρ 1 k ρ ( ˜ p 1 · k ) /bracketright.alt3A 4 . (B.12) \nThe action of the soft operators on the amplitude is given by \nκ p ( µ 1 J ν ) ρ 1 k ρ ( p 1 · k ) A 4 = iκ p 1 · k /parenleft.alt3 p ( µ 1 p ν ) 1 ( k · ∂ ∂p 1 ) - ( p 1 · k ) p ( µ 1 ∂ ∂p 1 ν ) /parenright.alt3A 4 = -iκ 3 2 q 2 ( k · p 1 ) p µ 1 /bracketleft.alt3( k · p 1 )( ˜ p ν 1 ( m 2 2 -p 2 · ˜ p 2 ) + p ν 2 ( ˜ p 1 · ˜ p 2 )) -p ν 1 (( k · ˜ p 1 )( m 2 2 -p 2 · ˜ p 2 ) + ( k · ˜ p 2 )( ˜ p 1 · p 2 ) +( k · p 2 )( ˜ p 1 · ˜ p 2 )) + ˜ p ν 2 ( k · p 1 )( ˜ p 1 · p 2 ) -1 2 q 2 /parenleft.alt3 p ν 1 ( k · ˜ p 1 + k · p 2 -k · ˜ p 2 ) - ( p 1 · k )( ˜ p ν 1 + p ν 2 -˜ p ν 2 )/parenright.alt3( 2 ( p 1 · p 2 ) 2 -m 2 1 m 2 2 )/bracketright.alt3 . (B.13) \nThe classical contribution is given by \nand \nκ p ( µ 1 J ν ) ρ 1 k ρ ( p 1 · k ) A 4 = -iκ 3 ¯ q 2 ( ¯ k · p 1 ) p ( µ 1 /bracketleft.alt3 p ν ) 2 ( ¯ k · p 1 )( p 1 · p 2 ) -p ν ) 1 ( ¯ k · p 2 )( p 1 · p 2 )/bracketright.alt3 . (B.14) \nκ ˜ p ( µ 1 ˜ J ν ) ρ 1 k ρ ( ˜ p 1 · k ) A 4 = -iκ ˜ p 1 · k /parenleft.alt3 ˜ p ( µ 1 ˜ p ν ) 1 ( k · ∂ ∂ ˜ p 1 ) - ( ˜ p 1 · k ) ˜ p ( µ 1 ∂ ∂ ˜ p 1 ν ) /parenright.alt3A 4 = iκ 3 2 q 2 ( k · ˜ p 1 ) ˜ p µ 1 /bracketleft.alt3( k · ˜ p 1 )( p ν 1 ( m 2 2 -p 2 · ˜ p 2 ) + p ν 2 ( p 1 · ˜ p 2 )) -˜ p ν 1 (( k · p 1 )( m 2 2 -p 2 · ˜ p 2 ) + ( k · ˜ p 2 )( p 1 · p 2 ) +( k · p 2 )( p 1 · ˜ p 2 )) + ˜ p ν 2 ( k · ˜ p 1 )( p 1 · p 2 ) -1 2 q 2 /parenleft.alt3 ˜ p ν 1 ( k · p 1 + k · ˜ p 2 -k · p 2 ) - ( ˜ p 1 · k )( p ν 1 + ˜ p ν 2 -p ν 2 )/parenright.alt3( 2 ( p 1 · p 2 ) 2 -m 2 1 m 2 2 )/bracketright.alt3 . (B.15) \nThe classical contribution is given by \nκ ˜ p ( µ 1 ˜ J ν ) ρ 1 k ρ ( ˜ p 1 · k ) A 4 = iκ 3 ¯ q 2 ( ¯ k · p 1 ) /bracketleft.alt3 p ( µ 1 p ν ) 2 ( ¯ k · p 1 )( p 1 · p 2 ) -p ( µ 1 p ν ) 1 ( ¯ k · p 2 )( p 1 · p 2 ) -1 2¯ q 2 ¯ q µ ¯ q ν ( ¯ k · p 1 )( 2 ( p 1 · p 2 ) 2 -m 2 1 m 2 2 )/bracketright.alt3 . (B.16) \nTherefore, substituting equations (B.14) and (B.16) in equation (B.12) the classical contribution of R µν ω, 1 is given by \nR µν ω, 1 = 1 m 1 m 2 /integral.disp ˆ d 4 ¯ q ¯ q 2 e -i ¯ q · b ˆ δ ( u 2 · ¯ q )/braceleft.alt3 ˆ δ ( u 1 · ¯ q )( i ¯ k · b ) - ( u 1 · ¯ k ) ˆ δ ' ( u 1 · ¯ q )/braceright.alt3 ×-2 iκ 3 ( ¯ k · p 1 ) /bracketleft.alt3 p ( µ 1 p ν ) 2 ( ¯ k · p 1 )( p 1 · p 2 ) -p ( µ 1 p ν ) 1 ( ¯ k · p 2 )( p 1 · p 2 ) + 1 2¯ q 2 ¯ q µ ¯ q ν ( ¯ k · p 1 )/bracketright.alt3 = κ 3 m 1 m 2 γ πγβ u ( µ 1 /bracketleft.alt3 u ν ) 2 -u ν ) 1 ( ¯ k · u 2 ) ( ¯ k · u 1 ) /bracketright.alt3 ωb log ( ωb ) + O( ω ) , (B.17) \nwhere we have used the integral result of equation (C.2). We are now left with computing one last term. \nR µν ω, 2 = 1 2 m 1 m 2 /integral.disp ˆ d 4 q 1 ˆ d 4 q 2 e iq 1 · b ˆ δ ( u 1 · q 1 ) ˆ δ ( u 2 · q 2 )( k · ∂ ) 2 ˆ δ ( 4 ) ( q 1 + q 2 ) S ( 0 ) ,µν A 4 = 1 m 1 m 2 /integral.disp ˆ d 4 q 1 ˆ d 4 q 2 e iq 1 · b /slash.left /uni0335 h ˆ δ ( u 1 · q 1 ) ˆ δ ( u 2 · q 2 )/braceleft.alt3 ˆ δ ( 4 ) ( q 1 + q 2 -k ) -ˆ δ ( 4 ) ( q 1 + q 2 ) +( k · ∂ ) ˆ δ ( 4 ) ( q 1 + q 2 )/braceright.alt3 S ( 0 ) ,µν A 4 . (B.18) \nIntegrating q 1 and relabelling q 2 → q , we have \nR µν ω, 2 = 1 m 1 m 2 /integral.disp ˆ d 4 qe -iq · b /slash.left /uni0335 h ˆ δ ( u 2 · q )/braceleft.alt3 ˆ δ ( u 1 · ( k -q )) e ik · b /slash.left /uni0335 h -ˆ δ ( u 1 · q ) + ˆ δ ( u 1 · q ) -ˆ δ ( u 1 · ( k -q )) e ik · b /slash.left /uni0335 h /braceright.alt3 S ( 0 ) ,µν A 4 , (B.19) \nwhere the ( k · ∂ ) term is written from the sub-leading distributional identity. Therefore, one should be careful and expand the last term to sub-leading order only. The (sub) 2 -leading contribution comes from expanding the first term to quadratic order in frequency. Therefore, \nR µν ω, 2 = 1 m 1 m 2 /integral.disp ˆ d 4 qe -iq · b /slash.left /uni0335 h ˆ δ ( u 2 · q )/braceleft.alt3 ˆ δ ( u 1 · q ) ( ik · b /slash.left /uni0335 h ) 2 2 -( u 1 · k )( ik · b /slash.left /uni0335 h ) ˆ δ ' ( u 1 · q ) + 1 2 ( u 1 · k ) 2 ˆ δ '' ( u 1 · q )/braceright.alt3 S ( 0 ) ,µν A 4 . (B.20) \nWe have, for particle 1 \nS ( 0 ) ,µν = 1 p 1 · k p ( µ 1 p ν ) 1 -1 ˜ p 1 · k ˜ p ( µ 1 ˜ p ν ) 1 = -¯ q ( µ p ν ) 1 p 1 · ¯ k + ( ¯ q · ¯ k ) p ( µ 1 p ν ) 1 ( p 1 · ¯ k ) 2 . (B.21) \nWe have the following integrals \nI µν 1 = -κ 3 ( 2 ( p 1 · p 2 ) 2 -m 2 1 m 2 2 ) ( ¯ k · b ) 2 4 /integral.disp ˆ d 4 ¯ q ¯ q 2 e -i ¯ q · b ˆ δ ( u 1 · ¯ q ) ˆ δ ( u 2 · ¯ q )/parenleft.alt3 -¯ q ( µ p ν ) 1 p 1 · ¯ k + ( ¯ q · ¯ k ) p ( µ 1 p ν ) 1 ( p 1 · ¯ k ) 2 /parenright.alt3 = iκ 3 ( 2 ( p 1 · p 2 ) 2 -m 2 1 m 2 2 ) ( ¯ k · b ) 2 8 πγβ /parenleft.alt3 b ( µ p ν ) 1 p 1 · ¯ k -( b · ¯ k ) p ( µ 1 p ν ) 1 ( p 1 · ¯ k ) 2 /parenright.alt3 , (B.22) \nwhere we have used the integral result of equation (C.3). Next, we have \nI µν 2 = κ 3 ( 2 ( p 1 · p 2 ) 2 -m 2 1 m 2 2 ) ( u 1 · ¯ k )( i ¯ k · b ) 2 /integral.disp ˆ d 4 ¯ q ¯ q 2 e -i ¯ q · b ˆ δ ' ( u 1 · ¯ q ) ˆ δ ( u 2 · ¯ q )/parenleft.alt3 ¯ q ( µ p ν ) 1 p 1 · ¯ k -( ¯ q · ¯ k ) p ( µ 1 p ν ) 1 ( p 1 · ¯ k ) 2 /parenright.alt3 = -κ 3 4 πγ 3 β 3 ( 2 ( p 1 · p 2 ) 2 -m 2 1 m 2 2 ) ωb log ( ωb )/parenleft.alt3( γu 2 -u 1 ) ( µ u ν ) 1 -(( γu 2 -u 1 ) · ¯ k ) u ( µ 1 u ν ) 1 ( u 1 · ¯ k ) /parenright.alt3 , (B.23) \nwhere we have used the integral result of equation (C.10) and \nI µν 3 = -κ 3 ( 2 ( p 1 · p 2 ) 2 -m 2 1 m 2 2 ) ( u 1 · ¯ k ) 2 4 /integral.disp ˆ d 4 ¯ q ¯ q 2 e -i ¯ q · b ˆ δ '' ( u 1 · ¯ q ) ˆ δ ( u 2 · ¯ q )/parenleft.alt3 ¯ q ( µ p ν ) 1 p 1 · ¯ k -( ¯ q · ¯ k ) p ( µ 1 p ν ) 1 ( p 1 · ¯ k ) 2 /parenright.alt3 = -κ 3 ( 2 ( p 1 · p 2 ) 2 -m 2 1 m 2 2 ) ( u 1 · ¯ k ) 2 γ 2 β 2 /integral.disp ˆ d 2 ¯ q ⊥ e -i ¯ q ⊥ · b 1 ( q 2 ⊥ ) 2 /parenleft.alt3 ¯ q ( µ ⊥ u ν ) 1 -( ¯ q ⊥ · ¯ k ) u ( µ 1 u ν ) 1 ( u 1 · ¯ k ) /parenright.alt3 →O( ω ) , (B.24) \nusing the integral result of equation (C.13). Therefore we collect the log terms and upon simplifying the ω log ω terms of radiation kernel w.r.t particle 1 from the quantum soft theorems is given by \nR µν ω log ω = κ 3 m 1 m 2 4 πγ 3 β 3 ( ωb ) log ( ωb ) γ ( 2 γ 2 -3 )/parenleft.alt3 u ( µ 1 u ν ) 2 -( u 2 · ¯ k ) ( u 1 · ¯ k ) u ( µ 1 u ν ) 1 /parenright.alt3 , (B.25) \nwhich matches with the the tree-level contribution to the ω log ω term and the log terms of (sub) 2 -leading order soft expansion of the radiation kernel. The rest of the terms obtained using quantum soft theorems also match with the soft expansion of the radiation kernel in equation (4.3).", 'C Evaluation of Integrals': 'In this appendix, we perform the integrals required to calculate the various terms of the soft radiation kernel that appear in the main text. The range of integration is ω < /divides.alt0 q ⊥ /divides.alt0 < b -1 in all the integrals, where k µ = ω ( 1 , ˆ n ) . \nWe start with the following integral: \nI 1 = /integral.disp ˆ d 4 ¯ qe -ib · ¯ q ˆ δ ( u 1 · ¯ q ) ˆ δ ( u 2 · ¯ q ) 1 ¯ q 2 = -1 γβ /integral.disp ˆ d 2 ¯ q ⊥ e ib · ¯ q ⊥ ¯ q 2 ⊥ . (C.1) \nThe two-dimensional integral over ¯ q ⊥ is easily done using polar coordinates. Let the magnitude of ¯ q ⊥ be r and orient the x and y axes so that b · ¯ q ⊥ = /divides.alt0 b /divides.alt0 r cos θ . Therefore the integral becomes \nI 1 = -1 2 π /integral.disp dr J 0 ( b /divides.alt0 r /divides.alt0) r = 1 2 π log ( ωb ) . (C.2) \nNext, we consider \nI µ 2 = /integral.disp ˆ d 4 ¯ qe -ib · ¯ q ˆ δ ( u 1 · ¯ q ) ˆ δ ( u 2 · ¯ q ) ¯ q µ ¯ q 2 = -1 γβ /integral.disp ˆ d 2 ¯ q ⊥ e ib · ¯ q ⊥ ¯ q µ ⊥ ¯ q 2 ⊥ = i γβ ∂ ∂b µ ⊥ /integral.disp ˆ d 2 ¯ q ⊥ e ib · ¯ q ⊥ ¯ q 2 ⊥ = i 2 πγβ b µ b 2 , (C.3) \nwhere equation (C.2) is used and ˆ b µ /divides.alt0 b /divides.alt0 = -b µ b 2 . We consider the integral \nI µ 3 = /integral.disp ˆ d 4 ¯ qe -ib · ¯ q ˆ δ ( n ) ( u 1 · ¯ q ) ˆ δ ( u 2 · ¯ q ) ¯ q µ ¯ q 2 , (C.4) \nwhere ( n ) denotes the number of derivatives acting over the on-shell delta function. To simplify this, we shall decompose the momentum ¯ q µ along u 1 , 2 and in the transverse direction \n¯ q µ = α 1 u µ 1 + α 2 u µ 2 + ¯ q µ ⊥ , u i · ¯ q ⊥ = 0 , (C.5) \nwhere the coefficients are given by \nα 1 = 1 γ 2 β 2 [ γx 2 -x 1 ] , α 2 = 1 γ 2 β 2 [ γx 1 -x 2 ] , (C.6) \nwith x 1 , 2 ∶= ( u 1 , 2 · ¯ q ) . Due to this change of variables, the measure transforms as follows \nˆ d 4 ¯ q = 1 γβ ˆ d 2 ¯ q ⊥ dx 1 dx 2 . (C.7) \nI µ 3 = 1 γβ /integral.disp ˆ d 2 ¯ q ⊥ ˆ dx 1 ˆ dx 2 e ib · ¯ q ⊥ ˆ δ ( n ) ( x 1 ) ˆ δ ( x 2 ) ¯ q µ ¯ q 2 . (C.8) \nIn terms of x 1 , 2 and ¯ q ⊥ variables, we rewrite \nIntegrating by parts, we have \nI µ 3 = (-1 ) n 1 γβ /integral.disp ˆ d 2 ¯ q ⊥ e ib · ¯ q ⊥ ∂ n ∂x n 1 /parenleft.alt4 ¯ q µ ¯ q 2 /parenright.alt4 /divides.alt3 x 1 = x 2 = 0 . (C.9) \nI µ 3 = /uni23A7 /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23A8 /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23A9 (-1 ) n n ! 2 γβ ∫ ˆ d 2 ¯ q ⊥ e ib · ¯ q ⊥ ¯ q µ ⊥ ¯ q 2 ⊥ /bracketleft.alt4/parenleft.alt3 1 /radical.alt1 ¯ q 2 ⊥ γ 2 β 2 /parenright.alt3 n +/parenleft.alt3 -1 /radical.alt1 ¯ q 2 ⊥ γ 2 β 2 /parenright.alt3 n /bracketright.alt4 , if n ≥ 2 . 1 2 πγ 3 β 3 log ( ωb )( γu 2 -u 1 ) µ , if n = 1 . i 2 πγβ b µ b 2 , if n = 0 . (C.10) \n∂ n ∂x n 1 /parenleft.alt4 1 ¯ q 2 /parenright.alt4 /divides.alt3 x 1 = x 2 = 0 = n ! 2¯ q 2 ⊥ /uni23A1 /uni23A2 /uni23A2 /uni23A2 /uni23A2 /uni23A3 /uni239B /uni239D 1 /radical.alt1 ¯ q 2 ⊥ γ 2 β 2 /uni239E /uni23A0 n + /uni239B /uni239D -1 /radical.alt1 ¯ q 2 ⊥ γ 2 β 2 /uni239E /uni23A0 n /uni23A4 /uni23A5 /uni23A5 /uni23A5 /uni23A5 /uni23A6 (C.11) \n∂ ∂x 1 ¯ q µ /divides.alt3 x 1 = x 2 = 0 = 1 γ 2 β 2 ( γu 2 -u 1 ) µ (C.12) \nwhere \nand \nTherefore the first integral of equation (C.10) is evaluated as \nI µ 3 , 1 = (-1 ) n (-i ) n ! 2 γβ ∂ ∂b µ /integral.disp ˆ d 2 ¯ q ⊥ e ib · ¯ q ⊥ 1 ¯ q 2 ⊥ /uni23A1 /uni23A2 /uni23A2 /uni23A2 /uni23A2 /uni23A3 /uni239B /uni239D 1 /radical.alt1 ¯ q 2 ⊥ γ 2 β 2 /uni239E /uni23A0 n + /uni239B /uni239D -1 /radical.alt1 ¯ q 2 ⊥ γ 2 β 2 /uni239E /uni23A0 n /uni23A4 /uni23A5 /uni23A5 /uni23A5 /uni23A5 /uni23A6 = (-1 ) n (-i ) 1 4 πγβ ∂ ∂b µ /uni23A1 /uni23A2 /uni23A2 /uni23A2 /uni23A2 /uni23A3 -ω -n Γ ( n ) /uni239B /uni239C /uni239D /uni239B /uni239D 1 /radical.alt1 γ 2 β 2 /uni239E /uni23A0 n + /uni239B /uni239D -1 /radical.alt1 γ 2 β 2 /uni239E /uni23A0 n /uni239E /uni239F /uni23A0 × /uni239B /uni239D ( ωb ) n 1 F 2 /parenleft.alt4n 2 ; 1 , 1 -n 2 ; -1 4 /parenright.alt4 -1 /uni239E /uni23A0 /uni23A4 /uni23A5 /uni23A5 /uni23A5 /uni23A5 /uni23A6 = (-1 ) n + 1 i n ! b µ 4 πγβb 2 /uni23A1 /uni23A2 /uni23A2 /uni23A2 /uni23A2 /uni23A3 b n /uni239B /uni239C /uni239D /uni239B /uni239D 1 /radical.alt1 γ 2 β 2 /uni239E /uni23A0 n + /uni239B /uni239D -1 /radical.alt1 γ 2 β 2 /uni239E /uni23A0 n /uni239E /uni239F /uni23A0 × 1 F 2 /parenleft.alt4n 2 ; 1 , 1 -n 2 ; -1 4 /parenright.alt4 /uni23A4 /uni23A5 /uni23A5 /uni23A5 /uni23A5 /uni23A6 , (C.13) \nwhere p F q ( a ; b ; z ) is the generalized hypergeometric function. \nLastly, we evaluate the integral \nI µ 4 = /integral.disp ˆ d 4 ¯ qe -ib · ¯ q ˆ δ ( n ) ( u 1 · ¯ q ) ˆ δ ( u 2 · ¯ q ) ¯ q µ ( ¯ q 2 ) m , (C.14) \nwhere ( n ) denotes the number of derivatives acting over the on-shell delta function and m ≥ 2 . In terms of x 1 , 2 and ¯ q ⊥ variables, we have the following integral \nI µ 4 = 1 γβ /integral.disp ˆ d 2 ¯ q ⊥ ˆ dx 1 ˆ dx 2 e ib · ¯ q ⊥ ˆ δ ( n ) ( x 1 ) ˆ δ ( x 2 ) ¯ q µ ( ¯ q 2 ) m , m ≥ 2 . (C.15) \nIntegrating by parts, we have \nI µ 4 = (-1 ) n 1 γβ /integral.disp ˆ d 2 ¯ q ⊥ e ib · ¯ q ⊥ ∂ n ∂x n 1 /parenleft.alt4 ¯ q µ ( ¯ q 2 ) m /parenright.alt4 /divides.alt3 x 1 = x 2 = 0 . (C.16) \nI µ 4 = /uni23A7 /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23A8 /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23A9 (-1 ) n n ! m /product.disp k = 2 ( n + 2 k -2 ) 2 m /product.disp k = 2 ( 2 k -2 ) γβ ∫ ˆ d 2 ¯ q ⊥ e ib · ¯ q ⊥ ¯ q µ ⊥ ( ¯ q 2 ⊥ ) m /bracketleft.alt4/parenleft.alt3 1 /radical.alt1 ¯ q 2 ⊥ γ 2 β 2 /parenright.alt3 n +/parenleft.alt3 -1 /radical.alt1 ¯ q 2 ⊥ γ 2 β 2 /parenright.alt3 n /bracketright.alt4 , if n ≥ 2 . b 2 m -2 4 π ( m -1 ) γ 3 β 3 1 F 2 /parenleft.alt1 1 -m ; 1 , 2 -m ; -1 4 /parenright.alt1 ( γu 2 -u 1 ) µ , if n = 1 . ib 2 m -4 2 πγβ 1 F 2 /parenleft.alt1 1 -m ; 1 , 2 -m ; -1 4 /parenright.alt1 b µ , if n = 0 . (C.17) \nwhere \n∂ n ∂x n 1 /parenleft.alt4 1 ( ¯ q 2 ) m /parenright.alt4 /divides.alt3 x 1 = x 2 = 0 = n ! m /product.disp k = 2 ( n + 2 k -2 ) 2 m /product.disp k = 2 ( 2 k -2 )( ¯ q 2 ⊥ ) m /uni23A1 /uni23A2 /uni23A2 /uni23A2 /uni23A2 /uni23A3 /uni239B /uni239D 1 /radical.alt1 ¯ q 2 ⊥ γ 2 β 2 /uni239E /uni23A0 n + /uni239B /uni239D -1 /radical.alt1 ¯ q 2 ⊥ γ 2 β 2 /uni239E /uni23A0 n /uni23A4 /uni23A5 /uni23A5 /uni23A5 /uni23A5 /uni23A6 . (C.18) \nTherefore the first integral of equation (C.17) is evaluated as \nI µ 4 , 1 = (-1 ) n (-i ) n ! m /product.disp k = 2 ( n + 2 k -2 ) 2 m /product.disp k = 2 ( 2 k -2 ) γβ ∂ ∂b µ /integral.disp ˆ d 2 ¯ q ⊥ e ib · ¯ q ⊥ 1 ( ¯ q 2 ⊥ ) m /uni23A1 /uni23A2 /uni23A2 /uni23A2 /uni23A2 /uni23A3 /uni239B /uni239D 1 /radical.alt1 ¯ q 2 ⊥ γ 2 β 2 /uni239E /uni23A0 n + /uni239B /uni239D -1 /radical.alt1 ¯ q 2 ⊥ γ 2 β 2 /uni239E /uni23A0 n /uni23A4 /uni23A5 /uni23A5 /uni23A5 /uni23A5 /uni23A6 = (-1 ) n (-i ) n ! m /product.disp k = 2 ( n + 2 k -2 ) 4 π ( 2 m + n -2 ) m /product.disp k = 2 ( 2 k -2 ) γβ ∂ ∂b µ /uni23A1 /uni23A2 /uni23A2 /uni23A2 /uni23A2 /uni23A3 -b 2 m + n -2 /uni239B /uni239C /uni239D /uni239B /uni239D 1 /radical.alt1 γ 2 β 2 /uni239E /uni23A0 n + /uni239B /uni239D -1 /radical.alt1 γ 2 β 2 /uni239E /uni23A0 n /uni239E /uni239F /uni23A0 × 1 F 2 /parenleft.alt4m -n 2 + 1; 1 , 1 -m -n 2 + 2; -1 4 /parenright.alt4 /uni23A4 /uni23A5 /uni23A5 /uni23A5 /uni23A5 /uni23A6 = (-1 ) n + 1 i n ! m /product.disp k = 2 ( n + 2 k -2 ) b µ 4 π m /product.disp k = 2 ( 2 k -2 ) γβ /uni23A1 /uni23A2 /uni23A2 /uni23A2 /uni23A2 /uni23A3 b 2 m + n -4 /uni239B /uni239C /uni239D /uni239B /uni239D 1 /radical.alt1 γ 2 β 2 /uni239E /uni23A0 n + /uni239B /uni239D -1 /radical.alt1 γ 2 β 2 /uni239E /uni23A0 n /uni239E /uni239F /uni23A0 × 1 F 2 /parenleft.alt4m -n 2 + 1; 1 , 1 -m -n 2 + 2; -1 4 /parenright.alt4 /uni23A4 /uni23A5 /uni23A5 /uni23A5 /uni23A5 /uni23A6 . \n(C.19) \nThe main text also involves higher-rank integrals of the following form which can be written in terms of derivatives w.r.t b µ : \nI µ 1 µ 2 /uni22EF µ r 5 = /integral.disp ˆ d 4 ¯ qe -ib · ¯ q ˆ δ ( n ) ( u 1 · ¯ q ) ˆ δ ( u 2 · ¯ q ) ¯ q µ 1 ¯ q µ 2 /uni22EF ¯ q µ r ( ¯ q 2 ) m = (-i∂ µ 1 b )(-i∂ µ 2 b )/uni22EF(i∂ µ r -1 b ) /integral.disp ˆ d 4 ¯ qe -ib · ¯ q ˆ δ ( n ) ( u 1 · ¯ q ) ˆ δ ( u 2 · ¯ q ) ¯ q µ r ( ¯ q 2 ) m . (C.20) \nThe results must lie in the plane orthogonal to both u 1 and u 2 . Therefore we use the projected metric [20, 101, 102] \n∂ ∂b µ b ν = Π µν = η µν + 1 γ 2 β 2 /parenleft.alt1 u µ 1 ( u 1 -γu 2 ) ν + u µ 2 ( u 2 -γu 1 ) ν /parenright.alt1 (C.21) \nto generate the integrals of any rank. For example, \nI µνρ 6 = /integral.disp ˆ d 4 ¯ qe -ib · ¯ q ˆ δ ( n ) ( u 1 · ¯ q ) ˆ δ ( u 2 · ¯ q ) ¯ q µ ¯ q ν ¯ q ρ ( ¯ q 2 ) m = (-i∂ µ b )(-i∂ ν b ) I ρ 4 . (C.22) \nTherefore, \nI µνρ 6 = /uni23A7 /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23A8 /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23AA /uni23A9 (-1 ) n + 2 i ( 2 m + n -4 ) n ! m /product.disp k = 2 ( n + 2 k -2 ) 4 π m /product.disp k = 2 ( 2 k -2 ) γβ /uni23A7 /uni23AA /uni23AA /uni23A8 /uni23AA /uni23AA /uni23A9 b 2 m + n -8 /parenleft.alt4/parenleft.alt3 1 /radical.alt1 γ 2 β 2 /parenright.alt3 n +/parenleft.alt3 -1 /radical.alt1 γ 2 β 2 /parenright.alt3 n /parenright.alt4 × 1 F 2 /parenleft.alt1m -n 2 + 1; 1 , 1 -m -n 2 + 2; -1 4 /parenright.alt1 /uni23AB /uni23AA /uni23AA /uni23AC /uni23AA /uni23AA /uni23AD /bracketleft.alt2( 2 m + n -6 ) b µ b ν b ρ + b 2 b ( µ Π νρ ) /bracketright.alt2 , if n ≥ 2 . -b 2 m -6 2 πγ 3 β 3 1 F 2 /parenleft.alt1 1 -m ; 1 , 2 -m ; -1 4 /parenright.alt1 ( γu 2 -u 1 ) ( µ /bracketleft.alt2( 2 m -4 ) b ν b ρ ) + b 2 Π νρ ) /bracketright.alt2 , if n = 1 . -i ( 2 m -4 ) b 2 m -8 2 πγβ 1 F 2 /parenleft.alt1 1 -m ; 1 , 2 -m ; -1 4 /parenright.alt1 /bracketleft.alt2( 2 m -6 ) b µ b ν b ρ + b 2 b ( µ Π νρ ) /bracketright.alt2 , if n = 0 . 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2023ApJ...954...31C

We present the survey design implementation and outlook for COSMOSWeb a 255 hr treasury program conducted by the James Webb Space Telescope in its first cycle of observations. COSMOSWeb is a contiguous 0.54 degSUP2SUP NIRCam imaging survey in four filters F115W F150W F277W and F444W that will reach 5 pointsource depths ranging 27.528.2 mag. In parallel we will obtain 0.19 degSUP2SUP of MIRI imaging in one filter F770W reaching 5 pointsource depths of 25.326.0 mag. COSMOSWeb will build on the rich heritage of multiwavelength observations and data products available in the COSMOS field. The design of COSMOSWeb is motivated by three primary science goals 1 to discover thousands of galaxies in the Epoch of Reionization 6 z 11 and map reionizations spatial distribution environments and drivers on scales sufficiently large to mitigate cosmic variance 2 to identify hundreds of rare quiescent galaxies at z gt 4 and place constraints on the formation of the universes mostmassive galaxies M SUBSUB gt 10SUP10SUP M SUBSUB and 3 directly measure the evolution of the stellarmasstohalomass relation using weak gravitational lensing out to z 2.5 and measure its variance with galaxies star formation histories and morphologies. In addition we anticipate COSMOSWebs legacy value to reach far beyond these scientific goals touching many other areas of astrophysics such as the identification of the first direct collapse black hole candidates ultracool subdwarf stars in the Galactic halo and possibly the identification of z gt 10 pairinstability supernovae. In this paper we provide an overview of the surveys key measurements specifications goals and prospects for new discovery.

2023-09-01T00:00:00Z

['10.3847/1538-4357/acc2bc', '2022arXiv221107865C', 'arXiv:2211.07865', '2023ApJ...954...31C', '10.48550/arXiv.2211.07865']

['Sky surveys', 'Large-scale structure of the universe', 'Galaxy evolution', 'Reionization', 'Weak gravitational lensing', '1464', '902', '594', '1383', '1797', 'Astrophysics - Astrophysics of Galaxies', 'Astrophysics - Cosmology and Nongalactic Astrophysics']

COSMOSWeb An Overview of the JWST Cosmic Origins Survey

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']

https://arxiv.org/pdf/2211.07865.pdf

{'COSMOS-Web: An Overview of the JWST Cosmic Origins Survey': "Caitlin M. Casey, 1, 2, ∗ Jeyhan S. Kartaltepe, 3, ∗ Nicole E. Drakos, 4 Maximilien Franco, 1 Santosh Harish, 3 Louise Paquereau, 5 Olivier Ilbert, 6 Caitlin Rose, 3 Isabella G. Cox, 3 James W. Nightingale, 7 Brant E. Robertson, 4 John D. Silverman, 8, 9 Anton M. Koekemoer, 10 Richard Massey, 11 Henry Joy McCracken, 5 Jason Rhodes, 12 Hollis B. Akins, 1 Natalie Allen, 2, 13 Aristeidis Amvrosiadis, 11, 7 Rafael C. Arango-Toro, 6 Micaela B. Bagley, 1 Angela Bongiorno, 14 Peter L. Capak, 2, 13 Jaclyn B. Champagne, 15 Nima Chartab, 16 ' Oscar A. Ch'avez Ortiz, 1 Katherine Chworowsky, 1, † Kevin C. Cooke, 17 Olivia R. Cooper, 1, † Behnam Darvish, 18 Xuheng Ding, 8 Andreas L. Faisst, 19 Steven L. Finkelstein, 1 Seiji Fujimoto, 1, ‡ Fabrizio Gentile, 20, 21 Steven Gillman, 2, 22 Katriona M. L. Gould, 2, 13 Ghassem Gozaliasl, 23 Christopher C. Hayward, 24 Qiuhan He, 7 Shoubaneh Hemmati, 19 Michaela Hirschmann, 25, 26 Knud Jahnke, 27 Shuowen Jin, 2, 22 Ali Ahmad Khostovan, 3 Vasily Kokorev, 28 Erini Lambrides, 29, § Clotilde Laigle, 5 Rebecca L. Larson, 1, † Gene C. K. Leung, 1 Daizhong Liu, 30 Tobias Liaudat, 31 Arianna S. Long, 1, ‡ Georgios Magdis, 2, 22, 13 Guillaume Mahler, 11, 7 Vincenzo Mainieri, 32 Sinclaire M. Manning, 33, ‡ Claudia Maraston, 34 Crystal L. Martin, 35 Jacqueline E. McCleary, 36 Jed McKinney, 1 Conor J. R. McPartland, 2, 13 Bahram Mobasher, 18 Rohan Pattnaik, 3 Alvio Renzini, 37 R. Michael Rich, 38 David B. Sanders, 39 Zahra Sattari, 18, 16 Diana Scognamiglio, 40 Nick Scoville, 41 Kartik Sheth, 42 Marko Shuntov, 5 Martin Sparre, 43, 44 Tomoko L. Suzuki, 8 Margherita Talia, 20, 21 Sune Toft, 2, 13 Benny Trakhtenbrot, 45 C. Megan Urry, 46 Francesco Valentino, 2, 13 Brittany N. Vanderhoof, 3 Eleni Vardoulaki, 47 John R. Weaver, 33 Katherine E. Whitaker, 33, 2 Stephen M. Wilkins, 48, 49 Lilan Yang, 8 and Jorge A. Zavala 50 \n1 The University of Texas at Austin, 2515 Speedway Blvd Stop C1400, Austin, TX 78712, USA \n2 Cosmic Dawn Center (DAWN), Denmark \n- 3 Laboratory for Multiwavelength Astrophysics, School of Physics and Astronomy, Rochester Institute of Technology, 84 Lomb Memorial Drive, Rochester, NY 14623, USA 4 Department of Astronomy and Astrophysics, University of California, Santa Cruz, 1156 High Street, Santa Cruz, CA 95064, USA 5 Institut d'Astrophysique de Paris, UMR 7095, CNRS, and Sorbonne Universit'e, 98 bis boulevard Arago, F-75014 Paris, France 6 Aix Marseille Universit'e, CNRS, CNES, LAM, Marseille, France 7 Department of Physics, Institute for Computational Cosmology, Durham University, South Road, Durham DH1 3LE, UK 8 Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo, Kashiwa, Chiba 277-8583, Japan 9 Department of Astronomy, School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan 10 Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD 21218, USA 11 Department of Physics, Centre for Extragalactic Astronomy, Durham University, South Road, Durham DH1 3LE, UK 12 Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91001, USA 13 Niels Bohr Institute, University of Copenhagen, Jagtvej 128, DK-2200, Copenhagen, Denmark 14 INAF-Observatory of Rome, via Frascati 33, 00074 Monteporzio Catone, Italy 15 Steward Observatory, University of Arizona, 933 N Cherry Ave, Tucson, AZ 85721, USA 16 The Observatories of the Carnegie Institution for Science, 813 Santa Barbara St., Pasadena, CA 91101, USA 17 Association of Public and Land-grant Universities, 1220 L Street NW, Suite 1000, Washington, DC 20005, USA 18 Department of Physics and Astronomy, University of California, Riverside, 900 University Avenue, Riverside, CA 92521, USA 19 Caltech/IPAC, 1200 E. California Blvd., Pasadena, CA 91125, USA 20 University of Bologna - Department of Physics and Astronomy 'Augusto Righi' (DIFA), Via Gobetti 93/2, I-40129 Bologna, Italy 21 INAF, Osservatorio di Astrofisica e Scienza dello Spazio, Via Gobetti 93/3, I-40129, Bologna, Italy 22 DTU-Space, Technical University of Denmark, Elektrovej 327, DK-2800 Kgs. Lyngby, Denmark 23 Department of Physics, University of Helsinki, P.O. Box 64, FI-00014 Helsinki, Finland 24 Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010, USA 25 Institute of Physics, GalSpec, Ecole Polytechnique Federale de Lausanne, Observatoire de Sauverny, Chemin Pegasi 51, 1290 Versoix, Switzerland 26 \nINAF, Astronomical Observatory of Trieste, Via Tiepolo 11, 34131 Trieste, Italy \n27 Max Planck Institute for Astronomy, Konigstuhl 17, D-69117 Heidelberg, Germany \nCorresponding author: Caitlin M. Casey, Jeyhan S. Kartaltepe \ncmcasey@utexas.edu, jeyhan@astro.rit.edu \n28 Kapteyn Astronomical Institute, University of Groningen, PO Box 800, 9700 AV Groningen, The Netherlands 29 NASA Goddard Space Flight Center, Code 662, Greenbelt, MD, 20771, USA 30 Max-Planck-Institut fur Extraterrestrische Physik (MPE), Giessenbachstr. 1, D-85748 Garching, Germany 31 Universit'e Paris-Saclay, Universit'e Paris Cit'e, CEA, CNRS, AIM, 91191, Gif-sur-Yvette, France 32 European Southern Observatory, Karl-Schwarzschild-Straße 2, D-85748 Garching bei Munchen, Germany 33 Department of Astronomy, University of Massachusetts Amherst, 710 N Pleasant Street, Amherst, MA 01003, USA 34 Institute of Cosmology and Gravitation, University of Portsmouth, Dennis Sciama Building, Burnaby Road, Portsmouth, PO13FX, UK 35 Department of Physics, University of California, Santa Barbara, Santa Barbara, CA 93109, USA 36 Department of Physics, Northeastern University, 360 Huntington Ave, Boston, MA 02115, USA 37 Istituto Nazionale di Astrofisica (INAF), Osservatorio Astronomico di Padova, Vicolo dell'Osservatorio 5, 35122, Padova, Italy 38 Department of Physics and Astronomy, UCLA, PAB 430 Portola Plaza, Box 951547, Los Angeles, CA 90095, USA 39 Institute for Astronomy, University of Hawai'i at Manoa, 2680 Woodlawn Drive, Honolulu, HI 96822, USA 40 Argelander-Institut fur Astronomie, Auf dem Hugel 71, D-53121, Bonn, Germany 41 Astronomy Department, California Institute of Technology, 1200 E. California Blvd, Pasadena, CA 91125, USA 42 NASA Headquarters, 300 Hidden Figures Way, SE, Mary W. Jackson NASA HQ Building, Washington, DC 20546, USA 43 Institut fur Physik und Astronomie, Universitat Potsdam, Karl-Liebknecht-Str. 24/25, 14476 Golm, Germany 44 Leibniz-Institut fur Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482 Potsdam, Germany 45 School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel 46 Physics Department and Yale Center for Astronomy & Astrophysics, Yale University, PO Box 208120, CT 06520, USA 47 Thuringer Landessternwarte, Sternwarte 5, 07778 Tautenburg, Germany 48 Astronomy Centre, University of Sussex, Falmer, Brighton BN1 9QH, UK 49 Institute of Space Sciences and Astronomy, University of Malta, Msida MSD 2080, Malta 50 National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan", 'ABSTRACT': "We present the survey design, implementation, and outlook for COSMOS-Web, a 255 hour treasury program conducted by the James Webb Space Telescope in its first cycle of observations. COSMOSWeb is a contiguous 0.54 deg 2 NIRCam imaging survey in four filters (F115W, F150W, F277W, and F444W) that will reach 5 σ point source depths ranging ∼ 27.5-28.2 magnitudes. In parallel, we will obtain 0.19 deg 2 of MIRI imaging in one filter (F770W) reaching 5 σ point source depths of ∼ 25.326.0 magnitudes. COSMOS-Web will build on the rich heritage of multiwavelength observations and data products available in the COSMOS field. The design of COSMOS-Web is motivated by three primary science goals: (1) to discover thousands of galaxies in the Epoch of Reionization (6 < ∼ z < ∼ 11) and map reionization's spatial distribution, environments, and drivers on scales sufficiently large to mitigate cosmic variance, (2) to identify hundreds of rare quiescent galaxies at z > 4 and place constraints on the formation of the Universe's most massive galaxies ( M glyph[star] > 10 10 M glyph[circledot] ), and (3) directly measure the evolution of the stellar mass to halo mass relation using weak gravitational lensing out to z ∼ 2 . 5 and measure its variance with galaxies' star formation histories and morphologies. In addition, we anticipate COSMOS-Web's legacy value to reach far beyond these scientific goals, touching many other areas of astrophysics, such as the identification of the first direct collapse black hole candidates, ultracool sub-dwarf stars in the Galactic halo, and possibly the identification of z > 10 pair-instability supernovae. In this paper we provide an overview of the survey's key measurements, specifications, goals, and prospects for new discovery.", '1. INTRODUCTION': "Designed to peer into the abyss, extragalactic deep fields have pushed the limits of our astronomical observations as far and as faint as possible. The first of these \n- ∗ First two authors are co-first-authors \n- ‡ NASA Hubble Fellow \ndeep fields imaged with the Hubble Space Telescope (the medium deep survey and the Hubble Deep Field North, or HDF-N; Griffiths et al. 1996; Williams et al. 1996) pushed three magnitudes fainter than could be reached with ground-based telescopes at the time. Their data revealed a surprisingly high density of distant galaxies, well above expectation. This surprise was due to highredshift galaxies' elevated surface brightness relative to nearby galaxies, likely caused by their overall higher star formation rates. It quickly became clear that 'the Uni- \nverse at high redshift looks rather different than it does at the current epoch' (Williams et al. 1996). \nThis unexpected richness found in these first deep fields marked a major shift in astronomy's approach to high-redshift extragalactic science, moving from specialized case studies scattered about the sky and instead placing more emphasis on statistical studies using multiwavelength observations in a few deep fields where the density of information was very high. Such a transformation had a major role in leveling access to the highredshift Universe for a wide array of researchers worldwide, regardless of their individual access to astronomical observatories. Several other deep fields were pursued in short order after the HDF-N with Hubble , the other Great Observatories, and ancillary observations across the spectrum from the ground and space (e.g., the HDFS, CDFN and CDFS, GOODS-N and GOODS-S, and the HUDF; Williams et al. 2000; Brandt et al. 2000; Giacconi et al. 2002; Giavalisco et al. 2004; Beckwith et al. 2006), complementing each other in depth and area and providing crucial insight into the diversity of galaxies from the faintest, lowest-mass systems to the brightest and most rare. \nIn parallel to the effort to push deep over narrow fields of view, another experiment with Hubble transformed our understanding of large scale structure (LSS) at high redshifts by mapping a contiguous two square degree area of the sky, ∼ 20 times larger than all other deep fields of the time combined. Through its large area and statistical samples (resolving over 2 × 10 6 galaxies from 0 < z < 6), the Cosmic Evolution Survey (COSMOS; Scoville et al. 2007) allowed the first in-depth studies linking the formation and evolution of galaxies to their larger cosmic environments across 93% of cosmic time. By virtue of its large area, COSMOS probed a volume significantly larger than that of 'pencil-beam' deep fields and thus substantially minimized uncertainties of key extragalactic measurements from cosmic variance. In addition, the diverse array of multiwavelength observations gathered in the COSMOS field (Capak et al. 2007; Ilbert et al. 2010; Laigle et al. 2016; Weaver et al. 2022a) made it possible to carry out a suite of ambitious survey efforts and understand the distribution of large scale structure at early cosmic epochs (Scoville et al. 2013; Darvish et al. 2015). \nDeep field images of the distant Universe - from the deepest, Hubble Ultra Deep Field (HUDF), to the widest, COSMOS - have transformed into rich laboratories for testing hypotheses about the formation and evolution of galaxies through time. These hypotheses initially encompassed the first basic cosmological models and ideas regarding the evolution of galaxy structure. \nThanks to the addition of multiwavelength observations in these deep fields, they expanded to include hypotheses about the formation of supermassive black holes, the richness of galaxies' interstellar media, the assembly of gas in and around galaxies, and the structure of large dark matter haloes. \nThese deep fields, initially motivated by Hubble but substantially enhanced with a rich suite of ancillary ground-based and space-based data, have deepened our understanding of the evolution of galaxies across cosmic time. They pushed the horizon of the distant Universe into the first billion years, a time marking the last major phase change of the Universe itself from a neutral to an ionized medium (known as the Epoch of Reionization, or EoR, at z > ∼ 6, e.g., Stanway et al. 2003; Bunker et al. 2003; Bouwens et al. 2003, 2006; Dickinson et al. 2004). They also enabled the detailed study of galaxy morphologies (e.g., Abraham et al. 1996; Lowenthal et al. 1997; Conselice et al. 2000; Lotz et al. 2006; Scarlata et al. 2007), stellar mass growth (e.g., Sawicki & Yee 1998; Brinchmann & Ellis 2000; Papovich et al. 2001), the impact of local environment (e.g., Balogh et al. 2004; Kauffmann et al. 2004; Christlein & Zabludoff 2005; Cooper et al. 2008; Scoville et al. 2013), the distribution of dark matter across the cosmic web (e.g., Natarajan et al. 1998; Mandelbaum et al. 2006; Massey et al. 2007a; Leauthaud et al. 2007, 2011), as well as the discovery of the tight relationship between galaxies stellar masses and star formation rates (e.g., the galaxies' 'star-forming main sequence,' Daddi et al. 2007; Noeske et al. 2007; Elbaz et al. 2007). \nHowever, due to the expansion of the Universe, the next leap forward required observations in the nearinfrared (NIR) part of the spectrum. That came with the installation of the WFC3 camera on Hubble during the 2009 servicing mission. WFC3 expanded Hubble 's deep field capabilities into the NIR at similar depths as was previously achieved in the optical, enabling a tenfold increase in the number of candidate galaxies identified beyond z > ∼ 6 (Robertson et al. 2015; Bouwens et al. 2015; Finkelstein et al. 2015; Finkelstein 2016; Stark 2016), from a few hundred to a few thousand as well as the study of galaxies' rest-frame optical light out to z ∼ 3 (e.g., Wuyts et al. 2011; Lee et al. 2013; van der Wel et al. 2014; Kartaltepe et al. 2015a). The Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey (CANDELS; Grogin et al. 2011; Koekemoer et al. 2011) was particularly pioneering as it imaged portions of five of the key deep fields (GOODS-N, GOODS-S, UDS, EGS, and COSMOS) with the F125W and F160W filters over a total area of ∼ 800 arcmin 2 . \nThe successful launch of the James Webb Space Telescope ( JWST ) now marks a new era for studying the infrared Universe and the distant cosmos. With six times the collecting area of Hubble and optimized for observations in the near- and mid-infrared, JWST is currently providing images with greater depth and spatial resolution than previously possible. This is beginning to enable a substantial improvement in our understanding of galaxy evolution during the first few hundred million years (the epoch of cosmic dawn, z > ∼ 6) to the peak epoch of galaxy assembly (known as cosmic noon, 1 ≤ z ≤ 3). Given the tremendous legacy value of the deep fields imaged by the Great Observatories, several JWST deep fields have been planned for the observatory's first year of observations. The largest program among these, in both area on the sky and total prime time allocation, is the COSMOS-Web 1 Survey (PIs: Kartaltepe & Casey), for which this paper provides an overview. \nCOSMOS-Web was designed to bridge deep pencilbeam surveys from Hubble with shallower wide-area surveys, such as those that will be made possible by facilities like the future Roman Space Telescope (Akeson et al. 2019) and Euclid (Euclid Collaboration et al. 2022). With its unique combination of contiguous area and depth, COSMOS-Web will enable countless scientific investigations by the broader community. It will forge the detection of thousands of galaxies beyond z > 6, while also mapping the environments of those discoveries on scales larger than the largest coherent structures in the cosmic web on > ∼ 10 Mpc scales. It will identify hundreds of the rarest quiescent galaxies in the early Universe ( z > 4) and place constraints on the formation mechanisms of the most massive galaxies. It will also directly measure the evolution of the stellar mass to halo mass relation (SMHR) out to z ∼ 2 . 5 as a function of various galaxy properties using weak lensing measurements to estimate halo mass. \nThis paper describes the motivation for the COSMOSWeb survey as well as the program's design, providing an initial overview of what is to come as the data are collected, processed, and analyzed. Section 2 presents the detailed observational design of the survey and Section 3 briefly describes the context of COSMOS-Web among other deep fields planned for the first year of JWST observations. Section 4 presents the scientific \nmotivation of the survey as the drivers for the observational design. In Section 5, we share other possible investigations and predictions for what will be made possible by COSMOS-Web, beyond the main science goals. We summarize our outlook for the survey in section 6. Throughout this paper, we use AB magnitudes (Oke & Gunn 1983), assume a Chabrier stellar initial mass function (Chabrier 2003), and a concordance cosmology with H 0 = 70 km s -1 Mpc -1 and (Ω tot , Ω Λ , Ω m ) = (1 , 0 . 7 , 0 . 3).", '2. OBSERVATIONAL DESIGN': 'The observational design of the COSMOS-Web survey is motivated by the requirements of the primary science drivers described in § 4 while also striving to maximize value for the broader community across a wide range of science topics, described in part in § 5. Here we describe the detailed layout of the COSMOS-Web survey and provide more detailed motivation for the design when discussing the science goals in § 4.', '2.1. Description of Observations': "COSMOS-Web consists of one large contiguous 0.54 deg 2 NIRCam (Rieke et al. 2022) mosaic conducted in four filters (F155W, F150W, F277W, and F444W) with single filter (F770W) MIRI (Wright et al. 2022) imaging observations obtained in parallel over a total non-contiguous area of 0.19 deg 2 . The NIRCam mosaic is spatially distributed as a 41.5 ' × 46.6 ' rectangle at an average position angle of 110 · ; the shorter side of the mosaic is primarily oriented in the east-west direction. The center of the mosaic is at α =10:00:27.92, δ =+02:12:03.5 and is comprised of 152 separate visits (where each visit observes a single tile in the mosaic 2 ) arranged in a 19 × 8 grid. The coverage of these visits overlaid on the COSMOS Hubble F814W imaging is shown in Figure 1. \nEach individual visit is comprised of eight separate exposures of ∼ 257 seconds each, split into two separate executions of the 4TIGHT dither pattern at the same position in the mosaic. Each 4TIGHT dither pattern contains four individual integrations; an illustration of this dither pattern in one standalone visit and embedded in the larger mosaic is shown in Figure 2. The first 4TIGHT dither executes two NIRCam filters - F115W at short wavelengths (SW) and F277W at long wave- \nFigure 1. A map of the COSMOS-Web tiling pattern embedded within the Hubble ACS F814W mosaic of the COSMOS field (Scoville et al. 2007; Koekemoer et al. 2007). The mosaic consists of 152 visits where NIRCam serves as the primary instrument (long wavelength detector coverage shown in blue) with MIRI in parallel (shown in orange). The entire NIRCam mosaic is centered on the position α =10:00:27.9, δ =+02:12:03.5 and is 41.5 arcminutes (in the east-west direction) × 46.6 arcminutes (in the north-south direction) in size. The entire mosaic has an average position angle of 110 · , with individual visit PAs equal to 293 o in the northern half and 107 · in the southern half. Three visits required slightly different position angles due to availability of guide stars; this includes the lone northern-most MIRI tile. The detailed coordinates and position angles of each visit are provided in the Appendix, § A. \n<!-- image --> \nTable 1. Summary of COSMOS-Web NIRCam Survey Depth \nNote -Depths quoted are average 5 σ point source depths calculated within 0 . '' 15 radius apertures on data from our first epoch of observations without application of aperture corrections. \nTable 2. Summary of COSMOS-Web MIRI Survey Depth \nNote -Depths quoted are average 5 σ point source depths calculated within 0 . '' 3 radius apertures on data from our first epoch of observations, without application of aperture corrections. \nlengths (LW) - and the MIRI F770W filter in parallel. The second execution of the 4TIGHT dither switches NIRCam filters - to F150W in SW and F444W in LW yet keeps the same MIRI filter, F770W, for added depth. \nThe northern half of the mosaic is observed at one position angle, 293 · , while the southern half of the mosaic is observed at another, 107 · . These position angles are relative to the NIRCam instrument plane and not V3 (which differ by < 1 · ); they are also not exactly a 180 · flip from one another. Instead they are staggered by ± 3 · to make scheduling more flexible while maintaining a contiguous mosaic using a slight jigsaw pattern to stitch adjacent visits together. The distribution of half of the mosaic at one position angle and the other half at another also makes it possible to fit most of the MIRI parallel exposures fully within the larger NIRCam mosaic. A few visits required further position angle modification due to limitations in guide star catalog availability at their initially intended angles. The Appendix ( § A) gives detailed information for each individual visit and a table of all visits. \nThe depth of the NIRCam observations varies based on the number of exposures at any position in the mosaic (see Table 1); of the total 1928 arcmin 2 ( ≈ 0.54 deg 2 ) area in the NIRCam SW mosaic, 71.3 arcmin 2 ( ∼ 3.7%) will be covered with only a single exposure per SW filter, 991.6 arcmin 2 ( ∼ 51.4%) will have two SW exposures, 60.0 arcmin 2 ( ∼ 3.1%) will have three SW exposures, and 805.2 arcmin 2 ( ∼ 41.8%) will have four SW exposures. The NIRCam LW mosaic covers a total area of 1924 arcmin 2 , of which 17.8 arcmin 2 ( ∼ 0.9%) has single exposure depth, 978.0 arcmin 2 ( ∼ 50.8%) has two exposure depth, 24.4 arcmin 2 ( ∼ 1.3%) has three exposure depth, and 904.3 arcmin 2 ( ∼ 47.0%) has four exposure depth. The most deeply exposed portions of the SW mosaic align with the deepest portions of the LW mosaic, \nthough the areas differ slightly based on the differences in detector size and gaps between SW detectors. \nDue to the design of the NIRCam mosaic as contiguous, the MIRI parallel observations are not contiguous but are distributed in 152 distinct regions corresponding to the 152 visits. MIRI coverage of each visit has an area of 4.2 arcmin 2 corresponding to the primary MIRI imager field of view, and 4.5 arcmin 2 when accounting for the additional area of the Lyot Coronographic Imager 3 . Of that area, 0.55 arcmin 2 (12%) has two MIRI exposures, 2.81 arcmin 2 (62%) has four, 0.21 arcmin 2 (5%) has six, and 0.96 arcmin 2 (21%) has eight MIRI exposures. The total area covered with MIRI in COSMOSWeb is 688 arcmin 2 or 0.19 deg 2 . Of the 152 MIRI visits, 143 (651 arcmin 2 , 95%) are fully contained within the NIRCam mosaic. Note that MIRI observations from PRIMER (GO #1837) add an additional 53arcmin 2 of (deeper) 7.7 µ m coverage (see § 3 for full details) contained within the NIRCam footprint, bringing the total MIRI coverage in COSMOS from these two Cycle 1 surveys to 742 arcmin 2 . \nTable 1 summarizes the characteristics of the NIRCam mosaic and the measured depths as a function of number of exposures. The NIRCam depths have been measured using data from the first epoch of COSMOSWeb observations, consisting of six visits (out of the total 152). These data are later described in § 2.7. These are broadly consistent with the expected performance of JWST in-flight (Rigby et al. 2022). These depths correspond to 5 σ point sources extracted within 0 . '' 15 radius circular apertures in each filter without any aperture corrections applied. Table 2 provides a summary for the MIRI exposures; similarly, these depths are measured directly using data from the first epoch of observations in COSMOS-Web using a a 0 . '' 3 radius circular aperture without aperture correction. We note that the measured MIRI depths are significantly better than expectation from the exposure time calculator. We conducted a number of tests to measure this depth accurately, including a direct comparison of IRAC 8 µ mflux densities with MIRI 7.7 µ mflux densities, measurement of depth within empty apertures in individual exposures, as well as measurement of the standard deviation in flux densities for individual sources in individual exposures. All tests give consistent results, showing F770W depths nearly a magnitude deeper than expectation. The depths of the sur- \nFigure 2. An illustration of the 4TIGHT dither pattern for NIRCam prime visits (top two panels) and MIRI parallel visits (bottom panel). The top panel shows the NIRCam SW exposure map for a single visit with coverage ranging from one (lightest) to four (darkest) exposure depth. Two of the four dither positions are outlined in color (red and blue) for clarity. The middle panel shows the NIRCam SW exposure map in the context of the larger COSMOS-Web mosaic. At bottom, the MIRI coverage is shown. The axes are positional offsets along the V3 and V2 angle (i.e., perpendicular and parallel to the PA) relative to the reference position, given for each visit in the Appendix, § A. \n<!-- image --> \nction of wavelength are shown in Figure 3 relative to other existing datasets available in the COSMOS field.", '2.2. Motivation for a Contiguous ∼ 0.5 deg 2 Area': "The contiguous, and roughly square, area of COSMOS-Web is driven by two of our primary science objectives. The first is to construct large scale structure density maps at 6 < z < 10 to address whether or not the most UV-luminous systems are embedded in overdense structures (see § 4.1 for details). Mapping the large scale environments of our discoveries and mitigating cosmic variance at these epochs (with cosmic variance less than 10%, i.e., σ 2 v < 0 . 10) requires contiguous solid angles larger than the expected size of reionization bubbles at these redshifts (Behroozi et al. 2019), \n> 0.3-0.4 deg 2 . Our 0.54 deg 2 program allows for some uncertainty in the scale of these reionization bubbles, as some simulations see bubbles extend on 40 ' scales (D'Aloisio et al. 2018; Th'elie et al. 2022). Our NIRCam mosaic maps to ∼ (114 Mpc) 2 between 6 < z < 8 and ∼ (122 Mpc) 2 between 8 < z < 10 projected on the sky at these epochs. We describe more about the expected cosmic variance in COSMOS-Web in § 2.6. \nThe second scientific driver for our contiguous area is the coherence we can achieve for the weak lensing measurement of galaxies' halo masses on scales < ∼ 10 Mpc in order to place constraints on the SMHR out to z ∼ 2 . 5 (see § 4.3 for details). This requires at least ∼ 5 dark matter halo scale lengths ( ∼ 3 proper Mpc across each) of contiguous coverage, for which our survey will provide ∼ 10 × 10 dark matter scale lengths to boost the signal-to-noise and allow splitting by galaxy type and by mass (Wechsler & Tinker 2018; Wang et al. 2018a; Debackere et al. 2020; Shuntov et al. 2022). Several smaller non-contiguous areas (of order 0.05 deg 2 ) would render the SMHR measurement and calibration of cosmological models severely hindered.", '2.3. Field on the Sky': "The COSMOS field was chosen for these observations for several reasons. First, the existing HST/ACS F814W coverage (Koekemoer et al. 2007) provides crucial value to our science goals of detecting galaxies beyond z > 6 using [F814W]-[F115W] colors. Second, COSMOS has the widest deep ancillary data coverage from X-ray to radio wavelengths (Ilbert et al. 2013; Laigle et al. 2016; Weaver et al. 2022a). Third, it is an equatorial field ( α = 150 · , δ = +2 · ), and thus accessible to all major existing and planned future facilities, essential for swift and efficient follow-up of JWST -identified sources. A sampling of the multiwavelength data already available in the COSMOS-Web footprint is shown in Figure 4. \nAdditionally, COSMOS has been selected or is a likely candidate to be a deep calibration field for future key projects including Euclid , the Roman Space Telescope , and the Vera Rubin Observatory LSST project. Over 250,000 spectra have been taken of > 100,000 unique objects in the COSMOS field at 0 < z < 7 (A. Khostovan et al. in preparation), including from large surveys such as zCOSMOS (Lilly et al. 2007, 2009), FMOS-COSMOS (Silverman et al. 2015; Kartaltepe et al. 2015b; Kashino et al. 2019), VUDS (Le F'evre et al. 2015), and many programs using Keck (e.g., Kartaltepe et al. 2010; Capak et al. 2011; Casey et al. 2012; Kriek et al. 2015; Hasinger et al. 2018), greatly enhancing the accuracy of photometric redshifts for all sources in the field. Lastly, the quality of photometric redshifts ∆ z/ (1 + z ) < 0 . 02 \nFigure 3. An illustration of the deepest filters available in COSMOS-Web and their depths across the spectrum. At top are the filter transmission profiles for existing COSMOS datasets that are ground-based (light gray), space-based (dark gray), and new additions from JWST for COSMOS-Web (blue). These filters are separated between those that have 5 σ point source depths between 24-26.5 magnitudes (top sub-panel), and those that reach depths beyond 26.5 magnitudes (bottom sub-panel; see Weaver et al. 2022a, for more details). We also include recent coverage from COSMOS-DASH at 1.6 µ m (Mowla et al. 2019; Cutler et al. 2022). Note that narrow-band and medium-band filters in the field are not shown (as they generally have depths shallower than ∼ 24 magnitudes). At bottom, we illustrate the 5 σ point source depths from these same filters, highlighting the depth of COSMOS-Web JWST observations in solid blue at full four-integration depth; dashed lines show half two-integration depth (covering approximately half the mosaic, as detailed in Tables 1 and 2). Overlaid are several galaxy templates: Lymanbreak galaxies at z ∼ 7 -13 (shades of lavender to dark purple), a M glyph[star] ∼ 10 10 M glyph[circledot] z = 4 passive galaxy (dark green), and a z = 2 dusty starburst (light green). \n<!-- image --> \nfor galaxies with i < 25 (Ilbert et al. 2013; Laigle et al. 2016; Weaver et al. 2022a) has facilitated the discovery and analysis of galaxies out to z ∼ 7 and beyond (Bowler et al. 2017, 2020; Stefanon et al. 2019; Kauffmann et al. 2022). The photometric redshifts will be further improved with the addition of COSMOS-Web (see § 2.5), dramatically improving the accuracy of the weak lensing measurement of galaxies' halo mass as well as galaxies' stellar masses and star formation rates across all epochs.", '2.4. Filter Optimization': "We simulated the effectiveness of many filter combinations to deliver the science objectives described in § 4 and determined that COSMOS-Web should be a four filter NIRCam survey with MIRI imaging conducted in parallel: F115W+F150W in SW, F277W+F444W in LW, and F770W with MIRI. Reionization science drives the choice of F115W and F150W to maximize \nFigure 4. The COSMOS-Web NIRCam (left) and MIRI (right) coverage shown together with the PRIMER NIRCam and MIRI coverage in a joint exposure map in grayscale. The two instruments' coverage are shown separately for clarity. We overlay maps of a number of multiwavelength datasets for context. The JWST coverage from COSMOS-Web and PRIMER are shown in shades of blue. The Hubble CANDELS survey (Grogin et al. 2011; Koekemoer et al. 2011) area is shown in dashed black, the ALMA extended MORA survey (Zavala et al. 2021; Casey et al. 2021, Long et al., in preparation) is shown in burnt orange and the deep SCUBA-2 450 µ m+850 µ m coverage area of the eS2-COSMOS survey and STUDIES survey (Wang et al. 2017) is shown in light orange (note that the entire COSMOS field is covered with 850 µ m SCUBA-2 coverage from the S2COSMOS Survey; Simpson et al. 2019). In the radio, we highlight the deep continuum coverage at 3 GHz and 10 GHz in dark and light green, respectively, from the COSMOS-XS survey (van der Vlugt et al. 2021), which complements the full-field 3 GHz VLA COSMOS Survey of Smolˇci'c et al. (2017). In the X-ray, we show the area covered by the Chandra C-COSMOS Deep survey (dashed magenta) as well as the medium depth survey (solid magenta), both summarized by Elvis et al. (2009), with full COSMOS field coverage extended by Civano et al. (2016). Finally, we note that the full COSMOS field has coverage with Subaru's Hyper Suprime-Cam (Aihara et al. 2022). \n<!-- image --> \ncoverage of the observed wavelength of a Lyman break from 6 < z < 13; we plan EoR source selection using a hybrid photometric redshift and dropout approach ( z ∼ 6 -7 galaxies drop out in HST -F814W, while z ∼ 8 -10 galaxies drop out in the F115W filter, and z > 12 will drop out in F150W). Weak lensing objectives are less sensitive to filter choice but benefit from tremendous depth in the NIR by increasing the background source density; we expect > 10 galaxies per arcmin 2 at z > 4 with measurable shapes, in other words, those found above a 15 σ detection threshold. We calculate the on-sky source density of galaxies above certain apparent magnitude thresholds from existing measurements of galaxy luminosity functions from 0 < z < 10 (e.g., Arnouts et al. 2005; Bouwens et al. 2015; Finkelstein et al. 2015). Indeed, preliminary sim- \nons show that galaxies at the 15 σ shape-detection threshold, F277W ∼ 26.8, with R eff < ∼ 0 . '' 3 ( ≈ 2-3 kpc), are recovered without bias introduced from the JWST point spread function (PSF; Liaudat & Scognamiglio et al., in preparation). We find F277W+F444W to be the most advantageous LW filter combination to improve the quality of photometric redshifts and mitigate lower redshift contaminants (more details discussed in § 4.1). The F444W filter is particularly useful for measuring the rest-frame optical morphologies of galaxies at z > 4, (e.g., Kartaltepe et al. 2022) and the rest-frame nearinfrared morphologies of lower redshift galaxies (e.g., Guo et al. 2022). The LW filters will be useful for the identification of very red z = 4 -6 quiescent galaxies and measuring their mass surface densities and morphologies at high signal-to-noise. \nThe choice of F770W for the MIRI parallel exposures is motivated by the need to constrain reliable stellar masses for z > 4 massive systems. F770W is roughly matched to the Spitzer 8.0 µ m filter (which has much shallower data in COSMOS; Sanders et al. 2007). F770W data will provide a factor of 50 × improvement in depth relative to Spitzer 8.0 µ m and a factor of 7.6 × improvement in the beam size, thus opening up detections to the z > 4 universe. Our MIRI data will cover an area ∼ 3.5 × larger than all other planned JWST MIRI deep fields from Cycle 1 combined (see § 3), making it particularly sensitive to rare, bright objects. F770W optimizes both sensitivity and the uniqueness of longer rest-frame wavelengths for high redshift galaxies. Longer wavelength filters would reduce the sensitivity by 10-30 × , and F560W does not provide a sufficient lever arm from F444W to measure highz galaxy stellar masses.", '2.5. Precision of Photometric Redshifts': "A crucial aspect of the design of COSMOS-Web was the selection of filters, largely driven by finding the most reliable selection of EoR sources from 6 < z < 11. We generated an empirical light cone of mock galaxies, populating it with galaxies following the galaxy luminosity function from the local Universe to z ∼ 4 from Arnouts et al. (2005); from z ∼ 4 -10, galaxies are drawn from the UV luminosity functions of Bouwens et al. (2015). From z ∼ 10 -12, the Bouwens et al. luminosity functions are extrapolated by fixing M glyph[star] and extrapolating trends in Φ glyph[star] and α measured at lower redshift (i.e., higher redshifts have lower densities and steeper faintend slopes). \nOnce the on-sky density of galaxies (as a function of rest-frame UV absolute magnitude, M UV ) is set, we assign a variety of spectral energy distributions (SEDs) to each galaxy. Given the focus on reliability of EoR targets, the SEDs we generate were of somewhat limited scope, focusing on three families of templates from Maraston (2005) with 61 ages for each. The primary difference between templates is the star-forming timescale, with exponentially declining star-formation histories of 0.25, 1, and 10 Gyr. Three attenuations were used with E ( B -V ) = 0 , 0 . 05, and 0 . 1 (not including very reddened sources). Both nebular line emission and IGM opacity (Madau 1995) were included. The choice of SED for a given galaxy was then assigned using a uniform distribution (with an allowable star-formation timescale). While there are clear limitations to this idealized, empirical calibration sample - such as the lack of more diverse SEDs, a mass- or redshift-weighted method of assigning SEDs, or using a wider set of templates to fit the ensuing photometric redshifts - it can still provide a useful first \npass at our photometric redshift precision, particularly for newly-discovered faint galaxies within the EoR. \nNoise is added to the mock observations according to the depth in each filter (to the greatest depth as quoted in Tables 1 and 2). Similarly, known noise characteristics of existing ground-based data have been added to the galaxies' mock photometry (details of those observations are provided by Weaver et al. 2022a). We use this mock sample to diagnose the contamination and precision of our photometric redshifts across all epochs, applying tools we will use for the real dataset. Specifically, here we use the LePhare SED fitting code to derive photometric redshifts (Arnouts et al. 2002; Ilbert et al. 2006), as implemented for the recent COSMOS2020 compilation by Weaver et al. (2022a). Note that in § 4.1.3 we explore the specific parameter space of EoR mock sources from this lightcone in more detail, and here we present the general characteristics of the expected photometric redshift quality across all epochs. \nFigure 5 shows the input 'known' redshift against the best measured output redshift for all mock sources from 0 < z < 12. The full simulation contains ∼ 3.3M sources, 13K of which ( ≈ 0 . 4%) are at 6 < z < 12. Given the sheer number of sources in the catalog, we split the simulation into two regimes: at z < 6 we only sample a random 1% subset of all sources for photometric redshift fitting (for computational ease); in other words, we fit photometric redshifts to ∼ 33K sources from 0 < z < 6. At z > 6 we fit all galaxies so that we adequately sample the full range of true EoR source properties. Thus, in Figure 5, there appears to be a dearth of sources at 4 < ∼ z < 6 due to this differential sampling of parameter space, but the apparent differential is simply visual (e.g., there are 1K sources modeled in the 4 < z < 6 bin). To understand the improvement in the photometric redshifts provided by COSMOS-Web data, we compare our inferred mock photometric redshift quality to those from the COSMOS2020 catalog. Specifically, Weaver et al. (2022a) find that sources with i -band magnitude between 25 < i < 27 have 5% photometric redshift precision. Over the same i -band magnitude range, we infer that these JWST data will improve that statistic to 2.5%. Both precisions are measured using the normalized median absolute deviation ( σ NMAD ) of ∆ z/ (1 + z ), a quantity analogous to the standard deviation of a Gaussian but less sensitive to outliers. \nWhile this direct comparison is useful, we also calculate σ NMAD for intrinsically fainter sources selected at longer wavelengths. We find that the median precision for sources with F277W magnitudes ranging 25-26.5 to be 2.3% across all epochs, and those with F277W magnitudes ranging 26.5-27.5 to be 4.2%. The right panel \n<!-- image --> \nFigure 5. Results of photometric redshift fitting to a set of mock galaxies; these mock galaxies have the full set of COSMOS photometry, spanning both existing ground and space-based photometry (drawn from the limits described in Weaver et al. 2022a), as well as model photometry in the JWST bands corresponding to COSMOS-Web. On the left, we show the known redshifts of mock galaxies vs. the best fit photometric redshifts, which are derived by performing spectral energy distribution (SED) fits with LePhare (Arnouts et al. 2002; Ilbert et al. 2006). Below z < 6, the purple heat map shows the density of sources down to F277W = 27.5 magnitudes in 1% of our simulation for clarity; the orange heat map shows all simulated sources above z > 6. The thick black lines enclose the average dispersion about the 1-to-1 line as a function of redshift, as measured using the normalized median absolute deviation ( σ NMAD ). The average precision across all redshifts and magnitudes (down to 27.5) is 3.3%. At the top right, we show how the σ NMAD statistic varies as a function of F277W magnitude; sources down to 27 th magnitude will have photometric redshifts precise to < 5%, and sources closer to the detection threshold will have 10-18% precision. At the bottom right, we show the anticipated outlier fraction ( η ) as a function of magnitude, defined as sources with photometric redshift precision worse than 15%. The outlier fraction is less than 5% down to ∼ 26.5, then increases toward 15% near the detection cutoff. Neither σ NMAD nor η show significant redshift dependence, other than slight spikes in the range 6 < z < 7 and 9 < z < 11. Gray bands show our 5 σ point source detection limits for two-integration and four-integration depths. \n<!-- image --> \nof Figure 5 shows how the photometric redshift precision is expected to degrade for sources as a function of F277W magnitude. Similarly, we investigate the outlier fraction, η , as a function of magnitude, where outliers are defined as sources with ∆ z/ (1 + z ) > 0 . 15. Outliers are below 10% for sources brighter than F277W < 27, increasing steeply to 20% near the 5 σ detection limit at ∼ 28.2. Note that we analyze both the photometric redshift precision and outlier fraction as a function of redshift as well as magnitude; overall, both quantities are somewhat constant with redshift, with slight spikes in both from 6 < z < 7 and 9 < z < 11, which is expected given the lack of complete filter coverage across expected break wavelengths at those redshifts. We analyze the efficacy of photometric selection for EoR galaxies further in § 4.1.3. \n2.6. Expected Cosmic Variance \nThe areal coverage of COSMOS-Web represents a real strength of the program in the reionization era. With claims of potential massive galaxies in the distant universe from smaller surveys that, if confirmed, may challenge our models of galaxy formation, representative samples of the distant galaxy population would help establish the true luminosity function shape and evolution at early times. \nFollowing Robertson (2010) (see also Trenti & Stiavelli 2008), we estimate the z ∼ 9 cosmic variance of the COSMOS-Web survey. We assume the survey area A = 0 . 54 deg 2 , a depth of 27.6 magnitudes, and the z ∼ 9 luminosity function parameters from Bouwens et al. (2021). We perform abundance matching between galaxies and the halo mass function, assigning the clustering strengths of halos to their hosted galaxies from the Tinker et al. (2010) peak background split model for the halo bias. We find that the cosmic sample variance uncertainty of COSMOS-Web at z ∼ 9 is σ v ≈ 16%, and \nPoisson uncertainty is σ p ≈ 8%, which sum in quadrature to a total expected uncertainty of σ tot ≈ 18% (giving a total variance σ 2 v = 0 . 03). \nHow does the cosmic variance of COSMOS-Web compare with the collection of smaller, deeper fields soon available with JWST coverage? Repeating our calculation for a single 100 arcmin 2 field to 29 th magnitude, we find such surveys have a z ∼ 9 cosmic sample variance uncertainty of σ v ≈ 34%. Five 100 arcmin 2 fields probing independent sight lines have a combined cosmic sample uncertainty of σ v ≈ 14%, a Poisson uncertainty of σ p ≈ 11% and a total expected variance uncertainty of σ tot ≈ 18%. Thus COSMOS-Web has comparable statistical power to the combined power of other JWST Cycle 1 programs conducted over a smaller area to greater depth. As discussed in Robertson (2010), by combining these wide area and pencil beam surveys the degeneracies in the constraints on luminosity function parameters, like M glyph[star] , Φ glyph[star] , or the faint-end slope α , can be broken or significantly ameliorated.", '2.7. Scheduling of the Observations and First Epoch of Data': 'COSMOS-Web was awarded a total of 208 hours, but due to changes in overhead and the dithering pattern described above, COSMOS-Web will take a total of 255 hours to execute. We requested relatively low zodiacal background observations ( < 10-20 th percentile) and to tile the mosaic at a nearly uniform position angle on the sky to avoid gaps within the mosaic. COSMOS-Web is observable in windows in April (PA ≈ 105) and December/January (PA ≈ 290) of each year. In order to maximize the amount of overlap between the prime (NIRCam) and parallel (MIRI) observations, we will observe roughly half of the mosaic in each window. \nThe first epoch of observations consists of six visits covering ∼ 77 arcmin 2 with NIRCam and was observed on 5-6 January 2023. Figure 6 shows the NIRCam mosaic of this region of the field, which is 4% of the final dataset. Figure 7 shows the six MIRI tiles from this epoch. As of this writing, 77 pointings are scheduled for April/May 2023 (roughly half the field) and the remaining 69 pointings in December 2023/January 2024. The COSMOS-Web team will release mosaics registered to Gaia astrometry after each subsequent epoch of observations is taken through the Mikulski Archive for Space Telescopes (MAST) and the NASA/IPAC Infrared Science Archive (IRSA); we will also make these mosaics accessible through the IRSA COSMOS cutout service 4 .', '3. CONTEXT OF COSMOS-WEB AMONG OTHER JWST DEEP FIELDS': "Several extragalactic deep field surveys will be conducted in the first year of JWST observations that span a range of areas, depths, and filter coverage; their approximate depths and areas are described in Table 3 for NIRCam programs and Table 4 for MIRI programs. Note that the NIRCam depths of other programs quote the pre-flight exposure time calculator (ETC) estimates and do not necessarily reflect the actual final measured depths of the data. For MIRI, we include the measured depths from COSMOS-Web, CEERS, and PRIMER observations along with the updated ETC estimates. The MIRI depth in COSMOS-Web is measured to be significantly deeper (by ∼ 1 magnitude) compared to the ETC estimates. We refer the reader to the recent review by Robertson (2022), their § 8.2, as well as their Figure 6, for a summary of many of the large extragalactic programs, and in particular their NIRCam coverage. These programs include the Guaranteed Time Observation (GTO) programs allocated to the instrument teams, the Director's Discretionary Early Release Science Programs (ERS), as well as the General Observer (GO) Cycle 1 programs. \nFigure 8 shows the relative depth and survey area of the major broad-band legacy extragalactic programs in Cycle 1, both for NIRCam imaging and MIRI imaging programs. To briefly summarize the relative scope of the NIRCam programs, the deepest surveys are NGDEEP 5 (GO #2079) and the JADES GTO Survey (in particular GTO #1180, 1210, & 1287). These collectively cover about ∼ 0.05 deg 2 to depths exceeding ∼ 29.5 mag in several broad-band filters. The medium depth programs JADES-Medium (including parts of GTO #1180, 1181, and 1286), CEERS (ERS #1345), and PRIMER (GO #1837) together cover a total of ∼ 0.18 deg 2 to a depth ∼ 28-29 mag. Note that the UDF Medium Band Survey (GO # 1963) achieves similar depths ∼ 28-29.8 mag in NIRCam medium bands over 10arcmin 2 in the HUDF (an area also covered by JADES-Deep in the broadbands). COSMOS-Web (GO #1727) is the shallowest but largest program to be observed, covering a total 0.54 deg 2 with NIRCam to a depth of ∼ 27.5-28.2 mag across the field. \nFigure 6. The first epoch of COSMOS-Web NIRCam observations obtained on 5-6 January 2023. These data cover six visits (or pointings) out of a total of 152. The total area covered by NIRCam here is ∼ 77 arcmin 2 . The relative position of this mosaic in the survey is shown at upper left. At lower left are several zoomed-in 10 '' × 10 '' cutouts (and one 16 '' × 16 '' cutout) of a handful of interesting objects, highlighting the level of detail revealed by these first data. \n<!-- image --> \nPlanned MIRI programs vary in depth more substantially, as the shorter wavelength filters achieve much deeper observations per fixed exposure time. The MIRI GTOprograms adopt two very different approaches: one (GO # 1283) goes quite deep in a single MIRI pointing in one filter, F560W. The other (GO # 1207) covers 30 arcmin 2 and uses all 8 broad-band MIRI filters and thus is significantly more shallow. MIRI imaging is obtained in parallel to much of the JADES program (from programs GTO #1180 and 1181) where a hybrid approach was adopted, going deep in one filter, F770W, \nover 10 arcmin 2 , and shallower in two filters, F770W and F1280W, over 15 arcmin 2 . CEERS similarly spans a broad range in depths over 13 arcmin 2 using 6 filters, and PRIMER covers much larger areas over ∼ 140 arcmin 2 in two filters. Similar to its NIRCam coverage, COSMOSWeb covers the largest area with MIRI, but with variable depth (based on the number of exposures) in F770W. We have shown the F770W depths of the MIRI surveys using a star in Figure 8 for more direct comparisons to the COSMOS-Web depths. \nFigure 7. The first epoch of COSMOS-Web MIRI observations obtained on 5-6 January 2023. Covering six visits, the MIRI data is distributed in six non-overlapping tiles and include data from both the MIRI imager and Lyot Coronograph field of view. At left is a comparison between Spitzer IRAC channel 4 (8 µ m) data and MIRI 7.7 µ m data in a 40 '' × 40 '' zoom-in panel, highlighting the increased sensitivity and resolution of MIRI observations over those previously obtained with IRAC. \n<!-- image --> \nThe total area covered by COSMOS-Web in NIRCam is roughly 2.7 × larger than the other planned JWST extragalactic deep fields combined . For MIRI, COSMOSWeb's coverage is 3.5 × larger than all other deep field programs combined . The extraordinary range of areas and depths of deep field surveys observed in JWST 's first year will be complementary, and enable a wide range of scientific studies, spanning the most distant and faintest galaxies ever detected to the most comprehensive environmental studies of the distant universe.", '4. SCIENTIFIC GOALS': "The scientific breadth of COSMOS-Web has the potential to be extraordinary, with an estimated ∼ 10 6 sources to be detected from z ∼ 0 . 1 to cosmic dawn. Nevertheless, the survey as proposed was motivated by \nthree key science areas that ultimately drive the design of the survey. The three primary goals of the program are to: \n- 1. forge the detection of thousands of galaxies in the Epoch of Reionization (6 < ∼ z < ∼ 11) and use their spatial distribution to map large scale structure during the Universe's first billion years,\n- 2. identify hundreds of the rarest quiescent galaxies in the first two billion years ( z > 4) to place stringent constraints on the formation of the Universe's most massive galaxies (with M glyph[star] > 10 10 M glyph[circledot] ), and\n- 3. directly measure the evolution of the stellar mass to halo mass relation (SMHR) out to z ∼ 2 . 5 and", 'Casey, Kartaltepe et al.': "Table 3. JWST Cycle 1 NIRCam Surveys \nNote -Depths quoted are 5 σ point source depths. NIRCam depths quoted have been drawn from the original proposals and pre-flight exposure time calculator estimates within 0 . '' 15 radius circular apertures. We have not adjusted for the in-flight calibration (Boyer et al. 2022) of the instruments; however, any differences with these figures is anticipated to be of order smaller than a 10% effect, smaller than the typical deviation across a mosaic stitched together with non-uniform depth, or from variation in depth filter-to-filter. Program IDs for these surveys are: NGDEEP (GO # 2079), UDF-Medium (GO # 1963), JADES-Deep (GTO # 1180, 1210, 1287), JADES-Medium (GTO # 1180, 1181, 1286), CEERS (ERS # 1345), PRIMER (GO # 1837), and COSMOS-Web (GO # 1727). † F070W in JADES-Medium imaging is only planned for parallel coverage areas currently lacking HST . \nTable 4. JWST Cycle 1 MIRI Surveys \nNote -Depths quoted are 5 σ point source depths within 0 . '' 3 radius circular apertures from the exposure time calculator. Program IDs for these surveys are: CEERS (ERS # 1345), MIRI-HUDF-Medium (GO # 1207), MIRI-HUDF-DEEP (GO # 1283), JADES-Medium (GTO # 1180 and 1181), PRIMER (GO # 1837), and COSMOS-Web (GO # 1727). † Note that the MIRIHUDF-Deep Program (GO # 1283) is nested within the MIRI-HUDF-Medium (GO # 1207) program, but both are spatially offset from the JADES-Medium HUDF coverage. ∗ Note that the deeper part of JADES-Medium HUDF F770W coverage is nested within the shallower JADES-Medium coverage. ‡ Note that the depth quoted here for COSMOS-Web differs from the reported measured depth as given in Table 2; similarly, PRIMER and CEERS measured depths differ from ETC estimates, with measured 7.7 µ m depths of those programs shown in Figure 8. What is quoted in this table is from the exposure time calculator. We expect the actual depth of all MIRI programs to differ from ETC estimates in a similar manner. \nits variance with galaxies' star formation histories and morphologies. \nBelow we detail the motivation and requirements of each science goal.", '4.1. Mapping the Heart of Reionization': "The first galaxies formed < 1 Gyr after the Big Bang are thought to drive the last major phase change of the \nUniverse from a neutral to ionized intergalactic medium (IGM). This reionization process (Robertson et al. 2015) most likely finished around z ∼ 6 (Zheng et al. 2011; Kakiichi et al. 2016; Castellano et al. 2016; Ouchi et al. 2020) and was halfway completed by z ∼ 7 -8, according to measures of the rest-frame UV galaxy luminosity function (UVLF; Finkelstein 2016). This is in broad agreement with the Planck constraint of the instanta- \nFigure 8. A comparison of several of the JWST Cycle 1 extragalactic survey programs in depth and area for NIRCam imaging (left) and MIRI imaging (right). The vertical bars bracket the survey depths across all filters. In the case of MIRI, the dynamic range of depths is large due to substantial depth differences by filter; the depths at F770W are marked with stars for each survey. In the case of COSMOS-Web that has a large dither, the vertical bars also capture the range of depths across the mosaic. Note that for MIRI we show the exposure time calculator (ETC)-predicted depths, while the measured 7.7 µ m depths for COSMOS-Web, CEERS, and PRIMER are shown with circles. We expect the depth of all programs to be similarly offset between ETC estimates and actual depth achieved. Depths have been converted to approximate 5 σ point source depths as detailed in Tables 3 and 4. \n<!-- image --> \ns reionization redshift z reion = 7 . 7 ± 0 . 8 (Planck Collaboration et al. 2020). However, neither the start and duration of reionization, nor the sources responsible - either intrinsically luminous galaxies or more intermediate mass galaxies (Naidu et al. 2020; Hutter et al. 2021) - are well-constrained due to the relative shortage of both bright and faint z ∼ 7 -11 galaxies known in the preJWST era. Additional complexity is introduced by potentially significant evolution in the nature of EoR galaxies themselves: their intrinsic star formation rates (SFR), ionizing power ( ξ ion ), ionizing radiation escape fraction ( f esc ), number density, physical distribution, and clustering. \nConstraining the physics of reionization requires identifying and characterizing the galaxies that are embedded deep within the predominantly neutral Universe at z > ∼ 8, though direct detection of EoR galaxies has been challenging to-date. Pioneering work with Hubble led to the discovery of ∼ 80 candidate Lyman Break Galaxies (LBGs) at z > 8 (see review by Finkelstein 2016). Despite the perceived rapid drop in the UVLF during this epoch (Oesch et al. 2014), there have been a few successful preJWST detections of surprisingly bright candidate LBGs out to z ∼ 11 (the most spectacular of which is GNz11 at z = 10 . 6, Oesch et al. 2016; Jiang et al. 2021; Bunker et al. 2023). Although these z > 8 galaxies are thought to reside in a predominantly neutral Universe, \nsomehow a number of them show Ly α in emission (e.g., Oesch et al. 2015; Zitrin et al. 2015; Hoag et al. 2018; Hashimoto et al. 2018; Pentericci et al. 2018). This is surprising given that those Ly α photons should have been resonantly scattered by the mostly neutral IGM (Dijkstra 2014; Stark 2016; Garel et al. 2021). Do these Ly α emitters at z > 8 live in special 'ionized' bubbles? If they are representative of the general population, are we missing some fundamental aspect of the first stage of reionization? These questions can only be answered with a large sample of bright z = 7 -11 sources across a range of large scale environments, only possible with a near-IR contiguous wide-area survey (Kauffmann et al. 2020). \nThe first candidate discoveries of unusually bright galaxy candidates identified in early JWST observations (e.g., Naidu et al. 2022a; Finkelstein et al. 2022a,b; Donnan et al. 2022; Harikane et al. 2022; Atek et al. 2022) suggest that these sources may not be as rare as our preJWST models of z > 7 galaxy formation would indicate (e.g., Mason et al. 2015; Yung et al. 2019, 2020; Behroozi et al. 2020; Wilkins et al. 2017, 2022). While we note that these early discoveries are still candidates that require spectroscopic confirmation (as of this writing only a few z > 9 systems have been spectroscopicallyconfirmed, Roberts-Borsani et al. 2022; Williams et al. 2022; Robertson et al. 2022a; Curtis-Lake et al. 2022), \nthe perceived wealth of bright candidates may be particularly relevant to understanding the distribution of galaxies within large scale structure at early times. These bright candidates theoretically occupy the rarest and most massive dark matter halos, which are thought to be more highly clustered, and as such, small area surveys (as has been carried out to-date with JWST ) would poorly constrain their volume densities and the environments in which they live. \nThe breadth of galaxies' environments at early times is closely related to how reionization propagated. It is thought that reionization was predominantly a patchy process, producing ionized bubbles in the surrounding IGM growing from 5 - 20 Mpc at z > 8 to 30 - 100 Mpc at z ∼ 7 (Furlanetto et al. 2017; D'Aloisio et al. 2018). This corresponds to angular scales of 10 - 40 arcmin across, much larger than all contiguous NIR deep fields from Hubble or other planned deep field areas from JWST (see Table 3. Furthermore, large variance in the IGM's opacity from 5 < z < 7 quasar sightlines (Becker et al. 2015) suggests that the patchiness exceeds theoretical expectation from the density field alone by factors of a few, exacerbating uncertainties in reionization constraints from cosmic variance in existing surveys. Follow-up studies around both transparent and opaque quasar sightlines indicate a wide variety of large scale environments (Becker et al. 2018; Davis et al. 2018). \nCOSMOS-Web will grow the census of EoR ( z > 6) galaxies beyond what is known from Hubble surveys by a factor of ∼ 5 and quantify the evolution of the UVLF, stellar mass function (SMF), and star formation histories of galaxies across the Universe's first billion years. By observing a large contiguous area, COSMOS-Web will detect a factor of ∼ 6-7 times more sources at or above the knee of the luminosity function, L glyph[star] , than expected from all other JWST deep field efforts combined. Figure 9 shows the aggregate UVLF measurements from the literature to-date from 6 < z < 13, combining Hubble samples with the most recent results from JWST . Table 5 gives statistics on the predicted number of EoR galaxies to be found in COSMOS-Web, calculated directly from the compiled UV luminosity functions, relative to other Cycle 1 medium and deep programs. \nMassive galaxies above L glyph[star] are most likely to trace the highest-density peaks from which the reionization process was likely to begin. In particular, the 0.54 deg 2 survey area of COSMOS-Web is sufficiently large to capture reionization on scales larger than its expected patchiness, minimizing the effect of cosmic variance. As a contiguous survey, COSMOS-Web will sample the full range of environments at this epoch, provided large scale structure is clustered on scales within an order of magni- \ntude of their predicted scales (Gnedin & Kaurov 2014). This contrasts with, for example, the innovative Hubble and JWST pure-parallel surveys (e.g., BoRG, Schmidt et al. 2014; Calvi et al. 2016 and PANORAMIC, GO #2514) that, by design, will sample a wide variety of environments but cannot directly map the large scale environments of their discoveries.", '4.1.1. Impact beyond z > 8': 'Beyond the halfway point of reionization, COSMOSWeb is likely to detect hundreds of intrinsically bright galaxies at 8 < z < 11 embedded deep in the predominantly neutral IGM. This will increase the number of known z > 8 galaxies from the preJWST era by a factor of 10 above a luminosity of L glyph[star] . Through such a transformative sample of luminous z > 8 candidates, these discoveries will allow the first constraints on the bright-end of the UVLF and SMF at z > ∼ 8 with minimal uncertainty from cosmic variance, minimized to < ∼ 10% on scales of 0.5 deg 2 at our detection threshold of ∼ 27.5 magnitudes (Behroozi et al. 2019). Table 5 shows the expected total number of sources COSMOSWeb will find, totaling to ∼ 600-900 above z > 8 and 12-25 from 11 < z < 13. \nOur NIRCam filter combination is specifically optimized for 8 < z < 11 galaxy selection above the F115W detection limit of ∼ 27.4 mag as shown in Figure 3. Such systems are expected to see a significant drop in the F115W filter. If we account for a possible deviation from a Schechter UVLF as measured by wide/shallow ground-based UVLF estimates at z > 8 (shown as double power laws in Figure 9), our detections will likely exceed 1000 sources above z > 8, sufficient to map their spatial distribution and trace large scale structure at such early times. Even with our fiducial expectations in 0.54 deg 2 coverage, we expect to see a factor of ∼ 7 improvement in the number of z > 8 candidate galaxies above L glyph[star] over all previous Hubble work and a factor 2 × larger samples at those luminosities than all other planned Cycle 1 programs combined.', '4.1.2. Inferring the bright-end shape of the UVLF and SMF': "While CANDELS found only ∼ 2 - 10 galaxies at z > 6 with M glyph[star] > 10 10 . 5 M glyph[circledot] , and none above 10 11 M glyph[circledot] (Grazian et al. 2015; Song et al. 2016), the wider Ultra-VISTA survey (Bowler et al. 2014, 2020) found a larger number of massive galaxies than expected based on an extrapolation of a Schechter function fit to the CANDELS-measured SMF. The recent candidate discovery of intrinsically bright z > 10 galaxies in small- \nFigure 9. Literature rest-frame UV luminosity functions from 6 < z < 13; both data points and functional fits are drawn directly from the literature to illustrate the range of predictions made to-date at each epoch. Data and fits are specifically drawn from McLure et al. (2013); Oesch et al. (2013); Bouwens et al. (2015); Finkelstein et al. (2015); Finkelstein (2016); McLeod et al. (2016); Stefanon et al. (2019); Bowler et al. (2020); Bouwens et al. (2021); Kauffmann et al. (2022); Naidu et al. (2022a); Donnan et al. (2022); Harikane et al. (2022). Gray regions mask out rest-frame UV magnitudes where COSMOS-Web will not be sensitive; the light gray region marks the limit corresponding to our two image depth while the dark gray corresponds to four image depth. The horizontal dashed line marks the rarity of galaxies at which we would expect only to see one in all of COSMOS-Web. The blue and green corners mark the sensitivity limits (in depth and source rarity) of all of the Cycle 1 medium-depth surveys combined and deep-depth surveys combined, respectively. \n<!-- image --> \nTable 5. Number of Sources Expected between 6 < z < 13 in Cycle 1 Programs \nNote -Here we refer to all 'medium' depth Cycle 1 programs ( † ) as surveys reaching ∼ 28.5-29.5 mags in broad-band filters from Table 3, including JADES-Medium, CEERS and PRIMER. The 'deep' Cycle 1 programs ( ‡ ) refer to JADES-deep and NGDEEP together, which will reach depths exceeding 29.5 mags. ∗ Note that the z ∼ 10 bin spans 9 . 5 < z < 11. \narea early release JWST observations (e.g., Naidu et al. 2022a; Castellano et al. 2022; Finkelstein et al. 2022a,b; Donnan et al. 2022; Atek et al. 2022) also hint at a possible overabundance of massive galaxies compared to a Schechter function expectation. This excess of bright sources could indicate that the most massive galaxies are highly clustered and/or that the SMF at \nz > 6 departs from Schechter (Bowler et al. 2017; Davidzon et al. 2017). COSMOS-Web will greatly improve the dynamic range of luminosities (and thus masses) probed beyond all other NIR surveys, detecting ∼ 280500 bright M UV < -21 galaxies at z ∼ 6 -8 and ∼ 30-80 at 8 < z < 13, corresponding to stellar masses > ∼ 4 × 10 9 M glyph[circledot] . We calculate these estimates using the \nliterature parameterized luminosity functions shown in Figure 9 integrated down to M UV = -21, significantly above our detection threshold as detailed in Table 5. Given these statistics, a Schechter UVLF will be distinguishable from a double power-law in this dataset at a minimum of ∼ 4 σ out to z = 9; this estimate is based on the Poisson uncertainties in the expected number of sources to-be-discovered in COSMOS-Web given a Schechter function and a conservative estimate on the bright-end slope of the UVLF in the case of a double power-law (using β = -4). Such a deviation could be indicative of a primordial galaxy formation stage with different star formation timescales (Finkelstein et al. 2015; Yung et al. 2019), a lack of dust, or before the onset of feedback from Active Galactic Nuclei (AGN).", '4.1.3. Selection of EoR Sources': "As discussed in § 2.5, we generate a mock lightcone of the COSMOS-Web field containing an idealized sample of 0 < z < 12 galaxies, and here we use that simulated photometric catalog to diagnose contamination and precision of our EoR photometric redshifts, applying tools we will use for the real dataset. \nFigure 10 highlights the distribution of mock galaxies in color-color space for z ∼ 6 -7 and z ∼ 8 -9 galaxies against potential contaminating populations. The primary contaminants in both redshift regimes are 1 < z < 4 faint galaxies ( ∼ 27 th mag). The F814W and F115W filters are effective drop out filters for the two redshift regimes, though small gaps in wavelength coverage between filters imply that photometric redshift precision in COSMOS-Web will be somewhat less accurate than in fields with more complete filter coverage. We find that contamination rates are most significant (up to ∼ 20%) within 0.5 magnitudes of our 5 σ point source detection limit, where the constraint on drop filters is slightly weaker. We also anticipate relatively elevated contamination ( ∼ 20%) in the redshift range 5 . 5 < z < 6 . 5 due to both the gap between F814W and F115W as well as the relative depth difference between the filters (where F814W is shallower but also serves as the drop out filter). For 6 . 5 < z < 9 . 5, we anticipate contamination rates below ∼ 10% with a photometric redshift precision of ∆ z/ (1+ z ) ≈ 0 . 02 -0 . 04. Above z ∼ 9 . 5, the precision of photometric redshifts is degraded substantially by the lack of coverage at 2 µ m (see Figure 3); while some candidate z > 12 sources may be identified, they would require spectroscopic follow-up to confirm their redshifts, as NIRCam photometry would not constrain them very precisely. We will present further analysis of photometric redshift precision, as well as EoR sample contamination and completeness, in a \nforthcoming COSMOS-Web paper on the rest-frame UV luminosity function (Franco et al., in preparation). \nAn important consideration for the selection of EoR galaxy candidates will also be contamination of samples with lower redshift strong nebular emission line sources. For example, an underlying dust obscured (and reddened) rest-frame optical continuum superimposed with strong emission lines can masquerade as a bluer rest-frame UV continuum in JWST 's broadband filters, as shown by Zavala et al. (2022) and Naidu et al. (2022b). In that case one might expect dust continuum emission at millimeter wavelengths, representing reprocessed emission from hot stars. However, Fujimoto et al. (2022a) demonstrates that even a lack of dust continuum emission cannot rule out possible contamination from type-II quasars or AGN (in this case, the area covered with MIRI in F770W could lead to the detection of AGN that satisfy LBG selection criteria; Fujimoto et al. 2022b). While these possible foreground contaminants have come to light with the identification of ultra highredshift sources ( z > 12), it is nevertheless an important consideration in the identification of all EoR candidates, given the relatively sparse broad-band sampling available to select such sources. Follow-up spectroscopy of many EoR candidates in the next year will elucidate the level of contamination present in such samples and play a crucial role in informing statistics about large samples selected in COSMOS-Web.", '4.1.4. The first maps of LSS during the EoR': 'The full 0.54 deg 2 COSMOS-Web survey will allow the direct construction of large scale structure density maps of galaxies spanning z ∼ 6 -10. Such snapshots of the density field will provide a direct test as to whether or not the brightest, most massive galaxies are indeed highly clustered, as suggested by cosmological simulations (e.g., McQuinn et al. 2007; Behroozi et al. 2013; Chiang et al. 2017). Though some massive galaxies have been identified at this epoch from Ultra-VISTA data (Bowler et al. 2020; Endsley et al. 2022b; Kauffmann et al. 2022), it remains to be seen whether or not they sit in overdense environments. Existing Hubble and other planned JWST surveys are insufficient to answer this question due to their limited areas, however, COSMOSWeb will have both the depth and area to enable this measurement. \nFigure 11 illustrates our approach by using a mock catalog from a cosmological simulation (GADGET2; Springel 2005) at z ∼ 7 with width ∆ z ≈ 0 . 5 (and z ∼ 9 with width ∆ z ≈ 0 . 7) to reconstruct the underlying density map from simulations using the weighted adaptive kernel smoothing technique (Darvish et al. 2015) on \n<!-- image --> \nFigure 10. Color-color diagrams of mock galaxies drawn from a semi-analytic model illustrating the selection of z ∼ 6 -7 (top panel) and z ∼ 8 -9 galaxies (bottom panel) using the COSMOS-Web filter-set. In the z ∼ 6 -7 panel, green points and contours illustrate the distribution of mock galaxies at all redshifts relative to those at 6 < z < 7 . 5, shown in purple points and contours. A strong drop in the [F814W]-[F115W] color and a blue [F115W]-[F150W] correlates strongly with galaxies at z ∼ 6 -7; z ∼ 1 sources serve as the major contaminant due to degeneracy with the Balmer break. In the z ∼ 8 -9 panel, green points and contours show galaxies with photometric redshift solutions above z = 5 and purple points highlight sources with known redshifts 8 < z < 9 . 5. At these redshifts, we expect the drop to migrate to the [F115W]-[F150W] color yet sources are still expected to be relatively blue in [F150W]-[F277W]. \n<!-- image --> \n5 Mpc scales. We have used this simulation to directly test our ability to reconstruct the density field of galaxies from detectable sources. We infer that we will be able to reconstruct ∼ 12 independent mappings of the full density field between 6 < z < 10 based on our simulated photometric precision (∆ z/ (1 + z ) ∼ 0 . 02 -0 . 04) and \nlow contamination rates using a combination of color cuts and photometric redshift fitting (see § 4.1.3). The smoothing scale of 5 Mpc is an ideal scale to achieve a S/N in the overdensity measurement of > 5 σ per beam and S/N > 20 σ per overdense structure (with > 25 galaxies per beam in the highest density regions). COSMOS-Web will provide the first direct measurement of the physical scale and strength of overdensities at these epochs for direct comparison to the hypothesized scale of reionization-era bubbles that theoretically emanate from them.', '4.1.5. Masses of EoR Galaxies': "COSMOS-Web will enable crucial stellar mass constraints for the most massive EoR galaxies via the detection of rest-frame optical light (e.g., Faisst et al. 2016). In particular, with the MIRI F770W observations covering 0.19 deg 2 , we expect to detect 90-130 galaxies at 6 < z < 8 and 2-3 galaxies at 8 < z < 10 based on estimates from the UVLF. This would double the expected number of EoR galaxies with rest-frame optical detections from the other Cycle 1 JWST programs. At these redshift regimes this corresponds to rest-frame 1 µ m and 7700 ˚ A light, respectively. This will place unique constraints on the physical characteristics of extremely rare M glyph[star] > ∼ 10 10 M glyph[circledot] galaxies in the Universe's first few hundred Myr for the subset of EoR sources that we expect to detect with MIRI, as no galaxies with these high masses are expected to be found in the other Cycle 1 JWST surveys that cover smaller areas (cf. the sources identified by Labbe et al. 2022, with extreme stellar masses established at z ∼ 7 -9; though those sources' stellar masses may yet be highly uncertain, e.g., Endsley et al. 2022a). The detection of ∼ 100 galaxies in this mass regime will provide important clues to the star formation histories of the Universe's most massive halos in the first Gyr after the Big Bang, which are currently unconstrained.", '4.1.6. Follow-up of EoR Sources': "Beyond the direct EoR discoveries that COSMOSWeb will make through its JWST imaging, follow-up observations will further enhance the impact of this program and shed light on key unknowns. These include (1) rest-frame UV diagnostics with JWST NIRSpec that will constrain ionizing photon production in z > ∼ 6 sources (i.e., constraints on f esc and ξ ion , e.g., Chisholm et al. 2020), (2) deep rest-frame UV observations of Ly α to infer local variations in the IGM neutral fraction with Keck, Subaru, VLT (which can typically reach line sensitivities of ∼ 10 -18 erg s -1 cm -2 ), and future 30 m-class telescopes (the extremely large telescopes, ELTs, that will push fainter), and (3) obscured star-formation and \n100 \n125 \n150 \n0 \n25 \n50 \n75 \n100 \n125 \n150 \n0 \n25 \n50 \n75 \n100 \n125 \n150 \n160 \n140 \n120 \n100 \nc \n80 \nMp \n60 \n40 \n20 \n0 \nc \nMp \n150 \n0 \n25 \n50 \n75 \n100 \n125 \n150 \n0 \n25 \n50 \n75 \n100 \n125 \n150 \n0 \n25 \n50 \n75 \n100 \n125 \n150 \nMpc \n0 \n25 \n50 \n75 \n100 \n125 \n150 \nMpc \nFigure 11. At left are two snapshots of a cosmological n-body simulation performed using GADGET-2 (Springel 2005) spanning a (100 h -1 Mpc) 3 volume (not the real COSMOS survey field). The cube is here projected from one side and has a thickness equivalent to δz = 0 . 5 at z = 7 and δz = 0 . 7 at z = 9. The underlying dark matter distribution is shown in blue (void indicated by darker blue) and the distribution of ionized hydrogen gas (H+) is shown in pink, whereas regions of neutral IGM have the underlying blue dark matter distribution visible. Galaxies that are detectable in COSMOS-Web are shown as white points (having F150W < 27.5). Larger points represent more luminous (with F150W < 26.5) galaxies. At right we show the recovered galaxy density maps inferred from the same simulation snapshots using the observational limits of our survey. The recovered maps use an adaptive kernel smoothing on a global 5 Mpc kernel scale (Darvish et al. 2015) and include a modeling of sources' incompleteness and photometric redshift uncertainties, demonstrating our ability to recover large scale structure at these redshifts. The galaxies responsible for reionization may be expected to be strongly clustered on 30 - 100 cMpc (10 - 40 arcmin) scales, much larger than all existing contiguous near-infrared Hubble deep surveys and other planned Cycle 1 JWST surveys. COSMOS-Web will span an area the size of the white box, about (46 arcmin) 2 , and will cover a mix of 4 - 16 independent reionization bubbles or neutral gas regions per dz = 0 . 3 slice across 12 independent slices. With ∼ 5000 sources detectable across 6 < z < 8 and ∼ 600 sources at 8 < z < 10, COSMOS-Web will be uniquely situated to gathering statistical samples of EoR density environments. Further follow-up observations in Ly α may then reveal the relationship between mass overdensities and ionized bubbles. \n<!-- image --> \nx \n[ \nMpc/h \nx \n[ \nMpc/h \n] \n] \n/uni223C \n9 \nMpc \nMpc \nc \nMp \n160 \n160 \n140 \n140 \n120 \n120 \n100 \n100 \n80 \n80 \n60 \n60 \n40 \n40 \n20 \n20 \n0 \n0 \n160 \n160 \n140 \n140 \n120 \n120 \n100 \n100 \n80 \n80 \n60 \n60 \n40 \n40 \n20 \n20 \n0 \n0 \nz \nz \n/uni223C \n7 \n160 \n140 \n120 \n100 \n80 \n60 \n40 \n20 \n0 \nz \n27 \n. \n7 \nReconstruction \n/uni223C \n7 \n27 \n. \n6 \n27 \n. \n5 \n27 \n. \n4 \n27 \n. \n3 \n) \n3 \nc \nMp \n/ \n2 \nh \n/circledot \nM \n/ \nρ \n( \n10 \nlog \nH \n] \n/ \nh \nc \nMp \nH \n+ \nMp \nc \n[ \ny \n] \n/ \nh \nc \nMp \n[ \ny \nc \nMp \n160 \n140 \n120 \n100 \n80 \n60 \n40 \n20 \n0 \n27 \n. \n7 \n27 \n. \n6 \n27 \n. \n5 \n27 \n. \n4 \n27 \n. \n3 \n0 \ncold ISM content of dust and metals from ALMA detections of the FIR continuum and the FIR fine-structure atomic cooling lines (Laporte et al. 2017; Hashimoto et al. 2018; Bakx et al. 2022; Fujimoto et al. 2022a), which will inform stellar population synthesis models of galaxies' first light, metals, and dust. COSMOS-Web, as a wide and shallow survey, will be particularly useful for the detection of bright, rare candidates that are well optimized for ground-based follow-up. These future observations will be crucial for detailed characterization of EoR overdensities, unlocking direct comparisons between mapped reionization bubbles (measured via Ly α follow-up) and JWST -measured density maps, as shown from a simulation in Figure 11.", '4.2. The Buildup of the Massive Galaxy Population': 'The wide-area coverage of COSMOS-Web, in particular the combination of the NIRCam LW (2.8 µ m and 4.4 µ m) and MIRI (7.7 µ m) observations with the already existing wealth of optical to NIR data in COSMOS, will allow us to take the first census of massive galaxies from the end of the EoR to the peak of galaxy assembly. Within the footprint of COSMOS-Web, we expect firm identification of half a million galaxies at all redshifts, ∼ 32,000 of which will be detected in MIRI F770W imaging, allowing us to constrain stellar masses, sizes, morphologies, star formation rates, and AGN activity for galaxies across a wide swath of cosmic time.', '4.2.1. The First Quiescent Galaxies': "The growing census of massive quiescent galaxies at early epochs ( M glyph[star] > ∼ 10 10 M glyph[circledot] out to z ∼ 3 -5, e.g., Straatman et al. 2014; Glazebrook et al. 2017; Schreiber et al. 2018; Merlin et al. 2019; Tanaka et al. 2019; Girelli et al. 2019; Valentino et al. 2020; Forrest et al. 2020; Carnall et al. 2022, 2023; Rodighiero et al. 2022) has presented a strong challenge to theoretical models of early massive galaxy formation (e.g., Feldmann et al. 2016; Steinhardt et al. 2016; Cecchi et al. 2019, see Figure 12). In order to build up their significant stellar masses and quench their star formation so early in the Universe's history, these galaxies must have formed their stars at exceptionally high rates ( glyph[greatermuch] 100 M glyph[circledot] yr -1 , comparable to luminous infrared galaxies; Sanders & Mirabel 1996, and dusty star-forming galaxies, DSFGs; Casey, Narayanan, & Cooray 2014) at very early times and then abruptly shut down the production of stars well within the Universe's first Gyr. The existence of these sources and their relative abundance provide important tests of the galaxy assembly process and the physical processes driving the quenching of star formation at this early epoch. \nThe quiescent galaxy mass function beyond z ∼ 4 is currently unconstrained, partly because of the difficulty \nFigure 12. The number density of quiescent galaxies (specific SFR, SFR/M glyph[star] < 10 -11 yr -1 ) as a function of redshift for galaxies with M glyph[star] > 10 10 M glyph[circledot] selected from Illustris TNG100 (blue) and extrapolated beyond z > 4 (blue dashed) where no quiescent galaxies are found in the Illustris TNG100 volume. Other simulation predictions for this population are shown from EAGLE (green), FLARES (purple), and the DREaM (red) semi-empirical model and predictions from the empirical model of Long et al. (2022) in black. Overplotted is a collection of number densities of quiescent galaxies from the literature, illustrating the wide range that both the observations and simulations span. The dashed lines correspond to the number density of one object in the NIRCam volume of COSMOS-Web (black), medium-volume JWST surveys such as CEERS and PRIMER (dark gray), and deep volume surveys such as JADES-Deep and NGDEEP (light gray). Only COSMOS-Web has the volume necessary to place strong constraints on the number densities of quiescent galaxies at z > 4 if they are indeed as rare as expected. \n<!-- image --> \nof detecting these rare galaxies in existing deep field observations (with volume densities < ∼ 10 -5 Mpc -3 ) and partly because such galaxies are particularly difficult to separate from DSFGs and post-starburst galaxies that can mimic the same red colors (see Figure 3). Detecting them requires deep rest-frame optical observations over wide areas of the sky. COSMOS-Web will provide the ideal dataset for identifying candidate quiescent galaxies and measuring (or placing constraints) on their number densities and relative abundances. Figure 12 highlights the expected number density of massive (M glyph[star] > 10 10 M glyph[circledot] ) quiescent (specific SFR, SFR/M glyph[star] < 10 -11 yr -1 ) galaxies from the cosmological hydrodynamical simulations IllustrisTNG100 (Pillepich 2018), EAGLE (McAlpine et al. 2016), and FLARES (Lovell et al. 2022), as well as \nthe DREaM semi-empirical model (Drakos et al. 2022) and predictions from the empirical model of Long et al. (2022), in comparison to some of the currently identified quiescent galaxy candidates in the literature (Muzzin et al. 2013; Straatman et al. 2014; Merlin et al. 2019; Schreiber et al. 2018; Girelli et al. 2019; Shahidi et al. 2020; Carnall et al. 2022; Weaver et al. 2022b; Gould et al. 2023; Valentino et al. 2023). Note that each study selects quiescent galaxies slightly differently, and the resulting samples span a range of stellar mass cuts, with the vast majority of candidates having M glyph[star] > 10 10 M glyph[circledot] . When multiple mass cuts are quoted by a given study, we show number densities above this mass limit for consistency. \nThe quiescent galaxy sample from IllustrisTNG100 was selected using the publicly available 6 star formation rate (Pillepich 2018; Donnari et al. 2019) and stellar mass value that corresponds to the mass within twice the half-mass radius of each object (Rodriguez-Gomez et al. 2016). Note that there are no quiescent galaxies in the IllustrisTNG100 volume beyond z > 4 using this definition. We similarly selected the quiescent galaxy sample from the public EAGLE galaxy database 7 (McAlpine et al. 2016) using the recommended aperture size of 30 physical kpc. These hydrodynamical simulations are calibrated to reproduce physical properties in the local Universe and predict the the SFR and M glyph[star] values at high redshift. For FLARES, we use the number densities measured by Lovell et al. (2022). These simulations generally underpredict the observed number densities of quiescent galaxies in the literature (though the observations span a wide range of values). On the other hand, semi-analytic models like DREaM are calibrated to match scaling relations at all redshifts. The DREaM number densities in Figure 12 are based on the SMF of Williams et al. (2018) and are a close match to the high end of the observed number densities. \nEven though true quiescent galaxies are expected to be rare at z > 4, with the large area of COSMOS-Web we will be able to identify massive quiescent galaxy candidates and place robust constraints on their abundances as a function of redshift if they are brighter than our detection limit with number densities ≥ 10 -7 Mpc -3 . This measurement will also be less impacted by the effects of cosmic variance than similar measurements from smaller area surveys (e.g., Carnall et al. 2022). \nThe combination of NIRCam and MIRI filters over 0.20 deg 2 (including the COSMOS-Web and PRIMER \nMIRI imaging that fall within the NIRCam footprint) will enable quiescent galaxies to be distinguishable from dusty star-forming interlopers via color-selection and SED analysis using the well-sampled rest-frame optical photometry. Additionally, the complementary (sub)millimeter observations over the COSMOS field (see Figure 4) will enable the direct identification of dusty galaxies at z > 4 and therefore disentangle them from quiescent and EoR galaxy candidates (see Zavala et al. 2022; Naidu et al. 2022b; Fujimoto et al. 2022a for detailed discussion of the difficulty in separating these populations using NIRCam colors alone). Specifically, the existing SCUBA-2 and future TolTEC observations will cover COSMOS to a depth of SFR > ∼ 50 M glyph[circledot] yr -1 , while the ALMA MORA survey (Zavala et al. 2021; Casey et al. 2021; Manning et al. 2022) and its continuation (ex-MORA; Long et al., in preparation) will cover 0.2 deg 2 ( ∼ 1/3 of COSMOS-Web), in addition to the public ALMA archival pointings from A3COSMOS (totalying 0.12 deg 2 across all of COSMOS; Liu et al. 2019) and will directly detect DSFGs at z > 4 in excess of SFR > ∼ 100 M glyph[circledot] yr -1 . \nIn addition to identifying the highest redshift quiescent galaxies, COSMOS-Web observations will allow us to study their properties in detail. MIRI 7.7 µ m observations (rest-frame 1.1-1.5 µ m at 4 < z < 6) for a subset will provide a long wavelength lever arm to accurately determine their masses. The full multiwavelength SED will enable us to measure their SFRs and constrain their star formation histories (SFHs) and dust attenuation, with improved uncertainties on the SFHs with constraints from JWST (e.g., Whitler et al. 2022). With the high resolution NIRCam and MIRI imaging we will be able to study their morphologies in great detail (see Figure 13) and robustly measure their rest-frame optical sizes as well as constrain the physical distribution of their SFR, mass, and dust content, giving insight into how these galaxies may have quenched. This will enable a detailed investigation of the galaxy size-mass relation for quiescent systems < 2 Gyr after the Big Bang, extending our understanding of size growth out to higher redshifts and less extreme massive galaxies than has been possible before (e.g., Toft et al. 2007; van der Wel et al. 2014; Straatman et al. 2015; Shibuya et al. 2015; Faisst et al. 2017; Kubo et al. 2018; Whitney et al. 2019) and a statistically robust study of their progenitors. \nWithin the COSMOS-Web footprint, we expect to detect ∼ 13,000 massive galaxies (M glyph[star] > 10 10 M glyph[circledot] ) between 4 < z < 6 ( ∼ 2,300 with MIRI coverage) of which we estimate there will be at least ∼ 350 quiescent candidates (in the NIRCam mosaic, and 120 with MIRI coverage, selected to have sSFR < 10 -11 yr -1 ) scaling the COS- \nFigure 13. A selection of three color NIRCam (F115W+F150W+F277W) cutouts of galaxies at z > 3 with varied morphologies selected from IllustrisTNG mock images (Rose et al. 2022) with noise added consistent with the COSMOS-Web depth. Each cutout is 3 '' on a side. These illustrate COSMOS-Web's ability to distinguish between different morphological types and detect a diversity of morphologies thanks to JWST 's sensitivity and resolution at these redshifts. \n<!-- image --> \nMOS2020 estimates of source counts at these redshifts (Weaver et al. 2022a); this will be ∼ 10 × improvement over current z > 4 quiescent galaxy candidate samples. Follow-up spectroscopic observations for subsamples of these quiescent galaxies (e.g., such as those by Schreiber et al. 2018; Valentino et al. 2020) will be able to confirm their redshifts, measure their velocity dispersions, and more fully characterize their ages and star formation histories, enabling us to separate true quiescent galaxies from post-starburst systems.", '4.2.2. Dusty Star Forming Galaxies': "DSFGs are an intrinsically rare population (with number densities < ∼ 10 -5 Mpc -3 ) whose individual discoveries, particularly at z > 5 test the limits of galaxy formation models (see reviews by Casey, Narayanan, & Cooray 2014; Hodge & da Cunha 2020). They are largely regarded as the dominant progenitor population of high-redshift quiescent galaxies, given their prodigious rates of star formation ( > ∼ 100-1000 M glyph[circledot] yr -1 ) and similar volume densities (though both are quite uncertain). While DSFGs are typically easily identified directly via FIR emission or their (sub)mm emission in single-dish or interferometric maps, often their more detailed physical characterization remains elusive. This may include the measurement of their redshifts or masses. It is difficult due to significant degeneracies in their submm emission with redshift and significant dust obscuration of the rest-frame UV and optical emission. Radio continuum emission can also be a vital tool in detecting DSFGs, and often facilitates quick multiwavelength identification via precise astrometric constraints (e.g., Algera et al. 2020; Talia et al. 2021; Enia et al. 2022). From ancillary FIR/submm data already in hand covering COSMOS-Web, we know of ∼ 1100 DSFGs at \nall redshifts detected by SCUBA-2 and Herschel that will be covered by the NIRCam mosaic with luminosities > ∼ 10 12 L glyph[circledot] ; many of these do not yet have spectroscopic redshifts and confirmed counterparts, though a significant fraction ( > ∼ 50%) have follow-up continuum ALMA observations providing precise astrometric constraints (Liu et al. 2019; Simpson et al. 2020). \nCOSMOS-Web will transform our understanding of the stellar content in DSFGs at all redshifts, but in particular shed light on the rarest DSFGs found at z > 5 (of which there are fewer than two dozen with spectroscopic redshifts). Based on recent models of the obscured galaxy luminosity function (Zavala et al. 2021), we estimate that ∼ 40-70 of the > 10 12 L glyph[circledot] DSFGs in the NIRCam mosaic will lie above z = 4, and ∼ 3-10 above z = 6. Including those with an order-of-magnitude lower luminosity ( > 10 11 L glyph[circledot] ), the statistics inflate by an order of magnitude. Roughly a third of DSFG samples, and especially those selected at longer wavelengths, are invisible even in deep Hubble imaging (Franco et al. 2018; Gruppioni et al. 2020; Casey et al. 2021; Manning et al. 2022). In contrast, JWST imaging (both with NIRCam and MIRI) pushes to depths sufficient to capture DSFGs' highly obscured stellar emission, enabling measurement of more precise photometric redshifts than are currently accessible, in addition to constraints on their morphologies and stellar masses. For example, the vast majority of DSFGs are detected in deep Spitzer /IRAC imaging (with [4.5 µ m] < 26); thus, we expect detection of all DSFGs in the NIRCam LW filters particularly because the median stellar mass of the population is expected to be high, ∼ 7 × 10 10 M glyph[circledot] (Hainline et al. 2011), roughly a factor of ∼ 150-200 × larger than the stellar masses of galaxies at the NIRCam LW detection limit at z ∼ 5. Through more reliable optical-IR pho- \nhifts, combined with additional (sub)mm constraints on their redshifts (Cooper et al. 2022), these data will unlock many unknowns about the evolution of and buildup of mass in such extreme star-forming galaxies at early times.", '4.3. Linking Dark Matter with the Visible': "The link between galaxies' dark matter halos and their baryonic content is of fundamental importance to cosmology. Yet directly observable tracers of halo mass are not available for the vast majority of galaxies, and in their place, either halo occupation distribution (HOD) modeling (Seljak 2000; Cowley et al. 2018) or abundance matching (Kravtsov et al. 2004; Conroy & Wechsler 2009; Behroozi et al. 2019) are used to infer halo mass from galaxies' stellar masses (via the stellar-mass-to-halo mass relation, SMHR; Croton et al. 2006; Somerville et al. 2008). However, the evolution of galaxies is direct evidence for the complexity of the halo-baryon relationship (Legrand et al. 2019; Shuntov et al. 2022). Halos provide the potential well for accretion of fresh gas, which in turn fuels stellar mass growth through star formation. Merging also substantially boosts stellar mass growth and relates directly to the physical interactions of halos which occurs on scales larger than individual galaxies. Indeed, it is thought that on such large scales, galaxies' halo mass growth should be independent of the baryonic processes within galaxies. If measurable, they could provide a direct path to constraining galaxy growth and their relationship to quenching mechanisms. Obtaining direct measurements of halo masses not only helps us to constrain the astrophysics of galaxies (Mandelbaum et al. 2006, 2014) but also gives independent measurements on cosmological parameters (Zheng & Weinberg 2007; Yoo et al. 2006, 2009). \nDirectly measuring halo masses out to large galactocentric radii ( ∼ 1 Mpc, needed to probe the underlying dark matter) can be done either with galaxy-galaxy lensing (Brainerd et al. 1996) or using kinematic tracers like satellite galaxies (McKay et al. 2002). Given the sparsity of bright satellites beyond the local Universe and rarity of strongly-lensed galaxies, weak lensing (WL) is the only tool that can be used as a direct probe of halo masses for a large sample of galaxies across cosmic time (Sonnenfeld & Cautun 2021). An innovative method combining galaxy clustering measures with HOD modeling and weak lensing was demonstrated by Leauthaud et al. (2011) and Leauthaud et al. (2012) using the COSMOS single band F814W Hubble imaging to measure SMHR evolution from 0 . 2 < z < 1 . 0 at M glyph[star] > 10 10 M glyph[circledot] . \nThese measurements are shown in the left panel of Figure 14. \nCOSMOS-Web's 4-band NIRCam imaging spanning > 0.5 deg 2 , joined with the high quality 40+ band imaging constraining galaxies' masses and photometric redshifts in COSMOS (Weaver et al. 2022a, see also Figure 5), will be the best available dataset from which high resolution weak lensing mass mapping measurements can be done. This will involve a careful reconstruction of the PSF for each exposure in each filter and measurements of source centroids, shapes, and orientations. These measurements will then be combined with the best possible photometric redshifts to infer evolution in the SMHR (Leauthaud et al. 2007, 2011). Extending from z ∼ 1 to z ∼ 2 . 5 and to depths an order of magnitude deeper in halo mass at fixed redshift is enabled by the significant boost in spatially-resolved background and foreground sources where the weak lensing signal goes roughly as the square-root of the foreground source density multiplied by the background source density, ∝ √ N fg ( z ) √ N bg ( > z ). The density of background sources will exceed 10 arcmin -2 out to z = 4, with ∼ 110,000 sources at z > 2 . 5 above 15 σ , which is the necessary detection threshold for adequate shape recovery (Jee et al. 2017). Furthermore, these data will push that deep in each independent filter , thus will be the first wide, deep multi-band survey from space; simultaneous weak lensing measurement in multiple bands will both let us go even deeper and provide independent crosschecks of instrumental effects like the PSF calibration. Euclid and Hubble cannot observe at such long wavelengths (Lee et al. 2018) and Roman will not achieve such high resolution. Contiguous, high resolution NIR imaging from JWST in COSMOS-Web will thus serve as a much needed absolute calibration of the SMHR relation out to z ∼ 2 . 5 that can be leveraged by other weak lensing surveys conducted on larger scales.", '4.3.1. Evolution in the SMHR': "Leauthaud et al. (2012) found unexpected evolution in the characteristic mass at which the SMHR is maximized (downsizing), in other words where the peak efficiency (around ∼ 10 12 M glyph[circledot] ) evolves downward from z ∼ 0 . 9 to z ∼ 0 . 3. Shuntov et al. (2022) similarly finds this downsizing trend with the COSMOS2020 catalog using clustering and constraints on the stellar mass function out to z ∼ 3. COSMOS-Web will significantly strengthen measurements to z ∼ 2 . 5 with the important addition of weak lensing constraints, facilitating a re-calibration of hydrodynamical simulations and semianalytic models that produce mock observables essential for much of cosmology and extragalactic astrophysics. \nHalo \n/uni22C6 \nFigure 14. At left, the stellar mass to halo mass ratio as a function of halo mass. Curves are shown for central galaxies only, not including satellites, though both relations may be constrainable with our dataset. Solid lines are measurements from COSMOS ACS weak lensing data (Leauthaud et al. 2012) at z = 0 . 3 (dark blue), z = 0 . 6 (light blue), and z = 0 . 9 (orange). In comparison, the SMHR from cosmological simulations is overplotted (gray dot-dashed line from Behroozi et al. 2010 and dashed line colored dashed lines from UniverseMachine, Behroozi et al. 2019) at matched redshifts extending out to z = 2 . 5 (red). The existing weak lensing measurements show evolution out to z ∼ 1 (black arrow noting SMHR peak evolving); COSMOS-Web weak lensing measurements will have the power to extend this analysis to z = 2 . 5, where the vertical red line will represent the lower mass limit at that redshift. At right, predictions from hydrodynamic simulations, specifically Illustris TNG100 (Pillepich 2018), suggest that ellipticals with blue or red cores experience different quenching mechanisms (gas stripping shown in blue, and gas exhaustion in red). These differences are reflected in how their M glyph[star] /M H evolves with time ( z ∼ 2 in solid; z ∼ 0 . 7 in dashed). COSMOS-Web data will have the potential to provide powerful constraints on the evolution of the SMHR for different galaxy populations and test fundamental galaxy quenching models. \n<!-- image --> \n<!-- image --> \n/uni2299 \n/uni2299 \nThis has important implications for how HOD modeling or abundance matching is used in the literature and how semi-analytic models and cosmological hydrodynamical simulations generate observables, on which much of extragalactic astrophysics relies. \nSuch weak lensing measurements rely on contiguous coverage over a large ( > ∼ 0.5 deg 2 ) area, otherwise they are substantially affected by edge effects (Mandelbaum et al. 2005; Massey et al. 2007b; Han et al. 2015). COSMOS-Web, with its large, deep, and contiguous coverage, will enable direct measurements of galaxies' halo masses out to z ∼ 2 . 5 down to M glyph[star] > few × 10 9 M glyph[circledot] (down to 10 8 M glyph[circledot] at z ∼ 1), well beyond current data limitations (above 10 10 M glyph[circledot] at z < ∼ 1) and future planned weak lensing measurements (e.g., from Euclid or Roman ). Extending weak lensing measurements to z ∼ 2 . 5 is essential for simulation calibration due to the significant evolution in galaxies' properties (e.g., SFRs; Noeske et al. 2007; Whitaker et al. 2014) in the past 11 Gyr from z = 0 -2 . 5. \nThe potential to extend SMHR constraints to higher redshifts is also possible using similar techniques to Shuntov et al. (2022). Such high redshifts and great mass depths can be reached due to the dramatic in- \ncrease in the number of background sources for weak lensing and sources at all epochs that will have highquality photometric redshifts.", '4.3.2. Constraining the Dependency of the SMHR on Resolved Baryonic Observables': "Given that COSMOS-Web data will be obtained in multiple filters, it will be the first sufficiently large dataset to test for alternate dependencies of resolved baryonic observables (e.g., color as a tracer of quenching mechanisms) on halo mass. This type of differential measurement with galaxy type is demonstrated by Tinker et al. (2013) out to z ≈ 1, who find that star-forming galaxies grow in lock-step with their dark matter halos, while quiescent galaxies have baryonic growth that is outpaced by dark matter growth. Higher redshifts can be reached by conducting the same experiment at longer wavelengths, boosting observed densities of highz sources. COSMOS-Web will push the limits of weak lensing's direct measurement of halo masses to z ∼ 2 . 5 with M glyph[star] > 10 10 M glyph[circledot] such that halo masses can be independently constrained as a function of galaxy type over a significant portion of the Universe's history. \nResolved color gradients in galaxies are thought to be the hallmark tracer of the quenching process. Bluer \ncores likely trace systems where cold gas has been stripped from the periphery (e.g., Meschin et al. 2014), while redder cores trace gas exhaustion, where gas at the galaxy core is not replenished (e.g., Kawata & Mulchaey 2008; Tacchella et al. 2015). Flat color gradients are expected for galaxy collisions, in which the gas supply is consumed quickly with no preferred radial distribution (e.g., Springel 2005; Sparre et al. 2015). While differential dust attenuation may complicate the interpretation of galaxies' color gradients, some independent observations at long wavelengths could break the degeneracy (see the discussion later in § 5.3). \nThe right panel of Figure 14 illustrates the expected difference between the SMHR of bluer-cored galaxies and redder-cored galaxies. Do galaxies with different gradients show different evolution in their SMHRs? Cosmological simulations predict that similar SMHRs may point to the significant role of major galaxy mergers in the quenching process, while different SMHRs would point to feedback quenching mechanisms. At z < ∼ 2 . 5, our weak lensing measurements of halo masses for large samples can be directly compared with assertions that massive > 10 10 M glyph[circledot] galaxies evolve from star-forming to quenched in ∼ 100 Myr (e.g., Barro et al. 2013).", "5. COSMOS-WEB'S IMPACT ON OTHER TOPICS": "The breadth of scientific studies that COSMOS-Web may advance is extraordinary and impossible to anticipate in full. Below we describe some key ancillary science cases that could make significant strides given the layout and plans for the COSMOS-Web Treasury program. We emphasize that COSMOS-Web's contribution to these areas will be powerful, though not made in isolation; much of the progress will be significantly aided by, if not fully dependent on, the legacy of data obtained in the COSMOS field from other observatories.", '5.1. Galaxy Morphologies and Sizes to z ∼ 8': "Over the age of the Universe, galaxies have undergone dramatic morphological transformations. Today's galaxies are a mix of well-formed spiral disk galaxies, ellipticals, and irregular galaxies and deep rest-frame optical images from Hubble have shown that the basis for what is known as the Hubble sequence was already in place by z ∼ 3 (e.g., Wuyts et al. 2011; van der Wel et al. 2014; Kartaltepe et al. 2015b). Early JWST studies (e.g., Robertson et al. 2022b; Ferreira et al. 2022; Kartaltepe et al. 2022) are finding that galaxies at even higher redshifts have a wide diversity of morphologies, and a significant fraction already show evidence for disks and spheroids. However, a large fraction of galaxies at high redshift also have irregular morphologies, some of \nwhich may be signatures of galaxy mergers and interactions (e.g., Kartaltepe et al. 2012) and some may be indicative of other physical processes such as disk instabilities (e.g., Kereˇs et al. 2009; Genzel et al. 2011). In order to quantify the morphological transformation of galaxies from very early epochs to today, and understand the physical drivers responsible, large samples at high redshift are required. \nCOSMOS-Web will spatially resolve the rest-frame optical emission of tens of thousands of galaxies from z = 3 -8, enabling a detailed morphological classification into spheroids/disks/irregulars and identification of interaction and merger signatures. These measurements will enable studies of morphological transformation as a function of environment and the relative roles of different physical processes responsible for enhanced star formation and black hole growth in the early universe. These large samples of morphology measurements will be essential training samples for machine learning algorithms (e.g., Snyder et al. 2019; Pearson et al. 2019; Hausen & Robertson 2020; ' Ciprijanovi'c et al. 2020; Rose et al. 2022) to classify galaxies, identify merger signatures, and identify unique morphologies that may otherwise be missed. \nThe evolution of galaxy sizes is also a useful tool for investigating the evolutionary history of galaxies and connecting the properties of today's galaxies to their progenitors in the early universe. Over the past decade, a number of studies have found evidence for strong evolution in the optical/UV sizes of galaxies, with effective radii growing by a factor of 2-7 since z ∼ 2 (e.g., van der Wel et al. 2008; Buitrago et al. 2008), suggesting that these massive galaxies have evolved through minor mergers in this time period (e.g., Naab et al. 2009; Bluck et al. 2012; Furlong et al. 2017). Both star-forming and quiescent populations of galaxies have been found to evolve in size, with samples of compact star forming (Barro et al. 2014a,b) and compact quiescent (e.g., Toft et al. 2007; van Dokkum et al. 2008; Bezanson et al. 2009; Barro et al. 2013) galaxies identified at cosmic noon. At even higher redshifts, z = 3 -7, significant, though less steep, evolution has been found by a number of studies (e.g., van der Wel et al. 2014; Straatman et al. 2015; Shibuya et al. 2015; Whitney et al. 2019), and a range of physical mechanisms driving this evolution have been suggested, including major and minor mergers (e.g., Bluck et al. 2012; Wellons et al. 2016), rejuvenated star formation in the galaxy's outer regions due to gas accretion (Conselice et al. 2013; Ownsworth et al. 2016; Dekel et al. 2020), quasar feedback (e.g., Fan et al. 2008; Dubois et al. 2016), and progenitor bias (van Dokkum & Franx 2001). \nWellons et al. (2016) used Illustris to track the evolution of a sample of compact quiescent galaxies at z ∼ 2 and found a diverse range of properties among their descendants, with very few remaining compact in the present day, in agreement with observations (Trujillo et al. 2009; Tortora et al. 2018; Scognamiglio et al. 2020). Most growth appears to be driven by the delivery of ex-situ mass and the impact of galaxy mergers and both are closely linked to a galaxy's environment (Trujillo et al. 2007; Song et al. 2021). The progenitors of these compact quiescent galaxies themselves could have formed through gas-rich major mergers (e.g., Hopkins & Hernquist 2009; Barro et al. 2013; Wellons et al. 2015) or through clump migration (e.g., Dekel & Mandelker 2014). Additionally, the extension of the size-mass relation into the EoR is currently not well-constrained. UV measurements from the HUDF and CANDELS have found a range of sizes for these early galaxies (0.3-1 kpc, Oesch et al. 2010; Ono et al. 2013; Curtis-Lake et al. 2016) where mass is not very well measured given the limited scope of detection bands to the rest-frame UV. \nRobust size measurements free from redshift bias are needed to adequately trace the evolution of galaxy sizes from the early Universe, which require deep rest-frame optical imaging of galaxies out into the EoR. In addition, the combination of analysis of galaxies' rest-frame optical sizes can be compared to the sizes of their dust and gas reservoirs (e.g., Hodge & da Cunha 2020) to further place constraints on the morphological transformation of galaxies across cosmic time.", '5.2. Spatially Resolved Galaxy SEDs': "The high resolution and deep images provided by COSMOS-Web NIRCam images will enable detailed pixel-by-pixel SED fitting of galaxies across a wide redshift range and down to lower stellar masses than has been possible to date (e.g., Wuyts et al. 2012; Jafariyazani et al. 2019; Abdurro'uf et al. 2021, 2023). The resulting mass, star formation rate, and dust attenuation maps can be used to study the star formation and quenching process in galaxies (e.g., Tacchella et al. 2015). The large number of sources in COSMOS-Web will enable the study of trends as a function of redshift, environment, and position relative to the star forming main sequence. \nMass maps that represent the overall resolved stellar mass of galaxies can be used for the morphological measurements described above. For example, Cibinel et al. (2015) show that morphological measurements using the mass maps of galaxies are better able to pick out features indicative of galaxy mergers than similar measures using standard light images. Similarly, precise size measure- \nments can be made using mass maps in comparison with standard measurements (e.g., Suess et al. 2019; Mosleh et al. 2020). Clumps can be more easily identified using stellar mass maps and star formation rate maps can be used to quantify the growth of stellar mass in galaxies as a function of their morphology and environment.", '5.3. Constraints on the Dust Attenuation Law': "The dust attenuation law plays a crucial role in SED modeling for galaxies at all redshifts (Salim & Narayanan 2020) but is heavily dependent on dust grain properties, total dust content, and dust geometry within galaxies' interstellar media. Without direct constraints, most SED fitting routines blindly adopt one of a few common dust attenuation curves, for example that of the Milky Way galaxy (Cardelli et al. 1989) or the 'Calzetti' curve (Calzetti et al. 2000). Such blind adoption of an attenuation law that may or may not be applicable can result in substantial systemic biases introduced to extrapolated dust emission models, mass estimates, and star-formation rates (Mitchell et al. 2013; Laigle et al. 2019). \nWell-sampled SEDs - from the rest-frame UV through the near-infrared - facilitate a direct measurement of the dust attenuation law (e.g., Kriek & Conroy 2013). This is done by construction of broad SEDs that can then be fit to stellar population synthesis models with a range of dust attenuation law prescriptions to infer the best-fit solutions. The broader COSMOS survey includes 40+ bands of coverage from the far-UV through the mid-infrared spanning both narrow and broad-band filters; such well-sampled SED coverage is sufficient to constrain some variation in the dust attenuation law, as has been measured in similar datasets (Pannella et al. 2015; Salmon et al. 2016; Reddy et al. 2018). However, a key limitation in constraining any possible evolution in the dust attenuation law comes from limited samples at higher redshifts, particularly at epochs where one might expect sufficiently different content and distribution of galaxies' dust reservoirs. The added nearinfrared (and mid-infrared) depth brought by COSMOSWeb will be crucial to dramatically increase the number of known, well-characterized galaxies out to z ∼ 4 whose photometry can then be extracted across all COSMOS datasets to piece together large statistical samples of SEDs. These SEDs can, in turn, be used to infer redshift evolution in dust attenuation. A presumption of energy balance - where absorbed rest-frame UV emission is re-emitted at long wavelengths - can then be directly tested against deep submillimeter observations in the field, stacked using single-dish datasets (e.g., Oliver et al. 2012; Simpson et al. 2019) or individual \nconstraints from galaxies observed by ALMA to much greater depths (e.g., the A3COSMOS project; Liu et al. 2019). \nIn addition to broad SED constraints, the ability to spatially resolve colors on kpc scales using NIRCam and Hubble /ACS imaging will allow direct measurement of the impact of dust geometry on galaxies' integrated SEDs. This will be particularly useful for galaxies already detected by ALMA, of which we estimate there are ∼ 1000 (from ALMA Cycles 0-9) across the COSMOSWeb mosaic footprint. Dust geometry in complex ISM environments has long been a nuisance to SED fitting, as it often results in decoupling of the stellar and dust SEDs (Lower et al. 2022). COSMOS-Web will allow direct constraints on the relative degree of decoupling and its correlation to galaxy morphology as a function of color.", '5.4. Finding & Characterizing Protoclusters': "Galaxy clusters represent the most massive gravitationally bound structures in the Universe, and yet the history of their assembly is observationally uncertain. Galaxy clusters are typically found at z < ∼ 1 . 5 thanks to thermal Bremsstrahlung radiation in the Xray (Kravtsov & Borgani 2012) or via the SunyaevZel'dovich effect in the millimeter (Menanteau & Hughes 2009; Vanderlinde et al. 2010) due to a hot ∼ 10 7 K intracluster medium. A complete catalog of X-ray groups identified in COSMOS is compiled by Gozaliasl et al. (2019). However, the progenitors of galaxy clusters called protoclusters - are observationally more elusive (Overzier 2016). They have not yet virialized, thus their intracluster medium is not yet substantially heated to be distinguishable from the surrounding IGM. Before virialization, at z > ∼ 2, overdense environments are extended in large filaments that may span up to > ∼ 10 comoving Mpc scales (Muldrew et al. 2015; Chiang et al. 2017). At z > ∼ 2, these physical scales span 10-30 arcminutes across, thus wide field-of-view surveys are needed to detect and characterize their spatial distribution. \nDue to its large solid angle and sufficient depth to detect structures at z > 2, COSMOS has served as a primary observational field used to detect and analyze protoclusters at high redshifts (Yuan et al. 2014; Casey et al. 2015; Diener et al. 2015; Chiang et al. 2015; Hung et al. 2016). Such works have highlighted some of the challenges in constraining the forward evolution of such diffuse structures, where it is particularly difficult to constrain protoclusters' halo masses, and yet total halo mass is crucial to the interpretation of their long-term evolutionary path (Sillassen et al. 2022). One particular structure, now dubbed 'Hyperion,' lies in the center of \nthe COSMOS field at z ∼ 2 . 5 with an estimated z = 0 halo mass exceeding 10 15 M glyph[circledot] ; a subcomponent of Hyperion has been discussed in the literature as a possible proto-virialized cluster core through the detection of associated extended X-ray emission (Wang et al. 2016, 2018b; Champagne et al. 2021). Its filamentary structures extend half a degree across and coincide well with the coverage of COSMOS-Web, which will allow a much richer mapping of its constituent galaxies at fainter luminosities. While spectroscopic follow-up will solve an essential piece of the puzzle in spatially mapping the full extent of known structures like Hyperion in COSMOS, the precise photometric redshifts provided by COSMOSWeb will dramatically improve the efficiency of followup. For example, reducing σ NMAD (∆ z/ (1 + z )) from ∼ 0.06 to 0.03 for ∼ 27 th magnitude sources reduces the uncertainty in line-of-sight projected distance by a factor of ∼ 2 to ∼ 100 Mpc from z ∼ 2 -5. While still significantly larger than the expected line-of-sight distances within protocluster environments, the increased precision will significantly improve the efficiency of follow-up spectroscopic campaigns targetting sources with photometric redshifts consistent with an overdensity of spectroscopic redshifts. \nAt higher redshifts, the prospect for discovering new protoclusters in COSMOS-Web is significant. Based on the z ∼ 0 cluster mass function (e.g., Bahcall & Cen 1993), we expect ∼ 30 structures between 2 < z < 8, ∼ 20 of which will be 4 < z < 8, that eventually collapse into > 5 × 10 14 M glyph[circledot] clusters at z = 0. Some of these we may have already found the first hints of based on ground-based data (e.g., Brinch et al. 2022), and the added depth and photometric redshift precision of COSMOS-Web will push the potential discovery space significantly. A more efficient mapping of such structures over an unbiased area will then allow more detailed investigations of the assembly history of protoclusters themselves (Casey 2016).", '5.5. Strong Lensing': "The past three decades have seen the discovery of hundreds of galaxy-scale strong lenses (e.g., Bolton et al. 2008; Gavazzi et al. 2012; Rojas et al. 2021). COSMOSWeb will better resolve the 40+ candidate strong lenses currently known in the COSMOS field from existing Hubble data and ground-based observations (Faure et al. 2008; Jackson 2008), and has the potential to discover many more previously unknown galaxy-galaxy lenses due to the survey's added depth in the near-infrared, sensitive to fainter, higher redshift background sources. \nWe perform a simple estimate of how many lenses will be in COSMOS-Web by first estimating the total num- \nber of galaxies acting as potential lenses. As intrinsically massive galaxies are needed to cleanly resolve the lensed system, we assume lensing galaxies are already part of the current COSMOS2020 catalog (Weaver et al. 2022a). We use the criterion 0 . 2 < z lens < 1 . 5, M glyph[star] > 10 9 M glyph[circledot] , and SFR < 10 -1 M glyph[circledot] yr -1 , resulting in a selection of ∼ 2 × 10 5 galaxies. We take the median of the distribution of the stellar-to-halo mass relation from Shuntov et al. (2022) to define four mass bins: 9 . 0 < log M glyph[star] < 9 . 5 ( M halo ∼ 2 × 10 11 M glyph[circledot] ), 9 . 5 < log M glyph[star] < 10 . 0 ( M halo ∼ 4 × 10 11 M glyph[circledot] ), 10 . 0 < log M glyph[star] < 10 . 5 ( M halo ∼ 7 × 10 11 M glyph[circledot] ), 10 . 5 < log M glyph[star] < 11 . 0 ( M halo ∼ 1 . 2 × 10 12 M glyph[circledot] ). We estimate the surface on the sky where multiple imaging occurs (2 < z source < 13) assuming lens mass profiles are isothermal spheres. We use the estimated number of ∼ 78 galaxies per arcmin 2 at 2 < z < 13 drawn from UV luminosity functions (compiled by Behroozi et al. 2019). This calculation does not account for factors hindering the confirmation of lens candidates (e.g., confusion with spiral arms) and therefore may overestimate the number of lenses confirmed. To bring the number of lenses closer to realistic numbers, we perform the same calculation in the larger COSMOS field and rescale our results to the size of COSMOS-Web, assuming that 70 lenses are confirmed. This yields an expected ∼ 90 new lenses will be found in COSMOSWeb. \nJWST 's unprecedented depth and resolution will lead to the discovery of the highest density of strong lenses per square degree, making it ideal for inferring line-ofsight shear with strong lenses (Fleury et al. 2021; Hogg et al. 2022) and complementing COSMOS-Web's weak lensing analysis (see also Kuhn et al. 2021). Uniform multiband imaging of every strong lens will be available, overcoming challenges with deblending the lens and source light (Etherington et al. 2022). The highly magnified source galaxy population will allow for studies of high redshift galaxy formation (e.g., Swinbank et al. 2015) as well as detailed studies of the central mass profile of lensing galaxies (e.g., Koopmans et al. 2009; Nightingale et al. 2019; Shajib et al. 2021; Etherington et al. 2022) and dark matter contents (e.g., Vegetti et al. 2014; He et al. 2022).", '5.6. Identifying Candidate Direct Collapse Black Holes': "Direct Collapse Black Holes (DCBHs) have been proposed to resolve the mysterious quick growth of the Universe's first supermassive black holes with M BH ∼ 10 9 M glyph[circledot] (Volonteri 2010, 2012; Natarajan 2011), found out to redshifts z ∼ 7 . 5 (e.g., Wang et al. 2021). DCBHs are hypothesized to form black hole seeds of significant mass ( ∼ 10 4 -6 M glyph[circledot] ; Shang et al. 2010; Johnson et al. \n2012) from the primordial collapse of an atomic-cooling halo whereby strong Lyman-Werner photons could dissociate H 2 and prevent gas fragmentation, allowing the formation of DCBHs with significant mass and growth rates possibly exceeding Eddington rates (e.g., Volonteri & Rees 2005; Alexander & Natarajan 2014). Though no DCBH candidates have been directly confirmed, they are thought to have significant infrared through submillimeter emission, resulting in a steep, red near-infrared spectrum; they are also expected to have X-ray emission (Natarajan et al. 2017). \nPacucci et al. (2016) present two possible candidate DCBHs from CANDELS data in the GOODS-S area, both of which have photometric redshifts larger than z ∼ 6 and robust X-ray detections as well as very steep infrared spectra. The improved depth of COSMOS-Web compared to CANDELS (out to 4.4 µ m or 7.7 µ m in the NIRCam mosaic or MIRI-covered subset) will allow more robust identification of fainter DCBH candidates with more robust photometric redshifts. The number density of the CANDELS-identified sources extrapolated to COSMOS-Web implies that we may find ∼ 20 such candidates in our full survey volume at z > 6. Such sources will then require spectroscopic follow-up with JWST to confirm their nature, assess their black hole masses, and to inform predictions for future deeper X-ray observations that may provide further confirmation.", '5.7. Supermassive Black Hole - Galaxy Coevolution': "COSMOS-Web will open new avenues to study AGN and quasars at high redshift. At z > 6, the black hole population with masses down to 10 6 M glyph[circledot] can be revealed through color-selection (see, for example, Goulding & Greene 2022) some of which may have resulted from an earlier DCBH event (see § 5.6). Using a semi-analytic model for the formation of the first galaxies and black holes (Trinca et al. 2022), we expect ∼ 50 black holes within the COSMOS-Web volume that have 7 < z < 10 and masses of 10 6 -10 8 M glyph[circledot] . With the spatial resolving power of JWST , morphologies can be decomposed into unresolved AGN emission and more diffuse host galaxy emission (Kocevski et al. 2022; Ding et al. 2022) on spatial scales of ∼ 1 kpc. With AGN-free host galaxy images, we can measure the mass relation between black holes and their hosts (M BH vs. M host ) beyond z ∼ 3 (Trakhtenbrot et al. 2015; Suh et al. 2020), carry out spatially-resolved studies of the stellar populations up to z ∼ 2, perform a quasar-galaxy cross correlation analysis (Garc'ıa-Vergara et al. 2017), and assess the influence of mergers (e.g., Mechtley et al. 2016; Shah et al. 2020). Furthermore, MIRI will aid in our ability to determine \nFigure 15. An illustration of the part of the COSMOSWeb area that will be covered by multi-epoch NIRCam observations, thanks to the PRIMER survey (GO #1837). Due to the scheduling of the PRIMER program primarily in Cycle 1, and this region of the COSMOS-Web mosaic during Cycle 2, a total of 133 arcmin 2 will see multiple visits; the first of these was observed in January 2023 (the PRIMER area covering both green and orange highlighted regions). The second will occur in ∼ April 2023 (PRIMER covering the purple and orange regions). The last epoch will occur in ∼ December 2023/January 2024 thanks to COSMOS-Web. Thus the purple region will have two epochs separated by ∼ 9 months (this corresponds to 36.5 arcmin 2 ), the green region will have two epochs separated by ∼ 1 year (this corresponds to 53.8 arcmin 2 ), and the orange area will have three epochs of separation ∼ 3 months followed by another ∼ 9 months, spanning a year in total (this corresponds to 42.6 arcmin 2 ). \n<!-- image --> \nthe demographics of the AGN population including the obscured population through detection of steep infrared (unresolved) sources with dominant torus emission.", '5.8. Searches for z > 10 Pair Instability Supernovae': 'COSMOS-Web sits in unique parameter space, able to search for intrinsically rare phenomena at sensitivities beyond most wide-field surveys. Very high-redshift ( z > 5) supernovae (SNe) in particular may provide a unique lens on the formation of the first massive stars by \nconstraining the high-mass end of the Population III initial mass function. Such a first generation of stars is indeed thought to be top-heavy (Bromm et al. 1999, 2002). Ranging in mass from 100-260 M glyph[circledot] , such stars are most likely to die as pair-instability supernovae (PISNe; Heger & Woosley 2002), which release up to 100 times more energy than Type Ia or Type II SNe (with intrinsic luminosities ∼ 10 47 -48 erg s -1 ). Their extreme energy release is a result of electron-positron pair creation via thermal heating after the cessation of carbon burning in the core, leading to collapse and thermonuclear burning of O and Si; the energy released unbinds the star without leaving a remnant. Thanks to their luminosity and relative longevity (lengthened by 1 + z time dilation) it is plausible to detect and identify PISNe brighter than NIR magnitudes ∼ 28 from z = 10 -30 in JWST NIRCam imaging surveys, particularly in the long wavelength channels. Whalen et al. (2013) present nearinfrared light curves of PISNe from radiation hydrodynamical simulations, suggesting PISNe with > ∼ 200 M glyph[circledot] progenitors at z ∼ 15 -30 may remain detectable with F444W < 28 (possibly as bright as ∼ 26 th magnitude) for 1-3 years, varying by ∼ 0.3 mags yr -1 . Hummel et al. (2012) present calculations of the expected number density of such JWST -detectable explosions at or below ∼ 0.02 arcmin -2 at any given time at z > ∼ 10. This could result in ∼ 40 such events sitting within the COSMOSWeb NIRCam footprint. \nThe primary challenge in identifying PISNe candidates in COSMOS-Web will come from distinguishing them from high-redshift galaxies; thus, multi-epoch observations (conducted on roughly a yearly timescale) become critical to measuring the time-variable fading of the explosion. While most prior near-infrared datasets in COSMOS reach depths of only 25-26 mags (and are limited by the poor spatial resolution of Spitzer or ground-based UltraVISTA data), there may be a handful of exceptionally bright PISNe at z > 5 whose transient nature can be constrained using existing observations on a ∼ 10 year cadence. Alternatively, over a smaller area, the CANDELS survey (Grogin et al. 2011; Koekemoer et al. 2011) conducted deep ∼ 28 imaging out to 1.6 µ m covering 200 arcmin 2 in late 2011 and early 2012; this provides a ∼ 10 year time baseline for potential z ∼ 5 -12 PISNe relative to COSMOS-Web observations detected in F150W (the redshift range limited directly by wavelength of deep imaging and the opacity of the IGM in absorbing photons shortward of 1216 ˚ A). \nOut to higher redshifts, it may be possible to detect PISNe candidates out to z ∼ 30 across a ∆ t = 1year timescale using imaging from the PRIMER JWST survey in conjunction with COSMOS-Web, as shown in \nFigure 16. Ultracool halo sub-dwarf templates projected in AB magnitudes at a distance of 2 kpc. Templates are taken from Saumon & Marley (2008) spanning effective surface temperatures of 900-2500K (purple to red). Synthetic photometry is calculated in the COSMOS-Web bands, whose depths are shown in a similar fashion as in Figure 3. Halo M-dwarfs would be detectable out to ∼ 10 kpc (with potential confusion with z ∼ 6 -7 galaxies between 3-10 kpc), while L-, T- and Y-dwarfs will be detectable to ∼ 2 kpc. \n<!-- image --> \nFigure 15. Assuming there are no significant changes to the JWST long-range plan as of this writing, the PRIMER survey (GO #1837) will obtain half of their COSMOS NIRCam imaging in late 2022 covering an area ∼ 96 arcmin 2 out to F444W, and the other half in April 2023. Both PRIMER regions of the field will then be covered in late 2023 by COSMOS-Web, allowing a careful comparison of differential photometry for a potential handful of PISNe candidates brighter than ∼ 28. The total area with deep, ∼ 28 th magnitude ∼ 4.5 µ m multi-epoch JWST observations is ∼ 133 arcmin 2 . Even with only a few detections, such PISNe candidates could potentially be extremely useful for constraining the nature of Population III stars formed shortly after the Big Bang.', '5.9. Ultracool Halo Sub-Dwarf Stars': 'Ultracool dwarfs (late M-dwarfs through Y-dwarfs) are the most abundant stellar population by number and their prominent emission in the near-infrared implies that deep field surveys from JWST are prone to detect them to significant distances in the Galactic halo (Ryan & Reid 2016). Indeed, mapping their number density to different distances in the outer halo may give unique constraints on the metal-poor initial mass function as well as the scale height of the Milky Way for lowmass objects (Burgasser et al. 2003; Carnero Rosell et al. 2019). Such discoveries are only enabled by deep near- \ninfrared imaging, and given the wide area of COSMOSWeb, we anticipate finding of order ∼ 1000 such dwarfs across the field at various distances. \nFigure 16 shows four ultracool dwarf templates from Saumon & Marley (2008) with varying effective temperature from 900 K, through the L-T transition at ∼ 1000 K up to late M-dwarfs at 2500 K. Models assume a 1000 m s -2 surface gravity, consistent with expectation for older halo stars that would likely be found in extragalactic fields like COSMOS; no cloud cover is assumed below 1000 K, above which models with a moderate amount of cloud cover are adopted (Bowler 2016). Significant absorption bands in ultracool T- and Y-dwarfs between 1.5-3.5 µ m, particularly at low < ∼ 1000 K temperatures, lead to very distinct near-infrared colors from galaxies in NIRCam bands provided they are located at distances ≤ 1 kpc. For example, a recent late T-dwarf candidate was identified in JWST imaging of Abell 2744 at a distance of ∼ 600 pc by Nonino et al. (2022). \nIn addition to the science questions addressed by the detection of ultracool dwarfs, the population has also been a dominant contaminating source for searches of high-redshift galaxies, particularly samples of z ∼ 6 -7 sources due to their lack of emission shortward of ∼ 1 µ m. While Hubble imaging was limited to the shorter wavelengths, the long wavelength channels of NIRCam are of particular use in breaking the color degeneracies for distinguishing ultracool dwarfs from compact high-redshift galaxies. In addition, those ultracool dwarfs that would be more consistent with high-redshift galaxy colors are expected to be significantly brighter (peaking in density around J ∼ 24; Ryan & Reid 2016). Dwarfs at considerable distances > 1 kpc, with apparent magnitudes fainter than ∼ 26 have the potential to contaminate z ∼ 6 -7 samples; however, their number density is expected to be relatively low relative to galaxies at similar magnitudes (fewer than ∼ 50 are expected across the COSMOS-Web mosaic fainter than J ∼ 26).', '6. SUMMARY': "We have presented the observational design and scientific goals of COSMOS-Web, the largest prime General Observer program in JWST 's Cycle 1 of observations. COSMOS-Web is a 0.54 deg 2 contiguous NIRCam survey imaged in four filters (F115W, F150W, F277W, and F444W) to depths of ∼ 27.5-28.2 magnitudes. In parallel, COSMOS-Web also includes 0.19deg 2 noncontiguous MIRI imaging in one filter (F770W) to a depth of ∼ 25.3-26.0 magnitudes. COSMOS-Web is approximately 2.7 × larger than all other Cycle 1 JWST NIRCam deep field efforts combined and 3.5 × larger than the combined MIRI deep field coverage. \nThe improvement in photometric redshift precision in COSMOS-Web will be substantial compared to the most recent catalogs compiled in the COSMOS field (Weaver et al. 2022a), with < 5% errors on photometric redshifts down to magnitudes ∼ 27 in F277W. \nThe primary science goals of COSMOS-Web are threefold. First, COSMOS-Web will detect thousands of new galaxies within the Epoch of Reionization (EoR, 6 < ∼ z < ∼ 11) and generate the largest number of galaxies at or above the knee of the UV luminosity function. Such intrinsically bright galaxies likely trace massive halos at early times at the nodes of the cosmic web. COSMOS-Web's large area will allow a detailed mapping of the galaxy density field within the EoR on physical scales ∼ 150 Mpc across, sufficiently large to minimize cosmic variance by exceeding the size of the largest cosmic structures at these redshifts. \nSecond, COSMOS-Web aims to detect the Universe's first massive quiescent galaxies that were likely in place between redshifts 4 < z < 6; such galaxies mark the extreme limits of galaxy evolution at early times by building their stellar reservoirs at extraordinary rates (exceeding ∼ 10 10 -10 11 M glyph[circledot] at z > 4). We will be able to distinguish them from their dust star forming counterparts, study their morphologies and star formation histories, and thus place constraints on their progenitors. \nLastly, COSMOS-Web will measure the evolution in the stellar mass to halo mass relation (SMHR) from 0 < z < 2 . 5 using weak gravitational lensing. The SMHR forms an essential anchor of cosmological simulations on large scales, and these data will extend its measurement from z ∼ 1 to z ∼ 2 . 5 in addition to allowing a detailed look at the SMHR by galaxy type and star-formation history (as probed by rest-frame optical colors and color gradients). \nBeyond these core science goals, COSMOS-Web's legacy value will extend to many subfields of extragalactic astronomy and beyond. We have summarized the potential impact of the survey on measuring galaxy morphologies, using spatially resolved SEDs to measure galaxy properties, placing constraints on the dust attenuation law, identifying and characterizing galaxy protoclusters, finding strong gravitational lenses, identifying direct collapse black hole candidates, studying the co-evolution of supermassive black holes and their host galaxies, searching for z > 10 pair instability supernovae, and identifying ultracool sub-dwarf stars in the Milky Way's halo. We hope the value of this survey continues to grow with time, as have many other deep-field observations before COSMOS-Web and JWST . \nWe thank the anonymous reviewer for helpful suggestions which greatly improved the manuscript. We thank the entire JWST team, including scientists, engineers, software developers, and the instrument and commissioning teams for making this amazing telescope a reality. We thank our program coordinator Christian Soto, our NIRCam reviewer Dan Coe, and our MIRI reviewer Stacey Bright for helping us to optimize our program and ensuring that the entire program is schedulable. We also thank the CEERS team for their quick release of simulations and observed data products, early testing and modification of the data reduction pipeline, and assisting with preparation for COSMOS-Web data reduction. \nSupport for this work was provided by NASA through grant JWST-GO-01727 and HST-AR-15802 awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. \nCMCthanks the National Science Foundation for support through grants AST-1814034 and AST-2009577 as well as the University of Texas at Austin College of Natural Sciences for support; CMC also acknowledges support from the Research Corporation for Science Advancement from a 2019 Cottrell Scholar Award sponsored by IF/THEN, an initiative of Lyda Hill Philanthropies. JSK acknowledges support from the College of Science and the Laboratory for Multiwavelength Astrophysics at the Rochester Institute of Technology. JSK acknowledges the important contributions to this paper, and the COSMOS-Web proposal, made by Shran and T'Pol, who attended every planning telecon and kept everyone's spirits up in the early days of the COVID-19 pandemic. \nThe Cosmic Dawn Center (DAWN) is funded by the Danish National Research Foundation under grant No. 140. JDR was supported by JPL, which is under a contract for NASA by Caltech. This research is also partly supported by the Centre National d'Etudes Spatiales (CNES). OI, CL, HHMCC acknowledge the funding of the French Agence Nationale de la Recherche for the project iMAGE (grant ANR-22-CE31-0007). BER was supported in part by NASA grant 80NSSC22K0814. MT and FG acknowledge the support from grant PRIN MIUR 2017 20173ML3WW 001. BT acknowledges support from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement 950533) and from the Israel Science Foundation (grant 1849/19) \nThe authors acknowledge Research Computing at the Rochester Institute of Technology (2019) for providing computational resources and support for the work reported in this publication. This work used the CANDIDE computer system at the IAP supported by grants from the PNCG, CNES and the DIM-ACAV and maintained by S. Rouberol. We acknowledge use of the lux supercomputer at UC Santa Cruz, funded by NSF MRI", 'REFERENCES': 'Zitrin, A., Fabris, A., Merten, J., et al. 2015, ApJ, 801, 44, \ndoi: 10.1088/0004-637X/801/1/44', 'A. DETAILS OF THE COSMOS-WEB MOSAIC VISITS': "Here we provide detailed information on the 152 visits that comprise the COSMOS-Web mosaic. Table 6 lists all of the individual visits, their reference positions, and their position angles. The observation number is given as in the COSMOS-Web Proposal (#1727) opened in the Astronomer's Proposal Tool (APT), and the visit name mirrors the target name in APT. The listed position angles are relative to the NIRCam frame (and differ from the V3 angle by 0.09 · ). The three visits that have position angles differing from the other visits in the mosaic are CWEBTILE-0-4, CWEBTILE-5-18, and CWEBTILE-7-15. Their angles are different due to availability of guide stars visible in the fine guidance sensor (FGS); no modification of their positions were required to keep the NIRCam mosaic contiguous. The positions as listed correspond to the reference position of NRCALL FULL and sit at the reference point (V2,V3) = ( -0.32, -492.59) with 4TIGHT dither offsets taken ± 24 . '' 7 along V2 and ± 3 . '' 00 along V3. The relative positions of single visit coverage with respect to this reference point are shown in Figure 2. \nTable 6 . COSMOS-Web Visit Positions \nTable 6 continued \nTable 6 (continued) \nTable 6 continued", 'The COSMOS-Web Survey': "Table 6 (continued)Table 6 continued \nTable 6 (continued) \nNote - The position angle (P.A.) of the visit is specified in the last column; only three visits have non-standard position angles caused by guide star catalog limitations and they are marked with a † . We quote 0 . '' 01 accuracy on tile positions."}

2024arXiv240913404B

Weak gravitational lensing WL surveys provide insight into the matter distribution over an extensive range of scales. Current WL results are in mild tension with cosmic microwave background measurements from the early Universe. Reconstructing the matter power spectrum from their measurements instead of condensing the information into a single cosmological parameter may help locate the origin of these differences. To investigate the cosmic shear measurements of Stage III WL surveys we compared their tomographic data by assuming a simple parametric model for the matter power spectrum. The model allows the comparison of surveys with different characteristics and in an agnostic approach gives insight into the shape of the matter power spectrum preferred by the data. For the matter power spectrum we assumed a double powerlaw model in scale factor and wavenumber. The bestfitting amplitude and exponents were inferred in an MCMC analysis. We identified the scales to which the data is most sensitive. We tested the sensitivity to different assumptions of the intrinsic alignment strength. We find that the constraining power of Stage III surveys on the power spectrum shape and evolution is still limited. Most information can be summarized as an overall amplitude at a pivot point in wavenumber and scale factor while constraints on the powerlaw indices are considerably weaker. Nevertheless all surveys show a weaker rate of growth from z 0.5 to 0.1 than predicted. The assumed intrinsic alignment strength is found to have no significant impact on the measured parameters and goodness of fit. Direct estimates of the matter power spectrum from Stage III weak lensing surveys can in principle be used to locate the physical origin of the observed S8 tension. We present a simple methodology for the first steps in this direction but find that current constraints are still weak.

2024-09-01T00:00:00Z

['2024arXiv240913404B', 'arXiv:2409.13404', '10.48550/arXiv.2409.13404']

['Astrophysics - Cosmology and Nongalactic Astrophysics']

First step toward matter power spectrum reconstruction with Stage III weak gravitational lensing surveys

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.13404.pdf

{'First step toward matter power spectrum reconstruction with Stage III weak gravitational lensing surveys': 'Jeger C. Broxterman 1 , 2 and Konrad Kuijken 1 \n- 1 Leiden Observatory, Leiden University, PO Box 9513, NL-2300 RA Leiden, The Netherlands e-mail: broxterman@strw.leidenuniv.nl\n- 2 Lorentz Institute for Theoretical Physics, Leiden University, PO Box 9506, NL-2300 RA Leiden, the Netherlands \nReceived September 20, 2024; accepted November 18, 2024', 'ABSTRACT': 'Context. Weak gravitational lensing (WL) surveys provide insight into the matter distribution over an extensive range of scales. Current WL results are in mild tension with cosmic microwave background measurements from the early Universe. Reconstructing the matter power spectrum from their measurements instead of condensing the information into a single cosmological parameter may help locate the origin of these di ff erences. \nAims. To investigate the cosmic shear measurements of Stage III WL surveys, we compared their tomographic data by assuming a simple parametric model for the matter power spectrum. The model allows the comparison of surveys with di ff erent characteristics and, in an agnostic approach, gives insight into the shape of the matter power spectrum preferred by the data without assuming a cosmological model. \nMethods. For the matter power spectrum, we assumed a double power-law model in scale factor and wavenumber. The best-fitting amplitude and exponents were inferred in a Markov chain Monte Carlo (MCMC) analysis. We identified the scales to which the data is most sensitive. We tested the sensitivity to di ff erent assumptions of the intrinsic alignment strength. \nResults. We find that the constraining power of Stage III surveys on the power spectrum shape and evolution is still limited. Most information can be summarized as an overall amplitude at a pivot point in wavenumber and scale factor, while constraints on the power-law indices are considerably weaker. Nevertheless, all surveys show a weaker rate of growth from z = 0.5 to 0.1 than predicted. The assumed intrinsic alignment strength is found to have no significant impact on the measured parameters and goodness of fit. \nConclusions. Direct estimates of the matter power spectrum from Stage III weak lensing surveys can, in principle, be used to locate the physical origin of the observed S 8 tension. We present a simple methodology for the first steps in this direction, but find that current constraints are still weak. \nKey words. Cosmology: theory - large-scale structure of Universe - Gravitational lensing: weak', '1. Introduction': "The concordance model of cosmology, the Lambda-cold dark matter ( Λ CDM) model, can fit an extensive range of observations with great accuracy (see, e.g., Lahav & Liddle 2022). Examples are the fluctuations in the cosmic microwave background (CMB, Planck Collaboration et al. 2020a, Planck ); baryon acoustic oscillations (BAOs; DESI Collaboration et al. 2024a); and cosmic shear, the slight distortion of galaxy images by weak gravitational lensing (WL) of the large-scale structure of the Universe (Kilbinger 2015). With the increasing accuracy of cosmological surveys, several tensions between different probes have emerged. Most notably, the tension in the value of the Hubble constant ( H 0) inferred from high-redshift Planck CMB measurements or local distance ladder measurements (Riess et al. 2021), and the value of S 8 ≡ σ 8 √ Ω m / 0 . 3. 1 Here the value of the Planck CMB measurements of the early Universe is in slight tension with the value measured from cosmic shear measurement of the local Universe (Hildebrandt et al. \n2017; Asgari et al. 2021; Secco et al. 2022; Dark Energy Survey and Kilo-Degree Survey Collaboration et al. 2023; Li et al. 2023). The main proposed solutions to solving the S 8 tension involve unrecognized systematics or new physics (see, e.g., Lucca 2021; Tanimura et al. 2023; Joseph et al. 2023). \nThe tension between the probes may have di ff erent physical origins. The CMB measures linear scales and the early Universe, whereas WL probes lower redshifts and nonlinear scales. Assuming a physical origin, the tension may originate from differences in either scale (high vs. low wavenumber; k ) or time (high vs. low redshift; z ). \nAmon & Efstathiou (2022) explore the first option, introducing a one-parameter phenomenological model that modifies the power spectrum by fixing it at linear scales to the Planck prediction and predicting the nonlinear suppression required to reconcile cosmic shear measurements from the Kilo-Degree Survey (KiDS) with Planck . They remain agnostic as to what causes the suppression: obvious candidates are misinterpreted baryonic components or new physics. However, the model predicts the suppression of the nonlinear power spectrum using only dark matter (DM), and it does not contain any baryonic-informed components. Although baryonic suppression and nonlinear collapse manifest themselves in roughly the same wavenumber \nFig. 1. Observational cosmological probes of the matter power spectrum and the wavenumber ( k ), scale factor ( a ), and redshift ( z ) ranges to which they are approximately sensitive. From early to late Universe: cosmic microwave background (CMB, purple), CMB lensing ( ΦΦ , light blue), the Lyman-alpha forest (Ly α , orange), quasi-stellar objects (QSOs, red), galaxy clustering ( gg , green), cosmic shear ( γγ , dark blue), and cluster abundance (CA, white). The black curve separates the linear and nonlinear collapse regimes. Constraining the matter power spectrum from di ff erent probes allows the localization of discrepancies in the two-dimensional ( k , a ) plane and provides insight into the physical origin of tensions between the measurements of di ff erent probes. \n<!-- image --> \nregime, the suppression they model is on slightly larger scales and has a di ff erent shape than predicted by state-of-the-art cosmological hydrodynamical simulation calibrated primarily to Xray data (see, e.g., Fig. 1 of Bigwood et al. 2024, and, e.g., McCarthy et al. 2017; Pakmor et al. 2023; Schaye et al. 2023; Schaller et al. 2024). \nPreston et al. (2023) extended the analysis to include the most recent cosmic shear measurement from the Dark Energy Survey (DES), and found that they require a somewhat lower suppression to resolve the tension with Planck . However, the current measurements are not precise enough to constrain the baryonic feedback models. Recent findings suggest that the simulations calibrated to X-ray data, which probe the central regions of haloes, underpredict baryonic feedback as they cannot reproduce Sunyaev-Zel'dovich (SZ) measurements, which are more sensitive to the outskirts of haloes (Amodeo et al. 2021; Bigwood et al. 2024; Hadzhiyska et al. 2024). Although it might help resolve the S 8 tension, it raises a di ff erent problem because the X-ray and SZ measurements are no longer consistent with the same baryonic feedback model. \nIn principle, we can distinguish tensions between measurements at di ff erent scales and times by directly constraining the matter power spectrum in the wavenumber ( k ) and scale factor ( a ) plane. Figure 1 illustrates the sensitivity of di ff erent observational cosmological probes in the ( k , a ) plane. The di ff erent probes are cosmic shear ( γγ ), galaxy clustering ( gg ; e.g., Reid et al. 2010; Gil-Marín et al. 2016), CMB, CMB lensing ( ΦΦ ; e.g., Planck Collaboration et al. 2020b), cluster abundance (CA; e.g., Tegmark et al. 2004), quasi-stellar objects (QSOs; e.g., DESI Collaboration et al. 2024b), and the Lyman alpha forest (Ly α ; e.g., Palanque-Delabrouille et al. 2015; Karaçaylı et al. 2024). The squares correspond to probes where the emitted radiation is directly observed, whereas the ellipses correspond to processes happening along the line of sight between emission and observation. The black curve separates the linear and \nnonlinear collapse regimes. The estimate is obtained from class (Blas et al. 2011) by determining the wavenumber for which the nonlinear estimate di ff ers by more than 5% from the linear prediction. The di ff erence between the di ff erent cosmic times and scales probed by the CMB and cosmic shear, as described above, is clearly visible in the figure. Comparing the di ff erent probes and their cross-correlations allows the localization of the tension, which should inform us regarding the origin of the observed discrepancies. \nHere, as a first step, we infer the best-fitting shape of the matter power spectrum using the latest cosmic shear measurement of Stage III WL surveys. We assume a simple parametric model that allows us to quantify the evolution of the matter power spectrum with k and a preferred by current data in an agnostic approach. The model allows the comparison of di ff erent surveys with di ff erent survey properties and does not assume a cosmological model for the matter power spectrum. We identify the scales to which the data is most sensitive, and test the impact of the assumed intrinsic alignment (IA) on the goodness of fit. \nPreston et al. (2024) propose constraining the matter power spectrum from cosmic shear surveys. They estimate the constraining power of the matter power spectrum using synthetic cosmic shear measurements for the final data release of Stage III and Stage IV WL surveys. They find that while current cosmic shear surveys should only marginally be able to distinguish di ff erences from a Λ CDMdark matter-only (DMO) model, nextgeneration surveys should provide clear constraints on the shape of the matter power spectrum at (mildly) nonlinear scales. \nIn Sect. 2 we summarize the relevant WL theory to estimate the measured cosmic shear two-point statistics from the matter power spectrum. Section 3 describes the data, and the assumed parametric model is introduced in Sect. 4. In Sect. 5 we present the measurements of the best-fitting parametric model of the matter power spectrum. We also compare our results to the bestfitting Λ CDMpredictions. The main findings are summarized in Sect. 6.", '2. Theory': "In this section we briefly summarize the theoretical framework for cosmic shear measurements as projected statistics of the matter power spectrum. Cosmic shear measures the systematic distortion of galaxy images from WL by the large-scale structure. The measured signal is comprised of the signal induced by weak gravitational lensing (G), the intrinsic alignment of the galaxies (I), and the cross-correlation of the two e ff ects. The observed cosmic shear angular power spectrum ( C ( i j ) ϵϵ ) between two populations of galaxies in di ff erent tomographic bins i and j , as a function of multipole moment ℓ , is given by \nC ( i j ) ϵϵ ( ℓ ) = C ( i j ) GG ( ℓ ) + C ( i j ) GI ( ℓ ) + C ( i j ) II ( ℓ ) . (1) \nCurrent cosmic shear surveys use B -mode statistics as null tests, and they find no evidence for nonzero B -modes or they limit their analysis to not include scales where the B -modes are nonzero due to unrecognized systematics. Here we assume that the B -modes are 0, and only use the E -mode signal such that C ( i j ) ϵϵ ( ℓ ) = C ( i j ) ϵϵ, E ( ℓ ). Assuming the modified Limber approximation, the angular power spectra are estimated from the matter power spectrum ( P m) as (Limber 1953; LoVerde & Afshordi 2008) \nC ( i j ) ab ( ℓ ) = Z χ hor 0 d χ W ( i ) a ( χ ) W ( j ) b ( χ ) f 2 K ( χ ) P m GLYPH<18> ℓ + 1 / 2 f K( χ ) , z ( χ ) GLYPH<19> , (2) \nwhere f K( χ ) is the angular diameter distance, χ the comoving distance, z the redshift, and a , b ∈ I , G. The integral runs until the comoving line of sight horizon that is the edge of the galaxy survey ( χ hor). The WL kernel W is given by (Kaiser 1992) \nW ( i ) G ( χ ) = 3 H 2 0 Ω m 2 c 2 f K( χ ) a ( χ ) Z χ hor χ d χ ' n ( i ) s ( χ ' ) f K( χ ' -χ ) f K( χ ' ) , (3) \nwhere c is the speed of light, a = 1 / (1 + z ) the scale factor, and n s( χ ) the comoving distance source distribution that relates to the source redshift distribution as n s( z )d z = n s( χ )d χ . For the IA kernel, we assume the nonlinear alignment (NLA) model (Bridle &King 2007) \nW ( i ) I ( χ ) = -A IA GLYPH<18> 1 + z ( χ ) 1 + z piv GLYPH<19> η IA C 1 ρ cr Ω m D ( a [ χ ]) n ( i ) ( χ ) , (4) \nwith A IA the IA amplitude; z piv a finite redshift pivot point; η IA the redshift dependence exponent, which we set to 0; C 1 ρ cr = 0 . 0134; and D the linear growth factor. We assume a flat Universe such that f K( χ ) = χ . We adopt either A IA = 0 or 1 to quantify whether including intrinsic alignment better fits the data. \nWeused three di ff erent cosmic shear observables in our analysis. Each observable is a projection of the angular power spectra, but is sensitive to di ff erent scales as they depend on di ff erent filter functions. The first observable is the shear two-point correlation functions (2PCFs, ξ + / -): \nξ ( i j ) + / -( θ ) = Z ∞ 0 d ℓℓ 2 π J 0 / 4( ℓθ ) C ( i j ) ϵϵ ( ℓ ) . (5) \nHere J 0 and J 4 are cylindrical Bessel functions of the first kind of the zeroth and fourth order, respectively. Second, we considered complete orthogonal sets of E / B-integrals (COSEBIs). An advantage of the COSEBIs is that nearly all cosmological information is contained in the first few n -modes, thus reducing the necessary computational time (Asgari et al. 2012). The COSEBIs ( En ) are estimated using (Schneider et al. 2010) \nE ( i j ) n = Z ∞ 0 d ℓ ℓ 2 π C ( i j ) ϵϵ ( ℓ ) Wn ( ℓ ) , (6) \nwhere the weight function, Wn ( ℓ ), is a Hankel transform given by Eq. 9 in Asgari et al. (2021). Similar to Asgari et al. (2021), we used the first five COSEBIs n -modes. \nFinally, we considered band power spectra ( C E , l , band powers). This statistic is an angular average of the two-point correlation function that has a weak correlation between values at di ff erent wavenumbers and is given by (Schneider et al. 2002) \nC ( i j ) E , l = 1 2 N l Z ∞ 0 d ℓ W l EE ( ℓ ) C ( i j ) ϵϵ ( ℓ ) , (7) \nwhere the normalisation N l and kernels W EE are respectively given by Eqs. 18 and 26 in Joachimi et al. (2021). We assume the best-fitting Λ CDM cosmology from Planck TT,TE,EE + lowE + lensing for the values of the cosmological parameters and to compute the redshift-comoving distance relation and linear growth factor (Planck Collaboration et al. 2020a).", '3. Data': "We used the latest public data releases of the two-point statistics of three Stage III WL surveys: the Kilo-Degree Survey data release 4 (KiDS-1000; Kuijken et al. 2019), the Dark Energy Survey year 3 (DES year 3; Abbott et al. 2022), and the Hyper \nSuprime-Cam Year 3 (HSC year 3; Dalal et al. 2023). Collectively, these surveys are referred to as Stage III WL surveys. We use each survey's summary statistics, source redshift distributions, and full covariance matrix, as described in Asgari et al. (2021), Hildebrandt et al. (2020), and Joachimi et al. (2021) for KiDS-1000; Abbott et al. (2022), Myles et al. (2021), and Friedrich et al. (2021) for DES year 3; and Shirasaki et al. (2019), Li et al. (2023), and Rau et al. (2023) for HSC year 3. We cut the data vectors at the same angular scales as the fiducial survey analyses. In Table 1 we summarize the main characteristics of the surveys and the statistics we used in this analysis. The table lists the sky area covered, e ff ective source number density ( n e ff ), standard deviation of the galaxy shape measurement error, and number of tomographic bins. As indicated in the final column, for KiDS-1000 we used the 2PCFs, COSEBIs, and band powers, whereas for HSC and DES year 3 we only used the real space 2PCFs from Li et al. (2023) and Abbott et al. (2022), respectively. The former allows us to quantify the di ff erence between the same measurements from a single survey and the scales that are probed by di ff erent statistics, whereas using all statistics allows us to quantify the variations between di ff erent surveys.", '4. Methodology': 'Section 2 provides the formalism for estimating the cosmic shear observables from the matter power spectrum. In this section we provide the model we used in our analysis. For the matter power spectrum, we assumed a double power law in wavenumber ( k ) and scale factor ( a ) as \nP m( k , a ) = A GLYPH<18> k k piv GLYPH<19> p GLYPH<18> a a piv GLYPH<19> m , (8) \nwhere the amplitude A , and exponents p and m are the parameters that will be fitted for. The pivot points k piv and a piv are free as they rescale the amplitude. We used two sets of pivots. First, we determined the optimal values of the pivot points for each survey and statistic by finding the values of k piv and a piv for which the covariance between the amplitude A and exponents p and m is zero. This method provides the power spectrum amplitude of the point on the ( k , a ) plane most tightly constrained by the statistic. We provided the best-fitting parameters of the double power-law parameters assuming these pivot points. After, to more straightforwardly compare the three surveys, we adopted a common pivot for all surveys: k piv = 0 . 5 h / Mpc and a piv = 0 . 75. We estimated the best-fitting parameters that maximize the log-likelihood by using the public Markov chain Monte Carlo (MCMC) sampler emcee (Foreman-Mackey et al. 2013). We used flat priors in the range log 10 A ∈ [0 , 10] , p ∈ [ -2 , 0 . 5], and m ∈ [ -5 , 5]. We computed the reduced chi-square value between the double power fits and measurements to quantify the goodness of fit.', '5.1. Double power-law constraints': "We first list the results of determining the optimal pivot points for each survey. Table 2 lists the best-fitting pivot points in wavenumber and scale factor for each of the five statistics considered in this work and for the choice of A IA = 0 or 1 in the third and fourth column, respectively. The fifth column gives \nTable 1. Survey characteristics.Notes. Listed for each survey: sky area covered; total e ff ective source number density, n e ff ; standard deviation of the error of galaxy shape measurements, σ e ; number of tomographic bins; and the statistics considered in this work. \nTable 2. Pivot points and best-fitting parameters for the double power-law model. \nNotes. Shown for each statistic and choice of intrinsic alignment amplitude ( A IA) are the values of the pivot points that minimize the covariance between the amplitude log 10 A , and exponents m and p in wavenumber ( k piv), scale factor ( a piv), and corresponding redshift ( z piv = 1 / a piv -1) and the best-fitting parameters for the double power-law parameters; amplitude, log 10 A ; wavenumber exponent, p ; scale factor exponent, m . The final column indicates the reduced chi-square value of the double power-law fit, χ 2 red . \nthe redshift corresponding to the scale factor pivot point ( z piv = 1 / a piv -1). \nThe values show that the HSC survey is the deepest, as the scale factor pivot point corresponds to z ≈ 0 . 4. Then, the three KiDS-1000 statistics, which have almost identical depths, probe slightly less deep. Finally, the DES year 3 2PCFs probe the least deep. The HSC 2PCFs probe the largest scales, whereas the KiDS-1000 2PCFs probe the smallest scales. \nTable 2 also includes the best-fitting parameter values and their 16th and 84th percentiles of the double power-law fit. For each fit the parameter best-fitting values correspond to an inference assuming the listed pivot point for that statistic and the value of intrinsic alignment amplitude. The best-fitting value of the amplitude parameter may therefore di ff er as this is the value of the double power-law model at ( k , a ) = ( k piv , a piv). In Appendix A we show an example of a double-fit power-law prediction to the KiDS-1000 COSEBIs. \nThe three KiDS-1000 statistics generally find the same values for the power-law exponents m and p , which are consistent with the HSC year 3 2PCFs estimates. The DES year 3 2PCFs show a weaker dependence on wavenumber and a stronger dependence on scale factor. \nFocussing first on the growth rate, quantified by the m exponent, we see that the constraints from WL are weak. Most statistics are consistent with no growth, with only DES year 3 2PCFs preferring a positive growth rate of m = 1 . 0 ± 0 . 5 irrespective of the choice of A IA. \nStage III WL surveys primarily probe the Λ -dominated era, during which linear growth eventually comes to a halt. However, this is a gradual process; in addition, nonlinear growth continues for a longer time: predictions obtained with class (Blas et al. 2011) indicate an increase in the power spectrum between redshift 0.5 and 0.1 of a factor of 1.5 (1.9) at wavenumber k = 0 . 1 (1.0) h / Mpc, corresponding to m values of 1.3 (2.1). Therefore, all the data sets we analyze here mildly favor a slower growth \nthan predicted by Λ CDM. This is most apparent in the HSC year 3 2PCFs, which favor m = -0 . 3 ± 0 . 4. \nComparing the three di ff erent KiDS-1000 estimates for p , we find that the 2PCFs provide the tightest constraint. The 1 σ uncertainty on p is a factor of two smaller than the uncertainty provided by the band powers estimate. This is not unexpected: for the chosen configuration, the 2PCFs probe the largest range of scales, as illustrated in Fig. 1 of Asgari et al. (2021). \nInterestingly, the assumed value for A IA does not markedly a ff ect the results. There are no significant di ff erences between the assumed values for A IA. The results using A IA = 0 or 1 are always consistent, and including the intrinsic alignment does not give a better fit, as indicated by the reduced chi-square value in the final column. The NLA model assumes that density perturbations are characterized by the Poisson equation. This assumption breaks down under nonlinear gravitational collapse, and therefore the model cannot physically capture the IA signal over the entire range of scales considered in this work (Krause et al. 2016; Asgari et al. 2021). However, as the current constraining power is still limited and the values we adopted for the IA strength correspond to those measured by current surveys, we expect that using a di ff erent model, such as the tidal alignment and tidal torquing model (Blazek et al. 2019), or including a redshift (e.g., Joachimi et al. 2011) or luminosity (e.g., Chisari et al. 2016) dependence will not change the insensitivity to IA strength. \nThe values of the reduced chi-square statistics are reasonable for the model's simplicity. For example, the model does not capture the BAO wiggles and can still fit the data over a broad range of wavenumbers and redshifts. The reduced chi-square values for the KiDS-1000 COSEBIs are 1.25 ( A IA = 0) and 1.26 ( A IA = 1). These values indicate the fits are of equal quality to the bestfitting Λ CDM prediction from Asgari et al. (2021), who quote a value of χ 2 red = 1 . 2. The same holds for the KiDS-1000 band powers and 2PCFs. \nFig. 2. Posterior contours for a double power-law model as indicated at the top. The results correspond to setting the pivot points to k piv = 0 . 5 h / Mpc and a piv = 0 . 75. Left: Assuming no contribution from intrinsic alignment. Right: Contours from assuming an intrinsic alignment strength set by A IA = 1. The panels show the posterior distribution of the inferences using the KiDS-1000 band powers (black), COSEBIs (green), 2PCFs (yellow), HSC year 3 2PCFs (blue), and DES year 3 2PCFs (red). The results are consistent at the ∼ 2 σ level. \n<!-- image --> \nIn Fig. 2 we show the best-fitting parameter constraints assuming a common pivot point for each survey. As the pivot point does not minimize the covariance between the di ff erent model parameters, clear correlations are visible between the amplitude parameter and p and m exponents. Even so, assuming the same pivot points allows for a direct comparison of the amplitudes measured by the di ff erent statistics at the same redshift and wavenumber. The left panel shows the results assuming an intrinsic alignment amplitude of 0, whereas the right panel shows the contours obtained for A IA = 1. All KiDS-1000 and HSC year 3 results are consistent at the 2 σ level. Comparing the 1D amplitude log 10 A posteriors, we see DES and HSC year 3 predict more power than the KiDS-1000 statistics at the 2 σ level.", '5.2. Comparison to Λ CDM predictions': 'Next, in Fig. 3, we compare our power-law prediction to the best-fitting Λ CDM predictions from the KiDS-1000 COSEBIs inference assuming A IA = 1. The green curve shows the double power prediction. The thin solid curves are all matter power spectrum predictions from the KiDS-1000 COSEBIs Λ CDM inference chains generated using class (Blas et al. 2011) and HMC ode (Mead et al. 2015). These curves are color-coded by their value of S 8, as indicated by the color bar. The solid and dashed gray curves respectively show the best-fitting and the 16th and 84th percentiles of the Λ CDM prediction from the KiDS-1000 COSEBIs (Asgari et al. 2021). The double powerlaw fit is consistent with the Λ CDMprediction over two decades in wavenumber. At the largest wavenumbers ( k > 0 . 5 h / Mpc), the double power law overestimates the power compared to the Λ CDM prediction. The model also overpredicts the power on the largest scales ( k < 10 -2 h / Mpc, not shown), which are be- \nFig. 3. Matter power spectrum at z = 0 . 35. The colored curves show the predictions from the KiDS-1000 COSEBIs chains from Asgari et al. (2021), and they are color-coded by their value of S 8 (see the color bar). The solid and dashed gray curves show the best-fitting and 16th and 84th percentiles of the KiDS-1000 inference, respectively. The green solid curve shows the best-fitting prediction of the A IA = 1 double power-law model assumed in this work. \n<!-- image --> \nthe scales to which cosmic shear is most sensitive. Within this range, the simplistic double power law is consistent with the KiDS-1000 COSEBIs Λ CDMprediction. The 16th and 84th percentile range typically covers half a dex, illustrating that the matter power spectrum is still poorly constrained. \nFig. 4. Suppression of the matter power spectrum compared to a DMO prediction at z = 0 . 35. The dashed green curve corresponds to the bestfitting prediction from the double power law model to the KiDS COSEBIs with A IA = 1. The solid gray curve shows the best fit of the KiDS1000 Λ CDM inference from Asgari et al. (2021), and the shaded area covers the 16th to 84th percentile region. The dotted red curve uses the parametric model from Amon & Efstathiou (2022) with A mod = 0 . 69. \n<!-- image -->', '5.3. Suppression of the matter power spectrum': 'Finally, we show the ratio of the matter power spectrum to a Planck DMO matter power spectrum as a function of wavenumber at z = 0 . 35 in Fig. 4. The dashed green curve shows the best-fitting double power-law estimate from this work. The solid gray curves and shaded area respectively show the best-fitting and the 16th and 84th percentiles of the KiDS-1000 COSEBIs Λ CDM chains. The dotted red curve shows the prediction assuming the one-parameter parametric model from Amon & Efstathiou (2022) and setting A mod = 0 . 69, which is the value found by Amon & Efstathiou (2022) to reconcile KiDS-1000 with Planck . \nThe simple parametric model from Amon & Efstathiou (2022), our double power law, and the DMO power spectrum are all within the 16th and 84th percentile prediction from the KiDS1000 COSEBIs, which reconfirms that current data are not able to constrain detailed deviations from Λ CDMpredictions or baryonic physics, as previously predicted in Preston et al. (2024).', '6. Conclusions': 'In this paper we have taken a first step toward reconstructing the shape of the matter power spectrum measured with Stage III WL lensing surveys. Reconstructing the matter power spectrum from di ff erent surveys that are sensitive to di ff erent redshift ranges and scales can give insights into the physical origins of observed tensions in cosmological parameters. We did not assume a cosmological model for the shape of the matter power spectrum, but instead remained agnostic to its shape by assuming a double power law in wavenumber and scale factor (Eq. 8). This allowed us to directly compare the results of di ff erent surveys and quantify the shape of the matter power spectrum preferred by the data. We identified the scales to which the data are most sensitive, and we tested di ff erent assumptions for the strength of the intrinsic alignment signal. The results highlight the potential of directly constraining the matter power spectrum to explore tensions in current data sets. We find the following: \n- 1. Current cosmic shear surveys do not provide tight enough constraints on the shape of the matter power spectrum to rule out deviations from Λ CDM or realistic baryonic feedback models as the results are still consistent with DMO predictions.\n- 3. Including an intrinsic alignment of galaxies does not significantly impact the fitting accuracy or the best-fitting parameters.\n- 2. Stage III WL surveys can only poorly constrain the shape and evolution of the matter power spectrum in the low-redshift Universe. Even so, they favor a weaker evolution in the redshift range of 0.5 - 0.1 than predicted by Λ CDM. \nMore accurate measurements by next-generation WL surveys carried out by Euclid (Euclid Collaboration et al. 2024), Roman (Spergel et al. 2015), and Rubin (LSST Science Collaboration et al. 2009) as well as more accurate modeling will help us understand the physical origin of the tensions between di ff erent observational cosmological probes.', 'References': "Abbott, T. M. C., Aguena, M., Alarcon, A., et al. 2022, Phys. Rev. D, 105, 023520 \nAmodeo, S., Battaglia, N., Schaan, E., et al. 2021, Phys. Rev. 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Rev. D, 105, 023515 Shirasaki, M., Hamana, T., Takada, M., Takahashi, R., & Miyatake, H. 2019, MNRAS, 486, 52 Spergel, D., Gehrels, N., Baltay, C., et al. 2015, arXiv e-prints, arXiv:1503.03757 Tanimura, H., Douspis, M., Aghanim, N., & Kuruvilla, J. 2023, A&A, 674, A222 Tegmark, M., Strauss, M. A., Blanton, M. R., et al. 2004, Phys. Rev. D, 69, 103501", 'Appendix A: KiDS-1000 COSEBI prediction': 'This Appendix shows the best-fitting prediction for the double power-law model to the KiDS-1000 COSEBIs measurements. The scatter points in Fig. A.1 are the KiDS-1000 measurements (Asgari et al. 2021), and the red and blue curves are the best-fitting double power-law fits with A IA = 0 and A IA = 1, respectively. The subpanels show the measured En as a function of the COSEBI n -modes for the combination of two tomographic bins, as indicated in the upper-right corner of each panel. There is no di ff erence in the fitting accuracy with or without including IA, as seen from the reduced chi-square values in Table 2. The double power-law estimate fits the same ranges well as the best-fitting Λ CDM prediction from Asgari et al. (2021). For example, the Λ CDM and our double power-law estimate both provide good fits to most of the data, but they underpredict the signal for the highn modes in the (1 -5) tomographic bin combination and across the entire range for the (2 -2) combination. \nFig. A.1. KiDS-1000 COSEBIs measurement (black scatter points) with best-fitting A IA = 1 and 0 double power law (solid blue and curves, respectively) for the first five COSEBIs n -modes. Each subpanel corresponds to a combination of two tomographic bins, as indicated in the upper right corner. \n<!-- image -->'}

2024ApJ...973L..49I

The James Webb Space Telescope JWST has unveiled numerous massive black holes BHs in faint broadline active galactic nuclei AGNs. The discovery highlights the presence of dustreddened AGN populations referred to as little red dots LRDs more abundant than Xrayselected AGNs which are less influenced by obscuration. This finding indicates that the cosmic growth rate of BHs within this population does not decrease but rather increases at higher redshifts beyond z 6. The BH accretion rate density deduced from their luminosity function is remarkably higher than that from other AGN surveys in Xray and infrared bands. To align the cumulative mass density accreted to BHs with the observed BH mass density at z 45 as derived from the integration of the BH mass function the radiative efficiency must be doubled from the canonical 10 value achieving significance beyond the gt3 confidence level. This suggests the presence of rapid spins with 96 of the maximum limit among these BHs under the thindisk approximation maintained by prolonged mass accretion instead of chaotic accretion with randomly oriented inflows. Moreover we derive an upper bound for the stellar mass of galaxies hosting these LRDs ensuring consistency with galaxy formation in the standard cosmological model where the host stellar mass is limited by the available baryonic reservoir. Our analysis gives a lower bound for the BHtogalaxy mass ratio that exceeds the typical value known in the nearby universe and aligns with that for JWSTdetected unobscured AGNs. Accordingly we propose a hypothesis that the dense dustrich environments within LRDs facilitate the emergence of rapidly spinning and overmassive BH populations during the epoch of reionization. This scenario predicts a potential association between relativistic jets and other highenergy phenomena with overmassive BHs in the early universe.

2024-10-01T00:00:00Z

['10.48550/arXiv.2402.14706', '2024arXiv240214706I', '10.3847/2041-8213/ad74e2', 'arXiv:2402.14706', '2024ApJ...973L..49I']

['Galaxy formation', 'High-redshift galaxies', 'Quasars', 'Supermassive black holes', '595', '734', '1319', '1663', 'Astrophysics - Astrophysics of Galaxies']

Birth of Rapidly Spinning Overmassive Black Holes in the Early Universe

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']

https://arxiv.org/pdf/2402.14706.pdf

{'Birth of Rapidly Spinning, Overmassive Black Holes in the Early Universe': '<!-- image --> \n1 Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, China \n- 2 Global Center for Science and Engineering, Faculty of Science and Engineering, Waseda University, 3-4-1, Okubo, Shinjuku, Tokyo 169-8555, Japan\n- 3 Department of Physics, School of Advanced Science and Engineering, Faculty of Science and Engineering, Waseda University, 3-4-1, Okubo, Shinjuku, Tokyo 169-8555, Japan', 'ABSTRACT': "The James Webb Space Telescope (JWST) has unveiled numerous massive black holes (BHs) in faint, broad-line active galactic nuclei (AGNs). The discovery highlights the presence of dust-reddened AGN populations, referred to as 'little red dots (LRDs)', more abundant than X-ray selected AGNs, which are less influenced by obscuration. This finding indicates that the cosmic growth rate of BHs within this population does not decrease but rather increases at higher redshifts beyond z ∼ 6. The BH accretion rate density deduced from their luminosity function is remarkably higher than that from other AGN surveys in X-ray and infrared bands. To align the cumulative mass density accreted to BHs with the observed BH mass density at z ≃ 4 -5, as derived from the integration of the BH mass function, the radiative efficiency must be doubled from the canonical 10% value, achieving significance beyond the > 3 σ confidence level. This suggests the presence of rapid spins with 96% of the maximum limit among these BHs under the thin-disk approximation, maintained by prolonged mass accretion instead of chaotic accretion with randomly oriented inflows. Moreover, we derive an upper bound for the stellar mass of galaxies hosting these LRDs, ensuring consistency with galaxy formation in the standard cosmological model, where the host stellar mass is limited by the available baryonic reservoir. Our analysis gives a lower bound for the BH-to-galaxy mass ratio that exceeds the typical value known in the nearby universe and aligns with that for JWST-detected unobscured AGNs. Accordingly, we propose a hypothesis that the dense, dust-rich environments within LRDs facilitate the emergence of rapidly spinning and overmassive BH populations during the epoch of reionization. This scenario predicts a potential association between relativistic jets and other high-energy phenomena with overmassive BHs in the early universe. \nKeywords: Galaxy formation (595); High-redshift galaxies (734); Quasars (1319); Supermassive black holes (1663)", '1. INTRODUCTION': "The cosmic evolution of massive black hole (BH) populations is predominantly driven by mass accretion, powering active galactic nuclei (AGNs) (e.g., LyndenBell 1969) with a certain level of merger contributions to BHs harbored in nearby massive ellipticals (e.g., McWilliams et al. 2014; Kulier et al. 2015). Multiwavelength observations have consistently shown that AGN activity peaks around z ∼ 2 and declines toward higher redshifts (e.g., Ueda et al. 2014; Delvecchio et al. \n2014). The analysis of AGN activity offers insights into the radiative efficiency of accreting BHs by comparing it with the local mass density of relic BHs (Soltan 1982; Yu & Tremaine 2002). \nRecent observations by the James Webb Space Telescope (JWST) have revealed a new category of dustreddened, broad-line AGNs, often referred to as 'little red dots' (hereafter LRDs, Matthee et al. 2024). These AGNs are characterized by their compact morphology and moderate dust obscuration ( A V ≈ 3) in the spectra (Barro et al. 2024; Kocevski et al. 2023; Harikane et al. 2023; Labbe et al. 2023). Investigations into the AGN luminosity function of LRDs at z = 4-8 found an abundance of Φ ∼ 10 -5 -10 -4 cMpc -3 mag -1 in the ob- \nFigure 1 presents the bolometric luminosity functions at z ≃ 5 (left) and z ∼ 7 (right), i.e., the number den- \n<!-- image --> \nFigure 1. Bolometric AGN luminosity functions at 4 . 5 < z < 6 (left) and 6 . 5 < z < 8 . 5 (right). The luminosity function data obtained from different surveys are shown: the rest-UV-selected quasars (Niida et al. 2020; Matsuoka et al. 2023), the X-ray selected AGNs (Ueda et al. 2014), and dust-reddened AGNs reported as 'little red dots (LRDs)' identified with JWST photometry and slitless spectroscopy (Matthee et al. 2024; Kokorev et al. 2024; Greene et al. 2024; Akins et al. 2024). The bright end slope of the LRD luminosity function is consistent with Φ ∝ L -1 bol at both redshifts. \n<!-- image --> \nd UV absolute magnitude range of -22 ≲ M UV ≲ -18 (e.g., Kokorev et al. 2024), significantly exceeding those predicted by extrapolations from the unobscured AGN luminosity function in ground-based surveys (e.g., Niida et al. 2020). Moreover, spectroscopic analysis of LRDs, facilitating direct measurement of broad H α emissions - a tracer of AGN activity (Greene & Ho 2005) - has led to the construction of the AGN bolometric luminosity function for LRDs. This result suggests that in contrast to expectations based on AGN surveys in the pre-JWST era, the cosmic growth rate of BHs within this AGN population does not decline but appears to increase at higher redshifts ( z > 6). \nIn this Letter , compiling data from extensive highz AGN and LRD surveys, we constrain the radiative efficiency of BHs in dominant LRD populations by comparing the observed BH mass density at z ≃ 4 -5 to the mass accreted to BHs calculated from the BH growth rate, i.e., the Soltan-Paczy'nski argument at the epoch of reionization. This analysis suggests a radiative efficiency higher than the canonical 10% value and favors rapid spins of these BHs under the thin-disk approximation (e.g., Shakura & Sunyaev 1973; Novikov & Thorne 1973). Furthermore, we establish an upper limit for the stellar mass of galaxies harboring these LRDs, and give a lower bound for the BH-to-galaxy mass that exceeds the typical value known in the nearby universe and aligns with that for JWST-detected unobscured AGNs. Consequently, we propose a hypothesis that the dust-rich environments in LRDs promote the emergence of rapidly spinning and overmassive BH populations. \nThroughout this paper, we assume a flat Λ cold dark matter (CDM) cosmology consistent with the constraints from Planck (Planck Collaboration et al. 2020); h = 0 . 6732, Ω m = 0 . 3158, Ω Λ = 1 -Ω m , Ω b = 0 . 04938, and σ 8 = 0 . 8102. It is important to note that observational studies referenced in our work have adopted a different set of cosmological parameters, h = 0 . 7 and Ω m = 0 . 3. However, the differences in parameter choice have a negligible impact on the results.", '2. SOLTAN ARGUMENT AT Z ≥ 5': 'Early studies of LRDs have indicated that the observed values of M UV are much fainter than the intrinsic UV magnitudes due to dust extinction (e.g., Kocevski et al. 2023; Matthee et al. 2024; Greene et al. 2024; Kokorev et al. 2024). To address the underlying AGN activity, these studies estimated bolometric AGN luminosity from rest-optical emissions. Matthee et al. (2024) and Greene et al. (2024) conducted JWST/NIRSpec observations on LRDs and converted the measured H α luminosity to bolometric luminosity, while Kokorev et al. (2024) and Akins et al. (2024) relied on continuum luminosity L 5100 and L 3000 for a bolometric luminosity estimator, respectively, by assuming that all the continuum flux originates from the AGN. The AGN bolometric luminosity derived from the H α luminosity, particularly its broad-line component emitted from fast-moving clouds near the AGN, is more accurate than using dustdereddened continuum (see Section 4.3). \nsity per unit comoving volume per log L bol in units of cMpc -3 dex -1 , combining data from various sources: rest-UV-selected unobscured quasars (Niida et al. 2020; Matsuoka et al. 2023), X-ray selected AGNs (Ueda et al. 2014), and LRDs identified through JWST photometry and spectroscopy (Matthee et al. 2024; Kokorev et al. 2024; Greene et al. 2024; Akins et al. 2024) 1 . For UV and X-ray selected AGNs, we adopt the bolometric correction factors calibrated by Duras et al. (2020). The abundance of unobscured quasars at z ∼ 5 aligns closely with that of X-ray detected AGNs for L bol ≳ 4 × 10 45 erg s -1 , while the X-ray AGN abundance further increases at lower luminosity regimes 2 . This suggests that the obscured AGN fraction increases toward the fainter end (Vito et al. 2018). In contrast, LRDs exhibit a substantially greater abundance at L bol ≲ 10 46 erg s -1 , nearly one order of magnitude above that of X-ray AGNs. While the LRD abundance data show some variations owing to differences in sample size and methods for estimating bolometric luminosity across the referred studies, the trend of over-abundance relative to the other AGN populations is consistently observed. The bright end slope of the LRD luminosity function is consistent with Φ ∝ L -1 bol (Kokorev et al. 2024; Akins et al. 2024). Therefore, it is ensured that a large fraction of the production of radiation (i.e., the amount of material accreted to BHs) is dominated by these bright populations. The over-abundance of LRDs and the bright-end slope of Φ ∝ L -1 bol hold even at z ≳ 7. \nIn our analysis below, we consider abundance data from luminosity bins where the sample size is N ≥ 2, but exclude bins with a sample size of N = 1. This approach aligns with Poisson statistical error estimates, where a single occurrence ( N = 1) is statistically indistinguishable from zero. Additionally, for the COSMOS-Web luminosity function data, we exclude the data at the faint end of L bol < 10 46 erg s -1 from Akins et al. (2024) since the COSMOS-Web survey is not deep enough to accurately measure the abundance of these faint populations and thus completeness correction matters. \nFigure 2 illustrates the BH accretion rate density (BHAD) across various redshifts, with each data point \nFigure 2. The cosmic BH accretion rate density (BHAD) as a function of redshift. Each data point and curve represent BHADs estimated under the assumption of a 10% radiative efficiency ( ϵ rad = 0 . 1) for the three different populations, including LRDs (Matthee et al. 2024; Kokorev et al. 2024; Greene et al. 2024; Akins et al. 2024), X-ray selected AGNs including Compton-thick populations (Aird et al. 2015; Ananna et al. 2019; Pouliasis et al. 2024), and mid-infrared selected AGNs (Delvecchio et al. 2014). For comparison, the cosmic SFRD scaled by a factor of 3,000 is overlaid (Harikane et al. 2022). The BHAD attributed to LRDs remains significantly dominant at z > 6. \n<!-- image --> \nand curve representing BHADs estimated under the assumption of a 10% radiative efficiency ( ϵ rad = 0 . 1). These include LRDs (Matthee et al. 2024; Kokorev et al. 2024; Greene et al. 2024; Akins et al. 2024), as well as Xray selected AGNs (Aird et al. 2015; Ananna et al. 2019; Pouliasis et al. 2024), and mid-infrared selected AGNs (Delvecchio et al. 2014). The BHAD estimated from X-ray sources including Compton-thick AGN contributions agrees well to that of mid-infrared AGNs at z ∼ 3. With the same value of ϵ rad = 0 . 1, the BHAD inferred from the bolometric luminosity function of LRDs indicates a persistent or even increasing trend toward higher redshifts (4 < z < 8 . 5), opposite to the declining trend of X-ray selected AGNs. Note that completeness corrections for some LRD samples we adopt (e.g., Matthee et al. 2024; Kokorev et al. 2024) have not been fully implemented in estimating the abundance of faint sources, thereby potentially leading to an increased abundance at the faint sources. Nevertheless, our finding is unlikely to alter because the brighter LRD populations dominantly contribute to the BHAD (i.e., d log Φ / dlog L bol ≃ -1). Note that the BHAD estimated from the COSMOSWeb result increases by < 20% when faint AGNs with L bol < 10 46 erg s -1 are included in our analysis. \n<!-- image --> \nFigure 3. Left : Cosmic evolution of the BH mass density in a comoving volume. At z ∼ 5, the BH mass density is derived from the integration of the BHMF for LRDs (red symbols). From this point, the mass density grows toward lower redshifts following the BHAD deduced from known AGN populations with a 10% radiative efficiency (solid curve; Ueda et al. 2014) and reaches the density of relic BHs in the nearby universe (Shankar et al. 2009). At z > 5, the cumulative mass accreted to BHs during the LRD phase, ∆ ρ · ≡ BHAD × ∆ t inferred from their bolometric luminosity function over a time span ∆ t for given redshift range based on the COSMOS-Web (magenta) and the other surveys (blue), assuming a 10% radiative efficiency, substantially exceed the observed mass density at z ≃ 5 as well as the predictions from a BH growth model calibrated with UV and X-ray selected AGN luminosity function (dotted curve; Li et al. 2024a). Data for LRDs are derived from luminosity functions and BH mass estimates provided in the literature (open symbols, Matthee et al. 2024; Greene et al. 2024; Kokorev et al. 2024; Akins et al. 2024) and the mean values for each group (filled symbols). Right : Summary of the BH mass density and the cumulative mass density during the LRD phase assuming a 10% radiative efficiency. Shaded areas indicate the BH mass density ρ · at z ≃ 5 (red) and the cumulative mass density accrued during the LRD stage calculated from the COSMOS-Web (magenta) and the other surveys (blue). The total sum of ∆ ρ · over the entire redshift range in each data group is shown with a star symbol. \n<!-- image --> \nFor comparison, we overlay the cosmic star-formation rate density (SFRD) scaled by a factor of 3,000 (Harikane et al. 2022). The scaled SFRD matches well the BHAD based on the mid-infrared selected AGNs at z ≲ 3 and appears to be consistent with the BHAD of LRDs at z ≃ 5 -6. On the other hand, the BHAD attributed to LRDs remains significantly dominant at z > 6. This finding, based on the assumption of ϵ rad = 0 . 1, indicates that rapid growth of BHs at these earlier epochs established a trend of overmassive BH in terms of the M · /M ⋆ ratio, as observed in recent JWST AGN studies at z > 6 (e.g., Harikane et al. 2023; Maiolino et al. 2023; Pacucci et al. 2023). \nIn the left panel of Figure 3, we show the evolution of the BH mass density within a comoving volume throughout cosmic time. The solid curve represents the cumulative BH mass density deduced from the AGN bolometric luminosity functions (primarily X-ray selected populations) at 0 < z < 5, under an assumed 10% radiative efficiency (Ueda et al. 2014). This projection agrees closely with the observed BH mass density at z ≃ 0 (Shankar et al. 2009), concluding the plausibility of the pre-assumed radiative efficiency ( ϵ rad = 0 . 1; Soltan 1982; Yu & Tremaine 2002). Additionally, we \npresent the BH mass density directly derived from the integration of the BHMF for LRDs at z ≃ 5, estimated to be ρ · ≃ 2 . 8 +2 . 2 -1 . 2 × 10 3 M ⊙ cMpc -3 (red symbols; Matthee et al. 2024; Greene et al. 2024; Kokorev et al. 2024) 3 . This estimate remarkably aligns with expectations based on the Soltan argument assuming the conventional radiative efficiency of 10% at 0 < z < 5, further reinforcing the consistency across the three distinct physical measures. \nNext, we extend our analysis to the universe at z > 5, focusing on the cumulative mass of BHs accreted during the LRD phase, ∆ ρ · ≡ BHAD × ∆ t , calculated from their bolometric luminosity function over a redshift interval. We categorize the LRD samples into two groups: those identified through the COSMOS-Web survey (magenta, Akins et al. 2024) and LRDs from other observational programs (blue, Matthee et al. 2024; Greene et al. 2024; Kokorev et al. 2024). This classification is \nTable 1. Significance of the difference between ρ · ( z ≃ 5) and ∆ ρ · ( z ≳ 5) for different radiative efficiencies. \nNote - Column (1): Survey. Column (2): Redshift ranges. Column (3): Cumulative mass density of BHs accreted during the LRD phase (in units of M ⊙ cMpc -3 ) with a 10% radiative efficiency. Column (4)-(7): the p -value evaluated in the t -test for the null hypothesis between ρ · ( z = 5) and ∆ ρ · at z > 5 for different values of the radiative efficiency (and the corresponding BH spin parameters). Here, we consider two cases with LRD data based on the COSMOS-Web survey (Akins et al. 2024) and other LRD surveys (Matthee et al. 2024; Greene et al. 2024; Kokorev et al. 2024) for calculating the total cumulative mass. The COSMOS-Web result requires ϵ rad ≥ 0 . 2 beyond the > 3 σ confidence level, while the confidence level is ≳ 2 σ with LRD samples from other surveys. \nbased on two considerations: (1) the redshift intervals different among the samples in the literature, requiring a uniform redshift bin size for comparative analysis; and (2) the need to evaluate the impact of wide-area surveys such as COSMOS-Web on the LRD Soltan argument. As seen in the left panel of Figure 3, one can find the cumulative mass of BHs at z > 5 substantially exceeds the observed mass density at z ≃ 5 as well as the predictions from a BH growth model calibrated with UV and X-ray selected AGN luminosity functions (dotted curve; Li et al. 2024a). This discrepancy raises concerns about a potential violation of the BH mass conservation law; namely, ρ · ( z ≃ 5) ≳ ∆ ρ · ( z > 5) needs to be hold. Thus, the possible inapplicability of ϵ rad = 0 . 1 is suggested for the early universe beyond z > 5. Adjusting the radiative efficiency upwards impacts the inferred BHAD, which follows ∝ (1 -ϵ rad ) /ϵ rad . For instance, adopting the theoretical upper limit of ϵ rad = 0 . 42 for an extreme Kerr BH with a spin parameter a · = 1 (e.g., Kerr 1963; Novikov & Thorne 1973) resolves the discrepancy between the integrated BHMF values and the cumulative mass derived from the BHAD. The relationship between the radiative efficiency and BH spin is well understood for geometrically-thin accretion disks, where a thermal equilibrium is maintained through efficient radiative cooling that balances with viscous heating (Shakura & Sunyaev 1973). However, this scenario changes in low-accretion-rate states, where the disk becomes geometrically thick due to inefficient cooling (e.g., Yuan & Narayan 2014). In such cases, the radiative efficiency substantially decreases from the values in the thin-disk approximation (Inayoshi et al. 2019). This reduction in ϵ rad enlarges the discrepancy between ρ · ( z ≃ 5) and ∆ ρ · ( z > 5), rather than mitigating it. \nThe right panel of Figure 3 provides a detailed quantitative comparison of BH mass density values from several studies, using the least-squares method for fitting \nρ · and ∆ ρ · at each redshift interval. The cumulative values of ∆ ρ · across the entire redshift range are denoted by star symbols for each LRD sample (see also Table 1). For the case without the COSMOS-Web survey (blue symbols), the accreted mass density at z ∼ 5 is found to be lower than that at z ∼ 7. This difference is primarily due to the finding of Kokorev et al. (2024), where N = 9 luminous LRDs with L bol ≃ 10 47 erg s -1 were identified at 6 . 5 < z < 8 . 5 but only one was reported at 4 . 5 < z < 6 . 5 within a large sample set of LRDs from multiple survey fields. We note that luminosity function bins with a sample size of N = 1 are excluded in our analysis as a single occurrence is statistically indistinguishable from zero. Therefore, the inclusion of these luminous populations substantially influences the BHAD estimate. Using only the COSMOSWeb result (magenta symbols), we consistently observe higher values of ∆ ρ · at the two redshift ranges, owing to the wide-area survey designed to identify more luminous and rarer populations. As a result, the total sum in each case reaches as high as ∆ ρ · ≃ 3 . 0 +2 . 2 -1 . 2 × 10 4 and 6 . 6 +6 . 3 -2 . 3 × 10 4 M ⊙ cMpc -3 , respectively (star symbol). With the mean values, the cumulative mass densities during the LRD phases over 5 < z < 9 appear to be ≳ 10 times higher than the BH mass density at z ≃ 5. However, there is a concern regarding the classification of both the LRD samples of Kokorev et al. (2024) and Akins et al. (2024), where all photometrically selected LRDs are considered as AGNs due to the lack of spectroscopic observations (see also Section 4.3). Due to these concerns, the cumulative mass density of BHs and their difference from the BH mass density are considered to be upper bounds. \nTo understand the influence of each contribution of ∆ ρ · on this analysis and the need for a radiative efficiency beyond the standard 10% value, we explore two scenarios: one considering the contribution from the \nCOSMOS-Web survey (magenta) and another one compiling LRD samples from other observational programs (blue). In this work, to assess the statistical difference between ρ · ( z ≃ 5) and ∆ ρ · for each of the two cases, we employ the t -test, which serves as an appropriate statistical method to determine whether there is a statistically significant difference in the mean values between two groups with unequal sample variances. The p -values, as summarized in Table 1, indicate that the hypothesis of agreement between the two quantities at ϵ rad = 0 . 1 is statistically rejected in both the two cases with a confidence level of > 99 . 7% ( p < 0 . 003). For the analysis with the COSMOS-Web survey result, a radiative efficiency greater than ϵ rad ≥ 0 . 3 is concluded with a confidence level of > 98%, while the case with other LRD survey data suggest ϵ rad ≥ 0 . 2 with a similar confidence level. This finding suggests that the majority of BHs within LRDs or similarly gas/dust-rich environments are likely to process rapid spins with an average ϵ rad ≥ 0 . 2 -0 . 3 (the corresponding BH spin is a · ≃ 0 . 96 -0 . 996), indicating a prevalent condition of rapid angular momentum in BH growing environments in the early universe. \nThis result suggests that BH growth at these high redshifts, especially in LRDs, is likely dominated by prolonged accretion episodes with coherent angular momentum directions or a modest degree of anisotropy (e.g., Volonteri et al. 2005; Dotti et al. 2013), unlike shortlived chaotic accretion with randomly oriented inflows that tend to spin BHs down (e.g., King et al. 2008). Our conclusion on rapid spins of the early BH population will be directly testable through future gravitationalwave observations with the space-based detectors such as LISA, TianQin, and Taiji (e.g., Amaro-Seoane et al. 2023; Torres-Orjuela et al. 2024). \nIntriguingly, clustering analyses of quasars and galaxies at z ≳ 6 suggest that the duty cycle of UV-bright quasars is as low as ≲ 1% (Eilers et al. 2024; Pizzati et al. 2024), corresponding to a quasar lifetime of 1 -10 Myr, which is significantly shorter than the e -folding time assuming the Eddington accretion rate. This finding implies that most of the BH mass growth would have occurred in highly (UV-)obscured environments and/or through episodic super-Eddington phases with a lower radiative efficiency (Davies et al. 2019; see also Inayoshi et al. 2016, 2022b). The first implication is consistent with the hypothesis that LRDs are moderately obscured AGNs (e.g., Li et al. 2024b). The second implication, concerning the potential mass contribution from radiatively inefficient super-Eddington growth, would be constrained by our findings in this work and is left for future investigations. \nFigure 4. Stellar mass density in galaxies hosting LRDs at various redshifts, calculated using Eq. (1) and assuming F ( ≡ f IMF f L ) = 1 . 0 (open symbols) at two redshift ranges of 4 . 5 < z < 6 . 5 and 6 . 5 < z < 8 . 5. For comparison, the stellar mass function derived from the DM halo mass function at 5 ≤ z ≤ 8 is shown with a 100% star formation efficiency. An upper bound of the stellar mass constrains F < 0 . 3 f ⋆ for 4 . 5 < z < 6 . 5 and F < 0 . 04 f ⋆ for 6 . 5 < z < 8 . 5 (filled symbols). \n<!-- image -->', '3. POTENTIAL OVERMASSIVE BH TRENDS IN LITTLE RED DOTS': "In this section, we explore the possibility that BHs within LRDs are overmassive relative to the mass correlation with their host mass, as implied from the BHADto-SFRD ratio shown in Figure 2. In general, estimating the stellar mass of dust-obscured sources poses a significant challenge in the absence of rest-frame near-infrared data provided by JWST MIRI (e.g., Williams et al. 2024; P'erez-Gonz'alez et al. 2024). Instead of examining the detailed spectral energy distribution fitting analysis, we focus on putting an upper bound for the stellar mass. This approach ensures that the observed abundance of LRDs does not exceed the theoretical upper bound in the standard ΛCDM model with a 100% conversion efficiency from gas to stars (e.g., Boylan-Kolchin 2023). \nThe stellar continuum for LRDs can be constrained by the dust-corrected continuum flux at 5100 ˚ A. Given that broad H α emission indicates AGN dominance in the continuum (see Section 4.3), we adjust L ⋆, 5100 = f L L 5100 , where f L is significantly less than unity. To estimate an upper bound of stellar mass, we employ the STARBURST99 population synthesis code (version 7.0.1; Leitherer et al. 1999), adopting a Kroupa IMF (Kroupa 2001; 0 . 1 -100 M ⊙ ), Padova isochrone models, constant star formation, and solar metallicity. This approach \nFigure 5. M · -M ⋆ distribution for high-redshift AGNs, including LRDs, JWST-detected unobscured ANGs at z = 4 -8 (purple, Maiolino et al. 2023; green, Harikane et al. 2023; cyan, Stone et al. 2024; and blue, Ding et al. 2023), and quasars identified in ground-based surveys (Izumi et al. 2021). For the LRDs at 4 . 5 < z < 6 . 5 (red) and 6 . 5 < z < 8 . 5 (orange), we derive the upper bound of the stellar mass based on the dust-corrected continuum flux measured by Greene et al. (2024) using Eqs. (1) and (2). Additionally, a z = 8 . 5 LRD with broad H β emission, for which the stellar mass is constrained by ALMA non-detections, is overlaid (blue, Kokorev et al. 2023). Two different mass correlations are overlaid: the local relationship (solid, Kormendy & Ho 2013) and the JWST-detected AGNs (dashed, Pacucci et al. 2023). \n<!-- image --> \nyields a galaxy mass -luminosity relation: \nM ⋆ 10 9 M ⊙ ≃ 1 . 3 f IMF ( L ⋆, 5100 10 43 erg s -1 )( t age 1 Gyr ) , (1) \nwhich is applicable for stellar ages of t age ∼ 0 . 3 -3 Gyr. This estimate is sensitive to the low-mass end of the stellar IMF ( f IMF = 1 for m ⋆, min = 0 . 1 M ⊙ ); for instance, setting the minimum mass up to m ⋆, min = 1 . 0 M ⊙ decreases the factor to f IMF ∼ 0 . 3. \nFigure 4 presents the stellar mass function (in units of M ⊙ cMpc -3 ) in galaxies hosting LRDs at various redshifts, calculated by using Eq. (1) and assuming F ( ≡ f IMF f L ) = 1 . 0 at two redshift ranges of 4 . 5 < z < 6 . 5 and 6 . 5 < z < 8 . 5 (open circles), based on the LRD luminosity function obtained by Kokorev et al. (2024). Since the luminosity function of LRDs follows Φ ∝ L -1 bol , the stellar mass density becomes flatter at the high-mass end when the stellar mass is translated from the luminosity with Eq. (1). We compare the results to the stellar mass function derived from the halo mass function at \n5 ≤ z ≤ 8, assuming M ⋆ = f ⋆ f b M h , where f b = 0 . 16 is the cosmic baryon fraction and f ⋆ is the star formation efficiency (see more details in Inayoshi et al. 2022a). Setting f ⋆ = 1 . 0 offers a theoretical upper limit on stellar mass in galaxies ( f ⋆ > 1; the forbidden region), highlighting a mismatch between the stellar mass density contained in LRDs with F = 1 . 0 and the ΛCDM upper limit. As a result, we deduce a stringent constraint denoted with filled circles, \nF < 0 . 3 f ⋆ at 4 . 5 < z < 6 . 5 , F < 0 . 04 f ⋆ at 6 . 5 < z < 8 . 5 . (2) \nFigure 5 shows the M · -M ⋆ distribution for high-redshift AGNs, including LRDs (square), JWSTdetected unobscured AGNs (cross), and quasars identified in ground-based surveys (circle). For the LRDs at 4 . 5 < z < 6 . 5 (red) and 6 . 5 < z < 8 . 5 (orange), we derive the stellar mass of those with broad H α emission from Greene et al. (2024), using Eq. (1). This calculation incorporates the upper limit for the stellar continuum ratio F to the observed continuum (see Eq. 2 and Figure 4), ensuring consistency with the theoretical upper bound of stellar mass density in the ΛCDM universe. The M · -M ⋆ values for LRDs tend to be overmassive compared to the local relationship (solid line; Kormendy & Ho 2013), and aligns well with other AGNs detected by JWST and follows the mass correlation inferred from JWST AGN data, excluding quasars from ground-based surveys (dashed line; Pacucci et al. 2023). Moreover, their distribution is consistent with the locus of a JWST/NIRSpec confirmed z = 8 . 5 LRD that exhibits broad H β emission, for which an upper limit on the stellar mass has been constrained by non-detection in ALMA observations (blue; Kokorev et al. 2023). \nThe mass ratios for these LRDs are also consistent with the BHAD/SFRD values at z > 6 shown in Figure 2. This suggests a model in which transient rapid growth phases during the LRD stages elevate these BHs into an overmassive state. This hypothesis is supported both theoretically (e.g., Inayoshi et al. 2022b; Hu et al. 2022) and observationally (Fujimoto et al. 2022), providing a comprehensive insight into the BH growth dynamics in the early universe.", '4.1. Missing X-ray radiation from LRDs': 'X-ray AGN surveys are generally effective in identifying obscured AGNs. However, in the case of LRDs observed with JWST, no X-ray counterparts have been reported in the early studies (e.g., Kocevski et al. 2023; Furtak et al. 2023; Matthee et al. 2024). X-ray weak- \nness has been consistently observed in LRDs as demonstrated by stacking analyses (Yue et al. 2024), and this phenomenon extends beyond LRDs to a more general category of unobscured broad-line AGNs (Maiolino et al. 2024). Further emphasizing the rarity of X-ray emissions, Kocevski et al. (2024) have identified only two X-ray detected LRDs at z = 3 . 1 and 4 . 66 among 341 examined objects, resulting in a detection fraction of less than 0 . 6 %. The optical continuum extinction measurements suggest a gas column density of N H ∼ 3 . 3 × 10 22 ( A V / 3 . 0) cm -2 (Maiolino et al. 2001), indicating that the column density outside the broadline region is too low to obscure X-rays. The column density estimate is broadly consistent with those measured from the X-ray spectral analysis for the two X-ray detected LRDs, N H ∼ (5 -20) × 10 22 cm -2 (Kocevski et al. 2024). \nConsidering the absence of X-ray counterparts for LRDs, we explore the possibility that their X-ray emission is intrinsically weak, as compared to typical Xray selected AGNs. Figure 6 presents the critical Xray luminosity for each bolometric luminosity, so that Φ X ( L X , crit ) ≥ Φ LRD ( L bol ), where we adopt the X-ray AGN luminosity function Φ X from Ueda et al. (2014) and the LRD bolometric luminosity function Φ LRD from Kokorev et al. (2024), respectively. This condition requires that LRDs must have a bolometric correction factor to X-rays that prevents them from being classified as X-ray AGNs and contributing to their abundance (red horizontal lines with arrows). The critical X-ray luminosity is limited below those derived from comparison between the X-ray and optical-based AGN luminosity functions at lower redshifts of z ≲ 2 (Ueda et al. 2003). \nTo quantify this intrinsic X-ray faintness in LRDs, we utilize the optical to X-ray spectral index, defined as α OX = log( L ν, 2keV /L ν, 2500 ) / log( ν 2keV /ν 2500 ), where L ν, 2keV and L ν, 2500 are the extinction corrected luminosity density at 2 keV and 2500 ˚ A. Our findings support a constant value of α OX lower than -1 . 8, rather than the luminosity-dependent α OX values observed in unobscured quasars across 0 ≲ z ≲ 6 (e.g., Steffen et al. 2006; Duras et al. 2020). The upper bound of α OX ≃ -1 . 8 is seen in the most luminous quasar populations with L bol ≳ 10 47 erg s -1 (Nanni et al. 2017), which does not apply to most LRDs studied in this paper. \nThe extensive LRD samples by Kokorev et al. (2024), consisting of 260 dust-reddened AGN candidates, are compiled from deep JWST/NIRCam fields totaling ∼ 340 arcmin 2 . This includes observations from the CEERS field, which spans ∼ 51 . 9 arcmin 2 and falls within the coverage area of the Chandra AEGIS-XD survey coverage (Nandra et al. 2015). The survey detection \nFigure 6. The critical X-ray luminosities for LRDs at z ∼ 5, determined by the requirement for LRDs not to be classified as X-ray AGNs nor contributing to the abundance of X-ray AGNs (red horizontal lines with arrows). For comparison, several models for bolometric correction to X-rays are shown (Ueda et al. 2003; Nanni et al. 2017; Duras et al. 2020), as well as the cases with constant optical to X-ray spectral indices of α OX = -1 . 5 (dashed) and -1 . 8 (solid). The L bol -L X values of the two LRDs detected in X-rays are shown with orange symbols; JADES 21925 (square) and PRIMER-COS 3982 (diamond) (Kocevski et al. 2024). The shaded area denotes X-ray luminosities below the detection threshold of current Chandra observations for JWST fields. \n<!-- image --> \nlimit reaches 1 . 5 × 10 -16 erg s -2 cm -1 at 0.5-10 keV, which corresponds to L X ≃ 1 . 1 × 10 43 erg s -1 at the restframe 2-10 keV for z ∼ 5 -7 sources assuming a photon index of Γ = -1 . 7 and a Compton-thin limit ( N H < 10 24 cm -2 ). Given the average surface density of these LRDs ( ≃ 0 . 77 arcmin -2 ), ∼ 40 LRDs in the CEERS field show no X-ray detection above this threshold. This suggests that non-detection of X-rays among LRDs can be explained by the intrinsic faintness of X-rays as shown in Figure 6. Nevertheless, the most luminous LRDs with L bol ≳ 10 47 erg s -1 might still be observed in X-rays unless classified as Compton-thick AGNs. Additionally, we note that the two LRDs detected in X-rays with a modest hydrogen column density of N H ∼ 10 23 cm -2 , show obscuration-corrected X-ray luminosities of L X ≃ 5 . 4 × 10 43 erg s -1 (for JADES 21925 at z photo = 3 . 1) L X ≃ 5 . 0 × 10 44 erg s -1 (for PRIMER-COS 3982 at z spec = 4 . 66), respectively (Kocevski et al. 2024). The bolometric luminosities calculated from the rest-optical fluxes are L bol ≃ 6 . 5 × 10 46 erg s -1 and 3 . 1 × 10 47 erg s -1 after dust attenuation correction. These findings also suggest a lower value of α OX ≃ -1 . 8 as shown in Figure 6 (orange symbols). \nAlternatively, if the X-ray emission was not intrinsically faint (except the two X-ray detected LRDs), most \nof these LRDs would be embedded in Compton-thick gas with N H ≫ 10 24 cm -2 , concealing the LRDs from deep X-ray observations. However, such high densities are not expected in the LRD rest-optical spectra, which show a modest extinction A V ∼ 3 mag, equivalent to N H ∼ 3 × 10 22 cm -2 . This discrepancy is also consistent with the lack of expected absorption features in their NIRSpec rest-frame UV spectra of LRDs with broademission lines (e.g., Greene et al. 2024).', '4.2. Dusty young starburst galaxies mimicking LRD-like AGNs?': 'The classification of LRDs as AGNs has relied on the detection of broad H α emission. However, such highvelocity gas can also originate from stellar processes, such as stellar winds or supernova explosions. WolfRayet (WR) galaxies, characterized by young and massive stellar populations, can exhibit broad H α emission as well as other high-ionization lines (e.g., Ho et al. 1995; Schaerer et al. 1999). Therefore, to conclusively confirm the AGN nature of LRDs, it is essential to perform line diagnostics that extend beyond simply identifying broad H α emission, e.g., detection of broad He II λ 4686 emission, a signature frequently associated with WR galaxies (e.g., NGC 4214 discussed in Sargent & Filippenko 1991). \nSpectroscopic observations provide valuable insights into the characteristics of emission lines in LRDs. Initial studies by Kocevski et al. (2023) and Greene et al. (2024) have found that the He II λ 4686 line, commonly associated with WR stellar activity, is absent in these LRD sources. Additionally, the spectral signatures typically linked to WR stars have not been observed within the LRD samples. The lack of He II and other WR indicative spectral lines suggests that the broad H α emissions detected in LRDs are not of stellar origin (note that detection of He II λ 4686 does not necessarily exclude the AGN possibility because the emission is also observed in AGNs). This finding supports the hypothesis that AGNs are responsible for these emissions. More detailed spectroscopic analyses of multiple emission lines would strength this conclusion (Greene et al. 2020; Reines 2022).', '4.3. Bolometric luminosity estimates': 'Our study is motivated by intriguing discoveries on the high abundance of LRDs, but an important consideration about the AGN luminosity estimate needs to be noted. While the UV luminosity functions show similar shapes across different studies, significant variations in the bolometric luminosity function arise due to the different methods used for bolometric luminosity estimates \n(see Figure 1). Since the observed rest-frame UV flux is heavily attenuated, earlier studies have employed either the rest-frame continuum flux at 5100 ˚ A (Kokorev et al. 2024), or the direct H α emission luminosity, when available, as a proxy for the AGN bolometric luminosity (Matthee et al. 2024; Greene et al. 2024). \nThe approach that relies on continuum flux introduces uncertainties of determining the AGN contribution to the total flux, which might result in overestimated luminosity. Furthermore, the selection of LRDs based solely on photometric criteria might include non-AGN sources such as Galactic brown dwarfs (Langeroodi & Hjorth 2023), thereby possibly overestimating the AGN abundance. Greene et al. (2024) reported the identification success rate of AGNs among LRDs in the UNCOVER field to be approximately 60%. Kokorev et al. (2024) studied LRDs based on the photometric data from multiple JWST survey areas, using color selection conditions provided by Greene et al. (2024), which effectively remove contaminants of Galactic brown dwarfs. To date, other types of low-z interlopers, such as Balmer break galaxies, for LRDs have not been reported via spectroscopic studies. \nIn contrast, spectroscopic data particularly with measurements of broad H α emission facilitate confirmation of the AGN presence and a more accurate determination of L bol . This method adopts an empirical relationship derived from local AGN observations (Greene & Ho 2005). For LRD sources with detected H α emissions, the continuum fluxes at 5100 ˚ A calculated through the two methods yield ratios of L 5100 , H α /L 5100 , c ≃ 1 . 2 ± 0 . 2 (Greene et al. 2024). This result supports the scenario that the continuum emission at 5100 ˚ A is substantially AGN-origin, not from dust-reddened stellar continuum of the host galaxy (i.e., f L ≪ 1). \nOne limitation in the work by Greene et al. (2024) is the use of low-resolution PRISM spectra for analyzing H α emissions, which complicates the decomposition of broad and narrow H α emissions in some cases. To address this challenge, we can use an empirical relationship obtained through higher-resolution spectroscopic observations by Matthee et al. (2024). This relationship examines the flux ratio between narrow and broad H α emissions in LRDs; namely, the ratio F H α, broad /F H α, tot exceeds 0.6 with a positive rest-frame optical spectral index, with the ratio approaching unity as the spectral index increases. Therefore, the potential systematic errors in estimating broad H α line luminosity due to incomplete spectral line decomposition could be alleviated by employing this relationship. \nDespite these complexities found in current analyses, the central conclusion of our discussion remains valid \non a qualitative level. Nevertheless, the development of more quantitative arguments will benefit from further observational explorations that refine the understanding of the AGN characteristics of LRDs and provide a more precise estimate of their cosmic abundance (e.g., Li et al. 2024b).', '4.4. Multi-messenger Counterparts': 'In this study, we propose a scenario where the dustrich environments within LRDs lead to the emergence of rapidly spinning and overmassive BH. The high spins of these BHs may yield a strong correlation between the presence of relativistic jets (e.g., Blandford & Znajek 1977), high-energy emissions and particles (e.g., Dai & Fang 2017; Murase et al. 2020), and transient bursts such as stellar tidal disruption events (e.g., Inayoshi et al. 2024) with early BHs being overmassive compared to the mass correlation observed in the nearby universe. \nMulti-wavelength surveys including radio, optical, and X-ray bands have reported that the radio powers (or radio loudness) of obscured AGNs initially identified by the VLA/FIRST survey as bright radio sources tend to increase with redshifts at 0 . 5 < z < 3 . 5, with jet powers exceeding P jet ≳ 10 46 erg s -1 (Ichikawa et al. 2021, 2023). The inferred jet production efficiency calculated from η jet ∼ ϵ rad P jet /L bol approaches unity, implying rapid spins of these nuclear BHs (e.g., Tchekhovskoy et al. 2011). These types of radio AGNs at intermediate redshifts may offer valuable insights into understanding the characteristics of LRDs at higher redshifts, thus further supporting our hypothesis of rapid BH spins.', '4.5. Implications to BH growth mechanisms at z ≳ 5': 'The spin of a massive BH is influenced by a range of physical processes including BH mergers and mass accretion. The coalescence of two non-spinning BHs results in a remnant with a significant spin, approximately a · ≃ 0 . 69 for an equal-mass merger (Berti et al. 2007), but the spin diminishes if the merging BHs have non-zero, misaligned spins relative to the orbital angular momentum (e.g., Berti & Volonteri 2008). Gas accreting onto BHs through a disk aligned with the angular momentum direction of the BH is expected to enhance the spin during mass accumulation (Bardeen et al. 1972; Thorne 1974). However, chaotic accretion characterized by short-lived episodes with random orientations tends to dampen the BH spin toward average values of a · ≃ 0 . 2 (e.g., King et al. 2008). \nChaotic accretion and related feeding mechanisms with low angular momentum gas are considered to promote the efficiency of BH mass growth in the early uni- \nverse, due to moderate centrifugal support (Eisenstein & Loeb 1995) and a lower radiative efficiency as a result of spin down (King et al. 2008). However, we show that the majority of the early BH populations growing in dust-rich environments tend to exhibit high spins with a · ≃ 0 . 9, corresponding to ϵ rad ≳ 0 . 2. This suggests that BH growth is likely dominated by prolonged accretion episodes with coherent angular momentum, consistent with an idea pioneered by Volonteri et al. (2005), or a modest degree of anisotropy (Dotti et al. 2013; Dubois et al. 2014). Cosmological simulations focused on galaxy assembly suggest that BHs retain high spins through coherent accretion modes. This occurs once BHs are settled down to the centers of their host galaxies and then the BH spin direction is well aligned with the angular momentum of the hosts (Peirani et al. 2024). As a consequence, a significant fraction of AGNs are expected to launch radio jets and influence the BH-galaxy coevolution (e.g., Beckmann et al. 2024, see also Section 4.4). \nOur conclusion regarding the rapid spins of BHs can be directly testable through future gravitational-wave observations with the space-based detectors such as LISA (e.g., Amaro-Seoane et al. 2023). Moreover, if these BHs frequently merge during galaxy coalescences leading to the LRD phase or similar activities observed in ultra-luminous infrared galaxies, such events could significantly contribute to a stochastic gravitationalwave background, detectable by pulsar-timing array experiments (Inayoshi et al. 2018; see also Agazie et al. 2023). Therefore, further exploration of the rapidly spinning, overmassive BHs in LRDs is needed in multiple aspects.', 'ACKNOWLEDGMENTS': "We greatly thank Luis C. Ho for emphasizing the significance of the Soltan-Paczy'nski argument and for dedicating time to an inspiring discussion during the spring festival holiday. We also thank Dale D. Kocevski for sharing their SED fit results for the two LRDs detected in X-rays, which are used in Figure 6. We also wish to thank Xiaoyang Chen, Anna-Christina Eilers, Joseph F. Hennawi, Haojie Hu, Kazumi Kashiyama, Vasily Kokorev, Masafusa Onoue, Elia Pizzati, and Yasushi Suto for constructive discussions. K. Inayoshi acknowledges support from the National Natural Science Foundation of China (12073003, 12003003, 11721303, 11991052, 11950410493), and the China Manned Space Project (CMS-CSST-2021-A04 and CMS-CSST-2021A06). This work is also supported by Japan Society for the Promotion of Science (JSPS) KAKENHI (20H01939; K. Ichikawa)."}

2023ApJS..267...44A

The eighteenth data release DR18 of the Sloan Digital Sky Survey SDSS is the first one for SDSSV the fifth generation of the survey. SDSSV comprises three primary scientific programs or Mappers the Milky Way Mapper MWM the Black Hole Mapper BHM and the Local Volume Mapper. This data release contains extensive targeting information for the two multiobject spectroscopy programs MWM and BHM including input catalogs and selection functions for their numerous scientific objectives. We describe the production of the targeting databases and their calibration and scientifically focused components. DR18 also includes 25000 new SDSS spectra and supplemental information for Xray sources identified by eROSITA in its eFEDS field. We present updates to some of the SDSS software pipelines and preview changes anticipated for DR19. We also describe three valueadded catalogs VACs based on SDSSIV data that have been published since DR17 and one VAC based on the SDSSV data in the eFEDS field.

2023-08-01T00:00:00Z

['2023ApJS..267...44A', 'arXiv:2301.07688', '10.48550/arXiv.2301.07688', '2023arXiv230107688A', '10.3847/1538-4365/acda98']

['Surveys', 'Astronomy databases', 'Astronomy data acquisition', 'Astronomy software', '1671', '83', '1860', '1855', 'Astrophysics - Astrophysics of Galaxies', 'Astrophysics - Cosmology and Nongalactic Astrophysics', 'Astrophysics - High Energy Astrophysical Phenomena']

The Eighteenth Data Release of the Sloan Digital Sky Surveys Targeting and First Spectra from SDSSV

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']

https://arxiv.org/pdf/2301.07688.pdf

{'The Eighteenth Data Release of the Sloan Digital Sky Surveys: Targeting and First Spectra from SDSS-V': "Andr'es Almeida, 1 Scott F. Anderson, 2 Maria Argudo-Fern'andez, 3, 4, 5 Carles Badenes, 6 Kat Barger, 7 Jorge K. Barrera-Ballesteros, 8 Chad F. Bender, 9 Erika Benitez, 8 Felipe Besser, 10 Jonathan C. Bird , 11 Dmitry Bizyaev, 12, 13 Michael R. Blanton, 14 John Bochanski, 15 Jo Bovy, 16, 17 William Nielsen Brandt, 18, 19, 20 Joel R. Brownstein , 21 Johannes Buchner, 22 Esra Bulbul, 22 Joseph N. Burchett, 13 Mariana Cano D'ıaz, 8 Joleen K. Carlberg , 23 Andrew R. Casey, 24, 25 Vedant Chandra , 26 Brian Cherinka, 23 Cristina Chiappini, 27 Abigail A. Coker , 21 Johan Comparat, 22 Charlie Conroy, 26 Gabriella Contardo, 28 Arlin Cortes, 10 Kevin Covey, 29 Jeffrey D. Crane, 30 Katia Cunha, 9 Collin Dabbieri, 11 James W. Davidson Jr., 1 Megan C. Davis, 31 Nathan De Lee , 32 Jos'e Eduardo M'endez Delgado, 33 Sebastian Demasi, 2 Francesco Di Mille, 10 John Donor, 7 Peter Dow, 1 Tom Dwelly, 22 Mike Eracleous , 18, 19 Jamey Eriksen, 12, 13 Xiaohui Fan, 9 Emily Farr, 34 Sara Frederick , 11 Logan Fries , 31 Peter Frinchaboy, 7 Boris T. Gansicke , 35 Junqiang Ge, 36 Consuelo Gonz'alez ' Avila, 10 Katie Grabowski, 12, 13 Catherine Grier, 37 Guillaume Guiglion, 38 Pramod Gupta, 2 Patrick Hall, 39 Keith Hawkins, 40 Christian R. Hayes , 41 J. J. Hermes, 42 Lorena Hern'andez-Garc'ıa, 43, 44 David W. Hogg, 14 Jon A. Holtzman , 13 Hector Javier Ibarra-Medel, 45, 46 Alexander Ji, 47, 48 Paula Jofre, 49, 50 Jennifer A. Johnson, 51, 52 Amy M. Jones, 23 Karen Kinemuchi, 12, 13 Matthias Kluge, 22 Anton Koekemoer, 23 Juna A. Kollmeier, 30, 53 Marina Kounkel, 11 Dhanesh Krishnarao, 54 Mirko Krumpe, 27 Ivan Lacerna, 46, 43 Paulo Jakson Assuncao Lago, 10 Chervin Laporte, 55 Chao Liu, 36 Ang Liu, 22 Xin Liu, 45, 56 Alexandre Roman Lopes, 57 Matin Macktoobian , 58 Steven R. Majewski, 1 Viktor Malanushenko, 12, 13 Dan Maoz, 59 Thomas Masseron, 60, 61 Karen L. Masters, 62 Gal Matijevic, 27 Aidan McBride , 21 Ilija Medan, 63 Andrea Merloni, 22 Sean Morrison, 45 Natalie Myers, 7 Szabolcs M'esz'aros, 64, 65 C. Alenka Negrete, 8 David L. Nidever, 66 Christian Nitschelm , 67 Daniel Oravetz, 12, 13 Audrey Oravetz, 12, 13 Kaike Pan, 12, 13 Yingjie Peng, 68, 69 Marc H. Pinsonneault, 51 Rick Pogge , 70 Dan Qiu, 36 Anna Barbara de Andrade Queiroz, 27 Solange V. Ramirez, 30 Hans-Walter Rix, 38 Daniela Fern'andez Rosso, 10 Jessie Runnoe, 11 Mara Salvato, 22 Sebastian F. Sanchez, 8 Felipe A. Santana, 71 Andrew Saydjari , 26 Conor Sayres, 2 Kevin C. Schlaufman , 72 Donald P. Schneider, 18, 19 Axel Schwope, 27 Javier Serna, 73 Yue Shen, 45 Jennifer Sobeck , 74 Ying-Yi Song, 16, 17 Diogo Souto, 75 Taylor Spoo, 7 Keivan G. Stassun , 11 Matthias Steinmetz, 27 Ilya Straumit, 51, 76, 59 Guy Stringfellow, 77 Jos'e S'anchez-Gallego, 2 Manuchehr Taghizadeh-Popp, 72 Jamie Tayar, 78 Ani Thakar, 72 Patricia B. Tissera, 79 Andrew Tkachenko, 76 Hector Hernandez Toledo, 8 Benny Trakhtenbrot , 59 Jos'e G. Fern'andez-Trincado, 80 Nicholas Troup, 81 Jonathan R. Trump, 31 Sarah Tuttle, 2 Natalie Ulloa, 10 Jose Antonio Vazquez-Mata, 8, 82 Pablo Vera Alfaro, 10 Sandro Villanova, 83 Stefanie Wachter, 30 Anne-Marie Weijmans, 84 Adam Wheeler, 51 John Wilson, 1 Leigh Wojno, 38 Julien Wolf, 22, 85 Xiang-Xiang Xue, 36 Jason E. Ybarra, 86, 87 Eleonora Zari, 38 and \n21 \nGail Zasowski \n1 Department of Astronomy, University of Virginia, Charlottesville, VA 22904-4325, USA 2 Department of Astronomy, University of Washington, Box 351580, Seattle, WA 98195, USA 3 Instituto de F'ısica, Pontificia Universidad Cat'olica de Valpara'ıso, Valpara'ıso, Chile 4 Universidad de Granada (UGR), Departamento de F'ısica Te'orica y del Cosmos, 18071, Granada, Spain 5 Instituto Universitario Carlos I de F'ısica Te'orica y Computacional, Universidad de Granada, 18071, Granada, Spain 6 PITT PACC, Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, USA 7 Department of Physics & Astronomy, Texas Christian University, Fort Worth, TX 76129, USA 8 Instituto de Astronom'ıa, Universidad Nacional Aut'onoma de M'exico, A.P. 70-264, 04510, Mexico, D.F., M'exico 9 Steward Observatory, University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721-0065, USA 10 Las Campanas Observatory, Ra'ul Bitr'an 1200, La Serena, Chile 11 Department of Physics and Astronomy, Vanderbilt University, VU Station 1807, Nashville, TN 37235, USA 12 Apache Point Observatory, P.O. Box 59, Sunspot, NM 88349 13 \nDepartment of Astronomy, New Mexico State University, Las Cruces, NM 88003, USA \ngail.zasowski@gmail.com \n- 14 Center for Cosmology and Particle Physics, Department of Physics, 726 Broadway, Room 1005, New York University, New York, NY 10003, USA \n15 \nRider University, 2083 Lawrenceville Road, Lawrenceville, NJ 08648, USA \n- 16 David A. Dunlap Department of Astronomy & Astrophysics, University of Toronto, 50 St. George Street, Toronto, Ontario M5S 3H4, Canada \n17 \nDunlap Institute for Astronomy & Astrophysics, University of Toronto, 50 St. George Street, Toronto, Ontario M5S 3H4, Canada \n18 \nDepartment of Astronomy & Astrophysics, The Pennsylvania State University, University Park, PA 16802, USA \n19 \n21 \n24 \nInstitute for Gravitation and the Cosmos, The Pennsylvania State University, University Park, PA 16802, USA \n20 \nDepartment of Physics, The Pennsylvania State University, University Park, PA 16802, USA \nDepartment of Physics and Astronomy, University of Utah, 115 S. 1400 E., Salt Lake City, UT 84112, USA \n22 \nMax-Planck-Institut fur extraterrestrische Physik, Giessenbachstraße 1, 85748 Garching, Germany \n23 \nSpace Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA \nSchool of Physics & Astronomy, Monash University, Wellington Road, Clayton, Victoria 3800, Australia \n25 \nCenter of Excellence for Astrophysics in Three Dimensions (ASTRO-3D), Australia \nCenter for Astrophysics \n| \nHarvard & Smithsonian, 60 Garden St, Cambridge, MA 02138, USA \n26 \n- 27 Leibniz-Institut fur Astrophysik Potsdam (AIP), An der Sternwarte 16, D-14482 Potsdam, Germany 28 SISSA, Scuola Internazionale Superiore di Studi Avanzati\n- 29 Department of Physics and Astronomy, Western Washington University, 516 High Street, Bellingham, WA 98225, USA 30 The Observatories of the Carnegie Institution for Science, 813 Santa Barbara Street, Pasadena, CA 91101, USA 31 Department of Physics, University of Connecticut, 2152 Hillside Road, Unit 3046, Storrs, CT 06269, USA\n- 32 Department of Physics, Geology, and Engineering Technology, Northern Kentucky University, Highland Heights, KY 41099 33 Astronomisches Rechen-Institut, Zentrum fur Astronomie der Universitat Heidelberg, Monchhofstr. 12-14, D-69120 Heidelberg, Germany\n- 34 Laboratory for Atmospheric and Space Physics, University of Colorado, 1234 Innovation Drive, Boulder, CO 80303, USA 35 Department of Physics, University of Warwick, Coventry CV4 7AL, UK\n- 36 National Astronomical Observatories, Chinese Academy of Sciences, 20A Datun Road, Chaoyang, Beijing 100101, China 37 Department of Astronomy, University of Wisconsin-Madison, 475N. Charter St., Madison WI 53703, USA 38 Max-Planck-Institut fur Astronomie, Konigstuhl 17, D-69117 Heidelberg, Germany \n39 \nDepartment of Physics and Astronomy, York University, 4700 Keele St., Toronto, Ontario M3J 1P3, Canada \n40 \nDepartment of Astronomy, University of Texas at Austin, Austin, TX 78712, USA \nNRC Herzberg Astronomy and Astrophysics Research Centre, 5071 West Saanich Road, Victoria, B.C., Canada, V9E 2E7 \n42 \nAstronomy Department, Boston University, 725 Commonwealth Ave, Boston, MA 02215, USA \nMillennium Institute of Astrophysics (MAS), Nuncio Monse˜nor S'otero Sanz 100, Providencia, Santiago, Chile \n43 \nInstituto de F'ısica y Astronom'ıa, Universidad de Valpara'ıso, Av. Gran Breta˜na 1111, Playa Ancha, Casilla 5030, Chile \n45 \nDepartment of Astronomy, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA \nInstituto de Astronom'ıa y Ciencias Planetarias, Universidad de Atacama, Copayapu 485, Copiap'o, Chile \n47 \nDepartment of Astronomy and Astrophysics, University of Chicago, Chicago, IL 60637, USA \n48 \nKavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA \n- 49 N'ucleo de Astronom'ıa de la Facultad de Ingenier'ıa y Ciencias, Universidad Diego Portales, Av. Ej'ercito Libertador 441, Santiago, Chile \n50 Millenium Nucleus ERIS \n51 Department of Astronomy, The Ohio State University, 140 W. 18th Ave., Columbus, OH 43210, USA 52 Center for Cosmology and AstroParticle Physics, Ohio State University, 191 West Woodruff Ave, Columbus, OH, 43210, USA 53 Canadian Institute for Theoretical Astrophysics, University of Toronto, Toronto, ON M5S-98H, Canada 54 Department of Physics, Colorado College, 14 East Cache la Poudre St., Colorado Springs, CO, 80903, USA 55 Institut de Ci'encies del Cosmos, Universitat de Barcelona, Mart'ı Franqu'es 1, 08028 Barcelona, Spain 56 National Center for Supercomputing Applications, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA 57 Departamento de F'ısica, Facultad de Ciencias, Universidad de La Serena, Cisternas 1200, La Serena, Chile 58 Electrical and Computer Engineering Department, University of Alberta, Edmonton, AB, Canada 59 School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel 60 nstituto de Astrof'ısica de Canarias, 38205 La Laguna, Tenerife, Spain 61 Departamento de Astrof'ısica, Universidad de La Laguna, 38206 La Laguna, Tenerife, Spain 62 Departments of Physics and Astronomy, Haverford College, 370 Lancaster Avenue, Haverford, PA 19041, USA 63 Department of Physics and Astronomy, Georgia State University, Atlanta, GA 30302, USA 64 ELTE Gothard Astrophysical Observatory, H-9704 Szombathely, Szent Imre herceg st. 112, Hungary 65 MTA-ELTE Lendulet 'Momentum' Milky Way Research Group, Hungary 66 Department of Physics, Montana State University, P.O. Box 173840, Bozeman, MT 59717, USA \n41 \n44 \n46 \n- 67 Centro de Astronom'ıa (CITEVA), Universidad de Antofagasta, Avenida Angamos 601, Antofagasta 1270300, Chile 68 Department of Astronomy, School of Physics, Peking University, Beijing 100871, China \n69 Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, China \n70 Department of Astronomy and Center for Cosmology and AstroParticle Physics, The Ohio State University, 140 W. 18th Ave, Columbus, OH, 43210, USA \n- 71 Departamento de Astronom'ıa, Universidad de Chile, Av. Libertador Bernardo O'Higgins 1058, Santiago de Chile \n72 William H. Miller III Department of Physics and Astronomy, Johns Hopkins University, 3400 N Charles Street, Baltimore, MD 21218, USA \n73 Instituto de Astronom'ıa, Universidad Nacional Aut'onoma de M'exico, Ensenada, Baja California, M'exico \n74 Maunakea Spectroscopic Explorer, CFHT, 65-1238 Mamalahoa Hwy, Kamuela, Hawaii 96743 \n- 75 Departamento de F'ısica, Universidade Federal de Sergipe, Av. Marechal Rondon, S/N, 49000-000 S˜ao Crist'ov˜ao, SE, Brazil \n76 Institute of Astronomy, KU Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium \n- 77 Center for Astrophysics and Space Astronomy, Department of Astrophysical and Planetary Sciences, University of Colorado, 389 UCB, Boulder, CO 80309-0389, USA\n- 78 Department of Astronomy, University of Florida, Bryant Space Science Center, Stadium Road, Gainesville, FL 32611, USA \n79 \nInstituto de Astrof'ısica, Pontificia Universidad Cat'olica de Chile, Av. Vicu˜na Mackenna 4860, 782-0436 Macul, Santiago, Chile \n80 \nInstituto de Astronom'ıa, Universidad Cat'olica del Norte, Av. Angamos 0610, Antofagasta, Chile \n81 Department of Physics, Salisbury University, Salisbury, MD 21801, USA \n- 82 Departamento de F'ısica, Facultad de Ciencias, Universidad Nacional Aut'onoma de M'exico, Ciudad Universitaria, CDMX, 04510, M'exico \n83 Departamento de Astronom'ıa, Universidad de Concepci'on, Casilla 160-C, Concepci'on, Chile \n84 \nSchool of Physics and Astronomy, University of St Andrews, North Haugh, St Andrews KY16 9SS, UK \n85 Exzellenzcluster ORIGINS, Boltzmannstr. 2, D-85748 Garching, Germany \n86 Department of Physics and Astronomy, West Virginia University, 135 Willey St, Morgantown, WV 26506, USA \n87 Center for Gravitational Waves and Cosmology, West Virginia University, Chestnut Ridge Research Building, Morgantown, WV 26505, USA", 'ABSTRACT': "The eighteenth data release of the Sloan Digital Sky Surveys (SDSS) is the first one for SDSS-V, the fifth generation of the survey. SDSS-V comprises three primary scientific programs, or 'Mappers': Milky Way Mapper (MWM), Black Hole Mapper (BHM), and Local Volume Mapper (LVM). This data release contains extensive targeting information for the two multi-object spectroscopy programs (MWM and BHM), including input catalogs and selection functions for their numerous scientific objectives. We describe the production of the targeting databases and their calibration- and scientifically-focused components. DR18 also includes ∼ 25,000 new SDSS spectra and supplemental information for X-ray sources identified by eROSITA in its eFEDS field. We present updates to some of the SDSS software pipelines and preview changes anticipated for DR19. We also describe three value-added catalogs (VACs) based on SDSS-IV data that have been published since DR17, and one VAC based on the SDSS-V data in the eFEDS field. \nKeywords: Surveys (1671), Astronomy databases (83), Astronomy data acquisition (1860), Astronomy software (1855)", '1. THE FIRST TWO DECADES OF THE SLOAN DIGITAL SKY SURVEYS': "This paper describes the eighteenth data release of the Sloan Digital Sky Survey (SDSS) and the first data release of SDSS-V. \nSince its operations began in 1998, SDSS has been taking near-continuous observations of stars, galaxies, and quasars, and other objects, spanning from our solar system to the early days of the Universe. The first phase, SDSS-I (York et al. 2000), imaged over 8000 deg 2 of the sky in the ugriz filters and collected optical spectra \nof more than 700,000 objects. SDSS-II completed the legacy imaging survey and added a dedicated supernova imaging survey (Frieman et al. 2008) and a spectroscopic survey of ∼ 230,000 Milky Way stars (SEGUE; Yanny et al. 2009). \nSDSS-III (Eisenstein et al. 2011) focused entirely on spectroscopy and comprised an extension of SEGUE (SEGUE-2; Rockosi et al. 2022); a radial-velocity exoplanet survey (MARVELS; Ge et al. 2008); a clustering survey of galaxies and intergalactic gas in the distant universe (BOSS; Dawson et al. 2013), which required \nan upgrade to the optical spectrographs (Smee et al. 2013); and an infrared survey of Milky Way and Local group stars (APOGEE-1; Majewski et al. 2017), which introduced infrared spectrographs to the suite of SDSS instrumentation (Wilson et al. 2019). SDSS-IV (Blanton et al. 2017) included a significant expansion of the APOGEE survey (APOGEE-2; Majewski et al.in prep), including the deployment of a new APOGEE-S spectrograph for observations with the 2.5m DuPont telescope at Las Campanas Observatory (LCO), as well as an extension of BOSS observations to previously understudied redshifts (eBOSS; Dawson et al. 2016), and an optical IFU survey of the gas and stellar properties of low-redshift galaxies (MaNGA; Bundy et al. 2015). \nContinuing in this tradition, SDSS-V (Kollmeier et al. 2017, Kollmeier et al. in prep.) constitutes a major innovation step in science scope and hardware. It comprises three primary scientific surveys, called 'mappers': the Milky Way Mapper (MWM; Johnson et al. in prep.), the Black Hole Mapper (BHM; Anderson et al. in prep.), and the Local Volume Mapper (LVM; Drory et al. in prep.). See § 2, § 6, and § 7 for more details. \nA hallmark of the SDSS family of surveys is the regular release of high-quality, well-documented, and ready-usable data to the entire world. Beginning with the Early Data Release (Stoughton et al. 2002), SDSS helped usher in the era of 'open science' through its regular data releases. This practice has proven fruitful for the astronomical community, enabling a very broad scientific reach and impact. The final data release of SDSS-IV was DR17, in December 2021 (Abdurro'uf et al. 2022). As of November 2022, SDSS data (across all survey phases) have been cited in more than 11,000 refereed papers, with over 650,000 citations. Stalzer & Mentzel (2016) named SDSS as one of the most influential data sets, even beyond astronomy and physics. SDSS data are also used in numerous educational contexts, from young schoolchildren to undergraduate and graduate students, especially through its Voyages and SciServer platforms 1 . \nOne of the primary reasons for the widespread use of SDSS data is the collaboration's commitment to highquality, user-friendly documentation to accompany each data release (e.g., Weijmans et al. 2019). SDSS-V continues that core practice with DR18, as summarized in this paper. In § 2, we briefly summarize SDSS-V's scientific and hardware components. In § 3-4, we outline the scope of DR18 and how to access the information. § 5 describes the multi-object spectroscopic (MOS) \ndatabases and targeting software, while § 6, § 7, and § 8 detail the MOS Milky Way Mapper, Black Hole Mapper, and 'open fiber' programs, respectively. § 9 describes the spectra that are released in DR18, and § 10 focuses on new and modified software. § 11 contains the Value-Added Catalogs released since DR17. Finally, § 12 summarizes all of this information and looks forward to the anticipated contents of DR19.", '2. SDSS-V: STATUS AND SCIENCE OBJECTIVES': "In this section, we provide a brief summary of the SDSS-V program at the time of this data release. Many of these elements have been described elsewhere (e.g., Kollmeier et al. 2017) and will be described in greater detail in companion papers, including Kollmeier et al. (in prep). We provide this information here as a standalone snapshot at this important survey milestone. \nSDSS-V sets out to be the first astronomical survey to provide all-sky, multi-epoch spectroscopy in the optical and infrared. Its scientific goals and targets are grouped into three top-level 'Mapper' programs. The Milky Way Mapper (MWM; § 6) is mapping the stellar populations and chemo-dynamics of the Milky Way to understand its evolution, and is probing stellar physics and stellar system architectures by collecting optical and infrared spectra of stars in the Milky Way and the Magellanic Clouds. The Black Hole Mapper (BHM; § 7) is studying the physics of black hole growth through timedomain spectroscopy and is providing spectra for extragalactic X-ray-luminous sources from eROSITA (Merloni et al. 2012; Predehl et al. 2021), using optical spectra across the sky. The Local Volume Mapper (LVM) is examining gas emission, star formation, and stellar/interstellar energy exchange processes in the Milky Way and beyond, at unprecedented scales using ultrawide-field optical IFU spectroscopy. \nMWMand BHM are designed to use the multi-object spectroscopy (MOS) infrastructure of SDSS-V, which includes the Sloan 2.5 m telescope at Apache Point Observatory (APO) in New Mexico, USA (Gunn et al. 2006) and the du Pont 2.5 m telescope at Las Campanas Observatory (LCO) in Chile (Bowen & Vaughan 1973). Obtaining homogeneous, all-sky spectral survey data at both optical and infrared wavelengths requires near-identical sets of optical and near-infrared fiber spectrographs in both hemispheres. For SDSSV, one of the optical BOSS spectrographs was moved from APO to LCO, so that each site is now equipped \nwith a pair of APOGEE and BOSS instruments 2 . The central new hardware component for MOS observations in SDSS-V is the implementation of a focal plane system (FPS) with robotic fiber positioners (Pogge et al. 2020), which replaces the plug-plate system and enables efficient, single-epoch pointings with ∼ 15 minute exposure times. The adoption of the FPS, in turn, required a new three-element corrector for the Sloan telescope (Barkhouser et al. 2022). \nFor the LVM, SDSS-V is building an entirely new facility at LCO to enable ultra wide-field IFU observations across ∼ 1,000 deg 2 . The Local Volume Mapper Instrument (LVM-i) builds upon replicated instrument concepts from DESI (Perruchot et al. 2018) and the successful IFU technology developed as part of SDSSIV/MaNGA. To meet the survey science requirements, LVM-i will comprise a set of four 160 mm telescopes coupled via fiber IFUs to three spectrographs providing full coverage of the optical waveband, all housed in a new dedicated enclosure (Herbst et al. 2020).", '3. SCOPE OF DATA RELEASE 18': "The main focus of DR18 is to lay the groundwork for future SDSS-V releases by presenting the updated access paths, data models, new targeting strategies, and other structures that will be used in DR19 and beyond (Table 1). \nDR18 consists primarily of targeting information (algorithms and databases) for the MWM and BHM programs ( § 5). These catalogs comprise 269.5 GB of data in MOS TARGET DIR (Table 2). They also introduce the concept of a 'carton', a new organizational unit for SDSS observations. A carton is a set of targets that results from a specific target selection algorithm, designed to advance certain science goals. Examples of cartons include white dwarfs selected from Gaia -SDSS catalogs for MWM, and reverberation mapping targets for BHM. Cartons will play a large role in understanding the SDSS-V targeting strategy, target selection algorithms, and eventual full sample. They are discussed in more detail in § 5.2, and lists of MWM and BHM cartons are given in Tables 3 and 4, respectively. Cartons are grouped into 'programs', which are also columns in the \n2 Details of APOGEE spectrographs: https://www.sdss. org/instruments/apogee-spectrographs/ and details of BOSS spectrographs: https://www.sdss.org/instruments/ boss-spectrographs/. Within SDSS-V, one will frequently encounter references to 'BOSS spectra'. This term always means 'spectra obtained with one of the two BOSS spectrographs', rather than 'spectra obtained as part of the SDSS-III BOSS (or SDSS-IV eBOSS) project'. \ntarget tables that refer to broader science cases whose goals will be met by targets from one or more cartons. \nA small number of new BOSS spectra are being made available as part of eFEDS (Brunner et al. 2022), an eROSITA follow-up program ( § 9.1). These add a volume of 301.9 GB to BOSS SPECTRO REDUX , including the ∼ 25k spectra. Accompanying these spectra is a new value added catalog (VAC), which contains updated redshifts and classifications (see § 11.2). DR18 also contains new software routines and updates to existing SDSS software ( § 10), particularly the BOSS spectral reduction package ( § 10.1). \nFinally, all previous SDSS data are also available through the SDSS portals without any changes or additional processing, essentially summarized in DR17 (Abdurro'uf et al. 2022). Newly available as part of DR17, coincident with DR18, are two new VACs based on DR17 data, as well as an earlier DR15-based VAC that has been updated with data from DR17. These VACs are described in greater detail in § 11.1.", '4. ACCESSING THE DATA': 'There are numerous ways to access the SDSS DR18 data products, summarized on the SDSS website 3 and on the data release website 4 . As in previous phases of SDSS, SDSS-V provides a searchable database for the Catalog Archive Server (CAS), using SkyServer 5 , with both SQL and Jupyter notebook interfaces. The Science Archive Server (SAS) is well-suited for directly downloading flat files, including spectra. Both the CAS and SAS are cumulative systems, with some data products replacing or extending data from previous releases, generally speaking. Examples of data products that can be replacements (not in DR18) include spectroscopic reductions and their associated parameters and Value-Added Catalogs. In these cases, only the latest version is included in the data release, but all previously released versions are available at their original locations. An example of data products that are cumulative is the raw data transferred from the observatories for each night of observations. \nThe best way to access a particular data product will depend on the data product itself and the anticipated use case. We recommended to use the CAS when accessing the MOS targeting data described in § 5.1, since this product was originally created as a relational database. \nT able 1. Summary of DR18 Con ten ts \nData Access \nDescription \nItem \nT argeting Information 1 \nrg/dr18 \ner.sdss.o \nttps://skyserv \nh \n) \n5.1 \n§ \n( \ncatalogs \nnput \ni \nof \nh \nCross-matc \ncatalog \nmos \ner.sdss.org/dr18 \nttps://skyserv \nh \ncatalogs) \ncore \nfor \n2 \nable \nT \n, \n5.1 \n§ \n( \ncatalogs \nInput \ncatalogs \nX \nmos \ner.sdss.org/dr18 \nttps://skyserv \nh \n) \n5.2 \n§ \n( \ncartons \nmore \nor \none \nof \ncriteria \nn \nselectio \nthe \ng \nmeetin \njects \nOb \ntarget \nmos \ner.sdss.org/dr18 \nttps://skyserv \nh \n) \n5.2 \n§ \n( \nts \nrequiremen \national \nobserv \nand \ncartons, \ntargets, \nlinking \nable \nT \nmos carton to target \ner.sdss.org/dr18 \nttps://skyserv \nh \n) \n8 \n§ \n, \n7 \n§ \n, \n6 \n§ \n, \n5.3 \n§ \n, \n5.1 \n§ \n( \ncartons \nSDSS-V \nof \nable \nT \ncarton \nmos \nectra \nSp \nDirectories and data mo dels are in § 9.1.2 \n) \n9.1 \n§ \nand \n7.3 \n§ \n( \nsources \nA-selected \nOSIT \neR \nof \nectra \nsp \nOptical \nectra \nsp \neFEDS', 'V alue-Added Catalogs (V A Cs) SDSS-IV V A Cs Based on SDSS-IV data ( § 11.1 , T able 6 ) SDSS-V V A C Based on SDSS-V (eFEDS) data ( § 11.2 , T able 6 )': 'Note - \n1 These are a subset of th e total individual targeting tables a v ailable f rom the CAS. The full list can b e see n in the CAS Sc hema \nBro wser: h ttps://skyserv er.sdss.org/dr18/MoreT o ols/bro wser . \nThe corresponding fits files on the SAS are primarily for archival purposes. Access to the eFEDS spectra is described in § 9.1.3; these spectra are also available for visual inspection in the Science Archive Webapp at https://dr18.sdss.org/optical/. \nEach type of data file on the SAS has an associated data model, which describes its format and content. SDSS-V is developing extensions to the existing data model products to include better accessibility in formats other than the current static html , such as yaml , json , and backward compatibility with classic html . For DR18, all data models can be found at https://data.sdss.org/datamodel/. \nThe SDSS DR18 website has numerous tutorials and examples available to access and interact with both the MOS targeting data and the eFEDS spectra released in DR18. See https://www.sdss.org/dr18/tutorials/ for more information.', '5. MULTI-OBJECT SPECTROSCOPY TARGETING': "The primary data products in DR18 are the underlying, cross-matched catalog of objects targeted by SDSSV and the list of targets. In this section, we describe how these data are organized into cross-matched catalog database tables and target tables. Throughout DR18, we use 0.5 to indicate the version of the catalog crossmatching process ( § 5.1). Variants of the target selection of individual cartons ( § 5.2) using this cross-match are versioned as 0.5.X , where ' X ' varies between cartons. A set of cartons and carton targeting versions is referred to as a 'targeting generation', which is tracked similarly as 0.5.X . Targeting generation 0.5.3 is the one released in DR18.", '5.1. Catalog Database Tables and Cross-Matching': "Previous generations of SDSS relied almost exclusively on a single imaging catalog for targeting. In the optical, the SDSS imaging survey (Abazajian et al. 2009; Padmanabhan et al. 2008) formed the basis for targeting, while in the H -band, the 2MASS all-sky survey was used (Skrutskie et al. 2006). For SDSS-V, this approach is no longer practical because of the need for all-sky, deep optical imaging and the desire to observe stars in both the optical and the infrared. Gaia imaging does not go deep enough in the extragalactic sky, while even the union of SDSS imaging, PanSTARRS imaging, and other wide-field imaging do not cover the full sky at the necessary wavelengths. \nTherefore, to ensure that each object in the sky of relevance to SDSS-V has a unique identifier, regardless of how many imaging catalogs it appears in, we created our own targeting database. All SDSS-V multi-object spectroscopy (MOS) targets (i.e., all targets for MWM and \nBHM) are stored in a suite of database tables, which contains both a unified list of all sources within a set of crossmatched input catalogs and the targeting classifications of those sources. This section describes how the catalogs are stored; § 5.2 describes how the targeting classifications are stored. \nFigure 1 shows a cartoon representation of the primary database tables and workflows presented in this section. The mos catalog table is the central unified list of sources. It was created from the set of spatially cross-matched core input catalogs, which are listed in Table 2 and shown in yellow in Figures 1 and 3. We have a set of additional catalogs used in targeting whose entries are associated with entries in the spatially crossmatched catalogs; these are the tables highlighted in blue in Figures 1 and 3. The database in DR18 stores this set of catalogs and the associations between objects within them. \nThe mos catalog table contains only a minimum of information for each object: only the positional information and the catalogid identifier, which is an internal ID used by the SDSS-V databases to uniquely identify each source. All other information about the sources is stored in the individual cross-matched and targeting catalogs. Each catalogid is associated with one or more entries within these original catalogs. For 17 of the original input catalogs, the association between mos catalog and the original catalog is expressed as shown in Figure 2, using a join table mos catalog to X for each input catalog ' mos X '. As above, the list of all original catalogs cross-matched in this fashion is in Table 2, along with the primary key used for each (i.e., the column labeled ' X id ' in Figure 2). \nThe initial basis for mos catalog is the TESS Input Catalog v8 (TIC; Stassun et al. 2019), which is all-sky and uses both Gaia DR2 and 2MASS to identify objects 6 . Then, cross-matching follows a three-phase procedure for each of the subsequent input catalogs listed in Table 2, which can include both point and extended sources. First, we use existing literature cross-matches included with the catalog information to identify physical targets that already exist in the mos catalog table. For those, a new entry is added to the appropriate mos catalog to X table that references the matched catalogid and the unique identifier in the input catalog. \nFigure 1. Cartoon diagram of the relationships between several key database tables and of the workflow between sky sources and observed targets ( § 5.1-5.2). The objects colored yellow, blue, pink, and purple correspond to the same types of tables in Figures 2-4. The green color has been repurposed to highlight the 'observational' section of the workflow, where candidate targets are down-selected and grouped into observable designs ( § 5.2). Left: the pink mos catalog is built as a superset of the yellow core input catalogs ( § 5.1, Table 2), whose circles are approximately scaled here to the size of the catalog. The blue 'additional catalogs' ( § 5.1) provide supplementary information to sources in mos catalog . Right: the small purple circles represent the many individual cartons ( § 5.2), which specify both target selection criteria and observational requirements. The potential targets satisfying the selection criteria, along with the observational requirements (stored in mos carton to target ) are used by the robostrategy code (Blanton et al., in prep) to produce observable designs, previously for plates (Covey et al., in prep) and now for the FPS robots. \n<!-- image --> \nFigure 2. Relationships between the cross-matched mos catalog table and the original source input catalogs ( § 5.1). There are 17 source input catalogs that are linked to mos catalog through the structure shown above, with 'X' representing the name of the original catalog (Table 2). \n<!-- image --> \nFor targets in the input catalog that do not have an available association with mos catalog , we perform a cone search around the coordinates of each target and find the associated mos catalog entries within the search radius. We use a default 1 arcsec search radius, but this value is sometimes modified to match the spatial resolution of the input catalog. All the \ncone-search-matched catalogid entries are associated with the input catalog target via the corresponding mos catalog to X table. The match with the smallest on-sky separation is marked as the 'best' match 7 by setting the best column in the mos catalog to X table to True . Finally, targets in the input catalog that cannot be spatially cross-matched are considered new physical objects and added to mos catalog and mos catalog to X with a new unique catalogid . \nFor some of these 17 catalogs, the database contains further associations between them and the additional \nTable 2. Cross-matched input catalogs for the catalog database ( § 5.1) \ncatalogs used in targeting. Figure 3 shows these other catalogs and how they are associated in the database with the cross-matched catalogs. \nDR18 contains the catalog cross-matching version v0.5 , which was used for targeting at the beginning of MOS observations in SDSS-V. Due to the size of the databases involved, the tables in the released DR18 version only contain sources that were ultimately identified with potential targets (i.e., are in one or more cartons). DR18 also excludes some X-ray catalogs from eROSITA that were used for targeting in v0.5 but not yet made public. Due to an error, DR18 also excludes tables associated with the TESS Objects of Interest, the Spitzer /MIPSGAL program (Gutermuth & Heyer 2015), and the Gaia ASAS-SN Classical Cepheid sample (Inno et al. 2021); these tables will be included in DR19.", '5.2. Target Database Tables and MOS Cartons': "Drawing on the catalog database and the information stored in the linked catalogs described above ( § 5.1), the target selection software determines which objects should be targeted for spectroscopy using the selection criteria and other properties of the target cartons. The target selection criteria are described in later sections, on the website, and in other program papers; here we only describe how the information is organized. \nAfter the target selection software is run, the targeting database contains a set of targets and a set of cartons. Figure 4 shows how this information is stored for a generalized subset of the database tables. The associations between targets and cartons are stored in the mos carton to target table. Each carton is associated with all of the candidate targets that satisfy the selection criteria for the carton. Each target is associated with one or more cartons. \nFor example, consider a star that has a detection in Gaia DR2. This star will appear in the TIC v8, and thus will have an entry in mos catalog . If this star satisfies the selection criteria for any carton, it will also appear in mos target , with a corresponding entry in mos carton to target . If this star satisfies the criteria for multiple cartons, then multiple entries in mos carton to target will be associated with the star's single entry in mos target . Each mos carton to target entry is also associated with an entry in the mos carton table, which gives the name of the carton and other information. \nEach mos carton to target entry has an associated set of entries cadence, value , and priority -that are used in the process of determining the survey strategy and fiber assignment process. The cadence describes how the target should be observed (number of epochs, number of observations per epoch, and observing condi- \nFigure 3. Relationships between cross-matched mos catalog table (pink), original source catalogs (yellow) and additional catalogs used for targeting (blue), for the cases where additional such catalogs exist. mos catalog contains the full crossmatched list of sources, and it is associated with the original source catalogs through join tables (green) in the manner described in Figure 2 and § 5.1. Additional catalogs can be joined as shown to the original source catalogs, and thereby to the mos catalog table, as shown. For clarity, only the columns necessary for table joins are shown in this diagram. \n<!-- image --> \ntions). The cadence pk values allow a join to a cadence table with the cadence name and other parameters associated with it. \nThe value expresses how important the target is to the overall objectives of the MOS program, which is used in the determination of the overall survey strategy. The priority expresses the order in which targets should be assigned to fibers during fiber assignment in a given pointing. These quantities are more fully explained in the robostrategy paper (Blanton et al. in prep). \nEach carton also has an assigned category , which can be one of science , sky boss , standard boss , sky apogee , or standard apogee . The first category indicates science targets, and the others are different types of calibration targets ( § 5.3-5.4). \nAmong the science target cartons, there are those whose observations are required for the stated success \nof SDSS-V science, as laid out in the survey's science requirements document ('SRD'). These cartons tend to have descriptive names. The cartons that arose from an initial call within the SDSS-V collaboration for 'open fiber' targets have carton names starting with opentargets ). There is also a set of filler targets for spare fibers. The released tables in DR18 do not unambiguously identify which cartons are SRD, open, or filler (although the information can often be approximately guessed from the carton names), because this identification is associated with the fiber assignment results, which will not be released until DR19.", '5.3. Standard star cartons': "5.3.1. APOGEE standard stars \nThe standard stars for APOGEE observations are used primarily to correct telluric absorption by Earth's \nFigure 4. Association of catalogid s with targeting information. Each catalogid that is selected by one or more cartons is given an entry in the mos target table. For each carton that selects it, there is a mos carton to target entry. Each entry is associated with one target and one carton, and specifies the cadence, instrument (BOSS or APOGEE), and other observing conditions. Further tables in the database define the properties of the cadences, the names of the instruments, and the names of the categories ( science , sky boss , standard boss , sky apogee , and standard apogee ). \n<!-- image --> \natmosphere. They are selected in a process very similar to that used for APOGEE-1 and APOGEE-2 observations, which aimed to select hot, blue stars (with few absorption lines of their own) distributed as evenly as possible across the field of view to allow for spatiallydependent telluric corrections (Zasowski et al. 2013; Zasowski et al. 2017). \nThe carton ops std apogee contains these calibration stars, which are drawn from the 2MASS PSC (Skrutskie et al. 2006). They are restricted to have magnitude 7 < H < 11, color -0 . 25 < ( J -K s ) < +0 . 5, and magnitude uncertainties (JERR, HERR, KERR) of ≤ 0.1 mag. In addition, we applied the requirements that the 2MASS read flag be equal to '1' or '2', the quality flag for J , H , and K s be equal to 'A' or 'B', the galaxy contamination flag be equal to '0', the confusion flag be equal to '000', the extkey ID (linking to the 2MASS Extended Source Catalog; Jarrett et al. 2000) be Null , and the star lie at least 6 '' from its nearest 2MASS neighbor. From this subset of potential standard stars, the five bluest sources are chosen from each HEALPix NSIDE = 128 pixel (0.21 deg 2 pix -1 ; G'orski et al. 2005) as telluric standard stars.", '5.3.2. BOSS spectrophotometric standard stars': 'Spectrophotometric standard stars are used by the BOSS pipeline to calibrate the absolute and relative throughput of the instrument during science observations. The strategy for selecting BOSS standards in SDSS-V builds upon past experience from the SDSSIII/IV BOSS and eBOSS projects (see, e.g., Dawson et al. 2013, 2016). Selecting standards for the SDSS- \nV program presents some new challenges: i) we need to select standard stars outside the footprint of the SDSS photometric imaging catalog, and ii) we need to reliably select standards along lines of sight that may be heavily reddened by Galactic dust. In order to satisfy the first challenge, we exploited wider area photometric and astrometric information from PanStarrs, the DESI Legacy Surveys, and Gaia. To mitigate the complications of extinction, we used the 3D reddening information provided by the TIC v8 catalog (Stassun et al. 2019). \nBelow we briefly describe the target selection criteria for the various BOSS standards cartons used in early SDSS-V operations, which are released as part of DR18 (targeting generation v0.5.3 ). \nThe ops std eboss carton is identical to the spectrophotometric standards used by the eBOSS survey (Dawson et al. 2016). These standards lie in the magnitude range 16 < r psf < 18 AB and so are most suitable for use in dark time. Use of these standards maintains an important continuity with archival SDSS spectroscopy, which is especially important when investigating the long term variations of QSOs ( § 7). \nThe ops std boss ps1dr2 , ops std boss lsdr8 , and ops std boss gdr2 standard star cartons are based on PanStarrs1-DR2, Legacy Survey DR8, and Gaia DR2 photometry, respectively. For each carton, we use the empirically determined location of eBOSS standards within the dereddened color space of the given photometric system to train our new selection locus. Additional cuts on data quality, and in the parallax vs magnitude plane (via Gaia DR2), are applied to further clean outliers from the sample. Magnitude limits are applied such that these standards are appropriate for BOSS dark-time observations (approximately 16 < r psf < 18 AB, or 16 < G < 18 Vega). Additionally, the ops std boss gdr2 carton is limited to Galactic latitudes of | b | > 10 · . We estimate SDSS g psf , r psf , i psf , z psf magnitudes for each BOSS standard star target, transferring their native multi-band photometry into the SDSS system via color transforms derived from the ops std eboss sample. \nAdditional BOSS standard cartons were developed to facilitate all-sky observations during bright time: ops std boss tic , ops std boss , and ops std boss red . These cartons fill in the regions of the sky that are not covered by the BOSS standard cartons described above, and they include brighter standards, to mitigate the potential impact of crosstalk from brighter science targets in the design. The ops std boss tic carton consists of likely F stars, selected via T eff and surface gravity cuts applied to TICV8 stellar parameters (Stassun et al. 2019). To provide \nuniform coverage over the sky, targets are sorted into a NSIDE=128 HEALpix grid (0.21 deg 2 pix -1 ), and the 10 highest-gravity stars in each healpix are retained for the final carton. The ops std boss carton similarly targets F stars, but selected via cuts on Gaia parallax, color, and absolute magnitude. Finally, the ops std boss red carton is intended to provide standards even in the most heavily extincted sections of the Galactic midplane. This carton uses cuts on observed and dereddened Gaia+2MASS colors, as well as parallax, proper motion and reduced proper motion criteria to reduce contamination from non-F stars whose observed colors can be dereddened into the relevant areas of color-magnitude space.', '5.4. Sky fiber locations': "The data reduction pipelines for BOSS and APOGEE both require a number of fibers to be placed at 'sky' locations, which do not contain light from astrophysical sources and thus allow for the correction of the observed spectra for contamination by emission processes in Earth's atmosphere. The number and 'quality' of sky fibers per configuration is dependent on the type of observation to be performed. For example, for SDSS-V observations targeting faint extragalactic populations in dark time, we reserve 20% of the BOSS fibers for sky observations, and those locations must be empty of astrophysical sources down to the magnitude limit of the deepest wide area imaging that we currently have available. In contrast, observations of bright targets in bright time generally require fewer sky fibers, and can sometimes tolerate mild contamination from faint astrophysical sources. However, even with such reduced criteria, suitable sky fiber locations can be challenging to find in the densest Galactic plane fields. \nWith these constraints in mind we have designed a hierarchy of sky fiber cartons, which collectively satisfy all SDSS-V FPS observational requirements. BOSS observations use three cartons: ops sky boss best , ops sky boss good , and ops sky boss fallback , and APOGEE observations draw on two cartons: ops sky apogee best , and ops sky apogee good . \nFirst, the sky is divided into HEALPix NSIDE = 32 pixel 'tiles', comprising roughly 3.4 deg 2 pix -1 , which is approximately the area spanned by a single FPS configuration. Each tile is further divided into HEALpix NSIDE = 32768 pixels (41 arcsec 2 pix -1 ), the centers of which are considered candidate sky locations 8 . Each \ncandidate sky location is then compared to a set of input catalogs (given below) and labeled valid (with respect to each comparison catalog, separately) if i) their NSIDE = 32768 HEALpixel contains no astrophysical objects in that catalog, and ii) they lie further than a magnitude-dependent separation from the nearest potentially contaminating astrophysical source. This minimum separation is computed as s ∗ = s + ( m thr -m ) β a , where s = 5 arcsec is a minimum radius floor and m thr is a fixed brightness threshold (14 mag). The a = 0 . 15 and β = 1 . 5 parameters control the scaling of the exclusion radius with brightness, and are set to conservative values. We consider the H , G , V total , r psf , and r model magnitudes for objects from the 2MASS PSC, Gaia DR2, Tycho2, Pan-STARRS1 DR2, and Legacy Survey DR8 catalogs, respectively (Skrutskie et al. 2006; Gaia Collaboration et al. 2018; Høg et al. 2000; Flewelling et al. 2020; Dey et al. 2019). Candidate sky locations lying near 2MASS extended sources (Jarrett et al. 2000) are considered to be valid if they lie outside the 'extrapolation/total radius' of the source. All of these candidate locations that are valid in one or more catalogs are stored in the catalogdb table skies v2 and serve as inputs for the cartons below. \nThe highest quality sky carton for BOSS (i.e., suitable for darkest observations) is ops sky boss best . For this carton, we require locations to be valid in each of the 2MASS (point- and extended sources), Tycho2, and Gaia DR2 catalogs. Furthermore, selected sky locations must be within the Pan-STARRS1 or Legacy Survey DR8 survey footprints, and must not be contaminated by sources from either catalog. Somewhat less restrictive is the ops sky boss good carton, which drops the requirement to avoid Pan-STARRS1 and Legacy Survey DR8 sources, and so extends the list of available sky locations to the entire sphere, excepting an area of the Galactic bulge and the inner parts of the Magellanic Clouds. In order to provide sky fiber locations in those remaining regions, the ops sky boss fallback carton loosens the radial exclusion criteria, preferring locations that lie furthest from Gaia DR2 sources but requiring only that the sky location lies more than 3 '' from a Gaia DR2 or Pan-STARRS1 source, more than 5 '' from any 2MASS source, and more than 15 '' from any Tycho2 star. \nFor APOGEE observations, the highest quality ops sky apogee best carton requires sky locations to be valid in each of the 2MASS (point- and extended sources), Tycho2, and Gaia DR2 catalogs. The lowerpriority ops sky apogee good carton is dedicated to finding the remaining least-bad sky locations in dense regions. These positions have a maximum nearest neigh- \nbor brightness of H = 10, and candidates lying farthest from 2MASS contaminants are preferred.", '6. MILKY WAY MAPPER: SCIENCE PROGRAMS AND TARGET SELECTION': "The Milky Way can serve as a 'model organism' for understanding the physical processes that shape galaxies over an enormous rage of temporal and physical scales in the context of hierarchical cosmogony. The Milky Way Mapper (MWM) takes advantage of our unique perspective within the Milky Way to create an unprecedented high-resolution map of our Galaxy's stellar populations. MWM is observing stars formed in the dawn of the Galaxy through the present day, thoroughly sampling the H-R diagram and chemical abundance space. Through multi-epoch, multi-wavelength observations of single stars and stellar systems, MWM seeks to understand the evolution of the luminous constituents of the Galaxy and detect unseen compact objects and substellar objects. \nMWM focuses on four connected themes: \n- · Galactic Archaeology : We can reconstruct the deep history of the Milky Way through observing the number, masses, composition, ages, and motions of its stars and the structures that they create. Luminous red giants cover a large range of ages, from ∼ 1 Gyr and older. However, samples of red giants are insensitive to the history of intermediate-mass stars older than ∼ 1 Gyr. Such stars have turned into stellar remnants, mostly white dwarfs. Other sites of Galactic archaeology studied in open fiber programs ( § 8) in targeting generation v0.5.3 include the Galactic halo and the Magellanic Clouds.\n- · The Young Galaxy : The programs under Galactic Archaeology investigate the cumulative history of star formation, chemical enrichment, and radial migration of the Milky Way, as told through evolved stars across a range of masses. Young stars (here ∼ 5 -100 Myrs) fill in the early phases of lowmass stellar evolution and the full life cycle of massive stars. They provide a detailed and complete snapshot of the youngest generation in the Milky Way. In MWM, we systematically target low-mass young stars (YSOs) within ∼ 500 pc and all luminous, hot stars throughout the Galactic disk. Dust is relevant to not only the current structure of the Milky Way, but also how metals are incorporated into the ISM and thence into the next generation of stars. MWM takes advantage of dust and extinction indicators along lines of sight to its ob- \nstars to map the dust and its properties in the disk. \n- · The High Energy Galaxy : Galactic sources of X-ray emission include compact object accretors, young stellar objects, and flaring stars, all objects of intense interest for the other themes here as well. The eROSITA mission is measuring X-ray fluxes for millions of point sources in the Galaxy. However, to identify and explore the physical nature of such objects, optical and/or infrared spectra rich in absorption and emission features are critical.\n- · Stellar Physics and Stellar Systems : Undergirding all of MWM's Galactic explorations are the properties of stars and stellar systems. We begin these investigation close to home - in the solar neighborhood, the only place in the universe where it is practicable to obtain a spectroscopic census of stars down to the hydrogen burning limit. \nTo reliably predict the age, evolutionary state, nucleosynthesis, internal mixing, and end state of stars, we need to understand their structure. Astereoseismology - the study of how waves propagate through stellar interiors - provides a powerful tool for this work, especially when combined with stellar and dynamical parameters from spectra. MWM is targeting oscillating red giant and hot stars. In the latter case, we are focusing on OBA stars in massive eclipsing binaries. \nMost stars orbit other stars. MWM is unique among current large spectroscopic surveys in targeting a vast range of stellar types for time-domain radial velocity observations, including YSOs, OB stars, WDs, and red giants. Particular attention is also paid to compact binaries, which are binary systems where at least one component is a WD, neutron star, or black hole, and to stars observed at high cadence for planet transits. By increasing the number of epochs, MWM is increasing the probability of orbit reconstruction with the goal of probing long baselines ( > 8 years) and the brown dwarf desert. \nThese top-level science goals are achieved by observing an anticipated ∼ 5 million stars, with the different target categories structured in a set of target cartons. Each carton has a well-defined selection function to enable subsequent population modeling (Rix et al. 2021). These cartons are summarized in Table 3, with complete selection functions provided in Appendix A and on the \nDR18 website 9 . It is important to note that these target selections are the input into the observational design code ( § 10). The final targets that will actually be observed are an as-yet-unknown subset of these selections. See Johnson et al. (in prep) for a complete description of MWM's goals and targeting. \nDR18 contains substantial targeting information for MWM, including the input catalogs used to generate potential target lists, the selection functions for the numerous cartons, and the data models for the anticipated output files. Users anticipating the first spectroscopic data release in DR19 can use this information to prepare for analysis of the DR19 data. We confine our discussion here to the v0.5.3 targeting scheme, deferring presentation of cartons added in subsequent targeting schemes (e.g., for validating stellar parameters, observing Gaia binaries, and other expansions and improvements) to the future MWM overview and DR19 papers. Thus, the subsections below enumerate the specific programs, and their constituent cartons, designed under the v0.5.3 targeting strategy to address the scientific goals above.", '6.1. Galactic Genesis': "Galactic Genesis is the flagship program of the MWM, with near-infrared (APOGEE) observations of over 4 million stars planned. Galactic Genesis's central science goal to obtain a fine sampling of the chemo-orbital distribution of stars across the entire radial extent of the Milky Way, probing the Galactic plane, the central regions of our Galaxy, and the far side of the disk. To do this, Galactic Genesis targets luminous red giants, above the red clump, where the multi-million sample size with dust-penetrating APOGEE observations provides a decisive advantage over other surveys. \nAll of the Galactic Genesis targets are contained in the mwm galactic core carton.", '6.2. White Dwarfs': 'White dwarfs (WDs) are tracers of Galactic star formation, progenitors of type Ia supernovae, important end products of both single and binary star evolution, and important cosmic physics laboratories (e.g., for the formation of strong magnetic fields). Gaia has recently produced enormous, well defined samples of WDs ( ∼ 250,000 objects). SDSS-V is enlarging the subset of these WDs that have complementary spectroscopy (especially in the southern hemisphere) and also providing multi-epoch observations for empirical input on the bi- \nry WD merging channels towards SNe Ia and other remnant products (e.g., Chandra et al. 2021). \nAll of the MWM WDs targeted for this program are contained in the mwm wd core carton.', '6.3. Solar Neighborhood Census': "MWM's Solar Neighborhood Census (SNC) program targets a volume-limited sample of stars with the goals of cataloging low-luminosity stellar populations. While not technically 'complete', the SNC will provide highquality, two-epoch BOSS or APOGEE spectra of > 10 5 stars within 100 pc and within 250 pc. The most complete census will be within 100 pc. However, there are few stars hotter than K types that close to the Sun, so a sample of stars out to 250 pc away is also targeted to enable a reliable match to higher luminosity populations. \nThe SNC targets span four cartons, all beginning with mwm snc : mwm snc 100pc apogee , mwm snc 100pc boss , mwm snc 250pc apogee , mwm snc 250pc boss , where the latter part of the carton names indicate the volume size and instrument.", '6.4. Young Stellar Objects (YSOs)': 'The MWM YSO program targets the pre-mainsequence phase of low-mass stars. These objects are key to understanding the early phases of low-mass stellar evolution, on what orbits stars are born, and how they disperse from their clustered birth sites to become field stars. The YSO program selects objects via their position in the color-magnitude diagram (CMD), optical/IR SED, or variability. APOGEE spectra then provide stellar parameters and velocities, while BOSS spectra provide indicators of youth, such as Li absorption and H α emission. Kounkel et al. (submitted) presents a complete discussion of the target selection rationale and on-sky validation. \nCartons that fall under the YSO program have names beginning with mwm yso , a third label indicating their selection method, and a fourth label indicating the instrument used (either apogee or boss ). The shorthand labels for the selection methods are cluster , cmz , disk , embedded , nebula , pms , and variable . For example, YSO targets selected based on their variability and observed with the APOGEE instrument are contained in the mwm yso variable apogee carton. The full selection functions for these methods are detailed in Appendix A.', '6.5. OB Stars': 'Massive stars on the main sequence are unambiguously young because of their short hydrogen-burning \nlifetimes. They are excellent tracers of the recent star formation throughout the disk (e.g., Zari et al. 2021) and they are most likely luminous companions to black holes and neutron stars, objects that were once even more massive stars. These hot stars are also the dominant ionizers of the interstellar medium. The MWM OB program targets all stars brighter than G = 16 that are also likely to be hotter than ∼ 8000 K. \nThe massive stars in this program fall into two cartons: mwm ob cepheids , targeting known Cepheid stars (Inno et al. 2021), and mwm ob core , targeting a larger sample of OBA stars across the MW and Magellanic Clouds.', '6.6. Galactic eROSITA Sources': 'In addition to the spectroscopic follow-up of highredshift X-ray sources from eROSITA ( § 7.2), the Galactic eROSITA sources program in MWM is targeting likely Milky Way X-ray sources, including accreting compact objects, YSOs, and flare stars. The optical and IR counterparts of these sources were identified by the eROSITA Stars and Compact Objects working groups and are being observed with the APOGEE and/or BOSS instrument(s), depending on the H -band or G -band magnitude, for source characterization. \nThe Galactic eROSITA program targets comprise three cartons: mwm erosita stars , for likely stellar coronal emitters, and mwm erosita compact gen and mwm erosita compact var for likely compact object accretors, using two different methods for finding the likeliest optical counterpart.', '6.7. Massive Eclipsing Binaries': 'The main-sequence structure of massive stars, including the size of their convective cores, is intimately linked to their (eventual) remnant mass and other properties. The Massive Eclipsing Binaries (EBs) program uses asteroseismic and photometric data to identify the likeliest double-lined spectroscopic EBs with pulsational variability, among OBA stellar types. \nThese targets are contained in the manual mwm tess ob carton.', '6.8. Binary Systems': 'The APOGEE-1 and -2 surveys contributed to binary studies both through statistical studies of large samples (e.g., Price-Whelan et al. 2020; Mazzola et al. 2020) and through reconstructing orbits (e.g., Washington et al. 2021). In MWM, the orbital size and mass range spanned by targets in the OB Stars and YSO programs will be increased by serendipitous observations of previously-observed stars that meet the Galactic Genesis criteria (see above). The Binary Systems program is \nfurther enhancing this sample of stellar and substellar companions of stars across the HR diagram by investing fibers to extending time baselines of red giants, subgiants, and M dwarfs. This program exclusively uses the APOGEE spectrograph because of its superior spectral resolution and improved RV precision. \nTargets in this program fall in one of two cartons: mwm rv long fps , for stars with existing multi-epoch observations from APOGEE, and mwm rv short fps , for stars without these earlier observations.', '6.9. Compact Binaries': 'Spectra of compact binary systems are frequently marked by non-stellar emission - such as X-ray, UV, or H α flux - that can probe mass transfer or other nonthermal processes. The Compact Binaries program is observing a large number of likely compact binary systems, identified through different combinations of CMD position, UV excess, and previous association with a cataclysmic variable (from the AAVSO 10 ). \nThese different combinations of selections are reflected in the many cartons contained in the Compact Binaries program. All of the carton names begin with mwm cb or manual nsbh , followed by additional labels indicating the selection method and (in some cases) the instrument used for the MWM observations. The shorthand labels for the selection methods are 300pc , cvcandidates , gaiagalex , and uvex[1-5] . The full selection functions for these methods are detailed in Appendix A.', '6.10. Planet Hosts': "Understanding the conditions that impact a star's likelihood of hosting planets, and the properties of the planetary orbits and the planets themselves, is essential for understanding planet formation and solar system evolution. The Planet Hosts program targets TESS stars with and without associated TESS Objects of Interest (TOIs). These will be observed with the APOGEE instrument to provide stellar parameters and detailed stellar abundances. \nThe single carton in this program is called mwm tess planet .", '6.11. Asteroseismic Red Giants': "For stars on the red giant branch (RGB), asteroseismology arguably provides the gold standard of stellar mass and surface gravity measurements, but maximizing its power requires additional input from spectroscopy (such as stellar metallicity). The goal of the Asteroseismic Red Giants program is to obtain spectra for stars \nwith asteroseismic signals, especially those observed by TESS. The start of SDSS-V overlapped with the nominal mission of the TESS satellite, so this program targets bright RGB stars throughout most of the sky (avoiding the Galactic plane), under the expectation that some large fraction of them would eventually have detectable oscillations. This simple selection function will enhance the legacy value of this program's sample. \nThe single carton in this program is called mwm tessrgb core .", '6.12. Dust': "Interstellar dust presents both a challenge and an opportunity in studying the structure and history of our Galaxy. For example, rigorous modeling of the Milky Way's structure, based on stellar observations, requires (approximate) knowledge of the 3D extinction distribution, as seen from the Sun. Such a map then permits dust-corrected estimates of stellar distances for stars with spectroscopic luminosity estimates but poor parallaxes, but also valuable information on the 3D structure of the cold interstellar medium. MWM's Dust program is obtaining spectra for mapping distribution of the density and properties of dust (such as R V ; e.g., Schlafly et al. 2016). Those spectra will of also provide information on the column density and kinematics of the ISM via the diffuse interstellar bands (e.g.. Zasowski et al. 2019). The targets for this program are RGB stars selected to 'fill in' the regions of low extinction avoided by the Galactic Genesis program, to achieve an even sampling of the Milky Way's dust with the total MWM sample. \nThe single carton in this program is called mwm dust core .", '6.13. Science Validation': 'The only carton in this program in the v0.5.3 targeting generation is mwm legacy ir2opt , which is a filler carton to obtain BOSS spectra of stars observed in APOGEE-1 and -2. The stellar parameters from the higher-resolution APOGEE spectrum of a given star are used to provide labels for the BOSS spectrum, facilitating data-driven modeling and cross-calibration of the BOSS spectra. Future targeting generations will include additional cartons targeting stars with known fundamental parameters and stars targeted by other large spectroscopic surveys.', '7. BLACK HOLE MAPPER: SCIENCE PROGRAMS AND TARGET SELECTION': "The Black Hole Mapper (BHM) sets out to better understand the growth of supermassive black holes at the centers of galaxies, through both time-domain spectroscopy of quasars and other AGN, and the first large-area optical spectroscopic follow-up of the newly available eROSITA X-ray survey. In its quasar timedomain program, SDSS-V will provide orders of magnitude advances in both time baseline and sample sizes. The BHM time-domain core programs ( § 7.1) will target about ∼ 10 4 . 5 previously known quasars with multiple additional optical spectral epochs. SDSS-V will also provide optical counterpart spectra for ∼ 10 5 . 5 Xray sources ( § 7.2), especially recently-discovered X-ray sources from SRG/eROSITA (Predehl et al. 2021). \nIn DR18, BHM is releasing two categories of data: i) initial catalogs of candidate targets for its main science programs, which may provide guidance to the community in planning for future SDSS-V data releases, and ii) optical BOSS spectra for ∼ 10 4 candidate counterparts to eROSITA X-ray sources from the eROSITA Final Equatorial Depth Survey field (eFEDS, § 7.3; Brunner et al. 2022). In the subsections below, we provide a high-level summary of those BHM science programs that directly flow down from the driving science goals of the project (i.e., the 'core' programs); these are summarized in Table 4. Further details of the target selection criteria for the full set of BHM target cartons released in DR18 can be found in Appendix B and on the SDSS DR18 website 11 . As in the MWM section above, these targeting selections provide input into the observational design software ( § 10). The actual targets that are finally observed are a subset of these selections, determined by the survey optimization software.", '7.1. Spectral time-domain programs': 'The time-domain BHM core programs will target about ∼ 10 4 . 5 previously known quasars with multiple additional optical spectral epochs. These spectral time-domain programs aim to sample a broad range of timescales and cadences, ranging from days to decades (when all SDSS data are combined), as different aspects of quasar physics result in variability on very different time scales. SDSS-V studies black hole masses via reverberation mapping, possible SMBH binarity, timeresolved accretion events, broad line region (BLR) dynamics, and Broad Absorption Line Quasi-Stellar Object (BALQSO) outflows. The SDSS-V quasar timedomain programs build on the results and experience of earlier generations of SDSS, e.g., the SDSS-III/IV rever- \nTable 3. Milky Way Mapper Cartons \nNote - \nberation mapping project (RM; Shen et al. 2015) and the Time Domain Spectroscopic Survey (TDSS; MacLeod et al. 2018), to enable long time baselines.', '7.1.1. All-Quasar Multi-Epoch Spectroscopy': 'The All-Quasar Multi-Epoch Spectroscopy (AQMES) program is core to the BHM science, with different tiers in survey area and number of epochs. It includes cartons that are aimed at wide areas and low cadence, and cartons aimed at a more modest area with higher cadence (see Table 4). Together these two tiers add new epochs in SDSS-V for ∼ 22,000 quasars that already have at least one previous epoch of spectroscopy from SDSSI-IV. These AQMES targets are selected from the SDSS DR16 quasar catalog (Lyke et al. 2020) and are readily observable from the Apache Point Observatory. \nFor ∼ 20,000 known quasars, the primary associated targeting carton is bhm aqmes wide2 . This carton aims to add ∼ 2 additional epochs of SDSS-V optical spectroscopy for each of these known SDSS quasars. When combined with their archival SDSS optical spectra these data will sample ∼ 1-25 yr timescales (observer frame). The primary science goals include probing the BLR dynamics of the most massive black holes, constraining the statistics of changing look quasars, and charting broad absorption line (BAL) disappearance/emergence. This is a relatively wide area, but low-cadence time-domain tier, encompassing ∼ 2000-3000 deg 2 of the sky. \nFor ∼ 2000 quasars, the bhm aqmes medium carton (Table 4) aims to add ∼ 10 optical spectral epochs in SDSSV, probing down to 1-month to 1-year timescales (in addition to the longer baseline timescales enabled with archival SDSS spectroscopy). The primary science goals of AQMES-Medium are to trace out BLR structural and dynamical changes, including BLR changes in modest mass SHMBs. This is a medium time-domain tier in area and cadence, encompassing ∼ 200-300 deg 2 .', '7.1.2. Reverberation Mapping': 'The Reverberation Mapping (RM) program is core to the BHM science of measuring black hole masses. \nFor ∼ 1000 quasars in 4-5 dedicated fields to be targeted under the BHM RM program, a set of cartons (with names starting with bhm rm , Table 4) aim to obtain optical repeat spectra with a high cadence of up to ∼ 174 epochs, which sample down to (observer frame) timescales of days to weeks. The time lags between the continuum and BLR emission, plus line velocity widths, yield virial estimates of the black hole mass, advancing RM measures yet further to a broad range of luminosity and redshift. This is a small-area but high-cadence time-domain tier, encompassing a total of ∼ 30 deg 2 .', '7.2. Spectroscopic follow-up of X-ray sources': 'In addition to the time-domain science projects described above, BHM is carrying out a program of spectroscopic characterization of counterparts to an unprecedentedly large sample of X-ray sources. In almost all cases this will be via optical spectroscopy. The broad science emphasis is to chart the astrophysics, and the growth and evolution, of SMBHs. The X-ray selection provides a probe of accretion onto SMBHs that is significantly less sensitive to intervening absorption, and that generally selects a broader range of AGN luminosities, than purely optical-based selections. BHM features a large core program aimed at eROSITA X-ray sources ( § 7.2.1) and a substantial complementary program following up Chandra archival X-ray source catalogs ( § 7.2.2).', '7.2.1. Spectroscopic Identification of eROSITA Sources': "The SPectroscopic Identification of ERosita Sources (SPIDERS) program is core to BHM science and will help reveal the connections between statistical samples of X-ray emitting quasars/AGN and clusters of galaxies, and the large mass structures that they trace. The BHM SPIDERS program expands greatly on the SDSSIV SPIDERS program (e.g., Clerc et al. 2016; Dwelly et al. 2017; Comparat et al. 2020). \nThe main goal of the SPIDERS program (sometimes referred to as the '(Southern) Hemisphere' survey) is to provide complete and homogeneous optical spectroscopic follow-up of ∼ 10 5 . 5 X-ray sources detected by eROSITA, across ∼ 10,000 deg 2 at high Galactic latitude (nominally | b | > 15 deg), within the German or 'DE' half of the sky. Ultimately, SPIDERS, via its wide-area survey, aims to obtain the optical spectroscopic identifications and redshifts, evolution, and astrophysics of ∼ 250,000 counterparts to X-ray point-like sources. It is anticipated that AGN/quasar identifications will form the large majority of the targets, but with a significant minority of X-ray emitting stars (from compact binaries to coronal emitters). Eventually, SPIDERS will use X-ray sources from eROSITA's first 1.5 years of survey operation (termed 'eRASS:3' because it contains data from three passes over the whole sky), which were completed in June 2021. The SPIDERS 'AGN' project cartons have names that start with bhm spiders agn (Table 4). However, note that the initial SPIDERS targeting information in v0.5.3 released in DR18 is derived from the somewhat shallower first 6 months of eROSITA-DE observations ('eRASS:1'). \nAn additional SPIDERS project is targeting ∼ 10 4 Xray emitting clusters of galaxies ( > 5 × 10 4 galaxy targets), selected from a combination of X-ray imaging and \nthe eROSITA red-sequence Matched-filter Probabilistic Percolation galaxy cluster finding algorithm (eROMaPPeR; Rykoff et al. 2014; Ider Chitham et al. 2020) applied to multi-band wide+deep optical/IR imaging catalogs. For the SPIDERS clusters cartons (with names including bhm spiders clusters ), the new SDSS-V spectroscopy will provide redshifts and confirmations for the candidate clusters, as well as constraints on the velocity dispersion of the member galaxies. See Bulbul et al. (2022) for additional discussion on the complexity of selecting galaxy clusters in X-ray data sets. \nA related pilot survey that began in SDSS-IV, and extended into early SDSS-V, has already largely completed BOSS spectroscopy for ∼ 10 4 candidate counterparts in the eROSITA-eFEDS mini-survey field, a ∼ 140 deg 2 region of the sky. The optical spectra in eFEDS are a centerpiece of the DR18 data release, and are detailed further in § 9.1.", '7.2.2. Chandra Source Catalog': 'The Chandra Source Catalog (CSC) program is a more modest, complementary program of mainly optical spectroscopy of X-ray counterparts (but including some IR APOGEE spectra as well). The SDSS-V/CSC program produces identifications, redshifts, and other properties of tens of thousands of X-ray source counterparts selected from the Chandra Source Catalog 12 (Evans et al. 2010); again many are likely to be verified as AGN in their SDSS-V spectra. The CSC sample is expected to probe fainter X-ray fluxes than the SPIDERS Hemisphere samples. The CSC target cartons released as part of DR18 are based on CSC2.0. In the future we plan to switch to the more recent CSC2.1 catalog in order to increase the available pool of targets.', '7.3. The SDSS-V/eFEDS Mini-survey': "The SDSS-V/eFEDS survey is a pathfinder component of the BHM survey program that exploits early performance validation observations from the SRG/eROSITA X-ray telescope in the eFEDS field (Brunner et al. 2022). The eFEDS X-ray footprint comprises 140 deg 2 centered near ( α, δ ) = (9 h , +1 · ), encapsulating the 'GAMA09' Field (Driver et al. 2009). We used plate observations at APO to collect optical BOSS spectroscopy for counterparts to both point-like and extended X-ray sources (Salvato et al. 2022; Klein et al. 2022). From past experience (e.g., Menzel et al. 2016; LaMassa et al. 2019; Clerc et al. 2016, 2020), one can \nexpect these X-ray populations to be numerically dominated by AGN and clusters of galaxies, but we also expect a substantial minority to be associated with stellar coronal emitters or accreting compact objects. \nBelow we give a summary of the observational goals, target selection, and survey strategy for the SDSSV/eFEDS observations. Details of the plate design and the scope and quality of the observed dataset can be found in § 9.1.", '7.3.1. SDSS-V/eFEDS Observational goals': 'The primary observational goal for this program was to achieve near-complete and reliable redshifts and classifications for optical counterparts to X-ray sources that were detected as part of the eROSITA/eFEDS performance verification survey, particularly for counterparts in the magnitude range 16 < r < 22 AB. On one hand, this goal was constrained by the number of hours of dark observing time available to the project, a general desire to minimize the total number of drilled plug plates, and the overall capabilities of the plate system to place fibers on naturally clustered targets. On the other hand, this goal of high completeness was assisted by the wealth of previous spectroscopic survey data in the eFEDS field, both from previous SDSS generations and from other telescopes and instruments. Our strategy was therefore to prioritize targets that did not have existing high quality spectroscopic observations.', '7.3.2. Target selection for SDSS-V/eFEDS plates': "The generation of targeting used for the eFEDS plates, during the Dec 2020-May 2021 run of plate operations predates the global v0.5.3 targeting information that is being released in DR18. The parent catalogs from which the eFEDS targets were selected were derived from early reductions of the eROSITA/eFEDS X-ray dataset, and are based on early attempts to match those Xray sources to longer-wavelength counterparts provided by several supporting optical/IR catalogs, including the DESI Legacy Survey (DR8; Dey et al. 2019), SDSS DR13 (Albareti et al. 2017), and the Hyper SuprimeCam Subaru Strategic Project (HSC-SSP DR2; Aihara et al. 2019). The eFEDS science target selection process concentrates on AGN candidates (one target carton) and galaxy cluster candidates (four cartons): \nbhm spiders agn-efeds : A carton that contains candidate AGN targets found in the eROSITA/eFEDS X-ray survey field. This carton provides optical counterparts to point-like (unresolved) X-ray sources detected in early reductions (eROSITA Science Analysis Software, eSASS, version 'c940/V2T') of the eROSITA performance \nTable 4. Black Hole Mapper Core Cartons 1 \nNote - \nvalidation survey data in the eFEDS field. The sample is expected to contain a mixture of QSOs, AGN, stars and compact objects. The X-ray sources have been cross-matched by the eROSITADE team to the Legacy Survey 13 optical/IR counterparts (Salvato et al. 2022). \nbhm spiders clusters-efeds-ls-redmapper : A carton that contains galaxy cluster targets found in the eROSITA/eFEDS X-ray survey field. The carton provides a list of galaxies that are candidate members of clusters selected from early reductions (eSASS version 'c940') of the eROSITA performance verification survey data. The parent sample of galaxy clusters and their member galaxies have been selected via a joint analysis of X-ray and (multiple) optical/IR datasets using the eROMaPPeR red-sequence finder algorithm (Rykoff et al. 2014; Ider Chitham et al. 2020). This particular carton relies on optical/IR data from the DESI Legacy Surveys (Dey et al. 2019).", 'bhm spiders clusters-efeds-sdss-redmapper :': 'Similar to the bhm spiders clusters-efeds-lsredmapper carton, except that the red sequence finder algorithm is run on the SDSS DR13 photometric catalogue (Albareti et al. 2017).', 'bhm spiders clusters-efeds-hsc-redmapper :': 'Similar to the bhm spiders clusters-efeds-lsredmapper carton, except that the red sequence finder algorithm is run on the HSC-SSP DR2 photometric catalogue (Aihara et al. 2019). \nbhm spiders clusters-efeds-erosita : This carton includes manually-identified counterparts to Xray extended sources that were not also selected by the eROMaPPeR algorithm, when applied to any of the DESI Legacy Survey, SDSS-DR13, or HSC-SSP datasets. \nAdditional details on these cartons can be found in Appendix B and at https://www.sdss.org/dr18/bhm/ programs/cartons/.', '8. MOS OPEN-FIBER PROGRAMS': "The hardware and the survey strategy for SDSS-V's MOS programs were designed around a set of core science cases that define this part of the survey. The underlying target densities in these science cartons vary dramatically across the sky, from star-rich Baade's Window towards the Galactic bulge to sparse high galactic latitudes. In addition, the FPS is subject to geometric constraints in placing fibers within a pointing's field of view: in a given individual exposure, fibers cannot be placed closer than ∼ 50 '' to each other, and given the limited patrol radius of each robot - the mean fiber density across the field of view cannot be \nchanged. These constraints imply that after survey optimization for SDSS-V's core science, many robots with either BOSS or APOGEE fibers remain unallocated. After simulating the entire survey, over three million anticipated fiber-exposures remained available. \nIn light of this, SDSS-V set up an 'open fiber' program, based on multiple collaboration-wide calls for ideas and proposals, that serves three purposes. First, and most immediately, to make sure that as few fibers as possible go scientifically unused. Second, to provide a mechanism to broaden the science case beyond the initial, defining science cases. And third, to provide a mechanism, through repeated open fiber proposal calls, to incorporate scientifically exciting targets that only become known or available as the survey goes on. \nIn 2020, SDSS-V issued a first call for open fiber proposals from the consortium, whose results have entered the targeting documented here in various ways. Boundary conditions for these proposals were that i) the science goals could be reached with single 15 minute exposures using either APOGEE or BOSS; ii) the targeting information was freely available to all consortium members, and a target selection function could be specified; iii) the existing core science goals were complemented or enhanced; and iv) the science goals had proponents within the SDSS-V consortium. \nThe project received nearly 30 proposals in late 2020. Much of the proposed science fell into one of three broad categories: 1) more extensive surveying of AGNs and Xray sources across the sky; 2) broader stellar physics, in particular binary star physics, across the CMD; and 3) studying the Milky Way's stellar halo and metal-poor population, which (intentionally) had not been a focus of the MWM Galactic Genesis program ( § 6). These programs were reviewed in early 2021 by the SDSS-V technical advisory group and the project leadership, who called on three consortium members not involved in the proposals to help with the review. The resulting target cartons are listed in Table 5. For the v0.5.X versions of the targeting, including the v0.5.3 generation released in DR18, we adopted these cartons as-is. In future versions of the target selection, a number of these cartons may be consolidated and incorporated into the versioned science programs of the BHM and MWM surveys.", '9. SPECTRA': 'This section describes the spectra being released in DR18, particularly the eFEDS survey and its data products.', '9.1. The SDSS-V/eFEDS mini-survey': 'The scientific objectives and carton design of the eFEDS program can be found in § 7.3. Below we describe the observational design details and the final data set.', '9.1.1. Tiling and plate design': 'Due to COVID-19-induced delays in the completion of the FPS systems, SDSS-V started observations at APOusing the existing fiber plugplate system, including the eFEDS observations. Modifications to the existing system included a new joint BOSS+APOGEE configuration, in which the focal plane was populated with 500 fibers feeding a BOSS optical spectrograph and 300 fibers feeding an APOGEE IR spectrograph. New procedures and software were put in place to manage target selection, fiber assignment and plate design for the single season of SDSS-V data that were obtained in this mode. These procedures will be described in more detail in a future publication (Covey et al. in prep). Here we provide a summary of special considerations that apply specifically to the SDSS-V/eFEDS plates. \nEach plate-based observing run comprises a set of plates created to observe a set of targets from a given field (a region of the sky) and drilled to observe around a specific hour angle (to account for atmospheric refraction effects). The targets on a given plate were assigned using a specific combination of cartons ( § 7.3.2). For each plate, we created a prioritized list of cartons (including cartons associated with calibration targets such as skies and spectrophotometric standards, § 5.2) to fill the fibers for each spectrograph. Individual fiber filling rules were shared by multiple plates in the same observing run or even across different runs; different fiber filling rules are distinguished by shifting in priority of a carton with respect to another, or an update on the version of the carton that was used. Each combination of fiber filling rule and unique set of targets was identified with a designID . Figure 5 shows the sky locations of the SDSS-V/eFEDS sources in relation to the other data sets available in the region. \nSDSS-V/eFEDS plates were designed within two independent iterations of the tiling and plate design process. The first of these plate runs ( 2020.11.a.bhm-mwm ) consisted of 31 initial plates spanning most of the ∼ 140 deg 2 eFEDS field, with an inner region (roughly corresponding to the GAMA09 field) targeted with two plates per sky position. The large fractional overlap between the eFEDS plates required special attention to ensure proper control over which targets were prioritized in which plates, as the standard SDSS platedesign software treats all plates independently. This complexity, and constraints on the fiber reach of the plate fibers, re- \nTable 5. Open Fiber Cartons \nsulted in only 20 of the 2020.11.a.bhm-mwm plates being available for observation. A second eFEDS plate run ( 2021.01.a.bhm-mwm ) was designed to recover eFEDS targets from the unobservable plate designs and to take advantage of updated projections of the available observing time and plug-plate manufacturing resources. A new heuristic tiling algorithm resulted in all 17 plates from this run being observed. \nFor each of these 37 SDSS-V/eFEDS plates, we reserved at least 80 BOSS fibers per plate for skies and 20 BOSS fibers for spectrophotometric standard stars, leaving up to 400 BOSS fibers for science targets 14 . We applied a bright limit for science targets of g psf , r psf , i psf > 16 . 5 AB, and we avoided placing fibers near brighter stars. These bright limits are imposed to allow observation of the faint end of the target population (Figure 6) without significant degradation from on-chip cross-talk between neighbouring BOSS fiber traces.', '9.1.2. Spectroscopic observations, reductions, and data quality': "The SDSS-V/eFEDS observing strategy prioritized coverage over depth, given the somewhat uncertain amount of good quality observing time that would be available whilst the eFEDS field was visible from APO. By the end of the visibility window (Dec 2020-May 2021), all 37 eFEDS plates had achieved a total fiducial g -band signal-to-noise ratio squared (SNR 2 g ) of at least 7.5. However, most of the plates are significantly deeper than that, with 33/37 of the eFEDS plates having SNR 2 g > 10 (comparable to the typical exposure depth in BOSS/eBOSS; Dawson et al. 2016), and more than half the plates having SNR 2 g > 20. The fiducial SNR 2 in the i -band is typically twice that measured in the g -band. An updated version of the BOSS idlspec2d data reduction pipeline (Bolton et al. 2012, see § 10.1 for updates) was used to generate a set of spectral data products per PLATE-MJD combination, with individual 1D spectra for each combination of PLATE-MJD-CATALOGID . \nMany science targets were observed on more than one PLATE-MJD combination. Therefore, a specialized coadding algorithm (within idlspec2d ) was additionally implemented for these eFEDS plates, collating spectroscopic data across plates and MJDs, in order to increase SNR in the 1D spectra ( § 10.1). This routine creates a single stacked spectrum per astrophysical object, using all suitable SDSS-V BOSS data. A total of 13269 specially coadded spectra from eFEDS science targets \nare released in DR18, along with 2608 skies, 671 spectrophotometric standards, and several hundred unvetted spectra not intended for scientific use ( § 9.2). \nRedshifts, classifications, and quality flags were computed automatically on the specially coadded spectra using the spec1d template fitting pipeline (Bolton et al. 2012). A warning flag was set for 669/13269 science spectra, indicating a problem with the data or the fit. Of the 12600 science spectra without warnings, the main pipeline gives source classifications of 'QSO' for 6204, 'GALAXY' for 4782 and 'STAR' for 1614 spectra. Pipeline redshifts for non-flagged 'QSO' and 'GALAXY' classified spectra span the range 0 . 0 < z < 4 . 5, with 90% of these objects falling in the range 0 . 14 < z < 2 . 55 (median 0.55). \nFigure 6 shows the derived redshift distribution for good quality ( ZWARNING =0) spectra obtained from the SDSS-V/eFEDS plates, grouped by their primary target selection criteria. The observed redshift distributions are broadly in line with expectations. SPIDERS AGN targets span a wide range of redshifts (0 < z < 4 . 5, median 0.8), including a small fraction (10.0%) that are revealed by spectroscopy to be stellar in nature. SPIDERS galaxy cluster targets are found at relatively lower redshift (0 . 1 < z < 0 . 6, median 0.28). \nIn order to increase the completeness and reliability of the redshifts and classifications derived from these spectra, we carried out visual inspections of a large fraction of the SDSS-V/eFEDS spectra. Those inspections have been combined with earlier SDSS spectroscopic redshift information and with non-SDSS redshift information gathered from the literature. Please see Merloni et al., (in prep.), and the associated Value Added Catalog 15 ( § 11.2) for further details of the eFEDS spectroscopic compilation. \nThis data set has enabled in-depth study of the clustering of X-ray selected AGN, resulting in new constraints on the AGN halo occupation distribution (Comparat et al. 2023).", '9.1.3. Accessing eFEDS Data': "To download the eFEDS spectra, go to https://dr18. sdss.org/sas/dr18/spectro/boss/redux/eFEDS/; the associated data models are at https://data.sdss.org/ datamodel/files/BOSS SPECTRO REDUX/RUN2D/ and subdirectories. Note that these spectra were generated using the specialized co-adding scheme described in § 9.1.2. The 'standard' coadds can be found at https: //dr18.sdss.org/sas/dr18/spectro/boss/redux/v6 0 4/. To download the eFEDS VAC of improved redshifts \nFigure 5. SDSS spectroscopic coverage in the eFEDS field ( § 9.1), color-coded by the SDSS phase when the spectrum was taken. Green points show the locations of individual SDSS/BOSS spectra made available in DR16 or before; orange corresponds to SDSS-IV/eFEDS BOSS spectroscopy released in DR17, and blue shows the new optical BOSS spectroscopy being released in DR18. X-ray-selected galaxy clusters, specifically targeted in DR17 and DR18, are readily visible as compact concentrations of points. The approximate bound of the eROSITA X-ray data is shown with the thick black line. The southwest (lower right) corner is empty of DR16 spectroscopy because it lies outside the nominal SDSS imaging footprint. The map is shown on a tangential projection, in Equatorial coordinates, with units of degrees. \n<!-- image --> \n( § 11.2), visit the DR18 VAC website: https://www. sdss.org/dr18/data access/value-added-catalogs/.", '9.2. Other Spectra': 'Users of DR18 may come across science spectra (i.e., not standard stars or sky targets) that were included in the eFEDS plates ( § 9.1) but that are not associated with eROSITA/eFEDS target cartons. These data are included in DR18 for logistical reasons but are not intended for scientific exploitation (e.g., they have not undergone any quality assurance testing). They can be identified by one of the following FirstCarton labels in the database: \n- · mwm cb cvcandidates\n- · mwm cb gaiagalex\n- · mwm cb uvex1\n- · mwm cb uvex2\n- · mwm cb uvex3\n- · mwm cb uvex4\n- · mwm cb uvex5\n- · mwm halo bb\n- · mwm halo sm\n- · mwm snc 100pc', '· mwm wd': 'Users can expect improved and vetted versions of these spectra in DR19.', '10. SOFTWARE': 'SDSS operations, data processing, and analysis rely on a large suite of software components. A limited number of these, new or updated for SDSS-V, have already been released or are being released as part of DR18.', '10.1. Updates to the BOSS Spectral Reduction Algorithms': "The optical BOSS data released in DR18 ( § 9.1) are processed with version v6 0 4 of the BOSS pipeline software idlspec2d (Bolton et al. 2012; Dawson et al. 2013, Morrison et al. in prep). This is the first official version of idlspec2d to be used for SDSS-V, with a few changes since the last full public release ( v5 13 2 ) of SDSS-IV/eBOSS. All SDSS-V versions of idlspec2d are available for download from the SDSS GitHub, with the version described here available at https://github. com/sdss/idlspec2d/releases/tag/v6 0 4. \nOne of the most significant operational changes to the pipeline is a modification to reduce 500 fibers from a sin- \nFigure 6. Upper panel: Optical magnitude distribution of BOSS science targets in the SDSS-V/eFEDS plates ( § 9.1). Note that target magnitudes in SDSS-V are synthesized from a range of input photometric systems; in this case we show the equivalent of an r -band 'fiber2mag' (i.e., the light falling into a 2 arcsec diameter fiber under nominal APO seeing conditions). Lower panel: Distribution of pipeline-determined redshifts for science targets in the SDSS-V/eFEDS plate reductions ( § 9.1.2). Spectra with redshift fitting warnings have been removed. Redshifts outside the range of the plots are included in the extremal bins. Target selection for SPIDERS AGN and Clusters is described in § 7.3.2, and for other targets in § 9.2. \n<!-- image --> \ngle BOSS spectrograph, rather then a combined reduction of 1000 fibers from the twin BOSS spectrographs. This change was made to support the move of the second BOSS spectrograph to LCO. \nA number of smaller changes were implemented into the pipeline to both improve ongoing reductions and facilitate future reductions of FPS/BOSS data. First, to improve the radial velocity precision of the extracted spectra, a refined arc lamp linelist (developed for the SDSS-IV MANGA project; Bundy et al. 2015) replaced the list from SDSS-IV, and skylines are utilized as a second-level calibrator. This second step required an adjustment to the outlier exclusion algorithm (designed to exclude cosmic rays) to preserve the peaks of the skylines by increasing the outlier rejection from 4 σ to 50 σ . \nThis change, in some rare cases, might leave the remnant of a cosmic ray in the extracted spectra. \nSecond, we modified the coadding scheme of targets between exposure frames and the red/blue cameras; individual spectra are now grouped according to the RA,Dec coordinates of the fiber(s) (or the SDSS-V CatalogID, if one exists), rather than on the combination of BOSS PLATE-MJD-FIBERID . The coadding scheme was also split into a two step process, in which the red and blue data for each exposure are combined, and then the all-exposures red and blue data are combined to produce coadds for the full observation epoch. These changes were implemented to improve the signal-to-noise ratio of the final reduced spectra. \nThird, v6 0 4 was modified to produce a special version of the DR18 SDSS-V/eFEDS spectra ( § 9.1.2), in addition to the existing standard pipeline results. This new combination includes every exposure of each target, regardless of the plate on which it was observed, to produce a set of spectra with maximum signal-to-noise. Both the standard pipeline, v6 0 4 , and the eFEDS special coadds are available through the SAS 16 ; and the CAS using SkyServer 17 . Future data releases will include variations of this special coadding routine, with the spectra included in each coadd chosen to fulfil the science requirements of the individual SDSS-V science programs. \nFourth, in addition to the changes in the reduction and coadding schemes, some modifications were made to the spec1d analysis part of idlspec2d . Utilizing 849 SDSS-IV RM quasars (Shen et al. 2019) and the Weighted empca package (Bailey 2012, 2016), a new set of QSO PCA templates are used for PCA reconstruction and redshift estimation of SDSS-V quasars. These new PCAtemplates are implemented with 10 eigenvectors instead of the 4 used in SDSS-I through SDSS-IV. To further support the MWM targets on the reduced eFEDs plates, on the observed MWM plates (not released in DR18 but documented here for completeness), and in future FPS observations, the SFD model for dust extinction (Schlegel et al. 1998; Schlafly & Finkbeiner 2011) for MWM plates was replaced by the 3D Bayestar2015 dust map (Green et al. 2015) to properly account for differential dust extinction within the Milky Way. The package pyXCSAO (Kounkel 2022), a python replication of the xcsao package (Kurtz et al. 1992; Mink & Kurtz 1998; Tonry & Davis 1979), was added as part of the spec1d analysis pipeline for cross-correlating a spectrum \nagainst template spectra of known velocities. While this step is run for all targets, the results are only valid for stellar targets. \nFinally, idlspec2d 's internal Python dependencies were updated to Python3, with the deprecation of Python 2.7. All of changes described here improve either data quality or spectroscopic classification success rates, or support the changes to the BOSS spectrograph for the SDSS-V plate program and FPS observations.", '10.2. Migration to GitHub': 'For SDSS-V, all of the SDSS software has been moved to a GitHub Organization 18 , from the previous Subversion private repository. With the exception of a few repositories containing proprietary code or credentials, all the code is publicly available under a BSD 3-Clause license. This includes most of the operational software, telescope and instrument control software, and data reduction pipelines.', '10.3. MOS Software for DR19': "The migration from plug-plates to robotic fiber positioners in SDSS-V required the development of a suite of new software for targeting and operations. Although this software was not used for the acquisition of DR18's spectra ( § 9.1), it is currently in use for FPS operations and will become relevant for DR19 and future data releases. Here we provide a short summary of FPS targeting and operations as a preview for those data releases. Further details are given in S'anchez-Gallego et al. (2020). \nCombining the complex set of observing constraints and epoch cadences defined by the various target selection cartons into a set to observable FPS configurations requires the use of an specialized algorithm. robostrategy is the pipeline that takes the outputs of target selection and determines how the targets should be observed. It determines the cadence with which each field will be observed, which includes the number of epochs, the number of observations ('designs') per epoch, and the desired observing conditions for each design (including, e.g., the sky brightness conditions and hour angle). At the beginning of each night of observations, the software roboscheduler selects the optimal list of designs to be observed based on cadence requirements and observing conditions, as well as on previous observations and achieved completion. \nOne major challenge of the SDSS robotic fiber positioner system derives from the fact that the patrol radius of each positioner overlaps with its neighboring \nones. For a given set of target-to-positioner assignments defined in a robostrategy design, an algorithm named Kaiju (Sayres et al. 2021) provides a deterministic trajectory for each robotic positioner from a starting 'folded' state, in which all robots are identically oriented, to the desired FPS configuration. Kaiju works following a reverse-solve method in which it is considered simpler to calculate collision-free trajectories for each robot from a complex, deployed state, to a latticelike folded configuration. To create a path between two observable configurations Kaiju calculates the reverse trajectories from each configuration to the folded state, and then applies the reverse trajectory. Following this approach Kaiju is able to determine deadlock-free trajectories between any two physically reachable robot configurations in over 99% of cases (Sayres et al. 2021).", '11. VALUE-ADDED CATALOGS': "In addition to its fundamental data products, SDSS regularly includes 'Value Added Catalogs' as part of its data releases 19 . These catalogs contain curated quantities from the data, usually to enable a specific scientific projects, but recognized as having value above and beyond the particular project for which the catalog was initially constructed.", '11.1. SDSS-IV VACs': "Table 6 includes the following VACs that rely on SDSS-IV data but are updated or published for the first time after DR17: \nBACCHUS Analysis of Weak Lines in APOGEE Spectra (BAWLAS) : Weak-lined species in APOGEE spectra are challenging to measure and in some cases, cannot be done with the standard ASPCAP pipeline. This VAC contains approximately 120,000 high-SNR giants from APOGEE DR17, for which Na, P, S, V, Cu, Ce, and Nd abundances and 12 C/ 13 C isotopic ratios have been derived with a specialized analysis (Hayes et al. 2022). An updated version of the code BACCHUS (Brussels Automatic Code for Characterizing High accUracy Spectra; Masseron et al. 2016) was used in the derivation, along with ASPCAP fundamental stellar parameters but also a dedicated set of individual line quality flags and a new prescription for identifying upper limits. Hayes et al. (2022) show how these newly derived abundances compare with literature values and provide examples of scientific projects that can be done with APOGEE and the new catalog, ranging from nuclear \nTable 6. New or Updated Value Added Catalogs \nphysics to Galactic chemical evolution and Milky Way population studies. \nMaNGA Dwarf Galaxy Sample (MaNDala): This VAC presents properties for a sample of 125 dwarf galaxies ( M ⋆ < 10 9 M ⊙ ) observed with MaNGA (Cano-D'ıaz et al. 2022), including newly derived photometric results, such as surface brightness, color, and position angle. Also measured are photometric profiles along with S'ersic fits and new estimations of effective radii for all of the galaxies, using DECaLS DR9 g, r, z images (Dey et al. 2019). Analysis of the MaNGA data provides estimates of M ⋆ , SFRs, metallicities, ages, and other properties, using MaNGA Pipe3D data products (S'anchez et al. 2016, 2018). Details for the MaNDala sample and analysis are given in Cano-D'ıaz et al. (2022). \nMaNGA Visual Morphology : This is an update of a VAC, originally released in DR17 (Section 5.5.2 in Abdurro'uf et al. 2022), that contains visual morphological classifications for MaNGA galaxies based on SDSS and DECaLs (Dey et al. 2019) images. See V'azquez-Mata et al. (2022) for details.", '11.2. SDSS-V VACs': 'The only VAC in DR18 derived from SDSS-V data is that providing supplementary properties for the eFEDS spectroscopic data set ( § 7.3). \nThis VAC provides updated redshift and classification information for optical counterparts to X-ray sources detected in the eFEDS field (Brunner et al. 2022; Salvato et al. 2022). We update the spectroscopic redshift and classifications (with respect to Salvato et al. 2022) using a large spectroscopic compilation dominated by SDSS optical spectroscopy. Most importantly, we include new information from the 37 dedicated SDSS-V/eFEDS plates, described in detail in § 9.1. We combine automated redshifts and classifications, derived from the BOSS idlspec1d pipeline ( § 10.1), with an extensive set of visual inspections, which together increase the reliability and completeness of the spectroscopic coverage. \nThe VAC includes three separate catalogs: \n- i) eFEDS Main speccomp - an update of redshift and classification information for the eFEDS Main (0.2-2.3 keV selection) source counterparts catalog;\n- ii) eFEDS Hard speccomp - an update of redshift and classification information for the eFEDS Hard band (2.3-5 keV selection) source counterparts catalog; and\n- iii) eFEDS SDSSV spec results - a catalog of spectroscopic redshifts and classifications derived solely from the SDSS-V/eFEDS plate data set, supplemented by the results of an extensive visual inspection process. \nA full description of this VAC will be provided in Merloni et al. (in prep).', '12. SUMMARY AND DR19 PREVIEW': 'In this paper, we have described the contents of the first data release of SDSS-V, which is also the eighteenth data release of the SDSS family of surveys. DR18 contains extensive targeting information for the Milky Way Mapper ( § 6) and Black Hole Mapper ( § 7), both compiled target catalogs and the selection algorithms for their numerous scientific programs. Nearly ∼ 25,000 new spectra for X-ray sources in the eROSITA eFEDS field are also made available, along with substantial supplementary information, including improved redshift and classifications based on visual inspections. Significant improvements to the BOSS data spectral reduction pipelines were also made, in support of these and other optical spectra. \nThe next SDSS-V data release will be DR19, anticipated in 2024. This will contain the first MWM spectra and derived stellar parameters & abundances, and the first spectra and derived properties from the primary BHM programs. Updates to the targeting catalogs presented in this paper will also be made available, along \nwith updated simulations that predict the number of candidate targets in a carton that are likely be to observed. Additional tools for accessing and exploring all of the data are also anticipated to be released. \nThe capabilities and flexibility of SDSS-V make it unique among recent, ongoing, or imminent surveys, with well-vetted hardware, software, and collaboration infrastructure serving as the foundation for new innovations in all three areas. SDSS-V looks forward to expanding the SDSS legacy with high-quality, optical IFU data of 1000 deg 2 and optical + infrared spectra of millions of sources across the entire sky.', 'ACKNOWLEDGEMENTS': "The documentation workshop for DR18 ('DocuLlama') was held virtually in September 2022, organized by Anne-Marie Weijmans and Gail Zasowski. This event was the main venue for the documentation of DR18, including significant progress on this paper and the website, and was attended by Scott Anderson, Joel Brownstein, Joleen Carlberg, Niall Deacon, Nathan De Lee, John Donor, Tom Dwelly, Keith Hawkins, Jennifer Johnson, Sean Morrison, Jordan Raddick, Jos'e S'anchezGallego, Diogo Souto, Taylor Spoo, Ani Thakar, Nick Troup, Anne-Marie Weijmans, Gail Zasowski, William Zhang, three llamas, and an elderly goat named Nibblets. \nFunding for the Sloan Digital Sky Survey V has been provided by the Alfred P. Sloan Foundation, the Heising-Simons Foundation, the National Science Foundation, and the Participating Institutions. SDSS acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. The SDSS web site is www.sdss.org. \nSDSS is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration, including the Carnegie Institution for Science, Chilean National Time Allocation Committee (CNTAC) ratified researchers, the Gotham Participation Group, Harvard University, The Johns Hopkins University, L'Ecole polytechnique f'ed'erale de Lausanne (EPFL), Leibniz-Institut fur Astrophysik Potsdam (AIP), Max-Planck-Institut fur Astronomie (MPIA Heidelberg), Max-Planck-Institut fur Extraterrestrische Physik (MPE), Nanjing University, National Astronomical Observatories of China (NAOC), New Mexico State University, The Ohio State University, Pennsylvania State University, Smithsonian Astrophysical Observatory, Space Telescope Science Institute (STScI), the Stellar Astrophysics Participation Group, Universidad Nacional Aut'onoma de M'exico, University of Arizona, University of Colorado Boulder, University of Illinois at \nUrbana-Champaign, University of Toronto, University of Utah, University of Virginia, and Yale University. \nThe Pan-STARRS1 Surveys (PS1) and the PS1 public science archive have been made possible through contributions by the Institute for Astronomy, the University of Hawaii, the Pan-STARRS Project Office, the MaxPlanck Society and its participating institutes, the Max Planck Institute for Astronomy, Heidelberg and the Max Planck Institute for Extraterrestrial Physics, Garching, The Johns Hopkins University, Durham University, the University of Edinburgh, the Queen's University Belfast, the Harvard-Smithsonian Center for Astrophysics, the Las Cumbres Observatory Global Telescope Network Incorporated, the National Central University of Taiwan, the Space Telescope Science Institute, the National Aeronautics and Space Administration under Grant No. NNX08AR22G issued through the Planetary Science Division of the NASA Science Mission Directorate, the National Science Foundation Grant No. AST-1238877, the University of Maryland, Eotvos Lorand University (ELTE), the Los Alamos National Laboratory, and the Gordon and Betty Moore Foundation. \nThis publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. \nThis work is based in part on observations made with the Spitzer Space Telescope, which was operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. \nThis publication makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. \nThis work presents results from the European Space Agency (ESA) space mission Gaia. Gaia data are being processed by the Gaia Data Processing and Analysis Consortium (DPAC). Funding for the DPAC is provided by national institutions, in particular the institutions participating in the Gaia MultiLateral Agreement (MLA). The Gaia mission website is https:// www.cosmos.esa.int/gaia. The Gaia archive website is https://archives.esac.esa.int/gaia. \nThe Legacy Surveys consist of three individual and complementary projects: the Dark Energy Camera Legacy Survey (DECaLS; Proposal ID #2014B-0404; PIs: David Schlegel and Arjun Dey), the Beijing- \nArizona Sky Survey (BASS; NOAO Prop. ID #2015A0801; PIs: Zhou Xu and Xiaohui Fan), and the Mayall z-band Legacy Survey (MzLS; Prop. ID #2016A-0453; PI: Arjun Dey). DECaLS, BASS and MzLS together include data obtained, respectively, at the Blanco telescope, Cerro Tololo Inter-American Observatory, NSF's NOIRLab; the Bok telescope, Steward Observatory, University of Arizona; and the Mayall telescope, Kitt Peak National Observatory, NOIRLab. Pipeline processing and analyses of the data were supported by NOIRLab and the Lawrence Berkeley National Laboratory (LBNL). The Legacy Surveys project is honored to be permitted to conduct astronomical research on Iolkam Du'ag (Kitt Peak), a mountain with particular significance to the Tohono O'odham Nation. \nNOIRLab is operated by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. LBNL is managed by the Regents of the University of California under contract to the U.S. Department of Energy. \nThis project used data obtained with the Dark Energy Camera (DECam), which was constructed by the Dark Energy Survey (DES) collaboration. Funding for the DES Projects has been provided by the U.S. Department of Energy, the U.S. National Science Foundation, the Ministry of Science and Education of Spain, the Science and Technology Facilities Council of the United Kingdom, the Higher Education Funding Council for England, the National Center for Supercomputing Applications at the University of Illinois at UrbanaChampaign, the Kavli Institute of Cosmological Physics at the University of Chicago, Center for Cosmology and Astro-Particle Physics at the Ohio State University, the Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M University, Financiadora de Estudos e Projetos, Fundacao Carlos Chagas Filho de Amparo, Financiadora de Estudos e Projetos, Fundacao Carlos Chagas Filho de Amparo a Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de Desenvolvimento Cientifico e Tecnologico and the Ministerio da Ciencia, Tecnologia e Inovacao, the Deutsche Forschungsgemeinschaft and the Collaborating Institutions in the Dark Energy Survey. The Collaborating Institutions are Argonne National Laboratory, the University of California at Santa Cruz, the University of Cambridge, Centro de Investigaciones Energeticas, Medioambientales y Tecnologicas-Madrid, the University of Chicago, University College London, the DESBrazil Consortium, the University of Edinburgh, the Eidgenossische Technische Hochschule (ETH) Zurich, Fermi National Accelerator Laboratory, the University \nof Illinois at Urbana-Champaign, the Institut de Ciencies de l'Espai (IEEC/CSIC), the Institut de Fisica d'Altes Energies, Lawrence Berkeley National Laboratory, the Ludwig Maximilians Universitat Munchen and the associated Excellence Cluster Universe, the University of Michigan, NSF's NOIRLab, the University of Nottingham, the Ohio State University, the University of Pennsylvania, the University of Portsmouth, SLAC National Accelerator Laboratory, Stanford University, the University of Sussex, and Texas A&M University. \nBASS is a key project of the Telescope Access Program (TAP), which has been funded by the National Astronomical Observatories of China, the Chinese Academy of Sciences (the Strategic Priority Research Program 'The Emergence of Cosmological Structures' Grant #XDB09000000), and the Special Fund for Astronomy from the Ministry of Finance. The BASS is also supported by the External Cooperation Program of Chinese Academy of Sciences (Grant #114A11KYSB20160057), and Chinese National Natural Science Foundation (Grant #12120101003, #11433005). \nThe Legacy Survey team makes use of data products from the Near-Earth Object Wide-field Infrared Survey Explorer (NEOWISE), which is a project of the Jet Propulsion Laboratory/California Institute of Technology. NEOWISE is funded by the National Aeronautics and Space Administration. \nThe Legacy Surveys imaging of the DESI footprint is supported by the Director, Office of Science, Office of High Energy Physics of the U.S. Department of Energy under Contract No. DE-AC02-05CH1123, by the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility under the same contract; and by the U.S. National Science Foundation, Division of Astronomical Sciences under Contract No. AST-0950945 to NOAO. \nThe national facility capability for SkyMapper has been funded through ARC LIEF grant LE130100104 from the Australian Research Council, awarded to the University of Sydney, the Australian National University, Swinburne University of Technology, the University of Queensland, the University of Western Australia, the University of Melbourne, Curtin University of Technology, Monash University and the Australian Astronomical Observatory. SkyMapper is owned and operated by The Australian National University's Research School of Astronomy and Astrophysics. The survey data were processed and provided by the SkyMapper Team at ANU. The SkyMapper node of the All-Sky Virtual Observatory (ASVO) is hosted at the National Computational Infrastructure (NCI). Development and support \nof the SkyMapper node of the ASVO has been funded in part by Astronomy Australia Limited (AAL) and the Australian Government through the Commonwealth's Education Investment Fund (EIF) and National Collaborative Research Infrastructure Strategy (NCRIS), particularly the National eResearch Collaboration Tools and Resources (NeCTAR) and the Australian National Data Service Projects (ANDS). \nThis paper includes data collected by the TESS mission. Funding for the TESS mission is provided by the NASA's Science Mission Directorate. \nWe acknowledge the use of public data from the Swift data archive. \nBased on observations obtained with XMM-Newton, an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA. This research has made use of NASA's Astrophysics \nData System Bibliographic Services. \nFacilities: Du Pont (APOGEE), Sloan, Spitzer, WISE, 2MASS, Gaia, GALEX, PS1, TESS, Swift, CXO, XMM, eROSITA", 'REFERENCES': "Perruchot, S., Guy, J., Le Guillou, L., et al. 2018, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 10702, Ground-based and Airborne Instrumentation for Astronomy VII, ed. C. J. Evans, L. Simard, & H. Takami, 107027K, doi: 10.1117/12.2311996 \n- Pogge, R. W., Derwent, M. A., O'Brien, T. 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B., Horne, K., et al. 2019, ApJS, 241, 34, doi: 10.3847/1538-4365/ab074f", 'A. DETAILS OF MWM V0.5.3 TARGET CARTONS': "In this Appendix we provide the detailed criteria used to select the MWM target cartons that are being released as part of SDSS DR18. We remind the reader that inclusion in a carton means that the target fits the carton selection criteria, not that it is guaranteed to be observed. \nFor each target carton, we provide the following information: The Description of selection criteria provides a short summary of the carton selection in human-readable terms. It also includes a list of the limits in color, magnitude, parallax, and other quantities applied to the carton's target candidates 20 . Data Sources gives the catalogs from which these quantities are drawn. In Target priority options , we indicate which priority is given to targets in this carton for observing; smaller priorities are more likely to be assigned fibers. The Cadence options describes which exposure time requirement(s) are assigned to sources in the carton. See Table 3 for each carton's instrument (BOSS or APOGEE) and the number of candidate targets that meet the carton's selection criteria. \nA.1. manual mwm tess ob \nDescription of selection criteria: A system selected from the Ijspeert et al. (2021) catalog of massive eclipsing binaries in TESS. Systems chosen were in the TESS Continuous Viewing Zones (CVZs) and had TESS lightcurves that show pronounced eclipses and intrinsic variability. \nData Sources: \nIjspeert et al. (2021) \nTarget priority options: \n2200 \nCadence options: bright 8x1, bright 8x2, bright 8x4", 'A.2. manual nsbh apogee': 'Description of selection criteria: List of targets drawn from public and private catalogs of binary systems with known or suspected black holes or neutron stars, such as X-ray binaries and pulsars. Must have a valid 2MASS H magnitude. \nData Sources: 2MASS PSC (H), Gaia DR2 (G) \nTarget priority options: \n1400 \nCadence options: bright 1x1', 'A.3. manual nsbh boss': 'Description of selection criteria: List of targets drawn from public and private catalogs of binary systems with known or suspected black holes or neutron stars, such as X-ray binaries and pulsars. Must have a valid Gaia DR2 G magnitude. \nData Sources: 2MASS PSC (H), Gaia DR2 (G) \nTarget priority options: \n1400 \nCadence options: bright 1x1, dark 1x2', 'A.4. mwm cb 300pc apogee': 'Description of selection criteria: Bright compact binary candidates within 300 pc. \n- • ( FUV - 5 log 10 ( r est / 10)) > 14( FUV - NUV ) - 46\n- • r est < 300 pc\n- • H < 11 \nData Sources: Bailer-Jones et al. (2018) distance catalog ( r est ), 2MASS PSC (H), GALEX (FUV, NUV) \nTarget priority options: 1400 \nCadence options: bright 1x1 \nA.5. mwm cb 300pc boss \nDescription of selection criteria: Faint compact binary candidates within 300 pc. \n- • ( FUV - 5 log 10 ( r est / 10)) > 14( FUV - NUV ) - 46\n- • r est < 300 pc \nData Sources: Bailer-Jones et al. (2018) distance catalog ( r est ), GALEX (FUV, NUV) \nTarget priority options: 1400 \nCadence options: bright 2x1, dark 2x1 \nA.6. mwm cb cvcandidates apogee \nDescription of selection criteria: Bright AAVSO cataclysmic variables \n- • Target in mos cataclysmic variables table\n- • H < 11 \nData Sources: 2MASS PSC (H) \nTarget priority options: 1400 \nCadence options: bright 1x1 \nA.7. mwm cb cvcandidates boss \nDescription of selection criteria: Faint AAVSO cataclysmic variables \n- • Target in mos cataclysmic variables table\n- • H ≥ 11 \nData Sources: 2MASS PSC (H) \nTarget priority options: 1400 \nCadence options: bright 2x1, dark 2x1 \nA.8. mwm cb gaiagalex apogee \nDescription of selection criteria: Bright compact binary candidates using Gaia G and GALEX FUV color cut \n- • ϖ/σ ϖ > 3\n- • G < 20\n- • FUV +5log1 10 ( ϖ/ 1000) + 5 > 1 . 5 + 1 . 28( FUV - G )\n- • H < 11 \nData Sources: Gaia DR2 (G, ϖ , σ ϖ ), 2MASS PSC (H), GALEX (FUV) \nTarget priority options: 1400 \nCadence options: bright 1x1', 'A.9. mwm cb gaiagalex boss': "Description of selection criteria: Faint compact binary candidates using Gaia G and GALEX FUV color cut \n- · ϖ/σ ϖ > 3\n- · G < 20\n- · FUV +5log 10 ( ϖ/ 1000) + 5 > 1 . 5 + 1 . 28( FUV -G )\n- · H ≥ 11 \nData Sources: Gaia DR2 (G, ϖ , σ ϖ ), 2MASS PSC (H), GALEX (FUV) \nTarget priority options: 1400 \nCadence options: bright 2x1, dark 2x1 \nA.10. mwm cb uvex1 \nDescription of selection criteria: Gaia and GALEX cross-match, keeping the nearest match within 5 '' and removing objects with small proper motion and parallax. Color cuts utilize both FUV and NUV magnitudes. 'AB' indicates magnitudes transformed to AB magnitudes, and M G is Gaia absolute magnitude calculated with r est . \n- · r lo ≤ 1500\n- · g vis per > 5\n- · Neither of the following two conditions are met\n- -(log 10 ( µ/σ µ ) < 0 . 301) AND ( ϖ/σ ϖ > -1 . 4996 log 10 ( µ/σ µ )-4 . 05) AND ( ϖ/σ ϖ < 1 . 4995 log 10 ( µ/σ µ ) + 4 . 05)\n- -(log 10 ( µ/σ µ )-0 . 301) 2 / 0 . 39794 2 +( ϖ/σ ϖ ) 2 / 4 . 5 2 ≤ 1\n- · NUV > -100\n- · FUV > -100\n- · σ NUV < 0 . 2\n- · σ FUV < 0 . 2\n- · M G > 4 . 09 OR M G > 4 . 5457( G BP -G RP ) + 4 . 0457\n- · M G > -1 . 11749253 × 10 -3 ( FUV -NUV ) 3 +1 . 53748615 × 10 -2 ( FUV -NUV ) 2 +3 . 66419895 × 10 -1 ( FUV -NUV ) + 2 . 20026639)\n- · ( FUV -G AB ) < 6 . 08 OR [( FUV -G AB ) < 11 . 82( G BP , AB -G RP , AB ) + 2 . 58 AND ( FUV -G AB ) < -0 . 79( G BP , AB -G RP , AB +9 . 21)] \nData Sources: Gaia DR2 ( M G , G BP , G RP , µ , σ m u , ϖ , σ ϖ , r lo , g vis per), Bailer-Jones et al. (2018) distance catalog ( r est ), GALEX (NUV, FUV, σ NUV , σ FUV ) \nTarget priority options: 1400 \nCadence options: bright 1x1, dark 1x2, dark 1x3", 'A.11. mwm cb uvex2': "Description of selection criteria: Gaia and GALEX cross-match, keeping the nearest match within 5 '' and removing objects with small proper motion and parallax. Color cuts utilize only NUV magnitudes. 'AB' indicates magnitudes transformed to AB magnitudes, and M G is Gaia absolute magnitude calculated with r est . \n- · r lo ≤ 1500\n- · g vis per > 5\n- · Neither of the following two conditions are met\n- -(log 10 ( µ/σ µ ) < 0 . 301) AND ( ϖ/σ ϖ > -1 . 4996 log 10 ( µ/σ µ )-4 . 05) AND ( ϖ/σ ϖ < 1 . 4995 log 10 ( µ/σ µ ) + 4 . 05)\n- -(log 10 ( µ/σ µ )-0 . 301) 2 / 0 . 39794 2 +( ϖ/σ ϖ ) 2 / 4 . 5 2 ≤ 1\n- · NUV > -100\n- · σ NUV < 0 . 2\n- · M G > 4 . 09 OR M G > 4 . 5457( G BP -G RP ) + 4 . 0457\n- · ( NUV -G AB < 2 . 25) OR [( NUV -G AB < 6 . 725( G BP , AB -G RP , AB) -1 . 735) AND ( NUV -G AB < -0 . 983( G BP , AB -G RP , AB +8 . 24)] \nData Sources: Gaia DR2 ( M G , G AB , G BP , G BP , AB , G RP , G RP , AB , µ , σ m u , ϖ , σ ϖ , g vis per), Bailer-Jones et al. (2018) distance catalog ( r lo , r est ), GALEX (NUV, σ NUV ) \nTarget priority options: 1400 \nCadence options: \nbright 1x1, dark 1x2, dark 1x3 \nA.12. mwm cb uvex3 \nDescription of selection criteria: Gaia and XMM-Newton Optical Monitor SUSS Catalog cross-match, keeping the nearest match with 3 '' and removing objects with small proper motion and parallax. Color and quality cuts utilize the XMM UVM2 band. \n- · r lo ≤ 1500\n- · g vis per > 5\n- · Neither of the following two conditions are met\n- -(log 10 ( µ/σ µ ) < 0 . 301) AND ( ϖ/σ ϖ > -1 . 4996 log 10 ( µ/σ µ )-4 . 05) AND ( ϖ/σ ϖ < 1 . 4995 log 10 ( µ/σ µ ) + 4 . 05)\n- -(log 10 ( µ/σ µ )-0 . 301) 2 / 0 . 39794 2 +( ϖ/σ ϖ ) 2 / 4 . 5 2 ≤ 1\n- · The following conditions are all NOT met\n- -qflag bit 1, 7, 8, or 9 is set\n- -qflag bit 2, or 3 is set AND UVM2 signif < 10\n- · M G > 4 . 09 OR M G > 4 . 5457( G BP -G RP ) + 4 . 0457\n- · UVM2 AB -G AB < 2 . 25 OR [(UVM2 AB -G AB < 6) AND (UVM2 AB -G AB < 5 . 57377( G BP , AB -G RP , AB ) + 0 . 2049)] \nData Sources: Gaia DR2 ( M G , G AB , G BP , G BP , AB , G RP , G RP , AB , µ , σ m u , ϖ , σ ϖ , g vis per), Bailer-Jones et al. (2018) distance catalog ( r lo , r est ), XMM OM SUSS v4.1(UVM2 AB , UVM2 signif , qflag) \nTarget priority options: 1400 \nCadence options: bright 1x1, dark 1x2, dark 1x3 \nA.13. mwm cb uvex4 \nDescription of selection criteria: Gaia and Swift UVOT Catalog cross-match, keeping the nearest match with 3 '' and removing objects with small proper motion and parallax. Color cuts utilize the UVW2 band, and quality cuts utilize both UVW2 and UVW1 \n- · r lo ≤ 1500\n- · g vis per > 5\n- · UVW1 AB > -100\n- · UVW2 AB > -100\n- · Neither of the following two conditions are met\n- -(log 10 ( µ/σ µ ) < 0 . 301) AND ( ϖ/σ ϖ > -1 . 4996 log 10 ( µ/σ µ )-4 . 05) AND ( ϖ/σ ϖ < 1 . 4995 log 10 ( µ/σ µ ) + 4 . 05)\n- -(log 10 ( µ/σ µ )-0 . 301) 2 / 0 . 39794 2 +( ϖ/σ ϖ ) 2 / 4 . 5 2 ≤ 1\n- · The following conditions are all NOT met\n- -qflag bit 0 or 6 is set\n- -qflag bit 1, 2, or 5 is set AND UVW2 signif < 10\n- · M G > 4 . 09 OR M G > 4 . 5457( G BP -G RP ) + 4 . 0457\n- · UVW2 AB -G AB < 2 . 25 OR [(UVW2 AB -G AB < 6) AND (UVW2 AB -G AB < 5 . 57377(( G BP , AB -G RP , AB ) + 0 . 2049)] \nData Sources: Gaia DR2 ( M G , G AB , G BP , G BP , AB , G RP , G RP , AB , µ , σ m u , ϖ , σ ϖ , g vis per), Bailer-Jones et al. (2018) distance catalog ( r lo , r est ), Swift UVOT(UVW1 AB , UVW2 AB , UVW2 signif , qflag2) \nTarget priority options: 1400 \nCadence options: bright 1x1, dark 1x2, dark 1x3", 'A.14. mwm cb uvex5': "Description of selection criteria: Gaia and GALEX cross-match, keeping the nearest match within 5 '' and removing objects with small proper motion and parallax. Only objects with FUV and NUV detections are kept, but the color cuts utilize only optical colors and magnitudes. Here, the expected Gaia Main Sequence locus (GMS) is defined by GMS = 0 . 00206868742 x 6 +0 . 0401594518 x 5 -0 . 842512410 x 4 +4 . 89384979 x 3 -12 . 3826637 x 2 + 17 . 0197205 x -3 . 19987835, where x = G -G RP . M G is the Gaia absolute magnitude calculated with r est . \n- · r lo ≤ 1500\n- · g vis per > 5\n- · Neither of the following two conditions are met\n- -(log 10 ( µ/σ µ ) < 0 . 301) AND ( ϖ/σ ϖ > -1 . 4996 log 10 ( µ/σ µ )-4 . 05) AND ( ϖ/σ ϖ < 1 . 4995 log 10 ( µ/σ µ ) + 4 . 05)\n- -(log 10 ( µ/σ µ )-0 . 301) 2 / 0 . 39794 2 +( ϖ/σ ϖ ) 2 / 4 . 5 2 ≤ 1\n- · NUV > -100\n- · FUV > -100\n- · σ NUV < 0 . 2\n- · σ FUV < 0 . 2\n- · M G > 4 . 0866\n- · H < 15\n- · | GMS -M G | ≤ 0 . 5\n- · r est < 0 . 51 × 10 0 . 2291 G \nData Sources: Gaia DR2 ( M G , G RP , µ , σ m u , ϖ , σ ϖ , r lo , g vis per), Bailer-Jones et al. (2018) distance catalog ( r est ), GALEX (NUV, FUV, σ NUV , σ FUV ), 2MASS PSC (H) \nTarget priority options: \n1400 \nCadence options: bright 1x1, dark 1x2, dark 1x3", 'A.15. mwm dust core': "Description of selection criteria: The dust carton selects bright, nearby, midplane giants with the same quality cuts as Galactic Genesis (GG) to supplement GG targets in regions of high reddening. The Galaxy within 5 kpc and with | z | < 0 . 2 kpc is divided into 100 pc 3 regions ('voxels') and the number of GG targets in each voxel are counted. In voxels with fewer than 10 GG stars, 10 -n GG stars are selected for inclusion in this carton. \n- · σ ϖ /ϖ < 0 . 2\n- · 1 /ϖ < 5 kpc\n- · | z | < 0 . 2 kpc (using ϖ -based distances)\n- · M K < 2 . 6, adopting A K = 0 . 918( H -4 . 5)\n- · ( J -K ) 0 > 0 . 5, where E ( J -K ) = 1 . 5 A K\n- · H < 11 . 2\n- · gal contam == 0\n- · cc flg == 0\n- · 0 < rd flag < = 3\n- · ph qual flag is A or B \nData Sources: Gaia DR2 ( ϖ , σ ϖ ), 2MASS PSC (J, H, K, gal contam, cc flag, rd flag, ph qual) \nTarget priority options: 2720 \nCadence options: bright 1x1 \nDescription of selection criteria: Optical counterpart in Gaia DR2 to a candidate compact binary detected by eROSITA, chosen as the possible counterpart with the smallest angular separation \n- · G > 16\n- · L > 8 in at least one of the three eROSITA energy bands \n- · Detections in three eROSITA energy bands\n- · log ( f x /f opt ) < 2 . 7\n- · radec err > 0 and s / radec err < 2 . 1, where s is the separation between the Gaia and eROSITA sources\n- · Neither of the following two conditions are met\n- -(log 10 ( µ/σ µ ) < 0 . 301) AND ( ϖ/σ ϖ > -1 . 4996 log 10 ( µ/σ µ )-4 . 05) AND ( ϖ/σ ϖ < 1 . 4995 log 10 ( µ/σ µ ) + 4 . 05)\n- -(log 10 ( µ/σ µ )-0 . 301) 2 / 0 . 39794 2 +( ϖ/σ ϖ ) 2 / 4 . 5 2 ≤ 1\n- · Is the optical source that meets the above criteria with the smallest separation s \nData Sources: eROSITA ( f x , L , radec err), Gaia DR2 ( G , µ , σ m u , ϖ , σ ϖ , f opt (from G )) \nTarget priority options: 1910, 2400 \nCadence options: bright 1x1, dark 1x2, dark 1x3 \nA.17. \nmwm erosita compact var \nDescription of selection criteria: Optical counterpart to a candidate compact binary detected by eROSITA, chosen \nas the possible counterpart with the largest variability in Gaia DR2 \n- · G > 16\n- · L > 8 in at least one of the three eROSITA energy bands\n- · Detections in three eROSITA energy bands\n- · log ( f x /f opt ) < 2 . 7\n- · radec err > 0 and s / radec err < 2 . 1, where s is the separation between the Gaia and eROSITA sources\n- · Neither of the following two conditions are met\n- -(log 10 ( µ/σ µ ) < 0 . 301) AND ( ϖ/σ ϖ > -1 . 4996 log 10 ( µ/σ µ )-4 . 05) AND ( ϖ/σ ϖ < 1 . 4995 log 10 ( µ/σ µ ) + 4 . 05)\n- -(log 10 ( µ/σ µ )-0 . 301) 2 / 0 . 39794 2 +( ϖ/σ ϖ ) 2 / 4 . 5 2 ≤ 1\n- · Is the optical source that meets the above criteria with the largest variability log 10 ( ( phot g n obs phot g mean flux over error ) 1 / 2 ) \nData Sources: eROSITA ( f x , L , radec err), Gaia DR2 ( G , µ , σ m u , ϖ , σ ϖ , f opt (from G ), phot g n obs, phot g mean flux over error) \nTarget priority options: 1900, 2400 \nCadence options: bright 1x1, dark 1x2, dark 1x3 \nA.18. mwm erosita stars \nDescription of selection criteria: Optical or infrared counterpart to a candidate stellar source detected by eROSITA. Counterpart identified using the techniques and data described in Freund et al. (2022). \nData Sources: eROSITA, Gaia DR2, 2MASS PSC \nTarget priority options: 1920, 2400 \nCadence options: bright 1x1, dark 1x2, dark 1x3 \nA.19. mwm galactic core \nDescription of selection criteria: A simple color-magnitude cut effectively targets luminous cool giant stars (median(log g ) ∼ 1 . 0 -1 . 5) with little ( < 6%) contamination from dwarf stars. \n- · H < 11\n- · G -H > 3 . 5 or Gaia non-detection\n- · gal contam == 0\n- · cc flg == 0\n- · 0 < rd flag < = 3\n- · ph qual flag is A or B \nGaia DR2 (G), 2MASS PSC (H, gal contam, cc flg, rd flag, ph qual) \nData Sources: Target priority options: 2710 Cadence options: bright 1x1 \nA.20. mwm legacy ir2opt \nDescription of selection criteria: This carton selects all bright Gaia objects that are in APOGEE DR16 (in particular the sdss apogeeallstarmerge r13 file) to be observed with BOSS. \n- · 3 < G < 18\n- · G BP > 13\n- · G RP > 13. \nData Sources: Gaia DR2 (G, G , G \nTarget priority options: 6100 \nCadence options: \nBP RP ), APOGEE DR16 bright 1x1 \nA.21. mwm ob cepheids \nDescription of selection criteria: Catalog of Cepheids compiled by Inno et al. (2021) \nData Sources: Inno et al. (2021) \nTarget priority options: \n2910 \nCadence options: bright 3x1 \nA.22. mwm ob core \nDescription of selection criteria: This carton uses color and magnitude cuts to select hot, young stars. \n- · ϖ < 10 (10 -K -0 . 61) / 5)\n- · G < 16\n- · J -K -0 . 25( G -K ) < 0 . 10\n- · J -K -0 . 25( G -K ) > -0 . 30\n- · J -H < 0 . 15( G -K ) + 0 . 05\n- · J -H > 0 . 15( G -K ) -0 . 15\n- · J -K < 0 . 23( G -K ) + 0 . 03\n- · G > 2( G -K ) + 3 . 0\n- · Cross match separation < 1 ''\n- · RUWE < 1 . 4 \nData Sources: Gaia DR2 (G, ϖ , RUWE), 2MASS PSC (J, H, K) \nTarget priority options: 2910 \nCadence options: bright 3x1 \nA.23. mwm rv long fps \nDescription of selection criteria: This carton selects stars previously observed at least 3 times with the APOGEE instrument if it was targeted as part of the main APOGEE sample or the APOGEE-2 binary program. \n- · Presence in sdss apogeeallstarmerge r13 file (previously observed with APOGEE 1 and/or 2\n- · APOGEE number of visits ≥ 3\n- · H < 11 . 5\n- · APOGEE TARGFLAGS includes one of the following: APOGEE SHORT, APOGEE INTERMEDIATE, APOGEE LONG, APOGEE2 BIN \n- · Gaia-based distance \nData Sources: \nAPOGEE, 2MASS PSC (H), Gaia DR 2 \nTarget priority options: 2510, 2520, 2530, 2540 \nCadence options: bright 6x1 bright 6x2, bright 9x1, bright 9x2, bright 12x1, bright 12x2, bright 15x1, bright 15x2", 'A.24. mwm rv short fps': "Description of selection criteria: This carton selects stars that were never previously observed with APOGEE and applies the same color and quality cuts as the main APOGEE surveys. \n- · H < 10 . 8\n- · J -K s -(1 . 5 · 0 . 918( H -W 2-0 . 05)) > = 0 . 05\n- · J msigcom,H msigcom,K s msigcom ≤ 0 . 1\n- · W 2 sigmpro ≤ 0 . 1\n- · 2MASS ph qual any of the following: AAA, AAB, ABA, BAA, ABB, BAB, BBA, BBB\n- · 2MASS gal contam = 0\n- · 2MASS cc flg = 0\n- · 2MASS rd flg any of the following: 111, 112, 121, 211, 122, 212, 221, 222\n- · 2MASS prox ≥ 6\n- · Gaia DR2 parallax exists\n- · Does not match a source in the 2MASS extended catalog \nData Sources: 2MASS PSC (J, H, K s , J msigcom, H msigcom, K s msigcom, ph qual, gal contam, cc flg, rd flg, prox), Gaia DR 2, AllWise (W2) \nTarget priority options: 2515, 2525, 2535, 2545 \nCadence options: bright 18x1 \nA.25. mwm snc 100pc apogee \nDescription of selection criteria: Infrared bright targets in the 100 pc volume limited region. \n- · ϖ -σ ϖ > 10\n- · H < 11\n- · astrometric excess noise < 2 if in one of the following regions\n- -l ≤ 180 AND b < -0 . 139 l +25 AND b > 0 . 139 l -25\n- -l > 180 AND b > -0 . 139 l +25 AND b < 0 . 139 l -25\n- -√ ( l -303 . 2) 2 +2 ∗ ( b +44 . 4) 2 < 5\n- -√ ( l -280 . 3) 2 +2 ∗ ( b +33 . 0) 2 < 8 \nData Sources: Gaia DR2 ( ϖ , σ ϖ , astrometric excess noise, l , b ), 2MASS PSC (H) \nTarget priority options: 1805 \nCadence options: bright 1x1 \nA.26. mwm snc 100pc boss \nDescription of selection criteria: Optically bright targets in the 100 pc volume limited region. \n- · ϖ -σ ϖ > 10\n- · astrometric excess noise < 2 if in one of the following regions\n- -l ≤ 180 AND b < -0 . 139 l +25 AND b > 0 . 139 l -25\n- -l > 180 AND b > -0 . 139 l +25 AND b < 0 . 139 l -25\n- -√ ( l -303 . 2) 2 +2 ∗ ( b +44 . 4) 2 < 5\n- -√ ( l -280 . 3) 2 +2 ∗ ( b +33 . 0) 2 < 8 \nData Sources: Gaia DR2 ( ϖ , σ ϖ , astrometric excess noise, l , b , G) \nTarget priority options: 1800 \nCadence options: bright 2x1, dark 2x1 \nA.27. mwm snc 250pc apogee \nDescription of selection criteria: Infrared bright targets in the 250 pc volume limited region. \n- · G +5log 10 ( ϖ/ 1000) + 5 < 6\n- · ϖ -σ ϖ > 4\n- · H < 11\n- · astrometric excess noise < 2 if in one of the following regions\n- -l ≤ 180 AND b < -0 . 139 l +25 AND b > 0 . 139 l -25\n- -l > 180 AND b > -0 . 139 l +25 AND b < 0 . 139 l -25\n- -√ ( l -303 . 2) 2 +2 ∗ ( b +44 . 4) 2 < 5\n- -√ ( l -280 . 3) 2 +2 ∗ ( b +33 . 0) 2 < 8 \nData Sources: Gaia DR2 ( ϖ , σ , astrometric excess noise, l , b , G), 2MASS PSC (H) \nTarget priority options: \nϖ 1815 \nCadence options: bright 1x1 \nA.28. mwm snc 250pc boss \nDescription of selection criteria: Optically bright targets in the 250 pc volume limited region. \n- · G +5log 10 ( ϖ/ 1000) + 5 < 6\n- · ϖ -σ ϖ > 4\n- · astrometric excess noise < 2 if in one of the following regions\n- -l ≤ 180 AND b < -0 . 139 l +25 AND b > 0 . 139 l -25\n- -l > 180 AND b > -0 . 139 l +25 AND b < 0 . 139 l -25\n- -√ ( l -303 . 2) 2 +2 ∗ ( b +44 . 4) 2 < 5\n- -√ ( l -280 . 3) 2 +2 ∗ ( b +33 . 0) 2 < 8 \nData Sources: Gaia DR2 ( ϖ , σ , astrometric excess noise, l , b \nϖ , G), 2MASS PSC (H) \nTarget priority options: 1810 \nCadence options: bright 2x1, dark 2x1 \nA.29. mwm tess planet \nDescription of selection criteria: Stars with TESS 2-minute cadence data taken during the nominal 2 year mission, prioritizing those that are either TESS Objects of Interest (TOIs) or Community TESS Objects of Interest (CTOIs) \n- · 7 < H < 12 \nData Sources: 2MASS PSC (H), TESS \nTarget priority options: 2600, 2605, 2610 \nCadence options: bright 1x1, bright 1x2, bright 1x3, bright 1x4, bright 1x5, bright 1x6 \nA.30. mwm tessrgb core \nDescription of selection criteria: This carton selects red giant candidates with a color cut and uses a magnitude cut to limit stars likely to have stellar oscillations detectable in TESS light curve data. \n- · J -K > 0 . 5\n- · H < 12\n- · T < 13\n- · H -10 + 5 log 10 ϖ < 1 \n- · | b | > 20 \nData Sources: Gaia DR2 ( ϖ \n), 2MASS PSC (J, H, K), TESS (T) \nTarget priority options: 2800 \nCadence options: bright 1x1, bright 1x2, bright 1x3, bright 1x4, bright 1x5, bright 1x6 \nA.31. mwm wd core \nDescription of selection criteria: All sufficiently bright white dwarf candidates, where P WD is the probability a target is a white dwarf \n- · G < 20\n- · P WD > 0 . 5 \nData Sources: Gaia DR2 (G), Gentile Fusillo et al. (2019, P WD ) \nTarget priority options: 1400 \nCadence options: dark 2x1 \nA.32. mwm yso cluster apogee \nDescription of selection criteria: Infrared bright YSO candidates associated with young moving groups, as identified by Kounkel et al. (2020). \n- · H < 13\n- · age < 10 7 . 5 years \nData Sources: 2MASS PSC (H), Kounkel et al. (2020, age) \nTarget priority options: 2705 \nCadence options: bright 3x1 \nA.33. mwm yso cluster boss \nDescription of selection criteria: Optically bright YSO candidates associated with young moving groups, as identified by Kounkel et al. (2020). \n- · G RP < 15 . 5\n- · age < 10 7 . 5 years \nData Sources: Gaia DR2 ( G RP ), Kounkel et al. (2020, age) \nTarget priority options: 2705 \nCadence options: bright 3x1, bright 4x1, bright 5x1, bright 6x1 \nA.34. mwm yso cmz apogee \nDescription of selection criteria: YSOs in the central molecular zone (CMZ), the most extreme star-forming environment in the Milky Way \n- · H < 13\n- · [8 . 0] -[24] > 2 . 5, in Gutermuth & Heyer (2015) catalog\n- · ϖ < 0 . 2 or not measured \nData Sources: Gaia DR2 ( ϖ ), 2MASS PSC (H), Spitzer ([8.0], [24]) \nTarget priority options: 2700 \nCadence options: \nbright 3x1 \nRP ), AllWise (W1, W2, W3, W4) 2705 \nA.35. mwm yso disk apogee \nDescription of selection criteria: YSO's expected to have protoplanetary disks, as identified by larger infrared excesses. \n- · H < 13\n- · W 1 -W 2 > 0 . 25\n- · W 2 -W 3 > 0 . 50\n- · W 3 -W 4 > 1 . 50\n- · ϖ > 0 . 3 mas \nData Sources: Gaia DR2 ( ϖ \nTarget priority options: \nCadence options: \n), 2MASS PSC (H), AllWise (W1, W2, W3, W4) 2705 bright 3x1 \nA.36. mwm yso disk boss \nDescription of selection criteria: Optically bright YSO's expected to have protoplanetary disks, as identified by larger infrared excesses. \n- · G RP < 8 . 5\n- · W 1 -W 2 > 0 . 25\n- · W 2 -W 3 > 0 . 50\n- · W 3 -W 4 > 1 . 50\n- · ϖ > 0 . 3 mas \nData Sources: Gaia DR2 ( G , ϖ \nTarget priority options: \nCadence options: bright 3x1, bright 4x1, bright 5x1, bright 6x1 \nA.37. mwm yso embedded apogee \nDescription of selection criteria: Deeply embedded YSO candidates too optically faint for parallax measurements have more stringent infrared color cuts to avoid contamination with reddened field stars. \n- · H < 13\n- · G > 18 . 5 or undetected\n- · J -H > 1\n- · H -K > 0 . 5\n- · W 1 -W 2 > 0 . 5\n- · W 2 -W 3 > 1\n- · W 3 -W 4 > 1 . 5\n- · W 3 -W 4 > 0 . 8( W 1 -W 2) + 1 . 1 \nData Sources: Gaia DR2 (G), 2MASS PSC (J, H, K), AllWise (W1, W2, W3, W4) \nTarget priority options: 2705 \nCadence options: bright 3x1 \nA.38. mwm yso nebula apogee \nDescription of selection criteria: YSO's with strong nebular emission should saturate in the long AllWise bands and are identified with short wavelength color cuts. \n- · H < 13\n- · (no W4 AND W 2 -W 3 > 4) OR (no W3, W4 AND J -H > 1 . 1)\n- · b < 5 ·\n- · b > -5 · OR l > 180 · \nData Sources: 2MASS PSC (J, H), AllWise (W1, W2, W3, W4) \nTarget priority options: 2705 Cadence options: bright 3x1 \nA.39. mwm yso pms apogee \nDescription of selection criteria: Infrared bright YSO candidates based on their location above the main sequence, as identified by Zari et al. (2018) or McBride et al. (2021). \n- · H < 13 \nData Sources: 2MASS PSC (H) \nTarget priority options: 2700 \nCadence options: bright 3x1 \nA.40. mwm yso pms boss \nDescription of selection criteria: Optically bright YSO candidates based on their location above the main sequence, as identified by Zari et al. (2018) or McBride et al. (2021). \n- · G RP < 15 . 5 \nData Sources: Gaia DR2 ( G RP ) \nTarget priority options: 2700 \nCadence options: bright 3x1, bright 4x1, bright 5x1, bright 6x1 \nA.41. mwm yso variable apogee \nDescription of selection criteria: YSO candidates that are variable in the Gaia bands, to be observed with APOGEE. Variability in a given band X is defined by V X = √ N obs σ I X / ( I X ), where I X is the mean flux and σ I X its associated error. Below, M X = G X -5(log 10 (1000 /ϖ ) -1) \n- · H < 13\n- · G < 18 . 5\n- · ϖ > 0 . 3\n- · V G > 0 . 02\n- · V BP > 0 . 02\n- · V RP > 0 . 02\n- · V 0 . 75 G < V BP < V G\n- · 0 . 75 V G < V RP < V 0 . 95 G\n- · G BP -G RP > 1 . 3\n- · M BP > 5 log 10 V BP +11\n- · 2 . 5( G BP -G RP ) -1 < M G < 2 . 5 ∗ ( G BP -G RP ) + 2 . 5 \nData Sources: Gaia DR2 (G, G RP , G BP , ϖ , I X , σ I X \nTarget priority options: 2705 \nCadence options: bright 3x1 \nA.42. mwm yso variable boss \nDescription of selection criteria: YSO candidates that are variable in the Gaia bands, to be observed with BOSS. \nVariability in a given band X is defined by V X = √ N obs σ I X / ( I X ), where I X is the mean flux and σ I X its associated error. Below, M X = G X -5(log 10 (1000 /ϖ ) -1) \n- · H < 13\n- · G < 18 . 5 \n), 2MASS PSC (H) \n- · ϖ > 0 . 3\n- · V G > 0 . 02\n- · V BP > 0 . 02\n- · V RP > 0 . 02\n- · V 0 . 75 G < V BP < V G\n- · 0 . 75 V G < V RP < V 0 . 95 G\n- · G BP -G RP > 1 . 3\n- · M BP > 5 log 10 V BP +11\n- · 2 . 5( G BP -G RP ) -1 < M G < 2 . 5 ∗ ( G BP -G RP ) + 2 . 5 \nData Sources: Gaia DR2 (G, G RP , G BP , ϖ , I X , σ I X ), 2MASS PSC (H) \nTarget priority options: 2705 \nCadence options: bright 3x1, bright 4x1, bright 5x1, bright 6x1", 'B. DETAILS OF BHM V0.5.3 TARGET CARTONS': "In this appendix we provide a more detailed description of the full set of BHM target cartons that are being released as part of SDSS DR18. We describe not only the 'core' BHM cartons in Table 4, but also all ancillary/supplementary BHM cartons, which collectively expand the scope, footprint, and depth of the project beyond the core science goals. We remind the reader that inclusion in a carton means that the target fits the carton selection criteria, not that it is guaranteed to be observed. \nFor each target carton, we provide the following information: The target selection plan and target selection tag codes give the software and configuration versions of the target selection code that were used in this carton instance. In Summary , we provide a short synopsis of the content of the target carton, followed by a Simplified description of selection criteria with a human-readable description of the specific selection criteria that have been applied to build the carton, including quantitative limits on colors, magnitudes, and other quantities. The Target priority options and Cadence options indicate which 'priority' and 'cadence' option(s) have been allocated to the targets in this carton. Targets with numerically smaller priority are more likely to be assigned fibers; the cadence describes a target's exposure time requirement. Under Implementation , we point to the specific section of the target selection Python code that implements this carton 21 . Finally, the Number of targets lists the number of targets which pass all of the carton selection criteria. Note that targets are often shared between two or more cartons. \nB.1. bhm aqmes med \ntarget selection plan: \n0.5.0 \ntarget selection tag: 0.3.0 \nSummary: Spectroscopically confirmed optically bright SDSS QSOs, selected from the SDSS QSO catalog (DR16Q, Lyke et al. 2020). Located in 36 mostly disjoint fields within the SDSS QSO footprint that were pre-selected to contain higher than average numbers of bright QSOs and CSC targets. The list of field centers can be found within the target selection repository. \nSimplified description of selection criteria: Select all objects from SDSS DR16 QSO catalog that have SDSS 16 . 0 < i psf < 19 . 1 AB, and that lie within 1.49 degrees of at least one AQMES-medium field location. \nTarget priority options: \n1100 \nCadence options: \ndark 10x4 4yr \nImplementation: \nbhm aqmes.py \nNumber of targets: \n2663 \nB.2. bhm aqmes med faint \ntarget selection plan: \n0.5.0 \ntarget selection tag: 0.3.0 \nSummary: Spectroscopically confirmed optically faint SDSS QSOs, selected from the SDSS QSO catalog (DR16Q, Lyke et al. 2020). Located in 36 mostly disjoint fields within the SDSS QSO footprint that were pre-selected to contain higher than average numbers of bright QSOs and CSC targets. The list of field centres can be found within the target selection repository. \nSimplified description of selection criteria: Select all objects from SDSS DR16 QSO catalog that have SDSS 19 . 1 < i psf < 21 . 0 AB, and that lie within 1.49 degrees of at least one AQMES-medium field location. \nTarget priority options: 3100 \nCadence options: \ndark 10x4 4yr \nImplementation: \nbhm aqmes.py \nNumber of targets: \n16853 \nB.3. bhm aqmes wide2 \ntarget selection plan: \n0.5.4 \ntarget selection tag: \n0.3.5 \nSummary: Spectroscopically confirmed optically bright SDSS QSOs, selected from the SDSS QSO catalog (DR16Q, Lyke et al. 2020). Located in 425 fields within the SDSS QSO footprint, where the choice of survey area prioritized field that overlapped with the SPIDERS footprint (approx 180 < l < 360 deg), and/or had higher than average numbers of bright QSOs and CSC targets. The list of field centers can be found within the target selection repository. \nSimplified description of selection criteria: Select all objects from SDSS DR16 QSO catalog that have SDSS 16 . 0 < i psf < 19 . 1 AB, and that lie within 1.49 degrees of at least one AQMES-wide field location. \nTarget priority options: \n1210, 1211 \nCadence options: \ndark 2x4 \nImplementation: \nbhm aqmes.py \nNumber of targets: \n24142 \nB.4. bhm aqmes wide2 faint \ntarget selection plan: \n0.5.4 \ntarget selection tag: \n0.3.5 \nSummary: Spectroscopically confirmed optically faint SDSS QSOs, selected from the SDSS QSO catalog (DR16Q, Lyke et al. 2020). Located in 425 fields within the SDSS QSO footprint, where the choice of survey area prioritized field that overlapped with the SPIDERS footprint (approx 180 < l < 360 deg), and/or had higher than average numbers of bright QSOs and CSC targets. The list of field centres can be found within the target selection repository. \nSimplified description of selection criteria: Select all objects from SDSS DR16 QSO catalog that have SDSS 19 . 1 < i psf < 21 . 0 AB, and that lie within 1.49 degrees of at least one AQMES-wide field location. \nTarget priority options: \n3210, 3211 \nCadence options: \ndark 2x4 \nImplementation: \nbhm aqmes.py \nNumber of targets: \n99586 \nB.5. bhm aqmes bonus core \ntarget selection plan: \n0.5.4 \ntarget selection tag: \n0.3.5 \nSummary: Spectroscopically confirmed optically bright SDSS QSOs, selected from the SDSS QSO catalog (DR16Q; Lyke et al. 2020). Located anywhere within the SDSS DR16Q footprint. \nSimplified description of selection criteria: Select all objects from SDSS DR16 QSO catalog that have SDSS 16 . 0 < i psf < 19 . 1 AB. \nTarget priority options: \n3300,3301 \nCadence options: \ndark 1x4 \nImplementation: \nbhm aqmes.py \nNumber of targets: \n83163 \ntarget selection plan: \n0.5.4 \ntarget selection tag: \n0.3.5 \nSummary: Spectroscopically confirmed optically faint SDSS QSOs, selected from the SDSS QSO catalog (DR16Q; Lyke et al. 2020). Located anywhere within the SDSS DR16Q footprint. \nSimplified description of selection criteria: Select all objects from SDSS DR16 QSO catalog that have SDSS 19 . 1 < i psf < 21 . 0 AB. \nTarget priority options: \n3302,3303 \nCadence options: \ndark 1x4 \nImplementation: \nbhm aqmes.py \nNumber of targets: \n424163 \ntarget selection plan: \n0.5.4 \ntarget selection tag: \n0.3.5 \nSummary: Spectroscopically confirmed, extremely optically bright SDSS QSOs, selected from the SDSS QSO catalog (DR16Q; Lyke et al. 2020). Located anywhere within the SDSS DR16Q footprint. \nSimplified description of selection criteria: Select all objects from SDSS DR16 QSO catalog that have SDSS 14 . 0 < i psf < 18 . 0 AB. \nTarget priority options: \n4040,4041 \nCadence options: \nbright 3x1 \nImplementation: \nbhm aqmes.py \nNumber of targets: \n10848 \ntarget selection plan: \n0.5.0 \ntarget selection tag: \n0.3.0 \nSummary: A supporting sample of candidate QSOs that have been selected by the Gaia-unWISE AGN catalog (Shu et al. 2019) and/or the SDSS XDQSO catalog (Bovy et al. 2011). These targets are located within five (+1 backup) well known survey fields (SDSS-RM, COSMOS, XMM-LSS, ECDFS, CVZ-S/SEP, and ELIAS-S1). \nSimplified description of selection criteria: Starting from a parent catalog of optically selected objects in the RM fields (as presented by Yang & Shen 2022), select candidate QSOs that satisfy all of the following: i) are identified via external ancillary methods (photo bitmask & 3 != 0); ii) have 15 < i psf < 21 . 5 AB (or 16 < G < 21 . 7 Vega in the CVZ-S/SEP field, photometry taken from Yang & Shen 2022); iii) do not have significant detections ( > 3 σ ) of parallax and/or proper motion in Gaia DR2; iv) are not vetoed due to results of visual inspections of recent spectroscopy; and v) do not lie in the SDSS-RM field. \nTarget priority options: \n900-1050 \nCadence options: \ndark 174x8, dark 100x8 \nImplementation: \nbhm rm.py \nNumber of targets: \n943 \nB.6. bhm aqmes bonus faint \nB.7. bhm aqmes bonus bright \nB.8. bhm rm ancillary \ntarget selection plan: \n0.5.0 0.3.0 \ntarget selection tag: \nSummary: A sample of candidate QSOs selected via the methods presented by Yang & Shen (2022). These targets are located within five (+1 backup) well known survey fields (SDSS-RM, COSMOS, XMM-LSS, ECDFS, CVZ-S/SEP, and ELIAS-S1). \nSimplified description of selection criteria: Starting from a parent catalog of optically selected objects in the RM fields (as presented by Yang & Shen 2022), select candidate QSOs that satisfy all of the following: i) are identified via the Skew-T algorithm (skewt qso == 1); ii) have 17 < i psf < 21 . 5 AB (or 16 < G < 21 . 7 Vega in the CVZ-S/SEP field, photometry taken from Yang & Shen 2022); iii) do not have significant detections ( > 3 σ ) of parallax and/or proper motion in Gaia DR2; iv) are not vetoed due to results of visual inspections of recent spectroscopy; v) have detections in all of the gri bands (a Gaia detection is sufficient in the CVZ-S/SEP field); and vi) do not lie in the SDSS-RM field. \nTarget priority options: \n900-1050 \nCadence options: \ndark 174x8, dark 100x8 \nImplementation: \nbhm rm.py \nNumber of targets: \n3721 \nB.10. bhm rm var \ntarget selection plan: \n0.5.0 \ntarget selection tag: \n0.3.0 \nSummary: A sample of candidate QSOs selected via their optical variability properties, as presented by Yang & Shen (2022). These targets are located within five (+1 backup) well known survey fields (SDSS-RM, COSMOS, XMM-LSS, ECDFS, CVZ-S/SEP, and ELIAS-S1). \nSimplified description of selection criteria: Starting from a parent catalog of optically selected objects in the RM fields (as presented by Yang & Shen 2022), select candidate QSOs that satisfy all of the following: i) have significant variability in the DES or PanSTARRS1 multi-epoch photometry (var sn[ g ] > 3 and var rms[ g ] > 0 . 05); ii) have 17 < i psf < 20 . 5 AB (or 16 < G < 21 . 7 Vega in the CVZ-S/SEP field, photometry taken from Yang & Shen 2022); iii) do not have significant detections ( > 3 σ ) of parallax and/or proper motion in Gaia DR2; iv) are not vetoed due to results of visual inspections of recent spectroscopy; and v) do not lie in the SDSS-RM field. \nTarget priority options: \n900-1050 \nCadence options: \ndark 174x8, dark 100x8 \nImplementation: \nbhm rm.py \nNumber of targets: \n934 \nB.11. bhm rm known spec \ntarget selection plan: \n0.5.0 \ntarget selection tag: \n0.3.0 \nSummary: A sample of known QSOs identified through optical spectroscopy from various projects, as collated by Yang & Shen (2022). These targets are located within five (+1 backup) well known survey fields (SDSS-RM, COSMOS, XMM-LSS, ECDFS, CVZ-S/SEP, and ELIAS-S1). \nSimplified description of selection criteria: Starting from a parent catalog of optically selected objects in the RM fields (as presented by Yang & Shen 2022), select targets which satisfy all of the following: i) are flagged as having a spectroscopic identification (in the parent catalog or in the bhm rm tweaks table); ii) have 15 < i psf < 21 . 7 AB (SDSS-RM, CDFS, ELIAS-S1 fields), 15 < i psf < 21 . 5 AB (COSMOS and XMM-LSS fields), 16 < G < 21 . 7 Vega in the CVZ-S/SEP field (photometry taken from Yang & Shen 2022); iii) have a spectroscopic redshift in the range 0.005 < z < 7; and iv) are not vetoed due to results of visual inspections of recent spectroscopy. \nTarget priority options: \n900-1050 \nCadence options: \ndark 174x8, dark 100x8 \nImplementation: \nbhm rm.py \nNumber of targets: \n3022 \nB.12. bhm csc apogee \ntarget selection plan: \n0.5.15 \ntarget selection tag: \n0.3.14 \nSummary: \nX-ray sources from the CSC2.0 source catalog with NIR counterparts in 2MASS PSC \nSimplified description of selection criteria: Starting from the parent catalog of CSC2.0 sources associated with optical/IR counterparts ( bhm csc v2 ). Select entries satisfying the following criteria: i) NIR counterpart is from the 2MASS catalog, ii) 2MASS H -band magnitude measurement is not null and in the accepted range for SDSS-V: 7 . 0 < H < 14 . 0. Allocate cadence (exposure time) requests based on H magnitude. \nTarget priority options: 2930-2939 \nCadence options: \nbright 1x1,bright 3x1 \nImplementation: \nbhm csc.py \nNumber of targets: 48928 \ntarget selection plan: \n0.5.15 \ntarget selection tag: \n0.3.14 \nSummary: X-ray sources from the CSC2.0 source catalog with counterparts in Panstarrs1-DR1 or Gaia DR2. Simplified description of selection criteria: Starting from the parent catalog of CSC2.0 sources associated with optical/IR counterparts ( bhm csc v2 ). Select entries satisfying the following criteria: i) optical counterpart is from the PanSTARRS1 or Gaia DR2 catalogs, ii) optical flux/magnitude is in the accepted range for SDSS-V: PanSTARRS1 g psf , r psf , i psf , z psf > 13 . 5 AB, and non-Null i psf (objects with PanSTARRS1 counterparts); G,G RP > 13 . 0 Vega (Gaia DR2 counterparts). Deprioritize targets which already have good quality SDSS spectroscopy. Allocate cadence (exposure time) requests based on optical brightness (PanSTARRS1 i psf or Gaia G ). \nTarget priority options: 1920-1939, 2920-2939 \nCadence options: \nbright 1x1,dark 1x2,dark 1x4 \nImplementation: \nbhm csc.py \nNumber of targets: \n122731 \nB.14. bhm gua bright \ntarget selection plan: \n0.5.0 \ntarget selection tag: \n0.3.0 \nSummary: A sample of optically bright candidate AGN lacking spectroscopic confirmations, derived from the parent sample presented by Shu et al. (2019), who applied a machine-learning approach to select QSO candidates from a combination of the Gaia DR2 and unWISE catalogs. \nSimplified description of selection criteria: Starting with the Shu et al. (2019) catalog, select targets that satisfy the following criteria: i) have a Random Forest probability of being a QSO of > 0.8, ii) are in the magnitude range suitable for BOSS spectroscopy in bright time ( G dered > 13 . 0 and G RP, dered > 13 . 5, as well as G dered < 18 . 5 or G RP, dered < 18 . 5 Vega), and iii) do not have good optical spectroscopic measurements from a previous iteration of SDSS. \nTarget priority options: \n4040 \nCadence options: \nbright 2x1 \nImplementation: \nbhm gua.py \nNumber of targets: \n254601 \ntarget selection plan: \n0.5.16 \ntarget selection tag: \n0.3.13 \nSummary: A supplementary magnitude limited sample of optically bright galaxies selected from the DESI Legacy Survey DR8 optical/IR imaging catalog. Selection is based on optical morphology, lack of Gaia DR2 parallax, and several magnitude cuts. \nSimplified description of selection criteria: Starting from the DESI Legacy Survey DR8 catalog (lsdr8), select entries satisfying all of the following criteria: i) lsdr8 morphological type != 'PSF', ii) zero or Null parallax in Gaia DR2, iii) z model , dered < 19 . 0 AB, and z fiber , dered < 19 . 5 AB, and z fiber < 19 . 0 AB, and r fiber > 16 . 0 AB, and G > 15 . 0 Vega, and G RP > 15 . 0 Vega (photometry from lsdr8). \nTarget priority options: \n7100 \nCadence options: \nbright 1x1, dark 1x1, dark 1x4 \nImplementation: \nbhm galaxies.py \nNumber of targets: \n7320203 \nB.17. bhm spiders agn lsdr8 \ntarget selection plan: \n0.5.0 \ntarget selection tag: \n0.3.0 \nSummary: This is the highest priority carton for SPIDERS AGN wide area follow up. The carton provides optical counterparts to point-like (unresolved) X-ray sources detected in early reductions of the first 6-months of eROSITA all sky survey data (eRASS:1). The sample is expected to contain a mixture of QSOs, AGN, stars and compact objects. The X-ray sources have been cross-matched by the eROSITA-DE team to DESI Legacy Survey DR8 (lsdr8) optical/IR counterparts. All targets are located in the sky hemisphere where MPE controls the data rights (approx. 180 < l < 360 deg). Due to the footprint of lsdr8, nearly all targets in this carton are located at high Galactic latitudes ( | b | > 15 deg). \nSimplified description of selection criteria: Starting from a parent catalog of eRASS:1 point source → legacysurvey.org/dr8 associations (method: NWAY assisted by optical/IR priors computed via a pre-trained Random Forest, building on Salvato et al. 2022), select targets which meet all of the following criteria: i) have eROSITA detection likelihood > 6.0, ii) have an X-ray → optical/IR cross-match probability (NWAY) of p any > 0.1, iii) have 13 . 5 < r fibertot < 22 . 5 or 13 . 5 < z fibertot < 21 . 0 AB, iv) are not saturated in Legacy Survey imaging, v) have at least one observation in r-band and at least one observation in g- or z-band, and vi) if detected by Gaia DR2 then have G > 13 . 5 and G RP > 13 . 5 Vega (photometry from lsdr8). We deprioritize targets if any of the following criteria are met: a) the target already has existing good quality SDSS spectroscopy, b) the X-ray detection likelihood is < 8.0, \ntarget selection plan: \n0.5.0 0.3.0 \ntarget selection tag: \nSummary: A sample of optically faint candidate AGN lacking spectroscopic confirmations, derived from the parent sample presented by Shu et al. (2019), who applied a machine-learning approach to select QSO candidates from a combination of the Gaia DR2 and unWISE catalogs. \nSimplified description of selection criteria: Starting with the Shu et al. (2019) catalog, select targets which satisfy the following criteria: i) have a Random Forest probability of being a QSO of > 0.8, ii) are in the magnitude range suitable for BOSS spectroscopy in dark time ( G dered > 16 . 5 and G RP, dered > 16 . 5, as well as G dered < 21 . 2 or G RP, dered < 21 . 0 Vega), and iii) do not have good optical spectroscopic measurements from a previous iteration of SDSS. \nTarget priority options: \n3400 \nCadence options: \ndark 1x4 \nImplementation: \nbhm gua.py \nNumber of targets: \n2156582 \nor c) the target is a secondary X-ray → optical/IR association. We assign cadences (exposure time requests) based on optical brightness. \nTarget priority options: \n1520-1523, 1720-1723 \nCadence options: \nbright 2x1, dark 1x2, dark 1x4 \nImplementation: \nbhm spiders agn.py \nNumber of targets: \n235745 \nB.18. bhm spiders agn ps1dr2 \ntarget selection plan: \n0.5.0 \ntarget selection tag: \n0.3.0 \nSummary: This is the second highest priority carton for SPIDERS AGN wide area follow up, included to expand the survey footprint beyond Legacy Survey DR8. The carton provides optical counterparts to point-like (unresolved) X-ray sources detected in early reductions of the first 6-months of eROSITA all sky survey data (eRASS:1). The sample is expected to contain a mixture of QSOs, AGN, stars and compact objects. The X-ray sources have been cross-matched by the eROSITA-DE team, first to CatWISE2020 mid-IR sources (Marocco et al. 2021), and then to optical counterparts from the PanSTARRS1 DR2 catalog. All targets are located in the sky hemisphere where MPE controls the data rights (approx. 180 < l < 360 deg), and at Dec > -30 deg, spanning a wide range of Galactic latitudes. Targets at low Galactic latitudes ( | b | < 15 deg) do not drive survey strategy. \nSimplified description of selection criteria: Starting from a parent catalog of eRASS:1 point source → CatWISE2020 → PanSTARRS1 associations (method: NWAY assisted by IR priors computed via a pre-trained Random Forest, building on Salvato et al. 2022), select targets which meet all of the following criteria: i) have eROSITA detection likelihood > 6.0, ii) have an X-ray → IR cross-match probability of p any > 0.1, iii) have PanSTARRS1 g psf , r psf , i psf , z psf > 13 . 5 AB and at least one of PanSTARRS1 g psf < 22 . 5, r psf < 22 . 0, i psf < 21 . 5 or z psf < 20 . 5 AB, iv) are not associated with a bad PanSTARRS1 image stack, v) have non-null measurements of g psf , r psf and i psf , and vi) if detected by Gaia DR2 then have G > 13 . 5 and G RP > 13 . 5 Vega. We deprioritize targets if any of the following criteria are met: a) the target already has existing good quality SDSS spectroscopy, b) the X-ray detection likelihood is < 8.0, or c) the target is a secondary X-ray → IR association. We assign cadences (exposure time requests) based on optical brightness. \nTarget priority options: 1530-1533,1730-1733,3530-3533,3730-3732 \nCadence options: \nbright 2x1, dark 1x2, dark 1x4 \nImplementation: \nbhm spiders agn.py \nNumber of targets: \n200681 \nB.19. bhm spiders agn gaiadr2 \ntarget selection plan: \n0.5.0 \ntarget selection tag: \n0.3.0 \nSummary: This is the third highest priority carton for SPIDERS AGN wide area follow up, included to expand the survey footprint to the full hemisphere where X-ray sources are available (beyond Legacy Survey DR8 and PanSTARRS1). The carton provides optical counterparts to point-like (unresolved) X-ray sources detected in early reductions of the first 6-months of eROSITA all sky survey data (eRASS:1). The sample is expected to contain a mixture of QSOs, AGN, stars and compact objects. The X-ray sources have been cross-matched by the eROSITA-DE team, first to CatWISE2020 mid-IR sources (Marocco et al. 2021), and then to optical counterparts from the Gaia-DR2 catalog. All targets are located in the sky hemisphere where MPE controls the data rights (approx. 180 < l < 360 deg). The targets in this carton distributed over a wide range of Galactic latitudes, but targets at low Galactic latitudes ( | b | < 15 deg) do not drive survey strategy. \nSimplified description of selection criteria: Starting from a parent catalog of eRASS:1 point source → CatWISE2020 → Gaia DR2 associations (method: NWAY assisted by IR priors computed via a pre-trained Random Forest, building on Salvato et al. 2022), select targets which meet all of the following criteria: i) have eROSITA detection likelihood > 6.0, ii) have an X-ray → IR cross-match probability of p any > 0.1, and iii) have G > 13 . 5 and \nG RP > 13 . 5 Vega. We deprioritize targets if any of the following criteria are met: a) the target already has existing good quality SDSS spectroscopy, b) the X-ray detection likelihood is < 8.0, or c) the target is a secondary X-ray → IR association. We assign cadences (exposure time requests) based on optical brightness. \nTarget priority options: 1540-1543,1740-1743,3540-3543,3740-3742 \nCadence options: \nbright 2x1, dark 1x2, dark 1x4 \nImplementation: \nbhm spiders agn.py \nNumber of targets: \n324576 \nB.20. bhm spiders agn skymapperdr2 \ntarget selection plan: \n0.5.0 0.3.0 \ntarget selection tag: \nSummary: This is a lower ranked carton for SPIDERS AGN wide area follow up, which supplements the survey in areas that rely on Gaia DR2 (beyond Legacy Survey DR8 and PanSTARRS1) by recovering extended targets (galaxies) that are missed by Gaia. The carton provides optical counterparts to point-like (unresolved) X-ray sources detected in early reductions of the first 6-months of eROSITA all sky survey data (eRASS:1). The sample is expected to contain a mixture of QSOs, AGN, stars and compact objects. The X-ray sources have been cross-matched by the eROSITADE team, first to CatWISE2020 mid-IR sources (Marocco et al. 2021), and then to optical counterparts from the SkyMapper-DR2 catalog (Onken et al. 2019b). All targets are located in the sky hemisphere where MPE controls the data rights (approx. 180 < l < 360 deg) and at Dec < 0 deg, spanning a wide range of Galactic latitudes. Targets at low Galactic latitudes ( | b | < 15 deg) do not drive survey strategy. \nSimplified description of selection criteria: Starting from a parent catalog of eRASS:1 point source → CatWISE2020 → SkyMapper-dr2 associations (method: NWAY assisted by IR priors computed via a pre-trained Random Forest, building on Salvato et al. 2022), select targets which meet all of the following criteria: i) have eROSITA detection likelihood > 6.0, ii) have an X-ray → IR cross-match probability of p any > 0.1, iii) have SkyMapper g psf , r psf , i psf , z psf > 13 . 5 AB and at least one of SkyMapper g psf < 22 . 5, r psf < 22 . 0, i psf < 21 . 5 or z psf < 20 . 5 AB, iv) are not associated with a bad SkyMapper source detection (flags < 4), v) have non-null measurements of g psf , r psf and i psf , and vi) if detected by Gaia DR2 then have G > 13 . 5 and G RP > 13 . 5 Vega. We deprioritize targets if any of the following criteria are met: a) the target already has existing good quality SDSS spectroscopy, b) the X-ray detection likelihood is < 8.0, or c) the target is a secondary X-ray → IR association. We assign cadences (exposure time requests) based on optical brightness. \nTarget priority options: \n1550-1553,1750-1753,3550-3553,3750-3752 \nCadence options: \nbright 2x1, dark 1x2, dark 1x4 \nImplementation: \nbhm spiders agn.py \nNumber of targets: \n82683 \nB.21. bhm spiders agn supercosmos \ntarget selection plan: \n0.5.0 \ntarget selection tag: \n0.3.0 \nSummary: This is a lower ranked carton for SPIDERS AGN wide area follow up, which supplements the survey in areas which rely on Gaia-DR2 (beyond DESI Legacy Survey DR8 and PanSTARRS1) by recovering extended targets (galaxies) that are missed by Gaia. The carton provides optical counterparts to point-like (unresolved) X-ray sources detected in early reductions of the first 6-months of eROSITA all sky survey data (eRASS:1). The sample is expected to contain a mixture of QSOs, AGN, stars and compact objects. The X-ray sources have been cross-matched by the eROSITA-DE team, first to CatWISE2020 mid-IR sources (Marocco et al. 2021), and then to optical counterparts from the SuperCosmos Sky Surveys catalog (derived from scans of photographic plates). All targets are located in the sky hemisphere where MPE controls the data rights (approx. 180 < l < 360 deg), spanning a wide range of Galactic latitudes. Targets at low Galactic latitudes ( | b | < 15 deg) do not drive survey strategy. \nSimplified description of selection criteria: Starting from a parent catalog of eRASS:1 point source → CatWISE2020 → SuperCosmos associations (method: NWAY assisted by IR priors computed via a pre-trained Random \nForest, building on Salvato et al. 2022), select targets which meet all of the following criteria: i) have eROSITA detection likelihood > 6.0, ii) have an X-ray → IR cross-match probability of p any > 0.1, iii) have B J , psf , R psf , I psf > 13 . 5 Vega and at least one of B J , psf < 22 . 5, R psf < 22 . 0, or I psf < 21 . 5 Vega (photometry from SuperCosmos), and iv) if detected by Gaia DR2 then have G > 13 . 5 and G RP > 13 . 5 Vega. We deprioritize targets if any of the following criteria are met: a) the target already has existing good quality SDSS spectroscopy, b) the X-ray detection likelihood is < 8.0, or c) the target is a secondary X-ray → IR association. We assign cadences (exposure time requests) based on optical brightness. \nTarget priority options: \n1560-1563,1760-1763,3560-3563,3760-3763 \nCadence options: \nbright 2x1, dark 1x2, dark 1x4 \nImplementation: \nbhm spiders agn.py \nNumber of targets: \n430780 \nB.22. bhm spiders agn efeds stragglers \ntarget selection plan: \n0.5.0 \ntarget selection tag: \n0.3.0 \nSummary: This is an opportunistic supplementary carton for SPIDERS AGN follow up, aiming, where fiber resources allow, to gather a small additional number of spectra for targets in the eFEDS field (which has been repeatedly surveyed in earlier iterations of SDSS). This carton provides optical counterparts to point-like (unresolved) X-ray sources detected in early reductions of the eROSITA/eFEDS performance validation field. The sample is expected to contain a mixture of QSOs, AGN, stars and compact objects. The X-ray sources have been cross-matched by the eROSITA-DE team to DESI Legacy Survey DR8 optical/IR counterparts. All targets in this carton are located within the eFEDS field (approx 126 < RA < 146, -3 < Dec < +6 deg). These targets do not drive survey strategy. \nSimplified description of selection criteria: Starting from a parent catalog of eFEDS point source → legacysurvey.org/dr8 associations (method: NWAY assisted by optical/IR priors computed via a pre-trained Random Forest, see Salvato et al. 2022), select targets which meet all of the following criteria: i) have eROSITA detection likelihood > 6.0, ii) have an X-ray → optical/IR cross-match probability (NWAY) of p any > 0.1, iii) have 13 . 5 < r fibertot < 22 . 5 or 13 . 5 < z fibertot < 21 . 0 AB, iv) are not saturated in Legacy Survey imaging, v) have at least one observation in r -band and at least one observation in g - or z -band, and vi) if detected by Gaia DR2 then have G > 13 . 5 and G RP > 13 . 5 Vega. We deprioritize targets if any of the following criteria are met: a) the target already has existing good quality SDSS spectroscopy, b) the X-ray detection likelihood is < 8.0, or c) the target is a secondary X-ray → optical/IR association. We assign cadences (exposure time requests) based on optical brightness. \nTarget priority options: 1510-1514, 1710-1714 \nCadence options: \nbright 2x1, dark 1x2, dark 1x4 \nImplementation: \nbhm spiders agn.py \nNumber of targets: \n15926 \nB.23. bhm spiders agn sep \ntarget selection plan: \n0.5.0 \ntarget selection tag: \n0.3.0 \nSummary: This special carton is dedicated to SPIDERS AGN follow up in the CVZ-S/SEP field. The carton provides optical counterparts to point-like (unresolved) X-ray sources detected in a dedicated analysis of the first 6-months of eROSITA scanning data near the SEP. The X-ray sources have been cross-matched by the eROSITA-DE team, first to CatWISE2020 mid-IR sources (Marocco et al. 2021), and then to optical counterparts from the Gaia DR2 catalog, using additional filtering (including Gaia EDR3 astrometric information) to reduce the contamination from foreground stars located in the LMC. All targets are located within 1.5 deg of the South Ecliptic Pole (RA,Dec = 90.0,-66.56 deg). Simplified description of selection criteria: Starting from a parent catalog of eRASS:1/SEP point source → CatWISE2020 → Gaia DR2 associations (method: NWAY assisted by IR priors computed via a pre-trained Random Forest, building on Salvato et al. 2022), select targets which meet all of the following criteria: i) have eROSITA detection likelihood > 6.0, ii) have an X-ray → IR cross-match probability of p any > 0.1, and iii) have Gaia G > 13 . 5 and \nG RP > 13 . 5 Vega. We deprioritize targets if any of the following criteria are met: a) the X-ray detection likelihood is < 8.0, or b) the target is a secondary X-ray → IR association. We assign cadences (exposure time requests) based on optical brightness. \nTarget priority options: \n1510,1512 \nCadence options: \nbright 2x1, dark 1x2, dark 1x4 \nImplementation: \nbhm spiders agn.py \nNumber of targets: \n697 \nB.24. bhm spiders clusters lsdr8 \ntarget selection plan: \n0.5.0 0.3.0 \ntarget selection tag: \nSummary: This is the highest priority carton for SPIDERS Clusters wide area follow up. The carton provides a list of galaxies which are candidate members of clusters selected from early reductions of the first 6-months of eROSITA all sky survey data (eRASS:1). The X-ray clusters have been associated by the eROSITA-DE team to DESI Legacy Survey DR8 (lsdr8) optical/IR counterparts using the eROMaPPeR red-sequence finder algorithm (Rykoff et al. 2014; Ider Chitham et al. 2020). All targets are located in the sky hemisphere where MPE controls the data rights (approx. 180 < l < 360 deg). Due to the footprint of lsdr8, nearly all targets in this carton are located at high Galactic latitudes ( | b | > 15 deg). \nSimplified description of selection criteria: Starting from a parent catalog of eRASS:1 → lsdr8 eROMaPPeR cluster associations, select targets which meet all of the following criteria: i) have 13 . 5 < r fibertot < 21 . 0 or 13 . 5 < z fibertot < 20 . 0 AB, ii) if detected by Gaia DR2 then have G > 13 . 5 and G RP > 13 . 5 Vega (photometry from lsdr8), and iii) do not have existing good quality SDSS spectroscopy. We assign a range of priorities to targets in this carton, with BCGs top ranked, followed by candidate member galaxies according their probability of membership. We assign cadences (exposure time requests) based on optical brightness. \nTarget priority options: 1501,1630-1659 \nCadence options: \nbright 2x1, dark 1x2, dark 1x4 \nImplementation: \nbhm spiders clusters.py \nNumber of targets: \n87490 \nB.25. bhm spiders clusters ps1dr2 \ntarget selection plan: \n0.5.0 \ntarget selection tag: \n0.3.0 \nSummary: This is the second highest priority carton for SPIDERS Clusters wide area follow up, designed to expand the survey area beyond the Legacy Survey DR8 footprint. The carton provides a list of galaxies that are candidate members of clusters selected from early reductions of the first 6-months of eROSITA all sky survey data (eRASS:1). The X-ray clusters have been associated by the eROSITA-DE team to the PanSTARRS1-DR2 catalog using the eROMaPPeR red-sequence finder algorithm (Rykoff et al. 2014; Ider Chitham et al. 2020). All targets are located in the sky hemisphere where MPE controls the data rights (approx. 180 < l < 360 deg). Nearly all targets in this carton are located at high Galactic latitudes ( | b | > 15 deg). \nSimplified description of selection criteria: Starting from a parent catalog of eRASS:1 → PanSTARRS1-DR2 eROMaPPeR cluster associations, select targets which meet all of the following criteria: i) have PanSTARRS1 r psf , i psf , z psf > 13 . 5 AB and at least one of r psf < 21 . 5, i psf < 21 . 0 or z psf < 20 . 5 AB, and ii) do not have existing good quality SDSS spectroscopy. We assign a range of priorities to targets in this carton, with BCGs top ranked, followed by candidate member galaxies according their probability of membership. We assign cadences (exposure time requests) based on optical brightness. \nTarget priority options: 1502,1660-1689 \nCadence options: \nbright 2x1,dark 1x2,dark 1x4 \nImplementation: \nbhm spiders clusters.py \nNumber of targets: \n86179 \nB.26. bhm spiders clusters efeds stragglers \ntarget selection plan: \n0.5.0 0.3.0 \ntarget selection tag: \nSummary: This is an opportunistic supplementary carton for SPIDERS Clusters follow up, aiming, where fiber resources allow, to gather a small additional number of spectra for targets in the eFEDS field (which has been repeatedly surveyed in earlier iterations of SDSS). The carton provides a list of galaxies which are candidate members of clusters selected from early reductions of the eROSITA performance verification survey in the eFEDS field. The X-ray clusters have been associated by the eROSITA-DE team to DESI Legacy Survey DR8 (lsdr8) optical/IR counterparts. All targets in this carton are located within the eFEDS field (approx 126 < RA < 146, -3 < Dec < +6 deg). \nSimplified description of selection criteria: Starting from a parent catalog of eFEDS → legacysurvey.org/dr8 cluster associations (eROMaPPeR, Rykoff et al. 2014; Ider Chitham et al. 2020; Liu et al. 2022), select targets which meet all of the following criteria: have 13 . 5 < r fibertot < 21 . 0 or 13 . 5 < z fibertot < 20 . 0 AB, ii) if detected by Gaia DR2 then have G > 13 . 5 and G RP > 13 . 5 Vega (photometry from lsdr8), and iii) do not have existing good quality SDSS spectroscopy. We assign a range of priorities to targets in this carton, with BCGs top ranked, followed by candidate member galaxies according their probability of membership. We assign cadences (exposure time requests) based on optical brightness. \nTarget priority options: \n1500,1600-1629 \nCadence options: \ndark 1x2,dark 1x4 \nImplementation: \nbhm spiders clusters.py \nNumber of targets: \n3060 \nB.27. bhm spiders agn-efeds \ntarget selection plan: \n0.1.0 \ntarget selection tag: \n0.1.0 \nSummary: A carton used during SDSS-V plate-mode observations that contains candidate AGN targets found in the eROSITA/eFEDS X-ray survey field. This carton provides optical counterparts to point-like (unresolved) X-ray sources detected in early reductions ('c940/V2T') of the eROSITA/eFEDS performance validation field. The sample is expected to contain a mixture of QSOs, AGN, stars and compact objects. The X-ray sources have been cross-matched by the eROSITA-DE team to DESI Legacy Survey DR8 (lsdr8) optical/IR counterparts. All targets in this carton are located within the eFEDS field (approx 126 < RA < 146, -3 < Dec < +6 deg). \nSimplified description of selection criteria: Starting from a parent catalog of eFEDS point source → lsdr8 associations (primarily via NWAY assisted by optical/IR priors computed via a pre-trained Random Forest, see Salvato et al. 2022, supplemented by counterparts selected via a Likelihood Ratio using r -band magnitudes), select targets which meet all of the following criteria: i) have eROSITA detection likelihood > 6.0, ii) have an X-ray → optical/IR cross-match probability of either p any > 0.1 (NWAY associations) or LR > 0.2 (Likelihood Ratio associations), iii) have r fiber > 16 . 5 and either r fiber < 22 . 0 or z fiber < 21 . 0 AB (photometry from lsdr8), and iv) did not receive high quality spectroscopy during the SDSS-IV observations of the eFEDS field (Abdurro'uf et al. 2022). We deprioritize targets if any of the following criteria are met: a) the target already has existing good quality SDSS spectroscopy in SDSS DR16, b) the X-ray detection likelihood is < 8.0, c) the target is a secondary X-ray → optical/IR association, or d) the optical/IR counterpart was only chosen by the LR method. All targets were assigned a nominal cadence of: bhm spiders 1x8 (8x15mins dark time). \nTarget priority options: \n1510-1519 \nCadence options: \nbhm spiders 1x8 \nImplementation: \nbhm spiders agn.py \nNumber of targets: \n12459 \nB.28. bhm spiders clusters-efeds-ls-redmapper", 'target selection tag: 0.1.0': "Summary: A carton used during SDSS-V plate-mode observations that contains galaxy cluster targets found in the eROSITA/eFEDS X-ray survey field. The carton provides a list of galaxies which are candidate members of clusters selected from early reductions ('c940') of the eROSITA performance verification survey in the eFEDS field. The parent sample of galaxy clusters and their member galaxies have been selected via a joint analysis of X-ray and (several) optical/IR datasets using the eROMaPPeR red-sequence finder algorithm (Rykoff et al. 2014; Ider Chitham et al. 2020). This particular carton relies on optical/IR data from DESI Legacy Survey DR8 (lsdr8). All targets in this carton are located within the eFEDS field (approx 126 < RA < 146, -3 < Dec < +6 deg). \nSimplified description of selection criteria: Starting from a parent catalog of eFEDS → optical/IR cluster associations, select targets which meet all of the following criteria: i) are selected by eROMaPPeR applied to lsdr8 photometric data, ii) have eROSITA X-ray detection likelihood > 8.0, iii) have r fiber > 16 . 5 and either r fiber < 21 . 0 or z fiber < 20 . 0 AB (photometry from lsdr8), and iv) do not have existing good quality (SDSS or external) spectroscopy. We assign a range of priorities to targets in this carton, with BCGs top ranked, followed by candidate member galaxies according their probability of membership. All targets were assigned a nominal cadence of: bhm spiders 1x8 (8x15mins dark time). \nTarget priority options: \n1500, 1511--1610 \nCadence options: \ndark 1x8 \nImplementation: \nbhm spiders clusters.py \nNumber of targets: \n4432 \nB.29. bhm spiders clusters-efeds-sdss-redmapper \ntarget selection plan: \n0.1.0 0.1.0 \ntarget selection tag: \nSummary: A carton used during SDSS-V plate-mode observations that contains galaxy cluster targets found in the eROSITA/eFEDS X-ray survey field. The carton provides a list of galaxies which are candidate members of clusters selected from early reductions ('c940') of the eROSITA performance verification survey in the eFEDS field. The parent sample of galaxy clusters and their member galaxies have been selected via a joint analysis of X-ray and (several) optical/IR datasets using the eROMaPPeR red-sequence finder algorithm (Rykoff et al. 2014; Ider Chitham et al. 2020). This particular carton relies on optical photometric data from SDSS DR13. All targets in this carton are located within the eFEDS field (approx 126 < RA < 146, -3 < Dec < +6 deg). \nSimplified description of selection criteria: Starting from a parent catalog of eFEDS → optical/IR cluster associations, select targets which meet all of the following criteria: i) are selected by eROMaPPeR applied to SDSS dr13 photometric data, ii) have eROSITA X-ray detection likelihood > 8.0, iii) have r fiber > 16 . 5 and either r fiber < 21 . 0 or z fiber < 20 . 0 AB (photometry from DESI Legacy Survey DR8), and iv) do not have existing good quality (SDSS or external) spectroscopy. We assign a range of priorities to targets in this carton, with BCGs top ranked, followed by candidate member galaxies according their probability of membership. All targets were assigned a nominal cadence of: bhm spiders 1x8 (8x15mins dark time). \nTarget priority options: \n1500, 1511-1610 \nCadence options: \ndark 1x8 \nImplementation: \nbhm spiders clusters.py \nNumber of targets: \n4304 \nB.30. bhm spiders clusters-efeds-hsc-redmapper \ntarget selection plan: \n0.1.0 \ntarget selection tag: \n0.1.0 \nSummary: A carton used during SDSS-V plate-mode observations that contains galaxy cluster targets found in the eROSITA/eFEDS X-ray survey field. The carton provides a list of galaxies which are candidate members of clusters selected from early reductions ('c940') of the eROSITA performance verification survey in the eFEDS field. The parent sample of galaxy clusters and their member galaxies have been selected via a joint analysis of X-ray and \n(several) optical/IR datasets using the eROMaPPeR red-sequence finder algorithm (Rykoff et al. 2014; Ider Chitham et al. 2020). This particular carton relies on optical/IR data from the Hyper Suprime-Cam Subaru Strategic Program (HSC-SSP). All targets in this carton are located within the eFEDS field (approx 126 < RA < 146, -3 < Dec < +6 deg). Simplified description of selection criteria: Starting from a parent catalog of eFEDS → optical/IR cluster associations, select targets which meet all of the following criteria: i) are selected by eROMaPPeR applied to HSCSSP photometric data, ii) have eROSITA X-ray detection likelihood > 8.0, iii) have r fiber > 16 . 5 and either r fiber < 21 . 0 or z fiber < 20 . 0 AB (photometry from DESI Legacy Survey DR8), and iv) do not have existing good quality (SDSS or external) spectroscopy. We assign a range of priorities to targets in this carton, with BCGs top ranked, followed by candidate member galaxies according their probability of membership. All targets were assigned a nominal cadence of: bhm spiders 1x8 (8x15mins dark time). \nTarget priority options: 1500, 1511-1610 \nCadence options: \ndark 1x8 \nImplementation: \nbhm spiders clusters.py \nNumber of targets: \n924 \nB.31. bhm spiders clusters-efeds-erosita \ntarget selection plan: \n0.1.0 0.1.0 \ntarget selection tag: \nSummary: A carton used during SDSS-V plate-mode observations that contains galaxy cluster targets found in the eROSITA/eFEDS X-ray survey field. The carton provides a list of galaxies which are candidate members of clusters selected from early reductions ('c940') of the eROSITA performance verification survey in the eFEDS field. The parent sample of galaxy clusters and their member galaxies have been selected via a joint analysis of X-ray and (several) optical/IR datasets. This particular carton includes counterparts to X-ray extended sources that were not selected by the eROMaPPeR red sequence finder algorithm when applied to any of the DESI Legacy Survey DR8, SDSS DR13, or HSC-SSP datasets (i.e., complementary to the cartons: bhm spiders clusters-efeds-ls-redmapper, bhm spiders clustersefeds-sdss-redmapper and bhm spiders clusters-efeds-hsc-redmapper). All targets in this carton are located within the eFEDS field (approx 126 < RA < 146, -3 < Dec < +6 deg). \nSimplified description of selection criteria: Starting from a parent catalog of eFEDS → optical/IR cluster associations, select targets which meet all of the following criteria: i) are identified as being X-ray extended but not selected via the eROMaPPeR algorithm, ii) have eROSITA X-ray detection likelihood > 8.0, iii) have r fiber > 16 . 5 and either r fiber < 21 . 0 or z fiber < 20 . 0 AB (photometry from DESI Legacy Survey DR8), and iv) do not have existing good quality (SDSS or external) spectroscopy. We assign a range of priorities to targets in this carton, with BCGs top ranked, followed by candidate member galaxies according their probability of membership. All targets were assigned a nominal cadence of: bhm spiders 1x8 (8x15mins dark time). \nTarget priority options: \n1500, 1511-1535 \nCadence options: \ndark 1x8 \nImplementation: \nbhm spiders clusters.py \nNumber of targets: \n15"}

2024MNRAS.533..120C

The observation data of blazar 1ES 142642.8 were obtained using the 1.02 m optical telescope of Yunnan Observatories during 2021 to 2023. Intraday variability IDV is detected on seven nights. We use the turbulent model to investigate the mechanism of IDV in 1ES 142642.8. The fitting light curves match the actual IDV curves well. Using this model we obtain the parameters such as the size of turbulent cells and the width of pulses in the jet. A possible shortlived quasiperiodic oscillation QPO of 58.55 pm 8.09 min was detected on 2022 April 26 whose light curve exhibits eight cycles at gt 3sigma global significance and confirmed by several different techniques. Through a more detailed analysis of the light curve of this night we find that the period is shortened from 54.23 min 4sigma to 29.71 min 3sigma. The possible QPO and period shortening phenomenon are best explained by the processes of magnetic reconnections.

2024-09-01T00:00:00Z

['2024MNRAS.533..120C', '10.1093/mnras/stae1839', '2024arXiv240906983C', '10.48550/arXiv.2409.06983', 'arXiv:2409.06983']

['Astrophysics - Astrophysics of Galaxies']

Optical intraday variability analysis for the BL Lacertae object 1ES 142642.8

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']

https://arxiv.org/pdf/2409.06983.pdf

{'No Header': '<!-- image --> \n<!-- image --> \nMNRAS 533, 120-130 (2024) \nAdvance Access publication 2024 August 7', 'Optical intraday variability analysis for the BL Lacertae object 1ES 1426 + 42.8': "X. Chang , 1 D. R. Xiong , 2 ‹ T. F. Yi , 3 , 4 ‹ C. X. Liu , 1 G. Bhatta , 5 J. R. Xu 6 ‹ and Y. L. Gong 7 \n- 1 South-Western Institute for Astronomy Research, Yunnan University, Kunming 650500, People's Republic of China\n- 2 Yunnan Observatories, Chinese Academy of Sciences, 396 Yangfangwang, Guandu District, Kunming 650216, People's Republic of China\n- 3 Key Laboratory of Colleges and Universities in Yunnan Province for High-energy Astrophysics, Department of Physics, Yunnan Normal University, Kunming 650500, People's Republic of China\n- 4 Guangxi Key Laboratory for the Relativistic Astrophysics, Nanning 530004, People's Republic of China\n- 5 Janusz Gil Institute of Astronomy, University of Zielona G · ora, ul. Szafrana 2, PL-65-516 Zielona G · ora, Poland\n- 6 Shandong Provincial Key Laboratory of Optical Astronomy and Solar-Terrestrial Environment, School of Space Science and Physics, Institute of Space Sciences, Shandong University, Weihai 264209, People's Republic of China\n- 7 Department of Astronomy, School of Physics and Astronomy, Yunnan University, Kunming 650091, People's Republic of China \nAccepted 2024 July 22. Received 2024 July 22; in original form 2024 May 10", 'A B S T R A C T': 'The observation data of blazar 1ES 1426 + 42.8 were obtained using the 1.02 m optical telescope of Yunnan Observatories during 2021 to 2023. Intraday variability (IDV) is detected on seven nights. We use the turbulent model to investigate the mechanism of IDV in 1ES 1426 + 42.8. The fitting light curves match the actual IDV curves well. Using this model, we obtain the parameters such as the size of turbulent cells and the width of pulses in the jet. A possible short-lived quasi-periodic oscillation (QPO) of 58 . 55 ± 8 . 09 min was detected on 2022 April 26 whose light curv e e xhibits eight c ycles at > 3 σ global significance and confirmed by several different techniques. Through a more detailed analysis of the light curve of this night, we find that the period is shortened from 54.23 min (4 σ ) to 29.71 min (3 σ ). The possible QPO and period shortening phenomenon are best explained by the processes of magnetic reconnections. \nK ey words: galaxies: acti ve -BL Lacertae objects: general -BL Lacertae objects: individual (1ES 1426 + 42.8) - galaxies: photometry.', '1 INTRODUCTION': 'Blazars are the most energetic populations of active galactic nuclei (AGNs) with relativistic jets pointing at a small angle to the line of sight. They are characterized by rapid flux variations throughout the entire electromagnetic spectrum, high luminosities, and strong polarizations (Urry & P ado vani 1995 ). Studies of blazars can help us understand the mechanisms that power the large-scale jets of AGNs as well as the physical properties of accretion discs near central supermassive black holes (SMBHs). Blazars consist of the BL Lacertae objects (BL Lacs) with almost featureless continuum spectra, and the flat spectrum radio quasars with prominent emission lines (Agarwal & Gupta 2015 ). \nThe study of variability is one of the most powerful tools for understanding the processes occurring in blazars. The rapid and significant variability observed in blazars can be utilized to investigate the internal physical processes of jets, including particle acceleration, the origin of flares, and the structure/dynamics of emission regions (Miller, Carini & Goodrich 1989 ; Gupta, Sri v astav a & Wiita 2009 ; \n- /star E-mail: xiongdingrong@ynao.ac.cn (DX); yitingfeng98@163.com (TY); xujingrichard@foxmail.com (JX) \nLiu & Bai 2015 ; Xiong et al. 2017 ). Extremely rapid variation o v er a few minutes to less than a day is often called intraday variability (IDV) or microvariability (Miller et al. 1989 ; Agarwal & Gupta 2015 ). Some blazars exhibit high amplitudes of variability across different wavebands, occurring on time-scales as short as several minutes (Sagar et al. 2004 ). Models related to jets and accretion discs have been proposed to explain the variability behaviors observed in blazars, but numerous details of these models are still under discussion (Bhatta 2021 ). Webb et al. ( 2021 ) present a theory to explain the IDV/microvariability. This model assumes that the observed flares are caused by shock waves propagating down the turbulent jet. Particles in each turbulence plasma cell are accelerated by the shocks and subsequently cooled by synchrotron radiation, resulting in pulses or flares in the light curves (Bła ˙ zejowski et al. 2000 ; Xu et al. 2019 ; Webb et al. 2021 ). The turb ulent cells ha ve specific sizes, densities, and magnetic field orientations. The flare duration indicates the size of the turbulent cells, and the amplitude correlates with the density and magnetic field properties of the cells. Solutions to the particle kinetic equations with synchrotron cooling can predict the shape of pulses emitted at specific frequencies as a result of the shock wave interacting with the turbulent cells (Kirk, Rieger & Mastichiadis 1998 ; Webb et al. 2017 ; Webb & Sanz 2023 ). By deconvolving the light curves into individual pulses and analysing the flare distribution characteristics, the turbulence \ncharacteristics in the jet can be studied (Bhatta et al. 2013 ; Webb et al. 2021 ). \nIn blazars, jets originate from the central engines of SMBHs. The IDVs are most likely produced in the vicinity of the SMBHs (e.g. Miller et al. 1989 ). Quasi-periodic oscillations (QPOs) on intraday time-scales are rarely detected in blazars (Hong, Xiong & Bai 2018 ). In the past decade, researchers have used data from different wavebands to detect QPOs in blazars across various time-scales. Additionally, the significance of the signals of QPOs still need to be further confirmed (Gupta et al. 2009 ; Lachowicz et al. 2009 ; Rani et al. 2010 ). Gupta et al. ( 2009 ) detected IDV QPOs with periods of ∼ 25 -73 min for S5 0716 + 714 with high probabilities (95 per cent -/greaterorsimilar 99 per cent ). Hong et al. ( 2018 ) confirmed their period estimation of the S5 0716 + 714 QPO to be ∼ 50 min at the 99 per cent significance level using the z-transformed discrete correlation function method and the Lomb-Scargle method. Rani et al. ( 2010 ) also disco v ered a QPO with a period of ∼ 15 min at the 3 σ confidence level in S5 0716 + 714 using several different techniques. The potential explanation for the observed IDV QPO is magnetic reconnection in the jet systems. In the inner regions of the jet flows, magnetic reconnection can be generated by magnetohydrodynamic turbulence, causing periodic instabilities and quasi-periodic fluctuations in the relativistic jets (Balbus & Ha wle y 1991 ; ˇ Cemelji · c et al. 2022 ). Jet systems are dominated by kink instabilities, which can more efficiently generate magnetic reconnection, and result in quasi-periodic releases of emission energy (Dong, Zhang & Giannios 2020 ; Jorstad et al. 2022 ; Raiteri et al. 2023 ). The observed periods can be drastically shortened by the relati vistic ef fects. \nThe blazar 1ES 1426 + 42.8 was disco v ered in the medium X-ray band (2 -6 keV ) with the Large Area Sky Survey experiment (LASS) on High Energy Astronomical Observatory (HEAO-1) (Wood et al. 1984 ). Abdo et al. ( 2010 ) classified 1ES 1426 + 42.8 ( z = 0 . 129) as a BL Lac object based on its featureless spectral energy distributions. In addition, it is an extreme TeV source with low power and high synchrotron peak ( νpeak > 100 keV ) (Costamante et al. 2003 ). Presently, most of the observations and explorations of 1ES 1426 + 42.8 focus on the high-energy bands, while the optical bands remain less studied. Gaur et al. ( 2010 ) collected a sample of blazars with IDV light curves that have at least four peaks to search for potential IDV QPOs. They found a possible QPO in one light curve of PKS 2155 -304, but no QPO was found in 1ES 1426 + 42.8. Ho we ver, Chang et al. ( 2023 ) detected a potential QPO with a period of 48.67 ± 13.90 min in 1ES 1426 + 42.8 on 2010 April 13, and this QPO is further confirmed at > 3 σ level on 2021 March 16. Unfortunately, the number of data points available for the QPO light curves in these 2 d is limited. With undersampled light curves, the confidence level is relati vely lo w. The moti v ation of this work is to search for IDV and further investigate the existence of QPO in 1ES 1426 + 42.8 with higher confidence levels, and to explore the physical mechanisms of the IDV and QPO. The study of the IDV and QPO can provide deeper insights of accretion discs and jets, as well as the dynamic processes that occur in the central engines of blazars (Li et al. 2023 ). Therefore, it would be beneficial to search for IDV and QPO in the light curves of blazars as many as possible. In this work, we conducted new optical monitoring for 1ES 1426 + 42.8 from 2021 to 2023 (16 nights), and present the results of our IDV and QPO study. \nThis paper is structured as follows. Section 2 describes the observations and data analyses. Section 3 describes the results of variability and periodicity analyses. Our discussions and conclusions are presented in Sections 4 and 5 .', '2 OBSERVATIONS AND DATA REDUCTION': 'Our observations of 1ES 1426 + 42.8 were carried out with the 1.02 m optical telescope administered by Yunnan Astronomical Observatories of China. During our observations, the 1.02 m telescope was equipped with an Andor Ikon XL CCD (4096 × 4112 pixels) camera at the Cassegrain focus ( f = 13 . 3 m ). The field of view is 15 × 15 arcmin 2 . The pixel scale is 0 . 238 arcsec pixel -1 . The standard Johnson broad-band filters are used. Our photometric observations were performed through the I -band filter. A 2-min exposure time was adopted to achieve a high signal-to-noise ratio (SNR) and ensure intensive sampling. Fiv e sk y flat frames were taken at dusk, and 10 bias frames were captured immediately before the scientific observations. The APPHOT task of the IRAF 1 software is used for aperture photometric measurements with flats and biases corrected. Aperture magnitudes are calculated through a set of radii. The aperture with a radius of twice of the full width at half-maximum (FWHM) of the point spread function, r = 2 × F W H M , is finally adopted for the best SNR. We measured the instrumental magnitudes of the comparison star and the check star in the same field of 1ES 1426 + 42.8. The comparison star and the check star are taken from the finding chart of Smith, Jannuzi & Elston ( 1991 ) 2 for flux calibration. The comparison star is required to have the colour and the brightness similar to 1ES 1426 + 42.8. The check star is chosen to be the one with the smallest variations in differential magnitudes with respect to the comparison star (Agarwal & Gupta 2015 ; Xiong et al. 2017 ). Then we chose to use Star 1 as the comparison star and measure Star 2 as the check star (see the finding chart of Smith et al. 1991 ). The apparent magnitude of 1ES 1426 + 42.8 can then be calculated from the comparison star and the check star using differential photometry (Bai et al. 1998 ; Fan et al. 2014 ). According to the magnitudes of the comparison star and the check star, the root-mean-square (RMS) errors of the photometry at this night can be calculated (Xiong et al. 2017 ): \nσ = √ ∑ ( m i -m ) 2 N -1 , (1) \nwhere m i represents the differential magnitude of the comparison star and check star, m represents the averaged differential magnitude for one night, and N represents the total number of observations on a given night. \nWe conducted the monitoring program for 1ES 1426 + 42.8 from 2021 to 2023 in I band with additional continuous observations on 2021 April 22 in C band (clear band). Excluding the nights with bad weather, the total number of nights with continuous observations for 1ES 1426 + 42.8 is 16 (1818 data points). They are January 25, March 16, 23, and 24, April 22 and 29, May 18 in 2021, April 24-27 in 2022, and April 17-21 in 2023. Details of the observations are listed in Table 1 . The light curves from 2021 to 2023 are presented in Figs 1 -3 , respectively. \nTable 1. The log of observations. Columns 1 to 5 represent the Universal Time ( UT ), Julian Day (JD), magnitudes, RMS errors, and the observed band, respectively. \nNote. (This table is available in its entirety in machine-readable form.)', '3.1 Variability analysis': "In order to quantify the IDV/microvariation, we employed the Howell statistical method (Howell, Mitchell & Warnock 1988 ) (hereafter Howell Test). The Howell Test compares the variances of the object to the variances of a comparison star and of a check star, accounting for the detector's instrumental noise properties and brightness differences between the source and comparison star, to determine whether observed variations are real or merely statistical noise (Howell et al. 1988 ; de Diego 2010 ; Webb et al. 2021 ). Based on the characteristics of CCD, it computes the probability that the detected variation in the variable is real, rather than random noise (Howell et al. 1988 ). The F value in the Howell Test is calculated as follows: \nF = S 2 ( BL -StarA ) /Gamma1 2 S 2 ( S tarA -S tarB ) , (2) \nwhere S 2 ( BL -StarA ) is the measured variance of the differential instrumental magnitudes between the blazar BL and comparison StarA , while S 2 ( S tarA -S tarB ) is the measured variance of the differential instrumental magnitudes between comparison StarA and check StarB . /Gamma1 2 is a statistical correction factor calculated based on the known properties of CCD, which is used to correct the different SNRs for the objects in CCD frame due to the photon noise (Howell et al. 1988 ). The F value is compared with the critical F value, F α νbl ,ν ∗ , where νbl and ν ∗ represent the number of degrees of freedom for the blazar and comparison star, respectively ( ν = N -1), and α represents the significance level set as 0.01 (2.6 σ ) (Xiong et al. 2017 ). If the F value exceeds the critical value, the blazar can be considered variable with a 99 per cent confidence level. Another alternative to the standard F-test is the one-way analysis of variance (ANOVA) test (de Diego 2010 ). de Diego ( 2010 ) reported that ANOVA is a powerful and robust estimator for microvariations. The ANOVA method derives the expected variance from subsamples of the data, rather than relying on error measurement. Considering the exposure time, we bin the data in groups of three or five observations (de Diego 2010 ; Xiong et al. 2017 ). If measurements in the last group are less than three or fiv e, then the y will be combined with the previous group. F α ν 1 ,ν 2 in the F-statistics can be used to obtain the critical value of ANOVA, where ν 1 = k -1( k is the number of groups), ν 2 = N -k ( N is the number of measurements), and α is the significance level (Hu et al. 2014 ). A blazar is classified as variable (V) if its light curve passes both tests on a given night. If one or all of the criteria are not met, the blazar is classified as non-variable (N). Table 2 displays the results of the variability analysis. The results show that IDVs were found on seven nights (2021 March 16 and 24, 2022 April 24-26, 2023 April 17 and 21). The IDVs in this object are intermittent, and there is no correlation between brightness level and the presence or absence of IDV.", '3.2 Periodicity analysis': "Blazars' QPO research holds significant values in terms of investigating the physical mechanisms as well as radiation processes in blazars (Gupta et al. 2009 ). Figs 1 , 2 , and 3 present the 16 light curves observed in the year of 2021, 2022, and 2023, respectively. The light curve on 2021 March 16 shows four obvious peaks (the upper right panel of Fig. 1 ). There are eight cycles in the light curve of 2022 April 26 (the bottom left panel of Figs 2 and 5 ). This light curve is considered to exhibit IDV (with an amplitude of 61.08 per cent) because it follows the normal distribution and it successfully passes both the F-test and the one-way ANOVA test (de Diego 2010 ; Joshi et al. 2011 ). Thus, we conduct a detailed QPO analysis on the light curve of 2022 April 26. \nWe first use Weighted Wavelet Z-transform (WWZ: see Section 3.2.1 ) method to search for QPO for the whole optical light curve. The analysis result is presented in Fig. 4 . Beyond the initial stable variations before JD (2459600 + ) 96.144, there is a bright red patch from JD(2459600 + ) 96.144 to 96.365 (segment 1) in Fig. 4 , which represents an obvious QPO signal. For the segment 1, signal of brightest red patch appears between JD(2459600 + ) 96.144 and 96.286 (segment 2). Thus, we select these two segments to explore QPOs with three methods: the Lomb-Scargle method, WWZ, and the REDFIT method.", '3.2.1 Lomb-Scargle periodogram and WWZ': 'Lomb-Scargle periodogram (LSP) (Lomb 1976 ; Scargle 1982 ) is a method widely used to search for and analyse QPOs. LSP competes the traditional Discrete Fourier Transform by employing a leastsquares fitting technique to approximate sinusoidal waves in the data to reduce the effect of uneven sampling (Lomb 1976 ; Scargle 1982 ; Bhatta 2017 ). The periodogram is a function of circular frequency f (Li et al. 2016 ). The power spectrum density function of a real signal exhibits a peak, and the maximum power at that peak frequency corresponds to the most probable period (Wang et al. 2014 ). The WWZ (Foster 1996 ; Bhatta 2017 ) method is utilized for detecting and quantifying QPOs in both the frequency and time domains. The frequencies and amplitudes of QPOs may vary o v er time in real astronomical systems. In such cases, the WWZ method is particularly ef fecti ve in the identifications of QPOs that develop and dissipate with time (Bhatta 2017 ). A set of Morlet wavelets with different combinations of positions τ (time shifts) and scale frequencies ω (test frequencies) are used to analyse and construct the WWZ spectra (Foster 1996 ; Wang et al. 2014 ; Bhatta 2017 ). The significance of a detected periodicity is estimated by the WWZ power in a statistical manner (Foster 1996 ). The WWZ power peaks can be used to determine periodic components and track the evolution of periodicities and/or amplitudes (Li et al. 2021 ). \nThe transitory apparently periodic fluctuations of blazars usually have higher level of red noises at lower frequencies or flicker noises, which can result in spurious periodicities generated in the periodograms (Press 1978 ). When analysing the periodicity of blazars, it is necessary to carefully consider the effects of frequencydependent red-noise or flicker-noise (Vaughan 2005 ; Fan et al. 2014 ; Sandrinelli et al. 2016 ; Bhatta 2017 ). This issue can be solved using the power response method, which characterizes the power spectral density (PSD) (Uttley, McHardy & Papadakis 2002 ). The random fluctuations of blazars are generally modelled as a power-law PSD with the form P ( f ) ∝ f -α , where P ( f ) represents the power at temporal frequency f and α is the spectral slope. Following Vaughan \nFigure 1. Light curves of 1ES 1426 + 42.8 in 2021. The black open circles are the light curves for the sources. The red open circles are the magnitude difference between comparison stars and check stars in the same period. \n<!-- image --> \n( 2005 ), we used the linear regression to estimate the power spectral slope α by fitting a linear function to the log-periodogram. \nThe left panels of Figs 6 and 7 display the best-fitting PSDs of segment 1 and segment 2 obtained from this process. The PSD analysis gives the slopes of α = 1 . 21 ± 0 . 02 and α = 1 . 02 ± 0 . 03. This value is then used to model the red noises in the optical variability of 1ES 1426 + 42.8. The next step is to assess the confidence level of the QPO. We generate 100 000 simulated light curves based on the bestfitting PSD model and then used the Monte Carlo method to establish the red-noise background as described in Timmer & Koenig ( 1995 ). These artificial light curves have identical sampling interval, standard deviation, and mean value. The LSP and WWZ power spectrum \ncan be obtained for each simulated light curve (Li et al. 2023 ). The confidence levels of 99.7 per cent, 99.99 per cent, and 99.994 per cent (4 σ ) can then be estimated using the power spectral distribution of the simulated light curves. The results of LSP and WWZ are shown in middle and right panels of Figs 6 and 7 , respectively. The red, blue, and purple lines in Fig. 6 represent confidences of 99 per cent, 99.7 per cent, and 99.9 per cent, respectively, while the lines in Fig. 7 represent 99.7 per cent, 99.9 per cent, and 99.994 per cent confidence, respectively. We use the FWHM of the peak as the uncertainty of the periodic modulation. As shown in middle panel of Fig. 6 , there is an obvious peak (marked by the red arrow) in the periodogram of LSP for segment 1. This suggests a potential QPO with a period \nFigure 2. Light curves of 1ES 1426 + 42.8 in 2022. \n<!-- image --> \nFigure 3. Light curves of 1ES 1426 + 42.8 in 2023. \n<!-- image --> \nTable 2. Results of IDV Observations. Column 1: the date of the observation; Column 2: the number of data points; Column 3: the F value of Howell Test; Column 4: the critical F value of Howell Test with 99 per cent confidence le vel; Column 5: the F v alue of ANOVA; Column 6: the critical F value of ANOVA with 99 per cent confidence level; Column 7: the variability status (V: variable, N: non-variable); Column 8: the daily average magnitudes and errors. \n<!-- image --> \nFigure 4. Left panel: the WWZ power for the whole light curve (right panel) on 2022 April 26. The bright red patch represents a possible QPO in the interval of JD(2459600 + ) 96.144 to 96.365 (segment 1). The brightest red patch represents a stronger signal between JD(2459600 + ) 96.144 and 96.286 (segment 2). Right panel: the light curve of 1ES 1426 + 42.8 on 2022 April 26. \n<!-- image --> \nFigure 5. The light curve in the interval from JD(2459600 + ) 96.144 to 96.365 (segment 1). The interval before the grey vertical dashed line is JD(2459600 + ) 96.144 -96.286 (segment 2), during which clear stable periodic modulation is seen. The red dash-dotted line represents the best-fitting curve of the sine function for this interval. The red arrows are plotted to indicate the modulation in this interval. \n<!-- image --> \nof 58 . 55 ± 8 . 09 min at > 99 . 9 per cent for 1ES 1426 + 42.8 on 2022 April 26. Similarly, the highest WWZ power peak in the right panel of Fig. 6 indicates a period of 58 . 34 ± 7 . 78 min at > 99 . 7 per cent (3 σ ). Fig. 7 suggests a similar QPO with higher confidence in segment 2 with a period of 54 . 23 ± 10 . 73 min ( > 4 σ ) using the LSP method, and a period of 53 . 88 ± 9 . 71 min at > 99 . 994 per cent (4 σ ) using the WWZ method. Both methods give consistent results.', '3.2.2 REDFIT method': 'The REDFIT program 3 (Schulz & Mudelsee 2002 ) based on the LSP is often used to estimate the QPOs and the red-noise levels in the light curves of blazars (Fan et al. 2014 ; Sandrinelli et al. 2016 ; \n<!-- image --> \n<!-- image --> \nFigure 6. Period analysis results of segment 1. Left panel: the PSD of 1ES 1426 + 42.8 on 2022 April 26. The black solid line is the best-fitting power-law model of the underlying coloured noise. The red solid line is the best-fitting P ( f ) ∝ f -α profile to the black solid line ( α = 1 . 21 ± 0 . 02). Middle panel: LSP of 1ES 1426 + 42.8. The black solid line represents the corresponding power spectrum of LSP. The red, blue, and purple dashed lines represent the confidence level of 99, 99.7 (3 σ ), and 99.9 per cent, respecti vely. The red arro w marks the period of the detected QPO. Right panel: 2D plane contour of the WWZ power. The black solid curve in the side panel represents the time-averaged WWZ power. The red, blue, and purple dashed lines represent the confidence level of 99, 99.7 (3 σ ), and 99.9 per cent, respectively. \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 7. Similar plots of Fig. 7 , but for segment 2. \n<!-- image --> \nGupta et al. 2019 ). The REDFIT program can ef fecti v ely remo v e the bias in the Fourier transform of unevenly spaced data by correcting for the correlation effect among Lomb-Scargle Fourier components. Additionally, the significances of peaks in the spectrum of unevenly spaced time series can be assessed against the red-noise background by fitting a first-order autore gressiv e (AR1) process (Xiong et al. 2017 ). The results of REDFIT are presented in the Fig. 8 , where the bias-corrected power spectrum is shown by the black line, and the theoretical red-noise spectrum is shown by the red line. The significance levels of 90 per cent, 95 per cent, and 99 per cent based on Monte Carlo simulations are presented by the blue, green, and purple dashed lines, respectively. From Fig. 8 , it can be seen that the peaks (58 . 54 ± 8 . 39 min for segment 1 and 55 . 42 ± 10 . 47 min for segment 2, highlighted by the red arrow) in the spectrum of a time series are significantly detected ( > 99 per cent ) against the red-noise background. \nIn addition to the first four peaks in the segment 2 (QPO with a period of ∼ 55 min), we further conduct QPO analysis on the remaining four peaks ranging from JD(2459600 + ) 96.286 to 96.365 (Fig. 9 ). The result from the LSP method shows that there is a possible QPO with a period of 29 . 71 ± 6 . 31 min (3 σ ). Therefore, the QPO period is shortened from about 55 to 29.71 min. Ho we ver, the remaining four peaks have lower significance and the shape of each cycle is more unstable compared to the first four peaks in segment 2.', '4 DISCUSSION': 'We use different statistical methods to analyse the IDVs in the light curves of 1ES 1426 + 42.8. Among these IDVs, we found a potential QPO in the light curve on 2022 April 26. In the following subsections, \nwe discuss the sub-hour IDV and possible QPO in terms of the potential explanations.', '4.1 Variability theoretical model of IDV': 'The observed IDV/microvariability can be interpreted as a result of a convolution of individual synchrotron pulses occurring in the turbulent jets (Bhatta et al. 2013 ; Webb et al. 2021 ; Webb & Sanz 2023 ). In the relativistic jet model, if a shock interacts with a turbulent cell, it causes electrons in the cell to accelerate. The electrons then cool by synchrotron emission, ultimately resulting in the flare production (Bhatta et al. 2013 ; Webb & Sanz 2023 ). The synchrotron emission depends on the orientation and strength of the magnetic field, as well as the density of electrons (Webb 2016 ; Webb et al. 2021 ). The equation of Kirk et al. ( 1998 ; hereafter KRM) calculates the particle distribution in the shock front for various orientations of the magnetic field and particle densities (Kirk et al. 1998 ; Webb, Bhatta & Hollingsworth 2010 ). The burst profiles as described by the KRM model matched the pulse profiles of IDV/micro-variability observed in the actual blazar light curves (Bhatta et al. 2011 ). Using the model originally developed by Bhatta et al. ( 2013 ) and later impro v ed by Xu et al. ( 2023 ), we recalculated the KRM burst profiles and compared them to our IDV/micro-variability light curves. The distribution of the particles is given by the diffusion equation: \n∂N ∂t + ∂ ∂γ [( γ t acc -βs λ 2 ) N ] + N t esc = Qδ ( γ -γ 0 ) , (3) \nwhere \nβs = 4 3 σt m e c ( B 2 2 µ 0 ) , (4) \n<!-- image --> \nFigure 8. The REDFIT analysis on segment 1 (left) and segment 2 (right). The black solid line is the bias-corrected power spectrum. The red solid line is the theoretical red-noise spectrum. The blue, green, and purple dashed lines represent the confidence levels of 90, 95, and 99 per cent, respectively. The red arrow marks the period of the detected QPO. \n<!-- image --> \nFigure 9. Left panel: the light curve of 1ES 1426 + 42.8 on 2022 April 26 from the interval JD(2459600 + ) 96.286 to 96.365. The blue dash-dotted line represents the best fit of the sine function for this interval. The blue arrows are plotted to indicate the modulation in this interval. Right panel: the LSP analysis of this interval. The black solid line represents the corresponding power spectrum of LSP. The blue and purple dashed lines represent the confidence level of 99 and 99.7 (3 σ ) per cent, respectively. \n<!-- image --> \nwhere N is the number density of the electrons as a function of time t and energy γ . t acc and t esc are the particle acceleration time and the particle escape time, respectively. The ratio between them controls the pulse shape. βs is the synchrotron emission, where B , σt , µ 0 , m e , and c are the magnetic field strength, Thomsons scattering cross-section, permeability of the free space, mass of an electron, and the speed of light, respectively. The amplitude is determined by the parameter Q and is related to the magnetic field strength B and orientation θ , as well as the enhanced electron density (Bhatta et al. 2013 ; Webb et al. 2021 ). KRM solves this equation for the case of a constant injection rate Q 0 after switch-on at t = 0. \nThe KRM solution determines the amplitude and shape of an individual pulse, which represents the emission from each individual turbulent cell (Webb et al. 2021 ). By adjusting the width and amplitude of the standard pulse to fit each significant pulse in the IDV light curves, the size of the emission region can be obtained (Webb et al. 2021 ; Webb & Sanz 2023 ). Fig. 10 shows the fitting results for the model. In each panel, the blue circles are observation data, the violet dashed lines indicate different flares obtained by the fit, and the solid black line is the total fit. The correlation coefficients of the light curves obtained by fitting the model were 0.81, 0.94, 0.92, 0.97,0.91, 0.89, and 0.87, respectively. The resulting parameters for the pulses used to model the light curves are listed in Table 3 . The \nfirst column shows the date of each IDV observation. Columns 2 and 3 give the number of pulses and the average amplitude. Column 4 shows the average width of the pulses ( τf lare ). Column 5 indicates the range of cell sizes in au based on the assumed shock speed of 0 . 1 c and the duration of the pulse. The last column lists the correlation coefficients. The results indicate that turbulence exists in most regions of the jet. All turbulent cell sizes are within the range of 2.58 to 54 . 52 au , and the distribution of cell sizes is continuous. The smallest cell size typically corresponds to the Kolmogorov scale length of the turbulent plasma. Most of the dissipation occurs in the non-relativistic plasma, where the turbulent kinetic energy is dissipated into heat at the Kolmogorov scale. The largest cell size corresponds to either the size of the plasma jet or the correlation length within the plasma. Turbulent cells exceeding this scale may become unstable (Bhatta et al. 2013 ; Meng et al. 2017 ; Webb et al. 2021 ; Webb & Sanz 2023 ).', '4.2 Magnetic reconnection model of QPO': "The magnetic reconnection model has been proposed to explain the periodic variability of blazars (Dong et al. 2020 ). Dong et al. ( 2020 ) found QPO signatures in both the light curve and the polarization degree variations during their relativistic magnetohydrodynamic \nFigure 10. IDV fitting results. In each panel, the blue cycle points are the original data, the violet dashed lines show the fitted individual flares, and the black solid line is the fitting result. \n<!-- image --> \nTo sum up, the magnetic reconnection model in jets provides a \nTable 3. Pulse fit parameters. \nsimulations. Recent observations from the Whole Earth Blazar Telescope also displayed intense polarimetry variability (Webb et al. 2021 ). These findings indicate significant periodic energy releases caused by kink instabilities occurring in a small subregion of the relati vistic jets. The current-dri ven kink instability will be triggered when the jet is perturbed by a lateral displacement (Dong et al. 2020 ; Jorstad et al. 2022 ). The distorted magnetic field lines in the kinks disrupt the magnetic field structure, causing magnetic reconnection, accelerating particles, and further disturbing the magnetic field (Dong et al. 2020 ; Acharya, Borse & Vaidya 2021 ). The increase in poloidal field components and the number of accelerated particles maintain the polarization oscillation and outburst of non-thermal radiation caused by the kink instability (Zhang et al. 2017 ; Bodo, Tavecchio & Sironi 2021 ; Jorstad et al. 2022 ). The kink in the jet naturally evolves into a quasi-periodic structure, consisting of twisted magnetic fields (kink nodes). The process leads to the moving region (plasma blob) of enhanced emission, which contains a few kink nodes (Barniol Duran, Tchekhovsk o y & Giannios 2017 ; Dong et al. 2020 ). Due to the quasi-periodic nature of the kink, this blob can generate rapid and distinct QPO radiation (Acharya et al. 2021 ; Jorstad et al. 2022 ; Raiteri et al. 2023 ). The period is associated with the kink growth time. The kink growth time can be estimated by the evolution of the kink's transverse motion (Mizuno et al. 2009 ). The kink's transverse displacement is roughly equal to the size of the emission blob (Jorstad et al. 2022 ). Then the period can be estimated as the ratio of the transverse displacement of the jet from its central spine over the av eraged transv erse v elocity. Based on this, the period of QPOs expected from a kink can be estimated as follows: P obs = R KI /v tr δ , where R KI is the size of the emission region in the comoving frame of the jet (transverse displacement of the strongest kinked region), and v tr is the average transverse velocity of the kink (Dong et al. 2020 ). Using the typical transverse velocity v tr ∼ 0 . 16 c (Dong et al. 2020 ), the Doppler factor of the jet δ = 27 . 3 (Wolter et al. 2008 ), along with our estimated upper limit of the emission region 8 . 15 × 10 12 m (from the autocorrelation function, ACF, analysis, see Alexander 1997 for more details of the ACF), the period should then be /lessorsimilar 103 . 66 min in the observer's frame. The possible QPO agrees well with this period upper limit. It should be noted that kink instabilities are not persistent physical processes in relativistic jets. The energy injected into the kinked jets can vary o v er time resulting in kink instabilities (Dong et al. 2020 ). The unchanged I -band flux intensity before the periodic behaviour of the light curve ( J D < 2459696 . 144, see the right of Fig. 4 ) could be the period before the formation of kink. Once kink produced in the jet, the periodic modulation becomes apparent (the first four peaks in the segment 2). As the propagated kink in the jet partially dissipates or changes, the size of the emission region R KI becomes smaller, and then the period appears to be shortening phenomenon and the quasi-periodic variability of flux could become less obvious (the remaining four peaks after J D = 2459696 . 286). \ngood explanation for the possible short-lived QPO and the period shortening phenomenon.", '5 CONCLUSION': 'Our main results are summarized as follows: \n- (1) We observed the BL Lac object 1ES 1426 + 42.8 for 16 nights from 2021 to 2023. The IDV is detected in 7 d. \n(2) We applied the turbulent cell model to the optical observations of blazar 1ES 1426 + 42.8. We fit the light curves using this model and determine the distribution of cell sizes, amplitudes, and the width of pulses. \n- (3) The analysis yielded 80 pulses. The size distributions of the radiation turbulent cells, calculated from the fitting results, range from 2 . 58 to 54 . 52 au , which show that the distribution of cell sizes is consistent with the Kolmogorov distribution.\n- (4) A possible QPO with a period shortening phenomenon is found on 2022 April 26, which can be best explained by the magnetic reconnection model in the jet.', 'ACKNOWLEDGEMENTS': "This work is supported by the National Natural Science Foundation of China (grants 11863007, 12063005, 12063007,11703078), the Yunnan Pro vince F oundation (2019FB004), the Program for Innov ati ve Research Team (in Science and Technology) in University of Yunnan Province (IRTSTYN), and Yunnan Local Colleges Applied Basic Research Projects (2019FH001-12). We acknowledge the science research grants from the China Manned Space Project with NO. CMS-CSST-2021-A06. XC, CXL, and XWL acknowledge supports from the 'Science & Technology Champion Project' (202005AB160002) and from two 'Team Projects' -the 'Innovation Team' (202105AE160021) and the 'Top Team' (202305AT350002), all funded by the 'Yunnan Revitalization Talent Support Program'. This work is partially supported by a program of the Polish Ministry of Science under the title 'Regional Excellence Initiative', project no. RID/SP/0050/2024/1.", 'DATA AVAILABILITY': 'The data underlying this article will be shared on reasonable request to the corresponding author.', 'REFERENCES': 'Abdo A. A. et al., 2010, ApJ , 716, 30 \nAcharya S. \n, Borse N. S., Vaidya B., 2021, MNRAS , 506, 1862 \nAgarwal A. , \nGupta A. C., \n2015, MNRAS , 450, 541 \nAlexander T. \n, \n1997, in Maoz D., Sternberg A., Leibowitz E. M., eds, \nAstronomical Time Series. Dordrecht, Kluwer \nBalbus S. A. , Ha wle y J. F., 1991, ApJ , 376, 214', 'SUPPORTING INFORMATION': 'Supplementary data are available at MNRAS online.', 'suppl data': 'Please note: Oxford University Press is not responsible for the content or functionality of any supporting materials supplied by the authors. Any queries (other than missing material) should be directed to the corresponding author for the article. \nThis paper has been typeset from a T E X/L A T E X file prepared by the author.'}

2024arXiv240908341M

For over two decades gammaray burst GRB prompt emission spectra were modelled with smoothlybroken power laws Band function and a positive and tight correlation between the spectral restframe peak energy Ep and the total isotropicequivalent luminosity Liso was found constituting the socalled Yonetoku relation. However more recent studies show that many prompt emission spectra are well described by the synchrotron radiation model hence significantly deviating from the Band function. In this work we test the impact of a more suited spectral model such as an idealized synchrotron spectrum from nonthermal electrons on the Yonetoku relation and its connection with physical parameters. We select GRBs with measured redshift observed by FermiGBM together with high energy observations gt30 MeV and perform spectral analysis dividing them in two samples the singlebin sample using the light curve peak spectrum of each GRB and the multiplebins sample where we explore the whole duration of 13 bright bursts with timeresolved spectral analysis. We observed that the Ep of synchrotron spectra in fastcooling regime numnucgg1 is generally larger than the one provided by the Band function. For this reason we do not find any EpLiso correlation in our samples except for the GRBs in an intermediatecooling regime 1ltnumnuclt3 namely where peak and break energies are very close. We instead find in both our samples a new tight correlation between the restframe cooling frequency nucz and Liso nucz propto Liso0.53 pm 0.06. These results suggest that assuming that prompt emission spectra are produced by synchrotron radiation the physical relation is between nucz and Liso. The fit of the Band function to an intrinsic synchrotron spectrum returns peak energy values EpzBand sim nucz.

2024-09-01T00:00:00Z

['2024arXiv240908341M', 'arXiv:2409.08341', '10.48550/arXiv.2409.08341']

['Astrophysics - High Energy Astrophysical Phenomena']

Gammaray burst spectralluminosity correlations in the synchrotron scenario

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.08341.pdf

{'No Header': '© ESO 2024', 'Gamma-ray burst spectral-luminosity correlations in the synchrotron scenario': "Alessio Mei 1 , 2 ⋆ , Gor Oganesyan 1 , 2 , and Samanta Macera 1 , 2 \n1 Gran Sasso Science Institute (GSSI), Via F. Crispi 7, 67100 L'Aquila, Italy \n2 INFN-Laboratori Nazionali del Gran Sasso, I-67100, L'Aquila (AQ), Italy", 'ABSTRACT': 'Context. For over two decades, gamma-ray burst (GRB) prompt emission spectra were modelled with smoothly-broken power laws (Band function), and a positive and tight correlation between the spectral rest-frame peak energy Ep , z and the total isotropic-equivalent luminosity Liso was found, constituting the so-called Yonetoku relation. However, more recent studies show that many prompt emission spectra are well described by the synchrotron radiation model, hence significantly deviating from the Band function. \nAims. In this work, we test the impact of a more suited spectral model such as an idealized synchrotron spectrum from non-thermal electrons on the Yonetoku relation and its connection with physical parameters. \nMethods. We select GRBs with measured redshift observed by Fermi / GBM together with high energy observations ( > 30 MeV), and perform spectral analysis dividing them in two samples: the single-bin sample, using the light curve peak spectrum of each GRB, and the multiple-bins sample, where we explore the whole duration of 13 bright bursts with time-resolved spectral analysis. \n, \nResults. We observed that the Ep , z of synchrotron spectra in fast-cooling regime ( ν m /ν c ≫ 1) is generally larger than the one provided by the Band function. For this reason, we do not find any Ep , z -Liso correlation in our samples except for the GRBs in an intermediatecooling regime (1 <ν m /ν c < 3), namely where peak and break energies are very close. We instead find in both our samples a new tight correlation between the rest-frame cooling frequency ν c z and Liso : ν c , z ∝ L (0 . 53 ± 0 . 06) iso . \nConclusions. These results suggest that, assuming that prompt emission spectra are produced by synchrotron radiation, the physical relation is between ν c , z and Liso . The fit of the Band function to an intrinsic synchrotron spectrum returns peak energy values E Band p , z ∼ ν c , z . This may explain why the systematic interpretation of prompt spectra through the Band function returns the Ep , z -Liso relation. \nKey words. Gamma-ray bursts - astroparticle physics - high energy astrophysics', '1. Introduction': 'Cosmic explosions triggered by collapses of massive stars (Woosley 1993; Woosley & Bloom 2006) or binary neutron star mergers (Eichler et al. 1989; Narayan et al. 1992; Berger 2014; Abbott et al. 2017) produce gamma-ray bursts (GRBs), the most luminous transient phenomena in the Universe (Paczynski 1986). \nThanks to several decades of observations, GRBs revealed to be associated to highly collimated ultra-relativistic jets launched as an aftermath of their progenitor\'s explosion (see Salafia & Ghirlanda 2022 for a review). \nThe very fast prompt emission variability (e.g. Bhat et al. 2012) clearly suggests an emission site internal to the jet (Sari & Piran 1997). A natural explanation for the jet kinetic energy dissipation is through internal shocks (Rees & Meszaros 1994). However, sub-photospheric dissipation (Rees & Mészáros 2005; Pe\'er 2008) or magnetic reconnection processes (Drenkhahn & Spruit 2002; Zhang & Yan 2011) are equally able to explain the GRB radiative output. \nThe early emission observed soon after the explosion, during the so-called prompt emission phase, shows a broad and peaked spectral energy density (SED), possibly associated with non-thermal radiative processes. Historically, prompt emission spectra were modelled by the phenomenological Band function (Band et al. 1993), namely two power laws smoothly connected around a peak at energy Ep . The systematic fit of this model to large samples of GRB spectra revealed that in some cases the low energy photon index α was consistent with the one predicted by synchrotron emission in a fast-cooling regime, whereas some other GRBs exhibited very hard α , pointing towards quasi-thermal processes (Ghisellini & Celotti 1999; Lazzati et al. 2000; Mészáros & Rees 2000). The vast majority of the GRBs showed an average behaviour, with spectra too hard to be produced by synchrotron processes, but too soft to advocate for thermal origins (Preece et al. 1998; Kaneko et al. 2006; Gruber et al. 2014). \nThis tension was partially mitigated when more complex models were employed and low energy spectral breaks were identified in some GRB spectra at energies Eb . It was shown that a single synchrotron spectrum in marginally fast-cooling regime (Oganesyan et al. 2017, 2018; Ravasio et al. 2019) or slow-cooling regime (Zhang et al. 2009; Burgess et al. 2020) are able to account for the \nentire prompt emission spectrum, from the optical (Oganesyan et al. 2019) up to the GeV band (Macera et al., article in preparation). Despite the synchrotron model seems to be a viable explanation for most of the considered spectra, GRBs show diverse spectral and temporal behaviours, preventing a convincing unified interpretation. \nA feature that appears to be shared among GRBs is that their prompt spectrum is harder when the GRB is brighter and more energetic. In fact, many sources show that the rest-frame spectral peak energy Ep , z is positively and tightly correlated with the isotropic-equivalent energy Eiso (Amati et al. 2002) and luminosity Liso (Yonetoku et al. 2004). These two relations span several decades in both energies and luminosities, and despite being a ff ected by selection and observational biases (Band & Preece 2005; Nakar & Piran 2005; Butler et al. 2007; Shahmoradi & Nemiro ff 2011), they seem to hold for most of the GRB class (Ghirlanda et al. 2008; Nava et al. 2008), also at di ff erent redshifts (Nava et al. 2012). \nIn particular, the Ep , z -Eiso relation (better known as the "Amati relation") was initially found for long GRBs, with durations T 90 > 2 s. Short GRBs ( T 90 < 2 s) exhibit a similar trend but occupy a parallel track in the Ep , z -Eiso plane (Ghirlanda et al. 2009), showing a systematically lower Eiso for the same Ep , z . On the other hand, both long and short GRBs follow the same Ep , z -Liso relation (better known as the "Yonetoku relation", Yonetoku et al. 2010). After their first discoveries, many outliers to these relations were found, however both the Amati and Yonetoku relations still hold when multiple time-bins from the same bright GRB are considered through time-resolved spectral fits (Ghirlanda et al. 2010, 2011a,b) . \nAt odds with the Amati relation, which is obtained comparing quantities averaged across the whole burst duration, the Yonetoku relation is found considering Liso (but also Ep , z , e.g. Tsvetkova et al. 2017) estimates only during the brightest pulse, i.e. at the peak of the light curve. This makes the Yonetoku relation inherently connected with the physics taking place in single GRB pulses, and it appears particularly suited to inquire prompt emission physics in depth. \nFor more than two decades, the observations carried out by gamma-ray instruments such as the Burst Alert Telescope (BAT, 15-150 keV) on board the Neil Gehrels Swift Observatory ( Swift ), Gamma-ray Burst Monitor (GBM, 8 keV-40 MeV) on board the Fermi satellite and Konus-Wind (KW, ∼ 20 keV-20 MeV) further corroborated the validity of these correlations (e.g. Nava et al. 2012; Tsvetkova et al. 2017; Minaev & Pozanenko 2020; Wang et al. 2024). \nDespite all these findings point towards a possible common emission mechanism occurring in GRB central engines, a clear physical interpretation of these correlations is still missing (see Parsotan & Ito 2022 for a review). \nIn this work, we carry out spectral fits of GRB samples detected by Fermi / GBM using not only the phenomenological Band function, but also a physical synchrotron model. Our goals are (i) to test whether using a physical model allows to derive the Ep , z -Liso (Yonetoku) relation and (ii) to explore any possible connection between the Yonetoku relation and physical synchrotron parameters such as the characteristic minimum frequency ν m and the cooling frequencies ν c . \nThe paper is organized as follows: in Section 2 we describe the sample selection and the spectral modelling; in Section 3 we show the results of the spectral fit to the whole sample of GRBs ( single-bin sample) and the time-resolved analysis of a sub-sample of bright GRBs ( multiple-bins sample); in Section 4 we discuss the results, and finally in Section 5 we summarize the conclusions. Throughout the paper we assume standard cosmology with h = ΩΛ = 0 . 7 and Ω m = 0 . 3. If not stated di ff erently, errors are reported at 68% confidence level. As a convention, the subscript z refers to rest-frame parameters.', '2.1. Sample selection': 'We analyse GRBs observed by the Fermi satellite before September 2023. We select the GRBs with a firm redshift estimate, as provided by the MPE GRB online catalog 1 . We consider only the GRBs that are bright enough during their main pulse to perform a robust spectral analysis in that temporal bin. Specifically, we include in our sample only the GRBs whose 1s peak photon flux in the energy range 50-300 keV P ≥ 3 . 5 photons s -1 cm -2 , which corresponds to nearly 7 times the flux sensitivity of Fermi / GBM. We exclude GRB221009A, GRB230307A and GRB130427A, the three brightest GRBs, since their emission during the peak are a ff ected by pile-up e ff ects and their GBM data cannot be used for spectral analysis. \nThis leads to a total sample of 74 GRBs, with a spectroscopic redshift estimate for 70 of them. Three GRBs (GRB100816A, GRB200826A and GRB211211A) have a redshift estimate through host galaxy association while one GRB (GRB200829A) has a photometric redshift estimate.', '2.2. Data reduction': 'Wedownloaded the Fermi / GBMdata (8 keV - 40 MeV) of all the 74 GRBs in our sample from the Fermi GBM Burst catalogue 2 , and performed a standard data reduction using the Fermi science tool GTBURST 3 . In particular, we consider data from the two NaI and one BGO detectors with best observational conditions (i.e. lowest viewing angles). In three GRBs (GRB121128A, GRB131231A, GRB140508A) the second best NaI was exhibiting artefacts in the low energy data ( < 30 keV) leading to spectral deviations with respect to the best NaI. For this reason, for these three GRBs we considered the best and third-best NaI, together with the best BGO detector. \nThe background analysis was performed through GTBURST by using, whenever the polynomial fit was converging, the background intervals reported by Fermi collaboration (von Kienlin et al. 2020). In case this selection was leading to a non convergent background \nγ \nm \nm \n, \nz \nc \n, \nz \nm \nc \npolynomial fit, larger custom background intervals were selected. \nWe selected the source interval to coincide with the peak of the GRB light curve. The time-bin is the one reported in the bcat file, a file containing basic burst parameters provided by Fermi collaboration for every GRB. The width of the time bin is 1 . 064 s for long GRBs ( T 90 > 2 s) and 256 ms for short GRBs ( T 90 < 2 s), in order to pinpoint their particularly short peak phase. \nThe outputs of the data reduction are the source, background and weighted response spectral files that are used for the spectral analysis through the Heasarc package XSPEC 4 (Arnaud 1996). \nThe addition of high energy ( > 30 MeV) data to the prompt emission proved to be crucial in determining spectral parameters otherwise inaccessible or di ffi cult to constrain. For this reason, we decided to include in our dataset Large Area Telescope (LAT) Low Energy (LLE, 30-100 MeV) data whenever available. We retrieved the data from the Fermi LAT Low-Energy Events Catalog 5 . LLE data are reduced using the same tools employed in the Fermi / GBManalysis, selecting the source in the same time-bin in order to perform a simultaneous joint time-resolved analysis during the brightest peak of the emission. In our current sample, only 22 GRB have available LLE data during their peak. For those GRBs we produce spectral files similarly to the GBM ones.', '2.3. Spectral fit routine': 'After the data reduction, we model the spectra in our sample. For the fitting process we use py XSPEC 6 , the Python interface to the XSPEC spectral-fitting program. We implemented the python-based software Bayesian X-ray Analysis (BXA, Buchner 2016), which allows for Bayesian parameter estimation and model comparison through nested-sampling algorithms in py XSPEC. \nWe ignored the energy channels outside 8-900 keV for the NaI detectors, as well as the 30-40 keV band in order to avoid the Iodine K-edge line at 33.17 keV (Meegan et al. 2009). We selected the energy range 300 keV - 40 MeV and 30-100 MeV for BGO and LLE data, respectively. For each dataset, consisting of GBM and LLE data (when available), we test two models: the phenomenological Band function ( grbm in XSPEC notation, Band et al. 1993) and an idealised physical Synchrotron model. We included the presence of a cross-calibration constant, constant in XSPEC notation, allowing for a 30% variation for each dataset. We applied a Poisson-Gaussian statistics ( pgstat ) to GBM data and Cash statistics ( cstat ) to LLE data. \nFor each model we measure the logarithm of the 10 keV - 10 MeV bolometric flux in the source rest frame log F using the convolutional model cflux in XSPEC. \nMoreover, in two GRBs (GRB090902B and GRB190114C) the prompt spectrum exhibits in NaI data a power law component (Abdo et al. 2009; Ajello et al. 2020; Ursi et al. 2020) in addition to the main emission one, also at low energies. Therefore, we add a power law in both the Band and Synchrotron models, powerlaw in XSPEC notation. In these two cases, we compute the flux of the main component only (Band or Synchrotron), since the physical nature of this additional power law was already discussed in the literature and it is outside the scope of this work. \nFor the Synchrotron case, we used the same table model presented in Oganesyan et al. (2019). Synchrotron spectra are produced by a population of non-thermal electrons injected as a power law d Ne / d γ ∝ γ -p with a characteristic minimum Lorentz factor γ m . The final spectrum is obtained by convolving the single electron spectrum with the electron population distribution after that particles cooled through synchrotron losses down to the cooling Lorentz factor γ c . Table spectra are produced by varying log ( γ m /γ c ) between -1 and 2, allowing for both fast ( γ m > γ c ) and slow ( γ m < γ c ) cooling regimes, and by varying p between 2 and 5 and the normalization NSync between 0 and 10 20 (in cgs units). However, this table model assumes a constant cooling frequency ν c = 1 keV. To let this parameter vary, we multiply the table model with the XSPEX convolutional model zmshift , which shifts the spectrum along the energy axes. The resulting photon spectrum depends on four parameters: the ratio between the Lorentz factors γ m /γ c , the power law slope p of the injected electron distribution, the spectral shift parameter zshi f t (univocally linked with the observed cooling frequency ν c ), and the normalisation NSync .', '2.4. Parameter estimation, model comparison and goodness-of-fit': "For each free parameter of the two models, we define broad and uninformative priors (Table 1). The analysis returns posteriors of the parameters fitted to our dataset together with the Bayesian evidence. \nBeing also interested in physical parameters, we derived them from the inferred ones. In the case of the Band model, we evaluate the rest frame peak energy defined as E Band p , z = (2 + α ) · Ech · (1 + z ), where α is the low-energy photon index and Ech is the Band characteristic energy. \nIn the case of the synchrotron model, we define the rest-frame synchrotron frequency associated to \nand to \nas \n= \n· \n( \n2 \n) \nand the rest frame peak energy as \nE \nSync \np \n, \nz \nm \n, \nz \nc \n, \nz \nIn both cases, we infer the logarithm of the 10 keV - 10 MeV rest frame flux log F , from which we compute the flux F = 10 log F and the total isotropic-equivalent luminosity Liso = 4 π D 2 L ( z ) · F , where DL ( z ) is the luminosity distance from the burst. \nFor each relevant parameter, we define the best fit value as the median on the posterior distribution, and lower / upper errors are derived from the 16th- and 84th-percentile of the posterior distribution, respectively. \nTo ensure the good convergence of the fit, we modified the prior of the cooling frequency for GRB170607A between ν c ∈ (8 , 500) keV, i.e. zshi f t ∈ ( -0 . 998 , -0 . 8). \nFrom the Bayesian fit of both models to each GRB dataset, we get the Bayesian evidence log Z . This provides a tool to compare which model describes better the data. In fact, the model that has a substantial larger evidence is the best-fit model. To assess this, \n= \nmax( \n). Here, \nzshi f t \nis the spectral shift parameter. \nγ \nc \nas \nν \nc \nz \n= \n(1 \n+ \nz \n) \n/ \n(1 \n+ \nzshi f t \n) \nν \nν \nγ \n/γ \nν \n, ν \n, \nTable 1: Free parameters and flat uninformative priors used for the spectral fit. Two main models, Band and Synchrotron, were fit to the dataset. For each of these models, we include cross-calibration constant and a measurement of the 10 keV - 10 MeV bolometric flux. \nwe define the Bayes factor for these two models as log B = log GLYPH<16> ZSync / ZBand GLYPH<17> , and we set the threshold to 0.5 according to Je ff rey's scale (Kass & Raftery 1995). If | log B | > 0 . 5 we can identify a best-fit model, being the synchrotron one if log B > 0 or the Band function if log B < 0. If | log B | < 0 . 5, the Bayesian evidence of one model is not large enough to statistically exclude the other. \nIn addition, we aim to establish whether or not the synchrotron model provides a good fit to the data. To do so, we performed Posterior Predictive Checks (PPC) for each synchrotron fit to our sample. \nBy fitting a given model to a dataset, the best-fit model is the realization that maximizes the relative likelihood, therefore returning the smallest statistics. To asses whether that model realization is a good fit to the data, we simulate 1000 fake spectra out of the best-fit model, using the XSPEC command fakeit . We estimate, for each fake spectrum, the likelihood of the best-fit model out of the fake spectrum dataset. \nBy repeating this computation for all the 1000 fake spectra, we can define a (Gaussian-like) probability density function relative to the statistics distribution. We measure the goodness-of-fit through the comparison between the simulated statistics distribution and the measured likelihood value. Specifically, we compute the p -value of the real statistics on the simulated distribution, and we consider it a good fit if p -value > 0 . 05. We define the p -value as the integral of the simulated statistics probability density function between the best-fit statistics value and + ∞ or -∞ depending if the measured statistics is greater or lower the maximum of the distribution, respectively.", '2.5. Linear fit routine': "To investigate the Yonetoku relation, we fit the following power law relation to the data of our samples: \nEp , z 100 keV = K · Liso 10 52 erg / s ! m (1) \nWhere m and K are the power law slope and the normalization, respectively. More practically, we use the logarithmic form of Eq. 1 in order to perform a linear fit: \nlog Ep , z = m · GLYPH<0> log Liso -52 GLYPH<1> + log K -2 (2) \nWe normalise both Ep , z and Liso in order to perform the linear fit closer to the data barycenter, providing better m and K estimates. Values and 1 σ errors for log Ep , z and log Liso are derived directly from Ep , z and Liso parameter distributions, respectively. \nWeperform the fit using a Bayesian approach. We define a likelihood that assumes Gaussian parameter distributions for both log Ep , z and log Liso and takes into account errors on both parameters with the addition of an intrinsic Gaussian noise term σ sc (D'Agostini 2005): \n-2 ln L GLYPH<16> m , K , σ sc | { xi , σ xi , yi , σ yi } GLYPH<17> = = N X i = 0 ln GLYPH<16> σ 2 sc + σ 2 yi + m 2 σ 2 xi GLYPH<17> + ( yi -m ( xi -52) -log K + 2) 2 ( σ 2 sc + σ 2 yi + m 2 σ 2 xi ) (3) \nwhere N is the number of GRBs in a given sample, ( xi , σ xi ) are log Liso mean and standard deviation relative to the i -th GRB and ( yi , σ yi ) are log Ep , z mean and standard deviation relative to the i -th GRB, respectively. In this approach, together with m and K we estimate the intrinsic scatter σ sc , which constitutes a new free parameter. \nGiven the shape of log Liso parameter distribution derived from the spectral fit, we note that the hypothesis of a Gaussian distribution constitutes a good approximation. For this reason, we define σ xi as the quadrature sum of its upper and lower errors. On the other hand, the shapes of log Ep , z parameter distributions are asymmetric and deviate from a classic normal distribution. To account for this inconsistency, we still approximate log Ep , z as a Gaussian, but we consider σ yi to be equal to the log Ep , z upper or lower error if the data point of the i -th GRB is above or below the best-fit line, respectively. \nWe adopt uniform priors for the three parameter, spanning the ranges m ∈ (0 , 5), K ∈ (0 . 01 , 100) and σ sc ∈ (0 , 100). We sampled the posterior probability density with a Markov chain Monte Carlo approach using the emcee Python package (Foreman-Mackey et al. 2013), employing Nwalk = 32 walkers. We initialized the walkers in a small 3-dimensional ball around a point in our parameter space ( m , K , σ sc ) = (0 . 5 , 1 . 8 , 10) and we performed Niter = 5000 iterations, for a total of Niter × Nwalk = 160000 samples.", '3. Results': 'The main results of the spectral analysis fit to our single-bin sample are shown in Tables 2 and 3. \nFrom the Bayes factor comparison, we see that 21 GRBs are best fitted by Synchrotron model, 40 GRBs are best fitted by Band model while the two GRBs with a second component during the prompt emission peak prefer a Band model together with a power law. For 11 GRBs, the Bayes factor modulus | log B | is not large enough to prefer one model to the other. \nHowever, the comparison between Band and the synchrotron models is not straightforward. Band is a phenomenological model, and it is able to accommodate a variety of spectra by providing measures of photon indices and peak energy. On the other hand, synchrotron model hinders many physical assumptions, providing precious insides on the nature of the emission, but still having less freedom in describing an observed spectrum. \nDespite the Band function being the best-fit model for most of the GRBs in our sample, we notice that the synchrotron model provides a good fit to the data in the majority of the cases. For this reason we split the sample in two groups: the "Synchrotron" sample (with 58 GRBs out of 74, Table 2) and the complementary "Band" sample (with 16 GRBs out of 74, Table 3). The latter includes all the GRBs that meet at least one of the following conditions: 1.) Synchrotron model provides a good fit of the data according to the PPC criterium; 2.) the best-fit model is Synchrotron; 3.) Bayes factor is not able to prefer one model to the other. For these GRBs we report parameter estimates obtained from the synchrotron fit. \nTable 2: List of bursts belonging to the single-bin sample, Synchrotron sub-sample. We show the results of the synchrotron fit for the burst that meet at least one of the following conditions: Synchrotron is the best-fit model, Synchrotron can not be excluded by the model comparison ( | log B | < 0 . 5), synchrotron model provides a good fit to the data. We report redshifts z (see Sec. 2.1), isotropicequivalent luminosities Liso , rest-frame peak energies Ep , z (namely the minimum frequency ν m , z ) and rest-frame cooling frequencies ν c , z obtained from the Synchrotron fit. We report errors at 68 per cent confidence level. We further report the p -value returned from the PPC analysis, the Bayes factor log B obtained comparing Band and Synchrotron models, and the associated best-fit model: S = Synchrotron, B = Band and ND = Not Defined, i.e. when | log B | < 0 . 5 \n. \nTable 2 continued \n. \n- \n5 \n74 \n- \n147 \n- \n79 \nTable 3: List of bursts belonging to the single-bin sample, Band sub-sample. The GRBs here listed are the ones which do not meet the condition of Table 2. Therefore, we report redshifts z (see Sec. 2.1), isotropic-equivalent luminosities Liso , rest-frame peak energies Ep , z and low energy photon indices α obtained from the Band fit. We report errors at 68 per cent confidence level. We further report the p -value returned from the PPC analysis, the Bayes factor log B obtained comparing Band and Synchrotron models, and the associated best-fit model: S = Synchrotron, B = Band and ND = Not Defined, i.e. when | log B | < 0 . 5 \n. \n. \nTable 4: Results of the statistical analysis on the Ep , z -Liso and ν c , z -Liso relations relative to the single-bin sample, to its Syncrotron sub-sample ("Sync"), to its Synchrotron sub-sample with intermediate-cooling spectra("Sync-Interm", i.e. 1 < ν m /ν c < 3) and to the multiple-bins Synchrotron sub-sample. We report the Spearman\'s rank correlation coe ffi cient ρ and its associated chance probability Pchance , the slope m and normalization K of the power law fit and the intrinsic scatter σ sc of the data points around the best-fit line. \nInterestingly, all the GRBs in this sample are in fast-cooling regime ( ν m /ν c > 1), therefore in Table 2 we report Liso , the peak energy Ep , z = ν m , z and the cooling frequency ν c , z , both in the rest-frame. On the other hand, the former sample includes all the GRBs where the synchrotron model does not provide a good fit to the data, and therefore we report parameter estimates obtained from the Band model fit. \nIn the Synchrotron sub-sample, all the spectra are well fitted by the Synchrotron model according to the PPC criterium except for three GRBs: GRB131231A, GRB150517A and GRB211211A. For the first two GRBs, p -value < 0 . 5 because the best-fit statistics is particularly smaller than the average simulated statistics, i.e. it has a significantly larger likelihood. In the case of GRB211211A, the Bayes factor prefers the Synchrotron model despite the very large statistics compared to the simulated ones. In our synchrotron analysis, we get a comparable statistics with respect to the one reported by (Gompertz et al. 2023) fitting a double smoothly broken power law function to the peak spectrum of this burst. \nBecause of model comparison, the results driven by analysing the single-bin synchrotron sub-sample are valid under the assumption of synchrotron emission producing the prompt emission spectra.', '3.1. Ep , z -Liso relation': "c \nm \n<!-- image --> \nFig. 1: Yonetoku relation obtained from the single-bin sample. Black stars represent GRBs from the Band sub-sample, while colored dots represent GRBs from the Synchrotron sub-sample. The color scale is associated to the frequency ratio ν m /ν c derived from the Synchrotron fit. We fit a power law to the whole single-bin sample (left-hand panel) and to the GRBs in the Synchrotron sub-sample in an intermediate cooling regime (right-hand panel). The black straight line represents the best-fit line from the linear fit, and the green-shaded area the relative 3 σ scatter region. For comparison, the black-dashed line represents the best-fit line from Yonetoku et al. 2004. \n<!-- image --> \nWefirst test the Ep , z -Liso relation obtained from the analysis of the whole sample. We computed the Spearman's rank correlation coe ffi cient ρ and its associated chance probability Pchance . \nWe report the results in Table 4, together with the ones of the linear fit. In Fig. 1 (left panel) we show that the data are scattered around the best-fit line, suggesting the absence of any correlation between Liso and Ep , z ( ρ ∼ 0 . 5). In addition, the slope we obtain deviates from the one expected in the Yonetoku relation. \nNonetheless, we observe that GRBs with di ff erent frequency ratios ν m /ν c populate di ff erently the Ep , z -Liso plane. In fact, GRBs with a high ratio (i.e. in a fast-cooling regime, ν m /ν c ≳ 10) tend to deviate from the Yonetoku relation, whereas GRBs with a lower ratio (i.e. in an intermediate -cooling regime, ν m /ν c ≲ 3) tend to cluster along the Yonetoku relation. \nBy restricting the synchrotron sample to the GRBs in the intermediate-cooling regime (1 < ν m /ν c < 3) and fitting the relation, we find that the data points are tightly correlated, and the linear fit returns values similar to the ones obtained in Yonetoku et al. 2004 (Fig. 1, right panel). The results of this linear fit are also shown in Table 4.", '3.2. A physical relation: ν c , z -Liso': "The Ep , z -Liso analysis previously discussed shows that, when the physical synchrotron model is used, the Yonetoku relation does not hold for the whole sample, but for only a subset of GRBs that show a synchrotron spectrum where 1 < ν m /ν c < 3 (Fig. 1, left panel). This constitutes an important connection between the Yonetoku relation and a physical parameter, namely the ratio between the rest-frame cooling frequency ν c , z and minimum frequency ν m , z . In particular, the GRBs exhibiting a relation between their Ep , z and Liso are the ones whose spectral break energy is particularly close to their peak energy. \nTherefore, we test the possibility to have a correlation between the break energy, namely the cooling frequency ν c , z , and the isotropic-equivalent luminosity Liso . We employ the same methodology used to investigate the Ep , z -Liso relation, but applied to the Synchrotron sub-sample of the single-bin sample (Table 2). We find that all the GRBs in the Synchrotron sub-sample, regardless \nc \nm \nFig. 2: ν c , z -Liso relation obtained from the Synchrotron single-bin sample. The color scale is associated to the frequency ratio ν m /ν c derived from the Synchrotron fit. The black straight line represents the best-fit line from the linear fit, and the green-shaded area the relative 3 σ scatter region. \n<!-- image --> \nfrom their ν m /ν c , show a tight ν c , z -Liso correlation (Fig. 2). Spearman's rank correlation test returns a coe ffi cient ρ = 0 . 81 with Pchance = 1 . 91 × 10 -14 . The slope of the power law is consistent with both the one expected from the Yonetoku relation and the one from the Ep , z -Liso relation of GRBs in the Synchrotron sub-sample. The results are reported in Table 4.", '3.3. Multiple-bins analysis': "Out of the 74 GRBs in our total sample, we select 15 GRBs with the highest fluence (as reported by Fermi / GBM catalog) in order to perform time-resolved spectral analyses across their whole duration. Among these 15 GRBs we find GRB090902B and GRB190114C, namely the two GRBs exhibiting a second power law spectral component in addition to the main one. We remove \nthese two GRBs from the sample, and the remaining 13 GRBs constitute the new multiple-bins sample. The time-resolved spectral analysis relative to this sample includes both Band and synchrotron models, similarly to the procedure described in Sec. 2. \nWe visually selected the time-bins from their GBM light curve in order to match as much as possible a single GRB pulse. The time-bin width is chosen in order to collect enough signal for the spectral analysis. In case a GRB shows a particularly bright pulse, we select multiple time-bins associate to a single pulse. We include LLE spectral data in any time-bin where they are available. \nThe multiple-bins sample covers a total of 125 spectra. The GRB list, their time-bins and the spectral fit results are reported in Table 5. \nAccording to the Bayes factor, 65 spectra are best-fitted by Synchrotron model, 48 spectra are best-fitted by Band and for 12 spectra the Bayes factor can not determine which is the best-fit model. For the 48 spectra best-fitted by Band, we report the Ep , z , Liso and α estimates obtained from the Band model, whereas for the remaining 77 spectra we report Ep , z , Liso and ν c , z obtained from the Synchrotron model. \nSimilarly to the analysis of the single-bin sample, we observe that the fit with the Synchrotron model is not returning any Ep , z -Liso relation. However, restricting the multi-bin sample to its Synchrotron sub-sample (i.e. the 77 GRBs for which we report synchrotron results), we observe again a very tight correlation between ν c , z and Liso (Fig. 3). The Spearman's rank correlation coe ffi cient is ρ = 0 . 83 with chance probability Pchance = 1 . 83 × 10 -20 (Table 4). We notice that the ν c , z -Liso relation derived in this way is \n] \nV \ne \nk \n[ \nFig. 3: ν c , z -Liso relation obtained from the Synchrotron GBRs in the multi-bin sample. We show multiple time-bins from di ff erent GRBs with circles of the same colors. The number inside the circles represents the bin number that data point is associated to. The black straight line represents the best-fit line from the linear fit while the dashed-line the best-fit line obtained from the ν c , z -Liso relation in the single-bin sample. The green-shaded area shows the 3 σ scatter region of the relation. \n<!-- image --> \ndescribed by a slightly flatter best-fit line, with slope m = 0 . 41 ± 0 . 03 and normalization K = 1 . 12 ± 0 . 09, whereas the intrinsic scatter is comparable ( σ sc = 0 . 31). It is worth to mention that, if GRB211211A time-resolved dataset is considered alone, is is consistent with a slope m ∼ 0 . 5. If excluded from the multiple-bins sample, the relative ν c , z -Liso relation has the same slope if the one obtained from the single-bin one. \nTable 5: List of bursts belonging to the multiple-bins sample. We report the bin number and the relative time interval with respect to the GBM trigger time. In column 4 we report the Bayes factor obtained comparing Synchrotron and Band models. In column 5 we report the best-fit model: S = Synchrotron, B = Band and ND = Not Defined, i.e. when | log B | < 0 . 5. We report the isotropicequivalent luminosities Liso and rest-frame peak energies Ep , z derived from the best-fit model. In column 8 we report the rest-frame cooling frequency ν c , z or the low-energy photon index α depending if the best-fit model is Synchrotron or Band, respectively. If the best-fit model is not defined, we report ν c , z . \nTable 5 continued \nTable 5 continued \n. \n.", '4. Discussion': 'We found a novel ν c , z -Liso relation by employing a physical synchrotron model to describe the prompt emission of GRBs observed by Fermi / GBM. The fact that this new relation is found with high significance in the single-bin and multiple-bins samples, where both the emissions during the peak and across the whole burst duration are explored, strongly points towards its possible physical nature. In the following, we provide an interpretation of these results. \nIn the internal shocks framework, synchrotron spectra have a peak energy Ep , z that depends on many other parameters, most relevantly the bulk Lorentz factor Γ and the dynamical timescale tvar (Rees & Mészáros 2005). The presence of a Ep , z -Liso correlation would imply that Γ and tvar are the same for all the GRBs. This condition in not supported by any observation on large GRB samples. Therefore, this scenario is not in accordance with the Yonetoku relation, and our findings point towards this conclusion. \nAs a matter of fact, when time-resolved spectral analysis is performed, Ep , z exhibits a strong spectral evolution, which can track the intensity of the pulse (e.g. Golenetskii et al. 1983) or show a hard-to-soft transition (e.g. Lu et al. 2012). The latter case is not consistent with the Yonetoku relation when multiple pulses are present. On the other hand, Ravasio et al. (2019) performed time resolved analysis of a sample of 10 bright long GRBs, fitting their spectra with a double smoothly broken power law (2SBPL) function, namely a Band function with two power laws before the peak. They find that the break energy, consistent with the synchrotron break in a marginally fast-cooling regime, does not vary much across the GRB duration. This finding suggests that the break energy may be more "standard" than the peak energy, or at least have minor dependencies on other parameters. However, the values of peak and break energies obtained through 2SBPL fits do not necessarily coincide with synchrotron breaks due to the shape of synchrotron spectra. \nThe question arises whether how the ν c , z -Liso relation can agree with almost two decades of observations pointing towards correlations with the peak energy Ep , z . \nAsynchrotron SED in fast-cooling regime ( ν m /ν c > 1) exhibits two typical energies: a peak around ν m and a break around ν c . Before the peak, we expect a hard photon index α 1 = -2 / 3 for E < h ν c , while for h ν c < E < h ν m the emission softens with a photon index α 2 = -3 / 2. The smaller the frequency ratio ν m /ν c is, the closer the two breaks are. \nHistorically, most of the spectral fits that lead to the Yonetoku relation were carried out using phenomenological models which allowed for only one power law segment at low energies, with a single spectral break coinciding with the peak. In the hypothesis that GRBs prompt spectra are produced by synchrotron processes, the fit of a single-break model (such as the Band function) to a spectrum that intrinsically presents two breaks (such as a synchrotron one) can lead to two main observational biases: \n- 1.) It appears that, when a single-break model like Band is fitted to an intrinsic double-break model, it returns Eb < E Band p < Ep , where E Band p is the Band peak energy whereas Eb and Ep are break and peak energies. Therefore, the fit of a two-breaks model to a suitable spectrum leads to systematically larger Ep , z .\n- 2.) The low-energy spectral index α that one obtains fitting a single-break model is a weighted average of the two spectral indices α 1 and α 2 before and after the break, respectively (To ff ano et al. 2021). Therefore, the hardness of the spectrum depends on the proximity of Eb to Ep . \nBy comparing results from the Band and Synchrotron models related to the single-bin Synchrotron sample, we observe a similar pattern. In fact, when the synchrotron fit returns a fast-cooling spectrum, its peak energy E Sync p , z is systematically larger than the value obtained by fitting the Band model E Band p , z . Conversely, when 1 <ν m /ν c < 3 the two frequencies are very close and break and peak \nenergies are hardly distinguishable, Band and synchrotron fits return similar peak energies, i.e. E Sync p , z / E Band p , z ∼ 1 (Fig. 4, left panel). In Fig. 4 (right panel) we show the distributions of the photon index α associated to the fast-cooling and intermediate-cooling spectra. While fast-cooling spectra span a large range of values pointing to softer spectra, the intermediate-cooling spectra exhibit harder photon indices close to α = -0 . 67, which is the value expected for E < Eb , z = h ν c , z . \nWe can conclude that the e ff ect of fitting a single-break model such as the Band function to a synchrotron spectrum leads to systematically lower values of Ep , z . These are the premises by which the Yonetoku relation was vastly studied in literature. According to these findings, it is not surprising that when a physical synchrotron model is used in testing the Yonetoku relation, the latter holds only for GRBs where 1 < ν m /ν c < 3, namely when peak and break energies nearly coincide and therefore Band fits return correct Ep , z estimates. This suggests that, at least for the GRBs that can be correctly described by an idealised synchrotron model, the underlying physical relation that holds is the ν c , z -Liso one. When spectra are analysed with the Band function, E Band p , z is not far from h ν c , returning the notorious Yonetoku relation. However, when a second break is accounted for, the Yonetoku relation does not hold anymore, and the new ν c , z -Liso appears to be the underlying relation. \nIn Fig. 5 we show an example of this phenomenon for GRB160625B, one of the brightest sources in the multiple-bins sample. Its time-resolved analysis shows that, during the time-bins where the spectrum is in fast-cooling regime, the corresponding Ep , z and Liso are outliers of the Yonetoku relation. As soon as its spectrum shows an intermediate-cooling state, its Ep , z approaches the relative ν c , z and the corresponding Ep , z and Liso follow the Yonetoku relation. \nNonetheless, Guiriec et al. (2013, 2015a) discuss an opposite scenario. They fit a two components model to time-resolved GRB prompt spectra. The presence of a second peaked component at lower energies resembles (in the photon flux representation) a total spectrum with two breaks. Despite finding systematically larger Ep , z with respect to single-break model fits, they claim a tighter Yonetoku relation when a second component is accounted for in the fit. This behaviour was proven in a sample of 5 GRBs (e.g. Guiriec et al. 2015b). However, the extension of this to larger samples would be in contrast with the findings of Ep , z -Liso relations in numerous samples when single-break models are used. In fact, if both the Ep , z increase and the Yonetoku relation from single-break models are valid observational evidences, the fit of a second thermal component (or a double-break model) would systematically bring GRBs out of the relation. In this scenario, the only way to have a tighter relation would be for these GRBs exhibiting breaks \nFig. 4: Ratio between the rest-frame peak energy from Synchrotron model fits E Sync p , z and Band model fits E Band p , z as a function of E Band p , z (left-hand panel) and low-energy break α histograms (right-hand panel), both relative to GRBs in the Synchrotron singlebin sample. Parameters from GRBs associated with fast-cooling spectra ( ν m /ν c > 3) are shown in red, while the ones associate to intermediate-cooling spectra (1 < ν m /ν c < 3) in blue. \n<!-- image --> \n1.50 \n1.25 \n1.00 \n0.75 \n0.50 \n0.25 \nto be all systematically beneath the best-fit line, i.e. to be scattered away from the best-fit relation with systematically lower Ep , z for the same Liso . Nonetheless, this hypothesis is hard to test in a small sample. In this work, testing a large sample of GRBs, we instead observed that GRBs in synchrotron sub-samples do not preferentially lay underneath the best-fit line, and this leads to the absence of the Yonetoku relation in both our samples. \nIt is important to stress that the results derived from the single-bin sample are obtained under the assumption of synchrotron being the main radiative process producing prompt emission. This hypothesis is verified through model comparison only for 21 / 74 GRBs. On the other hand, as also discussed in To ff ano et al. (2021), the comparison between physical and phenomenological models is quite tricky, being the peculiar synchrotron features lost in low signal-to-noise spectra, often favouring a Band function. However, the conclusions that we draw remain valid in this working hypothesis, caveat providing a good synchrotron fit therefore reliable estimates of ν m and ν c . Nevertheless, the results drawn from the multiple-bins sample go beyond this assumption, selecting only Synchrotron spectra through model comparison. \nIn this regards, one important result of this work is the fact that the synchrotron model we used to fit the data is not able to well describe all the spectra. The interpretation can be twofold: or it appears that prompt dissipation sites and / or radiative processes are not the same for all the GRBs (therefore synchrotron emission can not explain all GRBs spectra), or that the synchrotron model we use is too simplistic to account for the GRB spectra. \nFrom the single-bin Band sub-sample (Table 3), we notice that the spectra that reject the synchrotron model show a relatively hard spectral index α . The presence of these hard prompt emission spectra are in tension with a synchrotron interpretation, pointing to other emission mechanisms possibly produced in the optically thick region. However, we perform an additional fit of GRB spectra in the single-bin Band sub-sample restricting the dataset between 8-30 keV and fitting a power law model. We exclude from this analysis GRB190114C and GRB090902B, since their low energy spectrum at few keV is dominated by a second power law component. \nWe observe that the spectral indices obtained through the power law fit α PL is much smaller, i.e. the spectra are softer, than the ones obtained through Band fits α Band (Fig. 6). In particular, the α PL values obtained are close to the value α = -0 . 67 expected from synchrotron spectra at energies E < Eb . The reason of this mismatch can be due to the fact that the photon index α of the Band function is degenerate with the smoothness and position of the spectral peak. This finding suggests that α may be a biased indicator of GRB hardness (Burgess et al. 2015). \nStill, the synchrotron model used in this work is not able to provide a good representation on the data for 13 GRBs. We propose that this fact is not due to the violation of the single electron spectrum, as often stated in literature, but rather to the shape and smoothness of the spectral profile around the peak. In these regards, there are many process not accounted for in our modelling, such as inverse Compton scattering in Klein-Nishina regime (Derishev 2009; Bošnjak et al. 2009; Daigne et al. 2011) provided by decaying magnetic fields (Pe\'er & Zhang 2006; Derishev 2007; Zhao et al. 2014; Daigne & Bošnjak 2024), anisotropic pitch angle distribution (Medvedev 2000; Sobacchi et al. 2021) and so on. Accounting for all these process may lead to spectra more in accordance with the observations, without the need of introducing di ff erent radiative processes. \nWhen instead also a simple synchrotron model well describes the data, we see a particular trend. All the GRBs in the synchrotron sub-sample are in (marginally) fast-cooling regime, as expected from other works (Ravasio et al. 2019). \nInterestingly, the majority of them is in intermediate-cooling regime, which is a specific stage of marginally fast-cooling regime where the two main frequencies ν m and ν c are coincident. nearly 60% of GRBs in the single-bin Synchrotron sample are in this state, at odds with the ∼ 25% of GRBs in multiple-bins Synchrotron sample. This suggests that this state is reached more frequently \n8 \n6 \n4 \n2 \nm/ c > 3 \n1 < m/ c < 3 \n<!-- image --> \n] \nV \ne \nk \n[ \nE \nFig. 5: Fermi / GBM(8 - 900 keV) light curve and time-bins of the associated time-resolved analysis of GRB160625B (upper panel). In the lower panel, we show the relative Ep , z -Liso plane. The color-scale is associated to the time-bin in which the spectral analysis is performed. We show, for this GRB, the results of the Synchrotron fit, and we represent the bins with fast-cooling spectra as triangles, whereas the bins with intermediated-cooling spectra as squares. The black straight line represents the Yonetonu relation obtained from Yonetoku et al. (2004). \n<!-- image --> \nin the brightest pulses.', '5. Conclusions': 'In this work, we perform spectral analysis of GRBs in two samples: the single-bin sample (with 74 GRBs, one spectrum with 1s exposure per GRB during their peak) and the multiple-bins sample (with 13 bright GRBs, multiple spectra with few seconds exposures per GRB for a total of 125 spectra). \nIn both these samples we fit two main models. The first one is the Band function, which phenomenologically describes GRB spectra \nArticle number, page 16 of 19 \nFig. 6: Low energy spectral index α obtained by fitting a Band model (in red) or a power law model considering data between 8-30 keV (in blue). The fit is performed to GRBs in the single-bin Band sub-sample. The dashed vertical line corresponds to α = -0 . 67, the value expected in synchrotron spectra for E < Eb . Errors are reported with 68% confidence level. \n<!-- image --> \nwithout any assumption on the underlying physics. In addition, we used a simple physical model that accounts for power lawdistributed relativistic electrons cooling down through synchrotron radiation. \nWe aim to test the validity of the Ep , z -Liso (Yonetoku) relation using both Band and synchrotron models. While the former was the one historically used to derive GRB empirical relations, the introduction of a synchrotron model, being a viable explanation for GRB prompt spectra, allows to better investigate the nature of this emission and the relative empirical relations. \nWe summarize the main results of this work in the following: \n- a. We show that most of the GRBs in the single-bin sample are well fitted by the Synchrotron model (58 / 74). However, there are GRBs whose spectrum cannot be described by a Synchrotron model, preferring instead a Band function (16 / 74). Nonetheless, GRBs that prefer a Band model, when fitted with a power law at E < 30 keV, exhibit a photon index α PL ≲ -0 . 7, i.e. do not violate the single electron spectrum.\n- b. It appears that all the GRBs for which we report results from the synchrotron fit are in a fast-cooling regime ( ν m /ν c > 1). The majority of these ones show particular ν m /ν c values, where ν m (i.e. the spectral peak) is only slightly larger than ν c (i.e. the synchrotron break). We refer to these cases as GRBs in an intermediate-cooling state, a particular case of marginally fast-cooling spectra where the break and the peak energies are basically coincident (1 < ν m /ν c < 3).\n- c. We show that the fit of the Band model (single-break spectrum) to a possibly intrinsic synchrotron spectrum (double-break spectrum) leads to systematically lower Ep , z and the proximity of Eb , z to Ep , z influences the hardness of the spectrum, namely the low-energy spectral index α fitted with the Band model.\n- d. We observe that GRBs in the single-bin sample do not follow the Yonetoku relation unless they are in an intermediate-cooling state, i.e. peak and break energies are extremely close to each other.\n- e. We find a novel tight relation between the rest-frame cooling frequency ν c , z and the isotropic-equivalent luminosity : \nν c , z 100 keV = (0 . 97 ± 0 . 14) × Liso 10 52 erg / s ! (0 . 53 ± 0 . 06) \nThis relation is found in both single-bin and multiple-bins samples, with slightly di ff erent slopes. \nThe underlying assumption of this work is that prompt emission spectra are produced through synchrotron cooling of power-law distributed particles. Nonetheless, in the multiple-bins sample analysis we report synchrotron fit results whenever model comparison does not discard the synchrotron model. This points toward the physical nature of the ν c , z -Liso relation. \nWe show that, if GRB prompt emission is produced through synchrotron radiation, the fit using a Band function does not always represent a good spectral description. For example, Band fit would return a peak energy estimate E Band p , z in between h ν c and h ν m = \nE Sync p , z . This suggests that, for synchrotron spectra, the fundamental physical relation is between Liso and ν c , z . The Yonetoku relation, which was found through Band fits, can be the e ff ect of a single-break model fit to a double-break spectrum. Given the occurrence \nof GRB spectra in intermediate-cooling regimes (or in general in marginally fast-cooling regimes) E Band p , z provides a proxy of ν c , z . The higher the intrinsic ratio ν m /ν c , the higher the scatter of that GRB relative to the Yonetoku relation. \nThis novel interpretation of the Yonetoku relation constitutes a step forward in understanding the physics behind empirical relations \nand prompt emission. For the prompt synchrotron spectra, we found a new tight relation that connects the cooling regime with the \ntotal luminosity of the burst: the less it cools, the brighter it is. \nThis behaviour is counterintuitive. One would expect, in the optically-thin synchrotron framework, more radiative output when particles cool down interacting with stronger magnetic fields. This new relation can constitute a benchmark in testing GRB prompt emission models, and push forward the possibility of using GRBs in constraining cosmological properties. The recent launch of the SVOMand Einstein Probe missions can provide, in synergies with past observatories, a variety of GRB observations with measured redshift and extend the empirical correlations study with particularly soft burst emitting in the X-rays. \nAcknowledgements. The authors thank B. Banerjee, D. Bjørn Malesani, M. Branchesi, A. Celotti, G. Ghirlanda, G. Ghisellini, L. Nava and O.S. Salafia for the fruitful discussions about this work. AM thanks the Cosmic Dawn Center in Copenhagen and the Observatory of Brera in Merate for the hospitality during the development of this project.', 'References': "Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2017, ApJ, 848, L13 \nAbdo, A. A., Ackermann, M., Ajello, M., et al. 2009, ApJ, 706, L138 \nAjello, M., Arimoto, M., Axelsson, M., et al. 2020, ApJ, 890, 9 \nAmati, L., Frontera, F., Tavani, M., et al. 2002, A&A, 390, 81 \nArnaud, K. 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2024arXiv240907139L

The recent AMS02 measurements of cosmicray CR deuteron fluxes suggest the presence of primary deuterons in quantities far exceeding predictions from Big Bang nucleosynthesis. This poses a significant challenge to modern astrophysics as no known processes can account for such large amounts of deuterons without violating existing constraintsciteEpstein1976hq. In contrast it is recently proposed that the AMS02 measurements can be explained by a purely secondary origin when contributions from heavier nuclei are considered. In this study we recalculate the secondary deuteron flux using production cross sections updated with the latest collider data. We find that some of the deuteron production cross sections are overestimated in the widelyused calculation tools for CR propagation and a primary deuteron component is still necessary. We then propose a novel process for generating primary deuterons at CR sources through a fusion mechanism which is naturally unique to deuterons. This model could explain the observed deuteron excess while maintaining consistency with other CR measurements.

2024-09-01T00:00:00Z

['arXiv:2409.07139', '2024arXiv240907139L', '10.48550/arXiv.2409.07139']

['Astrophysics - High Energy Astrophysical Phenomena', 'High Energy Physics - Phenomenology']

Cosmicray deuteron excess from a primary component

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.07139.pdf

{'Cosmic-ray deuteron excess from a primary component': 'Xing-Jian Lv, 1, 2, ∗ Xiao-Jun Bi, 1, 2, † Kun Fang, 1, ‡ Peng-Fei Yin, 1, § and Meng-Jie Zhao 1, 3, ¶ \n1 Key Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China 2 \nUniversity of Chinese Academy of Sciences, Beijing 100049, China \n3 China Center of Advanced Science and Technology, Beijing 100190, China \n(Dated: December 16, 2024) \nThe recent AMS-02 measurements of cosmic-ray (CR) deuteron flux suggest the presence of primary deuterons in quantities far exceeding predictions from Big Bang nucleosynthesis. This poses a significant challenge to modern astrophysics, as no known processes can account for such large amounts of deuterons without violating existing constraints [1]. In contrast, it has recently been proposed that the AMS-02 measurements can be explained by a purely secondary origin when contributions from heavier nuclei are considered. In this study, we recalculate the secondary deuteron flux using production cross sections updated with the latest data. We find that the deuteron production cross sections are overestimated in the commonly used calculation tools for CR propagation, and a primary deuteron component is still necessary. We then propose a novel process for generating primary deuterons at CR sources through a fusion mechanism, which is naturally unique to deuterons. This model could explain the observed deuteron excess while maintaining consistency with other CR measurements.', 'I. INTRODUCTION': 'Big Bang nucleosynthesis, a cornerstone of the standard model of cosmology, predicts the formation of deuterium shortly after the Big Bang. During this epoch, temperatures had declined sufficiently to allow for the fusion of protons with neutrons, yet remained high enough to prevent further fusion of deuterium into 4 He. This fleeting window resulted in a predicted cosmic abundance of deuterium at D/H ≈ 3 × 10 -5 , which aligns well with observational data (e.g., Ref. [2]). \nInterestingly, the deuterons 1 are far more abundant within cosmic rays (CRs), comprising approximately 2% to 3% of the proton abundance. This observation initially appears paradoxical, as deuterium is expected to be depleted in stars, with no known mechanisms capable of reversing its decline to such a degree [3]. However, the theory of galactic CR propagation provides a natural explanation: the observed deuterons are produced through fragmentation processes involving more abundant primary CR nuclei, which are trapped in the Galactic magnetic field for millions of years [4, 5]. It has long been assumed that nearly all CR deuterons arise from collisions between 4 He ions and nuclei in the interstellar medium (ISM), a phenomenon that also generates CR 3 He. Consequently, it is expected that the CR spectra of deuterons and 3 He should exhibit similar energy dependencies [6, 7]. \nRecently, the AMS-02 collaboration published the precise measurement of the energy spectrum of deuterons, \n¶ \nbased on data from 21 million deuteron nuclei within the rigidity range of 1.9 to 21 GV, collected between May 2011 and April 2021 [8]. Notably, the measured deuteron flux is significantly higher than the secondary deuteron flux predicted by the CR propagation model above 5 GV. Moreover, there is a conspicuous misalignment in the energy dependences between the deuteron and 3 He spectra up to 20 GV. The AMS-02 collaboration has interpreted these novel phenomena as evidence for the presence of a primary-like deuteron component [8]. \nSubsequently, Ref. [9] highlighted the challenges of introducing a significant primary component of deuterons. Instead, the authors proposed that the observed deuteron flux could still be consistent with a secondary origin if the contributions from heavy incident nuclei with Z > 8 are taken into account. However, it is imperative to carefully scrutinize the production cross sections of deuterons in the evaluation of secondary CR fluxes. As we shall demonstrate later, the deuteron production cross sections employed by the commonly used calculation tools for CR propagation are overestimated in comparison to some of the latest measurements, particularly concerning heavy incident nuclei. Therefore, a purely secondary origin still struggles to fully explain the AMS-02 results. \nConsequently, the latest AMS-02 deuteron flux measurements necessitate a primary contribution. However, the production of substantial quantities of nonprimordial deuterons remains a conundrum with no recognized explanation in the literature. In this study, we propose a potential origin for primary 2 CR deuterons. We assume that the spectral hardenings observed in recent measurements of CR secondary-to-primary ra- \ntios by AMS-02 and, notably, by DAMPE at ∼ 100 GeV/n [10, 11], may be attributed to the production and acceleration of secondary particles at the source sites. In this scenario, deuterons produced by proton-proton fusion, which is a process unique to deuterons as the lightest composite nucleus, could account for the peculiar deuteron excess. \nThe paper is organized as follows. We revisit the deuteron excess after modifying its fragmentation cross sections in Section II, followed by the presentation of our model for primary CR deuterons in Section III. The results and discussions are detailed in Section IV, and conclusions are provided in Section V.', 'II. THE DEUTERON EXCESS REVISITED': "As introduced in Section I, Ref. [9] has clearly illustrated the importance of including nuclei with Z > 8 when calculating the contributions to secondary deuterons. For the production cross section into deuterons, which is crucial for accurately determining the yield fluxes of secondary particles, they utilized the parameterizations of deuteron production cross sections from Ref. [6] (hereafter CDMP12). The approach of CDMP12 for the fragmentation cross sections of nuclei with A > 4 is based on the following factorization: \nσ P p → F ( E k/n , A P ) = γ F P × f ( E k/n , A P ) × σ 4 He p → 3 He breakup ( E k/n ) , (1) \nwhere σ P p → F is the fragmentation cross section for the projectile P incident upon a target proton producing the fragment F, A P is the mass number of the projectile, f ( E k/n , A P ) is given by Eq. (B.2) of CDMP12, σ 4 He p → 3 He breakup is the breakup cross section of 4 He colliding on a proton to yield 3 He, and γ F P is an energy-independent factor determined from measured data points. \nFIG. 1: The revised γ D P used in this work (solid line) compared with that of CDMP12 [6](dotted line). These points are calculated according to the factorization of Eq. (1). Points labeled as [He06] are from Ref. [12]. Points labeled as [data] are from Ref. [13-16] and the data compilation provided by CDMP12. \n<!-- image --> \nIn determining the form of γ D P , CDMP12 did not include the cross-section measurements of incident nuclei heavier than oxygen, and provided γ D P = 0 . 28 A 2 . 1 P . Ref. [12] measured the cross section of deuterons produced in proton-induced fragmentation reactions with targets ranging from Al to Th. After including these data points, we find that CDMP12's γ D P factor is too high for heavy nuclei. Consequently, we introduce a refined factor form as follows: \nγ D P = 0 . 3 A 1 . 05 P . (2) \nA comparison of γ D P factor between CDMP12's and our proposed heavy-nuclei revision is shown in Fig. 1 alongside the measurements. It is evident that our revision provides a better fit to the data, and would predict less deuteron production. Additionally, we also find that CDMP12 overestimated the fragmentation cross sections for 3 He from heavy nuclei compared with the measurements from Ref. [12]. However, this issue lies outside the scope of our current work and will be addressed in future studies. \nAs of now, there has been no direct measurement of the deuteron cross section for 4 He-target channels. To determine these cross-sections, we conduct a two-step analysis. In the first step, we calculate some deuteron cross sections for 4 He-target channels with specific projectiles, using data obtained from other targets. Specifically, we estimate the deuteron production cross sections of 4 He, 12 C, and 16 O projectiles upon the 4 He-target based on measurements of [Li75] [17] and [Ab81] [18]. Subsequently, we extrapolate deuteron cross sections for other 4 He-target channels through interpolation. Further details on this analysis can be found in Appendix B. \nAs illustrated in Fig. 2, the three derived cross sections can be effectively represented by a simple relation σ = 0 . 6A 1 . 05 P σ 4 He+ p → 3 He , which scales Eq. (1) by a factor of 2 (compared with Eq. (2)). GALPROP v57 [19] calculates σ by scaling Eq. (1) with a factor of A 0 . 31 P , while USINE [20] scales it with a factor of 4 0 . 31 . In Fig. 2, the predictions from USINE are systematically lower than the derived cross section points, while GALPROP tends to provide higher predictions for projectiles heavier than 12 C compared to our results. The 4 He + 4 He channel, which contributes the most to the deuteron flux among all 4 Hetarget channels, exhibits a significant uncertainty and requires precise measurements for improved predictions in the future. Overall, our prediction of the deuteron flux from P + 4 He channels shows only a slight increase compared to GALPROP's prediction, representing a minor adjustment relative to the corrections made to the heavy nuclei fragmentation cross sections of P + p channels. \nThe modification of fragmentation cross sections results in a reduction in the deuteron flux, of approximately 15% around 10 GV when adjusting heavy-nuclei contributions, and an increment of about 1.5% when additionally adjusting 4 He-target contributions. Due to the poor constraints of the related measurements, non-negligible uncertainties of around 10% in the secondary deuteron \nFIG. 2: The production cross sections of deuterons from the various projectiles on the 4 He target at high energies. The derived data points are based on Ref. [17, 18]. The purple and green dotted lines are parametrizations used in GALPROP v57 [19] and USINE [20]. The brown solid line is the fitted result derived by this study. \n<!-- image --> \nproduction cross sections should be considered in the analysis, as would be mentioned in Section IV. We illustrate the difference of predicted deuteron flux between the previous and the updated cross sections in Fig. 3. The excess at the highest energy points appears modest with the previous cross sections, but the deficit becomes significant with the adoption of the revised cross sections at ∼ 10 GV. In addition, even with the utilization of the previous cross sections, the rigidity dependence of the predicted secondary deuterons remains harder than that observed in the AMS-02 data. This discrepancy is clearly delineated in the inset of Fig. 3, where linear scales are used to enhance clarity. In conclusion, our analysis suggests that the AMS-02 deuteron flux cannot be fully explained by a purely secondary origin. \nFIG. 3: Predicted fluxes of pure secondary deuterons using the previous and the updated cross sections compared with measurements [8, 21]. The inset displays the same plot in linear scale for a clear view of the discrepancy. The result derived with the previous cross sections is similar to that obtained by Ref. [9]. \n<!-- image -->", 'A. Primary Deuteron': 'The recent measurements of CR secondary-to-primary ratios by AMS-02 and DAMPE reveal significant spectral hardenings around 100 GeV/n [10, 11]. This phenomenon presents a challenge to conventional production and propagation models of CRs. These models typically posit that secondary particles are produced via the fragmentation of primary CRs interacting with ISM during their propagation through the Milky Way [22]. One plausible explanation for these spectral hardenings is the occurrence of similar particle interactions near CR sources, especially in regions where dense molecular clouds surround these sources. Such interactions could give rise to the production of primary boron nuclei, consequently leading to the observed hardenings in the B/C and B/O ratios [23-25]. This scenario could also account for the observed positron excess [26-28] and the excess in ultrahigh-energy diffuse γ -ray emission [23, 29, 30]. \n1 \nFIG. 4: The cross sections of two deuteron production channels provided in Ref. [6, 19]: one from the fragmentation of 4 He, and the other from the fusion of protons. \n<!-- image --> \nThe primary boron produced at sources through fragmentation is assumed to share the same injection spectrum as carbon and oxygen. As shown in Fig. 5, we determine the normalization of primary boron by adding the secondary boron flux to fit the overall B/C and B/O ratios. Notably, the production of primary boron through CR-gas interactions is always accompanied by the production of primary deuterons. Therefore, by utilizing their respective parent particle abundances and production cross sections as a scaling factor, we can calculate the primary deuteron flux from fragmentation of helium associated with the primary boron flux as follows: \nΦ LIS primary 4 He+ p → D = Φ LIS primary B × Φ LIS 4 He σ 4 He p → D Φ LIS C σ C p → B +Φ LIS O σ O p → B , (3) \nwhere the label LIS denotes the local interstellar flux under consideration. \nAs the lightest composite nucleus, the deuteron has a unique production channel unavailable to heavier nuclei such as 3 He or boron, namely, the coalescence of two protons to form a deuteron. Despite the cross section for this process being ten times smaller than that of other channels, the abundance of CR protons being approximately tenfold greater than that of 4 He makes this contribution to the secondary deuteron flux non-negligible [6]. This fusion process is also expected to occur near the source regions, following the same logic as fragmentation processes. Unlike fragmentation, this fusion cross section σ p + p → D is non-vanishing only within a very narrow energy range of around 600 MeV/n, as illustrated in Fig. 4. The contribution from primary fusion deuterons can be calculated as \nΦ fusion primary p + p → D = Φ LIS primary B × Φ LIS p σ pp → D Φ LIS C σ C p → B +Φ LIS O σ O p → B . (4) \nDespite the fusion process being confined to a narrow energy range around 600 MeV/n, the fusion deuterons originating in the vicinity of accelerating sources can undergo further acceleration by shocks near these sources [32, 33]. Consequently, the resulting spectrum is expected to exhibit a power law behavior akin to that of primary fragmentation deuterons. Accordingly, we reshape the contribution of primary fusion deuterons given by Eq. (4) to the LIS deuteron flux to match the shape of the LIS helium flux, incorporating a low-energy cutoff as \nΦ LIS primary p + p → D ∝ Φ LIS 4 He × exp[ -( E cut /E ) 3 ] . (5) \nIn the calculation, the integral intensity of deuteron is preserved as in Eq. (4). The low-energy cutoff is the sole free parameter, as all other parameters are fully constrained by the B/C and B/O ratios. We will discuss the justification for introducing this cutoff and examine its impact on the deuteron flux in Section IV.', 'B. Secondary Deuteron': 'We use the numerical tool GALPROP [34] v57 3 to calculate the production and propagation of secondary particles. We adopt the diffusion propagation framework with stochastic reacceleration in the ISM, which has been shown to accurately reproduce secondary-to-primary ratios [35, 36]. We use the production cross sections for secondary deuterons fitted to the latest measurements, \nas described in Section II, and include all nuclei up to Z = 28 (nickel) in our calculations. The contribution from the fusion of CR protons with ISM protons is also included in the calculation. For the injection spectra of primary CRs, we employ a broken power law to account for the hardenings observed in the primary CR spectra. The details regarding the propagation calculations and parameter determination are provided in Appendix A.', 'IV. RESULTS & DISCUSSION': 'The calculated flux of deuterons is shown in Fig. 6, along with the observational data from PAMELA [21] and AMS-02 [8]. Within the framework of this study, the deuteron flux includes three major components: i) a secondary component arising from the fragmentation of nuclei up to Z = 28 (nickel) during the propagation process; ii) a primary fragmentation component originating from the CR sources; iii) a primary fusion component also originating from the sources. To account for the solar modulation effect on the spectrum inside the heliosphere, we adopt the force-field approximation [37]. The modulation potential is taken to be 600 MV for all nuclei. A range of Φ = 600 ± 100 MV is considered to account for variations in data collection periods and the inherent limitations of the force-field approximation [38, 39], visually represented as a blue-shaded band in the figure. As discussed in Section II, there remain non-negligible uncertainties in the secondary deuteron production cross sections, particularly concerning uncertainties of heaviernuclei and 4 He-target modification ( ∼ 10% around 10 GV). The range of secondary deuteron flux allowed by the current cross-section data uncertainties is illustrated as an orange-shaded band. \nAs shown in Fig. 6, the predicted deuteron flux including three components aligns well with the measurements, taking into account the influences of solar modulation and uncertainties in cross section. Notably, the discrepancy between the AMS-02 data and the theoretical prediction at above ∼ 10 GV can be resolved by introducing a primary fusion component with a low-energy cutoff at 4 GeV/n. In comparison, the contribution of the primary fragmentation component to the deuteron flux is minor. \nAs the low-energy cutoff of the primary fusion component is the only free parameter in our model for calculating the deuteron flux, we further investigate the impact of varying E cut . Figure 7 illustrates the contributions of primary fusion deuterons and the total deuteron fluxes at Earth for several cutoff values - 0 . 8, 1 . 0, 2 . 0, 3 . 0, 6 . 0 GeV/n, in addition to the benchmark value of 4 . 0 GeV/n. As the cutoff energy decreases, the contribution of primary fusion deuterons to the high-energy end of the AMS-02 data diminishes, with a predominant flux shift towards lower energies. As a result, the primary fusion component is insufficient to explain the excess at high energies for E cut < 4 GeV/n, while the expected spectrum is harder than the actual measurement for E cut > 4 \n<!-- image --> \nFIG. 5: Best-fit results for B/C, B/O ratios, compared with measurements [11, 31]. In both panels, the solid line represents total boron, while the dashed line indicates contributions from primary boron. \n<!-- image --> \nFIG. 6: Expected deuteron flux in comparison with measurements [8, 21]. The solid blue line represents the total spectrum at the top of the atmosphere considering a solar modulation potential of Φ = 600 MV, with the blue-shaded band indicating the uncertainty range for Φ = 600 ± 100 MV. The orange-shaded band represents uncertainties due to secondary deuteron production cross sections. The green, orange, and red dashed lines correspond to the secondary, primary fragmentation, and primary fusion contributions to the deuteron flux, respectively. The red-shaded region represents the primary fusion deuteron spectrum before acceleration. The low-energy cutoff of the accelerated primary fusion spectrum is assumed to be 4 GeV/n. \n<!-- image -->', 'GeV/n.': 'We notice that a significant low-energy break at a few GeV/n in the injection spectrum for primary CR species has already been suggested [40-42] to explain the latest CR measurements. Ref. [42] found that for some species, \nthe spectra indices below this break are even smaller than 1. While the broken power-law form is milder than the cutoff feature adopted in this study, it is important to note that the seed spectra before acceleration for those primary CRs follow Maxwellian distributions [43], which provide an ample reservoir of low-energy particles. In contrast, the seed spectrum for primary fusion deuterons has a much sharper cutoff at low energies, as illustrated by the red-shaded region in Fig. 6. Hence, a sharp cutoff for the accelerated primary fusion deuterons appears to be a plausible scenario. \nFIG. 7: Effects of changing the low-energy cutoff for the accelerated primary fusion deuterons. The solid lines represent the total deuteron flux, while the dashed lines indicate contributions from primary fusion deuterons. The references are the same as in Fig. 6. \n<!-- image -->', 'V. CONCLUSION': 'In this study, we study the discrepancy between the theoretical predictions and the observed deuteron flux revealed by the AMS-02 results. Our analysis begins with \na thorough examination of the secondary deuterons flux. We find that the deuteron production cross sections utilized in commonly used calculation tools for CR propagation are overestimated compared to some latest measurements, especially for channels involving heavy incident particles. Consequently, considering the constraints on CR propagation parameters derived from secondaryto-primary CR flux ratios like B/C and B/O, we find that the flux of secondary CR deuterons is significantly lower than the deuteron spectrum measured by AMS02 above ∼ 10 GV. This discrepancy challenges previous findings based on the commonly used deuteron production cross sections and suggests that a substantial primary CR deuteron component is essential to account for the AMS-02 results. \nFor the possible origin of primary deuterons, we propose an innovative scenario in which deuterons generated through proton-proton fusion in the CR source region could offer a plausible explanation for the intriguing excess observed in the data. The observed hardening of the B/C and B/O spectra at ∼ 100 GeV/n could potentially be attributed to the superposition of secondary boron produced during propagation and primary boron generated at the sources. Building upon this hypothesis, we posit the presence of primary deuterons originating from heavy nuclei fragmentation and proton-proton fusion at the sources. These primary deuterons are subsequently accelerated and may exhibit similar spectral indices as 4 He at high energies. 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Fan, arXiv eprint (2024), arXiv:2406.19315 [astro-ph.HE].\n- [10] M. Aguilar et al. (AMS), Phys. Rept. 894 , 1 (2021).\n- [11] D. Collaboration (DAMPE), Sci. Bull. 67 , 2162 (2022), arXiv:2210.08833 [astro-ph.HE].\n- [12] C. M. Herbach et al. , Nucl. Phys. A 765 , 426 (2006). \nand boron, can naturally serve as a source of the observed excess in the deuteron flux. \nTo reproduce the AMS-02 data, our analysis indicates the necessity of implementing a low-energy cutoff at 4 GeV/n for the accelerated fusion deuteron spectrum. This introduction of a cutoff is phenomenological, not derived from first principles, due to the absence of studies on the acceleration mechanisms of the non-thermal seed spectrum. This parameter, serving as the sole free parameter in our model, plays a pivotal role in determining the flux of primary fusion deuterons above ∼ 10 GV. Current observations have suggested a significant suppression in the injection spectra of primary nuclei below several GV. However, the non-thermal nature of the seed spectrum of primary fusion deuterons before acceleration precludes direct comparisons. Hence, future investigations focusing on the acceleration mechanisms of this unique seed spectrum are essential to validate the proposed form of the low-energy spectral cutoff discussed in this study.', 'ACKNOWLEDGMENTS': "This work is supported by the National Key R&D Program Grants of China under Grant No. 2022YFA1604802, the National Natural Science Foundation of China under the Grants No. 12105292, No. 12175248, No. 12393853, and partially supported by the National Natural Science Foundation of China under grant No. 12342502. \n- [13] E. K. Bazarov, V. V. Glagolev, V. V. Lugovoi, S. L. Lutpullaev, K. Olimov, V. I. Petrov, A. A. Yuldashev, and B. S. Yuldashev, JETP Lett. 81 , 140 (2005).\n- [14] M. Enke et al. , Nucl. Phys. A 657 , 317 (1999).\n- [15] A. Letourneau et al. , Nucl. Phys. A 712 , 133 (2002).\n- [16] A. Korejwo, T. Dzikowski, M. Giller, J. Wdowczyk, V. V. Perelygin, and A. V. Zarubin, J. Phys. G 26 , 1171 (2000).\n- [17] P. J. Lindstrom, D. E. Greiner, H. H. Heckman, B. Cork, and F. S. Bieser, Physical Review Letters (1975).\n- [18] A. U. Abdurakhimov et al. (SKM-200), Nucl. Phys. A 362 , 376 (1981).\n- [19] T. A. Porter, G. Johannesson, and I. V. Moskalenko, Astrophys. J. Supp. 262 , 30 (2022), arXiv:2112.12745 [astro-ph.HE].\n- [20] D. Maurin, Comput. Phys. Commun. 247 , 106942 (2020), arXiv:1807.02968 [astro-ph.IM].\n- [21] W. Menn et al. (PAMELA), Astrophys. J. 862 , 141 (2018), arXiv:1806.10948 [astro-ph.HE].\n- [22] A. W. Strong, I. V. Moskalenko, and V. S. Ptuskin, Ann. Rev. Nucl. Part. Sci. 57 , 285 (2007), arXiv:astroph/0701517.\n- [23] P.-p. Zhang, B.-q. Qiao, Q. Yuan, S.-w. Cui, and Y.-q. Guo, Phys. Rev. D 105 , 023002 (2022), arXiv:2107.08280 [astro-ph.HE].\n- [24] P.-P. Zhang, X.-Y. He, W. Liu, and Y.-Q. Guo, JCAP 02 , 007, arXiv:2210.09591 [astro-ph.HE].\n- [25] P.-X. Ma, Z.-H. Xu, Q. Yuan, X.-J. Bi, Y.-Z. Fan, I. V. Moskalenko, and C. Yue, Front. Phys. (Beijing) 18 , 44301 (2023), arXiv:2210.09205 [astro-ph.HE].\n- [26] Y. Fujita, K. Kohri, R. Yamazaki, and K. Ioka, Phys. Rev. D 80 , 063003 (2009), arXiv:0903.5298 [astroph.HE].\n- [27] K. Kohri, K. Ioka, Y. Fujita, and R. Yamazaki, PTEP 2016 , 021E01 (2016), arXiv:1505.01236 [astro-ph.HE].\n- [28] R.-z. Yang and F. Aharonian, Phys. Rev. D 100 , 063020 (2019), arXiv:1812.04364 [astro-ph.HE].\n- [29] M. Amenomori et al. (Tibet ASgamma), Phys. Rev. Lett. 126 , 141101 (2021), arXiv:2104.05181 [astro-ph.HE].\n- [30] D.-X. Sun, P.-P. Zhang, Y.-Q. Guo, W. Liu, and Q. Yuan, arXiv eprint (2023), arXiv:2307.02372 [astro-ph.HE].\n- [31] M. Aguilar et al. (AMS), Phys. Rev. Lett. 120 , 021101 (2018).\n- [32] E. G. Berezhko, L. T. Ksenofontov, V. S. Ptuskin, V. N. Zirakashvili, and H. J. Voelk, Astron. Astrophys. 410 , 189 (2003), arXiv:astro-ph/0308199.\n- [33] P. Blasi, Phys. Rev. Lett. 103 , 051104 (2009), arXiv:0903.2794 [astro-ph.HE].\n- [34] A. W. Strong and I. V. Moskalenko, Astrophys. J. 509 , 212 (1998), arXiv:astro-ph/9807150.\n- [35] Q. Yuan, S.-J. Lin, K. Fang, and X.-J. Bi, Phys. Rev. D 95 , 083007 (2017), arXiv:1701.06149 [astro-ph.HE].\n- [36] Q. Yuan, C.-R. Zhu, X.-J. Bi, and D.-M. Wei, JCAP 11 , 027, arXiv:1810.03141 [astro-ph.HE].\n- [37] L. J. Gleeson and W. I. Axford, Astrophys. J. 154 , 1011 (1968).\n- [38] M. S. Potgieter, Living Reviews in Solar Physics 10 , 3 (2013), arXiv:1306.4421 [physics.space-ph].\n- [39] N. Tomassetti, Physical Review D 96 , 103005 (2017), arXiv:1707.06917 [astro-ph, physics:hep-ph].\n- [40] G. J'ohannesson et al. , Astrophys. J. 824 , 16 (2016), arXiv:1602.02243 [astro-ph.HE].\n- [41] M. J. Boschini et al. , Astrophys. J. Suppl. 250 , 27 (2020), arXiv:2006.01337 [astro-ph.HE].\n- [42] X. Pan and Q. Yuan, Res. Astron. Astrophys. 23 , 115002 (2023), arXiv:2403.06719 [astro-ph.HE].\n- [43] J. Park, D. Caprioli, and A. Spitkovsky, Phys. Rev. Lett. 114 , 085003 (2015), arXiv:1412.0672 [astro-ph.HE].\n- [44] Q. An et al. (DAMPE), Sci. Adv. 5 , eaax3793 (2019), arXiv:1909.12860 [astro-ph.HE].\n- [45] F. Alemanno et al. , Phys. Rev. Lett. 126 , 201102 (2021), arXiv:2105.09073 [astro-ph.HE].\n- [46] J. Wei, Properties of cosmic beryllium isotopes, https:// agenda.infn.it/event/28874/contributions/170166/ (2022), accessed: 2024-08-27.\n- [47] E. S. Seo and V. S. Ptuskin, The Astrophysical Journal 431 , 705 (1994).\n- [48] V. S. Berezinskii, S. V. Bulanov, V. A. Dogiel, and V. S. Ptuskin, Astrophysics of Cosmic Rays (1990).\n- [49] G. Di Bernardo, C. Evoli, D. Gaggero, D. Grasso, and L. Maccione, Astropart. Phys. 34 , 274 (2010), arXiv:0909.4548 [astro-ph.HE].\n- [50] V. H. M. Phan, F. Schulze, P. Mertsch, S. Recchia, and S. Gabici, Phys. Rev. Lett. 127 , 141101 (2021), arXiv:2105.00311 [astro-ph.HE].\n- [51] I. Angeli and K. P. Marinova, Atom. Data Nucl. Data Tabl. 99 , 69 (2013).", 'Appendix A: Propagation and injection spectra of cosmic rays': "The propagation equation of Galactic CRs is expressed as [22]: \n∂ψ ∂t = Q ( x , p ) + ∇· ( D xx ∇ ψ -V c ψ ) + ∂ ∂p [ p 2 D pp ∂ ∂p ( ψ p 2 )] -∂ ∂p [ ˙ pψ -p 3 ( ∇· V c ) ψ ] -ψ τ f -ψ τ r , (A1) \nwhere Q ( x, p ) represents the CR source term, ψ = ψ ( x, p, t ) denotes the CR density per unit momentum p at position x , ˙ p ≡ dp/dt is the momentum loss rate, and the time scales τ f and τ r characterize fragmentation processes and radioactive decays, respectively. In the framework of diffusive-reacceleration, the momentum space diffusion coefficient, D pp is related to the spatial diffusion coefficient D xx through the relation D pp D xx = 4 p 2 v 2 A / ( 3 δ ( 4 -δ 2 ) (4 -δ ) ) [47, 48], where v A is the Alfv'en velocity. \nThe spatial diffusion coefficient is parameterized as a function of rigidity as 4 \nThe injection spectrum is parameterized as: \nq i ( R ) ∝ ( R/R i br0 ) -ν i 0 , R ≤ R i br0 ( R/R i br1 ) -ν i 1 , R i br0 > R ≤ R i br ( R/R i br1 ) -ν i 2 , R > R i br , (A3) \nwhere i denotes the species of the nuclei. The low-energy break R br0 is introduced to account for the observed lowenergy spectral bumps observed in all nuclei [40, 50], while the high-energy break R br is introduced to account for the hardening observed in all primary nuclei [10]. \nFor interactions and energy losses, we include all relevant processes for the propagation of nuclei, such as fragmentation, momentum losses, radioactive decay, K capture, ionization losses, and Coulomb losses. In order to optimize computational efficiency, processes specific to lepton propagation, like synchrotron radiation, are not considered. For inputs such as the source distribution, gas model, and He/H ratio in the interstellar medium, we \nD xx = D 0 β η ( R R 0 ) δ , (A2) \nwhere β is the particle velocity in units of the speed of light, η is an empirical modification of the velocity dependence that allows for a better fit to low-energy data [49], and D 0 is the normalization at R 0 = 4GV. \nFIG. 9: Best-fit results for 10 Be/ 9 Be, compared with measurements [46]. \n<!-- image --> \nFIG. 8: Best-fit results for proton and helium fluxes, compared with measurements [10, 44, 45] \n<!-- image --> \n<!-- image --> \n1 \nadopted the default parameters from the 'Pulsar/XCO R-dep' model detailed in the latest GALPROP v57 release [19]. \nThe data used for the determination of injection and propagation parameters include: \n- · B/C: AMS-02 [31], DAMPE [11]\n- · B/O: AMS-02 [31], DAMPE [11]\n- · 10 Be/ 9 Be: AMS-02 preliminary [46]\n- · p & He: AMS-02 [10], DAMPE [44, 45] \nWe directly use the measurements from AMS-02 and DAMPE without introducing nuisance parameters, such as energy scale calibration, because their data is compatible within their respective error bars. To determine the total errors, we combine the systematic and statistical errors in quadrature. The best-fit propagation parameters are listed in Table I, while the best-fit injection parameters are listed in Table II. The comparison between the best-fit model results and the data of the proton and helium spectra and 10 Be/ 9 Be ratio are shown in Figs. 8 and 9. \nTABLE I: Propagation parameters.TABLE II: Injection parameters.", 'Appendix B: Cross sections for 4 He target channels': "The deuteron fragmentation cross sections of nuclei with A > 4 colliding with a target proton can be calculated through the factorization scheme provided by CDMP12 [6], utilizing a compilation of available crosssection data. The ISM is predominantly composed of 1 H and 4 He in a ratio of 9:1. The contributions of nuclei with A > 2 incident upon helium targets shall be considered, as they account for more than 20% of the total deuteron fluxes. \nCurrently, there has been no direct measurement of the deuteron cross section for 4 He-target channels. Our approach involves a two-step analysis to determine these cross sections. Firstly, we calculate some deuteron cross sections for 4 He-target channels with specific projectiles, based on data from other targets. Subsequently, we extrapolate deuteron cross-sections for other 4 He-target channels through interpolation. The available data for the first step are as follows. \n- 1. [Li75]: Lindstrom's group measured the deuteron cross section using a 2.1 GeV/n 16 O beam, along \nTABLE III: Production cross sections of deuterons for a 2.1 GeV/n 16 O beam and a 12 C beam upon different targets [17]. Additionally, the nuclear charge radii of these targets given by Ref. [51] are listed. \nwith 1.05 GeV/n and 2.1 GeV/n 12 C beams upon Be, CH 2 , C, Al, Cu, Ag and Pb targets [17]. The H-target data were obtained by C-CH 2 subtraction. Their measurements focused on deuterons associated with projectile fragments rather than target fragments. By extrapolating these measurements to the 4 He-target case, we can infer the cross sections for 16 O+ 4 He → D and 12 C+ 4 He → D. \n- 2. [Ab81]: The SKM-200 Collaboration measured the deuteron cross section using a 4.5 (GeV/c)/n 4 He beam upon Li, C, Al, Cu, and Pb targets [18]. The deuterons were observed within the momentum range of 6.5 GeV/c and 10.8 GeV/c, indicating that they exclusively originated from the projectile 4 He, rather than from the heavier target nuclei. By extrapolating these findings to the 4He 4 He-target case, we can infer the cross section for 4 He + 4 He → D. \nAccording to the analysis of [Li75], the production cross sections can be factored as σ = γ F P γ T , where γ F P depends on the projectile P and fragment F, while γ T mainly depends on the target T. The study revealed that γ T could be effectively modeled by γ T = A 1 / 4 T or γ T ∝ ( A 1 / 3 B + A 1 / 3 T -0 . 6). However, the inability to account for γ H and γ Be based on the available data suggests that a more precise fit can be achieved by utilizing the nuclear radius. \nBy analyzing the root-mean-square nuclear charge radii from Ref.[51], we find that the factor γ T for the deuteron cross section is proportional to the radius of the target nucleon, as listed in Table III. The relation γ T ∝ R T enables us to interpolate the unmeasured 4 He- \net cross sections. Using the radius of 4 He as 1.6755 fm, we derive the cross sections of 16 O+ 4 He → D and 12 C+ 4 He → D as 277.0 mb and 219.7 mb, respectively, with 10% uncertainties. \nTABLE IV: Production cross sections of deuterons for a 4.5 (GeV/c)/n 4 He upon different targets [18]. Additionally, the nuclear charge radii of these targets given by Ref. [51] are listed. \nIn Table IV we list the measurements from [Ab81] with charge radii of the targets. We assume that the relation γ T ∝ R T is valid for the helium projectile, and derive the cross section of 4 He + 4 He → D as 60.5 mb with a 30% uncertainty. Additionally, we calculate the cross section of 4 He+ p → D as 31.7 mb, which aligns closely with the value of ∼ 29 mb reported in GALPROP v57 [19] based on available measurements. \nBased on the derived cross sections for the three 4 Hetarget channels, we can calculate the cross section for other 4 He-target channels through interpolation. We assume that the deuteron cross sections of nuclei incident upon the 4 He-target as \nσ P+ 4 He → D ( E k/n , A P ) = γ D P 4 He × σ 4 He+ p → 3 He breakup ( E k/n ) . (B1) \nBy fitting the derived cross section values, we obtain γ D P 4 He = 0 . 6A 1 . 05 P , which is two times of the value in Eq. (2). This result implies that the reaction of P + 4 He → D can be equivalent to the reaction of the projectile upon two protons to produce deuterons."}

2024arXiv240910424R

Vmode polarization of the cosmic microwave background is expected to be vanishingly small in the LambdaCDM model and hence usually ignored. Nonetheless several astrophysical effects as well as beyond standard model physics could produce it at a detectable level. A realistic halfwave plate an optical element commonly used in CMB experiments to modulate the polarized signal can provide sensitivity to V modes without significantly spoiling that to linear polarization. We assess this sensitivity for some newgeneration CMB experiments such as the LiteBIRD satellite the groundbased Simons Observatory and a CMBS4like experiment. We forecast the efficiency of these experiments to constrain the phenomenology of certain classes of BSM models inducing mixing of linear polarization states and generation of V modes in the CMB. We find that newgeneration experiments can improve current limits by 1to3 orders of magnitude depending on the data combination. The inclusion of Vmode information dramatically boosts the sensitivity to these BSM models.

2024-09-01T00:00:00Z

['arXiv:2409.10424', '2024arXiv240910424R', '10.48550/arXiv.2409.10424']

['Astrophysics - Cosmology and Nongalactic Astrophysics']

Unveiling V Modes Enhancing CMB Sensitivity to BSM Physics with a NonIdeal HalfWave Plate

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.10424.pdf

{'N. Raffuzzi, a,b M. Lembo, a,b S. Giardiello, c,a M. Gerbino, b M. Lattanzi, b P. Natoli, a,b and L. Pagano a,b,d': "- a Dipartimento di Fisica e Scienze della Terra, Università degli Studi di Ferrara, via Saragat 1, I-44122 Ferrara, Italy\n- b Istituto Nazionale di Fisica Nucleare, Sezione di Ferrara, via Saragat 1, I-44122 Ferrara, Italy\n- c School of Physics and Astronomy, Cardiff University, The Parade, Cardiff, Wales CF24 3AA, United Kingdom\n- d Institut d'Astrophysique Spatiale, CNRS, Univ. Paris-Sud, Université Paris-Saclay, Bât. 121, 91405 Orsay cedex, France \nE-mail: nicolelia.raffuzzi@unife.it, margherita.lembo@unife.it, GiardielloS@cardiff.ac.uk \nAbstract. V-mode polarization of the cosmic microwave background is expected to be vanishingly small in the Λ CDM model and, hence, usually ignored. Nonetheless, several astrophysical effects, as well as beyond standard model physics could produce it at a detectable level. A realistic half-wave plate - an optical element commonly used in CMB experiments to modulate the polarized signal - can provide sensitivity to V modes without significantly spoiling that to linear polarization. We assess this sensitivity for some new-generation CMB experiments, such as the LiteBIRD satellite, the ground-based Simons Observatory and a CMB-S4-like experiment. We forecast the efficiency of these experiments to constrain the phenomenology of certain classes of BSM models inducing mixing of linear polarization states and generation of V modes in the CMB. We find that new-generation experiments can improve current limits by 1-to-3 orders of magnitude, depending on the data combination. The inclusion of V-mode information dramatically boosts the sensitivity to these BSM models.", '1 Introduction': 'Observations of anisotropies in the cosmic microwave background (CMB) radiation proved a major observational channel for modern cosmology. The Planck satellite observed them in temperature and polarization with unprecedented precision, providing state-of-the-art constraints on cosmology and fundamental physics [1]. Ground-based experiments have complemented these observations, especially targeting the polarization signal [2-5]. CMB E-mode polarization remains a key focus for next-generation experiments, aiming to achieve a cosmicvariance-limited estimate of the optical depth to reionization τ [6], the least constrained parameter in the Λ CDM model. Also, observations of large-scale B-mode polarization are crucial for improving sensitivity to, or detecting, the tensor-to-scalar ratio r [6-8], ac proxy for primordial gravitational waves. Enhanced sensitivity to small-scale E-mode and B-mode anisotropies will further constrain parameters of Λ CDM and its extensions [7, 8]. Additionally, improved observations of polarized CMB will help explore BSM physics, such as deviations from standard electromagnetism [9], neutrino properties [6], dark sectors [10], and supersymmetries [11, 12]. \nThe circular polarization of the CMB, also known as V modes, is expected to be small in the standard cosmological model, as it is not produced by Thomson scattering during recombination and reionization. Several standard and non-standard physical mechanism can however source a degree of circular polarization in CMB photons, by converting linear polarization generated at the last scattering surface. Among the standard mechanisms, photonphoton scattering via Heisenberg-Euler interaction at recombination produces the strongest circular polarization [13, 14]. For a comprehensive overview of conventional mechanisms that can source V-mode polarization in the CMB radiation see e.g. Ref. [15]. In the realm of non-standard physics, extensions of QED suggest that V-modes arise in the presence of Lorentz-violating operators [9, 16, 17]. Additionally, Refs. [18, 19] show that a pseudo-scalar field or axion inflation leads to V-mode generation. Magneto-optic effects are another class of phenomena capable of generating CMB circular polarization[20, 21]. The detection of circular \npolarization in the CMB has therefore the potential to offer further evidence for novel physics or more stringent constraints than those available from observations of linear polarization only. \n̸ \nVast improvements are expected in the observation of the CMB polarized signal both from ground-based experiments, such as the currently running Simons Observatory (SO) [7] and the next-generation CMB experiment CMB-S4 [8], and from satellites such as the LiteBIRD mission [6]. Achieving the promised sensitivity to detect the faint cosmological signals hidden in polarization requires monitoring and minimizing systematic effects. Upcoming observations will benefit from improved scanning strategies, including the use of a rotating half-wave plate (HWP) as a polarization modulator for SO and LiteBIRD. Previous studies have already demonstrated the effectiveness of a rotating HWP in mitigating 1/f noise, [22], and reducing temperature-to-polarization leakage resulting from pair-differencing of orthogonal detectors [23, 24]. However, despite the addition of a HWP is remarkably useful, the inclusion of another optical element in the acquisition chain brings along additional systematic effects, which in turn may compromise the final scientific product. One example is a HWP with a non-ideal phase shift, i.e., a HWP inducing a phase difference = 180 · between the two components of the electromagnetic wave traversing it. Interestingly enough, this deviation, possibly degrading the sensitivity to linear polarization, can also cause a coupling between total intensity and circular polarization. This would allow to investigate the presence of V modes in the CMB radiation [25]. In this paper, we exploit this possibility showing how forthcoming CMB experiments, equipped with HWPs, can be sensitive to circular polarization and, in turn, provide valuable information on a specific class of BSM physical models. We investigate a simple mechanism named the Generalized Faraday Effect (GFE), i.e. the mixing of CMB polarization states, including a partial conversion of linear into circular polarization [26, 27]. \nThe paper is structured as follows: Section 2 introduces the mathematical formalism (Jones calculus [28]) describing the optical behaviour of a (realistic) HWP, the instrumental systematic effect induced by a non-ideal phase shift, as well as the expected sensitivity to V modes; Section 3 illustrates the specific mechanism generating circular polarization and its cosmological phenomenology, while Section 4 describes the analysis method adopted in this work; in Section 5, we present the results of the analysis, forecasting the performance of future CMB experiments. Finally, Section 6 provides conclusions. Appendix A includes a full set of plots with posterior probability distributions of the key cosmological parameters for this work and for each experiment configuration considered in this analysis.', '2 Sensitivity to V modes': 'In this section, we give an overview of the mathematical framework to model a HWP with non-ideal phase shift β (for a detailed description, see, e.g., [29, 30]). We also provide an estimate of the sensitivity to V modes for a generic CMB experiment employing a non-ideal HWP.', '2.1 The non-ideal HWP': 'An ideal half-wave plate (HWP) is an optical device that induces a π -phase shift between the two orthogonal components of the incident wave. In the Jones formalism, the matrix describing the behavior of an ideal HWP is: \nJ Ideal HWP = ( 1 0 0 -1 ) . \nA realistic HWP can be described by the more general Jones matrix [29]: \nJ HWP = ( 1 + h 1 ζ 1 e iχ 1 ζ 2 e iχ 2 -(1 + h 2 ) e iβ ) , (2.1) \nwhere all but one entries of the matrix (e.g., J 11 ) are complex. The parameters h 1 , 2 are real and negative-defined, and indicate deviations from unitary transmission of the two polarized components, E x,y , of the incident light due to absorption. The parameters ζ 1 , 2 and χ 1 , 2 correspond to the amplitudes and phases of the off-diagonal terms that are responsible for cross-polarization, i.e., mixing between the two orthogonal components of the wave. The parameter β represents the departure from the ideal phase shift of π . In this work, we focus on the effects of a non-vanishing β and how it can be used to measure circular polarization. In the following, we assume for simplicity that the effects due to absorption and cross-polarization are negligible and fix the relative parameters in Eq. 2.1 to zero. The Jones matrix of a HWP with non-ideal phase shift and spinning with angular velocity ω is: \nJ HWP ( θ ) = R T ( θ ) J HWP R ( θ ) = ( J 11 ( θ ) J 12 ( θ ) J 21 ( θ ) J 22 ( θ ) ) , with R ( θ ) = ( cos θ sin θ -sin θ cos θ ) , (2.2) \nwhere θ = ωt is the time-dependent angle between the HWP fast axis and the x-axis in the chosen reference frame, and: \nJ 11 ( θ ) = cos 2 θ -e iβ sin 2 θ J 12 ( θ ) = J 21 ( θ ) = ( 1 + e iβ ) cos θ sin θ J 22 ( θ ) = -e iβ cos 2 θ +sin 2 θ. (2.3) \nIn the case under study, the complete optical chain traversed by the incoming signal includes a rotating HWP, two orthogonal polarizers, and the detector. Eventually, the Jones matrix of the full optical chain 1 is: \nJ x,y ( θ ) = J pol , ( x,y ) R T ( θ ) J HWP R ( θ ) , with J pol , x = ( 1 0 0 0 ) , J pol , y = ( 0 0 0 1 ) . (2.4) \nSo far, we have considered a scenario where the incoming wave is fully polarized and has a quasi-monochromatic frequency. However, the CMB signal is only partially polarized, and a CMB experiment measures the time-averaged incident intensity. Therefore, it is more natural to express the signal in terms of the Stokes vector s = ( T , Q , U , V ) , and employ the Müller formalism [31], which also simplifies tracking the impact on each Stokes parameter. \nGiven a Jones matrix, it is straightforward to compute the corresponding 4 × 4 Müller matrix. Explicitly, the Müller elements, for the x-oriented polarizer, are: \nM x TT = 1 2 ( | J 11 | 2 + | J 12 | 2 ) = 1 2 M x TQ = 1 2 ( | J 11 | 2 -| J 12 | 2 ) = 1 2 Cos ( β 2 ) 2 Cos(4 θ ) M x TU = Re[(J 11 J ∗ 12 )] = 1 2 Cos ( β 2 ) 2 Sin(4 θ ) M x TV = Im[(J 11 J ∗ 12 )] = -1 2 Sin( β ) Sin(2 θ ) . (2.5) \nThe y-oriented elements can be obtained by applying a 90-degree rotation ( 4 θ → 4 θ + π/ 2 ) to the above expressions. Finally, the total power collected by the detector is: \nd obs = Σ i = x,y ( M i TT T +M i TQ Q +M i TU U +M i TV V ) . (2.6) \nFrom Eqs. 2.5 - 2.6, it is clear that, when β = 0 , there is no contribution to the observed power from a non-vanishing V-mode signal. Conversely, the departure of the phase shift from the ideal value of π offers a handle to the detection of V modes.', '2.2 Sensitivity to Stokes parameters': "We now quantify the sensitivity to V modes of a CMB experiment equipped with a HWP with non-ideal phase shift. For the sake of simplicity, we consider Gaussian, stationary and uncorrelated instrumental noise of variance σ 2 pix in real space. This simplifies the expression of the noise covariance matrix in pixel space 2 . \nThe square root of the diagonal elements of the inverse noise covariance in pixel space provides an estimate of the experimental noise per pixel to each Stokes parameter. In the case of a HWP with non-ideal phase shift, the noise - which we normalize to the total intensity T one - is a function of β : \n( σ Q , U ( β ) σ T ) pix = √ 2 Cos 2 ( β/ 2) ( σ V ( β ) σ T ) pix = √ 2 Sin( β ) . (2.8) \nIn Figure 1, we show the sensitivity to the Stokes parameters Q , U , V , normalized to the sensitivity to the Stokes T , as a function of the phase shift β . The expected noise ratio for ideal CMB experiments is ( σ Q , U /σ T ) pix = √ 2 . The non-ideal phase shift leads to a mixing of power among the Q , U , V Stokes parameters. This can be understood from the fact that the sensitivity to Q and U decreases (i.e., the noise ratio increases, in red in Fig. 1) while that of V (blue) increases for larger values of β . However, while the sensitivity to V rapidly changes with β , the degradation of the sensitivity to Q , U significantly deviates from the ideal case only for fairly large values of β ≳ 20 · (for comparison, typical values for β vary from a few degrees up to a few tenths [32-34]). \nIt is also instructive to work out the sensitivity to V modes in harmonic space for a given experimental setup, as described in terms of the linear polarization noise level σ Q , angular resolution θ FWHM , observed sky fraction f sky and non-ideality HWP parameter β . \nˆ m pix = ( A T N -1 A ) -1 pix ( A T N -1 d ) pix , (2.7) \nwhere N = ⟨ nn T ⟩ is the noise covariance matrix in time domain and ( A T N -1 A ) -1 pix is the noise covariance matrix in pixel space. For the simple noise properties considered in this work, i.e. N = diag ( σ 2 ) , the noise covariance matrix in a pixel is simply given by σ 2 pix = σ 2 ( A T A ) -1 pix . This becomes σ 2 pix = 4 σ 2 N hit diag ( 1 , 2 / cos( β/ 2) 4 , 2 / cos( β/ 2) 4 , 2 / sin( β ) 2 ) , where N hit is the number of hits on that pixel, since we assume a very homogeneous and redundant scanning strategy. See, e.g. [30] and references therein for further details. \nFigure 1 : Noise ratio of the Stokes parameters for linear polarization σ Q , U , pix ( β ) (red) and circular polarization σ V , pix ( β ) (blue), normalized to the total intensity noise, as a function of the HWP non-ideal phase shift β . Typical values for β range from a few degrees to a few tenths [32-34]. For ideal CMB experiments, ( σ Q,U /σ T ) pix = √ 2 for polarization-sensitive detectors. In presence of a non-ideal phase shift, the noise level for circular polarization σ V , pix ( β ) decreases rapidly with increasing β , while the noise level for linear polarization σ Q , U , pix ( β ) does not increase significantly until very large values of β are reached. As a result, a slightly non-ideal HWP (small value of β ) can help gain sensitivity to V modes without significantly degrading the sensitivity to linear polarization. \n<!-- image --> \nNote that in the rest of this section we will denote with σ P the nominal linear polarization noise, corresponding to an ideal HWP plate, i.e. σ Q = σ Q ( β = 0) = √ 2 σ T . In this way factors of sin β and cos β will appear explicitly in the equations. \nThe observed power spectrum of the signal V modes ˆ C VV ℓ has an associated variance Σ 2 ℓ : \nΣ 2 ℓ = 2 (2 ℓ +1) f sky [ C VV , fid ℓ + N ℓ B 2 ℓ 1 Sin 2 ( β ) ] , (2.9) \nwhere C VV , fid ℓ is the true underlying power spectrum of the signal, N ℓ is the noise power spectrum of the linear polarization, assuming an ideal half-wave plate, and B ℓ is the harmonic equivalent of a Gaussian beam of width θ FWHM . We assume a white noise power spectrum N ℓ = σ 2 Q , and use a fiducial Gaussian approximation for the likelihood L ( C VV ℓ ) = Pr( ˆ C VV ℓ | C VV ℓ ) of a theoretical power spectrum C VV ℓ : \n-2 ln L ( C VV ℓ ) = ( C VV ℓ -ˆ C VV ℓ ) 2 Σ 2 ℓ , (2.10) \nFor the purpose of our analysis, we take ˆ C VV ℓ = ⟨ ˆ C VV ℓ ⟩ = C VV , fid ℓ . The inverse probability Pr( C VV ℓ | ˆ C VV ℓ ) is computed through the use of Bayes' theorem with a flat prior on Cℓ . \nArmed with the above, we compute, for each multipole ℓ , the 95% upper limit on C VV ℓ in the assumption of vanishing V-mode signal, i.e., for C VV , fid ℓ = 0 , and for different experimental configurations. The results are shown in Fig. 2. Here, we show the results for a LiteBIRD-like satellite (black solid curve) and for a ground-based Simons Observatory/SO SAT 3 experiment (magenta dashed line), assuming β = 10 · in both cases. The other parameters describing each configuration can be read in Tabs. 1 (noise and angular resolution) and 2 (sky coverage). For comparison, the current 95% upper bounds from the balloon-borne experiment SPIDER [25] (blue and red) and the ground-based CLASS [35] (yellow) are also shown. A LiteBIRDlike experiment would improve current bounds by several orders of magnitude and extend the sensitivity over a broader range of angular scales. On the other hand, a ground-based experiment like SO will dominate at smaller scales. \nIn Figure 3, we show iso-contours of constant integrated 95% sensitivity to a scaleinvariant VV spectrum, as a function of β and the instrumental noise level in linear polarization, σ Q . In the left (right) panel, the angular resolution and sky fraction are fixed to the values corresponding the LiteBIRD (SO) experiment. The two panels also correspond to different choices of the observed multipole range, again representatives of the aforementioned experiments: 2 ≤ ℓ ≤ 300 in the left panel and 50 ≤ ℓ ≤ 300 in the right panel. As expected, the sensitivity increases for smaller values of σ Q and large values of β . The stars in the plot indicate the expected performance of a LiteBIRD-like (left panel) and SO (right panel) experiment for β = 10 · .", '3 Production of V modes': 'In the previous section, we have quantified the sensitivity to V modes assuming a generic scaleinvariant spectrum. In the following, we work out the sensitivity of different combinations of CMB experiments to a V-mode signal generated in the phenomenological framework studied in [27]. We now briefly recall the basis of this formalism. \nWe consider a simple class of models allowing for the in vacuo mixing of CMB polarization states during propagation, including the conversion of linear into circular polarization. The mixing of polarization states in a weakly anisotropic medium is dubbed Generalized Faraday Effect (see sec. 14.3 of [36] for a description of the general framework, and [26] for an application to extragalactic synchrotron sources), or GFE. As the name suggests, the usual Faraday rotation is a particular case of GFE. The radiative transfer equation for the linearly polarized radiation propagating in a weakly anisotropic non-absorbing medium reads [36] \nd d s S = ρ × S (3.1) \nwhere s is an affine parameter following the direction of wave propagation, and S = ( Q,U,V ) is the polarization Stokes vector, precessing about the direction of ρ = ( ρ Q , ρ U , ρ V ) , which is determined by the dielectric tensor of the medium and thus encodes its optical properties. If ˆ ρ ≡ ρ / | ρ | is directed along the V -axis in the polarization space ( ρ Q = ρ U = 0 ), then the Stokes vector S precesses about it. The linear polarization component, i.e. the projection of S on the ( Q, U ) plane, will thus rotate while mantaining a constant magnitude. This is the usual \nV modes -by- sensitivity \nFigure 2 : The figure illustrates the ℓ -byℓ (i.e. no binning) 95% C.L. sensitivity for current and future CMB experiments to a VV-spectrum. Assuming a non-ideal HWP with a phase shift of β = 10 · , the black curve represents the sensitivity for a LiteBIRD-like experiment and the dark magenta curve corresponds to the sensitivity for a SO SAT experiment. Details on the full experimental setup adopted to obtain these curves are in Tab. 1. For comparison, the current bounds on V modes at different frequencies and ℓ -ranges from the balloon-borne SPIDER [25] (blue and red) and the ground-based CLASS [35] (yellow) are also shown. \n<!-- image --> \nFaraday rotation, where no circular polarization is generated, and only the Q and U states mix among themselves. On the other hand, if ρ has non-vanishing Q and U components, a partial conversion of linear into circular polarization occurs, generating non-zero V modes. \nIn Ref. [27], some of us have developed a formalism that allows to derive the modifications to the CMB spectra in temperature, linear and circular polarization, and their cross-correlations, given an effective dielectric tensor that describes the optical properties of the space traversed by CMB photons. This allows to be agnostic about the specific model beyond the mixing among CMB polarization components and just capture the basic phenomenology. Therefore, once the phenomenological parameters are constrained, it is possible to interpret the results in light of a specific physical model that could have induced GFE; see, e.g. [9] for a recent application. \nHere we consider the case of an effective dieletric tensor that is homogeneous and independent of both the direction and magnitude of the radiation wave vector. \nIn this specific case, parity-violating CMB cross spectra are vanishing, TB = EB = EV = BV = 0 (see Ref. [27] for details). \nThe remaining CMB power spectra C XY ℓ (with X, Y = T, E, B, V ) can be expressed \n4 \n4 \nFigure 3 : Contours of constant V-mode 95% sensitivity for a scale-invariant VV spectrum as function of non-ideal phase-shift β and nominal (i.e., assuming β = 0 ) linear polarization sensitivity σ Q . Left panel: the sensitivity is integrated over the multipole range 2 ≤ ℓ ≤ 300 for an angular resolution of 30 arcmin, similar to that achievable by LiteBIRD from space. For the LiteBIRD experiment, the expected noise level is σ Q ≃ 2 . 2 µ Karcmin , integrated over 22 frequency channels and 3-year observation time [6]. The star marks the combination of this noise level with a value of the nonideality parameter β = 10 · (chosen arbitrarily). Right panel: the sensitivity is integrated over 50 ≤ ℓ ≤ 300 for an angular resolution of 17 arcmin, similar to that achievable by the ground-based SO-SAT experiment. For the SO SAT, the noise level is σ Q ≃ 2 µ Karcmin , combining the 93 and 145 GHz bands [7]. The star marks the combination of this noise level with a value of the nonideality parameter β = 10 · (chosen arbitrarily). \n<!-- image --> \nin terms of the CMB spectra produced at the last-scattering surface ˜ C XY ℓ : \nC TE ℓ = ( 1 -β 2 V + β 2 E 8 π ) ˜ C TE ℓ C EE ℓ = ( 1 -β 2 V + β 2 E 4 π ) ˜ C EE ℓ + β 2 V 4 π [ W (1) ℓ ˜ C EE ℓ + W (1) ℓ +1 ˜ C BB ℓ +1 + W (1) ℓ -1 ˜ C BB ℓ -1 ] C BB ℓ = ( 1 -β 2 V + β 2 E 4 π ) ˜ C BB ℓ + β 2 V 4 π [ W (1) ℓ ˜ C BB ℓ + W (1) ℓ +1 ˜ C EE ℓ +1 + W (1) ℓ -1 ˜ C EE ℓ -1 ] C VV ℓ = β 2 E π [ W (2) ℓ +2 ˜ C BB ℓ +2 + W (2) ℓ +1 ˜ C EE ℓ +1 + W (2) ℓ ˜ C BB ℓ + W (2) ℓ -1 ˜ C EE ℓ -1 + W (2) ℓ -2 ˜ C BB ℓ -2 ] , (3.2) \nwhere W (1 , 2) are combinations of Wigner 3 j -symbols, and the phenomenological parameters β 2 E and β 2 V are, respectively, combinations of the diagonal and off-diagonal components of the effective susceptibility tensor [27]. \nFrom Eqs. 3.2 and Fig. 4, it is clear that the modification of the CMB polarization signal due to this particular realization of GFE has three main effects: \n) \n1 \n+ \n( \n- · rescaling of the amplitude of TE , EE , BB spectra via the combination β 2 V + β 2 E ;\n- · mixing between E and B modes (cosmic birefringence), proportional to β 2 V ;\n- · sourcing of V modes from E and B modes, proportional to β 2 E . \n1.5 \nFigure 4 : Impact of Generalized Faraday Effect (GFE) on polarization power spectra. The reference (blue) power spectra are computed using Planck best-fit values from TT, TE, EE, lowE + lensing [1] and vanishing GFE parameters, as reported in Tab. 3, while the modified spectra (orange) are calculated fixing β 2 V = 0 . 4 and β 2 E = 1 . 8 (chosen arbitrarily to make the GFE visible). \n<!-- image -->', '4 Analisys method and datasets': "In this section, we present methodology and data employed in the likelihood analysis. We simulate the performance of different upcoming CMB experiments (satellite and ground-based) and forecast their sensitivity to the phenomenological GFE parameters β 2 E,V introduced in the previous section. To this scope, we perform a Monte Carlo Markov Chain (MCMC) analysis to obtain forecasted constraints on β 2 E and β 2 V jointly with the six Λ CDM parameters \nand the tensor-to-scalar ratio r 4 . We use a customized version 5 of the CosmoMC [37] package to perform the analysis. We assume convergence of the MCMC chain by requiring that the Gelman-Rubin statistics R -1 ∼ 0 . 01 . \nWe run forecasts for five combinations of data sets which simulate the performance of different CMB experiments: \n- · a future LiteBIRD-like satellite experiment. We refer to this case as ' L ';\n- · LiteBIRD-like in combination with a simulated version of the Planck data, labeled ' LP ';\n- · a currently running Simons Observatory (SO) ground-based experiment, in combination with Planck, labeled ' PSO '. We include simulated performance of the observations with both the Large Aperture Telescope (LAT) which targets the smallest angular scales and the array of Small Aperture Telescopes (SATs) which target B modes at degree angular scales. The HWP is deployed on SATs only;\n- · Planck in combination with a next-generation ground-based experiment, i.e. CMBS4-like, labeled ' PS4 '. As for SO, we include information from both LAT and SATs. We note that the baseline design of CMB-S4, as described in [8, 38], does not include a HWP. Therefore, when referring to a CMB-S4-like experiment, we actually refer to the combination of expected sensitivity as determined by the sky coverage, angular resolution and noise properties reported in [8, 38], augmented with the use of a HWP in the telescopes targeting B modes to modulate the polarization signal;\n- · LiteBIRD-like in combination with CMB-S4-like, labeled ' LS4 '. \nThe instrument characteristics of each simulated experiment are reported in Tab. 1. In the following, when referring to individual experiments, we will omit the 'like' suffix for brevity. \nAs anticipated, for the sake of simplicity, we do not make use of the real data collected by Planck. We instead consider a simple white noise spectrum with 40 µ K-arcmin in temperature, and 80 µ K-arcmin in polarization [41], which reasonably reproduces the Planck performance. The noise spectrum for LiteBIRD is computed given the detector sensitivity and the angular resolution of LiteBIRD in the central frequency channels [6]. For SO [7], we make use of the publicly available noise curves 6 . The latter also include residual contributions from component separation, which dominates the cosmological signal both at the very small scales probed by the SO LAT, as well as at the intermediate scales probed by the SO SAT. CMB-S4 noise curves were computed referring to the central channels for both SAT and LAT [8]. \nWhen adding V-mode data in the analysis, the noise spectra of LiteBIRD, SO and CMB-S4 are further rescaled to account for the non-ideal phase shift, as detailed in Sec. 2. \nTable 1 : Instrumental specifications for each experiment: sensitivity to total intensity in µ Karcmin (linear polarization sensitivity is obtained by multiplying this value by a √ 2 factor, except for Planck which is 2 as only half the detectors are polarized); beam size FWHM in arcmin; non-ideal phase shift in degrees (for those experiments that will deploy - or we simulate will deploy - a HWP [7, 38, 39]). The noise levels for SO SAT and LAT are reported for reference. They are coadded over the central 90 and 150 GHz channels, and the FWHM corresponds to the 145 GHz channel. For more details, see [7, 40]. In this work, we make use of the official SO noise curves. \nIn practice, the rescaled noise curves become: \nN EE ℓ B 2 ℓ → 1 B 2 ℓ N EE ℓ Cos 4 ( β/ 2) N BB ℓ B 2 ℓ → 1 B 2 ℓ N BB ℓ Cos 4 ( β/ 2) N VV ℓ B 2 ℓ → 1 B 2 ℓ N VV ℓ Sin 2 ( β ) . (4.1) \nNote that Planck/CMB-S4 and LiteBIRD experiments observe largely overlapping sky fractions. While LiteBIRD dominates over Planck in B-mode sensitivity, the performance of the two experiments are different in T and E modes at different angular scales, depending on the respective noise level and angular resolution. A similar consideration applies to the combination ' LS4 ' as far as the T and E-fields ('TE' hereafter) are concerned. Therefore, for the 'TE' fields in the ' LP ' and ' LS4 ' cases, we use an effective inverse noise weighted curve: \nN eff ℓ = [ ( N ℓ B 2 ℓ ) -1 X + ( N ℓ B 2 ℓ ) -1 LiteBIRD ] -1 , (4.2) \nwhere X = Planck , CMB-S4. We can distinguish two regions where the two experiments perform differently: LiteBIRD dominates at larger scales, whereas Planck or CMB-S4 outperform LiteBIRD at smaller angular scales. \nDifferently, when Planck is combined with SO/CMB-S4, information from the former is only included in the sky fraction and/or at the angular scales not observed by SO/CMB-S4(LAT). As a result, in the ' PSO / PS4 ' case, we make use of the individual noise curves for each experiment. \nWe employ an exact likelihood in harmonic space, rescaled by a ℓ -independent sky fraction f sky to account for the experiment-specific sky coverage. \n-2 ln L ℓ = f sky (2 ℓ +1) ( Tr [ ˆ S ℓ S -1 ℓ ] +lndet[ S ℓ ] ) + const. (4.3) \nwhere ˆ S and S are matrices of the 'observed' (i.e., simulated, in our case) ˆ C ℓ and theoretical C ℓ power spectra, respectively (see, e.g., Sec. 2.2 of [42] for the definition of ˆ S and S ). When V modes are included in the likelihood analysis, the standard ˆ S and S matrices are extended to include the C VV ℓ . The sky fraction, range of angular scales and observed fields for each data combination 7 are reported in Tab. 2. \nWe simulate the observed spectra ˆ C ℓ as the sum of a reference set of CMB spectra obtained with the Boltzmann code camb [43] from a fiducial set of cosmological parameters and the noise power spectrum in harmonic space: ˆ C ℓ = C fid ℓ +N ℓ / B 2 ℓ (see, e.g., Sec. 3.3 of [42] for the extension of the definition of ˆ S and S in Eq. 4.3 in presence of noise and beam smearing). We assume the fiducial model to be Λ CDM, i.e., vanishing V-mode signal and vanishing primordial tensor modes. The cosmological parameters corresponding to the fiducial model are reported in Tab. 3. A major source of variance in the observation of primordial B modes is the BB lensing contribution, which can be significantly reduced via appropriate de-lensing procedures. In order to quantify the effect of delensing in our results, we consider two cases: a fully lensed fiducial BB, i.e., no de-lensing, and a partly de-lensed fiducial BB. The two cases are obtained by properly rescaling the BB lensing contribution (in both the fiducial and the theory BB spectra) by a factor A L = 1 (fully lensed) and A L = 0 . 3 (70% delensing). The GFE-modified (tensorial) B modes are later summed to the A L -rescaled lensed B modes. The underlying assumption, also used in Ref. [27], posits that GFE exclusively affects primordial B modes, even though GFE and lensing should occur simultaneously as integrated effects along the line of sight. This assumption is valid for LiteBIRD and SO, see, e.g., [27]. However, for fourth-generation ground-based experiments such as CMB-S4, caution is advised regarding the interplay between lensing and GFE. In order to assess the validity of the assumption for CMB-S4, we compute the difference between the BB spectrum obtained as detailed above and the BB spectrum obtained by applying GFE to the sum of tensor and lensing BB spectra. We then compare this difference with the CMB-S4 noise curve. We find that, for reasonably small values of GFE parameters ( β 2 E,V ≲ 0 . 1 ), the difference in the predicted CMB spectra between the two approaches is well below the noise contribution. Furthermore, other studies in the literature, such as Ref. [44], have confirmed that lensed B modes have a negligible impact on cosmic birefringence. \nTo better quantify the impact of a potential sensitivity to V modes, we compare results obtained with and without the inclusion of V modes in the datasets for each combination of experiments. We refer to the two cases as 'TEBV' and 'TEB', respectively. In the 'TEBV' case, we assume the non-ideal phase shift to be β = 10 · ; as already noted, this choice is in agreement with the expected values of the phase shift from simulations. In addition, as shown in Fig. 1, this value allows concurrently for a relatively acceptable noise level in V -Stokes parameter and negligible degradation of the noise level in Q / U -Stokes. In the 'TEB' case, we assume the phase shift to be β = 0 . \nTable 2 : Data combinations considered in this work. For each experiment, we report the observed fields (temperature T, linear polarization E and B, circular polarization V) and the corresponding range of angular scales (multipoles), as well as the sky fraction observed ( f sky ). Throughout the main text, we refer to each combination with the corresponding label in the first column. \nTable 3 : Fiducial values of cosmological parameters for the reference cosmological model assumed to simulate the observed CMB spectra in this analysis. They are taken from the Planck best-fit values from TT, TE, EE + lowE + lensing analysis [1].", '5 Results': "In this section, we present and discuss the results obtained for the various datasets and cases introduced in the previous section. These results are also compared with the current bounds \nreported in literature [27, 45]. We focus on a subset of cosmological parameters: the optical depth ( τ ), the tensor-to-scalar ratio ( r ), and the GFE parameters ( β 2 E and β 2 V ) 8 . The first two parameters are linked to the primary science goals of experiments observing large and intermediate-scale polarization, such as LiteBIRD, SO, and CMB-S4, while the latter two are the main focus of this study. The 68% C.L. constraints on τ and the 95% C.L. upper bounds on the remaining three parameters are in Tab. 4 for both the fully lensed and partially delensed scenarios. Marginalized 1D and 2D posterior probabilities for r , τ , β 2 E and β 2 V can be found in the Figures in Appendix A. We compare the results obtained with and without the inclusion of V-mode data. In doing so, we expect to place more stringent limits on β 2 E and β 2 V . In addition, we test whether the partial degradation of sensitivity to linear polarization resulting from a non-ideal phase shift affects the constraints on τ and r . \nFrom Tab. 4, we can draw the following considerations: \n- · as known, the constraints on τ and r are driven by the ability of a given data combination to recover the large-scale E-mode signal and the large-to-intermediate B-mode signal respectively. In Tab. 4, compare, e.g., the improvement of the constraints on τ from the combinations including LiteBIRD with respect to those including Planck. Similarly, compare the lack of improvement of the constraints on r from the ' L ' dataset with respect to the ' LP ' combination. \nIgnoring for a moment the impact of V-mode observations (but see the second bullet point below), the constraints on β 2 E are mostly driven by the ability to reconstruct the TE and EE spectra at intermediate and small scales, since β 2 E acts as an overall rescaling factor for the CMB spectra (see Eqs. 3.2). Indeed, by comparing the first rows of each Dataset box in Tab. 4, we note a steady improvement of the constraints on β 2 E when considering dataset which are more and more sensitive to small-scale E (compare, e.g., ' L ' and ' LP '; also, note the lack of improvement when comparing ' PS4 ' with ' LS4 ', as CMB-S4 dominates the sensitivity to E in both data combinations). \nThe constraints on β 2 V are mostly driven by the sensitivity of each dataset to B modes, both via the overall rescaling of the amplitude (first term on the RHS in Eq. 3.2) and via the mixing with E modes (second term on the RHS) 9 . \n- · the addition of V-mode data tightens the constraints on β 2 E by roughly an order of magnitude but has no effect on β 2 V (compare the two rows corresponding to each data label in Tab. 4; see also Appendix A). Indeed, C VV ℓ in Eq. 3.2 is solely affected by β 2 E that acts as an overall rescaling factor of a mixing of ˜ C EE ℓ and ˜ C BB ℓ . The inclusion of V modes, with the corresponding rescaling of the noise spectra as per Eq. 4.1 and Fig. 1, does not degrade the constraints on r and τ . This is one of the main findings of this work. This stands in contrast to the assertion made in [27] regarding the necessity of improving linear polarization for increased accuracy on GFE parameters. The inclusion of V-mode data proves to be an additional, significant source of constraining power, \nbeyond what is provided by linear polarization alone. The sensitivity (i.e. the low VV noise) is enhanced by the non-ideal phase shift of the HWP (as shown in Fig. 1), where a relatively conservative value of β = 10 · was employed; \n- · by reducing the contribution of signal variance in the observation of primordial B modes, the de-lensing procedure results in a factor of two improvement of the constraints on r and β 2 V (compare rows with the same data label in the top and bottom panels in Tab. 4). No improvement is seen for τ and β 2 E , as could be visually verified by comparing the left (no de-lensing) with the right (70% de-lensing) panel of triangular plots in Appendix A. This confirms that the constraining power on these parameters comes essentially from TE and EE spectra rather than from BB (in addition to VV for β 2 E ).\n- Improvements on r and β 2 V are expected, since they directly impact the C BB ℓ power spectrum as rescaling factors. In Eq. 3.2, we recognize an attenuating factor ( β 2 V + β 2 E ) / 4 π in front of the unmodified ˜ C BB ℓ . In the second term on the RHS, the amplitude of the combination of unmodified EE and BB spectra - with the ˜ C BB ℓ spectrum being subdominant compared to the unmodified ˜ C EE ℓ - is regulated by β 2 V . This requires β 2 V to suppress the power coming from the EE spectrum, thus restoring the expected amplitude of the observed tensorial B modes ( r = 0 ).\n- Also, de-lensing does not help improve the constraining power on the optical depth τ . In principle, information on τ can be also extracted from measurements of the reionization peak in the BB spectrum. However, the strong degeneracy between r and τ in that region as well as the residual lensing variance dominating also at lowℓ s for vanishingly small values of r severely limit the information content that can be extracted from BB to improve the constraints on τ . As a result, the constraining power for this parameter comes essentially from E modes only;\n- · when comparing the constraints on r from ' PSO ' and ' PS4 ' in the fully lensed case (top panel in Tab. 4), we note that the latter dataset provides - somehow unexpectedly - a looser bound on r than the former dataset (see also left panels of Figs. 7 and 8). This feature can be attributed to the fact that the two experiments driving the constraints on r in the two data combinations - namely, SO and CMB-S4 - are both cosmic variance limited in BB in the fully lensed scenario. With the instrumental noise being subdominant, the ability to jointly constrain the parameters affecting C BB ℓ is solely determined by volume effects in the parameter space explored in the MCMC analysis. More in detail, at first order, we can assume that the amplitude of C BB ℓ is determined by a combination of r , β 2 E and β 2 V (see Eqs. 3.2). When switching from ' PSO ' to ' PS4 ' data combination, the enhanced sensitivity of the latter to E modes leads to more stringent limits on β 2 E , thus allowing for a larger excursion of r in parameter space. For the same reason, the constraints on β 2 V from the two data combinations are virtually unchanged. In this case, the volume effect responsible for the degradation of the constraints on r is suppressed for β 2 V by the extra contribution of the E-B mixing term on the RHS in the equation for C BB ℓ . \nConversely, for partial de-lensing (bottom panel of Tab. 4; see also right panels of Figs. 7 and 8), we observe the expected improvement of the constraint on r when moving from ' PSO ' to ' PS4 '. While ' PS4 ' remains cosmic variance limited, the impact of instrumental noise in ' PSO ' cannot be neglected and washes out the volume effect described above. \nFinally, in Tab. 5 we quote again the 95% C.L. upper limits for the five analyzed cases combining TEBV modes, and compare them with current bounds. We note that current limits in the literature are obtained without a de-lensing procedure applied to data. Nevertheless, we report forecasts for both the fully lensed and the partially de-lensed scenarios, to also assess the benefit of de-lensing. The current upper limit on r is taken from the analysis of BICEP2, Keck Array, BICEP3, WMAP, and Planck data, marginalizing over a given model of Galactic dust and synchrotron contaminations [45]. As already mentioned, no de-lensing is applied to the data. Future experiments targeting r will see considerable improvement in the constraints by applying suitable de-lensing techniques (see, e.g., [8, 46, 47]). Our forecasts qualitatively confirm this expectation. Present bounds on β 2 E,V are taken from the analysis of [27] and were derived using temperature and linear polarization anisotropies from the Planck legacy release [48] and BICEP2/Keck 2015 [49] . The inclusion of current V-mode observations does not help improve the constraints on the GFE parameters [27]. Forecasted sensitivity to β 2 V is dominated by the sensitivity of upcoming experiments to B modes, with de-lensing playing a crucial role. As far as β 2 E is concerned, while de-lensing has minimal impact on the forecasted sensitivity, the inclusion of V modes from future experiments significantly improves the constraints. Our forecast shows that more stringent constraints ranging from 1 to 3 orders of magnitude can be achieved, depending on the considered parameter and dataset.", '6 Conclusions': "̸ \nIn this study, we investigated how the deployment of a realistic half-wave plate (HWP) inducing a non-ideal (i.e., = π ) phase shift, may affect the noise characteristics of CMB experiments, while providing sensitivity to the presence of a non-vanishing V-mode signal. We then forecasted the expected performance of new-generation CMB experiments, noting that our analysis did not consider foregrounds. \nDeviations from the ideal phase shift modify the noise properties of a CMB experiment deploying a HWP by degrading the sensitivity to Q / U polarization. This could potentially spoil the ability to reach the primary science goals of future CMB experiments, such as a CVL determination of the optical depth τ and a tight constraint on/high-significance detection of the tensor-to-scalar ratio r . However, at the same time, deviations from the ideal performance of a HWP allow for the experiment to become sensitive to possibly non-vanishing V modes in the observed signal. This work showed how even a significant deviation from the ideal phase shift (of the order of 10 · ) can open a window to the investigation of beyond-the-standardmodel (BSM) cosmological scenarios while having no impact on the sensitivity to τ and r . \nAs a test case, we focused on the phenomenology of a class of models leading to BSM electromagnetic effects on the CMB, the Generalized Faraday Effect (GFE) [27]. The GFE is responsible for E-B mixing as well as for sourcing circular polarization from the conversion of linear CMB polarization. These effects are modeled via two phenomenological parameters β 2 E and β 2 V - which modify the shape of the CMB spectra as they emerge from the last scattering surface (Eqs. 3.2). We forecasted the sensitivity to these parameters from currently running and future CMB experiments which will or might deploy a HWP. At the same time, we checked that the sensitivity to τ and r expected from these experiments is not degraded. We considered different combinations of simulated data from a LiteBIRD-like satellite experiment ( L ), from the ground-based Simons Observatory ( SO ) and a modified version (i.e., with the \nTable 4 : Constraints on a subset of cosmological parameters sampled in this analysis: 68% C.L. constraints for the optical depth ( τ ), and 95% C.L. upper limits for the tensor-to-scalar ratio ( r ) and the GFE parameters ( β 2 E , β 2 V ). The results are shown assuming a lensing factor A L = 1 (no de-lensing, top table) and A L = 0 . 3 (70% de-lensing efficiency, bottom table), for different data combinations ( L , LP , PSO , PS4 , and LS4 ). For each data combination, we compare the results obtained with and without the inclusion of V-mode observations. \ninclusion of a HWP) of a ground-based CMB-S4-like experiment ( S4 ) 10 . We also included information from Planck ( P ) observations where relevant. \nBy comparing the results with current limits in the literature [27], we forecasted a 1-to-3 order-of-magnitude improvement of the constraints on β 2 V and β 2 E , depending on the dataset, \nTable 5 : 95% C.L upper limits on the tensor-to-scalar ratio and GFE parameters ( β 2 E and β 2 V ) for the different data combinations considered in this analysis. These limits are obtained from the analysis of T, E, B and V modes, with a lensing factor A L = 1 (second, third and fourth columns) and A L = 0 . 3 (last three columns). In the last row, bounds on the same parameters from current data [27, 45] are also reported to ease the comparison, where no de-lensing procedures are included. \nwith the most stringent limits achieved with the ' LS4 ' combination (see Tab. 5). \nThe possibility to include observations of V modes, as enabled by the deployment of a realistic HWP, is key to obtain dramatic improvements in the constraints on β 2 E , ranging from a factor of a few for the combinations ' PSO ' and ' PS4 ' to an order of magnitude and more for the data combinations including LiteBIRD (see Tab. 4). Better sensitivity to intermediateto-small scale E modes is also an important driver of the improved constraints on β 2 E , as can be noted by comparing ' L ', ' LP ', ' PSO '/' PS4 '. Constraints on β 2 V are mostly insensitive to the inclusion of V modes in the analysis. Instead, they are driven by improved sensitivity to B modes. The constraints improve roughly by a factor of two when partial de-lensing (obtained as a 70% suppression of lensing power in the simulated BB data) is applied in the analysis (compare the top and bottom panels in Tab. 4). The stringent constraints on β 2 V and β 2 E will provide valuable insights on the fundamental physics governing the evolution of the Universe, particularly from the CMB last scattering to the present epoch. The GFE parameters can be mapped onto the elements of an effective 'cosmic susceptibility tensor', which describes the optical properties of the Universe seen as the medium through which CMB photons propagate. The mapping between β 2 V and β 2 E and the elements of the susceptibility tensor can be obtained once a specific physical model responsible for the GFE is specified. As such, constraints on β 2 V and β 2 E shed light on the nature of the GFE and its potential origins, such as magnetic fields [20, 21, 36, 50-52] or other BSM phenomena [9, 13, 14, 19, 53] that could have influenced the polarization state of the CMB. The methodology outlined in this work can be also applied to models capable of generating parity-breaking spectra (e.g. Ref. [9]). In conclusion, the results presented in this study indicate that the possibility of including observations of V modes in the cosmological analysis is a key tool to investigate more deeply specific aspects of fundamental physics which are otherwise less accessible, without spoiling the constraining power to other primary science targets. This emphasizes the necessity of fully exploit the potential of upcoming CMB experiments, including the extraction of V-mode data, which will provide unprecedented insights on the fundamental physics of the early Universe.", 'Acknowledgments': 'Wethank Alessandro Gruppuso for useful discussions during the preparation of the manuscript and for feedback on the final version of the paper. We acknowledge the financial support from the INFN InDark initiative and from the COSMOS network (www.cosmosnet.it) through the ASI (Italian Space Agency) Grants 2016-24-H.0 and 2016-24-H.1-2018, as well as 2020-9-HH.0 (participation in LiteBIRD, phase A). We acknowledge the use of camb [43], CosmoMC [37], GetDist [54], and the use of computing facilities at CINECA. MG is funded by the European Union (ERC, RELiCS, project number 101116027) and by the PRIN (Progetti di ricerca di Rilevante Interesse Nazionale) number 2022WJ9J33. SG acknowledges support from the Horizon 2020 ERC Starting Grant (Grant agreement No 849169) and from STFC and UKRI (grant numbers ST/W002892/1 and ST/X006360/1). This is not an official SO Collaboration paper.', 'References': "- [1] Planck collaboration, Planck 2018 results. VI. Cosmological parameters , Astron. Astrophys. 641 (2020) A6 [ 1807.06209 ].\n- [2] POLARBEAR collaboration, A Measurement of the Degree Scale CMB B -mode Angular Power Spectrum with POLARBEAR , Astrophys. J. 897 (2020) 55 [ 1910.02608 ].\n- [3] POLARBEAR collaboration, A Measurement of the CMB E -mode Angular Power Spectrum at Subdegree Scales from670 Square Degrees of POLARBEAR Data , Astrophys. J. 904 (2020) 65 [ 2005.06168 ].\n- [4] ACT collaboration, The Atacama Cosmology Telescope: DR4 Maps and Cosmological Parameters , JCAP 12 (2020) 047 [ 2007.07288 ].\n- [5] SPT-3G collaboration, Measurements of the E-mode polarization and temperature-E-mode correlation of the CMB from SPT-3G 2018 data , Phys. Rev. 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Gear, Achromatic half-wave plate for submillimeter instruments in cosmic microwave background astronomy: experimental characterization , Appl. Opt. 45 (2006) 6982. \n- [34] G. Savini, G. Pisano and P.A.R. Ade, Achromatic half-wave plate for submillimeter instruments in cosmic microwave background astronomy: modeling and simulation , Appl. Opt. 45 (2006) 8907.\n- [35] I.L. Padilla et al., Two-year Cosmology Large Angular Scale Surveyor (CLASS) Observations: A Measurement of Circular Polarization at 40 GHz , 1911.00391 .\n- [36] D.B. Melrose, Collective plasma radiation processes , Ann. Rev. Astron. Astrophys. 29 (1991) 31.\n- [37] A. Lewis and S. Bridle, Cosmological parameters from cmb and other data: A monte carlo approach , Phys. Rev. D 66 (2002) 103511.\n- [38] K. Abazajian et al., CMB-S4 Science Case, Reference Design, and Project Plan , 1907.04473 .\n- [39] LiteBIRD collaboration, LiteBIRD: JAXA's new strategic L-class mission for all-sky surveys of cosmic microwave background polarization , Proc. SPIE Int. Soc. Opt. Eng. 11443 (2020) 114432F [ 2101.12449 ].\n- [40] Simons Observatory collaboration, The Simons Observatory: Astro2020 Decadal Project Whitepaper , Bull. Am. Astron. Soc. 51 (2019) 147 [ 1907.08284 ].\n- [41] Planck collaboration, Planck 2013 results. I. Overview of products and scientific results , Astron. Astrophys. 571 (2014) A1 [ 1303.5062 ].\n- [42] M. Gerbino, M. Lattanzi, M. Migliaccio, L. Pagano, L. Salvati, L. Colombo et al., Likelihood methods for CMB experiments , Front. in Phys. 8 (2020) 15 [ 1909.09375 ].\n- [43] A. Lewis and A. Challinor, 'CAMB: Code for Anisotropies in the Microwave Background.' Astrophysics Source Code Library, record ascl:1102.026, Feb., 2011.\n- [44] T. Namikawa, Exact CMB B-mode power spectrum from anisotropic cosmic birefringence , 2404.13771 .\n- [45] BICEP, Keck collaboration, Improved Constraints on Primordial Gravitational Waves using Planck, WMAP, and BICEP/Keck Observations through the 2018 Observing Season , Phys. Rev. Lett. 127 (2021) 151301 [ 2110.00483 ].\n- [46] CMB-S4 collaboration, CMB-S4: Iterative Internal Delensing and r Constraints , Astrophys. J. 964 (2024) 148 [ 2310.06729 ].\n- [47] K. Wolz et al., The Simons Observatory: pipeline comparison and validation for large-scale B-modes , 2302.04276 .\n- [48] Planck collaboration, Planck 2018 results. V. CMB power spectra and likelihoods , Astron. Astrophys. 641 (2020) A5 [ 1907.12875 ].\n- [49] BICEP2, Keck Array collaboration, BICEP2 / Keck Array x: Constraints on Primordial Gravitational Waves using Planck, WMAP, and New BICEP2/Keck Observations through the 2015 Season , Phys. Rev. Lett. 121 (2018) 221301 [ 1810.05216 ].\n- [50] S. De and H. Tashiro, Circular Polarization of the CMB: A probe of the First stars , Phys. Rev. D 92 (2015) 123506 [ 1401.1371 ].\n- [51] N. Lemarchand, J. Grain, G. Hurier, F. Lacasa and A. Ferté, Secondary CMB anisotropies from magnetized haloes - I. Power spectra of the Faraday rotation angle and conversion rate , Astron. Astrophys. 630 (2019) A149 [ 1810.09221 ].\n- [52] A. Cooray, A. Melchiorri and J. Silk, Is the cosmic microwave background circularly polarized? , Physics Letters B 554 (2003) 1.\n- [53] T.-C. Li, T. Zhu, W. Zhao and A. Wang, Power spectra and circular polarization of primordial gravitational waves with parity and Lorentz violations , 2403.05841 .\n- [54] A. Lewis, GetDist: a Python package for analysing Monte Carlo samples , 1910.13970 .", 'A Complete set of triangular plots': 'We show here the triangular plots for the five cases analyzed, displaying 1D and 2D posteriors for the parameters we consider relevant in our analysis, i.e. r , τ , β 2 E and β 2 V . The general behaviour of all plots looks similar, the inclusion of V-mode polarization substantially enhances our constraining power improving the bound on β 2 E . Furthermore, the addition of the de-lensing procedure is effective in reducing the upper limits of r and β 2 V (see the right panel of Figs. 5, 6, 7, 8, 9). In contrast, as expected, bounds on τ were found to be unaffected by both the de-lensing and the addition of V-mode data. \nFigure 5 : One and two-dimensional posterior probability distributions for L dataset (see Tab. 2 for details). We report a subset of the total parameter space ( Λ CDM model + r + β 2 E + β 2 V ) showing TTTEEEBB in green and TTTEEEBBVV in magenta. The left panel omits de-lensing procedures ( A L = 1 ), while the right panel employs 70% de-lensing ( A L = 0 . 3 ). \n<!-- image --> \ne \nv \nr \n2 \ne \n2 \nv \nTEB \nTEBV \n2 \n0.060 \n0.055 \n0.050 \n0.15 \ne \n0.10 \n0.05 \n0.0010 \nv \n0.0006 \n0.0002 \n2 \n2 \n4 \n×10 \n4 \n0.05 \n0.06 \n0.05 \n0.15 \n3 \n9 \n×10 \n4 \n2 \n0.060 \n0.055 \n0.050 \n0.10 \ne \n0.05 \n0.0016 \nv \n0.0010 \n0.0004 \n2 \n2 \n5 \n8 \nr \nFigure 7 : One and two-dimensional posterior probability distributions for PSO dataset (see Tab. 2 for details). We report a subset of the total parameter space ( Λ CDM model + r + β 2 E + β 2 V ) showing TTTEEEBB in green and TTTEEEBBVV in magenta. The left panel omits de-lensing procedures ( A L = 1 ), while the right panel employs 70% de-lensing ( A L = 0 . 3 ). \n<!-- image --> \n0.05 \n0.06 \n0.03 \n0.10 \n2 \ne \n0.001 \n2 \nv \nFigure 6 : One and two-dimensional posterior probability distributions for LP dataset (see Tab. 2 for details). We report a subset of the total parameter space ( Λ CDM model + r + β 2 E + β 2 V ) showing TTTEEEBB in green and TTTEEEBBVV in magenta. The left panel omits de-lensing procedures ( A L = 1 ), while the right panel employs 70% de-lensing ( A L = 0 . 3 ). \n<!-- image --> \ne \nv \n0.002 \nr \n2 \n0.048 \n0.064 \n0.05 \n0.10 \ne \n0.003 \n2 \nv \nTEB \nTEBV \n×10 \n4 \nTEB \nTEBV \n2 \n0.06 \n0.05 \n0.10 \ne \n0.06 \n0.02 \n0.004 \n2 \nv \n0.002 \nFigure 9 : One and two-dimensional posterior probability distributions for LS4 dataset (see Tab. 2 for details). We report a subset of the total parameter space ( Λ CDM model + r + β 2 E + β 2 V ) showing TTTEEEBB in green and TTTEEEBBVV in magenta. The left panel omits de-lensing procedures ( A L = 1 ), while the right panel employs 70% de-lensing ( A L = 0 . 3 ). \n<!-- image --> \ne \nv \ne \nv \nFigure 8 : One and two-dimensional posterior probability distributions for PS4 dataset (see Tab. 2 for details). We report a subset of the total parameter space ( Λ CDM model + r + β 2 E + β 2 V ) showing TTTEEEBB in green and TTTEEEBBVV in magenta. The left panel omits de-lensing procedures ( A L = 1 ), while the right panel employs 70% de-lensing ( A L = 0 . 3 ). \n<!-- image --> \ne \nv \ne \nv'}

2023LRR....26....2A

The Laser Interferometer Space Antenna LISA will be a transformative experiment for gravitational wave astronomy and as such it will offer unique opportunities to address many key astrophysical questions in a completely novel way. The synergy with groundbased and spaceborn instruments in the electromagnetic domain by enabling multimessenger observations will add further to the discovery potential of LISA. The next decade is crucial to prepare the astrophysical community for LISAs first observations. This review outlines the extensive landscape of astrophysical theory numerical simulations and astronomical observations that are instrumental for modeling and interpreting the upcoming LISA datastream. To this aim the current knowledge in three main source classes for LISA is reviewed ultracompact stellarmass binaries massive black hole binaries and extreme or intermediate mass ratio inspirals. The relevant astrophysical processes and the established modeling techniques are summarized. Likewise open issues and gaps in our understanding of these sources are highlighted along with an indication of how LISA could help making progress in the different areas. New research avenues that LISA itself or its joint exploitation with upcoming studies in the electromagnetic domain will enable are also illustrated. Improvements in modeling and analysis approaches such as the combination of numerical simulations and modern data science techniques are discussed. This review is intended to be a starting point for using LISA as a new discovery tool for understanding our Universe.

2023-12-01T00:00:00Z

['2023LRR....26....2A', 'arXiv:2203.06016', '2022arXiv220306016A', '10.48550/arXiv.2203.06016', '10.1007/s41114-022-00041-y']

['Black holes', 'Gravitational waves', 'Stellar remnants', 'Multi-messenger', 'Extreme mass ratio in-spirals', 'General Relativity and Quantum Cosmology', 'Astrophysics - Cosmology and Nongalactic Astrophysics', 'Astrophysics - Astrophysics of Galaxies', 'Astrophysics - High Energy Astrophysical Phenomena', 'Astrophysics - Instrumentation and Methods for Astrophysics', 'Astrophysics - Solar and Stellar Astrophysics']

Astrophysics with the Laser Interferometer Space Antenna

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https://arxiv.org/pdf/2203.06016.pdf

{'Astrophysics with the Laser Interferometer Space Antenna': 'Amaro Seoane, Pau; Andrews, Jeff; Arca Sedda, Manuel; Askar, Abbas; Baghi, Quentin; Balasov, Razvan; [full author details at the end of the article]', 'Abstract': "The Laser Interferometer Space Antenna (LISA) will be a transformative experiment for gravitational wave astronomy, and, as such, it will offer unique opportunities to address many key astrophysical questions in a completely novel way. The synergy with ground-based and spaceborn instruments in the electromagnetic domain, by enabling multi-messenger observations, will add further to the discovery potential of LISA. The next decade is crucial to prepare the astrophysical community for LISA's first observations. This review outlines the extensive landscape of astrophysical theory, numerical simulations, and astronomical observations that are instrumental for modeling and interpreting the upcoming LISA datastream. To this aim, the current knowledge in three main source classes for LISA is reviewed; ultra-compact stellar-mass binaries, massive black hole binaries, and extreme or interme-diate mass ratio inspirals. The relevant astrophysical processes and the established modeling techniques are summarized. Likewise, open issues and gaps in our understanding of these sources are highlighted, along with an indication of how LISA could help making progress in the different areas. New research avenues that LISA itself, or its joint exploitation with upcoming studies in the electromagnetic domain, will enable, are also illustrated. Improvements in modeling and analysis approaches, such as the combination of numerical simulations and modern data science techniques, are discussed. This review is intended to be a starting point for using LISA as a new discovery tool for understanding our Universe.", 'General Introduction': "Gravitational wave (GW) observations have opened a new way to observe and characterize compact objects throughout the Universe and at all cosmic epochs. The Laser Interferometer Space Antenna (LISA Amaro-Seoane et al., 2017), with its low-frequency band coverage spanning nearly three decades, will allow the detection and study of signals from a strikingly large variety of sources, ranging from stellar-mass binaries in our own galaxy to mergers between nascent massive black holes (MBH), called black hole (BH) seeds , at high redshift. LISA is expected to revolutionize our understanding of these astrophysical sources by allowing reconstruction of their demographics and dynamical evolution, as well as discovery of new types of sources, including some that have been theorized but not yet detected by conventional means. Since the first detection of GWs by the Laser Interferometer GW Observatory (LIGO)/Virgo collaboration in 2015 (Abbott et al., 2016c), ground-based GW observations already had a remarkable impact on astrophysics. For instance, the gravitational-wave facilities LIGO and Virgo have observed the mergers of stellar BHs in the range ∼ 6-95 M ⊙ (Abbott et al., 2021), greatly expanding our knowledge of the mass spectrum of BHs. Recently, the first intermediate mass black holes (IMBH), with masses of ∼ 142 M ⊙ , has been discovered (GW190521, see Abbott et al., 2020c). The existence of stellar-mass BHs with masses higher than observed before, as well as the discovery of an IMBH, have fostered new exciting developments in theoretical models for the formation and evolution of stellar-origin black holes. The discovery of the double neutron star (NS+NS) merger GW170817 with accompanying electromagnetic observations (Abbott et al., 2017b) has had a great impact on our understanding of dense matter and the origin of heavy elements. These discoveries showcase the huge potential that gravitational wave astronomy has to revolutionize our understanding of astrophysical objects and processes. \nAt the lower frequencies in LISA's observing band, the stellar-mass systems, in binaries or multiples, provide a very rich source population. The population in the Milky Way is expected to consist of millions of double white dwarf (WD+WD) binaries, with a smaller population of neutron star (NS)/BH binaries, and possibly some of the heavy BHs that LIGO/Virgo have already detected. LISA observations of the BH populations will capture a snapshot of BH systems when their orbital periods are tens of minutes, a few years before their coalescence at the high frequencies observed by LIGO-Virgo. Overall, LISA observations of Galactic binaries will address many open questions in stellar astrophysics, such as the evolution of binary star systems, the origin of different transient phenomena, the origin of the elements and even the structure of the Galaxy. It should be noted that among the stellar-mass binaries in the Milky Way, a few are already known from electromagnetic observing campaigns, and can be used as LISA verification sources. While the vast majority of the stellar-mass binaries are expected to be too dim to be detected by electromagnetic instruments, there will be a substantive number that will be excellent targets for electromagnetic follow-up after LISA discovers them. \nThe observed BH mass spectrum spans ten orders of magnitude, ranging from a few M ⊙ for stellar-mass BHs to up to 10 11 M ⊙ for the most extreme MBHs. Many of the most massive MBHs, with M BH ≳ 10 8 M ⊙ , have been discovered in the high-redshift Universe, at z > 6 , powering some of the brightest quasars (Fan et al., 2003; Mortlock et al., 2011; Yang et al., 2020a). LISA will open up a wide discovery space for BHs. BH systems that merge at the millihertz frequencies, where LISA is most sensitive, are typically hosted in the most common type of galaxies, namely dwarf and massive spiral galaxies. The fact that LISA observations straddle the frequency bands of merging IMBHs and MBHs suggests that the potential impact on many fields of extragalactic astrophysics is huge. Such foreseen impact, however, relies heavily on our understanding of the astrophysical processes preceding and accompanying the evolution of the binaries during inspiral and into merger (De Rosa et al., 2019b). For MBHs, this knowledge is tightly entangled with the knowledge of the environments in which they evolve, namely their host galaxies and galactic nuclei. It follows then that LISA sources associated with MBH binaries cannot be understood without a robust knowledge of the landscape of galaxy formation and evolution, and in particular without a detailed knowledge of stellar dynamical processes and the interstellar medium inside galactic nuclei. There is thus an inherent multi-disciplinarity in the approach needed to understand these sources, which will naturally bring together various fields of galactic and extra-galactic astrophysics. Furthermore, since LISA will be able to detect MBH binary sources up to very high redshift ( z ∼ 10 -15 ), one also needs ancillary knowledge of cosmic structure formation, as galaxies, and thus their relevant environments, evolve significantly from high to low redshift (Woods et al., 2019). The endeavour then extends into cosmology, and hints at great possibilities for derivative knowledge, some already expected and others not, coming from the future discovery and characterization of LISA MBH binaries. \nThe stellar dynamics of the central cluster of stars at the galactic centre (the S-stars, or S0stars), provides compelling evidence for the existence of a MBH of mass ∼ 4 × 10 6 M ⊙ , Sgr A* (see for a review Genzel et al. 2010, and references therein). The stars in the centres of galaxies have the potential to interact with MBHs, but only if their pericentres are small enough. LISA will be able to observe the inspiral of a compact object such as a stellar-mass BH, a NS or a WD onto a (light) MBH, i.e., one with a mass between ∼ 10 4 M ⊙ and ∼ 10 7 M ⊙ . Because of the difference in mass between the MBH and the ≲ few -tens of solar masses of the compact object, we call these extreme mass-ratio inspirals (EMRIs)-where the mass ratio is 10 -8 ≲ q ≲ 10 -5 (AmaroSeoane et al., 2007). There is also a potential population of IMBHs with masses between 10 2 M ⊙ and 10 4 M ⊙ , which, through inspiral onto the central MBH, would generate GWs detectable by LISA, this being a class of sources dubbed intermediate mass-ratio inspirals (IMRIs) (AmaroSeoane et al., 2007). In principle, EMRIs and IMRIs could occur in the nuclei of any galaxy hosting a central MBH. They should be ubiquitous, since most galaxies host a central MBH and undergo a variety of merger events with other galaxies throughout their lives. IMRIs might \nalso occur outside galactic nuclei, for example in a star cluster cannibalizing its own population of compact objects. For EMRIs and IMRIs, astrophysical modelling of their origin are in their earliest theoretical stages; in recent years a number of new astrophysical scenarios have been proposed in which they could form even outside the conventional stellar-dynamical scenarios in the galactic centre or in star clusters. These scenarios have been, for the most part, detached from the notion that their host galaxies are highly dynamical systems with a diverse range of properties, at large scales as well as at the level of galactic nuclei and star clusters. From the astrophysical perspective, this is thus the least explored, albeit potentially most exciting, class of sources in the LISA band. An assessment of the current knowledge and upcoming developments in this area is of paramount importance, to propel new research on the astrophysical impact that the discovery of EMRIs and IMRIs by LISA can have. \nThe joint exploitation of LISA data with data from terrestrial GW detectors and electromagnetic observations across essentially all possible wavelengths, from infrared and radio to X-ray and gamma-rays, will further enhance its astrophysical impact (Mangiagli et al., 2020). Indeed, essentially all of LISA's individual sources have potential electromagnetic counterparts. Achieving a quantitative characterization of such counterparts, determining the feasibility of detecting them in one or more wavebands, and assessing the stage at which they would be detectable, relative to the inspiral and/or merger stage of the corresponding GW signal, are the main objectives ahead for current and upcoming research. An assessment of the current knowledge in this area is another important task. \nThe challenge to bring all these different pieces of knowledge into a coherent, robust picture within the next decade is huge, perhaps the most ambitious that the astrophysical community has ever faced. This review attempts to aid this ambitious, community-wide effort by assessing the status of knowledge in the modelling of LISA sources, and it summarizes our understanding of the astrophysical processes and environments relevant for the interpretation of the LISA data. Furthermore, it discusses the most important challenges ahead of us in the research of galactic binaries/multiples, massive and intermediate-mass black hole binaries, and EMRIs/IMRIs. Among these are the quest for identifying the different astrophysical formation channels for these various sources, including how these might be encoded in the LISA data stream, and the daunting multi-scale modelling needed to reconstruct the full dynamical history of such sources, from their emergence to the final inspiral phase and merger driven by GW radiation. The review material presented will help foster a critical discussion of the major gaps in our knowledge that need to be filled in the next decade, highlighting where disagreement exists between results, and what should be done next to reach beyond the current state of the art. This brings the discussion to important methodological tasks for the immediate future, from exploiting electromagnetic (EM) observations in the next decade, to improving simulation and semi-analytical techniques employed to build astrophysical models for the sources, and to refurbishing analysis and interpretation techniques for the models, for example by employing machine learning, neural networks and other modern inference strategies.", '1 Stellar Compact Binaries and Multiples': 'Section coordinators: Silvia Toonen, Tassos Fragos, Thomas Kupfer, Thomas Tauris', 'Contributors: Silvia Toonen, Tassos Fragos, Thomas Kupfer, Thomas Tauris': "Detection of GW emission from binary compact stars is one of the key drivers for the LISA mission. There are already at the time of writing (July 2022) about two dozen known Galactic sources, most of which are guaranteed to be detectable with LISA within a few years of its operation (Section 1.2). These are tight binaries (typically with orbital periods of P orb ≃ 5 -30 min ) of WD+WDs which give rise to continuous emission of GWs. Unlike binaries consisting of NSs and BHs, WD binaries (with their larger radii and thus lower orbital frequencies at merger) are not readily detectable by ground-based high-frequency (Hz-kHz) GW observatories, such as LIGO/Virgo/KAGRA, nor by the planned third generation of such detectors. These high-frequency detectors can observe the final a few - a few thousands orbits of inspiral (lasting a fraction of a second - minutes) and the merger event itself for NSs and BHs. Such merger events, however, are rare (of order a dozen events Myr -1 for a Milky Way equivalent galaxy) and therefore they are only anticipated to be detected as extra-galactic sources, across volumes that encompass large numbers of galaxies. A major advantage of LISA is that the inspiral phase (due to orbital GW damping in the compact binaries) of the vast population of tight Galactic double WDs, NSs and BHs is in the low-frequency ( ∼ mHz) GW window for up to ∼ 10 6 yr prior to their merger event. Thus a significant number of such local sources are anticipated to be detected by LISA, even though their emitted GW luminosity is relatively small compared to that of the final merger process. The possibility that LISA can measure sky locations of its sources will allow for EM follow-up observations which may result in much more precise compact object component masses, e.g. compared to high-frequency GW mergers. \nBinary population synthesis studies and early data-analysis work predicts of order 10 4 resolved Galactic WD+WD may be detected with LISA. This population includes both detached WD+WD and those undergoing mass transfer (the so-called AM Canum Venaticorum binaries or AM CVns, see Section 1.2.3.1). NS+NS systems are also expected to be detected by LISA. Based on the known Galactic population of tight-orbit radio pulsar binaries in combination with population synthesis predictions, an estimated number of 10 1 -10 2 NS+NS systems with a significant signal-to-noise ratio (SNR) may be detected by LISA within a 4-year mission (Section 1.2.2.3). An even larger number of NS+WD systems is expected to be detected too, including ultra-compact X-ray binaries (UCXBs, see Section 1.2.3.2, a sub-class of low-mass X-ray binaries, LMXBs). Binary BHs (BH+BH) detectable by LISA are strong candidates to become the first discoveries of such systems in the Milky Way. Given that LISA's volume sensitivity for a constant SNR scales with chirp mass to the fifth power, M 5 chirp , BH+BH sources may be detected in distant galaxies, located several hundreds of megaparsecs away (see examples in Fig. 1). Interestingly enough, this fortuitous condition will therefore allow LISA to discover extra-galactic BH+BHs several years before the final merger events that LIGO/Virgo/KAGRA or Einstein Telescope/Cosmic Explorer will detect. Finally, LISA is also expected to detect rare Galactic systems such as (see Section 1.2.4): triple stellar systems, tight systems of WDs with exoplanets, or helium star binaries. \nThe LISA mission will provide opportunities to learn new physics and answer key scientific questions related to formation and evolutionary processes of tight binary and multiple stellar systems containing compact objects. This includes questions related to the stability and efficiency of mass transfer, common envelopes (CEs), tides and stellar angular momentum transport, irradiation effects, as well as details of their formation and destruction in core-collapse supernovae (SNe) and Type Ia SNe (and related transients), respectively. Furthermore, information about the environments of these sources will be available too, and the number and Galactic distribution of LISA sources are excellent probes to gain new knowledge on the star formation history and the structure of the Milky Way. Finally, the sheer numbers of LISA sources will provide crucial knowledge concerning their formation and evolution processes and help to place constraints on \nkey physical parameters related to binary (and triple-star) interactions. \nThe current catalogue of known LISA 'guaranteed sources' consists of detached WD+WDs, accreting AM CVn binaries, a hot subdwarf binary, and an UCXB. Although the sample is still small and inhomogeneous, binary population synthesis predicts a large population of multimessenger sources that are EM bright and also detectable by LISA. This includes up to a few thousand detectable WD+WDs as well as a few tens of NS or BH binaries, with a population strongly peaking towards the Galactic Plane/Bulge. Many sources will be detected across different EM bands. Detached WD+WDs and NS+WDs are typically seen in optical and UV bands, whereas AM CVn systems and UCXBs are also seen in X-rays. NSs in compact binaries can potentially be detected as pulsars in the radio band. Therefore, in parallel with the LISA mission, we expect an EM bright future of thousands of resolved Galactic LISA binaries. \nSystems with orbital periods < 20 min will be the strongest Galactic LISA sources and will be detected by LISA within weeks after science operations begin. These verification binaries, as well as other so far unknown loud sources, are crucial in facilitating the functional tests of the instrument and maximize LISAs scientific output. Combined GW and EM multi-messenger studies of UCXBs will allow us to derive population properties of these systems with unprecedented quality including for the first time the effects of tides compared to GW radiation. Tides are predicted to contribute up to 10 % of the orbital decay. For accreting WDs as well as NS binaries, multi-messenger observations give us the possibility to study the angular momentum transport due to mass transfer. In particular for monochromatic GW sources, EM observations are required to break degeneracies in the GW data (e.g. between masses and distance). \nFigure 1: Distance to which LISA binaries can be detected as a function of GW frequency. The coloured lines represent the SNR threshold of 7 (here computed assuming a mission duration of 4 yr with 100% duty cycle) for (quasi-)stationary equal-mass circular binaries of different total masses in the distance-GW frequency parameter space. The shaded range represents angleaveraged curve limits for the optimal and worst binary orientation. The ticks on the curves represent binary merger times: for merger times ≫ 4 yr the binary will be seen by LISA as a monochromatic GW source, whereas for merger times < 4 yr the binary will be seen as evolving. Note in particular that evolving sources like GW190521 and GW150914 remain within the LISA band for less than the mission lifetime. Figure credit: Antoine Klein & Valeriya Korol. \n<!-- image -->", 'Contributors: Thomas Kupfer, Thomas Tauris, Silvia Toonen, Tassos Fragos': 'The most abundant sources in the LISA band will be binary stars with orbital periods < 60 min, so-called ultra-compact binaries (UCBs). They are a class of binary stars with ultrashort orbital periods, consisting of a WD or NS primary and a compact helium-star/WD/NS secondary. A subset of the known UCBs have predicted GW strains high enough that they will be individually detected due to their strong GW signals (e.g. Burdge et al., 2020b). These LISA guaranteed sources are termed verification binaries with some being expected to be detected on a timescale of weeks or a few months (Stroeer and Vecchio, 2006). Currently, we know of only about two dozen of these systems although hundreds are predicted by theory to be detectable in our Galaxy (e.g. Nelemans et al., 2004b; Timpano et al., 2006; Littenberg et al., 2013; Korol et al., 2017; Kremer et al., 2017; Kupfer et al., 2018; Lamberts et al., 2019). \nAt present, the catalogue of verification binaries include 13 WD+WDs, 11 semi-detached accreting WDs (AMCVn binaries, a subclass of Cataclysmic Variables, CVs), one hot subdwarf star with a WD companion, and one semi-detached UCXB. Table 2 and 3 present an overview of the known systems with observed EM properties. Figure 2 shows the characteristic strain of the known verification binaries which reach a predicted SNR ≥ 5 in LISA assuming an optimistic 10 year mission with an 80% duty cycle. So far large-scale searches for verification binaries have been conducted almost exclusively in the northern hemisphere, because large-scale survey instruments (e.g. the Sloan Digital Sky Survey, SDSS, and the Zwicky Transient Facility, ZTF) are located in the Northern Hemisphere, and mostly at high Galactic latitudes, to avoid stellar crowding. Fig. 3 shows the sky location of the known verification systems which presents the strong bias towards sources in the Northern Hemisphere. \nIn 2018, Gaia data release 2 (Gaia Collaboration et al., 2018a) announced parallaxes for ≈ 1 . 3 billion sources. The Gaia catalogue contains the distances of many of the known LISA verification binaries, allowing accurate prediction of their GW strains. Using the Gaia distances, Kupfer et al. (2018) found 13 sources will exceed an SNR of 5 after 4 yr of LISA observations. This sample consists of 13 verification binaries from the current, known list; it is strongly biased and incomplete. It includes AM CVn, CR Boo, V803 Cen and ES Cet, which were all found as outliers in surveys for blue, high-Galactic latitude stars. HM Cnc and V407 Vul are the most compact known AMCVn systems and were discovered during the course of the ROSAT All-Sky Survey showing an on/off X-ray profile modulated on a period of 321 and 569 s respectively. The known WD+WD verification binaries, such as SDSS J0651 and SDSS J0935, were found as part of the extremely low-mass (ELM) WD survey (Brown et al. 2020b and references therein). \nMore recently, more systematic searches for UCBs were performed. UCBs show up in lightcurves with variations on timescales of the orbital period (e.g. due to eclipses or tidal deformation of the components). Therefore, photometric surveys are well suited to identify UCBs in a homogeneous way. A number of fast cadence ground-based surveys, including the Rapid Temporal Survey (RATS; Ramsay and Hakala 2005; Barclay et al. 2011), OmegaWhite (Macfarlane et al. 2015) survey as well as the ZTF high-cadence Galactic plane survey (Masci et al., 2019; Kupfer et al., 2021), have been executed to study the variable sky down to a few minute period aiming to find UCBs and increase the number of known verification binaries. The ELM survey targets a colour-selected sample of B-type hypervelocity candidates from SDSS (Anderson et al., 2005; Roelofs et al., 2007c), which are being followed up systematically (Brown et al. 2020b and references therein). ELM WDs can be separated efficiently from the bulk of WDs with a colour selection (Brown et al., 2010). \nOver the last few years the number of known verification binaries has almost doubled thanks \n2 \nFigure 2: Sensitivity plot for LISA assuming 10 yr of observation with an 80% duty cycle showing the known binaries which reach a SNR ≥ 5 . Filled symbols represent eclipsing sources and open symbols represent non-eclipsing sources from Kupfer et al. (2018). The black lack solid line represents the LISA sensitivity curve. Acronyms for binaries: AM Canum Venaticorum (AM CVns), WD+WD (DWDs), subdwarf B-star (sdB) and ultracompact X-ray binary (UCXB). Figure credit: Thomas Kupfer. \n<!-- image --> \nto these large scale surveys. The two most significant contributors were the ELM survey (Brown et al. 2020b and references therein) and ZTF (Burdge et al., 2019a, 2020b,a). The ELM survey discovered six WD+WD verification binaries including SDSS J0651: a detached eclipsing system with an orbital period of 12 min. Most recently ZTF released seven new WD+WD verification binaries, five systems found as eclipsing sources. Remarkably, one of the first ZTF discoveries was the shortest orbital period eclipsing WD+WD known to date, ZTF J1539+5027, with an orbital period of just 6.91 min (Burdge et al., 2019a). \nFigure 3: Sky position of the verification binaries. The sky positions show a clear bias towards the northern hemisphere and to higher Galactic latitudes. The black line indicates the Galactic equator and | b | = 10 deg, with the Galactic Centre located at the black cross. See caption in Fig. 2 for explanation of acronyms. Figure credit: Thomas Kupfer. \n<!-- image --> \nTable 2: Physical properties (orbital periods, component masses, inclination angles) of the known verification binaries which reach a SNR > 5 after a 10 yr LISA mission with 80% duty cycle. Masses and inclination angles in brackets are assumed and based on evolutionary stage and mass ratio estimations. \n[1] Strohmayer (2005), [2] Roelofs et al. (2010), [3] Marsh and Steeghs (2002), [4] Espaillat et al. (2005), [5] Green et al. (2018a), [6] Skillman et al. (1999), [7] Roelofs et al. (2006), [8] Fontaine et al. (2011), [9] Kupfer et al. (2015), [10] Roelofs et al. (2007b), [11] Solanki et al. (2021), [12] Levitan et al. (2014), [13] Wevers et al. (2016), [14] Provencal et al. (1997), [15] Burdge et al. (2019a), [16] Burdge et al. (2020a), [17] Brown et al. (2011), [18] Hermes et al. (2012), [19] Burdge et al. (2020b), [20] Brown et al. (2016), [21] Kilic et al. (2014), [22] Brown et al. (2020a), [23] Burdge et al. (2019b), [24] Kilic et al. (2011), [25] Breedt et al. (2017), [26] Kilic et al. (2017), [27] Brown et al. (2010), [28] Geier et al. (2013), [29] Stella (1987), [30] Chen et al. (2020a) \nTable 3: Measured EM properties (Galactic coordinates, GW frequency, magnitudes and parallaxes from Gaia early data release 3 (Gaia Collaboration et al., 2020) of the known verification binaries which reach a SNR > 5 after a 10 yr LISA mission with 80% duty cycle.', '1.2.2 Detached binaries': 'Coordinators: Ashley Ruiter, Ross Church Contributors: Ashley Ruiter (1.2.2.1), Thomas Tauris (1.2.2.2-4), Jeff Andrews (1.2.2.3), Simone Bavera (1.2.2.4), Ross Church (1.2.2.2), Tassos Fragos (1.2.2.4), Gijs Nelemans (1.2.2.1), Milton Ruiz (1.2.2.3), Alberto Sesana (1.2.2.4), Antonios Tsokaros (1.2.2.3), Shenghua Yu (1.2.2.3)', '1.2.2.1 WD+WD systems': "For over three decades it has been known that WD+WD binaries will be the dominant contributor to signals detected by a space-based GW observatory (Hils et al., 1990). While most extra-galactic sources (Farmer and Phinney, 2003) as well as a significant fraction of those in the Galactic halo (Rosswog et al., 2009a) are likely too distant to be individually detected by LISA, a large portion of the frequency band observed by LISA will be swamped with GWs from millions of WD+WDs existing in the Galactic disc and bulge. At low frequencies, the combined signal of these millions of WD+WDs will populate just a few frequency bins and merge to form an unresolved confusion foreground (often referred to as the galactic foreground or the galactic confusion noise), with louder resolvable sources standing out above the confusion (see also Section 1.6.2). Together with high-frequency sources a large number these form ∼ 10 4 , (e. g., Nelemans et al. 2001c; Farmer and Phinney 2003; Ruiter et al. 2010; Korol et al. 2017) of resolved WD+WDs and we now discuss these key sources in more detail. \nWD+WDs were discovered in the late 1980s and initially were dominated by low-mass ( ≲ 0 . 4 M ⊙ ) helium-core (He-core) WDs that cannot be formed in single-star evolution within a Hubble time, and, thus, were the targets for radial velocity searches for binarity amongst known WDs (Marsh et al., 1995). Later, (more) unbiased surveys were done, e.g. the Supernova Ia Progenitor surveY (SPY, Napiwotzki et al., 2020, and references therein) and studies using SDSS, discovering also more massive WDs. Over the last decade, it has become more clear that previously-undetected WD+WD systems (and their progenitors, e.g. double-core planetary nebulae) are more easily detectable with today's sophisticated instrumentation (Wesson et al., 2018). A dramatic increase in the number of WD+WDs has come from the ELM WD survey (Kilic et al., 2012; Bell et al., 2017; Brown et al., 2020b) that targets a part of the parameter space in colour-colour diagrams that is occupied by (subdwarf) B-stars, but also by very low mass (below ∼ 0 . 3 M ⊙ ) proto-WDs that are still approaching the cooling track and are thus relatively large and bright (Istrate et al., 2014b, 2016). In total, the ELM survey alone has discovered 98 WD+WDs so far (Brown et al., 2020b). \nOver the past couple of years, ZTF (Bellm et al., 2019, [) has facilitated a rapid growth in the population of known WD+WDs with orbital periods under an hour. Three of the sources discovered by ZTF so far (Burdge et al., 2020a), the eclipsing WD+WDs: ZTF J1539+5027 ( P b = 6 . 91 min ), ZTF J2243+5242 ( P b = 8 . 80 min ) and ZTF J0538+1953 ( P b = 14 . 4 min ), should all be detected by LISA with a high SNR, enabling precise parameter estimation using GWs (Littenberg and Cornish, 2019). \nThe detached WD+WDs may consist of a pair of He-core WDs, carbon/oxygen-core (C/Ocore) WDs, oxygen/neon/magnesium-core (O/Ne/Mg-core) WDs, or any mixed combination thereof. For some systems, LISA measurements of the orbital-decay rate will yield the chirp mass for a given system, which can be combined with EM observations to reveal individual WD component masses. The distribution of WD masses (and their mass ratios), along with the number of detectable sources in a local volume, will provide important information to help understand their formation history (see Section 1.3). With enough detached WD+WDs in a sample, it may even be possible to set limits on mass-transfer efficiencies and CE physics (Section 1.7) through characterisation of chirp mass distributions (Ruiter et al., 2019). Furthermore, the detected WD+WDs will provide unique information on the formation of progenitors of R Coronae Borealis stars, thought to be formed by the merger of two WDs, e.g. (Tisserand et al., 2020), \nmassive carbon-enhanced WDs (Kawka et al., 2020), Type Ia SNe (see Section 1.7.1.6), and other transients. \nOver the last three decades, several works have made predictions about the scientific impact of LISA detections of Galactic WD+WDs. Different binary evolution population synthesis studies have uncovered how the WD+WD population will look to LISA in terms of characteristic strain amplitude (Nelemans et al., 2001b, 2004b; Yu and Jeffery, 2010; Lamberts et al., 2019; Korol et al., 2020), spectral density (Breivik et al., 2020b), as well as how different populations of WD+WDs contribute to the spectral amplitude signal (Rosswog et al., 2009a; Ruiter et al., 2010). \nDetached WD+WD binaries that are resolvable with LISA are expected to be on par with or slightly outnumber the resolvable interacting WD+WD binaries (Nelemans et al., 2001a, 2004b; Ruiter et al., 2010), and will be the sole WD+WD contributers to the LISA signal at GW frequencies below ∼ 2 × 10 -4 Hz . Kremer et al. (2017) found that a number of mass-transferring WD+WDs (Section 1.2.3) will be resolvable with LISA ( ∼ 200 -3000 for SNRs between 10 and 5, respectively), many of which are likely to exhibit a negative chirp (caused by orbital widening) - a diagnostic not applicable for detached WD+WDs. Finally, we expect that the number of detached WD+WDs, composed of a light He-core WD and a more massive C/O-core WD (or possibly an O/Ne/Mg-core WD) detected by LISA must be in accordance with the number of similar interacting AM CVn systems that LISA will detect, given their evolutionary connection (the detached systems being the precursors of the interacting WD+WDs, see Fig. 6). The transitional GW frequency between these two populations (detached and interacting) depends on the mass and temperature of the lighter (last-formed) WD, see examples in Figs. 4 and 5.", '1.2.2.2 NS+WD and BH+WD systems': 'The known population of Galactic NS+WD systems can be divided into two classes. Systems with: i) massive WDs (O/Ne/Mg-core or C/O-core WDs, typically more massive than 0 . 7 M ⊙ ), and ii) low-mass He-core WDs (typically less massive than 0 . 3 M ⊙ ). The massive NS+WD systems can again be subdivided into two populations, depending on the formation order of the WD and the NS. The NS+WD systems are observed as binary radio pulsars and the formation order can be clearly distinguished from the properties of the pulsar: if the pulsar has a strong B-field and an eccentric orbit (e.g. Tauris and Sennels, 2000; Church et al., 2006), it is the lastformed compact object, whereas if the pulsar is (mildly) recycled with a low-B-field and a fairly rapid spin, and in a near-circular orbit, it is the first-formed compact object (Tauris et al., 2012). For LISA detections, the formation order is irrelevant and among both types of systems examples are known to merge within a Hubble time, thus producing a bright LISA source well before their final merger. \nAmong the detached low-mass He-core WDs with NS companions, the systems in relatively tight orbits are completely dominated by millisecond radio pulsars. According to the ATNF Pulsar Catalogue (Manchester et al., 2005), there are about 120 such systems known in the MW disc, a handful of which will merge within a Hubble time, producing a bright detached LISA source (depending on their distance) for approximately the last several tens of Myr of the inspiral, before an UCXB is formed (Section 1.2.3.2). Based on the observed population of radio pulsars and their selection effects, Tauris (2018) argue that LISA could detect about 50 of these systems while still detached, before they become UCXBs and widen their orbits again, resulting in a negative chirp of the GW signal (see Figs. 4 and 5). \nAt present, we do not know of any detached BH+WD binaries. However, this is probably due to observational selection effects since the only EM radiation we would expect from such detached systems would be from the cooling of the WD companion - unlike the situation for semi-detached systems or systems containing NSs, which can be detected in X-rays and radio waves, respectively. Nevertheless, several Galactic LMXBs are known with low-mass donor stars and BH accretors (McClintock and Remillard, 2006) and thus we expect many of these systems \nto leave detached BH+WD systems, possibly (although still to be proven) in tight orbits that LISA will detect. A more viable formation channel for more massive WDs in tight orbits with BHs is formation via a CE. Optical follow-up observations of the WD companion, in combination with the measured chirp mass, will constrain the BH mass in these systems. Early simulations (Nelemans et al., 2001b) predict a Galactic merger rate of BH+WD binaries of order ∼ 100 Myr -1 and thus roughly ∼ 100 such systems detectable by LISA.', '1.2.2.3 NS+NS systems': "The known population NS+NS systems so far only manifest themselves as radio pulsars. The first one of these (PSR B1913 + 16, the Hulse-Taylor Pulsar ) was discovered in 1974 (Hulse and Taylor, 1975). According to the ATNF Pulsar Catalogue (Manchester et al., 2005), there are currently about 20 NS+NS systems detected in our Galaxy. Except for one case, the double pulsar PSR J0737 -3039 (Lyne et al., 2004), only one of the two NSs is detected - usually the recycled pulsar (Tauris et al., 2017). The other NS, is either not an active radio pulsar anymore or it is not beaming in our direction. \nGiven the small merger rate of NS+NS systems in our Galaxy 1 (most likely somewhere in the range from a few events up to a hundred events per Myr), it is statistically highly improbable that ground-based high-frequency detectors (LIGO-Virgo-KAGRA) will detect a NS+NS merger in the Local Group earlier than the LISA era. The advantage of LISA is that it can follow the inspiral of Galactic NS+NS systems up to ∼ 10 6 yr prior to their merger event, and thus a significant number of NS+NS sources are anticipated to be detected in GWs by LISA. \nAbout half of the 20 known NS+NS systems have orbital periods small enough (or eccentricities sufficiently large) to merge within a Hubble time. As an example, a 'standard' NS+NS system with NS masses of 1 . 35 M ⊙ and e.g. an orbital period of 16 hr will merge in: 11.8 Gyr, 4.4 Gyr or 0.35 Gyr for an initial eccentricity, e 0 of 0.1, 0.5 or 0.8, respectively. The number of NS+NS sources that LISA will detect can be evaluated, approximately to first order, from a combination of the Galactic NS+NS merger rate and the distribution of these sources within the Milky Way. The above three standard NS+NS systems will have a remaining lifetime of between ∼ 247 kyr ( ∼ 243 kyr for e 0 = 0 . 8 ) and 1 . 57 Myr ( 1 . 48 Myr for e 0 = 0 . 8 ) by the time they enter the LISA band, if this occurs at a GW frequency of about 2 mHz and 1 mHz, respectively. Thus, if the Galactic merger rate is, say, 10 Myr -1 , we can roughly expect to detect between a few and a dozen LISA sources. Of course, the details depend on the Galactic distribution of these sources, the SNR required for a detection, and the duration of the LISA mission. The merger rate can be estimated from population synthesis, but its value is uncertain by, at least, one or two orders of magnitude (Abadie et al., 2010). The merger rate derived from an extrapolation of the LIGO/Virgo empirical merger rate of NS+NSs still has very large error bars due to small number statistics. \nRecent works by Lau et al. (2020); Andrews et al. (2020) suggest that LISA may even detect up to ∼ 50 -200 Galactic NS+NS sources with a SNR greater than 7 within a 4 yr mission. Given that LISA's volume sensitivity for a constant SNR scales with M 5 chirp , unlike double BH sources, very few NS+NS sources are anticipated to be detected outside the Milky Way, although a few such binaries may be found in both the LMC and M31 (Seto, 2019). Applying a more conservative number, however, for the merger rate of Galactic NS+NS system of about 3 -14 Myr -1 (Kruckow et al., 2018) would lead to a substantial reduction in the predicted number of LISA detections. The Galactic merger rate is expected to be significantly better constrained in the coming decade such that we will have a clear idea about the expected number of NS+NS sources detected by LISA prior to its operation. Finally, an expected reduction in LISA SNR for detecting eccentric NS+NS systems, compared to circular NS+NS systems with similar orbital period and NS masses, should be noticed (Randall et al., 2021). \nLISA may give us the opportunity to probe a hidden subpopulation of NS+NS systems with different properties compared to those of the well-known radio pulsar NS+NSs. The nature of GW190425, a presumed NS+NS merger detected by the LIGO/Virgo network with a total mass of 3 . 4 M ⊙ (Abbott et al., 2020a), is still a mystery. With such a large total mass, GW190425 stands at five standard deviations away from the total mass distribution of Galactic NS+NSs detected in the Milky Way as radio pulsars (Farrow et al., 2019). If a subpopulation of heavy GW190425-like NS+NSs exists in our Galaxy, it is not yet clear why it should be radio-quiet (e.g. Safarzadeh et al., 2020b). Thus, LISA may actually be the most suited instrument for detecting the population of GW190425-like binaries (Galaudage et al., 2021; Korol and Safarzadeh, 2021). In particular, Korol and Safarzadeh (2021) demonstrated that if GW190425-like binaries constitute a fraction larger than 10% of the total Galactic population, LISA should be able to recover this fraction with better than ∼ 15 % accuracy, assuming the merger rate of 42 Myr -1 . \nAdditional recent investigations (Thrane et al., 2020; Kyutoku et al., 2019) have discussed the importance of sky-localization on LISA NS+NS sources for multi-messenger follow-ups that may allow to impose constraints on the equation-of-state of NSs by measuring the Lense-Thirring precession (Thrane et al., 2020) or test general relativity through the detection of radio pulses from Galactic NS+NS binaries in a very tight orbit with the period shorter than 10 min (Kyutoku et al., 2019). Sky-localization may also help disentangle NS+NS systems from others sources by either knowing their position in the Milky Way, or in nearby galaxies, thus enhancing the possibility of EM follow-ups (e.g. Lau et al., 2020). In particular, in addition to differences in chirp masses, it will allow us to distinguish between eccentric NS+NS and eccentric WD+WD systems - the latter only expected to be formed in globular clusters and ejected into the Galactic halo via dynamical interactions, while the former systems have an eccentricity encoded from the last SN explosion.", '1.2.2.4 BH+NS and BH+BH systems': 'For several decades, a number of high-mass X-ray binaries (HMXBs) containing BH accretors have been identified in the Milky Way and nearby galaxies (e.g. Cyg X-1, LMC X-1, LMC X-3, MCW656, M33 X-7, see van den Heuvel, 2019). It has been shown that a fraction of these known wind-accreting HMXBs may eventually form BH+BHs or BH+NS systems (Belczynski et al., 2012, 2013, see also Fig. 9), while others will merge in an upcoming CE phase (Section 1.7.1.3), once the companion star evolves to a giant-star size and possibly initiates dynamically unstable Roche-lobe overflow (RLO), depending on its stellar structure and the mass ratio between the two binary components. The masses of compact objects in X-ray binaries can be estimated with astrometry and the Galactic stellar-mass BHs are found to have masses between roughly 5 -21 M ⊙ (Gandhi et al., 2019; Arnason et al., 2021; van den Heuvel, 2019; Miller-Jones et al., 2021). The astrometric satellite Gaia, can also be used to detect optical emission from the HMXB companion stars (Barstow et al., 2014; Kawanaka et al., 2017; Mashian and Loeb, 2017; Breivik et al., 2017; Yamaguchi et al., 2018). Finally, non-interacting binaries with a BH component have also been discovered by combining radial velocity measurements with photometric variability data (Breivik et al., 2017; Thompson et al., 2019; Liu et al., 2019a), although their interpretations can, in some cases, be subject to alternative explanations (van den Heuvel and Tauris, 2020). \nThe LIGO-Virgo GW detectors have detected BH+BH mergers in distant galaxies, out to ∼ 5 Gpc . (Abbott et al., 2021). The inferred BH masses 2 of their inspiralling BH components potentially range all the way from ∼ 2 . 6 ± 0 . 1 M ⊙ (Abbott et al., 2020b) to ∼ 95 ± 10 M ⊙ (Abbott et al., 2021), thus significantly more massive than the known Galactic stellar-mass BHs. This difference is mainly attributed to the relatively high metallicity content of the Galaxy (Belczynski et al., 2016a; Kruckow et al., 2018). The LIGO-Virgo GW detectors have also identified two sources in close proximity or within the mass ranges expected for BH+NS binaries: GW190426 and GW190814. The first event has marginal significance (i.e. a high false-alarm \nrate, FAR = 1 . 4 yr -1 ) and the second is likely to be a BH+BH, not a BH+NS. Nevertheless, LISA like LIGO is much more sensitive to the masses (as opposed to matter content) of the binaries, so the presence of similar binaries suggests LISA will copiously find similar sources. \nInterestingly enough, although the LIGO-Virgo GW sources are located in distant galaxies at Gpc distances, their low-frequency GWs during the last few years of inspiral prior to the merger event is often so luminous that it allows for detection with LISA (Sesana, 2016). For example, the very first GW source (GW150914) would have been observable by LISA several years before its merger (see Fig. 4). Similarly, the extreme event GW190521 (Abbott et al., 2020c; Toubiana et al., 2020b), with a total stellar mass of ∼ 160 M ⊙ and located at a distance of about 5 Gpc, would also have been detected during its inspiral in the LISA band.', '1.2.2.5 Stochastic background': 'As discussed above, stellar-mass compact binaries (BH+BH, BH+NS, NS+NS) are one of the primary targets for LISA, with expected detection rates of between a few and a few thousands per year, as summarized in Sec 1.4. Nevertheless, many of these sources will not be detected, either because they are too distant and thus have a low SNR, or because the signals from multiple long-lived sources will overlap in time and will be difficult to disentangle. These unresolved signals will combine incoherently and produce a stochastic GW background (SGWB). Current predictions of the amplitude of this background typically rely on the merger rates measured in the local Universe by LIGO-Virgo (Abbott et al., 2016a, 2018b), but since most of the unresolved sources reside at higher redshifts, these predictions depend on the detailed population synthesis and galaxy evolution models. \nThe expected amplitude of the SGWB from BH+BH binaries in the LISA band (without source subtraction) varies in the range Ω GW ( f ) ∼ 10 -13 -10 -11 at f = 3 mHz (Sesana, 2016; Dvorkin et al., 2016a; Cusin et al., 2020; Périgois et al., 2021). Up to a few thousands of these binaries (likely responsible for about 10% of the background) will be individually detected by LISA (Sesana, 2016; Périgois et al., 2021). The prediction for the background from NS+NS is significantly lower at Ω GW ( f ) ∼ 10 -14 at f = 3 mHz (Périgois et al., 2021), and the background from BH+NS in the model of Périgois et al. (2021) is slightly higher but similar to that of NS+NS. The uncertainties on these rates will be significantly reduced in the coming years with more detections of stellar-mass binaries by ground-based detectors, and improved modelling of source formation and evolution. Detection of this background and in particular its shape, will provide important information about the population at periods too short to be directly observed, but before the merger phase probed by ground-based detectors (see Section 1.5.2.1).', '1.2.3 Interacting binaries': 'Contributors: Ashley Ruiter(1.2.3.1), Thomas Kupfer(1.2.3.1), Thomas Tauris(1.2.3.1- \nCoordinators: Shenghua Yu, Thomas Tauris 2), Gijs Nelemans (1.2.3.1), Shenghua Yu (1.2.3.2)', '1.2.3.1 AM CVn binaries (AM Canum Venaticorum binaries - accreting WDs)': "AM CVn binaries consist of a WD accreting from a hydrogen-deficient star (or WD) companion (Warner, 1995; Solheim, 2010). In their formation history (Fig. 6 and Section 1.3.1.1), AM CVns form after at least one CE phase of their progenitor system. The current RLO is initiated, due to orbital damping caused by GW radiation, at orbital periods of typically 5 -20 min (depending on the nature and the temperature of the companion star), and the mass-transfer rate is determined by a competition between orbital angular momentum loss through emission of GWs and orbital widening due to RLO from the less-massive donor star to the more-massive WD accretor. \nFigure 4: Characteristic strain amplitude vs GW frequency for LISA. Evolutionary tracks are for an UCXB (blue) and an AM CVn system (magenta) at a distance of d L = 1 kpc . Their slope on the inspiral leg is ∝ f 7 / 6 gw . The stars along the tracks represent (with increasing GW frequency) onset LMXB/CV stage, termination LMXB/CV stage, and onset UCXB/AM CVn stage. The evolutionary timescales along these tracks are shown in Fig. 5. The LISA sensitivity curve (red line, SNR = 1 ) is based on four years of observations. The grey curves are for the UCXB at d L = 15 kpc and 780 kpc (M31), respectively. Comparison tracks are shown for an MBH merger (green) and GW150914 (orange). Their inspiral slopes are ∝ f -1 / 6 gw . Data from LISA verification sources (Kupfer et al., 2018) include detached double WD binaries (solid squares), AM CVn systems (open circles), and a hot subdwarf binary (solid triangle). Figure from Tauris (2018). \n<!-- image --> \nIf the system survives the onset of the semi-detached phase, a stable accreting AM CVn binary is formed in which the orbital separation widens shortly after onset of RLO (Fig. 5), and the system evolves to longer orbital periods (see Fig. 5). When they reach an orbital period of ∼ 60 min . (after a few Gyr), the donor star has been stripped down to about 5 Jupiter masses ( 5 M J , Tauris, 2018). These systems have been hypothesised to be possible progenitors of faint thermonuclear explosions (flashes, or Type '.Ia' SNe, Nelemans et al., 2001a; Bildsten et al., 2007). \nFigure 4 shows examples of computed evolutionary tracks of AM CVn (and UCXB) systems in the characteristic strain amplitude vs GW frequency diagram. As can be seen, AM CVn systems are indeed anticipated to be detected by LISA - in some cases even with a SNR > 100 (for the sources located within 1 kpc). \nThough there are currently ∼ 65 AM CVn binaries known in the Galaxy (Ramsay et al., 2018), their formation pathways are still a puzzle (Green et al., 2018b). Their compact orbits and the lack of hydrogen in their spectra, led to three different proposed formation channels: i) the donor is a low-mass (likely He-core) WD (Paczyński, 1967); ii) the donor is a semi-degenerate hydrogen-stripped, helium-burning star (e.g. main-sequence helium star, or hot subdwarf); or iii) the donor is a helium-rich core of a main-sequence star that has not undergone helium-burning \nsince it had a rather low mass to begin with (see Solheim, 2010). Further discussions on their formation is given in Section 1.3.1.1. \nOnly for eclipsing AM CVns is it possible to fully determine all binary parameters and put constraints on the donor type. Recent results from eclipsing systems revealed that the donor stars are likely larger and more massive than previously assumed (Copperwheat et al., 2011; Ramsay et al., 2018), implying that a semi-degenerate donor is more likely for such systems, unless the donor star is a low-mass He-core WD which can remain bloated on a Gyr timescale (Istrate et al., 2014b). If that is the norm rather than the exception, it will lead to more AM CVn GW sources than previously predicted. \nBased on binary population synthesis, Nelemans et al. (2001a) predicted a space density of AM CVn stars in a range of 0 . 4 -1 . 7 × 10 -4 pc -3 and a number of resolvable AM CVn systems for LISA roughly equal to the number of detached WD+WDs (Nelemans et al., 2004b). More recently, Kremer et al. (2017) predicts that ∼ 2700 systems will be observable by LISA with a negative chirp of 0 . 1 yr -2 (i.e. ˙ f gw < 0 , resulting from orbital expansion due to mass transfer, see Figs. 4 and 5). Until very recently, when ZTF reported a large number of eclipsing WD+WDs (Burdge et al., 2019a, 2020a), the majority of known LISA verification binaries was dominated by AM CVn systems (Roelofs et al., 2007a, 2010; Kupfer et al., 2015; Green et al., 2018a; Kupfer et al., 2018). However, observational space density estimates from SDSS data are in strong disagreement with theoretical predictions from these binary population studies. Roelofs et al. (2007c); Carter et al. (2013) derived an observed space density about an order of magnitude below the prediction by Nelemans et al. (2001a); Kremer et al. (2017) which would result in only ≲ 1000 resolvable systems in the LISA band. The discrepancy could be real with the population synthesis predicting too many systems, related to assumptions in binary evolution physics (especially the treatment of mass transfer in close binaries), and/or possibly because some of the systems that are predicted to evolve into AM CVn binaries in fact merge in a CE shortly after the less-massive star fills its Roche lobe. On the other hand, it could also be that AM CVn stars are more difficult to find than expected (when not in outburst) or that they are distributed with relative high concentration in the thick disc (Nissanke et al., 2012). Ramsay et al. (2018) argued, based on Gaia data release 2 parallaxes, that a significant fraction of AM CVn systems, even within 100 pc, could still be undiscovered. Future transient sky surveys, such as LSST using the Vera C. Rubin Observatory, could have great success in detecting short-period binary systems with the implementation of appropriate cadence intervals (e.g. very short, ∼ 15 s sub-exposures). Indeed already some AM CVn systems are thought (or known) to be eclipsing (Burdge et al., 2020b). \nNonetheless, AM CVn binaries are expected to be extremely useful for LISA because they simultaneously provide EM information across different wavelengths, as well as being observable in LISA's GW frequency range. For this reason, AM CVn systems have been cited as being important verification sources for LISA (e.g. Kupfer et al., 2018). In other words, AM CVn binaries will be multi-messenger sources once LISA flies. See Section 1.5.1 for further discussion on the multi-messenger opportunities for LISA.", '1.2.3.2 UCXBs (ultra-compact X-ray binaries)': 'It has been known for many years that tight-orbit post-LMXB systems, leaving behind a NS+WD binary that spirals-in due to GW radiation, may avoid a catastrophic event, once the WD fills its Roche lobe. The outcome is expected to be a long-lived UCXB (Webbink, 1979; Nelson et al., 1986; Podsiadlowski et al., 2002; Nelemans et al., 2010; van Haaften et al., 2012; Heinke et al., 2013). These sources are tight X-ray binaries observed with an accreting NS and a typical orbital period of less than 60 min. Because of the compactness of UCXBs, the donor stars are constrained to be either a WD, a semi-degenerate dwarf or a helium star (Rappaport et al., 1982). UCXBs are not only excellent laboratories for testing binary-star evolution, but also important GW sources. Studies of their orbital parameters (mass, orbital \nperiod and eccentricity), chemical composition and spatial distribution may provide important information and clues to understand both the accretion processes of compact binaries (including spin-orbit and tidal interactions) and the long-term evolution of double compact object binaries. \nDepending on the mass-transfer rate, the UCXBs are classified in two categories: persistent and transient sources (e.g. Heinke et al., 2013). Only about 14 UCXBs have been confirmed so far (9 persistent, 5 transient), and an additional ∼ 14 candidates are known. Thus UCXBs are difficult to detect or represent a rare population. Earlier studies (e.g. Istrate et al., 2014a) have suggested the need for extreme fine tuning of initial parameters (stellar mass and orbital period of the LMXB progenitor systems) in order to produce an UCXB from an LMXB system. UCXBs are detected with different chemical compositions in the spectra of their accretion discs (e.g. H, He, C, O and Ne, see Nelemans et al., 2010). To explain this diversity requires donor stars which have evolved to different levels of nuclear burning and interior degeneracy, and therefore to different scenarios for the formation of UCXBs. Since a large fraction of the UCXBs are found in globular clusters, some of these UCXB systems could also have formed via tidal captures, direct collisions or stellar exchange interactions (Fabian et al., 1975; Sutantyo, 1975; Hut and Bahcall, 1983). \nFigure 5 displays the evolution of AM CVns and UCXBs in the GW frequency vs dynamical chirp mass diagram. These systems undergo stable RLO and will start to widen their orbits again within a few Myr after the onset of the mass transfer. LISA will detect such Galactic systems continuously both during the inspiral phase for a few tens of Myr, while the systems are still detached, and after the onset of RLO on a timescale of up to 100 Myr (depending on their distance). \nThe long-term stability of UCXBs has been a topic of debate. From an analytical investigation, van Haaften et al. (2012) argued that for a 1 . 4 M ⊙ NS accretor, only C/O-core WDs with a mass of ≲ 0 . 4 M ⊙ lead to stable UCXB configurations. Subsequent hydrodynamical simulations suggested that this critical WD mass limit could be lower (Bobrick et al., 2017). The first successful numerical calculations of RLO from a WD to a NS were presented by Sengar et al. (2017), and they were able to follow the entire evolution until the low-mass He-core WD donor star has become a ∼ 0 . 005 M ⊙ planet-like companion. These systems were further evolved (Tauris, 2018), including the finite-temperature (entropy) effects of the WD donor stars, and the first evolutionary tracks of such sources across the LISA GW band were produced, see e.g. Figs 4 and 5. Further independent studies on the detectability of UCXBs as LISA sources have been provided by e.g. Chen et al. (2020a); Yu et al. (2021). \nIt is also anticipated that LISA may detect interacting BH+WD systems (Bahramian et al., 2017; Sberna et al., 2020). It has been estimated in some studies (Yungelson et al., 2006) that the Galaxy contains some 10 4 of these systems. However, their formation process (especially those with low-mass WD companions) remains uncertain (Podsiadlowski et al., 2003).', '1.2.4 Other potential sources': 'Contributors: Camilla Danielski(1.2.4.1-2), Silvia Toonen(1.2.4.2), Thomas Kupfer(1.2.4.4), Jan van Roestel (1.2.4.3), Nicola Tamanini (1.2.4.1), Valeriya Korol (1.2.4.1)', '1.2.4.1 Helium-star binaries': 'Subdwarf B stars (sdBs) are stars of spectral type B with luminosities below that of mainsequence stars. The formation mechanism and evolution of sdBs are still debated, although most sdBs are likely He-burning stars with masses ∼ 0 . 5 M ⊙ , radii as small as ∼ 0 . 1 R ⊙ and thin hydrogen envelopes (Heber, 2016). A large fraction are found in binary systems and, due to their compact nature, the most compact ones have orbital periods ≲ 1 hr (Vennes et al., 2012; Geier et al., 2013; Kupfer et al., 2017, 2020b,a), making them potentially detectable sources for LISA. \nFigure 5: GW frequency vs dynamical chirp mass for an UCXB and two AM CVn systems, based on detailed mass transfer calculations (including finite-temperature effects of the WD donor star) using the MESA code (Tauris, 2018). The end points of the first mass-transfer phases (LMXB and CV) are indicated by red triangles; the starting points of the second mass-transfer phases (UCXB and AM CVn) are indicated by green circles. The time marks along the AM CVn tracks are for the same values as indicated for the UCXB system, unless stated otherwise (in Myr). Time zero is defined at the onset of the second mass-transfer phase. The maximum GW frequencies (strongest LISA signal) in these three examples are 5.45 mHz (UCXB), 5.64 mHz (AM CVn1), and 5.72 mHz (AM CVn2) corresponding orbital periods of 6.12 min, 5.91 min and 5.83 min, respectively. The frequency at the onset of the RLO (green circles) depends on the temperature of the low-mass He WD donor ( T eff =10850 K, 9965 K and 8999 K, respectively). \n<!-- image --> \nThe most compact systems have WD companions and as such they are prime progenitor systems for double detonation Type Ia SNe. In this scenario a WD is orbited by a core Heburning sdB star in an ultra-compact orbit ( P orb < 80 min ). Due to the emission of GWs, the binary shrinks until the sdB star fills its Roche lobe and starts mass transfer. He-rich material is then transferred to the C/O-core WD companion which will lead to the accumulation of a He-layer on top of the WD. After accreting about 0 . 1 M ⊙ , He-burning is predicted to be ignited in this shell. This in turn triggers the ignition of carbon in the core, even if the WD mass is significantly lower than the Chandrasekhar limit (Fink et al., 2010). So far, the only known candidate for this scenario is the ultra-compact sdB+WD binary CD -30 · 11223 with an orbital period of P = 71 min (Vennes et al., 2012; Geier et al., 2013). This system was also found to be detectable for LISA with an expected SNR of ∼ 5 after four years of LISA observations (Kupfer et al., 2018). More recently, the first members of ultra-compact sdB binaries which have started to transfer material to the WD companion were discovered. The most compact system, ZTFJ2130+4420, consisting of a low-mass sdB star with M sdB = 0 . 337 M ⊙ has an orbital period of 39 min. The system has well measured properties from EM observations and is expected to have a SNR of ∼ 3 after four years of LISA observations, adding to the growing number of LISA detectable He-burning stars. \nGötberg et al. (2020) modelled the Galactic population of stripped stars, which contain the low-mass sdB stars as well as more massive He-core burning stars, in tight orbits with compact companions, focusing on those that will be detectable by LISA. Their analysis predicts up to 100 stripped star + WD binaries and up to 4 stripped star + NS binaries with SNR > 5 after 10 years of observations with LISA. Although the expected numbers are significantly smaller than for WD+WDs or AM CVns, Götberg et al. (2020) finds that all of the LISA detectable sources are within 1 kpc and therefore bright in EM flux which makes them ideal targets for multi-messenger studies (see Section 1.5.1 for more details on multi-messenger opportunities).', '1.2.4.2 Period bouncing CVs': 'Period bouncing CVs are highly evolved cataclysmic variables where the donor has lost almost all of its mass and has become degenerate. These systems have reached the minimum orbital period for a hydrogen donor (70 min) and are evolving to longer orbital periods (up to 100 min). Model predictions are that 40-70% of all CVs are period bouncers (Kolb, 1993; Knigge et al., 2011). However, only a few have been identified so far because of the low accretion rate and low temperature of the WD and donor (e.g. Pala et al., 2018). While the donor-mass is low and the orbital periods are relatively long, nearby period bouncers are detectable with LISA. Given their high space density, a dozen of these systems are close enough to be detected by LISA.', '1.2.4.3 Exoplanets, brown dwarfs and substellar companions': 'In the Galaxy, due to the slope of the Salpeter-like initial mass function (IMF), more than 97% of all stars will terminate their lives as a WD, meaning that the vast majority of the known 4000+ planet-hosting stars will end their lives as WDs. In the last couple of decades, most of the attention in exoplanetary searches has been focused on the formation and characterisation of exoplanets orbiting host stars on the main sequence, but very little is known on planetary systems in which the host star evolves off the main sequence, to become a red giant. Theoretical models indicate that, if planets avoid engulfment and evaporation throughout the red-giant or/and the asymptotic-giant branch phases of the host star, they can survive (see e.g. Livio and Soker, 1984; Duncan and Lissauer, 1998; Nelemans and Tauris, 1998). This is expected to be the fate of the planet Mars, and other planets orbiting further out in our Solar System (Schröder and Smith, 2008). Observational evidence, in the form of photospheric contamination by the accreted debris (Zuckerman et al., 2010; Koester et al., 2014), dusty (Farihi et al., 2009) and gaseous circumstellar discs (Gänsicke et al., 2006; Manser et al., 2016), supports the existence of dynamically active planetary systems around WDs. Up to very recently, only two planetesimals had been observed \norbiting a WD (Vanderburg et al., 2015; Manser et al., 2019). However, within the last years, two giant planets have been detected orbiting single WDs (Gänsicke et al., 2019; Vanderburg et al., 2020), showing that planets can survive single host-star evolution. \nToday, over 1000 brown dwarfs (BDs) have been detected in the Solar neighbourhood (Burningham, 2018). Some of them have been discovered also around single WDs, and examples of BDs orbiting at distances beyond the tidal radius of the asymptotic-giant branch progenitor (but also within it, e.g. WD 0137 -349 B, Maxted et al., 2006), show that BDs can survive stellar evolution of their host star, whether or not they are engulfed by its expanding envelope. Farihi et al. (2005) predicted that few tenths of percent of Galactic single WDs hosts a BD. \nThe most straightforward way with which LISA could detect sub-stellar objects, such as planets or BDs, would be the direct detection of GWs emitted by a binary system composed of a sub-stellar object in an tight orbit around a single star. However, the absolute orbital period minimum for a hydrogen-rich body (i.e. a star, BD or a gas giant planet) in a binary system is about P orb ≃ 37 min (Rappaport et al., 2021). This corresponds to a GW frequency of at most f GW ≃ 0 . 9 mHz . Such a system could be detected only at close distances (say, within 1 kpc) and only for relatively high sub-stellar masses ( M ≳ 13 M J ), possibly excluding all exoplanets. Furthermore, the mass of the sub-stellar object cannot be directly inferred from direct detection, and at best only the chirp mass of the binary system can be retrieved. Further investigations and EM observations are necessary to better understand the detectability and the rates of these substellar objects, although at the moment it seems unlikely that a large number of these systems will be observed by LISA (Wong et al., 2019b). \nAnother option is to search for circumbinary planets around WD+WD through a modulation of the WD+WD signal (Tamanini and Danielski, 2019), that can probe regions of parameter space not probed by EM observations (far away and not towards the Galactic Centre). The discovery of evolved planetary systems will statistically increase the current sample of post-mainsequence planets, filling an area of the planetary Hertzsprung-Russell diagram that is currently not explored (Tamanini and Danielski, 2019). LISA will provide observational constraints on both planets that can survive two CE stellar evolution phases and on a possible second-generation planet population produced from CE ejecta material (Schleicher and Dreizler, 2014). Even in the case where LISA will prove no detection anywhere in the Milky Way, it will be possible to set strong unbiased constraints on planetary evolution and dynamical theories, and in particular on the fate of exoplanets bound to a binary that undergoes two CE phases.', '1.2.4.4 Triples and multiples': "LISA's stellar sources will also contain multiple body systems, such as triples and quadruples. Hierarchical systems that consist of nested orbits represent stable configurations that can remain intact for several Gyr and throughout (despite) the evolution of the stellar components, as evidenced by observations. Within 20 pc of the Sun, there are already two such systems that harbor close WD+WDs. These are WD 0326 -273 (Luyten, 1949; Poveda et al., 1994; Nelemans et al., 2005; Giammichele et al., 2012; Toonen et al., 2017) and WD 0101+048 (Saffer et al., 1998; Maxted et al., 2000a; Caballero, 2009; Giammichele et al., 2012; Toonen et al., 2017). The former is a triple that consists of a close WD+WD with a period of ∼ 1 . 8 d , and an M5 star in a wide orbit. The latter is a quadruple consisting of a close WD+WD (with a period of ∼ 1 . 2 d , but see (Maxted et al., 2000a) and a MS+MS binary. Two triple systems with three WD components are known as well, J1953 -1019 (Perpinyà-Vallès et al., 2019) and WD 1704+481 (Maxted et al., 2000b). The inner binary of the latter system has a period of ∼ 0 . 15 d , just inside the LISA frequency range. Even millisecond pulsars have been found to be part of triple-architectures; the PSR J0337+1715 system harbors a compact NS+WD (1.6 d orbital period) inner binary which is orbited by another (tertiary) WD every 327 d (Ransom et al., 2014; Tauris and van den Heuvel, 2014). The globular cluster (M4) pulsar B1620 -26 has a WD companion in a half-year orbit, and a planetary companion in a 100-yr orbit (Thorsett et al., 1999; Sigurdsson et al., 2003). \nTheory suggests that exoplanets (and BDs) also exist around WD+WDs in the Galactic disc, and that such objects are more likely to survive around evolving close binary stars than around evolving single stars (Kostov et al., 2016). The eclipse timing variation technique allowed the detection of a few post-CE systems (that is WD+low-mass star), and a BD companion(s) (see e.g. Goździewski et al., 2015; Beuermann et al., 2012; Almeida et al., 2019), nevertheless today no BDs or exoplanets orbiting WD+WDs have been observed yet. \nLISA will be able to detect outer companions to compact (inner) binaries when they impose eccentricity oscillations in the inner orbit due to three-body dynamics (von Zeipel, 1910; Lidov, 1962; Kozai, 1962; Naoz, 2016). In particular Hoang et al. (2019) showed such oscillations would be observable with LISA to distances up to a few Mpc for compact binaries near supermassive BHs, which can also be considered a three-body system. Furthermore, LISA can detect outer companions by exploiting the Doppler frequency modulation on the GW waveform due to their gravitational pull (Robson et al., 2018). The acceleration imparted by the hierarchical companions can be detected in the GW signal for outer periods as large as 100 yr (Robson et al., 2018; Tamanini et al., 2020). For systems with orbital periods that are shorter than, or comparable to, the mission lifetime, the perturbation allows for the determination of the orbital period, eccentricity, initial orbital phase and radial velocity parameter of the companion (Robson et al., 2018; Tamanini and Danielski, 2019). On a general level, the sensitivity of LISA will be able to detect WD+WDs companions with masses down to ∼ M J (Danielski et al., 2019), and therefore allow not only for the detection of stellar companion and compact objects, but also BDs and exoplanets. This being an indirect detection, i.e. the observation of a periodic Doppler shift modulation of an existing strong binary GW signal, we are able to probe a wider mass range, whose inferior limit also covers the giant planets range. The novelty of using LISA for the detection of planetary/low-mass companions is that GWs provide a much larger spatial coverage than the one provided by EM techniques, enabling us to probe regions of our Galaxy currently not accessible to other methods. More specifically, Danielski et al. (2019) showed that during a 4 yr nominal mission LISA will detect from 3 to 83 exoplanets, and from 14 to 2218 BDs everywhere in the Milky Way. The sensitivity of LISA is such that in the most optimistic cases exoplanets could be detected orbiting WD+WDs in the Milky Way's satellites, in particular in the Large Magellanic Cloud (LMC, Danielski and Tamanini, 2020). Such an observation could represent the first detection of an extra-galactic bound exoplanetary system.", '1.2.4.5 Capturing the inspiral of a CE system': 'It has been suggested by Renzo et al. (2021) that LISA may be able to detect the inspiral of binaries undergoing a CE phase. Depending on various assumptions, they anticipate that LISA could detect between 0.1 and 100 such GW sources in the Galaxy during the mission duration. Detecting this GW signal would provide direct insight into the gas-driven physics of CE evolution.', 'Coordinators: Katie Breivik': 'Contributors: Michela Mapelli (1.3.2-3), Simone Bavera, Katie Breivik, Martyna Chruslinska, Gijs Nelemans, Pau Amaro Seoane (1.3.2), Manuel Arca Sedda (1.3.23), Thomas Tauris, Silvia Toonen (1.3.2-3), Jeff Andrews, Tassos Fragos (1.3.1), Luca Graziani, Daryl Haggard (1.3.2), Melvyn B. Davies (1.3.2-3) \nIn the following section, we discuss the formation of LISA binaries. For a review and broader description of the many physical aspects of stellar evolution and binary star interactions that are referred to below in the context of the formation of compact object binaries, we refer to the \ntextbooks by, for example, Shore et al. (1994); Hilditch (2001); Eggleton (2006); Chaty (2022); Tauris and van den Heuvel (2023).', '1.3.1 Isolated binaries': "The formation pathways of isolated binaries observable by LISA are marked with several phases of mass loss or exchange. In the following Section, we refer to the initially more massive star as the primary and the initially less massive star as the secondary. Stable mass transfer can occur either through wind mass loss/accretion or RLO. Wind mass loss is generally assumed to be non-conservative across all phases of stellar evolution, with mass accretion efficiency ranging from ≲ 10% for the Bondi-Hoyle-Lyttleton mechanism and 20 -50% if the accretion is focused (de Val-Borro et al., 2017). In this case, the orbit widens as the mass lost from the system causes an increase of the remaining specific angular momentum. In practice, the dynamics are complicated and dependent on physics related to the geometry and structure of the wind, tidal effects, orbital characteristics and in some cases magnetic fields and radiation transport, thus calling for three-dimensional, multi-physics, hydrodynamical simulations (e.g. Saladino et al., 2018, 2019). \nRLO occurs when the donor star expands to the point that its radius exceeds the Roche radius (Eggleton, 1983). RLO mass transfer can proceed in a dynamically stable or unstable fashion, depending on the structure of the donor and accretor as well as their mass ratio. The stability of RLO mass transfer is commonly described using the Webbink radius-mass exponents (Webbink, 1985) that determine the timescales on which mass transfer will become unstable. In the case of dynamically stable mass transfer, the orbital evolution depends strongly on the mass ratio of the binary: if M donor /M accretor > 1 , the orbit tightens (for fully conservative masstransfer), while in the converse case the orbit widens. Mass-transfer efficiency, as well as the assumed specific angular momentum carried away from mass lost from the system, play a crucial role in mass-transfer stability and the orbital evolution of the binary. \nDynamically unstable mass transfer is believed to generate a CE phase where the donor star's core and companion are enshrouded in the donor's envelope (for a review see Ivanova et al., 2013). The precise dynamics of how CE proceeds are still not fully understood. In the context of compact object binary formation and population synthesis studies, energy budget arguments are most often employed to estimate the post-CE properties of a binary. In the ' α CE ' prescription, it is assumed that a fraction α CE of the released orbital energy is used to unbind the donor's envelope and eject it from the system (van den Heuvel, 1976; Webbink, 1984). Several recent studies have suggested that other sources of energy may be needed to successfully eject the envelope, including recombination energy (e.g. Zorotovic et al., 2014; Nandez and Ivanova, 2016) or jets launched by the companion (e.g. Shiber et al., 2019). Each of these will change the overall energy budget of the CE evolution and lead to differences in the final orbital separation (e.g. Iaconi et al., 2018). Alternatively, in the ' γ CE ' prescription, angular momentum conservation arguments, which lead to less dramatic inspiral, have been considered to explain the orbital period distribution of WD+WDs (Nelemans et al., 2000).", '1.3.1.1 WD+WD systems and AM CVn binaries': "The progenitors of isolated WD+WD and AM CVn binaries begin with zero age main sequence stars with masses below 8 -10 M ⊙ . The formation pathways of close WD+WD and AM CVn binaries contain several stages of stable and unstable mass transfer, or CE (see Fig. 6). The uncertain outcomes of these interactions determine whether the progenitor binary continues on in its evolutionary path toward becoming a LISA source or if it merges with its companion. Conversely, LISA observations of the populations of WD+WD and AM CVn sources will constrain these interactions. \nVirtually all close WD+WD and AM CVn progenitors experience an interaction as the primary star advances off the main sequence and fills its Roche lobe. This interaction can either \nFigure 6: Illustration of the formation of an AM CVn system and a detached WD+WD binary. LISA sources are indicated with waves. Acronyms. ZAMS: zero-age main sequence; RLO: Rochelobe overflow (mass transfer); CE: common envelope; He star: helium star; WD: white dwarf; CV: cataclysmic variable; AM CVn: AM Canum Venaticorum binary (Figure from Tauris and van den Heuvel, 2023). \n<!-- image --> \nproceed stably or unstably on dynamical timescales. In either case, the orbit will shrink because of the donor's higher mass relative to the accretor. For systems with late red giant and asymptotic giant branch donors, initially dynamically stable but thermally unstable mass transfer can produce mass loss from the L2 Lagrange point, which leads to a delayed dynamical instability and a CE phase (Ge et al., 2020; Misra et al., 2020). In the rare case of close WD+WDs where the more massive WD forms second (converse to stellar lifetime expectations), a phase of stable mass transfer, followed by a CE generated by the initially lower mass star could be necessary (Woods et al., 2012). \nAfter each interaction, the star which donates mass becomes stripped leaving behind a He core that can have varying structure depending on the evolutionary phase at which the donor filled its Roche lobe. Such stripped stars orbiting main sequence companions have been widely observed throughout the Galaxy (Rebassa-Mansergas et al., 2007) and been used to constrain CE ejection efficiencies (Zorotovic et al., 2010; Toonen and Nelemans, 2013). \nSince the previous interaction brings the two stars together, further interactions are likely. Interactions can occur while the secondary is still on the main sequence. In this case, if the mass transfer is stable, a cataclysmic variable is formed (Zorotovic et al., 2011). Conversely, mass transfer can also occur as the secondary advances up the giant branch. Due to previous envelope stripping, the mass ratio of the secondary donor to the WD accretor can be either greater or less than one. If the secondary is already less massive than the WD, stable mass transfer will occur and widen the orbit, removing the possibility of detection by LISA. However, if the secondary is more massive than the WD companion, the mass exchange will lead to orbital tightening. If the mass transfer is unstable, another CE phase takes place, potentially bringing the stars even closer together and leaving behind a WD with a stripped He core companion. The structure of the He core again depends on the evolutionary phase at which the secondary overflows its Roche lobe. At this point, a close WD+WD binary is assured and the slow evolution due to GW emission brings the WD+WD toward the LISA band. \nA key uncertainty in the formation pathways of AM CVn binaries is the nature of the donor \nFigure 7: Illustration of the formation of a detached NS+WD binary and an UCXB system. See Fig. 6 for details. Additional acronyms: SN: supernova; NS: neutron star; LMXB: low-mass X-ray binary; BH: black hole (Figure from Tauris and van den Heuvel, 2023). \n<!-- image --> \nstar. AM CVn binaries consist of a WD accreting He-rich material originating from a WD, semi-degenerate helium star, or evolved MS donor (Solheim, 2010). Indeed, it could be the case that the observed AM CVn population is a combination of all three with different relative contributions (Nelemans et al., 2004b). If the donor star is a non-degenerate evolved star, magnetic braking is required, along with GW emission, to maintain the ultra-short periods of observed AM CVn systems (van der Sluys et al., 2005). Magnetic braking is a process in which orbital angular momentum in a tight synchronized binary is converted into spin angular momentum via a magnetic stellar wind (a process that therefore requires a low-mass stellar component with a convective envelope). The ultra-compact orbital configuration is less problematic for semi-degenerate and fully-degenerate donors which originate from the ejection of a second CE, with tighter orbits allowed by more degenerate donors (Yungelson, 2008). In the case of fullydegenerate He-core WD donors, the orbit can become so small that the mass lost from the donor directly impacts the accretor leading to a rapid decrease in orbital size followed by a long-lived phase of accretion which widens the orbit (Nelemans et al., 2004b; Marsh et al., 2004; Deloye and Taam, 2006; Kremer et al., 2017). Regardless of the donor, a significant uncertainty still remains in how much He-rich material the accretor can handle until novae erupt on the WD's surface. While detailed binary evolution calculations (e.g. Tauris, 2018) have shown that RLO mass transfer in WD+WD can be stable, it has been suggested that interactions of the donor star with the expanding nova shells will likely lead to a rapid orbit shrinkage and eventually a merger (Shen, 2015).", '1.3.1.2 White-dwarf binaries with neutron-star or black-hole companions': 'Compared to WD+WDs, the formation of detached binaries with WD and NS or BH companions occur in binaries with stars that are massive enough to explode in a SN (see Fig. 7). Similar to WD+WD formation, the more massive primary evolves first and, because of the relatively large mass ratio, begins RLO mass transfer that is often expected to be unstable and lead to a CE. Soon after, the primary evolves to become a compact object, through either a supernova explosion (NS or BH) or direct collapse (BH only). Since a NS is thought to receive a kick during its formation, there is a significant probability that the binary disrupts at this point. \nFigure 8: Evolutionary sequence showing how ultra-compact X-ray binaries (prime LISA source candidates) are formed from merging NS+WD binaries, descending from LMXBs. Plotted here is mass-transfer rate of the donor star as a function of stellar age. The initial MS star + NS binary has components of 1 . 40 M ⊙ and 1 . 30 M ⊙ , respectively. The system evolves through two observable stages of mass transfer: an LMXB for 4 Gyr, followed by a detached phase lasting about 3 Gyr where the system is detectable as a radio millisecond pulsar orbiting the helium WD remnant of the donor star, until GW radiation brings the system into contact again, producing a UCXB. The colour bars indicate detectability in different regimes resulting in synergies between LISA and EM detectors (Figure from Tauris, 2018). \n<!-- image --> \nThe subsequent evolution, of a lower-mass non-degenerate star with a NS or BH, will typically go through a phase of stable mass transfer in which the binary becomes observable as X-ray binary, due to the strong heating of the accretion disc in the deep potential of the NS or BH. When the onset of the mass-transfer occurs after the secondary star has evolved past its main sequence, the core of the star has already contracted. Thus after the X-ray binary phase, when the envelope of the expanding star has been completely transferred, the NS or BH is left with a WD companion that was the core of the donor star. \nIn some cases, the NS/BH+WD binary is tight enough that angular momentum loss due to GW emmission will bring the two objects together as LISA sources (Fig. 8). At periods of ∼ 10 -20 min , i.e. within the LISA band, the WD will start to transfer mass to the NS/BH, forming an X-ray binary again, but now of ultra-short period, called an UCXB. Detailed numerical calculations, including finite-temperature (entropy) effects, have shown that UCXBs can indeed form via stable RLO from post-LMXBs systems (Sengar et al., 2017; Tauris, 2018).', '1.3.1.3 Double neutron star/black hole binaries': "NS+NS formation has been extensively discussed in the literature (see Tauris et al., 2017, for a review). The standard scenario (see Fig. 9 for a schematic diagram) involves several phases of interaction, starting with a stable RLO, during which the primary loses part of its envelope before it undergoes a SN to form a NS. The newly formed NS HMXB is likely too dim to be detectable in X-rays, as the orbital separation is still large and the NS may only be able to capture an appreciable fraction of the companion's stellar wind when the latter evolves to the giant phase. During the subsequent evolution the orbital separation needs to decrease from ∼ 10 3 R ⊙ to a few R ⊙ for the final binary to merge within the Hubble time. Significant tightening is typically achieved through a CE phase that occurs when the secondary this time \nFigure 9: Illustration of the formation of a tight BH+NS binary that evolves towards a merger. See Figs. 6 and 7 for explanation of acronyms (Figure from Tauris and van den Heuvel, 2023). \n<!-- image --> \nfills its Roche lobe. The post-CE binary is expected to encounter another phase of mass transfer, initiated by a stripped helium-burning secondary (i.e. so-called Case BB mass transfer, often initiated when a ∼ 2 . 5 -3 . 5 M ⊙ helium star expands during shell-helium burning, Habets, 1986). This leads to further orbital tightening, stripping of the secondary's envelope, and NS spin-up. If this last mass-transfer episode is unstable and leads to a second CE phase, a fast merging NS+NS will be formed; a scenario invoked to explain r-process element enrichment observed in some stellar systems (Safarzadeh et al., 2019; Zevin et al., 2019a). Such NS+NSs would be effectively unobservable with current radio surveys, and if they exist within the Galaxy, their presence will be revealed by LISA (Kyutoku et al., 2019; Andrews et al., 2020). However, recent detailed binary evolution calculations have shown that this last phase of Case BB mass transfer is expected to be stable (Tauris et al., 2015; Vigna-Gómez et al., 2018) and do not support the existence of the aforementioned fast-merging channel (in contrast to earlier works Ivanova et al., 2003; Dewi and Pols, 2003). \nBesides the pre-HMXB evolution, the most important and uncertain aspects of our current understanding of NS+NS and mixed BH+NS formation are related to: i) CE evolution and spiral-in of the NS, ii) momentum kicks (magnitude and direction) imparted onto newborn NSs, and iii) the mass distribution of NSs. \nFrom an energetics point of view, it has been shown that an inspiralling NS may indeed be able to eject the envelope of its massive star companion (e.g. Xu and Li, 2010; Loveridge et al., 2011; Wang et al., 2016a; Kruckow et al., 2016). However, predicting the final post-CE separation is difficult for several reasons, including: estimating the location of the bifurcation point within the massive star (Tauris and Dewi, 2001), separating the remaining core from the ejected envelope (Tauris and Dewi, 2001; Fragos et al., 2019), additional energy sources such as accretion energy (MacLeod and Ramirez-Ruiz, 2015; Murguia-Berthier et al., 2020), energy and radiation transport during the CE inspiral (Fragos et al., 2019) and the effect of an inflated envelope of the exposed naked helium core (Sanyal et al., 2015, see also Section 1.7.1.3). \nNewly formed NSs gain velocity (natal kicks; e.g. Gunn and Ostriker, 1970; Hobbs et al., 2005; Verbunt et al., 2017) due to asymmetries arising during their formation (e.g. Janka, 2012). The properties of the modelled NS+NS population (e.g. number of systems formed, orbital \nparameters, merger locations relative to formation site) are highly dependent on the adopted natal kick prescription (e.g. Portegies Zwart and Yungelson, 1998; Bloom et al., 1999; Chruslinska et al., 2018; Giacobbo and Mapelli, 2018; Andrews and Zezas, 2019). To match the current observational constraints on the NS+NS merger rate and parameters of several of the observed systems (e.g. van den Heuvel, 2007), it is necessary to assume that a fraction of NS forms with natal kicks smaller than typically found for young single pulsars. Some scenarios involve lowmass NS progenitors and electron-capture triggered explosions (e.g. Dessart et al., 2006; Jones et al., 2013). Others postulate a link between the natal kick magnitude and the mass of the NS progenitor and SN ejecta (e.g. Beniamini and Piran, 2016; Bray and Eldridge, 2016; Janka, 2017). These claims have been supported by 3D NS simulations of ultra-stripped stars (Müller et al., 2019). In fact, it has been demonstrated that close-orbit, low-eccentricity NS+NS and BH+NS systems most likely form via ultra-stripped SNe when the last star explodes (Tauris et al., 2013, 2015). The reason being that the last Case BB RLO mass-transfer phase causes the NS to significantly strip its evolved helium-star companion, almost to a naked metal core prior to its explosion, and thus there is very little SN ejecta (see also Section 1.7.1.7). \nFinally, a clear correlation has been predicted between the spin period of the recycled pulsar and the orbital period of the system after the second SN (Tauris et al., 2017). This correlation can be tested in LISA binaries, if the spin period is measured, since only short orbit systems will enter the LISA band within a Hubble time, and these binaries should therefore contain the most rapidly spinning NSs of this population. Another hypothesis that can be tested by LISA, is the resulting mass distribution among NS+NS systems (e.g. Özel and Freire, 2016, and references therein). \nMerging BH+BHs and NS+BHs in the field are thought to occur under some specific binary interactions which either (i) bring the parent stars closer together during their evolution or (ii) prevent stars in close obits from expanding. \nThe former one (i) occurs in a similar manner to the formation of NS+NSs described above, and involves many of the main uncertainties. In contrast to NS+NS formation, BH+BHs and to a lesser degree BH+NSs are sensitive to the metallicity of the progenitor stars, and they favor low-metallicity environments. In addition the second mass transfer episode, after the first compact object formation, can be either dynamically stable (e.g. van den Heuvel et al., 2017; Inayoshi et al., 2017a; Neijssel et al., 2019) or unstable (e.g., Smarr and Blandford, 1976; van den Heuvel, 1976; Tutukov and Yungelson, 1993; Kalogera et al., 2007; Postnov and Yungelson, 2014; Belczynski et al., 2016a). In the latter case this leads to a CE phase. The resulting tight system composed of a compact object and a Wolf-Rayet star can eventually undergo a tidal spin up of the star (Qin et al., 2018; Bavera et al., 2020). On the other end, if the second mass transfer is stable the binary will result in wider orbits compared to the evolution through CE and avoid a subsequent tidal spin up phase (Bavera et al., 2021). Eventually, following wind-driven mass loss, the secondary will collapse to a compact object. This leave us with either a BH+BH system or a NS+BH system with either a first- or second-born NS. \nThe latter possibility (ii) occurs when two massive stars are born in a tight orbit (orbital periods less than 4 days) in low-metallicity environments which due to their tidal interactions can maintain the stars at almost critical rotation. Such rapidly rotating stars develop a temperature gradient between the poles and the equator leading to chemical homogeneous evolution (e.g., de Mink et al., 2009; Mandel and de Mink, 2016; Marchant et al., 2016; du Buisson et al., 2020). In these stars meridional circulation transport hydrogen from the surface into the core and helium out into the envelope until nearly all the hydrogen in the star is fused into helium. At the end of their main sequence these stars are essentially Wolf-Rayet stars and do not expand, hence, avoiding any additional mass-transfer phase. \nLISA may answer whether or not mixed binaries of BHs and NSs, in which the NS formed first, are produced in the Galaxy. It is possible that the in-spiralling NS is unable to eject the envelope of the relatively massive BH progenitor star (Kruckow et al., 2018). Since we currently \nFigure 10: Illustration of the formation of LISA sources via examples of exchange encounters. See Figs. 6 and 7 for explanation of acronyms (Figure from Tauris and van den Heuvel, 2023). \n<!-- image --> \ndo not know of the existence of mixed BH+NS systems in the Galacy, any LISA detections of such systems, as well as double BH systems, will provide crucial information about their formation process.", '1.3.2 Sources in clusters': "The inner regions of stellar clusters are cosmic factories of compact binaries (i.e. binaries containing two compact objects; BHs, NSs or WDs), owing to the dominant role played by stellar dynamics in such environments. Massive stars in stellar clusters lose kinetic energy to lighter stars and accumulate into the cluster centre. In just a few Myr, these stars evolve into stellar BHs and NSs. \nThe production of compact binaries can take one of two routes. If only a small fraction of BHs are retained within a cluster, encounters between BHs and binary stars lead to dynamical exchanges, where the BH replaces a less massive star within the binary (Hills and Fullerton, 1980). Globular clusters have a considerably-enhanced population of X-ray binaries (Heinke et al., 2003), which might have formed when a NS or a BH exchanges into a binary star (e.g., Hills, 1976). After the first exchange, evolution of the stellar companion (which might also become a BH or a NS) or a second dynamical exchange can produce a compact binary. Binarysingle interactions represent an efficient mechanism to harden these binaries (Heggie, 1975) to the point where they can merge via the emission of gravitational radiation (see Fig. 10 for a schematic representation). \nAlternatively, if a large fraction of BHs are retained in the cluster, they can form a BH subsystem (Spitzer, 1969; Mackey et al., 2007, 2008; Arca Sedda et al., 2018; Kremer et al., 2020b, but see Breen and Heggie 2013). BHs in the subsystem tend to strongly interact with each other, undergoing frequent pairing, exchanges, and ejections. The most efficient mechanism driving binary formation in globular clusters is via three-body scatterings (e.g., Morscher et al., 2015). After formation, a binary can undergo dozens of interactions with passing stars and binaries, which can lead to the production of very hard binaries, capable of merging via the action of GW emission (Portegies Zwart and McMillan, 2000). If the star cluster centre harbours a BH subsystem, the BHs dominate the dynamics, quenching mass segregation and preventing the formation of binaries containing other compact objects (see e.g. Ye et al., 2020). However, dynamically evolved clusters can lose a substantial fraction of the BHs. In these BH-poor clusters, binarysingle interactions can allow the formation of binary NS and BH+NS binaries. Furthermore, it \nhas been proposed that a parabolic encounter between two compact objects could potentially lead to the formation of a binary due to an abrupt loss of energy emitted as gravitational radiation (e.g. Hansen, 1972; Quinlan and Shapiro, 1989; Kocsis et al., 2006; Hong and Lee, 2015). However, the event rate of this mechanism, which is often referred to as the 'gravitational brake' capture, is very likely to be negligible due to the small cross-section (Kochanek et al., 1990). \nOld BH-poor clusters may also be ideal for dynamical formation of WD+WD binaries as well as BH/NS+WD binaries (Kremer et al., 2020b), see Fig. 10. In old globular clusters, WDs are by far the most abundant type of compact object (roughly 10 5 WDs are expected in a 10 6 M ⊙ cluster). A number of analyses have studied ways WD binaries, both accreting and detached, may be dynamically assembled in stellar clusters (Grindlay et al., 1995; Ivanova et al., 2006; Belloni et al., 2016; Kremer et al., 2018b). Furthermore, a handful of the stellar-mass BH binary candidates observed in Galactic globular clusters are suspected to be ultra-compact accreting BH+WD binaries (Strader et al., 2012; Bahramian et al., 2017; Church et al., 2017). Overall, up to a few dozen dynamically formed WD binaries are expected to be resolved by LISA in the Galactic globular clusters, likely constituting the largest class of dynamically-formed LISA sources in the Galaxy (Willems et al., 2007; Kremer et al., 2019b). Currently two candidates to AM CVns in globular clusters have been identified (Zurek et al., 2016; Rivera Sandoval et al., 2018) and several more are expected to be discovered in upcoming globular cluster surveys. \nComparing nuclear stellar clusters with globular clusters, the former tend to have somewhat larger escape speeds (due in part to the presence of a central massive BH, MBH: Graham and Spitler, 2009). This means that a larger fraction of BHs are likely to be retained (e.g. Miller and Lauburg, 2009), while the higher dispersion velocity inhibits both exchange encounters and the dynamical formation of binaries (e.g. Heggie and Hut, 2003). The presence of a dense nuclear cluster surrounding the MBH can significantly affect the formation process of compact binaries in a number of ways. Dynamical three body encounters can form at least one compact BH+BH if the nuclear cluster-to-MBH mass ratio exceeds 10, whereas at lower values the reservoir of compact binaries might be replenished via star cluster inspiral (e.g. Arca Sedda, 2020a). The presence of an MBH can leave significant imprints on the BH+BH evolution, owing to the possible development of von Zeipel-Kozai-Lidov cycles (von Zeipel, 1910; Kozai, 1962; Lidov, 1962), which can boost the rate of BH+BH mergers (Blaes et al., 2002; Antonini and Perets, 2012; Hoang et al., 2018; Fragione et al., 2019; Arca Sedda, 2020a) and significantly affect the BH mass spectrum in these extreme environments (e.g. Arca Sedda, 2020a). \nYoung star clusters and open clusters, because of their relatively low total masses ( 10 2 -10 5 M ⊙ ), host a smaller population of BHs with respect to globular and nuclear clusters (e.g., Portegies Zwart and McMillan, 2000; Banerjee et al., 2010; Banerjee, 2017, 2021). BH+BHs in young/open star clusters mostly originate from dynamical exchanges or even from the evolution and hardening of primordial binaries (Ziosi et al., 2014; Di Carlo et al., 2019, 2020b; Kumamoto et al., 2019, 2020). Furthermore, dynamical exchanges favour the formation of BH+NS binaries in young and open clusters (Rastello et al., 2020). \nFinally, hierarchical mergers in globular/nuclear clusters (e.g. Miller and Hamilton, 2002a; Rodriguez et al., 2019; Antonini et al., 2019; Arca Sedda, 2020a; Arca Sedda et al., 2020b) or runaway collisions of massive stars (e.g. Portegies Zwart et al., 2004; Giersz et al., 2015; Mapelli, 2016; Rizzuto et al., 2020) and binary star mergers in young star clusters (Di Carlo et al., 2020a) might even lead to the formation of intermediate-mass BHs (Graham et al., 2019) and BHs with mass in the pair-instability gap (e.g. Arca Sedda et al., 2020b), similar to GW190521 (Abbott et al., 2020c,d).", '1.3.3 Triple stellar systems': "Some of the LISA sources may form as part of triples and higher-order multiples. This includes sources in the Galactic disc (formed through e.g. isolated triple evolution) as well as those in dense environments. Three-body (or more) interactions are important in the formation of compact \nFigure 11: Illustration of the formation of a LISA source in a triple system. See Figs. 6 and 7 for explanation of acronyms (Figure from Tauris and van den Heuvel, 2023). \n<!-- image --> \nsources in two ways: during short-lived dynamical interactions and in hierarchical triple systems. \nHierarchical triples, in which two bodies orbit each other, and a third body orbits the centre of mass of the inner orbit, can remain stable for secular timescales, and therefore stay intact for Hubble times (Kiseleva et al., 1994; Mardling and Aarseth, 1999; Georgakarakos, 2008; He and Petrovich, 2018). They may form in clusters (where they may interact with interloper stars in the densest environments) or exist in the Galactic disc (and evolve in pure isolation). Their evolution differs from that of isolated binaries due to three-body effects. Hence, triples that live their lives in isolation bridge the gap between classically isolated LISA sources and dynamicallyevolving sources (often used to mean cluster sources). The importance of three-body interactions in hierarchical systems has been recognised for the evolution of stellar triples (Thompson, 2011; Hamers et al., 2013; Silsbee and Tremaine, 2017; Antonini et al., 2017; Liu and Lai, 2017; Toonen et al., 2018; Fragione and Loeb, 2019; Fragione and Kocsis, 2019; Toonen et al., 2020), triples that consists of a combinations of stars and planets (Hamers and Portegies Zwart, 2016; Hamers, 2017; Veras et al., 2018; Stephan et al., 2018, 2020), as well as stellar binaries in dense environments (Antonini and Perets, 2012; Antonini et al., 2016; Petrovich and Antonini, 2017; Stephan et al., 2016, 2019; Hamilton and Rafikov, 2019b,a; Fragione et al., 2020; Martinez et al., 2020; Fragione et al., 2020). \nThe general formation of binaries and multiples (compact and wide) in clusters is boosted during the collapse of the dense cluster core, which is halted by frequent stellar interactions (Spitzer, 1987; Hut et al., 1992). The formation of the first binaries takes place most likely via three-body scatterings, involving three initially unbound objects (Goodman and Hut, 1993; Lee, 1995). As soon as binaries start forming, binary-single (Hut and Bahcall, 1983; Sigurdsson and Phinney, 1993) and binary-binary (Mikkola, 1983; Miller and Hamilton, 2002a) interactions take over and become the dominant dynamical processes at play. Even for relatively low triple fractions, dynamical interactions involving triples occur roughly as often as encounters involving either single or binary stars alone, particularly in low-mass star clusters (Leigh and Geller, 2013). When the objects involved in the interaction cross their mutual sphere of influence, a strong interaction can trigger the formation of a short-lived bound triple system (Goodman and Hut, 1993). During these chaotic resonances, a pair of objects has a non-negligible probability of experiencing a very close passage, triggering the formation of a compact binary and subsequent merger (Samsing et al., 2014). Depending on the cluster structure, binary mergers developing through resonant interactions can be highly eccentric at LISA frequencies and even still when entering the frequency range typical of ground-based detectors (Samsing, 2018; Samsing and \nD'Orazio, 2018; Arca Sedda et al., 2021b). Binary-binary interactions (Mikkola, 1984; McMillan et al., 1991; Miller and Hamilton, 2002a) represent another efficient mechanism to form triples, either in the form of short lived, resonant unstable triples (Hut and Bahcall, 1983; Zevin et al., 2019b; Arca Sedda et al., 2021b), or in a hierarchical configuration (Antonini et al., 2016; Zevin et al., 2019b; Arca Sedda et al., 2021b; Martinez et al., 2020; Fragione et al., 2020). \nThe best known manifestation of three-body dynamics are the von Zeipel-Kozai-Lidov cycles (von Zeipel, 1910; Lidov, 1962; Kozai, 1962) regime in which the inner orbit eccentricity and the inclination between the two orbits vary periodically. Strictly speaking this applies to the (inner) test particle regime, an axisymmetric outer potential and the lowest-order expansion of the Hamiltonian (i.e. quadrupole). Relaxing either one of these assumptions leads to qualitative different dynamical evolution, which include extreme eccentricity variations and orbital flips (see Naoz, 2016, for a review). The high eccentricities can lead to close passages between the bodies, mass transfer, and enhancement of dissipative processes such as from tides or by GW emission. Over time, the latter can lead to a significant reduction of the inner orbital separation during the nuclear burning stages or during the compact object phase of the stars (Mazeh and Shaham, 1979; Kiseleva et al., 1998; Fabrycky and Tremaine, 2007; Thompson, 2011). Through the internal stellar evolution, a triple may transition from one dynamical regime to another, enhancing (or diminishing) the three-body effects (Shappee and Thompson, 2013; Michaely and Perets, 2014; Toonen et al., 2016).", '1.4 Expected LISA observations: numbers and rates': 'Coordinators: Abbas Askar,Simone Bavera Contributors: Abbas Askar, Quentin Baghi, Simone Bavera, Tassos Fragos, Valeriya Korol, Kyle Kremer, Manuel Arca Sedda', "1.4.1 Binary's detectability": "The detectability of Galactic stellar binaries with LISA primarily depends on the parameters involved in the GW amplitude: \nA = 2( G M ) 5 / 3 ( πf ) 2 / 3 c 4 d , (1) \nwhere f is the binary's GW frequency, M = ( m 1 m 2 ) 3 / 5 / ( m 1 + m 2 ) 1 / 5 is the chirp mass (with m 1 and m 2 being the primary and secondary masses), d is the luminosity distance, G and c are respectively the gravitational constant and the speed of light. This follows from the fact that the signal-to-noise ratio scales linearly with amplitude ρ ∝ A √ T with T being the observation time (Babak et al., 2021). Besides, in contrast with electromagnetic observations, the observed GW signal scales as 1 /d , rather than 1 /d 2 . \nThe GW frequency, defined as f = 2 /P with P being the binary's orbital period, has the strongest impact on the signal's detectability. Binaries emitting at f > 3 mHz fall in the most sensitive part of the LISA frequency band (Amaro-Seoane et al., 2017). As a result, these highfrequency binaries - even if consisting of the lowest mass WD components - can be detected across the Milky Way also reaching satellite galaxies (cf. Figures 1 and 15). At a set frequency, the maximum distance at which the binary can be detected increases with its chirp mass, as shown in Figure 1.1. In addition, the chirp mass defines how fast the GW frequency changes during the in-spiral phase. This so-called chirp is given by \n˙ f = 96 5 π 8 / 3 ( G M c 3 ) 5 / 3 f 11 / 3 . (2) \nThe chirping rate for stellar binaries in the LISA band is generally very slow ( ˙ f < 10 -15 Hz 2 for typical WD+WD and NS+NS binaries emitting at frequencies lower than a few mHz). Note \nthat the frequency derivative (not the chirp mass) can be measured directly from GW data. The measurements is possible for binaries emitting at sufficiently large frequencies ( f > 3 mHz, Roebber et al. (2020)), such that during the observation time ( T ) with LISA the binary sweeps through at least a few frequency bins ( 1 /T ) . If ˙ f is measured and assuming that the inspiral is driven by radiation reaction only, the luminosity distance can be recovered by plugging in the measured value of the chirp mass into Eq.(1). At lower frequencies, binaries will be effectively 'seen' by LISA as monochromatic. In which case, only measurements of f and A will be possible, while the chirp mass estimate will be degenerate with that of the distance (cf. Eq. 1). \nUnlike circular binaries, eccentric ones emit GWs at multiple harmonics. Each signal can be thought as a collection of n signals from circular binaries emitting at f n = nf/ 2 and the amplitude A n = A (2 /n ) 5 / 3 g ( n, e ) 1 / 2 , where g ( n, e ) is given in Peters & Mathews (1963). The total signal-to-noise ratio can be estimated as the quadrature sum of the individual harmonics' signal-to-noise ratios. Thus, to measure the chirp mass in case of the eccentric binary, in addition to ˙ f , one needs to simultaneously measure the eccentricity. This is possible when at least two harmonics are detected (e.g. Seto, 2016), otherwise only an upper limit on the chirp mass can be derived. \nAnother aspect of Galactic binaries' detectability is that they will be observed continuously during the course of the mission. However, it is likely that the measurements will undergo occasional interruptions due to communication antenna re-pointing or spacecraft housekeeping. Additionally, spurious disturbances may affect the interferometer signals, such as the transient glitches observed in LISA Pathfinder (LISA Pathfinder Collaboration, 2022; Baghi et al., 2022), leading to corrupted data spans. These operating conditions may impact the duty cycle of the mission, with a minimum requirement of 89% at the time of writing. If the masking events are frequent, they could impact the detection of low-frequency sources (around 0.1 mHz and below). However, mitigation techniques have been developed and show promising results for restoring the optimal detectability (Baghi et al., 2019; Blelly et al., 2022).", '1.4.2 Detection and parameter estimation expectations': "It is expected that the Galactic stellar binaries will dominate the number of individually detected GWsources at mHz frequencies (Table 4). Out of ∼ O (10 7 ) stellar binaries emitting in the LISA frequency band, LISA is expected to deliver ∼ O (10 4 ) individual detections (e.g. Littenberg et al., 2020; Karnesis et al., 2021). Current estimates suggest that at frequencies > 3 mHz, Galactic WD+WD detectable by LISA will be counted in thousands, NS+NSs in few up to hundreds and BH+BHs in a few (see Tables 4 and 5). At frequencies < 3 mHz the number of stellar binaries is so large, that only a small fraction - the closest and more massive ones - will be individually detected, while the rest of the population will form an unresolved stochastic foreground (see Section 1.6.2). \nAmongst the resolvable systems, population synthesis simulations suggest that hundreds will be exceptionally strong LISA sources (e.g. Karnesis et al., 2021). These binaries will be detectable within weeks from the start of mission operations, and over the full mission lifetime can accumulate SNRs up to 10 3 . Based on an SNR evaluation via an iterative scheme for the estimate of the confusion foreground generated by Milky Way's GW sources (e.g. Littenberg et al., 2020; Karnesis et al., 2021), synthetic population analysis (e.g. Korol et al., 2017; Lamberts et al., 2019) yields about up to 10 4 binaries detectable with SNR > 20 , several 10 3 with SNR > 100 , a few with SNR > 1000 . For all binaries, frequencies predicted be measured very accurately with σ f /f < 10 -5 , which corresponds to σ P < 0 . 025 ms for a typical mHz WD+WD binary. Frequency derivative is estimated to be measured to better than 30% for up to 10 4 binaries, leading to the measurement of the chirp mass via Eq. (2). Consequently, also distances can be derived to better than 30% for several 10 3 binaries. The sky localisation uncertainty depends strongly on the SNR and GW frequency ( ∝ ρ -2 f -2 ). Additionally, it is also dependent on the ecliptic latitude: a source on the ecliptic has an order of magnitude more uncertainty than \na source at the poles (Roebber et al., 2020). On average, sky localisation error for Galactic WD+WD systems is expected to be of several deg 2 , improving with increasing GW frequency and SNR down to sub-square degree precision and below. These expectation are well supported by Bayesian-based data analysis exercises (Littenberg et al., 2020) with mock data simulating observations over one year, and including 30 millions of injected Galactic sources. Consolidation of these results are expected once more realistic data simulations featuring mixed source types are analysed (see https://lisa-ldc.lal.in2p3.fr). \nBeyond the Milky Way, the Local Group galaxies are expected to harbour from a few to a few hundreds LISA sources (mainly WD+WDs and some NS+NSs) depending on the total mass and the distance of the galaxy (Seto, 2019; Andrews et al., 2020; Korol et al., 2020; Lau et al., 2020) (see Fig. 1). For instance, the number of WD+WDs and NS+NS in the largest Milky Way satellites, the Large and Small Magellanic Clouds, will be high enough to overcome Galactic foreground and to unambiguously identify these galaxies in the LISA data (Roebber et al., 2020). Even further away, with the total mass comparable to the Milky Way's mass, the Andromeda galaxy could also be visible on the LISA sky as a group of GW sources (Korol et al., 2018). \nThe outside limits of the Local Group, LISA can access distances up to ∼ 1 Gpc through GW signal from stellar-mass BH+BHs, which will ultimately be observable during merger to groundbased detectors (see Section 1.5.2.1). Studies based on cosmological simulations of galaxies at z = 0 find that the present-day dwarf galaxies can accommodate a larger BH+BH merger rate compared to massive galaxies (O'Shaughnessy et al., 2017). Specifically, for massive BH+BHs similar to GW150914, about 40% of mergers are expected to be in galaxy progenitors of Milky Way-like systems and the rest in smaller satellite or isolated dwarf galaxies (Marassi et al., 2019). This translates into a large number of potential hosts within the estimated LISA horizon distance. Not only BH+BHs formed from the evolution of isolated binaries, but also BH+BHs formed in extragalactic globular clusters may be detectable by LISA, with initial studies predicting the number of such sources to be in the range of 1 10 2 (Kremer et al., 2018a). \nGiven LISA's selection effects, extra-galactic LIGO-like BH+BH are expected to have quite low signal-to-noise ratios ( ∼ 10 ). For this class of stellar binaries, Buscicchio et al. (2021) report sky localisation errors of a few tens of squared degrees and constrains on the detector-frame chirp mass down to ± 0 . 01 M ⊙ . They also show that at 10 mHz the eccentricity for these binaries can be measured down to 10 -3 , while the merger time can be determined within a time window of 1 hour. \nFor sources that form through isolated binary evolution, the rates (such as those mentioned above and in Table 4) are often estimated with the population synthesis approach. There are many uncertainties that affect the rate calculations. These uncertainties can be divided in five broad categories: (i) binary evolution physics (e.g., CE evolution, mass transfer stability, massaccretion efficiency, etc.), (ii) stellar evolution physics (e.g., metallicity dependent stellar winds, core-collapse mechanisms, natal kicks, pair-instability SN and pulsational pair instability, etc.), (iii) initial stellar and binary properties (e.g., initial mass function, binary fraction, initial distribution of separation, mass ratio, eccentricity, etc.), (iv) different assumptions for the Galactic spacial distribution (thin/thick disc and bulge) and star-formation history and (v) LISA selection effects (e.g., GW foreground, mission length, sensitivity curve and SNR detection threshold). \nAs discussed in Section 1.3.2, LISA sources may also form dynamically in dense stellar environments such as globular clusters (and may have markedly different features compared to sources that form through isolated binary evolution, Section 1.3.1). In Table 5, we show the estimated number of sources expected to be found in globular clusters in a Milky Way-like galaxy (Kremer et al., 2018a). The number of detectable LISA BH+BHs originating from young massive clusters and open clusters is expected to be several tens to ∼ 100 (or about 0 . 5 -3 times the density of those clusters in the local volume in units of Mpc -3 ; Banerjee, 2020). Merger rate estimates for GW sources produced in dynamical environments from cluster simulations, can \nTable 4: Estimated absolute number of compact binaries from isolated binary evolution in the Milky Way. The columns show the source type, the total number of binaries in the galaxy at any frequency and the total number of estimated sources detected by LISA. We report values from indipendent studies which assume different LISA mission lifetimes and SNR. WD+WD models assume a frequency range 0 . 5 -10 mHz while models for the other sources assume a frequency range 0 . 1 -10 mHz. At lower frequencies the total number of LISA sources is so high that it might become impossible to distinguish individual sources from the GW foreground. The ranges of the expected intrinsic number of each binary type are extracted from Nissanke et al. (2012); Breivik et al. (2020a); Belczynski et al. (2010a); Kruckow et al. (2018); Nelemans et al. (2001c); van Oirschot et al. (2014); Lamberts et al. (2018) while the ranges of the expected number of LISA sources are extracted from Nelemans et al. (2001b); Korol et al. (2017); Lamberts et al. (2019); Korol et al. (2018); Liu et al. (2010); Ruiter et al. (2010); Tauris (2018); Breivik et al. (2020a); Belczynski et al. (2010a); Kruckow et al. (2018); Lau et al. (2020); Andrews et al. (2020); Sesana et al. (2020). \n∼ \n- \nalso be influenced by many uncertain physical processes, some of which are common to the ones outlined for sources formed via isolated binary evolution. For instance, natal kicks for compact objects have a direct impact on retention of those objects in stellar clusters (Morscher et al., 2013; Arca Sedda et al., 2018; Webb et al., 2018; Pavlík et al., 2018; Banerjee et al., 2020) which influences the number of dynamically formed binary systems. The number of compact objects that form in dense environments also depends on their metallicity and the initial mass function of their stars which may vary with their formation environment (Dabringhausen et al., 2009; Geha et al., 2013; Krumholz, 2014; Chruślińska et al., 2020). Additionally, dissolved or tidally disrupted open and globular clusters can also contribute to the number of GW sources that may have dynamically formed (Muratov and Gnedin, 2010; Fragione et al., 2018; Giersz et al., 2019). \nApart from population synthesis-like simulations, another approach to predict the expected LISA rates is to derive empirical estimates from the already observed population of sources. For NS+NSs for instance, one can use the inferred merger rate coming from the known Galactic NS+NS population, and accounting for survey selection effects (Phinney, 1991; Kim et al., 2003), or the inferred merger rate from LIGO-Virgo (The LIGO Scientific Collaboration et al., 2020), to predict that LISA should be able to detect 50 -300 NS+NSs in the Milky Way (Andrews et al., 2020). In a similar manner, based on O1 LIGO-Virgo detections, it was estimated that LISA maybe able to detect up to ∼ 50 BH+BHs (Sesana, 2016), but the inferred BH+BH merger rate density decreased in the most recent O3 run by a factor of ∼ 2 . Empirical estimates of the Galactic NS+WD population have been derived, based on the observed pulsar population, with 100 -150 being predicted to be observable by LISA (Tauris, 2018).", '1.5.1 Synergies with EM observations': 'Contributors: Thomas Kupfer, John Tomsick, Nicole Lloyd-Ronning, John Quenby, \nCoordinators: Thomas Kupfer Thomas Tauris, Thomas J. Maccarone \nTable 5: Estimated absolute number of compact binaries in globular clusters in the Milky Way. The columns show the source type, the total number of binaries in the galaxy at any frequency and the total number of estimated sources detected by LISA assuming a frequency range of 10 -5 -1 Hz and a mission lifetime of 4 years with a SNR ranging between 2 and 7 (Kremer et al., 2018a). \n∼ \n× \n- \nMany new verification binaries are expected to be discovered before the launch of LISA, in particular with wide-field optical surveys such as ZTF, BlackGEM and LSST on the Rubin Observatory. Once flying, LISA will be complemented with surveys across different frequency bands (radio to gamma-rays). New sources discovered by LISA can be studied with the next generation of follow-up facilities such as ESO/ELT, CTA, SKA, ngVLA or even Athena as well as smaller space missions which could be approved, built and launched in the early 2030s (eXTP or STROBE-X). This provides a plethora of large-scale follow-up resources for detailed multimessenger studies but it requires a well planned follow-up strategy to generate the most useful results. \nThe large number of facilities running in the 2020s and 2030s will provide a bright future for the research on compact Galactic binaries, with hundreds of additional EM discovered systems ready to be studied in detail through EM+GW observations as soon as LISA is operational. A significant sample of binaries, observed with EM+GW observations, will open up possibilities to explore and study astrophysical phenomena which are crucial to our understanding of the Universe. This includes the long-standing questions of the progenitors of SNe Ia, the formation and evolution of compact objects in binaries and accretion physics under extreme conditions.', '1.5.1.1 UV/Optical/IR observations': 'Previous studies predict that we will be able to observe several thousand Galactic binaries in both GW and optical emission (Littenberg et al., 2013; Korol et al., 2017). A subset of the known UCBs have orbital periods that lie in the LISA band and these will be individually detected by LISA due to their strong GW signals, some on a timescale of weeks or a few months. Combined GW and EM multi-messenger studies of UCBs will allow us to derive population properties of these systems such as masses, radii, orbital separations, and inclination angles but in many cases EMobservations are required to complement GWs and break degeneracies in the GW data. Shah et al. (2012, 2013); Shah and Nelemans (2014b); Kupfer et al. (2018) present several studies on how EM observations can complement LISA GW data and vice versa. The GW amplitude and inclination is strongly correlated, but the GW amplitude can be improved by a factor of six when including EM constraints on the inclination and the sky position and inclination can reduce the uncertainty in amplitude by up to a factor of 60 (Shah and Nelemans, 2014b). \nAdditionally, knowing the distance from EM, e.g. from parallaxes measured by Gaia, will help to derive a chirp mass from GWs even for non-chirping sources because for many LISA sources only the frequency and amplitude will be measured. This leaves a degeneracy between chirp mass and distance and inclination, since a more massive binary further away can have the same amplitude as a lower mass one that is closer by and inclined systems closer by look like face on systems further away (Shah et al., 2012; Shah and Nelemans, 2014a). \nIf the chirp mass has been measured from GWs using e.g. Gaia parallaxes and the mass \nratio has been measured from EM, through radial velocities or ellipsoidal deformation of one component, both measurements can be combined to the measure the masses of both components with a few percent precision. Comparing the measured orbital decay with the predicted orbital decay from general relativity will allow us to measure the effects of tides compared to GWs. Tides are predicted to contribute up to 10 % of the orbital decay (Piro, 2011) but has not been measured so far and is very difficult with GW and EM alone. Some of the known AMCVn and detached WD+WD binaries have well constrained distances from Gaia (Kupfer et al., 2018; Ramsay et al., 2018). Measuring the distance from EM constrains the uncertainty in chirp mass to 20%, whereas adding the period derivative ˙ P reduces it to 0.1% (see e.g. Fig. 12). A GW chirp mass measurement would provide the first detection of tidal heating in a merging pair of WDs from the deviations in predicted ˙ P (Shah and Nelemans, 2014b). With a large enough sample of WD+WD binaries whose chirp masses can be determined, we can plausibly extract constraining information about CE phase evolution physics. In studying massive WD+WD binaries that are likely progenitors of merger-induced collapse NSs (Ruiter et al., 2019), chirp mass distributions had different shapes depending on the adopted CE phase prescription in the binary evolution population synthesis model. \nSeveral studies have shown that spectral and photometric analysis of detached WD+WD can provide precise sky positions, orbital periods and in some cases mass ratios, inclinations, and the rate of orbital period decay (e.g. Maxted et al., 2002; Brown et al., 2011; Hermes et al., 2012; Burdge et al., 2019a). All of the known verification binaries have precise sky positions and orbital periods and five systems have a measured orbital decay from photometric monitoring (Kupfer et al., 2018). SDSSJ0651, a 12 min orbital period detached WD+WD (Brown et al., 2011), and ZTFJ1539, a 7 min orbital period detached WD+WD (Burdge et al., 2019a), are prime example of what can be accomplished. Using only one year of eclipse timing measurements, Hermes et al. (2012) found an orbital decay of ˙ P = ( -9 . 8 ± 2 . 8) × 10 -12 s s -1 in SDSSJ0651 which has not been updated since then. Burdge et al. (2019a) used photometric data from PTF/iPTF and ZTF covering a total of ten years and measured a very precise orbital decay of ˙ P = ( -2 . 373 ± 0 . 005) × 10 -11 s s -1 . However, neither of the two systems has a precision of WD component masses good enough from optical observations to see if their ˙ P differs from the predictions of the theory of General Relativity (GR). GW observations can solve that. Tidal theory predicts a 10% deviation from GR if the WDs are tidally heating up. Which means that combining EM+GW observations from LISA will allow a measurement of the amount of tidal heating in these merging pairs of WDs for the first time. \nCombined EM+GW observations of the Galactic WD+WD population will help to solve another major problem in astrophysics: the SNeIa progenitor problem. Although only the thermonuclear explosion of a WD following the interaction with a binary companion can explain the observed features in the SN light, much less is known about their progenitors. Recent results have shown that SN Iae show a large range of explosion energies and decay times, photometric and spectroscopic signatures indicating different progenitor systems (Jha et al., 2019). Several different explosion scenarios are under discussion, including the merger and subsequent explosion of an ultra-compact WD+WD system (double-degenerate model) or the explosion triggered by ignition of an helium shell accreted from a helium star in a UCB (double-detonation scenario). However, the number of known progenitor systems is limited. Rebassa-Mansergas et al. (2019) studied the probability of finding WD+WD progenitors of SNe Ia using a binary population synthesis approach, and found that the chance of identifying such progenitors purely in EM data is ∼ 10 -5 . These include both double-lined spectroscopic binaries and the eclipsing systems. Even with the next generation of 30 m class telescopes, the probability for detection only goes up by a factor of ∼ 10 . Korol et al. (2017) predicts that LISA will individually resolve ∼ 25 , 000 detached WD+WD systems including the most massive systems. EM follow-up observations in combination with GW measurements will allow us to measure masses of individual systems and find and characterize the population of double degenerate SN Ia progenitors. \nKilonovae (see Metzger, 2019) are optical/IR emission accompanying the merger of NS+NS, possibly NS+BH, and in special cases, WD+WD mergers (Rueda et al., 2019). Although LISA is not sensitive to the actual merger that can produce a kilonova, it is sensitive to the GW emission from their progenitors. Therefore it is worthwhile to consider kilonova events in the nearby universe because they allow constraints on these degenerate stellar populations.', '1.5.1.2 X-ray observations': "Many of the LISA sources also emit X-rays, thus allowing for a number of joint LISA + Xray investigations. The donor stars in UCXBs appear to be a mixture of C/O-core and He-core WDs and abundance measurements can help identify their formation scenario (Nelemans et al., 2010). In the oxygen-rich systems, oxygen is the dominant coolant in the accretion discs, and the iron emission lines are suppressed; the strength of the iron lines is broadly in agreement with the thermonuclear burst properties of the sources, strengthening the case that this donor classification process works reasonably well even in its simplest form (Koliopanos et al., 2020). However, in some cases, the abundances of the WD inferred from X-ray data are at odds with those inferred from Type I burst properties. In a few cases, the inference has been made from neon-to-oxygen rations (Juett et al., 2001), and for this scenario, it has been shown that there is a channel of binary evolution that allows a He-core WD to have such neon-to-oxygen ratios (in't Zand et al., 2005). Alternatively, Bildsten et al. (1992) show that spallation of CNO elements in a NS atmosphere is quite likely. The combination of LISA measurements with X-ray (and optical) abundance measurements thus opens a window to determining which of these scenarios is correct. If the apparent C/O-core WD donors are paired with high mass NSs, then the spallation scenario is strongly preferred. \nMore detailed X-ray spectroscopy should potentially be able to make detailed abundance estimates for the donor stars, allowing, e.g., identification of systems in which the CNO processing may not have gone to completion, and perhaps estimating the time of formation through estimates of the abundances of non-CNO elements, which could yield the initial metallicity of the star. The X-ray measurements are essential given that a large fraction of the UCXBs are located in globular clusters, or deep in the Galactic Plane where ultraviolet and optical spectroscopy are more challenging. Substantive work has also been done in the optical wavebands (see e.g. Nelemans et al., 2004a, 2006). Furthermore, when combined with LISA data the UCBs then may provide a means of testing how conservative accretion onto NSs is; C/O-core WD donors must start at masses of at least ≈ 0 . 5 M ⊙ , but are generally observed with masses of 0 . 1 M ⊙ or less. If the early stages of mass transfer in these systems are conservative, the NSs should typically be about 1 . 8 M ⊙ , while if they show much lower masses, this implies that the mass transfer was strongly non-conservative. \nAdditionally, for 4U 1820 -30, X-ray measurements provide a straightforward way to monitor the source's period derivative (Tan et al., 1991), as the source is deep in the potential well of a globular cluster whose gravitational acceleration leads to its negative period derivative. It is easy to track the source's orbital period in the X-ray band (Stella et al., 1987; Tan et al., 1991), and hard in other bands, due to the crowding in the cluster. With more intensive X-ray data, the period derivative of 47 Tuc X-9, the best candidate BH+WD binary in the Milky Way (Bahramian et al., 2017), could be tracked. Imposing these constraints, which are likely to lie outside the range of normal templates, can help improve the quality of the LISA GW detection. Globular clusters are likely good hosts for LISA sources, but the globular cluster sources' periods have come from ultraviolet photometry (Dieball et al., 2005; Zurek et al., 2009) and the best non-cluster source's period comes from optical spectroscopy (Madej et al., 2013). Additional intensive monitoring campaigns would be required for the period derivative to be estimated. \nFor the AM CVn systems, X-ray emission is also valuable. The same abundance issues can be studied in the X-ray band in AM CVn systems, although primarily from the emission lines from the boundary layer of the accretor, rather than disc reflection. Relatively short period AM CVn \nsystems will be detectable to large distances, where reddening may be important, and in these cases, restricting the set of sources to those which are in a reasonable range of fluxes. For the faintest AM CVn sources with periods less than half an hour, the X-ray luminosities are typically about 10 31 erg s -1 (e.g. Nelemans et al., 2004b; Strohmayer, 2004; Ramsay et al., 2005, 2006; Zurek et al., 2016), meaning that eROSITA should detect them to distances of about 3 kpc. Combining with radio surveys to remove background Active Galactic Nuclei (AGN), and optical surveys to remove foreground stars and CVs should then yield a much more manageable list of candidates for high time resolution optical follow-up (which usually has limited fields of view) than without the X-ray data and potentially add more LISA verification sources. \nX-rays are also likely to provide the best EM distance estimators for many of these UCXBs (see Fig. 8) complementing LISA data. Some are located in globular clusters, where the cluster distance can be used. None of the Galactic field UCXBs is bright enough for Gaia in the optical, and most are also too faint for radio parallaxes with current facilities. Thermonuclear bursts with radius expansion can be used to estimate the Eddington luminosities (Kuulkers et al., 2003), and these can then be used in conjunction with the GW estimates of the masses to establish self-consistent properties for the sources. For the persistent sources that burst in an appropriate manner, these data are already in hand, but obtaining such data for transients would require instruments with large collecting area and small deadtime (e.g. NICER, STROBEX, eXTP Gendreau et al., 2016; Ray et al., 2019; Zhang et al., 2019). 3 The other approach that can be used to estimate distances is that of dust-scattering halos (Trümper and Schönfelder, 1973), something that requires good angular resolution, good collecting area, and the ability to observe bright sources; while Chandra has done some work in this area, Athena should be able to help dramatically (Corrales et al., 2019). \nSome UCXBs may contain BHs as well. The first strong globular cluster BH candidate (Maccarone et al., 2007) in NGC 4472 is an ultracompact system (Zepf et al., 2008), probably with an orbital period near 5 minutes, and 47 Tuc X-9 (Miller-Jones et al., 2015; Bahramian et al., 2017) is also a strong candidate ultracompact BH X-ray binary. At the shortest orbital periods, BH+WD binaries should be detectable by LISA to distances of a few megaparsecs. For these cases, imaging X-ray data would be needed, along with follow-up optical spectroscopy to look for [O III] nebulae as well as hydrogen emission similar to that in the globular cluster RZ 2109 in NGC 4472 (Zepf et al., 2008; Steele et al., 2014; Dage et al., 2019). In the Milky Way, UCXBs with BH accretors at relatively long orbital periods could be quite faint X-ray sources in quiescence (being considerably fainter than accreting NSs at the same mass transfer rate due to advection dominated accretion, Narayan and Yi 1994). They could also exhibit only rare outbursts, meaning that sensitive X-ray observations would be needed to detect their counterparts. Furthermore, these objects may be preferentially in globular clusters, meaning that excellent angular resolution, from Chandra or a Chandra successor mission like Lynx or AXIS would be needed to find their counterparts. If some new BH UCXBs are discovered with X-ray outbursts, they may become bright enough to make BH spin estimates using reflection and/or disc continuum modelling. \nFor most of the topics related to accretion, there is a need for developing better spectral models that treat unusual abundances. Development of reflection models that include both the reflection from the surface of the WD, and discs made from hydrogen-poor material is thus vital. It has already been found that details of how atomic physics is incorporated into the disc models can affect inferred spins and abundances (García et al., 2016; Tomsick et al., 2018). \nX-ray observations are also vital for understanding the detached systems with NS members. In most of these systems, the older NS will have experienced significant spin-up, and will be a millisecond pulsar. Pulsar beam opening angles are larger at high energy than at radio wavelengths (a phenomenon exhibited by objects like Geminga (Halpern and Holt, 1992) and many of the \nFermi-discovered pulsars), meaning that some fraction of these objects will be radio-quiet pulsars (Marelli et al., 2015). X-ray observations will then provide the most comprehensive means for estimating the spins of these systems and determining what fraction of them have become recycled. Furthermore, in combination with radio searches for pulsations, having a gravitationally-selected sample will allow a clean determination of the ratio of pulsars with radio and X-ray emission, allowing an important constraint for developing a full picture of the pulsar beam geometry. In an ideal case we may identify an object with thermal cap emission from the NS, such that pulse-profile measurement and modelling could be done to estimate its radius. If this comes in conjunction with sufficiently good LISA GW measurements to provide an independent, precise, estimate of the NS's mass, this would give a constraint on the equation of state for NSs, even from a single object (Watts, 2019).", '1.5.1.3 Radio observations': 'The synergies from joint, multi-messenger observations of radio pulsar binaries entering the LISA band are very promising and will provide significantly more information than observations in the EM or GW bands alone (see Fig. 8). Such benefits include better measuring the orbital inclination angle (Shah et al., 2012) and sky position (Shah et al., 2013) and potentially even constraining the NS mass-radius relation to within ∼ 0 . 2 % (Thrane et al., 2020). Additionally, radio astrometry can give parallax distances out to a few kpc already, and with ngVLA (Murphy et al., 2018), that should increase dramatically. Most LISA sources would be nearby enough that with ngVLA, one would be able to get 10% or better geometric parallaxes over about 3/4 of the sky. \nBinary NSs in NS+NS and NS+WD systems enter the LISA band at a GW frequencey of order 1 mHz (depending on their distance), corresponding to orbital periods of about 30 min . Doppler smearing of radio pulsations from pulsars in such tight binary orbits (Eatough et al., 2013) could cause a selection bias against detection of e.g., rapidly spinning millisecond radio pulsars in many previous and present day acceleration searches (at least for dispersion measures, < 100 cm -3 pc ). However, using neural networks, Pol et al. (2020) develop accurate modelling of the observed binary pulsar population and argue for a ∼ 50 -80 % chance of detecting at least one of these systems with P orb ≤ 15 min using data from surveys with the Arecibo radio telescope, and ∼ 80 -95 % using optimal integration times of ∼ 50 s in the next several years. The chances of a radio detection of a binary pulsar in the LISA GW band is expected to be significantly enhanced by the completion of the Square-Kilometre-Array (Keane et al., 2015). \nIt has been argued (Pol et al., 2020) that unequal mass NS+WD systems are easier to detect compared to the usually near-equal mass NS+NS systems. It should be kept in mind that RLO from these (often bloated) WD companions begins when P orb has decreased to 25 -15 min , depending on their temperature (Tauris, 2018) (see also Figs. 4 and 5). This will exclude radio detection of such pulsar binaries once accreted plasma enters the NS magnetosphere.', '1.5.1.4 Particle observations': "For high-frequency GW detections, there are prospects for detection of neutrino's or cosmic rays, from jets produced in mergers or from supernovae (Adrián-Martínez et al., 2016). For LISA, the prospects are not so clear, even though associations of LISA GW sources with AGN jets and tidal disruption events could be possible.", '1.5.2 Synergies with other GW detectors': 'Coordinators: Lijing Shao; Paul Groot \nContributors: Ilya Mandel (1.5.2.1), Alberto Sesana, Emanuele Berti, Lijing Shao (1.5.2.4), Davide Gerosa (1.5.2.1), Pau Amaro Seoane, Paul Groot, Thomas Tauris (1.5.2.2), Valeriya Korol (1.5.2.3) \nFigure 12: Upper panel : GW frequency derivative, | ˙ f GW | as a function of GW frequency, f GW for the UCXB shown in Figs. 4 and 5. The black coloured points correspond to the inbound leg (orbital shrinking) and the blue points correspond to the outbound leg (orbital expansion, after reaching the orbital period minimum) including mass transfer/loss from the system and finite-temperature effects of the WD donor. Each point represents a binary stellar MESA model. The green solid circle indicates the onset of the UCXB stage. (Before this point, the system is a detached NS+WD binary). Figure from Tauris (2018). Bottom panel : pure general relativistic (GR, blue) and astrophysical (orange) chirps as a function of GW frequency for WD+WD (AM CVn) systems Breivik et al. (Figure from 2018). The astrophysical contribution from mass transfer and tides leads to a significant deviation from the contribution from GR alone and will cause the systems to widen their orbit upon mass transfer. The decoupling of the astrophysical chirp from the GR chirp will be possible with combined measurements from LISA and Gaia. The error bars show the anticipated 1 σ measurement errors. On average, Breivik et al. (2018) find that about 50 AM CVn systems in the Mikly Way with P orb < 800 s have resolvable GR and astrophysically driven chirps. \n<!-- image -->', '1.5.2.1 High-frequency GW merger precursors seen by LISA': "LISA has a unique capability of covering the full frequency spectrum for stellar-mass binary BHs and NSs when combined with the higher-frequency ground-based GW detectors advanced LIGO (Aasi et al., 2015) and Virgo (Acernese et al., 2015), and their third generation successors such as the proposed Einstein Telescope (Punturo et al., 2010) and Cosmic Explorer (Abbott et al., 2017a). \nSome individual sources can be tracked on human timescales from the LISA band to the ≳ few Hz ground-based detector sensitive frequency band (Sesana, 2016). The GW driven merger timescale for a circular binary with components of equal mass m from a starting frequency f is (Peters, 1964a) \nτ GW ≃ 5 ( f 0 . 01 Hz ) -8 / 3 ( m 68 M ⊙ ) -8 / 3 yr . (3) \nThus, a signal like GW190521 (Abbott et al., 2020c) could be tracked from 10 mHz to merger across the the full range of frequencies with the combination of LISA and ground-based detectors. Rate estimates have been presented by Shannon et al. (2015); Kyutoku and Seto (2016); Sesana (2016, 2017); Gerosa et al. (2019); Moore et al. (2019), with predictions ranging from 0 to roughly a dozen detections during the LISA mission. \nThe high mass and correspondingly rapid orbital evolution of IMBHs with masses in the 100 -1000 M ⊙ range makes them particularly appealing targets for tracking across the LISA and ground-based detector frequency bands. IMBHs with these masses are a challenge for EM observations: their dynamical signature is relatively insignificant, while their X-ray emission can be confused with that of super-Eddington accretors (e.g., Miller and Colbert, 2004; Feng and Soria, 2011; Greene et al., 2019). On the other hand, IMBH mergers have been proposed in the context of both isolated binary evolution of very massive stars (Belczynski et al., 2014) and globular cluster dynamics (Amaro-Seoane and Freitag, 2006). The latter can also be responsible for intermediate-mass ratio inspirals of stellar-mass compact objects into IMBHs (Mandel et al., 2008; Haster et al., 2016a). Meanwhile, hierarchical mergers of few-hundred M ⊙ seeds have been proposed as seeds of today's massive BHs (Volonteri et al., 2003a, see Chapter 2). Joint observations with LISA and third-generation ground-based detectors (Sesana et al., 2011a; Gair et al., 2011) would provide the perfect tools for studying these elusive IMBHs. \nObservations of the same individual source across a broad range of frequencies can improve the accuracy of source parameter measurement. Some parameters are likely to be best measured at low frequencies. For example, sky localisation accuracy depends on timing precision (Fairhurst, 2009; Wen and Chen, 2010; Grover et al., 2014). The sky localization accuracy can be estimated as the timing accuracy divided by the light travel time across the detector baseline (Mandel et al., 2018), which is ∼ astronomical unit (AU) for LISA, yielding a relative position error of: \nσ θ ∼ 0 . 025 ( 0 . 01 Hz f ) ( 8 ρ ) , (4) \nwhere ρ is the detection signal-to-noise ratio. For heavier sources that can evolve faster than the LISA observing duration, the LISA frequency bandwidth f bandwidth should be used in place of f . The angular resolution scales inversely with baseline. Therefore, despite the lower observing frequency (lower bandwidth), for high SNR sources, LISA sky localisation is likely to be superior to the capabilities of ground-based detectors, whose baseline, even in a network, is limited by the size of the Earth (unless the signal is sufficiently long-lived to allow the effective baseline to be extended by the detector motion over the duration of the observation). \nOn the other hand, some source parameters will be better measured at higher frequencies, allowing ground-based detectors to provide complementary information to LISA observations. These include measurements of the ringdown of the post-merger BH, which yield the final mass and spin, and the tidal effects for NSs. \nYet other measurements could benefit from the joint constraints placed by low-frequency and high-frequency observations. These include measurements of spin magnitudes and spin-orbit misalignment angles, which could carry information about formation scenarios (e.g., Gerosa et al., 2013; Vitale et al., 2017; Stevenson et al., 2017a; Zevin et al., 2017; Farr et al., 2017; Gerosa et al., 2018a). Spin-orbit and spin-spin coupling enter the waveform at higher post-Newtonian orders in an expansion in the orbital frequency (Poisson and Will, 1995), and so may be better measured at higher frequencies by ground-based detectors. On the other hand, massive binaries like GW190521 may spend a million cycles in the LISA band (only 4 cycles were observed in the LIGO band when this signal was detected in 2019; Abbott et al. 2020c). Further analysis is necessary to explore the quantitative benefits of LISA for parameter estimation of such signals (but see e.g. Vitale, 2016; Moore et al., 2019; Mangiagli et al., 2019; Cutler et al., 2019) \nLower-mass GW sources such as double NSs (Lau et al., 2020; Andrews et al., 2020) will not be individually trackable on a human timescale from the LISA band to the band of ground-based detectors. However, they may still benefit from tracking the entire population of sources as the sources evolve from the LISA frequency band to the frequency band of ground-based detectors. For example, binaries circularise through GW emission (very roughly, the eccentricity scales inversely with the increase in frequency), making eccentricity challenging to observe with groundbased detectors (e.g., Romero-Shaw et al., 2019; Lenon et al., 2020). Thus, LISA observations at lower frequencies, where eccentricities are still significant, could help to distinguish compact binary formation scenarios (Breivik et al., 2016; Nishizawa et al., 2016, 2017). \nThis can be further aided by the detection of a stochastic background from a superposition of GWs emitted by multiple individually unresolvable binaries (see Section 1.2.2.5) . For circular binaries, the stochastic background should be a simple power-law in frequency, and any deviations from that could indicate the emergence of new binaries, particularly eccentric binaries, at high frequencies. Moreover, the combined low-frequency and high-frequency stochastic background observations may make it easier to subtract the astrophysical background and reveal a possible GW background of cosmological origin (e.g. Mandic et al., 2012; Lasky et al., 2016; Callister et al., 2016). \nLISA precursors to ground-based detector mergers could also have important repercussions in fundamental physics, allowing stringent tests of the BH no-hair theorems, as well as more stringent bounds on low-Post-Newtonian deviations from GR (Toubiana et al., 2020a; Barausse et al., 2016; Tso et al., 2019; Gnocchi et al., 2019; Carson and Yagi, 2020; Shao et al., 2017).", '1.5.2.2 Dual-line GW sources': "A possibility in upcoming GW astronomy will be the potential discovery of a dual-line Galactic GW source (Tauris, 2018), where ground-based detectors detect the continuous high-frequency GWemission from the rapid spinning (recycled) NS (Andersson, 2019) and LISA simultaneously detects the gravitational damping of the system's orbital motion via continuous low-frequency GW emission. Such a system could very well be a UCXB (e.g. van Haaften et al., 2012; Heinke et al., 2013). Combining the expressions for the strain amplitudes of the ground-based and LISA observations ( h spin and h orb respectively) yields (Tauris, 2018): \nI zz ε = √ 2 5 ( √ G 2 π ) 4 / 3 ( f 1 / 3 orb f spin ) 2 M 5 / 3 ( h spin h orb ) . (5) \nOnce the right-hand-side of this equation is determined observationally, and assuming that the NS mass, M NS can be determined from the chirp mass, M (see required assumptions on component mass determinations in Tauris, 2018), constraints can be made on the NS moment of inertia, I zz , and thus the NS radius (Ravenhall and Pethick, 1994). Suvorov (2021) recently examined the dual-line detectability of tight Galactic binaries, and found that at least two of the known systems (4U 1820-30 and 4U 1728-34) may be visible to both ground-based and space-based instruments simultaneously. Although only measuring the moment of inertia in combination \nwith the ellipticity, ε , it will still help in pinning down the long sought-after equation of state (EOS) of NS matter. The maximum spin rate and ε for accreting NSs (Andersson, 2019) remain to be constrained firmly.", '1.5.2.3 TianQin': 'TianQin is a space-based GW observatory conceived as an equilateral triangle constellation of three drag-free satellites with frequency sensitivity at 10 -3 -10 -1 Hz (Luo et al., 2016), between LISA and DECIGO. Unlike LISA, TianQin will follow a geocentric orbit with a radius of about 10 5 km (Hu et al., 2018; Ye et al., 2019). Its constellation plane will be nearly perpendicular to the ecliptic plane and will have a fix orientation pointing toward RX J0806.3+1527 (Strohmayer, 2005), a 5 min orbital period Galactic binary that is expected to be the strongest GW source among currently known systems (Kupfer et al., 2018). Planned for the launch around 2035, TinQin will see the same GW sources as LISA (Mei et al., 2020). Consequently, many synergies can be envisioned between the two missions. For instance, TianQin and LISA will simultaneously detect several thousand Galactic WD+WD binaries, which will improve the parameters estimation including the amplitude, inclination and sky localisation for these binaries (Huang et al., 2020).', '1.5.2.4 Mid-band observatories, e.g. DECIGO': 'After the discovery of GW150914 (Abbott et al., 2016c), it was realized that massive stellarmass BHs will be detectable in both LISA and LIGO/Virgo bands (Sesana, 2016; Amaro-Seoane and Santamaría, 2010). However, the SNRs are not expected to be large in the mHz band, and because of the use of template bank searching, in order to claim a confident detection, BH+BH signals in LISA require a larger SNR threshold than 15 (Moore et al., 2019). Fortunately, these sources will have large SNRs if they are seen in the decihertz band (Isoyama et al., 2018; Arca Sedda et al., 2020a; Liu et al., 2020). DECihertz laser Interferometer Gravitational wave Observatory (DECIGO) is a representative GW detector in the relevant frequency band (Yagi and Seto, 2011; Kawamura et al., 2011, 2020). Studies showed that, not only will observations in the decihertz band provide profound insights to astrophysics (see Arca Sedda et al., 2020a, for a comprehensive discussion), they will also provide unprecedented playgrounds for fundamental physics (e.g. testing the dipolar radiation (Barausse et al., 2016; Liu et al., 2020). The midband observations of decihertz frequency are natural means to bridge the gap between LISA and LIGO/Virgo observatories.', 'Coordinators: Irina Dvorkin': 'Contributors: Emanuele Berti, Sylvain Chaty, Astrid Lamberts, Alberto Sesana, Kinwah Wu (1.6.3), Shenghua Yu, Shane Larson, Irina Dvorkin (1.6.1, 1.6.3, 1.6.5), Pau Amaro Seoane, Giuseppe Lodato, Xian Chen, Valeriya Korol (1.6.2), Silvia Toonen (1.6.5)', '1.6.1 How to distinguish between different compact binaries?': 'One of the outstanding challenges of LISA will be to analyze a datastream that consists of multiple overlapping signals from astrophysical and possibly cosmological sources as well as instrumental noise. Since many LISA sources will remain in band for multiple orbits, from several days or months up to the entire duration of the mission, there will be an overlap between multiple sources in any given data stretch. Data analysis techniques suitable for this unique problem are currently under development, including in the context of the LISA Data Challenge (Babak et al., 2010; Cornish and Shuman, 2020; Littenberg et al., 2020). A standard procedure \nthat allows us to extract WD+WD signals from a noisy datastream uses waveform templates that span a large parameter space (Owen, 1996). The most studied case (and the only class detected so far by LIGO-Virgo) is that of isolated compact binaries (see Section 1.2.2), which are characterized by the component masses, the distance to the binary, its position on the sky and the orbital eccentricity, as well as the orbital frequency. Contrary to the case of groundbased interferometers, these binaries are long-lived LISA sources. In other words their orbital evolution timescale due to the emission of GW is very slow compared to the mission duration. This will allow us to collect data from many cycles of each binary, increasing the SNR, but also puts stringent requirements on the accuracy of the template waveform. The waveform of quasi-monochromatic sources, such as WD+WD binaries, is relatively simple and can be quite accurately described by the leading order terms in the orbital dynamics (Littenberg et al., 2020). On the other hand, binaries that evolve in the LISA band (such as BH+BH) require a more detailed computation to higher Post-Newtonian order (Mangiagli et al., 2019). Accurate waveform templates are crucial for measuring the source parameters and distinguishing between various source classes since any error in the predicted phase of the template waveform will accumulate over the many cycles the binary stays in band. \nThe main parameter that can help to distinguish the different classes of isolated compact binaries is the chirp mass of the system. In order to establish the class of a quasi-monochromatic source one may use the fact that the chirp mass distribution of WD+WD binaries peaks around ≃ 0 . 25 M ⊙ (Korol et al., 2017) with the tail up to 1 M ⊙ , while the chirp mass of NS+NS systems is expected to lie around ≃ 1 . 2 M ⊙ . BH+NS and BH+BH systems will have higher chirp masses. However, high-mass WD+WD systems at the tail of the distribution with chirp masses of up to ≃ 1 . 2 M ⊙ may be confused for a NS+NS or a NS+WD binary. Similarly, BH+NS binaries may be confused with NS+NS if the NS has an extremely high mass, or the BH has an extremely low mass. Indeed, the discovery by LIGO-Virgo of GW190814, a binary consisting of a 23 M ⊙ BH and a 2 . 6 M ⊙ compact object (Abbott et al., 2020b) is very difficult to interpret: the secondary component is either the lightest BH or the heaviest NS discovered to date. \nAdditional clues as to the identity of the source are somewhat model-dependent, although priors from copious ground-based observations will help with NS+NS, NS+BH, BH+BH scale events. Thus, it may be possible to use eccentricity measurements to distinguish between WD+WDs and NS+NSs (Lau et al., 2020). Since WD+WDs are expected to have formed via isolated binary evolution, their progenitors are expected to have circularised via multiple mass transfer episodes (see Sec. 1.3.1.1). This assumption is supported by the lack of observed eccentric galactic WD+WD binaries. On the other hand, NS+NSs in the LISA band could have measurable eccentricities: e.g. in the fiducial model by Lau et al. (2020) half of LISA NS+NS sources have eccentricities e > 0 . 1 . Thus, a detection of an eccentric source with chirp mass of around ≃ 1 . 2 M ⊙ can be interpreted as a likely NS+NS. Nevertheless, some rare eccentric WD+WD can be produced in globular clusters of the Milky Way, or via triple interactions (e.g. Kremer et al., 2018a). \nThe case of interacting binaries (see Sec. 1.2.3) is potentially even more complex, since their orbital evolution is influenced not only by gravity, but also mass transfer and magnetic braking, which lead to qualitatively different waveforms (such as anti-chirping phases) depending on the evolution stage of the binary (e.g. Kremer et al., 2017; Tauris, 2018). For example, as discussed in Sec. 1.2.3.2 and 1.2.3.1, AM CVns and UCXBs can be detectable by LISA either during their inspiral phase (when the binary components are detached and the orbits shrinks) or during mass transfer (when the orbit expands). The upside of this complexity is that anti-chirping signals are easier to distinguish from isolated compact binaries. \nClearly, an EM counterpart to a GW detection will help to identify the source. Indeed, as discussed in Sec. 1.5.1, EM observations can help in distinguishing between interacting and isolated sources, as well as identifying NS+NS or BH+NS binaries. For the technical aspects of EM synergies, see Sec. 1.5.1.', '1.6.2 Foreground sources': "The Milky Way hosts a large variety of stellar binaries (Section 1.2), numbering in the millions below mHz frequencies (Table 4). They will appear as nearly monochromatic (constant frequency) sources emitting over the whole duration of the mission (continuous GW sources). Up to tens of thousands - those with frequencies larger than a few mHz and/or located closer than a few kpc - will be individually resolvable. The rest of Galactic binaries will blend together into the confusion-limited foreground that is expected to affect the LISA data stream at frequencies below 3 mHz (e.g. Bender and Hils, 1997; Edlund et al., 2005; Ruiter et al., 2010; Cornish and Robson, 2017). The optimal detection, characterization, and subtraction of Galactic binaries from the data stream has been recognized as one of the fundamental tasks for the LISA analysis. Over-fitting the population of Galactic binaries can result in a large contamination fraction in the catalogue of detected sources, while under-fitting it can degrade the analyses of extra-galactic GW sources in the data due to the excess residual. \nThe waveforms for Galactic binaries are well predicted using only leading order terms for the orbital dynamics of the binary (Peters and Mathews, 1963) and can be computed at low computational cost using a fast/slow decomposition of the waveform combined with the instrument response (Cornish and Littenberg, 2007). Nevertheless, their identification in the LISA data will be laborious due to the sheer number of sources expected to be in the measurement band ( ∼ 10 4 ) and the large number of parameters required to model each source (between 5 and 10, depending on if the source is chirping and if spins are important in the modelling of one or both of the components). In addition, the high density of Galactic binaries in the LISA band (to the extent that sources are overlapping) and the modulation effects caused by LISA's orbital motion, which spreads a source's spectral power across multiple frequency bins, makes the true number of signals at a given frequency difficult to identify. Several techniques have been developed to address this challenge. \nA hierarchichal/iterative scheme . The detectable binaries can be identified by using an iterative process that utilizes a median smoothing of the power spectrum to estimate the effective noise level at each iteration, regresses binaries from the data with signal-to-noise ratios above the established threshold as detected sources, repeating the process until the convergence (Cornish and Larson, 2003; Timpano et al., 2006; Nissanke et al., 2012). However, each iteration can leave behind some residual due to 'imperfect subtraction' that can affect further analysis and bias parameter estimation. In practice, the stochastic signal is not actually subtracted but rather included in the covariance matrix during the likelihood calculation. \nGlobal fit . A number of studies show that a global fit to the resolvable binaries, while simultaneously fitting a model for the residual confusion or instrument noise and using Bayesian model selection to optimize the number of detectable sources can provide an effective solution to the Galactic binaries challenge (Cornish and Crowder, 2005; Umstätter et al., 2005). These global fit methods have been demonstrated on the Galactic binaries using data from the LISA Data Challenges (Littenberg et al., 2020), but work still needs to be done to extend this into a fully-developed analysis pipeline with the full variety of overlapping LISA sources. \nIt should be noted that there will possibly be SGWBs of unresolved extragalactic sources or of cosmological origin (see Section 1.2.2.5). Such backgrounds will similarly be a broadband confusion signal, and similar considerations to identify and characterize SGWB in LISA data may be needed in order to reveal some of the fainter signals from astrophysical and cosmological sources. Techniques to identify and subtract the SGWB in LISA data are currently being developed by several groups (Karnesis et al., 2019; Caprini et al., 2019; Pieroni and Barausse, 2020).", '1.6.3.1 Modelling isolated binary evolution and populations': 'Table 6: Population synthesis codes used by the community at the time of writing. Binary\\_c, COSMIC, MOBSE are based on the BSE code (Hurley et al., 2002). \nThe long-term evolution of stars and binaries is typically modelled in either of two methods; by solving the stellar structure equations, i.e. referred to as detailed calculations, or by faster approximate methods typically aimed at the population synthesis approach. The latter either interpolates in a grid of detailed calculations or uses parametrised stellar evolution tracks which are fitted to detailed calculations. The advantage of this method is the highly boosted computational speed (the simulation of the evolution of a single binary takes a fraction of a second in stead of hours or days), at the cost of detail; one only has access to those parameters included in the grid of tracks. Due to the speed, the effect of different assumptions for poorly understood stellar physics (e.g., stellar mass-loss, interaction physics, supernova kick physics) can be tested in a statistical way, which leads to a deeper understanding of the underlying physical processes involved. The population synthesis approach has proven to work well in retrieving the general characteristics of large binary populations (Toonen et al., 2014) and has led to many insights in binary evolution. \nPopulation synthesis codes are crucial for LISA science, both in order to make forecasts for source rates, but also to develop data analysis pipelines. Indeed, ongoing work on building fast and reliable waveforms relies on the knowledge of the expected properties of the sources (masses, spins, eccentricities) and the accuracies required to detect them and measure these properties. The codes currently in use by the LISA community are listed in Table 6. Further development of these codes, in particular the inclusion of additional physical processes as well as cross-checks between the codes will help to prepare for LISA observations.', '1.6.3.2 Modelling binary evolution in dense environments': 'Stellar-origin LISA GW sources can also be formed in dense stellar environments through dynamical interactions. Hence simulation codes that follow the evolution of dense stellar systems, either by direct integration or using Monte-Carlo techniques, are crucial for their study. Many of these codes follow simultaneously the evolution of single and binary stars within the dense stellar system, using one of the tools described in the previous Section. A list of stellar dynamics codes currently used in the community for the study of the formation of stellar-origin LISA GW sources can be fount in Table 7.', '1.6.3.3 GW signal tools': 'In order to calculate the GW signal of Galactic binary systems there are a number of approaches that can be used, ranging from detailed TDI based methods, e.g. via codes from the \nTable 7: N-body and few-body dynamics codes used by the community at the time of writing. \nLISA Data Challenges (LDC, https://lisa-ldc.lal.in2p3.fr ) to (more) analytic methods to calculate the signal and SNR of specific objects (e.g. Cornish and Larson, 2003; Robson et al., 2019; Korol et al., 2017; Kupfer et al., 2018; Smith and Caldwell, 2019). There are also some web-based tools to explore sensitivity of different GW detectors, including LISA and also detectability of sources in LISA, such as https://gwplotter.com (Moore et al., 2015) and the Gravitational Wave Universe Toolbox ( https://www.gw-universe.org , Yi et al. 2021)', 'Coordinators: Fritz Roepke, Alina Istrate': 'Contributors: Karel Temmink (1.7.1.1), Mike Lau (1.7.1.1, 1.7.1.7-8), Stephan Rosswog (1.7.1.2, 1.7.1.6), Vasileios Paschalidis (1.7.1.2), Alina Istrate (1.7.1.2), Fritz Roepke (1.7.1.3,1.7.1.6), Kinwah Wu (1.7.1.4-5), Stéphane Mathis (1.7.1.4), Thomas Tauris (1.7.1.5), Stéphane Blondin (1.7.1.6), Ashley Ruiter(1.7.1.6, 1.7.1.8), Chris Fryer (1.7.1.7), Thierry Foglizzo (1.7.1.7) Manuel Arca Sedda (1.7.1.8), Kyle Kremer (1.7.1.8), Simone Bavera (1.7.1.8), Abbas Askar (1.7.1.8), Silvia Toonen (1.7.1.8), Gijs Nelemans (refs for 1.7.1.8), Irina Dvorkin (refs for 1.7.1.8), Valerya Korol (refs for 1.7.1.8), Astrid Lamberts (refs for 1.7.1.8) \nThroughout this Section we will highlight science questions related to LISA that can/should be addressed before the launch of the mission with the label pre-LISA-launch objective , while science questions that can only be addressed by using LISA data will be highlighted with the label post-LISA-launch objective . It should be emphasized, that in most cases LISA detection of individual binaries is limited to GW frequency and a somewhat poor sky location. For a fraction of these thousands of systems, however, high SNR and/or high frequency will enable the measurement of the GW frequency derivative, a good sky localization, and possibly constraining the eccentricity of the binary system. The GW frequency derivative will reveal the chirp mass of the binary and thus the distance. Although these cases will be exceptional systems in the overall global perspective, they will be the key science drivers that deliver deep insight and new breakthroughs in our understanding of binary compact object systems - often because of the enhanced chances of finding the EM counterpart of these well-localized systems.', '1.7.1.1 Dynamical stability and efficiency of mass transfer in the formation of LISA sources': 'The formation of compact binary systems with two compact objects is still relatively poorly \nconstrained. Typically, at least two phases of mass transfer are required to form a general observable stellar LISA system: one for each component star to lose their hydrogen envelope, and additional phases are possible to remove the helium-rich envelope. To explain the compactness of the orbit, typically one or more of these mass-transfer phases are considered to proceed in an unstable fashion in order to get the necessary amount of orbital shrinkage i.e. a CE phase (e.g. Paczyński and Sienkiewicz, 1972; Paczynski, 1976; Webbink, 1984). Hence, it is crucial to understand for which binary configurations mass transfer proceed stably, and for which it will be unstable. \nPre-LISA-launch objective: The precise value of the stability boundary (i.e. a critical mass ratio, q between the two stellar components above which no stable mass transfer is possible) remains under debate. Theoretical work has shown that mass transfer can proceed significantly more stable than classical results have previously implied (e.g. Hjellming and Webbink, 1987; Chen and Han, 2008; Woods and Ivanova, 2011; Passy et al., 2012b; Pavlovskii and Ivanova, 2015; van den Heuvel et al., 2017; Misra et al., 2020). Similarly, the mass-retention fraction of the accreting companion remains relatively poorly understood (e.g. Paczyński and Sienkiewicz, 1972; Kato and Hachisu, 1999; Hachisu et al., 1999; Tauris et al., 2000; Hurley et al., 2002; Nomoto et al., 2007; Vinciguerra et al., 2020). These issues severely affect the predicted formation details of compact binaries, and determine which evolutionary pathways are dominating in the formation rate of LISA sources (Section 1.4), and hence would leave characteristic imprints on the numbers and properties (orbital periods, masses) of the resulting LISA population (e.g. Korol et al., 2017; Ruiter et al., 2019). For instance, whether or not the first phase of mass transfer in the formation of a WD+WD leads to a shrinkage or widening of the orbit (i.e. unstable or stable mass transfer) determines to which size the secondary star can evolve before filling its Roche lobe, which dictates the mass of its core at RLO, i.e. the mass of the resulting WD, and hence the mass ratio of the WD+WD (Nelemans et al., 2000, 2001c; van der Sluys et al., 2006; Toonen et al., 2012). \nStability of mass transfer depends rather sensitively not only on the intricate details of the structure of the donor star, but also on the transfer and potential loss of mass and angular momentum (Soberman et al., 1997). Pre-LISA-launch objective: Hydrodynamical simulations can help settle the question of dynamical stability (see Fig. 13). The mass that is transferred to the companion can not always be fully accreted by the companion star: spin-up to critical rotation rates and/or strong optically thick winds or outflows (including jets) can result in significant fractions of transferred material being lost from the accreting star. Since mass that is not accreted leaves the binary system, carrying with it an amount of orbital angular momentum, the efficiency of accretion and the stability of mass transfer are linked. In the case of compact object accretors, it is expected that less conservative mass transfer is typically more stable than mass transfer where all mass is accreted (Soberman et al., 1997). \nPost-LISA-launch objective: Observed samples are required to reverse-engineer the progenitor evolution, and constrain the evolutionary pathways (e.g. Nelemans et al., 2000; van der Sluys et al., 2006; Zorotovic et al., 2010; De Marco et al., 2011; Portegies Zwart, 2013). On the EM side, only relatively small samples exist currently, with relatively large biases towards the lower-mass and hotter WDs, since they have longer (observational) lifetimes and are brighter. However, LISA will be sensitive to WD+WDs throughout the whole Galaxy and will be able to provide properties of the entire WD+WD population with relatively few selection biases. This will allow for stronger and more meaningful statistical analyses. Additionally, LISA will be able to almost directly measure the Galactic rate and masses of merging WD+WDs, which is another useful tool in constraining progenitor evolution. \nThe detection of NS+NSs (and potentially NS+BH systems) with LISA allow for a direct view of their formation pathways through their eccentricity, that is induced by the supernova kick associated with the formation of the second NS (Vigna-Gómez et al., 2018; Lau et al., 2020). The population of eccentric LISA NS+NS binaries originate from NS+NS binaries born right in or near the sensitivity window of LISA, so that there is little time for GWs to circularise the \nFigure 13: Hydrodynamical investigation of dynamical stability in a UCXB with a 0 . 15 M ⊙ WD donor star and a 1 . 4 M ⊙ accreting NS, in an orbit with an initial eccentricity of 0.04. Plotted here is the mass density in the orbital plane after roughly 13 orbits of RLO. The density plot shows eccentric structures in the accretion disc, the complex character of the flow near the circularization radius and a strong density cusp near the NS. The envelope surrounding the binary is sparse but its total mass is significant compared to the disc. Figure from Bobrick et al. (2017). \n<!-- image --> \norbit. The tight orbit prior to the second SN, leading to NS+NS formation, is characterized by the last phase of mass transfer (see Section 1.3.1.3 and Fig. 1.3.1.3). This is Case BB mass RLO initiated by the expansion of the naked helium-star after core-helium depletion. The Case BB mass transfer episode is believed to be predominantly stable from detailed simulations (Tauris et al., 2015) and in order to match the observed period distribution of Galactic NS+NS systems (Vigna-Gómez et al., 2018). However, unstable Case BB RLO would lead to an additional CE phase that produces NS+NSs with sub-hour periods. Yet, because such NS+NSs that have gone through unstable Case BB RLO prior to the second SN are formed with higher GW frequencies, they also have a more rapid GW frequency evolution ( f GW / ˙ f GW ∝ f -8 / 3 GW ), which disfavours their detection by LISA (Lau et al., 2020; Andrews et al., 2020). Pre-LISA-launch objective: A deeper understanding of whether or not mass transfer is stable or unstable in Case BB RLO is needed, and should be investigated further.', '1.7.1.2 Dynamical stability and efficiency of mass transfer in accreting LISA sources': "A remarkable property of Roche-lobe filling stars is that, for mass ratios < 0 . 8 , their average density is related to their orbital period (see e.g. Frank et al., 2002). For mass ratios, M donor /M accretor < 0 . 8 the relation is given by: \nP orb = 10 . 5 hr ( ¯ ρ g cm -3 ) -1 / 2 , (6) \nAs an example, Roche-lobe filling stars with orbital frequencies of 0 . 1 mHz ( f GW = 0 . 2 mHz ) have ¯ ρ ∼ 10 g cm -3 , while those at 1 Hz possess an average density of ¯ ρ ∼ 10 9 g cm -3 . In other words, mass-transferring WDs are located right inside LISA's frequency band. For NS donors, \nmass transfer only sets in at much smaller separations when the frequencies are already close to the kHz-regime (Shibata and Taniguchi, 2011). \nFor a LISA stellar source, there are at least two possible scenarios for the subsequent evolution after the onset of mass transfer: a) after an initial brief phase of continued orbital shrinkage after RLO is initiated (Tauris, 2018), mass flows on a much longer timescale from the WD toward the accretor star while the binary separation increases. This process is commonly referred to as a form of stable mass transfer (see Fig. 14); b) The WD becomes tidally disrupted by the accretor, resulting in the binary merger. This process occurs on a dynamical (orbital) timescale. Pre-LISA-launch objective: Whether the binary undergoes stable mass transfer vs a merger may have important implications on the type of GW templates that are necessary to detect these binaries with LISA. \nThe above discussion makes it clear that for WD+WD, NS+WD, and BH+WD binaries the onset of mass transfer marks a turning point since the stability of mass transfer decides whether the binary can survive or will inevitably merge. As mentioned in the previous Sections, its fate depends sensitively on the internal structure of the mass-donating star, on the mass ratio and on the involved angular momentum exchange mechanisms, which here depend primarily on whether mass transfer takes place with or without an accretion disc around the accretor (Rappaport et al., 1982; Hut and Paczynski, 1984; Marsh et al., 2004; Gokhale et al., 2007; Motl et al., 2007; Paschalidis et al., 2009; Dan et al., 2011; Shen, 2015). Since fully degenerate WDs possess an inverted mass-radius relation, i.e. they grow in size as mass is removed, the mass-donating star will expand and thereby tend to speed up the mass transfer. However, since the mass is transferred to the heavier star, the orbit will tend to widen, and therefore stabilize the mass transfer (Soberman et al., 1997; Tauris and Savonije, 1999). \nThe way the transferred mass settles onto the accretor star has a decisive impact on the orbital evolution. If the circularization radius of the transferred matter is smaller than the radius of the accretor, it will directly impact onto the stellar surface and spin up the accreting star - this scenario is referred to as direct impact accretion. That means that orbital angular momentum is not fed back into the orbit, and therefore the orbital separation shrinks and mass transfer accelerates. If instead the circularization radius is larger than the radius of the heavier WD accretor, a disc can form and - via its large lever arm - the disc can feed back angular momentum into the orbital motion, increase the orbital separation, and thus stabilize the binary system (Iben et al., 1998; Piro, 2011; Paschalidis et al., 2009). \nThe majority of studies of mass-transfer stability to date assume that the mass-transferring WD is tidally locked. However, as pointed out by Webbink and Iben (1988), spinning up a WD while being tidally locked from some initial separation down to the Roche limit is accompanied by tremendous energy release that, depending on the dissipation mechanism, could potentially lift the degeneracy throughout the star. Given that the dissipation mechanisms in WD interiors are not well understood, this makes things even more complex: The tidal interaction between the stars can substantially heat up the mass-donating WD, change its internal structure, and thereby its response response to mass loss. \nFor LISA this means that the measured chirp of a mass-transferring binary is not only set by the decay of the orbit due to GW, but also due to mass transfer and tidal interactions, and therefore the chirp mass can not be directly measured as in the case of detached (chirping) binaries. However, for an assumed cold equation-of-state mass-radius relation of the donor star, both the mass of the donor star and the model mass-transfer rate can be derived, as these are fully set by the orbital period (Faulkner, 1971; Vila, 1971, see also Eq. 6). Pre-LISAlaunch objective: Future work should aim at including finite-temperature effects in the WD EOS consistently, e.g. by using detailed stellar structure calculations following the formation and evolution of the system - from the detachment of the CV/LMXB phase until onset of the AM CVn/UCXB phase (Fig. 8). \nPost-LISA-launch objective: We do not expect the second derivative ( f GW ) of the GW \nFigure 14: Stability regions in the donor mass-accretor mass plane of WD+WDs and UCXBs. Donor star masses are on the vertical axes, accretor star mass on the horizontal axes. Upper panel : analytical results of Marsh et al. (2004) where the solid line marks the transition between disc and direct impact accretion, and the other lines show how the strict stability limit of Nelemans et al. (2001a) is relaxed when dissipative torques feed angular momentum from the accretor back to the orbit. Bottom panel : ballistic calculation approach by Kremer et al. (2017) using zero-temperature mass-radius relations for the WD donor. Red systems are stable throughout their lifetimes, through stages of both direct-impact and disc accretion; black systems are unstable; and blue systems have an accretor that exceeds the Chandrasekhar limit during their evolution. The solid black line marks again the boundary between disc and direct-impact accretion for initially synchronous and circular binaries. The yellow region indicates systems with a total mass in excess of the Chandrasekhar limit, i.e. potential SN Ia progenitors. It is evident from these figures that UCXBs with low-mass ( ≲ 0 . 2 M ⊙ ) He WD donor stars, and NS accretors, are always dynamically stable. \n<!-- image --> \nfrequency an AM CVn system to be measurable with LISA, as the low mass ratios required from mass transfer stability considerations ( M donor ≪ M accretor ), imply a low-mass WD donor, whose large radius prevents the AM CVn system from penetrating into the highest frequency range where the second derivate is large (Nelemans et al., 2004b). The combination of LISA with EM surveys, such as Gaia, is particularly promising for AM CVn sources. If their distance is known, the chirp mass can be constrained, which allows for the orbital chirp to be decoupled into its different components (Breivik et al., 2018). \nApart from determining the orbital evolution, mass transfer in a WD+WD also leads to an accumulation of helium (possibly also carbon and oxygen) on the surface of the accreting WD. If (or when) nuclear fusion commences in this layer, rapid burning follows that causes a nova outburst or, in case of persistent fusion, X-ray emission may be observed as for supersoft X-ray sources (Kahabka and van den Heuvel, 1997). Rapid accretion during the last tens of orbits before a merger and the interaction with the incoming accretion stream can trigger surface detonations that cause weak SN Ia-like transients (Guillochon et al., 2010). A (tidal) disruption of a WD by a NS or BH could potentially lead to nuclear-dominated accretion flows (Metzger, 2012; Fernández and Metzger, 2013). \nPost-LISA-launch objective: The probability of detecting a Galactic WD+WD merger during the LISA mission is small (since the Galactic WD+WD merger rate is of order one per century (Nelemans et al., 2001b), let alone that this number includes all the WD+WD 'mergers' giving rise to stable RLO after contact, i.e. the AM CVn systems). Yet, the case of merging WD+WD binaries deserves a separate mention, because of the exciting possibility of multiband and/or multiwavelength observations: the inspiral phase would be detectable by LISA or a similar mission, and the merger phase by a future observatory such as DECIGO (Sato et al., 2017). If the merger gives rise to an optical transient (e.g. a SN of Type Ia/Iax), these can be observed by EM transients surveys such as ZTF and the Vera Rubin Observatory. For NS+WD, the postmerger phase would be detectable by ground-based high-frequency GW observatories, such as LIGO, Virgo, KAGRA and future facilities like Cosmic Explorer (Abbott et al., 2017a) and the Einstein Telescope (Punturo et al., 2010), due to either the eventual collapse of the NS core or post-merger oscillations of the remnant (Paschalidis and Stergioulas, 2017). Therefore, NS+WD systems offer the unique opportunity not only to study the dynamics/stability of mass-transfer, but also the potential to place constraints on the nuclear equation of state. Nevertheless, once again, we emphasize the small probability for a Galactic WD+WD merger event during the lifetime of the LISA mission. \nPost-LISA-launch objective: In summary, mass transfer is crucial for determining the final fate of interacting close binaries, but many details and many questions still remain unanswered related to e.g. formation and evolution of AM CVn and UCXB systems and their ultimate fates, and for related questions such as the progenitor systems of Type Ia/Iax supernovae. Observations with LISA may therefore bring a major leap forward in our understanding of the physics of this crucial evolutionary phase of close-orbit stellar binaries with compact objects.", '1.7.1.3 Common envelopes': 'CE phases are one of the greatest uncertainties in binary stellar evolution theory (Ivanova et al., 2020) and LISA will provide important measurements. Pre-LISA-launch objective: The inspiral of the secondary star into the envelope of the primary lacks obvious symmetries and is therefore not accessible to classical one-dimensional stellar evolution modelling approaches. At least some part of CE interaction takes place on a dynamical timescale. Therefore, parametrized prescriptions for CE evolution are used in stellar evolution modelling. Three-dimensional hydrodynamic simulations have been employed to study the process in more detail (Passy et al., 2012a; Iaconi et al., 2018; Sandquist et al., 2000; Ricker and Taam, 2012; Nandez et al., 2015; Nandez and Ivanova, 2016; Kuruwita et al., 2016; Ohlmann et al., 2016a,b; Chamandy et al., 2018; Reichardt et al., 2019; Rasio and Livio, 1996; Prust and Chang, 2019; Kramer et al., 2020; \nSand et al., 2020; Law-Smith et al., 2020), but they are numerically challenging due to the large dynamical range of spatial and temporal scales of the problem. Furthermore, most often threedimensional hydrodynamic simulations do not incorporate modelling of physical processes like convection and radiation transport, which are thought to be important especially in the later phases of the inspiral. Complementing the global three-dimensional CE simulations with local, wind-tunnel type, simulations that study the details of the flow around the inspiraling object (e.g. MacLeod et al., 2017; De et al., 2020; Everson et al., 2020), as well as one-dimensional but multi-physics hydrodynamic simulations (e.g. Clayton et al., 2017; Fragos et al., 2019) is a promising avenue to more physically accurate predictions of post-CE binary properties. Lastly, studies of post-CE binaries that are found observationally will provide insights into the CE phase. Valuable constraints on the CE mechanism have come from this method previously (Nelemans et al., 2000; van der Sluys et al., 2006; Zorotovic et al., 2010; Toonen and Nelemans, 2013). \nBecause the actual interaction is short (up to about 10 3 years), direct observations in optical astronomy are difficult. Some of the fainter optical transients (luminous red novae; Soker and Tylenda, 2003; Tylenda et al., 2011; Kulkarni et al., 2007; Howitt et al., 2020; Stritzinger et al., 2020) have been associated with CE events. The two fundamental, and to date unanswered, questions are: i) Which systems manage to eject the CE? ii) What is the final orbital separation of the two stellar cores in this case? Post-LISA-launch objective: The LISA mission is instrumental for clarifying many aspects of the physics of CE phases in two main directions:', '1. Direct observations of events related to CE interaction.': "The secondary star may be a compact object, but the primary (donor) is typically in a giant phase. Therefore, a generation of sufficiently strong GW signals during inspiral can only be expected if the core of the primary star is also relatively compact, and if the secondary (a compact object) comes close enough during the inspiral. \nThe prospects to observe GW signals from the inspiral phase, however, do not seem very promising in current models. While based on a parametrized description of CE inspiral, Ginat et al. (2020) predict about one detection in a few centuries with LISA. The full three-dimensional hydrodynamic CE simulations of Ohlmann (2016) find weaker signals. The rates for the slower self-regulated phase that proceeds on a thermal timescale are more promising, with a rate of ≈ 0 . 1 -100 events in the Galaxy during the LISA mission duration (Renzo et al., 2021). \nThe detectability, however, depends on how close the stellar cores come to one another during the evolution. This is uncertain and depends on the above questions i) and ii). Due to the inspiral of the secondary into the primary star's envelope, orbital energy and angular momentum are transferred to this (now) CE. Some material becomes unbound and is ejected from the system. Simulations, however, show that this process alone is inefficient and other energy sources (such as the ionization of envelope material, Nandez et al. 2015; Prust and Chang 2019; Sand et al. 2020) have to be tapped to achieve full envelope removal. The exact parameters allowing for such a successful CE ejection are still unknown, but it is likely that most initial configurations may fail. Such failed events, however, may produce stronger GW signals. Moreover, more exotic cases in which, for instance, triple systems enter CE evolution (Comerford and Izzard, 2020; Glanz and Perets, 2021), may potentially also be sources of detectable GW signals.", '2. Indirect information from detecting post-CE sources.': 'Since all stellar mass LISA sources have presumably gone through a CE phase (disregarding here sources produced in dense clusters via dynamical interactions), comparison of LISA populations with model predictions naturally test CE physics. Specifically, the occurrence (and detection) rates of these binary populations depend critically on the orbital separation of the stellar cores after the CE phase. In this sense, the LISA mission will statistically sample the outcome of CE events and the results provide valuable information for answering \nthe above question ii). Moreover, if CEs produce binaries with sufficiently short orbital periods, such that they are within the LISA band immediately at envelope ejection, they will act as a source term for the population of LISA binaries. In the absence of this injection of sources, evolution of the orbits due to GWs produce a predictable expectation for the orbital period distribution of binaries, any deviations from that expectation could be ascribed to an injected population of post-CE systems. By simultaneously modelling the LISA noise curve, the GW foreground from unresolved Galactic sources, and the effect of GWs on a binary population, the initial post-CE separations for the shortest period binaries can, in principle, be derived, and thereby deliver important insight into CE physics. \nContributing to these two aspects, the LISA mission will help to constrain physics of the mysterious, and yet crucial, CE phase in binary stellar evolution. Its results will be used to validate hydrodynamic simulations and to develop new efficient prescriptions of CE interactions that are to be used in binary stellar evolution and population synthesis studies.', '1.7.1.4 Tides and angular momentum transport': "The orbital evolution of detached binaries is practically determined by the loss of orbital angular momentum and the exchanges of angular momentum between the stars and the orbit, and also in some situations, the mass loss from the system. For binaries with only BH or NS components, tidal effects are completely insignificant (except for the few final orbits before a merger event) and the orbital evolution is entirely determined by GWs alone. An example of such a system is the radio pulsar binary PSR B1913+16 (see Weisberg and Huang, 2016). For binaries with non-degenerate stars, or tight systems with WDs, tides can be important, leading to measurable differences in the LISA signal. The orbital dynamics of the wide-orbit systems is relatively simple, as the orbital angular momentum is decoupled from the spin of the two stars. The orbital evolution of these binaries is therefore simply driven by the wind-mass loss of the stars and, in principle, GWs, which nevertheless is negligible in such wide systems. The situation is very different for the systems with a sufficiently short orbital period. First of all, the timescale for the loss of orbital angular momentum via GWs (Peters and Mathews, 1963) could be comparable to or shorter than the evolution of the stars and the tidal evolution of the system (Bildsten and Cutler, 1992). Secondly, the sizes of the stars are no longer negligible compared with the orbital separation, and torques can be exerted on the stars effectively (Lai et al., 1994; Hut, 1981). Hence, the structures and the hydrodynamical properties of the stars would play an important role in determining the orbital dynamics and the orbital evolution of these systems (Benacquista, 2011; Fuller and Lai, 2012; Shah and Nelemans, 2014a). These have, at least, two immediate consequences on the LISA science: i) the number density of persistent GW sources associated with these binaries observable within the LISA band, and ii) the event rate of burst sources associated with coalescence or merging, resulting from the orbital decay of these binaries, although these bursts are very rare in our Galaxy. \nThe exchange of angular momentum between the binary orbit and the spin of the stars can be facilitated by the viscous torque (Zahn, 1977), but in the compact star binaries this coupling is less efficient than the coupling caused by a stellar bulge cause by the tidal deformation of the star (Bildsten and Cutler, 1992; Dall'Osso and Rossi, 2013; Kochanek, 1992). The degree of tidal deformation of a star depends on its internal structure and dynamics (e.g. Flanagan and Hinderer, 2008; Ogilvie, 2014; Mathis, 2019). Thus, even in the absolute absence of viscosity, WDs and NSs would respond differently to tidal deformations (cf. the studies of Vick and Lai, 2019; Lai and Shapiro, 1995; Dall'Osso and Rossi, 2013; Bildsten and Cutler, 1992; Dall'Osso and Rossi, 2013; Kochanek, 1992). The presence of the close companion will trigger a large-scale flow induced by the hydrostatic adjustment of the studied primary to the tidal perturbation, the equilibrium tide (e.g. Zahn, 1966; Remus et al., 2012; Ogilvie, 2013), and a broad diversity of tidal waves (i.e. gravity waves, inertial waves, gravito-inertial waves), and the dynamical tide (e.g. Xu and Lai, 2017; Yu et al., 2020). Their dissipation and the quadrupolar moment they induce \nmodify the inspiral and cause changes in orbital frequency and phase shifts (e.g. Bildsten and Cutler, 1992; Wang and Lai, 2020; McNeill et al., 2020). Thus, WD binaries and NS binaries will behave and evolve differently, which will manifest in the LISA GW background. Their orbital evolution will also imprint signatures in the GWs that they emit (Vick and Lai, 2019; McNeill et al., 2020; Dall'Osso and Rossi, 2013, 2014; Shah and Nelemans, 2014b). The situation can be more complicated if the compact stars have a large magnetic moment, which is not uncommon among magnetic WDs (Ferrario et al., 2015). The magnetic moments of NSs may not be as large as those of WDs since NSs are more compact. Direct magnetic interactions between two magnetised components should be taken into account and may compete with tidal interactions in LISA sources. \nPost-LISA-launch objective: Our understanding of the interplay between tidal interaction, feedback magnetic-field amplification and orbital angular momentum extraction by GWs is currently very primitive. The observations of GWs from such systems will surely expand our knowledge on this subject substantially (e.g. King et al., 1990; Piro, 2012; Wu et al., 2002; Wang et al., 2018).", '1.7.1.5 Irradiation of companion star': 'Feedback irradiation effects on companion stars caused by the intense X-ray flux emitted from accreting compact objects may influence the evolution of the orbits of binary stars (Podsiadlowski, 1991; Benvenuto et al., 2014). In detached systems, energetic millisecond pulsars (MSPs, recycled to high spin frequencies from a previous recycling phase; Bhattacharya and van den Heuvel, 1991) may irradiate their companion star with a pulsar wind of relativistic particles and hard photons (Tavani and Brookshaw, 1992). Observations have revealed a growing number of such MSPs with a non- or semi-degenerate companion star which is being ablated by the pulsar wind, the so-called black widows and redbacks (Roberts, 2013). This is evidenced by the radio signal from the pulsar being eclipsed for some fraction of the orbit (Fruchter et al., 1988). Tidal dissipation of energy in the donor star envelope (Applegate and Shaham, 1994) may cause the companion star to be thermally bloated and thereby evaporate more easily. For LISA binaries, such mass loss via ablation/evaporation will modify their orbital evolution (e.g. Chen et al., 2013; Hui et al., 2018), which is otherwise dictated by GWs and tides. For this reason, we may gain new insight on irradiation efficiency from LISA detections of such systems and precise measurements of their orbital frequency. Pre-LISA-launch objective: The impact and the modelling of this effect, often leading to cyclic accretion, is still unclear needs to be improved before LISA flies. \nPost-LISA-launch objective: For accreting LISA sources, the irradiation will lead to disturbance of the thermal equilibrium of the companion star (Büning and Ritter, 2004) and, in the extreme situation, geometrical deformation (Phillips and Podsiadlowski, 2002), thereby affecting its mass-transfer rate and thus the orbital evolution of the binary. Such an effect may indeed be measured by LISA via its impact on the orbital frequency derivative, and thus the chirp mass of the system. Hence, detection of a number of mass-transferring UCXBs and AM CVn systems by LISA could provide us with unique ways of probing the physics governing close compact object binaries (Jia and Li, 2016; Kremer et al., 2017).', '1.7.1.6 Type Ia supernovae and other transients': "Stellar interactions in binary systems containing at least one WD are thought to trigger Type Ia supernovae (SN Ia) and likely a variety of other transients (see e.g. Wang and Han, 2012, for a review). SN Ia were of paramount importance for the discovery of the accelerated expansion of the Universe and they significantly contribute to cosmic nucleosynthesis, but the lack of a clear observational connection between a progenitor system and the observable phenomenon has made their understanding difficult. Without proper initial conditions their modelling remains uncertain. \nThe properties of the ensuing explosion are determined by the pre-explosion state of the WD, but is it triggered when approaching the Chandrasekhar-mass limit, or well before? The occurrence rate, the delay time between binary formation and SN explosion, and the ignition process are all determined by the nature of the progenitor system, and they have a strong impact on the contribution of thermonuclear SNe to galactic chemical evolution. A traditional broad classification is to distinguish between single-degenerate systems, where the companion of the exploding WD is a non-degenerate star - and the double degenerate systems - where the interaction of two WDs (mergers, or in rare cases collisions) triggers the SN explosion (see Ruiter, 2020, for a breakdown of binary star progenitor configurations). \nNone of the progenitors and explosion mechanisms is established beyond doubt. The singledegenerate Chandrasekhar-mass model, that served as a reference for a long time, has several shortcomings, but it seems to be needed to explain observed abundance trends (Seitenzahl et al., 2013). However, both population synthesis models and observations indicate that single degenerate explosions fall short of explaining the observed SN Ia rate. GW signals were derived from explosion simulations of near-Chandrasekhar mass WDs (Falta et al., 2011; Seitenzahl et al., 2015), but the prospect of measuring individual events is low. \nThe major competing double-degenerate scenario received increased attention over the past years and is of particular interest in the context of LISA. In this scenario, however, the process initiating the actual thermonuclear explosion is unclear. For massive WDs, the remnant can reach or exceed the Chandrasekhar-mass, but the explosion could also be triggered in the merger process itself while the more massive WD is well below the Chandrasekhar mass limit (Pakmor et al., 2010, 2012). Apart from GW-driven (close-to-circular) mergers, collisions can also (likely to a much smaller extent) contribute to the SN Ia rate (Rosswog et al., 2009a; Raskin et al., 2009). They may occur in locations with large stellar number densities such as globular cluster cores or galactic centres, but they are generally thought to occur too infrequently to explain the bulk of SN Ia (Toonen et al., 2018, but see Kushnir et al. 2013 for more optimistic claims). Such collisions, however, have the advantage of a very robust and physically understood explosion mechanism: WDs of the most common type ( ∼ 0 . 6 M ⊙ ), that collide with velocities given by their mutual gravitational attraction, cause strong shocks in the collision and nuclear burning occurs in the right density regime, so that the resulting explosions appear as rather common Type Ia SNe (Rosswog et al., 2009a). Before the final collision causes a thermonuclear explosion, the two WDs may undergo several close encounters causing a sequence of GW bursts in the LISA band of increasing amplitude. \nFurther clarification of the double-degenerate progenitor channel of Type Ia SNe requires the determination of the exact demographics of WD merger events - what is their occurrence frequency for different WD masses? It is further crucial to understand whether the explosion is triggered during the merger itself or, maybe, already during the inspiral phase when mass transfer between both WDs sets in. Even if only small amounts of mass are exchanged, the re-distribution of angular momentum can have a substantial impact on the orbital dynamics and therefore on the GW signal (Dan et al., 2011). Post-LISA-launch objective: The LISA mission has great potential to contribute here and to provide important clues to the mechanism of Type Ia SN explosions. Ruiter et al. (2010) found that on the order of ∼ 500 WD+WD pairs -whose total mass exceeds the Chandrasekhar mass limit and will merge within a Hubble time -could be resolvable by LISA in our own Galaxy. While most likely no such systems will merge and give rise to a SN Ia during LISA's operation, much can be learned about SN Ia (and more generally transient) demographics from detecting these plausible progenitor systems.", '1.7.1.7 Core-collapse and supernova kicks': "Observations of compact objects, from pulsar proper motions (Hobbs et al., 2005) to compact binary properties (Dewi et al., 2005; Mirabel, 2017), argue that many NSs and some BHs receive natal kicks during the collapse and explosion of the massive star that forms them. Asymmetries \nin the explosion mechanism, manifested either through asymmetries in the mass ejecta (Wongwathanarat et al., 2013) or neutrino emission, have been studied as a source of these kicks. The different mechanisms (Lai et al., 2001) produce different predictions for the distribution of their magnitude (Scheck et al., 2006), their orientation with respect to the orbital angular momentum (Blaauw, 1961), to the stellar spin (Wang et al., 2006; Noutsos et al., 2012; Wongwathanarat et al., 2013), and to the distribution of heavy elements (Wongwathanarat et al., 2013; Grefenstette et al., 2014). The asymmetries produced by strongly magnetized explosions are generally aligned with the angular momentum in the collapsing star (Sawai et al., 2008; Obergaulinger and Aloy, 2020; Kuroda et al., 2020) and these mechanisms will produce kicks with directions aligned with the rotation axis, which typically is also aligned with the orbital angular momentum axis. Mechanisms produced by the large-scale convective eddies in the neutrino driven mechanism can produce kicks that are distributed more isotropically (Wongwathanarat et al., 2013; Müller et al., 2019). \nDifferent kick mechanisms also predict different kick magnitudes as a function of the compact remnant mass (Tauris et al., 2017; Mandel and Müller, 2020, and references therein). These kick distributions, in turn, predict different properties in compact object binaries (Voss and Tauris, 2003; Lau et al., 2020). The NS kick properties can thus affect the number of NS+NS, BH+NS and BH+BH binaries detectable by LISA and LIGO/Virgo (Voss and Tauris, 2003; Belczynski et al., 2016b; Vigna-Gómez et al., 2018; Kruckow et al., 2018; Giacobbo and Mapelli, 2020; Lau et al., 2020), as well as EMRIs (Bortolas and Mapelli, 2019). \nFor example, it has been demonstrated (Tauris et al., 2013, 2015) that ultra-stripped SNe are at work in close-orbit NS+NS and BH+NS systems that LISA and LIGO will eventually detect. The reason being that extreme stripping of the companion star by the accreting NS or BH during the last mass-transfer stage (Case BB RLO), produces an almost naked metal core prior to the second SN. This has an important effect on the magnitude of the kick added onto the newborn (second) NS, which affects the survival probabilities. It was argued qualitatively and quantitatively (Tauris et al., 2017) that the resulting kicks are often, but not always, small - depending on the mass of the collapsing metal core and thus on the resulting NS mass which enhances the survival probability. \nPost-LISA-launch objective: The overall detection rate of Galactic NS+NS systems by LISA is thus directly affected by the magnitude of the kick, since a large kick can disrupt the binary during the SN. A large kick may also produce moderately more eccentric LISA NS+NS sources (Lau et al., 2020). The systemic velocity imparted by the two SN kicks displaces a binary from its birth position in the thin Galactic disc. LISA's ability to localise Galactic NS+NS sources on the sky to within a few degrees (Kyutoku et al., 2019; Lau et al., 2020) may therefore constrain the kick distribution by measuring the Galactic NS+NS scale height. By increasing the sample of observed compact binaries, LISA can thus be used to constrain the kick mechanism. In turn, this constrains the nature of SN explosions in binary system Podsiadlowski et al. (2004).", '1.7.1.8 Neutron star equation of state': "Matter in the interior of a NS is compressed to densities exceeding those in the centre of atomic nuclei, providing a unique possibility to probe the nature of the strong interaction and to determine the NS composition. Via the EOS, matter properties determine the star's radius for a given mass (Lattimer and Prakash, 2016; Özel and Freire, 2016). Candidate EOSs can be tested by measuring the mass and radius for a NS or via the accurate measurement of a NS with a high mass because each EOS has a corresponding maximum allowed mass. Thus, finding a NS with a mass above the maximum allowed for an EOS rules out that EOS, and the radio measurement that PSR J0740+6620 has a mass of 2.14 solar masses rules out many EOSs (Cromartie et al., 2020). In addition, the Neutron Star Interior Composition Explorer (NICER) has enabled the measurement of the mass and radius for PSR 0030+0451. In particular, M/R is measured to 5%, but M and R separately are known to ≈ 10 % (Riley et al., 2019; Miller et al., 2019), and \nthe uncertainties still do not allow for the determination of a unique EOS. STROBE-X and eXTP could do the same work as NICER to even to a larger distance. Masses provided by GW measurements would help dramatically, since for the pulsars observed using NICER the pulse profile fitting is mostly sensitive to M/R, and having data points with M and M/R measured well is much more valuable than just having M/R. With GW measurements, the determination of the tidal deformability for merging NSs is another measurable parameter that can constrain the EOS, as has been shown for GW170817 (Abbott et al., 2018a). Given the small number of constraining measurements to date, it is clear that additional EM measurements of pulsars and GW measurements of merging NSs are both necessary (Raaijmakers et al., 2020) to obtain conclusions that will affect our understanding of fundamental physics. \nPost-LISA-launch objective: Many binary systems with NSs produce GWs that will be detectable by LISA, leading to NS mass distributions for various binary populations, and some of these populations may have high mass NSs to further constrain the EOS. While NS+NSs are somewhat rare, binaries with a WD and a NS are expected to be plentiful. LISA will also detect binaries that are approaching mergers, and predicting mergers will allow for EM observations to be planned at the time of the merger. UV, optical, and near-IR observations to determine the remnant type and to constrain the mass and velocity of the ejecta will be very powerful for constraining the EOS (Coughlin et al., 2018; Margalit and Metzger, 2019), especially with a facility like STROBE-X.", '1.7.1.9 Disentangling formation environments based on LISA data': "One of the exciting prospects of LISA observations is the possibility to disentangle the formation channels from compact sources in different environments based on their distinctive properties/demographics; most importantly isolated binary evolution in the Galactic disc or dynamical interactions in dense environments (e.g. open, globular and nuclear star clusters) or isolated triple evolution (Section 1.3). Whereas the majority of LISA binaries are expected to form in isolation (Section 1.4), several key properties, in particular orbital eccentricity and component masses, can reveal deviating birth environments. If LISA is able to constrain these source properties from the GW signal for a given resolved system, the formation channel for that particular system may be inferred. Here, we describe briefly ways these properties may differ between different formation channels and describe applications to particular classes of binaries. \nIn general, BH and WD binaries that form as isolated systems through standard binary evolution processes are expected to be nearly circular by the time they enter the LISA frequency band. This is a consequence of the various dissipative forces expected to operate throughout the binary evolution that circularize the binary orbit, namely CE (Ivanova et al., 2013; Kruckow et al., 2016; Giacobbo and Mapelli, 2018; Vigna-Gómez et al., 2020) and tidal interactions (Zahn, 1977; Postnov and Yungelson, 2014; Belczynski et al., 2020). In contrast, LISA binaries that formed dynamically in dense stellar environments may have relatively high eccentricities. In the dense star clusters, frequent dynamical encounters impart large eccentricities to binaries (Heggie and Hut, 2003), whereas the formation of triple systems with an inner double compact object can reach high eccentricities via von Zeipel-Kozai-Lidov cycles (Antonini et al., 2016; Arca Sedda et al., 2021b; Rastello et al., 2019; Martinez et al., 2020), which induce a secular variation of the inner binary eccentricity (von Zeipel, 1910; Kozai, 1962; Lidov, 1962). \nDynamically formed binaries are expected to feature several distinct sub-classes of formation channels that may also be distinguished by their eccentricities. In order of increasing characteristic formation frequency, these dynamical sub channels include: binaries dynamically ejected from their host cluster that merge as isolated binaries ( f GW ≈ 10 -5 Hz ), binaries that merge in cluster between strong dynamical encounters ( f GW ≈ 10 -3 Hz ), and finally binaries that merge in cluster through GW capture during single-single ( f GW ≈ 10 -1 Hz ) or few-body dynamical encounters encounters ( f GW ≈ 1 Hz ) (Breivik et al., 2016; Banerjee, 2018; Kremer et al., 2018a; Samsing and D'Orazio, 2018; D'Orazio and Samsing, 2018; Arca Sedda et al., 2021b; Samsing \nand D'Orazio, 2019; Kremer et al., 2019b; Zevin et al., 2019b; Banerjee, 2020; Arca Sedda et al., 2020b). Post-LISA-launch objective: Binaries formed through the ejected and in-cluster merger channels are expected to have eccentricities at GW frequencies of 10 -2 Hz of roughly 10 -3 and 10 -2 , respectively, which are expected to be measurable by LISA (Nishizawa et al., 2016). Furthermore, the likelihood of an eccentric merger is dependent on the eccentricity and orbital separation of the outer perturber's orbit and the mutual orientation of the outer and inner orbit (Liu and Lai, 2018; Arca Sedda et al., 2021b). Since the typical binary architecture can be connected with the cluster structure, in terms of either mass and radius or velocity dispersion, the detection of binaries with given orbital properties can carry insights on the type of cluster that harboured the merger. \nIn nuclear star clusters and, more in general, galactic nuclei, the formation and evolution of compact binaries can be substantially affected by the presence of an MBH (Lee, 1995; Blaes et al., 2002; Miller and Lauburg, 2009; Arca Sedda, 2020a). MBHs are not only a common occurrence in nuclear star clusters (e.g., Graham and Driver, 2007; González Delgado et al., 2008), but their masses are correlated (Graham and Spitler, 2009; Scott and Graham, 2013; Graham, 2016a). The binary can develop ZKL oscillations as a result of secular perturbations exerted by the MBH tidal field (Antonini and Perets, 2012; Hoang et al., 2018; Fragione et al., 2019; Arca Sedda, 2020a). Up to 40% of binaries undergoing ZKL oscillations in galactic nuclei transit into the LISA band with an eccentricity > 0 . 1 (Arca Sedda, 2020a). LISA has the potential to measure the eccentricity oscillations driven by an MBH onto a stellar BH+BH binary out to a few Mpc (Hoang et al., 2019) thus offering a unique way to probe the KL mechanism in galactic nuclei and to disentangle this sub-channel of the dynamical formation scenario. Post-LISA-launch objective: Thus, if measurable by LISA, eccentricities (or lack thereof) may serve as a strong fingerprint pointing toward the specific formation channel (Breivik et al., 2016; Nishizawa et al., 2017; Randall and Xianyu, 2019a; Kremer et al., 2019b). \nIn the case of NS binaries, NS natal kicks (e.g., Hobbs et al., 2005) may result in higheccentricities for binaries that form through isolated binary evolution. In this case, eccentricity may no longer be useful for distinguishing between the dynamical and isolated formation channels. However, even in this case, dynamical and disc binary NSs may still have distinguishable eccentricity distributions that can potentially be differentiated with LISA (Andrews et al., 2020). \nAlthough rare, dynamically-formed NS+BH systems represent a class of GW sources that potentially offer the widest range of peculiarities compared to the isolated channel in terms of total mass, primary mass, and high eccentricity at mHz frequencies (Arca Sedda, 2020b). Isolated NS+BH systems (Kruckow et al., 2018) are mostly characterised by BHs with masses of 6 -10 M ⊙ at high, Milky Way-like metallicity ( Z = 0 . 0088 ), or BH masses of 10 -25 M ⊙ at low metallicity, ( Z = 0 . 0002 ), and nearly zero eccentricity at merger (Giacobbo and Mapelli, 2018). Dynamical formation of these systems is not generally relevant for isolated LISA sources in the Milky Way, but in globular and nuclear clusters up to 50% of dynamically formed compact NS+BH feature BH masses > 10 M ⊙ , and a large probability ( ∼ 50% ) will have an eccentricity > 0 . 1 when transiting into the mHz frequency band of LISA (Arca Sedda, 2020b). \nIn general, GW sources forming through dynamical channels may contain compact objects with masses that differ or that are even not expected to form at all from isolated binary evolution of Galactic disc sources. For instance, BHs with masses between ∼ 55 M ⊙ -120 M ⊙ are not expected to form from the evolution of single massive stars due to pair-instability SNe (Woosley et al., 2007; Fryer et al., 2012; Belczynski et al., 2016a; Spera and Mapelli, 2017; Farmer et al., 2019; Woosley and Heger, 2021). This range has been described as the upper-mass-gap for BHs. It may be possible to form binaries containing BHs in this mass range through dynamical processes in stellar clusters (Di Carlo et al., 2020a). One channel to form such BHs could be through hierarchical mergers of stellar-mass BHs in nuclear and globular clusters (Miller and Hamilton, 2002b; Rodriguez et al., 2019; Arca Sedda et al., 2020b; Arca Sedda, 2020a; Samsing and Hotokezaka, 2020; Mapelli et al., 2020). Hierarchical mergers are most likely to happen \nin the densest stellar clusters with the highest escape velocities, such as nuclear star clusters, as these clusters can retain the binary despite the GW recoil kick from the merger (Fragione et al., 2018; Antonini et al., 2019; Fragione and Silk, 2020; Neumayer et al., 2020; Arca Sedda, 2020a). In these extreme environments, binaries at formation are tighter, on average, than in normal clusters, and the interactions with flyby stars and the possible long-term effect of a central MBH can boost stellar collisions and BH mergers, thus possibly inducing a significant modification of the BH mass spectrum. Another possibility to form more massive BHs could be through collisional runaway mergers of BH progenitors in dense star clusters (Portegies Zwart and McMillan, 2002; Portegies Zwart et al., 2004; Freitag et al., 2006b,a; Giersz et al., 2015; Mapelli, 2016; Di Carlo et al., 2020a; Kremer et al., 2020a). \nPost-LISA-launch objective: Such runaway mergers and collisions in dense clusters can also lead to the formation of IMBHs in the mass range 10 2 -10 4 M ⊙ (Ebisuzaki et al., 2001; Portegies Zwart et al., 2006; Gürkan et al., 2006; Amaro-Seoane et al., 2007; MacLeod et al., 2016b; Arca Sedda and Mastrobuono-Battisti, 2019; Askar et al., 2021; Hong et al., 2020; Arca Sedda et al., 2020b; Mapelli et al., 2020), and mergers of IMBH+IMBH with component masses in the range 10 3 -10 4 M ⊙ can be observed with LISA up to redshift z ≲ 3 (Arca Sedda and Mastrobuono-Battisti, 2019; Arca Sedda et al., 2020a, 2021a; Jani et al., 2019). The number and characteristics of BHs in the upper mass-gap could shed a light on the relative contribution of dynamically-formed or isolated sources to the overall population of BH+BH mergers. \nLastly, the evolution of isolated triples can also lead to a mass distribution that deviates from that of isolated binary evolution. This is mainly due to two effects. Firstly, to form a LISA source, isolated binary evolution relies on one or more mass-transfer phases that reduce the orbital period (Section 1.3) down to the range observable by LISA. Mass transfer can also occur in triples (even a larger fraction of triples experiences RLO; Toonen et al., 2020) (Sections 1.2.4.4 and 1.3.3), however orbital shrinkage can also be achieved by the combination of three-body dynamics with dissipative processes. The increased eccentricities during von Zeipel-Kozai-Lidov cycles reduce the GW inspiral time (e.g. Thompson, 2011; Antonini et al., 2017; Rodriguez and Antonini, 2018; Fragione and Loeb, 2019). Moreover, if a star does not fill its Roche lobe, and does not lose its envelope prematurely, it typically will form a more massive remnant compared to the case of RLO mass stripping. Such triples will on average contain stars that are more massive than those formed through isolated binary evolution (Hamers et al., 2013; Toonen et al., 2018), and will not contain He-core WDs (which have masses ≲ 0 . 45 M ⊙ which can only be formed in a Hubble time through mass stripping). Secondly, similar to the evolution in star clusters, sequential mergers in multiples give rise to higher stellar masses (Safarzadeh et al., 2020a; Hamers and Safarzadeh, 2020; Lu et al., 2020). In addition, the effect of a tertiary perturber can induce precession of the spins and lead to spin misalignment (Antonini et al., 2018; Liu and Lai, 2018; Rodriguez and Antonini, 2018).", '1.7.2 LISA sources as galactic probes': "Contributors: Valeriya Korol, Raffaella Schneider, Luca Graziani, Astrid Lamberts, Samuel Boissier, Martyna Chruslinska, Alberto Sesana, Katie Breivik, Shane Lar- \nCoordinators: Valeriya Korol son, Michela Mapelli \nStellar binaries detectable by LISA bear the imprint of the properties of their native stellar environments (galaxies and stellar clusters) such as the total stellar mass, IMF, star formation history (SFH), age and metallicity ( Z ). These properties can be investigated by combining binary population synthesis (BPS) models (Section 1.6.3) with models of galaxy formation and evolution. Several methods have been developed to achieve this goal. The combination of BPS models with theoretical semi-analytic or observationally inferred cosmic star formation rate densities \nprovides a fast way of predicting the evolution of the overall birth and merger rates with redshift (Schneider et al., 2001; Regimbau, 2011; Marassi et al., 2011; Dominik et al., 2013; Belczynski et al., 2016a; Dvorkin et al., 2016b; Lamberts et al., 2016; Elbert et al., 2018; Chruslinska and Nelemans, 2019; Boco et al., 2019). In particular, observation-based approaches allow one to account for the current observational uncertainties on the birth metallicity distribution of stars forming over the cosmic history and evaluate the related uncertainty on the predicted properties of mergers (e.g. Chruslinska and Nelemans, 2019). A detailed understanding of the properties of galaxies hosting GW sources can be gained from cosmological simulations, which provide a detailed description of the cosmic star formation in a more accurate context of the galaxy evolution. Galaxy catalogues from the Illustris (Vogelsberger et al., 2014), GASOLINE (Stadel, 2001; Wadsley et al., 2004), EAGLE (Schaye et al., 2015) simulations have been used for predicting NS+NS and BH+BH mergers (e.g. Mapelli et al., 2017; O'Shaughnessy et al., 2017; Artale et al., 2019). Similarly, the Latte simulation of Milky Way-like galaxies of the FIRE hydrodynamical simulation project (Hopkins et al., 2014; Wetzel et al., 2016) was adopted to study the properties of Galactic WD+WDs and BH+BHs accessible to LISA (Lamberts et al., 2018, 2019). Alternative hybrid pipelines such as GAMESH (Graziani et al., 2015, 2017; Graziani, 2019), combining a dark matter simulation with semi-analytic star formation, chemical enrichment and numerical radiative transfer, represent an advantageous alternative to study the redshift evolution of compact binaries along the assembly of a Milky Way-like galaxy and in its local volume dwarf satellites (Schneider et al., 2017; Marassi et al., 2019; Graziani et al., 2020). \nEffect of the IMF. The IMF is one of the key ingredients in the BPS that sets the distribution of initial masses and the relative proportions of stars forming in different mass ranges. Therefore it has a direct impact on the observed merger rates and properties of the LISA sources. Studies often adopt the IMF inferred from the observations of stars in the local Galactic neighborhood (e.g. Kroupa, 2001; Chabrier, 2003). However the universality of this assumption is one of the fundamental open questions in astronomy and is still debated (e.g. Bastian et al., 2010). Theoretical studies show that with the assumption of the Milky Way-like IMF one may underestimate the number of WD and NS progenitors forming at redshifts ≲ 1, especially at low metallicities (Chruślińska et al., 2020), and, therefore, underestimate the predicted number of individual LISA detections and background/foreground noise. \nPost-LISA-launch objective: LISA's observations of stellar remnants - invisible to EM observatories - will offer us an alternative way of probing the IMF. For instance, hundreds of Galactic WD+WDs with measured chirp mass (Rebassa-Mansergas et al., 2019) can be used to constrain the low-mass end of the IMF in different Galactic habitats. In addition, numerous LISA detections in the Magellanic Clouds will enable the studies of the IMF with GWs in alternative environments (Korol et al., 2020). \nEffect of metallicity. The metallicity is another important assumption of the models that affects different types of stellar binaries in different ways (Chruslinska et al., 2019). The predicted metallicity dependence of the formation efficiency of merging BH/NS binaries is a complex function of numerous poorly constrained phases of binary evolution. Specifically, BH+BH mergers resulting from isolated stellar evolution are typically found to form much more efficiently at low metallicity ( ≲ 0 . 1 -0 . 3 Z ⊙ ) than at solar metallicity (Belczynski et al., 2010b; Eldridge and Stanway, 2016; Stevenson et al., 2017b; Schneider et al., 2017; Klencki et al., 2018; Giacobbo et al., 2018). The differences in formation efficiency reach up to two orders of magnitude and consequently, the size of the observable BH+BH population is sensitive to the amount of star formation happening at low metallicity (Dominik et al., 2013; Mapelli et al., 2017; Marassi et al., 2019; Chruslinska et al., 2019; Neijssel et al., 2019; Graziani et al., 2020; Santoliquido et al., 2020a,b). Furthermore, the most massive BH+BH are expected to form at low metallicity and their mass distribution could potentially be linked to the metallicity distribution of their progenitors. Metallicity dependence of the formation efficiency of NS+NS mergers is typically found to be much weaker than for BH+BH, with the mixed systems falling in between. For the case \nof WD+WD, the metallicity mainly changes the total number of WD+WDs by allowing lower masses for a star to reach the WD stage in a Hubble time with decreasing metallicity. This results in a moderate increase (few tens of percents) in the number of resolved LISA sources (Yu and Jeffery, 2010; Korol et al., 2020). \nEffect of star formation histories. The merger rate of compact stellar binaries across cosmic time is a direct consequence of the SFH (Madau and Fragos, 2017; Artale et al., 2019; Dominik et al., 2013; Mapelli et al., 2017; Mapelli and Giacobbo, 2018; Vitale et al., 2019; Neijssel et al., 2019; Santoliquido et al., 2020a). Together with BPS models, SFH regulates the content of stellar binaries in the LISA band at a given time. Galactic WD+WDs can be used as a tool to study the SFHs of the MW components: due to the different timescales to reach the mHz frequencies, WD+WDs of different core composition dominate different parts of the Galaxy due to their distinct SFHs. Specifically, double He-core WDs with formation times that can exceed 10 Gyr populate the Galactic bulge, thick disc and stellar halo; double C/O-core WDs, typically form on timescales shorter than 2 Gyr and are associated with a much younger populations present in the thin disc; mixed He-C/O-core binaries present an intermediate distribution (Yu and Jeffery, 2010; Lamberts et al., 2019). In addition, SFH has significant effects on the LISA detection rates in the Milky Way satellites (Korol et al., 2020). \nStructure of the Milky Way with resolved and unresolved sources. It is expected that the Galactic GW population at mHz frequencies will be largely dominated by WD+WDs and will have two components in the LISA data: population of high-frequency individually resolved binaries and unresolved stochastic foreground from low-frequency binaries (Section 1.6.2). Both resolved and unresolved WD+WDs encode global properties of Galactic stellar populations, and can thus be used as a tool to study the Milky Way's stellar content and shape. \nPost-LISA-launch objective: Affected by different selection biases than EM observatories, LISA can probe the entire volume of the Milky Way and therefore will facilitate detailed studies of its the far side (Fig. 15). Moreover, unaffected by the dust extinction and stellar crowding, LISA can also probe the inner Galaxy at all latitudes. For several thousands WD+WDs measurements of the sky positions and distances will enable the mapping of the Galaxy. Reconstructed density profiles of WD+WDs will provide unbiased constraints on the scale length parameters of Galactic bulge/bar and disc that are both accurate and precise, with statistical errors of a few % to 10% level (Adams et al., 2012; Korol et al., 2019; Wilhelm et al., 2020). The Galactic stellar halo is also expected to host up to a few thousand WD+WDs, and therefore can potentially be studied with WD+WDs in a similar way (Ruiter et al., 2009; Yu and Jeffery, 2010; Lamberts et al., 2019). Furthermore, the LISA sample is found to be sufficient to disentangle between different commonly used disc density profiles, by well covering the disc out to sufficiently large radii. The stellar bar will also clearly appear in the GW map of the bulge. LISA's WD+WDs can accurately characterise the bar's physical parameters: length, axis ratio and orientation angle with respect to the Sun's position (Wilhelm et al., 2020). However, because of the low density contrast compared to the background disc, the spiral arms will be elusive to LISA. Finally, building upon the analogy with simple stellar population models used for inferring stellar masses of galaxies based on their total light, the total stellar mass of the Galaxy can be estimated from the number of LISA events. Using a simplified example of Milky Way satellites, Korol et al. (2021) showed that based on BPS models of LISA sources satellite masses can be recovered within 1) a factor two if the SFH of the satellite is known and 2) within an order of magnitude even when marginalising over alternative SFHs. When also accounting for the unresolved Galactic foreground, this method could be extended for measuring the total stellar mass of the Milky Way. \nPost-LISA-launch objective: The power of constraining the overall properties of the Galactic potential will be significantly enhanced by using LISA detections in combination with EM observations of binaries motions. BPS studies forecast up to 150 detached and interacting WD+WDs detectable through both EM and GW radiation (e.g. Korol et al., 2017; Breivik et al., \nFigure 15: Numerous WD+WDs detectable by LISA will enable the mapping of our Galaxy. In the background the artist impression of our current view of the MW. Over-plotted in colour WD+WD with SNR > 7 from Wilhelm et al. (2020). LISA's position is at (0 , 0) . The selection effect due to GW frequency in visible in colour. Figure credit: Valeriya Korol. \n<!-- image --> \n2018, see also Section 1.4). For these multi-messenger binaries 3D positions provided by LISA can be combined with proper motions - for example provided by Gaia or Vera Rubin Observatory into the rotation curve, which allows the derivation of the stellar masses of the Galactic baryonic components (Korol et al., 2019). \nThe unresolved Galactic foreground will provide complementary constraints on the Galactic structure. For example, the Galactic foreground will show whether the WD+WD population traces the spatial distribution of young, bright stars (and thus do experience significant kicks), or traces a vertically heated spatial distribution associated with Galaxy's oldest stellar populations. This can be understood from the shape of Galactic power spectral density that depends on the characteristic scale height of the WD+WD population (Benacquista and Holley-Bockelmann, 2006). Post-LISA-launch objective: In addition, using the spherical harmonic decomposition of the LISA data streams, the structure of the disc population of Galactic WD+WDs can be constrained with an accuracy of 300 pc (Breivik et al., 2020b). The relative poor resolution compared with the resolved sources is a direct consequence of LISA's poor spatial resolution at low frequencies. Nevertheless, an independent measurement at low frequencies will either help to confirm the structure of the resolved sources or point to frequency-dependent Galactic structure.", '2 Massive Black Hole Binaries': 'Section coordinators: Elisa Bortolas, Pedro R. Capelo, Melanie Habouzit Section reviewers: Laura Blecha, Massimo Dotti, Zoltan Haiman', 'Contributors: Elisa Bortolas, Pedro R. Capelo, Melanie Habouzit': "The observed BH mass spectrum spans ten orders of magnitude, ranging from a few M ⊙ of stellar-mass BHs to more than 10 10 M ⊙ for the most extreme MBHs. Our knowledge of the mass spectrum has expanded over the past decade. On the low-mass end, the GW facilities Laser Interferometer Gravitational-wave Observatory (LIGO) and Virgo have observed the mergers of low-mass BHs in the range ∼ 6-80 M ⊙ (Abbott et al., 2020c). At the high-mass end, we have discovered in the high-redshift Universe extremely bright objects, called quasars, powered by MBHs with masses similar to those of the most massive MBHs around us ( M BH ⩾ 10 8 M ⊙ at z > 6 , e.g. Mortlock et al., 2011; Bañados et al., 2018b; Yang et al., 2020a). LISA has the ability to detect MBHs of M BH = 10 3 -10 7 M ⊙ through the last stages of in-spiral and merger up to z ∼ 20 , bridging these extremes of the mass spectrum. \nThe mass-redshift regime that LISA can probe is key to constraining the origin and growth of MBHs, and is one of LISA's main science goals. Considering the current state of observations, theory, and simulations, we still do not know how MBHs form and evolve in the early Universe. We do not know how they assemble with time and become present in almost all the galaxies in the local Universe, from dwarf galaxies (with stellar masses of ⩽ 10 9 . 5 M ⊙ , e.g. Mezcua and Domínguez Sánchez, 2020; Greene et al., 2019; Chilingarian et al., 2018; Mezcua et al., 2018; Baldassare et al., 2015; Reines et al., 2013) to large ellipticals (e.g. Magorrian et al., 1998; Gültekin et al., 2009; McConnell et al., 2011; Kormendy and Ho, 2013; Graham, 2016b; Davis et al., 2019a; Sahu et al., 2019a). LISA observations will play a key role in addressing these enigmas. In this Section, we only discuss MBHs, which we define as BHs with ≳ 100 M ⊙ . Additionally, we do not make an explicit distinction within that range, i.e. we do not distinguish between intermediate-mass BHs, massive black holes, and supermassive BHs. \nIn the hierarchical paradigm of galaxy formation, we expect central MBHs to coalesce after the merger of their host galaxies. As shown in Fig. 16, MBHs will have to cross an impressive range of scales, from when they are hosted in separate galaxies at early times to the end of their dance, when they coalesce with each other (Begelman et al., 1980; Milosavljević and Merritt, 2001; Dullo and Graham, 2014). Following a galaxy merger, while MBHs are still separated by kpc to tens of kpc scales, they will start losing orbital energy and angular momentum via gravitational drag from background gas and stars, causing them to sink to the centre of the remnant galaxy (a process referred to as dynamical friction). On ≲ pc scales, the MBHs will form a gravitationally-bound binary and evolve further via interactions with gas and individual stars (the so-called binary hardening phase). This may include interactions with a circumbinary disc on ∼ 10 -3 pc scales. Finally, the MBHs enter the last stage of the dance, i.e. the GW regime ( ⩽ 10 -5 pc scale). \nTo maximize the scientific return of LISA, advances are needed in our theoretical understanding of MBH formation, dynamics, and evolution, a field of research that started in the 1980s (Begelman et al., 1980). Building powerful tools such as semi-analytical models, N-Body and hydrodynamic simulations is crucial to predict the MBH mergers that LISA will detect as a function of the intrinsic properties that describe both the MBH and galaxy. Currently, the predicted MBH merger rate spans more than one order of magnitude, from a few LISA detections per year to tens. Rate predictions depend on computational methods, and on the modelling of the relevant physics. In the coming years we will improve on the dynamical range of scales that we can resolve, and address how different mechanisms of MBH formation, galaxy environments, MBH growth models, and MBH dynamics can shape the merger rates of MBHs, thus paving the way for the interpretation of LISA data. \nIn the near future, several space missions will be launched with the goal of constraining the formation and evolution of MBHs and their environments. These missions will complement LISA in the EM domain, and will provide unprecedented constraints on the entire population of \nCredit: Lupi et al. (2019) \n<!-- image --> \nCredit: Capelo et al. (2015) \n<!-- image --> \nFigure 16: A schematic view of the complex and multi-scale processes affecting the formation of a hard MBH binary system. Figure credit: Silvia Bonoli and Alessandro Lupi. \n<!-- image --> \n<!-- image --> \nMpcs: The large scale structure Influence of the large scale environment on: black hole seeding, frequency of mergers, galaxy transformation \n<!-- image --> \n1-100s kpcs: Galaxy interactions/merger \nDetails of the merger have influence on: black hole growth via gas accretion, formation of a black hole binary, galaxy transformation \n1-10s pc: Formation of a bound binary \nThe host properties have influence on: hardening of the binary, accretion episodes \n<1 pc: Hardening of the binary \nThe host properties have influence on: timescale of hardening Effect of circumbinary disc Three-body interactions (hyper-velocity stars) \nMBHs. The James Webb Space Telescope (JWST; Gardner et al., 2006) and the Roman-Wide Field Infra-red Survey Telescope (Spergel et al., 2015) will image the first galaxies (e.g. Williams et al., 2018), the cradles of the first MBHs. The assembly of galaxies will also be witnessed by the new thirty-meter telescopes such as E-ELT, TMT, and GMT. New X-ray facilities such as Athena (Nandra et al., 2013), as well as the LynX (The Lynx Team, 2018) and AXIS (Mushotzky, 2018) concept missions, will aim at uncovering the population of accreting young MBHs at high redshift ( z > 6 ). \nWith LISA and the aforementioned new instruments working in the EM domain, we will enter the new multimessenger era for MBHs. By performing synergistic observations that combine low-frequency GW signals with EM signals from the same source, we will uncover previously unavailable information. These combined observations will precede, accompany, or follow, the MBH merger events, helping us to constrain MBH activity, understand their immediate surroundings (e.g. the nature of the accretion disc, jets, and the accreted/ejected material), and its relation with the host galaxy. One challenge of multimessenger observations is the localization of the sources, and the confirmation that they are indeed MBH binaries. Before the launch of LISA, we will have to better understand, among other aspects, how the different potential observational EM signatures of coalescing systems are originated, and develop new analysis tools to identify these GW source candidates in large datasets. \nLISA will also constitute a bridge between the two GW frequency regimes that are already being investigated: the highest GW frequencies (LIGO/Virgo/KAGRA), and the lowest GW frequencies which build the GW background (observed by Pulsar Timing Arrays; PTAs). In the coming years, we will have to fully exploit these missions to, e.g. select, monitor, confirm and characterise MBH binaries (MBHBs), but also understand their small-scale to galactic and large-scale environments, and how they fit within the full MBH population. \nSections 2.2 and 2.3 review the theoretical background and highlight pre-launch objectives for the LISA community that can sharpen our preparation for the mission. Section 2.4 distills the theoretical picture into LISA's observables and highlights uncertainties. The pre-launch objective is to compare different approaches to obtain realistic predictions that can be used, post-launch, \n<!-- image --> \nCredit: Souza Lima et al. (2017) \n<!-- image --> \nCredit: Bowen et al. 2017 \n<!-- image --> \nto interpret LISA's data. The pre-LISA theoretical development is of paramount importance because the expectation is that LISA's event properties will be compared to theoretical models through a Bayesian framework in order to perform astrophysical inference (Sesana et al., 2011a). Section 2.5 focuses on EM signatures of MBHs, highlighting both pre-launch (improve theoretical models, search for EM emission from MBHBs) and post-launch (devise strategies for searches of EMcounterparts to MBHBs detected by LISA) objectives. Finally, Section 2.6 shows how LISA's results can be strengthened by complementary campaigns performed by different instruments and facilites, straddling pre-launch and post-launch objectives dependent on whether missions overlap or not.", '2.2 MBHs and their path to coalescence': "Coordinators: Matteo Bonetti, Hugo Pfister \nContributors: Emanuele Berti, Tamara Bogdanovic, Elisa Bortolas, Pedro R. Capelo, Monica Colpi, Pratika Dayal, Massimo Dotti, Alessia Franchini, Davide Gerosa, Zoltan Haiman, Peter Johansson, Fazeel Mahmood Khan, Giuseppe Lodato, Lucio Mayer, David Mota, Vasileios Paschalidis, Alberto Sesana, Nick Stone, Tomas Tamfal, Marta Volonteri, Lorenz Zwick \nThere is observational evidence that a significant fraction of galaxies host MBHs in their centres (Kormendy and Ho, 2013), and at least some of them harbour an MBH since the dawn of structure formation (e.g. Bañados et al., 2014; Wu et al., 2015; Bañados et al., 2018b). This, combined with the notion that galaxies aggregate via repeated mergers of smaller structures (Fakhouri et al., 2010; O'Leary et al., 2021), leads to the conclusion that a number of MBHBs should have formed across cosmic epochs, and that their ultimate coalescence phase could be observed by LISA (e.g. Klein et al., 2016; Dayal et al., 2019; Chen et al., 2020b; Barausse et al., 2020b; Valiante et al., 2021; Bonetti et al., 2019). \nThe exact number of detectable MBHB mergers and their properties (Section 2.4) will depend on still poorly understood parameters, such as the low-mass end of the MBH mass function and their seeding mechanism (Section 2.3), or the host galaxy structure and environment. However, as a start, we can try to address the following questions for MBHs in the LISA mass-redshift range: (i) What are the mechanisms which bring two MBHs in distinct galaxies separated by tens of kpc close enough, so that they emit GWs and merge when they are at separations of the order of their gravitational radii, ∼ 10 -6 ( M BH / 10 7 M ⊙ ) pc? (ii) Given the variety of galaxy types, MBH masses, and orbits they can have, are these mechanisms always efficient enough that a galaxy merger results in an MBH merger within the age of the Universe? We begin by considering two galaxies with a range of properties hosting an MBH in their centre. We will then describe the different steps that may or may not lead to the MBH merger following the merger of the two galaxies. \nIn a seminal work, Begelman et al. (1980) were the first to explore the dynamics of MBH pairs in merging galaxies. In their study they highlighted the occurrence of three steps, which we will use as the foundation of this section: the initial dynamical friction phase (Section 2.2.1; kpc scale), when MBHs and their hosts sink toward the centre of the remnant galaxy losing orbital energy and angular momentum until they are gravitationally bound and form a binary; the binary hardening phase (Section 2.2.2; pc scale), when the binary mainly interacts with single stars and/or gas; and finally the relativistic phase (Section 2.2.3; mpc scale), when the dynamics is dominated by GW emission. \nThroughout the years, this initial picture has been enriched by many different aspects, highlighting different astrophysical regimes for MBH orbital decay determined by the nature of the galactic environment, with associated different times-cales which depend on the MBHs mass, mass ratio, mass distribution and thermodynamics in the galactic nucleus etc. (see Fig. 17). Below we focus on the recent developments on the aforementioned stages of the orbital decay, \nFigure 17: Illustration of the physical processes affecting two coalescing MBHs after the merger of their host galaxies. The cartoon reports the typical physical scales associated to each process for a nearly equal mass MBH pair of about 10 6 M ⊙ . Scales vary depending on the exact mass, mass ratio and host galaxy properties. These physical processes are described in Sections 2.2.1.1, 2.2.1.2 (dynamical friction); Section 2.2.1.3 (clump scattering, effect of bars/spirals); Section 2.2.2.1 (stellar-driven hardening); Section 2.2.2.4 ( 3 rd incoming MBH); Section 2.2.2.2 (disc-driven migration torques; circumbinary disc and minidisc torques); Section 2.2.3 (gravitational waves). Figure credit: Elisa Bortolas. \n<!-- image --> \nand we highlight the prospects for future research in the context of the science relevant to LISA. We refer the reader to the many existing reviews for a more complete presentation of the topic (e.g. Mayer, 2013; Colpi, 2014; Dotti et al., 2012; De Rosa et al., 2019b).", '2.2.1 The galaxy merger and the large-scale orbital decay at kpc scales': "In order to make forecasts for the LISA event rates the first step is to quantify robustly the range of decay time-scales at kpc scales, where the BH pair is expected to spend most of its time. \n2.2.1.1 Dynamical friction in collisionless media When a massive perturber, such as an MBH, with mass M BH moves in a medium composed of collisionless particles (stars or dark matter, DM) with masses m ⋆ ≪ M BH , it deflects such particles from their unperturbed trajectories. As a result a trailing overdensity is generated, often referred to as 'wake', which then pulls the perturber towards it owing to its gravitational force, namely it causes a deceleration directed opposite to its motion. Such drag force is the so-called dynamical friction (Chandrasekhar, \n1943). Under the assumption of an infinite homogeneous medium with density ρ , if the background is characterized by an isotropic Maxwellian velocity distribution with velocity dispersion σ , Chandrasekhar (1943) showed the force acting on the perturbing body is: \n⃗ F DF ∝ -M 2 BH ρ G ( v σ ) ln Λ ⃗v v 3 , (7) \nwhere v is the perturber velocity relative to the surrounding background, ln Λ ∼ 10 is the Coulomb logarithm 4 and the function G ( x ) , x = v/σ depends on the underlying velocity distribution; if the latter is Maxwellian, as typically assumed, G ( x ) scales as x 3 for x ≪ 1 and as ∼ 1 for x ≳ 2 . When Eq. 7 is applied locally to the case of an MBH moving on a circular orbit of radius r in the stellar background of a singular isothermal sphere ( ρ ∝ σ 2 r -2 ), a calculation (Binney and Tremaine, 1987) shows that the orbital decay of M BH occurs on a time-scale \nτ DF ≈ 8 Gyr ln Λ ( r kpc ) 2 σ 200 km / s 10 7 M ⊙ M BH . (8) \nIf we assume MBHs at kpc scale separations and a 10 6 M ⊙ black hole in a galaxy with σ = 100km / s , this calculation shows that dynamical friction plays an important role in causing a rapid sinking of MBHs with masses in the range accessible to LISA as the process can take less than a Hubble time (in the early stage of a galaxy merger, M BH may be replaced by the mass of a residual galactic core embedding the MBHs, resulting in much shorter time-scales, Yu and Tremaine, 2002). Two 10 6 M ⊙ black holes are indeed expected to bind gravitationally and form a binary once their separation is reduced to a few pc. In the following we detail how this simplified picture is enriched when some of the assumptions made above are relaxed. Overall, more complex dynamics lead to a much broader range of time-scales than expected based on the previous discussion, and render the formation of a binary a more uncertain outcome.", '· Global asymmetries': "The description of dynamical friction given above implies that the drag is local, caused by the overdensity trailing the perturber, thus it neglects the global exchange of orbital angular momentum and energy between the MBH and the host system. Global asymmetries triggered in the mass distribution of the host system (called modes ; see, e.g. Tremaine and Weinberg, 1984; Weinberg, 1986, 1989) can give rise to global torques, and these can be enhanced at resonances between the perturber's orbital frequency and the orbital frequency of the background matter. Owing to new observational data (Gaia Collaboration et al., 2018b), as well as recent theoretical (Hamilton and Heinemann, 2020) and numerical work (Garavito-Camargo et al., 2019; Cunningham et al., 2020; Tamfal et al., 2021; Garavito-Camargo et al., 2020), the global halo mode theory has gained renewed attention. Accounting for the corrections to the dynamical friction time-scale introduced by global torques is likely important in order to provide robust estimates of the initial phase of black hole binary formation and sinking. Studies specifically for LISA MBHBs are required to ultimately assess the importance of these processes in the context of LISA's science.", '· Power-law density profiles: cusps and cores': 'The assumption of an isothermal sphere used to derive Eq. 8 is also a simplification: all real galaxies feature much more complex profiles. Even referring only to the DM distribution, \nits inner density profile is typically believed to behave as a Navarro-Frenk-and-White (NFW) profile ρ ∝ r -γ with γ = 1 , or even shallower in low-mass dwarf galaxies (see, e.g. the evidence on constant density cores in Oh et al., 2015). This shallower core could be the result of baryonic feedback effects (Governato et al., 2010), or of the phase-space density structure inherent to a specific DM model such as self-interacting DM or fuzzy DM (Hui et al., 2017). LISA will be particularly sensitive to MBHBs in the range of masses 10 4 -10 6 M ⊙ mainly hosted in low-mass dwarf galaxies; since many dwarfs appear to be DM cored (at least at low redshift; see, e.g. Moore 1994; Contenta et al. 2018; Leung et al. 2020), this motivates a thorough study of the dynamics in shallow inner density profiles. \nTamfal et al. (2018) modelled numerically the orbital dynamics of a pair of 10 5 M ⊙ BHs during the equal-mass merger of two dwarf galaxies. They showed that, if the merging galaxies have kpc-sized cores, or at least a profile shallower than NFW (inner slope γ ∼ 0 . 6 or lower), the pair of MBHs would stall at separations of 50-100 pc (i.e., when the bound binary is not formed yet) and the coalescence would be aborted. In a halo with an NFW profile, stalling does not occur, rather the MBHs sink very fast to sub-pc separations in less than a few 10 8 yr after the two host galaxies have merged. In self-interacting DM models, in which cores can be > 1 kpc in size assuming a large specific cross section of interaction of 10 cm 2 /g, and which are under-dense compared with Cold Dark Matter (CDM) control cases, an analogous suppression of dynamical friction was found to occur at even larger scales, when galaxies are still in the process of merging, leading to many wandering MBH pairs with few kpc separation (Di Cintio et al., 2017). This opens the possibility that the event rate of MBH mergers detected by LISA could constrain the density profile of dark matter halos of their host galaxies, which in turn can shed light on the physical properties of dark matter particles. \nGas dissipation and stellar feedback were not taken into account in the aforementioned studies; those could delay the binary formation even more. On the other hand, if at least one of the sinking MBHs is surrounded by a massive nuclear star cluster, as in the case of a captured ultra-compact dwarf galaxy, this may enhance the dynamical friction and aid the binary formation and shrinking even in cored DM profiles. These aspects should be investigated in detail in preparation for LISA.', '· MBHBs with very unequal mass ratio': "When the mass enclosed within the binary orbit becomes of order the mass of the secondary, dynamical friction is not effective anymore, and different processes are required to shrink the binary further (see §2.2.2). For equal mass binaries this critical separation roughly corresponds to the distance at which the binary becomes effectively bound, but for binaries in which the secondary MBH is much lighter than the primary, dynamical friction remains the main driver for the MBHB shrinking well below the separation at which the secondary becomes bound. In this situation, the fact that (Eq. 7) considers only the contribution of stars moving slower than the secondary MBH (Chandrasekhar, 1943) can be a major limitation. Antonini and Merritt (2012) found that, if the inner density profile scales as ρ ∝ r -γ , the conventional application of Chandrasekhar's formula works reasonably well if γ > 1 , but does not reproduce the inspiral of the secondary if the profile is very shallow ( γ ∼ 0 . 6 ). The reason is that, in the latter case, stars that move faster than the secondary MBH contribute to most of the force. As a consequence, conventional dynamical friction would predict stalling of the secondary MBH, while the orbit can keep shrinking, albeit at a much slower pace (Dosopoulou and Antonini, 2017). However, this has only been verified when the secondary MBH is significantly smaller than the primary ( q ≲ 0 . 01 , i.e. close to the IMRI regime) and orbits inside a nearly spherical and isotropic nucleus without net rotation. Assessing the outcome for a more realistic profile of the nucleus will be needed in the near future to prepare for LISA. \n2.2.1.2 Dynamical friction in a gaseous medium In the previous sections, we considered orbital decay of a pair of MBHs in a collisionless background of stars and dark matter. However, gas constitutes a significant fraction of mass in many galaxies, especially at high-redshift (Tacconi et al., 2018; Decarli et al., 2020). Similarly to stars, the gaseous wake lagging behind a massive perturber tends to slow it down but the details of the interaction depend on the geometry of the gas wake, which in the case of gas is subject to the additional effect of pressure. \nFor comparable densities gas-driven dynamical friction is larger by a factor of ∼ 5 than that of stars in the transonic regime, i.e., when the Mach number of the perturber is around unity (Ostriker, 1999), while it is of similar order in the supersonic regime, i.e., for Mach numbers much larger than unity, and it is suppressed in the subsonic regime, i.e., below Mach numbers of order unity. However, the overall contribution of the gas-driven component to the total drag force suffered by an MBH in a galactic nucleus is still debated, as it depends sensitively on the dynamical and thermodynamical state of the medium as well as on its density and cooling properties in the vicinity of the perturber. \nHot, low-density gas in gas-poor galaxies is virialized and thus gives little contribution to the drag. However, the central cold and dense region may play an important role. Using semianalytical models, Li et al. (2020a) find that galaxies with low gas fraction and a large stellar bulge favour the formation of binary MBHs, and that their dynamics is dominated by stellar dynamical friction. Hydrodynamical simulations find quite a range of results, whose often large differences are likely driven by the different setups, astrophysical as well as numerical, considered by different simulations. \nFor example, Pfister et al. (2017) also find that the contribution from gas friction is negligible compared to that from stars as in their case (i) the gas density is lower than stellar density; and (ii) the stellar density profile is more regular than the gas one, so that stars act as a smooth background, which is conceptually consistent with the theory of dynamical friction. On the other hand, numerical simulations and observations also show that stellar morphology in merging systems is often highly disturbed and rapidly varying, which complicates this picture and suggests that galactic substructure might be important to take into account. Chapon et al. (2013) find that the collision between the two massive equal mass gas-rich galactic discs drives rapid sinking, primarily owing to gas-driven friction, of the MBHs that pair into a binary. Note that, in equal mass galaxy mergers, funnelling of gas to the centre of the merger remnant via gravitational torques and shocks is maximized relative to the unequal mass merger case considered by Pfister et al. (2017), and this leads to a much higher central gas density. It is therefore not surprising that the two studies reach different conclusions on the relative importance on gas-driven and stellardriven friction. All this shows that the processes leading to the formation of LISA binaries in realistic gaseous and stellar environments deserves future investigations before LISA flies.", '2.2.1.3 More complex mass distributions and additional physical phenomena We': 'expect galaxies to not be realistically represented by the spherical, power-law, and smooth density profiles. As already mentioned in the previous section, global asymmetries affect the sinking time-scale. In order to prepare for LISA, we need to investigate the effects of more complex structures onto the dynamics of MBHs. We summarize below the recent results of several groups studying these effects and highlight some areas of particular interest for future study.', '· Effects of discs': "The question of the effects of large-scale galactic discs ( ∼ 1-10 kpc) and circumnuclear discs ( ∼ 100 pc) on the formation of gravitationally bound MBH pairs is closely related to the question of what types of galaxies are the most likely progenitors of LISA sources. Simulations have already addressed the effect of dynamical friction in composite, rotationally supported environments; they suggested that, quite independently of whether the background is mainly stellar or gaseous (Dotti et al., 2007), dynamical friction acting in rotating discs usually induces the circularization \n0.4 \n0.3 \n0.2 \n0.1 \n0 \n10 \n6 \n10 \n7 \nTHE PAIRING OF MBHS IN THE PRESENCE OF RADIATIVE FEEDBACK \n7 \nFigure 4. \n<!-- image --> \n<!-- image --> \nMBH pairing probability as a function of the host galaxy and MBH pair properties \n0.4 \nthe galactic gas and stellar disks. It is in nevertheless \n0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 0.1 0.2 -0.9 -0.6 -0.3 0 0.3 0.6 0.9 0 0.1 0.2 no RF with RF possible that some fraction of sMBHs evolve on orbits that are inclined relative to the galactic disk. sMBHs on inclined orbits on the one hand experience weaker DF from the gas and stellar disks, an effect that leads to longer inspiral times. On the other hand, perturbaof initially prograde and eccentric orbits, while it reverses the angular momentum of counterrotating trajectories, then again promoting circularization (see, e.g. Dotti et al., 2006; Callegari et al., 2011; Fiacconi et al., 2013; Bonetti et al., 2020a). \n0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 MBH pairing probability no RF with RF 0.5 0.6 0.7 0.8 0.9 1.0 MBH pairing probability left), q (top right), f g (bottom left), and v g (bottom right). We show the dependence on primary parameter with assumed equidistant values. cluster survives until late into the inspiral, it would lead to more efficient DF and a shorter orbital evolution time of the MBH pair (Dosopoulou & Antonini 2017). · The orbit of the sMBH is assumed to be co-planar with Figure 18: MBH pairing probability as a function of the MBH pair mass (left) and the gas fraction of the host galaxy (right) in models with and without radiative feedback (RF). In the presence of radiative feedback, the suppression of MBH pairing is most severe in galaxies with MBH pairs with mass < 10 8 M ⊙ and f g ≥ 0 . 1 . The pairing probability is calculated as a fraction of models in which the two MBHs reach a minimum separation of 1 pc within a Hubble time. Figure adapted from Li et al. (2020b). \nf \ng \nas a \n· The gas disk \nspiral arms or \nteractions \nlead to a \norbital evol \nbe ejected \n2017). \nWe find that, for \nerties, negative DF \n46%. In addition, \n· The effect of \nies with \ntermines \nample, in g \nin longer \nf g v g ( v c) Figure 4. MBH pairing probability as a function of the host galaxy and MBH pair properties with and without radiative feedback: M bin (top left), q (top right), f g (bottom left), and v g (bottom right). We show the dependence on f g as a histogram, since this is a derived, rather than a primary parameter with assumed equidistant values. cluster survives until late into the inspiral, it would lead to more efficient DF and a shorter orbital evolution time of the MBH pair (Dosopoulou & Antonini 2017). · The orbit of the sMBH is assumed to be co-planar with the galactic gas and stellar disks. It is in nevertheless possible that some fraction of sMBHs evolve on orbits that are inclined relative to the galactic disk. sMBHs on inclined orbits on the one hand experience weaker DF from the gas and stellar disks, an effect that leads · The gas disk in our model is smooth and devoid of spiral arms or gas clumps. When they are present, interactions between the sMBH and these structures can lead to a random walk of the sMBH, resulting in longer orbital evolution time. In some cases, when the inhomogeneities are large enough, the sMBH may even be ejected out of the galactic disk (Tamburello et al. 2017). 5. CONCLUSIONS We find that, for a wide range of galaxy and MBH proptions triggered by pericentric passages of the sMBH crossing the disk of the remnant galaxy can trigger the formation of a dense stellar cusp around the sMBH. This effect leads to the increase of stellar mass bound to the sMBH and can shorten the orbital evolution time (Van Wassenhove et al. 2014). Using a different approach, Li et al. (2020a) studied this aspect adopting a semi-analytical model to describe the orbital evolution of MBHs from separations of ∼ 1 kpc to ∼ 1 pc, under the influence of stellar and gaseous dynamical friction. Their study of the parameter space suggests that the dominant drivers of the MBH orbital evolution are the stellar bulge and galactic gas disc. They find that the chance of MBH pairing within a Hubble time is nearly 100 per cent in host galaxies with a gas fraction of < 0 . 2 , as shown in the right-hand panel of Fig. 18. They also find that the orbital evolution sensitively depends on the relative speed between the gas disc and the MBHs. Semi-analytical models, however, are quite limited in their predictive power when it comes to the effect of gas as they cannot account for the multi phase nature of the interstellar medium and the concurrent star formation in the galactic nucleus, which are bound to have an impact on both the local drag and the global torques, motivating the investigation of this problem with various approaches in order to assess the impact on LISA's MBH mergers. \nto longer inspiral times. On the other hand, perturba-", 'tions triggered by pericentric passages of the sMBH · Effects of feedback': "erties, negative DF reduces the MBH pairing probability by \n46%. In addition, we find that: \ncrossing the disk of the remnant galaxy can trigger the formation of a dense stellar cusp around the sMBH. This effect leads to the increase of stellar mass bound to the sMBH and can shorten the orbital evolution time (Van Wassenhove et al. 2014). · The effect of negative DF is most pronounced in galaxies with significant gas fractions, where gas DF determines orbital evolution of the MBH pairs. For example, in galaxies with f g ≥ 0 . 1 negative DF results in longer MBH inspiral times and reduces the pairing The distribution of gas in the host galaxy and its contribution to the total dynamical friction force on MBHs is likely to be strongly impacted by radiative feedback (e.g. Sijacki et al., 2011; Souza Lima et al., 2017). More specifically, it was recently shown for MBHs evolving in gasrich backgrounds that ionizing radiation that emerges from the innermost parts of the MBHs' accretion flows can strongly affect their gaseous dynamical friction wake and render gas friction inefficient for a range of physical scenarios. Combined with the effect of radiation pressure, the radiative feedback creates an ionized region larger than the characteristic size of the dynamical friction wake and a dense shell of gas in front of the MBH, as a consequence of the snowplow effect. In this regime, the dominant contribution to the MBH acceleration comes from the dense shell and such MBHs experience a positive net force, meaning that they speed up, contrary to the expectations for gaseous dynamical friction in absence of radiative feedback (Park and Bogdanović, 2017; Gruzinov et al., 2020; Toyouchi et al., 2020). This effect was dubbed 'negative dynamical friction'. \nIf prevalent in real merging galaxies, negative gaseous dynamical friction can lengthen the \n0.3 \nq \nMBH pairing probability \nM \nbin ( \nM \n) \nno RF \nwith RF \n10 \n8 \n1 \n2 \n0.4 \n0.3 \n0.2 \n0.1 \n0 \n1.0 \n0.9 \n0.8 \n0.7 \n0.6 \n0.5 \n0.4 \n0.3 \n0.2 \n0.1 \n0 \n1 \n9 \n1 \n8 \n-0.9 \n-0.6 \n1 \n7 \nno RF \nwith RF \n1 \n6 \n1 \n5 \nno RF \nwith RF \ninspiral time of MBHs and even offset the action of stellar dynamical friction. Its full implications for the formation and coalescence rate of MBHBs in galactic and cosmological settings for MBHs in the LISA mass range are however yet to be understood. Some early insights into this question are provided by Li et al. (2020b), who used a semi-analytical model to study the impact of negative dynamical friction on pairs of MBHs in merger remnant galaxies evolving under the combined influence of stellar and gaseous dynamical friction. They found that, for a wide range of galaxy mergers and MBH properties, negative dynamical friction reduces the MBH pairing probability to ∼ 50 per cent of that found in absence of radiative feedback (Fig. 18, left). This effect is particularly prevalent in systems with a gas fraction above 0 . 1 , especially if the disc rotational velocity is comparable to the circular velocity (Fig. 18, right). Importantly, the pairing probability in the presence of radiative feedback decreases five-fold (to ≲ 0 . 1 ) for MBHBs with mass ≲ 10 6 M ⊙ (Li et al., 2020b), implying that the pairing of the very population of MBHBs targeted by LISA may be greatly affected by it. \nDayal et al. (2019) on the other hand point out that many MBHs in the LISA mass range ( ≲ 10 6 M ⊙ ) MBHs would reside in low mass haloes, in which SN feedback and radiation background due to reionization will expel and photo-evaporate most of the gas, thus curbing the growth of the MBHs and suppressing the effect of gas on their orbital evolution (see Section 2.3.2.2 for a related discussion). It is therefore crucial to understand how feedback affects gas dynamical friction in realistic merger remnants. If pairing and merger rates of MBHBs in gas-rich environments are reduced, this has important implications for the likelihood of detection of multimessenger events with LISA and the contemporary EM observatories. For this reason, a much wider range of scenarios needs to be explored to investigate the complex role of radiative feedback and gaseous dynamical friction for MBHs in the LISA mass/redshift range, exploring the impact of a different feedback geometry, energetics, mass loading, momentum injection, etc.", '· Effects of bars': "A large fraction of disc galaxies at low redshift show clear deviations from axisymmetry in their stellar distribution. At least at low redshift, about half of massive ( M ∗ ≳ 10 10 M ⊙ ) discs (e.g. Consolandi, 2016, and references therein) host a prominent overdensity approximately symmetric with respect to the centre with constant phase, e.g. a bar, that can significantly affect the dynamical evolution of the different components of the host galaxies (Athanassoula, 2002; Sellwood, 2014). Quantifying the fraction of barred galaxies at high redshift is still challenging (see, e.g. Sheth et al., 2008; Melvin et al., 2014; Simmons et al., 2014), but (a) it has been observationally proposed that bars could be frequently hosted in massive galaxies at all redshifts, with an increasing mass threshold for entering in the bar unstable regime as a function of redshift (Gavazzi et al., 2015), and (b) bar formation has been found as early as at z ∼ 7 in cosmological zoom-in simulations (Fiacconi et al., 2017). \nBeing such strong perturbations to the host potential, bars could significantly affect the pairing of MBHs during galaxy mergers. The occurrence of such effect could be increased when the actual merger is responsible for the triggering of a bar (Byrd et al., 1986; Mayer and Wadsley, 2004; Romano-Díaz et al., 2008; Martinez-Valpuesta et al., 2016; Zana et al., 2018a,b; Peschken and Łokas, 2019), even when this is short-lived and not sustained by the galactic potential, e.g. if the galaxy stellar disc is below the threshold for bar instability. \nThe effect of a forming and growing bar on the pairing of MBHs within LISA's reach has been recently explored by Bortolas et al. (2020). They populated a main-sequence z ∼ 7 galaxy 5 with secondary MBHs at different radii and at different angles with respect to the forming bar, and found a stochastic behaviour in the pairing time-scales, with some of the secondary MBHs being pushed towards the centre of the main galaxy, and others being ejected by a slingshot with the bar. Noticeably, it was found that the orbital decay of the secondary MBHs was dominated by the global torque provided by the bar rather than by the local effect of dynamical \nfriction. This points to the need of including the effect of global torques in future recipes for sinking time-scales of MBH pairs at ∼ kpc distances. A first semi-analytical attempt to explore the broad parameter space of MBH pairs/bar interaction is currently ongoing (Bortolas et al., 2022), in which a time-dependent bar potential has been added to the integrator of orbits in disc galaxy potentials presented in Bonetti et al. (2020a, 2021). A much more thorough analysis, considering (a) different galactic properties, different bar potentials and bar precession velocities, (b) the dependence of the fraction of barred discs as a function of redshift and in recent mergers, and (c) the host galaxy evolution during the MBH pairing is needed to better evaluate the effect of bars on the population of MBH binaries in the LISA band.", '· Effects of clumps': "If an MBH happens to get close to a massive interstellar cloud or dense star cluster, its orbit can be severely affected, and this effect is particularly strong in a clumpy interstellar medium. The typical masses of the perturbers depend on the background gas density which determines the conditions of fragmentation in the framework of the Toomre instability. These masses are ∼ 10 5 -10 7 M ⊙ for giant molecular clouds in present-day galaxies; and 10 7 -10 8 M ⊙ for giant star-forming clumps in galactic discs at higher redshift, which have a much larger gas fraction (Tamburello et al., 2015, 2017a,b). As clumps have to be massive enough to have a dynamical impact on the MBH, this suggests that the effect of a clumpy medium is irrelevant for MBHs with masses > 10 8 M ⊙ (Fiacconi et al., 2013), but is likely relevant for the MBHs that are targeted by LISA. This is especially important considering that a significant fraction of galaxies at z ≈ 1-3 (i.e. an epoch in which galaxy mergers are supposedly frequent, Fakhouri et al. 2010) appears to be clumpy (Ceverino et al., 2010; Shibuya et al., 2016). Several numerical simulations show that the clumpiness of the gaseous medium renders the orbital decay highly stochastic (De Rosa et al., 2019b): in some situations, the MBH separation does not shrink (Roškar et al., 2015), in others the decay is promoted (Fiacconi et al., 2013; del Valle et al., 2015). \nWhen the decay stalls, it is often because the lighter secondary MBH is scattered away from the disc plane (galactic or circumnuclear), ending up in a region of much lower stellar and gas density, where dynamical friction becomes inefficient. This effect is even more emphasized when stellar and AGN feedback are included, as they can open cavities of low density gas (Souza Lima et al., 2017). Note that this is not a definitive effect, as a scattered MBH can eventually be dragged back to the disc: the net effect is to delay the formation of the binary which takes 10-100 times longer (Roškar et al., 2015), but this will contribute in shaping the redshift distribution of MBH coalescences (Volonteri et al., 2020). \nIn summary, while stalling of the MBH pair is an extreme outcome that cannot be verified due to the limited time-scales probed by current simulations, it is clear that the range of orbital decay time-scales of MBH pairs in a clumpy medium, from kpc scales to separations of order pc and below, can be widened by up to two orders of magnitude. In addition, the induced delay likely depends on the number and mass distribution of clumps within their hosts. Since LISA can detect MBHs up to high redshift, when clumpy galaxies were more common, future, better resolved observations of clumpy galaxies at z > 1 would be beneficial for the community. The latter, aided by more accurate simulations of the same systems, will help in better constraining the effect of clump-driven perturbations on the orbital decay of MBHs, especially their impact on the rates and properties of the MBHBs that LISA will detect. \n2.2.1.4 Is there a final kpc problem? In the previous sections we discussed several mechanisms that can cause complete stalling, or at least a significant delay of the orbital decay of a MBH pair. These mechanisms can essentially stifle the formation of an MBH binary in the first place, and the orbital evolution seems to be highly sensitive to the physical parameters involved. \nTo exemplify this, we show in Fig. 19 the outcome of two cosmological simulations (Pfister et al., 2019b) which only differ by the mass of the MBHs. In the case of a light MBH ( 10 4 M ⊙ \n<!-- image --> \nFigure 19: Distance between an MBH and a galaxy as a function of time for two cosmological simulations which only differ by the mass of the MBHs ( 10 4 M ⊙ in blue and 10 5 M ⊙ in pink). Left: We show the results for the central MBH of the galaxy. In both cases MBHs are smoothly offcentred due to inhomogeneities of the potential, but the more massive one remains in the centre, since dynamical friction is efficient, and the lighter one is instead displaced to kpc distances. Right: We show the results for the central MBH embedded in a satellite galaxy sinking towards the primary galaxy. The initial phase is similar in both cases as the whole satellite suffers dynamical friction, but as soon as material surrounding MBHs has been stripped, the light MBH stalls while the more massive one keeps sinking until it merges with the central MBH of the primary galaxy (subsequent evolution in dashed line). Adapted from Pfister et al. (2019b). \n<!-- image --> \nin blue), not only the sinking MBH stalls at ∼ kpc distances, similarly to Tamfal et al. (2018), but even the central one is smoothly off-centred due to inhomogeneties and never sinks back, as the dynamical friction time-scale is very long. While in this particular case, the massive MBH ( 10 5 M ⊙ in pink) behaves smoothly in agreement with the classic picture of dynamical friction, we recall that Bortolas et al. (2020) have shown that massive MBHs can wander for a long time because of bars. \nPerturbations, and the subsequent stalling, appear to be more relevant and likely to occur at higher redshift ( z > 1 ), as host galaxies have clumpier, more turbulent, and more inhomogeneous gas and stellar density profiles (Pfister et al., 2019b). At lower redshift, stalling is more likely to occur in low-mass/dwarf galaxies that have low background density (Tremmel et al., 2015; Bellovary et al., 2019) or cored DM profiles (Tamfal et al., 2018), or in high-mass galaxies with core-Sérsic profiles (Graham et al., 2003). Some isolated simulations of 1:4 massive spirals mergers (Callegari et al., 2009, 2011) also show a delayed inspiral with respect to the estimate of Eq. 8. However, the exact physics of the inspiral crucially depends on the details of gas accretion and star formation about the MBHs, as e.g. the formation of a dense stellar nucleus about the secondary may accelerate its orbital decay (Van Wassenhove et al., 2014; Ogiya et al., 2020). \nOn the observational side, recent radio observations of AGN in local dwarf galaxies (Reines et al., 2020) have highlighted that at least some of these objects are not located in the centre of their host, which is often not easy to define, due to irregular galactic morphologies, in line with the results of the simulations presented above. Known offsets of nuclear star clusters offer further insight (Binggeli et al., 2000). If the reason for the observed displacement is a failed inspiral (instead of e.g. the effect of a GW recoil following an MBH merger, or the interaction with a third MBH), it is easy to imagine that the formation of a bound MBH binary may become very unlikely. All the above suggests that the large-scale decay of MBHs is likely to be a stochastic process, highly dependent on the environmental conditions of the host galaxy nucleus, and on the orbital configuration of the MBH pair. \nModelling such stochasticity in a way simple enough that can be incorporated in population \nsynthesis models for LISA MBHs, but at the same time accurate enough to account for the relevant physical processes shaping the inspiral, is a key challenge ahead of us (see Barausse et al., 2020b, for an example investigating the effect on LISA's coalescences). Furthermore, it is practically hard to set the boundary between dynamical friction (thought as the response of the host to the perturber's passage) and different torquing mechanisms related to the galaxy mass distribution. For this, an effort towards a detailed and realistic characterization of MBHs, along with the effect of their feedback, in a variety of systems at all redshifts is vital to properly model the MBH merger population that LISA is going to probe. In order to interpret LISA's data it is crucial to develop well-motivated models that can be compared with the detected events in order to extract astrophysical information.", '2.2.2 Orbital decay after binary formation at pc scales': "As dynamical friction drives the orbital decay of the MBHs, they eventually find themselves inside their mutual sphere of influence 6 , resulting in the formation of an MBHB (Begelman et al., 1980; Milosavljević and Merritt, 2001). The subsequent evolution of the newborn binary can be driven by several processes. In general, the efficiency of these processes is critically connected to the characteristics of the environment surrounding the MBHB, and every MBHB will follow its own different evolutionary path. As for the larger-scale dynamics at kiloparsec scales, we can broadly identify two classes of physical processes that shape the further shrinking of the binary separation in galactic nuclei: those that operate in gas-poor stellar environments and those that instead work when a consistent reservoir of gas is present. The boundary between the two classes is definitely not strict, and although most studies available today focus on only one of the two environments at a time, both stellar and gaseous hardening can operate at the same time (see e.g. Kelley et al. 2017a; Bortolas et al. 2021). In the following we outline the key physical aspects featured by each shrinking mechanism, since they can all operate for LISA's MBHs, which are expected to dwell in environments rich in both gas and stars. \n2.2.2.1 Hardening in stellar environments As soon as two MBHs form a bound binary system, dynamical friction gradually ceases to be effective since at such small scales, of order a parsec, the surrounding background mass is too low to generate a significant back-reaction to the perturbation induced by the MBHB itself. In a galactic nucleus whose density is dominated by stars, then, the prevalent mechanism that can continue to shrink the orbit of the binary is threebody encounters of individual stars with the MBHB (Mikkola and Valtonen, 1992; Quinlan, 1996; Sesana et al., 2006). After a first rapid decay of the MBHB orbit in which dynamical friction and three-body encounters act in tandem, the binary shrinking starts to proceed at a slower but almost constant rate. The transition occurs around a separation commonly known as the hard binary separation, a h , corresponding to a semi-major axis (Merritt and Milosavljević, 2005) \na h ≤ Gµ 4 σ 2 , (9) \nwith µ denoting the reduced mass of the binary and σ the local velocity dispersion of the surrounding stellar distribution. Physically, this scale approximately denotes the point at which the binary orbital velocity exceeds the characteristic speed of the stellar background, therefore representing a sort of decoupling length below which the dynamics is strongly dominated by the self-gravity of the two black holes. During the process stars are generally ejected out of galaxy centre with high velocities as a result of the interaction with the binary. Therefore, ejections of stars by the MBHB result in a decrease of stellar density in the vicinity of the MBHB, with the damage extending typically up to a few influence radii (Ebisuzaki et al., 1991; Volonteri et al., 2003b; Khan et al., 2012) and effectively translating into less frequent stellar encounters. \nFigure 20: Evolution of an MBH pair in direct N -body simulations of a galaxy merger (obtained from cosmological simulations) at redshift z ∼ 3 . Left: MBH separation as a function of time; the small plot shows the time evolution of the Keplerian eccentricity past the binary formation. Right: Merger remnant axis ratios as a function of the radius for different simulation times. Figure adapted from Khan et al. (2016). \n<!-- image --> \nThis mechanism possibly justifies the almost ubiquitous presence of stellar cores at the centre of the most massive galaxies (i.e. the ones that likely experienced the largest number of mergers; Bonfini et al. 2018).", '· The final parsec problem': 'As the MBHB enters in the hard binary regime, it is expected to shrink at a rate determined by \nd dt ( 1 a ) = Gρ σ H, (10) \nwhere ρ is the density of the stellar background, σ is the velocity dispersion, a is the binary Keplerian semi-major axis and H ≈ 15 -20 is a numerical coefficient weakly dependent on the properties of the binary (mass, mass ratio, and eccentricity; Mikkola and Valtonen 1992; Quinlan 1996; Sesana et al. 2006, but see Ogiya et al. 2020). The equation above shows that the shrinking rate would be constant for fixed ρ and σ ; however, the binary surroundings get perturbed by its scouring action, resulting typically in a mildly declining hardening rate (Vasiliev et al., 2015; Bortolas et al., 2018a). Eq. 10 applies so long as the MBHB loss cone (the region of phase space containing stars with angular momentum low enough to interact with the binary) remains populated with stars. However, the loss cone is generally depleted within a typical stellar orbital period at the beginning of the hardening phase, and further MBHB shrinking crucially depends on the existence of processes able to repopulate it. In principle, the loss cone can be replenished by means of two-body relaxation. Unfortunately, this process acts on a time-scale much longer than a Hubble time if one considers the average properties of galactic nuclei, assuming a spherically symmetric potential (Binney and Tremaine, 1987, although it may be short enough for dwarf galaxies hosting low mass MBHs in the LISA band). For this, the possibility of an MBHB stalling at pc scales has been put forward from both numerical (Makino and Funato, 2004; Berczik et al., 2005) and theoretical grounds (Begelman et al., 1980), and has been referred to as the final parsec problem.', '· The final parsec problem is not a problem': 'Throughout the last decades, evidence has been building up that the final parsec problem would only occur in perfectly spherical, idealized galaxies; in fact, in these systems stars are bound to conserve all components of their specific angular momentum. If the stellar bulge is triaxial - as in real systems - stellar orbits can be torqued by the asymmetric mass distribution and their angular momentum does not have to be conserved in time, meaning that the loss cone can be easily repopulated in a collisionless fashion (Yu, 2002; Merritt and Poon, 2004; Merritt and Vasiliev, 2011). Berczik et al. (2006) first adopted numerical simulations to point out how rapid the MBHB coalescence can be in triaxial nuclei, which are themselves a natural outcome of the merger of two stellar bulges (Khan et al., 2011; Preto et al., 2011, see the right-hand panel in Fig. 20). Those findings were confirmed and extended to systems with different MBHB mass ratios, galaxy density profiles, and orbits, and were generalized to galaxies with realistic two body relaxation rates (Khan et al., 2012; Vasiliev et al., 2015; Sesana and Khan, 2015; Khan et al., 2016; Gualandris et al., 2017; Bortolas et al., 2018a). \nThe general consensus is that MBHBs in realistic merger remnants can reach the GW-driven coalescence through stellar hardening alone. The time-scale on which this happens, though, depends heavily on the details of the stellar density profile and the eccentricity growth of the binary, both of which are hard to pin down. For instance, (galaxy-morphology)-dependent scaling relations that correlate the MBH mass with many host galaxy quantities (e.g. Gültekin et al., 2009; McConnell et al., 2011; Kormendy and Ho, 2013; Reines and Volonteri, 2015; Graham and Scott, 2015; Davis et al., 2018) can be used to probe the binary lifetimes. Biava et al. (2019) found they can range between 10 -2 Gyr to more than 10 Gyr for MBHs of 10 5 -10 7 M ⊙ . When hosts are scaled to bulges of local galaxies, the merger times of MBHBs derived from simulations are typically less than 500 Myr (Khan et al., 2018b), but can reach ∼ 1 Gyr depending on central density, which is varied in a realistic range (Khan et al., 2018c, 2016). Time-scales, however, have also been shown to depend strongly on redshift because the scaling of mass density and velocity dispersion, which both affect hardening, is a rather steep function of redshift (Mayer, 2017). This is the reason behind the extremely short MBH merging time-scales found in cosmological simulations of massive galaxies at redshift ∼ 3.3 (Khan et al. 2016, see Fig. 20), and is naturally explained if one considers the scaling of structural properties of galaxies with respect to their host CDM halos as a function of redshift (Mayer, 2017). A detailed knowledge of the properties of stellar dominated galaxies hosting LISA MBHs at different redshifts appears thus to be important in order to derive a realistic distribution of hardening times for LISA MBHB evolving inside such hosts. \nAn acceleration of the MBHB shrinking can be induced by a non-zero orbital eccentricity during the hardening stage, as this would shorten the time-scale needed by the binary to enter the GW-dominated evolutionary stage (Peters, 1964a, but see Section 2.2.3). For example, Sesana and Khan (2015) find that the typical time spanning from the onset of the hardening to the GW-induced coalescence is ≈ 30 times shorter for binaries with e = 0 . 99 compared to circular ones. Eccentricity evolution in the hardening phase depends on a fine balance between energy and angular momentum exchange, and it is sensitive to a number of factors. Three-body scattering experiments in a non-rotating stellar system find that eccentricity tends to grow as the binary shrinks (Quinlan, 1996), and the growth is more prominent for binaries with moderately low mass ratio ( q ≳ 0 . 01 ) residing in steeper stellar density cusps (Sesana et al., 2008a). For binaries with even lower mass ratio, the evolutionary trend is less clear, and below q ∼ 0 . 001 the scattering process seems to even circularise binaries (Rasskazov et al., 2019; Bonetti et al., 2020b). Khan et al. (2012) also noticed that mergers of cuspy galaxies result in lower binary eccentricities at the time of binary formation, whereas galaxy mergers with shallower central density end up with higher eccentricity values. MBH mergers in the mass range 10 4 -10 7 M ⊙ , around the peak of the LISA sensitivity window, are generally observed to be hosted by galactic nuclei with steep density profiles, which might favour relatively low eccentricities. However, the situation may greatly vary once the hosts rotation is taken into consideration, as discussed be- \nexpected eccentricity of binaries close to merger is not only important to estimate the merger time-scale, but it has repercussions on what type of waveforms should be developed for LISA data analysis.', '· The host rotation': 'Recent numerical studies investigated the impact of rotation in galactic nuclei on the evolution of MBHBs, showing that it can profoundly affect the orbital parameters of the bound binary (Mirza et al., 2017; Rasskazov and Merritt, 2017). MBHs sink significantly faster in orbits corotating with galaxy rotation, because of the longer time for the encounter between the MBHs and the incoming stars, results in a more efficient extraction of energy from the orbit (HolleyBockelmann and Khan, 2015). Moreover, MBHs in co-rotating orbits circularise efficiently prior to binary formation, whereas those on counter-rotating ones tend to maintain their eccentricity, which starts to grow as the two MBHs approach the binary formation phase (Sesana et al., 2011b; Khan et al., 2020). Before a hard binary forms, MBHBs in counter-rotating orbits attain very high values of the orbital eccentricity ( e ≃ 1 ) and also flip their plane to align themselves with the orientation of the galactic angular momentum (Gualandris et al., 2012; Rasskazov and Merritt, 2017). This means that, in principle, MBHBs evolving in rotating environments may typically end up being close to co-rotating with their background and having a very low eccentricity; however, more studies on the modelling of MBHBs in rotating systems is needed in this direction in order to get a more complete picture of the phenomenon before LISA flies.', '· Substructure in galactic nuclei': "The evolution of an MBHB in its hardening stage can also be affected by the presence of perturbers near the galaxy nucleus. In the case of low-mass perturbers, such as the nearby stars, this results in Brownian wandering. The MBHB instantaneous centre-of-mass velocity gets continuously perturbed by gravitational three-body encounters with the nearby stars (Merritt, 2001), and is balanced by dynamical friction, which acts as a restoring force. As a result, the MBHB centre of mass wanders about the centre of the galaxy. As the MBHB gets displaced from the centre, the MBHB loss-cone re-population can be enhanced, resulting in a possible boost of the MBHB hardening rate in spherically symmetric systems (Quinlan and Hernquist, 1997; Chatterjee et al., 2003; Milosavljević and Merritt, 2003). For triaxial systems, where an almost full MBHB loss cone is usually expected (Gualandris et al., 2017), the MBHB shrinking rate can be impacted by the MBHB's wandering only if M BH /m ⋆ ≲ 10 3 (being m ⋆ the typical mass of stars, see Bortolas et al., 2016). This suggests that the MBHB's wandering would not significantly affect the hardening rate of LISA MBHBs. \nIn the case of perturbers with mass much larger than the stellar one, the effects can be more significant (Perets et al. 2007; Perets and Alexander 2008). Massive perturbers may be in the form of star clusters, giant molecular clouds or even a further inspiralling MBH, and may have masses up and above that of MBHs in the LISA band. Such objects can reach the galaxy centre and affect the binary inspiral in different ways. Due to their large mass, they scatter new stars into the loss cone, thus indirectly enhancing the MBHB shrinking rate, somehow acting as boosters for two body relaxation (Spitzer and Schwarzschild, 1951). In addition, if they come close enough to the binary, they may displace it from the galaxy centre, thus again affecting the flux of stars in the loss cone; furthermore, if the massive perturber is a stellar cluster, once the object reaches the binary, it delivers new stars onto it, thus directly promoting its shrinking (Bortolas et al., 2018b; Arca Sedda et al., 2019b). Thus, in principle, massive perturbers may have a significant effect on the orbital evolution of an MBHB. In order to properly model their impact on a population of LISA MBHBs, more studies are needed to pinpoint the rate at which different massive perturbers may interact with hardening binaries in different host environments. This regime bears similarities with the clumpy medium regime in gas-rich galaxies and in circumnuclear discs as far as the dynamical interaction with the MBHB is concerned. \nFigure 21: Illustration of the geometry of a circumbinary disc, with minidiscs surrounding each of the MBHs in the binary, a gap opened by the MBHs gravitational torques, and gas streams connecting the circumbinary disc to the minidiscs. Adapted from Bowen et al. (2018, the central larger black circle at the coordinate origin marks a central excision and is not physical). Image concept: Julian Krolik. Figure realization: Marta Volonteri. \n<!-- image --> \n2.2.2.2 Hardening in gaseous environments We now turn to the evolution of MBHBs embedded in a gas-dominated galactic nucleus. This is of great relevance for LISA, since it will detect low mass MBHs, which are expected to reside in high-redshift gas-rich galaxies. In Section 2.2.1 we have already discussed extensively the dynamics of MBH pairs, until MBHB formation, in gas-rich galaxies, from kpc scales in the galactic disc, to pc scale separations in the circumnuclear disc, showing how various effects can both hamper or promote the sinking of the MBHs. We now continue our investigation in gas-rich nuclei for smaller separations, namely following MBHB formation. \nOne first effect of gas which is relevant also at such small separations is related to accretion. In the limiting case that the gas accreting on to the MBHs has zero angular momentum, the gas inflow will be purely radial and accretion on to the binary will be Bondi-Hoyle-Lyttletonlike (hereafter Bondi; Hoyle and Lyttleton, 1939; Bondi and Hoyle, 1944; Bondi, 1952). Bondi accretion onto a binary has been studied in, e.g. Farris et al. (2010); Antoni et al. (2019); Comerford et al. (2019) and, with the inclusion of magnetic fields, in Giacomazzo et al. (2012); Kelly et al. (2017). The main conclusion is that the sinking time-scale of the binary caused by their distorted wakes remains comparable to the usual gaseous Bondi drag time-scale for a single compact object, only a factor of few smaller, at least in the parameter ranges studied in the above papers. \nIn reality, gas on large scales is likely to possess a specific angular momentum much larger than the one corresponding to the innermost stable circular orbit (ISCO) of the MBHs. Therefore, considerable loss of angular momentum is required to drive gas from kpc to sub-pc scales, and it is likely that some residual angular momentum remains on small scales, resulting in the formation of a disc surrounding the MBHB: the so-called circumbinary disc (for a single MBH the analogous structure is an accretion disc). An illustration of the geometry of a circumbinary disc is provided in Fig. 21. In this section we focus on the effect of the disc onto the MBH binary dynamics, while we refer to Section 2.2.2.3 for the implications on accretion.", '· The circumbinary disc': "Due to the computational burden, large scale simulations able to resolve ∼ 100 pc scales have \nnot yet managed to fully and self-consistently determine the properties of such circumbinary discs, with few exceptions (Souza Lima et al., 2020). To circumvent the limits of large scale simulations, Goicovic et al. (2016, 2017); Maureira-Fredes et al. (2018); Goicovic et al. (2018) detailed the properties of circumbinary discs through an extensive, though highly idealised, set of simulations, where the disc was built through a bombardment of gas clouds towards a central MBHB. These studies demonstrated that the detailed properties of circumbinary discs depend on the dynamical properties of the infalling material. \nWhen the MBHB reaches a critical small separation, its gravitational torque on the surrounding disc material becomes stronger than the angular momentum losses per unit of time due to the disc dissipative processes; at this point, depending on the mass ratio of the binary, either an annular gap centred on the secondary radius (MacFadyen and Milosavljević, 2008), or a large cavity encompassing both the MBHs can be opened (D'Orazio et al., 2016). It was initially suggested that the creation of such cavity would inhibit gas accretion onto the pair; more recent and resolved simulations seem instead to suggest that accretion may remain sustained through the inner edge of the disc (e.g., Farris et al., 2015a; Souza Lima et al., 2020). \nIf and when the binary reaches sufficiently small separations, the mass of the circumbinary disc enclosed within the MBHB orbit becomes much smaller than that of the binary itself, making the disc gravitationally stable against fragmentation (Goodman, 2003). The simplest expectation in this regime is that the gas disc will cause the binary to harden on a time-scale comparable to the viscous time-scale (in analogy with Type II planetary migration; Ward, 1997) down to the decoupling radius where GWs start dominating the MBHB dynamics. For typical, thin Shakura and Sunyaev discs (with ratio between the vertical length scale H and the radial extent R around H/R ≈ 0 . 05 ) and close to equal mass MBHBs, this occurs at ∼ 100 gravitational radii (Gold et al., 2014b). \nAt such close separations, for close to equal mass MBHBs ( q ∼ 1 ) the time-scale needed by the disc to refill the cavity would get longer than the GW-driven coalescence. If, on the other hand, q ≪ 1 , the ratio between the mass of the secondary MBH and the mass of the disc enclosed in the MBHB orbit, q 2 , disc ≡ M 2 /M disc , is a key parameter. A q ≪ 1 binary is expected to harden on the viscous time-scale of the surrounding disc, up to the binary separation when q 2 , disc > 1 , afterwards, the migration rate falls below the viscous rate. The MBHs separation at which q 2 , disc grows above unity can occur outside the region where the disc is stable against self-gravity-driven fragmentation (see figures 3 and 4 in Haiman et al. 2009 and Figure 6 in Lodato et al. 2009). The conclusion is that, if q 2 , disc ≫ 1 at large separations ( ≳ 0.1-1 pc), the ensuing slow-down would preclude the merger (Lodato et al., 2009), or else it would have to occur in a self-gravitating, clumpy disc. At smaller separations, the viscous time is generally short, and rapid merger can be promoted by a stable disc, despite the slow-down occurring when q 2 , disc ≫ 1 . \nAs commented earlier in the section, simulations have observed that gas continues to cross the inner edge of the circumbinary disc (e.g. D'Orazio et al., 2016), but in an unstable and strongly fluctuating fashion, and the spatial symmetry of the circumbinary gas is lost, resulting in a strongly lopsided, precessing disc, preventing analytical modelling of these processes. In the simplest case of an equal-mass binary on a circular orbit, surrounded by a locally isothermal but warm disc (with a low Mach number, or a high aspect ratio H/R = 0 . 1 ), several recent simulations have converged on the same conclusion: the disc causes the binary to outspiral (Tang et al., 2017; Moody et al., 2019; Muñoz et al., 2019, 2020). The outspiral rate is quite rapid, for accretion rates comparable to those of bright, near-Eddington quasars. This of course could represent an important obstacle to binary mergers, but more recent work suggests that this conclusion is peculiar, and holds only for the above, idealized, specific configuration. In particular, the sign of the torques and migration changes from positive to negative when the mass ratio is below q ≲ 0 . 05 (Duffell et al., 2019). This means that binaries below this mass ratio may migrate inwards - at least until the secondary accretes a sufficient mass to increase q above 0.05 (after which, in the absence of any other effects, the torque would change sign, causing the binary to \nmigrate outwards). More importantly, the disc torque has been found to strongly depend on the disc temperature (or equivalently Mach number or aspect ratio). Tiede et al. (2020) have emphasized that real AGN discs in the inner regions are expected to be thinner/colder. They measured the torques in simulations of such cooler discs, and have found that outspiral changes to inspiral, at a comparable rate, when H/R ≲ 0 . 04 . They attributed this to the importance of direct gravitational torques of the gas accumulating near the binary with an asymmetric distribution (as opposed to accretion torques). The dependence on the disc temperature was later confirmed by Heath and Nixon (2020), who found that binaries outspiral only for H/R ≳ 0 . 2 . However, Franchini et al. (2022) showed that, using high-resolution simulations, the result does not depend only on the disc temperature, but also on viscosity and argue that there is no threshold for expansion in terms of disc aspect ratios. However, these recent papers all agree on the same conclusion: binaries embedded in thin ( H/R ≲ 0 . 05 ) locally isothermal discs do inspiral as a result of the interaction with the gas. \nA good understanding of this gas-disc driven phase is important to better understand the properties of MBHBs when they enter in the LISA band. Assuming for simplicity that one can neglect accretion flows towards the binary, the viscous time-scale of the disc is the relevant evolution time-scale. This means that the MBHB will simply shrink as the disc material itself shrinks due to internal viscous stresses. Then one can define a decoupling radius by equating the viscous time-scale in the disc with the GW inspiral time-scale of the binary. \nFor an equal-mass binary at the decoupling radius, the GW observed frequency is given by \nf GW ∼ 10 -4 1 1 + z Hz ( H/R 0 . 05 ) 6 / 5 ( α 0 . 1 ) 3 / 5 ( M 10 6 M ⊙ ) -1 , (11) \nassuming the viscous time-scale t ν ∼ R 2 /ν follows the α -disc scaling. This suggests that for typical values of the Shakura and Sunyaev viscosity parameter α and sufficiently large H/R ratios, binary-disc decoupling may occur just inside the LISA band. Gas interaction becomes even more relevant in the mHz regime for binaries with smaller component masses or lower mass ratios. Also, the binary residual eccentricity when it enters the LISA band may be determined at binary-disc decoupling (Roedig et al., 2011), suggesting that residual eccentricities of up to 10 -2 in the LISA band are possible (see also Cuadra et al., 2009; Muñoz et al., 2019). \nIt is unclear, however, how realistic this way of reasoning is. Fully relativistic 3D magnetohydrodynamic (MHD) simulations (Farris et al., 2012; Gold et al., 2014b; Khan et al., 2018a) find that accretion on to the binary proceeds all the way through the binary merger, albeit at progressively slower rate, suggesting that there is never a true decoupling between disc and MBHB. These studies also showed that if the disc is cooler, then decoupling is more pronounced and the accretion on to the binary declines earlier than in hot discs. Recent 2D simulations (Farris et al., 2015a; Tang et al., 2018) are in agreement with the relativistic studies and suggest that angular momentum transport of the gas in the vicinity of the binary is driven by shocks, which enable it to flow inwards and follow the binary even well past the canonical decoupling radius. \nBefore the launch of LISA a number of improvements to these models are needed in order to develop tools (e.g., include eccentricity in waveforms and data analysis) and use them as guide for EM searches (see Section 2.5). Descriptions of the fuelling processes from large scale down to the central pc of galaxies, with a higher resolution than the one achieved in the current available studies, are needed to pin down the properties of circumbinary discs. Furthermore, current simulations of circumbinary discs are idealized in many ways (e.g. some simulations are in 2D rather than 3D, some do not include magneto-hydrodynamics, most have simplified equations of state and treatment of thermodynamics, all of them neglect radiative feedback, and disc self-gravity is rarely included except in Cuadra et al. 2009; Roedig et al. 2011, 2012; Roedig and Sesana 2014; Franchini et al. 2021. Moreover, although it is expected that discs at the decoupling radius are gravitationally stable, except for binaries too massive to be detected by LISA (Haiman et al., 2009), eccentricity evolution would be different in a self-gravitating regime, as the disc would \nbecome strongly distorted in response to its own self-gravity. Therefore, more sophisticated models of accretion in conjunction with future observations are necessary to properly predict the residual eccentricity of binaries and other aspects of their dynamics when they enter the LISA band.", '· The formation and evolution of mini-discs': "The matter that crosses the gap/cavity region (as discussed in the previous section) can form mini-discs around each MBH (see illustration in Fig. 21). Their presence may depend on the thermal state of the circumbinary disc, with colder and thinner discs producing lowermass and shorter-lived mini-discs than those in hotter and thicker circumbinary discs (Ragusa et al., 2016). While their masses may be small (Chang et al., 2010; Tazzari and Lodato, 2015; Fontecilla et al., 2017), they mediate the rate of accretion on to the MBHs (and determine their spin evolution, see the discussion in Section 2.3.2.4) and may play a role in the migration rate of the binary. Present 2D simulations find that the resulting disc torque that affects the binary evolution receives a dominant or significant contribution from the gas near the edge of the minidiscs, and, from the numerical point of view, therefore depends on the treatment of mini-discs and possibly even on the sink particles, and/or the inner boundary conditions that mimic MBHs in Newtonian simulations (Tang et al., 2017; Muñoz et al., 2019; Moody et al., 2019; Tiede et al., 2020). The importance of the mini-discs torques has also been confirmed with 3D numerical simulations by Franchini et al. (2022), therefore calling for comprehensive investigations about mini-discs modeling in Newtonian numerical simulations. \nCalculations partially involving relativistic corrections (Noble et al., 2012) or full GR (Farris et al., 2012; Gold et al., 2014b,a; Khan et al., 2018a) did not find persistent mini-discs. The more recent studies of Bowen et al. (2017); Bowen et al. (2018) initialized the simulations with mini-discs already in place, and found mini-discs which are more persistent, but also found that they undergo periods of depletion and replenishment. In Gold et al. (2014b), it was argued that the reason for the absence of persistent mini-discs in relativistic simulations at small orbital separations was due to the fact that the ISCO around the individual MBHs is larger than the corresponding Hill spheres, thereby any matter that is gravitationally bound to one MBH is immediately accreted. This hypothesis was recently confirmed in fully GR simulations in Paschalidis et al. (2021). \nDespite the progress made so far, studies of all these topics are at an infant state at the moment, and more sophisticated models are necessary to understand how the presence of minidiscs and a circumbinary disc affects the MBH spins, and the binary orbit as it evolves toward the LISA band. \n2.2.2.3 The effect of AGN feedback in the hardening phase Irrespective of the dominant hardening mechanism, AGN feedback, i.e. the energy injection from an accreting MBH, can affect the dynamics of an evolving MBHB, as it does in the phases before binary formation (see Section 2.2.1.3). For a binary migrating in a circumbinary disc, the effect of AGN feedback has been explored, with smoothed-particle hydrodynamics (SPH) simulations, only for binaries with parsec separation, i.e. in the early stages of binary evolution (del Valle and Volonteri, 2018). The effect of feedback in the late binary evolution has not been investigated explicitly yet. del Valle and Volonteri (2018) consider the two main regimes of binary evolution, one where the binary opens a gap in the disc and one where a gap does not form. As said, if viscous torques are inefficient in redistributing the angular momentum extracted from the binary, a low density cavity (gap) forms around the MBHs. In this situation, very little gas flows towards the MBHs, which have low accretion rates and AGN feedback is characterized by outflows carrying little mass and escaping through the cavity. They do not affect the binary orbital evolution which, however, is very slow exactly because of the presence of the cavities and inefficient torques. If the redistribution of angular momentum extracted from the binary is efficient, no gap forms, and the \nMBHs are embedded in a dense gas bath, leading to rapid migration. Under these conditions, however, gas accretion on the MBHs is also favoured. MBHs then produce mass-loaded winds that interact with the gas in the disc, shredding it and ejecting it in all directions. The ejection of gas leads to the formation of a hollow region ('feedback gap') around the MBHs, and the binary migration is stalled by the lack of gas with which to exchange torques. In these simulations feedback was injected isotropically, but outflows could be non-isotropic if launched by a disc, and the effect of a collimated outflow could be different and it would be worthwhile to explore this in future studies. These jets and collimated outflows could result in unique EM signatures, as discussed in Sections 2.5, 2.6. \nFor a binary evolving instead by stellar hardening, the effect of AGN feedback has not been explicitly studied. We can speculate that thermal or kinetic energy injection should have little effect on the distribution of existing stars, however, it can affect, and even suppress, the formation of new stars. If binary evolution is slower than star formation, persistent AGN feedback would prevent the formation of new stars that can repopulate the loss cone and further the shrinking of the binary. If the amount of gas present is very little, this effect is likely limited. If gas is copious, then this effect can become important, but then one has to consider that the dynamics of the binary will occur in a 'mixed environment', where both scattering with stars and gas torques contribute to the binary migration. \nIn summary, this regime is still largely unexplored, and may have important consequences for the orbital evolution and the EM counterparts of LISA's detections. In the near future, both simulations including isotropic and collimated AGN feedback in the late evolution of MBHs in circumbinary discs, and simulations of the stellar hardening phase including gas, star formation and AGN feedback, will need to be developed to address/understand the impact of feedback on MBH coalescence. \n2.2.2.4 The formation of triplets/multiplets of MBHs In the high-redshift Universe, the environment in which MBHs live is highly dynamical, as halo interactions and mergers are far more frequent (e.g. the Jackpot nebula, a system at z ∼ 2 containing several AGN in the same 400 kpc-wide Lyα nebula, see Hennawi et al., 2015). The outcome of these encounters could be either the formation of an MBHB or, at least temporarily, a wandering MBH, leading to multiple MBHs in the grown galaxy halo, each inherited from a different merger (Pfister et al., 2019b). Failures in the binary formation process affect the specific merger rate as a function of redshift, MBH mass, and mass ratio (Klein et al., 2016; Bonetti et al., 2019; Barausse et al., 2020b). \nIn these situations, the formation of MBH triplets or multiplets can arise, possibly triggering a richer and more complex range of few-body dynamics (Mikkola and Valtonen 1990; Heinämäki 2001; Blaes et al. 2002; Hoffman and Loeb 2007; Amaro-Seoane et al. 2010b; Kulkarni and Loeb 2012; Rantala et al. 2017; Ryu et al. 2018; Bonetti et al. 2018; Mannerkoski et al. 2021. Triplets of MBHs generally start their evolution as spatially hierarchical systems, i.e. systems where the hierarchy of orbital separations (or semi-major axes) defines an inner ( a in ) and an outer binary ( a out ), the latter consisting of the newly arrived MBH coming from ∼ kpc scales plus the former binary (viewed as an effective single body located at its centre of mass). Under certain circumstances, these MBH systems may undergo von Zeipel-Lidov-Kozai (ZKL) oscillations (von Zeipel 1910; Kozai 1962; Lidov 1962), in which secular exchanges of angular momentum between the two binaries periodically excite the inner binary's orbital eccentricity at the expense of the relative inclination (see also Sections ?? and 1.7.1.9 for the same process in the context of stellarmass compact objects), resulting in efficient GW emission (see Section 2.2.3). \nDespite the ZKL mechanism's efficiency in increasing the orbital eccentricity, it has been shown that relativistic precession (or other types of precession) can interfere with it. In practice, if the apsidal precession time-scale is shorter than that of the ZKL oscillations, then precession destroys the coherent accumulation of secular torques, hindering eccentricity growth (see e.g. \nFord et al. 2000; Naoz 2016; Lim and Rodriguez 2020). In the context of MBH triplets, the ZKL mechanism can be further re-enhanced by the orbital decay of the intruder MBH, due to its interaction with the host galaxy environment, which tends to shrink its separation from the inner binary (see e.g. Bonetti et al., 2018). This produces a shortening of the ZKL oscillation period and a strengthening of the perturbing force acting on the inner binary, again promoting the increase of eccentricity with subsequent GW emission and possible coalescence (Bonetti et al., 2018). \nAlthough ZKL oscillations may sometimes lead to a direct merger of the inner binary, there are many initial conditions under which no merger can occur during the secular evolution phase of MBH triplets. For example, the mutual inclination may not be high enough, the perturber may be too light, or the binary may be too wide for efficient emission of GWs. In this case, the triplet is likely to become Hill-unstable as the perturber's shrinking orbit brings it closer to the inner binary. The final fate of many MBH triplets is thus dynamical instability, wherein the secular interaction gives way to chaotic dynamics characterized by strong encounters, exchanges, and ejections. Again, this may not represent the end of the story, since in fact an ejected MBH may leave on a wide but bound trajectory, in which case it may return back and perturb the inner binary, this time through close energetic encounters, depending on the galactic potential (spherical, axisymmetric or triaxial), the specific outgoing trajectory and also on the dynamical friction efficiency. Repeated chaotic interactions between the ejected MBH and the leftover binary can increase the orbital eccentricity again, promoting coalescence in a non-negligible fraction of cases (see, e.g. Bonetti et al., 2018). Still, since this is not always effective, a considerable number of ejected MBHs may keep wandering inside galaxies. \nFinally, when the lifetime of hierarchical triplets is long enough, new galaxy mergers provide additional MBHs, forming hierarchical quadruplets and even higher-order multiplets. Considering quadruplets, a natural way in which they can form is when two merging galaxies each host MBHBs. In this particular case, the system can behave like a hierarchical triplet until the fourbody nature of the system becomes manifest, leading again to chaotic dynamics. The dynamics of MBHs multiplets can be highly stochastic and largely non-predictable, requiring therefore numerical investigations. Still, a likely signature of MBHB coalescence triggered by dynamical interaction is the very high acquired eccentricity, that will be retained (or at least, retained in residual form) well inside the GW-dominated phase (Ryu et al., 2018; Bonetti et al., 2019). \nIn the context of LISA, pre-launch more work is needed to generally include triple/multiple interactions in models of MBH evolution (as done in Bonetti et al., 2019), and to assess the consequences on the need of eccentric waveforms. Post-launch, detection of highly eccentric MBHBs would point to triple/multiple interactions as important drivers of MBHB coalescences.", '2.2.3 The GW-emission phase at mpc scale': 'As the MBHB continues to efficiently interact with the surrounding environment, which continuously drains energy and angular momentum from the MBHB system (e.g. Hills and Fullerton, 1980), it eventually enters into the gravitational radiation dominated phase. During this phase, the main parameters driving the evolution are the masses and spins of the MBHs, as well as the binary separation and eccentricity.', '· Relativistic evolution': 'Although relativistic effects can influence the binary evolution also in the previous hardening phase, at this stage we must necessarily take them into account. Relativistic effects can be introduced through spin-dependent post-Newtonian (PN) corrections in the equations of motion of the MBHs. \nSchematically, the PN-corrected acceleration can be written as \na = a N + a 1 PN + a 2 PN + a 3 PN + a 2 . 5 PN + a 3 . 5 PN + ..., (12) \nwhere, in the case of N -body numerical simulations, the Newtonian acceleration a N is usually computed including the surrounding stellar particles, whereas the PN-terms only include contributions from two MBHs (see, e.g. Will 2006; Kupi et al. 2006; Brem et al. 2013; Blanchet 2014;Mannerkoski et al. 2019). The PN-correction terms are labelled so that they are proportional to the corresponding power of the formal PN expansion parameter ϵ PN , i.e. \n| a i PN | ∝ ϵ i PN ∼ ( v c ) 2 i ∼ ( r g R ) i , (13) \nwhere v and R are the relative velocity and separation of the MBHB, while r g = GM/c 2 is the gravitational radius, with c the speed of light in vacuum, G the gravitational constant, and M the binary total mass. The PN terms of integer order are conservative, whereas the half-integer order terms are dissipative radiation reaction terms related to the emission of gravitational radiation. \nThe PN corrections, and thus the GW emission, are still negligibly small when the binary separation is of the order a ∼ a h (see Eq. 9). The PN radiative loss terms in the equations of motion start dominating the evolution when the binary separation drops down to a ∼ a GW ∼ 0 . 01 × a h (e.g. Quinlan 1996; Sesana et al. 2006; Rantala et al. 2018). This corresponds to a typical physical separation of a GW ∼ 10 -4 -10 -3 pc for equal-mass binaries with individual MBH masses of M BH ∼ 10 6 -10 7 M ⊙ , with the required separation being correspondingly smaller for lower-mass MBHs. However, it is also possible for the gas component to follow the binary essentially all the way down to merger, and thus gas can be present even in this GW-emission stage (see Section 2.2.2.2 and Farris et al., 2015a; Tang et al., 2018). In a novel attempt to quantify the effects of environmental perturbations (such as gas friction and torques) with respect to those due to PN corrections, Zwick et al. (2021) found simple analytical expressions for the regions of phase space wherein the two are comparable.', '· The GW-driven inspiral': "If we assume that the evolution of the system is purely driven by GW emission, to leading order, the secular evolution of the Keplerian orbital parameters of the isolated MBHB can be approximated following the seminal work by Peters (1964a). While the orbital period scales as t ∼ ( a/r g ) 3 / 2 (where a is the semi-major axis), the radiation reaction time-scale scales instead as t RR ∼ ( a/r g ) 4 . The inequality a ≫ r g implies that t orb ≪ t RR : the binary is thus approximately Keplerian, and the orbital parameters a and e change slowly. Using angular brackets to denote orbit averaging, the evolution of the binary's semi-major axis is described by (Peters and Mathews 1963; Peters 1964a) \n〈 da dt 〉 GW = -64 5 G 3 m 1 m 2 M c 5 a 3 (1 -e 2 ) 7 / 2 ( 1 + 73 24 e 2 + 37 96 e 4 ) = -64 5 G 3 m 1 m 2 M c 5 a 3 f ( e ) , (14) \nwhere f ( e ) = (1 + 73 e 2 / 24 + 37 e 4 / 96)(1 -e 2 ) -7 / 2 is the so-called eccentricity enhancement function, whereas m 1 , m 2 , and M denote the masses of two bodies and the binary total mass, respectively. The evolution of the eccentricity e is instead dictated by \n〈 de dt 〉 GW = -304 15 G 3 m 1 m 2 M c 5 a 4 (1 -e 2 ) 5 / 2 e ( 1 + 121 304 e 2 ) . (15) \nThe overall minus sign ensures that both the semi-major axis and the eccentricity decrease as the binary evolves, resulting in an increasingly tighter and more circular binary orbit. \nFor e ≪ 1 , Eq.s 14 and 15 imply that the eccentricity decays faster than the orbital separation. This causes a fast circularization of initially eccentric systems and, unless the initial eccentricity is extremely high, binaries in this GW-driven regime would mostly be circular. \nAn important caveat to the rather simplistic discussion presented above is that when all PN corrections up to a given order (e.g. 3.5 PN) are included in the motion of the MBHB, the standard Keplerian elements are no longer constant over an orbit, but rather they oscillate, \nespecially near the pericentre of an eccentric orbit (e.g. Will, 2006; Mannerkoski et al., 2019; Memmesheimer et al., 2004). When MBHs are spinning, their spins (both modulus and direction) also participate in shaping the dynamics of inspiraling binaries and profoundly affect the orbital motion (Cutler and Flanagan, 1994; Apostolatos et al., 1994; Kidder, 1995; Kesden et al., 2015; Gerosa et al., 2015a), as well the emitted GWs. \nDespite the inclusion of high PN order being able to describe very well the evolution down to a few gravitational radii, at binary separation of about a ∼ 6 r g and below, the strongly nonlinear gravitational field makes the PN expansion to become unreliable, and full GR simulations are necessary (see, e.g. Pretorius, 2005; Campanelli et al., 2006; Baker et al., 2006).", '· The GW inspiral time-scale': "A reasonable question to ask is the following: if the binary enters the GW-driven phase of its evolution with certain initial orbital parameters (semi-major axis and eccentricity), how much time will it take to merge? \nA proper answer requires the numerical integration of the evolution Equations 14 and 15, as discussed above. Still, a reasonable analytical approximation, valid for mildly eccentric binaries, is given by the so-called Peters' time-scale (Peters, 1964b) i.e. \nt P = 5 c 5 (1 + q ) 2 256 G 3 M 3 q a 4 0 f ( e 0 ) ≈ 0 . 32 (1 + q ) 2 qf ( e 0 ) ( a 0 AU ) 4 ( M 10 6 M ⊙ ) -3 yr . (16) \nwhere a 0 and e 0 are the initial semi-major axis and eccentricity, respectively. The interpretation of this time-scale is simple: the more massive and the more compact the binary is, the faster it will decay. Moreover, for a given semi-major axis, highly eccentric orbits decay much faster than circular ones, simply because the two MBHs at pericentre are closer to each other and the strong GW emission efficiently extracts a large amount of orbital energy. \nBecause of its simplicity, this formula has been widely used to estimate the decay time-scale of compact binaries, as done in many preceding Sections when the efficiency of GW-induced decay must be compared against other factors that affect the orbital evolution. \nWhile Peters's formula often suffices as an order-of-magnitude estimate for the decay timescale, it has two major limitations that are known but often overlooked. First of all, it is only a lower bound to the results of numerical integration, and it can underestimate the numerical time-scale by a factor of 1-8 (Peters, 1964a). In addition, Peters and Mathews' analysis assumes that the binary follows a Keplerian path, and that it only radiates according to the quadrupole formula: both of these assumptions are only true at the lowest order in the PN expansion. Corrections to the classic formula have been recently presented in Zwick et al. (2020) up to first order in PN theory. For orbits that are either eccentric or highly relativistic, one can expect errors of order ten to be accounted for by the correction factors. Recently, Zwick et al. (2021) expanded further on those results, obtaining a new spin-dependent correction. The corrected formula reads, for a given total mass and mass ratio, in the case of highly eccentric orbits: \nt PN ( a 0 , e 0 , s 1 ) = 5 c 5 (1 + q ) 2 256 G 3 M 3 q a 4 0 f ( e 0 ) ︸ ︷︷ ︸ Peters ' formula R ( e 0 ) exp ( 2 . 8 r S p 0 + s 1 0 . 3 r S p 0 + | s 1 | 3 / 2 ( 1 . 1 r S p 0 ) 5 / 2 ) ︸ ︷︷ ︸ eccentricity , spin and PN correction , (17) \nwhere p 0 = a 0 (1 -e 0 ) , r S = 2 GM/c 2 , R ( e 0 ) = 8 1 -√ 1 -e 0 , and s 1 ≡ S 1 cos θ , with S 1 being the magnitude of the spin of the most massive MBH and θ the angle between that MBH's spin vector and the orbital angular momentum vector. Adopting more accurate GW-emission time-scales in studies devoted to LISA would be beneficial to improve the investigations of MBHB dynamics and merger rates.", '· MBH coalescence and kicks/recoils': "When MBHs finally reach coalescence, the emitted GWs are responsible for dissipating not only energy and angular momentum (causing the shrinking of the orbit), but also linear momentum (Bonnor and Rotenberg, 1961; Peres, 1962; Bekenstein, 1973). Conservation of linear momentum implies that the MBH left behind following a merger has a non-zero recoil velocity (or 'kick'), which is independent of the MBH mass and depends only on the mass ratio, spins, and eccentricity of the merging binary. While energy and angular momentum are dissipated more gradually during the inspiral, linear-momentum emission is strongly peaked during the last few orbits prior to and at merger (e.g. Brügmann et al. 2008; Gerosa et al. 2018b). This implies that, although PN predictions are possible (Fitchett, 1983; Kidder, 1995; Blanchet et al., 2005), kicks can be modelled accurately only using numerical-relativity simulations (e.g. Campanelli et al. 2007b; González et al. 2007a; Tichy and Marronetti 2007; Lousto and Zlochower 2011). Such kind of (very expensive) simulations show that MBH recoils can reach velocities as large as ∼ 5000 km s -1 (the so-called 'superkicks'). A variety of tools, ranging from fitting formulae (Campanelli et al., 2007a; González et al., 2007b; Lousto and Zlochower, 2008, 2013; van Meter et al., 2010; Gerosa and Kesden, 2016) to full surrogate models (Gerosa et al., 2018b; Varma et al., 2019) calibrated on numerical-relativity results are now available to quickly estimate MBH kicks for large parameter-space explorations. \nKicks around 1000 km s -1 imply that MBH merger remnants might have velocities that exceed the escape speed of their galactic hosts (Redmount and Rees, 1989; Merritt et al., 2004; Gerosa and Sesana, 2015). The astrophysical consequences of recoils are several. Amongst them, we find that energetic kicks can critically modify the merger rate of MBHs, induce scatter in the correlations between MBHs and galaxy hosts, deplete low-mass galaxies of MBHs, create cores in the central stellar distribution, hinder the formation of > 10 9 M ⊙ MBHs powering z > 6 quasars, and generate a population of wandering MBHs and AGN. These possibilities were explored by various authors (e.g. Haiman, 2004; Boylan-Kolchin et al., 2004; Volonteri and Perna, 2005; Sesana, 2007; Gualandris and Merritt, 2008; Volonteri, 2007; Shields and Bonning, 2008; HolleyBockelmann et al., 2008; Blecha and Loeb, 2008; Blecha et al., 2011; Dunn et al., 2020; Sayeb et al., 2020). In the LISA context, the occurrence of kicks might have important consequences for the MBHB event rate, although the assessment of their impact depends very sensitively on the assumed spin directions that can be strongly affected by the interaction with the surrounding environment (Schnittman, 2007; Bogdanović et al., 2007; Kesden et al., 2010a,b; Berti et al., 2012; Miller and Krolik, 2013; Gerosa et al., 2015b, 2020; Dotti et al., 2010). Furthermore, recoiling MBHs would produce a post-merger EM signature that can aid in the identification of the merged MBH (Milosavljević and Phinney, 2005; Schnittman and Buonanno, 2007; Schnittman and Krolik, 2008; Lippai et al., 2008; Corrales et al., 2010; Rossi et al., 2010). \nPotential EM signatures of GW recoils are reviewed by Komossa (2012). If the recoiling MBHs carry the bound gas as they recoil, they would shine as off-nuclear AGN (Blecha and Loeb, 2008; Volonteri and Madau, 2008). The most characteristic signature is a set of broad emission lines, which led to the identification of several observational candidates (Komossa et al., 2008; Civano et al., 2012; Tsalmantza et al., 2011; Koss et al., 2014; Chiaberge et al., 2017; Kim et al., 2017; Kalfountzou et al., 2017) and the development of various detection strategies (Lena et al., 2014; Raffai et al., 2016; Blecha et al., 2016). Identification of such candidates is a particularly active field of research and is a difficult task (see Section 2.5); some candidates cited above have already been disproved in the recent years. Detection or confirmation of some candidates would prove that indeed MBHBs merge in the Universe, supporting LISA's science case. \nRecoiling systems are also expected to present GW signatures. These include a relative Doppler shift between inspiral and ringdown (Gerosa and Moore, 2016), different higher-order mode content (Calderón Bustillo et al., 2018), and statistical correlation with the spin properties (Varma et al., 2020). Gerosa and Moore (2016); Calderón Bustillo et al. (2018); Varma et al. (2020) all agree that the direct detectability of GW signatures from kicked MBHs is well within \nthe reach of LISA.", '2.3 MBH origin and growth across the cosmic time': 'Coordinators: Pratika Dayal, John Regan \nContributors: Pau Amaro-Seoane, Abbas Askar, Razvan Balasov, Emanuele Berti, Pedro R. Capelo, Laurentiu Caramete, Monica Colpi, Davide Gerosa, Melanie Habouzit, Daryl Haggard, Peter Johansson, Fabio Pacucci, Raffaella Schneider, Stuart L. Shapiro, Caner Unal, Rosa Valiante, Marta Volonteri \nMBHs are ubiquitous across space and time. Observations have revealed the likelihood that MBHs populate every massive galaxy in the Universe (e.g., Kormendy and Ho, 2013), with MBHs of upwards of 10 4 M ⊙ populating some, possibly large, fraction of dwarf galaxies (e.g. Baldassare et al. 2015; Chilingarian et al. 2018; Mezcua et al. 2018; Graham et al. 2019; Greene et al. 2019; Baldassare et al. 2020). At the massive end of the MBH mass function, MBHs are remarkably well-centred in the cores of galaxy bulges, and their mass is tightly correlated with many properties of the galaxy host, as the stellar mass of the bulge. Luminous quasars, powered by 10 8 -9 M ⊙ MBHs, were identified when the Universe was less than a billion years old ( z ∼ 7 . 5 , Bañados et al., 2019; Yang et al., 2020a; Wang et al., 2021a), evidence that MBH evolution started well before then. \nLISA will bring a wealth of new independent information on the population census and the ability of MBH mergers to contribute to the growth of MBHs, all the way to the realm of MBH "seeds" postulated by different formation models. No telescope can search for MBHs at redshifts as high as LISA can ( z > 10 ) , allowing us to observe an otherwise inaccessible region of the Universe. LISA will play a crucial role in, for instance, pinpointing the main formation channel of MBH seeds at high redshift, with light seeding models (numerous but low mass BH seeds) expected to drive a significantly higher merger rate at z ≳ 10 compared to heavy seeds (rare but massive BH seeds), provided that their dynamical decay is efficient (see Section 2.3.2.1). Several studies (e.g. Berti and Volonteri 2008; Sesana et al. 2011a; Barausse 2012; Amaro-Seoane et al. 2017; Dayal et al. 2019; Bonetti et al. 2019; Pacucci and Loeb 2020; Barausse et al. 2020b; Valiante et al. 2021) have shown that LISA will provide a unique view of the merger history of MBHs up to very high redshift ( z ∼ 20 ). \nIn this section, we first review our current understanding of the different seed MBH formation mechanisms, the resultant seed masses, and the outstanding questions in the field. In the second part, we examine the growth of MBHs across cosmic time under the assumption that a population of seeds, formed at z ∼ 10 -30, grow over cosmic time via the two key mechanisms of gas accretion and coalescence. Growth by accretion can typically occur by a stable influx of material, usually organised in a thin/thick accretion disc (e.g. Shakura and Sunyaev 1976; Jiang et al. 2014) or via chaotic accretion with cold gas raining from random directions (e.g. Gaspari et al. 2013, 2015; Voit et al. 2017). Coalescence between MBHs also contributes to their growth, with some small fraction of the total mass being radiated away via GWs. Through this section, we discuss how LISA will be crucial in shedding light on the origin and evolution of MBHs.', '2.3.1 MBH seeds: formation mechanisms': 'LISA will be sensitive to the detection of MBHs with masses in excess of a few thousand solar masses out to high redshifts, and therefore uniquely able to probe how MBHs form in the first galaxies. Theoretical models show that seeding mechanisms are crucial to make detailed predictions for the number density of MBH mergers and hence for predicted detections by LISA (see detailed description in Section 2.4). Correctly modelling the seeding of MBHs thus becomes of paramount importance to prepare and then interpret LISA data. \nFigure 22: Pathways towards the formation of MBHs are numerous, and include the collapse of first-generation stars ( Pop III BHs , M BH ≲ 10 3 M ⊙ ), the collapse and/or coalescence of massive stars formed in compact stellar clusters ( nuclear clusters , 10 2 M ⊙ ≲ M BH ≲ 10 4 M ⊙ ), the collapse of SMS formed in primordial environment ( direct collapse , M BH ≳ 10 3 M ⊙ ), and the collapse of cosmological density perturbations ( primordial BHs , 1M ⊙ ≲ M BH ≲ 10 10 M ⊙ ). The shaded orange region shows the redshift and MBH mass ranges of LISA, and the orange starburst symbols the LISA detections. LISA will significantly extend the current MBH EM detections, shown below the curved solid black line (from the local Universe at z ∼ 0 to the high-redshift quasars at z ⩾ 6 ). Figure credit: Melanie Habouzit \n<!-- image --> \nIn this section, we focus on formation pathways which can lead to the production of such seeds ( M BH = 10 2 -10 6 M ⊙ ) - see Fig. 22. This includes (a) seeds from metal-free Population III (Pop III) stars; (b) seeds originating from the dynamical processes in dense stellar clusters; (c) seeds born from the collapse of supermassive stars (SMSs); and (d) primordial MBH seeds. More detailed information on each of these scenarios is dealt with in other reviews (e.g. Volonteri 2010; Johnson and Haardt 2016; Valiante et al. 2017; Inayoshi et al. 2019; Volonteri et al. 2021). Here we outline the mechanisms behind each pathway as well as underlining outstanding issues in the field, especially those pertinent to LISA. The consequences on the detection rate and properties of mergers identified by LISA will be discussed in Section 2.4.2.2.', '· Formation of MBHs as Pop III remnants [M BH ≲ 10 3 M ⊙ ]': "One of the popular explanations behind the formation of high-redshift MBHs is related to Pop III stars, the hypothesized first-generation stars. Pop III stars are born in ∼ 10 5 -10 6 M ⊙ DM 'minihaloes'. The primordial gas in these first haloes is cooled primarily by H 2 , which allows the temperature of the gas to cool to approximately 200 K (Abel et al., 2002). This inefficient cooling channel leads to a top-heavy initial mass IMF expected for Pop III stars compared to present day star formation (Turk et al., 2009; Clark et al., 2011a,b), with mass values ranging from 10M ⊙ to 10 3 M ⊙ (Hirano et al., 2014). \nPop III stars with masses M ∗ ≳ 260M ⊙ will directly collapse into BHs, losing very little of their progenitor mass in the process (Heger et al., 2003). The retention of a significant amount of \nthe parent star mass is expected as a result of the weak stellar winds associated with metal-free stars. As a result, a large population of Pop III remnant BHs is expected to be left behind in these first minihaloes that are ubiquitous at early times. Less massive Pop III stars will explode as SNae, enriching their surroundings with metals. As metal enrichment is extended to nearby galaxies (e.g. Smith et al., 2015; Hicks et al., 2020) through both winds and halo mergers, the formation of Pop III stars declines severely and less massive Population-II stars begin to dominate the star formation history of the Universe (O'Shea et al., 2015; Xu et al., 2016). Nonetheless, this first generation of stars leaves in its wake a large number of Pop III remnant BHs, which may act as the seeds to future MBHs (Madau and Rees, 2001; Hirano et al., 2014). A key open question is therefore whether these Pop III remnants can grow into a population of MBHs, and under what conditions rapid growth can be achieved (this is particularly relevant for the high-z quasars) and their mergers be expected to be detected by LISA. We will explore research in this area and the significant challenges to their growth which must be overcome in Section 2.3.2. \n- · Formation of MBHs in dense stellar environments [ 10 2 M ⊙ ≲ M BH ≲ 10 4 M ⊙ ] Seed MBHs of 10 2 -10 4 M ⊙ can form in dense and massive stellar clusters of ∼ 10 5 M ⊙ through dynamical interactions (e.g. Omukai et al. 2008; Devecchi and Volonteri 2009; Reinoso et al. 2018; Schleicher et al. 2022). During the early evolution of star clusters with initial central densities ≳ 10 5 M ⊙ pc -3 , massive stars segregate to the cluster centre due to dynamical friction, where they may undergo runaway collisions resulting in the formation of very massive stars with masses of approximately 10 2 -10 3 M ⊙ (Portegies Zwart and McMillan, 2002; Portegies Zwart et al., 2004; Gürkan et al., 2004; Freitag et al., 2006b,a). In low-metallicity clusters, such massive stars may collapse into an MBH (Katz et al., 2015; Giersz et al., 2015; Mapelli, 2016; Sakurai et al., 2017; Giersz et al., 2015; Rizzuto et al., 2020). \nAnother possibility of forming an MBH in stellar clusters is through runaway mergers of stellar-mass BHs, which are expected to form from the evolution of massive stars. If stellar-mass BHs form with low-velocity natal kicks or are embedded in a dense gaseous halo (Belczynski et al., 2002; Mandel, 2016; Giacobbo and Mapelli, 2018; Davies et al., 2011), a significant fraction can be retained within the star cluster (Sippel and Hurley, 2013; Morscher et al., 2013, 2015; Wang et al., 2016b; Arca Sedda et al., 2018; Askar et al., 2018; Kremer et al., 2019a). While the mergers of these stellar mass sized BHs will generate GWs, their frequency ranges put them outside of the sensitivity range of LISA - they may however be detectable by future GW detectors like the Einstein Telescope (Valiante et al., 2021). A potential barrier to this formation scenario is that the retention of any MBH will depend on the GW recoil kicks that they receive during the merger process. If recoil kick velocities are larger than the escape speed of the cluster, then the seed MBH may be ejected out of the cluster (Holley-Bockelmann et al., 2008; Davies et al., 2011; Miller and Davies, 2012; Sesana et al., 2014; Morawski et al., 2018). \nIt may also be possible to grow stellar-mass BHs through gas accretion (rather or in conjunction with mergers) inside stellar clusters. Retained stellar-mass BHs could effectively grow and become MBHs by accreting the interstellar gas inside massive stellar clusters (Leigh et al., 2013; Natarajan, 2020). Moreover, BHs of ∼ 100M ⊙ can become more massive by growing through tidal capture and disruption of stars in dense nuclear star clusters (Stone et al., 2017a). Such runaway events can grow the mass of a BH from 10 2 -3 M ⊙ to up to 10 5 M ⊙ (Rosswog et al., 2009b; MacLeod et al., 2016a; Alexander and Bar-Or, 2017; Stone et al., 2017a; Boekholt et al., 2018; Sakurai et al., 2019). \n- · Formation of very massive seeds in atomic cooling haloes and primordial galaxies [ M BH ≳ 10 3 M ⊙ ] SMSs 7 were originally invoked to explain the existence of quasars prior to their \norigin being understood as the accretion of matter on to MBHs. More recently, SMSs have been 'reinvoked' as potential seeds for MBHs. SMSs are thought to form through the rapid accumulation of gas during the early stages of stellar evolution. If gas can be rapidly accreted with accretion rates in excess of 10 -3 M ⊙ yr -1 (Haemmerlé et al., 2018; Omukai and Palla, 2003), then the stellar envelope remains bloated and cool (with a temperature T eff ∼ 5000 K). Detailed numerical simulations have shown that such objects do not provide enough negative (radiative) feedback to halt accretion and the end result is an SMS (Sakurai et al., 2016; Chon et al., 2018; Sakurai et al., 2020). However, sustaining this accretion rate is nonetheless challenging, due to the complex dynamics between the gas and the stellar component (Chon and Omukai, 2020; Regan et al., 2020a). \nThe ideal environmental conditions for SMS formation can be achieved in so-called atomic cooling haloes (Tanaka and Haiman, 2009), where line-emission cooling due to neutral hydrogen allows the gas to cool and condense in a sufficiently massive halo (with a virial temperature T vir ∼ 8000 K and a virial mass M vir ∼ 5 × 10 7 M ⊙ at z ∼ 15 ). The larger mass of the atomic cooling halo, compared to the minihaloes in which Pop III stars are typically born, provides a larger baryonic reservoir for (metal-free) star formation. The key requirement for the development of an SMS is that the gas inflow onto the stellar surface remains high. Fragmentation of the gas into a (dense) stellar cluster must also be avoided for a truly SMS to form 8 (Regan et al., 2020a). Fragmentation may be avoided if the gas is sufficiently metal-poor (Chon and Omukai, 2020; Tagawa et al., 2020b), with metallicities not exceeding Z ≈ 10 -3 Z ⊙ , and perhaps also if the halo is not tidally disrupted (Chon et al., 2018). Given the difficulties in achieving monolithic SMS formation, the question of whether true SMS formation can be achieved remains an open and active research question. \nRadiative feedback in the Lyman-Werner (LW) band (in the energy range 11.2-13.6 eV) allows for the dissociation of H 2 , which suppresses Pop III star formation, allowing a halo to remain star-free (and hence metal-free). An attractive scenario here is the synchronised pair (Dijkstra et al., 2008; Regan et al., 2017) mechanism, whereby a pair of halos closely separated in time and space evolve together. The first of these haloes that forms stars could then provide the second halo with a strong enough LW background (Visbal et al., 2014); the key issue with this model is that the number density of such environments may be too rare to explain the number densities of expected MBHs. \nAlternative scenarios for avoiding premature star formation are to dynamically heat the gas (rather than photo-dissociating H 2 ). In this scenario, the gas can be shock-heated either through galactic collisions (Inayoshi et al., 2016) or through a rapid succession of minor and major mergers (Yoshida et al., 2003; Fernandez et al., 2014; Wise et al., 2019). The appeal of this scenario is that it arises more naturally through the mechanisms of DM structure formation and that the number density of MBH seed formation looks promising (e.g. Regan et al., 2020b) though further work on the expected number density of MBH seeds is required. \nFinally, the collisions of massive galaxies at moderately high redshifts ( z ∼ 8 -10) can lead to the direct formation of an MBH without any intermediate stage (Mayer et al., 2010, 2015). In this scenario (which, in stark contrast with the atomic cooling halo scenario, can occur also at solar metallicities) major mergers between the rare, most massive high-z galaxies funnel gas to their centre at rates exceeding 1000 M ⊙ /yr. The resulting accumulation of billions of solar masses of gas in a nuclear region less than a parsec in size could either induce the formation of a very large SMS, and hence a massive BH seed by direct collapse, or even directly form a large MBH via the radial general-relativistic instability of a supermassive protostellar precursor. Recent models show that an accreting SMS, owing to the much higher accretion rates occurring in the merger-driven scenario, can grow in mass much more than in the atomic cooling halo case, namely \nto > 10 7 M ⊙ in absence of rotation, before collapsing into an MBH seed (Haemmerlé et al., 2021).", '· Primordial Black Holes': "Primordial BHs are another plausible way to explain the formation of MBHs. Their abundance is constrained at various mass scales (Carr et al., 2021), but they can still form a considerable fraction of DM in mass ranges 1 -10 2 M ⊙ (Bird et al., 2016; Sasaki et al., 2016; Clesse and García-Bellido, 2017) and 10 -13 -10 -11 M ⊙ (Saito and Yokoyama, 2009; Garcia-Bellido et al., 2017; Domcke et al., 2017; Bartolo et al., 2019; Cai et al., 2019; Unal, 2019). Moreveor, primordial BHs of mass O (10 -10 5 )M ⊙ formed in the early universe (before recombination) could be the seeds of MBHs (Duechting, 2004; Belotsky et al., 2014; Clesse and García-Bellido, 2015; Nakama et al., 2016; Garcia-Bellido et al., 2016). The tail of their mass function 9 reaching a few hundred or thousand solar masses can grow many orders of magnitude (depending on formation mass) via accretion and mergers (Mack et al., 2007; Ali-Haïmoud et al., 2017; Raidal et al., 2019; Inman and Ali-Haïmoud, 2019; Serpico et al., 2020; De Luca et al., 2020). This claim has been studied and found to be consistent with the current cosmological probes of cosmological history. \nPrimordial BHs are formed by large density contrasts, and the most likely stage to produce these large perturbations is during inflation. Although cosmic microwave background-scale perturbations must be Gaussian and nearly scale invariant with a typical amplitude of 10 -5 , the fluctuations at smaller scales can be larger. There exist characteristic signatures of these enhanced fluctuations in various multimessenger probes, including cosmic microwave background distortions (Chluba et al. 2012; Ali-Haïmoud and Kamionkowski 2017; Aloni et al. 2017; Inomata et al. 2017; Garcia-Bellido et al. 2017; Nakama et al. 2018; Cappelluti et al. 2022) and secondary stochastic GWs resulting from the enhanced perturbations that re-enter the horizon in the radiation (or matter) dominated era (in particular enhanced inflationary perturbations that produce 1 -10 4 M ⊙ primordial BHs) which also produce stochastic GWs at Pulsar Timing Arrays (PTA) scales (Inomata et al., 2017; Garcia-Bellido et al., 2017; Vaskonen and Veermäe, 2020; Kohri and Terada, 2020; De Luca et al., 2020). The next generation PTAs, which can constrain the stochastic GW Background, as well as the cosmic microwave background experiments using spectral distortions, will probe inflationary fluctuations so sensitively that they could conclusively test the existence of primordial BHs from inflationary perturbations (Byrnes et al., 2019; Inomata and Nakama, 2019; Kalaja et al., 2019; Gow et al., 2020). We refer the reader to Auclair et al. (to appear) for more details on primordial BHs and LISA. \nResearch into the seeding of MBHs remains a highly active area of research. In an era where vigorous development of both semi-analytical models and full numerical calculations continues apace (see Section 2.4), understanding the mechanisms of MBH seeding becomes all the more important. A definitive pathway to forming MBHs remains an open question. An important metric of success for any formation model is to explain naturally the current abundance of MBHs of all masses in the nuclei of galaxies. These have a currently measured number density of n MBH ∼ (0 . 2 -1 . 0) × 10 -2 Mpc -3 at z = 0 , depending on how far down in mass function is integrated (e.g. Graham et al., 2007; Shankar, 2009; Terrazas et al., 2016). A key goal in any of the seeding models discussed above is therefore a calculation of the resulting number density of MBHs in the Universe as a function of redshift. So far, calculations within the community have varied significantly between approximately 10 -3 -10 -9 comoving Mpc -3 for the very massive seeds formed in atomic cooling halos (Dijkstra et al. 2008; Agarwal et al. 2012; Dijkstra et al. 2014; Agarwal et al. 2014; Habouzit et al. 2016; Wise et al. 2019), while we expect more seeds from e.g, the Pop III remnant formation mechanism. For reference, the number density of galaxies in the Universe today is ∼ 10 -1 comoving Mpc -3 , and the number density of quasars at z ∼ 6 is ∼ 10 -9 comoving Mpc -3 . MBH formation needs to explain both the population of high-redshift quasars, and the population of MBHs in the local Universe. \nA central challenge of models in the next decade leading up to LISA's launch will be to reduce the number density uncertainties associated with different models of MBH seed formation. A focal point of simulations in the next decade will be to accurately model the assembly of galaxies including modelling the environments, in a cosmological context, in which MBH seeds can form. Given the large dynamical range of nonlinear physical processes required to form MBH seeds, this is challenging. The use of focused, high-resolution and relatively large-scale numerical simulations with detailed (Pop III) star formation and MBH formation prescriptions will ultimately be required to break the current degeneracies between models which currently exist and provide models to be used for inference with LISA detections.", '2.3.2 MBH growth across time and space': "LISA will measure not only the masses but also the spins of massive black holes. In this section, we discuss three compelling open questions: how do MBH seeds grow across cosmic time? What is the impact of such growth on the spin of an MBH? What can the final spin reveal about its past accretion history? In the discussion of these issues, throughout this section, we will differentiate between light seeds and heavy seeds. Light seeds have masses of at most 10 3 M ⊙ and are typically those formed by the first generation of metal-free stars, while heavy seeds are those with higher masses that can result from stellar dynamical processes or from the direct collapse scenarios discussed above. \n2.3.2.1 How to grow light seeds Pop III remnants are predicted to have low mass, ⩽ 10 3 M ⊙ . In order for these seeds to grow massive enough to be in the LISA band, or to grow massive enough to even become the extremely massive quasars that we observe at z ∼ 67, they would need to clear two main hurdles. If not formed in the centre of their galaxies, these seeds must sink efficiently to the centre, but also sustain efficient accretion for a significant fraction of their lifetime. To produce the population of high-redshift quasars, they have to sustain near-Eddington accretion rates for nearly a Gyr (Haiman and Loeb, 2001). In the conventional picture of a spherically symmetric accretion flow whose energy loss is only controlled by radiation propagating isotropically, the Eddington limit expresses a maximum allowed accretion rate. Therefore, seeds would have to grow at near the maximum rate allowed for their entire lifetime, unless a mechanism for super-Eddington accretion is invoked by resorting to more complex configurations of the fluid flow and radiation field, and to a different energy transport mechanism. \nHigh accretion efficiency is challenging to explain physically, given that radiative feedback both from the surrounding stellar component and MBH growth can unbind gas in the vicinity of the seed BH, thus preventing further growth. These hurdles were first examined in the early 2000s (Omukai and Inutsuka, 2002; Oh and Haiman, 2003; Whalen et al., 2004), with each study finding that Pop III BHs initially find themselves in low-density environments within the galaxy, where they are unable to grow. Expanding on earlier studies, Smith et al. (2018) investigated the growth of more than 15,000 light-seed BHs using the Renaissance simulations and found that none were able to grow by more than 10 percent for the 300 Myr for which their growth was followed. This time period represents a significant fraction of the Hubble time at this epoch. These predominantly numerical works have been confirmed by semi-analytical approaches which also find that light seeds struggle to achieve significant growth (e.g. Valiante et al., 2016; Pacucci et al., 2017, and references therein). \nA mechanism of rapid growth may be required in order to grow light seeds, both to help stabilise their orbits (see Section 2.2) within the galactic centre and to allow them evolve into MBHs, as examined in Section 2.3.1. A number of studies have also shown that light seeds can grow through super-Eddington accretion given the correct environmental conditions (Alexander and Natarajan, 2014; Inayoshi et al., 2016, 2019; Lupi et al., 2016; Pezzulli et al., 2016, 2017). \nIn either case, a growing BH must reach a critical mass before it can sink to the centre of the potential and become a central MBH. Recent investigations by Pfister et al. (2019b) have shown that MBHs with masses M BH ≲ 10 5 M ⊙ are unable to sink via dynamical friction as the stellar component of high-redshift galaxies tends to be too irregular. This leads to a population of wandering MBHs that cannot efficiently accrete gas or merge with other BHs. The idea of a population of wandering MBHs is not new; this population has been previously associated with galaxy mergers which result in off-nuclear MBHs from the failure of them to reach the centre of the merger remnant (e.g. Volonteri, 2010). Nonetheless, more recent, high resolution, simulations have shown that wandering MBHs may result from seeds with masses M BH ≲ 10 5 M ⊙ that are unable to settle to the centre of the galactic potential. This result has been confirmed by other high-resolution simulations which show that a large population of wandering MBHs with M BH ≲ 10 5 M ⊙ is likely in most, if not all, galaxies (Tremmel et al., 2018a; Bellovary et al., 2019; Regan et al., 2020a). Interestingly, this result has also been tentatively confirmed by observations of off-center MBHs in galaxies (Reines et al., 2020). However, if associated with a compact massive star cluster, dynamical friction will be greater, and such off-centre MBHs may be transiting rather than stalled. Once MBHs exceed M BH ∼ 10 6 M ⊙ , they are less prone to 'jittering' (but see Ma et al., 2021), although they still remain susceptible to ejections (via triple interactions and GW recoils), the velocities of which depend on MBH mass ratios rather than absolute mass, although of course retaining the MBHs depends on the potential well of the galaxy. \nFigure 23: Predictions on the relative importance of MBH growth by gas accretion (blue shades) and mergers (red shades). ˙ ρ a and ˙ ρ m are the predicted mass growth rates by gas accretion and mergers, respectively. The contour where ˙ ρ a = ˙ ρ m is represented with a dash-dotted line. The region corresponding to a LISA signal-to-noise ratio ≥ 10 for MBHBs with mass ratio 0.2 is delimited within the two thick, dotted lines. The whited-out area on the top right corner indicates the region of the parameter space where no MBHs should be present (adapted from Pacucci and Loeb 2020). \n<!-- image --> \n2.3.2.2 Accretion versus MBH mergers Despite large uncertainties in the physical parameters that enable a mapping between luminosity and mass (e.g. duty cycle, matter-to-energy accretion efficiency, Eddington ratios, and bolometric corrections; see, e.g. Tanaka and Haiman 2009), a consistent picture is now emerging. Observations and theoretical models suggest that most of the mass growth over cosmic time occurred via gas accretion, and that more massive MBHs grew at earlier cosmic times, whereas lighter MBHs were still growing at z ≲ 1 (Soltan, 1982; Marconi et al., 2004; Merloni and Heinz, 2008; Shankar et al., 2009). Assuming a combination of light and heavy MBH seeds at z ∼ 20 -30, recent studies have confirmed that growth by gas accretion is dominant for most MBH masses during a large fraction of the evolution of the Universe ( 0 ≤ z ≤ 9 -10), especially for M BH > 10 6 M ⊙ and z < 8 (Pacucci and Loeb, 2020; Piana et al., 2021). \nGrowth by mergers - which we recall can at most double an MBH mass at each merger - can become dominant for M BH < 10 4 -5 M ⊙ at z > 6 (Dayal et al., 2019; Piana et al., 2021), and for M BH > 10 8 M ⊙ at z < 2 (Pacucci and Loeb, 2020). This is possible if one or more of the following conditions are met: (a) the number density of MBHs is large, and (b) the cold gas available for accretion is scarce given that the accreted mass fraction depends on the richness (over-density) of the environment (Dubois et al., 2014b). The first condition can be met at high redshifts if light seeding mechanisms are dominant, leading to a large number density of MBHs. This could in turn result in frequent mergers, although light seeds are unlikely to merge and sink to the centre as shown in Section 2.2. The second condition can be verified at z ≲ 1 (Power et al., 2010). Predictions on the contribution of mergers to the cosmic growth of MBHs strongly depend on a multitude of parameters, many of which are unknown or loosely constrained. For example, the number density of heavy MBH seeds can vary over ∼ 6 orders of magnitude (at a given redshift z ≳ 8 ) in modern cosmological simulations (see, e.g. Habouzit et al. 2016; Woods et al. 2019), with huge uncertainties being introduced by the time-scale on which MBHs can actually merge (e.g. Dayal et al., 2019; Barausse et al., 2020b). Nonetheless, the presence of partially-depleted cores in massive galaxies offers the promise of a substantial number of MBH mergers at least at late cosmic times (Begelman et al., 1980; Graham, 2004; Dullo and Graham, 2013). \nDespite significant unknowns, e.g. the contribution of obscured accretion, which is invisible in all bands apart from X-rays and higher energies, (e.g. Worsley et al., 2005; Fiore et al., 2009; Comastri et al., 2015), we now have a clear picture of growth by accretion (see Fig. 23). LISA, along with future third-generation GW observatories (e.g. the Einstein Telescope and/or Cosmic Explorer), is however the only way to actually measure the merger history of the full MBH mass spectrum, and this is what is expected to be delivered after its launch. However, as stressed already, to accurately assess the role that LISA will play in constraining the relative role of MBH mergers and accretion in MBH growth, theoretical models have to be refined in order to allow for inference on the astrophysical picture by comparing the data stream to predictions. \n2.3.2.3 Feedback as a barrier to MBH growth As detailed in Section 2.2, the ionizing radiation that emerges from the innermost parts of the MBHs' accretion flows can render gas dynamical friction inefficient for a range of physical scenarios (Park and Bogdanović, 2017, 2019), although this depends on the surrounding gas environment (Toyouchi et al., 2020). This can lengthen the inspiral time of MBHs and reduce the MBH pairing probability (Li et al., 2020a). The suppression of MBH pairing is most severe in galaxies with MBH pairs with mass < 10 8 M ⊙ and low mass ratio, which are direct progenitors of the merging binaries targeted by LISA. See Section 2.2.1.3 for additional details. \nSecondly, both hydrodynamic cosmological simulations with a physical model for light seed MBH formation (Habouzit et al., 2017) and semi-analytical models (Barausse et al., 2020b) converge on the fact that the number of MBHs growing enough to enter the LISA band depends on the strength of SN feedback. In case of strong feedback, SN winds can expel gas from the nuclear region of relatively low-mass galaxies ( M ⋆ ⩽ 10 10 M ⊙ ), depleting the gas reservoir of the \nMBHs (Dubois et al., 2015). This prevents the MBHs to grow in mass until the gravitational potential well of their host galaxies is deep enough to confine again the cold gas close to the central region. The MBHs may remain too light to be detected by LISA. \nIn addition to affecting the merger rates, strong feedback generated by the MBHs themselves can significantly slow down the growth of MBH seeds. As shown, e.g. in Regan et al. (2019) the strong outflows generated by the jets are able to deplete a region of ∼ 0 . 1 pc around the seed. Although the outflow generally does not reach the escape velocity from the host galaxy, it does suppress the growth for a time-scale comparable to the dynamical time. A super-Eddington ( ˙ M BH > ˙ M Edd = L Edd /c 2 ) accretion rate would then translate into a time-weighted, effective accretion rate of 0.1-0.5 the Eddington rate, significantly slowing down the growth of the MBH over ∼ 0 . 5 Gyr by factors ∼ 30 -3000 , when compared to the growth required to match the observations of z ∼ 7 quasars. While this is potentially an issue to explain the brightest high-z quasars, it could act to increase event rates in the LISA band at the highest redshifts as heavy BH seeds could remain longer within the mass range where LISA is most sensitive. \nFinally, galaxies can also experience external radiative feedback due to the heating background created by reionization photo-evaporating gas from the outskirts of low-mass galaxies in ionized regions (van Wassenhove et al., 2010; Dayal and Ferrara, 2018). However, this feedback has almost no effect on the mass build-up of MBHs in the early Universe since the MBHs of such reionization feedback affected galaxies are already accretion-starved due to SN feedback (Dayal et al., 2019). \nThe variety of astrophysical processes involved in modelling MBH growth, described in this Section, highlights that one of the challenges ahead of us to prepare for LISA is to assess degeneracies that can affect the interpretation of LISA's data. Overall, the large number of parameters and scales involved makes this a complex problem - at the same level of galaxy formation. Progress in delivering realistic models that can be compared to LISA's detection will require on the one hand more detailed investigations in all the subfields, and on the other hand a way to consolidate these results into coherent models. \n2.3.2.4 Spin evolution of MBHs under accretion and mergers LISA has a unique potential in providing measurements of the spins of merging MBHs: this means a theoretical understanding of how MBH spins evolve is necessary in order to be able to interpret LISA's results. Accretion and mergers establish profound links between the spin and the mass of the MBHs, which therefore have to be studied jointly. In the accretion process, the spin is a critical physical parameter, as it determines the radiative efficiency. For a geometrically thin accretion disc, the efficiency of converting mass into light varies from 0.057 for a non-spinning MBH to 0.43 for a maximally spinning MBH (e.g. Novikov and Thorne, 1973). This has a direct impact on the rate of MBH mass growth, on the amount of radiated energy, and on the spin magnitude and orientation at the end of an accretion episode. Also, a key manifestation of the spin when an MBH is accreting from a magnetized plasma is the launch of a collimated jet of matter and radiation which directly tracks its orientation (Blandford and Znajek, 1977). The link between spin, accretion and jet power/efficiency has started being compared to observations (Unal and Loeb, 2020) and being used to set lower bounds on AGN spins (Unal et al., 2020). \nSpins determine how efficiently the accreted matter is transformed into energy, but in turn the way in which MBHs accrete gas has a crucial bearing on their spins: depending on the accretion geometry, the resulting MBH spin's magnitude and direction can vary widely. Taking the limiting case of prolonged coherent accretion from a viscous disc, the spin can increase up to its limiting value of 0.998 (Thorne 1974; see also Popham and Gammie 1998; Gammie et al. 2004) after the MBH has accreted an amount of gas comparable to its initial mass, regardless of the flow being initially prograde or retrograde (Bardeen, 1970). The spin in this case gets aligned with the angular momentum of the disc from which it is fed (Bardeen and Petterson, 1975), and the time-scale for the alignment is short ( 10 5 yr) compared to the typical time for \nmass growth. At the other extreme, chaotic accretion, made up of randomly oriented smallmass accretion events, results instead in an erratic orientation of the spin and, in general, in a spun-down MBH (King and Pringle, 2006; King et al., 2008). Several semi-analytical models of MBH evolution have included either one (e.g. Volonteri et al., 2005) or both (e.g. Volonteri et al., 2007; Berti and Volonteri, 2008; Barausse, 2012) of these two limiting cases. More recent semianalytical models (Sesana et al., 2014) have included accretion flows that are neither perfectly coherent nor perfectly isotropic depending on the fuelling geometry (Dotti et al., 2013). These studies, together with numerical works that follow the evolution of the spin in relation to the dynamics of the accreting gas (e.g. Maio et al., 2013; Dubois et al., 2014b, 2015, 2020; Sayeb et al., 2020), have shown that the distribution of MBH spins depends on several quantities, such as host morphology, MBH mass, mass ratios, and redshift. \nIn principle in a binary all spin orientations and all spin magnitudes allowed by GR are possible. However, when an MBH binary, in its latest stages of evolution, is surrounded by a circumbinary disc, the interaction with the external gas leads both the binary orbital axis and the individual MBH spins to reorient their directions into a configuration of minimum energy where the two spins are aligned to a large degree with the orbital angular momentum axis, as discussed in Bogdanović et al. (2007) (see also Dotti et al., 2010; Miller and Krolik, 2013). This has a strong impact on the final spin of the new MBH and on the magnitude of the velocity acquired by gravitational recoil, which depends sensibly not only on the mass ratio, but also on the magnitudes and orientations of the spins (Kesden et al., 2010b; Lousto et al., 2012; Berti et al., 2012). Extrapolation of MBH coalescences with large initial spins (larger than ∼ 0.9) exactly aligned with the orbital angular momentum yields a final spin as large as ∼ 0 . 95 (Marronetti et al., 2008; Berti and Volonteri, 2008; Kesden et al., 2010a; Lovelace et al., 2011). \nIn gas-poor conditions, the potential lack of a massive circumbinary disc leads MBH binaries to have spins randomly oriented at the time of their coalescence relative to the orbital plane, with magnitudes determined by the previous accretion history. Statistically, when spins are equally distributed in all directions relative to the orbital axis, the remnant MBH spin depends on the binary's mass ratio: if an MBH merges with many lower-mass MBHs it tends to spin down, as the final spin is dominated by the orbit at plunge and retrograde accretion at larger radii reduces (on average) the spin of the larger MBH (Hughes and Blandford, 2003). If instead the mergers involve MBHs of comparable mass, on average the remant will have a spin ∼ 0 . 7 (Berti and Volonteri, 2008), consistent with the value of the final spin resulting from the merger of two equal-mass, nonspinning MBHs (Scheel et al., 2009). \nBerti and Volonteri (2008) studied the co-evolution of MBH masses and spins in a cosmological context, showing that in general accretion dominates over mergers in determining the spin evolution of the whole MBH population. While in prolonged accretion episodes spin-up is very efficient, with a large fraction of MBHs having individual spins in excess of 0.9, isotropic mergers reduce the fraction of high-spin MBHs and create a roughly uniform distribution. If accretion is chaotic, most MBHs have spins below 0.1 prior to merging. This demonstrates how spins offer the best diagnostics on whether MBHs before coalescence have experienced either coherent or chaotic accretion. These studies are important preparation for LISA as they provide insight for modelling realistic spin distributions to be used as priors during the analysis of waveforms to extract source parameters. \nIndeed, LISA will measure not only the MBH individual masses, but also a mass-weighted combination of the individual spins projected along the orbital angular momentum (the so-called 'effective spin' χ eff = ( M 1 χ 1 z + M 2 χ 2 z ) / ( M 1 + M 2 ) , where M 1 and M 2 are the MBH masses, χ 1 z and χ 2 z are the components of the spins along the orbital angular momentum) and possibly their precessional dynamics, which is encoded in the amplitude and phase of the waveform. A measurement of χ eff alone does not constrain the individual spins. For example, a small χ eff could result from both MBHs having small spins; from each MBH having significant spins in the angular momentum direction, but anti-aligned with each other; or from nonzero spins oriented along the \norbital plane. Through parameter estimation of precessing binaries, however, it is possible to infer posterior distributions for both spins. Preliminary work on simulated MBH populations has shown that the spin of the primary MBH can be measured by LISA with an exquisite accuracy ( ∼ 1 -10% ) for nearby, loud events. This precision in the measurement mirrors the fact that the primary MBH leaves a bigger imprint in the waveform through the mass-weighted χ eff . The measurement is more problematic for the spin of the secondary, that can be either determined to an accuracy of 0.1, or can remain completely undetermined, depending on the mass ratio and spin magnitude (Klein et al., 2016). \nIf LISA's detection rates will be at the high end of current estimates, it may be possible to learn about the statistical distribution of the spins, and therefore constrain the relative importance of mergers and accretion in shaping the MBH spin population in the mass range below 10 6 M ⊙ , which is poorly constrained by EM observations (see Section 2.6). Having a comparison between spin measurements from LISA and EM observations, which are tracing different populations, will be of paramount importance (Sesana et al., 2014). \nIn preparation for LISA, further improvements in numerical simulations are needed to make use of novel techniques to model physical processes below the resolution limits (e.g. Dubois et al., 2014b; Fiacconi et al., 2018), and to include changes in the spin directions that affect feedback (Bustamante and Springel, 2019; Cenci et al., 2020; Sala et al., 2021; Dubois et al., 2020). Semianalytical models are also needed to understand whether the interaction between MBHs and their accretion discs can lead to spin alignment (see e.g. Miller and Krolik (2013); Lodato and Gerosa (2013); Gerosa et al. (2015b, 2020)). Finally, an interesting possible outcome of MBH mergers is that in non-aligned conditions, the direction of the remnant's spin can flip with respect to those of the progenitors: this would leave an observed imprint in the surrounding medium in the form of a particular shape (X-shaped radio galaxies; Gergely et al., 2010), and observational searches for such systems (Roberts et al., 2015) can provide information on the spin properties of merging MBHBs complementary to those obtained from theoretical models.", '2.4 Statistics on MBH mergers': "Coordinators: Silvia Bonoli, Alessandro Lupi Contributors: Monica Colpi, Pratika Dayal, Massimo Gaspari, Melanie Habouzit, Chung-Pei Ma, Lucio Mayer, Sean McGee, Hugo Pfister, Raffaella Schneider, Alberto Sesana, Rosa Valiante, Marta Volonteri \nMBHs are not born nor evolve in isolation. The physical properties of the host galaxies are key not only to set the MBH initial mass (see Section 2.3), but also to modulate the subsequent growth and mergers. Indeed, most of the mass of today's MBHs is likely the result of multiple accretion episodes throughout their entire lifetime (Soltan, 1982), likely triggered by secular processes or during violent events, such as galaxy interactions. For this reason there are various aspects of galaxy formation and evolution that are indirectly very relevant to LISA, and which need to be well understood in order to enable predictions for observable MBH merger event rates as a function of key parameters, such as masses, mass ratios and spins of the MBHs, as well as their dependence on redshift. Likewise, the same deep understanding is required for post-launch interpretation of the LISA datastream. In the currently accepted cosmological framework, the Λ CDM model, galaxies are expected to experience a large number of interactions and mergers during their lifetimes (Lacey and Cole, 1993). Galaxy interactions not only likely foster the activation of accretion episodes (e.g. Kauffmann and Haehnelt, 2000; Di Matteo et al., 2005; Capelo et al., 2015), but also lead to the formation of binary MBH systems (Mayer et al., 2007; Tremmel et al., 2017; Volonteri et al., 2020). The creation of triplets and multiple MBH systems is also possible (Bonetti et al., 2019), in particular for galaxies in dense environments which generally experience more frequent mergers. The formation time-scale of an MBHB is, however, dependent on the properties of the host galaxies. As discussed in Section 2.3.1, the formation \nof a bound system is subject to the ability of the secondary MBH to sink towards the centre of the merger remnant, where the primary MBH is expected to reside. A substantial amount of orbital angular momentum needs to be transported away, with the distance between the two MBHs having to decrease by several orders of magnitude (from kpc to pc scales, see Fig. 16). The sinking process, driven by dynamical friction and global torques, depends non-trivially on the properties of the host, such as the overall galaxy structure, the gas fraction, the presence of clumps or stellar clusters and structures such as discs or bars (see Section 2.2). Once a bound binary forms, its ability to harden still depends on the properties of the surrounding medium (e.g. Sesana and Khan, 2015; Biava et al., 2019). The hardening time-scale is shorter in galaxies with a large amount of stars that can cross the binary 'loss-cone' and/or with enough gas in the centre to create a circumnuclear disc (e.g. Merritt and Poon, 2004; Dotti et al., 2007). \nGiven that galaxy properties are tightly connected to the large-scale environment, the frequency of MBH mergers that LISA will detect depends on the global cosmological evolution of the host galaxies. All these physical processes, highly non-linear, can only be studied via sophisticated theoretical models, either analytical, semi-analytical, or fully numerical. The main difficulty resides in the extremely wide range of physical processes and scales that need to be resolved simultaneously, from the Mpc cosmological scales to the sub-pc scales where GWs become dominant (see Fig. 16). We are currently unable to resolve the full dynamical range that would be required to predict the number and properties of MBH mergers for LISA. \nDespite such modelling difficulties, building the infrastructure for interpreting the LISA datastream in the context of structure formation and evolution is one of the key tasks for the LISA Consortium and the astrophysical community at large. In fact, LISA will provide a catalogue of MBHBs with posterior distribution of the parameters of each source, including masses, sky localization, distance, magnitude and orientation of individual MBH spins, and eccentricity of the orbit. The degree of precision of these measurements will obviously depend on the specific source. \nIndividual binary parameters and parameter distribution across the detected population encode important information about the physics underlying MBHB formation. For example, the mass function and redshift distribution of detected events strongly depend on the nature and efficiency of the seeding mechanism. The spins of individual MBHBs are strongly affected by their main accretion channel, whether this is accretion of cold gas, tidal disruptions of stars or capture of compact objects, or previous mergers with other MBHs. \nThe information encoded in LISA's catalogue of events has the potential to revolutionize our understanding of MBH formation and evolution, and the degree to which such potential can be exploited depends on the sophistication of the astrophysical inference models and pipelines at hand. Sesana et al. (2011a) conducted a pilot study demonstrating the power of inference on LISA data. They considered a number of different MBH cosmic evolution scenarios, encompassing different seeding models (Pop III versus direct collapse), accretion efficiency (Eddington versus sub-Eddington) and geometry (coherent versus chaotic), demonstrating that LISA will be able to discriminate among them with just a handful of detections. They also considered mixed models in which, for example, different seed populations were combined, and found that LISA could correctly recover the presence of multiple sub-populations and their relative abundance. \nAlthough this was a successful first step, ideally, the community should employ all the arsenal of analytical and numerical models to distill a meaningful mapping of key astrophysical processes into MBHB parameter distributions. The LISA catalogue can then be used to tackle the 'inverse problem' of reconstructing the cosmic history of MBHs from GW observations. A proof of concept example of such process can be found in Padmanabhan and Loeb (2020). They used a parametric toy model connecting the MBH properties to the host DM haloes, to demonstrate that LISA would be able to constrain the halo occupation fraction and the MBH-halo relation. \nAs we summarized above, the properties of MBHs are shaped by a number of physical ingredients that go beyond the host DM halo and involve a number of gas and stellar dynamical \nprocesses, and likely involve a non-negligible degree of stochasticity. In this respect, exascale numerical simulations combined with neural network techniques for model emulation, as we outline below, can be used to anchor and inform flexible semi-analytical models that can efficiently map a vast physical parameter space into a likelihood function of the MBHB population. Ideally, those models would be flexible enough to include information coming from future observations across the EM spectrum, including Rubin, JWST, Athena, and SKA, to name few notable examples, to enhance their constraining power (see Section 2.6). Last but not least, the ultimate LISA MBHB astrophysical inference pipeline will also take advantage of any EM counterpart to individual LISA sources, which will provide additional information about the environment of the merging binary. \nIn this section, we first review the state of the art of the models that attempt to connect the small-scale processes of MBH formation, growth, and dynamics with the broader cosmological context of galaxy evolution. We then discuss the current estimates for the number of events that LISA will be able to detect as predicted by both numerical and semi-analytical models. At the end of the section, we provide an outlook on the need of pushing these models to be progressively more and more accurate and flexible, taking advantage of both the progress in computational power and new computational and statistical techniques. The theoretical framework connecting MBH mergers with the broader cosmological picture will play a fundamental role in the data analysis and the physical interpretation of LISA events.", '2.4.1 Modelling MBH evolution in a cosmological context': "MBH assembly is considered an essential component of galaxy formation (e.g. Kormendy and Ho, 2013). As anticipated, while not specific to LISA science, this is a central topic to enable pre-launch studies, such as to inform models of LISA event rates, as well as instruct postlaunch studies by setting the framework for the interpretation of the data, allowing to generate astrophysical inference work. The inclusion of physical processes related to MBH growth into simulations of galaxy formation has been initially driven by the need of understanding the role of MBHs in shaping the host galaxies via processes such as AGN feedback. In particular, feedback from AGN has been invoked to explain the observed quenching of massive galaxies, which could not be explained by stellar feedback alone (e.g. Springel et al., 2005a; Croton et al., 2006; Bower et al., 2006). \nModels of galaxy formation and MBH assembly can be grouped into two categories: cosmological hydrodynamical simulations and semi-analytical models. In the former, the DM and baryonic components of the Universe are followed simultaneously, starting from given initial conditions set by the chosen cosmological model. These simulations are computationally expensive and, in their set-up, a trade-off has to be made between the size of the cosmological volume that needs to be probed and the mass and spatial resolution desired. Thanks to the fast advance of computational power, state-of-the-art simulations are able to encompass large volumes of ∼ 100 3 cMpc 3 with kpc resolution (e.g. Dubois et al. 2014a; Vogelsberger et al. 2014; Hirschmann et al. 2014; Schaye et al. 2015; Pillepich et al. 2019; Davé et al. 2019). While allowing to study the evolution of DM and baryons in a self-consistent way down to the resolved physical scales, astrophysical processes that act at smaller scales (e.g. gas cooling, star formation, stellar feedback, MBH seeding, MBH accretion, MBH feedback) need to be included via sub-grid recipes. \nSemi-analytical models, instead, follow the evolution of the baryonic component of the Universe through a series of differential equations which link the time-evolution of the baryons to that of the underlying DM haloes. While losing some level of self-consistency, this approach has the advantage of being able to statistically explore how different physical assumptions affect the global galaxy population or targeted sub-samples (see, e.g. the seminal paper by Kauffmann et al., 1993). The merger trees can either be derived via the Press-Schechter formalism (Press and Schechter, 1974) or using the outputs of N -body simulations. While the first approach is computationally less expensive and merger trees with a broad range of masses can be resolved \n(e.g. down to the mass of the first star-forming haloes), N -body simulations offer the advantage that the 3D spatial distribution of galaxies is fully tracked, as the dynamical evolution of the underlying DM haloes is properly followed. This allows the modelling and studying of the complex link between physical non-linear processes and the large-scale environment. \nIndependently of the adopted technique, the models that track the evolution of the MBH population need to use sub-grid assumptions derived from higher-resolution simulations or analytical derivations, whose parameters are calibrated using observed properties of MBHs and their host galaxies. The local stellar mass function (e.g. Baldry et al., 2012), the distribution of galaxy colours (e.g. Baldry et al., 2004), and the evolution of the star formation rate density (e.g. Madau and Dickinson, 2014) are some of the key observables used to calibrate the parameters regulating galaxy evolution, e.g., the galaxy and MBH sub-grid physics. \nModels including the growth and evolution of MBHs are typically anchored to local relationships between the MBH masses and host properties, such as stellar mass and velocity dispersion (Kormendy and Richstone, 1995; Magorrian et al., 1998; Ferrarese and Merritt, 2000; Gebhardt et al., 2000; Tremaine et al., 2002; Graham and Scott, 2015; Sahu et al., 2019a). Besides the calibration, the validation of the models is done by comparing the resulting MBH population to observational constraints. Typically, the AGN luminosity function from the local Universe to high redshifts (up to z ∼ 4 , Hopkins et al., 2007; Lacy et al., 2015; Shen et al., 2020), which constrains MBH accretion rates over cosmic time, is often used as diagnostics of the simulation or semi-analytical subgrid models. Additional diagnostics include the Eddington-ratio distribution of AGN (e.g. Hickox et al., 2009; Aird et al., 2018), the number density of the highest-redshift quasars (e.g. Fan et al., 2006; Mortlock et al., 2011; Decarli et al., 2018), and the clustering of active and luminous MBHs (e.g. Gilli et al., 2005; Ross et al., 2009). \nHere, we briefly review the different modelling aspects of MBHs - seeding, fuelling, feedback, and dynamics - and discuss different implementations and uncertainties among models, highlighting in particular those that are relevant for LISA. \n2.4.1.1 MBH seeding The first aspect that is crucial to determine the MBH occupation fraction in galaxies, and therefore MBH formation efficiency, is MBH seeding. Despite the strong effort by the community and the variety of proposed models (see Section 2.3 for a detailed review), MBH seeding mechanisms are still unconstrained. As in Section 2.3, we consider models assuming 'heavy' seeding, resulting in massive ( 10 4 -6 M ⊙ ) but rare seeds, as well as models assuming 'light' seeding, which results in less massive ( ⩽ 10 3 M ⊙ ) but more abundant seeds. LISA, by being sensitive to the mass of the merging MBHs that generate the GW signal, has the potential to constrain these models at statistical level through Bayesian analysis, because detection rates in the various mass intervals depend upon the seeding mechanism (Sesana et al., 2011a; Klein et al., 2016; Bonetti et al., 2019; Barausse et al., 2020b). Of course, right after their emergence, the mass growth of the seeds through various mechanisms will also affect that MBH merger mass distribution that LISA can detect at any given time, which may complicate the interpretation of the statistics (see next section). \nIn state-of-the-art cosmological hydrodynamical simulations of ∼ 100 3 cMpc 3 , the typical mass of DM particles is M DM ∼ 10 6 -8 M ⊙ . This is not enough to resolve the haloes where we expect MBH formation to happen, for example the atomic cooling haloes where we expect direct-collapse MBH seeds to form. These simulations also do not have enough resolution to model self-consistently some key physical processes required by the different channels of MBH formation described in Section 2.3. Instead, MBHs are commonly inserted as sink particles 'by hand', either in massive haloes of M h ⩾ 10 10 M ⊙ (e.g. Springel et al., 2005a; Hirschmann et al., 2014; Sijacki et al., 2015; Schaye et al., 2015; Davé et al., 2019), or in regions of the volume depending on the local properties of the medium, such as gas density (e.g.Taylor and Kobayashi 2014; Bonoli et al. 2016; Habouzit et al. 2017; Dubois et al. 2014a, 2020). \nIn smaller-volume cosmological simulations with higher resolution, it has become possible \nto start implementing more physical MBH formation recipes. For example, several simulations formed heavy seeds according to the local gas properties in high-redshift haloes (Tremmel et al., 2017; Bellovary et al., 2019). A model by Dunn et al. (2018) additionally includes LymanWerner flux as a seed formation criterion, most closely mimicking the direct collapse model. Other realistic seed formation mechanisms forming lighter MBHs in cosmological simulations were explored in Habouzit et al. (2017): seed MBHs were formed in dense metal-free collapsing regions to mimic the collapse of the first generation of stars or of dense nuclear star clusters. \nLeveraging the ability to efficiently probe larger effective volumes and smaller halo masses, semi-analytical models remain valuable for testing seeding models and statistically exploring the impact of seeding on multiple observables across cosmic times. Using a model connecting heavy MBH seeding to halo properties, Lodato and Natarajan (2006); Volonteri et al. (2008b); Volonteri and Natarajan (2009) explore, for example, the observational consequences of light seeding models compared to direct collapse models with varying efficiencies. These and other works (e.g. Bonoli et al., 2014) provided novel quantitative predictions on how seeding reflects on the galaxy-MBH correlation, e.g., M BH -σ , with σ the galaxy velocity dispersion. Other works have instead focused on the high-redshift universe, analyzing the ability of different seeding scenarios to lead to a population of z > 6 quasars consistent with current observational data (see the review of Valiante et al., 2017). As discussed in subsequent sections, semi-analytical models also predict clear seeding signatures in the mass function of GW event rates detectable by LISA. \nHowever, our current knowledge of the MBH population and their hosts across redshift is not sufficient to put tight constraints on seeding models, leaving predictions for the signatures of seeding on LISA events largely degenerate with other physical assumptions on MBH growth and dynamical evolution. This underlines on the one hand the importance of improving observational constraints of the key measurements in the seed mass regime to constrain models. This is needed to create reliable models to compare with LISA's event properties. On the other hand, LISA's results will likely provide the most stringent constraints on MBH seeds, since it can explore redshifts closer to seed formation ( z ≳ 10 ) than any EM facility can do. \n2.4.1.2 MBHfuelling After MBHs have formed, their growth is mainly driven by gas accretion, whose rate is determined by the efficiency of the fuelling process onto the MBH from galactic (kpc) scales down to the nuclear region. Because of the limited resolution, and the inability to properly track the angular momentum evolution of the inflowing gas and the formation of the accretion disc, cosmological simulations almost always describe the accretion process through some version of Bondi-Hoyle-Lyttleton accretion (hereafter Bondi; Hoyle and Lyttleton, 1939; Bondi and Hoyle, 1944; Bondi, 1952, accretion rate ∝ M 2 BH ), which assumes spherical symmetry and can be inaccurate in most realistic physical scenarios (e.g. Levine et al., 2010; Hobbs et al., 2012; Gaspari et al., 2017; Negri and Volonteri, 2017). In case of significant angular momentum of the MBH accreting material, the accretion rate may not be well represented by the Bondi model. As such, the EAGLE simulation employs a modified Bondi model that takes into account the circularization and subsequent viscous transport of the infalling material (Rosas-Guevara et al., 2015, 2016). The gravitational torque-driven model (Hopkins and Quataert, 2010), implemented in some recent cosmological simulations (e.g. Anglés-Alcázar et al., 2017; Davé et al., 2019), takes a different approach and since the accretion rate ∝ M 1 / 6 BH , low-mass MBHs initially grow more than in the Bondi model (e.g. Çatmabacak et al., 2022). This emphasizes the need for progress in bridging the gap between galactic and accretion disc scales. Indeed, zoom-in high-resolution simulations (from galactic down to sub-pc scales) show that the actual accretion flow often proceeds in the form of chaotic cold accretion (Gaspari et al., 2013, 2015), in which fractal clouds condense out of the turbulent hot halo, rain on to the nuclear region and, via frequent inelastic collisions, boost the accretion rate by 100 × over the simple Bondi rate (see also Section 2.6.1.2). \nSemi-analytical models often tie the growth of MBHs to galaxy mergers (e.g. Kauffmann and Haehnelt, 2000; Marulli et al., 2008), or events of starbursts or bulge growth, assuming some \nform of MBH-galaxy co-evolution model (e.g. Somerville et al., 2008; Shirakata et al., 2019). Models which do not track the full evolution of the galaxy population, have modelled similar co-evolution with the velocity dispersion derived directly from the DM halo (Volonteri et al., 2003a) or estimating the velocity dispersion of the galaxy based on empirical relations (Ricarte and Natarajan, 2018a). In this way, it is assumed that some combination of fuelling and feedback produces M BH -host relations. Other models directly relate the growth of MBHs to the evolution of different gas phases or different dynamical processes, such as disc instabilities (Gaspari et al., 2017; Dayal et al., 2019; Izquierdo-Villalba et al., 2020). The gravitational torque-driven model introduced by Hopkins and Quataert (2010) has also been implemented in semi-analytical models, together with analytic models for disc instabilities (Menci et al., 2014; Gatti et al., 2015). \nOn accretion disc scales, one of the most important physical ingredients is the Eddington limit, the accretion rate at which radiation pressure balances gravity for a spherical accretor (as defined in Section 2.3.2.3). Typically, MBH accretion rates are capped at Eddington, and indeed the overall quasar population appears to obey this limit (e.g. Wu et al., 2015). However, there are several theoretical motivations to consider relaxing this assumption. First, MBH accretion does not occur spherically, but rather through an accretion disc. State-of-the-art radiative MHD simulations have demonstrated that Super-Eddington flow regimes can be sustained for many disc orbits (Jiang et al., 2014; McKinney et al., 2015; Sądowski and Narayan, 2016; Dai et al., 2018). In addition, the existence of 10 9 -10 M ⊙ quasars at z ∼ 6 requires optimistic duty cycles to grow from the seed mass if an Eddington rate cap is assumed, even under a heavy seeding scenario (see the discussion in Section 2.3.2). \nBeing able to resolve the full journey of the gas inflow from galaxy scales down to the nuclear region is essential not only to properly address MBH accretion, but also to study formation of circumbinary discs (see Section 2.2.2.2, and below). Spin evolution is also connected to the frequency and properties of the accretion process. We refer the reader to Section 2.3.2.4 for a discussion of the physical approaches, and we only recall here that coherent accretion leads to maximally spinning MBHs, whereas randomly oriented accretion can spin MBHs down. MBH spins also evolve during coalescence, depending on the combination of the orbital angular momentum and the two initial spins. The latter part of the evolution is included in cosmological simulations by adopting fitting formulae to GR simulations (Rezzolla et al., 2008; Lousto et al., 2010; Barausse and Rezzolla, 2009; Hofmann et al., 2016). Only few semi-analytical models (e.g. Volonteri et al., 2005; Barausse, 2012; Volonteri et al., 2013; Izquierdo-Villalba et al., 2020) and cosmological simulations (Dubois et al., 2014b; Bustamante and Springel, 2019; Trebitsch et al., 2020; Dubois et al., 2020) follow MBH spin self-consistently with a sub-grid model. \nImprovements on the modelling of MBH fuelling, tighter observational constraints on MBH accretion rates across a wide range of masses and redshift, and direct estimates of MBH spins (see the review of Reynolds, 2019) will help discriminating between different spin evolution models on the run up to LISA to sharpen predictions. The spin of MBHs also bears a relation to the energy that can be released through AGN feedback. MBHs with high spins are predicted to release more specific energy than MBHs that have low spins or are non rotating (Dubois et al., 2014b; Bustamante and Springel, 2019). Constraining the spin distribution of MBHs with LISA could help us to better constrain AGN feedback models, which we discuss below. \n2.4.1.3 MBH feedback Feedback from AGN is arguably one of the most important and still open aspects of MBH-galaxy co-evolution. AGN are short-lived phases of MBHs evolution, and release substantial amounts of energy in their surroundings. The feedback drives winds, outflows, and jets, which create large-scale X-ray cavities in clusters and groups of galaxies. However, even if there is observational evidence for the role of AGN in quenching star formation and cooling flows in the host haloes, the exact mechanisms of such energetic processes are still unclear today (e.g. Fabian, 2012; Gaspari et al., 2020). Indeed, the range of physical scales ( ∼ 9 orders of magnitude) tied to gas inflows and negative/positive feedback processes in the host \nmake this a very challenging problem to solve (see also Section 2.6.1.2). \nCosmological simulations often partition feedback into two modes, depending on the efficiency of the accretion on to the MBHs (e.g. Merloni and Heinz, 2008). During accretion phases characterized by high Eddington ratios and thin accretion discs, often called 'quasar' mode, AGN feedback is strongly ejective and radiatively driven, whereas during phases of lower accretion rates, often called 'radio' mode and characterized by a thick disc or chaotic cold accretion, AGN feedback is mostly driven via radio jets or sub-relativistic outflows that maintain the macro-scale gaseous haloes in quasi-thermal equilibrium for several billion years. The transition between the two states likely occurs around an Eddington ratio of ∼ 0.01-0.1 (Yuan and Narayan, 2014). Based on analytic arguments, the MBH mass-velocity dispersion scaling relation ( M BH -σ ) may be a direct consequence of how these wind powers ought to scale with the host properties to curtail further accretion (Haehnelt et al., 1998; King, 2003; Zubovas and King, 2012). \nThe modelling of AGN feedback is one of the aspects that needs to be improved in cosmological simulations. Bridging the scale gap is unfeasible, and thus many sub-grid models often invoke simple direct heating mechanisms (either central or as pairs of hot bubbles) to quench local star formation (e.g. Springel et al., 2005b; Sijacki et al., 2007; Vogelsberger et al., 2014; Dubois et al., 2015; Schaye et al., 2015). Other approaches have included injection of kinetic energy via jets or winds (Dubois et al. 2012; Choi et al. 2012; Gaspari et al. 2012; Barai et al. 2016; Bourne and Sijacki 2017; Weinberger et al. 2017; Wittor and Gaspari 2020) instead of or jointly with heating. \nWhile sub-grid models are rooted in physical insight, they are still not able to follow the full range of processes related to MBH accretion and star formation in the interstellar/circumgalactic medium. Therefore, future improvements should take into account magnetic fields, employ at least some approximate radiative transfer, and consider more realistic models of stellar feedback and of the clumpy, multi-phase interstellar medium (e.g. Hopkins et al., 2018; Marinacci et al., 2019). One key aspect is the connection between AGN feedback and the spin of MBHs. Given that LISA will provide us with spin distribution of the merging systems, this is a direction that we need to address in the coming years. The strength of AGN feedback scales with the radiative efficiency, which is closely tied to MBH spin. Therefore, a self-consistent treatment of AGN feedback should account for the effect of spin on radiative efficiency (as in Trebitsch et al., 2020; Dubois et al., 2020, for example), which could become feasible if the spin distribution will be robustly constrained by future GW datasets. \n2.4.1.4 MBH dynamics MBH dynamics is key to model LISA's MBH mergers. Between when a galaxy merger begins and the final MBHs coalesce, an MBH must complete a journey of many orders of magnitude in spatial scales. We refer the reader to Section 2.2 for a detailed account of the orbital decay mechanisms acting on different scales, and in varying astrophysical environments. However, most cosmological simulations are unable to follow the dynamics of the infalling MBH down to the scale where the MBH binary system can form, because of the trade-off between the maximum resolution achievable and the simulation volume. For example, large-scale cosmological simulations like Horizon-AGN (Dubois et al. 2014a; Volonteri et al. 2016), Illustris (Sijacki et al., 2015), and Eagle (Schaye et al., 2015), with spatial resolutions of about 1 kpc, cannot follow MBH dynamics down to the centre of galaxies. The kpc-scale regime can now be directly probed with smaller volume cosmological simulations, in which multiple MBHs are allowed to co-exist within the same galaxy, although reaching the required resolution is very challenging. Taking care to correct the dynamical friction force onto MBHs lost due to gravitational softening, Tremmel et al. (2018b) find a wide range of delay times between galaxy merger and MBH pairing in the Romulus cosmological simulation, which can impact GW event rates (Barausse et al., 2020b). Even higher resolutions can be instead achieved by means of zoom-in simulations or isolated galaxy mergers, that allow to resolve the dynamics down to a few tens of pc (Van Wassenhove et al., 2014; Bellovary et al., 2019; Pfister et al., 2019b). \nWhen MBHs become gravitationally bound, their orbit must still shrink, the hardening phase, before GWs can act to bring about coalescence (see Section 2.2.2 for a description of the physical processes). The binary hardening phase can not be resolved in cosmological simulations, thus assumptions for the estimate of the hardening time-scale need to be adopted in the post-processing analysis of hydro-simulations or in semi-analytical models. Delay times can be assumed to be fixed (DeGraf et al., 2020) or to depend on some simplified way on the properties of the host galaxy (e.g. Izquierdo-Villalba et al., 2020) and/or of the circumbinary disc (e.g. Kelley et al., 2017b; Volonteri et al., 2020; Sayeb et al., 2020). Finally, in any situation characterised by moderately long binary hardening times, triple or even multiple MBH systems are also likely to form in galaxies experiencing frequent mergers, and cosmological models should also take those into account (see, e.g. Rantala et al., 2017; Ryu et al., 2018; Bonetti et al., 2019). It is clear that in preparation for LISA, significant developments are needed to both semi-analytical models and simulations to improve how dynamics is treated and obtain convergence in the predicted rates and MBHB properties.", '2.4.2 State of the art on MBH merger rates from cosmological simulations': "In what follows, we discuss how the modelling and assumptions for the processes mentioned above affect the predicted rate and properties of LISA events. The range of predictions for the merger rate of MBHs that LISA can detect currently spans a wide range, from about one to several hundreds per year. The reason for this large span lies both in different physical assumptions and in the different techniques used. To give a rapid overview, in terms of physical modelling the merger rate is high when MBHs are abundant in galaxies, hinging on the efficiency of the MBH formation model adopted, and when the dynamical evolution is fast. Then, the rate decreases as one or the other of these assumptions is relaxed. The rate of mass growth also is important, as it determines the redshift at which MBHBs enter and exit the frequency range accessible by LISA. There are also other subtleties that enter the models. For instance, the spins and mass ratios of the binaries at the time of coalescence, which are determined by formation, growth, and dynamical evolution all together, influence the speed of recoil kicks (e.g. Peres, 1962; Damour and Gopakumar, 2006), which in turn modulate further the merger rate by ejecting MBHs from galaxies. Thus, not only the merger rates but also mass and spin distributions of merging MBHs depend sensitively on the specific assumptions of the models in terms of seeding, accretion, feedback, and spin evolution, which are largely unconstrained by available observational datasets. The details of these physical aspects are described in Section 2.2 and Section 2.3 of this paper and below we discuss how they influence the statistics of LISA's merging MBHs also in dependence of the technique used. \nThe techniques adopted also have a bearing on the resulting merger rate. The main parameter in this context is the mass resolution of the cosmological simulation or the DM merger tree used to build a model Universe. LISA's MBHs have masses in the range 10 3 -10 7 M ⊙ and can be hosted in haloes with stellar masses as low as 10 6 M ⊙ (Bellovary et al., 2019; Volonteri et al., 2020). If the resolution of the model does not allow to resolve low-mass haloes, the merger history of MBHs in these haloes cannot be tracked and therefore the MBH merger rate obtained will be a lower limit to the real merger rate. The volume of the simulation also matters, and ideally a large diversity of environments is needed to accurately derive reliable MBH merger rates that can provide sensible predictions in preparation for LISA. Obviously, both high resolution and large volume requirements increase the computational cost. Therefore, models are currently a compromise and not an ideal set-up. \nIn the next two sections, we review the existing predictions for LISA and critically discuss their physical assumptions and technical approaches. \n2.4.2.1 Cosmological hydrodynamical simulations Cosmological simulations are a recent addition to predictions for LISA's merger rates and properties of merging MBHs, which \nstarted with analytical and semi-analytical models about 15-20 years ago (Haehnelt, 1994; Sesana et al., 2005). For LISA, these simulations are, in principle, the best tool, since they incorporate, to some extent, all the processes regarding MBH formation and evolution, and this in the context of an evolving population of host galaxies computed from high to low redshift. As a result, cosmological hydrodynamical simulations have the advantage over isolated merger simulations in that they naturally include a variety of mass ratios, orbital configurations, and galaxy structures. For instance, in isolated merger simulations the minimum mass ratio of a galaxy for MBHs to bind within ∼ 1 Gyr seems to be > 0 . 25 (Callegari et al., 2009; Van Wassenhove et al., 2014; Capelo et al., 2015), but cosmological simulations show (i) that also galaxy mergers with lower mass ratio contribute to the MBH merger rate (Tremmel et al., 2018b; Volonteri et al., 2020) and also (ii) the effect of irregular potentials in high-redshift and dwarf galaxies (Pfister et al., 2019b; Bellovary et al., 2019; Bortolas et al., 2020). \nCosmological hydrodynamical simulations have for the most part focused on the merger rates and mass ratio distributions of the merging events (Salcido et al., 2016; Katz et al., 2020; DeGraf and Sijacki, 2020; Volonteri et al., 2020). Spin has been for now included in post-processing in GW-related studies (Sayeb et al., 2020): although some simulations with spin evolution exist (Dubois et al., 2014b; Bustamante and Springel, 2019; Trebitsch et al., 2020; Dubois et al., 2020), for the moment the spins of merging MBHs has only been investigated in post-processing (Sayeb et al., 2020). \nMost cosmological simulations used to investigate statistically merging MBHs are largevolume ( ≥ 100 3 Mpc 3 ), low-resolution (DM particle mass ∼ 10 7 -10 8 M ⊙ , with the proviso that ∼ 50-100 particles are required to identify a halo; star particle mass ∼ 10 6 M ⊙ , spatial resolution 0.4-1 kpc) simulations (Salcido et al., 2016; Katz et al., 2020; DeGraf and Sijacki, 2020; Volonteri et al., 2020). However, such simulations are not suited for studying MBHs in the LISA mass range. Large volume is a positive aspect, improving statistics and capturing various environments in the large-scale structure. Mass resolution, as noted above, is a key point. LISA's MBHs have masses in the range 10 3 -10 7 M ⊙ , with some MBH formation models predicting MBHs with mass ∼ 10 4 M ⊙ in haloes with mass as low as 10 8 M ⊙ (see Section 2.3.2). Therefore, such low-mass haloes must be resolved in order to capture the full merger rate of LISA's MBHs. Most of the MBHs in well-resolved galaxies in low-resolution simulations are simply too massive and therefore merge outside the LISA band, at lower frequencies (they are better suited for PTA experiments, Kelley et al., 2017b). This means that we have to be aware that the merger rates predicted by current simulations - generally < 1 per year - could be a lower limit. \nVolonteri et al. (2020) present the first analysis of the merger rate and merging MBH properties in a high-resolution simulation ('NewHorizon', DM particle mass ∼ 10 6 M ⊙ , star particle mass ∼ 10 4 M ⊙ , spatial resolution 0.04 kpc) with a sufficiently large volume (a sphere of radius ∼ 10 cMpc) to have some statistics, while Bellovary et al. (2019) simulate a number of isolated dwarf galaxies at somewhat lower spatial resolution but higher mass resolution and Khan et al. (2016) simulate one single galaxy at similar resolution in a sphere with radius 13.5 kpc (they then extract and resimulate further the central nucleus of the galaxy at higher resolution, but without hydrodynamics). Volonteri et al. (2020) analyze in the same way a high-resolution, small-volume simulation and a low-resolution large-volume simulation and show explicitly that indeed the merger rate of the former is higher. This is because dwarf galaxies are resolved, and there are more dwarf galaxies than high-mass galaxies in the Universe. Furthermore, a significant fraction of dwarf galaxies host MBHs: in NewHorizon at z ∼ 0 . 5 , about 10 per cent of galaxies with mass 10 6 M ⊙ host an MBH, increasing to 100 per cent at 10 9 M ⊙ . Observationally, between 10 and 100 per cent of galaxies with mass ∼ 10 9 -10 10 M ⊙ at z = 0 appear to host an MBH (Greene et al., 2019), implying that MBH mergers in dwarf galaxies are indeed crucial for the low-mass MBHs relevant for LISA. Similar results have also been found by Bellovary et al. (2019), where, in addition, MBHs typically appear off-centred relative to the host. \nBesides mass resolution, how MBH dynamics is treated in simulations is also important, and \n<!-- image --> \nFigure 24: Comparison of merger rates from different cosmological hydrodynamical simulations, with (bottom) and without (top) the addition of a delay in post-processing. No SNR LISA cut has been applied: this is the merger rate of all MBHs independent of whether LISA can detect them or not, e.g. most MBHs in low-resolution simulations are too massive to enter the LISA band. NewHorizon and Horizon-AGN (Volonteri et al., 2020) include intrinsic delays from dynamical friction from gas, and additional below-resolution delays (bottom panels), and model MBHs above 10 4 and 10 5 M ⊙ , respectively. Illustris (Katz et al., 2020), with M BH ∼ 10 5 M ⊙ , does not implement any intrinsic delay and adds (bottom panel) physically motivated delays in post-processing. Romulus (Tremmel et al., 2018b), where M BH > 10 6 M ⊙ , includes intrinsic delays from dynamical friction from particles, and a fixed 0.1 Gyr below-resolution delay. Eagle (Salcido et al., 2016) seeds MBHs with M BH ∼ 10 5 M ⊙ and does not include any intrinsic delay, adding in post-processing fixed delays of 0.1 Gyr for gas-rich galaxies and 5 Gyr for gas-poor galaxies. Figure credit: Marta Volonteri. \n<!-- image --> \nthis circles back to spatial resolution. Typically, large-scale cosmological simulations do not have the sub-pc resolution needed to resolve MBH dynamics. Two approaches have been used in the literature. \nThe first is to not treat explicitly MBH dynamics: MBHs are repositioned at each timestep at the position of the lowest potential (gas) particle near the MBH. In a merger, an MBH would very rapidly be moved to the centre of the potential well, on time-scales much shorter than in reality. As a consequence, the merger rate of MBHs is increased. Some studies do not consider delays (DeGraf and Sijacki, 2020), in others delays have been added in post-processing, either using a fixed value (Salcido et al., 2016) or adopting a physical approach (Katz et al., 2020). (Salcido et al., 2016; Katz et al., 2020). \nThe second approach is to include sub-grid dynamical friction from gas (Dubois et al., 2013), from stars and DM (Tremmel et al., 2015), or all of the above (Pfister et al., 2019b). Adding dynamical friction in the code, however, poses an additional challenge: the ratio of MBH mass to the mass of star particles (or unsmoothed DM particles) must be > 10 to avoid spurious oscillations. This means a very challenging computational task when MBHs have mass < 10 5 M ⊙ . On-the-fly dynamical friction helps in having realistic dynamics down to the resolution of the simulations: the force acting on the MBHs captures the inhomogeneous, time-varying density distribution and irregular potential wells where MBHs, especially at high redshift, evolve. Still, this approach operates only down to the spatial resolution of the simulation, which is ∼ 10-50 pc in high-resolution cosmological simulations and 0.3-1 kpc in low-resolution simulations. Below this scale, statistical studies of MBH mergers can only rely on adding additional time-scales of binary evolution (stellar hardening, torques in circumnuclear discs and circumbinary discs; see Section 2.3.1) in post-processing (Katz et al., 2020; Volonteri et al., 2020; Sayeb et al., 2020), although there are prospects for a full on-the-fly treatment (Rantala et al., 2017). \nAn important point is that for the moment the mass ratio of merging binaries is based either on information obtained long before the MBH mergers (before including the dynamical delays) or on specific choices applied in post-processing (Sayeb et al., 2020), which may or may not capture how each of the MBHs grows in mass during the final phase of dynamical friction and during the hardening and circumbinary disc phase. Moreover, the limited resolution limits the ability to self-consistently follow the tidal stripping of the galaxy nucleus during the dynamical friction phase, and this affects the orbital decay. A comparison of the predictions obtained by different state-of-the-art simulations is reported in Fig. 24, with (bottom panel) and without (top panel) the inclusion of a post-processed delay between the time when MBHs merge in the simulation and the estimate of the coalescence time taking into account the expected, but unresolved, physical processes. \n2.4.2.2 Analytical and semi-analytical models Several studies have developed analytical and semi-analytical models to predict merger rates and chirp masses for LISA, with various assumptions on the main seeding mechanism for MBHs. Most of these studies pre-date the use of cosmological hydrodynamical simulations in the context of LISA, and have paved the ground for the latter. The predictions of these models can vary significantly, mostly because the physics of the formation and of the orbital shrinking of the MBHBs are thus far loosely constrained, although some advancements have been recently put forward. Analytical and semi-analytical models suggest that different seed populations have a different impact on the total number and mass distribution of potential LISA sources at different cosmic epochs (see, e.g. Volonteri et al., 2003a; Sesana et al., 2011a; Barausse, 2012; Klein et al., 2016; Bonetti et al., 2019; Dayal et al., 2019; Barausse et al., 2020b; Katz et al., 2020; Valiante et al., 2021). Generally speaking, all the models converge on predicting that the merger rates are significantly higher if seeding occurs mainly with light seeding mechanisms, e.g. MBHs are formed as remnants of Pop III stars, with a typical mass ≲ 10 3 M ⊙ (see, e.g. Ferrara et al., 2014; Valiante et al., 2016; Pacucci et al., 2018, for a description of initial mass functions for light and heavy seeds). Specifically, Ricarte \nand Natarajan (2018b) predict that LISA will observe ∼ 20 times more events if seeding occurred mainly from light seeds, with an upper limit of ∼ 300 events (over a 4-year mission duration) with a typical mass ∼ 10 3 M ⊙ in the most optimistic scenario. Similarly, Dayal et al. (2019) predict that light-seeding scenarios will drive the merger rates up, ending with a more conservative prediction of 12-20 mergers during a 4-year mission duration. Even when light and heavy seeds are combined in the same cosmological evolution history, as in Dayal et al. (2019) and Valiante et al. (2021), the number of predicted LISA events is dominated by (growing) light seed binary mergers, although the impact of feedback (reionization, SNae, AGN) by suppressing MBH growth or hindering dynamical friction, reduces the importance of the mergers of light and heavy seeds (Dayal et al., 2019; Barausse et al., 2020b; Li et al., 2020b). Notably, what drives significant differences in predictions is the probability that MBHs actually coalesce, once their host galaxies have merged (see a broad description of the issue in, e.g. Inayoshi et al. 2019 and in Section 2.2). Bonetti et al. (2019) predict a rate of ∼ 25 and ∼ 75 LISA events per year, respectively, in heavy and light seeding models, which is reduced to ∼ 10-20 yr -1 if MBHB mergers are efficiently driven only via triple interactions (i.e. if gas/stellar-driven shrinking mechanisms were to fail in driving the binary to coalescence). In addition, as the GWs emitted during the coalescence phase carry linear momentum, also the inclusion of gravitational recoil can impact the halo occupation fraction, hence the merger rates (see, e.g. Haiman, 2004; Tanaka and Haiman, 2009; Inayoshi et al., 2019; Izquierdo-Villalba et al., 2020). \nA comparison of the prediction by different semi-analytical models is reported in Fig. 25, for light seeds (bottom panel) and heavy seeds (top panel). In general, the predicted event rates span a wide range, from no detection to a few hundred events, depending on the adopted description of the multi-scale and complex processes regulating seed MBH formation, mergers, accretion, and dynamics, which are far from being fully understood. It is therefore important, to reliably predict the rate of MBH coalescences alongside the hierarchical assembly of galaxies, to get full control of the assumptions made to describe these processes, on different scales/times (see, e.g. Enoki et al., 2005; Sesana et al., 2011a; Klein et al., 2016; Tamanini et al., 2016; Ricarte and Natarajan, 2018a,b; Bonetti et al., 2019; Dayal et al., 2019; Volonteri et al., 2020; Barausse et al., 2020b; Valiante et al., 2021, and Sections 2.2 and 2.4). \nFrom a statistical point of view, LISA detections (or non-detections) may reflect more the dynamical properties and evolution of binary MBHs (i.e. their ability to form and merge) rather than their origin. For instance, heavy seeds are expected to form binaries more efficiently than the more common light seeds. Therefore, a low number (or even the lack) of detections of highredshift sources in the LISA band may indicate that heavy seeds are very rare and/or that they are not able to merge, after binding in binaries (because of inefficient hardening mechanisms in their host galaxies).", '2.4.3 How to advance and optimize the scientific return of LISA': "As we have seen above, predictions for LISA events depend in a complicated way on a large number of assumptions, from the seed mass to spin evolution and the dynamics of binary systems. In turn, these aspects are tightly linked to the properties of the host galaxies and of the environment. \nThe interplay between all these different non-linear physical processes leads to predictions for the merger rates that are highly degenerate. \nLearning about spin evolution, merger time-scales, accretion physics, and seed masses from the merger rates of LISA requires a data analysis process where the multi-dimensional parameter space can be quickly explored. By the time LISA launches, the community needs to be ready with a comprehensive and flexible set of theoretical models that can be efficiently confronted with the data. \nNumerical simulations of small-scale physical processes will need to be connected to simulations that trace the full cosmological evolution of structures and they will need to inform \n<!-- image --> \nFigure 25: Comparison of merger rates from different semi-analytical models, assuming heavy seeds (top panel) and light seeds (bottom panel). For all models, we employed the Science Requirement curve (Babak et al., 2021) applying an SNR cut of 8. Different assumptions for models by Barausse et al. (2020b) are shown, with or without SN feedback, and including or not delays. Dayal et al. (2019) include reionisation feedback and delays, whereas Ricarte and Natarajan (2018b) do not include delays. The still large uncertainties in the modelling result in significant variations, up to two orders of magnitude, with mergers between light seeds typically dominating the event rate, but for the case when SN feedback is included, as in Barausse et al. (2020b). Figure credit: Marta Volonteri. \n<!-- image --> \nanalytical or semi-anaytical models, that can scan and test the parameter space efficiently. This is central to quantify robustly the mapping between galaxy mergers and MBH mergers. Furthermore, we will need to understand which classes of galaxies, and in which environment, LISA events are most likely hosted. This aspect will be vital if an EM counterpart of a GW merger is to be discovered. Given the current capabilities of LISA to localize nearby sources (Mangiagli et al., 2020, and section 2.5.2), from a few to a few hundreds of galaxies could be in the field of view of, e.g. X-ray telescopes, depending on the loudness of the source. Thus, anticipating the characteristic properties of the host galaxies from simulations will help identifying the host and its redshift. \nIn the remaining of this section, we will give a brief outlook on the expected advances in 'traditional' techniques and on the possibility of using new statistical methods, and how those will be used to inform the LISA data processing. \n2.4.3.1 Improvements on current techniques An important role in building the theoretical framework will be played by the transition to the exascale computing, that will allow us to develop simulations of much larger portions of the observable Universe, of order comoving Gpc 3 (either as single simulations or as several smaller-volume ones targeting different environments), compared to the current ones (limited to a few hundreds of comoving Mpc), and to further increase the resolution in order to resolve ever smaller scales currently achieved only via dedicated idealised studies. The combination of ultra-large simulated cosmological volumes and very high resolution is the best strategy to enable astrophysical inference studies with the LISA datastream because the properties of the MBH binaries that enter the LISA band are determined by both large scale and small scale processes, which demands both large volumes and high resolution, and because LISA is an all-sky instrument that probes the Universe from low to very high redshift, which again calls for very large volumes. However, exascale computing is not a guaranteed solution, since even the best cosmological hydrodynamic codes available today are far from being able to scale on the billion-core platforms that will characterize such a computing phase over the next decade, when the resolution is increased beyond a certain threshold. This is because in simulations with very high resolution, reaching below tens of parsecs, load balancing on large core counts becomes a computational bottleneck which, unfortunately, gets worse as the number of computing cores is increased. This is an intrinsic challenge with modelling the non-linear process of gravitational collapse. Thus, unless a quantum leap occurs in the parallel architecture of simulation codes, for example owing to improvements in task-based parallelism, the larger cosmological volumes will still be limited in their ability to capture the small-scale stellar and gaseous processes (at sub-pc scales) that drive the hardening phase. This is why simulations will always need to be complemented by other techniques. Different techniques will also require different improvements, and should be combined together to exploit the respective advantages, i.e. the speed and the easy parameter exploration of semi-analytical models and the spatial information of hydrodynamic simulations. \nOn the side of semi-analytical models, more sophisticated and comprehensive assumptions for MBH seeding and the dynamics of binaries and multiplets will have to be included. Highresolution small-scale numerical simulations covering a wide range in the parameter space will be needed to create new parametric prescriptions for these physical processes. Moreover, semianalytical models can be combined together to offer a wide dynamic range: Press-Schechter-based models can be combined with models based on N -body simulations. Also, N -body simulations of different mass resolution and cosmological volume can be combined together. \nFor hydrodynamic simulations - including the so-called 'zoom-in' cosmological simulations which probe small volumes, often a single galaxy - the larger computational power will also allow to increase the resolution, reaching scales currently achievable only by dedicated simulations with idealized boundary conditions (down to sub-pc and AU scales). Since it is already clear from the vendors' strategic plans that exascale platforms will have 'fat nodes' with at least 128 cores, these \nsimulations, which are much smaller for number of compute elements compared to cosmological volumes, could fit on just a few nodes, partially resolving the load balancing issue mainly caused by communication between nodes. This also means that, with exascale computing, many more zoom-in simulations could be run with significantly less resources than today, allowing to probe a larger fraction of the parameter space. \nFor these improvements to be effective, a strong effort aimed at improving current sub-grid models of MBH formation, growth, and dynamics, and including new physical processes (e.g. magnetic fields, cosmic rays, non-equilibrium chemistry, radiation transport, and GR effects) is required. Furthermore, increasing resolution would ease the need for simplified prescriptions or post-processing models, but resolution cannot be increased ad libitum and a treatment of small-scale phases needs nevertheless to be added to simulations, based on the detailed results of smaller-scale simulations. This combination of different scales will be key to properly estimate the MBH spins, masses, and dynamics of MBHBs, and therefore the subsequent cosmic evolution of MBHs and sharpen predictions for, and intepretation of, LISA detections.", '2.4.3.2 New methodologies: artificial intelligence integrated with simulations De-': "spite the foreseen progress in simulations with the advent of exascale computing, the parameter space potentially probed by LISA will always be too large to be explored at the resolution needed to capture all the effects that determine the time-scales and occurrence rate of MBH mergers. Such effects represent a truly daunting computational challenge of global models. Stochastic processes play a role throughout, which implies that to derive truly robust quantitative models for LISA predictions, e.g. on the mapping between galaxy and MBH mergers, one would need to run a very large sample of simulations. Stochasticity applies both to scales that might be directly resolved in the next generation of cosmological hydrodynamic simulations (10-100 pc) and to scales that will not be resolved for long (parsec scales and below, into the circumbinary disc regime). While semi-analytical models could be used in complement, the complex dependencies of torques/drag regimes on the interstellar medium properties and the stochastic nature of the processes themselves conceptually speak against the use of deterministic phenomenological recipes, which is instead the standard approach of semi-analytical models. \nAn alternative, promising avenue, which is gaining increasing momentum in observational cosmology and in the analysis of large-scale structure statistics, is artificial intelligence (e.g. Fluri et al., 2019; Tsizh et al., 2020; Li et al., 2020c). This often entails using neural networks of varying complexity to recognize correlations and patterns, and subsequently produce many realizations of a given model (model emulator technique). One particular interesting class of such neural networks are generative adversarial networks. Such networks are at the base of modern facial recognition algorithms, which are becoming increasingly sophisticated, and are thus designed to work with an extremely large parameter space (each facial feature can be cast as a parameter, essentially). The networks are designed in such a way that they can be continuously updated to recognize deeper features and patterns without retraining, thus essentially allowing to tune the response based on the needs (namely based on the target, which would be determined by the scientific application). One can imagine training such algorithms to identify complex interstellar medium patterns and to correlate them with orbital decay regimes/time-scales for MBHs. Training would of course have to be done on small-scale simulations (non-cosmological, galactic and nuclear scale). For example, a first application of such techniques to galaxy dynamics is the morphological identification of merging versus isolated galaxies (Goulding et al., 2018; Nevin et al., 2019; Snyder et al., 2019; Pfister et al., 2020), which is becoming increasingly common in these years. An emulator of the 'small-scale dynamics' could be then be designed by integrating the sub-grid model computed via neural networks within a large-scale simulation, using a zoom-in simulation as intermediate step, to encapsulate their trends and results, and implant them in simulations of large cosmological volumes.", '2.4.3.3 Summary of LISA measurable quantities and how it will inform us on MBH physics': "- · LISA can determine the mass of merging MBHs at any time. We expect LISA to discover MBHs closest to the redshift of their formation. At such high redshift, the emitted radiation of these MBHs (that are likely low-mass objects) is too faint to be detected by current EM missions. On single events, the detection of MBHs with M MBH ⩽ 10 5 M ⊙ would confirm the existence of light MBH seeds, while not ruling out the existence of heavy ones (see Section 2.3.1 for a description of MBH formation channels). The detections of MBHs with M MBH ⩾ 10 5 M ⊙ cannot, however, validate the existence of heavy seeds as the MBHs could be grown light seeds (except if many of these detections take place at very high redshift). In case of a sufficiently large number of events, LISA will provide us with constraints on the most likely dominant MBH formation channels, as well as the first constraints on the low-mass end of the MBH mass function from low to high redshift.\n- · LISA can measure the effective spin of merging MBHs (see Section 2.3.2.4). Posterior distributions of the spins will be used to determine the spins of the two merging MBHs. For single events, it provides us with information on the dominant nature of the growth of these MBHs, i.e. whether their accretion histories were chaotic or coherent (in other words, whether MBH growth is accretion- or merger-dominated). Spin distributions for the population of MBHs detected by LISA will constrain the relative contributions of MBH growth channels as a function of MBH mass and redshift (see Sections 2.3.2.2, 2.3.2.3, and 2.3.2.4). In particular, spin measurements are likely going to be possible up to very high redshift ( z ∼ 10 ) with per-cent precision for nearly a third of the detections (Klein et al., 2016).\n- · LISA will measure on the full sky the merger rate of MBHs in the mass range 10 4 -10 7 M ⊙ . First, the observations of MBH mergers would be the evidence that these BHs dynamically pair and merge within relatively short time-scales, especially if observed at high redshift. Second, the merger rate of LISA will constrain a combination of MBH physical characteristics (MBH seeding, MBH dynamics, efficiency of MBHs to sink to galaxy centers, MBH growth) and characteristics of their host galaxies (Section 2.4.2). LISA will constrain the number density of merging MBHs, independently of their activity. As such, LISA could enable new investigations of the fraction of obscured AGN (by e.g. comparing LISA results to current and future AGN surveys).\n- · Localization of the LISA events on the sky will be crucial to enable multi-messenger science towards a full characterization of MBH physics and demographic evolution of MBHs. Among many new potential directions, LISA could open a new window on the origins of gamma-ray bursts (Section 2.5.1.3), jet (Sections 2.5.1.3 and 2.5.1.4) and cosmic ray astrophysics (Section 2.5.1.4), and MBH accretion (Section 2.4.1.2). Localization of the events in space and time could also help linking merging MBHs to their galactic and larger-scale environments and further disentangle MBH formation and growth channels as well as MBH and galaxy co-evolution (e.g. Section 2.6.1.2).\n- 2.5 Multimessenger on single events: What do we learn about BH physics from the multimessenger view of the coalescence of MBHs? \nCoordinators: Ioana Duţan, Delphine Porquet \nContributors: Imre Bartos, Tamara Bogdanovic, Federico Cattorini, Maria Charisi, Monica Colpi, Alessandra De Rosa, Daniel D'Orazio, Massimo Dotti, Massimo Gaspari, Alberto Mangiagli, Sean McGee, Vasileios Paschalidis, John Quenby, Milton Ruiz, Jessie Runnoe, Antonios Tsokaros, Rosa Valiante, Maurice van Putten, Silvia", 'Zane': 'The scientific exploitation of the LISA mission would be greatly increased by performing synergistic, multimessenger observations; that is, combining low-frequency GW observations by LISA with contemporary, prior, or follow-up observations of the same source by EM and astroparticle messengers. The overall goal of this section is to highlight the multimessenger view of single collisions between MBHs detected by LISA in the astrophysical environment posed by their host galaxies. We start this section by presenting the expected multimessenger signatures of coalescing MBHs (precursor, coincident, and afterglows observations). We then elaborate on the best observational strategies to maximize the multimessenger observations. Moreover, we present different inputs on what we need to prepare to improve estimations of the source parameters (e.g. sky position, luminosity distance, chirp mass, and mass ratio). Finally, at the end of this section, we present what is needed in the near future to maximize the scientific returns of LISA. In this section, particular attention is also given to the synergy between the LISA and Athena 10 missions, both of which will operate at the same time.', '2.5.1 The expected multimessenger signatures': "The stages which precede and follow the merger of an MBHB feature different spacetime geometries, and the ability to simultaneously detect both the GW and EM signals during each step differs as well. We distinguish between the pre-merger (late inspiral) phase, that could lead to the detection of an X-ray precursor signal , and the post-merger phase, that could lead to disc rebrightening, the formation of an X-ray corona , and that of an incipient jet . This subsection covers first pre-merger signatures, and subsequently possible signatures during merger and postmerger . There are then additional opportunities of multi-messenger observations associated with potential precursor objects of MBHs themselves, such as SMSs. \n2.5.1.1 Expected EM signatures of MBHB in-spirals at sub-pc scales In order to maximize the synergy between contemporaneous LISA and EM observations on single MBHB coalescence events, an understanding of the pre-merger population of MBHBs at sub-pc scales using EM observations is crucial. This section focuses on searches for MBHBs that are being carried out at the present time, and that can inform us of the expected LISA merger rate and possibly the expected orbital parameter distributions at merger time. The power of these predictions, however, will rely on how close to merger we can probe an EM identifiable MBHB population. After LISA detects MBHBs, interpretation of formation and evolution channels will rely on the characterization of EM identified populations. In such a case, population samples over the widest possible range of MBHB orbital parameters will be useful in piecing together the entire life stories of MBHBs. \nWhile multiple methods for EM identification of a population of MBHBs have been proposed and practiced over the last two decades (for more details, see Section 2.6), there is currently no definitive observational evidence for MBHBs with separations of order one parsec or smaller. Hydrodynamical simulations of circumbinary accretion show that the accretion rate onto an MBHB can be strongly modulated at multiples of the orbital periods (Haiman et al., 2009; MacFadyen and Milosavljević, 2008; D'Orazio et al., 2013). This has led to searches for sub-pc separation MBHBs manifesting as O ( yr ) time-scale periodicity in quasar light curves. Of order 100 such candidates exist to date (Graham et al., 2015; Charisi et al., 2016; Liu et al., 2019b). However, distinguishing the periodicities from the noise processes intrinsic to AGN variability remains a significant challenge (e.g. Vaughan et al., 2016; Zhu and Thrane, 2020). Signatures unique to MBHBs, with which to vet these periodic quasar candidates, have been proposed \nthrough the relativistic Doppler boost and binary self-lensing models for periodic variability and flares (D'Orazio et al., 2015; D'Orazio and Di Stefano, 2018; Hu et al., 2020; Charisi et al., 2018). \nMost of the effort has been focused on the exploration of large optical spectroscopic surveys (e.g. SDSS) using several approaches, searching for: \n- · large velocity differences between the narrow and broad emission lines, tracing the host galaxy and at least one of the two MBHs, respectively (Tsalmantza et al., 2011; Eracleous et al., 2012; Decarli et al., 2013; Liu et al., 2014; Runnoe et al., 2015, 2017),\n- · a time varying shift of the broad emission lines, tracing the highly accelerated motion of one of the two MBHs in a binary (Ju et al., 2013; Shen et al., 2013; Wang et al., 2017; Guo et al., 2019), or\n- · peculiar ratios between broad emission lines with different ionizing potentials due to the tidal effect of the other component of the candidate binary (Montuori et al., 2011, 2012). \nIf any of these systems are true MBHBs, then the modelling of their broad optical emission lines can in principle yield the properties of the binary, such as the minimum mass, separation, and mass ratio (e.g. Nguyen and Bogdanović, 2016; Bon et al., 2016; Runnoe et al., 2017; Nguyen et al., 2019, 2020). It is important to mention, however, that emission-line features mentioned above are not unique to MBHBs. As a consequence, searches like this can generate relatively large samples of MBHB candidates whose nature must be tested through continued follow-up or with help of other complementary observational techniques. For example, in the case of SDSS J0927+2943, multi-wavelength follow-up observations disproved both the binary and recoilingMBH hypotheses (Decarli et al., 2014). \nBecause they rely on the existing EM spectroscopic surveys, searches of this type are generally biased toward active MBHB candidates with masses ≳ 10 6 -7 M ⊙ and orbital separations ≳ 0 . 01 pc (Pflueger et al. 2018; Xin and Haiman 2021). Similarly, they are sensitive to MBHBs at redshifts z ≲ 1 -2 (Montuori et al., 2011, 2012; Nguyen and Bogdanović, 2016). Therefore, because of the observational selection effects, these widely used techniques may uncover a fraction of MBHB systems that are progenitors to binaries in the LISA frequency band but will not be detected by LISA because the coalescence time-scale is too long. These same techniques will miss low-mass and high-redshift systems, as well as the systems that do not show AGN activity intense enough to allow for a proper modelling of the broad emission lines. \nMore promising from the standpoint of the coincidental multimessenger detections are MBHBs with smaller orbital separations than those discovered by optical spectroscopic searches (the latter provides signatures too weak to detect if separation distances are smaller than the typical scales of the broad line regions). One can in principle search for such MBHBs using the broad iron fluorescence emission lines, observed at about 6.4 keV in the X-ray spectra of many individual AGN with masses as low as ∼ 10 6 M ⊙ (e.g. Reynolds, 2014). The broad iron emission lines are emitted by the parts of the accretion flow in the close proximity of the MBH (within ∼ 10-1000 gravitational radii). They can therefore trace the relative motion of the two MBHs even when they are well within the LISA band (Sesana et al., 2012; McKernan and Ford, 2015; Severgnini et al., 2018), as long as at least one of the MBHs exhibits observable AGN activity (see the discussion in Section 2.2). However, their current observations are limited to redshifts significantly smaller than those of the expected bulk of LISA MBHB coalescences, indicating a need for a high-sensitivity X-ray detector that will be able to measure broad iron emission lines in the spectra of AGN at higher redshift, such as the Athena mission. Additional signatures include modification of the disc emission caused by the presence of a gap, 'suppressing' emission from an annulus in the multi-colour black-body model, and shocks caused by matter hitting the minidiscs detectable in X-rays (Sesana et al., 2012; Roedig et al., 2014). In all cases, large enough signal-to-noise ratios (necessary in order to discriminate between single and double MBHs) will require long integration times, comparable to or longer than the orbital period of binaries in the LISA band. \nWhat about directly imaging and tracking the orbits of many MBHBs at sub-pc separation in the near future? Advances in Very Long Base Inteferometry (VLBI) at mm-wavelengths should make the direct imaging possible, and this would definitely be very complementary to the indirect methods described above. For example, the Event Horizon Telescope (EHT; Event Horizon Telescope Collaboration et al., 2019) has the angular resolution and sensitivity to astrometrically track the orbits of MBHBs separated by 0 . 01 pc at Gpc distances. Simple MBHB population models suggest that near-future mm-VLBI experiments could directly image and track the orbits of many such MBHB in, and possibly before, the LISA era (Johnson et al., 2019; D'Orazio and Loeb, 2018). \nThe next generation Very Large Array (ngVLA) will be added to this effort, with the ability to resolve MBHB pairs down to sub-10 pc separations and also track binary orbits through changing pc-scale jet morphology (Burke-Spolaor et al., 2018). \nIn summary, to prepare for LISA we must invest in theoretical understanding of accretion flows around MBHBs: \n- · to better understand what drives MBHB orbital evolution, and hence generate more accurate predictions for MBHB populations,\n- · to more accurately predict observational signatures generated before, during, and after merger, that we can reliably disentangle from AGN variability associated with single MBHs - this partly requires understanding such intrinsic variability better as well,\n- · to better model mm-wavelength emission from MBHB accretion for direct EM detection prospects. \nOn the observational side, we must: \n- · continue to extend time-domain surveys to longer baselines, in order to mitigate falseperiodicity detections due to AGN red noise,\n- · improve our understanding of intrinsic AGN noise processes and quasi-periodic oscillations,\n- · improve methods for detecting non-standard periodicity (e.g. variable accretion and selflensing induced periodic flares),\n- · advance MBHB-related science goals for VLBI experiments that could directly image MBHB orbits. \n2.5.1.2 Expected EM counterparts during the late inspiral and merger stages The multimessenger detection of MBHB inspirals and mergers will certainly establish unique breakthroughs in various fields of physics and astrophysics; yet, one shall be mindful of a series of caveats that make the concurrent observation of this class of events uncertain. Besides lacking firm predictions on the EM light-curves and spectra of coalescing MBHBs under a variety of conditions (see Section 2.5.3.1), the structure and properties of the astrophysical environment around MBHBs are uncertain as well and largely depend on the supply of gas for accretion in the aftermath of a galactic merger. \nGenerically, MBHBs can be surrounded by a circumbinary disc, and mini-discs can form around the two black holes (see Section 2.2.2.2). The accretion of mini-disc gas onto each MBH is expected to produce copious amounts of X-ray radiation. Analogously to what was discussed in Section 2.5.1.1, the orbital motion of the binary may imprint a modulation to the expected X-ray emission thanks to Doppler boosting or modulations in the accretion rate. The modulation is expected to be in phase with the GW incoming signal, allowing the correct identification of the host galaxy in the relatively large area provided by LISA (Haiman, 2017; Tang et al., 2018; Dal Canton et al., 2019). After the identification, alerts could be sent to other facilities in order to observe the very prompt emission. \nDynamical GR simulations of MBHBs in the force-free limit, which assumes that the plasma around the BHs is tenuous, suggest that two separate jets during the inspiral, one around each BH, could emerge from these systems (Palenzuela et al., 2010c; Mösta et al., 2012), providing a complementary way to search for MBHBs in the late inspiral phase. \nWhat happens at the time of merger is an active subject of research. The natal kick imparted by the GW recoil affects the properties of the accretion disc leading to modifications in the spectrum and light curve that can be non-universal depending on the orientations of the kick relative to the orbital plane pre-merger (Schnittman and Krolik, 2008; Rossi et al., 2010). The birth or rebrightening of a jet (Gold et al., 2014b; Khan et al., 2018a) are also possible outcomes. These studies have revealed new possibilities for EM counterparts from binary BHs that arise from jets in binary AGN. As a result, both non-thermal X-ray and gamma-ray signatures from these systems are expected, which would be of interest to Athena as well as other X-ray and gamma-ray satellites. \n2.5.1.3 Possible GW and EM signatures of MBH formation from the collapse of supermassive stars A widely accepted model of long gamma-ray bursts, with a typical duration of ∼ 30 s, is the so-called collapsar scenario. In this model, a BH accretion disc system forms after the core-collapse of a massive low-metallicity star, and launches a relativistic jet. The jet breaks through the stellar debris producing gamma-rays (Narayan et al., 1992; Paczynski, 1986; Woosley, 1993). \nSupermassive stars can be responsible for the formation of MBH seeds (see Section 2.3.1), hence they could have played a crucial role in generating the population of MBH binaries that LISA can detect out to very high redshift. If they were common at z > 10 , then a directcollapse population of high-redshift MBH binaries would have been prominent, leading to a very different population of GW sources detectable with LISA at high redshift relative to the case of light MBH seeds originating from Pop III stars (see Section 2.3.1). Hydrodynamic simulations in Shibata et al. (2016) found that the collapse of a ≳ 10 5 M ⊙ massive star at redshift z = 3 emits GWs, with a peak amplitude of 5 × 10 -21 at a frequency of ∼ 5 mHz . These GWs may be detectable by LISA (see also Liu et al., 2007; Sun et al., 2017, 2018). Simulations also found that after ∆ t ≈ 2000( M MBH / 10 6 M ⊙ ) s following the MBH formation a magnetically-driven jet is launched. The jet has a lifetime ∆ t ∼ 10 5 ( M MBH / 10 6 M ⊙ ) s, and the outgoing Poynting luminosity is L EM ∼ 10 51 -52 erg s -1 (Sun et al., 2017, 2018). These engines can shine for very long times compared to standard gamma-ray bursts and could be detected as ultra long gammaray bursts. The combination of GW and EM signals could help us constrain the origins of GRBs and MBHs. \n2.5.1.4 Expectations from astroparticle observations The most likely origin of cosmic rays of energy above 10 15 eV is in the jets of AGN. Shock acceleration is the popular explanation of the power law relativistic proton and electron energy spectra. A number of phenomena occurring in the dense hot plasma surrounding an MBH binary can affect the jet production and evolution, hence opening the possibility to use LISA sources, specifically merging MBH binaries, as novel laboratories for jet and cosmic ray astrophysics. A favourable observational window would seem to be during a merger where the separate MBH jets tend to co-align. The possibility of a spin flip turning two misaligned jets into one where a single enhanced jet is pointing close to the direction of the spin axis of the more massive of the two MBHs has been discussed in Gregely and Biermann (2009) and applies to mass ratios ≥ 0 . 1 . X-rays from the accretion disc relate to the seed particles for the accelerator and the source of p-nucleon or p γ neutrino production. The emergence of a gap in the circumbinary disc or lack of stars to be swallowed in the MBH could cause observable EM emission to cease. This situation is suggested from the lack of EM emission in the observation of the merger of stellar mass BHs. Correlated observation of GWs with those of the Athena X-ray mission and the Square Kilometer Array (SKA) in the radio bands could help \nin understanding cosmic ray origin. Specifically, data from Athena would reveal the amount of accreted gas available during the merger to power the jet, while simultaneous radio information would both yield the strength of the magnetic field associated with the jet and determine the spectrum of the accelerated relativistic electrons, which could then be directly related to the acceleration of protons. \nFor IceCube to detect neutrinos from p-nucleon collision, in the favourable case of a jet boosted flux and if 3 per cent of the accretion energy is available, requires that M 8 m e γ 4 10 D -2 4 ≥ 3 where M 8 is mass in units of 10 8 M ⊙ , m e is the ratio of the accretion rate to the Eddington rate, γ 10 is the jet Lorentz factor in units of 10 and D 4 is luminosity distance in units of 10 4 Mpc. However, the chance of seeing such a favourable geometry for the dominant jet in a merger is only 10 -3 γ -2 10 . Successful co-observation of neutrinos and GWs is yet to occur (Adrián-Martínez et al., 2016).", '2.5.2 Multimessenger observation strategy for MBHB mergers with LISA': "Figure 26: Time evolution of sky position uncertainties from Fisher Matrix simulations, from Mangiagli et al. (2020), for different source-frame MBHB total mass and redshift. Blue lines correspond to the median of distribution, whereas blue and green areas correspond to the 68 and 95 percentiles, respectively. Overall, lower-mass systems are localized better than more massive MBHBs. At z = 1 , systems with total mass of 3 × 10 5 M ⊙ are localized within 10 deg 2 ten hours before merger. The same accuracy is reached for MBHB with 10 7 M ⊙ total mass only 1 hour before merger. \n<!-- image --> \nMangiagli et al. (2020) recently demonstrated that overall the parameter estimation on the fly of light systems at z ∼ 1 and with total intrinsic mass ∼ 10 5 M ⊙ shows smaller uncertainties \nthan in heavy systems ( 10 7 M ⊙ ). The chirp mass and mass ratio are well constrained prior to the merger proper, with errors at the per cent level. In Fig. 26, we report LISA's abilities to constrain the sky position of the source. At z ≈ 1 , MBHBs with a total mass of 3 × 10 5 M ⊙ can be localized with a median precision of ∼ 100 deg 2 (1 deg 2 ) at 1 month (1 hour) before merger, whereas the sky position of 10 7 M ⊙ MBHBs can be determined to within 10 deg 2 only 1 hour before merger. Thus, only light and nearby sources can be traced during the inspiral phase. If the MBHs are embedded in a circumbinary disc, optical emission is predicted from the inner ring of the circumbinary disc and soft and hard X-rays from the mini-discs and the shock heated cavity (Tang et al., 2018; d'Ascoli et al., 2018). Modulation of the light curve is expected at frequencies commensurate to the fluid patterns (Bowen et al., 2017; d'Ascoli et al., 2018; Tang et al., 2018). Thus, observatories such as the Vera Rubin large synoptic telescope could detect the optical signal when the sky localization uncertainty falls below ∼ 10 deg 2 . Athena can strategically tile the optical field of view and then narrow down the sky position to detect a potential modulated X-ray chirp. This is possible for MBHBs in the near Universe ( z ≲ 0 . 3 ). \nAt merger, the sky localization improves down to ∼ 10 -1 deg 2 for all masses, giving us the chance to detect the post-merger emission by staring at the source for a sufficiently long time, from weeks to months, and witness a re-brightening of an AGN. Again, no definite spectral template exists to identify the source within the narrower error box indicated by LISA (but see Schnittman and Krolik, 2008; Rossi et al., 2010), and work in this direction should be performed before LISA flies. These multimessenger observations will be unique as for the first time and in real time it will be possible to correlate the masses and spins of the merging BHs with the EM emission by the surrounding gas to give quantitative estimates on the efficiency of the emission under extraordinary conditions, such as during the violence of a merger and from gas bound to a moving BH. \nThe exposure time needed to detect an Eddington-limited system varies with MBH mass, redshift, and can be more efficient in the soft or hard X-ray band depending on the obscuration of the source. For example, unobscured systems with M ∼ 10 6 -7 M ⊙ require an Athena exposure time of less than 1 kilosecond (i.e., a single pointing) up to z = 1 . 5 in the soft band, whereas systems of M ≈ 10 5 M ⊙ can be detected in a kilosecond up to z = 0 . 4 . Similarly, systems of M > 10 7 M ⊙ require less than kilosecond exposures at redshifts of z < 4 . 5 (McGee et al. 2020; Piro et al. 2021). For super-Eddington sources, shorter exposure times are expected, possibly through gas squeezing (Armitage and Natarajan, 2002; Cerioli et al., 2016). However, in the case of obscured sources (Gilli et al., 2022, whose fraction remains poorly constrained, and could increase with redshift), whose detections would be more efficient in the hard X-ray band, or objects accreting at low rates below the Eddington limit, the required exposure time of Athena can increase significantly, as described in McGee et al. (2020); Piro et al. (2021). A system with ∼ 10 6 M ⊙ and a luminosity of L = 0 . 1 L Edd at z ∼ 1 would require an exposure of more than 100 ks, against less than 10 ks for the same system with L = L Edd . \nIdentifying the best observational strategies to maximize the synergy between LISA and other missions such as Athena is a very recent and active field of research. Besides the detectability of the emission, in fact, matching the GW source to its EM counterpart requires the ability of identifying the host among a large number of potential candidates within the LISA error box. This aspect has recently been investigated by Lops et al. (2022), who considered the synergy between LISA and the future X-ray observatories LynX (The Lynx Team, 2018) and Athena (see below for the description of the missions). Assuming an active binary at merger, they found that most LISA sources with masses in the range 10 5 -10 7 M ⊙ at z < 2 will be detectable by those instruments within kiloseconds in a single pointing. However, the number of contaminating AGN unrelated to the GW event can be up to thousands for high-redshift signals, making it hard to pinpoint the correct host. Identification strategies need to be developed but require a better theoretical understanding of the peculiar features associated with the EM counterpart. For example, Tang et al. (2018) find that the EM luminosity of a merging binary is \nsuppressed in the last cycles prior to merger and enhanced after coalescence; if this is the case, a viable identification strategy would be to perform sequential pointings and search for a source displaying a monotonically increasing flux; in this case, the exposure time for each pointing might depend on the sky localization posterior distribution provided by LISA with longer exposure times for regions with higher probability to host the MBHB event. Identification of newborn jets powered by an highly spinning merger remnant (mentioned in Sec. 2.5.1.2) might offer another possibility for unambiguous counterpart identification. The best way to suppress the number of contaminants would be to improve the GW localization, which would be possible if LISA is joined by a second space-borne detector such as Taiji (Ruan et al., 2018) or TianQin (Luo et al., 2016). In particular, several works (Ruan et al., 2021; Wang et al., 2021b; Shuman and Cornish, 2022) showed that LISA-Taiji joint observations would improve the precision of the sky localization by three orders of magnitude. With this assumption, Lops et al. (2022) demonstrated that unambiguous identification of the active AGN related to the binary would be possible up to z = 2 . \nIn the context of sources identification, it is also important to mention that the astrophysical uncertainties on the population of merging MBHBs and on the type of EM emission strongly affect the number of expected EM counterparts. Recently, Mangiagli et al. (2022) computed the number of expected EM counterparts, starting from catalogs of merging MBHBs. Combining the information from radio, optical, and X-ray emission with the information from LISA sky localization, they estimated between 7 and 20 counterparts in 4 yr of LISA time mission. However, in the case of obscuration or collimated radio emission, the number of EM counterparts reduces to 2 or 3. This implies that a better understanding and modeling of the galaxies hosting MBHBs mergers are necessary to be ready for the LISA mission. \nIn general, it is clear that, in order to best prepare for LISA, we need to investigate better these multimessenger aspects, especially in view of forthcoming missions that could be operational when LISA will fly. AXIS (Mushotzky, 2018) and LynX (The Lynx Team, 2018) are two NASA concept X-ray observatories that could also fly simultaneously with LISA. Their flux sensitivity will be at least one order of magnitude better than Athena (while having smaller fields of view, see Section 2.6), making it possible to observe the X-ray emission from fainter AGN than achievable by current missions or Athena. Compared to the 5-10 arcsecond angular resolution of Athena, the high angular resolution of AXIS (sub-arcsecond resolution compared to the 5-10 arcsecond resolution of Athena), its fast slew rate and ToO response could be key for monitoring MBH binaries until coalescence. As mentioned above, further investigations are required in the near future to determine whether the sky position uncertainties of merging MBHB systems would be compatible with the characteristics of AXIS and LynX, and particularly their small fields of view. \nWe show in Fig. 27 how these X-ray missions will complement LISA by partially covering the same MBH mass and redshift ranges. The figure also illustrates the possible synergy between LISA and the Einstein Telescope (ET). As developed in Section 2.3, there are a lot of hurdles to grow light seeds efficiently in the high redshift Universe, and a population of long-living 'starved' (i.e.g, ungrown) merging MBH seeds could exist (Valiante et al., 2021). In this mass range coordinated multi-band GW observations are possible, with LISA having the capability to first follow the early inspiral of MBHBs, and tracking the merger phase. This unique combination will revolutionise our ability to carry out precise measurements of the source parameters also at z ∼ 5 (Jani et al., 2019). This mass and redshift range would also be covered by the X-ray missions LynX and AXIS. \nIn this section, we mainly discussed multimessenger observations with X-ray observatories. However, multimessenger observations with LISA span a large range of wavelengths. For example, we can learn about the physics of jets using radio and optical observations on LISA systems. Radio astronomy, for instance ngVLA as well as SKA, will allow us to observe MBH jets turning on. SKA should also be capable of detecting very luminous flares in the radio emitted by an equal \nFigure 27: LISA will be complemented by the X-ray mission Athena (launch expected in the early 2030s), and potentially by the NASA concept missions LynX and AXIS. These missions are shown in orange and black horizontal symbols, which indicate the sensitivity of the deepest pointing, in the [0.5-2] keV observed band, by Athena (orange) and LynX/AXIS (black). Waterfall plots show the average GW horizon computed for signal-to-noise ratio SNR = 10 and different BH mass ratios for the Einstein Telescope (red) and LISA (blue/green) bandwidth (Santamaría et al., 2010; Hild et al., 2011; Robson et al., 2019). For reference, main MBH formation mechanisms are shown with ellipses. The growth of some of MBH seeds could be stunted by several processes and could be detectable only at late times when merging with other MBHs at z ≤ 5 ('starved MBHs' in the white bottom ellipse, Valiante et al., 2021). Figure taken from Valiante et al. (2021). \n<!-- image --> \n/circledot \nmass MBH binary at merger time, and, through that, help with their sky localization (Tamanini et al., 2016). This can be complemented by observations of optical flares from e.g., the Roman Space Telescope or the Rubin Observatory (see Section 2.6 for a more complete descriptions of relevant instruments and space missions). Observational and theoretical constraints on these EM flares are still very poor, e.g., on the frequency of the peak emissions.", '2.5.3 The path towards LISA': 'In this subsection, we present several ideas for what we need to prepare to exploit the unique characteristics of LISA in the context of multimessenger study of MBHBs from the perspectives of theory, observations, and artificial intelligence.', '2.5.3.1 Theoretical and observational improvements in the multimessenger study': 'of MBHBs On the theoretical front, work is necessary to understand accretion on to binary MBHs and their EM signatures at various wavelengths; on the observational front, efforts are necessary to find and understand more MBHB candidates. Strengthening the collaborative studies between the EM and GW scientific communities is thus very important for scientific utilization of LISA data products.', '· Numerical simulations of EM counterparts to MBHB inspirals and mergers': "Over the last decade, several theoretical groups studied MBHBs in a circumbinary disc or more tenuous gas clouds (see Section 2.5.1.2), systematically adding the layers of physics necessary to investigate potential mechanisms for EM counterpart signals emerging during MBHB \ninspiral and merger. Newtonian viscous hydrodynamics (Farris et al., 2015b; Tang et al., 2017) and MHD (Shi and Krolik, 2016) simulations investigated the dynamics of the gas streams being stripped off the inner edge of circumbinary discs. MHD simulations over a post-Newtonian background spacetime explored the first stages of the strongly relativistic behaviour of MBHBs in circumbinary discs in the form of the disc's response to binary orbital evolution by GW emission (Noble et al., 2012) and, more recently, examined the mass-feeding mechanisms onto the individual mini-discs around the BHs (Bowen et al., 2017; Bowen et al., 2018) and the systems' radiative properties in the stage immediately prior to merger adopting ray-tracing techniques (d'Ascoli et al., 2018). \nThe first simulations in full, dynamical GR with resolved BH horizons and the MHD plasma from a circumbinary disc were performed in Farris et al. (2012) and Gold et al. (2014a), modelling the binary-disc pre-decoupling epoch, and in Gold et al. (2014b), modelling the post-decoupling, merger, and post-merger epochs. The inclusion of the BH horizons in these studies showed that powerful outflows and jets are launched from these systems even when the BHs are non-spinning. The more recent study in Khan et al. (2018a) found that accretion rates, temperatures, and jet launching from the interactions of the horizons with the magnetized medium exhibit modest dependence on the initial disc thickness. Jets in binary AGN would produce both non-thermal X-ray and gamma-ray signatures, which would be of interest to Athena as well as other Xray and gamma-ray satellites. However, modelling from first principles of such EM signals is currently absent. Therefore, efforts must be made towards adding radiation transport in dynamical-spacetime general relativistic MHD (GRMHD) simulations of accreting MBHBs. \nIn addition to GRMHD simulations in dynamical spacetime, dynamical GR simulations of MBHBs have been performed in the force-free limit, which assumes that the plasma around the BHs is tenuous (Palenzuela et al., 2009, 2010b,a,c; Mösta et al., 2010; Mösta et al., 2012). These studies showed how the orbital motion of the BHs alters magnetic and electric fields and leads to possible EM emissions. In particular, it was suggested that two separate jets during the inspiral, one around each BH, could emerge from these systems (Palenzuela et al., 2010c; Mösta et al., 2012). \nDynamical-spacetime simulations of MBHBs have also been performed in moderately magnetized clouds in Giacomazzo et al. (2012), showing a rapid amplification of the magnetic field over the last few orbits, leading to the creation of a post-merger magnetically dominated funnel aligned with the spin axis of the final BH, with properties relatively insensitive to aspects of the initial configuration (Kelly et al., 2017) \nThe future of numerical simulations will require an improved insight into the fuelling rate and the MHD properties of plasma accreting on to the BHs to sharpen the EM predictions. Furthermore, it will be necessary to match a range of different spatial scales in order to more properly address the evolution of the accreting gas during the early inspiral up to merger. These simulations will also need to account for radiation processes in order to correctly estimate EM light curves/spectra, as well as other modes of accretions (such as chaotic cold accretion Gaspari et al., 2013, 2015, Section 2.6.1.2) and radiation feedback (e.g. Sądowski and Gaspari 2017). The development of reliable radiation transport schemes in dynamical spacetime is therefore a high priority. \n2.5.3.2 Artificial intelligence: Deep learning methods to identify GW source candidates and to estimate LISA source parameters In recent years, artificial intelligence has been intensively applied in astronomy for a wide variety of tasks. As a sub-field of artificial intelligence, machine learning 11 has gained increasing popularity among astronomers, especially \nFigure 28: Schematic representation of a basic convolutional neural network architecture. Such numerical network can be trained on simulations, and later apply to observations to systematically identify MBHB candidates. Figure credit: Ioana Dutan \n<!-- image --> \nthrough utilization of one of its sub-sets, namely deep learning, when big data is involved. The most widely used deep learning algorithms are neural networks 12 containing multiple hidden layers that progressively extract higher-level features from input data.", '· Deep learning methods to identify GW source candidates from EM observations': 'An important issue in astronomy is to find astronomical sources in survey images in order to build source catalogues. These catalogues are valuable tools used for testing theories and numerical simulations against observational data. Convolutional neural networks are deep learning neural networks designed for processing structured arrays of data such as images (see Fig. 28). Convolutional neural networks are very good at learning features from images by hierarchical convolutional and pooling operations. \nMore precisely, convolutional neural networks algorithms have been already applied for image classification in order to find sources/objects in different EM wavebands. Among the uses in relation to LISA science we can mention detection and classification of quasars from light curves and identification of galaxy mergers (e.g., Pearson et al., 2019; Ackermann et al., 2018). Image classification employed in observational cosmology and in the analysis of large-scale structure statistics can set the stage for improving the estimations of time-scales and occurrence rate of MBH mergers via integration of artificial intelligence with simulations (see discussions in Section 2.4.3.2). \nMore importantly, the time spent to generate catalogues decreases dramatically when using deep learning algorithms instead of standard approaches. However, to reach a desired level of accuracy in image classification, training a deep learning algorithm can be costly in terms of duration and computational resources. Nevertheless, once properly trained, the algorithm can quickly classify thousands of GW source candidates (e.g. Pearson et al. 2019). Here, the training and validation/test samples can be either observational data from ongoing and upcoming galaxy imaging surveys or simulated data. Using such an approach, some possible biases in observations or additional requirements in simulations might be identified. In spite of current achievements, we need to further design algorithms that are able to learn representative features faster and achieve higher performance in image classification in order to better understand the AGN population and to find binarity signatures in the observational data.', '· Deep learning algorithms for estimation of LISA source parameters Standard al-': "12 Neural networks were inspired by the structure and the function of the brain, and they can be thought of as networks of neurons organised in layers: predictors (or inputs) form the bottom layer, forecasts (or outputs) form the top layer, and there may also be intermediate layers containing hidden neurons. \ngorithms used for estimation of the physical parameters that govern GW signals are effective, but the computations are time-consuming and they can take up to a few days. For example, in the case of synergistic observations with both LISA and Athena, a poor sky localisation of the source during the inspiral phase limits the possibility of performing concurrent observations. Moreover, the Athena capability of carrying out a target of opportunity is about 4 hours; that is, a low-latency alert should be released in less than 4 hours in order for Athena to be able to watch the merger phase. Therefore, reducing the computational time of source parameters is crucial for multimessenger studies. Over the past few years, deep learning algorithms have been employed for classification of glitches (non-Gaussian noise transients) in Advanced LIGO data (data referring here to the 'BH coalescence signal + noises', e.g. George et al. 2018; Razzano and Cuoco 2018). This allowed the identification of signals in Advanced LIGO data, where the training of the algorithm is performed on simulated stellar-mass BH merger signals in synthetic Gaussian noise representative to LIGO sensitivity (e.g. Gabbard et al. 2018), and for estimation of source parameters (e.g. Chua and Vallisneri 2020; Green and Gair 2020). Such models may have limited capacity as they do not currently account for an holistic approach to a quasirealistic GW data analysis specific to LISA, where tens of thousands of signals overlap with many gaps and glitches. Nevertheless, the current models represent a starting point from which novel architectures can be trained on non-stationary, non-Gaussian noise LISA-like data to conduct parameter estimation. Such development can allow us to perform time-sensitive multimessenger searches to greatly increase the science return of the LISA and other (future) experiments and observatories.", '2.6 Multimessenger view of MBH populations': 'Coordinators: Maria Charisi, Alessandra De Rosa Contributors: Stefano Bianchi, Tamara Bogdanovic, Monica Colpi, Pratika Dayal, Ioana Dutan, Saavik Ford, Massimo Gaspari, Melanie Habouzit, Albert Kong, Sean McGee, Barry McKernan, Francesca Panessa, Delphine Porquet, Raffaella Schneider, Stuart Shapiro, Rosa Valiante, Maurice van Putten, Cristian Vignali, Marta Volonteri, Silvia Zane \nLISA will bring crucial constraints on mass, redshift, and spin of merging BHs in the mass range ∼ 10 4 -10 7 M ⊙ . To achieve a complete understanding of the population of MBHs, from high redshift to the local Universe, from low to high mass, single and in binaries, the synergy of LISA with other missions will be key. In this section, we provide a global view on the facilities that complement LISA, or will complement it in the near future. We discuss how these missions will address different aspects of MBH physics and populations, but also how they will help us to understand the galactic and large-scale environments in which MBHs assemble, which is a major question in modern astrophysics.', '2.6.1 A landscape of new missions to understand MBH formation, growth, and environment': "Figure 29: Landscape of the upcoming and concept missions aiming at constraining the population of MBHs and their host galaxies, from the local to the high-redshift Universe. These missions will significantly increase current EM detections towards high redshifts ( z ∼ 10 ), while LISA will reach redshifts (e.g., z ⩾ 30 ) that will not be available with EM observations. We caution that the timelines reported in the figure are only indicative as delays in the launch of any of the missions, especially those a few years away from the time of writing, are always possible. In addition, at the time of writing the likely launch time-frame for ATHENA is set to slightly earlier than that of LISA. Characteristics of these and other missions are listed in Table 8. Figure credit: Melanie Habouzit. \n<!-- image --> \n2.6.1.1 A diversity of missions to complement LISA By exploring the 'light' MBHs (the low end of the mass distribution), LISA will open a new window on the GW spectrum, bridging the gap between high-frequency ground-based GW observations (e.g. by LIGO and Virgo), and the nano-hertz frequency observations by PTAs (Desvignes et al., 2016; Ransom et al., 2019; Perera et al., 2019; Kerr et al., 2020). PTAs are currently building up constraints on the GW background, generated by tight binaries of MBHs at the high-mass end ( M BH ∼ 10 7 -9 M ⊙ ) in the low-redshift Universe (Burke-Spolaor et al., 2019). LISA will be deaf to the population of even lower-mass seed mergers with 10 2 -10 3 M ⊙ , whose signal falls below the detection threshold (although some portion of the in-spiral might be accessible). These are potential sources for ground-based GW observatories, such as the Einstein Telescope (ET) and Cosmic Explorer. Space- and ground-based missions together will provide a complete census of MBHs, from the earliest seed BH mergers to the largest MBHs today. \nThe GW detections will be complemented by new observations of MBHBs from space- and ground-based facilities across the EM spectrum, as shown in Fig. 29. The ESA L2 mission Athena (Barcons et al., 2015), the proposed NASA missions AXIS (Mushotzky et al., 2019) and LynX (The Lynx Team 2018; Gaskin et al. 2019), and the ongoing eROSITA mission (Predehl et al., 2010) will probe the accretion properties of AGN in X-rays, while surveys like the Dark Energy Spectroscopic Instrument (DESI; DESI Collaboration et al. 2016), the James Webb Space Telescope (JWST; Gardner et al. 2006), the Nancy Grace Roman Space Telescope (Green et al., 2012) and Euclid (Amendola et al., 2013) in the optical and IR band will investigate galaxy hosts up to the highest redshifts. The next-generation ground-based optical telescopes, like the Extremely Large Telescope (ELT; Tamai et al. 2016) and the Thirty-Meter Telescope (TMT; Sanders 2013), will reveal the assembly of the first galaxies, and large-area photometric and spectroscopic surveys, like the Rubin Observatory Legacy Survey of Space and Time (LSST; Ivezić et al. 2019) and the Sloan Digital Sky Survey-V (SDSS-V; Kollmeier et al. 2017), are expected to discover a treasure trove of binary candidates. Wide-area and deep radio surveys will be available with radio interferometry provided by the Square Kilometre Array (SKA, Dewdney et al. 2009). \nIn Table 8, we summarize some of the existing and upcoming space- and ground-based telescopes, with their sky coverage and key science that they will address both leading up to LISA's launch and concurrently with LISA. We discuss in detail the role of all these missions in the following. \n2.6.1.2 The synergy of LISA and EM missions to answer major questions on MBHs and their host galaxies In the next decades, EM and GW messengers will work in concert, providing new knowledge of the galaxy and MBH assembly processes, as well as of the interplay between dynamic gravity and the relativistic plasma. Multimessenger observations are an emerging research domain of modern astrophysics. In the following, we detail how several missions can work in synergy with LISA to answer key scientific questions.", '· The formation of the first MBHs': "Thanks to its distant horizon and vast volume probed, LISA will be able to detect the first coalescing massive seeds of 10 4 -10 5 M ⊙ , witnessing the dawn of MBHBs at redshifts that are not reachable with EM observations. LISA will not, however, detect all the first MBHs, but only those that form binaries and merge, or those that form from the collapse of SMSs that provide a sufficiently high GW signal. EM observations, targeting a complementary population, supplement LISA's detections to provide an improved understanding of the fundamental question of MBH formation. \nCurrently, the only EM observational insights on low-mass MBHs can be obtained from relatively local (up to z ∼ 2 ) dwarf galaxies of total stellar mass M ⋆ = 10 7 -10 9 . 5 M ⊙ (e.g. Reines and Volonteri, 2015; Baldassare et al., 2015; Mezcua et al., 2016, 2018). Unlike massive local galaxies, which have experienced significant mass growth, local dwarfs have experienced less growth through cosmic history (Habouzit et al., 2017). Their MBHs are also expected to have experienced a similar limited growth. Properties of MBH formation could have been preserved in local low-mass galaxies (Volonteri et al. 2008b; Greene 2012). \nTo directly probe the properties of seed MBHs, before they grow significantly in mass via gas accretion (Valiante et al., 2018a), it is necessary to search for such sources at high redshifts ( z > 10 ). Theoretical models, including spectral-synthesis, predict that the emission from accreting heavy seeds (e.g., direct collapse MBH seeds), will be strong mainly in the IR-submm and X-ray bands, and thus could be detected by JWST up to z ∼ 15 and Athena up to z > 6 . By contrast, the emission from lighter accreting seeds ( ∼ 10 4 M ⊙ ) is expected to be weaker and difficult to observe with EM facilities at high redshifts (Pacucci et al., 2015; Natarajan et al., 2017; Valiante et al., 2018b; Barrow et al., 2018). The NASA concept mission Lynx, and to a lesser extend AXIS, which has lower sensitivity, directly aim at detecting these young, faint and faraway AGN. \nGiven the small fraction of the sky that EM missions such as JWST, Athena, and Lynx will cover (see Table 8), an optimized observational strategy is crucial to detect as many low-mass MBHs as possible. The MBH community is currently leading an effort to build up target selection criteria designed on the basis of the combined analysis of IR colours (colour-colour cuts), X-rays-to-optical flux ratios (rest frame), IR excess, and UV continuum slopes to efficiently detect and distinguish candidates (Natarajan et al., 2017; Valiante et al., 2018b).", '· The growth of MBHs': "MBH growth is one of the major open questions in astrophysics, and because of this, constraining it is one of the main goals for several of the upcoming EM surveys. Understanding the processes that determine the growth of MBHs from low-mass seeds to MBHs with mass 10 8 -10 10 M ⊙ requires observations of MBHs at different evolutionary stages over cosmic history. With the large samples of AGN/quasars discovered by these surveys, we expect significant developments before LISA flies. Having a better understanding of MBH growth will help us refine the theoretical models that will be confronted with LISA data and sharpen the astrophysical interpretation of LISA's detections. \nThe current population of rare bright high-redshift quasars ( z ∼ 6 -7, Mortlock et al., 2011; Bañados et al., 2016, 2018a; Matsuoka et al., 2019; Yang et al., 2020a) powered by MBHs of 10 8 -10 10 M ⊙ will be extended in the coming decade by several EM space and ground-based missions, and will provide us with a unique snapshot in the MBH growth timeline, offering a complementary view to LISA probing the low-mass end of the MBH mass spectrum. The Nancy Grace Roman Space Telescope (Fan et al., 2019) and the Euclid space telescope (Euclid Collaboration et al., 2019) are set to increase tenfold the number of high-redshift quasars discovered in the near-IR. By mapping large fractions of the sky, these surveys will identify quasar candidates, which will be confirmed with spectroscopic follow-up. At lower redshift, the SDSS-V Black Hole mapper program will deliver MBH mass measurements for about 1000-1500 quasars/AGN between redshift 0.1 and 4.5 (Kollmeier et al., 2017). \nX-ray observatories will also greatly enlarge the population of known AGN, in particular at high redshift. eROSITA, an all sky survey strong of an expected sample of about 3 million AGN, will study the accretion history of MBHs by measuring the luminosity-dependent fraction of obscured objects; studying the clustering properties of X-ray selected AGN at least up to z ∼ 2; and identifying rare AGN sub-populations such as high redshift, possibly highly obscured nuclei. Athena, with higher sensitivity than current missions, aims at detecting over 400,000 AGN, several hundred of which at z ≥ 6. With even higher sensitivity, the concept missions Lynx and AXIS aim to push the quest for faint AGN by two orders of magnitude in intrinsic luminosity. Besides probing high-redshift quasars, X-ray telescopes will complement the census of MBHs by discovering obscured AGN at the peak of the accreting MBH activity ( z ∼ 2 -4 ), which are inaccessible with optical/near-IR facilities but are crucial in order to obtain a complete census of MBH growth. We will be able to build the mass and spin distributions of a large population of MBHs that will provide crucial information on the growth process (e.g. merger versus accretion, Berti and Volonteri 2008, see also Section 2.3.2.4).", '· The co-evolution of MBHs and cosmic structures': "Over twenty years of observations have unveiled fundamental correlations between the properties of galaxies and the mass of their central MBH. This promoted important advancements in extragalactic astronomy, suggesting that the central MBHs and the host galaxies co-evolve from high to low redshift. Notable correlations are the M BH -σ and M BH -M bulge relations, which link the stellar velocity dispersion σ and the mass of the stellar bulge M bulge with the mass of the MBH(see Kormendy and Ho 2013, Graham 2016b for reviews). The correlation extends to haloes of galaxies, relating the MBH mass to the hot plasma halo temperature or luminosity (Gaspari et al. 2019; Bassini et al. 2019). These correlations indicate that the MBH, albeit tiny compared to the entire galaxy, is linked to the stellar component and the surrounding intracluster/intragroup medium (up to ∼ 10 per cent of the virial radius). The co-evolution between MBHs and galaxies/haloes is possible due to the self-regulation between feeding and feedback processes, from near the MBHs' horizon to the edge of the bound stellar and dark matter structure (see Gaspari et al. 2020 for a review). One way LISA will contribute to investigating the link between MBHs and their hosts is by offering completely independent mass measurements, allowing us to better calibrate the known correlations with galaxy host properties since currently mass measurements suffer from biases introduced by EM observations. Advanced optical/IR facilities will be instrumental in constraining the related stellar properties and evolution of the hosts, both for LISA sources and for MBHs with more uncertain mass measurements. Future galaxy surveys of JWST, Euclid, and Roman will include the host galaxies of LISA-band MBHs, i.e. lower-mass galaxies than possible to detect today. Respectively, these telescopes should uncover galaxies with stellar mass of ⩾ 10 7 , 10 9 . 5 , 10 8 M ⊙ at high redshift. Among others, the PRIMER JWST Treasury Program should detect about 120 000 galaxies out to z ∼ 12 (Dunlop et al., 2021), the FRESCO Program ∼ 1200 galaxies at z ∼ 5 -6.5 (Oesch et al., 2021) and ∼ 300 galaxies at z ∼ 7 -9), and the WDEEP Program ⩾ 1000 mostly low-mass galaxies with 10 6 -7 M ⊙ (Finkelstein et al., 2021). Further investigations in the community are required to assess whether the galaxies of these surveys could be later matched to LISA events. To connect MBH and galaxy mass with the feeding and feedback physics expected to establish their self-regulation, nextgeneration X-ray telescopes (Athena, LynX and XRISM; Tashiro et al. 2018) will constrain the inner hot accretion flows and surrounding plasma haloes, in particular by leveraging IFU instruments with high spectral and spatial resolution. Radio-mm telescopes (such as ALMA and LOFAR/SKA) will provide constraints on relativistic jets (especially at low frequencies), their launching mode, and duty cycle of AGN kinetic feedback, thus providing us with a comprehensive view of the role of feeding, feedback and self-regulation in the co-evolution of galaxies and MBHs. \n- · The impact of the cosmic large-scale structure on the MBH merger rate \nObservational studies of MBH scaling relations (eg, M BH -σ and M BH -M bulge ) have shown evidence for a dependence on large scale environment. Both central and satellite galaxies in galaxy groups and brightest cluster galaxies appear to have larger MBH masses given their galaxy velocity dispersion (McConnell and Ma, 2013; McGee, 2013; Dullo, 2019; Bogdán et al., 2012; McGee, 2013). These departures in the scaling relations remain controversial and could be due to selection effects in the observational samples. If the results are confirmed, the cause could be an enhanced galaxy and MBH merger rate in dense environment, but alternatively tidal effects from the host group/cluster could strip stellar material from the host galaxy (Volonteri et al., 2008a; Graham and Soria, 2019; van Son et al., 2019) or an additional channel of MBH growth could result from the host galaxy's interaction with the hot intragroup/cluster medium (Poggianti et al., 2017; Ricarte et al., 2020). \nIn group/cluster environments it is difficult to disentangle mergers and interactions from the abundant projection effects, so observational results on their relative rate are not firmly established (Edwards and Patton, 2012; Pipino et al., 2014). The low-surface brightness and wide-field capabilities of the Rubin Observatory will enable the identification of diffuse merger and tidal features which should narrow the observational uncertainty (Brough et al., 2020). The identification of large samples of stripped galaxies (e.g., Yagi et al., 2010) combined with AGN measures from time-varying photometric analysis or X-ray measurements will allow a robust quantification of this growth channel. The improvements in the understanding of this physics prior to the launch of LISA should enable stronger predictions for the environmental dependence of the MBH merger rate. If MBH mergers are more common in biased overdense regions , this will also help focusing the efforts for finding the EM counterpart of LISA sources.", '· Matter behaviour in the strong field gravity regime': "Astrophysical BHs span 10 orders of magnitude in mass, allowing for unique tests of the scale invariance of gravitational effects. LISA will detect in-spiraling and merging MBHs. LISA's ability to perform tests of gravity through BH coalescences and EMRIs as well as to provide spin measurements will be complemented by EM and GW observations that probe the behaviour of matter in different gravity regimes. Electromagnetic signatures of the inspiral and merger phases in the X-ray domain are produced so close to BHs that relativity enters into modelling their production. Spin measurements using EM observations include relativistic effects in modelling and data analysis. \nThe motion of accreting plasma near BHs provides a powerful diagnostic to study the very deep potential well generated by the central object. The infalling matter forms an accretion disc that may extend down to the ISCO, in the vicinity of which the bulk of the X-ray radiation is emitted. X-ray timing, spectroscopic and polarimetric techniques for probing matter flows into the strong gravity regime have been developed and, the first two have already been applied to real data, allowing us to infer the mass and spin of MBHs (Fabian et al., 2000; Remillard and McClintock, 2006; McClintock et al., 2011; Reynolds, 2014). Moreover, observations of matter orbiting a BH can be used to verify some of the key predictions of GR in a stationary spacetime metric, i.e a very different - and complementary - setting to that probed using GW measurements of merging BHs. \nThe relativistically broadened Fe lines observed in accreting BHs are direct diagnostics of matter behaviour in the strong-field gravity regime. In the standard scenario, the hot gas in the 'corona' produces thermal Comptonized emission that is reflected by the inner regions of the accretion disc, resulting in the Fe K α emission line. Special relativity (Doppler boost and relativistic aberration) and GR (gravitational redshift, light bending) affect the shape of the Fe line (Fabian et al., 2000). When line profile templates are fit to real data in both stellar mass BHs and MBHs, the disc inner radius and inclination can be measured. If the inner disc radius is assumed to be the ISCO, then the spin of BHs, which depends directly on the spin magnitude and whether the accretion disc is prograde or retrograde with respect to the BH rotation, can \nbe inferred (Brenneman and Reynolds, 2006; Miller, 2007; Reynolds, 2014). In the near future, the X-ray polarimetry mission IXPE (launched in December 2021, Weisskopf et al. 2016) will offer an independent method to measure inclinations and BH spins, mainly in stellar mass BH (Connors et al., 1980; Li et al., 2009; Schnittman and Krolik, 2009). Larger effective area Xray polarimetry missions, such as eXTP (Zhang et al., 2019), will extend the technique to the weakest AGN. \nIn order to further advance our understanding of the behaviour of matter around BHs, we need higher sensitivity (i.e. large effective area) and higher energy resolution, allowing a better characterization of the BH environment through the study of the narrow emission/absorption features in the X-ray spectrum. This will be achieved with the next generation of X-ray telescopes such as Athena, AXIS, Lynx, STROBE-X (Ray et al., 2018), eXTP, and XRISM, which are expected to produce unprecedented quality spectra with short exposures. Observations with such telescopes will minimize the modelling uncertainties (e.g. due to disc inclination, absorption properties, geometry), since these facilities will use different techniques (combining spectraltiming, and spectral-timing-polarimetry information) to measure the distinct physical quantities such as BH spin, accretion geometry, and BH mass (Dovčiak et al., 2008; Dovciak et al., 2013; De Rosa et al., 2019a). The sample of EM-measured MBH spins will also be enlarged.", '· Signatures of MBHB arising from circumbinary discs': "In contrast to earlier studies, recent simulations of GW-driven, nearly equal-mass binaries all the way to coalescence (Farris et al., 2015a; Tang et al., 2018; Bowen et al., 2018, 2019; Roedig et al., 2014) have shown that the gas is able to accrete on to the BHs all the way to the merger, despite the rapid contraction of the binary orbit and the formation of a central cavity (see Section 2.2.2.2). In the case of an optically thick flow, coronal emission around the two MBHs may give rise to hard X-ray emission at the mini-disc scales and soft X-ray emission from the inner rim of the circumbinary disc. Simulations suggest that the binaries can be very bright in hard X-rays when the spatial separation of the two MBHs is below about 100 gravitational radii, with thermal emission from the minidiscs dominating. The modulation of the X-ray emission might depend on the orientation of the binary orbital plane relative to the line of sight, Doppler beaming, and gravitational lensing (see Sections 2.2.2.2 and 2.5.1.2). \neROSITA can detect candidates through the hard X-ray binary signature of shocks in minidiscs (Krolik et al., 2019), while notch signatures (i.e. the lower thermal output at the frequencies that would have been radiated from the radii in the cavity) can be detected in optical surveys. For instance, the plan for SDSS-V is to acquire spectra for eROSITA's AGN, then joint signatures (notch and shock) can be looked for in the same sources. eROSITA and SDSS-V, however, have relatively shallow flux limits, therefore only rapidly accreting MBHs at the upper end of the masses of interest for LISA can be detected. \nAs discussed in Section 2.5.2, for sources with mass ∼ 3 × 10 5 M ⊙ at z < 0 . 5 LISA's sky localization can be of a few square degrees weeks to months prior to merger. This will allow widefield X-ray (and possibly optical) instruments to observe the EM precursor signal. Such sources, however, are expected to be few. For most sources a few square degrees in the sky localization uncertainty can be generally obtained only days/hours prior to merger, making pre-merger EM observations extremely challenging (if not impossible). In the post-merger phase (with ∼ 0.1-10 square degrees sky localization) we will have the chance to observe the disc re-brightening, the formation of an X-ray corona, and that of an incipient jet. In fact, in the post-merger phase, a relativistic jet may be launched by the newly born MBH, with production of gamma-ray emission and afterglow emission in its impact with the interstellar medium (Gold et al., 2014b).", '· Astrophysical neutrinos from MBHBs': 'Astrophysical neutrinos may originate in AGN jets, as supported by the detection of 10 15 eV neutrinos possibly associated with the blazar TXS 0506+056 (Aartsen et al., 2018). Since \nMBHBs are also likely associated with AGN, they may be promising sources for GWs+neutrinos observations. Coincident detections of GWs+neutrinos may be facilitated by the fact that neutrino observatories are all-sky detectors (like LISA) and do not need to be pointed towards a specific direction. If neutrinos are detected from a sizeable sample of LISA MBHBs, this will provide invaluable insights on the currently unexplored mechanisms of jet launching and acceleration in the presence of an MBHB.', '2.6.2 Preparing LISA using prior knowledge on MBHBs from current and upcoming missions': 'LISA detection rates are uncertain, varying between several to few hundreds over the planned 4-yr mission lifetime. Although the bulk of these events will involve MBHBs with M BH < 10 5 M ⊙ at z > 5 , more massive sources with M BH > 10 6 M ⊙ at lower redshift ( z < 3 ) might be detected at a rate of a few per year (see Section 2.4). A number of EM and GW facilities (already operating or upcoming) are expected to deliver significant results even before LISA flies, allowing us to tackle a number of key questions that will prepare the way for LISA. Detections by LISA will then complement these findings either through multimessenger observations or by opening a new window in the GW spectrum. Some of the main questions are as follows: How do MBHs pair following a galaxy merger? What role does the gas play during the MBH mergers and on which time-scale does coalescence occur? Can the merging MBHs shine down to the final coalescence? In this section, we summarize the EM and GW facilities that will operate before LISA, and discuss their main contribution to understanding MBHBs. LISA will bring unique and invaluable insights on this topic, enhancing the importance of the upcoming discoveries.', '2.6.2.1 Multi-band gravitational waves · Exploring the nHz GW band with Pulsar Timing Arrays to uncover the most massive MBH binaries in the local Universe': "The detection of GWs with LISA will expand the GW spectrum in the mHz regime, thus enhancing the discoveries of PTAs in the nHz window. PTAs systematically monitor stable millisecond pulsars over a long period of time, currently spanning almost two decades. GWs passing between a pulsar and the Earth change the time required by successive pulses to travel the path from the pulsar to the Earth. If such deviations in the travel time pulses are correlated over multiple pulsars in the array showing a characteristic quadrupolar correlation signature (Hellings & Downs curve; Hellings and Downs 1983), then GWs can be detected. PTAs are sensitive to nHz GWs and thus target MBHBs with masses of 10 8 -10 10 M ⊙ at z ∼ 1 -2 . PTAs are expected to detect primarily two signals: (1) the stochastic GW background from the superposition of many unresolved signals, and (2) continuous (monochromatic) GWs from individual sources that stand above the background. The former is expected to be detectable within the next few years, whereas GWs from individual binaries likely will follow soon after (Rosado et al., 2015; Taylor et al., 2016; Mingarelli et al., 2017; Kelley et al., 2018; Arzoumanian et al., 2020a). Recently, the North American Nanohertz Observatory for Gravitational Waves (NANOgrav) collaboration, based on their 12.5 years data release with a total of 47 pulsars studied with the Arecibo Observatory and Green Bank Telescope, showed that the stochastic GW background is consistent with predictions for the spectrum produced by SMBHs in the accessible frequency bands (Arzoumanian et al., 2020). However uncertainties remain large, and admit alternative explanations such as cosmic strings. which result in a slightly different spectral slope. Using a larger number of pulsars, longer observations time, and reducing systematic errors, will be needed to improve upon this latest result. \nBecause of large theoretical uncertainties in binary evolution, models with similar amplitudes for the GW background predict different merger rates for LISA. The amplitude of the GW background depends on how often galaxy mergers deliver sub-pc binaries, which in turn depends on how often galaxies merge and on how rapidly bound binaries to reach the GW regime. It further depends on the mass of the MBHBs in galaxies (Sahu et al., 2019a), with recent upper \nlimits placing constraints on the MBH scaling relationships (Simon and Burke-Spolaor, 2016). It is expected that, prior to LISA's launch, PTAs will constrain not only the GW background amplitude, but also the shape of the background spectrum, which encodes information about the binary eccentricities and/or environmental coupling (Sesana et al., 2009; Taylor et al., 2017; Kelley et al., 2017b; Taylor et al., 2019). \nConnecting the binary population at two cosmic epochs (i.e. the lower-mass binaries at higher redshifts observed with LISA and the higher-mass local binaries that dominate the GW background in the PTA band) will constrain the processes that drive the binary evolution following a galaxy merger, which have remained highly uncertain for several decades (Begelman et al., 1980), significantly improving our understanding of galaxy evolution, one of the most fundamental open questions in astronomy. Last but not least, individual MBHBs in the nHz band are candidates for multimessenger observations (Kelley et al., 2019; Arzoumanian et al., 2020b): the sky localization for PTA detections will be very poor (of order ∼ 100 deg 2 ), making the identification of the host galaxy challenging. Therefore, PTAs will develop and refine strategies for follow-up observations that will be invaluable for LISA.", '· Prospects from astrometry to reduce the gap between PTAs and LISA': 'Low-frequency GWs can also be detected with precise astrometry. GWs passing through the MWcanalter the apparent position of the stars on the sky, resulting in a characteristic oscillatory pattern. This requires long-term monitoring of the precise position of a large sample of stars. Fortunately, this can be achieved in the near future by Gaia, which at the end of its planned 5-year mission will provide precise astrometric measurements for billions of stars. It has been suggested that Gaia observations will provide high-quality data that would complement data from PTAs. This because, while the frequency domain is similar to that of PTAs, sensitivity is somewhat higher towards the high frequency tail accessible of the latter, around 300nHz (Moore et al., 2017). The sensitivity of Gaia at those frequencies, which corresponds to a strain of order 10 -14 , could perhaps allow to detect individual loud sources, such as a supermassive black hole binary with a mass of a few 10 8 M ⊙ , namely straddling between the typical PTA and typical LISA range, provided that the source is very nearby. Yet, even if detections would mainly occur for supermassive black hole binaries in the same range of masses of PTAs, the fact that the detection technique is completely different will, by itself, represent an important step forward. The three experiments (PTAs, Gaia, and LISA) together will consolidate our knowledge of the evolution of MBHBs through cosmic time.', '2.6.2.2 Multimessenger astrophysics · The Legacy Survey of Space and Time of Vera Rubin observatory to detect AGN binaries through photometric variability': 'The Vera C. Rubin Observatory will perform the Legacy Survey of Space and Time (LSST), which will provide time-domain observations of unprecedented quality and quantity (LSST Science Collaboration et al., 2009). This is particularly significant for EM searches of MBHBs, since they can be detected as AGN with periodic variability. LSST will monitor a large number of quasars (of order one million) providing multi-band observations with high cadence, and long baselines, extending up to 10 yr. Therefore, it is perfectly suited to detect the relatively short-lived and short-period MBHBs emitting GWs in the LISA band. \nAlready large numbers of binary candidates are identified in existing photometric datasets from the Catalina Real-time Transient Survey (CRTS; Graham et al. 2015), the Palomar Transient Factory (Charisi et al. 2016), the Panoramic Survey Telescope and Rapid Response System (PanSTARRS; Liu et al. 2019b), and the Dark Energy Survey (DES; Chen et al. 2020c). However, currently it is extremely challenging to distinguish the sources with genuinely periodic variability from the typical AGN that show intrinsic red noise variability (Vaughan et al., 2016). LSST will also facilitate binary searches from that perspective. The vast sample of AGN will allow an improved statistical description of the red noise properties of AGN, thus minimizing \nthe false periodic detections. \nThe upcoming detections with LSST , along with the current candidates, will illuminate the accretion processes in the presence of a binary, paving the way for multimessenger observations with LISA. More importantly, LSST will constrain the demographics of the population of GWemitting binaries, the distribution of periods, masses, and mass ratios. Additionally, LISA will provide independent measurements for the binary parameters, allowing us to examine potential biases in EM searches for binaries. \nAnother exciting possibility arises from the expectation that LSST will detect thousands of tidal disruption events (TDEs). The rate of TDEs depends on (i) the dynamics of stars surrounding MBHs; and (ii) the density surrounding MBHs. As orbits of stars can be perturbed by MBHB, it is expected that bound MBHB have a different rate than single MBHs, and N-body simulations actually find that galaxies hosting an MBHB should have a significantly higher rate of TDEs (Li et al., 2017). Therefore this is a possible explanation to the over-representation of TDEs in galaxies which undergone a starburst ∼ 1 Gyr ago but currently exhibit no sign of star formation (E+A galaxies; French et al., 2016). However, theoretical works have not converged on the origin of these post-starburst galaxies: galaxy mergers triggering nuclear star formation and enhancing the central stellar density (Stone and van Velzen, 2016; Pfister et al., 2019a, 2021a) provides a possible explanation, but the higher TDE rate could also be due anisotropy in the nuclear star cluster produced caused by the starburst (Lezhnin and Vasiliev, 2016) or a merger (Stone et al., 2018). In any case, the galaxies in which TDEs are detected may be more promising hosts of MBHBs or MBH pairs, and may serve as signposts for binary follow-up observations.', '· Spectroscopic search of MBHBs with the fifth Sloan Digital Sky Survey (SDDS-V)': 'Spectroscopic surveys, like SDSS, provide another potential route to detect sub-pc MBHBs with EM observations. If one of the MBHs in a binary system is surrounded by enough gas to produce a prominent broad line region, the motion of the MBH will result in detectable Doppler shifts in the broad emission lines (Nguyen et al. 2020 and references therein). The spectroscopic database of SDSS has already provided significant samples for spectroscopic searches of MBHBs and dozens of binary candidates have been identified from their large broad-line offsets (Eracleous et al., 2012). However, these are not unique signatures for binaries, and long-term spectroscopic follow-up is necessary in order to observe coherent changes in the broad emission lines and confirm the binary nature of the sources (Runnoe et al., 2017). \nSDSS-V will provide a promising sample for this type of search, since the BH mapper program will spectroscopically monitor thousands of AGN over multiple epochs (Kollmeier et al., 2017). This time-domain component to the spectroscopic survey will allow the detection of several more candidates. These candidates are likely progenitors of LISA sources before entering the GW-dominated phase of their evolution, since at mpc separations, the broad line region around individual MBHs cannot be that prominent. However, they can bridge the gap in our understanding of binary evolution in the sub-pc regime.', '· Identifying MBHBs through morphological and spectral investigations at radio wavelength': 'Radio emission in galaxies can directly mark the location of the MBH, since it is typically associated with active MBHs. In case of binary systems, if both nuclei are active, then a double radio core can be resolved. However, such systems are rarely found (Rodriguez et al., 2006). Sometimes, jets are produced and their associated synchrotron emission can help in tracing the past and current dynamics of MBHs in a merging system. Radio observations are crucial in identifying pairs via a morphological, spectral, and variability investigation. \nNowadays the highest spatial resolutions on ground are achieved by Global VLBI (Very Long Baseline Interferometry) network observations, that combine radio telescopes all over the world to synthesize an equivalent Earth size instrument, being able to reach angular resolution at \nmilli-arcsec scales, allowing to map the nuclear sub-pc scales for nearby sources (Venturi et al., 2020). \nFuture radio observatories such as Next Generation Very Large Array (ngVLA) and Square Kilometre Array (SKA) will work in excellent synergy with LISA on several grounds. On the one hand, they will be able to identify the radio EM counterparts to GWs due to MBHBs mergers, thanks to their high-resolution, sensitivity, and fast-mapping capabilities. On the other hand, the large-scale surveys on wide areas and with nearly µ Jy/beam sensitivity will significantly increase the dual AGN population at sub-kpc separation, by several orders of magnitude (see Paragi et al. 2015). For instance, SKA1-MID is expected to detect a few hundreds of dual AGN per square degree and probe scales of 1-100 kpc (Deane et al., 2014). In addition, precise measurements of AGN core positions will allow the investigation of offset MBH predicted by gravitational recoil. A combination of long baselines and high frequencies can ideally map and identify cores from MBHBs at sub-pc separations. The ngVLA and SKA with a VLBI expansion will allow to resolve the sub-mJy source population, tracking the orbital motions of radio cores for the most nearby GW candidates (Bansal et al., 2017; Burke-Spolaor et al., 2018). Jet precession might be due to the presence of an MBHB, potentially producing an X-shape morphology (Horton et al., 2020) that can be traced by high-sensitivity low surface brightness observations as offered by SKA. In addition, radio light curves of AGN can show periodic activity that can be associated to orbital precession. Candidate binaries, dual and offset MBH can be cross-matched with multi-frequency observations to confirm their nature (redshifts from the optical spectra, X-ray emission, e.g. from Athena, etc.).', '2.6.3 The path towards LISA': 'In the following, we inventory the different steps, studies, surveys, and developments, which to us seem crucial in view of LISA, and which are based on current and upcoming observational facilities. \n- · In the near future, the eROSITA X-ray survey will dramatically improve constraints on the MBHpopulation at the upper end of the LISA band and beyond, up to high redshift. While waiting for the new X-ray missions with better sensitivity and spatial resolution, such as Athena, AXIS, and Lynx, we should aim at exploiting the best capabilities of Chandra and XMM in order to characterize and confirm the candidate MBHB selected through optical variability (photometric variability). Moreover, we should improve modelling of intrinsic emission related to disc-corona in AGN, in order to reduce systematic uncertainties on the estimates of MBH spin and mass. This goal requires the use of spectral-timing (and spectral polarimetry in the future with, e.g. the eXTP mission) techniques that need deep observations of specific targets to investigate the variability properties.\n- · The upcoming surveys from DESI, JWST, Euclid, ROMAN, and the next phase of SDSS will provide massive catalogues of galaxies. It is imperative to enhance these catalogues with measurements of their MBHs (e.g. MBH mass) which will facilitate the identification of host galaxies for a large sample of LISA MBHB coalescences.\n- · The optical photometric surveys offer the possibility to select large samples of MBHB candidates (see previous section). These candidates should be further monitored with highly rewarding, albeit risky, X-ray observations in order to confirm or reject their binary nature. This will constrain the expected event rate for LISA. Moreover, X-ray observations with their ability to arbitrate between genuine MBHs and false positives will allow us to validate and refine the selection techniques. Such techniques will be widely used and improved in future facilities such as LSST. \n- · It is also crucial to improve the numerical simulations of inspiralling MBHBs embedded in gaseous discs, considering accretion properties and detailed thermodynamics of single MBHs and including radiative feedback. These studies will provide more reliable EM signatures that will allow the detection of LISA EM counterparts. They will also facilitate the efficient discovery (with low contamination of false detections) of binary candidates populations with current and future EM facilities.', '3 Extreme and Intermediate Mass-Ratio Inspirals': 'Chapter coordinators: Pau Amaro Seoane, Saavik Ford, Alejandro Torres-Orjuela, Martina Toscani, Lorenz Zwick', '3.1 Introduction': 'Coordinators: Pau Amaro Seoane, Saavik Ford and Cole Miller Contributors: Pau Amaro Seoane, Christopher Berry, Alvin Chua, Saavik Ford, Barry McKernan, Cole Miller, Carlos F. Sopuerta, Alejandro Torres-Orjuela and Veronica Vazquez-Aceves \nThanks to high-resolution observations of the kinematics of stars and gas we know that most nearby bright galaxies host a dark, massive, compact object (e.g., Kormendy, 2004; Genzel et al., 2010; Kormendy and Ho, 2013; Graham, 2016b). One of the most impressive cases is our own Galactic Centre. The stellar dynamics of the central cluster of stars (the S-stars, or S0-stars), provides compelling evidence for the existence of a massive BH (MBH) of mass ∼ 4 × 10 6 M ⊙ , Sgr A* (see for a review Genzel et al., 2010). In particular, the star S4714 in this cluster has an orbital eccentricity of 0 . 985 and moves at about 8% the speed of light at pericentre, with an orbital period of 9 . 9 years around the MBH in our Galactic centre. Another extreme case, S62, comes within 16 AU of Sgr A* (Peißker et al., 2020). Also very recently, Abuter et al. (2020) have presented the detection of pericentre precession in the orbit of the star S2. The best fit to a relativistic orbit yields a precession rate between 0.196 and 0.272 degrees per orbit, which is consistent with GR predictions at the ∼ 17 % level. Further compelling evidence for a MBH comes from the Event Horizon Telescope (EHT) observations of the centre of the galaxy M87. EHT measured the mass of its central MBH to be ∼ 6 . 5 × 10 9 M ⊙ , with an event horizon diameter of about 0 . 0013 pc (Akiyama et al., 2019). \nMore generally, it is believed that most large galaxies host a MBH. The currently highest inferred mass is 6 . 6 × 10 10 M ⊙ (Shemmer et al., 2004). The stars in the centres of such galaxies have the potential to interact with MBHs, but only if their pericentres are small enough. These orbits typically lead to the tidal disruption of an extended star; when these orbits are represented in phase-space by their energy and angular momentum, the section of phase space that leads to tidally disrupted systems is called the loss cone (Frank and Rees, 1976). \nThe range of frequencies that LISA will cover is ideally matched to the inspiral of a compact object such as a stellar-mass BH, a NS or a WD on to a (light) MBH; i.e., one with a mass between ∼ 10 4 M ⊙ and ∼ 10 7 M ⊙ . Because of the difference in mass between the MBH and the ∼ few -tens of solar masses of the compact object, we call these EMRIs-where the mass ratio q is 10 -8 < q < 10 -5 . There is also a potential population of BHs with masses between 10 2 M ⊙ and 10 4 M ⊙ , which are called intermediate-mass BHs (IMBHs). In principle, such BHs can be involved in intermediate mass-ratio inspiral (IMRI; 10 -5 < q < 10 -2 ) systems, either with a compact object inspiralling into them, or with them inspiralling into a MBH. These would fill the gap between EMRIs and comparable-mass binaries. IMRI observations are naturally complementary to EMRI observations, providing further insight into the development of MBHs \nand their surroundings and the possible evolutionary link between IMBHs and MBHs. Table (9) introduces the different acronyms used throughout this chapter, their meaning, mass ratio ranges, and configurations. \nTable 9: Nomenclature for the different types of mass ratio inspirals used in this chapter. \nThe frequencies of EMRIs are inaccessible to ground-based GW observatories, as are all but the lightest ( ≲ 10 3 M ⊙ ) IMRIs (Gair et al., 2011; Konstantinidis et al., 2013; Haster et al., 2016b; Amaro-Seoane, 2018a), yet their astrophysical production may be related to stellar-mass binary BHs (BH+BHs). Indeed, astrophysical mechanisms for generating EMRIs and IMRIs touch on an extremely diverse range of topics (see below), and LISA will lead the way in distinguishing between various formation channels, and furthering our knowledge in all of these areas. Both EMRIs and IMRIs have been reviewed in Amaro-Seoane et al. (2007) and more recently in Amaro-Seoane (2020, 2018b); Berry et al. (2019). Substellar objects, in particular brown dwarfs, with masses around 10 -2 M ⊙ form a third class of inspirals potentially observable in the LISA frequency range. These objects can last as many as 10 8 cycles before crossing the event horizon, due to their extremely large mass ratio, which is why they have been dubbed extremely large mass-ratio inspirals (XMRIs; q < 10 -8 ). XMRIs are particularly important in our own Galactic Centre, where a few of them should be present when LISA observations begin (Rubbo et al., 2006; Amaro-Seoane, 2019). \nOrdinary stars that are similar to our Sun would only complete a single periapsis cycle around a MBH before being tidally disrupted (for a close enough passage to enter the LISA frequency range). In contrast, compact stars can revolve around an MBH thousands or even hundreds of thousands of times, with the number of cycles roughly inversely proportional to the mass ratio. 13 Although the system is constantly emitting GWs, it is at the periapsis when the EMRI emits a strong burst of gravitational radiation. Since the orbit shrinks and precesses, we can envisage this as a probe taking pictures of spacetime around a MBH in the strong regime. \nObserving the large number of cycles of gravitational radiation emitted before disappearing into the event horizon has three main benefits: \n- · As an EMRI can spend up to hundreds of thousands of orbits within a few gravitational radii of the MBH, careful analysis promises to provide exceptionally precise tests of the nature of strong-field gravity and the Kerr nature of MBHs.\n- · Tracking the complicated orbit through many thousands of cycles yields outstanding measurements of parameters including the redshifted mass and spin of the MBH, without any modelling other than general relativity. In turn, this will give us hints about the evolution of MBHs, in a population that is typically inaccessible to EM observations, except, \nperhaps, for a limited number of local MBHs, e.g., with the Next Generation Event Horizon Telescope, but nevertheless with much less precision on mass and spin measurements compared to what LISA will do. \n- · If there is a correlation between the environment and the parameters of the EMRI or IMRI, we could reverse-engineer these to extract unique astrophysical information. \nHowever, these exciting prospects must be earned: the weakness of the signal from individual EMRI orbits means that detection, let alone parameter estimation, will require highly accurate computation of thousands of waveform cycles. EMRI waveform templates are challenging to model. Traditional computation techniques are not suitable because the post-Newtonian (PN) approximation (Blanchet, 2014) is inapplicable to these highly relativistic systems and numerical relativity calculations (Duez and Zlochower, 2019) are infeasibly computationally expensive because of the large difference in scales between the two binary components. Instead, templates can be calculated by treating the effects of the compact object as a perturbation to the background spacetime of the more massive MBH (Poisson et al., 2011; Barack and Pound, 2019). For IMRIs, the systems lie at the boundary of where perturbative methods may apply, where PN approximations may be used for the inspiral, and where numerical relativity simulations may be possible. Therefore, a combination of techniques will be needed to simulate IMRI templates. For EMRI and IMRI science, it will be essential to accurately compute these long waveforms in order to sift out these multi-year signals from the LISA data stream. \nIn this chapter we first give a summary about the formation mechanisms for EMRIs, which have received more detailed study than IMRIs or XMRIs, as well as the many different physical scenarios that play a crucial role in their event rate estimation. The fundamental theory on which these estimates rely, relaxation theory, is robustly understood and has yielded theoretical results which have been corroborated observationally. There remain astrophysical uncertainties which impact the EMRI rate, and there may be subtle effects that leave the theory incomplete. However, the theory has received extensive and detailed investigation in the context of EMRI formation and evolution in galactic nuclei over the last few decades. The narrative is complex. Here we will briefly summarise EMRI formation in the context of relaxation theory; for a detailed review see Amaro-Seoane (2020, 2018b). Assuming that at least one EMRI will be detected during the LISA mission, we lay out the anticipated science that is guaranteed, plausible, and speculative.', '3.1.1 Guaranteed science with the detection of EMRIs': "EMRIs are essentially guaranteed to happen in our Universe. The expected rates span a wide enough range that we cannot absolutely guarantee an observed EMRI in a 5-year mission (Mapelli et al., 2012). However, the uncertainties are such that LISA might also see multiple EMRIs-and if even one is observed, it is guaranteed to be a direct probe of the MBH spin. Currently our best measurements of MBH spin comes from studies of the broad component of the FeK α line in X-rays. The FeK α line is broadened due to relativistic effects (special and general) within a few 10 r g (here r g = GM MBH /c 2 , the gravitational radius) from the MBH. By assuming a compact X-ray illumination source close to the MBH, the FeK α line shape strongly constrains MBH spin (e.g. Reynolds and Nowak, 2003; Brenneman et al., 2011), but only for O (20) nearby AGN at present (Vasudevan et al., 2016), and both the statistical uncertainties and the possible systematic errors on the spin measurements are substantial. If we assume the Blandford-Znajek mechanism powers jets in radio-loud sources, it may also be possible to put constraints on MBH spin in a larger number of radio galaxies by measuring jet power (Daly, 2011). MBH spins have the power to reveal their growth history: what is the contribution due to mergers with other MBH versus gas accretion? The answers have implications for our understanding of galaxy assembly and evolution. In particular, near maximal spin would indicate that the most recent significant mass increase occurred via gas accretion, predominantly from a thin disc with coherent direction of the angular momentum; other individual spin measurements lead to less clear-cut \nconclusions but can permit constraints on the accretion history (see Section 2.3.2.4 for details). If many EMRIs are observed, we will have the opportunity to probe the distribution of MBH spins. Ideally, this would also enable us to probe a couple of decades in MBH mass, informing the underlying astrophysics of the M BH -σ relation. Second, GW inferences of MBH spin will allow us to test the assumptions underpinning EM measurements inferring spins (Daly, 2011; Vasudevan et al., 2016). \nEMRIs (and IMRIs) are also unique sources of GWs for studying fundamental physics with LISA-see more details in the white paper of the Fundamental Physics Working Group (Barausse et al., 2020a)-mainly because the small body spends a relatively high number of cycles (that scales with the inverse of the small mass ratio q ) very close to the horizon of the MBH, where precessional effects (periastron shift and orbital plane precession) become as strong as they can be (Babak et al., 2017; Berry et al., 2019). The orbital dynamics get imprinted in the GW signal by introducing the associated fundamental frequencies and their harmonics (Drasco, 2009). In the case of IMRIs there are extra timescales due to coupling of the small object spin with the orbital angular momentum and also with the MBH spin. These timescales evolve slowly due to gravitational backreaction of the small body gravitational field on its own trajectory. The orbital timescales depend on the MBH spacetime geometry (the Kerr geometry according to general relativity) and their evolution due to gravitational backreaction depends on the gravitational dynamics, which can be very sensitive to modifications to Einstein's equations of general relativity, such as modified gravity, extra fields, extra dimensions, etc. \nIt is expected (Babak et al., 2017) that EMRI and IMRI waveforms will be sensitive to both the parameters that describe the MBH geometry (mass, spin, and other gravitational multipole moments) and the parameters that describe deviations from the general relativity paradigm (coupling constants, extra dimension length scales, etc.). Therefore, they are unique probes of the geometry of the MBHs in galactic centres and of the particular details of the gravitational theory (and other non-gravitational fields that may affect the dynamics of EMRI/IMRIs) responsible for GW generation (Barack and Pound, 2019). \nHowever, in order to extract meaningful constraints from an EMRI (or IMRI) detection, it is essential to have reliable astrophysical predictions for the distribution of the key parameters of these systems. The values of such parameters determine to what level we can test the nohair conjecture and general relativity, and what kind of fundamental physics we can expect to carry out with LISA observations of EMRI/IMRIs. This leads us to discuss important plausible science, especially involving EMRIs in gas-rich environments.", '3.1.2 Plausible science with the detection of EMRIs': 'Identifying EMRIs in gas-rich environments could be an important observational result, as the effects of gas could in some cases mimic the effect of alternative theories of gravity. For AGNdriven EMRIs, we expect the gas to circularize prograde orbiters, and for the merger to occur in the equatorial plane of the MBH. For retrograde orbiters, however, the eccentricity could be driven to extremely large values (Secunda et al., 2019), and the interplay of gas- and GW-driven decay may be challenging to disentangle. By isolating the gas-driven mergers, we may also be able to directly probe important parameters of the gas, notably the density and viscosity-however, the detectability of the gas-driven phase shift requires substantial gas densities (Barausse and Rezzolla, 2008; Kocsis et al., 2011; Yunes et al., 2011a; Barausse et al., 2014; Derdzinski et al., 2019, 2020). \nAlthough there are no known IMRI systems, there are multiple plausible channels for their formation. In particular: \n- · Dwarf galaxies may contain an IMBH that could interact with stellar mass BHs (Koliopanos et al., 2017; Graham and Soria, 2019; Graham et al., 2019); \n- · Mergers of IMBH-bearing dwarf galaxies with larger MBH-bearing galaxies can produce systems with the relevant mass ratios;\n- · Globular clusters may contain IMBH that could (similar to dwarf galaxies) interact with stellar mass BH or decay into an MBH-bearing galactic nucleus;\n- · Finally, IMBHs could form through hierarchical mergers in a galactic nucleus-in particular in an AGN-which would continue to accrete stellar mass BH, while simultaneously creating a natural IMBH-MBH system. \nWe will discuss each of these formation scenarios in more detail in Section 3.2, and we further separate IMRIs into 2 classes: light IMRIs, where the primary mass is an IMBH and heavy IMRIs, where the primary mass is an MBH. Broadly, in the scenarios where we have the most theoretical confidence, the expected rates are small enough that there is a low probability of an event in the lifetime of a LISA mission (though as with EMRI rates, some uncertainties remain). In other scenarios, our uncertainty on many input parameters is such that the rate per galactic nucleus within the LISA observational horizon could be anywhere from zero to one per few years. However, this provides an excellent opportunity, even in the case of non-detections, to place important constraints on nuclear star clusters (NSCs) and their formation mechanisms, as well as on the structure and lifetime of AGN discs. In effect, LISA will enable us to reverse engineer important properties of AGN discs, including their radial surface density profiles and lifetimes. \nOne theoretical concern regarding IMRIs has recently been addressed: IMBH ( 10 2 -10 5 M ⊙ ) do exist in the relatively nearby universe. GW190521 demonstrated the formation of an IMBH ( > 100 M ⊙ ) at z < 1 (Abbott et al., 2020d). At the other end, some dwarf galaxies may host central IMBHs, at least at the lower end of their mass estimates (Moran et al., 2014; Koliopanos et al., 2017) and some of these may correspond to an extrapolation of the mass function and scaling relations towards low MBH mass ( 10 5 -10 6 M ⊙ ) (Reines et al., 2013; Graham and Scott, 2015). \nXMRIs, with q < 10 -8 are systems containing brown dwarfs (Amaro-Seoane, 2019). On account of their mass ratio, XMRIs would not evolve appreciably over the course of the LISA mission (Gourgoulhon et al., 2019). XMRIs are the GW event mostly likely to permit LISA to probe the Milky Way MBH and its nuclear star cluster. However, XMRI GWs would only be detectable from the Milky Way, or perhaps a few nearby galaxies. Nevertheless, given their expected rate from Sgr A*, and their possible interactions with the stellar cusp as they inspiral, XMRIs are an exciting probe of our nearest MBH and its environs. \nIn addition to the direct effects of fundamental physics on waveform generation, there are other effects that are accumulated during the propagation of the GWs from the source to the detector, such as those due to possible high-energy effects beyond general relativity: breaking of Lorentz invariance or of the weak equivalence principle, extra polarizations, gravitational parity violation, etc. (Barausse et al., 2020a). We can in principle detect those effects in EMRI/IMRI waveforms, but in the case of LISA, sources that are detectable at higher redshift, i.e. MBH binary coalescence, are more competitive in this regard. The test of the no-hair conjecture that EMRI/IMRIs provide are complementary to those that can be performed using quasinormal models excited in the ringdown of the final MBH after a MBH binary merger (Berti et al., 2018).', '3.1.3 Speculative science with the detection of EMRIs': 'EMRI/IMRI observations can also have impact on two other important subjects in fundamental physics. The first is the search for primordial BHs (Carr et al., 2021). Given the high precision expected for EMRI mass estimates, a detection would determine with confidence when the small compact object has a mass below what is reasonable from any astrophysical channel. This would be a strong indication of the primordial origin of that object. The other subject where \nEMRI/IMRIs can have an impact is in the understanding of dark matter. This is not independent from the previous subject since primordial BHs have been proposed to constitute all the dark matter in the observable universe (Carr and Kühnel, 2020). An example of how EMRI/IMRIs may probe the nature of dark matter is in the case where it is made by bosons that can form clouds around MBHs (see for our Galactic Centre Amaro-Seoane et al., 2010a), which would affect the orbital dynamics of EMRI/IMRIs and hence would leave an imprint in the waveforms (see, e.g. Hannuksela et al. (2019)). Moreover, EMRI/IMRIs could also contribute to the understanding of dark energy by adding significantly to the knowledge of the expansion history of the universe, assuming that we are able to determine their redshift, either from direct EM counterparts or via correlation with galaxy surveys (MacLeod and Hogan, 2008). \nFinally, the fundamental physics (also the cosmology) that we can investigate using some IMRI systems may be enhanced if we can do multiband GW astronomy, that is, by combining the information obtained with LISA with the information from detections with ground-based detectors, provided that the IMRI masses are such the system can enter the frequency band of the ground-detectors at a later time (Amaro-Seoane, 2018a; Datta et al., 2020).', '3.1.4 Data analysis & waveform modelling': "To unlock the rich scientific potential of EMRI and IMRI observations, we must be able to extract these signals from the LISA data stream. EMRI detection and characterisation is one of the most challenging problems in LISA data analysis (Amaro-Seoane et al., 2007, 2015; Amaro-Seoane, 2018b, 2020). There are three main sources of challenge: \n- · The complexity of EMRI signals-EMRI orbits are generically eccentric and precessing over a large number of cycles. The waveforms are thus extremely sensitive to the source parameters, and there is a gargantuan space of potential signals to be searched.\n- · The length of EMRI signals-EMRIs need to be tracked for an extended time in order to accumulate sufficient signal-to-noise ratio (SNR) to be detectable. If the phase cannot be accurately tracked, either due to hard-to-model effects, like transient resonances (Flanagan and Hinderer, 2012; Berry et al., 2016) or higher-order corrections due to nonlinear interactions of the compact object's gravitational field (Barack and Pound, 2019), or unaccounted for environmental effects, such as viscous drag (Barausse and Rezzolla, 2008; Kocsis et al., 2011; Barausse et al., 2014) or perturbations from a nearby object (Amaro-Seoane et al., 2012; Bonga et al., 2019), this will impact detectability.\n- · The number of EMRI signals-EMRIs are long-lived and possibly numerous. Thus there may be many EMRI signals in the LISA data stream at any given time, overlapping with one another as well as with signals from the multitude of other sources. This means that any data analysis strategy for EMRIs must be part of a global fit that analyzes all signals concurrently. \nThe first challenge means that, unlike when searching for LIGO-Virgo signals (e.g., Allen et al., 2012; Abbott et al., 2016b), it is computationally infeasible to perform a matched-filter search with a regular grid of templates: it has been estimated that ∼ 10 40 templates may be needed for this goal (Gair et al., 2004). Instead, we must trade search sensitivity for computational expediency. Multiple data analysis approaches have been explored following the initiation of the Mock LISA Data Challenges (Babak et al., 2008a,b, 2010). Techniques include identifying timefrequency tracks (Wen and Gair, 2005; Gair and Jones, 2007; Gair et al., 2008b,a), although this can be difficult in the presence of multiple signals, or using Monte Carlo techniques to stochastically search for signals, either using EMRI templates (Stroeer et al., 2006; Gair et al., 2008c; Babak et al., 2009; Cornish, 2011; Ali et al., 2012) or phenomenological waveforms (Wang et al., 2012). These techniques still do not extend to the full scope of global EMRI search, \nwhich must ultimately be conducted in a hierarchical fashion (Gair et al., 2004; Chua et al., 2017). Stochastically searching parameter space to fit for EMRI signals is especially challenging as there may be many regions in parameter space where there are good matches to the data, aside from the vicinity of the true parameters (Babak et al., 2008b, 2010); the full extent and severity of such parameter degeneracy is difficult to determine due to the size of the parameter space and the lack of tractable waveforms, and is currently being investigated (Chua and Cutler, 2021). Design of EMRI analyses hence remains an open area of research. IMRI detection is less well studied, but should be possible with a combination of the techniques designed for EMRIs and equal-mass binaries. \nEssential to measuring the properties of EMRIs and IMRIs is the modelling of the gravitational waveforms. Only with accurate models can the source properties be inferred. For EMRI waveforms, the highest accuracy waveforms are calculated using the self-force formalism (Poisson et al., 2011; Blanchet, 2019): the compact object's gravitational field is treated as a perturbation to the background spacetime of the larger black hole. For full characterisation of EMRI signals, we will require calculation of self-force effects to second order in the mass ratio for generic orbits in Kerr spacetime (Rosenthal, 2006). The self-force program is now well advanced (Barack and Pound, 2019); the self-force has been calculated to first order for a generic orbit in Kerr spacetime (van de Meent, 2018), and the foundations have been laid for a second-order calculation (Pound and Miller, 2014; Pound, 2014; Miller et al., 2016; Pound, 2017; Moxon and Flanagan, 2018; Pound et al., 2020). It is expected that concerted waveform development will lead to successful computation of EMRI waveforms ahead of LISA's launch. In the meantime, less accurate approximate waveforms are used for developing LISA data analysis. The most common approximations are the analytic kludge (Barack and Cutler, 2004b; Chua et al., 2017), based upon a Keplerian orbit augmented with relativistic corrections, and the numerical kludge (Babak et al., 2007), based upon Kerr geodesics mapped to flat spacetime for waveform generation. The low computational cost of these models makes them suitable for early stages of LISA data analysis, where we are looking for EMRI-like signals, but are not concerned about the precise parameter values. IMRI waveforms present a challenge as they lie at the boundary of different techniques for waveform generation (Hinderer and Flanagan, 2008; Blanchet, 2014). The self-force calculations may cover a large range of IMRI parameter space (van de Meent and Pfeiffer, 2020). These can be supplemented with calculations from PN theory (Blanchet, 2014; Buonanno et al., 2009) and effective one-body theory used for more equal mass binaries (Buonanno and Damour, 1999, 2000; Taracchini et al., 2014; Ossokine et al., 2020). Finally, numerical relativity can model IMRIs. Numerical relativity should give exact numerical solutions to the Einstein equations (Sperhake, 2015), but IMRIs require extremely high spatial and temporal resolution. Therefore, computation of high fidelity numerical relativity IMRI waveforms for LISA may require the development of a new generation of codes. The best IMRI waveform models should be produced through combining the strengths of each of these formalisms. Both EMRIs and IMRIs provide a valuable chance to validate our waveform calculation theory in new regions of parameter space.", '3.2 Formation channels': 'Coordinators: Manuel Arca Sedda, Xian Chen, and Andrea Derdzinski Contributors: Pau Amaro Seoane, Manuel Arca Sedda, Jillian Bellovary, Elisa Bortolas, Pedro R. Capelo, Xian Chen, Andrea Derdzinksi, Gaia Fabj, Saavik Ford, Jean-Baptiste Fouvry, Zoltan Haiman, Wen-Biao Han, Giuseppe Lodato, Barry McKernan, Syeda Nasim, Amy Secunda and Martina Toscani', '3.2.1 Gas-poor dynamics: Galactic nuclei including dwarfs, and globular clusters': 'Observations of galaxies and their nuclei have revealed close relationships between several galactic properties and the masses of their central MBHs (Seigar et al., 2008; Gültekin et al., 2009; Kormendy and Ho, 2013; Berrier et al., 2013; Reines and Volonteri, 2015; Graham, 2016b; Davis et al., 2017, 2018, 2019a; Sahu et al., 2019a,b; Davis et al., 2019b; Sahu et al., 2020). Extrapolating these relations to the lower mass end, one would expect 10 3 -10 5 M ⊙ IMBHs to exist in the centres of dwarf galaxies (Mezcua, 2017; Greene et al., 2019), as suggested by X-ray observations of low-mass AGN (Koliopanos et al., 2017; Mezcua et al., 2018; Graham and Soria, 2019; Graham et al., 2019; Reines et al., 2020). IMBHs may also form via collisions and mergers of stars and stellar-mass BHs in dense clusters (Portegies Zwart and McMillan, 2000; Miller and Hamilton, 2002b; Giersz et al., 2015; Mapelli, 2016; Di Carlo et al., 2021; Rizzuto et al., 2021; Arca-Sedda et al., 2021; González et al., 2021; Rizzuto et al., 2022). Dynamical friction can subsequently lead to the orbital decay of globular clusters into a galactic nucleus (Tremaine et al., 1975; Tremaine, 1976; Capuzzo-Dolcetta, 1993), allowing the formation of an IMBH-MBH system (Ebisuzaki et al., 2001; Matsubayashi et al., 2004; Portegies Zwart et al., 2006; Matsubayashi et al., 2007; Gualandris and Merritt, 2009; Arca-Sedda and Gualandris, 2018). Depending on the populations of stars and BHs in these environments, galactic nuclei (including dwarf nuclei) and globular clusters each provide plausible formation channels for EMRIs and IMRIs. We group these channels together because the underlying physical mechanisms for EMRI and IMRI formation is similar for all (gravitational interactions alone); in addition, these channels interact with one another astrophysically through mergers. There remain major astrophysical uncertainties in each formation channel, meaning that LISA observations (perhaps including non-detections) can crucially constrain the astrophysics that lead to their formation.', '3.2.2 Formation of EMRIs in gas-poor galactic nuclei': '3.2.2.1 Physics of EMRI formation I. Relaxation processes MBHs, often surrounded by nuclear star clusters, seem ubiquitous in the centre of nearby galaxies (Graham and Spitler, 2009; Genzel et al., 2010; Kormendy and Ho, 2013; Neumayer et al., 2020). Yet, the details of MBH formation, and their impact on both their surrounding NSC and their host galaxy, remain uncertain (Heckman and Best, 2014). Owing to the overwhelming mass of the central MBH and the steep potential well that it generates, NSCs encompass a wide range of dynamical processes that act on radically different timescales (Rauch and Tremaine, 1996; Hopman and Alexander, 2006; Alexander, 2017). The key dynamical processes that can generate an EMRI around a MBH hosted in an NSC are briefly illustrated in Fig. 30. Of these processes, the two-body relaxation time is the slowest, but also the main mechanism for producing EMRIs. We now discuss each timescale and their relevance for studying EMRIs with LISA. After that, we will give a description of how our understanding of EMRI formation has evolved in the last decades, and how LISA can help in that respect.', '(1) Dynamical time': "On account of its mass, the central MBH dominates the nucleus's mean potential, and imposes, at leading order, a Keplerian motion to any object orbiting within the MBH's sphere of influence. These motions take to leading order the form of closed ellipses, for example as currently monitored for the S cluster around Sgr A* (see, e.g., Fig. 31). A Keplerian orbit can be described by its orbital elements (Murray and Dermott, 1999), which we denote with ( M,ω, Ω , L c , L, L z ) . Here, M stands for the mean anomaly, ω is the argument of the pericentre, and Ω the longitude of the ascending node. An orbit is also characterised by three actions ( L c , L, L z )=( √ G M · a , L c √ 1 -e 2 , L cos( I )) . Here, M · is the mass of the central MBH, a the ellipse's semi major axis, e its eccentricity, I its inclination, L the norm of the angular momentum, L z its projection along a given axis, \nFigure 30: Illustration of the hierarchy of timescales in a NSC: (1) the dynamical time; (2) the precession time; (3) the vector resonant relaxation time; (4) the scalar resonant relaxation time; (5) the two-body relaxation time. \n<!-- image --> \nFigure 31: Keplerian orbits around an MBH. Left: Detailed observations of the Keplerian dynamics of the S-stars in the vicinity of Sgr A* ( ∼ 10 mpc ) from Gillessen et al. (2017). At leading order, orbits take the form of closed ellipses, because the central MBH dominates the gravitational potential. Right: Illustration of the Keplerian orbital elements from Murray and Dermott (1999). \n<!-- image --> \n���� \n[ \n�� \n] \nand finally L c the circular angular momentum. Describing dynamics in NSCs amounts then to describing the dynamics of these particular orbital elements. The dynamical time is associated with the motion ˙ M = √ G M · / a 3 . This dynamical time being so short, one is naturally led to performing an orbit-average over it, i.e. smearing out the orbiting objects along their Keplerian ellipses (see, e.g., Touma et al., 2009).", '(2) Precession time': 'On longer timescales, the gravitational potential self-consistently generated by the stellar cluster, as well as the relativistic corrections imposed by the MBH, namely the Schwarzschild precession (Merritt, 2013) cause the ellipses to precess in their planes. This drives the evolution of ω . Importantly, one can note that the relativistic precession frequency diverges as orbits get more and more eccentric, which ultimately leads to the breakdown of the orbit-averaged assumption. Such a relativistic precession has recently been observed for the S2 star around Sgr A* by the Gravity interferometer (Abuter et al., 2018). This is presently the best direct observational constraint on the metric in the vicinity of Sgr A*.', '(3) Vector resonant relaxation': "Subsequently, because of the non-spherical stellar fluctuations in the NSC, as well as the relativistic corrections induced by a spinning MBH, the Lense-Thirring precession (Merritt, 2013), the ellipses' orbital orientations get reshuffled. This process is called vector resonant relaxation (Kocsis and Tremaine, 2015). In that limit, the orbits' angular momenta change in their orientations, ̂ L = L / | L | , without changing in magnitude | L | (equivalently in e ), nor in energy L c (equivalently in a ). \nThe process of vector resonant relaxation has been studied, among others, through numerical simulations (Eilon et al., 2009), orbit-averaged simulations (Kocsis and Tremaine, 2015), as well as thermodynamical (Roupas et al., 2017; Takács and Kocsis, 2018) and kinetic theories (Fouvry et al., 2019). Vector resonant relaxation is essential to describe the warping of accretion (Bregman and Alexander, 2012) and stellar discs (Kocsis and Tremaine, 2015), and can enhance the rate of binary mergers in NSCs (which could naturally produce an IMBH-MBH binary). Furthermore, it can also explain the possible presence of stellar discs (Bartko et al., 2009; Yelda et al., 2014), or even strong anisotropic mass segregation of IMBHs discs (Szölgyén and Kocsis, 2018).", '(4) Scalar resonant relaxation': "On longer timescales, resonant torques between in-plane precessing orbits lead to an efficient diffusion of the ellipses' eccentricity. This process is called scalar resonant relaxation (Rauch and Tremaine, 1996) as the quantity that diffuses is the norm of the orbit's angular momentum. It is also said to be resonant, as only orbits that precess with matching in-plane precession frequencies will effectively and constructively interact with one another. \nThis relaxation process has been investigated through ad hoc methods (Hopman and Alexander, 2006; Eilon et al., 2009; Madigan et al., 2011; Antonini and Merritt, 2013; Merritt, 2015), as well as N -body simulations (Perets et al., 2009; Merritt et al., 2011; Antonini and Merritt, 2013; Hamers et al., 2014), and kinetic theories (Bar-Or and Alexander, 2014; Sridhar and Touma, 2016; Bar-Or and Fouvry, 2018). Scalar resonant relaxation may be paramount to explain the thermal distribution of stellar eccentricities around Sgr A* (Generozov and Madigan, 2020), while not necessarily mandatory (Chen and Amaro-Seoane, 2014). However, its efficiency drastically damps as orbits get very eccentric, an effect called the Schwarzschild barrier (Merritt et al., 2011). This particular problem has been addressed in detail by Bar-Or and Alexander (2014). They have shown that the divergence of the relativistic precession frequency as orbits get more and more eccentric is responsible for a drastic dampening of the efficiency of resonant relaxation. As such, the Schwarzschild \nbarrier corresponds to the abrupt transition from a relaxation dominated by resonant interactions (for not too eccentric orbits) to a relaxation dominated by non-resonant two-body scatterings (for eccentric enough orbits). Given that mainly highly eccentric stellar orbits undergo EMRIs, the total contribution of scalar resonant relaxation to EMRI event rates is small (Bar-Or and Alexander, 2016).", '(5) Two-body relaxation': 'Finally, on even longer timescales, rather than being driven by the interaction between Keplerian ellipses, an object will start to see its evolution being driven by nearby pairwise interactions, as a result of close two-body encounters. It is only through the cumulative contributions from these localised scatterings that objects can ultimately relax in their Keplerian energy (i.e. in a ), through a process called two-body relaxation (Bahcall and Wolf, 1976; Cohn and Kulsrud, 1978; Shapiro and Marchant, 1978; Bar-Or and Alexander, 2016; Bar-Or et al., 2013; Vasiliev, 2017; Amaro-Seoane, 2018b, 2020). It is generically the slowest relaxation timescale in NSCs. The main mechanism for producing EMRI in NSCs is two-body relaxation. This is because it allows for the orbits to become highly eccentric, where other resonant processes significantly damp eccentricity (Bar-Or and Alexander, 2014); the expected EMRI rates depend on the spin of the central MBH (Amaro-Seoane et al., 2013). We will elaborate on this later. \nOne of the first attempts to understand how to produce a successful orbit in a galactic nucleus that will lead to the formation of an EMRI goes back to the work of Sigurdsson and Rees (1997). By using standard relaxation and loss-cone theory (see Sec. 3.2), the authors derived the event rate for compact objects to merge with a MBH in a galactic nucleus. It is important to note that for MBHs above about a few 10 7 M ⊙ , the timescales for relaxation exceed a Hubble time; LISA is going to observe MBHs below this threshold, down to 10 5 M ⊙ . For lower masses, i.e. for IMBHs and hence IMRIs, we cannot further assume that the MBH is fixed at the centre of the stellar system and any analytical derivation becomes more difficult. \nWe define now a standard EMRI to consist of a stellar mass object of mass 10 M ⊙ and a MBH with a mass such as that of Sgr A*, the MBH in our own Galaxy. The event rate for this kind of EMRI, we obtain of the order of 10 -5 -10 -6 yr -1 (see e.g. Amaro-Seoane, 2018b, 2020, and references therein). This analytical result has been reproduced using numerical algorithms such as in the work of Freitag (2001). The properties of EMRIs formed via relaxation are such that they have large eccentricities at semi-major axis of about 0 . 1 -1 pc. They describe a random-walk-like evolution in phase-space, in particular in angular momentum, until one of two things happens: (i) the small mass deviates off the orbit which would evolved into an EMRI that inspirals into the MBH, or (ii) they cross a threshold in phase-space which separates orbital evolution dominated by dynamics into a regime where orbital evolution is driven only by the emission of GWs. When systems cross this line, which can be roughly derived by equating the relaxation timescale at pericenter to the associated timescale due to the emission of gravitational radiation, as derived by Peters (1964a), we can ignore any dynamical perturbation. \nThe increase in eccentricity during the EMRI formation can lead to a situation in which the eccentricity is so high that the smaller BH falls radially on to the MBH, and there is only one or a handful of gravitational radiation bursts before the source is lost. This can be regarded as a head-on collision. This is what is commonly referred to as a direct plunge in the related literature (not to be confused with the plunge when the EMRI crosses the event horizon after hundreds of thousands of orbits). Direct plunge sources are basically lost, because we can extract little or no information from it (but see Hopman et al., 2007; Berry and Gair, 2013c,a). It has been shown that the ratio between successful EMRIs and direct plunges could be of about 1 : 200 respectively (Amaro-Seoane, 2018b, 2020, and references therein). \nThis result led to an interesting new avenue in investigating the role of other types of relaxation. By getting closer and closer to the MBH, the stellar density drops, so that the danger of \nproducing direct plunges due to the accumulation of gravitational tugs of the orbit at apocentre is accordingly reduced. At the same time, the usual two-body relaxation time increases more and more. In addition, the process of scalar resonant relaxation was found to be inefficient in this region of phase-space. \nHowever, as we explain later in this section, direct plunges mostly occur in Schwarzschild MBHs. If the MBH has a spin, any direct-plunging orbit turns out to be a successful EMRI, meaning that it spends tens and up to hundreds of thousands of cycles in the LISA band, depending on the inclination and the spin of the MBH, as shown in Amaro-Seoane et al. (2013). This has an impact on the event rate, because the ratio of 1:200 that we mentioned before increases in favour of successful EMRI orbits. \nWe note that recently, Zwick et al. (2020) derived an improvement to the pioneering work of Peters (1964a), extending the timescale to be accurate to first post-Newtonian order. By taking into account this modification, the EMRI rates drop by at least one order of magnitude per nucleus. But then the role of the spin has another impact on the inspiraling timescale that might again enhance the event rate (Zwick et al. 2021; Vazquez-Aceves et al in prep 2020b). \nTo sum up, relaxation is a robust theory which has been tested in observations in dense stellar systems. In particular, recent results show that theory, numerical simulations and observational data agree on the existence of a segregated stellar cusp at our Galactic Centre (Baumgardt et al., 2018; Schödel et al., 2018; Gallego-Cano et al., 2018; Panamarev et al., 2019). This theory has been used in numerical simulations along with relativistic corrections (both PN and geodesic ones) to derive the event rate of EMRIs for a Milky Way-like nucleus (Amaro-Seoane and Preto, 2011; Brem et al., 2014; Arca-Sedda and Capuzzo-Dolcetta, 2019). \n3.2.2.2 Physics of EMRI formation II. Formation and disruption of binaries around a massive black hole Besides stellar relaxation, binary separation is another way of delivering stellar BHs to the vicinity of a MBH and forming EMRIs (Miller et al., 2005). In this model, a binary containing a stellar-mass BH could form relatively far away from the MBH and later be scattered by other stars to the vicinity of the MBH. If the periastron distance is smaller than the tidal-disruption radius of the binary, the most likely outcome is that the binary gets tidally disrupted, leaving the stellar-mass BH gravitationally bound to the MBH and the other binary component ejected from the system. The event rate is difficult to estimate because of the uncertainties in the physical properties of nuclear star clusters, but given the large cross section for tidal separation of binaries, it is considered that this channel could make a significant contribution to the total EMRI population. Unlike the EMRIs formed via stellar relaxation, the EMRIs produced by tidal separation have a much lower eccentricity in the LISA band because the captured stellar-mass BHs initially have a larger binding energy. As a result, these loweccentricity EMRIs are less susceptible to the perturbation by the stars around the MBHs and hence are more stable. \nThe contribution of LISA to the physics of EMRI formation As we have tried to convey in this section, there are a number of different processes that leave the expected EMRI event rate detectable by LISA substantially uncertain. It seems likely that many competing effects produce unique signatures in the LISA observables-either for individual events or in the distribution of their properties over multiple events. Theory work in advance of launch will help determine which effects may be dominant, and what observables are correlated with which effects. Hence, with detections in hand, we can hope to observe these phenomena and therefore address many open questions related to astrophysics, in addition to fundamental physics. \n- · LISA can determine the ratio of plunges (coalescence that involve tens or fewer orbits) to insprials (coalescence involving thousands or more orbits), since it can distinguish the waveforms of the two types of coalescences. Such a number can be used to test the predictions of stellar-dynamics models (Amaro-Seoane, 2018b, 2020, and references therein). \n- · LISA can measure the eccentricity of an EMRI to a precession of 10 -6 (Babak et al., 2017). Such a measurement can reveal those EMRIs formed by the binary-separation channel, since they have relatively mild eccentricities ( ∼ 0.1, Miller et al., 2005), while those produced by stellar relaxation preferentially have extreme eccentricities ( > 0.9).\n- · LISA can identify those EMRIs with zero eccentricities, and they are most likely produced in AGN accretion disks. Moreover, the small sky-localization error (on average smaller than 1 deg 2 , Babak et al., 2017) of LISA may help us identify the host AGN of the EMRI and understand the condition favorable to the formation of such wet EMRIs.', '3.2.3 Physics of IMRI formation: Dwarf galaxies, galactic nuclei and globular clusters': "3.2.3.1 Heavy IMRIs from galaxy mergers By providing the first access to the GW universe in the millihertz band, LISA will reveal the population of high-redshift MBHs coevolving with their host galaxies to ultimately assemble the galaxies of today's Universe. While much of our astrophysical understanding of BHs and galaxies through the cosmic dawn era, say 5 < z < 20 , has so far focused on the largest structures including massive highz quasars and their hosts, these are not the ancestors of today's typical galaxies. Galaxies like the MW were assembled from smaller, mainly dwarf, galaxies, and our galaxy's MBH grew through some combination of accretion and merger of smaller BHs hosted in these dwarfs, provided that orbital decay is efficient (see Section 2.2.1). EM observations of the smaller systems through this epoch will remain extremely challenging, making LISA's data crucial for understanding our cosmic history. \nRegardless of how IMBH seeds do form, it is likely that they form in low-mass ( ∼ 10 8 -10 9 M ⊙ ) galaxies at high redshifts. These galaxies merge over time to build up more massive galaxies, and any BHs they host will likely merge as well. Further evidence in the local universe supports this hypothesis: dwarf galaxies possibly hosting IMBHs/MBHs have been discovered recently, lending further support that low-mass galaxies can be IMBH hosts (Reines et al., 2013; Moran et al., 2014; Pardo et al., 2016; Ahn et al., 2017; Chilingarian et al., 2018). When these low-mass galaxies merge with larger halos, they are tidally stripped and disrupted, with their remnants joining the greater halo population. An IMBH orbiting in the halo will eventually spiral to the centre due to dynamical friction, and merge with the central MBH. \nCosmological simulations have shown that when dwarf galaxies hosting IMBHs merge with larger halos, they are tidally disrupted, leaving the IMBH to wander within the larger galaxy halo (Bellovary et al., 2010; Tremmel et al., 2017; Bellovary et al., 2019). Depending on their orbits and dynamical timescales, these IMBHs may spiral into the galactic centre and merge with the existing MBH. Bellovary et al. (2019) have shown that this event has a characteristic mass ratio of ∼ 20 : 1 , which is reflected in the large peak at low mass ratios in Fig. 32. \nDwarf galaxies are the most numerous in the Universe, and while the occupation fraction of BHs in dwarfs may be less than 1, because dwarf mergers with larger halos are common, this case cannot be ignored. This type of BH-BH merger is the most common interaction in low-mass galaxy environments. \n3.2.3.2 Heavy IMRIs from galactic nuclei assembly As discussed in the previous section, the mergers between dwarf satellites and their main galaxies can lead to the formation of IMBH-MBH pairs (corresponding to heavy IMRIs) in galactic nuclei. However, the corresponding merger timescale is long due to the large mass ratio between the merging galaxies, leaving this scenario unfavorable for the formation of IMRIs (Amaro-Seoane et al., 2007). For example, theoretical models show that hierarchical galaxy mergers produce < 10% of binary BHs with q < 0 . 01 (Volonteri et al., 2003a, 2020). For those IMRIs whose primary MBHs fall in the mass range of 10 5 -10 7 M ⊙ so that they can be detected by LISA, the event rate may be low because \nFigure 32: Peak in IMBH/MBH mergers at mass ratios ∼ 20 : 1 (Bellovary et al., 2019). \n<!-- image --> \nthe host galaxies are relatively small, and low-mass galaxies merge much less frequently than those heavier ones. \nAnother viable route to the formation of an IMBH-MBH binary is related to the possible formation of IMBHs via stellar collisions and accretion onto stellar-mass BHs in the nuclei of dense stellar clusters, i.e. globular clusters (Portegies Zwart and McMillan, 2002; Giersz et al., 2015; Mapelli, 2016; Arca Sedda et al., 2019a; Rizzuto et al., 2020). \nClusters forming sufficiently close to the centre of their host galaxy can migrate toward the galactic centre via dynamical friction (Tremaine et al., 1975; Capuzzo-Dolcetta, 1993). This mechanism is thought to contribute to the growth of galactic nuclei (Tremaine et al., 1975; Capuzzo-Dolcetta, 1993; Gnedin et al., 2014; Arca-Sedda and Capuzzo-Dolcetta, 2014; Antonini, 2013). Clusters harboring an IMBH can bring the black hole to the galactic innermost regions and release it in the galactic centre. Such a mechanism might contribute to the seeding and growth of MBHs (Ebisuzaki et al., 2001; Portegies Zwart et al., 2006; Arca-Sedda et al., 2015; Arca-Sedda and Capuzzo-Dolcetta, 2017; Askar et al., 2021). If one or more IMBHs reach the galactic centre after the MBH is fully grown, the subsequent interaction between the MBH and the IMBH can trigger the formation of a massive binary that might undergo coalescence within a Hubble time. Given the typical range of mass of IMBHs ( 10 2 -10 5 M ⊙ ) and MBHs ( 10 5 -10 10 M ⊙ ) these merging binaries would have mass ratios in the range of 10 -2 -10 -5 , typical of IMRIs. In massive elliptical galaxies, this mechanism can drive the formation of several IMRIs over a Hubble time, with an inferred rate of around 0 . 003 -0 . 03 Gpc -3 yr -1 (e.g. Arca-Sedda and Gualandris, 2018; Arca-Sedda and Capuzzo-Dolcetta, 2019). Upon the simplest assumption that these mergers are distributed uniformly through space and are all detectable with LISA within a redshift z < 1 (Sesana et al., 2021), we can derive an upper limit to the number of LISA detections of 2 -20 yr -1 . \nMassive ellipticals are not the only suitable nurseries for IMBH-MBH binaries. A number of IMBHs might be hiding in plain sight in our own Galaxy. The possible mass and location of such IMBHs can be constrained by the proper motion of Sgr A* and the kinematics of the S-stars close to it (Yu and Tremaine, 2003; Hansen and Milosavljević, 2003; Reid and Brunthaler, 2004), as well as the TDE rate in the Galactic Centre (Chen and Liu, 2013). According to these earlier studies, the possibility of an IMBH with a mass of ≲ 2000 M ⊙ and residing at a distance of ≲ 10 -3 pc from Sgr A* is not excluded. However, the motion of S-stars orbiting Sgr A* suggests that if the MW MBH has a companion IMBH, its mass should be most likely smaller than 10 3 M ⊙ if the IMBH-MBH binary orbital period exceeds 5 yr, or up to 10 5 M ⊙ if the binary separation falls in the range 0 . 1 -1 mpc (Gualandris and Merritt, 2009; Arca-Sedda and Gualandris, 2018). The recent measurements of relativistic precession in the S2 star (Abuter et al., 2018) helped in further constraining the phase space allowed for an IMBH, ruling out companions with a mass \n10 5 M ⊙ orbiting within 170 AU ( 0 . 8 mpc ) from Sgr A* (Naoz et al., 2020). Nonetheless, there is growing suspicion of IMBH candidates orbiting farther away, around 1 -10 pc from the MBH. These putative IMBHs are supposedly harboured in a handful of compact gaseous clouds, whose measured velocity dispersion is so high to suggest the presence in their centres of point-like objects with masses in the range 10 4 -10 5 M ⊙ (Oka et al., 2017; Takekawa et al., 2019, 2020). However, depending on their orbital properties, a population of IMBHs lurking at the Galactic Centre would affect significantly the motion of S-stars (Deme et al., 2020b) and the structure of the nuclear star cluster (Mastrobuono-Battisti et al., 2014). \nThe detection of such heavy IMRIs by LISA would have huge implications for our understandings of IMBH formation and evolution. \n3.2.3.3 Light IMRIs in stellar clusters and dwarf nuclei We can also imagine the formation of light IMRIs, where the IMBH is the more massive partner in a merger with a stellar mass BH. At first glance, one might expect these to be simply scaled down versions of the EMRI problem with a central MBH. However, the challenge of understanding stellar dynamics with an IMBH is that we cannot assume that the IMBH is fixed at the centre of the system, as we do with MBH in galactic nuclei. The wandering of the IMBH makes it demanding, to say the least, to attempt an analytical study, and hence we have to resort to numerical simulations to get an idea of what could be the IMRI event rate and the characteristic properties. Globular clusters and dwarf nuclei may also present quite different dynamical scenarios (higher escape velocities due to dark matter but lower stellar densities in dwarf galaxies, for example). \nThe first dynamical simulation addressing the evolution of a globular cluster harbouring an IMBH which successfully led to the formation of an IMRI was done by Konstantinidis et al. (2013). In this work they employed a direct-summation N -body code with relativistic corrections as presented by and a live treatment of the relativistic recoil. They find that IMBHs with masses 500 -1000 M ⊙ merge with stellar-mass BHs and escape the host globular cluster due to the low escape velocity of the system. The IMBH is in a binary in almost all cases. The companion is a stellar-mass BH of mass ∼ 20 -26 M ⊙ , and semi-major axis of about 5 -7 AU . Later, Leigh et al. (2014) found similar results for this mass range. In their simulations, the heaviest stellar-mass BH forms a tight binary with the IMBH in the system. The work of Haster et al. (2016a) is basically a reproduction of Konstantinidis et al. (2013), but using a different numerical scheme. This is interesting because it validates of the findings of Konstantinidis et al. (2013). Later, MacLeod et al. (2016b) explore lighter mass ranges for the IMBH, with masses at most of 150 M ⊙ . They confirm that the IMBH has a bound companion most of the time, with the probability distribution function for the semi-major axis maximised at 2 AU . \nRecently, Pestoni et al. (2021) performed a series of Fokker-Planck simulations to explore the occurrence of light IMRIs around IMBHs of 10 5 M ⊙ residing at the centre of massive starforming clumps in high-redshift galaxies, finding event rates of 10 -8 -10 -7 yr -1 , depending on the assumptions for the initial inner density profile. \nThe IMRI rate from globular clusters and detectable by LISA depends intrinsically on a number of unknown quantities, namely the fraction of clusters capable of nursing the IMBH seed and growth (Portegies Zwart and McMillan, 2002; Giersz et al., 2015), the amount of stellar-mass BHs and other compact objects lurking in the IMBH closest vicinity (MacLeod et al., 2016b; Arca Sedda et al., 2019a), the number of times the same IMBH can pair in an IMRI (MacLeod et al., 2016b), and the probability that upon merger an IMBH is ejected from the parent cluster due to anisotropic GW emission (Holley-Bockelmann et al., 2008; Fragione et al., 2018; Arca Sedda et al., 2021a). Depending on all these quantities, LISA might be able to detect 0 . 01 -60 IMRIs from globular clusters per year out to redshift z ∼ 2 (Arca Sedda et al., 2020a, 2021a). The number of light IMRIs from massive star-forming clumps detectable by LISA falls in the same ballpark: by integrating their computed IMRI event rate over z = 1 -3, when clumpy galaxies are more numerous, Pestoni et al. (2021) computed that LISA should be able to detect \n∼ 2 IMRIs per year, conservatively assuming that one star-forming clump per clumpy galaxy hosts a central IMBH. IMRIs with IMBHs in the mass range between 10 2 M ⊙ and a few 10 3 M ⊙ might be detected with LISA and provide advanced warning to ground-based detectors with a precision up to a second (Amaro-Seoane, 2018a). \nThe contribution of LISA to the physics of heavy- and light-IMRI formation in gaspoor environments In the processes we have just described with regard to the formation and evolution of heavy- and light IMRIs in gas-poor scenarios, our astrophysical theory so far only provides loose guidance about how the physics of the assembly process connects to the characteristics of this population. Current models suggest, though, that understanding heavy IMRIs will be an important discriminator between hierarchical formation models. \nIMRIs provoke more open questions than EMRIs, as the theoretical frameworks required to address IMRI formation are even more complex, due to the mobility of an IMBH, as compared to an MBH. Thus, many approximations used in relaxation theory to predict the gas-poor dynamics of EMRI formation cannot be applied to an otherwise similar IMBH system. In addition, there are several plausible channels of IMRI formation that involve theoretical frameworks that have far larger uncertainties than relaxation theory. \nAs for galactic nuclei assembly, several uncertainties might affect the formation of such heavy IMRIs, namely the number of clusters capable of reaching the galactic centre, the probability of IMBH formation, the relation between IMBH formation and the host cluster properties. Theoretical models suggest that star cluster infall and dispersal could bring 1-50 IMBHs in the MBH vicinity (Portegies Zwart et al., 2006; Mastrobuono-Battisti et al., 2014; Arca-Sedda and Gualandris, 2018; Arca-Sedda and Capuzzo-Dolcetta, 2019; Leveque et al., 2022; Fragione, 2022). If these models prove right, the delivery of IMBHs can give rise to up to 0.001-0.03 Gpc -3 yr -1 (Arca-Sedda and Capuzzo-Dolcetta, 2019; Fragione, 2022), corresponding roughly to 1 event per year within redshift 0.5-3 (Fragione, 2022). Interestingly, the possible ejection of the IMRI product from the parent nucleus owing to GW radiation could account for up to 10 5 Gpc -3 MBH wandering outside their host galaxies at redshift < 1 (Arca-Sedda and Capuzzo-Dolcetta, 2019). \nFinally, less consideration has been given to possible light IMRI rates in dwarf nuclei, although the presence of at least some IMBH in dwarf galaxies is more secure than in globular clusters. This is an important open question, especially since IMRI detection in dwarf nuclei could enable us to probe IMBH in quiescent dwarf galaxies. This in turn leads to a better understanding of the formation of seed MBHs. The detection of even a single light IMRI with LISA would incredibly improve our knowledge of how and where IMBHs form and grow.", '3.2.4 Formation of EMRIs and IMRIs in gas-rich galactic nuclei: AGN discs': '3.2.4.1 Gas-rich dynamics: Active galactic nuclei As discussed above, many galactic nuclei harbor a dense nuclear cluster of stellar-origin objects surrounding an MBH. Above we considered the state of the NSC as a result of stellar evolution, dynamical friction, secular evolution and minor mergers (Morris, 1993; Antonini, 2014; Generozov et al., 2018). Further complicating our picture, however, are the existence of AGN, which occur when low angular momentum gas forms a disc that accretes onto the MBH. As a result, a fraction of the nuclear cluster will end up embedded in the AGN disc via coincident orbits or capture (Syer et al., 1991; Artymowicz et al., 1993). While one might anticipate that AGN would be a subdominant mechanism for producing EMRIs or IMRIs, given that AGN represent only a fraction of all galactic nuclei (or, more likely, a relatively brief episode or series of episodes in the life of any given galactic nucleus), the presence of gas qualitatively changes the dynamics in the NSC. \nPrograde orbiters embedded in a disc are expected to have their eccentricities rapidly damped (Ward, 1988; Tanaka and Ward, 2004; Cresswell et al., 2007; Bitsch and Kley, 2010), although there is a strong dependence on the details and resolution of gas-flow on horse-shoe orbits (Bitsch \nand Kley, 2010). For plausible AGN disc densities, gas dynamical cooling dominates over spherical component dynamical heating, so prograde orbiters should experience very rapid ( < 0 . 1 Myr ) eccentricity damping (McKernan et al., 2012; Kennedy et al., 2016; MacLeod and Lin, 2020). \nThe majority of NSC objects begin as inclined orbiters not coincident with the disc. Stars experience geometric drag and BHs experience dynamical drag as they pass through the disc, causing a significant portion to be captured within a plausible range of AGN disc lifetimes ( 0 . 1 -100 Myr ). Stellar and BH orbiters not coincident with the disc experience geometric and dynamical drag forces, damping first the orbital eccentricity, followed by orbital inclination (Bitsch and Kley, 2011; Just et al., 2012; Kennedy et al., 2016; Panamarev et al., 2018; MacLeod and Lin, 2020; Fabj et al., 2020). Fabj et al. (2020) find O (10%) of prograde NSC BH are captured within a Sirko and Goodman (2003) type disc, for typical disc lifetimes, ignoring accretion. A much smaller fraction are captured by lower-density type Thompson et al. (2005) AGN discs. Stellar objects are primarily captured by the disc at small radii, losing around an order of magnitude in semi-major axis, whereas BHs are captured across the full range of disc radii but rapidly get delivered to the innermost disc (Fabj et al., 2020). The possibility of the accumulation of BHs at small radii across disc models has significant implications for the LIGO-Virgo merger detection rate (Fabj et al., 2020), but also for the possible EMRI/IMRI production rate. \nThe accumulation of prograde BH from the NSC in the inner disc leads to high interaction cross sections at low relative velocity. All embedded stellar-origin objects on prograde orbits should undergo mass-dependent Type I migration due to torques from gas at Lindblad resonances and co-rotating gas (Tanaka et al., 2002), enhancing pile-up of BHs in inner AGN discs, leading to a high merger rate (McKernan et al., 2012, 2014; Bartos et al., 2017; Stone et al., 2017b). \n3.2.4.2 Heavy IMRIs in AGN The rate of change of surface density in AGN disc models implies that we should expect the occurrence of locations in the discs where the outward and inward migration torques cancel (Bellovary et al., 2016). At such so-called migration traps, the local merger rate is significantly enhanced and IMBHs with masses ∼ 10 3 M ⊙ can quickly ( 1 Myr) be produced (Secunda et al., 2019, 2020a; Yang et al., 2019; McKernan et al., 2020b). The IMBHformation merger GW190521 detected by LIGO-Virgo (Abbott et al., 2020d) consisted of two BHs in the upper mass gap with mis-aligned spins, suggestive of a merger in a dynamically rich, deep gravitational potential well. If we assume this merger happened at a migration trap then the rate of such mergers inferred by LIGO-Virgo is ∼ 0 . 7 Gpc -3 yr -1 The LIGO Scientific Collaboration et al. (2020). From McKernan et al. (2020a), assuming O (15) mergers at migration traps over a Myr disc lifetime, and an AGN fraction of O (1%) of galactic nuclei (quasars and the brightest Seyfert nuclei), we find ∼ 1 Gpc -3 yr -1 mergers at migration traps, consistent with the observed rate of GW190521-like events. All of these effects taken together enhance stellar BH merger rates and encourage the formation of IMBH in AGN discs. IMBH formed in this manner automatically create an IMBH-MBH binary, which will typically decay due to GW emission on timescales of a few hundred Myr (Bellovary et al., 2016). \n3.2.4.3 Light IMRIs in AGN A large IMBH sitting in an AGN disc migration trap is an excellent site for the creation of light IMRIs-less massive BH will be delivered to the IMBH from more distant regions of the disc via disc migration torques, with an approximate merger rate of O (1) Gpc -3 yr -1 (McKernan et al., 2020a), assuming Sirko and Goodman (2003) type AGN disc models. LISA can detect IMBHs of several hundred M ⊙ out to ∼ 10 Gpc . So for migration trap mergers with an IMBH of few hundred M ⊙ , this suggests LISA can detect O (10 3 ) yr -1 light IMRI mergers from AGN discs. If migration traps are less common in AGN discs, the maximum IMBH masses from bulk disc mergers are much smaller ∼ 10 2 M ⊙ -and migration torques could drive objects rapidly onto the MBH, typically creating EMRIs with small eccentricities and inclination', 'angles. 14': "3.2.4.4 EMRIs in AGN One likely exception to the small ( e, i ) expectation due to AGN gas damping comes from retrograde orbiters. Approximately half of the initial (NSC) population that is geometrically coincident with the disc, should lie on retrograde orbits (Ivanov et al., 2015). Migration torques on retrograde orbiters in AGN discs are a small fraction of that on prograde orbiters (McKernan et al., 2014). However, retrograde orbiters experience eccentricity pumping at apocenter which rapidly drives them to very high eccentricities (Dunhill et al., 2013; Teyssandier and Ogilvie, 2016, 2019; MacLeod and Lin, 2020) and increases the decay rate of the semi-major axis (Secunda et al., 2020b). As a result we expect a significant population of retrograde orbiters in the innermost AGN disc. This population could yield a very high EMRI rate. The eccentricities of EMRIs from this population depend on the masses of the retrograde orbiters, since the GW circularization rate increases with mass (Peters, 1964a) and the eccentricity driving of the gas decreases with mass (Secunda et al., 2020b). For example, a 10 M ⊙ retrograde orbiter will likely have an eccentricity over 0 . 9 when it inspirals, whereas a 50 M ⊙ orbiter will circularize before inspiraling, making it much easier for LISA to detect over many cycles. However, the higher rate of eccentricity driving of lower mass retrograde orbiters also implies that their EMRI rates will be higher. Therefore, unlike prograde orbiters whose orbits tend to be circularized by the gas disc, retrograde orbiters could commonly produce highly eccentric EMRIs, though this effect is strongest at high (AGN disc) gas density and for low mass BH. \nThere may also be observable EM signals associated with smaller BHs in AGN discs. Stellarmass BH binaries (BH+BHs) can merge at high rates in AGN discs (McKernan et al., 2012, 2014; Bartos et al., 2017). Under these circumstances, since the BH+BH is surrounded by gas, there will always be an EM counterpart. Indeed, a candidate EM counterpart to GW190521 has recently been suggested in an AGN (Graham et al., 2020). Several key questions underpin the search for EM counterparts to BH+BH mergers in AGN discs: Is the EM counterpart detectable through a potentially large optical depth? Is the emission completely outshined by the AGN emission, and on what timescale? Does the radiation from the BH reduce the EM emission? \nAt merger, a remnant stellar-mass BH formed from the BH+BHs recoils with a kick velocity v k depending on the mass asymmetry and spin orientations of the progenitors. In an AGN disc, gas at distance R bound < GM BH+BH /v 2 k is bound to the merged BH+BH and attempts to follow the kicked merger product. In doing so, it collides with surrounding disc gas and a shock luminosity emerges on a time-scale of ∼ 20 days, with a bound gas energy of about 10 45 erg (depending on the BH+BH and gas properties). After the kicked stellar-mass BH has shed the bound gas, the passage of the stellar-mass BH through the gas in the accretion disc produces a shocked Bondi drag tail (e.g. Ostriker, 1999). This tail both decelerates the stellar-mass BH and accretes onto it, generating a potentially high luminosity. In order for enough radiation to escape to make for a bright flare against the AGN, a jet or collimated outflow is required. Such jets may also produce detectable X-ray or gamma-ray signatures. \nAlternatively, stellar-mass BHs in AGN discs may undergo substantial accretion even prior to merger (Yang et al., 2020b), which can lead to e.g., longer term X-ray emission. This scenario largely depends on the poorly understood accretion efficiency and gap opening by stellar-mass BHs in AGN discs. We recommend that future simulations of hyper-Eddington accretion establish whether there is an upper limit to accretion which can choke off jet formation and launching. This will help establish luminosity upper limits on any flares that originate from kicked stellarmass BH mergers in AGN discs and can guide searches for EM counterparts from AGN discs. We also recommend simulations of lightcurves from false-positive flaring events such as SNe and TDEs breaking out from within the AGN disc, or lightcurves of micro-lensing events. \n3.2.4.5 In situ formation of stars in AGN discs: a special population of EMRIs AGN discs are known to be prone to gravitational (Toomre) instability in the outer regions, and expected to form stars vigorously (Shlosman and Begelman, 1989; Goodman, 2003; Levin, 2007; Nayakshin et al., 2007). This expectation is supported by observations of nearby stellar discs in the nucleus of the MW (Levin and Beloborodov, 2003) and M31 (Bender et al., 2005) which, due to their large masses and orbital configurations, can be interpreted as remnants of a prior accretion episode from a gaseous disc (Levin, 2007). Numerous observational studies also suggest a broader connection between AGN activity and nuclear starbursts (e.g., Davies et al., 2007; Wild et al., 2010; Ishibashi and Fabian, 2016), although whether this connection is causal remains uncertain, given that AGN feeding occurs on scales difficult to resolve ( ≲ parsecs) and is often obscured (Alexander and Hickox, 2012). Theoretically, the stars formed in the outskirts of AGN discs are expected to be unusually massive because the disc material is much hotter and denser than star-forming regions in the galactic ISM. Once formed, a population of stars embedded in a gas disc will undergo a stellar evolution that is notably altered by their environment (Cantiello et al. 2021). Throughout their lifetime stars can grow by accretion, becoming even more massive (Goodman and Tan, 2004; Davies and Lin, 2020). In addition to experiencing drag and dynamical friction, they will excite perturbations in the disc that will exert torques on their orbit, typically causing inward migration (as discussed above). Furthermore, we expect an increased number of binaries within the stellar population in the AGN disc (e.g., Alexander et al., 2008), which can become harder due to disc-satellite interactions (e.g., Baruteau et al., 2011; Arca Sedda, 2020a). All this makes fertile ground for forming BH remnants in the disc, which can subsequently encounter each other in migration traps (described above) or migrate to the inner regions of the disc where their evolution becomes GW dominated. Initial estimates show that this process may be an efficient method for feeding MBHs at early times in a way that is not Eddington-limited (Dittmann and Miller, 2020). \nFuture numerical studies can help us narrow down the vast parameter space of accretion disc structures as a function of relevant parameters such as MBH mass, accretion rate, gas supply or redshift. As we improve our understanding of AGN discs, more investigations are needed to fully understand how embedded stars and BHs evolve over a range of system parameters and disc properties. Including more detailed physics (such as disc instabilities and stochastic torques, radiation transport or feedback from accretion, to name a few) may change the evolutionary outcome of these sources, the predicted rates and characteristic properties, as well as the precise waveform signatures and whether or not they are distinguishable amongst formation channels. At the same time, gas-embedded EMRIs/IMRIs also present a powerful opportunity to probe AGN properties in regions that are historically electromagnetically unresolvable, either with deviations in the GW waveforms that correlate with properties of the gas (see Sec. 3.4) or with populations statistics, if multiple events are detected. This is in addition to the possibility for multimessenger astrophysics with associated EM counterparts (e.g. variability in emission, see Section 3.3). \nAs a distinguishing characteristic between various formation channels, here we quote approximate expectations of eccentricity and inclination at late stages of the inspiral, which we define as approaching the central MBH ISCO. Gas-driven, prograde EMRIs are expected to have low e ( e ≲ 0 . 01 ) given that circularization by gas and GWs is efficient, but there remains a possibility of disc-driven eccentricity pumping for relatively massive secondaries or IMRIs (e.g., D'Angelo et al., 2006). These sources are also likely fully embedded in the disc with low inclinationhence if the orientation of the disc aligns with the spin of the central MBH (which may be true to varying degrees, see Volonteri et al. 2013), we expect the spins of the binary components to be closely aligned. Gas driven, retrograde EMRIs, on the other hand, may reach very high eccentricities even at the ISCO ( e ≲ 0 . 9 , Secunda et al., 2020b). They should also retain a low \ninclination, although the spin alignment of the BHs will depend on the accretion history of the embedded BH which remains to be investigated across the full range of parameter space. These estimates will be further constrained with future work that includes more realistic disc modelling and treatment of gas dynamics and accretion onto embedded BHs. \nThe contribution of LISA to the physics of formation of EMRIs and IMRIs in gasrich galactic nuclei From what we have presented, we can derive that depending on the (highly uncertain) duty cycle of AGN, an IMBH-MBH binary could correspond to a heavy IMRI in nearly every galactic nucleus. However, the absence of any such detection with LISA would seriously constrain the existence of a migration trap in a generic AGN disc. The absence of a migration trap implies the absence of strong changes in the disc surface density gradient and thus tightly constrains the transition between the radiation pressure and gas pressure dominated regions of AGN discs. \nAs we have mentioned, there is a clear correlation between the dynamical parameters of EMRIs formed in AGN discs and their detection. Thus the rate of EMRIs detected by LISA can put strong constraints on the populations and dynamics we expect to live in innermost AGN discs, a system lurking in a region inaccessible to spatially resolved EM observations. \nPrediction of rates and precise characteristics of disc-embedded EMRIs/IMRIs is a multifaceted problem that relies on details of gravitational instability, stellar evolution, nonlinear gas dynamics, and accretion physics. Given that MBHs spend 1 -10% of their evolution in an AGN phase (Shankar et al., 2013; Pardo et al., 2016) we expect at minimum the same fraction of EMRIs to occur in gas-rich environments. In-situ star formation likely leads to a population of compact remnants in addition to those that are captured from the nucleus. Thus we expect that dense accretion discs in near-Eddington AGN may not only boost EMRI rates, but also produce a population that is uniquely characteristic: with low eccentricity and (some degree of) spin alignment with the central MBH. These ( e, i ) expectations are strongest for EMRIs from objects formed in-situ. A single EMRI with low eccentricity will indicate a current (or recent) interaction with gas, showing that the host galaxy harbors an active (or recently active) MBH. The precise parameters (e.g. eccentricity, secondary BH mass) are intimately connected to prior evolution of the accretion disk, and thus these measurements will give constraints on the efficiency of gas-driven circularization and accretion disk structure. If a larger population of such EMRIs is detected, the distribution of orbital characteristics will tell us about the diversity of AGN disks. Rates and orbital characteristics will constrain various aspects of accretion onto MBHs - for example, the number of detected events will inform how many nearby AGN host disk-embedded BHs. If the number is high, it will challenge accretion models. Just as the migration rates of stars and compact remnants depend sensitively on characteristics of the disk (Baruteau and Masset, 2013; Duffell, 2015), the secondary masses will be a consequence of BH accretion or hierarchical mergers, all of which are tied to disk models. Overall, measurements that shed light on accretion disk structure and prevalence will improve our constraints on AGN duty cycles and MBH growth.", '3.2.5 Alternative formation scenarios': '3.2.5.1 XMRIs The possibility of observing an EMRI at our own galactic centre when LISA flies is basically zero, since the rates for an EMRI formed via relaxation in a MW-like galaxy are at most about 10 -6 yr -1 (e.g., Amaro-Seoane, 2018b, 2020). This means that about once every million years a stellar-mass BH plunges through the event horizon of our central MBH. Since the lifetime of LISA will be only a few years, the probability of detecting one at our galactic centre is negligible. \nWe can however find another class of EMRIs at our galactic centre. It has been recently put forward (Amaro-Seoane, 2019) that substellar objects, in particular brown dwarfs, stand very high chances of being in band of the detector when it is launched. The reason for this is very \nsimple: these substellar objects have mass ratios of about q ∼ 10 -8 as compared to the central MBH, Sgr A ∗ . Such XMRIs (extremely large mass ratio inspirals) can therefore cover up to ∼ 10 8 cycles before crossing the event horizon, since the number of cycles is roughly inversely proportional to the mass ratio. This means that they stay in band for millions of years. About 2 × 10 6 yr before merger they have an SNR at the galactic centre of 10 . Later, ∼ 10 4 yr before merger, the SNR reaches several thousands, i.e. they are at the level of the loudest MBH mergers. At the last stages of their evolution, some ∼ 10 3 yr before the merger, they can reach SNR as high as a few 10 4 (Amaro-Seoane, 2019; Gourgoulhon et al., 2019; Barack and Cutler, 2004b). \nThe work of Amaro-Seoane (2019) predicts that at any given moment there should be of the order of ≳ 5 XMRI that are highly eccentric and are located at higher frequencies, and about ≳ 15 are circular and are at lower frequencies. The mass ratio for an XMRI is about three orders of magnitude smaller than that of stellar-mass BH EMRIs. Since backreaction depends on q , the orbit closely follows a standard geodesic, which means that many approximations work better in the calculation of the orbit. XMRIs can be sufficiently loud so as to track the systematic growth of their SNR, which can be high enough to bury that of MBH binaries. \nIn addition, there are also plunge events during the formation of inspiralling sources. The GWs from low mass objects (brown dwarfs, primordial BHs, etc.) plunging into the central MBH are burst signals. For LISA, the SNRs of these bursts are quite high if they happen in our Galaxy. However, the event rates are estimated as ∼ 0 . 01 yr -1 for the Galaxy. If we are lucky, this kind of very extreme mass-ratio burst will offer a unique chance to reveal the nearest MBH and nucleus dynamics. The event rate could be as large as 4 -8 yr -1 within 10 Mpc , and because the signal is strong enough for observations by space-borne detectors, there is a good chance of being able to use these events to probe the nature of neighbouring BHs (Berry and Gair, 2013b). This kind of burst sources are called XMRBs (extreme mass ratio bursts) (Han et al., 2020). \n3.2.5.2 Binary and multiple EMRIs Recent theoretical studies pointed out the existence of a new type of EMRI in which the small body is a stellar-mass BH+BH. Such a triple system could form either due to tidal capture of a BH+BH by a MBH (Addison et al., 2019; Chen and Han, 2018) or the formation and migration of a BH+BH in the accretion disc of an AGN (Chen et al., 2019a). While the latter channel is considered to be more effective than the former one, both channels could deliver BH+BHs to a distance as small as tens of gravitational radii of the central MBH. As a result, the binary, as a single entity, spirals into the MBH due to GW radiation. For this reason, the source is referred to as a binary-EMRI, or b-EMRI. \nThe uniqueness of the b-EMRI lies in the fact that it simultaneously emits two kinds of GWs. One in the LISA band, due to the orbital motion of the BH+BH around the MBH, and the other in the ground-based detector band, when the BH+BH coalesces due to the tidal perturbation by the MBH. A coordinated observation by LISA and ground-based detectors would allow us to identify such interesting sources and, more importantly, constrain several aspects of fundamental physics to a precision more than one order of magnitude better than the current limit, including the loss of rest mass due to GW radiation, the recoil velocity of the merging BH+BH, and the dispersion of GWs of difference frequencies (Han and Chen, 2019). Due to the merger of the BH+BH, the remnant will obtain a recoil velocity which may be up to a few thousand kms -1 . This sudden kick will induce a glitch on the waveform of a b-EMRI (see Fig. 33). \n3.2.5.3 Supernova-driven EMRIs In addition to the aforementioned mechanisms, an EMRI can also be generated via the supernova explosion that accompanies the formation of a CO. When this happens, the velocity of the compact object gets almost instantaneously significantly perturbed, so that the compact object settles on a brand new trajectory: the timescale for the CO to coalesce with the MBH via GWs on this new orbit may be shorter than the timescale for two-body relaxation to perturb it, so that the CO is bound to evolve into an EMRI. Focusing on the Galactic Centre environment, Bortolas and Mapelli (2019) showed that one supernova out of \nFigure 33: Comparing the waveforms of the EMRIs with (red, solid) and without (blue, dashed) a glitch. The MBH has a mass of M = 10 6 M ⊙ and a spin parameter of 0 . 9 . The total mass of the BH+BH is m = 20 M ⊙ , and D refers to the luminosity distance. In this example, the centre of mass of the stellar-mass BH+BH initially is moving inside the equatorial plane of the MBH with an orbital eccentricity of e = 0 . 7 and a semilatus rectum of p = R (1 -e 2 ) = 17 r g . At the time t = 0 a kick to the centre-of-mass velocity of the binary happens, in the polar direction and with a magnitude of 1500 kms -1 . As a result, the orbital parameters change to p = 16 . 9990 r g and e = 0 . 7019 , and the orbital plane of the EMRI becomes inclined by ι = 0 . 5233 · relative to the equatorial plane of the MBH. \n<!-- image --> \n10 4 -10 5 occurring within the star forming structures present about the MW MBH will give rise to a supernova-driven EMRI. This result, coupled with the expected frequency of core-collapse supernovae explosions occurring in the Galactic Centre, implies a frequency of supernova-driven EMRIs up to 10 -8 yr -1 , i.e. an EMRI rate that is comparable or only mildly lower than the one associated to the standard two-body relaxation process (Bortolas and Mapelli, 2019). \nThe contribution of LISA to relativistic stellar dynamics and supernovae rates What we have described about XMRIs allows us to understand that these might be envisaged as a double-edged sword. From the one side we have a promising and strong source of GWs from an extreme-mass ratio which is easy to model. On the other hand they might pose a problem because their SNRs (as high as 10 4 for one-year observation if one resides in the Galactic Center) are such that can bury binaries of MBHs. Also, if they are present in most nuclei harbouring MBHs in the range of LISA, they might interact with EMRIs or even scatter them off from their inspiraling orbit. The detection of XMRIs will allow us to infer information on relativistic astrodynamics impossible to obtain otherwise. \nLISA can distinguish b-EMRIs from normal EMRIs by detecting the GWs from the small binary black hole months to years prior to its coalescence around the SMBH. If the frequency of the GWs from the small binary matches a fundamental frequency of the SMBH, the SMBH could be resonantly excited and the EMRI waveform could contain an enhanced quasi-normal mode (Cardoso et al., 2021). Moreover, the binarity of the small body also induces an addititional phase shift to the EMRI waveform, which can be used to identify b-EMRIs as well (Chen and Zhang, 2022). \nRegarding the detection of EMRIs formed via supernova, this result calls for a more extended analysis, in order to investigate this process in a wider range of galaxy environments and starformation rates, and exploring in more detail the waveform signatures associated to this kind of EMRI compared to other EMRI formation mechanisms. In this framework, LISA will thus help us shed light on the rates of supernovae near MBHs via the detection of EMRIs.', '3.3 Multimessenger prospects': 'Coordinators: Giuseppe Lodato and Martina Toscani \nContributors: Pau Amaro Seoane, Jillian Bellovary, Stefano Bianchi, Saavik Ford, Barry McKernan, Giuseppe Lodato, Tom Kimpson, Scott Noble, Martina Toscani,', 'Kinwah Wu, Ziri Younsi and Silvia Zane': 'We outline a variety of proposed EM counterparts of EMRIs and IMRIs, including TDE (in multiple configurations), AGN-related signatures, and pulsar EMRIs.', '3.3.1 Tidal disruption events': 'TDEs (Carter and Luminet, 1983; Rees, 1988; Phinney, 1989; Rossi et al., 2020, for a recent review) can be considered a particular type of EMRI, where the star is disrupted by the MBH tides during the first passage at the pericenter. For this to happen, the pericenter radius should be smaller (cf. Ryu et al., 2020) than the tidal radius \nr t ≈ R ∗ ( M · M ∗ ) 1 / 3 , (18) \nwhere R ∗ and M ∗ are the stellar radius and stellar mass respectively, while M · is the BH mass; r t is usually a factor of 10 -20 times the MBH Schwarzshild radius. For MBH masses larger than ∼ 10 8 M ⊙ ( 10 9 M ⊙ for rapidly spinning BHs), the tidal radius is within the event horizon and no TDEs can happen. \nTDEs are very luminous events over a broad range of EM bands. The first EM observations occurred in 1990s thanks to the ROSAT survey (Bade et al., 1996; Komossa and Greiner, 1999; Grupe et al., 1999; Greiner et al., 2000), which detected some bright flares from the cores of nonAGN galaxies. Since then, the number of X-ray detections has incresed. These observations seem to be in agreement with the theoretical expectation of X-ray emission from an accretion disc (e.g., Ulmer, 1999; Auchettl et al., 2017; Lodato and Rossi, 2011). Initially these flares are powered by a near-Eddington accretion, then the luminosity decreases over a period from months to years (Saxton et al., 2020, and references therein). Over the last decade, also a growing number of optical TDEs has been detected. It remains unclear what processes are at the origin of this optical emission. Some hypotheses concern the shocks from self-crossing debris (Piran et al., 2015; Shiokawa et al., 2015) or reprocessing in an outflow (e.g., Strubbe and Quataert, 2009; Lodato and Rossi, 2011; Metzger and Stone, 2016). Detailed reviews of optical TDEs are found in Wevers et al. (2019) and van Velzen et al. (2020). A small fraction of these jetted events has also shown significant radio emission (Alexander et al., 2020, and references therein). \nFurthermore, TDEs emit GWs. Their GW emission is produced by three different varying quadrupoles: (a) the star-BH quadrupole (Kobayashi et al., 2004; Toscani et al., 2022); (b) the stellar internal quadrupole (Guillochon et al., 2009; Stone et al., 2013) and (c) the quadrupole of the compact torus formed after disruption (e.g., van Putten, 2001, 2002; Kiuchi et al., 2011; Toscani et al., 2019; van Putten et al., 2019). The dominant contribution is the first term, that could be well described as a GW burst with strain (Kobayashi et al., 2004; Toscani et al., 2022) \nh ≈ β × r s r s ∗ r t d ≈ β × 2 × 10 -22 ( M ∗ M ⊙ ) 4 / 3 ( M · 10 6 M ⊙ ) 2 / 3 ( R ∗ R ⊙ ) -1 ( d 16 Mpc ) -1 , (19) \nand an associated Keplerian frequency of \nf ≈ β 3 / 2 2 π ( GM · r 3 t ) 1 / 2 ≈ β 3 / 2 × 10 -4 Hz × ( M ∗ M ⊙ ) 1 / 2 ( R ∗ R ⊙ ) -3 / 2 . (20) \nIn the above formulas we have introduced the Schwarzschild radius of the BH, r s = 2 r g , the Schwarzschild radius of the star, r s ∗ , and the penetration factor \nβ = r t r p , (21) \nwhere r p is the pericentre distance. A library of gravitational waveforms from TDEs has been presented in Toscani et al. (2022), generated using a general relativistic smoothed particle hydrodynamic code, phantom (Liptai and Price, 2019). To date, this numerical study models the star as a polytropic sphere with index γ = 5 / 3 , but we expect the strain to have a dependence on the internal structure of the star. This dependence needs to be further investigated. \nThe expected strain for a Sun-like star being disrupted by a 10 6 M ⊙ MBH at 15 Mpc distance (assuming β =1) is h ∼ 10 -22 , with a frequency f ∼ 10 -4 Hz . The two other contributions are expected to have similar frequency but are scaled down by some (two-five) orders of magnitude. \nPfister et al. (2021b) estimate the rate of TDEs which could be observed with different instruments. They find that for LISA it is unlikely to detect GWs from TDEs, unless BHs are surrounded by particularly massive stars. The next generation of detectors beyond LISA should however be able to detect GW from TDEs up to cosmological redshifts z ≥ 1 . \nAn interesting signal to study is the GW background from the entire cosmic population of TDEs. Details on this signal and its derivation may be found in Sec. 3.5.1. \n3.3.1.1 TDEs outside galactic nuclei In the majority of TDE studies the MBH that disrupts a star is implicitly taken to be the nuclear BH of the galaxy. The Swift transient source AT2018cow has many characteristics resembling a TDE (Kuin et al., 2019), but peculiarly the source is not located at the nuclear region of its suspected host galaxy Z 137-068. This leads to consideration of whether it is a TDE, with an MBH disrupting a WD, i.e. WD-TDE (Han and Fan, 2018), or alternatively a violent stellar explosion. The question is now: Can TDE involving a wandering 15 MBH occur? This question would be answered if there are mechanisms to populate a galaxy with massive BHs of non-stellar nature. Stellar systems are not stationary structures. A stellar cluster can dissolve on timescales of 10 -100 Myr (e.g., Gieles and Bastian, 2008). Globular clusters can survive longer but they can also be disrupted (e.g., Belokurov et al., 2006; Wan et al., 2020) or dissolved (see Baumgardt, 2009). Similarly, dwarf galaxies can be disrupted and dissolved (e.g., Li et al., 2018; Sanders et al., 2018) when they encounter and are accreted by a larger galaxy. The nuclear BHs, if present in these stellar systems, would be dispersed into the interstellar space of the cannibal galaxy. Stars are also carried along into the interstellar space by these BHs, and some of them will eventually spiral into their carrier BH and become a TDE.', '3.3.2 Electromagnetic counterparts of light IMRIs in AGN discs': 'Most BHs that merge in AGN discs (including IMBH-BH mergers) are expected to experience a GW recoil kick at the moment of merger with speeds v k of up to a few hundred kms -1 . Such merger kicks would happen for any comparable system, with or without gas; however, the consequences of such kicks for a gas-embedded merger may include a detectable EM counterpart. In an AGN disc, gas within \nR bound < GM BH+BH v 2 k (22) \nis bound to the merged BH+BH and attempts to follow the kicked merger product. In doing so, it collides with surrounding disc gas and, as long as the disc is geometrically thin or optically thin, a shock luminosity can emerge on a timescale t bound = R bound /v k = GM BH+BH /v 3 k (McKernan et al., 2019) or \nt bound ∼ 20 ( M BH+BH 100 M ⊙ ) ( v k 200 kms -1 ) -3 day . (23) \nThe total energy delivered to the bound gas is E bound = (1 / 2) M bound v 2 k = (3 / 2) Nk B T bound where M bound = Nm H is the mass of the bound gas expressed as N atoms of Hydrogen (mass m H ), k B \nis the Boltzmann constant, and T bound is the average temperature of the post-shock gas. This energy is \nE bound = 3 × 10 45 ( ρ 10 -10 g cm -3 )( M BH+BH 100 M ⊙ ) 3 ( v k 200 kms -1 ) -4 erg , (24) \nand the resulting average hot spot temperature is \nT bound ∼ 1 . 8 × 10 6 ( v k 200 kms -1 ) 2 K . (25) \nThe resulting UV/optical flare occurs between t = [0 , t ram ] , has an average (low) luminosity E bound /t ram and a shape given by sin 2 ( πt/ 2 t ram ) . \nOnce the kicked BH leaves behind originally bound gas, the disc gas it passes through is accelerated around the BH, producing an asymmetric low angular momentum Bondi tail inside the stagnation point (e.g., Ostriker, 1999; Antoni et al., 2019). This tail both acts as a drag on the BH and accretes onto it. The Bondi-Hoyle-Lyttleton luminosity is L BHL = η ˙ M BHL c 2 where η is the radiative efficiency and \n˙ M BHL = 4 πG 2 M 2 BH+BH ρ v 3 rel , (26) \nwith v rel = v k + c s and c s is the gas sound speed. In principle, hyper-Eddington accretion is allowed by this process. This should cause trapping of emergent radiation, unless a collimated outflow allows radiation to escape. However, if the kicked merger product travels out of the dense disc midplane into a more tenuous disc atmosphere, such signatures may be bright enough to be detected even against bright AGN hosts (Graham et al., 2020).', '3.3.3 FeK α lines (or other EM signatures) as probes of small separation MBHIMBH binaries': 'The relativistically broadened component of the fluorescent FeK α line (centered around 6 . 4 -7 keV source-frame) is believed to be a probe of material in the innermost accretion disc (Nandra et al., 1997; Fabian et al., 2000; Reynolds and Nowak, 2003). EMRIs and heavy IMRIs premerger will disrupt the flow of gas in the innermost disc, yielding flicker in the innermost disc (EMRIs) or carving gaps or a central cavity (MBH-MBH, MBH-IMBH binaries) with minidiscs. The resulting re-arrangement of gas yields signatures prior to GW merger events which may be detectable in the broad FeK α with high-throughput X-ray telescopes like Athena. A gap-opening secondary IMBH close to the primary MBH will leave an imprint in the broad component of the FeK α emission line, which varies in a unique and predictable manner (McKernan et al., 2013).', '3.3.4 EMRIs containing a pulsar': 'Pulsars are spinning NSs, mainly identified by their radio observations. To date there are about 2800 known pulsars in the Galaxy (Cameron et al., 2020), of which about 160 are found to be associated with globular clusters (see the ATNF pulsar catalogue, Manchester et al. (2005)). Among these radio pulsars, more than 170 are MSPs (e.g., Levin et al., 2013), and the majority of them actually reside in globular clusters. Although the number of known MSPs is growing, most MSPs are yet to be discovered. It has been suggested that the total population number of MSPs in the MW could be about 100 , 000 or even more (Levin et al., 2013). Radio pulsar timing is a relatively mature technique in astronomy, as researchers have accumulated experience in pulsar research over decades. The current developments in instrumentation and search techniques will enable us to detect radio pulsars outside the MW (Keane et al., 2015), and the detection range will be further extended by the time LISA is operating. \nEMRIs containing a radio pulsar are a special class of GW sources with a guaranteed EM counterpart, if they are close enough. The presence of a MSP provides researchers with several advantages to study these EMRI systems and their associated physics. NSs have a small mass range centred around 1 . 4 M ⊙ . Knowing the mass and the spin of one component in the EMRI system reduces the parameter space, hence easing the computational demands in the template matching and searching for establishing their GW properties, whereas other EMRI systems would require the determination of the system parameters simultaneously, relying solely on the GW signals. The availability of the EM signals with measurements at high precision will give another advantage. Both pulsar timing observations and GW experiments can obtain measurements to high precision; the accuracy and precision of pulsar timing is among the highest achievable in astrophysical time-domain analysis (Hartnett and Luiten, 2011). Radio pulsar timing and GW experiments employ different analysis techniques. As such, the orbital and spin dynamics, as well as the system parameters which they determine, will be independent, thereby giving us a means to understand certain systematic properties in the statistical and data analyses. \nThe contribution of LISA to multimessenger science The event rate of the potential emission of GWs by extended stars approaching a MBH will provide us with additional information about tidal disruption events. This combined with EM detections will deliver much more precise catalogues of disruptions and, within some limits, information about the star and MBH which is inaccessible via traditional telescopes. \nRegarding TDEs outside galactic nuclei, a WD-TDE system would emit GWs (Han and Fan, 2018) as well as bursts of EM radiation (Kuin et al., 2019). With the additional constraints provided by the GW observations, it would easily resolve the dispute about certain candidate TDE sources, such as AT 2018cow (Kuin et al., 2019; Perley et al., 2019). \nFrom what we have explained about EM counterparts of light IMRIs, we can conclude that a population of stellar-origin BH+BH in the LISA band that harden into the ground-based GW detector band in AGN discs can yield potentially detectable optical/UV counterparts. IMBHBH and IMBH-IMBH binary mergers in AGN discs are likely to occur at migration traps in the inner disc (Bellovary et al., 2016), so kicked merger products remain bound to the MBH. The kicked BH must splash back down into the AGN disc possibly yielding a repeat flare on half the orbital timescale. An off-center luminous flare should be detectable as an asymmetry in broad optical lines as the broad line region responds to non-central illumination (McKernan et al., 2019). \nDouble relativistic FeK α lines may be detectable from binary mini-disc emission, allowing us to localize LISA sources well before merger (Sesana et al., 2012). The barycenter of a MBH binary will lie outside the event horizon of the primary BH for modest values of mass ratio and binary separation. Analogous to the radial velocity method of planet detection, whereby the wobble of a star indicates the presence of a nearby Jupiter-sized planet, the radial velocity of the primary BH around the binary barycenter can leave a tell-tale oscillation in the broad component of FeK α emission (McKernan and Ford, 2015). Such oscillations are detectable by Athena for binaries with mass ratios q ≥ 0 . 01 , at binary separations of up to O (10 2 r g ) . Both the general-relativistic and Lense-Thirring precession of the periapse of the secondary orbit imprint a detectable modulation on these oscillations (McKernan and Ford, 2015). Athena is likely to detect O (30) FeK α broad lines at sufficient statistical significance in local AGN to carry out tests for ripples and oscillations of such binaries. O (100) AGN may have broad FeK α components that will allow us to search for double components (McGee et al., 2020). Hence, the input from LISA and Athena can be compounded to extract information about the separation of heavy IMRIs. \nLISA detection of EMRIs with a MSP will provide opportunities to investigate a variety of fundamental issues in gravitational physics. This is rooted in the extreme mass ratio between the MSP and the BH (the MSP being a test mass) and the ultra-fast rotation of the MSP (the MSP being an extreme gyro and a stable time-keeper). More specifically, the spin-spin, spin-orbit, and \nthe spin-curvature interactions (Chicone et al., 2005; Iorio, 2012; Remmen and Wu, 2013; Singh et al., 2014) between the MSP and the BH will manifest in the spin and orbital dynamics of the MSP (Li et al., 2019; Kimpson et al., 2020b), which will in turn modify the pulsar timing signals via modification of the pulse period and the pulse arrival time (Kimpson et al., 2019a, 2020a). Together with the information extracted from the GWs generated by the EMRI, researchers will be able to investigate how EM waves propagate in a non-vacuum space time (Kimpson et al., 2019b) or in a slightly perturbed space time, as well as having the opportunity to gain some understanding of certain fundamental issues, such as the gravitational self-force (e.g., Barack and Pound, 2019) in GW sources.', '3.4 Environmental effects on waveforms': 'Coordinators: Alvin Chua, Alejandro Torres-Orjuela and Lorenz Zwick Contributors: Pau Amaro Seoane, Manuel Arca Sedda, Emanuele Berti, Xian Chen, Alvin Chua, Andrea Derdzinski, Kyriakos Destounis, Wen-Biao Han, Kostas Kokkotas, Cole Miller, Scott Noble, Arthur Suvorov, Alejandro Torres-Orjuela and Lorenz Zwick \nWe know that EMRI/IMRI events can form in a variety of interesting astrophysical environments. Some of these environments may leave detectable imprints on the waveforms measured by LISA (though detecting the imprints may be challenging). We outline a wide variety of environmental effects, including gas-driven effects and many-body effects. Some effects may be degenerate with one another or with deviations from general relativity, even when an effect is detectable. Fortunately, complex dynamics sometimes lend themselves to breaking degeneracies through e.g., Doppler effects. We also use this section to address the possibility of detecting (and extracting astrophysical information from) the EMRI background, and possibly detecting the signatures of chaotic systems. Finally, we specifically consider the degeneracies between environmental effects and PN/self-force effects.', '3.4.1 Gas torques': 'For gas-embedded EMRIs/IMRIs, gas torques can speed up or slow down an inspiral while it is in the LISA band. The magnitude of the torques will scale with the disc density, and the precise value and direction of the torque will depend on the mass of the inspiralling CO and disc properties. The effect is several orders of magnitude weaker than GWs in this regime, but even a small dephasing over several thousand cycles may accumulate to a detectable phase shift (up to a few radians), depending on the density of the environment, as shown in analytic work (Yunes et al., 2011a; Kocsis et al., 2011; Barausse et al., 2014) as well as more recently in 2D hydrodynamical simulations (Derdzinski et al., 2019, 2020). Accretion discs in bright AGN are expected to be thin and dense, but their inner regions are hot and radiation pressuredominated. Analytical estimates of densities in the inner regions of such discs from simple models predict surface densities varying from ∼ 10 -10 7 g cm -2 (Shakura and Sunyaev, 1973; Frank et al., 2002). The wide range arises from our uncertainty on how viscosity scales with the (gas or total) pressure. State-of-the art 3D global magnetohydrodynamical disc simulations (Jiang et al., 2016, 2019; Jiang and Blaes, 2020) suggest that densities are between these values, somewhat closer to the lower end, which could make gas dephasing too small to detect, but this may change as we continue to explore the parameter space of MBH masses. \nWhether or not this effect is detectable will depend on the density of the environment as well as the SNR of the source. An EMRI embedded in a Shakura-Sunyaev alpha-disc with Σ ∼ 10 2 g cm -2 can accumulate a phase shift up to ≲ 10 -2 radians within 4 years, whereas if embedded in a beta-disc would dephase over 10 1 -10 2 radians over 4 years (Derdzinski et al., 2020). IMRIs, due to their higher mass, accumulate higher SNR and also feel stronger torques \n(since they scale with secondary mass), making them ideal events for producing detectable gas signatures. \nWhether or not this effect is distinguishable from other waveform deviations will depend on how well one can measure the phase shift and how this accumulates as the frequency evolves. Simulations suggest that the torque can be approximated by simple analytical formulae-torques are within an order of magnitude of the Type I torque derived by Tanaka et al. (2002), although variability in the torque can arise for sufficiently massive secondaries ( q ≳ 10 -3 ). This means that one can estimate how the deviation accumulates with frequency, assuming we know the disc density profile, and if f can be measured from the GW data, degeneracies between the GWwaveform distortions due to disc torques versus parameter variations or other effects can be disentangled in principle. In practice this may prove difficult given that the small mass ratio of these sources (and expected low eccentricity) means they will chirp slowly, and will more likely appear as near-continuous wave sources within a few year observation.', '3.4.2 Many-body interactions': "3.4.2.1 XMRIs and EMRIs As explained above, we expect a handful of XMRIs to be present in our own Galactic Centre (Amaro-Seoane, 2019). Since these systems are so loud, reaching SNRs of up to ∼ 20 , 000 , in principle we should be able to detect them in nearby galaxies harbouring MBHs in the mass range 10 5 -10 7 M ⊙ (i.e. nuclei for which the relaxation time is below a Hubble time Amaro-Seoane, 2020, 2018b; Preto, 2010). Farther away XMRI systems with much lower SNR will not be detected. However, XMRIs pose a problem for normal EMRI systems: Since XMRIs live in band for millions of years, and the estimated event rate for an EMRI is between 10 -5 -10 -6 yr -1 , the possibility that an EMRI encounters an XMRI on its way to the MBH is non-negligible. \n3.4.2.2 EMRIs interacting with a perturbing star Although unlikely, it is not ruled out that a star can be located close to an EMRI inspiraling towards the central MBH. In the work of Amaro-Seoane et al. (2012) the authors derive the shortest radius from the MBH within which one might expect to have at least one star. Then they run direct-summation N -body simulations with relativistic corrections following the first implementation as presented in Kupi et al. (2006) and find that periapsis shift along with gravitational-radiation effects induce nondeterminism in the evolution of the EMRI. This means that for two identical dynamical setups of an EMRI system with a perturbing star located at a distance of about ∼ 5 a EMRI , with a EMRI the semi-major axis of the EMRI, small changes of any dynamical parameter induces a different evolution of the EMRI. The presence of a perturbing star, therefore, can be misinterpreted as a deviation of general relativity, and this should be taken into account in the development of data analysis algorithms. \n3.4.2.3 Binary-EMRIs The formation rate of b-EMRIs for BH+BHs tidally captured by MBHs is equivalent to (10 -5 -10 -4 ) Gpc -3 yr -1 in the pessimistic case and 0.1 Gpc -3 yr -1 in the most optimistic one. However, due to the non-negligible lifetime of b-EMRIs, within a spherical volume of 1 Gpc 3 (corresponding to a radial distance of about 600 Mpc ), there are, on average, about 0 . 02 -20 b-EMRIs expected during LISA's mission duration. \nThe most exciting property of b-EMRIs is that they are multi-band GW sources, i.e., radiate low and high frequency GWs synchronously (Addison et al., 2019). Though the time scale of BH+BH merger is much shorter than the inspiral of binary into the MBH, LISA and the groundbased detectors (LIGO etc.) may observe the b-EMRI at the same time. The high-frequency GWs could be redshifted because they are generated close to a MBH (Chen et al., 2019a), providing an opportunity of studying the propagation of GWs in the regime of strong gravity.", '3.4.3 Moving sources': "Almost all astrophysical objects are moving relative to us and GWs sources are no exception. The motion of the centre of mass of a source is often related to the properties of its environment, e.g., the orbital motion induced by the interaction with other bodies (Wen, 2003; McKernan et al., 2012; Antonini and Perets, 2012; Naoz, 2016; Arca Sedda, 2020a; Stone et al., 2017b; Bartos et al., 2017; Tagawa et al., 2020a) and the motion of its host system like the peculiar velocity of galaxies (Zinn and West, 1984; Bahcall, 1988; Carlberg et al., 1996; Springel et al., 2001; Scrimgeour et al., 2016; Colin et al., 2017; Girardi et al., 1996; Ruel et al., 2014). Therefore, the detection of velocity can provide valuable and versatile information about the sources environment and its host system. \nIn the case of a constant velocity the Doppler effect changes the observed GW frequency f obs by a factor (Chen et al., 2019a) \nf obs = f (1 + z ) -1 , (27) \nand its derivative, ˙ f obs , by the same factor squared \n˙ f obs = ˙ f (1 + z ) -2 . (28) \nHere, f and ˙ f being, respectively, the GW frequency and its derivative in the source's rest frame. When only considering the dominant mode of GWs, these shifts lead to a wrong estimation for the sources actual chirp mass M , \nand actual luminosity distance, d L , \nM obs = M (1 + z ) , (29) \nd obs = d L (1 + z ) , (30) \nthus fundamentally affecting our interpretation of the source. An analogous effect appears for the cosmological redshift of GWs. Although the latter one is considered in current GW models and detections, the same is in general not true for the effect of velocity (Abbott et al., 2019). \nIf the velocity of the source varies in time, the previous picture changes significantly. The Doppler effect induces a time-dependent phase shift, proportional to the velocity of the source along the line of sight, which can be detected when having accurate models of the evolution of the phase of the source and the velocity profile (Inayoshi et al., 2017b; Meiron et al., 2017; Wong et al., 2019a). For a LISA mission of 4 years probably no accelerated sources could be detected. However, the number of detections could be increased to up to 3 when conducting joint measurements with a ground-based detector. For a LISA mission of 10 years up to 40 accelerated sources could be detected by LISA alone and up to 103 when conducting joint measurements with an earth based detector (Tamanini et al., 2020). Moreover, the aberration of the GWs rays affects the line of sight thus inducing an additional phase shift (Torres-Orjuela et al., 2020b). The magnitude of the aberrational phase shift is of the same order as the Doppler phase shift but proportional to the components of the velocity perpendicular to the line-of-sight. Therefore, considering the total phase shift, i.e. Doppler effect plus aberrational phase shift, the SNR required for the detection of the acceleration could be reduced by a factor of up to 1 . 8 , thus allowing the detection of up to 5 . 8 times more sources (Torres-Orjuela et al., 2020b).", '3.4.4 Dark matter as an environmental effect': 'The density profile of dark matter halos has a cusp at the centre of galaxies because of the large potential well there (Kuhlen et al., 2012). If a MBH resides at the centre of the galaxy, the strong gravity could lead to a significant increase of density in the central region and create a spike, which enhances the dark matter annihilation rate (Gondolo and Silk, 1999; Sadeghian et al., 2013). Similarly, IMBHs may have a smaller dark matter spike (Zhao and Silk, 2005; Bertone et al., 2005). The gravitational potential of the dark matter could impact the evolution \nof an EMRI/IMRI, particularly where enhancement of the density occurs due to spikes or a superradiant instability, leading to a detectable signature (Eda et al., 2013; Yue and Han, 2018; Hannuksela et al., 2020). However, dynamical events such as mergers of host galaxies can weaken the dark matter cusp (Ullio et al., 2001; Merritt et al., 2002; Merritt, 2004; Bertone and Merritt, 2005), which makes its presence harder to detect.', '3.4.5 Astrophysical chaos': "Owing to the huge mass disparity for the objects involved in an EMRI, the dynamics of the companion body can be modelled as a point particle traversing the gravitational field of the (super-massive) primary to high accuracy. Characteristics of GWs emitted during the inspiral are therefore dominated by the particulars of the metric geometry of the primary. Fundamental symmetries, or the absence thereof, associated with the spacetime geometry can therefore be probed by LISA. For a primary which is both stationary and axisymmetric-properties expected of astrophysically stable BHs-the energy and one component of the angular momentum of a companion are both constants of motion with respect to the orbital dynamics. The companion's Hamiltonian, H ∼ g µν p µ p ν for momentum p , provides a third constant of motion. In four spacetime dimensions, however, having only three conserved quantities implies that the equations of motion are not Liouville integrable (Contopoulos, 2002). In this context, a non-integrable system exhibits chaotic orbital phenomena, as is familiar from the three-body problem in (post)Newtonian gravity (Huang and Wu, 2014). The Kerr spacetime, which uniquely represents stable BHs in general relativity, also however admits a rank-two Killing tensor, which provides a fourth constant of motion in the form of the Carter constant (Carter, 1968). This implies the absence of astrophysical chaos in EMRIs within general relativity, at least for companions which are not themselves spinning rapidly (Kiuchi and Maeda, 2004; Lukes-Gerakopoulos et al., 2014; Piovano et al., 2020; Zelenka et al., 2020). \nIf, however, high-energy corrections to the field equations present themselves in nature, the particulars of gravitational collapse (e.g., Cembranos et al., 2012) and accretion (e.g., Harko et al., 2010) may be such that a non-Kerr object resides within galactic centres or elsewhere. There are many ways in which a hypothetical departure from a Kerr description may manifest within the spacetime metric, such as those described in Johannsen (2013). One interesting possibility is that the Carter symmetry is broken, thereby giving rise to a non-Kerr object, as opposed to a deformed-Kerr body which still forbids chaotic phenomena even if the spacetime is not exactly Kerr (Papadopoulos and Kokkotas, 2018; Destounis et al., 2020). \nIn general, sections of the inspiral that behave as bound orbits can be characterised by both radial ( ω r ) and angular ( ω θ ) libration frequencies, which describe the rate of transition from the periastron to the apastron of the orbit and longitudinal oscillations about the equatorial plane, respectively (Contopoulos, 2002). On the other hand, classical results from dynamical systems theory infer that small islands of stability (Birkhoff islands) form around periodic orbits in the phase space of non-integrable dynamical systems (Arnold, 1978). When an inspiralling orbit crosses an island, the ratio ω r /ω θ , which defines what is called the rotation curve , remains constant, while otherwise it behaves monotonically as a function of radius. Absence of islands therefore implies an everywhere monotonic rotation curve, while the dynamics display transient plateau features for non-Kerr spacetimes when orbits intersect with an island (Apostolatos et al., 2009). \nSeveral studies have shown that these transient plateaus also introduce features into the gravitational waveforms which are, in principle, discernible from deformed-Kerr features (Apostolatos et al., 2009; Lukes-Gerakopoulos et al., 2010; Contopoulos et al., 2011; Cárdenas-Avendaño et al., 2018; Destounis et al., 2020; Lukes-Gerakopoulos and Witzany, 2021). Figure 34 shows the fundamental frequency evolution of a gravitational waveform, associated with a particular non-Kerr spacetime (see Destounis et al. (2020); Destounis et al. (2021); Destounis and Kokkotas (2021) for details). However, non-integrable perturbations in the Arnold (1978) sense of the particle \nFigure 34: Periodogram of the fundamental frequency of a gravitational waveform associated with an inspiral on a particular non-Kerr spacetime. A sudden jump in the frequency evolution appears when the inspiralling object crosses a Birkhoff island, which is associated with a valley in the amplitude of the signal's frequencies (white line). \n<!-- image --> \nHamiltonian may also arise due to environmental effects within general relativity (Cardoso et al., 2022). For instance, N > 2 -body interactions (as in the Newtonian case; see also Sec. 3.4.2) (Barausse et al., 2007; Amaro-Seoane et al., 2012) or significant internal spins in the companion (Kiuchi and Maeda, 2004; Lukes-Gerakopoulos et al., 2014) can induce chaos.", '3.4.6 Environment versus PN/self-force degeneracies': "The secular evolution of EMRI orbital elements is intimately connected with the phase and the shape of the GWs that will be measured by LISA. In vacuum this evolution is fully described by general relativity, and it can in principle be computed to arbitrary precision with approximation schemes such as perturbation theory (an expansion in the small mass ratio q of the binary) or the PN expansion (an expansion in the small parameter v/c , where v is the orbital velocity and c is the speed of light). At leading order in perturbation theory, the system can be described as a point-like particle moving in a geodesic orbit around the large BH. The energy flux can be computed either numerically or analytically to varying degrees of accuracy. For example, Munna (2020) computed the energy radiated from eccentric orbits around nonrotating BHs up to 19PN order. Fujita (2015) computed the energy flux from a particle in circular orbit around a rotating BH up to 11PN order. Sago and Fujita (2015) computed the expansion for eccentric orbits around rotating BHs up to 4PN, and up to order e 6 in a small-eccentricity expansion. The PN expansion is an asymptotic series, and it is known to converge quite slowly for EMRIs (Yunes \nand Berti, 2008; Zhang et al., 2011). At higher orders in q , interaction of the particle with its own gravitational perturbation gives rise to gravitational self-force, which drives the radiative evolution of the orbit, and whose effects can be accounted for order by order in q (Barack and Pound, 2019). \nEMRIs are however by necessity embedded in astrophysical environments, and as such it is likely that their secular evolution will differ from the pure vacuum case. Of all the environmental factors, gravitational torques from accretion flows are likely to be most the significant (Cardoso and Maselli, 2020). Obtaining realistic estimates for the influence of accreting gas on the orbital evolution and phase of the binary is difficult because accretion dynamics is a largely unexplored 3D magnetohydrodynamics problem over a large dynamic range. The best theoretical prediction for the impact of gas dynamics on IMRI/EMRI phase errors are from 2D viscous hydrodynamics (Derdzinski et al., 2019, 2020). Phase errors grow with the surface density of the accretion disc. The sign in the tidal torque, i.e. whether the separation increases or decreases, depends on the mass ratio, the strength of the viscosity (the α parameter), and the rate of inspiral (Derdzinski et al., 2020). Some parameters even experience stochastic variability in the sign of the tidal torque for EMRIs; such variability would be in stark contrast with secular phase errors coming from truncating the PN expansion. Phase errors due to non-stochastic effects, however, will likely be comparable to PN errors for some disc configurations and orbital separations (Barausse et al., 2014; Cardoso and Maselli, 2020; Annulli et al., 2020). Understanding whether these situations are likely requires a combination of results from 2D viscous simulations (cf. Derdzinski et al., 2019, 2020) with EM surveys of AGN discs, population synthesis modelling of AGN/binary discs (e.g., Krolik et al., 2019), and estimates of the distribution of the observed binary parameters for LISA. Since all EMRI/IMRI simulations have used 2D Newtonian viscous hydrodynamics, it will be interesting to see how these results change when using more realistic 3D general-relativistic magnetohydrodynamical simulations. Unfortunately, performing a series of simulations with O (2000) orbits, which seems to be required to reach a steady-state in 2D simulations, is computationally prohibitive at present. \nAnother obvious effect that can spoil the vacuum evolution of an EMRI is the influence of a third gravitational body. In the case of a hierarchical triple, two effects can take place. First, the influence of the perturber can produce a shift in the binding energy of the inner binary. Will (2014) showed that this shift is constant even if the inner binary undergoes perihelion advance. Second, if the perturber is sufficiently inclined it can induce von Zeipel-Kozai-Lidov oscillations (von Zeipel, 1910; Kozai, 1962; Lidov, 1962) in the inner binary. This can in principle cause an enhancement in the eccentricity of the inner binary. In the case of EMRIs, however, we can expect gravitational perturbations in a hierarchical triple to be very weak. The pull of the third body essentially acts as a tidal force between the components of the inner binary. Therefore, it scales as ∼ aR -3 , where a is the typical separation of the inner binary and R is the distance of the perturber to its centre of mass. For EMRIs, a will generally be very small ( 10 to 10 3 Schwarzschild radii) and the third power of R will strongly suppress tidal forces. As an example, one can compare the von Zeipel-Kozai-Lidov oscillation timescale t KL with the gravitational radiation reaction timescale t GW : \nt GW ∼ t KL = 2 π √ GM Gm 3 R 3 a 3 / 2 . (31) \nBy using Peters' formula (Peters, 1964a) one can find the typical orbital separation at which GW emission and von Zeipel-Kozai-Lidov oscillations change the orbital elements on the same timescale. For circular orbits, this yields \nR KL / GW ≈ a ( m 3 qM ) 1 / 3 ( a r S ) 5 / 6 , (32) \nwhere q is the mass ratio and M the total mass of the inner binary, while r S is the Schwarzschild radius of the central MBH and m 3 the mass of the perturber. For EMRIs, it is clear that the two \ntimescales can be comparable only for very massive or very close perturbers. Nonetheless, recent works have used this result to compute event rates for binaries that are affected by the KozaiLidov eccentricity enhancement (Randall and Xianyu, 2019b; Deme et al., 2020a), and Yunes et al. (2011b) showed that a sufficiently large perturber ( ∼ 10 6 M ⊙ ) at sub-parsec distances can dephase the GW signal of an EMRI by a detectable amount. \nThe astrophysical community has mostly focused on understanding and modelling the influence of the environment on vacuum sources. Claims of detectability for any given effect are therefore often based on simple phenomenological criteria rather than complete signal injections and parameter inference methods. The simplest and most ubiquitous criterion is based on the concept of the SNR of a deviation from a vacuum waveform. While the SNR of a GW event can be estimated by the following formula: \nSNR = √ 2 · 4 ∫ f max f min df ' h 2 c (f ' ) S t (f ' )f ' 2 . (33) \nThe SNR of a deviation can be found by replacing h c with: \nh c exp( iϕ ) → h c exp( iϕ ) -h ' c exp ( iϕ ' ) , (34) \nwhere the primes denote the waveform of a source that is modified through the action of some environmental effect. For GW sources in general, an SNR value of ∼ 8 is chosen as a threshold required in order to claim detectability. The same is assumed to be true for a given deviation, δh c which is deemed detectable whenever ∆ SNR > 20 (at least for EMRIs). While such criteria can serve as a first order approximation, they do not take into account many of the complications that will plague data analysis procedures required to extract signal from LISA's datastream. Effects such as degeneracies, the lack of appropriate waveforms and subtleties of Bayesian analysis in parameter spaces with many dimensions are only a few of the many considerations that should in principle be taken into account when considering the detectability of environmental effects. Degeneracies are especially important, since the influence of many environmental effects might be misinterpreted as sources with different intrinsic parameters. As a simple example, consider the evolution of a BH binary in gas. The primary effect of gas would be to change the rate at which the binary chirps, i.e. ˙ f . However, in a blind waveform template search, the same variation in ˙ f could likely be accounted for by a slightly modified mass of the system, inducing a bias in parameter estimation. With sufficient SNR, it will be likely possible to break such degeneracies from leading order parameters such as chirp mass and distance, using e.g. the information contained in higher derivatives of the frequency, f . Further work should confirm that the same can be said for subtler GR vacuum effects, mainly the spin components and the eccentricity of the source. A possible way forward would be an increased collaboration between the data analysis and astrophysics working groups: the former can provide more sophisticated phenomenological detectability criteria, while the latter could direct the data analysis efforts towards those effects that are expected to be relevant. \nThe contribution of LISA to our understanding of the host environment The variety of gas torques suggests that, if chirping, EMRIs in gaseous environments will exhibit characteristic signatures that may allow us to probe the inner regions of AGN discs. If a phase shift is detected and confirmed to be of gas origin, its magnitude and evolution will be a direct consequence of interaction with local gas properties around the BHs. However, detecting such deviations will require improving models of environmental effects such that we can include them properly in parameter estimation. A critical step is to assess for which regions of parameter space these effects will be unique or degenerate with system parameters. In the latter case, neglecting them may induce biases in parameter estimation. We expect such signatures to arise in only a subset of systems, whereas deviations from general relativity would arise in all EMRIs (depending on the observed frequency). Gas effects can also lead to additional waveform implications: \nfor example prograde, disc-embedded sources will likely have low eccentricity and some degree of spin/inclination alignment with the central MBH. If deep within the potential well of the MBH, these effects may be complemented by a phase-shifting from the Doppler effect (Sec. 3.4.3). \nElaborating on what we have presented here but also in the previous sections, the presence of an XMRI can alter the evolution of an EMRI on its way to cross the event horizon of the MBH. This can lead to extreme situations in which the orbital dynamics of the EMRI is not just affected by the presence of the XMRI, but to the point of scattering off the EMRI from its inspiraling orbit towards the MBH (Vretinaris & Amaro Seoane, in prep.). Since our Galactic Centre and MBH might be envisaged as a typical target for LISA, this means that many, if not all nuclei in the LISA observational volume are prone to this problem. \nOur theory of how stars distribute around MBHs is more than four decades old and seems to be robust. However, at distances very close to the MBH, the power-law distribution of the stellar system (the 'cusp') is ill-defined, because the number density drops significantly. If a star happened to be close to an EMRI, in principle one could reverse-engineer the modulation induced in the waveform, in particular in the phase, to recover information about such perturbing stars from a region which is too obscured to be accessible to EM telescopes. \nThe merger in a b-EMRI system induces a kick to the BH remnant (Centrella et al., 2010). This kick causes a glitch in the EMRI waveform, which, through a careful analysis, is discernible in the data stream (Han and Chen, 2019). The b-EMRI can hence accurately weigh the mass loss due to the BH+BH merger, and offer an opportunity to test general-relativistic effects (in particular the dispersion relation of GWs and the weak equivalence principle). \nSince a significant fraction of EMRIs can be hosted in galaxies which move relative to us at very high speeds, the imprint of the aberration and beaming effects on the waveform can be crucial. A constant drift of the centre of mass of a source also can affect the higher multipoles of the gravitational waveform. This, in turn, affects the frequency and amplitude of the wave as seen by a distant observer (Gualtieri et al., 2008; Torres-Orjuela et al., 2020a). Therefore, higher modes can be used to break the aforementioned degeneracy between a constant velocity and the mass/distance of the source. Considering the change of the modes for EMRIs, LISA should be able to detect constant velocities of just 1000 kms -1 for an SNR of around 70 (Torres-Orjuela et al., 2021). Moreover, as mentioned in that work, we could use this information to obtain a detailed map of the relative speed distribution of galactic clusters out to distances unaccessible to EM observations. \nAnother interesting possibility is that dark matter minispikes could impact the gravitational waveform, inducing dephasings that could be detected by LISA (Eda et al., 2013; Yue and Han, 2018; Hannuksela et al., 2020; Kavanagh et al., 2020). Furthermore, the existence of dark matter halos around IMBHs could accelerate the formation of IMRIs (Yue and Han, 2018). Therefore, the event rates of IMRIs may be much higher than previous estimates, which did not include a DM halo (Yue et al., 2019). \nAs for astrophysical chaos, we can deduce that the detectability of the plateau scales with the magnitude of the non-Kerr parameters and the mass ratio of the EMRI: the 'islands' presented above become larger for greater non-Kerr parameters, and the system requires more time to cross an island for greater mass disparity (Lukes-Gerakopoulos et al., 2010). Crossing into an island leads to a period of frequency modulation, which can, in principle, be detected by LISA (Destounis et al., 2020). More careful data analysis (using, e.g., a Fisher matrix study) is therefore required to determine whether chaotic phenomena have a non-general-relativistic origin, since the frequency jumps described in Fig. 34 may be mimicked by environmental effects. \nThe research carried out so far demonstrates the difficulty of distinguishing gas-driven environmental effects from poorly modelled GR effects. Work to date has explored a relatively narrow range of parameter space for possible environmental effects, and more work should be done to understand the dominant effects, even within currently available 2D Newtonian hydrodynamical models. As computational resources increase, closer to launch, it would also be helpful to ex- \npand theoretical efforts to include at least some 3D general relativistic magnetohydrodynamical simulations. Though challenging, further work must also be done to examine the degeneracies between, e.g., additional PN terms and gas-driven departures from GR. The most important efforts are finding effects that share the same frequency or mass dependence-for effects that do not share dependencies, we can hope to distinguish the source from the GW observations themselves. Gas-driven sources should be a subset of all sources (and should have eccentricities and inclinations which help to distinguish them), while higher-order corrections should apply to every system (though the corrections may be mass- or frequency-dependent). Assuming we overcome the challenges listed here, disentangling various environmental signatures from GWs will give us access to unique measurements of MBH environments purely through GWs. Such measurements are inaccessible via EM observations. Characteristic deviations (or even a lack thereof) will provide constraints on gas densities, dark matter profiles, or the presence of external perturbers.", 'Coordinators: Pau Amaro Seoane, Andrea Derdzinski Contributors: Pau Amaro Seoane, Andrea Derdzinski, Giuseppe Lodato, Martina Toscani': "While the majority of this Chapter describes resolvable sources (as they are certainly the most interesting for guaranteed science), most EMRIs/IMRIs throughout the Universe will not be individually detectable, particularly if they are too distant, at earlier stages of their inspiral, or their GWs are too weak (which is moreso an issue for inspirals of WDs or NSs). The combined signal from the population of faint, unresolved sources will constitute a stochastic background. \nThe EMRI background may lie well below the LISA sensitivity or exceed it, contributing an additional confusion noise. Its amplitude scales with the EMRI rate, although not necessarily linearly (Barack and Cutler, 2004a), and its precise spectral shape will depend on the efficiency of various formation channels over cosmic time. Seminal predictions find that the background signal will only become comparable to the LISA noise if the EMRI rate is substantial: e.g., if the detection rate is as high as O (10 2 ) detections per year, the corresponding background may increase the LISA noise by a factor of nearly ∼ 2 (Barack and Cutler, 2004a). More recent estimates based on EMRI catalogues by Babak et al. (2017) use an updated version of the LISA sensitivity curve and find that, for a range of EMRI rates, the background may add considerable noise (attaining an SNR of a few to few-hundred) within the LISA sensitivity bucket around f ∼ 3 mHz (Bonetti and Sesana, 2020). Higher levels of confusion noise may compromise our ability to detect faint sources that fall into this frequency range, such as high redshift, low mass MBH mergers-although one could argue that this would be compensated for by the generous resolvable EMRI rate. \nA detectable background may provide additional information on the cosmological EMRI population and the efficiency of various formation mechanisms, if there exists robust differences in the spectrum between formation channels. The trick to detecting (and then hopefully characterizing) a background signal is to distinguish it from the instrumental noise as well as other confusion sources (see Romano and Cornish 2017 for a comprehensive review). For LISA, the main contributor to confusion noise is expected to arise from Galactic binaries: while many will be individually resolvable, the rest will form a confusion foreground (e.g., Nissanke et al., 2012, but see Chapter 1) that may overwhelm any extragalactic stochastic signal. Fortunately, LISA's orbital motion around the Sun introduces an annual modulation in the anisotropic galactic foreground, and this makes it possible to distinguish the astrophysical signal from the instrument noise. With a prior understanding of the LISA noise, knowing the distinct spectral shape of an astrophysical foreground further helps us separate the two, so there is hope for detecting an underlying stochastic background (Adams and Cornish, 2014). Such techniques were successfully \napplied in the LISA mock data challenge (Robinson et al., 2008). At the moment, predictions for the EMRI background signal suffer from the same uncertainties as detection rates (see Table 3.6), but these can be improved as we increase our understanding of formation mechanisms. Improving waveform modelling or finding other methods of accurate signal extraction will also be critical if we hope to detect an underlying signal. \nAn important step in this analysis would be to distinguish the EMRI background from other possible background sources. In addition to EMRIs, there may be characteristic background signals from extra-galactic WD, NS, or BH binaries (e.g., Chen et al., 2019b; D'Orazio and Samsing, 2018), TDEs (discussed below), phase transitions in the early universe (Maggiore, 2000; Giblin et al., 2012; Leitao et al., 2012), or cosmic strings (Siemens et al., 2007). If the spectrum is sufficiently constrained, then it is likely that the origin of the signal, whether from a large number of unresolved EMRIs or other extragalactic sources, can be determined (Barausse et al., 2020a).", '3.5.1 TDE background': 'A particular type of EMRI background is the one generated by the unresolvale GW signal from the cosmic TDE population. The calculation of such background has been performed by Toscani et al. (2020) for both main-sequence stars being disrupted by MBHs in galactic nuclei and WDs being disrupted by IMBHs in globular clusters. The signal has a characteristic spectral shape h c ∝ f -1 / 2 , due to the specific impulsive nature of these events. The predicted amplitude of the background is generally low, with WDs on IMBHs providing typical strains of ≈ 10 -23 -10 -21 and main-sequence stars on MBHs providing ≈ 10 -22 . \nThe contribution of LISA to our understanding of backgrounds of inspirals In summary, LISA will have the capability to detect a stochastic background signal, once the galactic foreground is subtracted. This measurement will improve throughout the mission lifetime as we constrain the instrument noise and resolve individual sources (Adams and Cornish, 2014). If EMRIs provide the dominant contribution within some frequency range-e.g., around 3 mHz , as predicted by Bonetti and Sesana (2020)-a measurement of the background spectrum can serve as an additional measurement of dynamics in galactic nuclei. For example, the amplitude and spectrum of the background are related to the number of EMRIs that are either at earlier inspiral stages or at higher redshifts, as well as the MBH mass function. If the background is above the LISA sensitivity and not well-characterised, it will contribute to the noise budget, possibly complicating the detection of other weak signals. To avoid this, it is important that we improve our predictions on the EMRI rate. Further work is also needed to constrain the expected spectrum of various background signals, in order to determine which will be dominant, distinguishable, and removable from the LISA data. If the EMRI rate is low enough such that the background falls below the LISA sensitivity, then it becomes possible to detect other stochastic signals, such as those predicted from extragalactic binaries or signatures from the early Universe. \nAs for the background from TDEs, its detection could provide interesting insights both on the distribution of quiescent MBHs (for main sequence stars tidally disrupted) and on the occupation fraction of IMBHs in globular clusters (for TDEs of WDs), up to redshift ≈ 3 . Yet, this detection seems to be very difficult. Indeed, the background produced by WD TDEs will lie in a high part of the frequency window (deciHertz to a few Hertz), where LISA will be less sensitive (yet, more sensitive interforemeters in this frequency interval are planned for the future). Instead, the background from MS TDEs is expected at lower frequency ( 10 -4 -10 -2 Hz), but will be still below the threshold LISA sensitivity. Hence, this detection seems unlikely (although some background signal below the strain sensitivity might still be visible in some cases, Sesana (see 2016, for more details)). \nTable 10: Rates and SNRs for inspirals. Note that the rates for XMRIs are at any given moment in the MW and, possibly, nearby galaxies (see reference).', 'Contributor: Saavik Ford': 'To summarise: the basic physics of EMRI mergers has been known for a long time. We can expect to find EMRIs in NSCs harboring an MBH, and can predict the dynamics of their formation and evolution using relaxation theory. The waveform modelling for EMRIs is also reasonably advanced, such that the path to detectability of such signals is understood (if challenging). However, there are substantial uncertainties in the astrophysical parameters that govern the rates of EMRI production, notably the spins of MBHs and, most critically, the radial mass distribution of NSCs. These astrophysical unknowns will change the ratio of plunging extreme mass ratio mergers to bona fide EMRIs - in the case of plunging mergers, an extreme mass ratio merger does occur, but the interactions of the merging object with the inner edge of the stellar cusp alters the trajectory of the low mass object after only a few orbits, and produce a rapid merger. Since LISA detection of EMRIs will depend on the buildup of a sufficient SNR over many orbits, plunges are rendered undetectable. \nHowever, assuming the inner edge of an NSC is typically sufficiently far from the MBH, and if MBH spins are typically non-zero, EMRIs can occur at a high enough rate that LISA would detect one or more over the lifetime of the mission. If an EMRI is detected, we will immediately obtain a wide variety of both fundamental physical and astrophysical information (including, implicitly, information about the mass distribution in NSCs and MBH spins). Because of the many-orbit nature of an EMRI, such events can provide a detailed map of the gravitational field in the vicintiy of the MBH, yielding exquisite measurements of the mass and spin of that MBH, and providing an opportunity to probe fundamental physics by testing for subtle departures from GR. \nGiven the current uncertainties on EMRI rates, it is most useful to proceed along multiple fronts: \n- · Theoretical work to understand higher-order dynamical effects which may preserve more EMRIs for sufficient cycles to allow detection by LISA (i.e. preventing plunges)\n- · Observational work constraining the inner edge of the NSC cusp in nuclei other than the MW (M32 would be notably useful)\n- · Theoretical and observational work constraining the binary fraction in typical NSCs (enabling better estimates of the EMRI rate due to binary tidal separation)\n- · Theoretical work to develop non-standard EMRI channels, especially AGN and SNe routes, to constrain rates and parameter distributions, and permit reverse engineering of astrophysical parameters \nIn addition, there is groundwork to be done on the waveform, data analysis and coordination front: \n- · Self-force calculations to second order in mass ratio for generic orbits in a Kerr metric to enable high precision waveform calculations\n- · Further data analysis work with updated waveforms to improve EMRI extraction from the LISA datastream\n- · Coordination of data analysis with radio and ground-based GW observatories in case of pulsar or b-EMRI detection \nIMRIs provide still more exciting science opportunities, but correspondingly more challenging uncertainties. Due to their larger mass ratio, IMRIs cannot be treated using the same theoretical mechanisms as EMRIs (i.e. as small perturbations), yet they are also not sufficiently large to be treated using the mechanisms that apply to near-equal-mass binaries. This reality provokes difficulties in several directions - we cannot generally apply the same relaxation theory strategies to predict IMRI formation dynamics, nor can we readily produce IMRI waveform models using numerical relativity without changing computational strategy. In addition, there are at least 2 types of IMRIs to be considered: 1) light IMRIs, where the more massive partner is an IMBH and the less massive partner is a stellar mass BH; and 2) heavy IMRIs, where the more massive partner is an MBH and the less massive partner is an IMBH. There are multiple formation channels for each, and thus large astrophysical uncertainties in predicting their rates. One substantial uncertainty has recently been removed: with the announcement of GW190521, we are certain that low-mass IMBHs do exist. Though their formation environment remains uncertain, work thus far points to some kind of dynamical origin, encouraging expectations that there may be environments conducive to the formation of at least some light IMRIs. \nBroadly, the channels for light IMRIs include: 1) formation in globular clusters (assuming the presence of an IMBH in the cluster); 2) formation in dwarf galaxies (assuming the presence of an IMBH in the galaxy); 3) formation in an AGN disc (assuming the formation of an IMBH at a disc migration trap). In each of these cases, the first uncertainty is the presence of an IMBH in the relevant environment. There are no universally accepted detections of IMBH in globular clusters. While there are some IMBH known in dwarf galaxies, their occupation fraction is not well-measured and depends on the still unknown physics of BH seed formation at high redshift. IMBH formation in AGN discs is expected to be nearly universal if such discs contain a migration trap and are sufficiently long-lived. Unfortunately neither condition is sufficiently theoretically or observationally well-constrained to make a confident statement on the rate of IMBH formation in AGN discs. \nSadly that is not the end of our uncertainties for the formation of these systems - the dynamics in each case are difficult to model, as noted above, and as with EMRIs, the systems might produce a beautiful, detectable inspiral, or a rapid plunge. Among the important open questions are the observational presence or absence of IMBH in each formation environment, and observations and theoretical investigation of the (extremely different) dynamical environment around each IMBH. \nFor heavy IMRIs, we have a similar diversity of formation channels: 1) globular clusters containing an IMBH infalling into a galactic nucleus containing a MBH; 2) dwarf galaxies infalling into a galactic nucleus containing a MBH; 3) AGN-produced IMBH falling into their host MBH (likely in the post-AGN phase). Globular clusters, being denser than dwarf galaxies, will deposit their IMBHs in galactic nuclei more rapidly than dwarf galaxies, and in general, are expected to dominate the rate in this formation channel. However, if globulars do not contain IMBHs, we should consider dwarf galaxy mergers quite carefully, since at least some dwarfs are known to harbor IMBHs. For low-mass galaxy groups, dwarf mergers will be the most common type of merger and do lead to the formation of bound IMBH-MBH systems in less than a Hubble time. However, theoretical astrophysical rate calculations for resulting IMRIs, over the volume probed by LISA, remain an important open question. Observations of dwarf galaxies and their evolution over cosmic time, as well as observations that inform the occupation fraction of globular clusters and dwarf galaxies are, consequently, critical unknowns. AGN production of IMBHs provide a \npotentially extremely high rate of IMRIs, given the formation location and expected GW inspiral time; such IMRI systems will not likely be disrupted by interactions with the NSC. However, as with the light IMRI channel, substantial uncertainties in the structure of AGN discs and their lifetimes lead to orders-of-magnitude uncertainties in the rate estimates for this channel. \nIMRIs have received less attention in the literature to date, and consequently the tasks to complete before LISA launches tend to be larger. These include: \n- · Determining the occupation fraction of globular clusters and dwarf galaxies\n- · Determining the contribution to the heavy IMRI rate from low mass galactic environments\n- · Determining the formation environment of GW190521-like sources\n- · IMRI waveform modelling and extraction (may require expensive numerical relativity modelling)\n- · Modelling of IMRI formation (and rates) in both gas-poor and gas-rich environments \nFinally, we consider the open questions related to the relatively new class of unequal mass ratio inspirals, XMRIs. Here, the physics of their formation and evolution is similar to that of EMRIs, but they are labelled eXtreme due to the very small mass of the secondary - for LISA frequencies, the secondary would typically be a brown dwarf. Similar to the situation for EMRIs, the uncertainties largely relate to our astrophysical ignorance; however, we are able to limit the locations we must investigate. Due to their low mass and consequently small strains, XMRIs will only be detectable within roughly our Local Group, meaning either from the MW or M31. In addition to the relevant questions for EMRIs (especially NSC radial mass distribution), we must understand the mass function of those NSCs. How many brown dwarfs are there in the galactic centre? If the IMF is top heavy, there may be very few - however, if we assume a more standard IMF, and the radial mass distribution and MBH spin are favorable, LISA can expect to detect one or more XMRIs from Sgr A* over the mission lifetime. \nFor XMRIs there are several useful items to work on as we proceed towards launch: \n- · Determine the low end mass function in the galactic centre\n- · Investigate possible interactions between XMRIs and EMRIs (and find distinguishing observables between interacting EMRI-XMRI systems and departures from GR such as the chaotic behaviour introduced by non-Kerr objects)\n- · Data analysis work to properly characterize potential loud XMRIs \nAs we have discussed, various mechanisms for producing EMRIs and IMRIs may have EM counterparts - this may enable independent rate constraints either prior to or concurrent with LISA; further, if specific counterparts are reliably identified, we can use the complementary information provided by each messenger to learn more about the astrophysics of the emitting system. Notable work to be done includes: \n- · Detailed hydrodynamical models of EM emission mechanisms for GW events in AGN discs\n- · Athena observations of candidate IMRI systems if Athena will not be flying simultaneously with LISA (or coordination if missions are concurrent) \nEach of these types of inspirals present a large parameter-space of possible waveforms, making detection itself a notable challenge. Narrowing the theoretical uncertainties for each channel, in advance of LISA, therefore also has implications for the detectability of their signals. However, work may also be done on the signal processing side to find new strategies for extracting the signals, and doing reliable parameter estimation on them, from the LISA data stream. In this \ncontext, there is currently substantial concern over possible degeneracies between gas-induced phase shifts and departures from general relativity; fortunately, if we have multiple events from multiple populations, departures from general relativity should be universal, while gas effects will impact only a subset of events. On the other hand, some signals containing environmental effects will be clearly identifiable (e.g., b-EMRIs). All of these channels may contribute to a detectable EMRI background; however, disentangling multiple populations from such an unresolved background will be extremely challenging and fundamentally requires more theoretical development from each contributing channel. From these areas, we would especially like to highlight the need for a thorough parameter space exploration with at least 3d Newtonian hydrodynamical simulations of the impact of gas on the inspiral waveforms. \nSubstantially unequal mass ratio inspirals of all types represent an important class of sources, uniquely detectable by LISA. In order to best exploit the astrophysical and fundamental physical science achievable by LISA using these types of events, in the years preceding launch, we will need to work primarily on developing more detailed models for each formation channel, and on observationally constraining the parameters used as inputs of those models, along the lines we have outlined above.', 'General Summary': "The decade prior to LISA's launch will be an exciting one for the astrophysics community, presenting unique challenges and opportunities in preparing for LISA's first observations. This review outlines the extensive landscape of astrophysical theory, numerical simulations, and current astronomical observations that will influence preparations for the pipelines that will deliver LISA data, and guide our interpretations of the first LISA observations and catalogues. \nThis review describes the current state of knowledge regarding three main source classes for LISA: ultra-compact stellar-mass binaries, massive black hole binaries, and extreme or intermediate mass ratio inspirals. For each of these three source classes, our current understanding of the astrophysical processes that create them and guide their ongoing evolution is a rich tapestry formed from extant observations (usually electromagnetic), numerical simulations and modelling, and theoretical considerations. LISA data will be added to this, providing new independent information that will help constrain the physics governing these systems, and opening up new avenues of investigation for future observations, theory, and simulations. \nAstronomy observations will continue to evolve and alter the scientific landscape prior to LISA's launch, and theory and modelling will become more refined. Such advances will inform our understanding of the ways in which LISA data can be used, and they can also sharpen the focus on the important ways in which gravitational wave data will expand and enhance our ability to understand astrophysical phenomena in many different environments and scales. This review endeavours to provide a framework within which to consider these possibilities, and should be a good starting point for those interested in using LISA as a new observational tool for understanding the Universe.", 'Acknowledgements': '- P. Dayal acknowledges support from the European Research council (ERC-717001) and from the Netherlands Research Council NWO (016.VIDI.189.162).\n- P.H. Johansson acknowledges the support from the European Research Council (ERC-818930).\n- S. Toonen acknowledges support from the Netherlands Research Council NWO (VENI 639.041.645 grant)\n- C. Unal is supported by European Structural and Investment Funds and the Czech Ministry of Education, Youth and Sports (Project CoGraDS - CZ.02.1.01/0.0/0.0/15\\_003/0000437).\n- S. Chaty acknowledges the LabEx UnivEarthS for the funding of Interface project I10 \'From binary evolution towards merging of compact objects".\n- A. De Rosa acknowledges financial contribution from the agreement ASI-INAF n.2017-14-H.O E.Berti is supported by NSF Grants No. PHY-1912550 and AST-2006538, NASA ATP Grants No. 17-ATP17-0225 and 19-ATP19-0051, NSF-XSEDE Grant No. PHY-090003, and NSF Grant PHY-20043.\n- D. Gerosa is supported by European Union\'s H2020 ERC Starting Grant No. 945155-GWmining, Leverhulme Trust Grant No. RPG-2019-350 and Royal Society Grant No. RGS-R2-202004.\n- T.Bogdanovic acknowledges support by the NASA award No. 80NSSC19K0319 and by the NSF award AST-1908042. D. Porquet acknowledges funding support from CNES.\n- C. Danielski acknowledges financial support from the State Agency for Research of the Spanish MCIU through the "Center of Excellence Severo Ochoa" award to the Instituto de Astrofísica de Andalucía (SEV-2017-0709)\n- B. L. Davis acknowledges support from Tamkeen under the NYU Abu Dhabi Research Institute grant CAP3.\n- F. Pacucci acknowledges support from a Clay Fellowship by the SAO and from the Black Hole Initiative, which is funded by grants from the John Templeton Foundation and the Gordon and Betty Moore Foundation.\n- A. J. Ruiter acknowledges support from the Australian Research Council Future Fellowship grant FT170100243.\n- V. Paschalidis is supported by NSF grant PHY-1912619 and NASA grant 80NSSC20K1542 to the University of Arizona, and NSF-XSEDE grant TG-PHY190020.\n- D. Haggard acknowledges support from the NSERC Discovery Grant and Canada Research Chairs programs, and the Bob Wares Science Innovation Prospectors Fund.\n- M. Toscani acknowledges European Union\'s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement NO 823823 (RISE DUSTBUSTERS project) and COST Action CA16104 - Gravitational waves, black holes and fundamental physics, supported by COST (European Cooperation in Science and Technology).\n- M. Chruslinska, A. Istrate and G. Nelemans acknowledge support from Netherlands Research Council NWO.\n- T. Fragos and S. Bavera acknowledge support from a Swiss National Science Foundation Professorship grant (project numbers PP00P2\\_176868 and PP00P2\\_211006)\n- V. 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Science China Physics, Mechanics, and Astronomy 62(2):29502. https://doi.org/10.1007/s11433-018-9309-2. arXiv:1812.04020 [astro-ph.IM]", 'Authors and affiliations': 'Amaro Seoane, Pau; Andrews, Jeff; Arca Sedda, Manuel; Askar, Abbas; Baghi, Quentin; Balasov, Razvan; Bartos, Imre; Bavera, Simone S.; Bellovary, Jillian; Berry, Christopher P. L.; Berti, Emanuele; Bianchi, Stefano; Blecha, Laura; Blondin, Ste\'phane; Bogdanović, Tamara; Boissier, Samuel; Bonetti, Matteo; Bonoli, Silvia; Bortolas, Elisa; Breivik, Katelyn; Capelo, Pedro R.; Caramete, Laurentiu; Cattorini, Federico; Charisi, Maria; Chaty, Sylvain; Chen, Xian; Chruślińska, Martyna; Chua, Alvin J. K.; Church, Ross; Colpi, Monica; D\'Orazio, Daniel; Danielski, Camilla; Davies, Melvyn B.; Dayal, Pratika; De Rosa, Alessandra; Derdzinski, Andrea; Destounis, Kyriakos; Dotti, Massimo; Duţan, Ioana; Dvorkin, Irina; Fabj, Gaia; Foglizzo, Thierry; Ford, Saavik; Fouvry, Jean-Baptiste; Franchini, Alessia; Fragos, Tassos; Fryer, Chris; Gaspari, Massimo; Gerosa, Davide; Graziani, Luca; Groot, Paul; Habouzit, Melanie; Haggard, Daryl; Haiman, Zoltan; Han, Wen-Biao; Istrate, Alina; Johansson, Peter H.; Khan, Fazeel Mahmood; Kimpson, Tomas; Kokkotas, Kostas; Kong, Albert; Korol, Valeriya; Kremer, Kyle; Kupfer, Thomas; Lamberts, Astrid; Larson, Shane; Lau, Mike; Liu, Dongliang; Lloyd-Ronning, Nicole; Lodato, Giuseppe; Lupi, Alessandro; Ma, Chung-Pei; Maccarone, Tomas; Mandel, Ilya; Mangiagli, Alberto; Mapelli, Michela; Mathis, Stéphane; Mayer, Lucio; McGee, Sean; McKernan, Berry; Miller, M. Coleman; Mota, David F.; Mumpower, Matthew; Nasim, Syeda S; Nelemans, Gijs; Noble, Scott; Pacucci, Fabio; Panessa, Francesca; Paschalidis, Vasileios; Pfister, Hugo; Porquet, Delphine; Quenby, John; Ricarte, Angelo; Röpke, Friedrich K.; Regan, John; Rosswog, Stephan; Ruiter, Ashley; Ruiz, Milton; Runnoe, Jessie; Schneider, Raffaella; Schnittman, Jeremy; Secunda, Amy; Sesana, Alberto; Seto, Naoki; Shao, Lijing; Shapiro, Stuart; Sopuerta, Carlos; Stone, Nicholas C.; Suvorov, Arthur; Tamanini, Nicola; Tamfal, Tomas; Tauris, Thomas; Temmink, Karel; Tomsick, John; Toonen, Silvia; Torres-Orjuela, Alejandro; Toscani, Martina; Tsokaros, Antonios; Unal, Caner; Vázquez-Aceves, Verónica; Valiante, Rosa; van Putten, Maurice; van Roestel, Jan; Vignali, Christian; Volonteri, Marta; Wu, Kinwah; Younsi, Ziri; Yu, Shenghua; Zane, Silvia; Zwick, Lorenz; Antonini, Fabio; Baibhav, Vishal; Barausse, Enrico; Bonilla Rivera, Alexander; Branchesi, Marica; Branduardi-Raymont, Graziella; Burdge, Kevin; Chakraborty, Srija; Cuadra, Jorge; Dage, Kristen; Davis, Benjamin; de Mink, Selma E.; Decarli, Roberto; Doneva, Daniela; Escoffier, Stephanie; Fragione, Giacomo; Gandhi, Poshak; Haardt, Francesco; Lousto, Carlos O.; Nissanke, Samaya; Nordhaus, Jason; O\'Shaughnessy, Richard; Portegies Zwart, Simon; Pound, Adam; Schussler, Fabian; Sergijenko, Olga; Spallicci, Alessandro; Vernieri, Daniele; Vigna-Gómez, Alejandro \n- P. Amaro Seoane: Institute of Multidisciplinary Mathematics, Universitat Politécnica de Valéncia, Spain; Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics of Gansu Province, School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, People\'s Republic of China; Institute of Theoretical Physics & Research Center of Gravitation,Lanzhou University, Lanzhou 730000, People\'s Republic of China; Kavli Institute for Astronomy and Astrophysics, Beijing, China; amaro@riseup.net;\n- J. Andrews: Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA); Department of Physics and Astronomy, Northwestern University, 1800 Sherman Ave, Evanston, IL 60201, USA; jeffrey.andrews@northwestern.edu;\n- M. Arca Sedda: Astronomisches Rechen Instituy (University of Heidelberg); m.arcasedda@gmail.com; A. Askar: Lund Observatory, Department of Astronomy, and Theoretical Physics, Lund University, Box 43, SE-221 00 Lund, Sweden; askar@astro.lu.se;\n- Q. Baghi, IRFU, CEA, Université Paris-Saclay, F-91191 Gif-sur-Yvette, France; quentin.baghi@cea.fr R. Balasov: Institute of Space Science, Romania; Faculty of Physics, University of Bucharest,\n- Romania; rabalasov@spacescience.ro;\n- I. Bartos: Department of Physics, University of Florida, PO Box 118440, Gainesville, FL 32611-8440, USA; imrebartos@ufl.edu;\n- S. Bavera: Geneva Observatory, University of Geneva, Chemin Pegasi 51, 1290 Versoix, Switzerland; Gravitational Wave Science Center (GWSC), Université de Genève, CH1211 Geneva, Switzerland; simone.bavera@unige.ch;\n- J. Bellovary: CUNY - Queensborough Community College; American Museum of Natural History; CUNY Graduate Center; jbellovary@amnh.org;\n- C. Berry: Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA), Department of Physics and Astronomy, Northwestern University, 1800 Sherman Ave, Evanston, IL 60201, USA; SUPA, School of Physics and Astronomy, University of Glasgow, Kelvin Building, University Ave, Glasgow G12 8QQ, UK; christopher.berry.2@glasgow.ac.uk;\n- E. Berti: Johns Hopkins University; berti@jhu.edu;\n- S. Bianchi: Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre, via della Vasca Navale 84, I-00146 Roma, Italy; bianchi@fis.uniroma3.it;\n- L. Blecha: Department of Physics, University of Florida; lblecha@ufl.edu;\n- S. Blondin: Aix Marseille Univ, CNRS, CNES, LAM, Marseille, France; stephane.blondin@lam.fr; T. Bogdanović: School of Physics and Center for Relativistic Astrophysics, 837 State St NW, Georgia Institute of Technology, Atlanta, GA 30332, USA; tamarab@gatech.edu;\n- S. Boissier: Aix Marseille Univ, CNRS, CNES, LAM, Marseille, France; samuel.boissier@lam.fr;\n- M. Bonetti: Dipartimento di Fisica G. Occhialini, Universita\' degli studi di Milano-Bicocca, Piazza della Scienza 3, 20126 Milano Italy; matteo.bonetti@unimib.it;\n- S. Bonoli: Donostia International Physics Centre (DIPC), Paseo Manuel de Lardizabal 4, 20018 Donostia-San Sebastian, Spain; IKERBASQUE, Basque Foundation for Science, E-48013, Bilbao, Spain; silvia.bonoli@dipc.org;\n- E. Bortolas: Dipartimento di Fisica \'G. Occhialini\', Universitá degli Studi di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy; INFN, Sezione di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy; elisa.bortolas@unimib.it;\n- P. Capelo: Center for Theoretical Astrophysics and Cosmology, Institute for Computational Science, University of Zurich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland; pcapelo@physik.uzh.ch;\n- L. Caramete: Institute of Space Science, Magurele, Romania; lcaramete@spacescience.ro;\n- F. Catorini: DiSAT, Universitá degli studi dell\'Insubria, Via Valleggio, 11, I-22100 Como, Italy; INFN, Sezione di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy; fcattorini@uninsubria.it;\n- M. Charisi: Department of Physics & Astronomy, Vanderbilt University, 2301 Vanderbilt Place, Nashville, TN 37235, USA; maria.charisi@nanograv.org; \n- S. Chaty: Université de Paris, CNRS, AstroParticule et Cosmologie, F-75013 Paris, France; sylvain.chaty@u-paris.fr;\n- X. Chen: Astronomy Department, School of Physics, Peking University, Beijing 100871, P.R. China; xian.chen@pku.edu.cn;\n- M. Chruślińska: Department of Astrophysics/IMAPP, Radboud University, PO Box 9010, 6500 GL, The Netherlands; mchruslinska@mpa-garching.mpg.de;\n- A. Chua: Theoretical Astrophysics Group, California Institute of Technology, Pasadena, CA 91125, U.S.A.; achua@caltech.edu;\n- R. Church: Lund Observatory; ross@astro.lu.se;\n- M. Colpi: Department of Physics, University of Milano Bicocca, Milan, Italy; monica.colpi@unimib.it; D. D\'Orazio: Niels Bohr International Academy, Niels Bohr Institute, Blegdamsvej 17, 2100\n- Copenhagen, Denmark; daniel.dorazio@nbi.ku.dk;\n- C. Danielski: Instituto de Astrofísica de Andalucía (IAA-CSIC), Glorieta de la Astronomía S/N, 18008 Granada, Spain; cdanielski@iaa.es;\n- M. Davies: Centre for Mathematical Sciences, Lund University, Box 118, 221 00 Lund, Sweden; melvyn\\_b.davies@math.lu.se;\n- P. Dayal: Kapteyn Astronomical Institute, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands; p.dayal@rug.nl;\n- A. De Rosa: INAF - Istituto di Astrofisica e Planetologia Spaziali, via Fosso del Cavaliere, I-133 Roma, Italy; alessandra.derosa@inaf.it;\n- A. Derdzinski: Center for Theoretical Astrophysics and Cosmology, Institute for Computational Science, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland; andrea@ics.uzh.ch;\n- K. Destounis: Theoretical Astrophysics, IAAT, University of Tübingen, 72076 Tübingen, Germany; kyriakos.destounis@uni-tuebingen.de;\n- M. Dotti: Dipartimento di Fisica \'G. Occhialini\', Universitá degli Studi di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy; INFN, Sezione di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy; INAF, Osservatorio Astronomico di Brera, Via E. Bianchi 46, I-23807, Merate, Italy; massimo.dotti@unimib.it;\n- I. Duţan: Institute of Space Science, Atomiştilor 409, RO-077125 Măgurele, Romania; idutan@spacescience.ro;\n- I. Dvorkin: Institut d\'Astrophysique de Paris, Sorbonne Université & CNRS, UMR 7095, 98 bis bd Arago, F-75014 Paris, France; dvorkin@iap.fr;\n- G. Fabj: Astronomisches Rechen-Institut, Zentrum für Astronomie, Universität Heidelberg, 69120 Heidelberg, Germany; Dept. of Astrophysics, American Museum of Natural History, New York, NY 10024 USA; gaia.fabj@stud.uni-heidelberg.de;\n- T. Foglizzo: AIM, CEA, CNRS, Université Paris-Saclay, Université Paris Diderot, Sorbonne Paris Cité, F-91191 Gif-sur-Yvette, France; thierry.foglizzo@cea.fr;\n- S. Ford: Department of Astrophysics, American Museum of Natural History, New York, NY 10024, USA; Center for Computational Astrophysics, Flatiron Institute, New York, NY 10010, USA; Graduate Center, City University of New York, 365 5th Avenue, New York, NY 10016, USA; Department of Science, BMCC, City University of New York, New York, NY 10007, USA; sford@amnh.org;\n- J. Fouvry: CNRS and Sorbonne Université, UMR 7095, Institut d\'Astrophysique de Paris, 98 bis Boulevard Arago, F-75014 Paris, France; fouvry@iap.fr;\n- T. Fragkos: Département d\'Astronomie, Université de Geneve, Chemin Pegasi 51, CH1290 Versoix, Switzerland; Gravitational Wave Science Center (GWSC), Université de Genève, CH1211 Geneva, Switzerland; anastasios.fragkos@unige.ch;\n- A. Franchini: Dipartimento di Fisica \'G. Occhialini\', Universita degli Studi di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy; INFN - Sezione di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy; alessia.franchini@unimib.it \n- C. Fryer: Center for Theoretical Astrophysics, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA; fryer@lanl.gov;\n- M. Gaspari: INAF - Osservatorio di Astrofisica e Scienza dello Spazio, via P. Gobetti 93/3, I-40129 Bologna, Italy; Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08544-1001, USA; massimo.gaspari@inaf.it;\n- D. Gerosa: Dipartimento di Fisica \'G. Occhialini\', Universitá degli Studi di Milano-Bicocca, Piazza della Scienza 3, 20126 Milano, Italy INFN, Sezione di Milano-Bicocca, Piazza della Scienza 3, 20126 Milano, Italy School of Physics and Astronomy & Institute for Gravitational Wave Astronomy, University of Birmingham, Birmingham, B15 2TT, United Kingdom; davide.gerosa@unimib.it;\n- L. Graziani: Dipartimento di Fisica, Sapienza, Universit \' a di Roma, Piazzale Aldo Moro 5, 00185, Roma, Italy; INFN, Sezione di Roma I, P.le Aldo Moro 2, 00185 Roma, Italy; INAF/Osservatorio Astrofisico di Arcetri, Largo E. Femi 5, 50125 Firenze, Italy; luca.graziani@uniroma1.it;\n- P. Groot: Department of Astrophysics/IMAPP, Radboud University, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands; Department of Astronomy, University of Cape Town, Private Bag X3, Rondebosch, 7701, South Africa; South African Astronomical Observatory, P.O. Box 9, Observatory, 7935, South Africa; The Inter-University Institute for Data Intensive Astronomy, University of Cape Town, Private Bag X3, Rondebosch, 7701, South Africa; p.groot@astro.ru.nl;\n- M. Habouzit: Max-Planck-Institut für Astronomie, Königstuhl 17, D-69117 Heidelberg, Germany; Zentrum für Astronomie der Universität Heidelberg, ITA, Albert-Ueberle-Str. 2, D-69120 Heidelberg, Germany; habouzit.astro@gmail.com;\n- D. Haggard: McGill Space Institute and Department of Physics, McGill University, 3600 rue University, Montréal, Québec, H3A 2T8, Canada; daryl.haggard@mcgill.ca;\n- Z. Haiman: Columbia University; zoltan@astro.columbia.edu;\n- W. Han: Shanghai Astronomical Observatory, CAS, 80 Nandan Road, Shanghai, China 200030; wbhan@shao.ac.cn;\n- A. Istrate: Department of Astrophysics/IMAPP, Radboud University, PO Box 9010, 6500 GL, The Netherlands; a.istrate@astro.ru.nl;\n- P. Johansson: Department of Physics, Gustaf Hällströmin katu 2, FI-00014, University of Helsinki, Finland; Peter.Johansson@helsinki.fi;\n- F. Khan: Department of Space Science, Institute of Space Technology, Islamabad 44000, Pakistan; khanfazeel.ist@gmail.com;\n- T. Kimpson: Mullard Space Science Laboratory, University College London, Holmbury St. Mary, Dorking, Surrey, RH5 6NT, UK; tom.kimpson.16@ucl.ac.uk;\n- K. Kokkotas: Theoretical Astrophysics, University of Tuebingen, Germany; kostas.kokkotas@unituebingen.de;\n- A. Kong: Institute of Astronomy, National Tsing Hua University, Hsinchu 30013, Taiwan; akong@gapp.nthu.edu.tw;\n- V. Korol: Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Straße 1, 85741 Garching, Germany; Institute for Gravitational Wave Astronomy & School of Physics and Astronomy, University of Birmingham, Birmingham, B15 2TT, UK; korol@mpa-garching.mpg.de;\n- K. Kremer: TAPIR, California Institute of Technology, Pasadena, CA 91125, USA; The Observatories of the Carnegie Institution for Science, Pasadena, CA 91101, USA; kkremer@caltech.edu;\n- T. Kupfer: Department of Physics and Astronomy, Texas Tech University, PO Box 41051, Lubbock, TX 79409, USA; tkupfer@ttu.edu;\n- A. Lamberts: Université Côte d\'Azur, Observatoire de la Côte d\'Azur, CNRS, Laboratoire Lagrange, Laboratoire Artémis, Bd de l\'Observatoire,CS 34229, 06304 Nice cedex 4, France.; astrid.lamberts@oca.eu;\n- S. Larson: Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA); Department of Physics and Astronomy, Northwestern University, 1800 Sherman Ave, Evanston, IL 60201, USA; s.larson@northwestern.edu; \n- M. Lau: Monash Centre for Astrophysics, School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia; OzGrav, Australian Research Council Centre of Excellence for Gravitational Wave Discovery, Australia; ;\n- D. Liu: National Astronomical Observatories, Chinese Academy of Sciences; dlliu@bao.ac.cn;\n- N. Lloyd-Ronning: Los Alamos National Lab and The University of New Mexico; lloydronning@lanl.gov;\n- G. Lodato: Universitá degli Studi di Milano; giuseppe.lodato@unimi.it;\n- A. Lupi: Dipartimento di Fisica \'G. Occhialini\', Università degli Studi di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy; INFN - Sezione di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy; alessandro.lupi@unimib.it;\n- C. Ma : Department of Astronomy, Department of Physics, University of California at Berkeley, CA 94720 USA; cpma@berkeley.edu;\n- T. Maccarone: Department of Physics & Astronomy, Texas Tech University, Box 41051, Lubbock TX 79409-1051; thomas.maccarone@ttu.edu;\n- I. Mandel: Monash Centre for Astrophysics, School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia; OzGrav, Australian Research Council Centre of Excellence for Gravitational Wave Discovery, Australia; Institute of Gravitational Wave Astronomy and School of Physics and Astronomy, University of Birmingham, Birmingham, B15 2TT, United Kingdom; ilya.mandel@monash.edu;\n- A. Mangiagli: APC, AstroParticule et Cosmologie, Université de Paris, CNRS, F-75013 Paris, France; mangiagli@apc.in2p3.fr;\n- M. Mapelli: Physics and Astronomy Department Galileo Galilei, University of Padova, Vicolo dell\'Osservatorio 3, I-35122, Padova, Italy; INFN-Padova, Via Marzolo 8, I-35131 Padova, Italy; INAF-Osservatorio Astronomico di Padova, Vicolo dell\'Osservatorio 5, I-35122, Padova, Italy; michela.mapelli@unipd.it;\n- S. Mathis: Département d\'Astrophysique-AIM, CEA/DRF/IRFU, CNRS/INSU, Université Paris-Saclay, Université Paris-Diderot, Université De Paris, F-91191 Gif-sur-Yvette, France; stephane.mathis@cea.fr;\n- L. Mayer: Center for Theoretical Astrophysics and Cosmology, Institute for Computational Science, University of Zurich, Winterthurerstrasse 190, 8057, Zurich, Switzerland; lmayer@physik.uzh.ch; S. McGee: School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK; smcgee@star.sr.bham.ac.uk;\n- M. Miller: University of Maryland, Department of Astronomy, College Park MD 20742-2421; miller@astro.umd.edu;\n- D. Mota: Institute of Theoretical Astrophysics, University of Oslo, PO Box 1029, Blindern 0315, Oslo, Norway; d.f.mota@astro.uio.no;\n- M. Mumpower: Los Alamos National Laboratory; mumpower@lanl.gov;\n- S. Nasim: Missouri University of Science and Technology & Rolla, MO (USA); American Museum of Natural History & New York, NY (USA); ssnkct@mst.edu;\n- G. Nelemans: Department of Astrophysics/IMAPP, Radboud University, PO Box 9010, 6500 GL, The Netherlands; SRON, Netherlands Institute for Space Research, Sorbonnelaan 2, NL3584 CA Utrecht, The Netherlands; Institute of Astronomy, KU Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium; nelemans@astro.ru.nl;\n- S. Noble: Gravitational Astrophysics Laboratory, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA; scott.c.noble@nasa.gov;\n- F. Pacucci: Center for Astrophysics | Harvard & Smithsonian, Cambridge, MA 02138, USA; Black Hole Initiative, Harvard University, Cambridge, MA 02138, USA; fabio.pacucci@cfa.harvard.edu;\n- F. Panessa: INAF - Istituto di Astrofisica e Planetologia Spaziali, via Fosso del Cavaliere 100, I-00133 Roma, Italy; francesca.panessa@inaf.it;\n- V. Paschalidis: Departments of Astronomy and Physics, University of Arizona, Tucson, AZ 85719, USA; vpaschal@email.arizona.edu; \n- H. Pfister: Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China; DARK, Niels Bohr Institute, University of Copenhagen, Jagtvej 128, 2200 København, Denmark; pfisterastro@gmail.com;\n- D. Porquet: Aix Marseille Univ, CNRS, CNES, LAM, Marseille, France; delphine.porquet@lam.fr;\n- J. Quenby: Imperial College London UK; j.quenby@imperial.ac.uk;\n- F. Röpke: Zentrum für Astronomie der Universität Heidelberg, Institut für Theoretische Astrophysik; Heidelberg Institute for Theoretical Studies; friedrich.roepke@h-its.org;\n- J. Regan: Department of Theoretical Physics, Maynooth University, Maynooth, Ireland; john.regan@mu.ie;\n- S. Rosswog: The Oskar Klein Centre, Department of Astronomy, Stockholm University; stephan.rosswog@astro.su.se;\n- A. Ruiter : University of New South Wales (Canberra); ashley.ruiter@adfa.edu.au;\n- M. Ruiz: Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801; ruizm@illinois.edu;\n- J. Runnoe: Vanderbilt University, Department of Physics & Astronomy, 6301 Stevenson Center, Nashville, TN 37235, USA; jessie.c.runnoe@vanderbilt.edu;\n- R. Schneider: Dipartimento di Fisica, Universita di Roma La Sapienza, Piazzale Aldo Moro 2, 00185 Roma Italy; Sapienza School for Advanced Studies, Viale Regina Elena 291, 00161 Roma Italy; Istituto Nazionale di Astrofisica/Osservatorio Astronomico di Roma, via Frascati 33, 00078 Monte Porzio Catone, Roma Italy Istituto Nazionale di Fisica Nucleare, Sezione di Roma1, Piazzale Aldo Moro 2, 00185 Roma Italy; raffaella.schneider@uniroma1.it;\n- J. Schnittman: NASA Goddard Space Flight Center, 8800 Greenbelt Rd, Greenbelt, MD 20771; jeremy.schnittman@nasa.gov;\n- A. Secunda: Department of Astrophysical Sciences, Princeton University, Peyton Hall, Princeton, NJ 08544, USA; asecunda@princeton.edu;\n- A. Sesana: Dipartimento di Fisica \'G. Occhialini", Università degli Studi di Milano-Bicocca, Piazza della Scienza 3, IT-20126 Milano, Italy; alberto.sesana@unimib.it;\n- N. Seto: Department of Physics, Kyoto University, Kyoto 606-8502, Japan; seto@tap.scphys.kyotou.ac.jp;\n- L. Shao: Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, China; lshao@pku.edu.cn;\n- S. Shapiro: University of Illinois at Urbana-Champaign; slshapir@illinois.edu;\n- C. Sopuerta: Institut de Ciéncies de l\'Espai (ICE, CSIC), Campus UAB, Carrer de Can Magrans s/n, 08193 Cerdanyola del Vallés, Spain; Institut d\'Estudis Espacials de Catalunya (IEEC), Edifici Nexus, Carrer del Gran Capitá 2-4, despatx 201, 08034 Barcelona, Spain; carlos.f.sopuerta@csic.es;\n- N. Stone: The Hebrew University of Jerusalem; nicholas.stone@mail.huji.ac.il;\n- A. Suvorov: Theoretical Astrophysics, IAAT, University of Tübingen, Tübingen 72076, Germany; arthur.suvorov@tat.uni-tuebingen.de;\n- N. Tamanini: Laboratoire des 2 Infinis - Toulouse (L2IT-IN2P3), Université de Toulouse, CNRS, UPS, F-31062 Toulouse Cedex 9, France; nicola.tamanini@l2it.in2p3.fr;\n- T. Tamfal: Center for Theoretical Astrophysics and Cosmology, Institute for Computational Science, University of Zurich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland; tomas.tamfal@uzh.ch;\n- T. Tauris: Department of Materials and Production, Aalborg University, Denmark; tauris@mp.aau.dk;\n- K. Temmink: Department of Astrophysics/IMAPP, Radboud University Nijmegen; Karel.Temmink@ru.nl;\n- J. Tomsick: Space Sciences Laboratory, 7 Gauss Way, University of California, Berkeley, CA 94720-7450, USA; jtomsick@berkeley.edu;\n- S. Toonen: Anton Pannekoek Institute for Astronomy, University of Amsterdam, 1090 GE Amsterdam, The Netherlands; toonen@uva.nl; \n- A. Torres-Orjuela: MOE Key Laboratory of TianQin Mission, TianQin Research Center for Gravitational Physics & School of Physics and Astronomy, Frontiers Science Center for TianQin, CNSA Research Center for Gravitational Waves, Sun Yat-Sen University (Zhuhai Campus), Zhuhai 519082, China; atorreso@mail.sysu.edu.cn;\n- M. Toscani: Laboratoire des 2 Infinis - Toulouse (L2IT-IN2P3), Université de Toulouse, CNRS, UPS, F-31062 Toulouse Cedex 9, France; Dipartimento di Fisica, Universitá Degli Studi di Milano, Via Celoria, 16, Milano, 20133, Italy; martina.toscani@l2it.in2p3.fr;\n- A. Tsokaros: University of Illinois at Urbana-Champaign; tsokaros@illinois.edu;\n- C. Unal: CEICO, Institute of Physics of the Czech Academy of Sciences, Na Slovance 1999/2, 182 21 Praha 8, Czechia; unalx005@umn.edu;\n- V. Vázquez-Aceves: Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100190 Beijing, China; veronica@nao.cas.cn;\n- R. Valiante: INAF-Osservatorio Astronomico di Roma, via di Frascati 33, I-00078 Monteporzio Catone, Italy; INFN, Sezione di Roma I, P.le Aldo Moro 2, I-00185 Roma, Italy; rosa.valiante@inaf.it;\n- M. van Putten: Physics and Astronomy, Sejong University, 209 Neungdong-ro, Gwangjin-gu. Seoul 143-747; mvp@sejong.ac.kr;\n- J. van Roestel: Caltech, 1201 E. California Blvd, Pasadena, CA 91125, California (CA) United States; jvanroes@caltech.edu;\n- C. Vignali: Dipartimento di Fisica e Astronomia \'Augusto Righi", Universitá degli Studi di Bologna, Via Gobetti 93/2, I-40129 Bologna, Italy; INAF - Osservatorio di Astrofisica e Scienza dello Spazio di Bologna (OAS), Via Gobetti 93/3, I-40129 Bologna, Italy; cristian.vignali@unibo.it;\n- M. Volonteri: Sorbonne Université, CNRS, UMR 7095, Institut d\'Astrophysique de Paris, 98 bis bd Arago, 75014 Paris, France; martav@iap.fr;\n- K. Wu: Mullard Space Science Laboratory, University College London, Holmbury St Mary, Surrey RH5 6NT, United Kingdom; kinwah.wu@ucl.ac.uk;\n- Z. Younsi: Mullard Space Science Laboratory, University College London, Holmbury St. Mary, Dorking, Surrey, RH5 6NT, UK; z.younsi@ucl.ac.uk;\n- S. Yu: National Astronomical Observatories, Chinese Academy of Sciences; shenghuayu@bao.ac.cn;\n- S. Zane: Mullard Space Science Laboratory, University College London, Holmbury St Mary, Dorking, Surrey, RH5 6NT, UK; s.zane@ucl.ac.uk;\n- L. Zwick: University of Zürich; Centre for Theoretical Astrophysics and Cosmology; zwicklo@ics.uzh.ch;\n- F. Antonini: Gravity Exploration Institute, School of Physics and Astronomy, Cardiff University, Cardiff, CF24 3AA, United Kingdom; antoninif@cardiff.ac.uk;\n- V. Baibhav: Department of Physics and Astronomy, Johns Hopkins University, 3400 N. Charles St, Baltimore, Maryland 21218, USA; baibhavv@gmail.com;\n- E. Barausse: SISSA, Via Bonomea 265, 34136 Trieste, Italy and INFN Sezione di Trieste; IFPU - Institute for Fundamental Physics of the Universe, Via Beirut 2, 34014 Trieste, Italy; barausse@sissa.it;\n- A. Bonilla Rivera: Departamento de Física, Universidade Federal de Juiz de Fora, 36036-330, Juiz de Fora, MG, Brazil.; alex.acidjazz@gmail.com;\n- G. Branduardi-Raymont: Mullard Space Science Laboratory - University College London; g.branduardi-raymont@ucl.ac.uk;\n- K. Burdge: Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, CA 91125, USA; kburdge@caltech.edu; S. Chakraborty: Scuola Normale Superiore; srija.chakraborty@sns.it;\n- J. Cuadra: Departamento de Ciencias, Facultad de Artes Liberales, Universidad Adolfo Ibáñez, Padre Hurtado 750, Viña del Mar, Chile; jcuadra@npf.cl;\n- K. Dage: Department of Physics, McGill University, 3600 University Street, Montréal, QC H3A 2T8, Canada; McGill Space Institute, McGill University, 3550 University Street, Montréal, QC H3A 2A7, Canada; dagek@physics.mcgill.ca; \n- B. Davis: Center for Astro, Particle, and Planetary Physics, New York University Abu Dhabi; ben.davis@nyu.edu;\n- S. de Mink: Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Straße 1, 85741 Garching, Germany; sedemink@mpa-garching.mpg.de;\n- R. Decarli: INAF - Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, via Gobetti 93/3, I-40129, Bologna, Italy; roberto.decarli@inaf.it;\n- D. Doneva: University of Tuebingen; daniela.doneva@uni-tuebingen.de;\n- S. Escoffier: Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France; escoffier@cppm.in2p3.fr;\n- G. Fragione: Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA); Department of Physics and Astronomy, Northwestern University, 1800 Sherman Ave, Evanston, IL 60201, USA; giacomo.fragione@northwestern.edu;\n- P. Gandhi: School of Physics & Astronomy, University of Southampton, Southampton, SO17 1BJ, UK; poshak.gandhi@soton.ac.uk;\n- F. Haardt: Dipartimento di Scienza e Alta Tecnologia, Università degli Studi dell\'Insubria, via Valleggio 11, I-22100 Como, Italy; haardt@uninsubria.it;\n- C. Lousto: CCRG, Rochester Institute of Technology; colsma@rit.edu;\n- J. Nordhaus: Center for Computational Relativity and Gravitation, Rochester Institute of Technology, Rochester, NY 14623, USA; nordhaus@astro.rit.edu;\n- A. Pound: School of Mathematical Sciences and STAG Research Centre, University of Southampton, SO17 1BJ, United Kingdom; a.pound@soton.ac.uk;\n- R. O\'Shaughnessy: Center for Computational Relativity and Gravitation, Rochester Institute of Technology; richard.oshaughnessy@ligo.org;\n- S. Portegies Zwart: Leiden Observatory, Leiden, the Netherlands; spz@strw.leidenuniv.nl;\n- F. Schussler: IRFU, CEA, Université Paris-Saclay, F-91191 Gif-sur-Yvette, France; fabian.schussler@cea.fr;\n- O. Sergijenko: Astronomical Observatory of Taras Shevchenko National University of Kyiv,\n- Observatorna str., 3, Kyiv, 04053, Ukraine; Main Astronomical Observatory of the National Academy of Sciences of Ukraine, Zabolotnoho str., 27, Kyiv, 03680, Ukraine; olga.sergijenko.astro@gmail.com;\n- A. Spallicci: Université d\'Orléans - Centre National de la Recherche Scientifique; spallicci@cnrsorleans.fr;\n- D. Vernieri: Dipartimento di Fisica \'E. Pancini\', Università di Napoli \'Federico II\' and INFN, Sezione di Napoli, Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy.; daniele.vernieri@unina.it;\n- A. Vigna-Gómez: DARK, Niels Bohr Institute, University of Copenhagen, Jagtvej 128, 2200, Copenhagen, Denmark; avignagomez@nbi.ku.dk'}

2024arXiv240906141M

Cauchycharacteristic evolution CCE is a powerful method for accurately extracting gravitational waves at future null infinity. In this work we extend the previously implemented CCE system within the numerical relativity code SpECTRE by incorporating a scalar field. This allows the system to capture features of beyondgeneralrelativity theories. We derive scalar contributions to the equations of motion Weyl scalar computations Bianchi identities and balance laws at future null infinity. Our algorithm tested across various scenarios accurately reveals memory effects induced by both scalar and tensor fields and captures Prices powerlaw tail ul2 in scalar fields at future null infinity in contrast to the t2l3 tail at future timelike infinity.

2024-09-01T00:00:00Z

['10.48550/arXiv.2409.06141', '2024arXiv240906141M', 'arXiv:2409.06141']

['General Relativity and Quantum Cosmology']

EinsteinKleinGordon system via Cauchycharacteristic evolution Computation of memory and ringdown tail

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.06141.pdf

{'Einstein-Klein-Gordon system via Cauchy-characteristic evolution: Computation of memory and ringdown tail': "Sizheng Ma , 1, ∗ Kyle C. Nelli , 2 Jordan Moxon , 2 Mark A. Scheel , 2 Nils Deppe , 3, 4, 5 Lawrence E. Kidder , 5 William Throwe , 5 and Nils L. Vu 2 1 Perimeter Institute for Theoretical Physics, Waterloo, ON N2L2Y5, Canada 2 TAPIR 350-17, California Institute of Technology, 1200 E California Boulevard, Pasadena, CA 91125, USA 3 Laboratory for Elementary Particle Physics, Cornell University, Ithaca, New York 14853, USA 4 Department of Physics, Cornell University, Ithaca, New York 14853, USA 5 Cornell Center for Astrophysics and Planetary Science, Cornell University, Ithaca, New York 14853, USA (Dated: September 11, 2024) \nCauchy-characteristic evolution (CCE) is a powerful method for accurately extracting gravitational waves at future null infinity. In this work, we extend the previously implemented CCE system within the numerical relativity code SpECTRE by incorporating a scalar field. This allows the system to capture features of beyond-general-relativity theories. We derive scalar contributions to the equations of motion, Weyl scalar computations, Bianchi identities, and balance laws at future null infinity. Our algorithm, tested across various scenarios, accurately reveals memory effects induced by both scalar and tensor fields and captures Price's power-law tail ( u -l -2 ) in scalar fields at future null infinity, in contrast to the t -2 l -3 tail at future timelike infinity.", 'I. INTRODUCTION': 'The LIGO, Virgo, and KAGRA collaboration [1-3] has detected about 100 gravitational wave (GW) events [4-7], providing us with unprecedented opportunities to explore the strong gravitational fields surrounding compact binary systems. This marks a new era in testing general relativity (GR) [4, 8-28]. \nThe progress mentioned above primarily centers on the Cauchy formalism, where the equations of motion are formulated as initial boundary value problems. Since it is not feasible to simulate an infinitely large space, the spatial Cauchy domain is typically truncated at a finite distance from the source, which prevents direct access to GWs at future null infinity. One approach to address this limitation is to measure waveform quantities at finite radii within the Cauchy domain and then extrapolate them to infinity [52]. However, this extrapolation does not guarantee that the Einstein equations are satisfied, making it fail to capture memory effects [53]. A more rigorous solution is to use Cauchy-characteristic evolution (CCE) and Cauchy-characteristic matching (CCM) [54], which can faithfully extract GWs at future null infinity up to Bondi-Metzner-Sachs freedom. Recently, a CCE [55, 56] and a CCM [57] algorithm for GR have been built in a NR code SpECTRE [58]. However, a robust CCE/CCM code for beyond-GR theories is still missing. \nTo rigorously test GR, it is crucial to have precise GW predictions for both GR and alternative theories of gravity. Numerical relativity (NR) stands as the only ab initio method capable of producing complete inspiral-merger-ringdown waveforms to achieve this goal. Although two decades have passed since the first numerical simulation of a binary black hole (BBH) merger in GR [29], evolving modified theories of gravity remains challenging, as reviewed in [30]. The primary difficulty lies in the potential modification of the principal part of the Einstein equations, rendering the original numerical methods for GR ill-posed. Several approaches have been proposed to address this issue. The first approach involves directly formulating well-posed schemes for evolving the full set of evolution equations in certain specific theories, such as weakly-coupled Horndeski gravity theories (e.g. Einstein-scalar-Gauss-Bonnet) [31-35] based on the modified generalized harmonic formulation [36, 37], as well as Damour-Esposito-Farèse scalar-tensor theories [38-42]. The second approach is to expand the equations perturbatively in the coupling constant and solve them order by order. This order reduction scheme has been applied to the evolution of dynamical Chern-Simons [43-45] and scalar Gauss-Bonnet gravity [46, 47]. The final treatment introduces auxiliary variables to fix high-energy (short length/timescale) degrees of freedom, while leaving low-energy parts unchanged, see [48-50]. Recently, Corman et al. [51] compared the results of these three approaches under various physical setups. \nIn this paper, we take a step to let the SpECTRE CCE system capture beyond-GR features. Specifically, we choose to include a (massless) scalar field, which serves as an important ingredient in theories such as Horndeski [59] and \ndynamical Chern-Simons gravity [60, 61]. We will be testing the accuracy of our code and demonstrating its ability to reveal ringdown tails [62, 63] in the scalar field, as well as the memory effects sourced by the scalar field. \nThis paper is organized as follows. In Sec. II, we outline the details of CCE for an Einstein-Klein-Gordon system. In particular, we show how the equations of motion, Weyl scalar computations, and Bianchi identities are modified in the presence of a scalar field. Next in Sec. III, we focus on the memory effects sourced by the scalar field and the corresponding balance laws. We then test our code by evolving scalar fields on two prescribed spacetime backgrounds (Sec. IV) and performing the full CCE procedure to compute tails and memory effects (Sec. V). Finally, we summarize the results in Sec. VI. Throughout this paper, complex conjugates are represented by overlines.', 'II. EINSTEIN-KLEIN-GORDON SYSTEM VIA CCE': 'The Einstein-Klein-Gordon action offers the most basic description for a system containing both scalar and tensor fields. For instance, after performing a conformal transformation [64], Damour-Esposito-Farèse scalar-tensor theories [64-66] in the Einstein frame are governed by: \nS = ∫ d 4 x √ -g [ R 16 π -1 2 ∇ µ ψ ∇ µ ψ + V ( ψ ) ] , (1) \nwhere the real-valued scalar field ψ is minimally coupled to the metric. The action leads to \nR µν = 8 π ∇ µ ψ ∇ ν ψ, □ ψ + ∂V ∂ψ = 0 . \n(2) \nHere, we see that the Einstein field equations obtain a source term, while the scalar field obeys the Klein-Gordon (KG) equation. In our following discussions, we assume V ( ψ ) = 0 for simplicity (rendering ψ massless), though these discussions can be easily extended to a complex-valued massive scalar field with an arbitrary potential.', 'A. Equations of motion': 'By adopting the Bondi-Sachs metric of an asymptotically flat spacetime [67-69] \nds 2 = -e 2 β ( rW +1) du 2 -2 e 2 β dudr + r 2 h AB ( dx A -U A du )( dx B -U B du ) , (3) \nwith x A = ( θ, φ ) standing for angular coordinates, the Einstein field equations can be converted to a set of hypersurface equations [70, 71] \n∂ r β = S β ( J ) + 2 πr ( ∂ r ψ ) 2 , ∂ r ( r 2 Q ) = S Q ( J, β ) + 16 πr 2 ∂ r ψ ð ψ, ∂ r U = S U ( J, β, Q ) , ∂ r ( r 2 W ) = S W ( J, β, Q, U ) + 2 πe 2 β [ J ( ¯ ð ψ ) 2 + ¯ J ( ð ψ ) 2 -2 K ¯ ð ψ ð ψ ] , ∂ r ( rH ) + L H ( J, β, Q, U, W ) H + L ¯ H ( J, β, Q, U, W ) ¯ H = S H ( J, β, Q, U, W ) + 4 π e 2 β r ( ð ψ ) 2 , (4) \nwith U = U A q A , Q = r 2 e -2 β q A h AB ∂ r U B , H = ∂ u J , and q A = ( -1 , -i csc θ ). The expressions of S β , S Q , S U , S W , S H , L H , and L ¯ H as functions of ( J, β, Q, U, W ) can be found in Sec. IV of [55]. The spin-weighted derivative operators ð and ¯ ð are defined to be \nð ψ = q A D A ψ, ¯ ð ψ = ¯ q A D A ψ (5) \nwhere D A is the angular covariant derivative compatible with the unit sphere metric. On the other hand, the KG equation for the scalar field ψ becomes [70] \n2 r∂ r Π+2Π = 1 r ∂ r [ r 2 ( rW +1) ∂ r ψ ] + e 2 β r ( N ψ 1 -N ψ 2 + N ψ 3 ) -N ψ 5 -r 2 N ψ 4 , (6) \nwhere we have defined an auxiliary variable Π = ∂ u ψ , and \nN ψ 1 = 2 K Re ( ð β ¯ ð ψ ) + K ¯ ðð ψ, N ψ 2 = Re ( ¯ J ðð ψ +2 ¯ ð β ¯ ð ψJ + ¯ ð J ¯ ð ψ ) , N ψ 3 = Re ( ð K ¯ ð ψ ) , N ψ 4 = 2Re ( ¯ ð ψ∂ r U + ð ¯ U∂ r ψ +2 ¯ U ð ∂ r ψ ) , N ψ 5 = 2Re ( U ¯ ð ψ ) . (7) \nwith \nΛ ˘ Π = 1 2 (1 -˘ y ) ∂ ˘ y ˘ W∂ ˘ y ˘ ψ + (1 -˘ y ) 2 4 R wt ∂ 2 ˘ y ˘ ψ + ∂ ˘ u R wt R wt (1 -˘ y ) ∂ 2 ˘ y ˘ ψ + 1 2 (1 -˘ y ) ˘ W∂ 2 ˘ y ˘ ψ + 1 2 ˘ W∂ ˘ y ˘ ψ -1 2 Re ( ¯ ð ˘ ψ∂ ˘ y ˘ U + ð ¯ U∂ ˘ y ˘ ψ +2 ˘ U ¯ ð ∂ ˘ y ˘ ψ +2 ð R wt R wt ¯ ˘ U∂ ˘ y ˘ ψ ) . (12) \nHere we have used two identities: \nð ∂ r = (1 -˘ y ) 2 2 R wt ( ð ∂ ˘ y + ˘ ð R wt R wt ∂ ˘ y ) , ∂ r = (1 -˘ y ) 2 2 R wt ∂ ˘ y . (13) \nNotice that in Eqs. (11), (12), and (13) the spin-weighted derivative ð is still evaluated with respect to the original angular system x A . It is related to the numerically adapted version ˘ ð by the following relation [55]: \nð ˘ ψ = ˘ ð ˘ ψ -(1 -˘ y ) ˘ ð R wt R wt ∂ ˘ y ˘ ψ. (14) \nTo compute ˘ ð ˘ ψ , we follow [56] and use libsharp routines [72, 73] to obtain a modal decomposition of ˘ ψ in terms of spin-weighted spherical harmonics; subsequently, we apply the differential matrix. \nIn Eq. (4), the presence of ψ does not alter the left-hand side of the evolution equations, making their hierarchical structure unchanged: the right-hand side of the β equation solely comprises J, ψ and their radial derivatives, the right-hand side of the Q equation comprises solely J, ψ, β and their derivatives, and likewise for the other evolved variables. To evolve the system, we provide initial data for J and ψ on the first u = const null slice, then radially integrate Eqs. (4) and (6) by order to determine β, Q, U, W, H , and finally Π. Following this, we advance J and ψ to the subsequent null slice based on the values of H = ∂ u J and Π = ∂ u ψ .', 'B. SpECTRE CCE system': 'A spectral CCE system has been implemented for GR in SpECTRE [55, 56]. To facilitate spectral implementations, the authors introduced numerically adapted coordinates (˘ u, ˘ y, ˘ x ˘ A ): \n˘ u = u, ˘ y = 1 -2 R wt r , ˘ θ = θ, ˘ φ = φ, (8) \nwhere R wt is the Bondi radius of a worldtube for CCE, and it is not to be confused with the Ricci scalar R . On a ˘ u = const null slice, the simulation domain spans radially from the worldtube at ˘ y = -1( r = R wt ) to future null infinity at ˘ y = 1( r = ∞ ). With this new coordinate system, the time (˘ u ) derivative of J , ˘ H = ∂ ˘ u J , is related to the original one H , via [55] \n˘ H = H +(1 -˘ y ) ∂ ˘ u R wt R wt ∂ ˘ y J. (9) \nThe second term arises as the Jacobian of the coordinate transformation. Similarly, the time derivative of the scalar field, ˘ Π = ∂ ˘ u ψ , transforms in the same way: \n˘ Π = Π + (1 -˘ y ) ∂ ˘ u R wt R wt ∂ ˘ y ψ. (10) \nThe values of other variables are not affected by the coordinate transformation, namely ˘ F = F , for all F ∈ { ψ, J, β, Q, U, W } . \nInserting Eq. (10) into Eq. (6) yields the hypersurface equation for ˘ Π: \n(1 -˘ y ) ∂ ˘ y ˘ Π+ ˘ Π = -Re ( ˘ U ¯ ð ˘ ψ ) +(1 -˘ y ) [ Λ ˘ Π + e 2 ˘ β 4 R wt ( ˘ N ψ 1 -˘ N ψ 2 + ˘ N ψ 3 ) ] , (11) \nR wt \nThe transformation of the metric sector in Eq. (4) has been extensively discussed in [56]. Here we simply present additional terms contributed by ˘ ψ \n˘ = ( ˘ ) + 2 (1 ˘ )( ˘ ) \n∂ ˘ y β S ˘ β J π -y ∂ ˘ y ψ 2 , (1 -˘ y ) ∂ ˘ y ˘ Q +2 ˘ Q = S ˘ Q ( ˘ J, ˘ β ) + 16 π (1 -˘ y ) ð ˘ ψ∂ ˘ y ˘ ψ, (1 -˘ y ) ∂ ˘ y ˘ W +2 ˘ W = S ˘ W ( ˘ J, ˘ β, ˘ Q, ˘ U ) + πe 2 ˘ β (1 -˘ y ) R wt ( ˘ J ( ¯ ð ˘ ψ ) 2 + ¯ ˘ J ( ð ˘ ψ ) 2 -2 ˘ K ð ˘ ψ ¯ ð ˘ ψ ) , (1 -˘ y ) ∂ ˘ y ˘ H + ˘ H + L ˘ H ( ˘ J, ˘ β, ˘ Q, ˘ U, ˘ W ) ˘ H + L ¯ ˘ H ( ˘ J, ˘ β, ˘ Q, ˘ U, ˘ W ) ¯ ˘ H = S ˘ H ( ˘ J, ˘ β, ˘ Q, ˘ U, ˘ W ) + 2 π (1 -˘ y ) e 2 ˘ β ( ð ˘ ψ ) 2 . (15)', 'C. Worldtube transformation': "In order to integrate the hypersurface equation for ˘ Π in Eq. (11), we need to supply a boundary condition at a worldtube R wt . Normally it comes from a separate Cauchy evolution of the KG equation, e.g., see [74]. Since the two simulation methods adopt different gauges, a worldtube transformation is needed to construct the value of ˘ Π at the worldtube out of Cauchy variables. \nIt turns out that the transformation has a concise expression: \n˘ Π | wt = ∂ t ' ψ +Re ( U (0) ¯ ˘ ð ψ ) . (16) \nDerivation details can be found in Appendix A. In Eq. (16), ∂ t ' ψ is the Cauchy time derivative of ψ ; ¯ ˘ ð ψ is the angular derivative of ψ [see Eq. (5)] with respect to the numerically adapted coordinates; and U (0) is defined in Eq. (34b) of [55], with a spin weight of 1. Physically speaking, the numerically adapted angular coordinates ˘ x ˘ A on the worldtube evolve over time with respect to the Cauchy coordinates, via [Eq. (28) of [55]] \n∂ t ' ˘ x ˘ A = -1 2 ( U (0) ¯ ˘ q ˘ A + ¯ U (0) ˘ q ˘ A ) , (17) \nwhere the quantity U (0) captures the rate of the coordinate motion. Therefore, U (0) corresponds to the shift vector pulled back 1 to the 3D timelike worldtube. On the other hand, the numerically adapted time ˘ u is chosen to flow at the same rate as Cauchy's time t ' - the lapse is manually set to 1 [55]. As a result, the worldtube transformation in Eq. (16) is nothing but the Lie derivative of ψ . \nAfter setting the boundary condition ˘ Π | wt , we integrate the hypersurface equation in Eq. (11) and evolve the scalar field forward in time. On the worldtube, the scalar field computed by the Cauchy system, ψ Cauchy , should agree with the one obtained from the characteristic system ψ Char . This yields a worltube constraint \nSince ψ is an evolved variable of the characteristic system, we can directly compute the value of ¯ ˘ ð ψ on the characteristic grid using the libsharp routines, without taking information from the Cauchy side. Meanwhile, the spin-weighted variable U (0) is already available while evolving the GR system. Therefore, in order to perform the worldtube transformation in Eq. (16), we simply need to communicate ∂ t ' ψ from the Cauchy to the characteristic system. \nC wt ≡ ψ Char -ψ Cauchy . (18) \nIn actual numerical simulations, the constraint is normally nonvanishing since the two fields are evolved independently; and it should converge to 0 in the continuum limit. Therefore, tracking its evolution provides a diagnosis of our simulations. In fact, C wt offers a specific examination for the worldtube transformation in Eq. (16), since the characteristic evolution of ψ Char is fully controlled by the boundary condition ˘ Π | wt - a wrong transformation would lead to a wrong characteristic evolution of ψ Char , which drives it away from ψ Cauchy .", 'D. Computation of Weyl scalars at future null infinity': "The SpECTRE CCE system [55, 56] uses the Newman-Penrose (NP) formalism to compute Weyl scalars. The relevant equations are 2 [75] \nΨ 0 = Dσ -δκ -σ ( ρ + ¯ ρ +3 /epsilon1 -¯ /epsilon1 ) + κ ( τ -¯ π NP + ¯ α +3 β NP ) , (19a) \nΨ 1 = Dτ -∆ κ -( τ + ¯ π NP ) ρ -(¯ τ + π NP ) σ -( /epsilon1 -¯ /epsilon1 ) τ +(3 γ + ¯ γ ) κ -Φ 01 , (19b) \nΨ 2 = Dµ -δπ NP -(¯ ρµ + σλ ) -π NP (¯ π NP -¯ α + β NP ) + µ ( /epsilon1 +¯ /epsilon1 ) + νκ -2Λ , (19c) \nΨ 3 = ¯ δµ -δλ + ν ( ρ -¯ ρ ) + π NP ( µ -¯ µ ) + µ ( α + ¯ β NP ) + λ (¯ α -3 β NP ) + Φ 21 . (19d) \nwhere D = l a ∇ a , ∆ = n a ∇ a , δ = m a ∇ a are the derivative operators used in the NP formalism. The tetrad ( l a , n a , m a ) is chosen to be [55] \nl a = 1 √ 2 ∂ a r , n a = √ 2 e -2 β ( ∂ a u -V 2 r ∂ a r + 1 2 ¯ Uq a + 1 2 U ¯ q a ) , m a = -1 √ 2 r ( √ K +1 2 q a -J √ 2(1 + K ) ¯ q a ) . (20) \nCompared with their vacuum GR counterparts, the equations gain additional source terms Φ 01 , Λ, and Φ 21 . They are related to the scalar field ψ through [75] \nΦ 01 = 1 2 R ml = 4 πδψDψ, Φ 21 = 1 2 R n ¯ m = 4 π ¯ δψ ∆ ψ, Λ = 1 24 R = -1 12 ( R ln -R m ¯ m ) = -2 π 3 [ Dψ ∆ ψ -δψ ¯ δψ ] . (21) \nThe asymptotic expansion of these quantities at future null infinity I + is of particular interest. Hereafter, we denote the terms in the expansion with the following notation: \nf = f ( n ) r n + f ( n +1) r n +1 + O ( 1 r n +2 ) , (22) \nwhere f represents any variable under investigation, and n is an integer. As for the massless scalar field ψ , its value at I + , namely ψ (0) , is a constant and can be set to 0 without changing physical aspects of the system. Therefore, its radial falloff obeys ψ ∼ r -1 . Substituting the asymptotic behavior of ψ into Eq. (20) leads to \nDψ = -1 √ 2 ψ (1) r 2 + O ( 1 r 3 ) , ∆ ψ = √ 2 e -2 β (0) Π (1) r + O ( 1 r 2 ) , δψ = -1 √ 2 ð ψ (1) r 2 + O ( 1 r 3 ) , (23) \nwhere we have used the fact that U (0) = J (0) = 0, K (0) = 1, and V ∼ r . The radial falloff rate for Φ 01 , Λ, and Φ 21 can be obtained accordingly: \nΦ 01 = 2 π ψ (1) ð ψ (1) r 4 + O ( 1 r 5 ) , Λ = 2 π 3 e -2 β (0) Π (1) ψ (1) r 3 + O ( 1 r 4 ) , Φ 21 = -4 πe -2 β (0) Π (1) ¯ ð ψ (1) r 3 + O ( 1 r 4 ) . (24) \nIn the code, we compute the value of Π (1) , ψ (1) , and ð ψ (1) via \nΠ (1) = -2 R wt ( ∂ ˘ y ˘ Π+ ∂ ˘ u R wt R wt ∂ ˘ y ˘ ψ )∣ ∣ ∣ ∣ ˘ y =1 , ψ (1) = -2 R wt ∂ ˘ y ˘ ψ, ð ψ (1) = -2 ( ð R wt R wt ∂ ˘ y ˘ ψ + ð ∂ ˘ y ˘ ψ )∣ ∣ ∣ ∣ ˘ y =1 . (25) \nThe asymptotic expansions of Wely scalars in GR have been shown in Eq. (93) of [55]. With the presence of ψ , their expressions are modified to \nΨ (4) 1 = Ψ (4)GR 1 -Φ (4) 01 , Ψ (3) 2 = Ψ (3)GR 2 -2Λ (3) , Ψ (2) 3 = Ψ (2)GR 3 . (26) \nNotice that the expression for Ψ (2) 3 is the same as its GR counterpart, since the additional source term Φ 21 ∼ O ( r -3 ) decays faster than that of Ψ 3 . The computation of Ψ 0 requires more attention. Although the formula in Eq. (19a) \nis valid geometrically and does not rely on the detail of a gravity theory, the asymptotic expansion in [55] [see their Eq. (93a)] implicitly assumes the vanishing of the stress-energy tensor. Instead, here we directly expand Eq. (19a) and obtain: \nΨ (5) 0 = β (2) J (1) + ¯ J (1) J (1)2 4 -3 2 J (3) . (27) \nThe value of β (2) can be obtained by expanding the rr component of Einstein's equations [see Eq. (2.9c) in [69] or Eq. (2.9a) in [71]] \nβ (2) = -J (1) ¯ J (1) 16 -πψ (1)2 . (28) \nFinally, the Weyl scalar Ψ (1) 4 and the Bondi-Sachs News take the same forms as in GR, as shown in Eq. (93e) and (42) of [55], respectively.", 'E. Bianchi identities at future null infinity': 'One way to validate our CCE code is to examine Bianchi identities at future null infinity, which establish connections between Weyl scalars Ψ (5) ... (1) 0 ... 4 and the strain h (1) . Their expressions in GR can be found in, e.g. [52]. To extend the identities to our case, we consider the following NP equations [76]: \n-∆ λ + ¯ δν -λ ( µ + ¯ µ ) -(3 γ -¯ γ ) λ +(3 α + ¯ β NP + π NP -¯ τ ) ν -Ψ 4 = 0 , (29a) \n- -∆Ψ 3 + δ Ψ 4 +(4 β NP -τ )Ψ 4 -2(2 µ + γ )Ψ 3 +3 ν Ψ 2 -¯ δ Φ 22 +∆Φ 21 +(¯ τ -2 ¯ β NP -2 α )Φ 22 +2( γ + ¯ µ )Φ 21\n- +2 λ Φ 12 -2 ν Φ 11 -¯ ν Φ 20 = 0 , \n(29b) \n- -∆Ψ 2 + δ Ψ 3 +2 ν Ψ 1 -3 µ Ψ 2 +2( β NP -τ )Ψ 3 + σ Ψ 4 -D Φ 22 + δ Φ 21 +2(¯ π NP + β NP )Φ 21 -2 µ Φ 11 -¯ λ Φ 20\n- +2 π NP Φ 12 +(¯ ρ -2 /epsilon1 -2¯ /epsilon1 )Φ 22 -2∆Λ = 0 , \n(29c) \n- -∆Ψ 1 + δ Ψ 2 + ν Ψ 0 +2( γ -µ )Ψ 1 -3 τ Ψ 2 +2 σ Ψ 3 +∆Φ 01 -¯ δ Φ 02 +2(¯ µ -γ )Φ 01 -2 ρ Φ 12 -¯ ν Φ 00 +2 τ Φ 11 \n+(¯ τ -2 ¯ β NP +2 α )Φ 02 +2 δ Λ = 0 , (29d) \n- \n∆Ψ \n0 \n+ \nδ \nΨ \n1 \n+(4 \nγ \n- \nµ \n- +(¯ ρ +2 /epsilon1 -2¯ /epsilon1 )Φ 02 = 0 . \n(29e) \nwhere the source terms Φ 22 , Φ 00 , Φ 11 , Φ 02 , and Φ 20 are given by \nΦ 22 = 1 2 R nn = 4 π (∆ ψ ) 2 = 8 π ˙ ψ (1) 2 r 2 + O ( 1 r 3 ) , Φ 00 = 1 2 R ll = 4 π ( Dψ ) 2 = 2 π ψ (1) 2 r 4 + O ( 1 r 5 ) , Φ 11 = 1 4 ( R ln + R m ¯ m ) = 2 π ( Dψ ∆ ψ + δψ ¯ δψ ) = -2 π ψ (1) ˙ ψ (1) r 3 + O ( 1 r 4 ) , ¯ Φ 20 = Φ 02 = 1 2 R mm = 4 π ( δψ ) 2 = 2 π ( ð ψ (1) ) 2 r 4 + O ( 1 r 5 ) . (30) \nIn Eq. (30) we have used \nδ = -1 √ 2 r ð + O ( 1 r 2 ) , ∆ = √ 2 ∂ u + O ( 1 r ) , (31) \nto obtain the asymptotic expansions. The overhead dots denote the time derivative ∂ u . Meanwhile, the Bianchi identities involve NP spin coefficients, whose expressions in terms of Bondi-Sachs variables can be found in Eq. (86) of \n)Ψ \n0 \n- \nτ \n2(2 \n+ \nβ \nNP \n)Ψ \n1 \n+3 \nσ \n2 \nΨ \n- \nD \n02 \nΦ \n+ \nδ \n01 \nΦ \nπ \n+2(¯ \nNP \n- \nβ \nNP \n)Φ \n01 \n- \n2 \nκ \n12 \nΦ \n- \n¯ \nλ \n00 \nΦ \n+2 \nσ \n11 \nΦ \n[55]. Near future null infinity, their asymptotic behaviors read 3 \nJ = ¯ h (1) r + O ( 1 r 2 ) , σ = ¯ h (1) 2 √ 2 r 2 + O ( 1 r 3 ) , µ = -1 √ 2 r + O ( 1 r 2 ) , β NP = cot θ 2 √ 2 r + O ( 1 r 2 ) , λ (1) = 1 √ 2 ˙ h (1) + O ( 1 r 2 ) , ρ = -1 √ 2 r , κ = 0 , τ ∼ O ( 1 r 2 ) , /epsilon1 ∼ O ( 1 r 2 ) , π NP ∼ O ( 1 r 2 ) , ν ∼ O ( 1 r 2 ) , γ ∼ O ( 1 r 2 ) . (32) \nPlugging Eqs. (30), (31), and (32) into Eq. (29), we obtain the following Bianchi identities at future null infinity \nΨ (1) 4 = -h (1) , (33a) \n˙ Ψ (3) 2 = -1 2 ð Ψ (2) 3 + 1 4 ¯ h (1) Ψ (1) 4 + 8 π 3 ˙ ψ (1) 2 -4 π 3 ψ (1) ¨ ψ (1) , (33c) \n˙ Ψ (2) 3 = -1 2 ð Ψ (1) 4 , (33b) \n˙ Ψ (4) 1 = -1 2 ð Ψ (3) 2 + 1 2 ¯ h (1) Ψ (2) 3 -8 π 3 ˙ ψ (1) ð ψ (1) + 4 π 3 ψ (1) ð ˙ ψ (1) , (33d) \n˙ Ψ (5) 0 = -1 2 ð Ψ (4) 1 + 3 4 ¯ h (1) Ψ (3) 2 +2 π ( ð ψ (1) ) 2 -πψ (1) ðð ψ (1) -π ˙ ¯ h (1) ψ (1) 2 -√ 2 π ¯ h (1) ψ (1) ˙ ψ (1) , (33e) \nWe will use the PYTHON package scri [79-83] to verify the identities in our following calculations. The package adopts a different convention (hereafter MB) [83, 84] compared to SpECTRE CCE. The conversion factors read [52] \nΨ [MB] n = ( -√ 2) 2 -n Ψ n , ð [MB] = 1 √ 2 ð , σ (2)MB = √ 2 σ (2) . (34) \nThen Eqs. (33) become \nΨ (1)[MB] 4 = -¨ ¯ σ (2)[MB] , (35a) \n˙ Ψ (2)[MB] 3 = ð [MB] Ψ (1)[MB] 4 , (35b) \n˙ Ψ (4)[MB] 1 = ð [MB] Ψ (3)[MB] 2 +2 σ (2)[MB] Ψ (2)[MB] 3 + 16 π 3 ˙ ψ (1) ð [MB] ψ (1) -8 π 3 ψ (1) ð [MB] ˙ ψ (1) , (35d) \n˙ Ψ (3)[MB] 2 = ð [MB] Ψ (2)[MB] 3 + σ (2)[MB] Ψ (1)[MB] 4 + 8 π 3 ˙ ψ (1) 2 -4 π 3 ψ (1) ¨ ψ (1) , (35c) \n˙ Ψ (5)[MB] 0 = ð [MB] Ψ (4)[MB] 1 +3 σ (2)[MB] Ψ (3)[MB] 2 +8 π ( ð [MB] ψ (1) ) 2 -4 πψ (1) ð [MB] ð [MB] ψ (1) -4 π ˙ σ (2)[MB] ψ (1) 2 -4 πσ (2)[MB] ψ (1) ˙ ψ (1) . (35e)', 'III. SCALAR-INDUCED MEMORY EFFECTS': 'A massless scalar field can propagate to future null infinity [85] and leave nontrivial imprints on memory effects. Here we extend the discussions in [53, 86] and derive the memory contribution from the scalar field. Since most of our discussions in this section adopt only the leading-order asymptotic behavior near future null infinity, we suppress the superscript defined in Eq. (22) for conciseness, unless stated otherwise. To be consistent with previous discussions, we still use the convention of SpECTRE CCE, as opposed to MB [Eq. (34)]. \nThe computation of memory effects can be achieved via the balance laws [69] [see their Eq. (2.11a) and (C6)] \n1 1 \n˙ ˆ N A = -6 π ˙ ψD A ψ +2 πψ∂ A ˙ ψ +4 πuD A ˙ ψ 2 + u 8 D A ( N BC N BC ) -3 8 N AB D C C BC + 3 8 C AB D C N BC + 1 8 N BC D B C AC -1 8 C BC D B N AC + 1 4 D B D A D C C BC -1 4 D 2 D C C AC -1 4 uD A D B D C N BC , (36b) \n˙ m = -4 π ˙ ψ 2 -8 N AB N AB + 4 D A D B N AB , (36a) \nwhere m = -1 2 W (2) is the Bondi mass aspect; ˆ N A is related to the angular momentum aspect N A via \nˆ N A = N A -uD A m -1 16 D A ( C BC C BC ) -1 4 C AB D C C BC . (37) \nFinally, C AB is the O ( r -1 ) order of the angular metric h AB defined in Eq. (3), and N AB = ∂ u C AB is the Bondi News tensor. \nFollowing [69, 86], we decompose C AB into an electric (Φ) and a magnetic (Ψ) potential \nC AB = ( D A D B -1 2 q AB D 2 ) Φ+ /epsilon1 C ( A D B ) D C Ψ , (38) \nwhere /epsilon1 CA = i 2 q C ∧ ¯ q A is the volume form compatible with the unit sphere metric.', 'A. Electric memory': 'We insert Eq. (38) into Eq. (36a) and obtain [86] \nΦ = D -1 [ m + ∫ ( 4 π ˙ ψ 2 + 1 4 ˙ h ˙ ¯ h ) du + α ( θ, φ ) ] . (39) \nHere α ( θ, φ ) is an arbitrary function on a two-sphere and D ≡ 1 8 D 2 ( D 2 +2) is an angular differential operator. We see that the null (nonlinear) memory obtains an additional contribution from the scalar energy flux ∼ ∫ ˙ ψ 2 du , compared to its GR counterpart [86]. On the other hand, the Bondi mass aspect contributes to the ordinary (linear) memory, and is given by [78] \nm = -Re ( Ψ 2 + 1 4 ˙ h ¯ h ) -4 π 3 ψ ˙ ψ. (40) \nNotice that the scalar ordinary memory vanishes in a nonradiative regime ( ˙ ψ ∼ 0). \nThe potential Φ is related to the electric memory J E via [86] \nJ E = 1 2 ¯ ð 2 Φ = J ET o + J ET null + J ES o + J ES null , (41) \nwhere the scalar pieces read \nJ ES o = 1 2 ¯ ð 2 D -1 ( -4 π 3 ψ ˙ ψ ) , J ES null = 1 2 ¯ ð 2 D -1 (∫ 4 π ˙ ψ 2 du ) . (42) \nThe tensor contributions J ET o and J ET null are the same as GR, and we provide their expressions below for completeness \nJ ET o = 1 2 ¯ ð 2 D -1 [ -Re ( Ψ 2 + 1 4 ˙ h ¯ h )] , J ET null = 1 2 ¯ ð 2 D -1 [∫ 1 4 ˙ h ˙ ¯ hdu + α ( θ, φ ) ] . (43)', 'B. Magnetic memory': 'The magnetic memory can be obtained by projecting both sides of Eq. (36b) with /epsilon1 AE D E , and the corresponding expression for Ψ in GR is provided in Eq. (45) of [86]. With the presence of the scalar field, a convenient way to compute Ψ is by replacing ˙ ˆ N = q A ˙ ˆ N A in the GR version with the following combination: \n˙ ˆ N → ˙ ˆ N +6 π ˙ ψ ð ψ -2 πψ ð ˙ ψ, (44) \nwhich yields \nD 2 D Ψ = -Im [ ¯ ð ˙ ˆ N +8 π ¯ ð ˙ ψ ð ψ + 1 8 ð ( 3 h ¯ ð ˙ ¯ h -3 ˙ h ¯ ð ¯ h + ˙ ¯ h ¯ ð h -¯ h ¯ ð ˙ h ) ] . (45) \nTo obtain ˆ N , we first contract Eq. (37) with q A and get \nˆ N = N -u ð m -1 8 ð ( h ¯ h ) -1 4 ¯ h ð h. (46) \nIn GR, the angular momentum aspect N is solely determined by Ψ 1 , e.g., see Appendix B of [86]. As for the Einstein-Klein-Gordon system, an extra term shows up. To see this, we use Eqs. (B2), (B5) and (B6) in [86], namely \nU (3) = -1 3 ( Q (2) -¯ h ¯ Q (1) ) , U (3) = -2 3 N + 1 8 ð ( h ¯ h ) + 1 2 ¯ h ð h, ¯ Q (1) = ð h, (47) \ntogether with the asymptotic expansion for Ψ 1 in Eq. (26) and the one for β in Eq. (28), then we arrive at \nN = 2Ψ 1 -2 πψ ð ψ. (48) \nPlugging Eqs. (46) and (48) into Eq. (45), and by virtue of the Bianchi identity in Eq. (33d), we find the scalar contribution to Ψ vanishes identically. The corresponding magnetic memory J B = -i 2 ¯ ð 2 Ψ reads [86] \nJ B = J B o + J B null , (49) \nwith \nJ B o = i 2 ¯ ð 2 D -1 D -2 Im { ¯ ð [ -ð Ψ 2 + 1 4 ( ¯ h ð ˙ h -˙ ¯ h ð h ) ]} , J B null = i 2 ¯ ð 2 D -1 D -2 Im { 1 8 ð ( 3 h ¯ ð ˙ ¯ h -3 ˙ h ¯ ð ¯ h + ˙ ¯ h ¯ ð h -¯ h ¯ ð ˙ h ) } . (50) \nThe expression can be further simplified by replacing the term ¯ ð ( ¯ h ð ˙ h -˙ ¯ h ð h ) in J B o with its negative complex conjugate 4 . After combing with J B null , we obtain \nJ B = -i 2 ¯ ð 2 D -1 D -2 Im ¯ ðð [ Ψ 2 + 1 4 ˙ h ¯ h ] = i 8 ¯ ð 2 D -1 Im ð 2 h, (51) \nwhere we have used the fact that the Bondi mass aspect is real-valued, namely [77] \nIm ( Ψ 2 + 1 4 ð 2 h + 1 4 ˙ h ¯ h ) = 0 . (52) \nNotice that the right-hand side of Eq. (51) is linear in h , as opposed to the electric memory in Eq. (41) that depends on energy fluxes, the balance law for the angular momentum aspect in Eq. (36b) in fact defines a projection operator for the magnetic component of h . In particular, Eq. (51) implies that \nJ B lm ∼ h lm -( -1) m ¯ h l, -m . (53) \nIt corresponds to the radiative current moment, see e.g., Eq. (30) in [87], which plays an important role in building up gravitational recoils [82, 87, 88] and the excitation of quadratic quasinormal modes [89, 90]. Under parity conjugation, J B lm transforms as follows \nJ B lm → ( -1) l +1 J B lm , (54) \nwhere we have used Eq. (C10d) in [82]. Therefore, J B lm carries odd parity. In the context of Schwarzschild perturbation theory, the dynamics of J B is described by the Regge-Wheeler equation [91].', 'IV. TESTING THE CHARACTERISTIC EVOLUTION': 'Having outlined CCE for the Einstein-Klein-Gordon system, we now proceed to test our code built in SpECTRE. Given that the metric evolution in GR has been thoroughly examined in [56], our tests primarily focus on the new \nfeatures introduced by the scalar field. This involves two main parts: solving the KG equation in Eq. (11) and conducting the full CCE procedure for the coupled (scalar+tensor) system. We address these two parts separately in this and the subsequent sections. \nBelow in this section, we test the implementation of the KG equation by evolving scalar fields on two prescribed spacetime backgrounds (namely without scalar-induced backreaction into the metric sector). We pay particular attention to two aspects: (a) the correct volume integration of the KG equation, and (b) the accurate implementation of the worldtube transformation in Eq. (16). For (a), we compare the computed scalar modes at future null infinity to expected (analytical) behaviors. For (b), we monitor the worldtube constraint defined in Eq. (18), as discussed in Sec. II C.', 'A. Bouncing Schwarzchild BH': "Following [56, 92], we consider a Schwarzchild BH in an oscillating coordinate system - a time-dependent transformation is applied to Kerr-Schild coordinates { t KS , x KS , y KS , z KS } [56, 92]: \nt ' = t KS , x ' = x KS -a sin 4 ( 2 πt KS b ) , y ' = y KS , z ' = z KS , (55a) \nwhere a and b are two constants. Since the transformation is a pure gauge effect, one expects no gravitational waves at future null infinity, even though each metric component evolves with time t ' nontrivially. The whole characteristic system must be properly established to cancel out the gauge effect at future null infinity. This makes the bouncing BH system the most demanding test [56]. \nFor our testing purposes, it is preferable to construct a scalar profile that can be controlled analytically, allowing us to compare numerical results with the expected analytical expression. To do this, we first work in the outgoing Eddington-Finkelstein coordinate system { u, r, θ, φ } . The retarded time u is related to the Kerr-Schild time t KS via \nu = t KS -r -4 M ln ( r 2 M -1 ) , (56) \nwith M being the mass of the BH. The KG equation □ ψ = 0 in the Eddington-Finkelstein frame reads \n2 ∂ r ∂ u ( rψ ) = ∂ r [( 1 -2 M r ) ∂ r ( rψ ) ] -2 M r 3 ( rψ ) + ð ¯ ð ( rψ ) r 2 , (57) \nwhere ð and ¯ ð are derivatives associated with the outgoing Eddington-Finkelstein angular coordinates { θ, φ } . Considering a spherical profile, namely ð ¯ ð ( rψ ) = 0, the solution to the KG equation has the following form \nrψ = ∞ ∑ n =0 ψ n ( u ) r n . (58) \nThe coefficients ψ n 's are functions of the retarded time u . Plugging Eq. (58) into (57) yields a three-term recurrence relation \ndψ 1 du = 0 , 2 dψ n +1 du = 2 M n +1 n 2 ψ n -1 -nψ n , n > 0 . (59) \nIt has a solution of (assuming M = 1) \nψ 0 = sin u, ψ 1 = 0 , ψ 2 = -1 2 cos u, ψ 3 = 1 2 sin u, ψ 4 = 3 4 cos u -9 8 sin u, ψ 5 = -77 20 cos u -3 2 sin u, ψ 6 = 15 16 cos u + 51 4 sin u, ψ 7 = 1287 28 cos u -1809 80 sin u, ψ 8 = -12579 80 cos u -19857 128 sin u, ψ 9 = -73557 160 cos u + 133813 140 sin u, ψ 10 = 49797063 8960 cos u + 1272267 1600 sin u, ψ 11 = -116136241 24640 cos u -57286503 1792 sin u, ψ 12 = -16472195091 89600 cos u + 419653067 5120 sin u, ψ 13 = 197057851611 232960 cos u + 517853793843 492800 sin u, ψ 14 = 23027722022071 3942400 cos u -2420206444191 313600 sin u, ψ 15 = -166944468961581 2464000 cos u -218435295225339 7321600 sin u. (60) \nFigure 1. The convergence of numerical error with angular resolution ( l max ) in the bouncing Schwarzschild BH test. The left panel shows the difference between the computed ψ at future null infinity and the expected value ψ 0 ( u ) = sin u , while the right panel displays the worldtube constraint as defined in Eq. (18). \n<!-- image --> \nHere we truncate the solution at n = 15, which is sufficient for our following tests. At future null infinity, the solution is given by \nrψ | I + = ψ 0 ( u ) = sin u. (61) \nAs mentioned in Sec. II C, we need to provide a boundary condition for ∂ t ' ψ at a worldtube to perform CCE. This can be obtained by differentiating Eq. (58): \n∂ t ' ψ = ∞ ∑ n =0 ˙ ψ n ( u ) r n +1 du dt ' -∞ ∑ n =0 ( n +1) ψ n ( u ) r n +2 dr dt ' , (62) \nwith \ndu dt ' = 1 -r +2 M r -2 M dr dt ' , dr dt ' = 8 πa br [ x ' + a sin 4 ( 2 πt ' b )] sin 3 ( 2 πt ' b ) cos ( 2 πt ' b ) , (63) \nwhere we have used Eq. (56) and \nr 2 = [ x ' + a sin 4 ( 2 πt ' b )] 2 + y ' 2 + z ' 2 . (64) \nIn our tests, we follow [56] and set the oscillation period b to 40 M and the amplitude a to 2 M . Four worldtube radii, 15 M, 20 M, 25 M, 30 M , are used for CCE. We choose the absolute tolerance of SpECTRE 's time stepper residual to be 10 -12 to ensure that differences between our results and the expected one ψ 0 ( u ) = sin u are dominated by spatial resolution. Figure 1a provides the differences with angular resolution l max ranging from 8 to 22. We can see that they converge exponentially to the level of 10 -11 -10 -10 . Similarly, as shown in Fig. 1b, the worldtube constraint defined in Eq. (18) follows a similar convergence behavior and finally levels off at ∼ 10 -12 .", 'B. Linearized Bondi-Sachs metric': "In the second test, we choose the linearized Bondi-Sachs metric [93] as our spacetime background. The corresponding worldtube data read [92] \nJ lin lm = √ ( l +2)! ( l -2)! 2 Z lm Re [ J l ( r ) e iνu ] , U lin lm = √ l ( l +1) 1 Z lm Re [ U l ( r ) e iνu ] , (65a) \nβ lin lm = 0 Z lm Re [ β l ( r ) e iνu ] , W lin lm = 0 Z lm Re [ W l ( r ) e iνu ] . (65b) \nFigure 2. The linearized Bondi-Sachs test. The left panel shows the amplitude of ψ l =2 ,m =0 (blue dots) and ψ l = m =2 (orange dots) as a function of C 2 a . They depend quadratically and linearly on C 2 a , as expected. The right panel provides the worldtube constraint as defined in Eq. (18). \n<!-- image --> \nFollowing [56], we consider a l = m = 2 angular profile, with s Z lm given by 5 \ns Z lm ( θ, φ ) = 1 √ 2 [ s Y lm ( θ, φ ) + ( -1) m s Y l -m ( θ, φ )] . (66) \nThe expression of J l ( r ) , U l ( r ) , β l ( r ) and W l ( r ) can be found in Eqs. (139) and (140) of [92]. In particular, we have \nJ l =2 ( r ) = 24 B 2 +3 iνC 2 a -iν 3 C 2 b 36 + C 2 a 4 r -C 2 b 12 r 3 . (67) \nHere B 2 , C 2 a and C 2 b are three free complex constants. Below we set B 2 to 0 for simplicity, which subsequently yields 3 C 2 a = ν 2 C 2 b due to the asymptotic flatness condition J l =2 ∼ O ( r -1 ). Therefore, the size of the linearized Bondi-Sachs wave is controlled by a single parameter C 2 a . As for the scalar sector, we choose the worldtube data to be \nψ ( u ) | worldtube = sin u r , ∂ t ' ψ | worldtube = cos u r . (68) \nWhen the linearized Bondi-Sachs wave vanishes ( C 2 a = 0), the scalar field propagates through flat spacetime. The consequent wavefunction at future null infinity is simply given by rψ ( u ) | I + = sin u . However, for a finite C 2 a , an exact analytical solution is not available. Instead, we can consider a perturbative expansion: \nrψ ( u, θ, φ ) | I + = sin u + C 2 a ψ (1) ( u, θ, φ ) + C 2 2 a ψ (2) ( u, θ, φ ) + O ( C 3 2 a ) . (69) \nTo the first order in C 2 a , ψ (1) ( u, θ, φ ) is generated by the coupling between metric components (characterized by the angular profile s Z l = m =2 ) and the leading-order evolution of ψ , namely sin u (characterized by the angular profile 0 Z l = m =0 ). The angular selection rule immediately leads to \nψ (1) ( u, θ, φ ) ∼ 0 Z 22 ( θ, φ ) . (70) \nIn other words, the amplitude of the l = m = 2 harmonic of the scalar field is expected to scale linearly with C 2 a . By contrast, other harmonics of the scalar field are generated by nonlinear couplings, making their amplitudes depend at least quadratically on C 2 a . \nTo perform our tests, we fix the frequency of the linearized Bondi-Sachs wave at ν = 0 . 2, while varying the wave amplitude C 2 a from 10 -6 to 1. To ensure the accuracy of our analysis, we still set the absolute tolerance of the time \nstepper to 10 -12 . Meanwhile, we find different choices of angular resolution l max lead to only a ∼ 10 -12 difference in final products. Figure 2a shows the amplitude of ψ l =2 ,m =0 (blue dots) and ψ l = m =2 (orange dots) as a function of C 2 a , measured at future null infinity. As expected, they depend quadratically and linearly on C 2 a , respectively. In Fig. 2b we provide the worldtube constraint [Eq. (18)] associated with ( l = 2 , m = 0) and ( l = m = 2). We see that the constraint violation for ψ l =2 ,m =2 is always on the order of 10 -15 , whereas for ψ l =2 ,m =2 it scales quadratically with C 2 a , consistent with the behavior at future null infinity.", 'V. CAUCHY CHARACTERISTIC EVOLUTION': 'After examining the characteristic evolution of the KG equation in Eq. (11), we now proceed to test the full CCE procedure for the Einstein-Klein-Gordon system, which involves both Cauchy and characteristic evolutions. Here we consider a simple setup where a scalar pulse strikes a BH. To perform the simulation, we first construct initial data by solving the extended conformal thin-sandwich equations with the spectral elliptic solver in SpEC [94, 95]. Next, we evolve the system nonlinearly using a first-order generalized harmonic formulation [96]. Our Cauchy evolution for the scalar field follows the method in [74]. Finally, we use the dumped worldtube data for CCE, as outlined in [55, 56] and earlier in this paper. For conciseness, we always set the initial BH mass to unity and use it as our code unit. \nBelow we will consider two distinct features for tests. (a) It is expected that the scalar emission from a perturbed BH could undergo a tail phase at late times, where the scalar field decays as a power law [62, 63]. In particular, the power law has different exponents at timelike infinity and null infinity [62, 63, 97-99]. The late-time tail at timelike infinity was investigated by Scheel et al. [74]. Below in Sec. V A, we focus on the behavior at null infinity to demonstrate the accuracy of our CCE system. (b) As discussed in Sec. III, the GW (tensor emission) of a BH stirred by a scalar field consists of memory effects. They are governed by the balance laws in Eqs. (41) and (49). In the second part of this section (Sec. V B), we will examine the deviations from the balance laws, as well as the Bianchi identities derived in Sec. II E.', 'A. Late-time tail': "In the first test, we place a Schwarzschild BH at the coordinate center. Initially, the scalar field profile is constructed as follows: \nψ = A e -( r ' -r ' 0 ) 2 /w 2 r ' Y l = m =1 ( θ ' , φ ' ) , (71) \nwith A = 0 . 1 , r ' 0 = 50 and w = 5. The angular dependence is set to be the l = m = 1 spherical harmonic. Meanwhile, we choose Π ' = ∂ r ' ψ at t ' = 0, where 6 \nΠ ' ≡ -1 α ' ( ∂ t ' ψ -β i ' ∂ i ' ψ ) , (72) \nis an auxiliary variable introduced by Scheel et al. [74]. The choice of Π ' represents an infalling scalar pulse into the BH. \nWe can see that ψ 11 at both timelike and null infinity evolves similarly during the excitation and ringdown phases, whereas their late-time tails exhibit distinct decay rates. Specifically, in Figs. 3b and 3c we fit the late-time portion of ψ 11 (600 < time < 1050, i.e., the gray-shaded regions) to a power law A ( t + t 0 ) -µ , with A,t 0 , and µ being three constants. We consider the following cost function for the fit: \nAs shown in Fig. 3a, the blue curve corresponds to the l = m = 1 spherical harmonic of the extracted scalar field as a function of the retarded time u , with the CCE worltube placed at a radius of 75. The evolution consists of three stages: the excitation phase (time ≲ 125), the quasinormal ringing phase (125 ≲ time ≲ 250), and the tail phase (time ≳ 250). We note that the scalar field is extracted at future null infinity, where the Bondi radius r approaches ∞ while the retarded time u remains fixed. By comparison, we also obtain the scalar radiation at an approximate timelike infinity by measuring late-Cauchy time ψ at a fixed Cauchy radius, as done by Scheel et al. [74]. The orange curve in Fig. 3a shows the corresponding l = m = 1 scalar harmonic. To make a fair comparison with CCE, our extraction radius is still at 75. \n∑ t i [log | ψ 11 ( t i ) | -log A + µ log( t i + t 0 )] 2 , (73) \n(a) Null infinity versus timelike infinity \n<!-- image --> \nFigure 4 shows the l = m = 1 harmonic of the scalar field measured at future null infinity. Since the initial data is time-symmetric, it radially contains an ingoing and an outgoing component. The outgoing piece reaches the CCE worldtube and appears at future null infinity at a time of r ' wt -r ' 0 = 80 M ; whereas the ingoing wave first strikes the Kerr BH at r ' 0 = 20 M , the excited quasinormal modes are then transmitted to future null infinity by CCE at \n<!-- image --> \n(b) Null infinity \n<!-- image --> \n(c) Timelike infinity \nFigure 3. Resolving ringdown tail in the scalar emission from a Schwarzschild BH. The BH is struck by a scalar field composed of a l = m = 1 spherical harmonic [Eq. (71)]. Figure 3a shows the extracted ψ 11 at null infinity (in blue, computed with CCE) and at an approximate timelike infinity (in orange, measured at a fixed Cauchy radius). In Figs. 3b and 3c, the late-time portion (in gray) of ψ 11 is fitted to a power law. The best fits are plotted as two orange dashed curves. The green curves represent the difference between ψ 11 and the best fits. \nwhere t i 's are time samples. To improve the fit performance, we notice that for a given nonlinear parameter t 0 , two linear parameters log A and µ can be determined uniquely using ordinary least squares linear regression. This reduces the fit to a one-dimensional minimization problem, and we deal with it using the Nelder-Mead method built in SciPy [100]. We have checked that the results are insensitive to the initial guess. The best fits are plotted as orange dashed lines in Figs. 3b and 3c. We find the tail exponents at null infinity ( µ n ) and timelike infinity ( µ t ) to be 2.93 and 4.92, respectively. They are consistent with the predictions of BH perturbation theory [62, 63, 97-99]: µ n = l +2 and µ t = 2 l +3.", 'B. Memory effects': "In the second test, we consider a Kerr BH with a dimensionless spin χ = 0 . 7, initially surrounded by a time-symmetric scalar field [Π ' = 0, defined in Eq. (72)]. The initial profile of ψ is chosen to be the same as Eq. (71), with A = 1 , r ' 0 = 20 and w = 2. The CCE worldtube is placed at r ' wt = 100 M . \nFigure 4. The l = m = 1 spherical harmonic of the scalar emission from a Kerr BH, extracted at future null infinity. \n<!-- image --> \nr ' wt + r ' 0 = 120 M . Due to the no-hair theorem [26, 38, 101-103], a Kerr BH does not carry a scalar charge. Consequently, the scalar field vanishes when the BH settles to stationary. \nAs discussed in Sec. II E, the Bianchi identities in Eq. (35) yield relations between the Weyl scalars, the strain, and the scalar field. Meanwhile, Eq. (52) imposes a further constraint between Ψ 2 and h . Figure 6 shows the L 2 -norms of deviations from these constraints, computed at three numerical resolutions. In the first two rows, the convergence of the deviations with increasing resolution validates our CCE procedure. In the bottom row, the constraint violations for Ψ 4 [(35a)] and Ψ 3 [(35b)] do not converge, presumably because they are already small ( ∼ 10 -7 ). \nDuring the dynamical process, the scalar field stirs spacetime nonlinearly [Eq. (4)] and excites GWs. The blue curves in Figs. 5a and 5b represent the corresponding ( l = 2 , m = 0) and ( l = m = 2) harmonics, respectively. Both curves exhibit a permanent jump after the passage of the scalar wave, with a quasinormal-mode ringing superimposed. As discussed in Sec. III, a strain can be decomposed into an electric and a magnetic memory. In this case, we find that the magnetic sector vanishes. Three dominant electric components [see Eq. (41)] are plotted in orange, green, and red in Figs. 5a and 5b. The scalar-driven null memory J ES null contributes to the overall jump, while the tensor-driven ordinary piece J ET o contributes to the quasinormal-mode ringing. The scalar-driven ordinary memory J ES o only results in a small pulse when the initial outgoing scalar wave reaches future null infinity. Unlike in BBH systems, where the tensor-driven null component J ET null induces a strong memory effect [86], its contribution is negligible in this process. Two lower panels of Figs. 5a and 5b present the difference between the total strain and the sum of all the individual memory contributions. We see that the difference is overall smaller than the numerical error (in yellow) except for the initial junk-radiation regime, thereby justifying our CCE code. To examine the magnetic memory, we consider the ( l = 3 , m = 2) harmonic in Fig. 5c. For the current system, it is solely contributed by the ordinary memory associated with the angular momentum aspect. The difference between J B o and the total strain is again smaller than the numerical error. Finally, Figure 5d shows the ( l = 3 , m = 2) harmonic of the spin memory [104], as a time integration of h 32 .", 'VI. CONCLUSION': "In this paper, we have implemented a CCE algorithm for the Einstein-Klein-Gordon system. Compared to GR, we presented the additional terms contributed by the scalar field in the equations of motion (Sec. II A), the computation of Weyl scalars (Sec. II D), the Bianchi identities (Sec. II E), as well as the memory effects (Sec. III). In addition, we reformulated the characteristic form of the KG equation using the numerically adapted coordinates [Eq. (8)], facilitating its implementation in our numerical relativity code SpECTRE . We also derived a concise worldtube transformation to construct the boundary condition for the characteristic KG equation from a generic Cauchy evolution (Sec. II C). \nThe tests demonstrate that our CCE code can effectively reveal scalar-induced memory effects. In the future, a direct application of this work is to extract GWs emitted by binary mergers in alternative theories of gravity. For \nTo evaluate the accuracy of our code, we designed various test systems. We first focused on the implementation of the KG equation and the worldtube transformation by evolving scalar fields on two prescribed spacetime backgrounds. As expected, we found that the numerical error decreases exponentially with increasing simulation resolution. We then examined the full CCE procedure by striking a BH with a scalar pulse. During the late-time evolution, we observed that the scalar field decays as a power law u -l -2 , consistent with Price's law at future null infinity. Furthermore, we verified that the tensor emissions extracted with CCE are consistent with the balance laws and the Bianchi identities. \n1e 1 \n1e 1 \nFigure 5. Memory effects in the GWs emitted by a Kerr BH, perturbed by a scalar field. Figures 5a and 5b: the blue curves show the ( l = 2 , m = 0) and ( l = m = 2) harmonics of the full strain. They are compared to the three dominant electric memory components (in orange, green, and red), as evaluated based on Eqs. (42) and (43). Two lower panels display the difference between the total strain and the sum of all electric memory contributions (in black), which is further compared to numerical error (in yellow). Figures 5c and 5d: the blue curves (overlapped with the orange curves) illustrate the ( l = 3 , m = 2) harmonic of the full strain and its time integration (spin memory). They are solely contributed by the magnetic ordinary memory J B o (in green), as evaluated based on Eq. (50). \n<!-- image --> \nFigure 6. The L 2 -norms of the constraint violations given in Eq. (52) ( C Im ψ 2 , top left) and Eq. (35), computed at three numerical resolutions. \n<!-- image --> \nexample, a recent study simulated a black hole - neutron star merger in scalar-tensor theory [42], where the neutron star underwent spontaneous scalarization. Working within the Einstein frame, the current CCE algorithm can faithfully compute memory effects in both tensor and breathing modes [105-107], allowing us to make comparisons with the post-Newtonian approximation [108-110]. These numerical studies will improve our understanding of asymptotic symmetries [71, 111-113]. In addition, our implementation lays a foundation for performing CCE simulations in other modified theories of gravity, e.g., [35, 50], paving the way for studying the associated memory effects [107]. \nOur simulations show tails in scalar fields. A possible avenue for future work is to look for tails in breathing modes (e.g. [42]) and tensor modes [114-116] emitted by binary systems. The accurate simulation of these tails is sensitive to the choice of boundary conditions [117]. Incorrect boundary conditions can introduce nonoscillatory numerical artifacts during the ringdown phase, which might be mistaken for physical tails. This issue can be avoided by either positioning the outer boundaries far enough away to remain causally disconnected from the system or by adopting CCM [57]. \nFinally, it would also be interesting to apply our CCE algorithm to extract scalar emissions from intermediatemass-ratio inspirals. Recently, a worldtube excision method [118, 119] was developed to simulate a small scalar charge orbiting around a BH. By matching a perturbative description around the scalar charge to a Cauchy simulation, the method enables the evolution of a binary system in the intermediate-mass-ratio regime. The accuracy of this method was evaluated by comparing scalar modes extracted at a finite radius to perturbative calculations. Our CCE algorithm could further improve this comparison. Additionally, the algorithm allows for verifying balance laws at future null infinity.", 'ACKNOWLEDGMENTS': 'S.M. would like to thank Vijay Varma and Leo C. Stein for useful discussion. Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development and by the Province of Ontario through the Ministry of Colleges and Universities. This material is based upon work supported by the National Science Foundation under Grants No. PHY-2407742, No. PHY- 2207342, and No. OAC-2209655 at Cornell. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. This work was supported by the Sherman Fairchild Foundation at Cornell.', 'Appendix A: Worldtube transformation': "As discussed in [55], in SpECTRE, five coordinate systems are involved in worldutbe transformations, including: \n- · Cauchy coordinates r ' , x ' A ' , t ' .\n- · Null-radius coordinates λ, x A , u\n- · Bondi-like coordinates r, x A , u .\n- · Partially flat Bondi-like coordinates ˆ r, ˆ x ˆ A , ˆ u .\n- · Numerically adapted coordinates ˘ y, ˘ x ˘ A , ˘ u \nThe numerically adapted coordinates are used for characteristic evolution, namely our target system. The Jacobians between these coordinate systems are summarized in Ref. [57]. Here we shall not go into full details. Instead, we list only necessary components in the following discussions.", '1. Obtaining ˘ Π | wt : from numerically adapted to partially flat Bondi-like coordinates': 'To obtain the boundary condition ˘ Π | wt , we need the following Jacobian [see Eq. (46b) of [55]] \n∂ ˆ u = ∂ ˘ u -(1 -˘ y ) ∂ ˘ u ˆ R wt ˆ R wt ∂ ˘ y , (A1) \nwhere ˆ R wt is the radius of the worldtube in the partially flat Bondi-like coordinate system. Applying the Jacobian to ψ on the worldtube ˘ y = -1, we obtain \n˘ Π | wt = ( ∂ ˘ u ψ ) ˘ y = ( ∂ ˆ u ψ ) ˆ r + ∂ ˘ u ˆ R wt ˆ R wt ∂ ˘ y ψ. (A2) \nThe second term ∂ ˘ u ˆ R wt ˆ R wt ∂ ˘ y ψ is already available in the current characteristic system. Therefore, we are left to obtain the value of ( ∂ ˆ u ψ ) ˆ r from the Cauchy system. Below we will transform ( ∂ ˆ u ψ ) ˆ r to the rest of coordinates.', '2. Obtaining ( ∂ ˆ u ψ ) ˆ r : from partially flat Bondi-like to Bondi-like coordinates': "To obtain the value of ( ∂ ˆ u ψ ) ˆ r from the Bondi-like coordinates, the necessary Jacobian is \n( ∂ u ) r = ( ∂ ˆ u ) ˆ r + ( ∂ u ˆ x ˆ A ) r ∂ ˆ A + ˆ r ∂ u ˆ ω ˆ ω ∂ ˆ r = ( ∂ ˆ u ) ˆ r -Re U (0) ¯ ˆ ð +(1 -˘ y ) ∂ u ˆ ω ˆ ω ∂ ˘ y , (A3) \nwhere the first line comes from Eq. (4.20) of [57]. The second line uses the evolution equation of ˆ x ˆ A : \n( ∂ u ˆ x ˆ A ) r = -1 2 ( U (0) ¯ ˆ q ˆ A + ¯ U (0) ˆ q ˆ A ) , (A4) \nsee Eq. (17) with t ' replaced by u and ˘ x ˘ A by ˆ x ˆ A ; as well as an identity [Eq. (46a) of [55]] \nˆ r∂ ˆ r = (1 -˘ y ) ∂ ˘ y . (A5) \nFinally, ˆ ω in Eq. (A3) is the conformal factor of the angular transformation ˆ x ˆ A ( x A ): \nˆ q ˆ A ˆ B ∂ A ˆ x ˆ A ∂ B ˆ x ˆ B = 1 ˆ ω 2 q AB , (A6) \nwhere q AB (ˆ q ˆ A ˆ B ) is the unit sphere metric of the (partially flat) Bondi-like coordinates. Applying the Jacobian in Eq. (A3) to ψ and restricting it to the worldtube, we obtain \n( ∂ u ψ ) r = ( ∂ ˆ u ψ ) ˆ r -Re U (0) ¯ ˆ ð ψ +2 ∂ u ˆ ω ˆ ω ∂ ˘ y ψ. (A7) \nIn practice, our CCE system does not directly provide the value of ∂ u ˆ ω . Instead, it gives us ∂ ˆ u ˆ ω . The two are related by applying the Jacobian in Eq. (A3) to ˆ ω , which yields \n( ∂ u ˆ ω ) r = ( ∂ ˆ u ˆ ω ) ˆ r -Re U (0) ¯ ˆ ð ˆ ω. (A8) \nNote that we have used the fact that the conformal factor ˆ ω does not depend on ˘ y , i.e., ∂ ˘ y ˆ ω = 0. Plugging Eq. (A8) into Eq. (A7), we get \n( ∂ ˆ u ψ ) ˆ r = ( ∂ u ψ ) r +Re U (0) ¯ ˆ ð ψ -2 ( ∂ ˆ u ˆ ω ) ˆ r -Re U (0) ¯ ˆ ð ˆ ω ˆ ω ∂ ˘ y ψ. (A9) \nThe only remaining unknown variable on the right-hand side is ( ∂ u ψ ) r . Therefore, our next task is to compute its value from Cauchy variables.", '3. Obtaining ( ∂ u ψ ) r : from Bondi-like to Cauchy coordinates': "To compute ( ∂ u ψ ) r , we need the following Jacobians \n( ∂ u ) λ = ( ∂ u ) r +( ∂ u r ) ∂ r , ( ∂ t ' ) r ' = ( ∂ u ) λ , (A10) \nsee Eqs. (4.6) and (4.3) of [57]. Restricting them to the worldtube, we obtain \n( ∂ u ψ ) r = ( ∂ t ' ψ ) r ' -2 ∂ u R wt R wt ∂ ˘ y ψ, (A11) \nwhere R wt is the radius of the worldtube in the Bondi-like coordinate system. It is related to ˆ R wt via \nR wt = ˆ R wt ˆ ω . (A12) \nThe first term ( ∂ t ' ψ ) r ' in Eq. (A11) is a dynamical variable of the Cauchy system, and the second term ∂ u R wt R wt ∂ ˘ y ψ is already available in the CCE system. Therefore, once we have collected the value of ( ∂ t ' ψ ) r ' from a Cauchy evolution, we can retrace our derivations presented in Secs. A 1, A 2 and A 3 to construct the boundary condition ˘ Π | wt .", '4. Simplication': "Combining Eqs. (A2), (A9) and (A11), we obtain a lengthy expression for ˘ Π | wt : \n˘ Π | wt = ( ∂ t ' ψ ) r ' -2 ∂ u R wt R wt ∂ ˘ y ψ +Re U (0) ¯ ˆ ð ψ -2 ( ∂ ˆ u ˆ ω ) ˆ r -Re U (0) ¯ ˆ ð ˆ ω ˆ ω ∂ ˘ y ψ + ∂ ˘ u ˆ R wt ˆ R wt ∂ ˘ y ψ. (A13) \nA pivotal step in simplifying it is to leverage an identity, see Eq. (15b) of [56] 7 : \n∂ ˆ u ˆ R wt ˆ R wt = ∂ u R wt R wt + ∂ ˆ u ˆ ω ˆ ω +Re U (0) ˆ ð R wt R wt . (A14) \nwhich leads to \n˘ Π | wt = ( ∂ t ' ψ ) r ' +Re [ U (0) ( ¯ ˆ ð ψ +2 ∂ ˘ y ψ ¯ ˆ ð ˆ R wt ˆ R wt )] . 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2024AJ....168..219X

Extending geodetic and astrometric Very Long Baseline Interferometry VLBI observations from traditional centimeter wavebands to millimeter wavebands offers numerous scientific potentials and benefits. However it was considered quite challenging due to various factors including the increased effects of atmospheric opacity and turbulence at millimeter wavelengths. Here we present the results of the first geodeticmode VLBI experiment simultaneously observing 82 sources at 224388132 GHz KQWD bands using the Korean VLBI Network KVN. We introduced the frequency phase transfer FPT method to geodetic VLBI analysis an approach for calibrating atmospheric phase fluctuations at higher frequencies by transferring phase solutions from lower frequencies. With a 2 minute scan FPT improved the signaltonoise ratio of most fringes some by over 100 thereby enhancing the detection rate of weak sources at millimeter wavebands. Additionally FPT reduced systematic errors in group delay and delay rate with the weighted root mean squares WRMS of the postfitting residuals decreasing from 25.0 to 20.5 ps at the W band and from 39.3 to 27.6 ps at the D band. There were no notable differences observed in calibrating atmospheric phase fluctuations at the K band WRMS 12.4 ps and Q band WRMS 11.8 ps. This experiment demonstrated that the millimeter waveband can be used for geodetic and astrometric applications with high precision.

2024-11-01T00:00:00Z

['10.48550/arXiv.2409.07309', 'arXiv:2409.07309', '2024arXiv240907309X', '10.3847/1538-3881/ad7af0', '2024AJ....168..219X']

['Quasars', 'Very long baseline interferometry', 'Radio source catalogs', 'Radio astrometry', '1319', '1769', '1356', '1337', 'Astrophysics - Instrumentation and Methods for Astrophysics', 'Astrophysics - Earth and Planetary Astrophysics', 'Astrophysics - Astrophysics of Galaxies']

A Geodetic and Astrometric VLBI Experiment at 224388132 GHz

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']

https://arxiv.org/pdf/2409.07309.pdf

{'A Geodetic and Astrometric VLBI Experiment at 22/43/88/132 GHz': "Shuangjing Xu , 1 Taehyun Jung , 1 Bo Zhang , 2 Ming Hui Xu , 3, 4 Do-Young Byun , 1 Xuan He , 2, 5 Nobuyuki Sakai , 6 Oleg Titov , 7 Fengchun Shu , 2 Hyo-Ryoung Kim, 1 Jungho Cho, 1 Sung-Moon Yoo, 1 Byung-Kyu Choi, 1 Woo Kyoung Lee , 1 Yan Sun , 2 Xiaofeng Mai , 2, 5 and Guangli Wang 2 \n1 Korea Astronomy and Space Science Institute, 776 Daedeok-daero, Yuseong-gu, Daejeon 34055, Republic of Korea \n2 Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030, People's Republic of China 3 GFZ German Research Centre for Geosciences, 14473 Potsdam, Germany \n4 Aalto University Metshovi Radio Observatory, Metshovintie 114, 02540 Kylml, Finland \n5 University of Chinese Academy of Sciences, No.19 (A) Yuquan Rd, Shijingshan, Beijing 100049, People's Republic of China 6 National Astronomical Research Institute of Thailand (Public Organization), 260 Moo 4, T. Donkaew, A. Maerim, Chiang Mai, 50180, \nThailand \n7 Geoscience Australia, PO Box 378, Canberra 2601, Australia", 'ABSTRACT': 'Extending geodetic and astrometric Very Long Baseline Interferometry (VLBI) observations from traditional centimeter wavebands to millimeter wavebands offers numerous scientific potentials and benefits. However, it was considered quite challenging due to various factors, including the increased effects of atmospheric opacity and turbulence at millimeter wavelengths. Here, we present the results of the first geodetic-mode VLBI experiment, simultaneously observing 82 sources at 22/43/88/132 GHz (K/Q/W/D bands) using the Korean VLBI Network (KVN). We introduced the frequency phase transfer (FPT) method to geodetic VLBI analysis, an approach for calibrating atmospheric phase fluctuations at higher frequencies by transferring phase solutions from lower frequencies. With a 2-minute scan, FPT improved the signal-to-noise ratio (SNR) of most fringes, some by over 100% , thereby enhancing the detection rate of weak sources at millimeter wavebands. Additionally, FPT reduced systematic errors in group delay and delay rate, with the weighted root-mean-squares (WRMS) of the post-fitting residuals decreasing from 25.0 ps to 20.5 ps at the W band and from 39.3 ps to 27.6 ps at the D band. There were no notable differences observed in calibrating atmospheric phase fluctuations at the K band (WRMS = 12.4 ps) and Q band (WRMS = 11.8 ps) with the KVN baselines. This experiment demonstrated that the millimeter waveband can be used for geodetic and astrometric applications with high precision. \nKeywords: reference systems / astrometry / techniques: interferometric / quasars: general / galaxies: nuclei/radio continuum: general', '1. INTRODUCTION': "Geodetic and astrometric Very Long Baseline Interferometry (VLBI) observations have made significant contributions to astronomy and geodesy over the past 40 years, particularly in the areas of the terrestrial reference frame (TRF), the celestial reference frame (CRF), and the earth orientation parameters (EOPs) (Sovers et al. 1998). The basic concept of geodetic and astrometric VLBI involves using pairs of radio telescopes to observe the signals of compact extra-galactic objects that emit radiation in the radio frequency regime. By analyzing the differences in the arrival times of the same wavefront between two telescopes, geodetic parameters such as telescope coordinates, positions of celestial objects, and EOPs can be inferred for various scientific and practical applications (Nothnagel 2019). In \naddition to the traditional S/X band (2.3/8.4 GHz), multiple frequency bands have recently been employed to enhance the potential of geodetic VLBI observations. \nThe accuracy of geodetic and astrometric measurements relies on the precision of derived group delays, baseline length, and systematic delay errors. The broadband geodetic VLBI system at 2-14 GHz, known as VLBI Global Observing System (VGOS), has improved the precision of group delays to a few picoseconds (ps; 1 ps = 10 -12 s) (Niell et al. 2018), however, uncompensated systematic errors at the level of 20 ps still dominate the error budget (Xu et al. 2021a). Further efforts are required to calibrate systematic errors originating from tropospheric delay (Petrov 2024) and source structure (Xu et al. 2022) to achieve the VGOS's goal of 1 mm position accuracy. \nHigher frequency bands offer advantages such as achieving higher-resolution imaging of radio sources, mitigating source structure effects (de Witt et al. 2023), measuring frequency-dependent position shifts in Active Galactic Nuclei (AGN) jets (i.e., core-shift) (Blandford & Konigl 1979; Hada et al. 2011), and reducing interference from scattering (Koryukova et al. 2022) and ionospheric plasma effects (Fomalont et al. 2009). The K band (24 GHz) ， X/Ka band (8.4/32 GHz), and Q band (43 GHz) (Lanyi et al. 2010; Charlot et al. 2020) have been used for establishing a multifrequency International Celestial Reference Frame (ICRF). In the meantime, the Gaia satellite mission realized the first extra-galactic frame at optical wavelengths (Gaia Collaboration et al. 2022). How to realize a fully consistent and integrated multi-waveband celestial reference frame becomes an important issue (Charlot 2022). In addition, the use of millimeter wavebands for geodesy is valuable for determining the station coordinates of antennas without receivers operating at lower frequencies. \nThe independent geodetic VLBI programs operating at different frequencies may have astrometric limitations in detecting the core-shift in most ICRF sources (Petrov 2024). As frequency increases, the signal-to-noise ratio (SNR) of VLBI fringes is affected by decreased flux densities of sources, shorter coherence times, increased atmospheric absorption, and higher receiver temperatures, limiting the precision of group delays at millimeter wavelengths. The single K band observation also has difficulty in calibrating the ionospheric delay (Lanyi et al. 2010). \nThe Korean VLBI Network (KVN) has the capability of simultaneously observing at multiple frequencies (Han et al. 2013), including K band at 18-26 GHz, Q band at 35-50 GHz, W band at 85-116 GHz, and D band 1 at 125-142 GHz. A similar K/Q/W band system is being developed globally (Dodson et al. 2023). This kind of system is particularly useful for extending the coherence time at millimeter wavelengths using the frequency phase transfer (FPT) technology (Rioja & Dodson 2020). Specifically, it achieves this by calibrating the tropospheric phase at higher frequencies through transferring phase solutions from lower frequencies. This enables the observation of more sources and facilitates the measurement of core shift effects using source-frequency phase-referencing (SFPR) astrometry (Rioja et al. 2015; Jung et al. 2015). The global K/Q/W band system may also benefit the ICRF by employing geodetic and astrometric VLBI across a broad frequency range from 20 to over 100 GHz. This approach offers several advantages: 1) simultaneous multi-frequency investigation of core shift effects and source structures; 2) monitoring numerous ICRF source images with resolutions of a few tens of microarcseconds; 3) overcoming limitations in group delay precision at millimeter wavelengths through very broad bandwidth synthesis. \nIn this paper, we present the first geodetic VLBI observation at 22/43/88/132 GHz simultaneously using the KVN as a pilot experiment for future broad bandwidth synthesis from 20 to over 100 GHz.", '2. OBSERVATIONS AND GENERAL DATA ANALYSIS': 'We conducted the first geodetic and astrometric VLBI experiment observed at K/Q/W/D bands (22/43/88/132 GHz) simultaneously using the KVN under the East Asian VLBI Network (EAVN) (Akiyama et al. 2022) program a2129a . The KVN consists of three 21-m antennas: KVN-Yonsei (KYS), KVN-Ulsan (KUS), and KVN-Tamna (KTN), with baseline lengths ranging from 305 to 476 km. We used a 24-hour track for the session from 2021-Dec-07/15:25:00 to 2021-Dec-08/15:25:00. The received signals were recorded with four 512 MHz base-band channels (BBCs) and recorded right (for K band) or left (for Q/W/D bands) circularly polarized signals with Nyquist sampling and 2 bits per sample for a total sampling rate of 8 Gbps. A summary of the observation is listed in Table 1.', 'GeoVLBI at 22/43/88/132 GHz': 'Table 5 (continued) \nNote -The flux density values are obtained from KVN baseline ( < 500 km) amplitude using AIPS , while the SNR with 2-min scans is produced by fourfit . These values (weighted averages and standard deviations) are provided as a reference for conducting observations and may not be accurate for AGN astrophysics studies. The different standard deviations observed in the baseline amplitudes and SNRs may be attributed to variations in weather conditions, elevation angles, and pointing accuracy. \nThis work utilized the KVN under the EAVN program. We are grateful to all staff members in KVN and EAVN who helped to operate the array. The KVN and a high-performance computing cluster are facilities operated by the KASI (Korea Astronomy and Space Science Institute). The KVN observations and correlations are supported through the high-speed network connections among the KVN sites provided by the KREONET (Korea Research Environment Open NETwork), which is managed and operated by the KISTI (Korea Institute of Science and Technology Information). The presented figures were generated using Matplotlib (Hunter 2007) and Astropy (Astropy Collaboration et al. 2013). This research was supported by the National ResearchCouncil of Science & Technology(NST) grant by the Korea government (MSIT) (No. CAP22061-000). BZ was supported by the National Natural Science Foundation of China (Grant No. U2031212 and U1831136), and Shanghai Astronomical Observatory, Chinese Academy of Sciences (Grant No. N2020-06-19-005). SX thanks Dr. John Barrett and Dr. Daniel Hoak for solving the technical problems with HOPS , and thanks to Dr. Sergei Bolotin for the guide on nuSolve . \nFacilities: KVN, EAVN \nSoftware: HOPS (Hoak et al. 2022), nuSolve (Bolotin et al. 2014), SKED (Gipson 2018), SCHED (Walker 2022), DiFX (Deller et al. 2011), AIPS (Greisen 2003), ParselTongue (Kettenis et al. 2006).', '2.2. Correlation and Fringe fitting': "The observations were correlated using the DiFX software correlator (Deller et al. 2011) in Daejeon, South Korea. The output of the correlator was converted to Mark 4 format in order to be compatible with the Haystack Observatory Processing System ( HOPS ) suite of programs (Whitney et al. 2022) and converted to FITS format for imaging with NRAO Astronomical Image Processing System ( AIPS ) (Greisen 2003). \nThe HOPS main tool, fourfit , was used to estimate group and phase delays, phase, delay rate, and cross-correlation amplitude for each observation using the Mark 4 data. As the hardware phase calibration system to cover the entire frequency range of KVN (18 - 142 GHz) is under development, we implemented manual phase calibration through a single scan of the bright source OJ287, to align the delays and phases among different frequency bands. Contrary to the standard practices in geodetic VLBI, we introduced the utilization of FPT to address the effects of atmospheric turbulence in millimeter wavebands. This method is elucidated in Section 3. A database in vgosDb format (Bolotin et al. 2016) was finally produced in three steps: ' vgosDbMake ' produced the skeleton database for all observables (version 1); ' vgosDbCalc ' added the apriori values and partial derivatives (version 2), and ' vgosDbProcLogs ' added meteorological information (version 3; no cable cal in KVN log file)(Bolotin et al. 2014).", '2.3. Geodetic data analysis': "The geodetic analysis was conducted using the nuSolve program (Bolotin et al. 2014). This program operates on the vgosDb database to perform least-squares estimation of various geodetic, geophysical, astronomical, and instrumental \nparameters. We disassembled the database into individual bands (e.g., K, Q, W, D) and dual-bands (e.g., Q/K, W/K, D/K) by manually editing the 'wrapper' file in the vgosDb database. \nObtaining the final geodetic estimate utilized multiple steps proceeding from least precise to most precise. The detailed steps can be found in the User Guide of nuSolve and Niell et al. (2021). External files containing a priori information are used in the analysis, such as updated station coordinates from the multi-epoch EAVN geodetic observations (Xu et al. 2021b), source positions from the ICRF3 K band catalog (Charlot et al. 2020), and earth rotation parameters from the VLBI solution provided by NASA Goddard Space Flight Center (GSFC). The group delays with the single 512 MHz channel in our data are unambiguous. We start with simple parameterization, only clock shifts and rates, and perform an analysis of the group delays. At this stage, we use the KYS station as the clock reference and have not identified any clock breaks at any of the stations. Then we add zenith delays and station positions to the list of estimated parameters. \nAlso, we test ionospheric corrections for dual-frequency (Q/K, W/K, D/K) band data. Usually, geodetic VLBI uses multi-frequency channels for each frequency band, i.e., the bandwidth synthesis technique, to improve the group-delay measurement precision. In this case, an 'effective frequency' is calculated and assigned to ionosphere group delays as the approximate reference frequency (Bohm & Schuh 2013). We calculated the central frequencies of each band (22240 MHz for K band, 42876 MHz for Q band, 88192 MHz for W band, 132160 MHz for D band) as the effective frequency and used them latter in ionospheric calibration. \nTime-varying models of clock and tropospheric parameters are introduced in the latter stage. They are modeled as continuous piece-wise linear (PWL) functions with incremental rates. For such a PWL model, the estimated values are, for each parameter, an initial value and rate for the first interval and a new rate for each of the successive equalduration intervals. The results reported in the remainder of this paper are based on a PWL interval of 30 min for the troposphere and 60 min for the clock. The daily averaged atmospheric gradients (MacMillan 1995) are estimated with constraints in this independent solution. In the last stage of data processing, additional parameters such as the rate of Earth rotation and angles of nutation are included. We also re-weight the observations by examining the additive noise required to achieve a chi-squared per degree of freedom (chi2pdof) of approximately 1. Any post-fit delay residuals greater than 3.5 times their re-weighted uncertainty are marked for exclusion. We iterate through the estimation, re-weighting, and outlier-check sequence until no outliers are detected, resulting in the final solution. To efficiently compare results from different frequency bands and/or parameters, we use the script mode with nuSolve (Bolotin et al. 2023).", '3. APPLICATION OF FPT IN GEODETIC VLBI': "The inherent challenges of geodetic VLBI in millimeter wavebands, such as sensitivity, can be effectively addressed due to the impressive performance of the KVN telescopes (Lee et al. 2014), characterized by their high aperture efficiency, precise pointing accuracy, receivers with low noise temperature, wide-band digital backend, and rapid slewing speed. Of particular significance is the telescopes' ability to operate at multiple frequencies, which enables us to mitigate atmospheric phase fluctuations using the FPT method. In the FPT method, high-frequency (target frequency) observations are calibrated using scaled solutions obtained from a lower, more easily manageable frequency (Jung et al. 2011; Rioja & Dodson 2020). \nThe majority of mm-VLBI observations serve imaging purposes, wherein the rapid nonlinear phase (atmospheric phase fluctuations) is usually estimated per scan using self (on-source) detections from a single reference station to other stations (Blackburn et al. 2019). The correlated signal must have a high SNR so that the atmospheric phase can be estimated on a short timescale (a few seconds). Therefore, this method is limited to using bright sources or using a high-sensitivity station as the phase reference. In addition, it is difficult to distinguish nonlinear atmospheric phase from the linear phase drift due to delay rate, and can lead to the loss of frequency-dependent information in the phases. The FPT method can be effectively employed to overcome these limitations, either simultaneously or through fast frequency switching. \nWe effectively employed the FPT method using the packets HOPS for geodetic data, drawing from the guidance provided in the fourfit user's manual and a tutorial (Fish 2015). This implementation involved the following steps: \n- · Initiating fringe fitting using the fourfit program and employing the alist program to track the fringe phase, SNR, and other information for each observation, leading to the creation of a corresponding text file.\n- · Employing the fringex program to segment data from a single scan time ( ∼ 2 min) into a few seconds. This step utilized the aforementioned text file ( alist ) as input and provided atmospheric phase fluctuations using \nFigure 1. Comparison atmospheric phase fluctuations with self detection ( ϕ Self ) and FPT ( ϕ FPT ) method using a strong source 3C279 on KTN-KYS baseline. The four panels represent distinct frequency bands. The 2 π ambiguity of phase is ignored for clarity. \n<!-- image --> \nself detection at each bands (e.g., ϕ Self K , ϕ Self Q , ϕ Self W , and ϕ Self D ). Choosing a reference station and an appropriate cadence for segmented phases is crucial; it should be short enough to capture the atmospheric phase fluctuations while maintaining enough SNR. In this experiment, we used KYS as the reference station and examined cadences of 1 second, 3 seconds, 5 seconds, and 10 seconds. Subsequently, we selected the smallest cadence with an SNR exceeding 10. Given the high SNRs in the K band, ∼ 90% of the scans adopted a 1-second cadence. However, for a small number of weaker scans with an SNR below 10 at the 10-second cadence, we refrained from applying FPT to prevent the potential deterioration of results. \n- · Calculating atmospheric phase fluctuations at higher frequencies by transferring phase solutions from a lower frequency (i.e., K band), based on frequency ratios. As illustrated in Table 1, the frequency ratios are 1.93 (approximately equal to 2) for Q/K bands, 4 for W/K bands, and 6 for D/K bands, resulting in ϕ FPT Q = ϕ Self K × 2, ϕ FPT W = ϕ Self K × 4, and ϕ FPT D = ϕ Self K × 6. We have applied phases from one polarization (LCP of K band) to the other (RCP of Q/W/D band), as atmospheric phases remain unaffected by polarization.\n- · Within the HOPS framework, the solutions were based on baselines. We designated KYS as the reference station and 'viewed' the atmospheric phases ( ϕ Self K , ϕ FPT Q , ϕ FPT W , ϕ FPT D ) originating from KUS and KTN stations. Finally, we integrated this ' ad hoc ' phase information into fourfit and re-performed the process of fringe fitting. \nFigure 2. Same as Figure 1, but for a weaker source 0642+449. \n<!-- image --> \nAs shown in Figure 1 and 2, the FPTed phases exhibit close agreement with the trends observed in self detected phases at each K/Q/W/D bands. The clear linear phase differences ( Diff : ϕ Self -ϕ FPT in Figure 1) between self detection and FPT method indicate the different delay rates. In Table 2, it is noteworthy that the fringe quality ('Qcodes') of FPT-derived detection is comparable to those of self-detection. The term 'Qcodes' represents the fringe quality code as defined by HOPS-fourfit , where higher Qcodes indicate better quality. Subsequent to the application of FPT, the proportion of all usable observables with Qcodes of 9 has improved from 79% to 99%, characterized by heightened SNR and improved delay/rate precision. The improvements in group delay and rate accuracy are presented in Section 4. After applying FPT, the presence of phase differences among different bands in Table 2 necessitates further investigation through precise instrumental phase calibration. However, their influence on group delay and delay rate measurements is negligible in the following analysis.", '4. RESULTS': "The weather conditions were clear during the observation period, with median system temperature (Tsys) ranges as follows: 87-131 K for the K band, 96-103 K for the Q band, 189-193 K for the W band, and 186-270 K for the D band at the three KVN stations. These favorable conditions have led to excellent detection (SNR > = 7), with the detection rate of 99.8% at K band, 99.8% at Q band, 95.5% at W band (or 96.3% with FPT), and 68.2% (or 70.9% with FPT) at D band for 485 scans on KYS-KTN baseline. However, the Q band data from the KUS station exhibited a rare \nTable 2. The fringe information with observables in Figure 1 and 2 \nlower SNR compared to other stations, for unknown instrumental reasons at this time. Therefore, we didn't use the Q band data of KUS station in the final results.", '4.1. The group delay and delay rate measurements at 22/43/88/132 GHz': 'As shown in Figure 3, FPT improved the SNR of most fringes, some by more than 100% with a 2-minute scan length, resulting in a higher detection rate (over 100 observables) for weak sources at millimeter wavebands. Note that most sources in this experiment are bright. Detection can be further improved in experiments with weaker sources and longer scan lengths. √ \nThe theoretical uncertainties of group delays produced by fourfit are calculated using 12 / (2 π *SNR*512 MHz) for the single 512 MHz channel data. Figure 4 shows the measurement noise of group delays on KYS-KTN baseline for each band. The median formal errors of group delays are 3.5 ps at K band, 5.0/4.7 ps at Q band, 22.5/19.5 ps at W band, and 49.7/43.0 ps at D band without/with FPT. Similarly, the median formal errors of delay rates are 6.6E-4 ps/s at K band, 5.0E-4/4.8E-4 ps/s at Q band, 1.1E-3/9.8E-4 ps/s at W band, and 1.7E-3/1.5E-3 ps/s at D band without/with FPT. \nThe closure group delay for a triangle of three stations simultaneously observing the same source is given by the sum of the three baseline group delays going around a closed loop of the triangle (Xu et al. 2016). In this summation, the effects of station-based delays (tropospheric delays, ionospheric delays, station position errors, station thermal deformation errors, clock offset errors, cable delay errors, EOP errors, errors from pointing offsets, and so on) cancel exactly. The major error terms at centimeter wavebands that remained in the closure quantities, the so-called nonclosing errors, are source structure and measurement noise (Anderson & Xu 2018). However, in millimeter wavebands, atmospheric phase fluctuations become significant within seconds and differ at each triangle baseline, leading to errors in baselinedependent fringe fitting when employing a 2D linear model (phase vs. time, phase vs. frequency) over a 2-minute \nFigure 3. Comparison of the SNR with and without FPT within a 2-minute scan length. The left figure shows the percentage increase in SNR after using FPT, and the right figure compares the SNR of weak sources (SNR < 15). The dashed lines indicate the fourfit detection limit (SNR = 7). For clarity, the results with an SNR below 6 using FPT are not shown. \n<!-- image --> \nFigure 5 and Table 3 show the closure observables in the KVN triangle baseline (KTN-KUS-KYS, with SNR > = 7 on each single baseline). With the exception of notable errors in the Q band attributed to the KUS station issue, the weighted standard deviations of closure group delays are 3.7 ps at K band, 19.9/15.7 ps at W band, and 47.2/36.7 ps at \n<!-- image --> \nDelay Rate Precision \nWithout FPT \n10 \n4 \nDelay Rate Precision \nWith FPT \n10 \n4 \n10 \n10 \n3 \n3 \nThe formal error of the delay rate (ps/s) \nFigure 4. The histogram of measurement noise for group delay and delay rate produced by fourfit on the KYS-KTN baseline. Different bands are represented by different colors. \nscanning duration. As a result, uncalibrated atmospheric phase fluctuations will contribute to non-closure errors in millimeter wavebands when using HOPS-fourfit . \n40 \n35 \n30 \n25 \n20 \n15 \n10 \n5 \n0 \n40 \n35 \n30 \n25 \n20 \n15 \n10 \n5 \n0 \nCount \nCount \n22 GHz \n43 GHz \n88 GHz \n132 GHz \n10 \n2 \n22 GHz \n43 GHz \n88 GHz \n132 GHz \n10 \n2 \nD band without/with FPT. These values commendably align with their corresponding formal errors, as illustrated in Figure 4. Moreover, the application of FPT noticeably enhances the accuracy of the closure group delays at the W and D bands. Similarly, the weighted standard deviations of closure delay rates are 1.7E-3 ps/s at K band, 3.5E-2/1.3E-3 ps/s at W band, and 6.1E-2/1.3E-3 ps/s at D band without/with FPT. And the weighted standard deviations of closure phases are 2.1 · at K band, 24.4 · /3.1 · at W band, and 55.9 · /4.4 · at D band without/with FPT. The FPT method significantly enhances the closure delay rate and closure phase by over an order of magnitude at the W and D bands. \nTable 3. Statistics of the closure observables in KVN triangle baselines', '4.2. The tropospheric and ionospheric effects with KVN': "The path delay of radio waves caused by the troposphere is one of the major error sources. Modeling the tropospheric delay is generally divided into hydrostatic and wet parts, each of which is the product of the zenith delay and the corresponding mapping function dependency on elevation angle (Bohm & Schuh 2013). In nuSolve , the zenith hydrostatic delay (ZHD) is modeled as a function of the surface pressure, and the zenith wet delay (ZWD) is calculated with the relative humidity and temperature at the surface (Saastamoinen 1972; Davis et al. 1985). However, the uncertainty of ZWD model is far larger than ZHD model due to high spatial and temporal variability and unpredictability of the amount of water vapor. Therefore, the residual ZWD is then parameterized as a PWL function of time (e.g., the interval of 30 min in this experiment) in the data analysis. We compared the solutions with two different sets of priori ZHD+ZWD: one using meteorological data (pressure, relative humidity, temperature) from the station's log files and the other using a constant value from the standard model in nuSolve . The use of meteorological data resulted in increased weighted root-mean-squares (WRMS) of the post-fitting residuals from ∼ 12 ps to ∼ 15 ps with K band data. In this experiment, the temperature or water vapor content at the KVN site, including their temporal changes, might not accurately reflect the conditions of the air masses above. Therefore, we did not adopt the KVN meteorological data in the following analysis. \nThe KVN system provides the greatest spanned frequency range yet used to calibrate the ionosphere. The ionospheric delays of single K band or Q band geodetic observations were usually calibrated using the global vertical total electron content (TEC) map derived from global navigation satellite system (GNSS) observations provided by the International GNSS Service (IGS) with a temporal resolution of 120 minutes (Lanyi et al. 2010; Charlot et al. 2020). We also tested the regional vertical TEC map provided by the Korea Astronomy and Space Science Institute GNSS network (KASINet) (Jeong et al. 2022), which has a temporal resolution of 5 minutes and includes observing data from co-located GNSS stations at each KVN site. The differential slant TEC (dTEC) for KVN derived with either IGS or KASINet TEC maps is only a few TEC units (TECU), except for the noon time. One TECU corresponds to an ionospheric delay \nFigure 5. The closure group delay, closure delay rate, and closure phase in the KVN triangle baseline (KTN-KUS-KYS), with different frequency bands represented by dots of different colors. Up panels: closure group delay without/with FPT; Middle panels: closure delay rate without/with FPT; and Bottom panels: closure phase without/with FPT. The 43 GHz result is not presented and has large uncertainties due to the low SNR issues on KUS station. \n<!-- image --> \nTable 4. Geodetic results \nof 2.5 ps, 0.6 ps, 0.15 ps, and 0.07 ps at 22 GHz, 43 GHz, 88 GHz, and 132 GHz, respectively. Consequently, the ionospheric delays in our KVN observations are at the 10 ps level for the K band and are negligible for the Q, W, and D bands. \nWith the single KTN-KYS baseline, the post-fit delay residuals have a WRMS of 13.2 ps at K band and 11.6 ps at Q band. This discrepancy could potentially stem from reduced ionospheric effects at the Q band. For the W and D bands, larger WRMS values are observed due to relatively larger delay uncertainties. The utilization of either GNSS TEC maps or dual-band VLBI combinations did not yield noticeable improvements in WRMS with this data, likely due to the negligible ionospheric effects on the short baselines. Notably, considering that global vertical ionospheric effects typically vary from a few to dozens of TECUs (Nothnagel 2019), the multi-band KVN system holds promise for obtaining ionospheric-free delays across the entire broad bandwidth from 20 to over 100 GHz with long baselines in the near future.", '4.3. Geodetic results': 'We performed a comparative analysis of WRMS post-fit delay residuals (pfdr) and estimated baseline lengths, as detailed in Table 4 and Figure 6, considering various frequency bands and the presence or absence of FPT. The baseline lengths estimated and reported in Table 4 are consistent with their respective uncertainties. \nThe post-fit delay residuals exhibit a WRMS of 12.4 ps at K band and 11.8 ps at Q band. When employing FPT, there is no noticeable difference in WRMS at K and Q bands. This suggests that the impact of atmospheric phase fluctuations at K and Q bands, particularly for baselines of less than 500 km and 2-minute scans, is minimal. This finding aligns with the similar closure group delays observed in Figure 5 and Table 3. Notably, the delay accuracy at the Q band is similar with that at the K band, and the slightly smaller WRMS at the Q band may be attributed to reduced ionospheric effects. \nIn the case of W and D bands, the visibility phases are significantly affected by atmospheric turbulence. As depicted in Figure 3, FPT enhances fringe SNR, leading to an additional 100 delay observables at the D band. It is worth noting that the W band already boasts a very high detection rate ( ∼ 95%). Additionally, FPT reduced systematic errors in group delay and delay rate. The WRMS of the post-fitting residuals decreased from 25.0 ps to 20.5 ps at the W band and from 39.3 ps to 27.6 ps at the D band. This indicates that the errors introduced by atmospheric phase fluctuations contributing to the WRMS are ∼ 14 ps for the W band and ∼ 28 ps for the D band in this experiment. This finding aligns with the effects observed in closure group delays in Figure 5 and Table 3. The baseline lengths can be different at 4 mm between without and with FPT. And using FPT, the baseline lengths at W and D bands are generally closer to that of the K band. Therefore, calibration for atmospheric phase fluctuations is necessary in geodetic VLBI at millimeter wavebands. \nThe source positions estimated in geodetic analysis for this experiment achieve milliarcsecond precision, posing challenges in investigating sub-milliarcsecond frequency-dependent position offsets. Nevertheless, this can be achieved with multiple epochs, extended baselines, and/or including the SFPR method. \nFigure 6. Post-fitting residuals at different bands \n<!-- image -->', '5. CONCLUSION AND OUTLOOK': "We have successfully conducted the first simultaneous geodetic and astrometric VLBI experiment at 22/43/88/132 GHz with KVN. Our achievement includes a high detection rate of approximately 95% at the W band and about 70% at the D band. Moreover, we have obtained competitive accuracy when compared to traditional centimeter wavebands, with the WRMS of the post-fitting residuals measuring 12.4 ps at K band, 11.8 ps at Q band, 20.5 ps at W band, and \n27.6 ps at D band. This experiment demonstrates that the millimeter waveband can be used for geodetic applications with high precision. \nFor the first time, we introduced the FPT method to geodetic VLBI analysis, an approach for calibrating atmospheric phase fluctuations. Our results demonstrated that FPT improves fringe detection and enhances the accuracy of delay measurements at the W and D bands. We found that atmospheric phase fluctuations, prevalent with baselines under 500 km and 2-minute scans, contribute to errors of approximately 14 ps at the W band and 28 ps at the D band. These fluctuations represent a major error source in millimeter-wave geodetic VLBI and can be effectively mitigated through the FPT method for general sources. It is important to emphasize that FPT proves highly beneficial for geodetic mm-VLBI observations. This extends beyond merely detecting weaker sources, encompassing the precise measurement of frequency-dependent offsets in source positions at millimeter wavebands (Rioja & Dodson 2011). \nThe challenges we previously believed regarding geodetic VLBI at the millimeter waveband might be surmountable. The high detection rate at 22/43/88/132 GHz for 82 sources has encouraged us to complete an all-sky distributed source catalog for geodetic VLBI and mm-VLBI studies in astrophysics (e.g., Dodson et al. 2017). The number of detectable ICRF sources at K/Q/W bands is expected to exceed several hundred with KVN (Xu et al. 2024). Typical mm-VLBI observations (above 80 GHz) require reference pointing scans to ensure accurate antenna pointing. In this experiment, such scans were not implemented due to the lack of support in SKED . Nevertheless, the results were satisfactory, owing to the high pointing accuracy of KVN. Further efforts to incorporate reference pointing scans into geodetic mm-VLBI scheduling are currently underway. The precision of group delay with broad bandwidth synthesis from 20 to 100 GHz with KVN can be comparable to or better than VGOS, as demonstrated by a fringe in Xu et al. (2024). The KVN phase-cal system is also being tested to determine the delay and phase offsets among different bands and monitor the effect of changes in ambient temperature on components of the signal chain. While the current KVN baseline is relatively short and limits our ability to explore the ionosphere, source structure, and frequency-dependent offsets in source positions, it's worth noting that simultaneous tri-band (K/Q/W) receivers are undergoing global development (e.g., Dodson et al. 2023). We anticipate that we will have access to longer baselines in the coming years, enabling more comprehensive investigations in these areas as mentioned in Section 1.", 'A. SOURCE CATALOG': 'The 82 sources used in this experiment are listed in Table 5. \nTable 5 . 82 sources for geodetic VLBI at K/Q/W/D band \nXu et al. \nTable 5 (continued) \nTable 5 continued on next page', 'REFERENCES': 'Akiyama, K., Algaba, J.-C., An, T., et al. 2022, Galaxies, \n10, 113, doi: 10.3390/galaxies10060113 \nAnderson, J. M., & Xu, M. H. 2018, Journal of Geophysical \nResearch (Solid Earth), 123, 10,162, \ndoi: 10.1029/2018JB015550 \nAstropy Collaboration, Robitaille, T. P., Tollerud, E. J., \net al. 2013, A&A, 558, A33, \ndoi: 10.1051/0004-6361/201322068 \nBlackburn, L., Chan, C.-k., Crew, G. B., et al. 2019, ApJ, 882, 23, doi: 10.3847/1538-4357/ab328d \nSovers, O. J., Fanselow, J. L., & Jacobs, C. S. 1998, Reviews of Modern Physics, 70, 1393, doi: 10.1103/RevModPhys.70.1393 Walker, R. 2022, The sched user manual Whitney, A. R., Cappallo, R., Aldrich, W., et al. 2022, HOPS: Haystack Observatory Postprocessing System, Astrophysics Source Code Library, record ascl:2205.019. http://ascl.net/2205.019 Xu, M. H., Anderson, J. M., Heinkelmann, R., et al. 2021a, Journal of Geodesy, 95, 51, \ndoi: 10.1007/s00190-021-01496-7 \nXu, M. H., Heinkelmann, R., Anderson, J. M., et al. 2016, AJ, 152, 151, doi: 10.3847/0004-6256/152/5/151 Xu, M. H., Savolainen, T., Anderson, J. M., et al. 2022, A&A, 663, A83, doi: 10.1051/0004-6361/202140840 Xu, S., Jung, T., & Byun, D.-Y. 2024, Geodetic and Astrometric VLBI at K/Q/W/D Bands with the KVN, Zenodo, doi: 10.5281/zenodo.10902979 \nXu, S., Jike, T., Jung, T., et al. 2021b, in 25th European VLBI Group for Geodesy and Astrometry Working Meeting, ed. R. Haas, Vol. 25, 71-73'}

2024arXiv240303325B

Even though subNeptunes likely represent the most common outcome of planet formation their natures remain poorly understood. In particular planets near 1.52.5Roplus often have bulk densities that can be explained equally well with widely different compositions and interior structures resulting in grossly divergent implications for their formation. Here we present the full 0.65.2mu mathrmm JWST NIRISSSOSSNIRSpecG395H transmission spectrum of the 2.2Roplus TOI270d 4.78Moplus Tmathrmeq350380 K delivering unprecedented sensitivity for atmospheric characterization in the subNeptune regime. We detect five vibrational bands of CH4 at 1.15 1.4 1.7 2.3 and 3.3mum 9.4sigma the signature of CO2 at 4.3mum 4.8sigma water vapor 2.5sigma and potential signatures of SO2 at 4.0mu mathrmm and CS2 at 4.6mumathrmm. Intriguingly we find an overall highly metalrich atmosphere with a mean molecular weight of 5.471.141.25. We infer an atmospheric metal mass fraction of 58128 and a CO of 0.470.190.16 indicating that approximately half the mass of the outer envelope is in highmolecularweight volatiles H2O CH4 CO CO2 rather than H2He. We introduce a subNeptune classification scheme and identify TOI270d as a miscibleenvelope subNeptune in which H2He is wellmixed with the highmolecularweight volatiles in a miscible supercritical metalrich envelope. For a fully miscible envelope we conclude that TOI270ds interior is 9043wt rockiron indicating that it formed as a rocky planet that accreted a few wt of H2He with the overall envelope metal content explained by magmaoceanenvelope reactions without the need for significant ice accretion. TOI270d may well be an archetype of the overall population of subNeptunes.

2024-03-01T00:00:00Z

['2024arXiv240303325B', 'arXiv:2403.03325', '10.48550/arXiv.2403.03325']

['Astrophysics - Earth and Planetary Astrophysics']

JWST Reveals CH4 CO2 and H2O in a Metalrich Miscible Atmosphere on a TwoEarthRadius Exoplanet

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2403.03325.pdf

{}

2024arXiv240903803S

We present OGRePy the official Python port of the popular Mathematica tensor calculus package OGRe ObjectOriented General Relativity a powerful yet userfriendly tool for advanced tensor calculations in mathematics and physics especially suitable for general relativity. The Python port uses the same robust and performanceoriented algorithms as the original package and retains its core design principles. However its truly objectoriented interface enabled by Python is more intuitive and flexible than the original Mathematica implementation. It utilizes SymPy for symbolic computations and Jupyter as a notebook interface. OGRePy allows calculating arbitrary tensor formulas using any combination of addition multiplication by scalar trace contraction partial derivative covariant derivative and permutation of indices. Transformations of the tensor components between different index configurations andor coordinate systems are performed seamlessly behind the scenes as needed eliminating user error due to combining incompatible representations and guaranteeing consistent results. In addition the package provides facilities for easily calculating various curvature tensors and geodesic equations in multiple representations. This paper presents the main features of the package in great detail including many examples of its use in the context of general relativity research.

2024-09-01T00:00:00Z

['arXiv:2409.03803', '10.48550/arXiv.2409.03803', '2024arXiv240903803S']

['General Relativity and Quantum Cosmology', 'Computer Science - Mathematical Software', 'Computer Science - Symbolic Computation', 'Mathematics - Differential Geometry', 'G.4', 'I.1', 'J.2']

OGRePy An ObjectOriented General Relativity Package for Python

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.03803.pdf

{'Barak Shoshany': 'bshoshany@brocku.ca | ORCID 0000-0003-2222-127X | https://baraksh.com/ \nDepartment of Physics, Brock University \n1812 Sir Isaac Brock Way, St. Catharines, Ontario, L2S 3A1, Canada \nSeptember 4, 2024', 'Abstract': 'We present OGRePy, the official Python port of the popular Mathematica tensor calculus package OGRe (Object-Oriented General Relativity) - a powerful, yet userfriendly, tool for advanced tensor calculations in mathematics and physics, especially suitable for general relativity. The Python port uses the same robust and performance-oriented algorithms as the original package, and retains its core design principles. However, its truly object-oriented interface, enabled by Python, is more intuitive and flexible than the original Mathematica implementation. It utilizes SymPy for symbolic computations and Jupyter as a notebook interface. OGRePy allows calculating arbitrary tensor formulas using any combination of addition, multiplication by scalar, trace, contraction, partial derivative, covariant derivative, and permutation of indices. Transformations of the tensor components between different index configurations and/or coordinate systems are performed seamlessly behind the scenes as needed, eliminating user error due to combining incompatible representations, and guaranteeing consistent results. In addition, the package provides facilities for easily calculating various curvature tensors and geodesic equations in multiple representations. This paper presents the main features of the package in great detail, including many examples of its use in the context of general relativity research.', 'Summary': 'OGRePy is a modern Python package for differential geometry and tensor calculus, designed to be both powerful and user-friendly. It can be used in a variety of contexts where tensor calculations are needed, in both mathematics and physics, but it is especially suitable for general relativity. \nTensors are abstract objects, which can be represented as multi-dimensional arrays once a choice of index configuration and coordinate system is made. OGRePy stays true to this definition, but \ntakes away the complexities that come with combining tensors in different representations. This is done using an object-oriented programming approach, taking advantage of principles such as encapsulation and class invariants to eliminate the possibility of user error. \nThe user initially defines each tensor in OGRePy using its explicit components in any single representation. Operations on this tensor are then done abstractly, without needing to specify which representation to use. Possible operations include addition of tensors, multiplication of tensor by scalar, trace, contraction, and partial and covariant derivatives. \nOGRePy will automatically choose which representation to use for each tensor based on how the tensors are combined. For example, if two tensors are added, then OGRePy will automatically use the same index configuration (upper and lower indices) for both. Similarly, if two tensors are contracted, then OGRePy will automatically ensure that the contracted indices are one upper (contravariant) and one lower (covariant). OGRePy will also automatically transform all tensors being operated on to the same coordinate system. \nTransformations between representations are done behind the scenes; all the user has to do is specify which metric to use for raising and lowering indices, and how to transform between the coordinate systems being used. This information only needs to be given once and for all when first defining the tensors and coordinate systems, and will be used automatically from that point on. \nThis also means that there is no room for user error. The user cannot mistakenly perform "illegal" operations such as . Instead, the user simply inputs the names of the tensors, the order (but not the configuration) of indices for each, and the operations to perform - and the correct combination will be automatically deduced. 2 A μν + B μλ C λν 2 A μν + B μ λ C λν \nOGRePy is a Python port of the popular Mathematica package OGRe [1], first released in February 2021, used by many general relativity researchers worldwide. The Python port uses the same robust and performance-oriented algorithms, and retains the package\'s core design principles. It was made to be as flexible and powerful as possible, while also being simple to learn and easy to use, and suitable for both experienced and novice researchers. OGRePy uses SymPy [2] to facilitate symbolic computations and Jupyter [3][4] as a notebook interface. \nThe Python port was specifically designed to mimic as much of the original Mathematica package\'s syntax as possible, while also greatly improving on that syntax in many ways due to the fact that Python, unlike Mathematica, is a truly object-oriented language. The documentation for both packages was also kept as similar in structure and scope as possible, with the same practical examples. This means that anyone who is familiar with the Mathematica version should easily be able to use the Python version, and vice versa.', 'Features': '- · Define coordinate systems and the transformation rules between them. The Jacobians are automatically calculated. Tensor components are then transformed automatically between coordinates behind the scenes as needed.\n- · Each tensor is associated with a specific metric. Tensor components are then transformed automatically between different index configurations, raising and lowering indices behind the scenes as needed.\n- · Display any tensor in any index configuration and coordinate system, either in vector/matrix form or as a list of all unique non-zero elements. Metrics can also be displayed as a line element.\n- · Automatically simplify tensor components, optionally with a user-defined simplification function.\n- · Easily calculate arbitrary tensor formulas using any combination of addition, multiplication by scalar, trace, contraction, partial derivative, covariant derivative, and permutation of indices.\n- · Built-in methods for calculating the Christoffel symbols (Levi-Civita connection), Riemann tensor, Ricci tensor and scalar, Einstein tensor, Kretschmann scalar, curve Lagrangian, and volume element from a metric.\n- · Calculate the geodesic equations in terms of an affine curve parameter, in two different ways: from the Christoffel symbols or from the curve Lagrangian. For spacetime metrics, the geodesic equations can be calculated in terms of the time coordinate.\n- · Easily keep track of all tensors created in a notebook session, including the relations between them - for example, see which metrics were created and which tensors are associated with each metric.\n- · Export tensor components in TeX or Mathematica format.\n- · Designed with speed and performance in mind, using optimized algorithms developed specifically for this package.\n- · Clear and detailed documentation, with many examples, in both Markdown and Jupyter notebook format.\n- · Open source. The code is extensively documented; please feel free to fork and modify it as you see fit.\n- · Under continuous and active development. Bug reports and feature requests are welcome, and should be made via GitHub issues.', 'The object-oriented design philosophy': "Object-oriented programming refers to a paradigm where a program's code is organized around objects. An object belongs to a user-defined type, called a class . The class defines the data that the object stores, as well as methods or member functions that read or manipulate that data. One of the fundamental principles of object-oriented programming is encapsulation , which means that the user may only access an object's data using the methods defined by the class, and is unable to access the object's data directly. \nImportantly, encapsulation allows for the preservation of class invariants . An invariant is a condition of validity that can always be assumed to be satisfied by the data stored in each object. If the methods make sure to preserve the invariant whenever they store or manipulate the data, and the user is prevented from changing the data manually and thus potentially violating the invariant, then the implementation of the class can be greatly simplified, and performance can be \nimproved, because the class will not need to verify that the data is valid every time it performs an operation. \nThe main idea behind OGRePy is to simplify the use of tensors by encoding all the information about a tensor in a single, self-contained object. As I mentioned above, a tensor is an abstract object. One can find components which represent this abstract entity in a particular coordinate system and index configuration, but the tensor is not its components. In OGRePy, a tensor object is initially defined (or constructed ) by providing the components of the tensor in a particular representation - but once this is done, the user does not need to worry about coordinates or indices anymore, or even remember which coordinates and indices were initially used. The abstract tensor object will automatically transform the initial data to a different coordinate system or index configuration as needed, based on the context in which it was used. \nAs a tensor object holds the components of the same tensor in many different representations, the most important class invariant is the assumption that the different components indeed represent the same tensor. This is achieved using encapsulation; the object's data can only be modified by private methods that preserve the invariant, and thus the user cannot accidentally cause a violation of the invariant by assigning components to one representation that are not related to the components of all other representations by the appropriate coordinate and/or index transformation. \nSince Mathematica is not an object-oriented language, the original OGRe package merely simulated classes and objects using associative arrays, resulting in a somewhat awkward syntax. Python, on the other hand, is an inherently object-oriented language, and the Python package takes full advantage of that. Tensors are objects, and the various tensor operations are done directly on these objects using methods and overloaded operators. Class invariants and encapsulation guarantee that the different representations of the tensor objects are always consistent, and the correct representation is chosen on demand for each calculation using intelligent algorithms.", 'Global installation': 'To install OGRePy from PyPI using pip , simply run the following command in the terminal: \npip install OGRePy \nThe current version of OGRePy officially supports only Python v3.12 and above . It may also work with older versions of Python 3, but this is not guaranteed, as development and testing was only done with the indicated Python version. If your global Python installation is an older version, and you cannot upgrade it, consider using pyenv or pyenv-win to install multiple Python versions in parallel, or use a portable local installation to run OGRePy. \nInstalling OGRePy using pip will also automatically install its dependent packages, ipykernel and sympy, if they are not already installed. The current version of OGRePy officially supports only ipykernel v6.29 and above and sympy v1.13 and above , so if you are still using older versions, you should upgrade these packages using the command pip install --upgrade ipykernel sympy .', 'Installing in a virtual environment': 'Advanced users may wish to install OGRePy inside a Python virtual environment in order to avoid potential dependency conflicts with other packages. To do this, first open the directory where you would like to store your new virtual environment in the terminal, and run: \n- · python -m venv .OGRePy-env --upgrade-deps on Windows,\n- · python3 -m venv .OGRePy-env --upgrade-deps on WSL/Linux/macOS. \nThis will create a virtual environment under the .OGRePy-env subdirectory. The --upgradedeps flag automatically upgrades pip to the latest version. To activate the virtual environment, run: \n- · .OGRePy-env\\Scripts\\activate.bat on Windows (Command Prompt),\n- · & .OGRePy-env\\Scripts\\Activate.ps1 on Windows (PowerShell),\n- · source .OGRePy-env/bin/activate on WSL/Linux/macOS. \nIf this worked correctly, you will see the text (.OGRePy-env) at the beginning of the terminal prompt. Now you can install OGRePy using pip as above. To deactivate the virtual environment, simply run the command deactivate in the terminal.', 'Creating a Jupyter notebook': 'OGRePy is designed to run within a Jupyter notebook. It is also possible to run it from within a Python script, usually for automation purposes, but Jupyter is required for interactivity and for displaying tensors and their components as rendered TeX equations. \nOGRePy supports two Jupyter notebook interfaces: \n- · Visual Studio Code: This is the officially recommended way to use OGRePy, due to helpful features such as IntelliSense, tooltips, and type checking. Download and install from the official website. Run VS Code, then create a new file with the .ipynb extension and open it, or press F1 to open the Command Pallette and choose the option "Create: New Jupyter Notebook". This will prompt you to automatically install the required VS Code extensions and Python packages if they are not already installed.\n- · JupyterLab: Install with pip install jupyterlab . Run by executing jupyter-lab in the terminal, and then create a new notebook in the web browser. Please note that JupyterLab is not officially supported, as development and testing was only done with VS Code, although I have verified that the package does work in JupyterLab. \nIf you are running OGRePy in a virtual environment: \n- · With Visual Studio Code, open the folder where you create the virtual environment, press F1 to open the Command Pallette, choose the option "Python: Select Interpreter", and select the .OGRePy-env environment. The interpreter can also be selected for individual Jupyter notebooks in VS Code using the "Select Kernel" button at the top right of the notebook.\n- · With JupyterLab, first activate the virtual environment in the terminal as explained above, and then run jupyter-lab from the same terminal.', 'Importing the package': 'To load OGRePy, type the following code in a Jupyter notebook cell and execute it using Shift+Enter: \n```\nimport OGRePy as T In [ ]:\n``` \nOGRePy: An Object-Oriented General Relativity Package for Python \nBy Barak Shoshany (baraksh@gmail.com) (baraksh.com) v1.0.1 (2024-09-04) GitHub repository: https://github.com/bshoshany/OGRePy \nAll of OGRePy\'s functions are now accessible via the T namespace. While it is not common practice in Python to import packages as single letters, OGRePy uses this convention because in the original Mathematica version of OGRe, all module names started with a capital T (which stands for "Tensor"). However, you can change that to another namespace if you prefer, for example import OGRePy as gr . \nIf desired, the welcome message can be disabled by defining OGREPY\\_DISABLE\\_WELCOME = True in the notebook before importing the package. If you changed your mind later and you want to see the welcome message (for example, if you want a link to the GitHub repository), use the welcome() function: \n```\nT . welcome() In [ ]:\n``` \nOGRePy: An Object-Oriented General Relativity Package for Python By Barak Shoshany (baraksh@gmail.com) (baraksh.com) v1.0.1 (2024-09-04) GitHub repository: https://github.com/bshoshany/OGRePy', 'Getting help': 'One of the reasons I recommend Visual Studio Code as the preferred notebook interface for this package is the IntelliSense feature, which displays a helpful popup with suggestions and information about various language components. To test this feature, once OGRePy is loaded in the notebook, create a new code cell and start typing T. - once you write the dot character, you \nwill see a popup menu listing all the functions contained in the T namespace. \nBrowse the menu using the arrow keys. There will be an additional popup next to this menu with the documentation for each function. If you do not see the documentation, press Ctrl+Space. You can also start typing to filter the options in the menu. For example, if you type w , the welcome() function will be selected, and you will see the documentation for that function. In the same way, you can view the documentation and usage instructions for all OGRePy functions. \nPress Tab to complete the code and write down the full function welcome() . Once the code is written, the popup will disappear, but it will reappear again after you write ( to display the parameters that should go into the parentheses. You can also hover with the mouse over any function to read its documentation. \nIf you are using JupyterLab instead of VS Code, the popups will not be displayed automatically by default, but you can press Ctrl+, to go to the settings, then click on "Code Completion" and check "Show the documentation panel" and "Enable autocompletion". (However, note that the documentation will not be formatted as nicely on JupyterLab.) \nYou can also view the documentation for a particular OGRePy function using the function doc() . For example: \nT . doc(T . welcome) welcome() -> None Print the welcome message. doc() itself also has documentation: T . doc(T . doc) doc(obj: Callable[..., Any] | type) -> None Print the documentation for an object as a Markdown-formatted notebook cell. If the object is a class, print the documentation for its constructor. Parameters: · obj : The object to print the documentation for. Exceptions: · OGRePyError : If no documentation exists. In [ ]: In [ ]: \nIn [ ]:', 'Defining coordinates': 'To define tensors, we first need to define the manifold on which they reside. Since we are focusing on general relativity, we will use 4-dimensional spacetime manifolds in the following examples, but this package works equally well with manifolds that are purely spatial and/or have a different number of dimensions. \nThe first step is to define the coordinate system. We can represent a coordinate system as a vector (or a tensor of rank 1) defining a point in space(time). In OGRePy, coordinates are represented as objects of the class Coordinates . Therefore, defining a coordinate system is a simple matter of constructing a new Coordinates object. The constructor for this class is defined as follows: x μ \nT \n. \ndoc(T \n. \nCoordinates) \nCoordinates(*components: s.Symbol | str) -> None \nConstruct a new coordinate object.', 'Parameters:': '- · components : One or more strings or SymPy Symbol objects specifying the coordinates. Strings should be in the same format as the argument to SymPy\'s symbols() , e.g. "t x y z" , and it is possible to enter just one string for all the coordinates. \nFor example, let us create an object for the Cartesian spacetime coordinates . First we will need some SymPy Symbol objects to represent the individual coordinates , , , and . Conveniently, OGRePy contains a module, OGRePy.abc , which contains SymPy symbols for all English and Greek letters, both lowercase and uppercase. Note that the Greek letter lambda (lowercase , uppercase ) is accessed via the symbols lamda and Lamda respectively, since lambda is a reserved keyword in Python. ( t , x , y , z ) t x y z λ Λ \nFor users familiar with SymPy: OGRePy.abc is similar to sympy.abc , except that OGRePy.abc explicitly assumes that all symbols are real, and also contains uppercase Greek letters. If complex symbols are desired, they should be imported from sympy.abc or created directly via sympy.Symbol or sympy.symbols() instead. \nWe import the symbols as follows: \n```\nfrom OGRePy.abc import t, x, y, z\n``` \nNow we have direct access to the symbols t , x , y , and z in our notebook. Let us use them to \nconstruct our Cartesian coordinate system: \n```\nCartesian = T . In [ ]:\n``` \n```\nCoordinates(t, x, y, z)\n``` \nHere is a breakdown of the code: \n- · Cartesian is the name of the new object we are creating.\n- · T is the namespace we chose for OGRePy when we imported it via import OGRePy as T .\n- · Coordinates is the name of the class we want to construct an instance of. This class represents a coordinate system in OGRePy.\n- · Coordinates() is the constructor , that is, the function that creates a new Coordinates object representing a particular coordinate system.\n- · We can pass any number of arguments to the constructor. Usually these will be SymPy symbols representing the coordinates (but it is also possible to pass strings, which will be converted to symbols automatically).\n- · t , x , y , and z are the symbols we exported above. \nWe can similarly define the spherical spacetime coordinates : ( t , r , θ , ϕ ) \n```\nfrom OGRePy.abc import phi, theta r = T . sym("r", nonnegative =True ) Spherical = T . Coordinates(t, r, theta, phi) In [ ]:\n``` \nNote that Greek letters are imported using the full name of the letter: theta stands for . Similarly, Theta will be the uppercase . One thing you should be aware of is that the letters and are imported as lamda and Lamda respectively, because lambda (with a b ) is a reserved keyword in Python. θ Θ λ Λ \nAnother thing to note here is that we defined the coordinate manually as a SymPy Symbol object using OGRePy\'s sym() function instead of importing it from OGRePy.abc . The reason for defining separately this way is that we get more control over the properties of this coordinate. As mentioned above, any symbol imported from OGRePy.abc is automatically assumed to be real. However, for , we also want to indicate that it is a non-negative symbol. This signals to SymPy to treat as non-negative when doing calculations or performing simplifications. r r r r \nTo illustrate this point, consider that , defined above using from OGRePy.abc import t , is a real coordinate that can be positive, negative, or zero. Therefore, when we try to simplify , we get the absolute value of : t √ t 2 t \n```\nT . s . simplify(T . s . sqrt(t ** 2)) In [ ]: Out[ ]: | t |\n``` \nOn the other hand, when we do the same to , which is designated as non-negative, we simply get back, without an absolute value: r r \n```\nT . s . simplify(T . s . sqrt(r ** 2)) In [ ]: Out[ ]: r\n``` \nIn these examples, note that SymPy is automatically imported into the OGRePy namespace as s , which means we can access the entire SymPy namespace as T.s . This is done purely for convenience, so you don\'t have to import SymPy to the notebook separately. However, you could also import sympy directly if you prefer. Because SymPy is available as T.s , we could access the SymPy simplify() function directly via T.s.simplify() . \nOGRePy offers two functions that can be used to create your own symbols: sym() , which is the preferred alternative to calling SymPy\'s Symbol() constructor, and syms() , which is the preferred alternative to calling SymPy\'s symbols() function. The main differences between OGRePy\'s sym() and syms() and SymPy\'s Symbol() and symbols() are: \n- 1. OGRePy\'s functions always add the assumption that the symbols are real, which helps with simplification.\n- 2. OGRePy\'s functions always convert strings to TeX codes. This is important, because in SymPy, Symbol("mu") != Symbol(r"\\mu") , even though they are both displayed using the same symbol. On the other hand, in OGRePy, sym("mu") == sym(r"\\mu") , which prevent errors.', 'Error checking': 'OGRePy contains robust error checking. If you call the constructor with invalid input, the construction will fail and you will get an error message telling you what to fix. For example, if you try typing T.Coordinates(42) you will get the following friendly error message: \n- The components must be either a SymPy Array object or a list. The object 42 is of type int . \nIf you are an advanced user who prefers to see the full traceback and/or catch the exceptions and handle them on your own, you can set T.options.friendly\\_errors = False to turn off the friendly error messages and raise exceptions instead. Set it back to True to re-enable the friendly error messages.', 'Defining metrics': 'To finish defining a (Riemannian or pseudo-Riemannian) manifold, we need to define its metric tensor. Like any other tensor in OGRePy, the metric tensor is an abstract tensor that has multiple representations. We "jump start" the tensor by providing its components in one particular representation, and all the other representations will be calculated automatically. \nIn the case of a metric tensor, the defining representation must always be the one with two indices down: . However, it can be given in any coordinate system. In OGRePy, metrics are represented as objects of the class Metric . Therefore, as with coordinates, defining a metric is a simple matter of constructing a new Metric object. The constructor for this class is defined as follows: g μν \n```\nT . doc(T . In [ ]:\n``` \n- Metric) Metric(*, coords: Coordinates, components: list[Any] | s.NDimArray | s.Matrix, symbol: str | s.Symbol = g) -> None Construct a new metric object. Parameters: · coords : An OGRePy Coordinates object specifying the coordinate system of the representation of the initialization components. Will also be designated the default coordinate system of the metric. · components : The components with which to initialize the metric. Can be a list, a SymPy Array object, or a SymPy Matrix object. · symbol (optional): A string or a SymPy Symbol object designating the symbol to be used when displaying the metric. The string can include any TeX symbols, e.g. r" T " (note the r in front of the string, indicating that the \\ in the string is not an escape character). Exceptions: · OGRePyError : If the metric components are not an invertible symmetric matrix. \nLet us create a tensor object for the Minkowski metric , specifying the components in Cartesian coordinates: \n```\nMinkowski = T . Metric( coords = Cartesian, components = T . diag( -1, 1, 1, 1), symbol = "eta", ) In [ ]:\n``` \nTo define the components we used the OGRePy diag() function, which generates a diagonal matrix (a SymPy Matrix object) with the given components on the diagonal. OGRePy\'s diag() is a convenient shorthand for SymPy\'s Matrix.diag() . \nFor the symbol, we used the string "eta" , which will be displayed as the Greek letter . Alternatively, we could have used any TeX string, such as r"\\eta" . (Note the r in front of the string, indicating that it is a "raw" string literal, so the \\ in the string is treated as an actual \\ and not an escape character.) Internally, the string "eta" is actually converted to r"\\eta" . The η \nsymbol argument also accepts SymPy Symbol objects, in which case it extracts the TeX code from the object, so we could have also used from OGRePy.abc import eta and then entered eta as the symbol, but that is more cumbersome. \nSimilarly, let us define the Schwarzschild metric , this time specifying the components in spherical coordinates: \n```\nfrom OGRePy.abc import M Schwarzschild = T . Metric( coords = Spherical, components = T . diag( -(1 -2 * M / r), 1 / (1 -2 * M / r), r ** 2, r ** 2 * T . s . sin(theta) ** 2, ), ) In [ ]:\n``` \nHere we imported the symbol M to use as the mass. Be careful not to write something like 2M instead of 2 * M . While 2M makes sense mathematically, it is not a legal Python expression. Note that we did not specify a symbol, so the symbol will be used by default. g', 'Displaying tensors': 'In OGRePy, the term tensor object refers to any object of the Tensor class or its derived classes, which include Metric (but not Coordinates , which is not a tensor, just a list of symbols) Every tensor object in OGRePy has a method called show() , which shows the symbol, indices, coordinates, and components in those indices and coordinates, in vector or matrix form when applicable. Let us try it for the two metrics we created: \n```\nMinkowski . show() In [ ]:\n``` \n```\nSchwarzschild . show() η μν ∣ ∣ ∣ ( t , x , y , z ) = ⎛ ⎜ ⎜ ⎜ ⎝ -1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ⎞ ⎟ ⎟ ⎟ ⎠ In [ ]: g μν ∣ ∣ ∣ ( t , r , θ , ϕ ) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ -1 0 0 0 0 0 0 0 0 r 2 0 0 0 0 r 2 sin 2 ( θ ) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ 2 M r 1 -+1 2 M r\n``` \nIn fact, calling the show() method explicitly is not necessary. If the output of a notebook cell is a tensor object, the output of the show() method will be displayed automatically: \nMinkowski In [ ]: Out[ ]: \nη μν ∣ ∣ ∣ ( t , x , y , z ) = ⎛ ⎜ ⎜ ⎜ ⎝ -1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ⎞ ⎟ ⎟ ⎟ ⎠ \nA coordinate system is not a tensor, but it does have a show() method as well, and it is also executed automatically if it\'s the output of a notebook cell: \nCartesian . show() In [ ]: \n( t x y z ) \nSpherical In [ ]: \nOut[ ]: \n( t r θ ϕ ) \nThe other method available for displaying the contents of tensors is list() , which lists all of the unique (up to sign), non-zero components of the tensor. It is usually the best option for higherrank tensors, which cannot be displayed in vector or matrix form, such as the Christoffel symbols or Riemann tensor (see below). For example, let us list the components of the Minkowski metric: \nMinkowski . list() In [ ]: \nη tt = -η xx = -η yy = -η zz = -1 \nThere is a convenient shortcut for calling list() : simply use the ~ (invert) operator in front of the tensor. For example: \n~ Schwarzschild In [ ]: \ng tt = -2 M \ng \n1 g rr = g θθ = r 2 ϕϕ = r 2 sin 2 ( θ ) r 1 -+1 2 M r \nIn [ ]: \nIn [ ]: \ncomponents in this manner. \nIf, as in the examples above, no additional arguments are given to show() and list() , they display the tensors in their default indices and default coordinates, which are the ones first used to define the tensor (unless you change them later). So, for example, the default indices of the Minkowski metric are two lower indices, and its default coordinates are Cartesian. We will show later how to change these defaults, and how to display any tensor in any index configuration and coordinate system. Note that if a tensor object is displayed automatically as the output of a cell, or using the ~ shortcut for list() , it will always be displayed in its default indices and coordinates. \nA good practice when using OGRePy is to set up the notebook so that the result of the last assignment in the cell is automatically printed out. This will save us the trouble of writing an extra line every time we want to print out tensors we assign to variables. This is achieved by executing the following command: \n```\nfrom IPython.core.interactiveshell import InteractiveShell\n``` \nInteractiveShell . ast\\_node\\_interactivity = \n```\n"last\\_expr\\_or\\_assign"\n```', 'Changing the output style': 'The options object of the OGRePy package is used to set various options, which will then be respected by all functions and classes in the package. We already saw above that we can use it to turn off the friendly error message by setting T.options.friendly\\_errors = False . \nTo control the style of the output, you can change the property T.options.css\\_style to any string of your choice. The default is just an empty string, but we can change this to any CSS style we want. For example: \nT \n. \noptions \n. \ncss\\_style \n= \n"background-color: #000; color: #fff; font-size: 20px; padding: 5px" \n~ \nSchwarzschild \nTo reset the style to the default value, we simply "delete" the property using del : \n<!-- image --> \n```\ndel T . options . css\\_style Now the style is back to normal: ~ Schwarzschild In [ ]: In [ ]:\n``` \nThis is common to all properties of options ; the del operator does not delete the property, it simply resets it to the default value. Line and volume elements In the case of metrics, we can also display them as a line element using the method g tt = -1 g rr = g θθ = r 2 g ϕϕ = r 2 sin 2 ( θ ) 2 M r 1 -+1 2 M r \nline\\_element() . For example, here are the line elements for our two metrics: \n```\nMinkowski . line\\_element() Schwarzschild . line\\_element() In [ ]: Out[ ]: -d t 2 +d x 2 +d y 2 +d z 2 In [ ]: Out[ ]: d ϕ 2 r 2 sin 2 ( θ ) + d θ 2 r 2 + +d t 2 ( -1 ) d r 2 -+1 2 M 2 M r\n``` \n```\nNote that these are standard SymPy expressions, so they can be manipulated like any other expressions, including operations such as simplifying or factoring. As an example of a more r\n``` \ninteresting (non-diagonal) line element, consider the Alcubierre warp drive metric : \n```\nv\\_t = T . func("v")(t) f\\_t\\_x\\_y\\_z = T . func("f")(t, x, y, z) Alcubierre = T . Metric( coords = Cartesian, components = [ [ -1 + f\\_t\\_x\\_y\\_z ** 2 * v\\_t ** 2, 0, 0, -f\\_t\\_x\\_y\\_z * v\\_t], [0, 1, 0, 0], [0, 0, 1, 0], [ -f\\_t\\_x\\_y\\_z * v\\_t, 0, 0, 1], ], )\n``` \n```\nIn [ ]:\n``` \ng μν ∣ ∣ ∣ ( t , x , y , z ) = ⎛ ⎜ ⎜ ⎜ ⎝ f 2 v 2 -1 0 0 -fv 0 1 0 0 0 0 1 0 -fv 0 0 1 ⎞ ⎟ ⎟ ⎟ ⎠ \nHere we used OGRePy\'s func() function, which is a wrapper around SymPy\'s Function class which also defines the function to be real. Note that the metric was automatically printed in matrix form, since we configured the notebook to print out the result of the last assignment. Here is a list of its non-zero components: \n```\n~ Alcubierre In [ ]:\n``` \ng tt = f 2 v 2 -1 g tz = g zt = -fv g xx = g yy = g zz = 1 \nis a form function which is equal to 1 inside a "warp bubble" of finite radius and 0 outside it, and is the velocity of the bubble, which can be faster than the speed of light ( ). Note that for and we used a new type of object: a SymPy Function object. This represents a function of the elements given as the arguments to the constructor, so is a function of while is a function of all of the coordinates. f v v > 1 v f v t f \nIt is easy to see that the metric is flat where , that is, outside the bubble. Its line element is: f = 0 \nAlcubierre . line\\_element() In [ ]: \nOut[ ]: \nd \nt \n2 \n( \nf \n2 \n( \nt \n, \nx \n, \ny \n, \nz \n) \nv \n2 \n( \nt \n) \n- \n1 \n) \n- \n2d \nt \nd \nzf \n( \nt \n, \nx \n, \ny \n, \nz \n) \nv \n( \nt \n) + d \nx \n2 \n+d \ny \n2 \n+d \nWe can simplify it as follows. First, we expand the parentheses: \nAlcubierre . line\\_element() . expand() In [ ]: \nOut[ ]: d t 2 f 2 ( t , x , y , z ) v 2 ( t ) -d t 2 -2d t d zf ( t , x , y , z ) v ( t ) + d x 2 +d y 2 +d z 2 \nUsing the args method, we can split this expansion into individual terms (we put the result inside a SymPy Array so the terms will be properly displayed as SymPy expressions in the notebook): \nargs = Alcubierre . line\\_element() . expand() . args T . s . Array(args) In [ ]: \nOut[ ]: [ d x 2 d y 2 d z 2 -d t 2 d t 2 f 2 ( t , x , y , z ) v 2 ( t ) -2d t d zf ( t , x , y , z ) v ( t ) ] \nNow we can factorize the third, fifth, and sixth terms together, then add the rest: (recall that indices start from zero!) \nz \n2 \nargs[0] + args[1] + args[3] + T . s . factor(args[2] + args[4] + args[5]) In this form, it is immediately clear that the metric is flat outside the warp bubble (where is ), and inside the warp bubble (when is ) it is a flat metric translated by an amount in the direction. Another thing we can do with a metric is calculate its volume elements squared, which is simply the determinant of the metric, using the method volume\\_element\\_squared() . For example: Minkowski . volume\\_element\\_squared() Schwarzschild . volume\\_element\\_squared() Alcubierre . volume\\_element\\_squared() As with the line elements, these are SymPy expressions, so they can be modified just like any other expression. Therefore, to calculate the volume element itself, we can just take the square root (adding a minus sign if the metric is Lorentzian): T . s . simplify(T . s . sqrt( -Schwarzschild . volume\\_element\\_squared())) In [ ]: Out[ ]: -d t 2 +d x 2 +d y 2 +(d tf ( t , x , y , z ) v ( t ) -d z ) 2 f 0 f 1 v ( t ) d t z In [ ]: Out[ ]: -1 In [ ]: Out[ ]: -r 4 sin 2 ( θ ) In [ ]: Out[ ]: -1 In [ ]: Out[ ]: r 2 |sin ( θ )|', 'Choosing index letters': 'By default, the show() method uses Greek letters for the indices, in a specific pre-determined order. The letters can be changed by setting the property T.options.index\\_letters to a list of symbols. The default letters are: \n```\nT . options . index\\_letters In [ ]:\n``` \n```\n[\'\\\\mu\', \'\\\\nu\', \'\\\\rho\', \'\\\\sigma\', \'\\\\kappa\', \'\\\\lambda\', \'\\\\alpha\', \'\\\\beta\', \'\\\\gamma\', \'\\\\delta\', \'\\\\epsilon\', \'\\\\zeta\', \'\\\\epsilon\', \'\\\\theta\', \'\\\\iota\', \'\\\\xi\', \'\\\\pi\', \'\\\\tau\', \'\\\\phi\', \'\\\\chi\', \'\\\\psi\', \'\\\\omega\'] Out[ ]:\n``` \nAs you can see, they are given as strings containing TeX symbols. We can display these symbols more nicely in the notebook using the IPython package: \n```\nfrom IPython.display import Math Math("," . join(T . options . index\\_letters)) In [ ]: Out[ ]: μ , ν , ρ , σ , κ , λ , α , β , γ , δ , ϵ , ζ , ϵ , θ , ι , ξ , π , τ , ϕ , χ , ψ , ω\n``` \nThis means that the letter will be used for the first index, for the second, and so on. However, sometimes we want to use different letters. T.options.index\\_letters can accept a list of TeX symbols, SymPy symbols, and/or strings in the same format as SymPy\'s symbols() function, that is, a space- or comma-separated list of one or more letters or TeX codes - or a mix and match of all of the above, as long as it\'s inside a list. Ranges of letters can be indicated using a colon, so for example, here is how to change the indices to lowercase English letters in alphabetical order: μ ν \n```\nT . options . index\\_letters = ["a:z"] show() will now use these letters - in this particular order - when displaying tensors: Minkowski In [ ]: In [ ]: Out[ ]:\n``` \nη ab ∣ ∣ ∣ ( t , x , y , z ) = ⎜ ⎜ ⎜ 0 1 0 0 0 0 1 0 \n```\n⎛ ⎝ -1 0 0 0 0 0 0 1 ⎞ ⎟ ⎟ ⎟ ⎠\n``` \n```\nAs always with the object, to reset the property to its default value,\n``` \noptions index\\_letters we "delete" it using del : \n```\ndel T . options . index\\_letters In [ ]:\n``` \nNote that list() always uses the coordinate symbols themselves for the indices (e.g. , , etc.), so it is not affected by T.options.index\\_letters . η tt η xx', 'Creating tensors in a given manifold': 'Any tensors other than coordinates and metrics are created as objects of the OGRePy class Tensor . The constructor for this class is defined as follows: \n```\nT . doc(T . In [ ]:\n``` \n```\nTensor) Tensor(*, metric: Metric, indices: IndexConfiguration, coords: Coordinates, components: list[Any] | s.NDimArray | s.Matrix, symbol: str | s.Symbol = \\\\square, simplify: bool = False) -> None Construct a new tensor object. Parameters:\n``` \n- · metric : An OGRePy Metric object specifying the metric which will be used to raise and lower indices for this tensor.\n- · indices : A tuple of integers specifying the index configuration of the representation of the initialization components. Each integer in the tuple can be either +1 for an upper index or -1 for a lower index. Will also be designated the default index configuration of the tensor.\n- · coords : An OGRePy Coordinates object specifying the coordinate system of the representation of the initialization components. Will also be designated the default coordinate system of the tensor.\n- · components : The components with which to initialize the tensor. Can be a list, a SymPy Array object, or (for rank 2 tensors) a SymPy Matrix object.\n- · symbol (optional): A string or a SymPy Symbol object designating the symbol to be used when displaying the tensor. The string can include any TeX symbols, e.g. r" T " (note the r in front of the string, indicating that the \\ in the string is not an escape character). If omitted, the placeholder ( r"\\square" ) will be used. □\n- · simplify (optional): Whether to simplify ( True ) or not simplify ( False , the default) the components before storing them. \nIn OGRePy, all tensor objects must have an associated metric - except coordinate objects, and the metric tensors themselves. This is because OGRePy automatically raises and lowers indices as \nappropriate for various operations such as adding and contracting tensors, and it cannot do so without knowing which metric to use. Even scalars, which have no indices, should still be associated to a specific metric - since they can multiply other tensors, and you cannot multiply tensors from different manifolds together. \nThe index configuration of the tensor is a tuple. The number of indices is the rank of the tensor. Each element in the tuple corresponds to one index, with +1 specifying an upper index and -1 specifying a lower index. For example, (-1, -1) corresponds to a tensor such as the metric , which has two lower indices, while (1, -1, -1, -1) corresponds to a tensor such as the Riemann tensor , which has one upper index followed by three lower indices. g μν R ρ σμν \nThe components of the tensor can be given in several equivalent forms: a list, a SymPy Array object, or (for rank 2 tensors) a SymPy Matrix object. Usually, a list is the simplest option if we are specifying the components explicitly. (For advanced users: The components can, more generally, be any SymPy NDimArray , including mutable and/or sparse arrays, but OGRePy always stores the components as an immutable dense array, no matter what form the input was originally in.) \nThe components are the representation of the new tensor in the given index configuration and coordinate system. If a coordinate system is not specified, the default coordinate system of the associated metric will be used - but it is recommended to always specify the coordinate system explicitly, to avoid accidentally defining the tensor with the wrong components. The components will be automatically converted to different indices or coordinates later as needed, as we will demonstrate below. \nTo create a scalar , or a tensor of rank 0 (with no indices), we must input an empty tuple () for the indices, and a list with exactly one item for the components. Note that a "bare" expression, not inside a list, will not work. For example, let us define the Kretschmann scalar in the Schwarzschild spacetime (below we will show how to calculate it directly from the metric): \n```\nSchwarzschildKretschmann = T . Tensor( metric = Schwarzschild, coords = Spherical, indices = (), components = [(48 * M ** 2) / r ** 6], symbol = "K", ) In [ ]:\n``` \nOut[ ]: \nK ∣ ∣ ∣ ( t , r , θ , ϕ ) = 48 M 2 r 6 \nSimilarly, we can create a vector , or a tensor of rank 1 (with one index). For example, let us create a vector for the 4-velocity of a particle moving at 3-velocity along the direction in Minkowski space. Since the 4-velocity has an upper index by definition, we make sure to define the components in the representation of the tensor with an upper index by specifying the index v x \n- configuration as (1,) : from OGRePy.abc import v FourVelocity = T . Tensor( metric = Minkowski, coords = Cartesian, indices = (1,), components = T . s . Array([1, v, 0, 0]) / T . s . sqrt(1 -v ** 2), ) There are a few important things to note here: 1. In Python, a tuple of one element must be specified with a comma, i.e. (1,) , because (1) would be interpreted as an integer. 2. We used a SymPy Array object to define the components since this allowed us to divide each component by the square root . This would not be possible with a normal Python list. 3. Since we did not specify a symbol for this tensor, its symbol is just a placeholder . We will give it a proper symbol below. Finally, as an example of a tensor of rank 2 (with two indices), let us define the stress-energy tensor for a perfect fluid, using its matrix representation with two upper indices by specifying the index configuration (1, 1) : from OGRePy.abc import p, rho PerfectFluid = T . Tensor( metric = Minkowski, coords = Cartesian, indices = (1, 1), components = T . diag(rho, p, p, p), symbol = "T", ) In [ ]: Out[ ]: □ μ ∣ ∣ ∣ ( t , x , y , z ) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ 1 √ 1 -v 2 v √ 1 -v 2 √ 1 -v 2 □ T μν In [ ]: Out[ ]: T μν ∣ ∣ ∣ ( t , x , y , z ) = ⎛ ⎜ ⎜ ⎜ ⎝ ρ 0 0 0 0 p 0 0 0 0 p 0 0 0 0 p ⎞ ⎟ ⎟ ⎟ ⎠ \nmost often derived by operating on lower-rank tensors, rather than defined manually via their components. We will see an example of such a derivation when we derive the Christoffel symbols and Riemann tensor from the metric below.', "Changing a tensor's symbol": 'If we ever want to change the symbol used to display a tensor, we can simply change the property symbol to any string, TeX code, or SymPy Symbol . For example, let us give the symbol to the four-velocity: u \nFourVelocity . symbol = "u" In [ ]: \nNow, when we display the tensor using show() or list() , this is the symbol that will be used: \nFourVelocity In [ ]: \nOut[ ]: \nu μ ∣ ∣ ∣ ( t , x , y , z ) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ 1 √ 1 -v 2 v √ 1 -v 2', 'Raising and lowering indices': 'Raising and lowering indices is one of the most basic tensor operations. For example, if we have a vector represented with one upper index, , we can turn it into a covector, which is represented with one lower index, by contracting it with the metric: v ν \nv μ = g μν v ν . \nThis is called "lowering an index". Here and in the rest of this documentation, we will be using the Einstein summation convention , where the same index repeated exactly twice , once as an upper index and once as a lower index, implies summation over that index. In this case, the implied summation is over : ν ∈ 0, 1, 2, 3 \nv μ = 3 ∑ ν =0 g μν v ν = g μ 0 v 0 + g μ 1 v 1 + g μ 2 v 2 + g μ 3 v 3 . \nSuch a sum over an index is called a contraction, and it is a generalization of the inner product, as we will describe in more details below. Conversely, if we have a covector , we can raise its index by contracting it with the inverse metric: w μ \nw μ = g μν w ν . \nThis works the same for indices of higher-rank tensors. For example, if we have a tensor of rank 2 represented with two upper indices, , we can lower either one or both of its indices: T μλ \nT μ ν = g νλ T μλ , T μν = g μρ g νλ T ρλ . \nIn OGRePy, since tensor objects are abstract tensors , independent of any specific index configuration, there is no notion of raising or lowering the indices of a tensor object . Instead, one simply request to display the components of the tensor with the desired index configuration, without modifying the object itself. This works with both the show() and list() methods, by simply providing as an argument the list of indices in the format , as when we created a new tensor. (±1, ±1, . . . ) \nAs an example, let us use show() to display the vector FourVelocity with a lower index, that is, with index configuration (-1,) : \nFourVelocity . show(indices = ( -1,)) In [ ]: \nu μ ∣ ∣ ∣ ( t , x , y , z ) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ -0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ 1 √ 1 -v 2 v √ 1 -v 2 \nOGRePy automatically knows to use the Minkowski metric to lower the index, which means that a minus sign has been added to the first component, as expected. Similarly, here is PerfectFluid with just the second index lowered, this time displayed using list() : \nPerfectFluid . list(indices = (1, -1)) In [ ]: \nT t t = -ρ T x x = T y y = T z z = p \nThe components of the representation of the metric with two upper indices are the components of the inverse metric, since \ng μλ g λν = δ ν μ . \nTherefore, a quick way to show the components of the inverse metric is to display it with the index configuration (1, 1) : \ng μν ∣ ∣ ∣ ( t , r , θ , ϕ ) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ -0 0 0 0 0 0 0 0 0 0 0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ r -2 M + r -2 M + r r 1 r 2 1 r 2 sin 2 ( θ ) \nFor the same reason, the metric with one upper and one lower index is just the identity matrix: \nSchwarzschild . list(indices = (1, -1)) In [ ]: \ng t t = g r r = g θ θ = g ϕ ϕ = 1 \nAs explained above, if show() or list() are called without any arguments, the tensor is displayed in its default index configuration , which is the one first used to define the tensor. So the 4-velocity has one upper index by default, and the stress tensor has two upper indices by default, because that is how we initially defined them. However, the default indices can be changed by setting the property default\\_indices . For example, let us change the default indices of the perfect fluid stress tensor to two lower indices: \nPerfectFluid . default\\_indices = ( -1, -1) In [ ]: \nNow, when we display the tensor using show() or list() without any arguments, this is the index configuration that will be used: \nPerfectFluid In [ ]: \nOut[ ]: \nT μν ∣ ∣ ∣ ( t , x , y , z ) = ⎛ ⎜ ⎜ ⎜ ⎝ ρ 0 0 0 0 p 0 0 0 0 p 0 0 0 0 p ⎞ ⎟ ⎟ ⎟ ⎠', 'Coordinate transformations': 'The components of any tensor may be transformed from one coordinate system to another coordinate system using the following prescription: x μ x μ \' \n- · For every lower index , add a factor of (i.e. the derivative of the old coordinates with respect to the new, or the Jacobian ). μ ∂ x μ / ∂ x μ \'\n- · For every upper index , add a factor of (i.e. the derivative of the new coordinates with respect to the old, or the inverse of the Jacobian). μ ∂ x μ \' / ∂ x μ \nFor example, given a tensor with components in a coordinate system , we can transform to components in another coordinate system as follows: T αβ x μ T α \' β \' x μ \' \nIn [ ]: \nT α \' β \' ( x μ \' ) = / ∂ x α ∂ x α \' ∂ x β ∂ x β \' \nFor a general rank tensor with upper indices and lower indices , the transformation takes the form ( p , q ) p α 1 , …, α p q β 1 , …, β q \nT α \' 1 ⋯ α \' p β \' 1 ⋯ β \' q ( x μ \' ) = ( ⋯ )( ⋯ ) T α 1 ⋯ α p β 1 ⋯ β q ( x μ ) ∂ x α \' 1 ∂ x α 1 ∂ x α \' p ∂ x α p ∂ x β \' 1 ∂ x β 1 ∂ x β \' q ∂ x β q \nAs a mnemonic for this formula, recall that two indices may only be contracted if one of them is an upper index and the other is a lower index. If an index is in the denominator of a derivative, then its role is reversed (upper lower). Thus the old (non-primed) and new (primed) indices can only be in places that allow properly contracting the Jacobian or inverse Jacobian with the tensor. For example, is an upper index in and therefore must be contracted with a lower index. Thus, must be in the denominator, to lower its index and allow it to be contracted with the tensor. ↔ α 1 T ∂ x α 1 \nAs we saw above, OGRePy automatically knows how to raise or lower indices as needed using the appropriate metric. Similarly, any operation that requires transforming to another coordinate system will preform the transformation automatically behind the scenes. However, for this to happen, OGRePy needs to know the appropriate transformation rules. These are defined between the tensor objects representing the coordinates, which were created as Coordinates objects. The rules for transforming from a source coordinate system to a target coordinate system are stored within the tensor object representing the source. This is done using the method set\\_coord\\_transformation() . To illustrate, let us define transformations from Cartesian to Spherical and back: \n```\nCartesian . set\\_coord\\_transformation( target = Spherical, rules = { x: r * T . s . sin(theta) * T . s . cos(phi), y: r * T . s . sin(theta) * T . s . sin(phi), z: r * T . s . cos(theta), }, ) Spherical . set\\_coord\\_transformation( target = Cartesian, rules = { r: T . s . sqrt(x ** 2 + y ** 2 + z ** 2), theta: T . s . acos(z / T . s . sqrt(x ** 2 + y ** 2 + z ** 2)), phi: T . s . atan2(y, x), }, )\n``` \nAs you can see, the rules are supplied as a dictionary specifying the transformation from each source coordinate to the target coordinates. Note that we did not have to input a rule for t , since it stays the same in both cases; the transformation is in the spatial coordinates only. \nNow OGRePy knows how to convert back and forth between these two coordinate systems - and this will happen automatically whenever required. We just needed to provide these rules once and for all, and any tensor initially defined in one coordinate system can now be automatically converted to the other. \nAs in the case of raising and lowering indices, displaying a tensor in a different coordinate system is a simple matter of calling the methods show() or list() with an additional argument specifying the coordinate system to use. For example, let us show the Minkowski metric in spherical coordinates: \nMinkowski . show(coords = Spherical) In [ ]: \nη μν ∣ ∣ ∣ ( t , r , θ , ϕ ) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ -1 0 0 0 0 1 0 0 0 0 r 2 0 0 0 0 r 2 sin 2 ( θ ) ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ \nWe can also ask to see a tensor in a specific index configuration and a specific coordinate system: \nPerfectFluid . show(coords = Spherical, indices = In [ ]: \n(1, 1)) \nT μν ∣ ∣ ∣ ( t , r , θ , ϕ ) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ρ 0 0 0 0 p 0 0 0 0 0 0 0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ p r 2 p r 2 sin 2 ( θ ) \nThe method list() works in exactly the same way, for example: \nSchwarzschildKretschmann . list(coords = In [ ]: \nCartesian) \nK = 48 M 2 ( x 2 + y 2 + z 2 ) 3 \nJust as with default indices, every tensor has a default coordinate system, which is, initially, the one we used to create it. We can change it by setting the property default\\_coords , and then whenever we display the tensor, it will be displayed in that coordinate system if no other coordinate system is specified. For example, let\'s change the default coordinates of the perfect fluid stress tensor to spherical coordinates: \nNow, when we display the tensor using show() or list() without any arguments (or with just a choice of indices), this is the coordinate system that will be used: \n```\n~ PerfectFluid In [ ]:\n``` \nT tt = ρ T rr = p T θθ = pr 2 T ϕϕ = pr 2 sin 2 ( θ ) \nNote that the coordinate transformation we defined is only invertible for . However, since we defined the coordinate above as a non-negative symbol, this is already taken care of by SymPy behind the scenes. To illustrate this, let us define a new scalar in Minkowski space, which is equal to the spatial distance from the origin: r ≥ 0 r \n```\nSpatialDistance = T . Tensor( metric = Minkowski, coords = Cartesian, indices = (), components = [T . s . sqrt(x ** 2 + y ** 2 + z ** symbol = "d", ) In [ ]:\n``` \nOut[ ]: \n```\n2)],\n``` \nd ∣ ∣ ∣ ( t , x , y , z ) = √ x 2 + y 2 + z 2 \nWhen we convert this scalar to spherical coordinates, we get , as expected: r \n```\nSpatialDistance . show(coords = Spherical) In [ ]:\n``` \nd ∣ ∣ ∣ ( t , r , θ , ϕ ) = r \nHowever, if we did not define as a non-negative symbol, we would have obtained instead. r | r |', 'Replacing symbols in the tensor components': 'By using the replace argument of list() and show() , we can replace symbols in the tensor components with other symbols or numerical values. The replacement must be in the form of a dictionary, where each key in the dictionary will be replaced with its value. Each of the keys and the values of the dictionary can be either a SymPy Symbol object or a SymPy Expr object. The components will then be simplified, and the tensor will be displayed with the new components. Note that this only applies to displaying the components; the tensor data itself will not change. \nFor example, perhaps we would like to display the value of the Kretschmann scalar for a particular \nSchwarzschildKretschmann . show(replace = In [ ]: \n{M: 1, r: 10}) \nK ∣ ∣ ∣ ( t , r , θ , ϕ ) = 3 62500 \nOr maybe we would like to display the perfect fluid stress tensor with equal to : p ρ \nPerfectFluid . list(replace = {p: rho}) In [ ]: \nT tt = T rr = ρ T θθ = ρ r 2 T ϕϕ = ρ r 2 sin 2 ( θ ) \nThe replacement can, of course, also be combined with a choice of indices and/or coordinates: \nPerfectFluid . list(coords = Cartesian, indices = (1, 1), replace = {p: rho}) In [ ]: \nT tt = T xx = T yy = T zz = ρ \nAnother, more advanced, thing we can do with list() and show() is to pass a function to be executed on each tensor component before printing it. We will see an example below, in the "Geodesic equations from the Lagrangian" section.', 'Customizing the simplification function': "Whenever OGRePy performs an operation that creates or modifies tensor components, such as converting between index representations or coordinate systems, it automatically simplifies the result using SymPy's simplify() . However, advanced users may want to have more control over this simplification process. This can be done using by setting T.options.simplify\\_func to a function of your choice. \nFor example, you may want to customize the arguments passed to simplify (such as ratio or inverse , see here for more information), or you may want to use specific SymPy simplification functions such as powsimp() or logcombine() in a specific combination, or even refine() with specific assumptions. \nIn extreme situations, you may even want to cancel simplification altogether, if it is taking too long, which can be achieved using T.options.simplify\\_func = lambda x: x - that is, replacing the simplification function with the identity function. \nNote that changing the simplification function will not automatically apply it to any existing tensors. The reason is that when OGRePy calculates the components of a tensor in a particular \nrepresentation, it calculates them once and for all , and then saves them in the object's data to be reused later. This is done to improve performance, so that the components don't have to be recalculated every time they are needed. \nWe can force re-simplification of the stored components of a specific tensor using the method simplify() . As usual with the options object, you may restore the simplification function to the default, SymPy's simplify() , with the command del T.options.simplify\\_func .", 'Getting information about tensors': "The info() method can be used to display the information encoded in a tensor object in human-readable form. Here is an example: \n```\nMinkowski . info() In [ ]:\n``` \n- · Name : Minkowski · Class : Metric · Symbol : · Rank : 2 · Dimensions : 4 · Default Coordinates : Cartesian · Default Indices : (-1, -1) · Associated Metric For : FourVelocity , PerfectFluid , SpatialDistance As for and , OGRePy defines a convenient shortcut for calling : use the η μν \nshow() list() info() + (unary plus) operator in front of the tensor. For example: \n```\n+ PerfectFluid In [ ]:\n``` \n- · Name : PerfectFluid\n- · Class : Tensor\n- · Symbol : T μν\n- · Rank : 2\n- · Dimensions : 4\n- · Default Coordinates : Spherical\n- · Default Indices : (-1, -1)\n- · Metric : Minkowski \nA Coordinates object also has an info() method, and it can be used to check which tensors use this coordinate system as their default: \n```\n+ Cartesian In [ ]:\n``` \n- · Name : Cartesian\n- · Class : Coordinates\n- · Dimensions : 4\n- · Default Coordinates For : Minkowski , Alcubierre , FourVelocity , SpatialDistance \nIt is also possible to get each of these properties of the tensor individually, using the properties symbol , default\\_indices , and default\\_coords and the methods rank() , dim() , and metric() . Note that the symbol, default indices, and default coordinates are properties that can be changed, but rank() , dim() , and metric() are read-only properties obtained using methods, as it doesn't make sense to change these properties. Here are some examples of using these properties and methods. The symbol is a bit cryptic: \n```\nPerfectFluid . symbol 'T[0][1]' In [ ]: Out[ ]:\n``` \nThe purpose of the [0][1] is to serve as a placeholders for indices, since the actual letters that will be used as the indices can be different each time. (These placeholders are added automatically when we create the tensor, there is no need to specify them manually, although you can if you want.) To get the symbol as a TeX string, we can use the tex\\_symbol() method, and pass its output to the IPython Math() function to display it in the notebook: \nMath(PerfectFluid . tex\\_symbol()) Similarly, we can use the default\\_indices and default\\_coords properties to obtain the default indices and coordinates: In [ ]: Out[ ]: T μν \n```\nPerfectFluid . default\\_indices (-1, -1) PerfectFluid . default\\_coords In [ ]: Out[ ]: In [ ]: Out[ ]:\n``` \n( t r θ ϕ ) \nAnd we can use the metric() method to obtain the associated metric: \nIn [ ]: \nOut[ ]: \nη μν ∣ ∣ ∣ ( t , x , y , z ) = ⎛ ⎜ ⎜ ⎜ ⎝ -1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ⎞ ⎟ ⎟ ⎟ ⎠ \nIn the last two examples, default\\_coords and metric() , notice that the output directly shows the tensors used as the default coordinates and associated metric respectively. This is because default\\_coords and metric() return a reference to the relevant Coordinates or Metric object respectively, and that object then gets displayed in the notebook using the show() method, as it is the output of the cell. \nHowever, since we are working inside a notebook, it would be helpful to know the name of the notebook variable referring to this Coordinates or Metric object. It turns out that is not at all straightforward to obtain this information in Python, since an object reference might not even be associated to any specific variable, or it may be associated to more than one variable. Luckily, OGRePy comes with a special algorithm to figure out which notebook variables refer to which objects. We already saw that algorithm in action when we used the info() method above. However, we can also obtain the name of the variable by simply converting the object to a string using the str constructor. This works on both Coordinate and Metric objects: \n```\nstr(PerfectFluid . default\\_coords) 'Spherical' str(PerfectFluid . metric()) 'Minkowski' In [ ]: Out[ ]: In [ ]: Out[ ]:\n``` \nThat same algorithm powers the module function info() , which lists all the tensors created so far, including the names of the variables used to define these tensors. Here are all the tensors we defined so far in this notebook: \n```\nT . info() In [ ]:\n``` \n9 tensor objects created: 2 coordinates, 3 metrics, 4 tensors. \nCoordinate systems: \n- 1. Cartesian (id: 0x21c55f5d760 ), default for: Minkowski , Alcubierre , FourVelocity , SpatialDistance\n- 2. Spherical (id: 0x21c55f96430 ), default for: Schwarzschild , SchwarzschildKretschmann , PerfectFluid \nMetrics and associated tensors: \n- 1. Minkowski (symbol: ) (id: 0x21c56751bd0 ), used by: FourVelocity , PerfectFluid , SpatialDistance η μν\n- 2. Schwarzschild (symbol: ) (id: 0x21c56881b20 ), used by: SchwarzschildKretschmann g μν\n- 3. Alcubierre (symbol: ) (id: 0x21c56882150 ) g μν", 'Tensors:': '- 1. SchwarzschildKretschmann (symbol: ) (id: 0x21c57c4db70 ) K\n- 2. FourVelocity (symbol: ) (id: 0x21c57c69c10 ) u μ\n- 3. PerfectFluid (symbol: ) (id: 0x21c57c69620 ) T μν\n- 4. SpatialDistance (symbol: ) (id: 0x21c580709f0 ) d \nWe see that we created 9 tensors in total so far: 2 coordinate systems, 3 metrics, 3 tensors associated with the Minkowski metric, and 1 tensor associated with the Schwarzschild metric.', 'Getting the components of a tensor': 'Sometimes you may want to extract the components of a tensor in a specific representation as a list, so you can use them outside of this package, as regular SymPy expressions rather than tensor objects. This is done using the components() method. For example, we can retrieve the components of the inverse Schwarzschild metric (with two upper indices): \nInverseSchwarzschild \n= \nSchwarzschild \n. \ncomponents(coords \n= \nSpherical, indices \n= \n(1, 1)) \n```\n⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ -0 0 0 0 0 0 0 0 0 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ r -2 M + r -2 M + r r 1 r 2 1 r 2 sin 2 ( θ )\n``` \nWe can now treat InverseSchwarzschild as any other SymPy Array - for example, extract the element at a particular position: \nIn [ ]: \nOut[ ]: \nOut[ ]: -r -2 M + r \nIf the desired index configuration and/or coordinate system are not specified, the default ones will be used. However, it is important to always know exactly which representation the components are in, to avoid confusion. Thus, you will be notified which representation was used: \nSchwarzschild . In [ ]: \n```\ncomponents()\n``` \nOGRePy : Using default coordinate system Spherical and default index configuration (-1, -1). \nOut[ ]: \n⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ -1 0 0 0 0 0 0 0 0 r 2 0 0 0 0 r 2 sin 2 ( θ ) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ 2 M r 1 -+1 2 M r \nThis warning can be disabled by adding the argument warn=False . \nSince components() returns a SymPy Array , we can use the subs() method to perform replacements, just like the replace argument of show() and list() (see above). For example, here are the components of the Schwarzschild metric on the hypersurface with : θ = π /2 \nSchwarzschild . components() . subs({theta: T . s . pi / 2}) In [ ]: \nOGRePy : Using default coordinate system Spherical and default index configuration (-1, -1). \nOut[ ]: \n⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ -1 0 0 0 0 0 0 0 0 r 2 0 0 0 0 r 2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ 2 M r 1 -+1 2 M r \nIn the case of a coordinate system, that is, a Coordinates object, components() takes no arguments, since a coordinate system cannot have multiple representations: \nSpherical . In [ ]: \n```\ncomponents()\n``` \nOut[ ]: [ t r θ ϕ \n]', 'Calculations with tensors': 'Now that we have all the bookkeeping of tensors out of the way, we can finally discuss how to use those tensors in calculations. In OGRePy, all tensor calculations are performed by simply using normal operations such as addition and multiplications on the tensors. However, this does not work the same as operating, for example, on integers; in most tensor operations, we also have to specify indices . Some of these indices will be *free indices , which will remain in the final result, while others may be **contraction indices , which will be contracted upon. \nOGRePy supports a comprehensive collection of tensor operations. A tensor calculation in OGRePy can involve any number of tensor objects and can contain any combination of addition, multiplication by scalar, trace, contraction, partial derivative, and covariant derivative. The result will be stored in a new tensor object. Let us now go over these operations one by one, and give some examples.', 'Addition of tensors': 'Addition of tensors in OGRePy is represented by a sum of the form tensor1(index1, index2, ...) + tensor2(index1, index2, ...) , where tensor1 and tensor2 are the tensor objects to be added, and (index1, index2, ...) are the index specifications for each tensor, given as SymPy symbols. (Note that an index specification is not the same as an index configuration , which is a tuple of the form (±1, ±1, ...) specifying which indices are up (+1) and which are down (-1).) \nNote that you do not specify the position (upper or lower) of the indices. Furthermore, just like in any tensor equation, the index letters themselves have no meaning ; they are just placeholders. Therefore, (a, b, c) , (X, Y, Z) , and (alpha, beta, gamma) are all completely equivalent. The only requirement is that the indices are consistent ; in the case of addition, this means that both tensors being added must have the same indices up to permutation . \nThe following constraints apply to addition of tensors: \n- · You may not add a tensor representing a coordinate system to any other tensor, since coordinates do not transform like tensors.\n- · You may not add two tensors associated with different metrics, since their sum would have undefined transformation properties.\n- · You may not add two tensors with different ranks, since that is not a well-defined operation.\n- · As stated above, both tensors must have the same indices up to permutation. and (with inverted indices on the second tensor) are both okay, but doesn\'t make sense, as it has more free indices than the rank of the result (that is, the result will be of the form instead of ). A μν + B μν A μν + B νμ A μν + B αβ T μναβ T μν \nAs an example, let us add the Minkowski metric and the perfect fluid stress tensor . First we import symbols from OGRePy.abc to use as indices, then we perform the actual sum: η μν T μν \n```\nfrom OGRePy.abc import mu, nu result = Minkowski(mu, nu) + PerfectFluid(mu, nu) In [ ]:\n``` \n```\nOut[ ]: η μν + T μν ∣ ∣ ∣ ( t , x , y , z ) = ⎛ ⎜ ⎜ ⎜ ⎝ ρ -1 0 0 0 0 p +1 0 0 0 0 p +1 0 0 0 0 p +1 ⎞ ⎟ ⎟ ⎟ ⎠\n``` \nIn [ ]: \nOut[ ]: \nNotice that the addition operation returned a new tensor object. This tensor\'s symbol has been automatically set to reflect the formula that was used to create it. However, often we want the new tensor to have its own single-letter symbol. To do that, we can use the symbol property: \nresult \n. \nsymbol \n= \n"S" \nresult \nS μν ∣ ∣ ∣ ( t , x , y , z ) = ⎛ ⎜ ⎜ ⎜ ⎝ ρ -1 0 0 0 0 p +1 0 0 0 0 p +1 0 0 0 0 p +1 ⎞ ⎟ ⎟ ⎟ ⎠ \nWith this symbol, the tensor equation we calculated becomes: \nS μν = η μν + T μν . \nThe order of indices we specify for each tensor matters. To give an example, let us define the following non-symmetric tensor: \n```\nNonSymmetric = T . Tensor( metric = Minkowski, coords = Cartesian, indices = ( -1, -1), components = [[0, 0, 0, 1], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]], symbol = "N", ) In [ ]:\n``` \nOut[ ]: \nIn [ ]: \nOut[ ]: \nN μν ∣ ∣ ∣ ( t , x , y , z ) = ⎛ ⎜ ⎜ ⎜ ⎝ 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 ⎞ ⎟ ⎟ ⎟ ⎠ \nIf we add it to the Minkowski metric, we get: \nMinkowski(mu, nu) \n+ \nNonSymmetric(mu, nu) \nη μν + N μν ∣ ∣ ∣ ( t , x , y , z ) = ⎛ ⎜ ⎜ ⎜ ⎝ -1 0 0 1 0 1 0 0 0 0 1 0 0 0 0 1 ⎞ ⎟ ⎟ ⎟ ⎠ \nNote that in this example we did not save the new tensor object in a variable, we just showed the result. However, if we flip its index specification from (mu, nu) to (nu, mu) , then we instead get: \nη μν + N νμ ∣ ∣ ∣ ( t , x , y , z ) = ⎛ ⎜ ⎜ ⎜ ⎝ -1 0 0 0 0 1 0 0 0 0 1 0 1 0 0 1 ⎞ ⎟ ⎟ ⎟ ⎠ \nTo stress an important point, there is no difference between NonSymmetric(mu, nu) and NonSymmetric(nu, mu) on its own, as the index labels themselves are meaningless unless there is some context in which they obtain meaning - as is always the case for tensor expressions. However, there is a big difference between, for example, Minkowski(mu, nu) + NonSymmetric(mu, nu) and Minkowski(mu, nu) + NonSymmetric(nu, mu) , as the indices have a different order, and thus the two expressions refer to adding different components. \nOf course, any number of tensors can be added, not just two - and the same tensor can be added multiple times, with different index specifications each time. For example, we can calculate the following expression: \nMinkowski(mu, nu) + PerfectFluid(mu, nu) + NonSymmetric(mu, nu) + NonSymmetric(nu, mu) In [ ]: \nOut[ ]: \nIn [ ]: \nOut[ ]: \nη μν + T μν + N μν + N νμ ∣ ∣ ∣ ( t , x , y , z ) = ⎛ ⎜ ⎜ ⎜ ⎝ ρ -1 0 0 1 0 p +1 0 0 0 0 p +1 0 1 0 0 p +1 ⎞ ⎟ ⎟ ⎟ ⎠', 'More on index specifications': 'For calculations that involve many indices, it may be more convenient to specify the indices as a string instead of individual symbols. This also saves us the trouble of importing or defining those symbols explicitly. This string must be given in the same format as SymPy\'s symbols() function, that is, a space- or comma-separated list of one or more letters or TeX codes. It is also possible to provide a list of strings, or even mix and match symbols and strings. For example, the previous calculation can also be written as follows: \nMinkowski(mu, nu) \n+ \nPerfectFluid("mu nu") \n+ \nNonSymmetric("mu", nu) \n+ \nNonSymmetric("nu", \nη μν + T μν + N μν + N νμ ∣ ∣ ∣ ( t , x , y , z ) = ⎛ ⎜ ⎜ ⎜ ⎝ ρ -1 0 0 1 0 p +1 0 0 0 0 p +1 0 1 0 0 p +1 ⎞ ⎟ ⎟ ⎟ ⎠ \nIndex specifications have a use even if we are not doing a calculation: they change the indices that appear when show() is called, instead of the default index letters (as specified using T.options.index\\_letters ). For example, with the default index letters, NonSymmetric will \nbe displayed with the indices : μν \nNonSymmetric In [ ]: Out[ ]: \nN μν ∣ ∣ ∣ ( t , x , y , z ) = ⎛ ⎜ ⎜ ⎜ ⎝ 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 ⎞ ⎟ ⎟ ⎟ ⎠ \nHowever, if we want to display it with the indices instead, we can simply indicate these indices in parentheses: αβ \nNonSymmetric("alpha beta") In [ ]: \nOut[ ]: \nN αβ ∣ ∣ ∣ ( t , x , y , z ) = ⎛ ⎜ ⎜ ⎜ ⎝ 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 ⎞ ⎟ ⎟ ⎟ ⎠ \nAnother alternative syntax is available for those who prefer the index specification format from the Mathematica version of OGRe: a string where each letter is a separate symbol, with no spaces between the letters, e.g. "abc" corresponds to (a, b, c). This format is less useful in the Python version since there is no easy way to enter Greek indices as individual letters; in Mathematica it\'s easy to write e.g. "μν" using escape sequences, but in Python it\'s easier to write "mu nu" or use symbols named mu and nu explicitly. The Mathematica format is accessible via square brackets, e.g.: \nNonSymmetric["ab"] In [ ]: Out[ ]: \nN ab ∣ ∣ ∣ ( t , x , y , z ) = ⎛ ⎜ ⎜ ⎜ ⎝ 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 ⎞ ⎟ ⎟ ⎟ ⎠', 'Multiplication of tensor by scalar': 'Multiplication of tensor by scalar in OGRePy is represented by a product of the form scalar * tensor(index1, index2, ...) , where tensor is the tensor object to be multiplied, (index1, index2, ...) is an index specification as for addition, and scalar is the scalar to multiply by. Note that scalar should be a number or SymPy expression, and not a tensor object of rank 0. To multiply a tensor by a tensor of rank 0, use contraction instead, as detailed in the next section. \n2 * Minkowski(mu, nu) In [ ]: Out[ ]: \n2 η μν ∣ ∣ ∣ ( t , x , y , z ) = ⎛ ⎜ ⎜ ⎜ ⎝ -2 0 0 0 0 2 0 0 0 0 2 0 0 0 0 2 ⎞ ⎟ ⎟ ⎟ ⎠ \nWhile in this example the indices seem redundant, they are necessary because in most non-trivial situations we would like to combine multiplication with other operations, such as addition or contraction, in which the order of indices matters. For example, consider: \n2 * t * Minkowski(mu, nu) -3 * x * PerfectFluid(mu, nu) + 4 * y * NonSymmetric(mu, nu) In [ ]: \nOut[ ]: \n2 t η μν -3 xT μν +4 yN μν -5 zN νμ ∣ ∣ ∣ ( t , x , y , z ) = ⎛ ⎜ ⎜ ⎜ ⎝ -3 ρ x -2 t 0 0 4 y 0 -3 px +2 t 0 0 0 0 -3 px +2 t 0 -5 z 0 0 -3 px +2 t', 'Taking traces and contracting tensors: theoretical review': 'The most complicated tensor operation is contraction , a generalization of the vector inner product. This is done by summing over one or more disjoint pairs of indices, with each pair containing exactly one upper index and one lower index. Raising and lowering indices is one example of contraction: the metric (or its inverse) is contracted with a tensor. Coordinate transformations are another example, where we contract the Jacobian (or its inverse) with a tensor. \nThe simplest example of contraction is the vector inner product , which is defined as the contraction of a vector (one upper index) with a covector (one lower index): \nv μ w μ = g μν v μ w ν = g ( v , w ). \nThe middle part of this equality comes from the fact that, as explained above, when we lower an index on , we use the metric: w ν \nw μ = g μν w ν . \nThis, in turn, justifies the notation on the right-hand side, as this is, in fact, an inner product of two vectors using the metric (in index-free notation). g ( v , w ) g \nContraction of indices in higher-rank tensors is simply a generalization of the inner product, for example: \nA μα B αν = g αβ A μα B β ν . \nWe can also contract more than one index: \nA μν B μν = g μα g νβ A μν B αβ . \nThis simply amounts to the fact that lowering both indices of involves contracting each index with the metric. We can even contract two indices of the same tensor : B αβ \nA μ μ = g μν A μν . \nThis is called taking the trace . Furthermore, it is also possible to contract pairs of indices from more than two tensors at the same time: \nA μν B νρ C ρσ = g να g ρβ A μν B αβ C ρσ . \nHowever, such operations can always be broken down into individual contractions of pairs of tensors. For example, in this case, one could first contract with and then contract the result with - which is indeed how this kind of contraction will be performed in OGRePy in practice: B νρ C ρσ A μν \nA μν B νρ C ρσ = A μν ( B νρ C ρσ ) . \nIn a contraction, there are two types of indices: contracted indices , which are summed upon, and free indices , which are not summed upon. The rank of the tensor that results from the contraction is the number of free indices. So for example, in the expression we have one contracted index, , and two free indices, and . Therefore, the resulting tensor is of rank two: A μα B αν α μ ν \nT μ ν = A μα B αν .', 'Taking traces and contracting tensors: OGRePy syntax': 'Contraction of tensors in OGRePy is represented by an expression of the form tensor1(index1, index2, ...) @ tensor2(index1, index2, ...) , where tensor1 and tensor2 are the tensor objects to be contracted, and (index1, index2, ...) are the index specifications for each tensor. Any matching indices in both index specifications will be contracted. This means that, for example, is calculated using v(mu) @ w(mu) and is calculated using A(mu, nu) @ B(nu, rho) @ C(rho, sigma) . Note that the user doesn\'t need to worry about the contracted indices being one upper and one lower, which is a common source of errors when contracting tensors by hand; the order of the indices, and whether the same index repeats twice, is all that matters. v μ w μ A μν B νρ C ρσ \nAs a first example, let us create the stress-energy tensor for a perfect fluid with a 4-velocity . This is defined as follows: u μ \nT μν = ( ρ + p ) u μ u ν + pg μν . \n<!-- image --> \nEven though this does not involve any contractions, it still counts as a "trivial" contraction, since two tensors (the 4-velocities) are juxtaposed next to each other to create another tensor. This is also known as an outer product . Therefore, it uses the same @ operator syntax as any other \ncontraction, except that there are no matching indices . Note that this expression involves not just contraction (in the first term), but also multiplication by scalar (in both terms), and addition of the two terms together. Again, OGRePy takes care of everything behind the scene, so this just works: \nPerfectFluidFromVelocity = (rho + p) * FourVelocity(mu) @ FourVelocity(nu) + p * Minkowski PerfectFluidFromVelocity . symbol = "T" PerfectFluidFromVelocity In [ ]: \nOut[ ]: \nT μν ∣ ∣ ∣ ( t , x , y , z ) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ -0 0 -0 0 0 0 p 0 0 0 0 p ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ -ρ -pv 2 v 2 -1 v ( ρ + p ) v 2 -1 v ( ρ + p ) v 2 -1 -ρ v 2 -p v 2 -1 \nIndeed, for we get the previously defined stress tensor: v = 0 \nPerfectFluidFromVelocity . show(replace = In [ ]: \n{v: 0}) \nT μν ∣ ∣ ∣ ( t , x , y , z ) = ⎛ ⎜ ⎜ ⎜ ⎝ ρ 0 0 0 0 p 0 0 0 0 p 0 0 0 0 p ⎞ ⎟ ⎟ ⎟ ⎠ \nMultiplying a tensor by a scalar (i.e. a tensor of rank 0) is also done using a "trivial" contraction with no contracted indices. For example: \n(SpatialDistance() @ Minkowski(mu, nu)) . show(coords = Spherical) In [ ]: \nd η μν ∣ ∣ ∣ ( t , r , θ , ϕ ) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ -r 0 0 0 0 r 0 0 0 0 r 3 0 0 0 0 r 3 sin 2 ( θ ) ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ \nNote the empty index specification () , which is mandatory in order for OGRePy to recognize that the scalar is involved in a tensor calculation. We can also multiply a scalar by another scalar: \nSpatialDistance() @ SpatialDistance() In [ ]: Out[ ]: \ndd ∣ ∣ ∣ ( t , x , y , z ) = x 2 + y 2 + z 2 \nNow let us demonstrate some non-trivial contractions. First, we have the inner product of vectors in this case, we get the norm (squared) of the 4-velocity, since we are contracting it with itself: \nFourVelocity(mu) @ FourVelocity(mu) In [ ]: \nOut[ ]: \nu μ u μ ∣ ∣ ∣ ( t , x , y , z ) = -1 \nWe can also contract several tensors together, with two matching pairs of indices: \nFourVelocity(mu) @ PerfectFluidFromVelocity(mu, nu) @ NonSymmetric(nu, rho) In [ ]: \nOut[ ]: \nu μ T μ ν N ν ρ ∣ ∣ ∣ ( t , x , y , z ) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 0 0 0 -⎞ ⎟ ⎟ ⎟ ⎟ ⎠ ρ √ 1 -v 2 \nFinally, to take the trace of a tensor, we simply match pairs of indices in that tensor\'s index specification: \nMinkowski(mu, mu) In [ ]: Out[ ]: \nPerfectFluid("mu mu") In [ ]: \nOut[ ]: \nT μ μ ∣ ∣ ∣ ( t , r , θ , ϕ ) = -ρ +3 p \nOf course, this also works for tensors with more than two indices, as we will see below. Any combination of indices can be used, with no limit on the number of traces taken for each tensor.', 'Derivatives and curvature tensors': 'The partial derivative is represented in OGRePy using the class PartialD . It can be contracted with other tensors using the usual OGRePy contraction notation - including an appropriate index specification - to calculate gradients and divergences. ∂ μ \nThe gradient of a tensor is the partial derivative acting on the tensor with a free index, e.g. for a tensor, for a vector, or for a rank-2 tensor, resulting in a tensor of one rank higher (due to the extra index). In OGRePy, this is done by contracting the PartialD object from the left with the tensor, using the contraction operator @ . For example, we can calculate the gradient of the Kretschmann scalar as follows: ∂ μ ∂ μ ϕ ∂ μ u ν ∂ μ T νλ ∂ μ K \nη μ μ ∣ ∣ ∣ ( t , x , y , z ) = 4 \nT . PartialD(mu) @ SchwarzschildKretschmann() In [ ]: \nOut[ ]: \n∂ μ K ∣ ∣ ∣ ( t , r , θ , ϕ ) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 0 -0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ 288 M 2 r 7 \nAnd here is the gradient of the Schwarzschild metric: \n~ (T . PartialD(mu) @ Schwarzschild("alpha beta")) In [ ]: \n∂ r g tt = -∂ r g rr = -∂ r g θθ = 2 r ∂ r g ϕϕ = 2 r sin 2 ( θ ) ∂ θ g ϕϕ = 2 r 2 sin ( θ ) cos ( θ ) 2 M r 2 2 M r 2 ( -+1 ) 2 2 M r \nThe divergence of a tensor is the contraction of the partial derivative with one of the tensor\'s indices, e.g. for a vector or for a rank-2 tensor, resulting in a tensor of one rank lower . To illustrate, let us create the position vector of a particle in Minkowski space: ∂ μ ∂ μ u μ ∂ μ T μν \n```\nPosition = T . Tensor( metric = Minkowski, coords = Cartesian, indices = (1,), components = [t, x, y, z], symbol = "x", ) In [ ]:\n``` \nOut[ ]: \nIts gradient is: \nT . PartialD(mu) @ In [ ]: \n```\nPosition(nu)\n``` \nx μ ∣ ∣ ∣ ( t , x , y , z ) = ⎛ ⎜ ⎜ ⎜ ⎝ t x y z ⎞ ⎟ ⎟ ⎟ ⎠ \nOut[ ]: \n∂ μ x ν ∣ ∣ ∣ ( t , x , y , z ) = ⎛ ⎜ ⎜ ⎜ ⎝ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ⎞ ⎟ ⎟ ⎟ ⎠ \nAnd its divergence is: \nT . PartialD(mu) @ Position(mu) In [ ]: \nOut[ ]: \n∂ μ x μ ∣ ∣ ∣ ( t , x , y , z ) = 4 \nAs you can see, the syntax for both the gradient and divergence is the same; if the index specification of PartialD matches one of the indices of the tensor to its right, then the divergence will be calculated, otherwise the gradient will be calculated. \nWARNING: When applying partial derivatives to tensors, the result generally does not transform like a tensor under a coordinate transformation. For this reason, in general relativity we normally use the covariant derivative instead of a partial derivative. However, there are three important exceptions, where partial derivatives must be used: in the covariant derivative itself, the Levi-Civita connection , and the Riemann tensor , all of which will be discussed below. \nOf these three special cases, the covariant derivative and the Riemann tensor turn out to nonetheless transform like tensors under coordinate transformations, due to cancellations. However, the Levi-Civita connection, whose components are called the Christoffel symbols , has a special transformation rule, which is used automatically by OGRePy, as we will show. \nIn all other cases, if the user creates an arbitrary tensor using partial derivatives, the result will generally transform incorrectly under a coordinate transformation in OGRePy. Therefore, it is highly recommended to avoid using partial derivatives in OGRePy unless you really know what you\'re doing.', 'The Christoffel symbols': 'The Christoffel symbols are a very important tensor-like object in differential geometry. They are the components of the Levi-Civita connection , which is the unique torsion-free connection that preserves the metric. The Christoffel symbols are defined as follows: \nΓ λ μν = g λσ ( ∂ μ g νσ + ∂ ν g σμ -∂ σ g μν ) . 1 2 \nEach of the terms inside the parentheses is a gradient of the metric, with different indices. For example, the first term is represented in OGRePy as T.PartialD(mu) @ metric(nu, sigma) where metric is the tensor object representing the metric. Since OGRePy allows us to ∂ μ g νσ \neasily perform an arbitrary number of contraction, addition, multiplication by scalar, and partial derivative operations, we can calculate the Christoffel symbols of the Schwarzschild metric directly as follows: (We used SymPy\'s Rational class to create a symbolic 1/2 in the front, otherwise it would have been a numeric 0.5) \n```\nfrom OGRePy.abc import lamda, sigma WrongSchwarzschildChristoffel = T . s . Rational(1, 2) * Schwarzschild(lamda, sigma) @ (T . PartialD WrongSchwarzschildChristoffel . symbol = "Gamma" WrongSchwarzschildChristoffel . default\\_indices = (1, -1, -1) ~ WrongSchwarzschildChristoffel In [ ]:\n``` \nΓ t tr = Γ t rt = Γ r tt = Γ r rr = Γ r θθ = 2 M -r Γ r ϕϕ = (2 M -r ) sin 2 ( θ ) Γ θ r θ = Γ θ θ r = Γ ϕ r ϕ = Γ ϕ ϕ r = Γ θ ϕϕ = -Γ ϕ θϕ = Γ ϕ ϕθ = M r ( -2 M + r ) M ( -2 M + r ) r 3 M r (2 M -r ) 1 r sin (2 θ ) 2 1 tan ( θ ) \nHowever, there is a problem; as we mentioned above, the Christoffel symbols are not the components of a tensor , meaning that the Levi-Civita connection does not transform as a tensor does under a coordinate transformation. Indeed, by transforming the metric in the definition, one can show that \nΓ λ \' μ \' ν \' = Γ λ μν + . ∂ x μ ∂ x μ \' ∂ x ν ∂ x ν \' ∂ x λ \' ∂ x λ ∂ x λ \' ∂ x λ ∂ 2 x λ ∂ x μ \' ∂ x ν \' \nThe first term is the familiar transformation rule for a tensor, with one factor of the Jacobian per index as usual. However, there is also an extra second term, meaning that the Christoffel symbols do not transform like a tensor. \n(Similarly, you are also not supposed to raise or lower indices in the Christoffel symbols, but in practice, you can do that as long as you make it clear that it\'s just an abuse of notation - you are only adding factors of the metric, not creating a new tensor representation with different transformation properties.) \nDue to the extra transformation term, the tensor object WrongSchwarzschildChristoffel we calculated manually above must not be used ! Instead, we should use the method \nchristoffel() of the Metric class, which not only performs the calculation automatically for us, but also marks the result as a special tensor object with special transformation properties (more precisely, it will be an instance of the Christoffel subclass): \n```\n~ Schwarzschild . christoffel() In [ ]:\n``` \nΓ t tr = Γ t rt = Γ r tt = Γ r rr = Γ r θθ = 2 M -r Γ r ϕϕ = (2 M -r ) sin 2 ( θ ) Γ θ r θ = Γ θ θ r = Γ ϕ r ϕ = Γ ϕ ϕ r = Γ θ ϕϕ = -Γ ϕ θϕ = Γ ϕ ϕθ = M r ( -2 M + r ) M ( -2 M + r ) r 3 M r (2 M -r ) 1 r sin (2 θ ) 2 1 tan ( θ ) \nThese are the same components we got before, but now they will transform properly. In addition, the tensor object automatically has the correct index configuration (1, -1, -1) . \nFor maximal clarity, let us demonstrate the discrepancy in the coordinate transformation with a simple test metric: \n```\nx, 1, 1, 1),\n``` \n```\nSimpleMetric = T . Metric( coords = Cartesian, components = T . diag( -) In [ ]:\n``` \nOut[ ]: \ng μν ∣ ∣ ∣ ( t , x , y , z ) = ⎛ ⎜ ⎜ ⎜ ⎝ -x 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ⎞ ⎟ ⎟ ⎟ ⎠ \nWe calculate its Christoffel symbols in two ways. First manually, as we did above for the Schwarzschild metric: \n```\nWrongSimpleMetricChristoffel = T . s . Rational(1, 2) * SimpleMetric(lamda, sigma) @ (T . PartialD WrongSimpleMetricChristoffel . symbol = "Gamma" WrongSimpleMetricChristoffel . default\\_indices = (1, -1, -1) ~ WrongSimpleMetricChristoffel In [ ]:\n``` \nIn [ ]: \nIn [ ]: \nΓ t tx = Γ t xt = Γ x tt = 1 2 x 1 2 \nThen, with the built-in christoffel() method: \n~ (SimpleMetricChristoffel := SimpleMetric . christoffel()) \nΓ t tx = Γ t xt = Γ x tt = 1 2 x 1 2 \nNote that in this example we used Python\'s "walrus operator" := , which is an assignment operator which returns the result of the assignment (this is also called an "assignment expression"). This allowed us to easily call list() on the result by prepending the ~ operator, instead of having to write an additional line. \nThe two results have the same components, as expected. But now, let us now transform them to spherical coordinates. First, we transform the tensor object obtained using christoffel() : \nSimpleMetricChristoffel . list(coords = Spherical) \nΓ t tr = Γ t rt = Γ t t θ = Γ t θ t = Γ t t ϕ = Γ t ϕ t = -Γ r tt = Γ r θθ = -r Γ r ϕϕ = -r sin 2 ( θ ) Γ θ tt = Γ θ r θ = Γ θ θ r = Γ ϕ r ϕ = Γ ϕ ϕ r = Γ θ ϕϕ = -Γ ϕ tt = -Γ ϕ θϕ = Γ ϕ ϕθ = 1 2 r 1 2 tan ( θ ) tan ( ϕ ) 2 sin ( θ ) cos ( ϕ ) 2 cos ( ϕ ) cos ( θ ) 2 r 1 r sin (2 θ ) 2 sin ( ϕ ) 2 r sin ( θ ) 1 tan ( θ ) \nThis is the correct representation of the Christoffel symbols in spherical coordinates, as the extra term in the transformation was taken into account. However, if we transform the Christoffel symbols we obtained by manual calculation, we get: \n```\nWrongSimpleMetricChristoffel . list(coords = Spherical) In [ ]:\n``` \nΓ t tr = Γ t rt = Γ t t θ = Γ t θ t = Γ t t ϕ = Γ t ϕ t = -Γ r tt = Γ θ tt = Γ ϕ tt = -1 2 r 1 2 tan ( θ ) tan ( ϕ ) 2 sin ( θ ) cos ( ϕ ) 2 cos ( ϕ ) cos ( θ ) 2 r sin ( ϕ ) 2 r sin ( θ ) \nThis is not the correct result, since the transformation did not take into account the extra term. To verify that the former result is indeed the correct one, let us change the default coordinate system of SimpleMetric to spherical: \nSimpleMetric . default\\_coords = Spherical In [ ]: \nNow, when we calculate the Christoffel symbols manually from this metric, we will get their correct representation in spherical coordinates. This is because OGRePy always performs the calculations internally in the default coordinates of the first tensor in any operation (e.g. A for the contraction A @ B ), so the result will be calculated from scratch in spherical coordinates, instead of being calculated first in Cartesian coordinates and then transformed: \n```\nWrongSimpleMetricChristoffel2 = T . s . Rational(1, 2) * SimpleMetric(lamda, sigma) @ (T . PartialD WrongSimpleMetricChristoffel2 . symbol = "Gamma" WrongSimpleMetricChristoffel2 . default\\_indices = (1, -1, -1) ~ WrongSimpleMetricChristoffel2 In [ ]:\n``` \nΓ t tr = Γ t rt = Γ t t θ = Γ t θ t = Γ t t ϕ = Γ t ϕ t = -Γ r tt = Γ r θθ = -r Γ r ϕϕ = -r sin 2 ( θ ) Γ θ tt = Γ θ r θ = Γ θ θ r = Γ ϕ r ϕ = Γ ϕ ϕ r = Γ θ ϕϕ = -Γ ϕ tt = -Γ ϕ θϕ = Γ ϕ ϕθ = 1 2 r 1 2 tan ( θ ) tan ( ϕ ) 2 sin ( θ ) cos ( ϕ ) 2 cos ( ϕ ) cos ( θ ) 2 r 1 r sin (2 θ ) 2 sin ( ϕ ) 2 r sin ( θ ) 1 tan ( θ ) \nIndeed, this is the exact same result we got when we transformed SimpleMetricChristoffel to spherical coordinates. We have learned an important lesson: since the Christoffel symbols do not transform like a tensor, we should always use the built-in method christoffel() of the Metric class to calculate them, which ensures that they transform properly. (Of course, this method is also much more convenient than writing the explicit definition...) \nFor future use, let us define the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric , which describes an expanding universe: \na\\_t = T . func("a")(t) FLRW = T . Metric( coords = Spherical, components = T . diag( -1, a\\_t ** 2, a\\_t ** 2 * r ** 2, a\\_t ** 2 * r ** 2 * T . s . sin(theta) ** 2), ) Here, is the scale factor . This metric has the line element: FLRW . line\\_element() In [ ]: Out[ ]: g μν ∣ ∣ ∣ ( t , r , θ , ϕ ) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ -1 0 0 0 0 a 2 0 0 0 0 a 2 r 2 0 0 0 0 a 2 r 2 sin 2 ( θ ) ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ a ( t ) In [ ]: \nOut[ ]: d ϕ 2 r 2 a 2 ( t ) sin 2 ( θ ) + d θ 2 r 2 a 2 ( t ) + d r 2 a 2 ( t ) -d t 2 \nand the volume element squared: \nFLRW . volume\\_element\\_squared() In [ ]: \nOut[ ]: -r 4 a 6 ( t ) sin 2 ( θ ) \nIts Christoffel symbols can be easily calculated using christoffel() : \n~ FLRW . christoffel() In [ ]: \nΓ t rr = ∂ t aa Γ t θθ = ∂ t aar 2 Γ t ϕϕ = ∂ t aar 2 sin 2 ( θ ) Γ r tr = Γ r rt = Γ θ t θ = Γ θ θ t = Γ ϕ t ϕ = Γ ϕ ϕ t = Γ r θθ = -r Γ r ϕϕ = -r sin 2 ( θ ) Γ θ r θ = Γ θ θ r = Γ ϕ r ϕ = Γ ϕ ϕ r = Γ θ ϕϕ = -Γ ϕ θϕ = Γ ϕ ϕθ = ∂ t a a 1 r sin (2 θ ) 2 1 tan ( θ )', 'The Riemann tensor': 'The Riemann curvature tensor can be calculated from the Christoffel symbols using the definition: R ρ σμν \nR ρ σμν = ∂ μ Γ ρ νσ -∂ ν Γ ρ μσ + Γ ρ μλ Γ λ νσ -Γ ρ νλ Γ λ μσ \nEven though it contains partial derivatives, it nonetheless transforms like a tensor under a change of coordinates, because the extra transformation terms exactly cancel each other. To calculate this tensor, we can simply write down the formula explicitly with the correct indices contracted: \n```\nSchwarzschild In [ ]:\n``` \n```\nSchwarzschildRiemann = ( T . PartialD(mu) @ Schwarzschild . christoffel(rho, nu, sigma) -T . PartialD(nu) @ )\n``` \n<!-- image --> \nNotice that this time we used the christoffel() method with arguments corresponding to an index specification; this is simply a shortcut for using the () operator on the resulting tensor, as we have done above. \nIt would be more instructive to use list() to display the components, but we should change the symbol to first, since the current symbol contain the entire formula and would be very cumbersome to display multiple times: R \n```\nSchwarzschildRiemann . symbol = "R" ~ SchwarzschildRiemann In [ ]:\n``` \nR t t rr = R t t θθ = R r r θθ = -R θ t t θ = -R θ r r θ = -R t t ϕϕ = R r r ϕϕ = -R ϕ t t ϕ = -R ϕ r r ϕ = -R t r rt = R t θ θ t = R t ϕ ϕ t = R r t tr = R r r tt = R r θ θ r = R r ϕ ϕ r = R θ θ tt = R ϕ ϕ tt = R θ θ rr = R ϕ ϕ rr = R θ θ ϕϕ = -R ϕ θ θϕ = R θ ϕ ϕθ = -R ϕ ϕ θθ = -2 M r 2 ( -2 M + r ) M r M sin 2 ( θ ) r 2 M ( -2 M + r ) r 4 M (2 M -r ) r 4 2 M r 2 (2 M -r ) 2 M (2 M -r ) r 4 M r 2 ( -2 M + r ) M ( -2 M + r ) r 4 M r 2 (2 M -r ) 2 M sin 2 ( θ ) r 2 M r \nHere we run into another issue: we wanted , but what we actually got was , since this is the order of indices from left to right in the definition. There are two ways to fix this in OGRePy. One is to use the permute() method. We simply need to call permute() with as the old indices and as the new indices to fix the issue: R ρ σμν R ρ μνσ μρνσ ρσμν \nR t rtr = R t rrt = R t θ t θ = -R t θθ t = R r θ r θ = -R r θθ r = -R t ϕ t ϕ = -R t ϕϕ t = R r ϕ r ϕ = -R r ϕϕ r = -R r ttr = R r trt = R θ tt θ = R ϕ tt ϕ = R θ t θ t = R ϕ t ϕ t = R θ rr θ = R ϕ rr ϕ = R θ r θ r = R ϕ r ϕ r = R θ ϕθϕ = -R θ ϕϕθ = R ϕ θθϕ = -R ϕ θϕθ = -2 M r 2 ( -2 M + r ) 2 M r 2 (2 M -r ) M r M sin 2 ( θ ) r 2 M ( -2 M + r ) r 4 2 M (2 M -r ) r 4 M (2 M -r ) r 4 M ( -2 M + r ) r 4 M r 2 ( -2 M + r ) M r 2 (2 M -r ) 2 M sin 2 ( θ ) r 2 M r \nNow we have obtained the correct expression for the Riemann tensor of the Schwarzschild metric. In fact, we did not have to specify the old indices explicitly; since SchwarzschildRiemann is the result of a tensor calculation, it actually remembers the index specification it obtained as a result of the calculation, and this will be used automatically if the old argument is not specified. \nThe other way to fix this is to wrap the original calculation inside the calc() function, which is simply a convenience function that allows us to calculate a tensor, change its symbol, and permute its indices in just one function call. We will show examples of its usage below.', 'Exact sign checks with list()': "The sharp-eyed reader may have noticed that, when we used list() on the Schwarzschild Riemann tensor above, it listed, for example, the components and separately, even though they are the negatives of each other. Unfortunately, SymPy's comparison operation is very rudimentary, comparing the general structure of the expression rather than an actual mathematical comparison. This can be seen on even simpler comparisons - for example, the following comparison will yield False even though the two expressions are clearly mathematically equal: R t rtr R t rrt \nexpr1 = 1 / (1 -x) expr2 = -(1 / (x -1)) expr1 == expr2 In [ ]: \nFalse Out[ ]: \nThis can be resolved by noticing that if and only if . So if we subtract one expression from the other, simplify the result, and compare to zero, we will get a correct answer: a = b a -b = 0 \nIn [ ]: \nT . s . simplify(expr1 -expr2) == 0 \nTrue Out[ ]: \nNormally, list() doesn't do this for every single component of the tensor, since that could be a very time-consuming task for large tensors with complicated components. However, we could ask list() to perform these more precise comparisons by adding the exact=True option: \nSchwarzschildRiemann . list(exact =True ) In [ ]: \nR t rtr = -R t rrt = R t θ t θ = -R t θθ t = R r θ r θ = -R r θθ r = -R t ϕ t ϕ = -R t ϕϕ t = R r ϕ r ϕ = -R r ϕϕ r = -R r ttr = -R r trt = R θ tt θ = -R θ t θ t = R ϕ tt ϕ = -R ϕ t ϕ t = R θ rr θ = -R θ r θ r = R ϕ rr ϕ = -R ϕ r ϕ r = R θ ϕθϕ = -R θ ϕϕθ = R ϕ θθϕ = -R ϕ θϕθ = -2 M r 2 ( -2 M + r ) M r M sin 2 ( θ ) r 2 M ( -2 M + r ) r 4 M (2 M -r ) r 4 M r 2 ( -2 M + r ) 2 M sin 2 ( θ ) r 2 M r \nYou can see that now list() correctly identifies all of the components that are negatives of each other, resulting in a much shorter list. If you're wondering why this option only applies to comparing components that are the negative of each other, rather than all comparison - that is because OGRePy automatically simplifies all tensor components in advance, so if two components are the same, they should already be simplified to the exact same expression. \nThe Riemann tensor with all its indices lowered satisfies the following symmetry and antisymmetry relations: \nR ρσμν = -R σρμν = -R ρσνμ = R μνρσ \nIn [ ]:", 'SchwarzschildRiemann . list(indices = ( -1, -1, -1, -1), exact =True )': 'R trtr = -R trrt = -R rttr = R rtrt = -R t θ t θ = -R t θθ t = -R θ tt θ = R θ t θ t = R t ϕ t ϕ = -R t ϕϕ t = -R ϕ tt ϕ = R ϕ t ϕ t = R r θ r θ = -R r θθ r = -R θ rr θ = R θ r θ r = R r ϕ r ϕ = -R r ϕϕ r = -R ϕ rr ϕ = R ϕ r ϕ r = R θϕθϕ = -R θϕϕθ = -R ϕθθϕ = R ϕθϕθ = 2 Mr sin 2 ( θ ) 2 M r 3 M ( -2 M + r ) r 2 M ( -2 M + r ) sin 2 ( θ ) r 2 M 2 M -r M sin 2 ( θ ) 2 M -r \nThis shows that the symmetry and anti-symmetry relations are indeed satisfied.', 'The riemann() method and caching': "Don't worry - you don't need to write the explicit definition of the Riemann tensor every time you want to calculate it. Instead, OGRePy offers the method riemann() of the Metric class. For example, for the FLRW metric we get: \nFLRW . riemann() . list(exact =True ) In [ ]: \nR t rtr = -R t rrt = -R t θ t θ = -R t θθ t = -R t ϕ t ϕ = -R t ϕϕ t = -R r ttr = -R r trt = -R θ tt θ = -R θ t θ t R r θ r θ = -R r θθ R r ϕ r ϕ = -R r ϕϕ R θ rr θ = -R θ r θ \nNotice two things here. First, we did not save the result in a variable. The reason is that the results of the riemann() method, and in fact all similar methods such as christoffel() , are cached. This means that the next time we call FLRW.riemann() , we will get the exact same tensor indeed, it won't just be another tensor with the same components, it will be a reference to the exact same object we got the first time. \nSecond, calculating the Riemann tensor requires first calculating the Christoffel symbols, which we did not do. Behind the scenes, the riemann() method obtains the Christoffel symbols by calling the christoffel() method. In turn, the christoffel() checks if the Christoffel symbols have already been calculated. If so, it returns the cached results, and if not, it calculated, caches, and returns the results. \nAs a result, even though we did not call FLRW.christoffel() before, the Christoffel symbols have in fact already been calculated and cached for us, so if we call it now we will get the result immediately: \n~ FLRW . In [ ]: \n```\nchristoffel()\n``` \nΓ t rr = ∂ t aa Γ t θθ = ∂ t aar 2 Γ t ϕϕ = ∂ t aar 2 sin 2 ( θ ) Γ r tr = Γ r rt = Γ θ t θ = Γ θ θ t = Γ ϕ t ϕ = Γ ϕ ϕ t = Γ r θθ = -r Γ r ϕϕ = -r sin 2 ( θ ) Γ θ r θ = Γ θ θ r = Γ ϕ r ϕ = Γ ϕ ϕ r = Γ θ ϕϕ = -Γ ϕ θϕ = Γ ϕ ϕθ = ∂ t a a 1 r sin (2 θ ) 2 1 tan ( θ ) \nThe same principle also applies to the other built-in methods for calculating curvature tensors, which we will present below; they always calculate and cache any intermediate tensors in their definitions automatically as needed. \nStandard practice when using OGRePy is to never save the Christoffel symbols, Riemann tensor, etc. in their own variables. Instead, you must call the christoffel() , riemann() , etc. methods every time you want to access these tensors. \nThe reason behind this is to maintain consistency between the metric and its curvature tensors. For example, let's say we decided to redefine the FLRW metric. Since tensor components in OGRePy are immutable, meaning they cannot be changed after the tensor object is created, this means we actually create a new Metric object and save it in the same FLRW variable. If we previously calculated the Christoffel symbols and saved them in a variable called FLRWChristoffel , that variable now stores the Christoffel symbols for the old FLRW metric, and is therefore inconsistent with the new one. On the other hand, if we simply use the FLRW.christoffel() method, we are guaranteed to always get the correct Christoffel symbols for the metric stored in the FLRW variable. \nIn this documentation, we will continue to create variables for curvature tensors because we will be calculating them explicitly and therefore they are not cached, but in normal use you should instead simply rely on OGRePy's comprehensive caching algorithms.", 'The Kretschmann scalar': 'The Kretschmann scalar is a curvature invariant calculated by contracting all the indices of the \nIn [ ]: \nOut[ ]: \nRiemann tensor with itself: \nK = R ρσνμ R ρσνμ . \nRecall that above, we gave the Kretschmann scalar for the Schwarzschild metric as an example of a scalar. Now that we have the Riemann tensor, and the ability to contract tensors, we can actually calculate the Kretschmann scalar from scratch: \nSchwarzschild . riemann(rho, sigma, mu, nu) @ Schwarzschild . riemann(rho, sigma, mu, nu) \nR μνρσ R μνρσ ∣ ∣ ∣ ( t , r , θ , ϕ ) = 48 M 2 r 6 \nNote that like the christoffel() method, the riemann() method allows you to pass an index specification for use in calculations. As usual, OGRePy allows you to calculate this curvature tensor directly, using the method kretschmann() of the Metric class, without typing the formula explicitly. \nThis method follows the same caching algorithm as christoffel() and riemann() : it will calculate the Riemann tensor (and as a side effect, the Christoffel symbols) if they have not already been calculated, otherwise it will use the cached results, and it will cache its own result for later use.', 'The Ricci tensor and scalar': 'The Ricci tensor is the trace of the first and third indices of the Riemann tensor: R μν \nR μν = R λ μλν . \nTherefore, we can calculate it by taking the trace, with the usual OGRePy syntax. For the Schwarzschild metric, the Ricci tensor vanishes: \n~ Schwarzschild . In [ ]: \n```\nriemann(lamda, mu, lamda, nu)\n``` \nNo non-zero elements. \nWe can also use the convenience method ricci\\_tensor() of the Metric class. For example, here is the Ricci tensor for the FLRW metric: \n~ FLRW . In [ ]: \n```\nricci\\_tensor()\n``` \nR tt = -R rr = 2 ∂ t a 2 + a ∂ t a R θθ = r 2 ( 2 ∂ t a 2 + a ∂ t a ) R ϕϕ = r 2 ( 2 ∂ t a 2 + a ∂ t a ) sin 2 ( θ ) 3 ∂ t a d d t a d d t d d t d d t \nThe Ricci scalar is the trace of the Ricci tensor: \nR = R λ λ = g μν R μν \nWe can calculate it from the Ricci tensor by taking the trace: \nFLRW . ricci\\_tensor(mu, mu) In [ ]: \nOut[ ]: \nR μ μ ∣ ∣ ∣ ( t , r , θ , ϕ ) = 6 ( ∂ t a 2 + a ∂ t a ) d d t a 2 \nOr, as usual, we can simply use the shorthand method ricci\\_scalar() to calculate it directly from the metric: \nFLRW . ricci\\_scalar() In [ ]: \nOut[ ]: \nR ∣ ∣ ∣ ( t , r , θ , ϕ ) = 6 ( ∂ t a 2 + a ∂ t a ) d d t a 2', 'The Einstein tensor': 'The Einstein tensor is given by: G μν \nG μν = R μν -Rg μν . 1 2 \nAs with all other curvature tensors, we can calculate it by combining the previously calculated tensors with the usual syntax: \n~ (FLRW . ricci\\_tensor(mu, nu) -T . s . Rational(1, 2) * FLRW . ricci\\_scalar() @ FLRW(mu, nu)) In [ ]: \nR tt -Rg tt = R rr -Rg rr = -∂ t a 2 -2 a ∂ t a R θθ -Rg θθ = r 2 ( -∂ t a 2 -2 a ∂ t a ) R ϕϕ -Rg ϕϕ = r 2 ( -∂ t a 2 -2 a ∂ t a ) sin 2 ( θ ) 1 2 3 ∂ t a 2 a 2 1 2 d d t 1 2 d d t 1 2 d d t \nOr we can use the built-in module einstein() : \n~ FLRW . einstein() In [ ]: \nG tt = G rr = -∂ t a 2 -2 a ∂ t a G θθ = r 2 ( -∂ t a 2 -2 a ∂ t a ) G ϕϕ = r 2 ( -∂ t a 2 -2 a ∂ t a ) sin 2 ( θ ) 3 ∂ t a 2 a 2 d d t d d t d d t', 'Covariant derivatives': 'The partial derivative has limited use in general relativity, as it does not transform like a tensor . Therefore, it is only used in special cases, such as calculating the Christoffel symbols and the Riemann tensor. The covariant derivative is a generalization of the partial derivative, which does transform like a tensor (as long as it acts on a proper tensor). It is defined as follows: ∇ μ \n- · On a scalar , the covariant derivative acts as . Φ ∇ μ Φ = ∂ μ Φ\n- · On a vector , the covariant derivative acts as . v μ ∇ μ v ν = ∂ μ v ν + Γ ν μλ v λ\n- · On a covector , the covariant derivative acts as . w μ ∇ μ w ν = ∂ μ w ν -Γ λ μν w λ \nMore generally, on a rank tensor with components , the covariant derivative is defined as follows: ( p , q ) T ν 1 ⋯ ν p σ 1 ⋯ σ q ∇ μ T ν 1 ⋯ ν p σ 1 ⋯ σ q \n- · The first term will be the partial derivative . ∂ μ T ν 1 ⋯ ν p σ 1 ⋯ σ q\n- · We add one term for each upper index . Γ ν i μλ T ν 1 ⋯ λ ⋯ ν p σ 1 ⋯ σ q ν i\n- · We subtract one term for each lower index . Γ λ μσ i T ν 1 ⋯ ν p σ 1 ⋯ λ ⋯ σ q σ i \nNote that even though the covariant derivative is made from ingredients that do not transform like tensors - the partial derivative and the Christoffel symbols - the unwanted terms in the transformations of these ingredients cancel each other exactly, so that in the end, the entire sum does transform like a tensor. \nAs usual, we can, of course, write down the covariant derivative manually. For example, the covariant divergence of the metric is: \n∇ μ g αβ = ∂ μ g αβ -Γ λ μα g λβ -Γ λ μβ g αλ . \nIt should vanish, by definition, for any metric; this is what we meant when we said the Levi-Civita connection preserves the metric. Indeed, we have for the Schwarzschild metric: \nfrom OGRePy.abc import alpha, beta ~ (T . PartialD(mu) @ Schwarzschild(alpha, beta) -Schwarzschild . christoffel(lamda, mu, alpha In [ ]: \nNo non-zero elements. \nMuch more conveniently, the covariant derivative is represented in OGRePy using the class CovariantD . It will automatically add the correct terms, as detailed above, for each of the tensor\'s indices, using the (possibly cached) Christoffel symbols of the tensor\'s associated metric. To use it, simply contract it with any tensor, just like PartialD . For example, we can check that the covariant derivative of the FLRW metric also vanishes: \n~ (T . CovariantD(mu) @ In [ ]: \nFLRW(mu, nu)) \nNo non-zero elements. \nThe covariant divergence of the Einstein tensor is: \n∇ μ G μν = ∂ μ G μν + Γ μ μλ G λν + Γ ν μλ G μλ . \nNote that it involves a contraction in the index , which becomes a trace in the first Christoffel symbol. This expression vanishes because of the Bianchi identity : μ \n∇ μ R μν = ∇ ν R ⟹ ∇ μ G μν = 0. 1 2 \nTo calculate it in OGRePy, we simply write: \n~ (T . CovariantD(mu) @ FLRW . In [ ]: \neinstein(mu, nu)) \nNo non-zero elements. \nFinally, for a non-trivial result, let us recall that the stress-energy tensor should be conserved : \n∇ μ T μν = ∂ μ T μν + Γ μ μλ T λν + Γ ν μλ T μλ = 0. \nThis follows from the fact that , combined with the Einstein equation : ∇ μ G μν = 0 \nG μν = κ T μν , \nwhere and is Newton\'s gravitational constant. However, unlike , the relation is not an identity; it is an energy-momentum conservation equation . To derive the equation for the FLRW metric, let us first define the rest-frame fluid 4-velocity in this spacetime: κ = 8 π G G ∇ μ G μν = 0 ∇ μ T μν = 0 \n```\nRestVelocity = T . Tensor(metric = FLRW, coords = Spherical, indices = (1,), components = [1, 0, 0 In [ ]: Out[ ]:\n``` \nu μ ∣ ∣ ∣ ( t , r , θ , ϕ ) = ⎛ ⎜ ⎜ ⎜ ⎝ 1 0 0 0 ⎞ ⎟ ⎟ ⎟ ⎠ \nUsing the 4-velocity and the metric, we redefine the perfect fluid stress tensor in the FLRW spacetime using the formula , and give and spacetime dependence: T μν = ( ρ + p ) u μ u ν + pg μν ρ p \n```\nrho\\_t\\_r\\_t\\_p = T . func("rho")(t, r, theta, phi) p\\_t\\_r\\_t\\_p = T . func("p")(t, r, theta, phi) PerfectFluidFLRW = T . calc( formula = (rho\\_t\\_r\\_t\\_p + p\\_t\\_r\\_t\\_p) * RestVelocity(mu) @ RestVelocity(nu) + p\\_t\\_r\\_t\\_p symbol = "T", ) In [ ]:\n``` \nOut[ ]: \nT μν ∣ ∣ ∣ ( t , r , θ , ϕ ) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ρ ( t , r , θ , ϕ ) 0 0 0 0 0 0 0 0 0 0 0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ p a 2 p a 2 r 2 p a 2 r 2 sin 2 ( θ ) \nFinally, we take the covariant derivative of the stress tensor: \n~ (T . CovariantD(mu) @ In [ ]: \n```\nPerfectFluidFLRW(mu, nu))\n``` \n∇ μ T μ t = ∇ μ T μ r = ∇ μ T μθ = ∇ μ T μϕ = ∂ t ρ ( t , r , θ , ϕ ) a +3 ∂ t a ( p + ρ ( t , r , θ , ϕ )) a ∂ r p a 2 ∂ θ p a 2 r 2 ∂ ϕ p a 2 r 2 sin 2 ( θ ) \nFrom demanding that the component vanishes, we get the following equation: t \n˙ ρ = -3( ρ + p ) . ˙ a a \nWe see that in an expanding universe, energy is not conserved, but rather, the energy density changes with time in a way that depends on the scale factor. If the universe is not expanding, that is, , then energy will be conserved. ˙ a = 0', 'The curve Lagrangian': 'Consider a curve , which is a function on the manifold where is called the curve parameter . The curve Lagrangian of a metric is defined as the norm-squared of the tangent to the curve: x μ ( λ ) λ \nL = g μν ˙ x μ ˙ x ν , \nwhere is the first derivative of with respect to the curve parameter (in Newton dot notation). We can calculate it using the method lagrangian() of the Metric class. For example: ˙ x μ x μ λ \nMinkowski . lagrangian() In [ ]: \nOut[ ]: \nSchwarzschild . lagrangian() In [ ]: \nOut[ ]: \nL ∣ ∣ ∣ ( t , r , θ , ϕ ) = -˙ r 2 r 2 + ˙ t 2 (2 M -r ) 2 + r 3 (2 M -r ) ( ˙ ϕ 2 sin 2 ( θ ) + ˙ θ 2 ) r (2 M -r ) \nFLRW . lagrangian() In [ ]: \nOut[ ]: \nL ∣ ∣ ∣ ( t , r , θ , ϕ ) = ˙ ϕ 2 a 2 r 2 sin 2 ( θ ) + ˙ θ 2 a 2 r 2 + ˙ r 2 a 2 -˙ t 2 \nAlcubierre . lagrangian() In [ ]: \nOut[ ]: \nL ∣ ∣ ∣ ( t , x , y , z ) = ˙ t 2 ( f 2 v 2 -1 ) -2 ˙ t ˙ zfv + ˙ x 2 + ˙ y 2 + ˙ z 2 \nNotice how show() (and list() as well) use Newton dot notation for the derivatives of the \nL ∣ ∣ ∣ ( t , x , y , z ) = -˙ t 2 + ˙ x 2 + ˙ y 2 + ˙ z 2 \nIn [ ]: \ncoordinate functions, for improved readability. To get the full expressions with the explicit derivatives, we can use components() . For example: \nMinkowski . lagrangian() . components() \nOGRePy : Using default coordinate system Cartesian and default index configuration (). \nOut[ ]: [ -( t ( λ ) ) 2 + ( x ( λ ) ) 2 + ( y ( λ ) ) 2 + ( z ( λ ) ) 2 ] d d λ d d λ d d λ d d λ', 'Geodesic equations from the Lagrangian': "By applying the Euler-Lagrange equations to the curve Lagrangian: \n( ) -= 0, d d λ ∂ L ∂˙ x μ ∂ L ∂ x μ \nwe can obtain the geodesic equations for our spacetime. This is done using the method geodesic\\_from\\_lagrangian() of the Metric class. For the Minkowski metric, the geodesic equations are: \n~ Minkowski . geodesic\\_from\\_lagrangian() In [ ]: \n0 t = -∂ λ ( -˙ t ) 0 x = -x 0 y = -y 0 z = -z \nNote that this method only calculates the left-hand side of the Euler-Lagrange equations; if we equate the result to zero, we will get the actual geodesics equations. This is hinted at visually by setting the resulting tensor's symbol to 0, so that you actually see the equations when using list() . It is trivial to see that the solution to these equations is simply a curve with a constant velocity; in a flat Minkowski spacetime, particles experience no gravitational force, and thus no acceleration (unless some other force acts on them, of course). \nThe derivatives with respect to the curve parameter are kept unevaluated in the output of geodesic\\_from\\_lagrangian() , by using the SymPy Derivative class and passing doit=False to simplify() . This simplifies the equations, and can sometimes help solve them by inspection. In this simple example, since SymPy simplifies the second derivatives even if doit=False is used, the second derivatives are actually evaluated (except from the first one, due to the minus sign), but in more complicated metrics they will remain unevaluated. λ \nIf we want to activate the derivatives, we simply need to apply the doit() method to them. Recall that list() and show() can apply a function to the tensor's components before displaying them, so we just need to pass a lambda function that executes the doit() method on each component: \n0 t = t 0 x = -x 0 y = -y 0 z = -z \nNow the derivatives have been activated. \nAs with the Lagrangian itself, the geodesic equations are displayed in compact notation when using list() . If we want the full expressions with the explicit derivatives, for example in order to pass them to dsolve() and actually solve the equations, we can use components() -remembering to apply doit() to activate the derivatives: \nMinkowski . geodesic\\_from\\_lagrangian() . components() . doit() In [ ]: \nOGRePy : Using default coordinate system Cartesian and default index configuration (1,). \nOut[ ]: [ t ( λ ) -x ( λ ) -y ( λ ) -z ( λ ) ] d 2 d λ 2 d 2 d λ 2 d 2 d λ 2 d 2 d λ 2 \nThis is a SymPy Array where each of the 4 components is a differential equation (with assumed). It can be easily solved by passing it to SymPy's dsolve() : = 0 \nT . s . Array(T . s . dsolve(Minkowski . geodesic\\_from\\_lagrangian() . components() . doit())) In [ ]: \nOGRePy : Using default coordinate system Cartesian and default index configuration (1,). \nOut[ ]: [ t ( λ ) = C 1 + C 2 λ x ( λ ) = C 3 + C 4 λ y ( λ ) = C 5 + C 6 λ z ( λ ) = C 7 + C 8 λ ] \nNote that this will return a list of solutions, so we converted it back to a SymPy Array so it will be displayed nicely in the notebook. \nWe can similarly find the geodesic equations of other metrics. For example, here they are for the Schwarzschild metric: \n~ Schwarzschild . geodesic\\_from\\_lagrangian() In [ ]: \n( \n0 t = -∂ λ ( ) 0 r = ) ˙ t (2 M -r ) r -4 M 3 ˙ t 2 +4 M 2 ˙ ϕ 2 r 3 sin 2 ( θ ) + 4 M 2 ˙ θ 2 r 3 +4 M 2 ˙ t 2 r -4 M 2 ∂ λ ( -) r 2 -4 M ˙ ϕ 2 r 4 sin 2 -∂ λ ( -) r 4 ˙ rr 2 M -r ˙ rr 2 M -r r 2 (4 M 2 -4 Mr + 2 \n0 θ = -∂ λ ( ˙ θ r 2 0 ϕ = -∂ λ ( ˙ ϕ r 2 sin 2 ( θ ) ) ˙ ϕ r 2 sin (2 θ ) 2 \n(Note that the component is very long, ugly, and complicated. In the Mathematica version of OGRe, we get a much shorter and nicer expression, but if the two expressions are compared by exporting this expression from SymPy to Mathematica (which can be done using the mathematica() method), it turns out that the SymPy expression is in fact correct, just not simplified properly. This appears to be an issue with SymPy's simplify() function, but it could perhaps be resolved by using specific SymPy simplification functions, and it is possible that in the future SymPy's simplification algorithms will improve.) r \nAs another example, here are the geodesic equations for the FLRW metric: \n~ FLRW . geodesic\\_from\\_lagrangian() In [ ]: \n0 t = ˙ ϕ 2 ∂ t aar 2 sin 2 ( θ ) + ˙ θ 2 ∂ t aar 2 + ˙ r 2 ∂ t aa -∂ λ ( -˙ t ) 0 r = ˙ ϕ 2 a 2 r sin 2 ( θ ) + ˙ θ 2 a 2 r -∂ λ ( ˙ ra 2 ) 0 θ = -∂ λ ( ˙ θ a 2 r 2 ) 0 ϕ = -∂ λ ( ˙ ϕ a 2 r 2 sin 2 ( θ ) ) ˙ ϕ 2 a 2 r 2 sin (2 θ ) 2 \nAnd finally, here they are for the Alcubierre metric: \n~ Alcubierre . geodesic\\_from\\_lagrangian() In [ ]: \n0 t = ˙ t 2 fv ( ∂ t fv + ∂ t vf ) -˙ t ˙ z ∂ t fv -˙ t ˙ z ∂ t vf -∂ λ ( ˙ t ( f 2 v 2 -1 ) - ˙ zfv ) 0 x = -x + ˙ t 2 ∂ x ffv 2 -˙ t ˙ z ∂ x fv 0 y = -y + ˙ t 2 ∂ y ffv 2 -˙ t ˙ z ∂ y fv 0 z = ˙ t 2 ∂ z ffv 2 -˙ t ˙ z ∂ z fv -∂ λ ( -˙ t fv + ˙ z ) \nThe latter is a good example of how we can solve the geodesic equations by inspection. Indeed, it is easy to see that \n˙ x μ = (1, 0, 0, vf ) \nis a solution to this system of equations, since then we have and , and both terms in each equation vanish (the last term in the first equation will reduce to , which is of course zero). We can check this solution by replacing the coordinate functions with their solutions; since we will be left with in the first equation, we must also activate the derivative. ˙ x = ˙ y = 0 ( fv ˙ t - ˙ z ) = 0 ∂ λ ( -1) ∂ λ ( -1) \nHowever, for this we have to write the coordinates explicitly as functions of the curve parameter, even when they are arguments of a function; for example, should be instead be . Luckily, OGRePy offers several ways to simplify this process. The Coordinates class contains the method of\\_param() , which returns the coordinates as functions of the curve parameter: v ( t ) v ( t ( λ )) \n```\nCartesian . of\\_param() [t(\\lambda), x(\\lambda), y(\\lambda), z(\\lambda)] However, what we really want is an easy way to replace the coordinates with functions of the curve parameter. We can obtain a list of such replacements using the method of\\_param\\_dict() : param = Cartesian . of\\_param\\_dict() {t: t(\\lambda), x: x(\\lambda), y: y(\\lambda), z: z(\\lambda)} Similarly, of\\_param\\_dot() returns the first derivatives of the coordinates: Cartesian . of\\_param\\_dot() [Derivative(t(\\lambda), \\lambda), Derivative(x(\\lambda), \\lambda), Derivative(y(\\lambda), \\lambda), Derivative(z(\\lambda), \\lambda)] But again, what we really want is a dictionary of replacements, obtained using of\\_param\\_dot\\_dict() : dot = Cartesian . of\\_param\\_dot\\_dict() {t: Derivative(t(\\lambda), \\lambda), x: Derivative(x(\\lambda), \\lambda), y: Derivative(y(\\lambda), \\lambda), z: Derivative(z(\\lambda), \\lambda)} We can now use the param dictionary as an argument to the subs() method to replace the arguments in the function and : display(v\\_t . subs(param)) display(f\\_t\\_x\\_y\\_z . subs(param)) In [ ]: Out[ ]: In [ ]: Out[ ]: In [ ]: Out[ ]: In [ ]: Out[ ]: v f In [ ]: v ( t ( λ ))\n``` \nf ( t ( λ ), x ( λ ), y ( λ ), z ( λ )) \nThe explicit solution is given by \n```\n˙ t ( λ ) = 1, ˙ x ( λ ) = 0, ˙ y ( λ ) = 0, ˙ z ( λ ) = v ( t ( λ )) × f ( t ( λ ), x ( λ ), y ( λ ), z ( λ )).\n``` \nLet us define a dictionary of replacements which maps each derivative of the coordinates to its solution: \n```\nsolution = {dot[t]: 1, dot[x]: 0, dot[y]: 0, dot[z]: v\\_t . subs(param) * f\\_t\\_x\\_y\\_z . subs(param {Derivative(t(\\lambda), \\lambda): 1, Derivative(x(\\lambda), \\lambda): 0, Derivative(y(\\lambda), \\lambda): 0, Derivative(z(\\lambda), \\lambda): f(t(\\lambda), x(\\lambda), y(\\lambda), z(\\lambda))*v(t (\\lambda))} Now all we need to do is perform these substitution, and then simplify. We can do this by passing , and setting will also In [ ]: Out[ ]:\n``` \nthe dictionary as the value of the replace argument to instruct list() simplify=True to the expression after doing the replacement. Note that simplify() automatically call doit() to evaluate the derivatives with respect to : λ \n```\nAlcubierre . geodesic\\_from\\_lagrangian() . list(replace = solution, simplify =True In [ ]:\n``` \n```\n)\n``` \nNo non-zero elements. \nSince this solution zeros all the elements, it must be the correct solution to the geodesic equations. We could use a substitution procedure similar to the one we used here to verify solutions to any geodesic equations. \nThis solution indicates that we are traveling with velocity in the direction; the warp bubble (inside which, as you recall, ) moves whatever is inside it, such as a spaceship, through space at the velocity , but there is no limit on - it can even be faster than light! v z f = 1 v v", 'Geodesic equations from the Christoffel symbols': 'Another way of obtaining the geodesic equations is using the covariant derivative, and thus the Christoffel symbols: \n˙ x ρ ∇ ρ ˙ x σ = 0 ⟹ x σ + Γ σ μν ˙ x μ ˙ x ν = 0. \nIn OGRePy, we can calculate the left-hand side of this equation using the geodesic\\_from\\_christoffel() method of the Metric class. For example:', '~ Schwarzschild . geodesic\\_from\\_christoffel()': '~ FLRW . geodesic\\_from\\_christoffel() \n~ Alcubierre . geodesic\\_from\\_christoffel() \n0 t = t \n0 x = x \n0 y = y \n0 z = z \n0 t = -2 M ˙ r ˙ t + t r (2 M -r ) r (2 M -r ) \n0 r = M ˙ r 2 r 2 -M ˙ t 2 (2 M -r ) 2 + r 3 (2 M -r ) ( r + ˙ ϕ 2 (2 M -r ) sin 2 ( θ ) + ˙ θ 2 (2 M -r ) ) r 3 (2 M -r ) \n2 \n0 θ = ¨ θ -+ 0 ϕ = ¨ ϕ + + ˙ ϕ sin (2 θ ) 2 2 ˙ θ ˙ r r 2 ˙ ϕ ˙ θ tan ( θ ) 2 ˙ ϕ ˙ r r \n0 t = t + ˙ ϕ 2 ∂ t aar 2 sin 2 ( θ ) + ˙ θ 2 ∂ t aar 2 + ˙ r 2 ∂ t aa 0 r = r -˙ ϕ 2 r sin 2 ( θ ) -˙ θ 2 r + 0 θ = ¨ θ -+ + 0 ϕ = ¨ ϕ + + + 2 ˙ r ˙ t ∂ t a a ˙ ϕ 2 sin (2 θ ) 2 2 ˙ θ ˙ r r 2 ˙ θ ˙ t ∂ t a a 2 ˙ ϕ ˙ θ tan ( θ ) 2 ˙ ϕ ˙ r r 2 ˙ ϕ ˙ t ∂ t a a \n0 t = t + ˙ t 2 ∂ z ff 2 v 3 -˙ t ˙ x ∂ x ffv 2 -˙ t ˙ y ∂ y ffv 2 -2 ˙ t ˙ z ∂ z ffv 2 + ˙ x ˙ z ∂ x fv + ˙ y ˙ z ∂ y fv + ˙ z 2 ∂ z fv \n0 x = x -˙ t 2 ∂ x ffv 2 + ˙ t ˙ z ∂ x fv \n0 y = y -˙ t 2 ∂ y ffv 2 + ˙ t ˙ z ∂ y fv \n0 z = z + ˙ t 2 ( -∂ t fv -∂ t vf + ∂ z ff 3 v 4 -∂ z ffv 2 ) -˙ t ˙ x ∂ x fv ( f 2 v 2 +1 ) -˙ t ˙ y ∂ y fv ( f 2 v 2 +1 ) -2 ˙ t \nOften, you will find that the Lagrangian method produces simpler equations, which can even be solved by inspection, as we did for the Alcubierre metric. This is due to the possibility of leaving the derivative unevaluated. However, in other cases, the Christoffel method might produce simpler equations. We can clearly see that by comparing the geodesics equations for the Schwarzschild metric obtained via the Lagrangian vs. Christoffel methods. λ \nThe best thing to do is to try both methods and see which one produces simpler or nicer results for the specific metric in question. Note that the system of equations obtained using geodesic\\_from\\_lagrangian() will often be different from the one obtained using geodesic\\_from\\_christoffel() , but both systems will always have the same solutions.', 'Geodesics equations in terms of the time coordinate': 'If the metric is a spacetime metric, it is often convenient to obtain the geodesic equations in terms of the time parameter, instead of an affine curve parameter. It can be shown that the geodesic equations in terms of the time coordinate are given by \n+ ( Γ σ μν -Γ 0 μν ) = 0, d 2 x σ d t 2 d x σ d t d x μ d t d x ν d t \nwhere we are assuming the time coordinate is and it is the first (zero) coordinate. These equations can be obtained using the geodesic\\_time\\_param() method of the Metric class. Note that geodesic\\_time\\_param() assumes time is the first coordinate, but the coordinate does not need to have the symbol . As an example, the equations for the FLRW metric in Cartesian coordinates in terms of a curve parameter are: t t \nFLRW . geodesic\\_from\\_christoffel() . list(coords = Cartesian) In [ ]: \n0 t = t + ˙ x 2 ∂ t aa + ˙ y 2 ∂ t aa + ˙ z 2 ∂ t aa 0 x = x + 0 y = y + 0 z = z + 2 ˙ t ˙ x ∂ t a a 2 ˙ t ˙ y ∂ t a a 2 ˙ t ˙ z ∂ t a a \nBut in terms of , we only need 3 equations: t \nFLRW . geodesic\\_time\\_param() . list(coords = Cartesian) In [ ]: \n0 x = x - ˙ x 3 ∂ t ( a ) a - ˙ x ˙ y 2 ∂ t ( a ) a - ˙ x ˙ z 2 ∂ t ( a ) a + 0 y = y - ˙ x 2 ˙ y ∂ t ( a ) a - ˙ y 3 ∂ t ( a ) a - ˙ y ˙ z 2 ∂ t ( a ) a + 0 z = z - ˙ x 2 ˙ z ∂ t ( a ) a - ˙ y 2 ˙ z ∂ t ( a ) a - ˙ z 3 ∂ t ( a ) a + 2 ˙ x ∂ t ( a ) a 2 ˙ y ∂ t ( a ) a 2 ˙ z ∂ t ( a ) a \nThese equations are easier to solve. For simplicity, assume that we are only moving along the x coordinate. Then we only have one equation to solve: \nFLRW\\_eq = FLRW . geodesic\\_time\\_param() . components(coords = Cartesian, indices = (1,))[1] In [ ]: Out[ ]: 2 \n-a ( t ) a ( t ) ( x ( t ) ) 3 -a ( t ) a ( t ) x ( t ) ( y ( t ) ) 2 -a ( t ) a ( t ) x ( t ) ( z ( t ) ) 2 + x ( t ) + d dt d dt d dt d dt d dt d dt d dt d dt d 2 dt 2 \nWe are assuming that , so let us remove them from the equation. First, let us get ˙ y ( t ) = ˙ z ( t ) = 0 \ndictionaries mapping the coordinates to functions of time. This is identical to what we did above for the Alcubierre metric, except that this time we pass to the of\\_param functions so we get functions of instead of : t t λ \nparam = Cartesian . of\\_param\\_dict(t) In [ ]: \n{t: t(t), x: x(t), y: y(t), z: z(t)} Out[ ]: \ndot = Cartesian . of\\_param\\_dot\\_dict(t) In [ ]: \nOut[ ]: \n{t: Derivative(t(t), t), \nx: Derivative(x(t), t), \ny: Derivative(y(t), t), \nz: Derivative(z(t), t)} \nIf we now substitute in the equation, it simplifies considerably: ˙ y ( t ) = ˙ z ( t ) = 0 \nFLRW\\_eq . subs({dot[y]: 0, dot[z]: 0}) In [ ]: \nOut[ ]: -a ( t ) a ( t ) ( x ( t ) ) 3 + x ( t ) + d dt d dt d 2 dt 2 2 a ( t ) x ( t ) d dt d dt a ( t ) \nThe solution can be obtained using dsolve() in terms of an integral over , passing in the second argument as the function to solve for. The command to do that is all\\_solutions = T.s.Array(T.s.dsolve(FLRWEq.subs({dot[y]: 0, dot[z]: 0}), param[x])) , but I did not include it in the notebook because it takes a very long time to run. The solution found by SymPy is (after some beautification): a ( t ) x ( t ) \nx ( t ) = C 1 ± ∫ d t 1 a ( t ) √ 1 + C 2 a 2 ( t ) \nWe can use the rhs property to obtain the right-hand side of the equation, selecting the positive solution (at position 1 of the array): all\\_solutions[1].rhs . By taking the derivative with respect to , all\\_solutions[1].rhs.diff(t) , we can get the coordinate velocity along : t ˙ x x \n˙ x ( t ) = 1 a ( t ) √ 1 + C 2 a 2 ( t )', 'Changing the curve parameter': 'By default, the curve parameter is . However, sometimes we want to use another parameter - for example for proper time. To change the parameter, we can set T.options.curve\\_parameter to a symbol of our choice, whether as a string or a SymPy Symbol . As an example, let us change it to : λ τ τ \npreviously calculated . The curve parameter should instead be changed at the beginning of the session, before calculating any tensors which use it, such as Lagrangians or geodesic equations. Indeed, the curve parameter for the Minkowski Lagrangian, which we already calculated, is still : λ \nMinkowski . lagrangian() . In [ ]: \n```\ncomponents()\n``` \nOGRePy : Using default coordinate system Cartesian and default index configuration (). \nOut[ ]: [ -( t ( λ ) ) 2 + ( x ( λ ) ) 2 + ( y ( λ ) ) 2 + ( z ( λ ) ) 2 ] d d λ d d λ d d λ d d λ \nBut for newly calculated Lagrangians, it will be : τ \n```\nT . Metric( coords = Cartesian, components = T . diag( -1, 1, 1, 1), ) . lagrangian() . components() In [ ]:\n``` \nOGRePy : Using default coordinate system Cartesian and default index configuration (). \nOut[ ]: [ -( t ( τ ) ) 2 + ( x ( τ ) ) 2 + ( y ( τ ) ) 2 + ( z ( τ ) ) 2 ] d d τ d d τ d d τ d d τ', 'Bug reports and feature requests': 'This package is under continuous and active development. If you encounter any bugs, or if you would like to request any additional features, please feel free to open a new issue on GitHub and I will look into it as soon as I can.', 'Contribution and pull request policy': "Contributions are always welcome. However, I release my projects in cumulative updates after editing and testing them locally on my system, so my policy is to never accept any pull requests . If you open a pull request, and I decide to incorporate your suggestion into the project, I will first modify your code to comply with the project's coding conventions (formatting, syntax, naming, comments, programming practices, etc.), and perform some tests to ensure that the change doesn't break anything. I will then merge it into the next release of the project, possibly together with some other changes. The new release will also include a note in CHANGELOG.md with a link to your pull request, and modifications to the documentation in README.md as needed. \nTo create a development environment for this package, download the source code directly from the GitHub repository, then create a virtual environment in the root folder of the repository as explained above, activate it, and run pip install ipykernel sympy to install the dependent packages. \nFor your convenience, a PowerShell script, update\\_packages.ps1, is provided in the GitHub repository to allow easily updating all outdated packages. Another script, compile\\_docs.ps1, is used to compile the documentation in README.md to a Jupyter notebook and run the notebook. Finally, cleanup.ps1, is used to clean up Python and Jupyter cache folders. \nThis package was developed in Visual Studio Code. The .vscode folder is provided in the GitHub repository for your convenience, including tasks for running the above PowerShell scripts. It is highly recommended to install the following linters: \n- · Pyright: install the Pylance VS Code extension.\n- · Ruff: pip install ruff and install the VS Code extension.\n- · Pylint: pip install pylint and install the VS Code extension. \nConfigurations for all 3 linters are included in the pyproject.toml file in the GitHub repository.", 'Starring the repository': 'If you found this project useful, please consider starring it on GitHub! This allows me to see how many people are using my code, and motivates me to keep working to improve it.', 'Acknowledgements': 'I would like to thank my student Jared Wogan, whose undergraduate research project, a preliminary Python port of my Mathematica package OGRe, motivated and inspired me to eventually write my own port, OGRePy. I acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), RGPIN-2024-04063.', 'Copyright and citing': 'Copyright (c) 2024 Barak Shoshany. Licensed under the MIT license. \nIf you use this package in software of any kind, please provide a link to the GitHub repository in the source code and documentation. If you use this package in published research, please cite this arXiv paper.', 'Other projects to check out': 'This package is a Python port of OGRe: An Object-Oriented General Relativity Package for Mathematica [1]. You may also be interested in BS::thread\\_pool : a fast, lightweight, and easyto-use C++17 thread pool library for high-performance scientific computing [5].', 'References': '[1] Barak Shoshany, "OGRe: An Object-Oriented General Relativity Package for Mathematica", Journal of Open Source Software, 6(65), 3416 (2021), doi:10.21105/joss.03416, arXiv:2109.04193 \n(2021) \n- [2] Meurer A, Smith CP, Paprocki M, Č ertík O, Kirpichev SB, Rocklin M, Kumar A, Ivanov S, Moore JK, Singh S, Rathnayake T, Vig S, Granger BE, Muller RP, Bonazzi F, Gupta H, Vats S, Johansson F, Pedregosa F, Curry MJ, Terrel AR, Rou č ka Š, Saboo A, Fernando I, Kulal S, Cimrman R, and Scopatz A., "SymPy: Symbolic Computing In Python", PeerJ Computer Science 3:e103, doi:10.7717/peerjcs.103 (2017)\n- [3] Fernando Perez and Brian E. Granger, "IPython: A System for Interactive Scientific Computing", Computing in Science & Engineering, vol. 9, no. 3, pp. 21-29, May-June 2007, doi:10.1109/ MCSE.2007.53 (2007) \n[4] Marijan Beg, Juliette Taka, Thomas Kluyver, Alexander Konovalov, Min Ragan-Kelley, Nicolas M. Thiéry, and Hans Fangohr, "Using Jupyter for Reproducible Scientific Workflows", Computing in Science & Engineering 23, no. 2 (2021): 36-46, doi:10.1109/MCSE.2021.3052101 (2021) \n[5] Barak Shoshany, "A C++17 Thread Pool for High-Performance Scientific Computing", SoftwareX 26 (2024) 101687, doi:10.1016/j.softx.2024.101687, arXiv:2105.00613 (2024)'}

2024MNRAS.534.1700E

Stellar mass and specific angular momentum are two properties of a galaxy that are directly related to its formation history and hence morphology. In this work the tight planar relationship between stellar specific angular momentum inlineformulatexmath idTM0001 notationLaTeXjtexmathinlineformula mass inlineformulatexmath idTM0002 notationLaTeXMtexmathinlineformula and mean effective surface brightness inlineformulatexmath idTM0003 notationLaTeXleftlangle mu mathrmeffrightrangle texmathinlineformula that was recently constrained using ALFALFA Arecibo Legacy Fast ALFA galaxies is measured more accurately using galaxies from the SIMBA cosmological simulation. The distribution of 179 SIMBA galaxies in inlineformulatexmath idTM0004 notationLaTeXlog 10j log 10Mtexmathinlineformulainlineformulatexmath idTM0005 notationLaTeXleftlangle mu mathrmeffrightrangle texmathinlineformula space is shown to be very tightly planar with inlineformulatexmath idTM0006 notationLaTeXjpropto M0.694texmathinlineformula and the distribution of perpendicular distances between the galaxies and the plane being approximately Gaussian with inlineformulatexmath idTM0007 notationLaTeXmathrmRMS0.057texmathinlineformula dex. The parametrized distribution is used with existing inlineformulatexmath idTM0008 notationLaTeXjtexmathinlineformula and inlineformulatexmath idTM0009 notationLaTeXleftlangle mu mathrmeffrightrangle texmathinlineformula measurements of 3607 ALFALFA galaxies and 84 SPARC Spitzer Photometry and Accurate Rotation Curves galaxies to reliably predict their published stellar masses to within inlineformulatexmath idTM0010 notationLaTeXsim 0.1texmathinlineformula0.2 dex over several decades of stellar mass. Thus this work presents a new method of easily generating accurate galaxy stellar mass estimates for latetype galaxies and provides a new measurement of the fundamental link between galaxy morphology mass and angular momentum.

2024-11-01T00:00:00Z

['arXiv:2409.08076', '10.48550/arXiv.2409.08076', '2024MNRAS.534.1700E', '2024arXiv240908076E', '2024MNRAS.tmp.2103E', '10.1093/mnras/stae2145']

['Astrophysics - Astrophysics of Galaxies']

Using the SIMBA cosmological simulations to measure the planar relation between stellar specific angular momentum mass and effective surface brightness

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']

https://arxiv.org/pdf/2409.08076.pdf

{'E. Elson, 1 ⋆': '1 Department of Physics & Astronomy, University of the Western Cape, Robert Sobukwe Rd, Bellville, 7535, South Africa \nAccepted 2024 September 11. Received 2024 September 11; in original form 2024 June 6', 'ABSTRACT': 'Stellar mass and specific angular momentum are two properties of a galaxy that are directly related to its formation history, and hence morphology. In this work, the tight planar relationship between stellar specific angular momentum ( j ∗ ), mass ( M ∗ ) and mean effective surface brightness ( ⟨ µ eff ⟩ ) that was recently constrained using ALFALFA galaxies is measured more accurately using galaxies from the Simba cosmological simulation. The distribution of 179 Simba galaxies in log 10 j ∗ -log 10 M ∗ -⟨ µ eff ⟩ space is shown to be very tightly planar with j ∗ ∝ M 0 . 694 ∗ and the distribution of perpendicular distances between the galaxies and the plane being approximately Gaussian with rms = 0 . 057 dex. The parameterised distribution is used with existing j ∗ and ⟨ µ eff ⟩ measurements of 3 607 ALFALFA galaxies and 84 SPARC galaxies to reliably predict their published stellar masses to within ∼ 0 . 1 to 0.2 dex over several decades of stellar mass. Thus, this work presents a new method of easily generating accurate galaxy stellar mass estimates for late-type galaxies and provides a new measurement of the fundamental link between galaxy morphology, mass and angular momentum. \nKey words: galaxies: evolution - galaxies: kinematics and dynamics', '1 INTRODUCTION': "In a Λ CDM Universe, the hierarchical buildup of galaxies implies that a galaxy's angular momentum content is primarily acquired through the tidal torques exerted during mergers and interactions with other structures (e.g., Hoyle 1951; Peebles 1969). This process results in the transfer and redistribution of angular momentum, influencing the rotation and spin of galaxies as they evolve and grow. For dark matter haloes in a standard Λ CDM Universe, j h ∝ λM 2 / 3 h , where j h is a halo's specific angular momentum and M h its mass. By considering the ratio of a galaxy's stellar mass to that of its halo, as well as the ratio of the average specific angular momentum in stars in the galaxy to that of its halo, its total stellar specific angular momentum can be shown to be j ∗ ∝ M 2 / 3 ∗ , where M ∗ is its stellar mass 1 . \nMore than 40 years ago, Fall (1983) provided empirical evidence for the existence of a tight power-law relationship between a galaxy's stellar specific angular momentum and its mass: j ∗ = βM α ∗ , where α ≈ 0 . 6 and β is related to galaxy morphology. Since then, many other studies have confirmed the power-law relationship and have further constrained its parameters (e.g., Romanowsky & Fall 2012a; Fall & Romanowsky 2013; Obreschkow & Glazebrook 2014; Cortese \n- ⋆ E-mail: eelson@uwc.ac.za (EE) \net al. 2016; Elson 2017; Posti et al. 2018b, and others). The value of α is usually close to 2/3 while galaxy morphology or suitable proxies thereof - seems to affect the value of β . \nSeveral investigators have explicitly studied how the scatter seen in the j ∗ -M ∗ relation is related to various proxies of galaxy morphology. Considering a galaxy's gas fraction, f gas 2 , Mancera Piña et al. (2021b) fitted 2D planes to the distributions of galaxies in log 10 j ∗ -log 10 M ∗ -log 10 f gas space and reported that at fixed M ∗ galaxies with higher f gas typically also have higher j ∗ . More recently, Elson (2024) used a large sample of 3607 galaxies from the Arecibo Legacy Fast ALFA (ALFALFA) survey (Haynes et al. 2018) to demonstrate for them the existence of a very tight planar distribution in log 10 j ∗ -log 10 M ∗ -〈 µ I eff 〉 space, where 〈 µ I eff 〉 - the magnitude measure (in the SDSS I -band) of the average flux surface density of a galaxy within its effective radius - was used as a proxy for galaxy morphology. Projecting 〈 µ I eff 〉 -selected subsamples of the 3607 galaxies onto log 10 j ∗ -log 10 M ∗ space yielded the tightest j ∗ -M ∗ relations measured to date over the stellar mass range 8 ≲ log 10 ( M ∗ /M ⊙ ) ≲ 11 . 3 \nThe goal of the present study is to use galaxies from \n- 2 f gas = 1 . 33 M HI /M ∗\n- 3 Note that the Elson (2024) study was based on spatiallyunresolved H i imaging while the studies of Posti et al. (2018b) and Mancera Piña et al. (2021b) were based on spatially-resolved H i maps. \nthe Simba cosmological galaxy formation simulations (Davé et al. 2019) to better constrain the distribution of galaxies in log 10 j ∗ -log 10 M ∗ -⟨ µ eff ⟩ space. A parameterisation of the planar distribution of Simba galaxies in the 3D space is shown to serve as a useful tool for accurately estimating the stellar masses of real galaxies for which there are available measurements of their global properties that can be used to constrain their j ∗ and ⟨ µ eff ⟩ quantities. \nThe layout of this paper is as follows. Section 2 briefly introduces the Simba simulations. The data products generated and used in this study are discussed in Section 3. The sample of Simba galaxies used in this study is discussed and presented in Section 4. The study's main results appear in Section 5 while a demonstration of their utility to predict the stellar masses of real galaxies is given in Section 6. Finally, Section 7 offers a summary of this study.", '2 SIMBA SIMULATIONS': "The Simba cosmological galaxy formation simulations provide all of the data used in this study. Simba is run with Gizmo's meshless finite mass hydrodynamics. The simulation includes prescriptions for various star formation processes including blackhole growth. This work utilises the z = 0 simulation box of the highest resolution, consisting of 512 3 dark matter particles and 512 3 gas elements with periodic volume of size 25 h -1 Mpc 3 . The associated mass resolutions are 1 . 2 × 10 7 M ⊙ and 2 . 28 × 10 6 M ⊙ for the dark matter particles and gas elements, respectively. Rather than explicitly model physical processes that control the cold phase of the interstellar medium, Simba uses various prescriptions to transform ionised gas into atomic and molecular phases. Star formation is modelled using an H 2 -based Schmidt (1959) relation that uses H 2 density and dynamical time. To model stellar evolution, a Chabrier (2003) stellar initial mass function is assumed and a 6D friends-of-friends algorithm is used to group star particles into galaxies. \nA total of 5218 galaxies are associated with the 25 h -1 Mpc 3 simulation volume. For Simba , these galaxies have had their properties computed using Caesar 4 , which is a particle-based extension to YT 5 which itself is a Python package for visualising volumetric data. Photometry for Simba galaxies is performed using PyLoser 6 , which is a Python version of Loser package (Davé et al. 2017) that computes line-of-sight dust column densities through the gas of a given galaxy.", '3 MEASUREMENTS': "This study aims to measure the distribution of galaxies in log 10 j ∗ -log 10 M ∗ -⟨ µ eff ⟩ space. A galaxy's stellar specific angular momentum is approximated as \n˜ j ∗ = 2 V circ R d , (1) \nwhere ˜ j ∗ is the approximation of j ∗ , V circ is the circular velocity and R d is the exponential disc scale length. If a spiral \ngalaxy's stellar mass distribution is well-described by a thin exponential disc, and if a radially constant rotation curve is assumed, ˜ j ∗ provides an excellent approximation of j ∗ . This fact is well-demonstrated by Romanowsky & Fall (2012b) for a sample of nearly 100 nearby galaxies. From this point forward, j ∗ = ˜ j ∗ is assumed. \nAssuming a galaxy's stellar mass to be exponentially distributed with radius, the disc scale length is taken to be R d = R eff / 1 . 68 , where R eff is its effective radius from the Caesar catalogue. Circular velocity is calculated as V c = W 50 / 2 sin i , where W 50 is a measure of the velocity width of its H i line profile, in units of km s -1 , and i is the H i disc inclination angle. This conversion of velocity width to circular velocity assumes corrections for pressure-supported motions to be negligible. Mancera Piña et al. (2021a) showed that for the mass regime of the galaxy sample used in this work, circular velocity is very close to the rotation traced by the gas. \nIn this study, each galaxy's W 50 is measured from an H i line profile that is generated by spatially integrating the flux in the channels of an H i data cube generated for the galaxy. The MARTINI (Mock APERTIF-like Radio Telescope Interferometry of the Neutral ISM) package (Oman et al. 2019; Oman 2019, 2024) is used in this work to create synthetic resolved H i line data cubes of the Simba galaxies. All cubes are given an RA-DEC pixel size of 4 arcsec and a channel width of 5 km s -1 . A distance of 4 Mpc is used for all galaxies while a Gaussian point-spread function of halfpower width 12 arcsec is used to spatially smooth the data. To produce cubes essentially free of noise, Gaussian noise with rms = 3 × 10 -9 Jy arcsec -2 is added before beam convolution. In all cubes, the H i disc of the galaxy is set to be inclined by 60 degrees to the y-axis. MARTINI attempts to identify a preferred disc plane for the galaxy based on the angular momenta of the central 1/3 of particles in the galaxy. Once an H i data cube is created for a galaxy, it is further smoothed to a spatial resolution of 3.5 arcmin and then has the flux in each channel spatially integrated to generate the H i line profile. W 50 is measured as the width of a spectrum at a flux density level equal to half of the peak value. \nThe stellar masses of the galaxies are taken straight from the Caesar catalogue and are used to calculate the magnitude measure of a galaxy's mean flux surface density within its effective radius, ⟨ µ eff ⟩ . This quantity is used in this work to quantify the degree to which a galaxy's light distribution is centrally concentrated. As will be explained in Section 6, parameterised distributions of Simba galaxies in log 10 j ∗ -log 10 M ∗ -⟨ µ eff ⟩ space are used to predict the derived stellar masses of galaxies taken from the ALFALFA survey and the Spitzer Photometry and Accurate Rotation Curves (SPARC, Lelli et al. 2016) database. To this end, ⟨ µ eff ⟩ measurements for the Simba galaxies are needed for the SDSS I -band and the Spitzer 3.6 µ mband. These measurements are henceforth referred to as 〈 µ I eff 〉 and 〈 µ 3 . 6 eff 〉 . \nTo convert a galaxy's mean stellar mass surface density within its effective radius to 〈 µ I eff 〉 and 〈 µ 3 . 6 eff 〉 in units of apparent magnitudes per arcsec 2 , an assumed stellar massto-light ratio is required for each band. For the 3.6 µ m band, 0.5 M ⊙ / L ⊙ is used for all galaxies. This is the value adopted by Lelli et al. (2016) to convert total 3.6 µ m luminosities of SPARC galaxies to total stellar masses. For the SDSS I -band, 1.60 M ⊙ / L ⊙ is used. This value comes from considering the absolute magnitudes of all galaxies, taken from PyLoser \nwhich computes the dust extinction to each star particle in a galaxy based on its metal content. The PyLoser magnitudes are converted to luminosities 7 and then compared to the total stellar masses of the galaxies to yield their I -band stellar mass-to-light ratios. For the galaxies used in this study, these ratios are approximately Gaussian-distributed from 0.45 to 2.80 M ⊙ / L ⊙ , with a median value of 1.60 M ⊙ / L ⊙ and an RMSvalue of 1.67 M ⊙ / L ⊙ . Finally, the assumed stellar massto-light ratios can be combined with the assumed distance of 4 Mpc for all galaxies to generate measurements of their mean effective surface brightnesses. Most galaxies have 〈 µ I eff 〉 in the range 21 to 23 mag arcsec -2 , while 19 to 21 mag arcsec -2 is typical for 〈 µ 3 . 6 eff 〉 . Distributions of these quantities are shown in Figure 1. \nTo generate a realistic uncertainty estimate for each galaxy's stellar specific angular momentum measurement, 8 values of j ∗ are calculated by considering upper and lower uncertainty limits for each of its constituent factors. The range of these 8 j ∗ values is then used as the uncertainty on j ∗ . To generate an uncertainty range for a galaxy's W 50 measurement, the half-power width of the H i spectrum is calculated for the peak flux density measured on either side of the galaxy's systemic velocity. The difference between these two W 50 measurements is taken as the uncertainty. An inclination uncertainty of 7.5 · is used for all galaxies. The uncertainty on effective radius is set equal to 10 per cent of the R eff measurement taken from the Caesar catalogue. Using this approach, the median relative uncertainty on j ∗ is 0.19.", '4 GALAXY SAMPLE': 'To generate a sample of Simba galaxies suitable for the current study, various cuts are applied to the full set of 5218 galaxies from the Caesar catalogue. Firstly, mass cuts are applied: M HI ≥ 1 . 25 × 10 8 M ⊙ and M ∗ ≥ 7 . 25 × 10 8 M ⊙ . Mass cuts such as these ensure the baryonic mass distributions of the galaxies to be reliably simulated. To ensure galaxies have their dynamics dominated by rotation, a cut on dynamical morphology is made such that the fraction of energy invested in the ordered gas rotation is above 0.75 8 . To avoid galaxies with morphologies that are disturbed due to interactions, systems that have galaxy neighbours within a 30 kpc sphere are eliminated. Additionally, each galaxy has an ellipse fitted to a thin flux annulus in the 12-arcsec version of its H i total intensity map consisting of pixels with mass surface densities in the range 0.9 to 1.1 M ⊙ pc -2 . The mean separation between the pixels and the fitted ellipse is calculated in units of kpc. Galaxies with a mean separation greater than 5 per cent the size of the semi-major axis of their ellipse are eliminated. This further ensures the sample consists of galaxies that have neat, rotating H i discs. Finally, galaxies with a relative j ∗ uncertainty greater than 0.25 are removed. The final sample used in this study consists of 179 Simba galaxies. Figure 1 shows the distributions of various measured properties: H i mass, stellar mass, H i -to-stellar mass, velocity width of H i line profile, stellar disc effective radius, effective surface brightness in the \nSDSS I -band and Spitzer 3.6 µ m band, and stellar specific angular momentum.', '5.1 j ∗ -M ∗ relation': 'The last panel of Figure 1 shows the distribution of log 10 j ∗ measurements for the sample of galaxies used in this study. Figure 2 shows log 10 j ∗ as a function of log 10 M ∗ . Over nearly two decades in stellar mass, the M ∗ dependence of j ∗ is wellmodelled by a single power law given by \nlog 10 ( j ∗ kpc km s -1 ) = α log 10 ( M ∗ M ⊙ ) + β, (2) \nwith α = 0 . 38 ± 0 . 01 and β = -0 . 960 ± 0 . 162 . Hardwick et al. (2022) found α = 0 . 47 ± 0 . 02 for a sample of 564 nearby galaxies from the eXtended GALEX Arecibo SDSS Survey. More recently, Elson (2024) measured α = 0 . 404 ± 0 . 003 for a sample of 3607 ALFALFA galaxies spanning the mass range ∼ 10 8 -10 11 M ⊙ .', '5.2 The log 10 j ∗ -log 10 M ∗ - < µ eff > plane': "Several authors have shown the j ∗ -M ∗ relation to vary with morphological type and/or with properties that serve as proxies for morphology. Fall (1983) and Romanowsky & Fall (2012a) showed spirals and ellipticals to follow parallel j ∗ -M ∗ tracks with slopes close to 0.6, with ellipticals containing roughly 3 to 4 times less angular momentum than spirals of equal mass. Spatially resolved H i imaging of 16 THINGS galaxies was used by Obreschkow & Glazebrook (2014) to show that bulge-to-disk mass significantly controls the normalisation of the j ∗ -M ∗ relation. For a large sample of galaxies from the SAMI Galaxy Survey, Cortese et al. (2016) show j ∗ -M ∗ scatter to be strongly correlated with optical morphology (as determined visually and according to Sérsic index). More recently, Mancera Piña et al. (2021b) and Hardwick et al. (2022) showed the H i gas fraction of the interstellar medium to be fundamentally linked to the stellar, baryonic and gas angular momenta of galaxies. \nGiven that the Simba galaxies used in this study are morphologically diverse 9 , the scatter in their j ∗ -M ∗ relation is likely driven significantly by morphology. Accounting for this should yield tighter, more reliable correlations between j ∗ and M ∗ . Elson (2024) showed the distribution of 3607 ALFALFA galaxies in log 10 j ∗ -log 10 M ∗ -〈 µ I eff 〉 space to be very well modelled by a 2D plane that yielded j ∗ ∝ M 0 . 589 ± 0 . 02 ∗ ⟨ µ eff ⟩ 0 . 193 ± 0 . 002 . The scatter of the galaxies about that best-fitting plane was only 0.089 dex. Thus, the degree to which a galaxy's light is centrally concentrated serves as an effective predictor (in addition to its stellar mass) of its stellar specific angular momentum content. \nThe primary aim of the current study is to use the Simba data to better constrain the planar relation between log 10 j ∗ , log 10 M ∗ and ⟨ µ eff ⟩ found by Elson (2024). A 2D plane of the \nFigure 1. Distributions of various measured and derived properties of the 179 Simba galaxies used in this study. From left to right, top to bottom, the panels show the distributions of HI mass, stellar mass, H i gas fraction, velocity width of the H i profile, stellar disc effective radius, effective surface brightness in the SDSS I -band and Spitzer 3.6 µ m band, and stellar specific angular momentum. \n<!-- image --> \nFigure 2. Stellar specific angular momentum as a function of stellar mass for the 179 Simba galaxies used in this study. The solid black line with a slope of 0.38 represents a linear fit to all of the data points. \n<!-- image --> \nform \nlog 10 ( j ∗ kpc km s -1 ) = α log 10 ( M ∗ M ⊙ ) + β ( < µ eff > mag arcsec -2 ) + γ (3) \nis fit to the 179 Simba galaxies for which j ∗ , M ∗ and ⟨ µ eff ⟩ measurements have been generated in this work. The IDL function MPFIT2DFUN, which is part of the MPFIT package of curve fitting and function optimisation routines (Markwardt 2009), is used to perform a Levenberg-Marquardt leastsquares fit of Equation 3 to the data (incorporating the j ∗ uncertainties discussed at the end of Section 3). \nAs previously mentioned, ⟨ µ eff ⟩ measurements for the galaxies are produced for the SDSS I -band and Spitzer 3.6 µ m band by assuming a constant mass-to-light ratio in each band. Therefore, the distributions of the Simba galaxies in log 10 j ∗ -log 10 M ∗ -〈 µ I eff 〉 space and log 10 j ∗ -log 10 M ∗ -〈 µ 3 . 6 eff 〉 space are exactly parallel to one another, offset only along the ⟨ µ eff ⟩ axis. For both distributions, α = 0 . 694 ± 0 . 046 and β = 0 . 190 ± 0 . 026 are the best-fitting plane parameters. The distribution of measured perpendicular distances between the galaxies and either of the two best-fitting planes is very well approximated by a Gaussian, with a median separation of 0.007 dex and an RMS separation of only 0.057 dex. The best-fitting γ parameters for the I -band and 3.6 µ mband planes are -8 . 111 ± 1 . 009 and -7 . 691 ± 0 . 953 , respectively. \nWhile the best-fitting β and γ parameters found for the Simba galaxies (for the 〈 µ I eff 〉 case) are well-consistent within 1 σ of the best-fitting values from Elson (2024), the α value found in the current study is consistent within ∼ 2 . 5 σ . Compared to α = 0 . 58 ± 0 . 002 from Elson (2024), the best-fitting α parameter for the Simba galaxies is ∼ 20 per cent higher. This could be due to uncertainties of up to a few tenths of a dex on the stellar masses from Durbala et al. (2020) that were used to measure α in the ALFALFA study. It could indicate that the approximation used for a galaxy's j ∗ content (in both studies) might be less accurate than expected under certain conditions. The 3607 ALFALFA galaxies from the Elson (2024) study likely have the properties of their H i discs being affected by environmental factors in ways that the Simba galaxies from the present study do not. However, this study's best-fitting α parameter is very close to the value of \nα = 0 . 67 ± 0 . 03 from Mancera Piña et al. (2021b) who fitted planes to the distributions of galaxies in log 10 j ∗ -log 10 M ∗ -log 10 f gas , where f gas is that ratio of total gas mass to total baryonic mass. It is also very close to the theoretical expectation of α = 2 / 3 for galaxies in a Λ CDM cosmology 10 . The very small amount of scatter (0.057 dex) of the Simba galaxies about the best-fitting plane suggests it represents a fundamental relationship between j ∗ , M ∗ and ⟨ µ eff ⟩ , at least for late-type galaxies dominated by rotation. To test this idea, the following section is dedicated to using the plane to predict the stellar masses of real galaxies.", '6 PREDICTING STELLAR MASSES': "Given the very tight planar distributions of the Simba galaxies in log 10 j ∗ -log 10 M ∗ -⟨ µ eff ⟩ space, this study's parameterisations of the best-fitting planes for the 〈 µ I eff 〉 and 〈 µ 3 . 6 eff 〉 cases can be used to accurately predict the stellar masses of real galaxies for which there exist j ∗ and ⟨ µ eff ⟩ measurements (determined independently of M ∗ ).", '6.1 ALFALFA': "The stellar masses of the 3607 ALFALFA galaxies from Elson (2024) are predicted using their j ∗ and 〈 µ I eff 〉 measurements from that study, and compared to the stellar masses from Durbala et al. (2020). \nFigure 3 shows in its top-left panel the 2D distribution of the predicted M ∗ values 11 as a function of those from the Durbala et al. (2020) catalogue. The solid grey line is a fit to the median predicted M ∗ values in bins of width 0.1 dex along the x-axis, while the red line is that of equality. The predictive ability of this study's fitted plane is seen to vary with stellar mass. While galaxies with derived stellar masses close to ∼ 10 9 . 5 M ⊙ are very well predicted, masses are under/over predicted by up to a few tenths of a dex at lower/higher masses. For all 3607 galaxies, the top-right panel in Figure 3 shows the distribution of the differences between predicted and derived stellar masses. Over ∼ 3 decades of stellar mass, the median discrepancy is 0.056 dex while the RMS discrepancy is 0.187 dex. \nThe above-mentioned discrepancies are small compared to uncertainties typically associated with derived stellar masses. Figure 4 considers the ALFALFA galaxies from Elson (2024) that have more than one derived stellar mass available and shows the discrepancies between them. These masses are all taken from the Durbala et al. (2020) catalogue and are derived either from SDSS optical photometry, infrared unWISE photometry and/or ultraviolet imaging from GALEX (McGaugh & Schombert 2015), or multi-wavelength spectral energy distribution fitting from the GALEX-SDSS-WISE Legacy Catalog 2 (Salim et al. 2016, 2018). Very clear is the fact that all distributions are well-approximated by a Gaussian with a mean significantly offset from zero and with RMS \nranging from ∼ 0 . 18 to ∼ 0 . 35 dex. Thus, this study's parameterisation of the distribution of Simba galaxies in log 10 j ∗ -log 10 M ∗ -〈 µ I eff 〉 space is generally able to predict the stellar masses of real galaxies with more reliability and less uncertainty than the differences between the various derived stellar masses that are based on sophisticated modelling processes. Any SDSS galaxy with available W 50 and inclination measurements can have its stellar masses accurately predicted.", '6.2 SPARC galaxies': "As a second test of the predictive ability of this study's results, galaxies from the database of the Spitzer Photometry and Accurate Rotation Curves (SPARC, Lelli et al. 2016) are considered. The data gathered in Table1.mrt on the SPARC website 12 is used to generate j ∗ measurements for the SPARC galaxies: \nj SPARC ∗ = 2 V flat R eff / 1 . 68 , (4) \nwhere V flat is the asymptotically flat rotation velocity in units of km s -1 and R eff is the effective radius in units of kpc as measured in the 3.6 µ m band of the Spitzer telescope. A 〈 µ 3 . 6 eff 〉 measurement for each SPARC galaxy comes from using half of its total luminosity at 3.6 µ m and dividing it by the area within its effective radius, and then taking into account its distance. Only those SPARC galaxies with a quality flag of 1 are considered. Additionally, the relative uncertainty in V flat must be less than 0.1. These cuts yield a sample of 84 SPARC galaxies. \nLelli et al. (2016) convert total 3.6 µ m luminosity in L ⊙ units to total stellar mass in M ⊙ units by assuming a constant stellar mass-to-light ratio of 0.5 M ⊙ /L ⊙ . These stellar masses are compared to those predicted using this study's 2D plane together with its j ∗ and 〈 µ 3 . 6 eff 〉 measurements for the SPARC galaxies. Figure 3 shows in its bottom-left panel the predicted M ∗ values 13 as a function of those based on the information from the SPARC website. The grey line represents a linear fit to the data while the red line is that of equality. Very clear is the fact that the current study's bestfitting log 10 j ∗ -log 10 M ∗ -〈 µ 3 . 6 eff 〉 plane has a very high level of predictive accuracy that is constant with stellar mass. The bottom-right panel of Figure 3 shows the difference to be approximately Gaussian distributed with a median value of -0.086 dex and an RMS of only 0.111 dex. These results again demonstrate the best-fitting planes from this study to serve as effective tools for accurately predicting the stellar masses of real galaxies.", '7 SUMMARY': "This work aims to use galaxies from the Simba simulation to study their distribution in log 10 j ∗ -log 10 M ∗ -⟨ µ eff ⟩ space to better constrain the dependence of stellar specific angular momentum ( j ∗ ) on stellar mass ( M ∗ ) and the central concentration of a galaxy's light - as quantified by the magnitude measure of the average flux within its effective radius ( ⟨ µ eff ⟩ ). \nSpanning a stellar mass range 8 . 5 ≲ log 10 ( M ∗ /M ⊙ ) ≲ 11 , \nFigure 3. Comparisons between predicted and derived stellar masses for ALFALFA galaxies (top row) and SPARC galaxies (bottom row). Left panels: predicted stellar masses as a function of derived stellar mass. Grey lines represent a first-order polynomial fit to the data, red lines represent equality. The 2D histogram for the ALFALFA galaxies is based on square-shaped bins of side length 0.075 dex. Right panels: distribution of differences between predicted and derived stellar masses. Shown in each panel are the median and RMS of the distribution. These results demonstrate the best-fitting planes from this study to serve as effective tools for accurately predicting the stellar masses of real galaxies. \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 4. For the 3607 ALFALFA galaxies used in Elson (2024), the panels show differences in logarithmic stellar masses based on SDSS optical photometry and infrared unWISE photometry and/or ultraviolet imaging from GALEX (left panel), SDSS optical photometry and measurements from GALEX-SDSS-WISE Legacy Catalog 2 (middle panel), measurements from GALEX-SDSS-WISE Legacy Catalog 2 and infrared unWISE photometry and/or ultraviolet imaging from GALEX (right panel). Differences in derived stellar masses from these three sources are typically larger than the error with which the current study's parameterisation of the distribution of Simba galaxies in log 10 j ∗ -log 10 M ∗ -⟨ µ eff ⟩ space can predict them. \n<!-- image --> \n179 Simba galaxies are selected to have their j ∗ , M ∗ and ⟨ µ eff ⟩ quantities measured. A galaxy's gas particles are used to produce an H i data cube from which a measure of the velocity width of its H i line profile is extracted. The star particles of a galaxy are used to directly acquire measurements of disc scale length and total stellar mass. These measurements are all combined to approximate j ∗ . \nThe distribution of Simba galaxies in log 10 j ∗ -log 10 M ∗ -⟨ µ eff ⟩ space is found to be very well-approximated by a 2D plane that yields j ∗ ∝ M 0 . 694 ∗ µ 0 . 190 eff . This measured mass dependence of j ∗ on M ∗ is very close to what is predicted by Λ CDMtheory. The RMS value of the perpendicular distances \nof the galaxies about the best-fitting plane is only 0.057 dex, thereby making the planar distribution of Simba galaxies much tighter than the distribution of ALFALFA galaxies from Elson (2024). \nThe utility of the parameterised planar distribution of Simba galaxies in log 10 j ∗ -log 10 M ∗ -⟨ µ eff ⟩ space as a tool for accurately predicting the stellar masses of real galaxies with SDSS I -band or Spitzer 3.6 µ m photometry is demonstrated. Over a large mass range, derived stellar masses of galaxies from the ALFALFA survey and the database of the Spitzer Photometry and Accurate Rotation Curves (SPARC) are accurately predicted to within ∼ 0 . 1 to 0.2 dex. Consid- \nrge number of ongoing H i and optical/infrared galaxy surveys, the planar relations presented in this study make it possible to accurately estimate the stellar masses of their detected galaxies without the need for detailed modelling and acquisition of additional multi-wavelength imaging. \nThis study serves as a theoretical demonstration of the intrinsic relationship between stellar specific angular momentum, mass and morphology for late-type galaxies, and presents yet another galaxy scaling relation from the Simba simulations that is highly consistent with empirical results.", 'ACKNOWLEDGEMENTS': 'Sincere thanks are extended to the anonymous referee for providing truly insightful comments that improved the quality of this work.', 'DATA AVAILABILITY': 'Upon reasonable request, the author is willing to make available the measurements generated in this study.', 'REFERENCES': 'Posti L., Fraternali F., Di Teodoro E. M., Pezzulli G., 2018b, A&A, 612, L6 \n```\nRomanowsky A. J., Fall S. M., 2012a, ApJS, 203, 17 Romanowsky A. J., Fall S. M., 2012b, ApJS, 203, 17 Salim S., et al., 2016, ApJS, 227, 2 Salim S., Boquien M., Lee J. C., 2018, ApJ, 859, 11 Schmidt M., 1959, ApJ, 129, 243\n``` \nThis paper has been typeset from a T E X/L A T E X file prepared by the author.'}

2022ApJ...940L..14N

The first few 100 Myr at z gt 10 mark the last major uncharted epoch in the history of the universe where only a single galaxy GNz11 at z 11 is currently spectroscopically confirmed. Here we present a search for luminous z gt 10 galaxies with JWSTNIRCam photometry spanning 15 m and covering 49 arcminSUP2SUP from the public JWST Early Release Science programs CEERS and GLASS. Our most secure candidates are two M SUBUVSUB 21 systems GLASSz12 and GLASSz10. These galaxies display abrupt 1.8 mag breaks in their spectral energy distributions SEDs consistent with complete absorption of flux bluewards of Ly that is redshifted to z12.40.30.1 and z10.40.50.4 . Lower redshift interlopers such as quiescent galaxies with strong Balmer breaks would be comfortably detected at gt5 in multiple bands where instead we find no flux. From SED modeling we infer that these galaxies have already built up 10SUP9SUP solar masses in stars over the 300400 Myr after the Big Bang. The brightness of these sources enable morphological constraints. Tantalizingly GLASSz10 shows a clearly extended exponential light profile potentially consistent with a disk galaxy of r SUB50SUB 0.7 kpc. These sources if confirmed join GNz11 in defying number density forecasts for luminous galaxies based on Schechter UV luminosity functions which require a survey area gt10 larger than we have studied here to find such luminous sources at such high redshifts. They extend evidence from lower redshifts for little or no evolution in the bright end of the UV luminosity function into the cosmic dawn epoch with implications for just how early these galaxies began forming. This in turn suggests that future deep JWST observations may identify relatively bright galaxies to much earlier epochs than might have been anticipated.

2022-11-01T00:00:00Z

['10.3847/2041-8213/ac9b22', '10.48550/arXiv.2207.09434', '2022arXiv220709434N', '2022ApJ...940L..14N', 'arXiv:2207.09434']

['James Webb Space Telescope', 'Galaxy evolution', 'Early universe', 'High-redshift galaxies', 'Galaxy formation', '2291', '594', '435', '734', '595', 'Astrophysics - Astrophysics of Galaxies']

Two Remarkably Luminous Galaxy Candidates at z 1012 Revealed by JWST

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']

https://arxiv.org/pdf/2207.09434.pdf

{'No Header': '8', 'Two Remarkably Luminous Galaxy Candidates at z ≈ 10 -12 Revealed by JWST': "Rohan P. Naidu, 1, 2, ∗ Pascal A. Oesch, 3, 4 Pieter van Dokkum, 5 Erica J. Nelson, 6 Katherine A. Suess, 7, 8 Gabriel Brammer, 4 Katherine E. Whitaker, 9, 10 Garth Illingworth, 11 Rychard Bouwens, 12 Sandro Tacchella, 13, 14 Jorryt Matthee, 15 Natalie Allen, 4 Rachel Bezanson, 16 Charlie Conroy, 1 Ivo Labbe, 17 Joel Leja, 18, 19, 20 Ecaterina Leonova, 21 Dan Magee, 22 Sedona H. Price, 23 David J. Setton, 16 Victoria Strait, 4 Mauro Stefanon, 24, 25 Sune Toft, 4 John R. Weaver, 9 and Andrea Weibel 3 \n1 Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA \n2 \nMIT Kavli Institute for Astrophysics and Space Research, 77 Massachusetts Ave., Cambridge, MA 02139, USA \n3 Department of Astronomy, University of Geneva, Chemin Pegasi 51, 1290 Versoix, Switzerland \nCosmic Dawn Center (DAWN), Niels Bohr Institute, University of Copenhagen, Jagtvej 128, København N, DK-2200, Denmark \n5 Astronomy Department, Yale University, 52 Hillhouse Ave, New Haven, CT 06511, USA \n6 Department for Astrophysical and Planetary Science, University of Colorado, Boulder, CO 80309, USA \n- 7 Department of Astronomy and Astrophysics, University of California, Santa Cruz, 1156 High Street, Santa Cruz, CA 95064 USA \nKavli Institute for Particle Astrophysics and Cosmology and Department of Physics, Stanford University, Stanford, CA 94305, USA \n9 Department of Astronomy, University of Massachusetts, Amherst, MA 01003, USA \n10 Cosmic Dawn Center (DAWN), Denmark \n11 Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA \n12 Leiden Observatory, Leiden University, NL-2300 RA Leiden, Netherlands \n- 13 Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, UK \n14 Cavendish Laboratory, University of Cambridge, 19 JJ Thomson Avenue, Cambridge, CB3 0HE, UK \n15 Department of Physics, ETH Zurich, Wolfgang-Pauli-Strasse 27, 8093 Zurich, Switzerland \n16 Department of Physics and Astronomy and PITT PACC, University of Pittsburgh, Pittsburgh, PA 15260, USA \n- 17 Centre for Astrophysics and Supercomputing, Swinburne University of Technology, Melbourne, VIC 3122, Australia \n18 \nDepartment of Astronomy & Astrophysics, The Pennsylvania State University, University Park, PA 16802, USA \n19 \nInstitute for Computational & Data Sciences, The Pennsylvania State University, University Park, PA, USA \n- 20 Institute for Gravitation and the Cosmos, The Pennsylvania State University, University Park, PA 16802, USA \n21 GRAPPA, Anton Pannekoek Institute for Astronomy and Institute of High-Energy Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands \n22 UCO/Lick Observatory, University of California, Santa Cruz, CA, 95064 \n- 23 Max-Planck-Institut fur extraterrestrische Physik (MPE), Giessenbachstr. 1, D-85748 Garching, Germany \n24 Departament d'Astronomia i Astrof'ısica, Universitat de Val'encia, C. Dr. Moliner 50, E-46100 Burjassot, Val'encia, Spain 25 Unidad Asociada CSIC 'Grupo de Astrof'ısica Extragal'actica y Cosmolog'ıa' (Instituto de F'ısica de Cantabria - Universitat de Val'encia)", 'ABSTRACT': 'The first few hundred Myrs at z > 10 mark the last major uncharted epoch in the history of the Universe, where only a single galaxy (GNz11 at z ≈ 11) is currently spectroscopically confirmed. Here we present a search for luminous z > 10 galaxies with JWST /NIRCam photometry spanning ≈ 1 -5 µ m and covering 49 arcmin 2 from the public JWST Early Release Science programs (CEERS and GLASS). Our most secure candidates are two M UV ≈ -21 systems: GLASS-z12 and GLASS-z10. These galaxies display abrupt glyph[greaterorsimilar] 1 . 8 mag breaks in their spectral energy distributions, consistent with complete absorption of flux bluewards of Lymanα that is redshifted to z = 12 . 4 +0 . 1 -0 . 3 and z = 10 . 4 +0 . 4 -0 . 5 . Lower redshift interlopers such as quiescent galaxies with strong Balmer breaks would be comfortably detected at > 5 σ in multiple bands where instead we find no flux. From SED modeling we infer that these galaxies have already built up ∼ 10 9 solar masses in stars over the glyph[lessorsimilar] 300 -400 Myrs after the Big Bang. The brightness of these sources enable morphological constraints. Tantalizingly, GLASS-z10 shows a clearly extended exponential light profile, potentially consistent with a disk galaxy of r 50 ≈ 0 . 7 kpc. These sources, if confirmed, join GNz11 in defying number density forecasts for luminous galaxies based on Schechter UV luminosity functions, which require a survey area > 10 × larger than we have \n4 \nstudied here to find such luminous sources at such high redshifts. They extend evidence from lower redshifts for little or no evolution in the bright end of the UV luminosity function into the cosmic dawn epoch, with implications for just how early these galaxies began forming. This, in turn, suggests that future deep JWST observations may identify relatively bright galaxies to much earlier epochs than might have been anticipated. \nKeywords: High-redshift galaxies (734), Galaxy formation (595), Galaxy evolution (594), Early universe (435)', '1. INTRODUCTION': "When and how the first galaxies formed remains one of the most intriguing questions of extragalactic astronomy and observational cosmology (see Dayal & Ferrara 2018; Robertson 2021, for recent reviews). Although deep observations with the Hubble Space Telescope (HST) have pushed our cosmic horizon to within the first 400 Myr of the Big Bang, galaxies at z glyph[greaterorsimilar] 12 cannot be observed with HST due to the limit of its wavelength coverage at 1.6 µ m. \nWith the advent of JWST, we now have an unprecedented view of the Universe at ∼ 2 -5 µ m thanks to the extremely sensitive NIRCam instrument (see, e.g., Rieke et al. 2005). The extended wavelength coverage enables the study of rest-frame optical wavelengths up to z ∼ 10 and allows for rest-frame UV selections of galaxies out to much higher redshifts. \nHere we present first results from a search for particularly luminous z > 10 sources across the two JWST Early Release Science deep fields. The most luminous galaxies are of particular importance. They may trace overdensities and thus pinpoint where galaxy formation first started in the early Universe (e.g., Leonova et al. 2021; Endsley et al. 2021; Larson et al. 2022). Furthermore, they provide the most stringent constraints on early galaxy build-up and promise rich scientific returns. \nOne particular example of this is provided by GN-z11 (Oesch et al. 2016) that was detected with HST. Its discovery in the two CANDELS/GOODS fields that cover a search volume of only ∼ 10 6 Mpc 3 was initially quite surprising. Theoretical and empirical models of early galaxy formation predicted that a 10-100 × larger survey would have been required to find one such bright galaxy at z = 11 (e.g. Waters et al. 2016; Mutch et al. 2016). This highlights the potential of the brightest galaxies at the cosmic frontier to set unique constraints on the physics of galaxy formation (see also Behroozi & Silk 2018). In particular, the number density of such bright sources, i.e. the bright end cutoff of the UV luminosity function, is a very powerful tool to test the efficiency of star-formation and potential feedback mechanisms in the very early Universe (Bowler et al. 2014; Tacchella et al. 2018; Bowler et al. 2020). \nThese results have been extended over the last few years, and evidence is emerging for a differential evolution of the galaxy population during the reionization epoch at z > 6. While the number densities of fainter galaxies continue to decline with redshift, the most UVluminous sources seem to be in place rather early (e.g., Stefanon et al. 2019; Bowler et al. 2020; Morishita et al. 2020; Harikane et al. 2022; Bagley et al. 2022). Furthermore, several authors found evidence for pronounced Balmer breaks in bright z ∼ 8 -10 galaxies, which would indicate a very early formation epoch with intense starformation (e.g., Hashimoto et al. 2018; Roberts-Borsani et al. 2020; Laporte et al. 2021). However, others find very young ages for the average population (Stefanon et al. 2022a,b). These inferences, at the moment, are also highly sensitive to the prior adopted on the starformation history (e.g., Tacchella et al. 2022; Whitler et al. 2022). Timing the onset of first star-formation in bright galaxies out to z ∼ 10 is thus still highly uncertain, and probing the number density of the most luminous sources at even higher redshifts is the most direct way of addressing this. \nThe brightest galaxies are also the most amenable to follow-up studies through spectroscopy. For instance, GN-z11 had its redshift confirmed both through grism spectroscopy with HST and with emission lines from ground-based Keck observations (see Oesch et al. 2016; Jiang et al. 2021). With the combination of NIRCam and NIRSpec, we enter a new era - the rest-frame optical features of galaxies in the Epoch of Reionization ( z ≈ 6 -9, e.g., Mason et al. 2019) will come fully into view. However, the strongest emission lines of galaxies at z > 10 will still remain out of reach. Spectroscopically confirming such sources will require measuring order of magnitude fainter lines or strong continuum breaks. Identifying luminous systems at z > 10 to facilitate these measurements is therefore a crucial step in fulfilling JWST's mission of charting cosmic dawn. \nThis paper is organized as follows -§ 2 describes the datasets analyzed in this work, § 3 presents our sample selection, and in section § 4 we show our results. A discussion on the implications follows in § 5, before we conclude with a summary and an outlook in § 6. \nMagnitudes are in the AB system (e.g., Oke & Gunn 1983). For summary statistics we report medians along with 16 th and 84 th percentiles. We adopt a Planck Collaboration et al. (2015) cosmology. \nFigure 1. Summary of photometry and redshift solution for GL-z12. Top: 4.5' × 4.5' images spanning ≈ 0 . 9 -4 . 5 µ m centered on the z ≈ 12 candidate highlighted with white crosshairs. The source is well-detected ( > 20 σ ) in F200W and all redder bands, and abruptly drops out in the bluer filters. Bottom left: Photometry for the source is shown in purple, with upper limits for non-detections plotted at the 1σ level. The best-fit spectral energy distribution (SED) template from EAZY is shown in dark orange - a Lyman-break galaxy (LBG) at z = 12 . 2. The best-fit SED from EAZY constrained to lie at z < 6 is plotted in silver, which corresponds to a quiescent galaxy at z ≈ 3 . 5 whose Balmer break produces a drop-off across F200W and F150W. However, such a quiescent galaxy is predicted to be detected ( > 5 σ ) in bluer bands, and is at odds with the dramatic > 1 . 8 mag break observed. Bottom right: Probability distributions for the source redshift derived using EAZY (solid orange) and Prospector (dashed orange). We adopt a flat prior across the redshift range depicted ( z = 0 -20). The derived distributions are in excellent agreement and suggest a redshift of z ≈ 12, with negligible ( EAZY ) or no ( Prospector ) support for solutions at z < 10 \n<!-- image --> \n.", '2.1. Early Extragalactic JWST Observations': "Our analysis is based on some of the first JWST/NIRCam datasets that have been observed and released over extragalactic fields. In particular, we analyze the two Early Release Science programs GLASS and CEERS. \nThe first is the NIRCam parallel field of the ERS program GLASS (PID: 1324, Treu et al. 2022). This program obtained a single NIRCam pointing in seven wide filters F090W, F115W, F150W, F200W, F277W, F356W, and F444W, observed for 3.3, 3.3, 1.7, 1.5, 1.5, 1.7, and 6.6 hrs, respectively. \nWe also analyzed the first four NIRCam pointings from the ERS program CEERS (PID: 1345, Finkelstein et al., in prep.) that have been observed to date. The CEERS images include F115W+F277W, F115W+F356W, F150W+F410M, and F200W+F444W short- (SW) and long-wavelength (LW) exposure sets, for a typical integration time of 0.8 hr per filter, except for F115W that obtained double this integration time. \nThe combined area of these five NIRCam fields used in our analysis amounts to 49 arcmin 2 ( ≈ 10 arcmin 2 in GLASS, and ≈ 40 arcmin 2 in CEERS) reaching an unprecedented 5 σ depth that ranges between 28.6 and 29.6 AB mag at 4 µ m as measured in 0 . '' 32 diameter apertures (see Table 1).", '2.2. Data Reduction': "The NIRCam images were reduced with the standard JWST pipeline up to stage 2 using the reference files from jwst 0942.pmap, as well as our own sky flats. Additional, chip-dependent zeropoint offsets were applied based on our own reductions of the flux standard star J1743045 as well as observations of the Large Magellanic Cloud. For some filters, the individual corrections in different modules of the NIRCam detector can reach up to 30% 1 . The resulting images were then processed with the grizli 2 pipeline for proper alignment onto a com- \nFigure 2. Summary of photometry and redshift solution for GL-z10, similar to Figure 1. Top: GL-z10 is well-detected in all but the two bluest bands. Bottom left: The best-fit lowz solution (quiescent galaxy at z ≈ 2 . 5) is disfavored by the F115W image, where a > 5 σ detection is expected. In addition to the JWST data (dark purple), we measure HST photometry (light purple) for this source from data acquired by the BUFFALO program (Steinhardt et al. 2020). The HST data are fully consistent with the JWST data as well as the best-fit SED. Bottom right: The EAZY and Prospector posteriors agree on a z ≈ 10 galaxy. \n<!-- image --> \nmon WCS that was matched to the Gaia DR3 catalogs. To do this, grizli re-computes the traditional SIP distortion headers that were common for HST data. This allows us to use drizzlepac as with HST images to combine the individual frames and produce distortioncorrected mosaics. Additionally, grizli mitigates the 1/f noise and masks 'snowballs' that are most prominent in short-wavelength filters. \nThe pipeline was used to drizzle images at 20mas / 40mas pixels for the SW and LW data, respectively. In the following, our analysis is based on pixel-matched 40mas images, however. For more information on the grizli processing see Brammer et al., (in prep).", '2.3. Multi-Wavelength Catalogs': "After processing the new NIRCam images, we produced photometric catalogs in all fields using the SExtractor software (Bertin & Arnouts 1996). Sources were detected in dual mode with two different detection images: F200W or a weighted combination of all the LW filters. Fluxes were measured in small circular apertures of 0 . '' 32 diameter and were corrected to total using the AUTO flux measurement from the detection image. An additional correction of typically a few percent only was applied for remaining flux outside of this aperture based on the predicted encircled energy for the JWST point-spread functions. Flux uncertainties were estimated based on sigma-clipped histograms of circular apertures placed throughout the images in random sky \npositions. These were then used to rescale the drizzled rms maps. Thus, our uncertainties are as close to the data as possible. The 5 σ depths derived in this way are listed in Table 1.", '2.4. Quality Control': 'Given that JWST is a completely new facility for which calibration is still ongoing, it is important to test the resulting images for any issues. Indeed, the NIRCam data revealed several features that are not accounted for in the standard pipeline. In particular, the SW data suffer from significant scattered light, if there are bright stars in the vicinity. This is particularly pronounced in the GLASS parallel field where a 10th mag star just outside of the field seems to cause artificial images across the field. This is particularly pronounced in the F090W filter in the B4 detector. However, also other detectors and filters seem to be affected, albeit to a lower extent. While this issue could be overcome with improved processing, or with additional observations at a different roll angle, this does not significantly affect the current analysis. Since we are searching for very high redshift galaxies that disappear at shorter wavelengths, these data issues only introduce some level of incompleteness. Most importantly, the areas of the two candidate sources presented later in this paper are not affected. \nWhere available, we compared our JWST photometry in the shorter wavelength filters with existing HST data to check both for issues with residual distortion or \nFigure 3. Absolute UV magnitude vs. Redshift for a representative sample of known galaxies in the first billion years of the Universe. Galaxies with photometric redshifts, sourced from Bouwens et al. (2022), are shown as points, and those with spectroscopic redshifts compiled from the literature as squares. The candidates presented in this work are depicted as purple stars, and populate a hitherto unoccupied region of parameter space. The brightness of these sources present a unique opportunity to efficiently extend the spectroscopic frontier to the first few hundred Myrs after the Big Bang. \n<!-- image --> \nwith magnitude zeropoint offsets. In particular, for the GLASS parallel field, we used HST images made available by Kokorev et al. (2022) from the BUFFALO survey (Steinhardt et al. 2020). However, only a small portion of the field is covered by both HST and JWST. For the CEERS data, we make use of re-reductions from the imaging taken by the CANDELS survey (e.g., Koekemoer et al. 2012). No significant offsets or issues were detected. Zeropoint corrections remained small ( < 15%; see next section).', '3. SAMPLE SELECTION & METHODS': "Photometric redshifts form the basis of our search for bright z > 10 galaxies. We fit redshifts using EAZY (Brammer et al. 2008) adopting the 'tweak fsps QSF 12 v3' template set derived from FSPS (Conroy et al. 2009, 2010; Conroy & Gunn 2010a). The allowed range is 0 . 1 < z < 20 adopting a flat luminosity prior, after applying modest ( < 15%) zero-point corrections that are derived iteratively. Candidates of interest are selected to have best-fit redshifts z > 10 along with > 84% of their derived redshift probability distribution function, p ( z ), lying at z > 10. \nTable 1. 5 σ Depth of JWST Data in this Analysis \nNote -Measured in 0 . '' 32 diameter circular apertures. \nWe further require > 10 σ detections in both F356W and F444W. These bands sample the rest-frame UV at z > 10 and are critical in establishing the flux levels with respect to which we seek strong Lyman breaks. \nWe inspect images of every candidate source for data quality issues (e.g., contamination from neighbors, \nTable 2. Photometry in units of nJy \nNote -We set an error floor of 10% on our measured fluxes for EAZY and Prospector fits to account for systematic uncertainty not reflected in the errors stated above. \nTable 3. Summary of properties. \nNote -SED fitting assumes a continuity prior on the starformation history and a Chabrier (2003) IMF. \ndiffraction spikes, location on the edge of the detector). In tandem, we examine plausible lowz solutions for the candidates by running EAZY constrained to z < 6 (primarily dusty, quiescent galaxies with Balmer breaks masquerading as Lyman breaks). We inspect lowz solutions with the understanding that the errors on fluxes in the dropout band may be underestimated in some cases (in e.g., image areas with residual striping), which can have an important impact on the p ( z ). For instance, we find one bright source in CEERS with a confident z EAZY ≈ 17 and z Prospector ≈ 17 whose z > 10 solution assumes a secure non-detection in F150W - unfortunately, the source falls in a low-SNR region of the \nF150W image and it is difficult to judge the reality of the non-detection. \nWe find 5 plausible z > 10 candidates that survive all our conservative checks - 3 in CEERS, and 2 in GLASS. Of these, the two identified in GLASS - GLz10 ( z EAZY = 10 . 4 +0 . 2 -0 . 7 ) and GL-z12 ( z EAZY = 12 . 4 +0 . 2 -0 . 2 ) - stand out as being particularly luminous and secure. No other objects when fit with the Prospector SEDfitting code (see following Section) have a p ( z ) with all modes contained at z > 10. Further, the GLASS candidates are among the most luminous found - GL-z10, in particular, is by far the brightest of all sources (by > 2 × in F444W). For the rest of this work, we focus on these two particularly luminous candidates and we defer the rest of the sources to future papers that present an analysis of the full z > 10 galaxy population in these fields.", '4.1. Two Luminous z > 10 Galaxy Candidates': "We confirm the photometric redshifts for the two GLASS candidates and derive stellar population properties using the Prospector SED fitting code (Leja et al. 2017, 2019; Johnson et al. 2021). The SED parameter space explored by Prospector is more expansive than EAZY 's linear template combinations, and therefore it acts as an important check on our derived redshifts. We use FSPS (Conroy et al. 2009, 2010; Conroy & Gunn 2010a) with the MIST stellar models (Choi et al. 2017). We adopt the 19-parameter physical model and parameter choices described in Tacchella et al. (2022) that fits for the redshift, stellar and gas-phase metallicities, stellar mass, star-formation history, dust properties, AGN emission, and scaling of the IGM attenuation curve. We make slight modifications to their setup - in particular we explore a broader redshift range of z = 0 . 1 -20 and keep two bins fixed at lookback times of 0 -5 Myr and 5 -10 Myrs in the star-formation history following Whitler et al. (2022) to capture recent bursts that may be powering extreme nebular emission expected to occur generically at the redshifts of interest (e.g., Labb'e et al. 2013; De Barros et al. 2019; Endsley et al. 2019). We adopt a 'continuity' prior on the star-formation history, which limits the amount of variance across consecutive time-bins resulting in smooth histories (Leja et al. 2019; Tacchella et al. 2022). For further details we direct readers to Table 1 and § 3.4 of Tacchella et al. (2022). \nThe redshift fits from Prospector are in excellent agreement with EAZY - we find z Prospector = 10 . 4 +0 . 4 -0 . 5 for GL-z10 and z Prospector = 12 . 4 +0 . 1 -0 . 3 for GL-z12. The photometry and redshift inference for these sources are summarized in Figures 1 and 2, with fluxes listed in Table 2. We confirm that no significant data quality issues affects the z > 10 candidacy of the sources in the imaging. We derive p ( z > 10) ≈ 100% for GL-z12 and p ( z > 9 . 4) ≈ 100% for GL-z10, with their dramatic \nglyph[greaterorsimilar] 1 . 8 mag breaks explained by total absorption of photons bluewards of Lymanα by neutral Hydrogen in the intergalactic medium. \nBoth galaxies are detected at very high significance in all filters longward of their break, by virtue of our selection. While they appear very luminous in the JWST data, these sources have UV absolute magnitudes ( M UV ≈ -21) that correspond to L ∗ UV at z ∼ 8 -10 (see, e.g., Bouwens et al. 2021). This also makes them 1 mag fainter than GN-z11 and even 2.5 mag fainter than the possible z ∼ 13 galaxy candidate HD1 (Harikane et al. 2022). Hence, these sources are not really extreme outliers (see also Fig. 3). Nevertheless, it is interesting that the first few images with JWST already reveal two such bright sources. We will discuss their implications on the UV LF in a later section.", '4.2. Possible Lower Redshift Contamination': "The non-detections of both sources in deep, shorter wavelength images essentially rules out a lower redshift solution. Nevertheless, it is interesting to explore the nature of possible contaminants. We thus rerun our photometric redshift codes and force them to find lower redshift fits. The best z < 6 solutions in our lowz EAZY runs for these sources are ≈ 10 8 -9 M glyph[circledot] quiescent galaxies at z ≈ 3 . 4 ( z ≈ 2 . 5) with Balmer breaks straddling the dropout filter (silver SEDs in Figs. 1, 2). However note that Balmer breaks, even in the most pathological cases (e.g., 4000 ˚ A falls just redward of the dropout filter in a super-solar metallicity galaxy as old as the age of the Universe at z ≈ 2 -3 . 5), can only produce drops of glyph[lessorsimilar] 1 . 5 mag (assuming no attenuation). The best-fit lowz solutions have A V ≈ 0 . 1 - stronger attenuation that deepens the break is disfavored by the blue continuum slope at wavelengths longer than the break. In other words, the best-fit lowz solutions predict > 5 σ detections in bands where we find no flux, and continuum slopes redder than we observe. \nIn order to allow for possible systematic effects in the new JWST data, we perform further testing. We refit redshifts to multiple versions of photometry for these sources - e.g., by adding PSF corrections using WebbPSF , by increasing the error floor on the photometry, by extracting photometry using different apertures and detection bands. The only test that produces viable lowz solutions is when we set a 10 nJy error-floor on all photometry - this is roughly the level in the SW filters at which the strongest Balmer breaks at z ≈ 2 -3 can no longer be ruled out (see open silver squares in bottomleft panels of Figures 1 and 2). This test is a vivid demonstration of why the sensitivity of JWST/NIRCam is required to identify objects like GL-z10 and GL-z12 with confidence. \n4.3. Physical Properties - Billion M glyph[circledot] Galaxies within ≈ 400 Myrs of the Big Bang \nWhile the discovery of GN-z11 has already demonstrated that the formation of billion solar mass galaxies was well underway at ∼ 400 Myr after the Big Bang, the discovery of these two new sources allows us to derive further constraints on the physical properties of galaxies at this very early epoch of the Universe. The Prospector results are summarized in Table 3. In order to efficiently sample the redshift range of interest, we assume a tighter redshift prior (a Gaussian centred on the EAZY p ( z ) with width set to the 84 th - 16 th percentile) than in our previous runs when fitting for the redshift. \nThe stellar mass for both objects is constrained to be ≈ 10 9 M glyph[circledot] , comparable to GNz11 (Oesch et al. 2016; Johnson et al. 2021; Tacchella et al. 2022). We have verified the stellar mass is stable to changes in the starformation history prior by also testing the 'bursty' prior from Tacchella et al. (2022) which allows more rapid fluctuations in the SFH from time-bin to time-bin than the fiducial model. The star-formation rates averaged over the last 50 Myrs (SFR 50 ) are typical for galaxies of comparable mass at z ≈ 7 -10 (e.g., Stefanon et al. 2022a). The SEDs are consistent with negligible dust attenuation and have blue UV slopes, β glyph[lessorsimilar] -2. We note that all these derived properties from the SED fits are collectively consistent with a z > 10 interpretation for these galaxies.", '4.4. Galaxy-galaxy Lensing': "Galaxy-galaxy lensing may be particularly important at the redshift frontier where flux-limited surveys may be preferentially sampling magnified sources (e.g., Wyithe et al. 2011). Here we make a simple estimate of how lensed our sources are by assuming their neighbors are singular isothermal spheres (e.g., Fort & Mellier 1994; Schneider et al. 2006; Treu 2010) following e.g., McLure et al. (2006); Oesch et al. (2014); Matthee et al. (2017). For this estimate, galaxy redshifts and stellar masses are based on our EAZY fits. Velocity dispersions that trace the underlying dark matter halos are inferred from the stellar mass by extrapolating the empirical scaling relation in Zahid et al. (2016) that is fit to z < 0 . 7 quiescent galaxies that span M glyph[star] ≈ 10 9 -10 12 M glyph[circledot] . We choose a local relation to cover the low masses relevant to the most massive neighbors at < 10 '' that are likely to produce significant magnification, while noting that the redshift evolution of such relations at least for M glyph[star] glyph[greaterorsimilar] 10 11 M glyph[circledot] is expected to be gradual - e.g., ≈ 20% higher dispersion at fixed stellar mass at z ≈ 2, (e.g., Mason et al. 2015b). \nFor both GL-z10 and GL-z12 we find negligible lensing ( µ < 1 . 1) from all foreground sources within 10 . '' . GL-z12 has two relatively massive M glyph[star] ≈ 10 9 M glyph[circledot] neighbors apparent in the bottom-left quadrant of the stamps in Fig. 1. Even for these two nearby neighbors ( z ≈ 2 at a separation of 0 . '' 8, and z ≈ 3 at 2 . '' 0) the lensing is modest. We further note that GL-z13 has a compact morphology ( § 4.5) that does not show elongation along \nimage \nmodel \nresidual \nFigure 4. Results of the GALFIT morphology analysis for our two sources (GL-z10 top and GL-z12 bottom). The different columns from left to right correspond to the original data (in the F444W filter), the model, and the residual. The sizes and Sersic profiles of both sources are well constrained. GL-z10 shows some clear extension, consistent with a disk galaxy of 0.7 kpc at z ∼ 11. GL-z12 appears quite compact with an estimated size of 0.5 kpc. \n<!-- image --> \n1 kpc \n1 kpc \nany particular direction that would hint at strong magnification. Based on these considerations we conclude that the observed luminosities of GL-z10 and GL-z12 are likely to be their intrinsic luminosities.", '4.5. The Sizes of Luminous z ≈ 10 -12 Galaxies': "We fit the sizes of both candidates in the F444W imaging ( λ rest ∼ 3500 ˚ A) using GALFIT (Peng et al. 2010). We create 100-pixel cutouts around each galaxy, then use photutils and astropy to create a segmentation map to identify nearby galaxies. We simultaneously fit any sources with magnitudes (estimated from the segmentation map) up to 2.5 magnitudes fainter than the target galaxy that have centers within 3 '' of the galaxy; we mask fainter or more distant galaxies. In our fits, we constrain the center of the target galaxy to be within 10 pixels (0 . '' 4) of the input value, the Sersic index n to be between 0.01 and 8, the magnitude to be between 0 and 45, and the half-light radius r e to be between 0.3 and 200 pixels (0 . '' 012 - 8 . '' 0). We calculate and subtract off a scalar sky background correction from each cutout, estimated from the masked, sigma-clipped cutout, then fix the sky background component in GALFIT to zero. We use a theoretical PSF model generated from WebbPSF \nat our 0 . '' 04 pixel scale; we oversample the PSF by a factor of 9 in order to minimize artifacts as we rotate the PSF to the GLASS observation angle calculated from the APT file, then convolve with a 9x9 pixel square kernel and downsample to the mosaic resolution. \nWe find reliable Sersic fits for both galaxies, with halflight radii of 0.5 and 0.7 kpc, respectively, and disk-like profiles ( n = 1 and n = 0 . 8, respectively). The models are shown in Fig. 4, and the size and Sersic profile estimates are listed in Table 3. \nThe resulting sizes of 0.5 and 0.7 kpc are typical for luminous L ∗ galaxies at z ∼ 6 -9, where measurements have been possible to date (e.g., Holwerda et al. 2015; Shibuya et al. 2015; Bowler et al. 2017; Kawamata et al. 2018; Yang et al. 2022). They are also consistent with expectations from simulations for z > 9 galaxies (e.g., Roper et al. 2022; Marshall et al. 2022). However, at z ∼ 7, the most luminous sources often break up in multiple clumps (Bowler et al. 2017). This is not the case for these two sources, at least down to the resolution limit of order 500 pc for the F444W bandpass. Interestingly, GL-z10 even shows tantalizing evidence for being an ordered disk galaxy at z ∼ 10, based on \nthe exponential light profile and elliptical morphology. If we interpret GL-z10's projected axis ratio of 0.65 using a sample of randomly oriented axisymmetric oblate rotators (following e.g., Holden et al. 2012, Chang et al. 2013, van der Wel et al. 2014) and adopt c/a ≤ 0 . 4 as a threshold for disks, we find that the observed axis ratio implies P(disk) ∼ 0 . 5. Our analysis shows the unparalleled power of JWST to provide accurate profile measurements of early Universe galaxies.", '5.1. Caveats': 'The key caveat, as well as the key animating spirit of this work, is that these data are among the first deep extragalactic fields collected by a new Great Observatory. Systematic uncertainties (e.g., zero-point corrections, treatment of artefacts) can still be significant. We have tested for zero-point offsets by comparing HST and JWST photometry for brighter sources where possible and have not found any major issues at the glyph[lessorsimilar] 10% level. Nevertheless, we have attempted to account for remaining uncertainties with conservative choices - e.g., a 10% error floor on all fluxes and focusing on bright galaxies whose > 2 mag breaks are robust to even major uncertainties. \nAnext caveat applies to the SED models underpinning the stellar population parameters and photoz fitting. Important details about the nebular emission and nature of massive stars at low metallicities, which dominate the light in these few hundred Myr old star-forming systems, remain unconstrained (e.g., Stanway et al. 2020). These uncertainties directly translate to the parameters we recover from SED fitting and the sources for which we are able to fit high quality redshifts. Fortunately, for our redshift range of interest, from the perspective of continuum fitting for the stellar population analyses, extreme nebular emission from strong rest-frame optical lines is shifted out of all NIRCam bands and the most important feature for the photoz s in the bright galaxies we study is the Lyman break. Further, there exists a range of plausible, but hitherto unconstrained physical ingredients that are unaccounted for in our models (e.g., primordial AGN, top-heavy IMFs, super-luminous Pop III stars; Windhorst et al. 2018; Pacucci et al. 2022; Steinhardt et al. 2022). Some of these ingredients may potentially produce large UV luminosities in the absence of substantial stellar mass. Spectroscopic follow-up is therefore essential to confirm the nature of these sources.', '5.2. Implication: The Number Density of Luminous Galaxies in the Early Universe': 'These two galaxies enable a first estimate of the number densities of relatively luminous sources at z = 10 -13. Given the large uncertainties in redshift, we conservatively estimate a selection volume across this full redshift range, i.e., ∆ z = 3. Given that the depth of all \nthe fields we studied is far deeper than the bright magnitude of these two sources, these galaxies would have been identified over the full area, without foreground galaxies. This amounts to a volume of 2 . 2 × 10 5 Mpc 3 . The detection of two galaxies with M UV = -21 thus results in an estimated UV LF point of log φ [Mpc -3 mag -1 ] = -5 . 05 +0 . 37 -0 . 45 . In § 4.4 we conclude that this UV LF point likely reflects the intrinsic luminosity of the sources, and is unaffected by lensing. \nIn Figure 5, we compare this estimate with previous UV LF determinations and with extrapolations from lower redshifts. In particular, we show that an extrapolation of the Schechter function trends estimated at z = 3 -10 results in an LF at M UV = -21 that is a factor > 10 × lower than our estimate. Interestingly, however, when extrapolating the trends in the double-power law LFs from Bowler et al. (2020) to z ∼ 11 . 5 (the mean redshift of our sources), we find relatively good agreement. In fact, GN-z11 also lies on this extrapolated LF. However, this would indicate very little evolution in the bright galaxy population at z > 8. Indeed, our estimate is in good agreement with previous z ∼ 10 UV LF determinations and constraints at the bright end from Oesch et al. (2018); Bouwens et al. (2019); Stefanon et al. (2019); Morishita et al. (2020); Finkelstein et al. (2022); Bagley et al. (2022); Leethochawalit et al. (2022). \nFinally, we also briefly compare our estimates with simulated LFs from the UniverseMachine model (Behroozi et al. 2019) and from the Delphi model (Dayal et al. 2014, 2022). While our estimate is in rough agreement with these prediction at z ∼ 11, the model LFs evolve very rapidly at these early times, such that the z ∼ 12 LF is already > 30 × below our estimate. This is a general trend of model predictions: a relatively rapid evolution of the LF at z > 10, driven by the underlying evolution of the dark matter halo mass function (see also Oesch et al. 2018; Tacchella et al. 2018; Bouwens et al. 2021). However, the handful of bright galaxies that have been found at z ∼ 10 -13 to date appear to oppose this trend. \nIt is still unclear what the physical reason for this might be. Combined with the discovery of GN-z11 (Oesch et al. 2016), and taking our SED fits at face value (see § 5.1 for caveats), evidence is mounting that the star-formation efficiency in the early Universe may be much higher than expected (e.g., Tacchella et al. 2013; Mason et al. 2015a; Tacchella et al. 2018) in at least a few sources, thus resulting in the early appearance of UV-luminous galaxies with stellar masses as high as ≈ 10 9 M glyph[circledot] already a few hundred Myrs after the Big Bang. The existence of these massive galaxies at such early times raises interesting questions about just how early such galaxies began forming, potentially earlier than current expectations. Wider area datasets will be required to increase the search volume, for more reliable constraints on the number densities of luminous sources. \n4.0 \nFigure 5. Constraints on the bright end of the UV LF at z ∼ 10 -13. The current JWST data allow us to derive a first estimate of the number density of galaxies with M UV ∼ -21 at these redshifts (purple star). While this estimate lies a factor ∼ 10 × above the extrapolation of Schechter function constraints to z = 11 . 5 from Bouwens et al. (2021, dashed black line), they are in very good agreement with extrapolated double-power law LFs from Bowler et al. (2020, black solid line). In fact, GN-z11 (orange star) is also consistent with the double-power law LF. Other LF estimates and upper limits at z ∼ 10 are shown as open symbols (see legend for references). Simulated predictions are shown from the UniverseMachine models at z ∼ 11 and z ∼ 12 (dotted lines) and from the Delphi model at z ∼ 11 (darker gray shaded region). \n<!-- image -->', '6. SUMMARY & OUTLOOK': "This paper presented a search for luminous z > 10 galaxies across the two JWST Early Release Science programs in extragalactic fields. We find the following - \n- · We identify two particularly luminous sources in the GLASS ERS program. These sources, GLz10 and GL-z12, have continuum magnitudes of ∼ 27 at 2 µ m and display dramatic > 1 . 8 mag breaks in their SEDs that are best fit as Lyman breaks occurring at redshifts of z ≈ 11 and z ≈ 13 respectively. [Fig. 1, Fig. 2, § 4.1]\n- · SED modeling of these sources shows they have properties (e.g., β slopes, specific star-formation rates) expected of z > 10 galaxies. These systems are a billion solar mass galaxies, having built up their mass only < 300 -400 Myrs after the Big Bang. [Table 3, § 4.3]\n- · The brightness of these objects present a unique opportunity for detailed spectroscopic and morphological follow-up at z > 10. As a demonstra- \non, we model the morphology of both galaxies finding that they are well-described by disk-like profiles with small sizes (half-light radii ∼ 0 . 6 kpc). GL-z10, in particular, shows an extended exponential light profile, that may be tantalizing evidence for a disk already in place at z ≈ 10. [Fig. 4, § 4.5] \n- · These two objects already place novel constraints on galaxy evolution in the cosmic dawn epoch. They indicate that the discovery of GN-z11 was not simply a matter of good fortune, but that there is likely a population of UV luminous sources with very high star-formation efficiencies capable of compiling > 10 9 M glyph[circledot] at z > 10. [Fig. 3, § 5.2]\n- · The inferred number-density of M UV ≈ -21 sources from our search (log φ [Mpc -3 mag -1 ] = -5 . 05 +0 . 37 -0 . 45 ) strongly supports a significant deviation from the Schechter UV luminosity function at the bright end, and is consistent with the doublepower law evolution reported at lower redshifts. The physical mechanisms driving this departure \nare yet to be definitively established. These luminous sources highly conducive to NIRSpec spectroscopy may hold the key. [Fig. 5, § 5.2] \nIf these candidates are confirmed spectroscopically, and indeed two z ≈ 10 -12 candidates lie awaiting discovery in every ∼ 50 arcmin 2 extragalactic field, it is clear that JWST will prove highly successful in pushing the cosmic frontier all the way to the brink of the Big Bang. \nFacilities: \nJWST , HST \nSoftware: IPython (P'erez & Granger 2007), matplotlib (Hunter 2007), numpy (Oliphant 2015), scipy (Virtanen et al. 2020), jupyter (Kluyver et al. 2016), Astropy (Astropy Collaboration et al. 2013, 2018), grizli (v1.5.0; Brammer 2018; Brammer et al. 2022), MIST (Choi et al. 2017), Prospector (Leja et al. 2017, 2019; Johnson et al. 2021), FSPS (Conroy et al. 2009, 2010; Conroy & Gunn 2010a,b; Foreman-Mackey et al. 2014), EAZY (Brammer et al. 2008), SExtractor (Bertin & Arnouts 1996), GALFIT (Peng et al. 2002, 2010)", 'ACKNOWLEDGMENTS': 'We thank the referee for insightful comments that strengthened this manuscript. We are grateful to the CEERS and GLASS teams for planning these early release observations. \nRPN acknowledges funding from JWST programs GO-1933 and GO-2279. Support for this work was provided by NASA through the NASA Hubble Fellowship grant HST-HF2-51515.001-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555. We acknowledge support from: the Swiss National Science Foundation through project grant 200020 207349 (PAO, AW). The Cosmic Dawn Center (DAWN) is funded by the Danish National Research Foundation under grant No. 140. RJB and MS acknowledge support from NWO grant TOP1.16.057. \nCloud-based data processing and file storage for this work is provided by the AWS Cloud Credits for Research program. \nThis work is based on observations made with the NASA/ESA/CSA James Webb Space Telescope. The data were obtained from the Mikulski Archive for Space Telescopes at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-03127 for JWST. These observations are associated with programs # 1324 and # 1345.', 'REFERENCES': 'Whitler, L., Stark, D. P., Endsley, R., et al. 2022, arXiv e-prints, arXiv:2206.05315. \nhttps://arxiv.org/abs/2206.05315 \nWindhorst, R. A., Timmes, F. X., Wyithe, J. S. B., et al. 2018, ApJS, 234, 41, doi: 10.3847/1538-4365/aaa760 Wyithe, J. S. B., Yan, H., Windhorst, R. A., & Mao, S. 2011, Nature, 469, 181, doi: 10.1038/nature09619 \nYang, L., Leethochawalit, N., Treu, T., et al. 2022, \nMNRAS, 514, 1148, doi: 10.1093/mnras/stac1236 Zahid, H. J., Geller, M. J., Fabricant, D. G., & Hwang, H. S. 2016, ApJ, 832, 203, doi: 10.3847/0004-637X/832/2/203 \nFigure 6. Comparison of SEDs (left) and p ( z ) distributions (right) derived based on initial NIRCam calibrations available at the time of the ERS data release (light blue) with results based on updated calibrations used in this paper (orange). Photometry from the initial calibrations is shown in the left panels in transparent purple, and from the updated calibrations in solid purple. The SEDs are broadly similar, with the updated calibrations implying slightly lower redshifts for both sources. The high redshift nature of these sources remains secure. \n<!-- image -->', 'A. COMPARISON WITH INITIAL REDUCTION AND CALIBRATION': 'In Figure 6 we compare the results presented in this paper with fluxes, SEDs, and p ( z ) based on the initial NIRCam calibrations (e.g., zero-points, flats, darks) available with the ERS data release in July 2022. Relative to results based on these calibrations, the p ( z ) distribution has shifted slightly towards lower redshifts for both galaxies. GL-z10 now shows a stronger hint of rest-optical lines or a Balmer break in its F 444 W -F 356 W color, along with a slightly redder β UV slope ( -1 . 9 vs. -2 . 1) which does not require as much damping by the IGM to explain the F150W flux. GL-z12 is now detected at higher significance in F 150 W due to better overall handling of background features (e.g., wisps) in the SW filters, ruling out the secondary peak in the Prospector p ( z ) distribution centered at z ≈ 14.'}

2024MNRAS.534.2037S

Twentyonecentimetre signals from the Epoch of Reionization EoR are expected to be detected in the lowfrequency radio window by the nextgeneration interferometers particularly the Square Kilometre Array SKA. However precision data analysis pipelines are required to minimize the systematics within an infinitesimal error budget. Consequently there is a growing need to characterize the sources of errors in EoR analysis. In this study we identify one such error origin namely source blending which is introduced by the overlap of objects in the densely populated observing sky under SKA1Lows unprecedented sensitivity and resolution and evaluate its twofold impact in both the spatial and frequency domains using a novel hybrid evaluation HEVAL pipeline combining endtoend simulation with an analytic method to mimic EoR analysis pipelines. Sky models corrupted by source blending induce small but severe frequencydependent calibration errors when coupled with astronomical foregrounds impeding EoR parameter inference with strong additive residuals in the twodimensional power spectrum space. We report that additive residuals from poor calibration against sky models with blending ratios of 5 and 0.5 per cent significantly contaminate the EoR window. In contrast the sky model with a 0.05 per cent blending ratio leaves little residual imprint within the EoR window therefore identifying a blending tolerance at approximately 0.05 per cent. Given that the SKA observing sky is estimated to suffer from an extended level of blending strategies involving deblending frequencydependent error mitigation or a combination of both are required to effectively attenuate the calibration impact of sourceblending defects.

2024-11-01T00:00:00Z

['10.48550/arXiv.2409.11691', '2024MNRAS.534.2037S', '10.1093/mnras/stae2168', '2024arXiv240911691S', 'arXiv:2409.11691', '2024MNRAS.tmp.2193S']

['Astrophysics - Instrumentation and Methods for Astrophysics', 'Astrophysics - Cosmology and Nongalactic Astrophysics', 'Astrophysics - Astrophysics of Galaxies', '85-04', '85-08', '85-10', 'D.2.0', 'J.2', 'I.1.2', 'I.6.5', 'I.6.6']

An evaluation of sourceblending impact on the calibration of SKA EoR experiments

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']

https://arxiv.org/pdf/2409.11691.pdf

{'An evaluation of source-blending impact on the calibration of SKA EoR experiments': 'Chenxi Shan, 1 ★ Haiguang Xu, 1 ★ Yongkai Zhu, 1 Yuanyuan Zhao, 1 Sarah V. White, 2 Jack L. B. Line, 3 , 4 Dongchao Zheng, 1 Zhenghao Zhu, 5 Dan Hu, 6 Zhongli Zhang, 5 , 7 and Xiangping Wu 8 \n- 1 School of Physics & Astronomy, Shanghai Jiao Tong University, Shanghai, China\n- 2 Department of Physics and Electronics, Rhodes University, PO Box 94, Makhanda, 6140, South Africa\n- 3 International Centre for Radio Astronomy Research, Curtin University, Perth, WA 6102, Australia\n- 4 ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO-3D)\n- 5 Shanghai Astronomical Observatory, CAS, 80 Nandan Road, Shanghai, China\n- 6 Department of Theoretical Physics and Astrophysics, Faculty of Science, Masaryk University, Kotlářská 2, Brno, 611 37, Czech Republic\n- 7 Key Laboratory of Radio Astronomy and Technology, Chinese Academy of Sciences, 20A Datun Road, Beijing 100012, China\n- 8 National Astronomical Observatories, Chinese Academy of Sciences, 20A Datun Road, Beijing 100012, China \nAccepted 2024 September 08. Received 2024 September 05; in original form 2023 December 30', 'ABSTRACT': "Twenty-one-centimetre signals from the Epoch of Reionization (EoR) are expected to be detected in the low-frequency radio window by the next-generation interferometers, particularly the Square Kilometre Array (SKA). However, precision data analysis pipelines are required to minimize the systematics within an infinitesimal error budget. Consequently, there is a growing need to characterize the sources of errors in EoR analysis. In this study, we identify one such error origin, namely source blending, which is introduced by the overlap of objects in the densely populated observing sky under SKA1-Low's unprecedented sensitivity and resolution, and evaluate its two-fold impact in both the spatial and frequency domains using a novel hybrid evaluation (HEVAL) pipeline combining end-to-end simulation with an analytic method to mimic EoR analysis pipelines. Sky models corrupted by source blending induce small but severe frequency-dependent calibration errors when coupled with astronomical foregrounds, impeding EoR parameter inference with strong additive residuals in the two-dimensional power spectrum space. We report that additive residuals from poor calibration against sky models with blending ratios of 5 and 0.5 per cent significantly contaminate the EoR window. In contrast, the sky model with a 0.05 per cent blending ratio leaves little residual imprint within the EoR window, therefore identifying a blending tolerance at approximately 0.05 per cent. Given that the SKA observing sky is estimated to suffer from an extended level of blending, strategies involving de-blending, frequency-dependent error mitigation, or a combination of both, are required to effectively attenuate the calibration impact of source-blending defects. \nKey words: instrumentation: interferometers - dark ages, reionization, first stars - techniques: interferometric - software: simulations - radio continuum: general", '1 INTRODUCTION': "The low-frequency radio window opens up new opportunities for probing the high-redshift Universe through its unique capability to receive the redshifted 21-cm hyperfine line emission of neutral hydrogen from the early stages of the Universe, which can probe the untethered Cosmic Dawn ( 𝑧 ∼ 15 to 30) and the Epoch of Reionization (EoR; 𝑧 ∼ 6 to 15) with unprecedented precision (see Furlanetto et al. 2006; Pritchard & Loeb 2012; Zaroubi 2013; Furlanetto 2016, for reviews). The faint nature of the 21-cm signal demands instruments with extremely high sensitivity, wide field-of-view (FoV), and wideband coverage, making radio interferometers favoured for statistical detection of the 21-cm signal (Liu & Shaw 2020), presumably in the power spectrum (PS) space, given their flexibility to be designed \nand constructed as an array that meets these technical demands while also offering precise baseline coverage of EoR-specific scales (0 . 1 ≲ 𝑘 ≲ 2 Mpc -1 ). Major interferometers in the low-frequency band include the Giant Metrewave Radio Telescope (GMRT 1 ; Swarup 1991), the Murchison Widefield Array (MWA 2 ; Bowman et al. 2013; Tingay et al. 2013), the LOw Frequency ARray (LOFAR 3 ; van Haarlem et al. 2013), the MIT Epoch of Reionization (MITEoR 4 ; Zheng et al. 2014), the Precision Array for Probing the Epoch of Reionization (PAPER 5 ; Parsons et al. 2010), and next-generation \n- 1 http://www.ncra.tifr.res.in/ncra/gmrt\n- 2 https://www.mwatelescope.org\n- 3 https://www.astron.nl/telescopes/lofar\n- 4 https://space.mit.edu/home/tegmark/main\\_omniscope.html \narrays, the Hydrogen Epoch of Reionization Array (HERA 6 ; DeBoer et al. 2017) and the Square Kilometre Array (SKA 7 ; Mellema et al. 2013; Koopmans et al. 2015). However, owing to the complex baseline designs and complicated instrumental effects of these advanced arrays, precision data analysis pipelines are required for EoR experiments to overcome calibration, imaging, and analysis challenges and to minimize exposure to systematics, such as calibration biases (Datta et al. 2009; Grobler et al. 2014; Wijnholds et al. 2016; Patil et al. 2016; Gehlot et al. 2021), polarization leakages (Moore et al. 2017; Hales 2017; Dillon et al. 2018; Gehlot et al. 2018), wide-FoV and wide-band imaging errors (Bhatnagar et al. 2013; Jagannathan et al. 2015; Rau et al. 2016; Ye et al. 2022) and foreground residuals (Trott et al. 2012; Chapman et al. 2016; Nasirudin et al. 2020; Hothi et al. 2021). Therefore, it has become clear that EoR experiments must operate under an infinitesimal error budget with rigorous systematic characterizations to detect 21-cm signals successfully (see Liu & Shaw 2020; Shaw et al. 2023, for reviews). \nAdvanced low-frequency interferometers, such as LOFAR and SKA, can reach new detection limits that transform the occupation of the observed low-frequency radio sky from sparsely distributed, relatively rare sources (e.g. bright radio quasars and radio galaxies) to densely distributed populations constituting the majority of the extragalactic sky. In particular, the upcoming SKA1-Low is estimated to probe extragalactic discrete radio source (EDRS) populations deep into the faint sub-mJy radio sky ( ∼ 4400 deg -2 above 0 . 1 mJy at 150 MHz) and detect the bulk of galaxies with radio power originating from star-forming processes, the central active galactic nucleus (AGN), and a composite of both (Padovani 2016). This increase in sensitivity comes with increasing chances of overlap between the surface brightness distributions of sources (known as blending), which introduces systematic effects due to contaminated measurements (see Melchior et al. 2021, for a review). Without proper identification and mitigation strategies, these blended measurements will bias the interferometric calibration and, therefore, contribute to the overall EoR error budget by introducing imperfections during the construction of sky models, which are required to overcome line-of-sight (LoS) effects and imperfect instrumental responses (Liu et al. 2010; Li et al. 2018; Liu & Shaw 2020; Shaw et al. 2023) by most, if not all, calibration strategies of EoR experiments, including sky-based (Pearson & Readhead 1984; Rau et al. 2009), array-based (Noordam &de Bruyn 1982; Wieringa 1992), and hybrid calibration methods (Sievers 2017; Kern et al. 2020; Zhang et al. 2020; Byrne et al. 2021). \nOf all the contributions to the tight EoR error budget, calibration errors, especially chromatic residual gain errors originating from sky-model defects, are of great importance because biases from poor calibration 8 couple with contamination from strong astronomical foregrounds ( ∼ 10 4 -10 5 brighter than the 21-cm signal), propagate further down the analysis pipeline by leaving additive residuals in the measurement space, and ultimately impede the parameter inference of the EoR signal (Datta et al. 2010; Trott & Wayth 2016; Barry et al. 2016; Ewall-Wice et al. 2017; Byrne et al. 2019; Mazumder et al. 2022). Given the propagation nature of calibration errors, quantification of the impact of sky-model defects is achieved by mimicking full EoR analysis pipelines (see He et al. 2023, for an overview on EoR pipelines) and inferring the propagated residuals \n- 6 http://reionization.org\n- 7 https://www.skao.int \n8 Weuse the term 'poor calibration' to describe scenarios where an inaccurate or imperfect calibration is performed, in contrast to a perfect calibration. Poor calibrations result in an ill-calibrated instrument, thereby introducing calibration errors. \nin the measurement space, particularly in the two-dimensional (2D) PS space. Although the exact path for establishing the calibration propagation effects in the measurement space varies among studies, we can primarily sort the existing efforts into two types based on the approach used to mimic an EoR analysis pipeline. The first type uses an end-to-end approach to simulate propagation effects (e.g. Barry et al. 2016, hereafter B16), whereas the second type directly drives the final PS biases via analytic analysis (e.g. Ewall-Wice et al. 2017, hereafter E17). Despite the different approaches adopted, both types of methods have demonstrated that chromatic residual gain errors originating from sky-model imperfections severely damage the detectability of the 21-cm signal by introducing strong contaminations owing to poor calibration (one to two orders of magnitude brighter than the EoR signal, E17) and require an extremely small tolerance for model defects ( ∼ 10 -5 of spectral features, B16). Therefore, even though sky models can never be perfect, defects should always be evaluated and factored into the overall EoR error budget. \nWith its unprecedented sensitivity and baseline coverage in the low-frequency window, the SKA1-Low will contribute primary sky models for its observing sky (Trott & Wayth 2017). Thus, it is crucial to identify sky-model defects for the SKA and properly evaluate the impact of these defects on SKA EoR experiments under the telescope's observation specifications and conditions. Given the expected observing depth enabled by the SKA1-Low ( ∼ 0 . 1 mJy at 150 MHz, Prandoni & Seymour 2015), source blending will undoubtedly introduce one such defect to sky-model constructions of SKA EoR experiments and impact the EoR detections via blending-induced calibration errors. Since LOFAR, a pathfinder of SKA1-Low with comparable spatial resolution but lower sensitivity, has already suffered from source-blending effects to an extended level ( ∼ 3 per cent under a source density ∼ 3100 deg -2 , Kondapally et al. 2021), we expect the fraction of blended sources observed by the SKA1-Low to be at the same level or higher than that of the LOFAR observations, but lower than that of the optical surveys. \nTo date, there is still a lack of investigations that evaluate the impact of source blending on interferometric calibration, estimate the blending-induced calibration residual error, and identify blending tolerance for SKA sky-model construction. In this study, we present the first systematic assessment of source blending in the low-frequency radio window for the upcoming SKA1-Low, identify a largely two-fold effect of blending-induced uncertainties in both the spatial and frequency domains, and investigate the impact of frequency-dependent errors originating from calibrations against a sky model corrupted by source blending (hereafter blending-corrupted sky model) for SKA EoR experiments. In contrast to the simulation and analytic approaches used to infer calibration effects by B16 and E17, respectively, we propose a novel hybrid evaluation (HEVAL) approach combining custom end-to-end simulations and analytic calibration analysis mimicking SKA EoR experiment pipelines: (i) the end-to-end modules of our hybrid evaluation pipeline simulate radio maps of the low-frequency radio sky down to the smallest scales resolvable by the SKA1-Low ( ∼ 6 arcsec at 196 MHz for the foregrounds and ∼ 0 . 245 cMpc for the EoR signal), introduce sky-model defects into skymodel construction, include SKA1-Low instrumental response with visibility synthesis, and reconstruct the radio sky with map-making; (ii) the analytic fraction of the HEVAL pipeline adopts the widely used logarithmic implementation of the calibration equations (Pearson & Readhead 1984; Wieringa 1992; Liu et al. 2010, hereafter the PWL method) to mimic the calibration processes of EoR experiments and estimates blending-induced calibration errors and propagation biases. By utilizing a pair of sky models, consisting of a perfect sky model and a corrupted one, HEVAL decouples the instrumental noise bias \nfrom the calibration solution and derives the relative errors from poor calibration due to sky-model defects. The combination of end-to-end simulation and analytic analysis enables HEVAL to isolate the impact originating solely from source-blending defects and to quantify the blending-induced error budget under realistic SKA observations. \nThis paper is organized as follows. Section 2 introduces the definition of source blending adopted for this study, identifies its underlying impact on sky-model construction, and evaluates the blending ratio expected for the SKA1-Low. Section 3 presents our HEVAL pipeline and the 2D PS measurement space for evaluating the impact of source-blending defects on the calibration of SKA EoR experiments. We introduce both the key components of our custom end-to-end simulation suite and the analytic formalism of blendinginduced residual gain error and its propagation under a sky-based calibration scheme. In Section 4, we present the propagated residual power in the 2D PS space and identify the blending tolerance of the SKA sky-model construction based on the residual power contamination. We then discuss the implications of these results for EoR experiments and provide potential strategies for mitigating blending-induced bias in Section 5. Throughout this work, we adopt a flat Λ CDM cosmology with 𝐻 0 = 100 ℎ = 67 . 66 km s -1 Mpc -1 , Ω 𝑚 = 0 . 3096, Ω Λ = 1 -Ω 𝑚 = 0 . 6904, Ω 𝑏 = 0 . 0489, 𝑛 𝑠 = 0 . 9665, and 𝜎 8 = 0 . 8102.", '2 SOURCE BLENDING': "Blending of astronomical sources often refers to the overlapping of structural components in the projected sky. To properly evaluate and simulate the source-blending effect, a clear definition of blending is required. In this study, source blending is defined as independent, unrelated objects (including compact sources, elements of multicomponent sources, and side lobes) distributed close enough in the projected sky plane so that they cannot be accurately measured independently under the telescope's resolving condition. Based on the spatial relationship between the two objects in the image space and the existing blending definitions (Arcelin et al. 2021; Sanchez et al. 2021), we identify three different blending cases: \n- · Stacked : The projected radio emission contours of two objects coincide entirely, such that 100 per cent of the flux density distribution overlays each other.\n- · Overlapping : The projected radio emission contours of two objects partially intersect, with greater than 0 per cent but less than 100 per cent of the flux density distribution of one object overlapping the other.\n- · Proximity : The projected radio emission contours of two objects are close to each other but have a 0 per cent overlap in their flux density distribution. The angular separation of the peak flux density centroids can be up to several times the telescope's resolving beam. \nIn the most common form, source blending in the radio domain occurs when two individual objects blend into an apparently elongated or reshaped source. For simplicity, we only consider the blending of two objects in our study. Although we note the existence of multisource blends, multi-source blending cases can be considered as the superposition of two-object blends. Using the observation sensitivity estimated by Prandoni & Seymour (2015) and the simulated source model from the SKA Designed Study Simulated Skies ( 𝑆 39 ; Wilman et al. 2008, hereafter W08), we can roughly estimate the fraction \nof sources that are blended with other sources for upcoming SKA1Low observations. By calculating the per unit-area density of all the sources ( 𝐷 , arcsec -2 ) above the SKA1-Low's detection limit and projected extension of each source ( 𝑎 , arcsec 2 ), the likelihood of blending with another source can be inferred as 𝐿 = 𝑚𝑖𝑛 ( 𝑎 / 𝐷 / 𝑇, 1 ) , where 𝑇 ( 𝑇 ∈ (0, 1]) is the dimensionless threshold of blending. Subsequently, a quick and dirty estimation of the SKA blending ratio can be achieved by calculating the blending likelihood for each source above the SKA1-Low's detection limit. The total blending ratio of all the sources for the SKA1-Low is estimated to be approximately 5 -28 per cent considering different thresholds 𝑇 and projected extensions 𝑎 . As expected, this level of blending falls shy of the blending ratio of those optical surveys (see Melchior et al. 2021, and references therein), but still poses a significant challenge for SKA sky-model construction. In Fig. 1, we present the cumulative distribution function of the blending ratio under SKA1-Low's observation specifications and conditions at 150 MHz with a 𝑇 = 0 . 68 threshold. Given that the underlying W08 source-count models are characterized by Schechter parametrization, blending sources are unsurprisingly dominated by sources at the faint end. \nFor the sky-model construction, defects due to source blending are introduced during the detection and measurement phases. In a typical catalogue-building pipeline, source-detection methods decompose radio emission into Gaussian components and group the associated components as detected sources (Mohan & Rafferty 2015; Hancock et al. 2018). The presence of source blending often leads to biased detection because the source finder may struggle to decompose the correct Gaussian components owing to blended structures. In addition, component associations can also lead to source-blending defects due to the complex morphology of EDRS populations (e.g. see the G4Jy Sample, White et al. 2020b,a). Without physical constraints from the host galaxy (links to optical counterparts through crossmatch techniques, such as Line et al. 2017), distinctively separated components, such as isolated lobes and nearby faint compact sources, are easily associated as one blended source. Therefore, source blending significantly affects the accuracy of sky-model construction and can cause uncertainties with a largely two-fold impact: (i) the spatial domain flux density and position error 10 , which originate from a mixture of originally independent flux density distributions, and (ii) the frequency-domain spectral index deviation , which is caused by the blending of intrinsically different spectral components. As most limiting factors of sky models (e.g. instrumental noise and defects from sky-model incompleteness) are often coupled with one another, methods with the ability to isolate the error contribution of one error of origin at a time are required to evaluate the impact of each sky-model defect properly. Section 3 introduces one such method to evaluate the sole impact of calibration against sky models with source-blending defects for SKA EoR experiments.", '3 METHODOLOGY': "This section presents a hybrid approach for evaluating the calibration impact of sky-model defects originating from source blending for future SKA EoR experiments. Fig. 2 outlines the data and software blocks of our HEVAL pipeline, in which the solid blue and dashed pink lines mark the end-to-end and analytic data flows, respectively. \nFigure 1. The cumulative distribution function for blending ratio of extragalactic discrete objects under the SKA1-Low observation specifications and conditions. The x-axis represents the 150 MHz flux density of the discrete radio sources of the SKA1-Low observing sky, plotted on a logarithmic scale in the unit of Jy, whereas the y-axis represents the cumulative probability. The distribution is calculated by dividing the cumulative number of blended sources for each flux-density bin by the total number of blended sources. The shadow marks the 3-sigma uncertainty estimated using a Jackknife resampling technique with 60 subsamples. The lower and upper flux density are limited by the SKA1-Low's detection limit and the W08 simulation, respectively. For this estimation, the SKA observation condition is set using the synthesized beam size (8 arcsec) of the SKA1-Low at 150 MHz; the threshold of blending is set as 0.68; the total blending ratio of all the sources is estimated to be 7.25 per cent. \n<!-- image --> \nWe introduce the complete set of HEVAL modules 11 , including sky-model building, visibility synthesis, calibration error estimation, and map-making, and eventually arrive at the measurement space to quantify the blending-induced impact through 2D PS analysis. To better put our method and results into context, we attempt to summarize the assumptions adopted by the HEVAL pipeline in Table 1. We list the data-related mathematical notions in Table 2 and present the space transformation of the data cubes in Fig. 8. \nFirstly, the HEVAL pipeline starts with the simulation in the image space by presenting the construction of sky models along with the introduction of sky-model defects, resulting in a pair of sky models, one with source-blending defects and one without, and the simulation of the low-frequency radio sky, which includes extragalactic discrete foregrounds, Galactic diffuse foregrounds, and 21-cm signals. All maps are simulated across frequency channels forming sky-map cubes [ 𝑆 ori ( 𝑙, 𝑚, Δ 𝑓 ) ] along the frequency axis. The modules used are from our custom F/g.pc21S/i.pc/m.pc+ 12 simulation suite, which is developed based on our previous simulation efforts (Wang et al. 2010, 2013; Li et al. 2019), notably the /f.pc/g.pc21/s.pc/i.pc/m.pc 13 simulator. As our next-generation simulation suite, F/g.pc21S/i.pc/m.pc+ is designed to simulate the low-frequency radio sky in the image space down to the arcsec level to cover the smallest angular scales resolved by high spatial resolution instruments, such as LOFAR and SKA1-Low. In addition, F/g.pc21S/i.pc/m.pc+ can introduce imperfections into the simulation, making it particularly useful for evaluating sky-model defects within data analysis pipelines. \nSecondly, we detail image-to-visibility space transitions within the \nHEVAL pipeline. Cross-correlation synthesis with the SKA1-Low array configuration is achieved by feeding the F/g.pc21S/i.pc/m.pc+-generated sky-map cubes of the sky-model pairs, foreground emissions, and the EoRsignals to the OSKAR array simulator, where the output visibility cubes mimic realistic instrumental responses under the SKA1-Low observation conditions. The array configuration and corresponding SKA1-Low specifications are summarized in Tables 3 and 4, respectively. Within the visibility space, we estimate the propagation effects during interferometric calibration under the sky-based calibration scheme by presenting an analytic analysis to infer the relative residual gain errors from poor calibration owing to sky-model defects and their subsequent propagation visibility biases. We use visibilities of the simulated sky model pairs to evaluate the blending-induced per-frequency per-antenna relative residual gain error and, subsequently, apply the relative error solutions to the visibilities of each sky component to infer the per-frequency per-baseline propagation visibility biases. Although no explicit foreground-removal module is implemented by HEVAL, the derived visibility biases are similar to the residual visibilities generated by pipelines explicitly performing foreground removal, owing to the nature of the relative calibration. Detailed analytic derivations are introduced in Section 3.4. In the visibility space, HEVAL produces two sets of visibility cubes: (i) the sky emission visibility cubes [ 𝑉 ori ( 𝑢, 𝑣, Δ 𝑓 ) ] representing the true response of the extragalactic discrete foregrounds, Galactic diffuse foregrounds, and EoR signals and (ii) the blending-induced visibility-bias cubes [ 𝑉 res ( 𝑢, 𝑣, Δ 𝑓 ) ] for each sky component. \nThirdly, the HEVAL pipeline converts the visibilities back to the image space to reconstruct the radio sky through the map-making processes presented in Section 3.5, given our aim of inferring the residual PS impact under the 'reconstructed'-sky approach (Morales et al. 2019). The WSC/l.pc/e.pc/a.pc/n.pc imager is used for the Fourier transform, convolution, and de-convolution of the simulated visibilities. Both sets of visibility cubes [ 𝑉 ori ( 𝑢, 𝑣, Δ 𝑓 ) and 𝑉 res ( 𝑢, 𝑣, Δ 𝑓 ) ] are converted \nFigure 2. The HEVAL pipeline for sky-model defect evaluation. The solid-blue lines mark the end-to-end simulation faction for sky-map generation, SKA1-Low visibility realization, and map making. The dashed pink lines mark the analytic analysis approach for the gain error estimation and visibility bias calculation. The solid blue lines mark the end-to-end simulation faction, including sky-map generation, SKA1-Low visibility realization, and map making, whereas the dashed pink lines mark the analytic analysis approach for the gain error estimation and visibility bias calculation. Although typical foreground removal is not explicitly included in the pipeline, this process is implicitly handled through our analytical fraction of the hybrid HEVAL pipeline directly in the visibility space, and the analytic derivation of visibility biases is analogous to the visibility residuals produced by pipelines subtracting the foregrounds. Coloured double diamonds and rounded grey squares represent data blocks and software blocks, respectively. From the data flow perspective, the upper and lower fractions denote the ill-calibrated residual and sky emission data flow, respectively. The simulation of sky model pairs, using ES/i.pc/m.pc (labelled 1) and B/l.pc/e.pc/n.pc/d.pcS/i.pc/m.pc (labelled 2), is introduced in Section 3.1; the generation of evaluation data, using ES/i.pc/m.pc (labelled 1), GS/i.pc/m.pc (labelled 3), and 21/c.pc/m.pcS/i.pc/m.pc (labelled 4), are detailed in Section 3.2; the visibility realization, using OSKAR (labelled 5), is described in Section 3.3; the analytic analysis of the source-blending impact within the visibility space is introduced in Section 3.4; the step-by-step analytic derivation of gain error (labelled 6) and the analytic calculation of the propagation visibility bias (labelled 7) are detailed in Appendix B1 and Appendix B3, respectively; and the convolution and deconvolution of the radio maps, using WSC/l.pc/e.pc/a.pc/n.pc (labelled 8), are described in Section 3.5; assumptions adopted by the HEVAL pipeline are organized in Table 1; data-related mathematical notions are noted in Table 2; the array configuration and the simulation specifications used for this study are listed in Tables 3 and 4, respectively. \n<!-- image --> \ninto two sets of image cubes [ 𝐼 ori ( 𝑙, 𝑚, Δ 𝑓 ) and 𝐼 res ( 𝑙, 𝑚, Δ 𝑓 ) ] for each sky component, respectively. \nFinally, we establish the PS measurement space for statistical EoR detection and transform the two sets of image cubes into two sets of three-dimensional (3D) PS [ 𝑃 ori ( 𝑘 𝑥 , 𝑘 𝑦 , 𝑘 𝑧 ) and 𝑃 res ( 𝑘 𝑥 , 𝑘 𝑦 , 𝑘 𝑧 ) ] and calculate the cylindrical-averaged 2D PS [ 𝑃 ori ( 𝑘 ⊥ , 𝑘 ∥ ) and 𝑃 res ( 𝑘 ⊥ , 𝑘 ∥ ) ] for each sky component, respectively. Being a key EoR figure of merit, 2D PS, along with EoR windows, is used to evaluate the eventual impact of blending-induced poor calibration. \nThroughout this paper, we denote the F/g.pc21S/i.pc/m.pc+-generated sky emission, OSKAR-simulated cross-correlations, WSC/l.pc/e.pc/a.pc/n.pcconvoluted, and -deconvoluted images as sky maps, 'observed' visibilities, dirty maps, and clean maps, respectively. To clearly distinguish between the terms, we label the terms of sky emission with 'original' (e.g. 'original' image cubes and 'original' visibility cubes). In contrast, we label all the poor-calibration related terms with 'residual' (e.g. 'residual' visibility-bias cubes and 'residual' powers).", '3.1 Simulation of sky models and source-blending effect': "Wedetail the realization of sky-model construction in the image space. This study follows the convention and includes only EDRS populations in the sky model using the ES/i.pc/m.pc module, because most commonly used sky models for interferometric calibration are constructed using \nEDRSpopulations, often from high-quality EDRS catalogues (Carroll et al. 2016; Procopio et al. 2017; White et al. 2020b,a; Kondapally et al. 2021) and dedicated surveys (Norris et al. 2011; Lane et al. 2014; Wayth et al. 2015; Heald et al. 2015; Jackson et al. 2016; Zheng et al. 2016; Lacy et al. 2016; Shimwell et al. 2017; Intema et al. 2017; Hurley-Walker et al. 2022). While diffuse emission contributes power to the radio sky at larger angular scales, the limited existing data and algorithmic challenges leave discrete sources as the primary ingredients for modelling the sky (Liu & Shaw 2020). \nOur next-generation F/g.pc21S/i.pc/m.pc+ suite offers the flexibility to add defects to sky models in the image space through defect-simulation modules, such as B/l.pc/e.pc/n.pc/d.pcS/i.pc/m.pc, which introduces additional sourceblending effects to the ES/i.pc/m.pc-simulated sky models. By combining ES/i.pc/m.pc and B/l.pc/e.pc/n.pc/d.pcS/i.pc/m.pc, our simulation of sky-model construction and blending-induced defects arrives at a pair of sky models: (i) an ' ideal ' sky model representing the true flux density distribution of discrete sources without any blending of sources and (ii) a ' blended ' counterpart with some of the sources blended together resulting in flux density distribution deviations with respect to (w.r.t) their ' ideal ' counterparts. The data flow is reflected in the upper-left part of Fig. 2. With the ' ideal ' and ' blended ' sky models, our HEVAL pipeline can utilize a paired sky model estimation approach to derive the sole contribution of source blending during poor calibration, which is achieved by isolating the impact of blending defects from any \nTable 1. Summary of assumptions adopted by the HEVAL pipeline. \nTable 2. Data-related mathematical notions used in the paper. \nother limiting factors of sky-model construction (e.g. baseline noises and other sky-model defects) and inferring the blending-induced relative residual gain errors. Given that the relative residual gain error and propagation bias are subject to the sky model differences in the visibility space, the simulation of sky models with end-to-end control and clearly defined procedures is crucial for estimating the source-blending effect properly.", '3.1.1 Simulation of EDRS populations': "EDRSpopulations (see Padovani 2016; Panessa et al. 2019; Blandford et al. 2019; O'Dea & Saikia 2021, for reviews) are key objects of interest in the radio window, not only due to their diversity in terms of morphological types but also due to the underlying physical processes, black-hole accretion and star formation, which govern how galaxies evolve. As the bulk of the extragalactic sky, EDRSs dominate the radio sky in terms of both number and brightness, making them a key foreground component of EoR experiments. Consequently, EDRS populations are critical for calibration, imaging, and subtraction in the field of 21-cm science. In light of this, we compose our sky model simulation by considering only extragalactic radio sources with radio emission originating from supermassive black hole (SMBH) activity, which comprises compact radio core emission and structural features (such as relativistic jets, plumes, lobes, and hotspots), and star \nformation processes, which produce synchrotron radiation through relativistic plasma within the supernova remnants, associated with galaxies. \nThe EDRS populations included in the ES/i.pc/m.pc module consist of (i) bright extended radio galaxies (RGs), which are Fanaroff-Riley (FR) type I and type II galaxies (Fanaroff & Riley 1974), (ii) compact radio-loud (RL) AGNs, which consist of both compact steep-spectrum (CSS) and peaked-spectrum (PS) AGNs, (iii) radio-quiet (RQ) AGNs, and (iv) star-forming (SF) and starburst (SB) galaxies. The underlying source count distribution follows the commonly used W08 source catalogue simulation. To generate sky maps of individual sources, we simulated the brightness distribution of compact sources (i.e. compact AGNs, SF, and SB galaxies) with a single 2D Gaussian model and extended sources (i.e. FR-I and FR-II RGs) with multipleGaussian models by adopting morphological properties from the W08 catalogue (e.g. angular size and position angle). In particular, the extended sources are created from the convolution of a 2D Gaussian kernel with a radial profile along the jet axis, which results in a superposition of multiple Gaussian distributions, representing the extended structures, such as the jet and lobes of radio galaxies (Fig. 3). For the spectral models, ES/i.pc/m.pc utilizes a power-law spectrum for most radio spectra, whereas a curvature model from Jarvis & Rawlings (2000) and spectral turnover discovered by O'Dea (1998) are implemented for compact core emission and PS sources, respectively. \n<!-- image --> \n<!-- image --> \nFigure 3. Illustration of ES/i.pc/m.pc-generated radio galaxies using a multiple-Gaussian model. (Upper) ES/i.pc/m.pc-simulated flux-density distributions of typical (a) FR-I and (b) FR-II sources. The two flux density distributions are presented using the same colour bar. The dashed white line marks the jet axis. (Lower) The corresponding radial profiles along the jet axis of the two sources above. The blue line marks the radial profile of the extended structures, whereas the red line marks the overall radial profile of the whole source. The two figures from the same column are plotted with the matching pixel coordinates. \n<!-- image -->", '3.1.2 Simulation of the blending effect': "The B/l.pc/e.pc/n.pc/d.pcS/i.pc/m.pc module is used to introduce source-blending effects in both the spatial (Fig. 4) and spectral domains (Fig. 5). The implementation of B/l.pc/e.pc/n.pc/d.pcS/i.pc/m.pc contains all the EDRS populations in Section 3.1.1, resulting in 15 different blending types, such as RG-RQ AGN, AGN-SF, and RG-SB. To mimic the spatial displacement of brightness distribution, a beam with a limited resolution is used to convolute the ideal brightness distribution of the selected source pairs in the 'stacked', 'overlapping', or 'proximity' case by adjusting the distance between the centres of two objects. The width of the convolution beam should not be larger than that of the SKA synthesized beam, and we set the beam size to 6 arcsec for this study. To consider only spatial distribution deviations rather than flux density errors, B/l.pc/e.pc/n.pc/d.pcS/i.pc/m.pc employs a blending convolution procedure that utilizes a normalized kernel to retain the same total flux density of the source pair before and after blending. \nAfter the source pair is blended in the spatial domain, B/l.pc/e.pc/n.pc/d.pcS/i.pc/m.pc adds the blending-induced error in the spectral domain. Because our sky-model building does not include spectral-index spatial variations for individual sources, the spectral domain error is introduced via the spectral index deviations of the integrated spectral indices of the blended pair. Given that most of the blended sources present either \npower-law or power-law-like spectra within our simulated frequency band, a power-law 14 is used to model the integrated spectral index of the blended spectra (e.g. the blended spectra presented in Fig. 5). After the spectral model is assigned, B/l.pc/e.pc/n.pc/d.pcS/i.pc/m.pc generates multi-frequency sky models using the ' blended ' monochromic sky model with the estimated blended spectral indices. \nFor the simulation of blending defects, it is important to note that B/l.pc/e.pc/n.pc/d.pcS/i.pc/m.pc does not introduce flux density errors, which are most likely unavoidable for blending sources in real data considering image noise and artefacts. Flux density errors will not only introduce their own impact on calibrations but also bias spectral modelling, leading to additional spectral index errors. The coupling of different error sources hinders the final interpretation of the impact in the measurement space. Given that our aim is to decouple the sole impact of blending, we explicitly choose not to simulate any flux density errors to ensure that the evaluation and interpretation target only the source-blending impact. While this omission of flux density errors in the simulation does mean that the errors of blending are most likely \n14 Unlike spectral fitting with real data, our modelling processes are ideal cases with no noise or errors that may cause spectral fluctuations or distortions. A power-law model approach to mimic spectral index deviation is sufficient for this study. \nFigure 4. Mostcommonlyoccurredblendingtypesbetweentwocomponentsin the image space (the cases of two extended structures fully stacked are omitted due to the rare occurrence). The rule-of-thumb illustrations demonstrate the possible blending-induced error in the image space during source detection with the presence of noises. The elongated and round shapes indicate extended and compact features, respectively. The colour-filled shapes represent the ground truth of the components, whereas the dashed black lines mark the detected components by source finders. For bright sources, the detection errors are dominated by blending. For faint sources, image noises contribute to flux density distribution bias in addition to blending-induced uncertainties. \n<!-- image --> \nunderestimated compared to real-world scenarios, it allows us to directly attribute the propagated biases to blending defects alone. This approach provides a clean and controlled evaluation, isolating the effects of a single source of origin, without the confounding influence of other error sources.", "3.1.3 The 'ideal' and 'blended' sky model": "To generate the sky map of the ' ideal ' sky model and its ' blended ' counterpart, a basis sample of 4,000 sources, ranging from 0.1 to 1000 mJy, are sampled from the W08 catalogue satisfying its underlying source count model with a minimum separation requirement of 100 arcsec within the estimated SKA1-Low FoV of 5 · . Therefore, the sparsely distributed sources are not biased by blending effects. With the sky model basis at hand, a sub-sample of the basis is selected as the host of the blended source pair and assigned to another source with a maximum separation of 6 arcsec, corresponding to the synthesized beamsize of the SKA1-Low. Using the ideal flux density distribution and the true spectral index of both the source pair and the basis sample, sky-map cubes of the ' ideal ' multi-frequency sky models are generated. As for the counterpart, the blended flux density distribution and deviated spectral index are simulated using B/l.pc/e.pc/n.pc/d.pcS/i.pc/m.pc. By replacing the ideal model of the source pair with the blended model, we obtain sky-map cubes of the ' blended ' multifrequency sky models. In Fig. 6, we present a small patch of a pair of ' ideal ' and ' blended ' sky models. Adopting the 15 types of blending pairs described in Section 3.1.2, B/l.pc/e.pc/n.pc/d.pcS/i.pc/m.pc assigns the types based on the likelihood using the W08 number count model. \n(a) Compact-Compact Blend \nFrequency [Hz] \n<!-- image --> \n(b) Compact-Extended Blend \nFigure 5. Demonstration of blending defects in the spectral domain. (Top) Blended spectra of the two compact radio sources. (Bottom) Blended spectra of compact and extended sources. The two dashed lines mark the original spectra of the individual sources and the purple line marks the spectra of the blended pair. The green shaded area indicates the frequency range used in this study. In both cases, an apparent spectral index deviation exists after the two individual sources are blended. \n<!-- image --> \nFor a systematic estimation of the source-blending impact on the SKA EoR experiments, a set of 2, 20, and 200 blended source pairs are generated for the basis sample of 4,000 sources, representing source-blending levels of 0.05 per cent, 0.5 per cent, and 5 per cent, respectively. Because we aim to find an expectedly modest blending tolerance of sky-model construction for SKA EoR experiments, the 5 per cent minimum of the estimated SKA blending ratio is used as the maximum of the three levels. In the latter part of the manuscript, we refer to the three blending levels as 'mild', 'moderate', and 'high' blending ratios, respectively. For simplicity, sky models affected by the three blending ratio levels are marked as 'mildly'-, 'moderately'-, and 'highly'-corrupted sky models, respectively.", '3.2 Simulation of the evaluation data': "Evaluation data cubes are required to represent the low-frequency radio sky, including the EoR signal and its foregrounds, to evaluate the propagation effect of residual gain errors. The SKA1-Low has baselines across an extensive range of lengths that can probe the EoR signal across approximately three decades in scale. The ability to detect the EoR signal at a substantial number of scales, both large and small, gives the edge of the SKA1-Low over other EoR-aimed \nFigure 6. Demonstration of a monochromic sky model pair simulated using our ES/i.pc/m.pc and B/l.pc/e.pc/n.pc/d.pcS/i.pc/m.pc models. (Left) A small patch (1 . 5 arcmin × 1 . 5 arcmin) of an ' ideal ' sky model with the ideal source model pair. (Right) The same sky patch of the corresponding ' blended ' sky model with the blended source model pair. For a sky model pair, the differences between the ' blended ' and ' ideal ' sky models are the flux density distributions of these source model pairs marked by the dashed white lines, whereas the rest of the sources have exactly the same flux density distributions. Since B/l.pc/e.pc/n.pc/d.pcS/i.pc/m.pc normalizes the blending convolution procedure, no flux density uncertainties are introduced for source model pairs. The two figures are plotted using the same colour bar. Multi-frequency sky model pairs are simulated by supplying the monochromatic version with spectral models. For this particular pair, the spectral models were similar to those presented in Fig. 5 (b). \n<!-- image --> \ninstruments. Even though the total power of the 21-cm signal is expected to be constituted mainly from large-scale contributions, the EoR signal at the medium and small scales pokes unique aspects of the Universe, particularly during the later stages of the EoR. For one thing, the 21-cm signal can trace the small-scale structure and its coupling to the large-scale structure (Lidz et al. 2007). For another, the EoR signal is expected to be highly non-Gaussian during the later stages of reionization (Mondal et al. 2016; Majumdar et al. 2018), and its non-Gaussianity is probed at the smaller spatial scales. Therefore, to thoroughly investigate the foreground coupling effect of the blending-induced calibration error for SKA EoR experiments, it is crucial to simulate the radio sky containing physical scales that are spatially resolvable by the SKA1-Low baselines. In the remainder of this section, we introduce the F/g.pc21S/i.pc/m.pc+ modules used to generate evaluation data containing simulated scales from the degree level down to the smallest scales resolvable by the SKA1-Low. The data flow of the low-frequency radio sky simulation is illustrated in the lower left part of Fig. 2. The corresponding F/g.pc21S/i.pc/m.pc+-generated sky maps are presented in Fig. 7.", '3.2.1 Extragalactic foregrounds': 'The extragalactic foregrounds, dominated by EDRS populations (see Section 3.1.1), predominate the contamination of the EoR signal mainly on smaller scales because they are discrete sources with flux density fluctuations at the arcsec to arcmin scales. Although other extragalactic sources, such as diffuse radio emission associated with galaxy clusters, present unique challenges to the EoR experiment and impose contamination at the arcmin scale (Li et al. 2019; Zhou et al. 2022), they are ultimately excluded from our model because of both the relatively faint flux, which is three to four orders of magnitude lower, and a significantly lower number density compared to the EDRS populations. We use the ES/i.pc/m.pc module to generate sky maps of the extragalactic foregrounds, adopting the simulation specifications from Section 3.3.2. Here, the full EDRS source-count model of the \nW08 catalogue is implemented, in contrast to the sampling procedure when simulating sky models detailed in Section 3.1.', '3.2.2 Galactic foregrounds': 'The Galactic foregrounds mainly contain the diffuse Galactic synchrotron and free-free emission, with the synchrotron component being the dominant among the two. Our GS/i.pc/m.pc treats both the synchrotron and free-free components separately and simulates a patch of the sky using basis template maps: (i) For Galactic synchrotron emission, the basis synchrotron template map deployed by GS/i.pc/m.pc is the reprocessed Haslam 408 MHz all-sky map 15 , which is referred to as the HAS14 map, by Remazeilles et al. (2015) with significantly improved removal of both extragalactic sources and instrument artefacts. To calculate the sky maps in the SKA1-Low frequency via extrapolation with a power-law model from the 408 MHz, the all-sky spectral index map ( 𝛼 ∈ [2.50, 3.20], Giardino et al. 2002) of the Galactic synchrotron emission is used. (ii) As for the Galactic freefree emission, the basis template map is simulated through indirect estimations, as direct observations of the free-free component, which is overwhelmed by the strong synchrotron counterpart, is not possible. Owing to the common origin of free-free and H 𝛼 emission, a tight relation between the two has been demonstrated. It can be utilized to calculate the free-free emission (see Dickinson et al. 2003, and references therein). The GS/i.pc/m.pc module largely follows the prescription of Dickinson et al. (2003), employs the H 𝛼 survey from Finkbeiner (2003), corrects the dust absorption with a full-sky 100µ mdust map (Schlegel et al. 1998), and subsequently, simulates the free-free emission. Because the free-free and H 𝛼 relation is frequency dependent, the conversion from the H 𝛼 map to the free-free map is calculated for the desired SKA1-Low frequency window.', '(a) Galactic Emission (Synchrotron)': 'Figure 7. Simulated sky maps of evaluation data representing the low-frequency radio sky: the Galactic diffuse (a) synchrotron and (b) free-free emission, the (c) extragalactic discrete emission, and the (d) EoR signal at 158 MHz. All the images are zoomed in to the central 1 · × 1 · to show the small-scale structures. Each figure uses its own colour bar for the display. In particular, the EoR signal is plotted using unit [mK] different from the rest. \n<!-- image --> \n[K] \n[mK] \nLimited by existing observations, Galactic contributions to the radio sky are primarily at large scales. Under the scope of this work, flux density fluctuations at small scales are also required to properly estimate the impact at the longer baselines of the SKA1-Low. Our implementation in GS/i.pc/m.pc also includes the addition of Galactic small-scale fluctuations using the realization of a Gaussian random field (GRF). Our implementation of adding small-scale fluctuations is similar to the existing efforts based on the realization of GRFs (Miville-Deschênes et al. 2007; Delabrouille et al. 2013; Remazeilles et al. 2015; Hervías-Caimapo et al. 2016; Thorne et al. 2017) but with some enhancements. The fundamental idea is to extrapolate the angular PS ( C ℓ , parametrized by 𝛾 ) to the required small scales ( ℓ ) and generate the small-scale fluctuations ( 𝐺 ss , parametrized by 𝛼 and 𝛽 ) using a GRF corresponding to the extrapolated C ℓ . We detail the step-by-step realization of adding small-scale fluctuations in Appendix A. \nTo generate sky maps of the galactic foregrounds, we set the simulation centre of sky maps at a high galactic latitude centre position (R.A., Dec. = 0 · , -27 · ) since future SKA observations are expected to point at high galactic latitudes (e.g. | 𝑏 | > 60 · ) to minimize contamination. As for the small-scale fluctuations, we use the best-fitting of the three required parameters: (i) the 𝛾 index of synchrotron and free-free component is fitted as -2.220 and -2.426, respectively, using the full sky template maps; (ii) the 𝛼 and 𝛽 index of the small-scale maps are fitted for the synchrotron ( 𝛼 = 0 . 0342, \n𝛽 = 0 . 227) and free-free emission ( 𝛼 = 0 . 00785, 𝛽 = 0 . 526) using a patch of the template map centred at the selected simulation centre.', '3.2.3 EoR signal': "The final component of the radio sky is the 21-cm signal. To compensate for the need for high spatial resolution and a relatively large FoV, a fast semi-numerical approach is used in the 21/c.pc/m.pcS/i.pc/m.pc module by adopting the 21/c.pc/m.pcFAST 16 (Mesinger et al. 2011; Murray et al. 2020) package. The cosmic reionization process is simulated using the same physical parameters (A. Mesinger, priv. comm.) as the 'faint galaxies' model of the Evolution Of 21 cm Structure project 17 (Mesinger et al. 2016) with a box of 294 comoving Mpc and 1200 cells along the side. Our simulation run resolves the smallest physical scale at 0.245 cMpc, making the finest-resolution 21-cm simulation to date using 21/c.pc/m.pcFAST. From the simulated light-cone object, the designated sky maps of each frequency band are sliced w.r.t the corresponding redshift and tiled according to the simulation specifications listed in Table 4 to form sky-map cubes. \n- 16 21/c.pc/m.pcFAST: https://github.com/21cmfast/21cmFAST\n- 17 Evolution Of 21 cm Structure project: http://homepage.sns.it/ mesinger/EOS.html \nTable 3. Array specifications adopted for the SKA1-Low.", '3.3 The realization of simulated SKA observations': "This section presents the transformation from the image space to the visibility space. First, we detail the SKA array configuration used and the corresponding technical specifications of the given array. Then, we describe the simulation specifications under limited computational powers. Finally, we introduce the realization of visibility synthesis using an array simulator which outputs 'observed' visibilities with instrumental responses of the SKA1-Low. In the end, we transform all the sky-map cubes into visibility cubes for both the sky model pairs and the sky components, as presented in the middle-left part of Fig. 2.", '3.3.1 Array configuration': "As the most advanced and transformational project in radio astronomy to date, the upcoming SKA will be constructed in two phases, owing to challenges and difficulties in designing, constructing, and operating extremely large telescopes. Among the two telescopes of Phase 1, SKA1-Low in Australia and SKA1-Mid in South Africa, the low-frequency aperture array SKA1-Low operating in the 50 MHz to 350 MHz frequency range is the instrument intended for EoR detection and 21-cm cosmology. \nBoth the SKA1-Low and SKA1-Mid instruments are yet to be fully built, although pathfinders, precursors, and verification systems have been built and operated as stepping stones for the operation of SKA1. For this study, we use the array configuration from the SKA1 System Baseline Design (Dewdney et al. 2013, 2016, 2019). These reports detail that SKA1-Low will contain 131072 log-period dipole antennas with band coverage from 50 MHz to 350 MHz, assembling 512 stations of diameter 35 -40 m each consisting of 256 antennas. Among all the stations, 224 stations are placed randomly within the 'core' region of 1000 m in diameter, and 288 stations are distributed into 'clusters' along the three array spiral arms, which extend the baselines up to 65 km. Following the SKA1-Low Design Baseline, the estimated SKA1-Low specifications are summarized in Table 3.", '3.3.2 Simulation specification estimation': 'The high spatial resolving power of the SKA1-Low advocates the capability of SKA EoR experiments to probe the structural scale ranging from several arcsecs to degrees via 21-cm signals. In accordance with the estimated spatial resolution of the SKA1-Low ( ∼ 6 arcsec at 196 MHz), the simulated sky maps are required to contain the smallest scale at the 6 arcsec level ( ∝ λ ) at the corresponding frequency, which expects the pixel size of the sky map to be approximately 2 arcsec. \nWithin the designed SKA1-Low frequency window, three sparsely selected frequency bands, with a bandwidth of 8 MHz are chosen to limit the possible cosmological evolution of the EoR signal (e.g. Thyagarajan et al. 2013; Li et al. 2019), covering 120 - 128 MHz, 154 - 162 MHz, and 188 - 196 MHz. \nSimulating the radio sky with both high spatial and frequency resolution across extensive sky coverage presents significant computational challenges. This is because each stage of the process requires substantial resources: (i) the cosmological simulation of the 21-cm signal is memory-intensive, (ii) the generation of multi-frequency sky maps demands a high-performance GPU, and (iii) the visibility synthesis and subsequent map-making processes require substantial CPU power. Consequently, the combined high demands on CPU, GPU, and memory inherently restrict the practical specifications of our simulations. Given these limitations, it is impractical to simulate the SKA observing sky in a way that achieves a large field of view (FoV) and high spatial resolution simultaneously. Since spatial resolution plays a more critical role under the scope of this study (e.g. EDRS-only sky models and determination of blending of sources), we prioritize spatial resolution over areal coverage. However, to still adequately cover the key EoR scales (0 . 1 ≲ 𝑘 ≲ 2 Mpc -1 ), we chose a reasonable balance to simulate sky patches of up to 2 · × 2 · . The simulation specifications for both sky models and evaluation data are presented in Table 4.', '3.3.3 Array simulator': "The realization of visibilities with practical instrumental effects of the SKA1-Low is archived with the OSKAR 18 array simulator (Mort et al. 2010) by generating SKA1-Low 'observed' visibilities with the SKA1-Low specification and configuration using GPU accelerations. The specific input telescope model is created using the SKA1-Low array layout with antenna coordinates 19 from Dewdney & Braun (2016). The three frequency bands (Section 3.3.2) are divided into 51 channels with a channel width of 160 kHz. To minimize the pointing effect, we choose to centre all the sky maps at the sky position of (R.A., Dec.) = (0 · , -27 · ), which is the expected SKA1-Low zenith. \nSky-map cubes of both the ' ideal ' and ' blended ' sky models are directly passed to the OSKAR simulator without pre-processing processes under a sky coverage of 5 · × 5 · and a pixel size of 1 arcsec with a single 2-min snapshot exposure time. The sky-map cubes of the evaluation data, extragalactic discrete emission, Galactic diffuse emission, and EoR signal, are simulated with a sky coverage of 2 · × 2 · and a pixel size of 2 arcsec with deep observations for 6 h. For the EoR foregrounds, pre-processing progresses are made before the visibility synthesis. Being extended in nature, the Galactic components, synchrotron and free-free, are combined and tapered, which reduces the window effect of sky maps containing large-scale structures with a limited size before they are 'observed' by the simulator. Since the brightest sources of EDRS are already known to strongly bias the EoR observations (e.g. Datta et al. 2010), we assume these sources are adequately dealt with and removed during data analysis pipelines. Hence, our simulations mask the EDRS populations with a 158 MHz flux density above 50 mJy, as indicated \nhttps://astronomers.skatelescope.org/wp-content/uploads/ 2016/09/SKA-TEL-SKO-0000422\\_02\\_SKA1\\_ \nLowConfigurationCoordinates-1.pdf (released on 2016 May 31) \nTable 4. Simulation specifications of the sky models and evaluation datasets. \nin Fig. 2. All the specifications and pre- and post-processing processes are listed in Table 4.", '3.4 Evaluation of blending-induced calibration errors': "This section introduces the analytic fraction of the HEAVL pipeline, which is shown in the central part of Fig. 2. A sky-based per-frequency per-antenna calibration scheme is first introduced as the basic framework. We then present the derivation of the frequency-dependent per-antenna gain error caused by a blending-corrupted sky model w.r.t the ' ideal ' sky model counterpart through analytic analysis based on the PWL logarithmic method. Unlike traditional methods aimed at a direct antenna-based calibration solver to infer the solution of the complex gain factors (Thompson & D'Addario 1982), we use an alternative approach to estimate the blending-induced relative gain error using a pair of ' ideal ' and ' blended ' sky models. By adopting the PWL logarithmic implementation, the analytic expressions arrive at two simple sets of linear systems of equations, resulting in an overdetermined linear regression problem mathematically. To solve for the estimated gain error, we use a singular value decomposition (SVD)-based method. With the solved relative residual gain error, we can derive an analytic expression of the propagation visibility bias of the sky signal in the visibility space induced by an ill-calibrated instrument. For conciseness, we state the critical steps in this section and leave the step-by-step mathematical derivation in Appendix B. A full description of the calibration-related mathematical notations is presented in Table B1.", '3.4.1 Calibration in a sky-based scheme': 'Considering an array with 𝑁 antennas, the 𝑖 th antenna measures the unpolarized voltage 𝑣 𝑖 at the antenna end at a single frequency channel 𝑓 within a limited time span Δ 𝑡 , and can be estimated as \n𝑣 𝑖 = 𝑔 𝑖 𝑠 𝑖 + 𝑛 𝑖 , (1) \nwhere 𝑔 𝑖 is the antenna complex gain factor, 𝑛 𝑖 denotes the intrinsic antenna noise, and 𝑠 𝑖 represents the true sky signal. As interferometers measure cross-correlations, each baseline correlates measurements from a pair of antennas, resulting in a time-averaged visibility of the baseline 𝑩 𝑖 𝑗 in the form of \n𝑉 𝑖 𝑗 ≡ GLYPH<10> 𝑣 𝑖 𝑣 𝑗 GLYPH<11> \n≊ 𝑔 𝑖 𝑔 𝑗 GLYPH<10> 𝑠 𝑖 𝑠 𝑗 GLYPH<11> + 𝑛 𝑖 𝑗 . \n∗ (2a) = 𝑔 ∗ 𝑖 𝑔 𝑗 GLYPH<10> 𝑠 ∗ 𝑖 𝑠 𝑗 GLYPH<11> + 𝑔 ∗ 𝑖 GLYPH<10> 𝑠 ∗ 𝑖 𝑛 𝑗 GLYPH<11> + 𝑔 𝑗 GLYPH<10> 𝑛 ∗ 𝑖 𝑠 𝑗 GLYPH<11> + GLYPH<10> 𝑛 ∗ 𝑖 𝑛 𝑗 GLYPH<11> (2b) ∗ ∗ (2c) \nBy assuming that only sky radio signals are correlated, the last three terms of equation (2b) can be reduced to noise 𝑛 𝑖 𝑗 specific to the \nbaseline 𝑩 𝑖 𝑗 . Since each antenna gain is a complex factor, it can be parametrized by its amplitude 𝜂 and phase 𝜙 as \n𝑔 𝑖 ≡ 𝑒 𝜂 𝑖 + i 𝜙 𝑖 . (3) \nIf we use the parametric form of each antenna complex gain, the baseline measurement question, equation (2c), takes a new form: \n𝑉 𝑖 𝑗 = exp GLYPH<2> GLYPH<0> 𝜂 𝑖 + 𝜂 𝑗 GLYPH<1> + i GLYPH<0> 𝜙 𝑗 -𝜙 𝑖 GLYPH<1> GLYPH<3> GLYPH<10> 𝑠 ∗ 𝑖 𝑠 𝑗 GLYPH<11> + 𝑛 𝑖 𝑗 . (4) \nIdeally, the calibration process solves both the true complex gain factor 𝑔 𝑖 of each antenna and the true sky signal cross-correlation 𝑆 𝑖 𝑗 ≡ GLYPH<10> 𝑠 ∗ 𝑖 𝑠 𝑗 GLYPH<11> of each baseline simultaneously. An array of 𝑁 antennas measures 𝐶 2 𝑁 visibilities, whereas the variables to solve for are 𝑁 true antenna gain amplitude 𝜂 𝑖 items, 𝑁 true antenna phase 𝜙 𝑖 items, and 𝐶 2 𝑁 ≡ 𝑁 ( 𝑁 -1 )/ 2 true sky cross-correlations 𝑆 𝑖 𝑗 . Therefore, the measurements are insufficient to directly solve for 𝐺 𝑖 𝑗 ≡ 𝑔 ∗ 𝑖 𝑔 𝑗 and recover 𝑆 𝑖 𝑗 . However, by adopting a sky-based calibration scheme, which supplies a sky model to replace the unknown sky cross-correlation, the calibration of 𝐺 𝑖 𝑗 becomes an overdetermined problem that can be solved using a least-squares solution. The sky cross-correlations, 𝑆 𝑖 𝑗 , can then be recovered by applying the solution of the estimated gain values. Calibrating all frequencies separately can recover a set of sky cross-correlations within the observed band.', '3.4.2 Analytic analysis of the calibration error': "As stated in Section 1, sky-based calibration techniques are ultimately restricted by the supplied sky model. Consequently, the precision and fidelity of these models becomes critical. Source blending, along with various limiting factors that couple with one another [such as model completeness (e.g. Barry et al. 2016; Gehlot et al. 2021), polarized emission (e.g. Moore et al. 2017; Asad et al. 2018), and diffuse components (e.g. Lanman et al. 2022; Sims et al. 2022)], collectively impacts the accuracy of calibration processes and subsequently dictates the detection ability of the EoR signal. To individually evaluate the source-blending impact on the overall calibration process, we propose a theoretical estimation that confines blending defects as the only sky model limiting factor using a pair of sky models consisting of an ideal sky model and its blended counterpart. The sky model pairs include only discrete sources and their simulation realization is presented in Section 3.1. Under this paired sky model estimation approach, we can use the measured visibility equation (equation 4) to infer the per-frequency per-antenna gain error induced by a ' blended ' sky model w.r.t its ' ideal ' counterpart within the sky-based calibration scheme. We refer to this type of calibration error as the relative residual gain error. \nTo begin with, by using an ' ideal ' sky model 𝑉 ideal to calibrate the 𝑁 antenna composed array, we have the measured visibility and true \ngain amplitudes and phases ( 𝜂 and 𝜙 , respectively) for the baseline 𝑩 𝑖 𝑗 in the form of \n𝑉 𝑖 𝑗 = exp GLYPH<2> GLYPH<0> 𝜂 𝑖 + 𝜂 𝑗 GLYPH<1> + i GLYPH<0> 𝜙 𝑗 -𝜙 𝑖 GLYPH<1> GLYPH<3> 𝑉 ideal 𝑖 𝑗 + 𝑛 𝑖 𝑗 . (5) \nThen, by supplying a ' blended ' sky model ' 𝑉 blend , the same baseline measurement, albeit ill-calibrated, will take the form: \n𝑉 𝑖 𝑗 = exp GLYPH<2> GLYPH<0> ' 𝜂 𝑖 + ' 𝜂 𝑗 GLYPH<1> + i GLYPH<0> ' 𝜙 𝑗 -' 𝜙 𝑖 GLYPH<1> GLYPH<3> ' 𝑉 blend 𝑖 𝑗 + 𝑛 𝑖 𝑗 , (6) \nwhere ' 𝜂 and ' 𝜙 are the gain factors a ' blended ' sky model solves. Subsequently, by combining equations (5) with (6) under the PWL logarithmic method (see the step-by-step derivation in Appendix B1), we can obtain two sets of linear equations with the form \nΔ 𝜂 𝑖 + Δ 𝜂 𝑗 = 𝑅 𝑖 𝑗 , (7a) \nΔ 𝜙 𝑗 -Δ 𝜙 𝑖 = 𝐼 𝑖 𝑗 , (7b) \nwhere both the relative gain errors ( Δ 𝜂 𝑖 ≡ 𝜂 𝑖 -' 𝜂 𝑖 and Δ 𝜙 𝑖 ≡ 𝜙 𝑖 -' 𝜙 𝑖 , respectively) and the visibility difference in terms of amplitude and phase ( 𝑅 𝑖 𝑗 ≡ ln GLYPH<12> GLYPH<12> GLYPH<12> ' 𝑉 blend 𝑖 𝑗 GLYPH<12> GLYPH<12> GLYPH<12> -ln GLYPH<12> GLYPH<12> GLYPH<12> 𝑉 ideal 𝑖 𝑗 GLYPH<12> GLYPH<12> GLYPH<12> and 𝐼 𝑖 𝑗 ≡ arg GLYPH<12> GLYPH<12> GLYPH<12> ' 𝑉 blend 𝑖 𝑗 GLYPH<12> GLYPH<12> GLYPH<12> -arg GLYPH<12> GLYPH<12> GLYPH<12> 𝑉 ideal 𝑖 𝑗 GLYPH<12> GLYPH<12> GLYPH<12> , respectively) are substituted for simplicity. For each baseline, both 𝑅 𝑖 𝑗 and 𝐼 𝑖 𝑗 are only subject to the visibility difference between the pair of sky models. Because we supplied simulated sky model pairs with source blending as the only limiting factor, the error contribution of a ' blended ' sky model w.r.t an ' ideal ' counterpart can be inferred by solving the amplitude and phase differences separately with those two simple sets of equations. In this way, we decouple the source-blending impact on the calibration of EoR experiments from both the baseline noises and other limiting factors of sky-model constructions. \nAlthough recent efforts have been made to demonstrate solving the complex gain factor as a complex optimization for polarization and direction-dependent calibration (e.g. Smirnov & Tasse 2015; Grobler et al. 2018), we adopt the conventional approximation and treat the real and imaginary parts separately for our unpolarized case. Therefore, we can determine the blending-induced calibration error individually by solving the two overdetermined systems of equations and derive the per-frequency Δ 𝜂 𝑖 and Δ 𝜙 𝑖 for each antenna. A Python-based SVD solver is implemented to derive the solution following the exact derivation presented in Appendix B2. Because there is no contribution of baseline noise or possible biases from specific antenna-based calibration solvers, the solutions should be considered as a theoretical upper-limit estimation of the gain errors. \nFinally, by supplying the per-frequency per-antenna relative residual gain errors back to equation (4) for each sky component (we note true visibility 𝑆 true 𝑖 𝑗 and ill-calibrated visibility ' 𝑆 ill 𝑖 𝑗 ), we can infer the per-frequency per-baseline propagation bias of the sky signal in the visibility space as: \nΔ H 𝑖 𝑗 = GLYPH<2> exp GLYPH<0> Δ 𝜂 𝑖 + Δ 𝜂 𝑗 GLYPH<1> -1 GLYPH<3> GLYPH<12> GLYPH<12> GLYPH<12> 𝑆 true 𝑖 𝑗 GLYPH<12> GLYPH<12> GLYPH<12> , (8a) \nΔΦ 𝑖 𝑗 = Δ 𝜙 𝑗 -Δ 𝜙 𝑖 , (8b) \nwhere Δ H 𝑖 𝑗 ≡ GLYPH<12> GLYPH<12> GLYPH<12> ' 𝑆 ill 𝑖 𝑗 GLYPH<12> GLYPH<12> GLYPH<12> -GLYPH<12> GLYPH<12> GLYPH<12> 𝑆 true 𝑖 𝑗 GLYPH<12> GLYPH<12> GLYPH<12> and ΔΦ 𝑖 𝑗 ≡ arg GLYPH<12> GLYPH<12> GLYPH<12> ' 𝑆 ill 𝑖 𝑗 GLYPH<12> GLYPH<12> GLYPH<12> -arg GLYPH<12> GLYPH<12> GLYPH<12> 𝑆 true 𝑖 𝑗 GLYPH<12> GLYPH<12> GLYPH<12> denote the amplitude and phase bias, respectively. Based on equation (8), the derived amplitude and phase biases can be viewed as residuals after the removal of the corresponding foreground components without explicitly performing foreground subtraction. Because we use the true visibility of the foreground, the visibility bias is additive in nature, even after a perfect foreground removal. The detailed derivation is presented in Appendix B3. By utilizing the relative gain errors that are independently estimated for each frequency, the propagation bias can be inferred for all channels within each observation band. Thus, with the input visibility cubes [ 𝑉 ori , gal ( 𝑢, 𝑣, Δ 𝑓 ) , 𝑉 ori , ext ( 𝑢, 𝑣, Δ 𝑓 ) , and \n𝑉 ori , eor ( 𝑢, 𝑣, Δ 𝑓 ) ], we can explicitly derive the visibility-bias cubes for each sky component, namely, 𝑉 res , gal ( 𝑢, 𝑣, Δ 𝑓 ) , 𝑉 res , ext ( 𝑢, 𝑣, Δ 𝑓 ) , and 𝑉 res , eor ( 𝑢, 𝑣, Δ 𝑓 ) , respectively.", '3.5 Deconvolution and imaging': "As the final fraction of the HEVAL pipeline, this section details mapmaking, which is a crucial final step to enter the measurement space under the 'reconstructed'-sky approach. To deal with the challenge of both wideband and wide-FOV imaging, the HEVAL pipeline uses the WSC/l.pc/e.pc/a.pc/n.pc 20 imager (Offringa et al. 2014) tailored for map-making in the low-frequency radio sky. \nTo reconstruct 3D representations of the radio sky, image cubes along an observation frequency band are required. This is achieved by converting the 3D visibility cubes back to the image space. Both the 'original' visibility cubes (the lower part of Fig. 2) and the propagated 'residual' visibility-bias cubes (the upper part of Fig. 2) of each sky component are converted directly into 'original' [ 𝐼 ori ( 𝑙, 𝑚, Δ 𝑓 ) ] and 'residual' image cubes [ 𝐼 res ( 𝑙, 𝑚, Δ 𝑓 ) ], respectively. In particular, the direct imaging of foreground visibility-bias cubes, instead of ill-calibrated visibility cubes, allows an efficient map-making phase, given that imaging fidelity in the spatial domain is less a priority than spectral fidelity. Because we aim to evaluate the impact of frequency-dependent calibration errors, any spectral variations during the deconvolution phase need to be dealt with. The wideband deconvolution mode with multi-frequency weighting of the WSC/l.pc/e.pc/a.pc/n.pc (Offringa & Smirnov 2017) under a Briggs robustness weighting set to -0 . 5 (Briggs 1995) is used to account for spectral variations by griding the multi-frequency imaging weights together and cleaning all frequency channels jointly. In addition, a polynomial is fitted to the full bandwidth, reducing possible imaging noise and spectral artefacts that are typically caused by cleaning false peaks impersonated by side lobes or noise peaks. These measures allow HEVAL-produced foreground image cubes to mitigate possible spectral fluctuations, which might be coupled with the effect of frequency-dependent calibration errors in the measurement space. For the 21-cm signal, dirty images are used directly for analysis due to the insufficient deconvolution effect of the CLEAN algorithm to diffuse faint signals. All the final images are cropped to retain only the central 1 · × 1 · regions to mitigate imaging errors due to insufficient CLEANing of the marginal regions.", '3.6 The 2D power spectrum & EoR window': "As a redshifted line emission, observations of the EoR signal within a frequency span are three-dimensional, including two spatial dimensions across the sky and one vertical frequency dimension to the LoS. Statistical detection of the EoR signal under the 'reconstructed'-sky approach is expected to be achieved using the 3D nature of the EoR observation by converting reconstructed image cubes to 3D Fourier representations [ 𝑃 ( 𝑘 𝑥 , 𝑘 𝑦 , 𝑘 𝑧 ) ] in the measurement space via Fourier transform. The data flow from interferometric observations to the PS measurement space is presented in Fig. 8. \nWithin the measurement space, PS analysis can be achieved by averaging 𝑃 ( 𝑘 𝑥 , 𝑘 𝑦 , 𝑘 𝑧 ) either in spherical shells of radii 𝑘 , due to its spherical symmetry, resulting in the 1D PS 𝑃 ( 𝑘 ) , or over the angular annuli of radii 𝑘 ⊥ ≡ √︃ 𝑘 2 𝑥 + 𝑘 2 𝑦 for each LoS plane 𝑘 ∥ ≡ 𝑘 𝑧 , owing to the independence between the spatial and frequency dimensions, \nyielding the 2D PS 𝑃 ( 𝑘 ⊥ , 𝑘 ∥ ) . Theoretically, the cylindrical-averaged 𝑃 ( 𝑘 ⊥ , 𝑘 ∥ ) confines spectrally-smoothed foregrounds to be distributed within the low𝑘 ∥ region of the ( 𝑘 ⊥ , 𝑘 ∥ ) plane, leaving the rest of the space relatively free of foreground contamination. Thus, the 2D PS analysis is widely adopted as the most crucial figure of merit for experiments aimed to detect the EoR signal. \nHowever, the reality of complicated instrumental and observational effects, such as chromatic primary beams, redistributes the foreground power to higher 𝑘 ∥ dimensions via mode mixing, resulting in an expanded wedge-shaped region known as the foreground wedge (Datta et al. 2010; Morales et al. 2012; Pober et al. 2014). Beyond the foreground wedge, a region known as the EoR window (for more details on the topic, please refer to Liu et al. 2014; Dillon et al. 2014), is relatively free of foreground power contamination and can be located at \n𝑘 ∥ ≥ 𝐻 ( 𝑧 ) 𝐷 𝑀 ( 𝑧 ) ( 1 + 𝑧 ) 𝑐 GLYPH<20> 𝑘 ⊥ sin Θ + 2 𝜋𝑤 𝑓 21 -rf ( 1 + 𝑧 ) 𝐷 𝑀 ( 𝑧 ) 𝐵 GLYPH<21> (9) \n(Thyagarajan et al. 2013; Li et al. 2019), where 𝐻 ( 𝑧 ) denotes the Hubble parameter at redshift 𝑧 , 𝐷 𝑀 ( 𝑧 ) measures the transverse comoving distance, 𝑐 is the speed of light, 𝐵 is the bandwidth of the frequency window, 𝑓 21 -rf denotes the frequency of the 21-cm line emission in the rest-frame, 𝑧 = 𝑓 21 -rf / 𝑓 𝑐 -1 is the signal redshift of the central frequency 𝑓 𝑐 within the measured frequency band, 𝑤 is the number of characteristic convolution widths of the spillover region due to frequency response variations, and Θ is the foreground source angular separation from the primary beam centre. Based on the angular separation Θ , commonly used EoR window boundaries include 'FoV'- and 'horizon'-bound windows, which are defined by supplying the FoV and horizon of the instrument. \nFrequency-dependent calibration errors due to source blending are expected to aggravate the effect of mode mixing for 'reconstructed'sky PS, which will cause further foreground contamination by leaving additive 'residual' powers within EoR windows. Thus, to fully evaluate the impact of source blending on the EoR experiments, we discuss the 3D imprint (both the spatial and spectral deviations) of the blending-induced visibility bias with the 'original' [ 𝑃 ori ( 𝑘 ⊥ , 𝑘 ∥ ) ] and 'residual' 2D PS [ 𝑃 res ( 𝑘 ⊥ , 𝑘 ∥ ) ] for each sky component, which is calculated using the 'original' [ 𝐼 ori ( 𝑙, 𝑚, Δ 𝑓 ) ] and 'residual' image cubes [ 𝐼 res ( 𝑙, 𝑚, Δ 𝑓 ) ], respectively.", "4.1 'Residual' powers of EoR foregrounds": "We calculate the cylindrical-averaged 2D PS using image cubes produced by the HEVAL pipeline. In Fig. 9, we present the 'residual' 2D PS [ 𝑃 res , ext ( 𝑘 ⊥ , 𝑘 ∥ ) and 𝑃 res , gal ( 𝑘 ⊥ , 𝑘 ∥ ) ] transformed from foreground 'residual' image cubes [ 𝐼 res , ext ( 𝑙, 𝑚, Δ 𝑓 ) and 𝐼 res , gal ( 𝑙, 𝑚, Δ 𝑓 ) ] for each component owing to the propagation effects of blending-induced calibration errors. Given that these powers will still be present in the PS space even after the perfect removal of spectrally-smoothed foregrounds, we refer to them as 'residual' powers. 'Residual' powers originating from the 'highly'-corrupted sky model (5 per cent blending ratio) show a substantial occupation of the low 𝑘 ∥ modes for both the Galactic and extragalactic foregrounds. The strongest Galactic 'residual' powers reside at larger spatial scales ( 𝑘 ⊥ ≲ 0 . 2), while the strong extragalactic 'residual' powers spread across the large and intermediate spatial scales. For the high 𝑘 ∥ modes, we can see a clear excess of powers within the EoR window above both the dashed black 'FoV' line and solid black 'horizon' line. These 'residual' powers contaminate the expected foreground-free \nregions. Hence, they pose a threat to the parameter inference of the EoR signal. For the 'moderate' case (0.5 per cent blending ratio), the 'residual' powers, albeit weaker, distribute similarly to that of the 'high' blending ratio case in the 2D PS space. Unlike the prior two cases, the 'mildly'-corrupted sky model (0.05 per cent blending ratio) introduces much weaker 'residual' powers with a slightly different vertical distribution, as there is no characteristic distribution of 'residual' powers occupying low 𝑘 ∥ modes for both the Galactic and extragalactic foregrounds. However, the contamination of the 'residual' power within the EoR window remains. \nBytaking the 2D PS ratio between the ill-calibrated 'residual' power [ 𝑃 res , ext ( 𝑘 ⊥ , 𝑘 ∥ ) and 𝑃 res , gal ( 𝑘 ⊥ , 𝑘 ∥ ) ] and 'original' sky emission power [ 𝑃 ori , ext ( 𝑘 ⊥ , 𝑘 ∥ ) and 𝑃 ori , gal ( 𝑘 ⊥ , 𝑘 ∥ ) ] for each foreground component, we can identify the region with excess power caused by blending-induced calibration errors. Given the nature of frequencydependent calibration errors that cause relatively small fluctuations across the frequency channels, the region with excess power should be located at higher 𝑘 ∥ modes. In Fig. 10, we can see excess powers locate at the region covering important high 𝑘 ∥ modes ( 𝑘 ∥ ≳ 0 . 2), as expected. Among the three cases, the 'mild' blending case has the lowest excess power. All the 'residual' powers introduced by the 'mildly'-corrupted sky model are below 10 per cent of the 'original' sky emission powers. In contrast, the other two cases pose significant contamination by introducing excess foreground 'residual' powers, which are 10 to over 1000 times the 'original' sky emission powers, covering most of the EoR window. Outside the 'FoV' window, foreground 'original' powers still dominate the wedge region for all three cases. Given that we treat the two foreground components separately, we can also calculate the 2D PS ratio between the extragalactic and Galactic 'residual' powers to infer the dominant modes of each foreground component. Fig. 11 illustrates the spatial scales that divide the dominance. The Galactic 'residual' errors dominate the spatial scales larger than 𝑘 ⊥ ∼ 0 . 2, while the extragalactic 'residual' errors dominate the remaining 𝑘 ⊥ modes.", '4.2 Blending impact on the EoR PS and blending ratio tolerance': "The direct quantification of the blending-induced 'residual' power impact on EoR detection is achieved by calculating the 2D PS ratio between EoR 'residual' powers and 'original' powers. We can infer the specific impact by analysing the contamination within the estimated foreground-free EoR window. We plot the 𝑅 res , fg / ori , eor ( 𝑘 ⊥ , 𝑘 ∥ ) ratio for the two EoR foregrounds for the three blending cases. From the plotted 𝑅 res , gal / ori , eor ( 𝑘 ⊥ , 𝑘 ∥ ) in Fig. 12, the Galactic 'residual' powers dominate the large spatial scales, polluting the most important EoR 𝑘 ⊥ modes significantly. For both the 'high' (5 per cent) and 'moderate' (0.5 per cent) blending ratio cases, strong Galactic 'residual' powers, which are at least 1000 times the corresponding EoR powers, cover almost all the modes within the 'horizon'-bound region. Unanticipatedly, Galactic 'residual' powers also contaminate the EoR powers at the large 𝑘 ⊥ end with a wedge-shaped region at 𝑘 ∥ ≳ 7 and 𝑘 ∥ ≳ 9 for the 'high' and 'moderate' cases, respectively. Recall that the simulated Galactic diffuse foregrounds include small-scale fluctuations down to the SKA-Low's angular resolution, which enables PS analysis to infer the Galactic impact below the arcmin scale for the first time. Although the extragalactic 'residual' powers dominate at intermediate and small spatial scales, their impact within the EoR window remains significant viewing from the plotted 𝑅 res , ext / ori , eor ( 𝑘 ⊥ , 𝑘 ∥ ) in Fig. 12. For the 'high' blending ratio case, all of the modes within the 'FoV'-bound region are not recoverable under the impact of either of the foreground components. Interestingly, there is a small window of opportunity for the 'moderate' case since there are modes, \nFigure 8. The data flow of our 'reconstructed'-sky PS analysis pipeline. The multi-frequency visibilities of a given band are transformed into image cubes reconstructing the sky emission through the Fourier transform in the sky coordinates. Subsequently, the reconstructed image cubes are transformed into the measurement space through the Fourier transform in both the sky coordinates and the frequency dimension to analyse the spatial features of each sky component. Both 'original' visibility and 'residual' visibility-bias cubes for each sky component are transformed following the data flow. The 𝑉 ori 'original' visibility cubes are converted to 𝐼 ori and eventually transform into 𝑃 ori . The 𝑉 res 'residual' visibility-bias cubes follow the same trajectory to the image and PS space as 𝐼 res and 𝑃 res , respectively. This plot is inspired by fig. 2 of Morales & Hewitt (2004). \n<!-- image --> \nResidual Power of Extraglactic and Galactic ForegroundsFigure 9. (Left, a-i) Extragalactic 'residual' powers 𝑃 res , ext ( 𝑘 ⊥ , 𝑘 ∥ ) and (right, j-i) Galactic 'residual' powers 𝑃 res , gal ( 𝑘 ⊥ , 𝑘 ∥ ) originating from blending-induced calibration errors. Within each 3 × 3 grid, the three rows mark the 'high' (5 per cent), 'moderate' (0.5 per cent), and 'mild' (0.05 per cent) blending scenarios, respectively; the three columns mark the 188 - 196 MHz, 154 - 162 MHz, and 120 - 128 MHz frequency band, respectively. Setting 𝑤 as 2, the black lines define the EoR windows with the dashed and solid lines marking the 'FoV' and the 'horizon' boundary, respectively. All the plots share the same logarithmic colour bar using unit [mK 2 Mpc 3 ]. \n<!-- image --> \nwhich is located at 0 . 6 ≲ 𝑘 ⊥ ≲ 5 . 0 and 0 . 2 ≲ 𝑘 ∥ ≲ 1 . 0, relatively free of Galactic 'residual' power. However, this small area vanishes after combining the Galactic and extragalactic foregrounds. For the 'mild' (0.05 per cent) blending ratio case, significant less 'residual' powers are present in the measurement space. For one thing, only \nthe largest spatial scales, which are located at 0 . 04 ≲ 𝑘 ⊥ ≲ 0 . 08 and 0 . 3 ≲ 𝑘 ∥ ≲ 1 . 0, are affected by the Galactic 'residual' power, leaving most of the 'FoV'-bound region relatively free of impact. For another, the extragalactic 'residual' power leaves all 'FoV'-bound regions free of contamination by polluting only the smaller spatial scales outside", "Extragalactic 'Residual' Power / Extragalactic 'Original' Power": "Figure 10. The 154 - 162 MHz 2D PS ratio between foreground 'residual' and 'original' powers 𝑅 res , fg / ori , fg ( 𝑘 ⊥ , 𝑘 ∥ ) . (Top) 𝑅 res , ext / ori , ext ( 𝑘 ⊥ , 𝑘 ∥ ) ratios under (a) 'high' (5 per cent), (b) 'moderate' (0.5 per cent), and (c) 'mild' (0.05 per cent) blending scenarios. (Bottom) Corresponding 𝑅 res , gal / ori , gal ( 𝑘 ⊥ , 𝑘 ∥ ) ratios under (d) 'high' (5 per cent), (e) 'moderate' (0.5 per cent), and (f) 'mild' (0.05 per cent) blending scenarios. Setting 𝑤 as 2, the black lines define the EoR windows with the dashed and solid lines marking the 'FoV' and the 'horizon' boundary, respectively. Figs. 10 to 15 share the same logarithmic colour bar, showing only six orders of magnitude for easy viewing. \n<!-- image -->", "Extragalactic 'Residual' Power / Galactic 'Residual' Power": "Figure 11. The 154 - 162 MHz 2D PS ratios between extragalactic and Galactic 'residual' powers 𝑅 res , ext / res , gal ( 𝑘 ⊥ , 𝑘 ∥ ) under (a) 'high' (5 per cent), (b) 'moderate' (0.5 per cent), and (c) 'mild' (0.05 per cent) blending scenarios. Setting 𝑤 as 2, the black lines define the EoR windows with the dashed and solid lines marking the 'FoV' and the 'horizon' boundary, respectively. Figs. 10 to 15 share the same logarithmic colour bar, showing only six orders of magnitude for easy viewing. \n<!-- image --> \nthe EoR window. Although caution should be exercised when dealing with Galactic 'residual' pollution, the impact of 'residual' powers on EoR detection is insignificant. \nThe HEVAL pipeline is designed to isolate source blending as the sole origin of errors to evaluate its impact on SKA EoR experiments under a sky-based calibration scheme. In the image space, the implementation of B/l.pc/e.pc/n.pc/d.pcS/i.pc/m.pc simulates defects that can be attributed solely to source blending. In the visibility space, our analytical modules employ a relative calibration scheme to further segregate the calibration errors of blending-induced origin. During the map-making phase, wideband, multi-frequency, and spectral fitting procedures mitigate possible spectral artefacts that may cloud the impact evaluation in the measurement space. Given these measures of impact isolation, we can confidently conclude that the inference in the 2D PS space presented in this section originates from blending defects and propagates along the entire EoR analysis pipeline. Consider that the 'residual' powers are propagated from 'residual' visibility-biases that are present even after a perfect foreground subtraction, we can draw a blending ratio tolerance to guide the sky-model construction for the calibration of SKA EoR experiments. While caution should be taken when addressing Galactic 'residual' pollution, the blending ratio tolerance for the SKA EoR experiment can be drawn at or slightly below the 0.05 per cent level of the 'mildly'-blended case, which translates to containing only 5 pairs of blended sources per 10000 sources for sky models aimed at calibration.", '5.1 Limitations of the study': "Although this paper offers the first systematic insight into source blending in the upcoming SKA era, there are limitations to our study. \nNoise consideration The simulation of the source-blending effect should be considered as the best-case scenario, as the noise impact of the sky-model construction is not considered. Given that noise plays a vital role during the detection phase of sky-model construction (i.e. the noise level determines the detection criteria), a mixture of components often results in a measurement error in the total flux density. Considering the accuracy of flux density measurements as its own source of sky-model defects, our implantation of source-blending effects only considers the flux density reallocation without affecting the total flux density of the blended source. Hence, the total flux density of the ' ideal ' and ' blended ' from the same pair remains the same. \nEffects Decoupling One of the key considerations of this study is to infer the calibration errors solely from source-blending defects, given that sky-model defects often couple with each other, instrumental effects, or a combination of both in practical calibration pipelines. Hence, the realization of the HEVAL pipeline in this study decouples source-blending defects from other factors, such as polarimetric and side-lobe impact. In particular, source-blending defects from sources in the side lobes may introduce additional calibration-induced errors in the 2D PS space (see Li et al. 2019, for a discussion on side-lobe effects in the 2D PS space), which may contaminate different parts of the EoR window. We will consider these topics in future studies because of the high computational expenses required to evaluate these coupled effects. \nTemporaleffects Themainaimofthisstudyistoinferthe frequencydependent gain error. The temporal effects of the complex gain are its \nown topic (see Kumar et al. 2020, 2022, for detailed discussions on time-correlated gain effects). We omit consideration of the temporal effect of the complex gain while considering the calibration process. Therefore, our results should be viewed as a time-averaged scenario. \nSmall-scale simulations The low-frequency radio data used in this study is simulated to include physical scales down to the estimated SKA1-Low spatial resolution. Although a commonly used practice, which effectively redistributes powers to the simulated small scales, is applied to add small-scale fluctuations to our Galactic components, Galactic observations to constrain the simulation at such scales are still lacking. For extragalactic foregrounds, our implemented Gaussian models of the EDRS populations indicate that fine structures below 6 arcsec are excluded from this study. We consider the fine structures of the extended sources as their own topic (i.e. the fidelity of the extended source model) and will discuss the impact in an upcoming project. Interested readers can refer to Procopio et al. (2017) for a discussion of foreground spatial fidelity impact on EoR foreground subtraction. As for the small-scale fluctuations of the EoR simulation, the accuracy of our data is subject to the limitations of the semi-analytic models used in 21/c.pc/m.pcFAST.", "5.2 Impact of EoR 'residual' powers": "Similar to the calculation of the 2D PS ratio between the foreground 'residual' and the EoR signal, we can plot the EoR 'residual' and 'original' power ratio 𝑅 res , eor / ori , eor ( 𝑘 ⊥ , 𝑘 ∥ ) and infer the impact of EoR 'residual' powers on the EoR signal itself. As demonstrated in Fig. 13, only the 'high' (5 per cent) blending ratio case has introduced small excess 'residual' power within the EoR window. However, these powers are expected to have little impact on the EoR detection because they are smaller than one-tenth of the EoR signal and can be considered to be below the noise level. The only strong impact is outside the EoR window down to smaller scales, translating to little effect under the foreground avoidance strategy. Therefore, unlike the foreground 'residual' powers, the propagated EoR 'residual' powers originating from the 'highly'-corrupted sky model have little to no direct impact on the EoR detection. However, there might be impact on the accuracy of the EoR parameter inference (such as parameters related to the non-Gaussianity), depending on how precise the model can be constrained under the influence of the EoR 'residual' powers.", '5.3 Impact of imaging weighting on calibration errors': "Map-making is a critical step in the 'reconstructed'-sky PS analysis pipeline. Even with the advanced baseline coverage of the upcoming SKA1-Low, the interferometer can only discretely sample the sky without unlimited baselines. Thus, image weights are required to determine the filling of the telescope's sampling gap during the mapmaking phase. We discuss the impact of imaging-weight choice on the propagation effects of blending-induced calibration errors. We consider the 'moderate' (0.5 per cent) blending case, which is above the blending ratio tolerance, and add Briggs-weighted clean maps with different robustness, including -1, 0, 0.5, and 1, in addition to the -0.5 robustness used in Section 3.5. We retain the remaining imaging settings and change only the weight. Thus, we have 5 differently weighted maps for the 'residual' and 'original' image cubes of each foreground component. Fig. 14 plots the 2D PS ratio between foreground 'residual' powers and EoR 'original' powers across the 5 different weighting options. There are clear trends across the different imaging options: (i) there is a reduction in 'residual' power at the", "Extragalactic 'Residual' Power / EoR 'Original' Power": "Figure 12. The 154 - 162 MHz 2D PS ratios between foreground 'residual' powers and EoR 'original' powers 𝑅 res , fg / ori , eor ( 𝑘 ⊥ , 𝑘 ∥ ) . (Top) 𝑅 res , ext / ori , eor ( 𝑘 ⊥ , 𝑘 ∥ ) ratios under (a) 'high' (5 per cent), (b) 'moderate' (0.5 per cent), and (c) 'mild' (0.05 per cent) blending scenarios. (Bottom) Corresponding 𝑅 res , gal / ori , eor ( 𝑘 ⊥ , 𝑘 ∥ ) ratios under (d) 'high' (5 per cent), (e) 'moderate' (0.5 per cent), and (f) 'mild' (0.05 per cent) blending scenarios. Setting 𝑤 as 2, the black lines define the EoR windows with the dashed and solid lines marking the 'FoV' and the 'horizon' boundary, respectively. Figs. 10 to 15 share the same logarithmic colour bar, showing only six orders of magnitude for easy viewing. \n<!-- image --> \nEoR 'Residual' Power / EoR 'Original' PowerFigure 13. The 154 - 162 MHz 2D PS ratios 𝑅 res\\_eor / ori\\_eor ( 𝑘 ⊥ , 𝑘 ∥ ) (between EoR 'residual' and 'original' powers) under (a) 'high' (5 per cent), (b) 'moderate' (0.5 per cent), and (c) 'mild' (0.05 per cent) blending scenarios. Setting 𝑤 as 2, the black lines define the EoR windows with the dashed and solid lines marking the 'FoV' and the 'horizon' boundary, respectively. Figs. 10 to 15 share the same logarithmic colour bar, showing only six orders of magnitude for easy viewing. \n<!-- image --> \nsmallest spatial scales with robustness varying from -1 to 1; (ii) the Galactic 'residual' power at the intermediate spatial scales increases with robustness varying from -1 to 1; and (iii) the 'residual' power at the largest spatial scales increases as the robustness varies from -1 to 1. Given these trends, we can identify a potential mitigation strategy for blending-induced 'residual' powers by choosing the suitable imaging weight for the particular observing sky during the map-making phase. As shown in Fig. 15, a viable approach is using the robustness -1 option for this particular simulated sky patch considering the reduced impact in the measurement space. There is an evident boost in the detection possibility at the largest spatial scales, which can be inferred from the weighting ratio figures in the second row of Fig. 15. From a general EoR detection point-of-view, a more holistic approach should be considered by balancing between different sources of errors across the analysis pipeline with each individual bias factor identified, evaluated, and mitigated.", '5.4 Mitigation of blending-induced effects': "To reduce the propagated 'residual' power discussed in this paper, upcoming SKA EoR experiments require dedicated de-blending efforts built directly within the sky-model construction pipeline. Current de-blending efforts based on LOFAR observations heavily involve human input (see Williams et al. 2019; Kondapally et al. 2021, for the LOFAR de-blending workflow). With the upcoming SKA1Low's improved sensitivity compared to the current observations, source blending could only be exacerbated. Therefore, automatic de-blending approaches, such as those applied in the optical domain (see Melchior et al. 2021, for a review), are required to address the blending issue both in decomposition and association, but the implementation is still an open question. To aid the development of de-blending pipelines and strategies, the developer community needs proper datasets through observations and simulations. From an observational perspective, obtaining higher-resolution radio observations of the SKA EoR fields through the international LOFAR Telescope (Vermeulen & van Haarlem 2011; Jackson et al. 2016) would both offer a direct blending effect deduction and help with the development of de-blending methods during decomposition. Multi-frequency synergies, such as joint observations with JWST 21 (McElwain et al. 2023), the Large Synoptic Survey Telescope (LSST 22 ; Ivezić et al. 2019), or the Chinese Survey Space Telescope (CSST 23 ; Zhan 2011), of the SKA observing sky, would also be crucial for the SKA to battle source blending during the source-association phase. From a simulation perspective, a high-fidelity multi-frequency (across the radio to infrared and optical bands) EDRS simulation is required to develop de-blending methods. An alternative approach to direct de-blending efforts is to treat blending-induced 'residual' errors in a similar manner to other frequency-dependent errors. General frequency-dependent error-mitigation strategies include the usage of spectrally-smoothed antennas (Barry et al. 2016), baseline weighting (Ewall-Wice et al. 2017), and simultaneous multi-channel calibration (Yatawatta 2015). In practice, a combination of both a direct de-blending pipeline and indirect error-mitigation strategies might be used to achieve an optimal solution that effectively attenuates the blending-induced impact.", '6 CONCLUSION': "Wemake the first attempt to systematically assess the impact of source blending in the low-frequency radio band for the upcoming SKA1Low. This study introduces a clear definition of source blending in the radio window, identifies the two-fold blending defect in both the spatial and frequency domains, and investigates the underlying impact of source blending on interferometric calibrations for SKA EoR experiments. We summarize our work as follows: \n- (i) Using the quick-and-dirty method presented in Section 2, which utilizes the W08 source model and the latest SKA1-Low array configuration, we estimate an extended level of blending ( ∼ 5 -28 per cent sources are blended) for the upcoming SKA1-Low observing sky and identified source blending as one of the key imperfections impacting the sky-model construction for SKA EoR experiments.\n- (ii) ByusingourHEVALpipeline(whichsimulatesthelow-frequency radio sky containing physical scales spatially resolvable by the SKA1Low) to quantify the blending-induced calibration impact in a skybased scheme for EoR detections, we find that frequency-dependent calibration errors from poor calibration against blending-corrupted sky models coupled with strong foregrounds leave additive 'residual' powers within the EoR window in the 2D PS space and may significantly impede EoR detections. Our findings corroborate with both the B16 simulation approach and the E17 analytic analysis results, albeit focusing on three different sky-model defects. Furthermore, by considering the extragalactic and Galactic foregrounds separately, we conclude that the extragalactic and Galactic 'residual' powers contaminate EoR scales at larger 𝑘 ⊥ and smaller 𝑘 ⊥ , respectively, with a clear division around 𝑘 ⊥ ∼ 0 . 2. \n(iii) To determine the blending tolerance for SKA EoR experiments, we perform three tests with relatively low levels of blending ratios (5, 0.5, and 0.05 per cent). The tests show that the blending defects at the former two levels introduce strong 'residual' powers in the measurement space, seriously contaminating the key EoR scales (0 . 1 ≲ 𝑘 ≲ 2 Mpc -1 ) within the EoR window, whereas sky models with a blending ratio at 0.05 per cent leave little impact in the measurement space. Hence, a blending ratio of approximately 0.05 per cent is identified as the blending tolerance for the upcoming SKA1-Low sky-model construction. \n- (iv) Given that our estimated 5 -28 per cent blending ratio for the SKA1-Low has quite a large margin with a tolerance of 0.05 per cent, we discuss the possibility of mitigating and suppressing the blendinginduced impact through different imaging weights. For the designed baseline of the SKA1-Low, Briggs weighting with robustness -1 would be a better choice to mitigate the blending-induced impact at key EoR scales. \n(v) To directly mitigate the impact of source-blending defects, deblending techniques must be involved in both the SKA source detection and sky-model construction to pass the required blending tolerance. From a software perspective, the source detection community should prioritize including de-blending or blending flagging during both the component-decomposition and -association phases. Additionally, priority should be set to port and adapt existing automatic de-blending methods, such as those used in the optical bands, to the radio bands. From an observation perspective, adding higher-resolution or lessblended observations to the SKA EoR fields via synergies with the international LOFAR Telescope and instruments from other wavelength bands will be crucial for the SKA to reduce the blending level through cross-match assessments directly. In practice, joint efforts from the software and observation perspectives are most likely required to contain blending-induced errors within the necessary overall EoR error budget. \nFigure 14. The 154 - 162 MHz 2D PS ratios between foreground 'residual' powers and EoR 'original' powers 𝑅 res , fg / ori , eor ( 𝑘 ⊥ , 𝑘 ∥ ) under the 'moderate' (0.5 per cent) blending scenario using different Briggs weighting options. (Top) 𝑅 res , ext / ori , eor ( 𝑘 ⊥ , 𝑘 ∥ ) ratios using (a) -1, (b) -0.5, (c) 0, (d) 0.5, and (e) 1 robustness options. (Bottom) Corresponding 𝑅 res , gal / ori , eor ( 𝑘 ⊥ , 𝑘 ∥ ) ratios using (f) -1, (g) -0.5, (h) 0, (i) 0.5, and (j) 1 robustness options. Setting 𝑤 as 2, the black lines define the EoR windows with the dashed and solid lines marking the 'FoV' and the 'horizon' boundary, respectively. Figs. 10 to 15 share the same logarithmic colour bar, showing only six orders of magnitude for easy viewing. \n<!-- image -->", 'ACKNOWLEDGEMENTS': "We thank Weitian Li for valuable discussions regarding /f.pc/g.pc21/s.pc/i.pc/m.pc, Andrei Mesinger, Yuxiang Qin, and Steven Murray for their help with 21/c.pc/m.pcFAST, Fred Dulwich for technical support in resolving issues with OSKAR, and André Offringa for guidance on map-making with WSC/l.pc/e.pc/a.pc/n.pc. We thank Hekun Lee, Benjamin McKinley, and Junhua Gu for scientific discussions, Sanjay Bhatnagar for reading the manuscript and providing insightful comments, and Bingxue Yang's help with the preparation of the final draft of this manuscript. This work was performed on the Gravity Cluster of Department of Astronomy (DOA) at Shanghai Jiao Tong University (SJTU). Some of the results in this paper have been derived using the healpy and HEALPix packages. This work is supported by the Ministry of Science and Technology of China (Grant No. 2020SKA0110201), and the National Natural Science Foundation of China (Grant Nos. 11973033 and 11835009). ZZH acknowledges the support from the National Science Foundation of China (Grant No. 12203085). SVW acknowledges the financial assistance of the South African Radio Astronomy Observatory (SARAO; https://www.sarao.ac.za). Parts of this research were supported by the Australian Research Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project number CE170100013.", 'DATA AVAILABILITY': 'The end-to-end simulate suite F/g.pc21S/i.pc/m.pc+ is an open-source project developed at https://github.com/Fg21Sim/ . Owing to the large \ndata volume, the data generated by this work will be shared under a reasonable request at https://github.com/Fg21Sim/Data . Datasets with the same specifications as in this study can be simulated using F/g.pc21S/i.pc/m.pc+ with the parameters provided by the authors.', 'AUTHOR CONTRIBUTION STATEMENT': "Here is the list of contributions: \n- · C.Shan: project proposal, coding (blending ratio estimation and the majority of HEVAL pipeline), methodology, testing, validation, result analysis, and writing (original draft and modification).\n- · H.Xu: scientific feedback, results discussion, writing (feedback and modification), and funding support.\n- · Y.Zhu: coding (contribution to the ES/i.pc/m.pc and B/l.pc/e.pc/n.pc/d.pcS/i.pc/m.pc modules), validation, results analysis (feedback and discussion), and writing (feedback and modification).\n- · Y.Zhao: scientific feedback, results analysis (feedback and discussion), and writing (feedback and modification).\n- · S.White: scientific feedback, figure modification, and writing (feedback and modification).\n- · J.Line: scientific feedback, results analysis (feedback and discussion) and writing (feedback).\n- · D.Zheng: methodology (feedback and discussion) and coding (contribution to the GS/i.pc/m.pc).\n- · Z.Zhu: scientific feedback and writing (feedback).\n- · D.Hu: scientific feedback and writing (feedback).\n- · Z.Zhang: scientific feedback.\n- · X.Wu: scientific and funding support. \nFigure 15. (Top) The 154 - 162 MHz 2D PS ratios between combined foreground 'residual' powers and EoR 'original' powers 𝑅 res , ext + gal / ori , eor ( 𝑘 ⊥ , 𝑘 ∥ ) under the 'moderate' (0.5 per cent) blending scenario using (a) -1, (b) -0.5, (c) 0, (d) 0.5, and (e) 1 Briggs robustness options, respectively. (Bottom) The 154 - 162 MHz 2D PS ratios between combined foreground 'residual' powers using various Briggs robustness options and the Briggs robustness 0 option 𝑅 res , Briggs ( i )/ res , Briggs ( 0 ) ( 𝑘 ⊥ , 𝑘 ∥ ) under the 'moderate' (0.5 per cent) blending scenario. The various robustness options include (f) -1, (g) -0.5, (h) 0, (i) 0.5, and (j) 1. Setting 𝑤 as 2, the black lines define the EoR windows with the dashed and solid lines marking the 'FoV' and the 'horizon' boundary, respectively. Figs. 10 to 15 share the same logarithmic colour bar, showing only six orders of magnitude for easy viewing. \n<!-- image -->", 'REFERENCES': "Arcelin B., Doux C., Aubourg E., Roucelle C., LSST Dark Energy Science Collaboration 2021, MNRAS, 500, 531 \nAsad K. M. B., Koopmans L. V. E., Jelić V., de Bruyn A. G., Pandey V. N., Gehlot B. K., 2018, MNRAS, 476, 3051 \nBarry N., Hazelton B., Sullivan I., Morales M. F., Pober J. C., 2016, MNRAS, 461, 3135 \nBhatnagar S., Rau U., Golap K., 2013, ApJ, 770, 91 \nBlandford R., Meier D., Readhead A., 2019, ARA&A, 57, 467 \nBowman J. D., et al., 2013, Publ. Astron. Soc. Australia, 30, e031 \nBriggs D. S., 1995, PhD thesis, New Mexico Institute of Mining and Technology \nBrunton S. L., Kutz J. N., 2019, Singular Value Decomposition (SVD). Cambridge University Press, p. 3-46 \nByrne R., et al., 2019, ApJ, 875, 70 \nByrne R., Morales M. F., Hazelton B. J., Wilensky M., 2021, MNRAS, 503, 2457 \nCarroll P. 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L., Nunhokee C. D., Smirnov O. M., van Zyl A. J., de Bruyn A. G., 2014, MNRAS, 439, 4030 \nGrobler T. L., Bernardi G., Kenyon J. S., Parsons A. R., Smirnov O. M., 2018, MNRAS, 476, 2410 \nHales C. A., 2017, AJ, 154, 54 \nHancock P. J., Trott C. M., Hurley-Walker N., 2018, Publ. Astron. Soc. Australia, 35, e011", 'APPENDIX A: SMALL-SCALE FLUCTUATION OF GALACTIC FOREGROUNDS': 'In this section, we introduce our method of adding small-scale fluctuations to the Galactic emission in detail. First, GS/i.pc/m.pc extrapolates the angular PS of the basis templates to the smaller scales and generates GRFs, 𝐺 grf , based on the extrapolated angular PS. We adopt the angular power spectrum model for the synchrotron and free-free emission as \nC sync ℓ = ℓ 𝛾 h 1 -exp GLYPH<16> -ℓ 2 𝜎 2 temp GLYPH<17> i (A1) \n(Remazeilles et al. 2015) and \nC free ℓ = ℓ 𝛾 h exp GLYPH<16> -ℓ 2 𝜎 2 sim GLYPH<17> -exp GLYPH<16> -ℓ 2 𝜎 2 temp GLYPH<17> i (A2) \n(Delabrouille et al. 2013), respectively, where 𝜎 temp and 𝜎 sim are the full width at half-maximum (FWHM) beam of the basis template and simulated sky maps, respectively, and the 𝛾 index is the power law index of the basis template. By adopting the two models, the GRF maps ( 𝐺 grf ) of the two Galactic diffuse components can be simulated using GS/i.pc/m.pc by converting C ℓ to maps with angular scales from degree level down to the required ℓ . To compensate for the high demand of both memory and CPU for generating high spatial resolution GRFs, we employ the GS/i.pc/m.pc module with a custom /c.pc/l.pc2/a.pc/l.pc/m.pc tool modified from the /h.pc/e.pc/a.pc/l.pc/p.pc/y.pc 24 package (Zonca et al. 2019) and /d.pc/u.pc/c.pc/c.pc0 25 with multi-threaded support. The former calculates the spherical harmonic coefficients ( 𝑎 𝑙𝑚 ) from the angular PS, and the latter offers map realization from the coefficients. Then, the small-scale fluctuation map ( 𝐺 ss ) is generated with a whitened 𝐺 grf , which has zero mean and unit variance, using \n𝐺 ss = 𝛼𝐺 grf 𝐺 𝛽 temp , (A3) \nwhere 𝛼 and 𝛽 , which dictate the mean and variance of the small-scale maps, respectively, are custom parameters to be determined. Finally, the simulated Galactic emission with the addition of small-scale fluctuations ( 𝐺 sim ) can be inferred with \n𝐺 sim = 𝐺 temp + 𝐺 ss , \n= 𝐺 temp + 𝛼𝐺 grf 𝐺 temp . \n(A4a) 𝛽 (A4b) \nFor each simulation run, GS/i.pc/m.pc offers on-the-fly fitting of the three key parameters, 𝛼 , 𝛽 , and 𝛾 , to prescribe the generation of sky maps. The fitting process is based on the /e.pc/m.pc/c.pc/e.pc/e.pc 26 (Foreman-Mackey et al. 2013) Python library of the Markov Chain Monte Carlo (MCMC) method. To fit the power law index 𝛾 , GS/i.pc/m.pc employs the angular PS analysis of the Galactic all-sky basis template on a Hierarchical Equal Area and isoLatitude Pixelization (HEALPix 27 , Górski et al. 2005) sphere using the /a.pc/n.pc/a.pc/f.pc/a.pc/s.pc/t.pc 28 module. Subsequently, the 𝛾 index is inferred by fitting a power law to the large angular scale ( ℓ ≲ 500) of the computed C ℓ . As for the fitting of 𝛼 and 𝛽 , we use a set of statistical constraints, which require the preservation of the underlying statistical properties of the basis map with the addition of small-scale \nfeatures (equation A4b). To achieve these constraints, GS/i.pc/m.pc uses a model that combines (i) an equivalence test, which is implemented through a two one-sided t-tests (TOST, Schuirmann 1987) procedure, (ii) a maximum mean discrepancy (MMD, Fortet & Mourier 1953) test, and (iii) difference minimization of the mean, variance, skewness, and kurtosis between the basis template ( 𝐺 temp ) and the feature-added map ( 𝐺 temp + 𝛼𝐺 grf 𝐺 𝛽 ). \ntemp \nFor the purpose of this work, the 𝛾 index of synchrotron and free-free component is fitted as -2.220 and -2.426, respectively, using C ℓ with ℓ ranging from 30 to 90. However, due to the high spatial resolution requirement of the small-scale features (a HEALPix map with Nside = 32,768 meets the SKA spatial resolving power), we fit 𝛼 and 𝛽 using partial sky coverage instead of the full sky map to avoid the extremely high CPU and memory costs. By fitting to the aimed sky coverage (R.A., Dec. = 0 · , -27 · ), we find the best-fitting result for the synchrotron ( 𝛼 = 0 . 0342, 𝛽 = 0 . 227) and free-free emission ( 𝛼 = 0 . 00785, 𝛽 = 0 . 526).', 'APPENDIX B: DERIVING THE GAIN ERROR SOLUTION & PER-BASELINE BIAS': 'This section details our formalism of the analytic analysis to evaluate the blending-induced calibration impact. Recall that we have established the sky-based per-frequency per-antenna calibration scheme in Section 3.4.1 and presented key equations in Section 3.4.2. Here, we start by deriving the linear systems of equations for the per-antenna gain amplitude and phase error in Appendix B1 using the PWL logarithmic implementation 29 . Then, Appendix B2 presents the SVDbased method used to infer the least-squares solution. Finally, the per-frequency per-baseline propagation visibility bias is estimated in Appendix B3.', 'B1 Deriving the per-antenna gain amplitude and phase error': "With our paired sky model approach, we can realize two equations using the measured visibility equation (equation 4) for applying an ' ideal ' sky model (equation 5) and a ' blended ' (equation 6) sky model, respectively. For each cross-correlation 𝑉 𝑖 𝑗 , we can determine the calibration error of the ill-calibrated scenario by combining equations (5) and (6) and solve \nexp GLYPH<2> GLYPH<0> 𝜂 𝑖 + 𝜂 𝑗 GLYPH<1> + i GLYPH<0> 𝜙 𝑗 -𝜙 𝑖 GLYPH<1> GLYPH<3> 𝑉 ideal 𝑖 𝑗 + 𝑛 𝑖 𝑗 = exp GLYPH<2> GLYPH<0> ' 𝜂 𝑖 + ' 𝜂 𝑗 GLYPH<1> + i GLYPH<0> ' 𝜙 𝑗 -' 𝜙 𝑖 GLYPH<1> GLYPH<3> ' 𝑉 blend 𝑖 𝑗 + 𝑛 𝑖 𝑗 . (B1) \nHere, we effectively eliminated the impact of the baseline noise, since the noise item 𝑛 𝑖 𝑗 is specific to the baseline 𝐵 𝑖 𝑗 under the assumption made in equation (2c). By taking the logarithmic form of equation (B1), we have \nGLYPH<0> 𝜂 𝑖 + 𝜂 𝑗 GLYPH<1> + i GLYPH<0> 𝜙 𝑗 -𝜙 𝑖 GLYPH<1> + ln 𝑉 ideal 𝑖 𝑗 = GLYPH<0> ' 𝜂 𝑖 + ' 𝜂 𝑗 GLYPH<1> + i GLYPH<0> ' 𝜙 𝑗 -' 𝜙 𝑖 GLYPH<1> + ln ' 𝑉 blend 𝑖 𝑗 . (B2) \nBy substituting the gain error ( Δ 𝜂 𝑖 ≡ 𝜂 𝑖 -' 𝜂 𝑖 and Δ 𝜙 𝑖 ≡ 𝜙 𝑖 -' 𝜙 𝑖 , respectively) and the visibility difference in terms of amplitude and \nphase ( 𝑅 𝑖 𝑗 ≡ ln GLYPH<12> GLYPH<12> GLYPH<12> ' 𝑉 blend 𝑖 𝑗 GLYPH<12> GLYPH<12> GLYPH<12> -ln GLYPH<12> GLYPH<12> GLYPH<12> 𝑉 ideal 𝑖 𝑗 GLYPH<12> GLYPH<12> GLYPH<12> and 𝐼 𝑖 𝑗 ≡ arg GLYPH<12> GLYPH<12> GLYPH<12> ' 𝑉 blend 𝑖 𝑗 GLYPH<12> GLYPH<12> GLYPH<12> -arg GLYPH<12> GLYPH<12> GLYPH<12> 𝑉 ideal 𝑖 𝑗 GLYPH<12> GLYPH<12> GLYPH<12> , respectively), equation (B2) can be decoupled as two sets of linear equations presented in equation (7).", 'B2 Gain error solution': 'To solve the two sets of linear systems of equations (equation 7), we proceed by rewriting equation (7a) and equation (7b) in two matrix systems. Taking equation (7a) as an example, we have \n<!-- image --> \nwhere A 𝑅 , which will be referred to as the array configuration matrix, is a 𝑁 × 𝐶 2 𝑁 matrix, 𝒙 𝑅 is a 𝑁 × 1 matrix, and 𝒃 𝑅 is a 𝐶 2 𝑁 × 1 matrix. In general, there will not be enough degrees of freedom in 𝒙 𝑅 when solving 𝑁 unknown Δ 𝜂 with 𝐶 2 𝑁 measurements. Thus, the least-squares solution is solved and satisfies the optimization in equation (B3b). In a similar fashion, we can also write the set of the phase difference equations in a matrix form and solve the least-squares solution: \nA 𝐼 𝒙 𝐼 = 𝒃 𝐼 , (B4a) \nargmin ˜ 𝒙 𝐼 ∥ A 𝐼 ˜ 𝒙 𝐼 -𝒃 𝐼 ∥ 2 . (B4b) \nTo solve those overdetermined systems of equations, we use the SVD (see Stewart 1993, for a review) to perform the pseudoinverse of the array configuration matrix and find the least-squares solution (see Brunton & Kutz 2019, for detailed demonstrations). According to the Moore-Penrose left pseudoinverse (Penrose 1955, 1956; Rohde 1965; Zlobec 1970), A matrix (referring to for both the amplitude and phase array configuration matrix) has a pseudoinverse A † , and the least-squares solution of the blending-induced amplitude and phase error, ˜ 𝒙 𝑅 and ˜ 𝒙 𝐼 , can be solved via \n˜ 𝒙 = A † 𝒃 . (B5)', 'B3 Deriving the per-baseline propagation error': "The aftermath of the calibration against a sky model with sourceblending defects is further discussed here. As mentioned in Section 3.4.1, after solving each antenna gain factor, 𝑔 𝑖 , the calibration solution will be applied to the observed visibility to recover the true cross-correlation of the sky signal. In the ill-calibrated scenario, the blending-induced antenna gain errors, Δ 𝜂 s and Δ 𝜙 s, will propagate to the calibrated visibilities of the sky signal for each baseline and leave a per-frequency per-baseline propagation bias within the visibility space. \nUsing the least-squares solution of both the amplitude and phase error matrix ( 𝒙 𝑅 and 𝒙 𝐼 , respectively) of the blending-induced antenna gain, we can further infer the per-frequency per-baseline propagation \nbias upon the visibilities of the sky signal. To infer the poor-calibration impact on the SKA EoR experiments, a set of simulated data with instrumental responses and resolution fidelity, including extragalactic foregrounds, Galactic foregrounds, and the 21-cm signal, are utilized to estimate the propagation bias for each baseline in a realistic fashion. Here, taking Galactic foregrounds as an example, we can rewrite the measured visibility equation (equation 4) and its logarithmic form using the true visibility 𝑆 gal -true 𝑖 𝑗 and ill-calibrated visibility ' 𝑆 gal -ill 𝑖 𝑗 of Galactic emission as \nexp GLYPH<2> GLYPH<0> 𝜂 𝑖 + 𝜂 𝑗 GLYPH<1> + i GLYPH<0> 𝜙 𝑗 -𝜙 𝑖 GLYPH<1> GLYPH<3> 𝑆 gal -true 𝑖 𝑗 + 𝑛 𝑖 𝑗 = exp GLYPH<2> GLYPH<0> ' 𝜂 𝑖 + ' 𝜂 𝑗 GLYPH<1> + i GLYPH<0> ' 𝜙 𝑗 -' 𝜙 𝑖 GLYPH<1> GLYPH<3> ' 𝑆 gal -ill 𝑖 𝑗 + 𝑛 𝑖 𝑗 (B6) \nand \nGLYPH<0> 𝜂 𝑖 + 𝜂 𝑗 GLYPH<1> + i GLYPH<0> 𝜙 𝑗 -𝜙 𝑖 GLYPH<1> + ln 𝑆 gal -true 𝑖 𝑗 = GLYPH<0> ' 𝜂 𝑖 + ' 𝜂 𝑗 GLYPH<1> + i GLYPH<0> ' 𝜙 𝑗 -' 𝜙 𝑖 GLYPH<1> + ln ' 𝑆 gal -ill 𝑖 𝑗 , (B7) \nrespectively. Similar to the decoupling of equation (B2), we further decompose equation (B7) into the real and imaginary parts separately and substitute the calibrated gain errors. Therefore, equation (B7) takes the form of \nln GLYPH<12> GLYPH<12> GLYPH<12> ' 𝑆 gal -ill 𝑖 𝑗 GLYPH<12> GLYPH<12> GLYPH<12> -ln GLYPH<12> GLYPH<12> GLYPH<12> 𝑆 gal -true 𝑖 𝑗 GLYPH<12> GLYPH<12> GLYPH<12> = Δ 𝜂 𝑖 + Δ 𝜂 𝑗 , (B8a) \narg GLYPH<12> GLYPH<12> GLYPH<12> ' 𝑆 gal -ill 𝑖 𝑗 GLYPH<12> GLYPH<12> GLYPH<12> -arg GLYPH<12> GLYPH<12> GLYPH<12> 𝑆 gal -true 𝑖 𝑗 GLYPH<12> GLYPH<12> GLYPH<12> = Δ 𝜙 𝑗 -Δ 𝜙 𝑖 . (B8b) \nSince all the Δ 𝜂 s and Δ 𝜙 s were solved for each frequency individually using methods described in Appendix B2, the per-frequency perbaseline propagation amplitude and phase bias of Galactic emission can be inferred using equation (B8) by taking the exponential form of equation (B8a): \nGLYPH<12> GLYPH<12> GLYPH<12> ' 𝑆 gal -ill 𝑖 𝑗 GLYPH<12> GLYPH<12> GLYPH<12> -GLYPH<12> GLYPH<12> GLYPH<12> 𝑆 gal -true 𝑖 𝑗 GLYPH<12> GLYPH<12> GLYPH<12> = GLYPH<2> exp GLYPH<0> Δ 𝜂 𝑖 + Δ 𝜂 𝑗 GLYPH<1> -1 GLYPH<3> GLYPH<12> GLYPH<12> GLYPH<12> 𝑆 gal -true 𝑖 𝑗 GLYPH<12> GLYPH<12> GLYPH<12> , (B9a) \narg GLYPH<12> GLYPH<12> GLYPH<12> ' 𝑆 gal -ill 𝑖 𝑗 GLYPH<12> GLYPH<12> GLYPH<12> -arg GLYPH<12> GLYPH<12> GLYPH<12> 𝑆 gal -true 𝑖 𝑗 GLYPH<12> GLYPH<12> GLYPH<12> = Δ 𝜙 𝑗 -Δ 𝜙 𝑖 . (B9b) \nSimilarly, we also derive the amplitude and phase bias for the other components. Hence, the per-frequency per-baseline propagation bias of all the sky components in the visibility space can be estimated in the general form, as shown in equation (8). \nThis paper has been typeset from a T E X/L A T E X file prepared by the author. \nTable B1. Calibration-related mathematical notions used in the paper."}

2024ApJS..271...25C

We present the first catalog of veryhighenergy and ultrahighenergy gammaray sources detected by the Large High Altitude Air Shower Observatory. The catalog was compiled using 508 days of data collected by the Water Cherenkov Detector Array from 2021 March to 2022 September and 933 days of data recorded by the Kilometer Squared Array from 2020 January to 2022 September. This catalog represents the main result from the most sensitive large coverage gammaray survey of the sky above 1 TeV covering decl. from 20 to 80. In total the catalog contains 90 sources with an extended size smaller than 2 and a significance of detection at gt5. Based on our source association criteria 32 new TeV sources are proposed in this study. Among the 90 sources 43 sources are detected with ultrahigh energy E gt 100 TeV emission at gt4 significance level. We provide the position extension and spectral characteristics of all the sources in this catalog.

2024-03-01T00:00:00Z

['10.48550/arXiv.2305.17030', '10.3847/1538-4365/acfd29', '2024ApJS..271...25C', '2023arXiv230517030C', 'arXiv:2305.17030']

['Gamma-ray astronomy', 'Gamma-ray observatories', 'Catalogs', '628', '632', '205', 'Astrophysics - High Energy Astrophysical Phenomena', 'High Energy Physics - Phenomenology']

The First LHAASO Catalog of GammaRay Sources

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']

https://arxiv.org/pdf/2305.17030.pdf

{'No Header': 'Typeset using L A T X preprint style in AASTeX62 \nDraft version November 28, 2023 E', 'The First LHAASO Catalog of Gamma-Ray Sources': "Zhen Cao, 1, 2, 3 F. Aharonian, 4, 5 Q. An, 6, 7 Axikegu, 8 Y.X. Bai, 1, 3 Y.W. Bao, 9 D. Bastieri, 10 X.J. Bi, 1, 2, 3 Y.J. Bi, 1, 3 J.T. Cai, 10 Q. Cao, 11 W.Y. Cao, 7 Zhe Cao, 6, 7 J. Chang, 12 J.F. Chang, 1, 3, 6 A.M. Chen, 13 E.S. Chen, 1, 2, 3 Liang Chen, 14 Lin Chen, 8 Long Chen, 8 M.J. Chen, 1, 3 M.L. Chen, 1, 3, 6 Q.H. Chen, 8 S.H. Chen, 1, 2, 3 S.Z. Chen, 1, 3 T.L. Chen, 15 Y. Chen, 9 N. Cheng, 1, 3 Y.D. Cheng, 1, 3 M.Y. Cui, 12 S.W. Cui, 11 X.H. Cui, 16 Y.D. Cui, 17 B.Z. Dai, 18 H.L. Dai, 1, 3, 6 Z.G. Dai, 7 Danzengluobu, 15 D. della Volpe, 19 X.Q. Dong, 1, 2, 3 K.K. Duan, 12 J.H. Fan, 10 Y.Z. Fan, 12 J. Fang, 18 K. Fang, 1, 3 C.F. Feng, 20 L. Feng, 12 S.H. Feng, 1, 3 X.T. Feng, 20 Y.L. Feng, 15 S. Gabici, 21 B. Gao, 1, 3 C.D. Gao, 20 L.Q. Gao, 1, 2, 3 Q. Gao, 15 W. Gao, 1, 3 W.K. Gao, 1, 2, 3 M.M. Ge, 18 L.S. Geng, 1, 3 G. Giacinti, 13 G.H. Gong, 22 Q.B. Gou, 1, 3 M.H. Gu, 1, 3, 6 F.L. Guo, 14 X.L. Guo, 8 Y.Q. Guo, 1, 3 Y.Y. Guo, 12 Y.A. Han, 23 H.H. He, 1, 2, 3 H.N. He, 12 J.Y. He, 12 X.B. He, 17 Y. He, 8 M. Heller, 19 Y.K. Hor, 17 B.W. Hou, 1, 2, 3 C. Hou, 1, 3 X. Hou, 24 H.B. Hu, 1, 2, 3 Q. Hu, 7, 12 S.C. Hu, 1, 2, 3 D.H. Huang, 8 T.Q. Huang, 1, 3 W.J. Huang, 17 X.T. Huang, 20 X.Y. Huang, 12 Y. Huang, 1, 2, 3 Z.C. Huang, 8 X.L. Ji, 1, 3, 6 H.Y. Jia, 8 K. Jia, 20 K. Jiang, 6, 7 X.W. Jiang, 1, 3 Z.J. Jiang, 18 M. Jin, 8 M.M. Kang, 25 T. Ke, 1, 3 D. Kuleshov, 26 K. Kurinov, 26 B.B. Li, 11 Cheng Li, 6, 7 Cong Li, 1, 3 D. Li, 1, 2, 3 F. Li, 1, 3, 6 H.B. Li, 1, 3 H.C. Li, 1, 3 H.Y. Li, 7, 12 J. Li, 7, 12 Jian Li, 7 Jie Li, 1, 3, 6 K. Li, 1, 3 W.L. Li, 20 W.L. Li, 13 X.R. Li, 1, 3 Xin Li, 6, 7 Y.Z. Li, 1, 2, 3 Zhe Li, 1, 3 Zhuo Li, 27 E.W. Liang, 28 Y.F. Liang, 28 S.J. Lin, 17 B. Liu, 7 C. Liu, 1, 3 D. Liu, 20 H. Liu, 8 H.D. Liu, 23 J. Liu, 1, 3 J.L. Liu, 1, 3 J.Y. Liu, 1, 3 M.Y. Liu, 15 R.Y. Liu, 9 S.M. Liu, 8 W. Liu, 1, 3 Y. Liu, 10 Y.N. Liu, 22 R. Lu, 18 Q. Luo, 17 H.K. Lv, 1, 3 B.Q. Ma, 27 L.L. Ma, 1, 3 X.H. Ma, 1, 3 J.R. Mao, 24 Z. Min, 1, 3 W. Mitthumsiri, 29 H.J. Mu, 23 Y.C. Nan, 1, 3 A. Neronov, 21 Z.W. Ou, 17 B.Y. Pang, 8 P. Pattarakijwanich, 29 Z.Y. Pei, 10 M.Y. Qi, 1, 3 Y.Q. Qi, 11 B.Q. Qiao, 1, 3 J.J. Qin, 7 D. Ruffolo, 29 A. S'aiz, 29 D. Semikoz, 21 C.Y. Shao, 17 L. Shao, 11 O. Shchegolev, 26, 30 X.D. Sheng, 1, 3 F.W. Shu, 31 H.C. Song, 27 Yu.V. Stenkin, 26, 30 V. Stepanov, 26 Y. Su, 12 Q.N. Sun, 8 X.N. Sun, 28 Z.B. Sun, 32 P.H.T. Tam, 17 Q.W. Tang, 31 Z.B. Tang, 6, 7 W.W. Tian, 2, 16 C. Wang, 32 C.B. Wang, 8 G.W. Wang, 7 H.G. Wang, 10 H.H. Wang, 17 J.C. Wang, 24 K. Wang, 9 L.P. Wang, 20 L.Y. Wang, 1, 3 P.H. Wang, 8 R. Wang, 20 W. Wang, 17 X.G. Wang, 28 X.Y. Wang, 9 Y. Wang, 8 Y.D. Wang, 1, 3 Y.J. Wang, 1, 3 Z.H. Wang, 25 Z.X. Wang, 18 Zhen Wang, 13 Zheng Wang, 1, 3, 6 D.M. Wei, 12 J.J. Wei, 12 Y.J. Wei, 1, 2, 3 T. Wen, 18 C.Y. Wu, 1, 3 H.R. Wu, 1, 3 S. Wu, 1, 3 X.F. Wu, 12 Y.S. Wu, 7 S.Q. Xi, 1, 3 J. Xia, 7, 12 J.J. Xia, 8 G.M. Xiang, 2, 14 D.X. Xiao, 11 G. Xiao, 1, 3 G.G. Xin, 1, 3 Y.L. Xin, 8 Y. Xing, 14 Z. Xiong, 1, 2, 3 D.L. Xu, 13 R.F. Xu, 1, 2, 3 R.X. Xu, 27 W.L. Xu, 25 L. Xue, 20 D.H. Yan, 18 J.Z. Yan, 12 T. Yan, 1, 3 C.W. Yang, 25 F. Yang, 11 F.F. Yang, 1, 3, 6 H.W. Yang, 17 J.Y. Yang, 17 L.L. Yang, 17 M.J. Yang, 1, 3 R.Z. Yang, 7 S.B. Yang, 18 Y.H. Yao, 25 Z.G. Yao, 1, 3 Y.M. Ye, 22 L.Q. Yin, 1, 3 N. Yin, 20 X.H. You, 1, 3 Z.Y. You, 1, 3 Y.H. Yu, 7 Q. Yuan, 12 H. Yue, 1, 2, 3 H.D. Zeng, 12 T.X. Zeng, 1, 3, 6 W. Zeng, 18 M. Zha, 1, 3 B.B. Zhang, 9 F. Zhang, 8 H.M. Zhang, 9 H.Y. Zhang, 1, 3 J.L. Zhang, 16 L.X. Zhang, 10 Li Zhang, 18 P.F. Zhang, 18 P.P. Zhang, 7, 12 R. Zhang, 7, 12 S.B. Zhang, 2, 16 S.R. Zhang, 11 S.S. Zhang, 1, 3 X. Zhang, 9 X.P. Zhang, 1, 3 Y.F. Zhang, 8 Yi Zhang, 1, 12 Yong Zhang, 1, 3 B. Zhao, 8 J. Zhao, 1, 3 L. Zhao, 6, 7 L.Z. Zhao, 11 S.P. Zhao, 12, 20 F. Zheng, 32 B. Zhou, 1, 3 H. Zhou, 13 J.N. Zhou, 14 M. Zhou, 31 P. Zhou, 9 R. Zhou, 25 X.X. Zhou, 8 C.G. Zhu, 20 F.R. Zhu, 8 H. Zhu, 16 K.J. Zhu, 1, 2, 3, 6 and X. Zuo 1, 3 \nThe LHAASO Collaboration \nCorresponding author: S.Q. Xi, S.C. Hu, S.Z. Chen, M. Zha xisq@ihep.ac.cn, hushicong@ihep.ac.cn, chensz@ihep.ac.cn, zham@ihep.ac.cn", 'ABSTRACT': 'We present the first catalog of very-high energy and ultra-high energy gamma-ray sources detected by the Large High Altitude Air Shower Observatory (LHAASO). The catalog was compiled using 508 days of data collected by the Water Cherenkov Detector Array (WCDA) from March 2021 to September 2022 and 933 days of data recorded by \nthe Kilometer Squared Array (KM2A) from January 2020 to September 2022. This catalog represents the main result from the most sensitive large coverage gamma-ray survey of the sky above 1 TeV, covering declination from -20 · to 80 · . In total, the catalog contains 90 sources with an extended size smaller than 2 · and a significance of detection at > 5 σ . Based on our source association criteria, 32 new TeV sources are proposed in this study. Among the 90 sources, 43 sources are detected with ultra-high energy ( E > 100 TeV) emission at > 4 σ significance level. We provide the position, extension, and spectral characteristics of all the sources in this catalog.', '1. INTRODUCTION': "Gamma rays, located at the highest energy band of the cosmic electromagnetic radiation, serve as a powerful probe for astrophysics and fundamental physics under extreme conditions. Most gamma rays are produced through the acceleration and propagation of relativistic particles, such as protons or electrons, in astrophysical sources. Relativistic electrons can scatter low-energy photons, including star light, dust scattered light and the Cosmic Microwave Background (CMB), to the gamma-ray band via the inverse Compton process.The whole universe is filled with low energy photons, especially the CMB. Therefore, gamma rays are a unique tool to probe the relativistic electrons when the surrounding magnetic field is weak and the synchrotron radiation is undetectable. Relativistic protons interact with the surrounding medium to create hadronic cascades, which include secondary π 0 mesons that quickly decay into gamma-rays. Hence, gamma rays are also an important tool to study the origin, acceleration and propagation of cosmic rays (CRs). \nThanks to the advancements in space-based and ground-based gamma-ray detectors, our knowledge about the high energy gamma-ray universe has made impressive progress over the past two decades. At high energy (HE, E > 0 . 1 GeV), the Fermi Large Area Telescope ( Fermi -LAT, Atwood et al. 2009) has been surveying the whole sky since 2008 and detected 6658 Galactic and extragalactic gamma-ray sources using the first twelve years of observations (Abdollahi et al. 2022). Compared to its predecessor, EGRET (Hartman et al. 1999), the number of detected sources has increased by a factor of more than 20, and some new categories of gamma-ray emitters have been revealed. Important evidence about the acceleration of GeV-TeV CRs were found in two ancient supernova remnants (Ackermann et al. 2013). The Dark Matter Particle Explorer (DAMPE) collaboration also reported the detection of more than 200 gamma-ray sources above GeV (Duan et al. 2021). \nAt Very High Energy (VHE, E > 0 . 1 TeV), ground-based gamma-ray detectors are necessary due to their large area requirements. The successful operation of the second generation Imaging Atmospheric Cherenkov Telescopes (IACTs), such as H.E.S.S. (Aharonian et al. 2006), MAGIC (Aleksi'c et al. 2016), and VERITAS (Meagher & VERITAS Collaboration 2015), has significantly increased the number of detected VHE gamma-ray sources from about 10 to over 200 since 2004. Several categories of VHE gamma-ray emitters have been firmly established, including Active Galactic Nuclei (AGNs), pulsars and Pulsar Wind Nebulae (PWNe), Supernova Remnants (SNRs), binaries, starburst galaxies, Gamma-Gay Bursts (GRBs) and others. However, due to their narrow Field Of View (FOV, 3 · -5 · ) and low duty cycle (10% -15%), IACTs are not suitable for performing long-term comprehensive sky surveys. Most VHE sources are detected while searching for counterparts of sources observed at lower energies, and only certain parts of galactic plane have been surveyed by the IACTs, such as H.E.S.S. (H. E. S. S. Collaboration 2018) and VERITAS (Staszak & VERITAS Collaboration 2015). \nTo achieve an overall view of the VHE universe, a roughly unbiased sky survey is needed, similar to that carried out by Fermi -LAT in the GeV band. Ground-based Extensive Air Shower (EAS) arrays, with their large field of view and high duty cycle, are ideal detectors for this scientific goal. Several VHE gamma-ray surveys with positive results have been conducted to date, including those by Tibet AS γ (Amenomori et al. 2005), Milagro (Abdo et al. 2007), and ARGO-YBJ (Bartoli et al. 2013). However, due to the limitations of detector sensitivity, only a handful of sources have been observed. Nonetheless, impressive progress has been made in observing some typical extended sources (Bartoli et al. 2014) and variable AGNs (Bartoli et al. 2016, 2012), highlighting the invaluable role of EAS arrays in VHE gamma-ray observation. Recently, the sensitivity of EAS arrays has greatly improved thanks to the successful operation of the new generation arrays, such as HAWC and LHAASO. The HAWC Observatory has detected 65 VHE sources including 20 new ones using five years of data in their third catalog (3HWC; Albert et al. 2020). In particular, HAWC has revealed a new source category,the TeV pulsar halo (Abeysekara et al. 2017), which is a useful tool to probe the CR diffusion in the interstellar medium (ISM) near PWNe. \nAnother crucial characteristic of the EAS array is that it can extend the observation to the UltraHigh Energy (UHE, E > 0 . 1 PeV) range due to its large detector area and long duty cycle. The Tibet AS γ collaboration first reported a UHE gamma-ray source, the Crab Nebula (Amenomori et al. 2019), followed by another three sources reported by the HAWC collaboration (Abeysekara et al. 2020). Recently, the LHAASO collaboration has made exciting progress by reporting the detection of 12 UHE gamma-ray sources with a statistical significance of over 7 σ and the maximum energy up to 1.4 PeV (Cao et al. 2021a). Additionally, some sources were observed with VHE emission for the first time (Cao et al. 2021b,c). These observations provide crucial candidates to explore the origin of PeV CRs within the Galaxy. LHAASO also detected PeV gamma-ray emission from the Crab Nebula, revealing an extreme electron accelerator with unprecedented high accelerating efficiency (LHAASO Collaboration 2021). These observation involving the highest gamma-ray energy also shed light on exploring the Lorentz Invariance Violation (Cao et al. 2022a) and dark matter (Cao et al. 2022b). It is worth noting that these results were achieved using only 1 year of data and half of the LHAASO array prior to completion of its construction. \nThis paper is structured as follows. Section 2 briefly describes the LHAASO detector and the data set. Additionally, it presents the background estimation and significance sky maps. In Section 3, the methods of searching for sources and constructing the catalog are introduced. The characteristics of WCDA and KM2A source components are briefly described and compared. In Section 4, a final source catalog is compiled, including relevant positional, spatial, and spectral information. The results of some typical source populations are discussed in Section 5. Section 6 provides a summary.", '2.1. The LHAASO Detector and Data': "LHAASO is a multi-purpose and comprehensive EAS array, designed for the study of CRs and gamma rays across a wide energy range, from sub-TeV to beyond 1 PeV (Ma et al. 2022). It consists of three interconnected detector arrays, a 1.3 km 2 array (KM2A) for gamma-ray detection above 10 TeV, a 78,000 m 2 Water Cherenkov Detector Array (WCDA) for TeV gamma-ray detection, and a Wide Field-of-view Cherenkov Telescopes Array (WFCTA) mainly for CR physics. When a highenergy extraterrestrial particle, gamma ray or CR, enters Earth's atmosphere, it initiates a cascade \nconsisting of secondary hadrons, muons, leptons, and photons known as an air shower. The WCDA and KM2A detectors record different components of these air showers, which are used to reconstruct the types, energies, and arrival directions of the primary particles. The WCDA consists of three ponds: WCDA-1 and WCDA-2, both measuring 150 m × 150 m , and WCDA-3 measuring 300 m × 110 m. The total area of the array is 78,000 m 2 . WCDA-1, WCDA-2, WCDA-3 are composed of 900, 900, 1320 detector units, respectively. Each detector unit is 5 m × 5 m separated by nonreflecting black plastic curtains and is equipped with two upward-facing PMTs (8-inch and 1.5 inch PMT combination) on the bottom at the center of the unit. To further lower the threshold energy, WCDA-2 and WCDA-3 employ a 20-inch and 3-inch PMT combination. Each pond is filled with purified water up to 4 m above the PMT photo-cathodes. A closed recycling system is implemented to maintain water purity and achieve an attenuation length for near-ultraviolet light longer than 15 meters. \nThe results presented here for WCDA were obtained using the full-array configuration from 2021 March 5 to 2022 September 30. For each PMT, a hit is formed with the threshold of 1/3 photoelectron (PE) for an 8-inch PMT, and 1 PE for a 20-inch MCP-PMT. A trigger algorithm was implemented to record CR air showers by requiring at least 30 PMTs fired among 12 × 12 PMT arrays simultaneously within a window of 250 ns. To ensure a reliable data sample, data quality checks were performed based on the DAQ data status and reconstruction quality. The total effective live time used in the data analysis is around 508 days with a trigger rate around 35 kHz. The number of gammalike events recorded by WCDA is around 1.29 × 10 9 events after certain data selection and gammaray/background discrimination cuts. More details about the array and the reconstruction can be found in Aharonian et al. (2021a). \nKM2A is composed of 5195 electromagnetic particle detectors (EDs) and 1188 muon detectors (MDs) distributed over an area of 1.3 km 2 . Each ED consists of 1-m 2 plastic scintillator covered by a 0.5-cm thick lead plate and a 1.5-inch photomultiplier tube (PMT). The typical detection efficiency is about 98% and time resolution is about 2 ns. The response time and signal amplitude of each ED is calibrated using offline methods (Lv et al. 2018; Aharonian et al. 2022). A trigger is generated when 20 EDs are fired within a 400 ns window. The signals recorded by EDs are used to determine the energy and arrival direction of the primary particles. Each MD consists of a cylindrical water tank, with a diameter of 6.8 m and a height of 1.2 m, and a 8-inch PMT, which is filled with pure water and buried under 2.5 m of soil. The MDs are designed to detect the muon component of showers, which is used to discriminate between gamma-ray and hadron-induced showers. The performance of KM2A for the gamma-rays with energies from 10 TeV to 1.6 PeV has been thoroughly tested using detector simulations and observations of the Crab Nebula (Aharonian et al. 2021b). The resolution is energy- and zenith-dependent. For showers with a zenith angle less than 20 ° , the angular resolution ranges from 0.5 · at 20 TeV to 0.2 · at 100 TeV. The energy resolution is about 24% at 20 TeV and 13% at 100 TeV. The rejection power of the hadron-induced showers is about 1000 at 20 TeV and greater than 4000 at energies above 100 TeV. \nThe KM2A detectors were constructed and merged into the data acquisition system (DAQ) in stages. The first half of the KM2A consisting of 2365 EDs and 578 MDs started operating in December 2019. This partial array was expanded to a 3/4 array, comprising 3978 EDs and 917 MDs, in December 2020. The whole KM2A was completed and operated starting 2021 July 19. The KM2A data collected from 2019 December 27 to 2022 July 31 were used for the analysis in this work. The \ntotal live time is 933 days, corresponding to 730 days of complete KM2A detector. The pipeline of KM2A data analysis presented in Aharonian et al. (2021b) is directly adopted in this analysis with the same event selection conditions. After the pipeline cuts, the number of events used in this analysis is ∼ 1 . 35 × 10 7 with reconstructed energies above 25 TeV.", '2.2. Data Binning and Background Estimation': "We use only events with zenith angles less than 50 · in this analysis, corresponding to the survey region in the declination band from -20 · to 80 · . For the WCDA data, events are divided into five groups according to the number of hits ( N hit ), i.e., 100-200, 200-300, 300-500, 500-800, ≥ 800. For Crab-like sources, the corresponding energies roughly range from 1 TeV to 25 TeV. It should be noted that events in the same bin for a source with a harder spectrum, or at a larger declination, will tend to have a larger average energy. The PSF width of WCDA data depends on the shower size which is closely related to N hit . Therefore, it is insensitive to depend on declination or spectrum. The Crab observation provides a clear measurement of the angular resolution (denoted as ϕ 68 ), which is 0 . 73 · ,0 . 46 · ,0 . 37 · ,0 . 29 · and 0 . 25 · in the five bins, respectively. For the KM2A data, events are divided into five groups per decade according to reconstructed energy ( E rec ), i.e, 25-40 TeV, 40-63 TeV, 63-100 TeV, 100-160 TeV, 160-250 TeV, 250-400 TeV, 400-630 TeV, 630-1000 TeV, 1000-1600 TeV, > 1600 TeV. The median energy and the angular resolution in each E rec bin slightly vary with the declination and spectrum of the source. The properties of each bin are shown in Table 1. \nThe sky maps in celestial coordinates (right ascension and declination) are divided into 0 . 1 · × 0 . 1 · pixels, and each pixel is filled with the number of the detected events according to their arrival direction. To obtain the excess of gamma-ray induced showers in each pixel, the 'direct integration method' (Fleysher et al. 2004) is adopted to estimate the number of background events. This method uses events with the same direction in local coordinates but different arrival times to estimate the background. In this work, the integrated time is 10 hours and the events within the regions of the Galactic plane ( | b | < 10 · ) and VHE gamma-ray sources (with space angle less than 5 · ) are excluded to estimate the background which is dominated by the residual CRs. The isotropic diffuse gamma rays and electrons may contribute slightly to the background. \nThe diffuse Galactic gamma-ray emission (GDE) resulting from the interaction of CRs with the interstellar medium (ISM) and background photons is an essential component of the gamma-ray sky. In Galactic plane, the VHE and UHE GDE has already been clearly detected (Abramowski et al. 2014; Amenomori et al. 2021; Cao et al. 2023). Therefore, the GDE is also an essential background to take into account for detecting and characterizing gamma-ray sources. In VHE and UHE band, the GDE have been interpreted to be mainly pion decay photons generated from hadronic interactions of CR with ISM. Approximately 99% of the ISM mass is gas. Modeling the VHE and UHE GDE requires knowledge of CR intensities and spectra, along with the distributions of gas, throughout the Galaxy. However, due to the variation of gas density and CR density across the Galaxy, it is not possible to completely disentangle the GDE. To simplify the analysis, we ignore the variation of CR density. The flux morphology of GDE is assumed to follow the spatial distribution of gas, which coexist with dust grains. Observations have shown that the gas-to-dust ratio leads to a mass ratio of M gas /M dust ∼ 100. The dust column density can therefore provide a template for the GDE morphology. In this work, we use the GDE template derived by Planck maps of dust optical depth (Planck Collaboration et al. 2014, 2016). The spectral function is assumed to be identical in all regions. To avoid the influence of the gamma-ray sources, the spectrum of GDE is estimated using the region of Galactic latitude \nTable 1. Property of each Analysis Bin \nNote -ϕ 68 is the 68% containment radius of angular resolution. E MC γ is median energy in each bin. We consider a reference source with a broken power-law spectrum of an index of 2.5 in 1-25 TeV and an index of 3.5 above 25 TeV, at a declination of 30 · . \n5 · < | b | < 10 · and then extrapolating it to other Galactic plane regions ( | b | < 5 · ). Note that the region of several VHE sources in the region of 5 · < | b | < 10 · is also masked for the measurement of GDE, effectively excluding the contribution of both Galactic and extragalactic sources. The GDE spectrum for this region is found to be well reproduced by the diffuse Galactic gamma-ray emission model, which takes into account the local CR spectrum and the gas column density. We add the extrapolated GDE maps from the measurements in the region of 5 · < | b | < 10 · to the background maps for subsequent analysis.", '2.3. Significance Map': "We can calculate the significance of excess centered at each pixel using the event and background maps as described in previous section, taking the detector responses into consideration. The WCDA and KM2A data are separately analyzed. To combine the data bins of each instrument, the 3Dlikelihood method is used. A likelihood ratio test is performed between the background-only model and the one point source model. The likelihood calculation assumes that the number of counts in each pixel is distributed according to a Poisson distribution, with the mean given by the number of background counts for the background-only model and the number of background plus expected gamma-ray counts for the source model (Stewart 2009). The test statistic (TS), defined as TS = 2 ln( L / L 0 ), where L 0 is the maximum likelihood value for the null hypothesis and L is the maximum \nFigure 1. Significance maps of the region monitored by LHAASO. A point test source with a spectral index of 2.6 for WCDA data and 3.0 for KM2A data is used. \n<!-- image --> \nvalue for the source hypothesis, is used to estimate the significance. In this work, a power-law spectrum is assumed with an index of 2.6 for WCDA data in the energy range 1 -25 TeV and 3.0 for KM2A data at energies E > 25 TeV as initial conditions. This leaves only one free parameter for the likelihood calculation. According to Wilks' Theorem, the TS is distributed as χ 2 with one degree of freedom (dof), and the significance can be estimated with S = √ TS. Figure 1 shows the significance maps obtained in the energy bands 1 TeV < E < 25 TeV and E > 25 TeV in Galactic coordinates. The signals are clearly visible. However, most sources in the Galactic plane are nearby and overlapping. Hence, further analysis is needed to derive each source separately.", '3. CONSTRUCTION OF THE CATALOG': 'The identification of point-like gamma-ray sources and their corresponding significance can be roughly derived from Figure 1. However, it is important to note that the significance may be overestimated due to the overlap with nearby sources. Conversely, in the search for point sources, a significant portion of the sources may actually be extended, resulting in an underestimation of their significance. To improve source detection, the significance of a given source is reassessed by coupling the fitting of localization, extension and spectrum, and new potential sources are also explored. In \nthe first step, the WCDA and KM2A data are analyzed separately to achieve two source component catalogs, one for energies ranging from 1 -25 TeV and another for energies above 25 TeV. These two source component catalogs are merged into the final source catalog following a specific procedure. The sources with clear E > 100 TeV emission will also be identified in the final catalog.', '3.1. All Sky Fitting and Source Component Detection': 'The WCDA data and KM2A data are analyzed following the same strategy to fit the entire sky region and yield the source components with the characteristics of localization, extension and spectrum.', '3.1.1. Determination of Seeds and ROIs': "The signals shown in Figure 1 are potential point sources, denoted as 'seeds'. To identify these seeds in the maps, all local maxima within a 0.5 · region centered around the candidate, with a significant above 4 σ , are selected. Most of the seeds are concentrated in the inner Galactic plane, while in the outer Galactic plane, the seeds are clustering in the region near Geminga. In other areas of sky region, a few individual seeds stand out. \nTo facilitate the fitting process, all sky data are split into different regions of interest (ROI) based on the seeds and their clustering characteristics. For the inner Galactic plane region, a sliding window 20 · × 20 · ROI is adopted along Galactic longitude with a step length of 10 · . For the Geminga region, a ROI with the radius of 12 · centered at right ascension ( α 2000 ) 100 . 2 · and declination ( δ 2000 ) 16 . 2 · is used. For isolated seeds, the ROIs are circular regions with a 3 · radius, and the overlapping ROIs are merged into one.", '3.1.2. Source Component Detection Procedure': 'The seeds obtained in Section 3.1.1 may not correspond to a real source component because the spectrum and extension characteristics are not fully considered. Moreover, the position can be influenced by nearby sources with small angular separations. Thus, an analysis pipeline based on a three-dimensional maximum likelihood algorithm is developed for further analysis. Using this analysis pipeline, the spectral information and spatial morphology and significance for individual source components can be determined simultaneously. Some technical details are described below: \n- · The first step of the 3D maximum likelihood algorithm is to build a source model characterized by a 2D-Gaussian morphology based on the initial positions of the seeds. The source spectrum is assumed to follow a power-law shape dN/dE = N 0 ( E/E 0 ) -Γ , where N 0 is the differential flux at E 0 , E 0 is the reference energy which is set to 3 TeV for WCDA data and to 50 TeV for KM2A data, respectively, and Γ is the spectral index.\n- · Starting with one Gaussian component, we add Gaussian components successively to the source model of each ROI. A likelihood ratio test is used to compare between two models with N and N +1 Gaussian components. We define TS = 2 ln( L N +1 / L N ), where L N and L N +1 represents the maximum likelihood of the model with N and N + 1 source components, respectively. During this iterative process, an additional source characterized by 5 free parameters ( α 2000 , δ 2000 , extension, differential flux and spectral index) is added to the model if TS > 25 (3.8 σ for 5 dof). \n- · The iteration process is terminated once no significant residual is left in the TS map for each ROI. Here, no significant residual means that no pixel is with TS > 16 (4 σ for 1 dof) in the residual TS map, which is calculated by adding the current sources to the background model and using the method similar to that in Sec. 2.3.\n- · During the fitting process, all of the parameters are optimized simultaneously. The Spectral parameter of the GDE is always keep to fixed.\n- · After completing the previous steps, the WCDA and KM2A source component lists are formed, respectively, by merging all the list determined in each ROI. Due to the overlap ROIs, the same source component may appear in two or more ROIs. In such cases, the one that was closest to the center of the ROI is selected. \nThis analysis pipeline yields a preliminary source component list, in which most components have good estimates of parameters. However two possible biases may exist for two cases: 1) the target Gaussian component is close to the ROI edge or a bright source; 2) the iteration fitting scheme described above could converge toward a local maximum. Additional checks have been implemented for these two cases using a new ROI centered on the position of each component and re-analyzing the target source component with free parameters for all components in the ROI. If new parameters of the target source component do not significantly differ from the previous iteration fitting, the new parameters of this source component are considered reliable and are adopted. The significant difference refers to the situation where the variance of a single parameter is greater than the corresponding statistical error. If significant variations are observed, a manual fitting process is conducted. This involves adjusting the starting values of the parameters and/or the size of the ROI until the parameters of the target source component converge. This convergence is determined through a visual inspection of likelihood profiles. Once the parameters are obtained, the TS value of each source component is evaluated within each ROI. In this evaluation, the positions and extensions of the other components are fixed, while the spectral parameters are left free for optimization.', '3.1.3. False Positive Expectation': 'The trial factor can be roughly estimated as f t = Ω / [2 π (1 -cos 0 . 5 · )] ∼ 3 . 5 × 10 4 , where Ω is the solid angle of the LHAASO survey sky within the declination ranging from -20 · to 80 · , and 0 . 5 · is the minimum location searching radius for seeds. Since the TS value for the above components follows the χ 2 distribution with 5 degrees of freedom, TS = 25 corresponds to the p -value of 6 . 9 × 10 -5 . Based on this, we can expect that 2.4 sources in our survey are false detections due to background fluctuations.Thus, we set a higher threshold of TS = 37 (5 σ for 5 dof) for reported sources, corresponding to an expectation of 0.01 false-positive sources. For sources with both WCDA and KM2A components, the summed TS value of 50 (5.1 σ for 10 dof) corresponds to the comparable false-detection to that of the single component of TS = 37. Hence, we also report the sources of which two components are with 25 < TS < 37. Furthermore, due to the mismodeling GDE background, the sources in the gas-rich region with a low significance level might still be potential false detection. Therefore, further study is required to determine whether or not they are real sources. \n̸ \nThe localization of each source component, represented by α 2000 and δ 2000 , is given by the procedure in Sec 3.1.2. The statistical errors of the localization ( σ α 2000 ,stat , σ δ 2000 ,stat ) are also estimated by analyzing the shape of the likelihood function with the HESSE tool in the Minuit2 package. It is worth noting that the localization of a faint or soft source component is more sensitive to the influence of diffuse emission and nearby sources. As a result, we may find asymmetric position error circles for these components, where σ α 2000 ,stat cos δ 2000 = σ δ 2000 ,stat . To account for this, we conservatively consider the bigger error as the one-dimensional 1 σ position error, i.e., x stat = max [ σ δ 2000 ,stat , σ α 2000 ,stat cos δ 2000 ]. In this work, we report the position uncertainties at the 95% confidence level as σ p, 95 ,stat = 2 . 45 × x stat , where the 2.45 is the factor √ -2 ln(1 -0 . 95) by which the 95% confidence level position uncertainty is related to x stat for a two-dimensional axisymmetric Gaussian. Figure 2 illustrates the position uncertainties as a function of the TS values. The relatively large dispersion seen at a given TS is due to variations in local conditions , the source extension and the source spectrum. Considering the systematic error, the error radius (95% confidence) of the LHAASO sources ranges from ∼ 0 . 04 · to ∼ 0 . 8 · . \nFigure 2. Source component location uncertainties ( σ p, 95 ) as a function of TS values. σ p, 95 is defined by √ σ 2 p, 95 ,stat + σ 2 p,sys , where σ p,sys is the systematic positional error which is 0 . 03 · for the KM2A component and 0 . 04 · for the WCDA component, respectively, considering the pointing error only (Detailed in Section 3.6). The dashed line is a trend for reference, considering a point source at a declination of 20 · with the index of 2.5 for WCDA component and of 3.5 for KM2A component. The marker size is scaled by the source radius. The marker color represents the photon spectral index. The source components with the significance of TS > 37 and the extension size of r 39 < 2 · are plotted. \n<!-- image --> \nThe extension size of each source component is described by a two-dimensional Gaussian σ (namely r 39 in this work) corresponding to 39% of the source flux. A statistical error for the measured extension ( σ r 39 ,stat ) at the 1 σ level can also be determined. The measurement of r 39 is invalid for the point-like components unresolved by LHAASO. Thus, for all source components, the extension Test Statistic (TS ext ) is performed. Here TS ext = 2 ln( L ext / L ps ), where L ps and L ext are the maximum-likelihood values assuming the source spatial model is a point-like and 2D-Gaussian model, respectively. When the TS ext is lower than 9 corresponding to a significance smaller than 3 σ level, we change the 2D-Gaussian spatial model to a point-like model and re-optimize the position and spectral parameters. The upper limits on the extension ( r 39 ,ul,stat ) at the 95% confidence level for \nthese point-like components are derived from the shape of the likelihood function. Figure 3 shows the extension size of each source component. Roughly 1/3 are unresolved by LHAASO and modeled by a point-like morphology, and roughly 2/3 are modeled by a 2D-Gaussian morphology with an extension size ranging from ∼ 0 . 2 · to > 2 · . \nFigure 3. The distribution of the extension sizes ( r 39 ) of the source components detected by LHAASO. The size of the point-like source component is set to 0. Due to the limitation of statistical level and angular resolution, the measured minimum extension is approximately 0 . 17 · , leading to a gap of around 0 . 1 · . \n<!-- image --> \nThe spectral energy distribution (SED) of each source component is modeled by a simple power law (PL). Figure 4 shows the distribution of the SED parameters of the source components. The differential flux has a mean of ∼ 3 × 10 -12 TeV cm -2 s -1 for WCDA components at 3 TeV and ∼ 4 × 10 -13 TeV cm -2 s -1 for KM2A components at 50 TeV. The photon index distributions are obviously different, with an average of ∼ 2 . 5 for WCDA components and ∼ 3 . 5 for KM2A components, implying that a single power law shape cannot describe the overall SED for the majority of 1LHAASO catalog sources. \nWe compare the flux measurements with the sensitivity of source derived by our used data set. The flux sensitivity is defined as the flux normalization required to have 50% probability of detecting a source at 5 σ level. As shown in Figure 5, the sensitivity dependents on the declination, spectral index and source size. Sources passing directly overhead, with a declination of 30 · for LHAASO (20 · for HAWC), exhibit the highest sensitivity. The sensitivity for a source with a softer spectrum or larger size, or at a larger declination, will tend to decrease. Thus, about 10 WCDA sources, which have fluxes above the 3HWC point source sensitivity, have not been included into the 3HWC catalog due to their large size and/or soft spectrum. In addition, it is possible that the source searching strategy employed in the 3HWC catalog results in the failure to detect several of these sources. \nFigure 6 shows the 1LHAASO source fluxes related to source sizes or source photon spectral index. In the flux-size panel, we illustrate the approximate flux sensitivity limit of the 1LHAASO source as a function of source size. It can be observed that the sensitivity worsens as the source size increases, as mentioned earlier. For extended sources with sizes larger than 1 · , we may find a possible lack of sources with flux F 3 > 4 × 10 -13 TeV -1 cm -2 s -1 or F 50 > 4 × 10 -16 TeV -1 cm -2 s -1 . In the index-flux panel, the photon index in the WCDA band appears to be uncorrelated with the flux F 3 . However, there might be a broken power-law correlation between the photon index in the KM2A \nFigure 4. The distribution of the SED parameter of the source components. Left: the distribution of the differential flux ( E 2 0 N 0 ). The reference energy E 0 is 3 TeV or 50 TeV for the WCDA or KM2A component, respectively. Right: the distribution of the photon spectral index (Γ) for WCDA and KM2A components. \n<!-- image --> \nFigure 5. 1LHAASO source differential flux at 3 TeV ( F 3 ) or 5 TeV ( F 50 ) as a function of declination. The marker size represents the source size. The color indicates whether or not these sources are the new sources (seen in Section 5.2). Left: the integral sensitivity is shown at 3 TeV for three hypotheses: 1) point source with spectrum of E -2 . 0 ; 2) point source with spectrum of E -2 . 5 ; 3) 0 . 5 · gaussian source with spectrum of E -2 . 5 . The 3HWC sensitivity corresponds to the point-source search with the 3HWC data set (Albert et al. 2020). Right: the integral sensitivity is shown at 50 TeV for three hypotheses: 1) point source with spectrum of E -3 . 0 ; 2) point source with spectrum of E -3 . 5 ; 3) 0 . 5 · gaussian source with spectrum of E -3 . 5 . \n<!-- image --> \nband and the logarithm of F 50 , with a break occurring at a flux of F 50 ∼ 4 × 10 -17 TeV -1 cm -2 s -1 . It is not yet clear whether this phenomenon is caused by an instrument selection effect or a physical relation.', '3.3. Source Detection Above 100 TeV': 'An exciting characteristic of LHAASO is that it can extend the observation to the UHE regime owing to the large detector area. The UHE sources are crucial to identify the emission mechanism and explore the maximum acceleration energy within the sources, and they are also very important \nFigure 6. Flux vs. source size and flux vs. photon spectral index. Top: 1LHAASO source differential flux ( F 3 or F 50 ) as a function of source size. The error bar of r 39 represents the statistical uncertainty at a 95% confidence level. The dashed lines are integral sensitivity shown at 3 TeV or 50 TeV, assuming a point-like source. Bottom: 1LHAASO source photon spectral index as a function of the differential flux ( F 3 or F 50 ). The definition of the new sources is seen in Section 5.2. \n<!-- image --> \ncandidates to explore the origin of PeV CRs within the Galaxy. To explore UHE sources, the sources detected at energies above 25 TeV are selected. The SED parameters of these sources are re-optimized while freezing the spatial model parameters, i.e., the position and extension size, to the values measured at E > 25 TeV. The TS values above 100 TeV (denoted by TS 100 ) are calculated. Sources are with TS 100 > 20 ( > 4 σ significance level based on the χ 2 distribution with two degrees of freedom) are claimed as UHE sources in this work.', '3.4. Merging of the Source Components': 'A WCDA component and a KM2A component are considered to be the same source according to the agreement in position, i.e., r 2 d < ( σ p, 95 ,W ) 2 + ( σ p, 95 ,K ) 2 + δ 2 p , where r d is the positional offset between the WCDA component and the KM2A component, and σ p, 95 ,W and σ p, 95 ,K are the position uncertainties at the 95% confidence level for the WCDA and the KM2A components, respectively. δ p is the position-offset correction of the components, possibly due to the physical position variation or the changes in morphology at different energy bands. Additionally, incomplete descriptions of \nthe GDE and/or of nearby sources in the analysis can also contribute to the positional offset. It is difficult to determine the value of the δ p for each source. We expect that δ p is related to the source extension. As a simple criterion, we adopt a loose condition with δ p = ( r 39 ,W + r 39 ,K ), where r 39 ,W and r 39 ,K are the extensions of the WCDA and KM2A components, respectively. We merge the closest pairs of the WCDA and KM2A components that satisfy this condition. Ultimately, 54 pairs of components are merged.', '3.5. Known TeV Source Association': 'We have conducted a preliminary association of LHAASO sources with known TeV sources based on their position. A searching radius is defined by r sr = √ σ 2 p, 95 + r 2 39 +(0 . 3) 2 , where σ p, 95 and r 39 are the position error and extension size of the source component. For the LHAASO sources with KM2A and WCDA components, the position and extension of the component with higher significance is used. For the point-like source components, r 39 is set to the extension upper limit. The average value of the position errors for the known TeV sources detected by EAS arrays is about 0.3 · . While for the sources detected by IACTs, the average position error is around 0.05 · . Roughly, we used the factor of 0.3 degree for the searching radius. If r sr > 1 · , a maximum searching radius of 1 · is applied. \nThe vast majority of the known TeV sources from ground based observatories are included in the TeV online catalog, i.e., TeVCat 1 (Wakely & Horan 2008). It is frequently updated, containing 252 TeV source entries to date. When associating LHAASO sources with known TeV sources, we used the canonical name and position reported in TeVCat. We exclude the TeV pulsars for the associations, since the pulsed TeV emission is expected to be null in our source searching procedure. If no TeV source in TeVCat is found for association, we further compare 1LHAASO sources with 3HWC sources, which include some sources not yet included in TeVCat so far. We list the closest TeV counterparts within the searching radius for each LHAASO source, as shown in table 2.', '3.6. Systematic Uncertainties and Caveats': 'Based on the measurement of the standard candle, the Crab Nebula, with a position of α 2000 = 83 . 633 · and δ 2000 = 22 . 015 · for the systematic error estimation, a systematic pointing error can be evaluated as 0 . 03 · for KM2A data. For the WCDA data, an investigation of the pointing systematic error has been conducted using three point sources: Crab Nebula, Mrk 421, and Mrk 501. The maximum error is from Mrk 501 at about ∼ 0 . 04 · , which is taken as the pointing error of the WCDA array in this work. It is important to note that no clear zenith angle dependence of the systematic error is found according to the observation of the Crab Nebula using events at different zenith angles, ranging over 0 · -15 · , 15 · -30 · , and 30 · -50 · . Furthermore, no systematic location bias related to declination has been identified based on the observation of the CR Moon shadow at different declination bands. \nThe systematic uncertainty of the size of 1LHAASO sources could be contributed by the uncertainties of the Point Spread Function (PSF). For KM2A data, we obtained a systematic bias for the ϕ 68 at the order of ∼ 0 . 08 · by comparing the Crab measurement with our simulated data. For WCDA data, we compared the observed events profile among Crab, Mrk 421, and Mrk 501. The uncertainties can also yield a systematic bias of ∼ 0 . 05 · . Thus, we can conservatively estimate the systematic error to be at the order of σ r 39 ,sys ∼ 0 . 05 · for WCDA and σ r 39 ,sys ∼ 0 . 08 · for KM2A component. \nThe systematic errors affecting the spectrum have been investigated in Aharonian et al. (2021a) and in Aharonian et al. (2021b). The main systematic error is contributed by the atmospheric model in the Monte Carlo simulations. The total systematic uncertainty is estimated to be 7% on the flux and 0.02 on the spectral index for KM2A SED measurement. In the case of WCDA data, the overall systematic uncertainty can be as large as +8% -5% on the flux, which is estimated by the same method in Aharonian et al. (2021a). The power-law spectral shape can adequately describe the WCDA components, while it is not suitable for about 1/3 KM2A components which shown an evident curved shape at energies above 25 TeV. A detailed spectrum study needs to be carried out in the future. \nMismodeling of GDE could affect the fitted locations, extensions, and SEDs of several source components, especially for those with lower fluxes and larger sizes. A rough assessment of the impact of the GDE was performed by excluding it from the background maps for the source components in the Galactic plane region. As a result, 11 KM2A source components and 10 WCDA source components exhibited changes (in terms of location, flux, index, or extension) exceeding three times the statistical errors. Source components that were significantly affected by the GDE were tentatively labeled as such. The GDE test is not thorough because we cannot determine all the sources affected by the GDE mismodeling. As a conservative approach, source components with an extended size greater than 2 · were excluded due to the growing impact of the GDE with increasing component size. Further understanding of very extended sources or obvious GDE-impacted sources requires deeper study of the GDE model, which is beyond the scope of this work. \nAs shown in Figure 7, at 95% confidence level, approximately 40% of the merged sources exhibit a noticeable bias in position or differences in extension. The shift could be an indication of an energydependent morphology of these sources. However, it should be noted that due to the poor angular resolution of LHAASO, the offset in position and extension could also be influenced by emission from nearby unresolved sources or mismodeling of GDE. There is a possibility of a false merged source, which could be a combination of WCDA and KM2A components that are physically unrelated. To further examine this, we have roughly conducted individual investigations on the surface brightness distribution and spectrum of each merged source. During this process, conflicts were found in 11 merged sources, which are tentatively labeled as dubious mergers. More comprehensive studies are needed to confirm the physical association between the two components of the dubious merged sources.', '4. RESULTS': 'Following the above procedure, the source catalog of LHAASO has been constructed. Overall, 90 sources with extension < 2 · are found over the whole LHAASO survey sky. Among them, 65 sources exhibit extended morphology with a confidence level greater than 3 σ . A total of 54 sources have been simultaneously detected by both WCDA and KM2A. Among all the sources, 43 UHE sources have been detected at > 4 σ confidence level when E > 100TeV. \nTable 2 presents a comprehensive list of all LHAASO sources obtained through the above procedure, ordered by α 2000 2 . To illustrate these catalog sources, a detailed view of 82 sources with Galactic latitude | b | < 12 · are shown in Figures 11-12, in which most sources are concentrated in the inner Galactic plane. Eight individual sources are detected at Galactic latitude | b | > 12 · , as shown in \nFigure 7. The position and extension comparison between the WCDA and KM2A components.The six most significant sources are highlighted in red. Left: the position offsets ( r d ) relative to the declinations. In the case where the declination of WCDA component is smaller than that of the KM2A component, we take the opposite value. The error bar is defined as √ σ 2 p,stat, 95 ,W + σ 2 p,stat, 95 ,K , where σ p,stat, 95 ,W and σ p,stat, 95 ,K are the statistical positional error of WCDA and KM2A components, respectively. The orange represents the systematic error just including the pointing error. Right: the extensions of the WCDA components relative to that of the KM2A components. The error bar represents the statistical uncertainty at a 95% confidence level. The orange band represents the systematic error just considering that of PSF. \n<!-- image --> \nFigure 13. Among them, 1LHAASO J1653+3943 and 1LHAASO J1104+3810, which correspond to Mrk 421 and Mrk 501 respectively, have been significantly detected by the WCDA detector, however, no gamma-ray emission has been found at E > 25 TeV by KM2A detector, due to significant absorption from interstellar radiation fields (ISRFs) and CMB. In addition to Mrk 421 and Mrk 501, there are another 34 sources detected by only one detector. This finding is reasonable considering the curved spectral shape, as illustrated in Figure 8. The different shapes of the spectra can result in various relationships in the TS value between WCDA and KM2A detections. \nFigure 8. TS value of KM2A component versus that of WCDA component. The reference dashed lines indicate the expected TS value for each detector. These values are calculated based on a point source which has a broken power-law spectral shape with a break at an energy of 25 TeV. \n<!-- image -->', '5.1. PeVatrons': 'The search for and identification of galactic sources capable of accelerating CRs up to PeV energies, known as PeVatrons, is of great importance. UHE gamma rays serve as a crucial and highly promising tool for achieving this target. With its unprecedented sensitivity at UHE, LHAASO represents the best instrument to survey the PeVatrons. In fact, using approximately 300 days of data collected by the half KM2A, LHAASO has already detected 12 UHE sources with significance above 7 σ and maximum energies reaching up to 1.4 PeV (Cao et al. 2021a), providing crucial candidates for hadronic PeVatron exploration. In this paper, 43 sources with significance above 4 σ at energy beyond 100 TeV are listed in Table 2. Since these sources are well detected at E > 25 TeV with much higher significance, they are significant enough to be identified as UHE sources. Among the 43 sources, 22 sources are detected with significance above 7 σ (TS 100 ≈ 54) at E > 100 TeV, improving by a factor of 2 the previous number of sources in Cao et al. (2021a). It is worth noting that 57% (43 out of 75) E > 25 TeV sources are UHE sources. According to Figure 9, most of these UHE sources have higher significance or harder spectral index than the other E > 25 TeV sources, which indicates that the remaining E > 25 TeV sources may also be detected as UHE sources by LHAASO in the future with further accumulation of data. This provides important evidence for the exciting fact that the Milky Way is full of UHE sources and full of PeV particle accelerators. Further deep analysis focusing on these sources one by one or type by type to identify the emission mechanism is a straightforward task to explore and identify the hadronic PeVatrons. \n<!-- image --> \nFigure 9. Distribution of the spectral index and the significance for sources at energy range from 1 -25 TeV (left) and E > 25 TeV (right), respectively. The red points indicate the sources also with significance above 4 σ at E > 100 TeV. The extragalactic sources are excluded in these figures. The six most significant sources are labeled by their name. \n<!-- image --> \nAccording to Table 2, 76% (33 out of 43) UHE sources are detected at energies 1 -25 TeV by WCDA. Figure 9 shows the spectral index versus significance for these sources at energies 1 -25 TeV. For comparison, other sources detected at energies 1 -25 TeV are also presented in the figure, excluding the extragalactic sources listed in Table 2. The UHE sources have harder spectral index or higher significance than the others in the energy band 1-25 TeV. Up to now, more than 250 VHE \nsources have been detected, and more than 100 sources are within the Galaxy with a significant fraction being located in the southern sky, which is out the FOV of LHAASO. These Galactic sources would be important UHE candidates and may be revealed with UHE emission in the future observations. \nIt is worth noting that 10 out of 43 UHE sources were not detected by WCDA in 1-25 TeV band. These sources exhibit dominant gamma-ray emissions at energies around a few tens of TeV or E > 100 TeV. Among them, sources like 1LHAASO J0216+4237u possibly belong to a new population of gamma-ray sources. These would demonstrate the distinctive importance of the UHE window at higher energy, which could explore new phenomena and new extreme celestial bodies of the Universe. With more accumulation of data, LHAASO will discover more sources with such features in the future. These sources should be also important candidates for PeVatron exploration and identification.', '5.2. TeV Sources Discovered by LHAASO': "As shown in Table 2, there are 25 sources without any known TeV source association. Additionally, 7 sources have conflicting extensions (with a difference of more than 0.5 · ) compared to the known TeV sources within the searching region. Therefore, tentatively, these 32 sources are announced as new TeV sources detected by LHAASO in this catalog. To determine the physical nature of these sources it is often necessary to analyze their spectral and morphological features, as well as the information about multi-wavelength associations. In this section, we conduct a preliminary analysis of the associations for each new TeV source. The majority of these newly discovered LHAASO sources are expected to be Galactic sources, due to their extended properties or their KM2A component detection. However, several point-like sources detected only by WCDA could potentially have an extra-galactic origin. It is known that there are a number of types of Galactic gamma-ray emitters, such as SNRs, PSRs and their PWNe, massive star clusters (MSCs), star-forming regions(SFRs), superbubbles, binaries, etc. The vast majority of detected Galactic TeV sources are likely to be associated with SNRs and PWNe (H. E. S. S. Collaboration 2018). To search for SNR and PWN associations, we take into account the most complete catalog of SNRs and PWNe to date, SNRcat 3 (Ferrand & Safi-Harb 2012). Although we do not expect pulsed TeV emission from a pulsar to be detected by LHAASO, the pulsar is a probe to illustrate the characteristic properties of an association SNR and/or PWN, and is also an indicator of a possible unseen SNR and/or PWN. For the pulsar associations, we used the web-based Australia Telescope National Facility Pulsar catalog (ATNF pulsar catalog 4 ). There is less association between low spin-down luminosity pulsars and known TeV sources, hence we apply a cut of ˙ E > 10 34 erg s -1 . Additionally, we search for associated high-energy (HE) gamma-ray sources in the 4FGL catalog (Abdollahi et al. 2020), which covers the sources at energies ranging from 0.1 GeV to > 500 GeV. Note that in this section we claim an association or a counterpart based on a position less than 0 . 5 · away. For the likely extragalactic sources, the AGN catalog is adopted to search for associations within the position error. The association results are listed in Table 3. \nSeven of the new TeV sources do not have any associations. They are tentatively classified as 'dark' sources in this work. We list and discuss these 'dark' sources briefly here: \n- · 1LHAASO J0007+5659u is a point-like source detected only by KM2A. It is an UHE source with a significance of TS 100 = 43.5 at energies above 100 TeV, implying a Galactic origin.\n- · 1LHAASO J0206+4302u and 1LHAASO J0212+4254u are the point-like sources only detected by KM2A. Due to both of them being UHE sources, we suggest a Galactic origin, although these two sources are obviously far from the Galactic plane with Galactic latitude b ∼ -17 · . As shown in Figure 13, these two sources and 1LHAASO J0216+4237u are close by and likely to be connected by bridges. Considering that they have a similar spectral shape (with the index of ≈ 2 . 5), we favor a physical association among these three 1LHAASO sources.\n- · 1LHAASO J1937+2128 is an extended source with the size of ≈ 1 . 3 · with the WCDA and KM2A components. Although 3HWC J1935+213 (0 . 36 · away) and 3HWC J1936+223 (0 . 85 · away) are found in point searches and are within the 1 · region around the center position of this 1LHAASO source, we report this source as a new detection due to its larger extension size.\n- · 1LHAASO J1959+1129u is a point-like TeV source detected above 25 TeV with TS = 90.2, and also detected as an UHE source with TS 100 = 59 . 3. We cannot find any pulsar, SNR or PWN counterparts within a 0 . 5 · region around its position. Interestingly, it is 0 . 23 · from the low mass X-ray binary (LXB) 4U 1957+11 which is a black hole candidate with the simplest and cleanest soft, disk-dominated spectra (Nowak et al. 2012) in the X-ray regime. The TeV emission from 11LHAASO J1959+1129u is possibly associated with LXB 4U 1957+11. A dedicated study of 1LHAASO J1959+1129u from our LHAASO group is ongoing.\n- · 1LHAASO J2200+5643u is an extended TeV source with the size of ∼ 0 . 5 · detected by WCDA and KM2A simultaneously. It is an UHE source with the significance of TS 100 = 38. The spectral index is ≈ 1 . 77 at energies ranging from ∼ 10 TeV to 30 TeV as measured by WCDA and is ≈ 3 . 44 at energies above 25 TeV as detected by KM2A, implying a peak or a break energy at a few tens of TeV.\n- · 1LHAASO J2229+5927u is a large extended source with the size ∼ 1 . 8 · detected by WCDA and by KM2A. Within the 1 · region centered at the position of the WCDA component, we cannot find any counterpart. The famous SNR G106.3+2.7, which has been extensively studied as a candidate PeVatron, is located at the edge of the extended region of 1LHAASO J2229+5927u. It is possible that this 1LHAASO source is the product of CRs escaping from SNR G106.3+2.7. More detailed study for this 1LHAASO source is ongoing by our collaboration. \nEight of the new TeV sources have GeV counterparts only. The details of these sources are as follows: \n- · 1LHAASO 0056+6346u is an extended source ( r 39 ≈ 0 . 3 · ) detected by WCDA and KM2A detectors simultaneously. It is also a UHE source with TS 100 = 94. The spectral shape is obviously curved with the index ≈ 2 . 35 by WCDA and ≈ 3 . 33 by KM2A. The unidentified point-like GeV source 4FGL J0057.9+6326 is 0.38 · from this 1LHAASO source.\n- · 1LHAASO J0500+4454 is an extended source with a size of ∼ 0 . 4 · , only detected by WCDA in this work. The extension characteristic supports a Galactic origin. An unidentified point-like GeV source 4FGL J0501.7+4459 is 0.32 · from this 1LHAASO source. \n- · 1LHAASO J1858+0330 is an extended source with a size of ≈ 0 . 5 · detected by WCDA with a significance of TS = 114 and by KM2A with a significance of TS = 299. It is located in the Galactic plane ( l = 36 . 8 · and b = 0 . 1 · ), towards which a complex gas distribution exists. The closest GeV source is 4FGL J1857.9+0313c which is identified as the a blazar candidate of uncertain type by the Fermi -LAT group. We exclude that the 1LHAASO source is associated with this extragalactic GeV source due to the extension characteristic. Another unidentified point-like GeV source 4FGL J1858.0+0354 is 0 . 41 · from this 1LHAASO source, and an association cannot be ruled out.\n- · 1LHAASO J1902+0648 is a point-like source with a significance of TS = 46.2 only detected by WCDA. The unidentified GeV source 4FGL J1902.5+0654 is 0 . 12 · from this 1LHAASO source.\n- · 1LHAASO J1924+1609 is an extended source with r 39 ≈ 1 . 3 · detected by WCDA and KM2A simultaneously. Although 3HWC J1923+169, found in a point search by the HAWC group, is 0 . 86 · from the center of the WCDA component, we announce 1LHAASO J1924+1609 as a new TeV source due to its large extended size. Within the 0 . 5 · region around the centroid of the WCDA component, there are three unidentified point-like GeV sources, i.e., 4FGL J1924.3+1601c, 4FGL J1925.4+1616 and 4FGL J1924.3+1628. Note that the positional offset between the WCDA component and the KM2A component is about 0 . 69 · with an uncertainty 0 . 66 · (at 95% confidence level). The pulsar PSR J1921+1544 ( ˙ E = 1 . 31 × 10 34 erg s -1 , τ c = 2320 . 0 kyr, d = 9 . 04 kpc) is 0 . 1 · from the position of the KM2A component.\n- · 1LHAASO J1931+1653 is a point-like source with a significance of TS = 51.8, only detected by KM2A. The unidentified GeV source 4FGL J1931.1+1656 is 0 . 05 · from this 1LHAASO source.\n- · 1LHAASO J2027+3657 is an extended source with r 39 ≈ 0 . 23 · only detected by KM2A. It is resolved from the previously published LHAASO source LHASSO J2018+3651, which is one of the brightest sources observed by LHAASO. The point-like GeV source 4FGL J2026.5+3718c is located within a 0 . 5 · region around this 1LHAASO source.\n- · 1LHAASO J2047+4434 is an extended TeV source with TS = 62.4, only detected by KM2A. A weak WCDA component with TS ≈ 20 is found but is not included in this catalog. The unidentified point-like GeV source 4FGL J02049.3+4440c is 0 . 32 · from this 1LHAASO source. \nSixteen of the new TeV sources have pulsar or PWN/SNR associations. Each source is briefly described as follows: \n- · 1LHAASO J0216+4237u is a point-like source located at high Galactic latitude b ≈ -17 · . Interestingly, 0 . 32 · from this source, an energetic millisecond pulsar PSR J0216+4238 ( ˙ E = 2 . 44 × 10 35 erg s -1 , τ c = 476 Myr, d = 3 . 15 kpc, p 0 = 2 . 3 ms) is located. The large position offset reduces the possibility of an association between 1LHAASO J0216+4237u and PSR J0216+4238. As mentioned above, we favor a physical association between 1LHAASO J0216+4237u, 1LHAASO J0206+4302u and 1LHAASO J0212+4254u. More details will be discussed in a forthcoming report.\n- · 1HAASO J0249+6022 is an extended source with r 39 ≈ 0 . 71 · for the WCDA component and with r 39 ≈ 0 . 38 · for the KM2A component. We can find that the KM2A component overlaps \n- the extended region of the WCDA component, with a positional offset of ∼ 0 . 45 · . The WCDA component at energies 1 -20 TeV could plausibly include the emission of two TeV sources, resulting in the large position offset compared to the KM2A component. In our searching radius, we just find one pulsar counterpart (PSR J0248+6021, d = 2 . 0 kpc ˙ E = 2 . 13 × 10 35 erg s -1 , τ c = 62 . 4 kyr, p 0 = 217 . 1 ms), 0 . 16 · from the position of the KM2A component. The 1LHAASO source is likely to correspond to an astrophysical system associated with PSR J0248+6021, such as a composite SNR.\n- · 1LHAASO J0359+5406 is an extended source with a size of ≈ 0 . 3 · detected by WCDA and KM2A simultaneously. Two energetic pulsars, PSR B0355+54 ( d = 1 kpc, ˙ E = 4 . 45 × 10 34 erg s -1 , τ c = 564 kyr, p 0 = 156 . 4 ms) and PSR J0359+5414 ( d is unknown, ˙ E = 1 . 32 × 10 36 erg s -1 , τ c = 75 . 2 kyr, p 0 = 79 . 4 ms) are 0 . 12 · and 0 . 16 · away, respectively, from the position of the KM2A component. The X-ray observations have revealed the presence of a ∼ 30 '' compact nebula and a fainter tail of emission visible up to ∼ 6 ' southwest of PSR B0355+54 (Klingler et al. 2016), and of ∼ 30 '' diffuse X-ray emission extended roughly along the SE-NE direction around PSR J0359+5414 (Zyuzin et al. 2018). Thus, this 1LHAASO source is likely to be a TeV PWN powered by PSR B0355+54 or PSR J0359+5414. On the other hand, it could be a TeV halo associated with the middle-aged pulsar PSR B0355+54. It is worth noting that the HAWC collaboration also reported their detection at the same time (Albert et al. 2023).\n- · 1LHAASO J0428+5531 is an extended source detected by WCDA and KM2A detectors simultaneously. The WCDA component is with r 39 ≈ 1 . 18 · . We can find that the shell-type SNR G150.3+04.5, with a size of 3 . 0 · × 2 . 5 · (Gao & Han 2014), is 0 . 26 · from the center of the WCDA component. In the GeV band, an extended source 4FGL J0426.5+5522e modeled as a 1 . 515 · disk is also associated with SNR G150.3+04.5 (Ajello et al. 2017). The WCDA component of 1LHAASO J0428+5531 is possibly of SNR origin due to the comparable location and extension size in the multi-wavelength observation. The extended KM2A component ( r 39 ≈ 0 . 23 · ) is ≈ 0 . 97 · from the central position of the WCDA component and ≈ 0 . 1 · away from the radio west shell of SNR G150.3+04.5 (with a size of 64 . 1 ' × 18 . 8 ' , also named SNR G150.8+03.8 in Gerbrandt et al. 2014). A point-like GeV source 4FGL J0426.5+5434, which is possibly a pulsar candidate due to the curved SED and the cutoff energy at a few GeV, is 0 . 06 · from the position of KM2A component. Whether the KM2A component of 1LHAASO J0428+5531 is associated with the SNR G150.8+03.8 needs more study due to the obvious offset of the position and the extension size.\n- · 1LHAASO J0534+3533 is a point-like source detected by WCDA and KM2A in this work, which is near the centroid position of the shell-type SNR G172.8+01.5, 0 . 3 · away. However, the radio shell size of SNR G172.8+01.5 is estimated as 4 . 4 · × 3 . 4 · , which is much larger than the TeV emission size of 1LHAASO J0534+3533. If the radio shell is physically associated with 1LHAASO J0534+3533, the TeV emission is likely to be a product of the central pulsar wind nebula.\n- · 1LHAASO J1814-1636u is an extended UHE source with the size of ≈ 0 . 68 · , only detected by KM2A. Only two SNRs, i.e., G14.1-00.1 (0 . 43 · away) and SNR G014.3+00.1 (0 . 31 · away) are found in our searching catalog. However, both SNR candidates are with < 0 . 2 · radio shell \nsize, which is inconsistent with that of gamma-ray emission. An unidentified GeV source 4FGL J1816.2-1654c is found at 0 . 4 · away from the position of the TeV emission. \n- · Source 1LHAASO J1831-1028 only has an extended KM2A component and with the size of ≈ 0 . 94 · . In our association procedure, we just find three SNR candidates, i.e., G021.0-00.4 (0 . 3 · away), G021.5-00.1 (0 . 36 · away) and G021.8-00.6 (0 . 49 · away), all of which are with a radio size of < 0 . 2 · . Due to the larger conflict between the size of TeV emission and that of radio shell, we disfavor that the same population of electrons produces the TeV emission and the radio emission.\n- · 1LHAASO J1852+0050u is an extended source with a ∼ 0 . 64 · WCDA component and ∼ 0 . 85 · KM2A component. The nearest known TeV source is 2HWC J1852+013* (0 . 55 · away), which is suffers from GDE impact and is without an extended size measurement. We claim this 1LHAASO source as a new TeV source due to its large extension. The middle-aged pulsar, PSR J1853+0056 ( d = 3.84 kpc, ˙ E = 4 . 03 × 10 34 erg s -1 , τ c = 204 kyr) is the nearest pulsar, 0 . 31 · from the position of the KM2A component. The 1LHAASO source is possibly a TeV halo associated with this middle-aged pulsar. On the other hand, an extended GeV source 4FGL J1852.4+0037e (namely Kes 79), modeled by 0 . 63 · disk shape and identified as the SNR or PWN type by the Fermi -LAT group, is in agreement with the position and size of the 1LHAASO source. In addition, the SNRcat source G033.6+00.1, which is a small extended radio source with the size of 10 ' , is suggested to be associated with the GeV source.\n- · 1LHAASO J1906+0712 is an extended source only detected by WCDA with the size of ∼ 0 . 21 · . A gamma-ray pulsar PSR J1906+0722 ( ˙ E = 1 . 02 × 10 36 erg s -1 , τ c = 49 . 2 kyr, d is unknown) is 0 . 19 · from this 1LHAASO source. 0 . 34 · away, a shell-type SNR G041.1-00.3 (3C 397) is detected in the radio band with the size of 4 . 5 ' × 2 . 5 ' , which is also detected by Fermi -LAT in the GeV band. The pulsar associated with SNR 3C 397 is not identified. An unidentified GeV source 4FGL J1906.9+0712 is also found within the 0 . 5 · region of 1LHAASO J1906+0712.\n- · 1LHAASO J1928+1813u is an extended source with r 39 ≈ 0 . 63 · only detected by KM2A. It is resolved from the UHE source LHAASO J1929+1745 seen in previous LHAASO results. Within the 0 . 5 · region around the centroid of 1LHAASO J1928+1813u, we can find the source SNR G053.4+00.0 (0 . 39 · away), which is a shell-type SNR with the radio size of 5 ' at a distance 5 . 6 -6 . 4 kpc. Two GeV sources listed in 4FGL are found within the association region. At ∼ 0 . 47 · away, there is an energetic pulsar PSR J1928+1746 ( d =4.34 kpc, ˙ E = 1 . 6 × 10 36 erg s -1 , τ c = 82 . 6 kyr).\n- · 1LHAASO J1954+3253 is an extended TeV source ( r 39 ≈ 0 . 17 · ), only detected by WCDA with the significance of TS = 144. SNR G069.0+02.7 (also named CTB 80) with the size ∼ 80 ' overlaps this 1LHAASO source. SNR CTB 80 is an old SNR with a kinematic distance of ∼ 1 . 5 kpc. At the center of SNR CTA 80, the pulsar PSR B1951+32 ( ˙ E = 3 . 74 × 10 36 erg s -1 , τ c = 107 . 0 kyr, d = 3 . 00 kpc) is generally regarded as the compact object associated with this old SNR. A X-ray PWN around the pulsar PSR B1951+32 has been identified by ROSAT, which is shown as a 5 arcmin extended nebula. The pulsar and the SNR/PWN are also detected by Fermi -LAT, which are 4FGL J1952.9+3252 and 4FGL J1955.1+3321, respectively. We note that the position of TeV emission is ∼ 0 . 33 · from that of the pulsar and its X-ray PWN, for \nwhich more study is needed to confirm the physical association between the 1LHAASO source and CTB 80. \n- · 1LHAASO J1956+2921 is a large extended source with an r 39 ≈ 0 . 99 · WCDA component and r 39 ≈ 0 . 78 · KM2A component. It is resolved from the published LHAASO source LHAASO J1958+2845. At 0 . 36 · from the position of the WCDA component, a shell type SNR with radio size of 31 ' × 25 ' is found.\n- · 1LHAASO J1959+2846u is an UHE TeV source with the extension size of r 39 ≈ 0 . 3 · , only detected by KM2A, which is also resolved the from previously published source LHAASO J1956+2845. The pulsar PSR J1959+2846 (0 . 1 · away, ˙ E = 3 . 42 × 10 35 erg s -1 , d = 1 . 95 kpc, τ c = 21 . 7 kyr) is the only pulsar counterpart in our searching radius. SNR G065.8-00.5 and SNR G066.0-00.0 are found at 0 . 16 · and 0 . 39 · from the position of 1LHAASO J1959+2846u, with radio sizes of 10 ' × 6 ' and 30 ' × 25 ' , respectively.\n- · 1LHAASO J2002+3244u is a point-like source with TS=74.0 as detected by WCDA and with TS = 43.6 as detected by KM2A. It is also an UHE source with TS 100 = 28.1. 4FGL J2002.3+3246 is spatially coincident with this source, which is identified as a potential association with a SNR or PWN by the Fermi -LAT group. A shell type SNR G069.7+01.0 is possibly associated with 1LHAASO 2002+3244u (0 . 04 · away). We favor that 1LHAASO 2002+3244u has an SNR origin because the position and size of TeV emission agree with the radio shell of SNR G069.7+01.0.\n- · 1LHAASO J2028+3352 is a large extended source with r 39 ≈ 1 . 6 · , only detected by KM2A. 0 . 36 · from their centroid position, a middle-aged pulsar PSR J2028+3332 ( ˙ E = 3 . 48 × 10 34 erg s -1 , τ c = 576 . 0 kyr) is found, implying a possible TeV halo identification. Two GeV sources are found within the 0 . 5 · region around the 1LHAASO source, of which one is the pulsar PSR J2028+3332 and the other is an unidentified point-like source.\n- · 1LHAASO J2238+5900 is an extended source with the size of ≈ 0 . 51 · as detected by WCDA with TS = 110.2 and with the size of ≈ 0 . 44 · detected by KM2A with TS= 361. Just one pulsar PSR J2238+5903 ( ˙ E = 8 . 99 × 10 35 erg s -1 , d = 2 . 83 kpc, τ c = 26 . 6 kyr) is found within 0 . 5 · region of this source, at 0 . 07 · from the centroid position of the KM2A component. The young pulsar and the size of TeV emission shrinking with increasing energy support that 1LHAASO J2238+5900 has a PWN origin. \nOne possible new extragalactic source is as follows: \n- · 1LHAASO J1219+2915 is a point-like source, only detected by WCDA with a significance of TS = 49.2, and located at high Galactic latitude ( b ∼ 82 . 5 · ). We cannot find any pulsar or SNR/PWN counterpart associated with this source. It is a likely extragalactic source due to the high Galactic latitude and null detection by KM2A. The closest AGN counterpart is the LINER-type AGN NGC 4278, 0 . 05 · from the 1LHAASO source. It is the most possible association even though we can not firmly identify this 1LHAASO source at present.", '5.3. Pulsar Wind Nebulae or TeV Halos in the 1LHAASO Catalog': "Pulsars left behind from supernova explosions are rapidly spinning neutron stars with extremely strong magnetic fields. The pulsed gamma-ray emission dominates at GeV energies, and only a few \ncases can extend to VHE. At VHE, the steady emission is from a PWN produced by the termination of the ultra-relativistic wind. PWNe constitute one of the largest VHE source populations within the Galaxy. The Crab nebula PWN has also been identified as an electron PeVatron (LHAASO Collaboration 2021). The diffusion of escaping particles from a PWN would leads to extended VHE gamma-ray emission, which has been extensive discussed as a 'TeV halo' after the discoveries of several cases (Abeysekara et al. 2017; Aharonian et al. 2021c). TeV halos are thought to form around middle-aged pulsars with ages of at least several tens of thousands of years. PWNe or TeV halos should also contribute a significant fraction of the 1LHAASO sources, and an effective method to check for this is to search for spatial coincidence with the known pulsars. \nTo search for pulsars associated with 1LHAASO sources, the ATNF pulsar catalog is adopted. For the hunting of associations in astronomy, an important work is to estimate the possibility of spatially coincidence by accident. A similar method as that used in Mattox et al. (1997) is adopted here. According to a previous empirical result at VHE, the identified PWN or halo type VHE sources are always associated with pulsars with high spindown power. Therefore, for each pulsar within 0.5 · of a 1LHAASO source, the chance probability P c is estimated according to the space angle r and pulsar spindown power ˙ E using \nP c = 1 -e r 2 /r 2 0 , (1) \nWhere r 0 is the characteristic angle between confusing sources, \nr 0 = [ πρ ( ˙ E )] -1 / 2 , (2) \nand where ρ ( ˙ E ) is the number density of pulsars with ˙ E not lower than that of the candidate pulsar. The ρ ( ˙ E ) is counted using the pulsars nearby the candidate pulsar with Galactic latitude | b -b c | < 2 . 5 · and longitude | l -l c | < 10 · , where ( b c , l c ) denotes the Galactic coordinates of the candidate pulsar. \nWith this searching, 65 1LHAASO sources are found with at least one pulsar nearby within 0.5 · . To decrease the fake association by accident, the pulsars with chance probability higher than 1% are excluded. After this filter, 35 1LHAASO sources are found with one associated pulsar each and 2 1LHAASO sources, 1LHAASO J0359+5406 and 1LHAASO J1929+1846, are found with two associated pulsars each. For the source with two associations, the associated pulsar with a lower chance probability is listed. There are also two pairs of 1LHAASO sources, 1LHAASO J1848-0001u vs. 1LHAASO J1850-0004 and 1LHAASO J2020+3638 vs. 1LHAASO J2020+3649u, associated with the same pulsar. Again, the 1LHAASO source with a lower chance probability is listed. Finally, 35 associated pulsars are derived for the 1LHAASO sources. Detailed information about these associations is listed in Table 4. \nFigure 10 shows the spindown power versus age of the associated pulsars. For comparison, all the pulsars of the ATNF catalog within LHAASO FOV are also shown in the figure. As expected, the TeV-gamma associated pulsars are among that with the most energetic spindown power ( ˙ E > 10 34 erg s -1 ). Among these associated pulsars, the ages of 24 pulsars are less than 100k years. The corresponding 1LHAASO sources have a high possibility to be PWNe. The ages of 11 pulsars are older than 100k years, and this marks the corresponding 1LHAASO sources to possibly be PWNe/TeV halos. It is worth noting that some of them have already been identified as PWNe or PWN/TeV halos in the TeVCat, as marker in the Table 4. An exciting result is that one pulsar, i.e., PSR J0218+4232, is a millisecond pulsar, which has been expected to produce VHE emission but there has been a lack of observation evidence. The spatial coincidence between this millisecond pulsar and 1LHAASO \nJ0216+4237u is notable, with a confidence level of 2.9 σ . This finding favor the existence of VHE emission around millisecond pulsars, although a non-physical association cannot be definitively ruled out. \nFigure 10. Pulsar spindown power ˙ E versus age for all pulsars of the ATNF catalog within the FOV of LHAASO and the 1LHAASO-associated pulsars with the chance probability P c less than 0.01. \n<!-- image --> \nAccording to Figure 10, most of the energetic pulsars with ˙ E higher than 10 36 erg s -1 . within the FOV of LHAASO are associated with 1LHAASO sources. These shows that the PWNe of energetic pulsars are promising as effective to VHE gamma-ray emitters. It is worth noting that no VHE or UHE emission was found from a handful of energetic pulsars with ˙ E higher than 10 37 erg s -1 . according to Figure 10. This shows that emission of PWNs may also be affected by other parameters of the pulsars or the surrounding environment, besides the spindown power. Among the 35 1LHAASO sources with pulsar associations, twenty-two are labeled as UHE sources. This result suggests that PWNe also contribute a significant portion of the UHE sources and acceleration of electrons close to or up to 1 PeV is common for energetic PWNe. In addition to the Crab Nebula, more electron PeVatrons may be revealed by LHAASO in the future.", '6. SUMMARY': "The 1LHAASO catalog, with 90 VHE gamma-ray sources, is the first comprehensive search conducted using data from the LHAASO observatory. The catalog utilizes 508 days of data from the full WCDA and 933 days of data from the full KM2A and its partial array. With 32 sources discovered, this survey is the most sensitive and roughly unbiased investigation of large sky regions in the VHE band, covering a declination range from -20 · to 80 · . Seven new sources do not have counterparts in the GeV gamma-ray, pulsar, and SNR catalogs, and are referred to as 'dark sources'. Among the 90 1LHAASO sources, sixty-nine are detected with emissions in the energy range 1-25 TeV, while seventy-five are detected at emission energy of E > 25 TeV, both with a significance above \n5 σ . The catalog provides information on the extension and spectrum of each source, with 65 sources exhibiting clear extension. To be conservative and avoid confusion with diffuse gamma rays from the Galactic plane, only sources with extensions less than 2 · are included in the 1LHAASO catalog. Additionally, 43 sources exhibit clear UHE emission, with a significance above 4 σ at E > 100 TeV. This significantly expands the current number of known UHE sources, by a factor of 3, and suggests that most Galactic VHE sources are potential UHE sources, indicating a prevalence of PeV particle accelerators in the Milky Way. Ten out of the forty-three UHE sources are not detected at the 1-25 TeV range, possibly representing a new class of gamma-ray sources dominated by emission above tens of TeV. Thirty-five 1LHAASO sources are associated with energetic pulsars, with a chance coincident probability of less than 1%. Apart from sources already identified as PWNe or TeV halos, the remaining sources associated with energetic ( ˙ E > 10 36 erg s -1 ) pulsars are likely PWNs or TeV halos. The majority of energetic pulsars within the LHAASO field of view are associated with 1LHAASO sources, and a significant fraction of these sources are also UHE sources. This suggests that PWNe associated with energetic pulsars are effective emitters of VHE and UHE radiation and have a general capability to accelerate electrons close to or up to PeV energies. \nWith a large FOV and full duty cycle, the full LHAASO detector configuration has been monitoring the sky since July 2021 and will continue operations for the next 20 years. With the the accumulation of data, the sensitivity in both the VHE and UHE bands will improve. Further optimization of data analysis and refining detector response simulation are ongoing, which may also expand its capacity. Deep and multi-wavelength analysis focusing on the 1LHAASO sources one by one is also ongoing. Therefore, LHAASO is expected to unveil more new discoveries and provide a deeper understanding of the VHE and UHE universe in the near future.", '7. ACKNOWLEDGMENTS': 'Wewould like to thank all staff members who work at the LHAASO site above 4400 meters above sea level year-round to maintain the detector and keep the water recycling system, electricity power supply and other components of the experiment operating smoothly. We are grateful to Chengdu Management Committee of Tianfu New Area for the constant financial support for research with LHAASO data. This research work is supported by the following grants: The National Key R&D program of China No.2018YFA0404201, No.2018YFA0404202, No.2018YFA0404203, No.2018YFA0404204, National Natural Science Foundation of China No.12022502, No.U1831208, No.12205314, No.12105301, No.12261160362, No.12105294, No.U1931201, No.12005246, No.12173039, Department of Science and Technology of Sichuan Province, China No.2021YFSY0030, Project for Young Scientists in Basic Research of Chinese Academy of Sciences No.YSBR-061, and in Thailand by the NSRF via the Program Management Unit for Human Resources & Institutional Development, Research and Innovation (No. B37G660015).', '8. AUTHOR CONTRIBUTIONS': 'S.Q. Xi and S.Z. Chen led the drafting of text and performed the data analysis of KM2A, S.C. Hu and M. Zha conducted the data analysis of WCDA. Y.Y. Guo and J.Y. He provided the cross-check. G.M. Xiang illustrated some figures in this article. Zhen Cao, the spokesperson of the LHAASO Collaboration, coordinated the specific working group for this paper involving all corresponding authors. All other authors participated in data analysis, including detector calibration, data processing, \nevent reconstruction, data quality checks, and various simulations, and provided comments on the manuscript.', 'REFERENCES': "Table 2 . 1LHAASO source catalog \nTable 2 continued on next page \nTable 2 (continued)Table 2 continued on next page \nTable 2 (continued)Table 2 continued on next page \nTable 2 (continued) \nTable 2 continued on next page \nTable 2 (continued) \nNote -The first column lists the LHAASO catalog name. The name designation is 1LHAASO JHHMM+DDMM, where the 1 refers to this being the first LHAASO catalog, JHHMM+DDMM is according to the source location. For a source observed by both WCDA and KM2A, the source name corresponds to the position of that with higher significance level. If the source is an UHE source with TS 100 > 20, we add 'u' at the end of the name, i.e., 1LHAASO JHHMM+DDMMu. For dubiously merged sources, we label the names with '*'. \nNote -The 2nd column lists the component name, represented by the detector name. If the component suffers a significant GDE impact in our test, we label the component with '*'. \nNote -The 3rd -5th columns list the position parameters for the source components. α 2000 and δ 2000 are the right ascension and declination at J2000.0 epoch. σ p, 95 ,stat is the 95% statistical uncertainty of the source component position. The units are degrees. \nNote -The 6th column lists the extension of the source components. r 39 is the 39% containment radius of the 2D-Gaussian model. For the point-like source components, the 95% statistical upper limits (i.e., < r 39 ,ul,stat ) are shown. \nNote -The 7th column lists the TS values of the source components. It is distributed as χ 2 with 5 dof. \nNote -The 8th -9th columns list the parameters of the power-law SED of the source components. The power-law shape is defined by dN/dE = N 0 ( E/E 0 ) -Γ , where N 0 is the differential flux, E 0 is the reference energy which is 3 TeV for the WCDA component and 50 TeV for the KM2A component, and Γ is the photon spectral index. N 0 is in a units of 10 -13 cm -2 s -1 TeV -1 and 10 -16 cm -2 s -1 TeV -1 for the WCDA the KM2A components, respectively. For a source with single component detection, we give the 95% statistical upper limits of N 0 of the non-detected component with the same position, extension and photon index. \nNote -The 10th column lists the TS values of the UHE sources at energies E > 100 TeV. \nNote -The 11th column lists the closest known TeV source counterpart within the searching region (as described in Sec. 3.5). The angular separation between the TeV counterpart and the LHAASO source is shown in parentheses. \nFigure 11. LHAASO significance map within the region 10 · ≤ l ≤ 115 · , | b | ≤ 12 · . Top: WCDA (1 TeV < E < 25 TeV) significance map. Middle: KM2A ( E > 25 TeV) significance map. Bottom: KM2A ( E > 100 TeV) significance map. In this figure and following, the LHAASO source are represented by gray crosses and white labels. The LHAASO sources for which the WCDA component has higher significance are plotted in the top panel. The LHAASO sources for which the KM2A component has higher significance are plotted in the middle panel. Meanwhile, UHE sources are shown again in the bottom panel. \n<!-- image --> \nl [ \n° \n] \nFigure 12. LHAASO significance map within region 115 · ≤ l ≤ 220 · , | b | ≤ 12 · .Top: WCDA (1 TeV < E < 25 TeV) TeV significance map. Middle: KM2A ( E > 25 TeV) significance map. Bottom: KM2A ( E > 100 TeV) significance map. \n<!-- image --> \nFigure 13. LHAASO significance map for eight sources with | b | > 12 · . For each source, WCDA (1 TeV < E < 25 TeV) and KM2A ( E > 25 TeV) significance map are shown in top and bottom map, respectively. \n<!-- image --> \n2000[ \n] \nTable 3 . Associations of the New TeV Sources in 1LHAASO Catalog \nTable 3 continued on next page \nTable 3 (continued) \nNote -In the description of the 4FGL counterparts, 'unk' represents | b | < 10 sources solely associated with the likelihood-ratio method from large radio and X-ray surveys, 'bcu' is a blazar candidate of uncertain type, 'spp' is a supernova remnant or pulsar wind nebula, 'PSR' is a gamma-ray pulsar identified by pulsations, and 'snr' is the Supernova remnant (Abdollahi et al. 2022). \nTable 4 . 1LHAASO sources associated pulsars"}

2024arXiv240913184G

Spontaneous scalarization of black holes typically occurs through the condensation of a scalar field with the field evolving from a U1symmetric phase into a symmetrybreaking one with lower energy. We show that there exist symmetrybreaking phases which are themselves unstable to the formation of an additional scalar condensate or cloud which is partly accreted into the black hole. By studying the fully nonlinear dynamical evolution of the process we find that symmetry breaking causes the accretion channels of scalar clouds to be nondegenerate favoring a dominant channel for evolution. Additionally the final states form a characteristic energy band due to varying amounts of radiation emitted by clouds in different channels.

2024-09-01T00:00:00Z

['arXiv:2409.13184', '2024arXiv240913184G', '10.48550/arXiv.2409.13184']

['General Relativity and Quantum Cosmology', 'High Energy Physics - Theory']

Black hole accretion of scalar clouds with spontaneous symmetry breaking

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.13184.pdf

{'Black hole accretion of scalar clouds with spontaneous symmetry breaking': "Sebastian Garcia-Saenz, 1 Guangzhou Guo, 1 Peng Wang, 2 and Xinmiao Wang 1 \n1 Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China 2 Center for Theoretical Physics, College of Physics, Sichuan University, Chengdu 610064, China \nSpontaneous scalarization of black holes typically occurs through the condensation of a scalar field, with the field evolving from a U (1)-symmetric phase into a symmetry-breaking one with lower energy. We show that there exist symmetry-breaking phases which are themselves unstable to the formation of an additional scalar condensate, or 'cloud', which is partly accreted into the black hole. By studying the fully nonlinear dynamical evolution of the process, we find that symmetry breaking causes the accretion channels of scalar clouds to be non-degenerate, favoring a dominant channel for evolution. Additionally, the final states form a characteristic energy band due to varying amounts of radiation emitted by clouds in different channels.", 'I. INTRODUCTION': "The past decade has witnessed spectacular progress in the field of black hole physics, driven primarily by the detection of gravitational waves from binary black hole mergers, which opened up new avenues for exploring the properties of black holes [1]. Prime among these is the testing of the validity of the black hole uniqueness theorems of general relativity (see [2] for a review), particularly through the measurement of quasinormal modes during the ringdown phase [3, 4]. The Event Horizon Telescope has similarly revolutionized our understanding of black holes by capturing the images of M87* and Sgr A* [5, 6], revealing a striking feature: a luminous ring encircling a dark shadow. These distinctive signatures have been attributed to the intense light deflection occurring near unstable bound photon orbits, providing insights into black hole physics in the strong field regime, e.g. the structure of luminous accretion disks [7-11]. \nWhile these phenomena align with the predictions of general relativity [12-14], future observations of increased precision offer the exciting prospect of measuring deviations from Einstein's gravity interacting with standard matter [15]. For instance, the no-hair theorem [16, 17] may be circumvented in the presence of a scalar field that couples non-minimally either to gravity or to other matter fields [18-20]. In this well-studied setup, black holes are subject to the growth of 'hair' in the form of a scalar cloud, i.e. a nonstatic, but potentially longlived, condensate of the field in the vicinity of the event horizon. 1 Although originally studied in the context of neutron stars [25], this so-called spontaneous scalarization phenomenon has been shown to also arise in black \nhole spacetimes within scalar-tensor theories of gravity, the best-known model being scalar-tensor-Gauss-Bonnet theory and its extensions [26-30]. \nOn the other hand, it has been appreciated that scalarization may occur in a simpler class of models, which feature no higher curvature terms and only minimal gravitational couplings, but they require the presence of an additional matter field with which the scalar interacts non-minimally. A very natural choice in this class is the Einstein-Maxwell-scalar (EMS) theory where the scalar field Φ couples to the standard Einstein-Maxwell Lagrangian through the term f ( | Φ | ) F µν F µν , with F µν the electromagnetic field strength and f some function [31] (see also [32] for earlier work in the context of nonlinear electrodynamics). The spontaneous scalarization effect in the EMS model has been further studied in [33-36], while various properties of these scalarized black holes have been investigated in [37-40], along with their spinning counterparts [41, 42]. \nScalarization in the EMS model occurs dynamically through a tachyonic (i.e. negative effective potential) instability of an initial scalar field perturbation on the background of a hairless black hole, driving the system into a new configuration with lower energy accompanied by a non-trivial scalar field distribution. In this paper we ignore spin, so the initial state is a Reissner-Nordstrom (RN) black hole; notice that electric charge is needed to trigger the destabilization effect [31]. The appearance of a nontrivial scalar profile, or condensate, may be naturally understood as a process of spontaneous symmetry breaking. In the original setup, the scalar field is real, so the symmetry that is broken by the scalarization effect is Z 2 , or Φ ↦→ -Φ. Here we focus instead on the case of a complex scalar as it provides a more interesting symmetry breaking pattern, namely that of a (global) U (1) symmetry. The study of spontaneous scalarization in this scenario has been pioneered in [43, 44]. \nIn this paper, we employ numerical relativity to investigate the fully nonlinear dynamics of the EMS model, specifically the spontaneous scalarization of a charged black hole and the subsequent evolution of the scalar cloud. After its initial formation following the tachyonic destabilization, we observe that the cloud continues to \nevolve, on a timescale of order the distance between the cloud and the event horizon, in a process of energy loss both to accretion into the black hole and ejection to spatial infinity. We find essential differences between the scenario where scalarization occurs starting in a symmetric phase, i.e. without further instabilities upon formation of a scalar condensate, versus the case where a scalar cloud forms from a symmetry-broken phase. Unlike with scalarization from the symmetric phase, the breaking of U (1) symmetry allows for multiple accretion channels for the cloud, each leading to different amounts of energy loss and distinct final states. Notably, a dominant accretion channel emerges due to symmetry breaking.", 'II. SCALARIZATION': 'We consider the EMS model of [31], extended here to a complex scalar field Φ = ϕ 1 + iϕ 2 , which is non-minimally coupled to the electromagnetic field A µ . The complete action is (we use geometrized units with G = c = 1) \nS = 1 16 π ∫ d 4 x √ -g [ R -2 ∂ µ Φ ∗ ∂ µ Φ -f ( | Φ | ) F µν F µν ] , (1) \nwhere F µν = ∂ µ A ν -∂ ν A µ . In this paper, we adopt f ( | Φ | ) = e α | Φ | 2 for the non-minimal coupling function, with α a real constant, a choice which admits an electrovacuum solution given by a RN black hole and Φ = 0. 2 Note that we do not include a minimal coupling of the scalar field to electromagnetism, i.e. Φ is uncharged under the electromagnetic U (1), unrelated to the global U (1) symmetry Φ ↦→ e iφ Φ. The electrovacuum solution preserves the U (1) symmetry, so we refer to it as the symmetric phase. \nA small scalar perturbation δ Φ on the RN black hole background obeys the equation ( ✷ -µ 2 eff ) δ Φ = 0, where µ 2 eff ≡ -αQ 2 /r 4 ( r is the radius in standard spherical coordinates) and Q is the charge of the hole. If α > 0, the negative value of µ 2 eff introduces a tachyonic effect, potentially destabilizing the background and triggering scalarization. \nMore in detail, and considering now a general static and spherically symmetric ansatz for the metric, ds 2 = -e -2 δ ( r ) N ( r ) dt 2 + dr 2 /N ( r )+ r 2 d Ω 2 , one finds the equation d 2 Ψ dx 2 + ( ω 2 -V eff ( x ) ) Ψ = 0 obeyed by the spherical, or monopole, perturbation Ψ ≡ rδ Φ, where ω is the frequency, x is defined by dx/dr ≡ e δ ( r ) /N ( r ) and \nFIG. 1. Scalar field profiles (dashed lines) and effective potentials (solid) for Q/M = 0 . 55 RN (green) and Q/M = 1 . 29 scalarized (purple) black holes, respectively. The event horizons are set at r h = 0 . 2. Both the RN and scalarized black holes (SBH) possess negative potential wells. In the latter case, the well is separated from the horizon by a potential barrier, and its location coincides with the node of the scalar field. \n<!-- image --> \n̸ \nV eff is the effective potential. 3 In Fig. 1, the effective potential for the U (1)-symmetric phase is seen to display a negative well near the event horizon, and a numerical calculation shows that a tachyon mode with ω = 0 . 2454 iM is present. In agreement with expectations, this mode is responsible for triggering spontaneous scalarization [31, 45]. Intriguingly, we find that there exist U (1)-broken phases (scalarized black holes with Φ = 0), which also exhibit a negative potential well, a feature that is linked to the presence of a node in the scalar profile. This suggests that these symmetrybreaking phases are unstable to the formation of a scalar cloud localized around the potential well. Our aim in this paper is to advance the investigation of the nonlinear dynamics of scalar clouds to encompass both U (1)symmetric and U (1)-broken phases.', 'III. DYNAMICAL FORMATION OF SCALAR CLOUDS': "To simulate the fully nonlinear evolution in the EMS model, we use the Baumgarte-Shapiro-ShibataNakamura (BSSN) formulation implemented in BlackHoles@Home [46]; see [47-52] for references and Appendix A for some technical details and numerical convergence tests. To ensure consistent numerical results, \nV eff = N r 2 e 2 δ 1 -N -2 r 2 ∣ ∣ Φ ' ∣ ∣ 2 -Q 2 ( 1 + α -2 | α Φ -r Φ ' | 2 ) r 2 e α | Φ | 2 . \nFIG. 2. Dynamical formation of a scalar cloud for the initial perturbation with θ 0 = π/ 2 in the U (1)-symmetric ( M = 2 . 5, leftmost column) and U (1)-broken (Φ h = 0 . 19 and M = 0 . 7744, right columns) phases. A scalar cloud develops within the negative potential well due to scalarization and is subsequently accreted onto the black hole, terminating in the formation of a stable cloud structure around it. \n<!-- image --> \nwe also perform nonlinear simulations with the Einstein Toolkit [53]. Given our focus on spherically symmetric evolution, we have the electromagnetic field A µ = A ( t, r ) δ t µ (choosing the gauge A r = 0), and from the equations of motion one infers ∂ µ ( √ -ge α Φ ∗ Φ ∂ r A t ) = 0, indicating that the black hole charge Q , defined by the integration constant, is conserved during temporal evolution. For our numerical study, we set units such that Q = 1 and α = 160. 4 For the initial data, we introduce a scalar perturbation δ Φ = pe -( r -r 0 ) 2 ∆ 2 e iθ 0 around both the U (1)-symmetric and U (1)-broken phases. Here, θ 0 is the phase factor of the perturbation, the amplitude is chosen as p = 10 -6 , the location r 0 is set around the negative potential well, and the width is chosen as ∆ = 1. For latter \nFIG. 3. Dynamical evolution of the field norm | Φ | for initial perturbations with θ 0 = 0 (upper row) and θ 0 = π (lower) in the symmetry-broken phase (Φ h = 0 . 19 and M = 0 . 7744). The scalar cloud formed from the positive perturbation ( θ 0 = 0) can induce nonlinear effects at late times, causing the scalar field node to propagate towards spatial infinity. Conversely, the negative perturbation ( θ 0 = π ) leads to the node of the field moving into the event horizon at late times, resulting in a stable, nodeless scalar distribution outside the black hole. \n<!-- image --> \nuse, we define Φ h to be the (time-dependent) value of the scalar field Φ at the event horizon. At the background level, i.e. prior to the introduction of the above perturbation δ Φ, the U (1)-breaking system is chosen, without loss of generality, to have a real and positive value of Φ. \nIn Fig. 2, an initial wavepacket with phase θ 0 = π/ 2, perturbing only the field component ϕ 2 , is placed within the negative potential well. In the symmetric phase, the field amplitude | Φ | grows near the event horizon, forming a scalar cloud within the potential well. Ultimately, a long-lived cloud accretes onto the black hole, with the evolution terminating in a scalarized black hole, consistent with the results of [31, 45]. In the U (1)-broken phase, the negative potential well is separated from the horizon of the scalarized black hole (with Φ h = 0 . 19 in our simulation). The perturbation grows via tachyonic instabilities, forming a ring structure for ϕ 2 around the well (see the rightmost column). After approximately t ≈ 20 in our numerics, the cloud becomes massive enough and is drawn into the black hole by the gravitational pull. As a result, the cloud component ϕ 2 is gradually accreted onto the hole, forming a lighter cloud that remains near the horizon in equilibrium, i.e. a hairy black hole. During this process, the node of the scalar field | Φ | gradually fades as the ϕ 2 cloud forms and eventually disappears. \nWhile in the symmetric phase the dynamics remain U (1)-equivalent, in the symmetry-broken phase the accretion process can vary significantly based on the initial perturbation phase. In Fig. 3, we exhibit the evolution of the scalar field for perturbations with θ 0 = 0 and θ 0 = π , neither of which induces a scalar cloud in the ϕ 2 direction. For the θ 0 = 0 perturbation, a cloud of positive ϕ 1 accumulates within the potential well. As the cloud grows, non-linear effects cause the node of Φ to propagate towards spatial infinity. A similar scenario occurs for the θ 0 = π perturbation, now with a cloud of nega- \nFIG. 4. Channels of scalar cloud accretion, represented by the value of the scalar field at the event horizon, Φ h . Upper row: In the U (1)-symmetric phase (Φ = 0 and M = 2 . 5), the accretion dynamics and final states remain U (1)-symmetric with respect to the initial perturbation phase θ 0 . Lower: In the U (1)-broken phases, using here Φ h = 0 . 1, M = 1 . 8744 (left panel) and Φ h = 0 . 19, M = 0 . 7744 (right panel), the accretion paths no longer respect the U (1) symmetry. Cloud accretion tends to follow a dominant path, a phenomenon that becomes more pronounced for larger Φ h values of the initial field: in the right panel, the black and colored lines almost completely overlap (the visible purple and orange lines cover all other colored as well as black lines). As a result, the final states are not U (1)-symmetric, as indicated by the varying values of | Φ h | shown in the lower right corners. \n<!-- image --> \n, driving the node of Φ inward towards the black hole. In both cases, the formation and accretion of scalar clouds effectively peel off the negative potential well and 'radiate' the node of the field. \nTo further explore the relationship between the dynamical accretion of scalar clouds and the initial perturbation phase θ 0 , we systematically examine the evolution of the scalar field at the event horizon, Φ h ( t ), for a range of θ 0 values. In Figs. 4 and 5, 24 uniformly spaced values (black lines) within the interval (0 , 2 π ), as well as a set of specific values (colored lines) within the narrower intervals ( -π/ 12 , π/ 12) and (11 π/ 12 , 13 π/ 12). In the symmetric phase, we observe that the accretion process of the cloud manifests the U (1) symmetry, as depicted in the upper row of Fig. 4, which shows in particular the conservation of the phase θ during time evolution. On the other hand, in the U (1)-broken phase, the symmetry breaking causes the phase factor θ to become a non-trivial dynamical variable during the accretion, as illustrated in the lower panels of Fig. 4. We observe a manifest loss \nof U (1) invariance in the process, although it remains symmetric with respect to the ϕ 1 -axis due to a residual Z 2 symmetry. 5 Interestingly, the symmetry breaking introduces a preferred path for scalar cloud accretion, acting as an attractor that draws in the majority of accretion channels. Fig. 2 illustrates one such typical path ( θ 0 = π/ 2). Furthermore, the final equilibrium state will not be U (1)-symmetric for scalar clouds in the symmetrybroken phase. As shown in the lower right corner of the lower panels, we find that ∣ ∣ ∣ Φ θ 0 h ∣ ∣ ∣ , which represents the value of | Φ h | at the end state for a given initial perturbation phase θ 0 , is maximal for θ 0 = 0 and minimal for θ 0 = π (see also Fig. 5). \nThe existence of a dominant path in field space may be understood from the fact that one expects most of the energy of the field to go into the gapless degrees of freedom, in this case the angular component of Φ, with a comparatively smaller fraction in the radial field. 6 This picture is consistent with the observed trajectory, which appears to be roughly circular, at least qualitatively. This intuition is corroborated by an explicit comparison of the energy densities stored in the radial and angular components of the field; see Appendix B for details.", 'IV. ENERGY LOSS': 'A noteworthy phenomenon during scalar cloud accretion is that the cloud reaches equilibrium after a portion of its energy is radiated to spatial infinity (see Figs. 2 and 3), potentially causing the final state to have less energy than the initial state. In Fig. 5, we calculate the Arnowitt-Deser-Misner mass [54] of the initial and final states to quantify the energy loss during accretion in both the U (1)-symmetric and U (1)-breaking phases. In both cases, a discernible energy gap exists between the initial and final states, confirming that the formation of a stable scalar cloud involves the radiation of energy to spatial infinity. \nIn the symmetric phase, the end states maintain the U (1)-symmetry and are degenerate with with respect to the final energy ( M 1 = 2 . 4816 in our numerics). The energy loss, calculated as M 1 -M 0 (displayed in the table of Fig. 5), indicates a transition from the U (1)-symmetric phase to a lower-energy final state. On the other hand, in the U (1)-broken phase, symmetry breaking lifts this degeneracy and leads to an energy band for the final states, bounded by the cases with initial phases θ 0 = 0 and θ 0 = π . Notably, the θ 0 = 0 perturbation induces the formation of a positive ϕ 1 cloud, with most of its energy \n<!-- image --> \nM \nFIG. 5. Upper panel: Energy of initial (dashed lines) and final (solid) states. Energy loss in the U (1)-symmetric phase is illustrated by the purple lines, and we consider two U (1)breaking models (orange and blue). In the latter cases, the colored bands represent the range of final values of the energy. The bands are bounded by the cases with θ 0 = 0 (squares) and θ 0 = π (circles). The zoomed-in insets allow one to discern the energy gap between initial and final states. Lower: The table provides explicit data for both the initial and end states in our numerics, in particular the energy loss. Here, M 0 represents the energy of the initial state, M 1 is the energy of the end state, and ∆ M 1 denotes the width of the energy band for the final states. \nradiating to spatial infinity. It raises a backreaction on the field itself, which pushes the node of Φ outward from the potential well towards spatial infinity, as mentioned previously. As a result, the end state for θ 0 = 0 has the minimal energy. Conversely, for the θ 0 = π perturbation, most of the negative ϕ 1 cloud propagates toward the black hole, driving the node of Φ into the hole to reach equilibrium. In this scenario, a minimal amount of energy from the cloud radiates to infinity, leading to the end state having maximal energy. For detailed numerical data, refer to the table in Fig. 5.', 'V. CONCLUSIONS': 'In the EMS model, the non-minimal coupling between the complex scalar and electromagnetic fields can induce a negative potential well near a RN black hole ( U (1)symmetric phase), triggering the onset of scalarization and leading to the accretion of a scalar cloud onto the event horizon. During dynamical evolution, the accretion of scalar clouds preserves the global U (1) symmetry, re- \nsulting in U (1)-symmetric final states with energy that is degenerate with respect to the initial perturbation phase θ 0 . Remarkably, even in the U (1)-breaking phase, a negative potential well may exist in the presence of a nontrivial scalar field. Perturbations with different initial phases can induce the formation of scalar clouds within this well, which subsequently accrete through the event horizon, bringing energy to the black hole and radiating some energy to spatial infinity. Unlike the symmetrypreserving case, symmetry breaking allows for multiple accretion channels for scalar clouds with different initial phases. This phenomenon leads to varying amounts of energy radiated to spatial infinity, resulting in a lifting of degeneracy and the appearance of a band of possible final states. Moreover, symmetry breaking establishes a dominant accretion channel, acting as an attractor which draws in the majority of field trajectories. \nThis paper highlights the significant impact of spontaneous symmetry breaking on the dynamics of scalar cloud accretion. Our studies reveal distinctive signatures of symmetry breaking in black holes, providing compelling motivation for their exploration in astrophysical observations. The assumption of spherical symmetry in our setup is admittedly simplistic, and implies that neither electromagnetic nor gravitational radiation will be emitted during the evolution. It would be therefore interesting to relax this assumption in order to investigate these radiation channels and how the properties of the system may be assessed through their observation. Future research could also extend these findings to more realistic systems, such as rotating black holes, where the superradiance phenomenon is likely to produce richer scalar cloud structures and dynamics. \nAcknowledgements. We are grateful to Yupeng Zhang and Shenkai Qiao for useful discussions and valuable comments. SGS, GG and XW are supported by the NSFC (Grant Nos. 12250410250 and 12347133). PW is supported in part by the NSFC (Grant Nos. 12105191, 12275183, 12275184 and 11875196).', 'Appendix A: Numerical scheme for black hole evolution': 'In this paper, we simulate the black hole evolution using the 3 + 1 decomposition of the metric, expressed as \nds 2 = -N 2 0 dt 2 + γ ij ( dx i + N i dt ) ( dx j + N j dt ) . (A1) \nFor the gravity sector, we apply the BSSN formulation, using the 1+log slicing and Gamma-driver shift conditions [47-52]. \nTo model the scalar field, we employ dynamical variables Φ and Π, where Π = n µ ∇ µ Φ, where n µ is given by n µ = ( -N 0 , 0 , 0 , 0), as the 4-vector orthogonal to the spatial hypersurface. For the electromagnetic field, we decompose F µν into the 3-dimensional electric and magnetic components, defined as E i = γ µ i n ν F µν and \nFIG. 6. Convergence test of the numerical scheme, illustrated by the event-horizon value | Φ h | of the scalar field as function of the number N R of grid points. The blue dots represent the final state of spontaneous scalarization from a RN black hole with Q/M = 0 . 5 and α = 160. As the grid number increases, | Φ h | converges, indicating convergence of the simulation. \n<!-- image --> \nB i = γ µ i n ν ∗ F µν , respectively, where ∗ denotes the Hodge dual, and γ µ i projects 4-dimensional vectors onto the spatial hypersurface. Since we focus on spherically symmetric evolution, the magnetic field B i vanishes during the simulation. The evolution equations for the matter fields read \n∂ ⊥ Φ = N 0 Π , ∂ ⊥ Π = D i ( N 0 χ i ) + N 0 Π K -1 2 αN 0 Φ e α | Φ | 2 F 2 , ∂ ⊥ E i = αKE i -2 αN 0 E i ( ϕ 1 Π 1 + ϕ 2 Π 2 ) , (A2) \nwhere the time derivative is defined by ∂ ⊥ ≡ ∂ t - L N , and L N is the Lie derivative along N i ; K is the trace of the extrinsic curvature, F 2 = -2 E i E i , and ϕ 1 , 2 and Π 1 , 2 denote respectively the real and imaginary parts of ϕ and Π. \nDuring the evolution, the gravity-matter interaction is described by the stress-energy tensor T µν , given by \nT µν = 1 8 π ( 2 ∂ ( µ Φ ∗ ∂ ν ) Φ -g µν | ∂ Φ | 2 ) + 1 8 π e α | Φ | 2 ( 2 F µρ F ν ρ -1 2 g µν F 2 ) . (A3) \nTo incorporate the matter contribution into the BSSN formalism, we project the stress-energy tensor onto 3 + 1 \nvariables, \nρ = n µ n ν T µν = 1 8 π γ ij ( ∂ i ϕ 1 ∂ j ϕ 1 + ∂ i ϕ 2 ∂ j ϕ 2 ) + 1 8 π ( Π 1 Π 1 +Π 2 Π 2 e α | Φ | 2 E 2 ) , J i = -γ µ i n ν T µν = -1 4 π ( ∂ i ϕ 1 Π 1 + ∂ i ϕ 2 Π 2 ) , S ij = γ µ i γ ν j T µν = 1 4 π ( ∂ i ϕ 1 ∂ j ϕ 1 + ∂ i ϕ 2 ∂ j ϕ 2 -e α | Φ | 2 E i E j ) -1 8 π γ ij γ kl ( ∂ k ϕ 1 ∂ l ϕ 1 + ∂ k ϕ 2 ∂ l ϕ 2 ) + 1 8 π γ ij ( Π 1 Π 1 +Π 2 Π 2 + e α | Φ | 2 E 2 ) . (A4) \nIn our numerical simulations, we adopt a spherical-like coordinate system, introducing a dimensionless radial coordinate R to replace the standard spherical radial coordinate r . The relationship between r and R is defined as \nr = r max ( R 0 + e R/a -e -R/a e 1 /a -e -1 /a ) , (A5) \nwhere r max denotes the outer boundary, and R 0 and a are scaling constants that map between r and R . Since we focus on a spherically symmetric spacetime, the radial direction R is discretized uniformly with N R grid points, while N θ = N φ = 1 for the angular directions ( θ, φ ). \nIn Fig. 6, we present a convergence test for the simulation of spontaneous scalarization of a RN black hole, evaluated across different grid resolutions N R . In the simulation setup, the radial coordinate parameters are specified as r max = 30000, r 0 = 0 . 00012 and a = 0 . 07 (recall that we use units such that Q = 1). The integration time step ∆ t is chosen to match the smallest spatial ∆ r min (i.e., ∆ t = ∆ r min ) to avoid numerical instabilities. Our numerical results indicate that the simulation converges with increasing grid number N R .', 'Appendix B: Radial-angular scalar field decomposition': "In this Appendix we expand on the relationship between the dominant path observed in the scalar cloud accretion channels and the question of energy distribution among the field components. At a qualitative level, and at first approximation, the attractor trajectory in Fig. 4 (black curves in bottom left panel) appears to correspond to circular motion, suggesting that the energy density of the scalar field is predominantly in the angular field. This agrees with the expectation that most of the energy should go to the gapless degrees of freedom in the system, with a comparatively smaller amount going to the radial part of the scalar field. \nWe can confirm this intuition through an explicit calculation of the energy density of each field component, \nFIG. 7. Time evolution of the energy densities ρ σ and ρ θ , evaluated at the event horizon, and using the initial values σ 0 = 0 . 19 and σ 0 = 0 . 10. One observes that both the radial and angular fields experience a peak in the energy density corresponding to the process of accretion onto the black hole. \n<!-- image --> \nwith the radial and angular fields defined via \nΦ = σ ( x ) e iθ ( x ) . 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Abbott et al. GW170608: Observation of a 19solar-mass Binary Black Hole Coalescence. Astrophys. J. Lett. , 851:L35, 2017.\n- [14] Benjamin P. Abbott et al. GW170104: Observation of a 50-Solar-Mass Binary Black Hole Coalescence at Redshift 0.2. Phys. Rev. Lett. , 118(22):221101, 2017. [Erratum: Phys.Rev.Lett. 121, 129901 (2018)].\n- [15] R. Abbott et al. Tests of General Relativity with GWTC3. 12 2021.\n- [16] Jacob D. Bekenstein. Nonexistence of baryon number for static black holes. Phys. Rev. D , 5:1239-1246, 1972.\n- [17] J. D. Bekenstein. Novel ''no-scalar-hair'' theorem for black holes. Phys. Rev. D , 51(12):R6608, 1995.\n- [18] Carlos A.R. Herdeiro and Eugen Radu. Asymptotically flat black holes with scalar hair: a review. Int. J. Mod. Phys. D , 24(09):1542014, 2015.\n- [19] Thomas P. Sotiriou and Shuang-Yong Zhou. Black hole hair in generalized scalar-tensor gravity. Phys. Rev. Lett. , 112:251102, 2014.\n- [20] G. Antoniou, A. Bakopoulos, and P. Kanti. Evasion of No-Hair Theorems and Novel Black-Hole Solutions in Gauss-Bonnet Theories. Phys. Rev. Lett. , 120(13):131102, 2018.\n- [21] Jun Zhang and Huan Yang. Dynamic Signatures of Black Hole Binaries with Superradiant Clouds. Phys. Rev. D , 101(4):043020, 2020. \nFIG. 8. Trajectory of the horizon value of the scalar field, Φ h , in the ϕ 1 -ϕ 2 plane, using the initial values σ 0 = 0 . 19 and σ 0 = 0 . 10. Dots are separated by equal time intervals, and their color indicates the ratio of ρ θ to its maximum value. \n<!-- image --> \n- [22] Daniel Baumann, Horng Sheng Chia, John Stout, and Lotte ter Haar. The Spectra of Gravitational Atoms. JCAP , 12:006, 2019.\n- [23] Daniel Baumann, Gianfranco Bertone, John Stout, and Giovanni Maria Tomaselli. Sharp Signals of Boson Clouds in Black Hole Binary Inspirals. Phys. Rev. Lett. , 128(22):221102, 2022.\n- [24] Marco O. P. Sampaio, Carlos Herdeiro, and Mengjie Wang. Marginal scalar and Proca clouds around Reissner-Nordstrom black holes. Phys. 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D , 87(4):044026, 2013.\n- [53] Zachariah Etienne, Steven R. Brandt, and Peter Diener et al. The Einstein Toolkit, 2021, To find out more, visit http://einsteintoolkit.org.\n- [54] Charles W. Misner, K. S. Thorne, and J. A. Wheeler. Gravitation . W. H. Freeman, San Francisco, 1973."}

2021AJ....161..147B

Stellar distances constitute a foundational pillar of astrophysics. The publication of 1.47 billion stellar parallaxes from Gaia is a major contribution to this. Despite Gaias precision the majority of these stars are so distant or faint that their fractional parallax uncertainties are large thereby precluding a simple inversion of parallax to provide a distance. Here we take a probabilistic approach to estimating stellar distances that uses a prior constructed from a threedimensional model of our Galaxy. This model includes interstellar extinction and Gaias variable magnitude limit. We infer two types of distance. The first geometric uses the parallax with a directiondependent prior on distance. The second photogeometric additionally uses the color and apparent magnitude of a star by exploiting the fact that stars of a given color have a restricted range of probable absolute magnitudes plus extinction. Tests on simulated data and external validations show that the photogeometric estimates generally have higher accuracy and precision for stars with poor parallaxes. We provide a catalog of 1.47 billion geometric and 1.35 billion photogeometric distances together with asymmetric uncertainty measures. Our estimates are quantiles of a posterior probability distribution so they transform invariably and can therefore also be used directly in the distance modulus 5mathrmlog10r5 . The catalog may be downloaded or queried using ADQL at various sites see httpwww.mpia.decaljgedr3distances.html where it can also be crossmatched with the Gaia catalog.

2021-03-01T00:00:00Z

['2021AJ....161..147B', 'arXiv:2012.05220', '10.3847/1538-3881/abd806', '10.48550/arXiv.2012.05220', '2020arXiv201205220B']

['Catalogs', 'Galaxy structure', 'Bayesian statistics', 'Parallax', 'Stellar parallax', 'Photometric parallax', 'Distance indicators', 'Astrometry', 'Markov chain Monte Carlo', 'Absolute magnitude', '205', '622', '1900', '1197', '1618', '1231', '394', '80', '1889', '10', 'Astrophysics - Solar and Stellar Astrophysics', 'Astrophysics - Astrophysics of Galaxies']

Estimating Distances from Parallaxes. V. Geometric and Photogeometric Distances to 1.47 Billion Stars in Gaia Early Data Release 3

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']

https://arxiv.org/pdf/2012.05220.pdf

{'No Header': 'Estimating distances from parallaxes. V: \nGeometric and photogeometric distances to 1.47 billion stars in Gaia Early Data Release 3 \nC.A.L. Bailer-Jones, 1 J. Rybizki, 1 M. Fouesneau, 1 M. Demleitner, 2 and R. Andrae 1 \n1 Max Planck Institute for Astronomy, Heidelberg, Germany \n2 Astronomisches Rechen-Institut, Zentrum fur Astronomie der Universitat Heidelberg, Germany \n(Received 9 December 2020; Revised 30 December 2020; Accepted 31 December 2020)', 'ABSTRACT': "Stellar distances constitute a foundational pillar of astrophysics. The publication of 1.47 billion stellar parallaxes from Gaia is a major contribution to this. Yet despite Gaia's precision, the majority of these stars are so distant or faint that their fractional parallax uncertainties are large, thereby precluding a simple inversion of parallax to provide a distance. Here we take a probabilistic approach to estimating stellar distances that uses a prior constructed from a three-dimensional model of our Galaxy. This model includes interstellar extinction and Gaia's variable magnitude limit. We infer two types of distance. The first, geometric, uses the parallax together with a direction-dependent prior on distance. The second, photogeometric, additionally uses the colour and apparent magnitude of a star, by exploiting the fact that stars of a given colour have a restricted range of probable absolute magnitudes (plus extinction). Tests on simulated data and external validations show that the photogeometric estimates generally have higher accuracy and precision for stars with poor parallaxes. We provide a catalogue of 1.47 billion geometric and 1.35 billion photogeometric distances together with asymmetric uncertainty measures. Our estimates are quantiles of a posterior probability distribution, so they transform invariably and can therefore also be used directly in the distance modulus (5 log 10 r -5). The catalogue may be downloaded or queried using ADQL at various sites (see http://www.mpia.de/ ∼ calj/gedr3 distances.html) where it can also be cross-matched with the Gaia catalogue. \nKeywords: catalogs - Galaxy: structure - methods: statistical - stars: distances - parallax", '1. INTRODUCTION': "There are various ways to determine astrophysical distances. Near the base of the distance ladder on which almost all other distance measures are built are geometric parallaxes of stars. In recognition of this, the European Space Agency (ESA) implemented the Gaia mission to obtain parallaxes for over one billion stars in our Galaxy down to G glyph[similarequal] 20 mag, with accuracies to tens of microarcseconds (Gaia Collaboration 2016a). The first two data releases (Gaia Collaboration 2016b, 2018) presented a significant leap forward in both the number and accuracy of stellar parallaxes. The recently published early third release (Gaia Collaboration 2020a) (hereafter EDR3) reduces the random and systematic errors in the parallaxes by another 30%. \nWhile parallaxes ( glyph[pi1] ) are the basis for a distance determination, they are not themselves distances ( r ). This is due to the nonlinear transformation between them ( glyph[pi1] ∼ 1 /r ) and the presence of significant noise for more distant stars. Small absolute uncertainties in parallax \ncan translate into large uncertainties in distance, and while parallaxes can be negative, distances cannot be. Thus for anything but the most precise parallaxes, the inverse parallax is a poor distance estimate. An explicit probabilistic approach to inferring distances may instead be taken. This has been discussed and applied to parallax data in various publications in recent years; a recent overview is given by Luri et al. (2018). The simplest approach uses just the parallax and parallax uncertainty together with a one-dimensional prior over distance. This yields a posterior probability distribution over distance to an individual star (Bailer-Jones 2015). A suitable prior ensures that the posterior converges to something sensible as the precision of the parallax degrades. This is important when working with Gaia data, because its truly revolutionary nature notwithstanding, in EDR3 43% of the sources have parallax uncertainties greater than 50% (63% greater than 20%), and a further 24% have negative parallaxes. The shape and scale of the prior distribution should reflect the expected \ndistribution of stars in the sample, including observational selection effects such as magnitude limits. The prior's characteristic length scale will typically need to vary with direction in the Galaxy (Bailer-Jones et al. 2018). More sophisticated approaches use other types of data, such as the star's magnitude and colour (Astraatmadja & Bailer-Jones 2016a; McMillan 2018; Anders et al. 2019; Leung & Bovy 2019), velocity (Schonrich & Aumer 2017; Zucker et al. 2018), or spectroscopic (Sanders & Das 2018; Queiroz et al. 2020) or asteroseismic (Hall et al. 2019) parameters. In order to exploit such additional data, these methods must make deeper astrophysical assumptions than parallax-only approaches, and may also have more complex priors. The benefit is that the inferred distances will usually be more precise (lower random errors), and hopefully also more accurate (lower systematic errors) if the extra assumptions are correct. \nIn the present paper, the fifth in a series, we determine distances for sources in EDR3 using data exclusively from EDR3. The resulting catalogue should be more accurate and more useful than our earlier work, on account of both the more accurate parallaxes in EDR3 and improvements in our method. We determine two types of distance. The first, which we call 'geometric', uses only the parallaxes and their uncertainties. We explored this approach in detail in the first two papers in this series (Bailer-Jones 2015; Astraatmadja & BailerJones 2016a) (hereafter papers I and II), and applied it to estimate distances for 2 million stars in the first Gaia data release (Astraatmadja & Bailer-Jones 2016b) (paper III) and 1.33 billion stars in the second Gaia data release (Bailer-Jones et al. 2018) (paper IV). Both papers used a (different) direction-dependent distance prior that reflected the Galaxy's stellar populations and Gaia's selection thereof. \nOur second type of distance estimate uses, in addition to the parallax, the colour and magnitude of the star. We call such distances 'photogeometric'. As well as the distance prior, this uses a model of the directiondependent distribution of (extincted) stellar absolute magnitudes. \nWe construct our priors from the GeDR3 mock catalogue of Rybizki et al. (2020). This lists, among other things, the (noise-free) positions, distances, magnitudes, colours, and extinctions of 1.5 billion individual stars in the Galaxy as a mock-up of what was expected to appear in EDR3. GeDR3mock is based on the Besan¸con Galactic model and PARSEC stellar evolutionary tracks. We exclude stars from GeDR3mock that simulate the Magellanic Clouds ( popid=10 ) and stellar open clusters ( popid=11 ). We divide the sky into the \n12288 equal-area (3.36 sq. deg.) regions defined by the HEALpixel scheme 1 at level 5, and fit our prior models separately to each. In doing this we only retain from GeDR3mock those stars that are brighter than the 90th percentile of the EDR3 magnitude distribution in that HEALpixel (Rybizki & Drimmel 2018; Gaia Collaboration 2020b). This is done to mimic the variable magnitude limit of Gaia over the sky, and varies from 19.2 mag around the Galactic centre to 20.7 mag over much of the rest of the sky (the median over HEALpixels is 20.5 mag). \nWe apply our inference to all sources in EDR3 that have parallaxes. As our prior only reflects single stars in the Galaxy, our distances will be incorrect for the small fraction of extragalactic source in the Gaia catalogue, and may also be wrong for some unresolved binaries, depending on their luminosity ratios. \nAs some readers may be familiar with our previous catalogue using GDR2 data (paper IV), here is a summary of the main changes in the new method (which we describe fully in section 2). \n- 1. We update the source of our prior from a mock catalogue of GDR2 (Rybizki et al. 2018) to one of EDR3 (Rybizki et al. 2020).\n- 2. We replace the one-parameter exponential decreasing space density (EDSD) distance prior with a more more flexible three-parameter distance prior (section 2.3).\n- 3. We again fit the distance prior to a mock catalogue, but we no longer use spherical harmonics to smooth the length scale of the prior over the sky. We instead adopt a common distance prior for all stars within a small area (level 5 HEALpixels).\n- 4. We introduce photogeometric distances (section 2.4) using a model for the (extincted) colourabsolute magnitude diagram, also defined per HEALpixel (section 2.5).\n- 5. In paper IV we summarized each posterior with the mode and the highest density interval (HDI). The mode has the disadvantage that it is not invariant under nonlinear transformations. This means that if we inferred r mode as the mode of the posterior in distance, then 5 log 10 r mode -5 would not, in general, be the mode of the posterior in distance modulus. This is also the case for the mean. The quantiles of a distribution, in contrast, \nare invariant under (monotonic) nonlinear transformations. We therefore provide the median (the 50th percentile) of the posterior as our distance estimate. To characterize the uncertainty in this we quote the 14th and 86th percentiles (an equaltailed interval, ETI). These are therefore also the quantiles on the absolute magnitude inferred from the distance. \nIn the next section we describe our method and the construction of the priors. In section 3 we apply our method to the GeDR3mock catalogue, giving some insights into how it performs. We present the results on EDR3 in section 4, and describe the resulting distance catalogue in section 5 along with its use and limitations. We summarize in section 6. Auxiliary information, including additional plots for all HEALpixels, for both the prior and the results, can be found online 2 .", '2. METHOD': 'For each source we compute the following two posterior probability density functions (PDFs) over the distance r \nGeometric: P ∗ g ( r | glyph[pi1] , σ glyph[pi1] , p ) Photogeometric: P ∗ pg ( r | glyph[pi1] , σ glyph[pi1] , p, G, c ) \nwhere glyph[pi1] is the parallax, σ glyph[pi1] is the uncertainty in the parallax, p is the HEALpixel number (which depends on Galactic latitude and longitude), G is the apparent magnitude, and c is the BP -RP colour. The parallax and apparent magnitude will be adjusted to accommodate known issues with the EDR3 data, as detailed below. The star ∗ symbol indicates that we infer unnormalized posteriors. The geometric posterior uses just a distance prior. The photogeometric posterior uses this distance prior as well as a colour-magnitude prior that we explain below. The posteriors are summarized using quantiles computed by Markov Chain Monte Carlo (MCMC) sampling.', '2.1. Geometric distance': 'The unnormalized posterior PDF is the product of the likelihood and prior: \nP ∗ g ( r | glyph[pi1] , σ glyph[pi1] , p ) = P ( glyph[pi1] | r, σ glyph[pi1] ) P ( r | p ) . (1) \nThe likelihood is conditionally independent of p . We chose to make the second term, which we define in section 2.3, independent of σ glyph[pi1] .', '2.2. Likelihood': 'Under the assumption of Gaussian parallax uncertainties the likelihood is \nP ( glyph[pi1] | r, σ glyph[pi1] ) = 1 √ 2 πσ glyph[pi1] exp [ -1 2 σ 2 glyph[pi1] ( glyph[pi1] -glyph[pi1] zp -1 r ) 2 ] (2) \nwhere glyph[pi1] zp is the parallax zeropoint. In paper IV we adopted a constant value of -0 . 029 mas for this zeropoint, as recommended in the GDR2 release. For EDR3 the Gaia team has published a more sophisticated parallax zeropoint based on analyses of quasars, binary stars, and the Large Magellanic Cloud (LMC) (Lindegren et al. 2020a). This is a function of G , the ecliptic latitude, and the effective wavenumber used in the astrometric solution. Ideally this last term was derived from the BP -RP colour, and this is the case for the standard 5-parameter (5p) astrometric solutions used for 585 million sources (Gaia Collaboration 2020a). But where BP -RP was unavailable or deemed of insufficient quality, the effective wavenumber was derived as a sixth parameter in the astrometric solution (6p solutions) (Lindegren et al. 2020b), which is the case for 882 million sources. Overall the zeropoint ranges between about -0 . 150 and +0 . 130 mas (it is narrower for the 5p solutions), although the RMS range is only 0.020 mas. We use this zeropoint correction in equation 2. Our geometric distances are therefore weakly conditioned also on G and c , but we omit this in the mathematical notation for brevity. For the 2.5 million sources that have parallaxes but no G (strictly, no phot g mean mag ), we use the EDR3 global zeropoint of -0 . 017 mas (Lindegren et al. 2020b).', '2.3. Distance prior': 'In paper IV we used the one-parameter EDSD distance prior, which models the space density of stars as dropping exponentially away from the Sun according to a (direction-dependent) length scale. Here we adopt the more flexible, three-parameter Generalized Gamma Distribution (GGD), which can be written as \nP ( r | p ) = 1 Γ( β +1 α ) α L β +1 r β e -( r/L ) α if r ≥ 0 0 otherwise (3) \nfor α > 0, β > -1, and L > 0. Γ() is the gamma function. This PDF is unimodal with an exponentially decreasing tail to larger distances. The mode is L ( β/α ) 1 /α for β > 0, and zero otherwise. The EDSD is a special case of the GGD with α = 1 , β = 2. We fit the GGD prior for each HEALpixel separately via maximum likelihood using stars from the mock catalogue. \nFigure 1. Distance priors for two HEALpixels, number 6200 at high latitude (left) and number 7593 at low latitude (right). The histograms show the distributions of the data in the mock catalogue. The smooth curves are the fit of the Generalized Gamma Distribution (GGD; equation 3) to these data, which defines the distance prior P ( r | p ) with the parameters L , α , and β . Similar plots for all HEALpixels are available with the auxiliary information online. \n<!-- image --> \nFigure 2. The variation of the median of the distance prior over the sky shown in Galactic coordinates on a Mollweide equal-area projection. The LMC/SMC are excluded from our prior. \n<!-- image --> \nThe HEALpixel ( p ) dependency on the left side of equation 3 is equivalent to a dependency on α, β, L . \nExample fits for two HEALpixels, one at low Galactic latitude and one at high Galactic latitude, are shown in Figure 1. Although the GGD prior provides a better fit than the EDSD prior - which is why we use it - the parameter L may no longer be interpreted as a meaningful length scale, because it varies from 3e-7 to 1e4 pc over all HEALpixels. The appropriate characteristic scale of the GGD prior in this work is its median, for which which there is no closed-form expression. The median varies between 745 and 7185 pc depending on HEALpixel (Figure 2). Fits for each HEALpixel can be found in the auxiliary information online. \nIn the limit of uninformative parallaxes, the geometric posterior converges on the GGD prior, and so the median distance converges on the median of this prior. In paper IV this convergence was on the mode of the EDSD prior. For the prior fits used in the present pa- \nper, the ratio of the GGD median to the EDSD mode ranges from 1.17 to 1.57. There are potential improvements one could make to the prior to give a better convergence in the limit of poor data. Some considerations are in appendix A.', '2.4. Photogeometric distance': "We define the quantity Q G as \nQ G ≡ M G + A G = G -5 log 10 r +5 . (4) \nThe equality (=), which is a statement of flux conservation, holds only when all the quantities are noise-free. If we knew Q G for a star, then a measurement of G gives us an estimate of r . Given that the uncertainties on G in EDR3 are generally less than a few millimagnitudes (0.3 to 6 mmag for G < 20 mag; Gaia Collaboration 2020a), this would be a reasonably precise estimate. We do not know Q G , but we can take advantage of the fact that the two-dimensional colourQ G space for stars is not uniformly populated. This space (e.g. Figure 3) which we call the CQD - in analogy to the CMD (colour-magnitude diagram) - would be identical to the colour-absolute-magnitude diagram if there were no interstellar extinction. Thus if we know the BP -RP colour of the star, this diagram places limits on possible values of Q G , and therefore on the distance to the star. We will use the mock catalogue to model the CQD (per HEALpixel) and from this compute a prior over Q G given the magnitude and colour of the star. \nThe formal procedure is as follows, initially making no assumptions about G . We assume the colour to be effectively noise-free. This is reasonable given the relatively low noise for most sources (13 to 120 mmag for G < 20 mag; Gaia Collaboration 2020a), and the fact \n<!-- image --> \nFigure 3. CQDs for HEALpixels 6200 (left) and 7593 (right) in the mock catalogue. The density of stars is shown on a logarithmic colour scale relative to the maximum density in each HEALpixel (so the zero point of the density scales are not the same in the two panels). The text at the top of each panel gives the Galactic longitude and latitude ( l, b ) of the centre of the HEALpixel in degrees, the number of stars, and the faintest magnitude. The vertical lines identify particular Q G -models that are shown in Figures 4 and 5. Similar plots for all HEALpixels are available with the auxiliary information online. \n<!-- image --> \nthat the prior is anyway imperfect (see section 2.5). Using Bayes' theorem, the unnormalized posterior we want to estimate can be decomposed into a product of two terms \nP ∗ pg ( r | glyph[pi1] , σ glyph[pi1] , G, c, p ) = P ( glyph[pi1] | r, σ glyph[pi1] ) P ( r | G,c, p ) . (5) \nThe first term on the right side is the parallax likelihood (section 2.2). It is independent of G , c , and p once it is conditioned on σ glyph[pi1] , which is estimated in the Gaia astrometric solution using quantities that depend on the magnitude, colour, scanning law, etc. (Lindegren et al. 2020b). The second term is independent of the parallax measurement process and thus of glyph[pi1] and σ glyph[pi1] . We may write this second term as a marginalization over Q G and then apply Bayes' theorem as follows \nP ( r | G,c, p ) (6) = ∫ P ( r, Q G | G,c, p ) d Q G = ∫ 1 P ( G | c, p ) P ( G | r, Q G ) P ( r, Q G | c, p ) d Q G = P ( r | c, p ) P ( G | c, p ) ∫ P ( G | r, Q G ) P ( Q G | r, c, p ) d Q G . \nIn the last line, the first term under the integral is formally the likelihood for G (and is conditionally independent of c and p due to equation 4). However, as G is measured much more precisely than the intrinsic spread in Q G - that is, the second term under the integral is a much broader function - we can consider G to be noise-free to a good approximation. This makes the first term a delta function and so the integral is non-zero only when equation 4 is satisfied. \nWe make two further assumptions about the terms in the last line of equation 6. The first is to make the distance prior independent of colour, i.e. P ( r | c, p ) → P ( r | p ). This is now the same distance prior as used in the geometric posterior (equation 1). The second is to assume that the CQD is independent of distance, i.e. P ( Q G | r, c, p ) → P ( Q G | c, p ). This is not true in general, but we chose not to add this extra layer of dependence on GeDR3mock (see section 2.5). \nWith these assumptions, the (unnormalized) posterior in equation 5 can now be written as \nP ∗ pg ( r | glyph[pi1] , σ glyph[pi1] , G, c, p ) glyph[similarequal] P ( glyph[pi1] | r, σ glyph[pi1] ) P ( r | p ) × P ( Q G = G -5 log 10 r +5 | c, p ) . (7) \nThe missing normalization constant, 1 /P ( glyph[pi1] , G | c, p ), is not required. This posterior is simply the geometric posterior (equation 1) multiplied by an additional prior 3 over Q G .", '2.5. Q G prior': 'We construct the prior P( Q G | c , p ) from the mock catalogue. Given the complexity of the CQD and its variation over the sky, we do not attempt to fit the prior as a continuous 3D (position and colour) parametric function. We instead compute a CQD for each HEALpixel, two examples of which are shown in Figure 3. Within each we compute a series of one-dimensional functions at a series of colours in the following way. We divide', 'Bailer-Jones et al.': 'Figure 4. Q G prior models constructed from the CQD of HEALpixel 6200. Each of the six panels shows a fit to the mock data at a different BP -RP colour, corresponding to the six vertical stripes shown in Figure 3 (left panel). Model fits using smoothing splines are plotted as black lines with the degrees of freedom (df) as indicated and the (binned) data in the fit show as red circles. Model fits using one or two Gaussian components are plotted as orange and blue lines respectively, with the data in the fit shown as black circles and the mean and standard deviation of the fit components indicated in parentheses at the top of each panel. These density functions show the prior PDF P ( Q G | c, p ) at discrete colours before imposing the minimum threshold which ensures the prior density is always greater than zero. Similar plots for all colour strips in HEALpixels are available with the auxiliary information online. \n<!-- image --> \nFigure 5. As Figure 4 but now for HEALpixel 7593, the CQD of which is in the right panel of Figure 3. \n<!-- image --> \nthe full colour range of a given HEALpixel into strips of 0.1 mag width in colour, then for each strip fit a model to the stellar number density as a function of Q G (now ignoring the colour variation in each strip). If there are more than 40 stars in a strip, we bin the data into bins of 0.1 mag and fit a smoothing spline with min( glyph[floorleft] N/ 4 glyph[floorright] , 50) degrees of freedom (df), where N is the number of stars in the strip (which can be many thousands). If there are fewer than 40 stars we cannot fit a good spline. This generally occurs at the bluest and reddest ends of the CQD. Here the Q G distribution is often characterized by two widely-separated components, either the main sequence (MS) and white dwarf (WD) branches, or the MS and giant star branches (see Figure 3). Thus when N < 40 we instead fit a two-component Gaussian mixture model, with the constraint that the minimum and maximum standard deviation of each component be σ min = 0 . 08 mag and σ max = 1 . 0 mag respectively. A full fit requires at least five stars, so if there are as few as two stars we constrain the solution to first have equal standard deviations and then to have standard deviations of σ min . If N = 1 our model is a one-component Gaussian with mean equal to the Q G of the star and standard deviation equal to σ min . If there are no stars the model is null. Examples of the fits are shown in Figures 4 and 5. \nAs a smoothing spline can give a negative fit, and both these and the Gaussian models can yield very small values for the density, we impose that the minimum density is never less than 10 -3 of the integrated density (computed prior to fitting the model). Thus our prior is nowhere zero, meaning that even if the data indicate a Q G in the regions where the mock catalogue is empty, the posterior will not be zero. This allows sources to achieve distances that place them outside the occupied regions of the mock CQD. \nFor a given HEALpixel, each prior model refers to a specific colour, namely the centre of a 0.1 mag-wide strip. This is larger than the uncertainty in the colour for all but the faintest EDR3 sources. When evaluating the prior during the inference process, we compute Q G from equation 4, evaluate the densities of the two priors that bracket its colour, then linearly interpolate. This ensures that our prior is continuous in colour. If one of the models is null we use the other model as is. If both models are null, or if the source is outside of the colour range of the mock CQD, we do not infer a photogeometric distance. The flag field in our catalogue indicates what kind of Q G models were used (see section 5). \nThe computation of Q G in equation 4 requires the G-band magnitude of the source. For this we use the phot g mean mag field in EDR3 corrected for the processing error described in section 8.3 of Riello et al. \n(2020). This correction, which is a function of magnitude and colour, can be as large as 25 mmag.', '2.6. Posterior sampling and summary': 'The posteriors are formally the answer to our inference process. The geometric posterior has a simple parametric form which may be computed by the reader using the data in the EDR3 catalogue and the parameters of our prior (available with the auxiliary information online). The photogeometric posterior is generally nonparametric. Both posteriors are asymmetric and not necessarily unimodal (section 2.6.2). \nThere are a variety of statistics one could use to summarize these PDFs, such as the mean, median, or mode. There is no theoretically correct measure, and all have their drawbacks. We use quantiles, primarily because they are invariant under nonlinear transformations, and so are simultaneously the quantiles of the posterior in distance modulus, 5 log 10 r -5. We use the three quantiles at 0.159, 0.5, and 0.841, which we label r lo , r med , and r hi respectively. The central quantile is the median. The outer two quantiles give a 68% confidence interval around the median. The difference between each quantile and the median is a Gaussian 1 σ -like estimate of the uncertainty. Due to the intrinsic asymmetry of the posteriors we report the lower and upper values separately.', '2.6.1. Markov Chain Monte Carlo': 'Neither the geometric nor photogeometric posteriors have closed-form expressions for their quantiles so we must compute these numerically. We do this using Markov Chain Monte Carlo (MCMC), specifically the Metropolis algorithm. \nWe adopt the following scheme for the MCMC initialization and step size. We first compute the geometric distance posterior using the EDSD prior from paper IV. The length scale of this prior is set to 0.374 r med , where r med is the median distance of the stars in the mock catalogue for that HEALpixel. 4 We use the mode of this posterior, r EDSD mode , which has a closed-form solution (paper I), as the initialization for the geometric posterior. The initialization scheme for the photogeometric posterior is more complicated, in accordance with its more complicated shape, and depends on r EDSD mode , fractional parallax uncertainty (fpu, σ glyph[pi1] /glyph[pi1] ), and the characteristic length scale of the Q G prior model(s). \n4 In paper IV we used (1 / 3) r med , as the maximum likelihood fit of the length scale is a third of the mean. However, the median is a slightly biased estimator of the mean for the EDSD. For the typical length scales involved we found empirically that the mean is about 12% (0.374/0.333) larger than the median. \nFor both types of posterior the step size needs to be adapted to the characteristic width of the posterior, which is generally wider the larger the fpu. We found a suitable step size to be (3 / 4) r init × min( | σ glyph[pi1] /glyph[pi1] | , 1 / 3), where r init is the initialization value. \nThis scheme allows relatively short burn-ins: we use just 50. We experimented with chains of various chain lengths, employing various tests of convergence. Longer chains are always better, but as we need to sample around three billion posteriors, some parsimony is called for. We settled on 500 samples (post burn-in). Although the chains are not always settled, they are generally good enough to compute the required quantiles with reasonable precision. To quantify this we obtained 20 different MCMC chains and computed the standard deviation of the median distance estimates and half the mean of the confidence intervals. The ratio of these is a measure of the convergence noise. Doing this for thousands of stars we find this to be between 0.1 and 0.2 in general. For the geometric posteriors in particular it can be larger for fractional parallax uncertainties larger than 0.3.', '2.6.2. Multimodality': 'The posteriors can be multimodal. This is more likely to be the case for the photogeometric posterior at large fpu, as its prior can be multimodal. Multimodality is very rare for the geometric posterior. \nAlthough multimodality is a challenge for MCMC sampling methods, we find that even widely-separated modes can be sampled in our scheme. Our 68% confidence interval often encompasses the span of such multimodality. This is a blessing and a curse: the distance precision in a single mode may be quite good, yet a large confidence interval is obtained due to the presence of a second mode. To assist in identifying possible multimodality we perform the Hartigan dip test (Hartigan & Hartigan 1985). This is a classical statistical test in which the null hypothesis is a unimodal posterior, i.e. a small p-value suggests the distribution may not be unimodal. We select a threshold of 10 -3 and set a flag to 1 if the p-value is lower than this, thereby suggesting possible multimodality. If the p-value is above this threshold or the test does not work for any reason, the flag is 0. The test is not particularly accurate and should not be over-interpreted. Furthermore, it is done on the MCMC samples, not on the true posterior, so tends to be raised more often than expected due to the intrinsic noise of MCMC sampling.', '3. PERFORMANCE ON THE MOCK CATALOGUE': "Before looking at the results on EDR3, we evaluate the performance of our method using the mock catalogue, \nas here we know the true distances. In doing this we add Gaussian random noise to the parallaxes using the parallax error field in GeDR3mock, which is a model of the expected uncertainties in the EDR3 parallaxes. As the data are drawn from the same distance distribution and CQD from which the prior was constructed, this is a somewhat optimistic test, despite the noise. Unless noted otherwise, throughout this section the term 'fpu' refers to the true fractional parallax uncertainty, i.e. that computed using the true parallax", '3.1. Example posteriors': 'Figure 6 shows examples of both types of posterior compared to their priors. At small fpu, e.g. panels (a) to (c), the two posteriors are very similar, with a median (and mode) near to the true distance, shown as the vertical line. As long as the fpu is not too large, the prior plays little role and the posterior can be quite different, e.g. panel (d), although this can also occur at larger fpu, e.g. panels (i) and (l). Panel (f) shows a multimodal photogeometric prior and posterior. The two types of prior sometimes disagree, as can the posteriors. In panel (h), which is for a 30% parallax uncertainty, the geometric posterior is more consistent with the true distance. Note that the parallax that the algorithm sees does not correspond to the vertical line, so for large fpu we cannot expect either posterior to peak near this. Panel (k) shows a multimodal posterior in which the true distance is close to a smaller mode. This happens here because the parallax has 50% noise, so the measured parallax corresponds to a smaller distance (where both geometric and photogeometric posteriors peak). At larger fpu - the bottom row is all for more than 1.0 - the photogeometric prior is often more consistent with the true distance than the geometric one.', '3.2.1. Qualitative analysis': 'Distance inference results for two HEALpixels are shown in Figures 7 and 8. We see a good correlation between the inferred and true distances out to several kpc (left columns). The degradation at larger distances is mostly due to stars with larger fpu, as can be seen in the middle columns of these figures. The fractional residual is defined as the estimated minus true distance, divided by the true distance. Note that these middle columns show the true fpu, i.e. as computed from the noise-free parallax, which is not the same as the measured (noisy) fpu that the inference algorithm encounters. (See section B for a consequence of this difference.) At large fpu the photogeometric distances perform better than the geometric ones, because even when the parallax is \nFigure 6. Example normalized posteriors (solid lines) and corresponding normalized priors (dashed lines) for geometric distances (blue) and photogeometric distances (orange) for various stars in the mock catalogue (one per panel). These have been selected to show the variety; they are not a random subset. The vertical solid line is the true distance. The inverse of this is not the parallax seen by the inference, because noise was added. All stars are from HEALpixel 6200, so the distance prior (blue dashed line) is the same in all panels. The four numbers in the top-right corner of each panel are, from top to bottom, glyph[pi1] , true fpu, G , and BP -RP. Stars are ordered by increasing fpu. The two posteriors coincide in the top-left panel. \n<!-- image --> \nof limited use there is still distance information from the colour and magnitude via the Q G model. For geometric distances, in contrast, as the measured fpu increases, the distance prior dominates the likelihood, so the median of the posterior is pushed towards the median of the prior. Hence at large fpu, the geometric distances to stars that are truly more distant than the median of the prior will generally be underestimated. Faraway stars tend to have larger fpu than nearby stars, because they have both smaller parallaxes and larger parallax \nuncertainties (as they are fainter). Thus as a whole, any underestimation of geometric distances to stars that are beyond the median of the prior will tend to be larger than the overestimation of the geometric distances to stars that are closer than the median of the prior. This explains why the distribution in the top-left panels of Figures 7 and 8 flatten at larger distances. This feature is suppressed in the photogeometric distances (bottomleft panels) because for large fpu, the Q G prior can overrule the geometric prior. We also see more flattening for \nFigure 7. Results of the distance inference on mock catalogue HEALpixel 6200. The top row shows geometric distances, the bottom row photogeometric ones. The left column compares the inferred distances (vertical axis) to the true distances for all sources. This cover the full range of fractional parallax uncertainties, which has a median of 0.20 and central 90% range of 0.03-1.08. The middle column shows the fractional distance residuals as a function of the true fractional parallax uncertainty (fpu). In these first four panels the colour scale is a logarithmic density (base 10) scale relative to the highest density cell in each panel. The right column shows the normalized residuals: the difference between the inferred and the true value, divided by an uncertainty measure. The three colours refer to three uncertainty measures: orange is r med -r lo , blue is r hi -r med , black is (1 / 2)( r hi -r lo ). The blue and black lines virtually coincide. The smooth red curve is a unit Gaussian for comparison. \n<!-- image --> \nFigure 8. As Figure 7 but now for HEALpixel 7593. The median fpu is 1.18 and the 90% range is 0.21-3.57. \n<!-- image --> \nthe low latitude HEALpixel in Figure 8 than the high latitude HEALpixel in Figure 7 because the low latitude HEALpixel has larger fpus on average. \nThe right columns of Figures 7 and 8 assess how well the estimated distance uncertainties explain the residuals, by plotting the distribution of residual / uncertainty. This is shown using three different representations of the uncertainty. The upper uncertainty, r hi -r med , and symmetrized uncertainty( r hi -r lo ) / 2, shown in blue and black respectively, yield almost identical distributions. For the high latitude HEALpixel 6200 (Figure 7) they are quite close to a unit Gaussian, in particular for the photogeometric estimates. The lower uncertainty, r med -r lo , shown in orange, is negatively skewed (larger tail to negative values), suggesting that the lower uncertainty measure, r lo , is slightly underestimated. This is more noticeable in the low latitude HEALpixel 7593 (Figure 8), where we also see that the photogeometric estimates are slightly more skewed than the geometric ones.', '3.2.2. Quantitative analysis': "To quantify the accuracy of our results we use the median of the fractional distance residual, which we call the bias , and the median absolute of the fractional distance residual, which we call the scatter . These are robust versions of the mean and standard deviation, respectively. For normally-distributed residuals the mean equals the median, and the standard deviation is 1.48 times the median absolute deviation. \nFor HEALpixel 6200 the bias and scatter for the geometric distances over all stars are +0.29e-3 and 0.10 respectively. If we limit the computation of these metrics to the 50% of stars in this HEALpixel with 0 < σ glyph[pi1] /glyph[pi1] < 0 . 20, the bias is +5.3e-3 and the scatter is 0.037. The scatter in this subsample is smaller, as expected. The bias is larger because stars with small fpu tend to be nearer stars, whereas the distance prior is characteristic of all the stars, which are more distant on average. Hence the prior pulls up the distances for the small fpu subsample, leading to a more positive bias. \nFor the photogeometric distances, the bias and scatter over all stars are +5.7e-3 and 0.059 respectively, and for the 0 < σ glyph[pi1] /glyph[pi1] < 0 . 20 subsample are +2.5e-3 and 0.032 respectively. The scatter over the full sample is smaller for the photogeometric estimates than for the geometric ones, because the former benefit from the additional information in the stars' colours and magnitudes. The situation is particularly fortuitous here because of the near-perfect match between the Q G models and the actual distribution of Q G in the data. For the full sample the bias is larger for the photogeometric distances \nthan for the geometric ones, although still small on an absolute scale. For the small fpu subsample the photogeometric distances are not much more accurate than the geometric ones, because the parallax dominates the distance estimate. \nTurning now to the low latitude HEALpixel 7593 (Figure 8), the bias and scatter in the geometric distances over all stars are -0 . 16e-3 and 0.27 respectively. There are two reasons for the larger scatter in this HEALpixel. The first is that the parallax uncertainties are larger: the median parallax uncertainty is 0.32 mas, as opposed to 0.15 mas in HEALpixel 6200. This in turn is because the stars are on average 0.9 magnitude fainter in HEALpixel 7593 (one reason for which is the larger extinction, as is apparent from Figure 3). The second reason is that the median true distance to stars is larger in this low latitude HEALpixel than in the high latitude one (4.0 kpc vs 1.2 kpc; see Figure 1). This may seem counter-intuitive, but is a consequence of distant disk (and bulge) stars at low latitudes that remain visible to larger distances despite the higher average extinction. At higher latitudes, in contrast, there are no distant disk stars, and hardly any halo stars (which are scarce in Gaia anyway). Both of these facts contribute to the larger fpu in the low latitude pixel - median of 1.18, central 90% range of 0.21-3.57 - than in the high latitude HEALpixel median of 0.20, central 90% range of 0.03-1.08. Even if we look at just the 9% of stars in the low latitude HEALpixel with 0 < σ glyph[pi1] /glyph[pi1] < 0 . 20, we get a bias and scatter of +25e-3 and 0.069 respectively, which are still significantly worse than the higher latitude HEALpixel for the same fpu range. \nConcerning the photogeometric distances in HEALpixel 7593, the bias and scatter for all stars are -3 . 8e-3 and 0.17 respectively, and for the 0 < σ glyph[pi1] /glyph[pi1] < 0 . 20 subsample are +20e-3 and 0.062 respectively. For the full sample we again see a significant decrease in the scatter compared to the geometric distances. In a real application we may get less benefit from the Q G prior at low latitudes because our model CQD may differ from the true (unknown) CQD more than at high latitudes, on account of the increased complexity of the stellar populations and interstellar extinction near the Galactic plane.", '3.3. Inferred CQDs': 'We can also assess the quality of our distance estimates by computing Q G = G -5 log 10 r med + 5 and plotting the resulting CQD. We do this for both the geometric and photogeometric distances, for three ranges of fpu, for HEALpixel 6200 in Figure 9 and HEALpixel 7593 in Figure 10. These can be compared to the CQD \nFigure 9. The CQD inferred for mock catalogue HEALpixel 6200 using the median geometric distance (top row) and median photogeometric distance (bottom row) for three ranges of the true fractional parallax uncertainty (fpu): all (left), 0-1.0 (middle) and 0-0.2 (right). The colour scale is a logarithmic (base 10) density scale relative to the highest density cell in each panel. \n<!-- image --> \nFigure 10. As Figure 9 but now for HEALpixel 7593. \n<!-- image --> \nfor the same HEALpixels constructed using the true distances shown in Figure 3. Imperfect distance estimates can only move sources vertically in this diagram as the BP -RP colours are not changed. We see how the inferred main sequence is wider for the larger fpu samples for the geometric distances (left two columns in both plots), but much less so for the photogeometric distances. This is again due to the stablizing influence of the Q G prior. Both distance estimates are able to recover the primary structures: the main sequence, white dwarf sequence, giant branch, and horizontal branch. These plots will be useful when it comes to analysing the results on the real EDR3 data, because they do not involve the truth as a reference.', '4. ANALYSIS OF DISTANCE RESULTS IN EDR3': "We applied our inference code (written in R) to the 1.47 billion sources in Gaia EDR3 that have parallaxes. This required 1 . 6 × 10 12 evaluations of the posteriors and took 57 000 CPU-core-hours. Throughout this section the term 'fpu' of course refers to the measured fractional parallax uncertainty, as we do not know the true parallax.", '4.1.1. Distance distributions and uncertainties': 'Results for our two example HEALpixels are shown in Figures 11 and 12. The two panels in the left column compare the two types of distance estimates. As expected, the photogeometric estimates extend to larger distances (see section 3.2.1 for an explanation). The middle columns plot the ratio of the inferred distance to the inverse parallax distance (corrected for the zeropoint). The latter is of course generally a poor measure of distance because it is not the true parallax, and this is the whole point of using an appropriate prior (see section 1 and references therein). We see that both of our distance estimates converge to 1 /glyph[pi1] in the limit of small fpu. Although the apparent lack of sources at large fpu in the lower middle panels is primarily a plotting artefact (due to the finite density scale), the two samples in the upper and lower panels are not identical, because not all sources have photogeometric distances. For HEALpixel 6200 there are 24 007 sources with geometric distances and 23 829 with photogeometric distances. For HEALpixel 7592 these numbers are 385 902 and 369 608 respectively. \nThe panels in the right columns of Figures 11 and 12 show how the fractional symmetrized distance uncertainty varies with fpu. At small (positive) fpu they are nearly equal for both geometric and photogeometric dis- \ntances, because here the likelihood dominates the posterior. At larger fpu the geometric distances become more uncertain, which is commensurate with their lower expected accuracy. For very large fpu ( glyph[greatermuch] 1) the geometric distances and their uncertainties will be dominated by the prior, which for HEALpixel 7593 has a median of 3.98 kpc and lower (16th) and upper (84th) quantiles of 2.06 kpc and 6.74 kpc respectively (corresponding to a fractional distance uncertainty of 0.59). The photogeometric fractional distance uncertainties tend to be smaller than the geometric ones. This is because the Q G prior (section 2.5) is usually more informative than the distance prior. \nWe extend the axes in the right panels of Figures 11 and 12 to negative fpu, which occur when sources have negative parallaxes. One of the advantages of probabilistic inference is to provide meaningful distances for negative parallaxes (a quarter of all parallaxes in EDR3). Negative observed parallaxes ususally correspond to sources with small true parallaxes, and although such measurements generally have reduced impact on the posterior, they do carry information. They do not yield precise distances, but insofar as the prior can be trusted the posterior and resulting confidence intervals are meaningful. We see from the figures that the precisions are low for both types of distance, but sometimes more constrained for the photogeometric ones due to the additional use of colour and magnitude. In some senses the negative fpu regime is a continuation of the σ glyph[pi1] /glyph[pi1] >> 1 regime (see Figures 3 and 6 of paper II).', '4.1.2. ColourQ G diagrams': "From the inferred median distances we can compute the median Q G via equation 4 and then plot the CQD. This is shown in Figure 13 for HEALpixel 6200 for the geometric distance (top row) and photogeometric distance (bottom row) for three different ranges of the fpu. As interstellar extinction should be low towards this high latitude field (around 0.15 mag in GeDR3mock), Q G glyph[similarequal] M G so this CQD is similar to the colour-absolute magnitude diagram. In all of the panels we see a welldefined main sequence and giant branch, as well as a white dwarf sequence in some of the panels. Comparing the upper and lower panels we see how the photogeometric distances constrain the Q G distribution more than the geometric distance do. The puffing-up of the geometric CQD is due to sources with large fpu: their distances tend to be underestimated (see section 3.2.1) so Q G becomes larger - intrinsically fainter - for a given G (see equation 4). This puffing-up diminishes as we successively reduce the range of fpu, as shown in the middle and right columns of Figure 13. \nFigure 11. EDR3 distance results for HEALpixel number 6200 at ( l, b ) = (285 . 7 · , 34 . 8 · ). The colour scale in the density plots is logarithmic (base 10) relative to the highest density cell in each panel. The top-left panel compares the median geometric and photogeometric distances. The bottom-left panel shows normalized histograms on a linear scale of the median geometric (blue) and photogeometric (orange) distances, compared to the distance prior (black). The middle column shows the ratio of the inferred distance to the inverse parallax distance as a function of the measured fractional parallax uncertainty (fpu). Note that the apparent lack of sources in the lower panel at fpus above about 1.0 is mostly a plotting artefact: regions with too-low a density of sources are white. The two panels in the right column show the fractional symmetrized distance uncertainty also as a function of fpu (note the different scales). This plot is available for all HEALpixels with the auxiliary information online. \n<!-- image --> \nFigure 12. As Figure 11 but for HEALpixel number 7593 at ( l, b ) = (29 . 0 · , 7 . 7 · ). \n<!-- image --> \nFigure 13. The CQD inferred for EDR3 HEALpixel 6200 using the median geometric distance (top row) and median photogeometric distance (bottom row) for three ranges of the measured fpu: all (left), 0-1.0 (middle) and 0-0.2 (right). In total there are 24 007 sources with geometric distances and 23 829 with photogeometric distances. No other filtering has been applied. The colour scale is a logarithmic (base 10) density scale relative to the highest density cell in each panel, so is not comparable across panels. This plot (including also a comparison with the prior CQD) is available for all HEALpixels with the auxiliary information online. \n<!-- image --> \nFigure 14. As Figure 13 but now excluding the 54% of sources in this HEALpixel with G > 19 . 0 mag. \n<!-- image --> \nThe photogeometric CQD for the full fpu range (bottom left panel of Figure 13) shows a conspicuous blob of sources at BP -RP glyph[similarequal] 0 . 5 mag between the MS and WD sequences. These are sources with spuriously large parallaxes, well known from GDR2 (Arenou et al. 2018) and still present, if less so, in EDR3 (Fabricius et al. 2020; Gaia Collaboration 2020b). They are usually close pairs of sources that receive incorrect astrometric solutions, as the EDR3 astrometric model is only suitable for single stars (Lindegren et al. 2020b). Figure 13 shows that spurious parallaxes are less common among the smaller fpu subsample. The Q G prior will often help to constrain the distance of these spurious solutions and thus place them on the correct part of the CQD. This is only partially successful at around BP -RP glyph[similarequal] 0 . 5 mag in this HEALpixel, however, because the distance prior may still be pulling truly very distant sources with larger fpu towards us. \nSources with spurious parallaxes are preferentially faint. To quote from Gaia Collaboration (2020a): 'For faint sources ( G > 17 for 6-p astrometric solutions and G > 19 for 5-p solutions) and in crowded regions the fractions of spurious solutions can reach 10 percent or more.' This can be seen in Figure 14 where we replot the CQD only for sources with G < 19 . 0 mag. This also reduces the puffing-up of the geometric CQD, although some of this reduction is simply because magnitude is correlated with fpu, so a magnitude cut also lowers the fpu. \nThese effects can be seen more prominently in the low latitude HEALpixel 7593, shown in Figures 15 and 16. Due to the larger mean distance of stars at low latitudes (see section 3.2.2), as well as the more complex stellar populations and larger mean extinction (up to 3.5 mag), the CQD is more complex. For the full fpu range, the geometric CQD in Figure 15 is quite washed out, due in part to large fpus and spurious parallaxes, although an extincted red clump is visible. The photogeometric CQDs are cleaner, with a better defined main sequence. The CQD for the G < 19 . 0 mag subsample (Figure 16) again shows the removal of spurious sources. Section 3.2 of Fabricius et al. (2020) analyses spurious astrometric solutions and offers more sophisticated ways of identifying them than a simple magnitude cut.", '4.2. All sources': 'We now look at a representative sample of the entire catalogue. All plots and analyses in this section use a random selection of 0.5% of all sources from each HEALpixel. This has 7 344 896 geometric and 6 739 764 photogeometric distances. \nFigure 17 shows the distribution of distances. As expected, the photogeometric distances extend to larger distances that the geometric one. The fractional symmetrized distance uncertainties as a function of distance are shown in Figure 18 for three different magnitude ranges. As noted earlier, the photogeometric distance uncertainties are generally smaller than the geometric ones, at least for fainter sources. This plot also shows again that photogeometric estimates extend to larger distances.', '4.2.1. ColourQ G diagrams': 'Figure 19 shows the CQD over the whole sky. Because the sample is a constant random fraction per HEALpixel it is numerically dominated by sources at low latitude Galactic latitudes where there can be significant interstellar extinction. This is apparent from the upper diagonal feature - especially clear in the photogeometric panel - which is the red clump stretched by extinction/reddening. The white dwarf sequence appears clearly in the photogeometric CQD. Although some white dwarfs are correctly placed in the CQD by the geometric distances, they are not visible here due to the finite dynamic range of the plotted density scale. Furthermore, for reasons explained in section 3.2.1, faint nearby sources with large fpu tend of have their geometric distances overestimated and therefore their Q G underestimated, thereby pushing them up from the true white dwarf sequence. These plots have not filtered out spurious sources, some of which are clearly visible in the photogeometric CQD as the blob between the upper MS and the white dwarf sequence. Other broad differences between the geometric and photogeometric CQDs were explained in section 3.3.', '4.2.2. Distribution on the sky': 'Figure 20 shows the mean distance of sources (i.e. mean of r med ) in each HEALpixel in our catalogue, as well as the ratio of these in log base 2. Over all HEALpixels the 5th, 50th, and 95th percentiles of the mean of the geometric distances are 1.3, 2.1, and 4.4 kpc respectively. The percentiles for the mean of the photogeometric distances are 2.2, 3.3, and 5.0 kpc. These translate into low ratios of geometric to photogeometric distances in general. Only in the Galactic plane and the bulge are the two mean distances comparable. At high Galactic latitudes the photogeometric average is easily twice as large as the geometric average.', '4.2.3. Galactic spatial distribution': "Figure 21 shows the projected distribution of stars in EDR3 in the Galaxy using our distance estimates. The Sun is at the origin, and we see the expected larger \nFigure 15. As Figure 13 but now for HEALpixel 7593. All sources are shown (no magnitude cut). In total there are 385 902 sources with geometric distances and 369 608 with photogeometric distances. \n<!-- image --> \nFigure 16. As Figure 15 but now excluding the 70% of sources in this HEALpixel with G > 19 . 0 mag to remove spurious sources. \n<!-- image --> \nFigure 17. Distribution of inferred geometric and photogeometric median distances, r med , in EDR3. This plot uses a random sample of 0.5% of all sources in each HEALpixel. \n<!-- image --> \ndensity of sources in the first and fourth Galactic quadrants. Finer asymmetries in the distribution projected onto the Galactic plane (upper panels) are presumably due to both a genuine asymmetry in the Galactic population and Gaia's scanning law. These, as well as nearby dust clouds, also explain the various radial lines pointing out from the origin. The lack of sources in the fan around the positive x-axis in the lower panels is due to extinction in the Galactic plane. The overdensity in the same direction in the upper panels is the projection of the bulge. The lower panels demonstrate the point made earlier (section 3.2.2) about being able to see sources to larger mean distances at lower Galactic latitudes. \nThe high density rays extending below the Galactic plane (lower panels of Figure 21) are in the directions of the Magellanic Clouds. Many stars in these satellite galaxies are in EDR3 - they are some of the densest HEALpixels - yet they are so far away (50-60 kpc) that most have poor (and often negative) parallaxes, such that the inferred geometric distances are dominated by the prior (see appendix B for further discussion). Our photogeometric distances are similarly poor, because we excluded the Magellanic Clouds from the mock CQD out of which our Q G priors are built. This was intentional: anyone interested in estimating distances to sources in the Magellanic clouds can do better than just use Gaia parallaxes and photometry. \nFigure 22 shows the fractional distance uncertainties also in Galactic projection. As expected, the uncertainties generally increase with distance from the Sun, but there are exceptions due to bright distant stars having more precise distances than faint nearby ones. The rays \ntowards the Magellanic clouds also stand out as having larger uncertainties on the whole.", '4.3. Validation using clusters': 'Figures 23 and 24 show our geometric and photogeometric distances and their uncertainties for members of various star clusters. The membership lists have been drawn from paper IV. NCG6254 (= M10) and NGC6626 (=M28) are globular clusters; the rest are open clusters. Recall that our prior does not include star clusters. The horizontal dashed line in each panel shows the inverse of the variance-weighted mean parallax of the members, i.e. a pure parallax distance for the cluster. Both of our distance estimates congregate around this for small, positive fpu, but deviate for large or negative fpu, as one would expect. We generally see a larger deviation and/or scatter for the geometric distance: compare in particular the panels for NGC2437 (=M46) and NGC6254. Despite this, the weighted mean of our distances is often quite close to the pure parallax distance, even for clusters up to several kpc away. \nWe nevertheless emphasise that the inverse of the variance-weighted mean parallax will usually be a better estimate for the distance to a cluster than the mean of our distances. This is because any combination of our individual distances will re-use the same prior many times. If stars have large fpus, this product of priors will dominate and introduce a strong bias into the combined distance. This would particularly affect clusters beyond a few kpc.', '4.4. Comparison to other distance estimates': 'Figure 25 compares our distance estimates for 36 858 red clump (RC) stars with those estimated by Bovy et al. (2014) using high-resolution APOGEE (Majewski et al. 2017) DR16 spectra. This method selects sources using colour, effective temperature, metallicity, and surface gravity, and is calibrated via stellar evolution models and high-quality asteroseismology data. Given the narrowness of the red clump locus in the parameter space, their distances are expected to be precise to 5% with a bias of no more than 2%. \nThe 5th, 50th, and 95th percentiles of fpu for this sample are 0.01, 0.05, and 0.27 respectively, and of G are 10.4, 13.4, and 16.2 mag respectively. The fractional bias and rms of the deviations of our estimates relative to those of Bovy et al. are +0 . 05 and 0.31 respectively for the geometric distances, and +0 . 03 and 0.29 respectively for the photogeometric distances. For reference, the fractional bias and rms of the deviations of the APOGEE red clump estimates relative to the StarHorse (Queiroz et al. 2020) estimates (see next paragraph) for \nFigure 18. Fractional symmetrized distance uncertainty, ( r hi -r lo ) / 2 r med , vs distance for the geometric distance estimates (top) and photogeometric distance estimates (bottom) for the three different G ranges. The colour scale is a logarithmic density (base 10) scale relative to the highest density cell in each panel. This plot uses a random sample of 0.5% of all sources in each HEALpixel. \n<!-- image --> \nthe same sample are +0 . 05 and 0.21 respectively. The parallaxes for this sample are mostly of such high quality that the prior does not strongly effect our posteriors, although we still see a slight improvement in the photogeometric distances over the geometric ones. When counting the percentage of sources where the Bovy et al. estimate is within our upper and lower bounds (+ 7% error margin from Bovy et al.) we find that 65% are compatible with the geometric distances and 69% with photogeometric (we expect 68% to be within 1 σ ). If we do the same for the StarHorse estimates (which also have upper and lower percentiles) for the red clump sample we see that 84% of the StarHorse estimates are within 1 σ pf the Bovy et al. estimates. \nFigure 26 compares our distance estimates for 307 105 stars with those estimated by Queiroz et al. (2020) using their StarHorse method, which uses APOGEE DR16 spectra, multiband photometry, and GDR2 parallaxes. This sample comprises around 1 / 3 main sequence stars; the rest are turnoff star and giants, excluding the red clump stars used in the previous comparison. StarHorse estimates a posterior probability distribution which the authors likewise summarize with a median, so our distance estimates are directly comparable. They report achieving typical distance uncertainties of 11% for giants and 5% for dwarfs. \nThe 5th, 50th, and 95th percentiles of fpu for this sample are 0.002, 0.02, and 0.46 respectively, and of G are 10.2, 13.3, and 16.6 mag respectively. The fractional bias and rms of the deviations of our distance estimates relative to the StarHorse estimates are 0 . 00 and 0.30 respectively for the geometric distances, and -0 . 01 and 0.23 respectively for the photogeometric distances. As this sample extends to larger distances (and larger fpu) than the sample in Figure 25, we begin to see that our geometric distances (and to a lesser extent our photogeometric distances) are smaller than the Starhorse distances beyond about 6 kpc, which is where some of the large fpu sources will have true distances beyond the median of the distance prior.', '5.1. Content': "The distance catalogue includes an entry for all 1 467 744 818 sources in EDR3 that have a parallax. All of these have geometric distances and 1 346 621 631 have photogeometric distances. In comparison there are 1 347 293 721 sources in EDR3 that have defined G-band \nTable 1. The format of the distance catalogue showing results on five fictitious sources. The source id is the same as in EDR3. r med geo , r lo geo , and r hi geo are the median, 16th percentile, and 84th percentile of the geometric distance posterior in parsec. r med photogeo , r lo photogeo , and r hi photogeo are the median, 16th percentile, and 84th percentile of the photogeometric distance posterior in parsec. Flag is defined in Table 2. The distances are shown here rounded to three decimal places, but are provided in the catalogue with 32-bit floating point precision, which guarantees a precision of at least 1 part in 2 24 (17 million). The photogeometric fields can be missing, indicated here with NA . \nFigure 19. The EDR3 CQD over the whole sky using the geometric distances (top) and photogeometric distances (bottom). This plot uses a random sample of 0.5% of all sources in each HEALpixel. These plots include sources of all magnitude and fpu, and so include sources with spurious parallaxes. \n<!-- image --> \nFigure 20. The mean distance of sources per HEALpixel (level 5) for our median geometric distances (top) and median photogeometric distances (middle), and the log 2 ratio of these (bottom), i.e. log 2 (geo/photogeo). This plot uses a random sample of 0.5% of all sources in each HEALpixel. \n<!-- image --> \nFigure 21. Projected distribution of EDR3 stars in the Galaxy using our geometric distances (left) and photogeometric distances (right). The projections are in Galactic Cartesian coordinates with the Sun at the origin. The Galactic North Pole is in the positive z direction and the Galactic centre is at around (+8 , 0 , 0) kpc. Galactic longitude increase anticlockwise from the positive x-axis. The top plots are the view from the Galactic North Pole. The bottom plots are a side view. This plot uses a random sample of 0.5% of all sources in each HEALpixel. \n<!-- image --> \nFigure 22. As Figure 21 but now showing the fractional symmetrized distance uncertainties, i.e. ( r hi -r lo ) / 2 r med . \n<!-- image --> \nFigure 23. Validation of the geometric distance estimates using star clusters (one per panel). Each panel shows the estimated distance, r med , of the cluster members as open circles, as a function of the fractional parallax uncertainty σ glyph[pi1] /glyph[pi1] . The error bars show the lower ( r lo ) and upper ( r hi ) bounds of the confidence intervals. The distance range spans everything in the plotted fpu range, but a few stars lie outside of the plotted fpu range for some clusters. The dashed horizontal line is the inverse of the variance-weighted mean parallax for all cluster members (including any beyond the fpu limits plotted). The solid horizontal (blue) line is the weighted mean geometric distance for the same stars, where the weight is the inverse square of the symmetrized distance uncertainty. The clusters are ordered by increasing parallax distance. \n<!-- image --> \nFigure 24. As Figure 23 but now for photogeometric distances. The solid horizontal (orange) line is the weighted mean photogeometric distance. \n<!-- image --> \n<!-- image --> \nFigure 25. Comparison of APOGEE DR16 red clump star distance estimates from Bovy et al. (2014) to our geometric estimates (top panel) and to our photogeometric estimates (bottom panel) for a common sample of 36 858 sources. \n<!-- image --> \nmagnitudes 5 , BP -RP colours, and parallaxes, and so could in principle have received a photogeometric distance estimate, but did not due to missing Q G prior models. \nThe fields in our catalogue are defined in Table 1. 3% of the sources have changed their source id identifier from GDR2 to EDR3 (Fabricius et al. 2020), so the source id cross-match table dr2 neighbourhood provided with EDR3 should be used to find the best match before doing source-by-source comparisons between the two releases. r med geo in Table 1 is the median ( r med ) of the geometric distance posterior and should be taken as the geometric distance estimate. r lo geo ( r lo ) and r hi geo ( r hi ) are the 16th and 84th percentiles of the posterior and so together form a 68% confidence interval \n<!-- image --> \nFigure 26. Comparison of StarHorse distance estimates from Queiroz et al. (2020) to our geometric estimates (top panel) and to our photogeometric estimates (bottom panel) for a common sample of 307 105 sources. \n<!-- image --> \naround the median. r hi -r med and r med -r lo are therefore both 1 σ -like uncertainties on the distance estimate, and are generally unequal due to asymmetry of the posterior. The fields r med photogeo , r lo photogeo , and r hi photogeo are defined in the same way for the photogeometric distance posterior. \nWe cannot overstate the importance of the uncertainties provided. They reflect the genuine uncertainty in the distance estimate provided by the median. As r hi -r lo is a 68% confidence interval, we expect the true distance to lie outside of this range for a third of the sources. This is the nature of statistical uncertainty and should never be ignored. \nThe field flag is a string of five decimal digits defined in Table 2. Flag A is set to 2 if the source is fainter than the faintest mock source used to make the prior for that HEALpixel. The estimated distances can still be used. Faint stars tend to have poor parallaxes so the distance uncertainties will generally be larger in these cases. The \nTable 2. The flag field in the catalogue is a string of five decimal digits ABBCC. \ntwo digits of flag B refer to the Hartigan dip test, as explained in section 2.6.2. We find that 2% of geometric posteriors and 3% of photogeometric posteriors may not be unimodal according to this test, although this test is not particularly accurate, so this is only a rough guide. Even when the sampled posterior shows a true, significant bimodality (or even multimodality), the 68% confidence interval sometimes spans all modes. \nThe two digits of flag C indicate the nature of the two Q G models that were used to construct the Q G prior. If both numbers are between 1 and 3 then two models bracket the source's colour and were combined by linear interpolation, as explained in section 2.5. If only one of them is 0 then only a single model was used. If both flags are 0 then there is no non-null model within 0.1 mag colour of the source, so the photogeometric posterior is not computed. There are is one special value of this flag: 99 means the star lacked the necessary data to compute the photogeometric distance. \nWe provide additional information on the prior for each HEALpixel in the auxiliary information online, including plots like Figures 1, 3, and 4, and a table with the three parameters of the geometric prior (equation 3).", '5.2. Filtering': 'We have not filtered out any results from our catalogue. Parallaxes with spurious parallaxes remain, as do sources with negative parallaxes (the latter is no barrier to inferring a sensible distance; Bailer-Jones 2015). Any filtering should be done with care, as it \noften introduces sample biases. The flag field we provide is for information purposes; we do not recommend to use it for filtering. Lower quality distances will arise from lower quality input data. These can be identified using the various quality fields in the main Gaia catalogue of EDR3, which is easily crossmatched to our catalogue using the source id field, as shown in the example in section 5.4. Useful quality metrics may be ruwe , parallax over error , and astrometric excess noise , as defined in the EDR3 documentation, where users will also find advice on their use. See in particular section 3.2 of Fabricius et al. (2020) for suggestions for filtering spurious parallaxes. \nParallaxes from the 6p astrometric solutions (identified by astrometric params solved = 95 ) are not as accurate as those from the 5p solutions (Lindegren et al. 2020b), because they were normally used in more problematic situations, such as crowded fields, and are also fainter on average than the 5p solutions. Sources with 6p solutions should not be automatically removed, however. Their larger parallax uncertainties reflect their lower quality. In some applications users may want to filter out sources with large absolute or relative distance uncertainties. One must exercise caution here, however, because uncertainty generally correlates with distance and/or magnitude (among other things), so filtering on these quantities will introduce sample biases.', '5.3. Use cases': "For stars with positive parallaxes and σ glyph[pi1] /glyph[pi1] < 0 . 1, the inverse parallax is often a reasonably good distance estimate for many purposes (when using a suitable parallax zeropoint). This applies to 98 million sources in EDR3. For sources with negative parallaxes or σ glyph[pi1] /glyph[pi1] > 1 (704 million sources), our distances will generally be prior dominated, and while the photogeometric distances could still be useful, the geometric ones are probably less so. The sweet spot where our catalogue adds most value is for the remaining 665 million sources with 0 . 1 < σ glyph[pi1] /glyph[pi1] < 1. \nThe choice of whether to use our geometric or photogeometric distance depends on the specific situation and what assumptions you are willing to accept. In the limit of negligible parallax uncertainties they will agree. At large fractional parallax uncertainties our photogeometric distances will generally be more precise than geometric ones, because they use more information and have a stronger prior (see Figures 11 and 12). Whether they are also more accurate depends on how well the Q G prior matches to the true (but unknown) Q G distribution. The Q G model reflects the stellar population and interstellar extinction in a small patch of sky (HEALpixel \nof area 3.36 sq. deg). The GeDR3mock catalogue and our prior should model these reasonably well at higher Galactic latitudes, but may be less accurate at lower latitudes where extinction is higher and the stellar populations along the line-of-sight are more complicated. If you do not want to rely on colour and magnitude information in the distance inference, use the geometric distance, as the distance prior is less sensitive to the exact stellar population in GeDR3mock. \nSome example use cases are as follows. \n- 1. Look-up of distance (or distance modulus) for particular sources of interest using their source id or other identifier matched to this. EDR3 includes a crossmatch to many existing catalogues. Positional crossmatches can also be done on the EDR3 data site or using TAP uploads, and at other sites that host our catalogue.\n- 2. Identification of sources within a given distance (or distance modulus) range. The confidence intervals should be used to find all sources with a distance r satisfying k ( r med -r lo ) < r < k ( r hi -r med ), where the size of k will depend on the desired balance between completeness and purity of the resulting sample. A better approach would be to use the actual posterior to get a probability-weighted sample. For the geometric distances our posterior can be reconstructed using the geometric distance prior provided for each HEALpixel in the auxiliary information online. Readers interested in using our photogeometric priors should contact the authors.\n- 3. Construction of absolute-colour-magnitude diagrams. One of the reasons that we provide quantiles for our distance estimates is that 5 log 10 ( r med ) -5 is the median of the distance modulus posterior. (This would not be the case if we provided the mean or mode, for example.) Using G from EDR3 one can then compute Q G , and from this the absolute magnitude M G , if the extinction is zero or otherwise known. The same can be done for any photometric band from any other catalogue. When computing Q G in this way with equation 4, the user should remember to apply the correction to the EDR3 G-band magnitude as described in section 8.3 of Riello et al. (2020).\n- 4. For constructing the three-dimensional spatial distribution of stars in some region of space. This may also assist selection of candidates in targeted follow-up surveys. \n- 5. As a baseline for comparison of distance or absolute magnitude estimates obtained by other means.\n- 6. Our distances could be used for another layer of inference, such as computing transverse velocities using also the EDR3 proper motions, although users will need to consider the appropriate error propagation. In particular, if the error budget is not dominated by a single source (e.g. not just the distance), users are advised to infer their desired quantities directly from the original parallaxes, perhaps using the priors provided here. \nUsers should realise that uncertainties in the parallaxes in EDR3 are correlated between different sources to a greater or less degree depending on their angular separations (Lindegren et al. 2020b; Fabricius et al. 2020). Caution must therefore be exercised when combining either the parallaxes or our distances, e.g. averaging them to determine the distance to a star cluster. In such a case the simple 'standard error in the mean' may underestimate the true uncertainty, and the same prior would be used multiple times. One should instead set up a joint likelihood for the sources that accommodates the between-source correlations and solve for the cluster distance directly.", '5.4. Access': 'Our distance catalogue is available from the German Astrophysical Virtual Observatory at http://dc.g-vo. org/tableinfo/gedr3dist.main where it can be queried via TAP and ADQL. This server also hosts a reduced version of the main Gaia EDR3 catalogue (and GeDR3mock). Typical queries are likely to involve a join of the two catalogues. By way of example, the following query returns coordinates, our distances, BP -RP, and the two Q G values using the median distances, for all stars with a low ruwe in a one-degree cone in the center of the Pleiades. This should run in about one second and return 22 959 sources. \n```\nSELECT source\\_id, ra, dec, r\\_med\\_geo, r\\_lo\\_geo, r\\_hi\\_geo, r\\_med\\_photogeo, r\\_lo\\_photogeo, r\\_hi\\_photogeo, phot\\_bp\\_mean\\_mag-phot\\_rp\\_mean\\_mag AS bp\\_rp, phot\\_g\\_mean\\_mag-5*LOG10(r\\_med\\_geo)+5 AS qg\\_geo, phot\\_g\\_mean\\_mag-5*LOG10(r\\_med\\_photogeo)+5 AS gq\\_photogeo FROM gedr3dist.main JOIN gaia.edr3lite USING (source\\_id) WHERE ruwe<1.4\n``` \nAND DISTANCE(ra, dec, 56.75, 24.12)<1 \nA bulk download for the catalogue is also available at the URL given above. Our catalogue will also become available soon together with the full EDR3 catalogue hosted at https://gea.esac.esa.int/archive/ and its partner data centers. At these sites the table names gedr3dist.main and gaia.edr3lite may well be different.', '5.5. Limitations': "When using our catalogue users should be aware of its assumptions and limitations. \n- 1. We summarize the posteriors using only three numbers (quantiles), which cannot capture the full complexity of these distributions. This is more of a limitation for the photogeometric posteriors. The confidence intervals should not be ignored.\n- 2. Most sources in EDR3 have large fractional parallax uncertainties and our distances correspondingly have large fractional uncertainties, especially for the geometric distances.\n- 3. The poorer the data, the more our prior dominates the distance estimates. Our prior is built using a sophisticated model of the Galaxy that includes 3D extinction, but it will not be perfect. If the true stellar population, extinction, or reddening law are very different in reality, our distances will be affected. In section 3.2.1 we explained, using results on simulated data, what biases can occur and why.\n- 4. Sources with very large parallax uncertainties will have a posterior dominated by the prior. The median of this varies between 745 and 7185 pc depending on HEALpixel (Figure 2). Stars with large fpus that truly lie well beyond the prior's median will have their geometric distances underestimated; stars with large fpus that lie closer than the prior's median will have their geometric distances overestimated. As distant stars generally have larger fpu than nearby stars, and distant stars are more numerous, the former characteristic will dominate among poor quality data. This leads to a bias in distance estimates, one that is probably unavoidable (see appendix A). Poor data remain poor data.\n- 5. Our prior is spatially discretized at HEALpixel level 5, i.e. in patches of 3.36 sq. deg. on the sky. The distance prior and CQD change discontinuously between HEALpixels, and this may be visible in sky maps of posterior distances. The Q G priors (constructed from the CQD) are formed by a linear interpolation over colour whenever possible, so in these cases there should be no discontinuity of distance with colour within a HEALpixel.\n- 6. Our inferred distances retain all of the issues affecting the parallaxes, some of which have been explored in the EDR3 release papers (Lindegren et al. 2020b; Fabricius et al. 2020). We applied the parallax zeropoint correction derived by Lindegren et al. (2020a), which is better than no correction or a single global correction, but is not perfect. Any error in this will \npropagate into our distance estimates. The published parallax uncertainties are also probably also underestimated to some degree (Fabricius et al. 2020). Gaia Collaboration (2020a) and Riello et al. (2020) report some issues with the EDR3 photometry, such as biased BP photometry and therefore BP -RP colours for very faint sources, which could affect our photogeometric distances. These distance estimates additionally suffer from any mismatch between the published EDR3 photometry and the modelling of this - in particular the passbands - used in the GeDR3mock catalogue, which forms the basis for our Q G priors. 6 Note that we applied the G-band magnitude correction to the EDR3 photometry as described by Riello et al. (2020). \n- 7. We implicitly assume that all sources are single stars in the Galaxy. Our distances will be incorrect for extragalactic sources. The geometric distances will be wrong for unresolved binaries if the parallax for the composite source is affected by the orbital motion. Even when this is not the case the photogeometric distance may still be wrong, because the G-band magnitude will be brighter than the Q G prior expects (binaries were not included in the prior).\n- 8. By design we infer distances for each source independently. If a set of stars is known to be in cluster, and thus have a similar distance, this could be exploited to infer the distances to the individual stars more accurately than we have done here. In its most general form this involves a joint inference over multiple sources. Various methods exist in the literature for doing this, such as Palmer et al. (2014), CantatGaudin et al. (2018), and Olivares et al. (2020). Likewise, in order to estimate the distance to the cluster as a whole, one should be aware that averaging our individual distances will compound the prior. If the fpu of the individual sources is large, this product of priors would dominate the distance estimate more than desired. A joint inference can easily be set up overcome this.", '6. SUMMARY': "We have produced a catalogue of geometric distances for 1.47 billion stars and photogeometric distances for 92% of these. These estimates, and their uncertainties, can also be used as estimates of the distance modulus. Geometric distances use only the EDR3 parallaxes. Photogeometric \n- 6 We compared simulations of the G-band magnitude and the BP-RP colour between the GeDR3mock passbands and those published for EDR3, using isochrones at 4 Myr and 1 Gyr. The differences in the G magnitudes are below 6 mmag, except for sources bluer than -0 . 15, where it can be as high as 700 mmag. For BP -RP using the BP bright band (in GeDR3mock), the difference is around 10 mmag, but up to 25 mmag for sources with BP -RP > 1 . 2 mag and up to 100 mmag for sources with BP -RP < -0 . 2 mag. For BP faint, the BP -RP difference is around 20 mmag, but up to 60 mmag for sources with BP -RP > 0 . 5 mag and up to 100 mmag for sources with BP -RP < -0 . 15 mag. \ndistances additionally use the G magnitude and BP -RP colour from EDR3. Both types of estimate involve directiondependent priors constructed from a sophisticated model of the 3D distribution, colours, and magnitudes of stars in the Galaxy as seen by Gaia, i.e. accommodating both interstellar extinction and a Gaia selection function. Tests on mock data, but moreover validation against independent estimates and open clusters, suggest our estimates are reliable out to several kpc. For faint or more distant stars the prior will often dominate the estimates. We have identified various use cases and limitations of our catalogue. \nOur goal has been one of inclusion: to provide distances to as many stars in the EDR3 catalogue as possible. This has required us to make broad, general assumptions. If one focuses on a restricted set of stars with some approximately known properties, it will be possible to construct more specific priors, and to use these to infer more precise and more accurate distances. Better distances may also be achievable by using additional data, such as spectroscopy or additional photometry. \nWe thank the IT departments at MPIA and ARI for computing support. This work was funded in part by the DLR (German space agency) via grant 50 QG 1403. It has made use of data from the European Space Agency (ESA) mission Gaia (http://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, http://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. This research made use of: TOPCAT, an interactive graphical viewer and editor for tabular data (Taylor 2005); Vaex, a tool to visualize and explore big tabular data (Breddels & Veljanoski 2018); matplotlib, a Python graphics library (Hunter 2007); HEALpix (G'orski et al. 2005) and healpy (Zonca et al. 2019); the NASA Astrophysics Data System; the VizieR catalogue access tool, CDS, Strasbourg. \nFacility: Gaia", 'A. THOUGHTS ON A BETTER DISTANCE PRIOR': "The strong dependence of the geometric posterior on the distance prior in the limit of large parallax uncertainties is an unavoidable consequence of inference with noisy data. We saw something similar in paper IV. This leads to a distance bias mostly for distant stars with large fpu. Could this be avoided? Conceptually one would like a distance prior that depends on the true fpu, but this is impossible because the true parallax is not known. One may be tempted to use the measured fpu instead, but this is not what we want: a star with a large true fpu could have a small measured fpu due to noise, and thereby be treated incorrectly. Its use is also be theoretically dubious because it places the parallax - a measurable - in the prior, as well as in the likelihood. We experimented with using a prior conditioned on σ glyph[pi1] , but found that this did not help (see the technical note GAIA-C8-TN-MPIA-CBJ-089 with the auxiliary information online). One may achieve something close to what is desired by simply shifting the distance prior to greater distances, so that it better represents stars with a larger true fpu, which is where the prior is needed more. Yet this would detrimentally affect the distance estimates for nearby stars. It seems a poor trade-off to sacrifice accuracy on high-quality data for a better prior on low-quality data. Conditioning the prior on the star's magnitude may help, and this is what our photogeometric distances do (section 2.4).", 'B. THE LIMIT OF POOR PARALLAXES': "We tend to think that a large fpu means that the likelihood is uninformative and that the posterior converges towards the prior. Consider a red clump star in the LMC with a true parallax of 0.02 mas and a typical parallax uncertainty of 0.2 mas for a star with G = 19mag. The true fpu is 0 . 2 / 0 . 02 = 10. Let's assume initially that we actually measure a parallax of 0.02 mas, i.e. we have an measured fpu of 10. (Of course in this lucky case the inverse parallax would be the correct distance, but it's very rare in practice.) In the LMC HEALpixel 8275 our distance prior has a median of 1.2 kpc because we exclude the LMC from our prior, so we might expect to see many sources with this inferred distance. In fact we see many sources with larger inferred distances (see the plot with the auxiliary online information). The reason is that the likelihood of a measurement of 1 mas (corresponding to a distance of 1 kpc) is still at 4.9 σ and therefore quite unlikely. This shows that even when the fpu is large the parallax can be quite informative. \nOne should remember, however, that our inference never sees the true parallax but only the measured parallax, which is normally distributed around the unknown true value (with a standard deviation which is also only estimated). So it is quite likely that our measurement of the above red clump star gives us a parallax measurement of, say, 0.4 mas. In that case the measured fpu is 0.5 and the likelihood of 1 mas, i.e. a 1 kpc distance, is only 3 σ away from this measurement. Taking the parallax measurement into account essentially redistributes probability mass into the wings of the likelihood and therefore to higher and lower (also negative) parallax values. Given the truncation of negative parallaxes when calculating the posterior, this implies that the median distance estimate is lower for the true measurements, compared to the idealised inference using the true parallax. Similarly, one should be careful not to interpret plots involving the measured fpu as though it were the true fpu.", 'REFERENCES': 'Anders, F., Khalatyan, A., Chiappini, C., et al. 2019, A&A, \n628, A94, doi: 10.1051/0004-6361/201935765 \nArenou, F., Luri, X., Babusiaux, C., et al. 2018, A&A, 616, \nA17, doi: 10.1051/0004-6361/201833234'}

2024arXiv240706166S

Hopes are being widely expressed that C2023 A3 could become a nakedeye object about the time of its perihelion passage in late 2024. However based on its past and current performance the comet is expected to disintegrate before reaching perihelion. Independent lines of evidence point to its forthcoming inevitable collapse. The first issue which was recently called attention to by I. Ferrin is this Oort cloud comets failure to brighten at a heliocentric distance exceeding 2 AU about 160 days preperihelion accompanied by a sharp drop in the production of dust Afrho. Apparent over a longer period of time but largely ignored has been the barycentric original semimajor axis inching toward negative numbers and the mean residual increasing after the lightcurve anomaly suggesting a fragmented nucleus whose motion is being affected a nongravitational acceleration and an unusually narrow teardrop dust tail with its peculiar orientation implying copious emission of large grains far from the Sun but no microscopic material recently. This evidence suggests that the comet has entered an advanced phase of fragmentation in which increasing numbers of dry fractured refractory solids stay assembled in dark porous blobs of exotic shape becoming undetectable as they gradually disperse in space.

2024-07-01T00:00:00Z

['arXiv:2407.06166', '2024arXiv240706166S', '10.48550/arXiv.2407.06166']

['Astrophysics - Earth and Planetary Astrophysics']

Inevitable Endgame of Comet TsuchinshanATLAS C2023 A3

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2407.06166.pdf

{'INEVITABLE ENDGAME OF COMET TSUCHINSHAN-ATLAS (C/2023 A3)': 'Zdenek Sekanina La Canada Flintridge, California 91011, U.S.A.; ZdenSek@gmail.com \nVersion July 9, 2024', 'ABSTRACT': "Hopes are being widely expressed that C/2023 A3 could become a naked-eye object about the time of its perihelion passage in late 2024. However, based on its past and current performance, the comet is expected to disintegrate before reaching perihelion. Independent lines of evidence point to its forthcoming inevitable collapse. The first issue, which was recently called attention to by I. Ferrin, is this Oort cloud comet's failure to brighten at a heliocentric distance exceeding 2 AU, about 160 days preperihelion, accompanied by a sharp drop in the production of dust ( Af ρ ). Apparent over a longer period of time, but largely ignored, has been the barycentric original semimajor axis inching toward negative numbers and the mean residual increasing after the light-curve anomaly, suggesting a fragmented nucleus whose motion is being affected by a nongravitational acceleration; and an unusually narrow, teardrop dust tail with its peculiar orientation, implying copious emission of large grains far from the Sun but no microscopic dust recently. This evidence suggests that the comet has entered an advanced phase of fragmentation, in which increasing numbers of dry, fractured refractory solids stay assembled in dark, porous blobs of exotic shape, becoming undetectable as they gradually disperse in space. Subject headings: individual comets: C/2023 A3; methods: data analysis", '1. INTRODUCTION': "Initially discovered at the Purple Mountain Observatory's XuYi Station, China, on 9 January 2023, then lost and discovered a second time at the Asteroid TerrestrialImpact Last Alert System (ATLAS) search project's station at Sutherland, South Africa, 44 days later (Green 2023), comet C/2023 A3 has been monitored for nearly 17 months at the time of this writing and it still has more than two months to go before reaching perihelion. Since its minimum distance from the Sun is to be merely 0.39 AU, the comet has been widely predicted to become a very bright, possibly naked-eye object around perihelion, especially in early October, in part because of effects of forward scattering of sunlight by microscopic dust at phase angles near 180 · . \nIn the past, such predictions for other comets with similar parameters (discovered long before perihelion as a fairly bright object, approaching the Sun to a small fraction of the Sun-Earth distance) failed miserably on a number of occasions, but proved correct in some cases. Performance prognostication for individual objects is a high-risk branch of cometary science, which is very slowly being mastered in the process of our gradually learning to understand the whims of comets. Yet, it appears that at least some comets do signal early messages that correlate with their subsequent behavior. Timely recognition and proper interpretation of these messages should pave the way for a more successful future in this unusual field of scientific endeavor. \nComet C/2023 A3 appears to be a member of the class of undersized Oort cloud objects in orbits with small perihelion distances, many of which turned out to be unable to cope with the physical conditions they were subjected to in the inner Solar System and disintegrated as a result of progressing fragmentation. In the following I address the traits of comet C/2023 A3 that appear to be of the same or similar nature.", '2. THE LIGHT CURVE, DUST CONTENT, AND PRODUCTION OF WATER': "With all pre-discovery images the comet's light-curve database covers about 27 months. The dependence of the total brightness, normalized to a unit geocentric distance, /Ifractur ∆ , on the heliocentric distance, r , has often been expressed by a power law, /Ifractur ∆ = /Ifractur 0 r -n , whose parameters are the absolute magnitude H 0 = -2 . 5 log /Ifractur 0 and exponent n , which determines the slope d log /Ifractur ∆ /d log r . 1 For example, S. Yoshida 2 has used H 0 = 4 . 5 and n = 4 to fit all magnitude observations made between 2 March 2023 and 24 January 2024. Similarly, A. Kammerer 3 has found H 0 = 4 . 7 and n = 3 . 96 from 384 data reported by 64 observers until the end of April 2024. \nBecause of time constraints, I have not investigated the entire light curve of the comet. From the standpoint of predicting the comet's evolution in the near future, it is the recent activity that is most critical. To assess it, I deemed it sufficient to inspect a representative light curve based on the data reported by a single observer, but on the condition of their maximum possible homogeneity to mitigate a contamination as much as possible. \nFrom the Comet Observation Database 4 (COBS) website I eventually selected a set of 48 unfiltered CCD totalmagnitude observations made by A. Pearce with a 35-cm f/5 Schmidt Cassegrain between 21 January and 13 June 2024. Although the contribution from the outer coma may be unaccounted for, this is not a major problem, as the emphasis is on rather abrupt brightness variations that begin at the nuclear condensation. 5 \n- 2 See http://www.aerith.net/comet/catalog/search.cgi .\n- 3 See https://fg-kometen.vdsastro.de .\n- 4 See https://cobs.si . \nFigure 1. Light curve of comet Tsuchinshan-ATLAS between 250 and 106 days before perihelion (21 January through 13 June 2024), as portrayed by a set of 48 CCD total-magnitude observations made by A. Pearce with his 35-cm f/5 Schmidt-Cassegrain. The ordinate is the total observed CCD magnitude of the comet normalized to a unit geocentric distance, H ∆ . Surprisingly, the comet was brightening at an accelerating rate before 165 days preperihelion (15 April 2024), but began to fade afterwards. \n<!-- image --> \nTo obtain an insight into the comet's activity variations over the past five months covered by Pearce's observations, I plotted the magnitudes, normalized to 1 AU from the Earth, H ∆ , as a function of time. This plot, displayed in Figure 1, has immediately raised the red flag because it shows a dramatic change at 165 days before perihelion, on 15 April 2024, when the intrinsic brightness reached a peak: the comet was brightening at a rate that appears to have been increasing with time before this date, but began to fade afterwards. This finding supports recent concerns expressed by I. Ferrin 6 (e.g., his message #32307). \nTo further investigate the comet's activity, I examined the same dataset on a plot that displays the normalized magnitude H ∆ as a function of heliocentric distance r (on a logarithmic scale). This plot, presenred in Figure 2, suggests that the early segment of the light curve, before the brightness peaked, may have consisted of two parts. Up to 190 days before perihelion (21 March 2024), when the comet was 3.3 AU from the Sun, its light curve was described by the absolute magnitude H 0 = 5 . 6 ± 0 . 4 and the slope parameter n = 3 . 5 ± 0 . 3. Hence, the absolute brightness was only about 1 mag fainter than given by both Yoshida and Kammerer, as mentioned above, while the slope was slightly lower than the results of the two authors indicated. This suggests that up to 190 days before perihelion the comet showed only a moderate rate of progressive fading compared to earlier times. \nThe latter part of the period of brightening in Figures 1 and 2, starting about 190 days before perihelion and 3.3 AU from the Sun, shows a rapid reactivation of the comet. One cannot call it an outburst because the rate of reactivation was not steep enough. However, compared to the previous part of the light curve, n doubled to n = 7 . 2 ± 0 . 3 and H 0 shot up by about 5 magnitudes to H 0 = 0 . 8 ± 0 . 4. The episode could perhaps be described as a surge of activity. It is proposed that, if not \nFigure 2. Intrinsic brightness of comet Tsuchinshan-ATLAS as a function of heliocentric distance. The set of magnitude data from Figure 1 now shows three likely stages of evolution, as the comet's heliocentric distance was decreasing from 4 AU to 2 AU. At > 3.3 AU the light curve followed a law that did not differ too much from the law that applied far from the Sun. Between 3.3 AU and 3 AU the comet underwent a surge of activity, caused apparently by prolific fragmentation of the nucleus, which resulted in doubling the value of the slope parameter n and a jump in the absolute brightness by 5 magnitudes. By the time the comet reached 3 AU the out-of-control fragmentation of the nucleus ceased on the global scale, with further fracturing proceeding locally. This trend is seen to have continued for more than two months and is expected to continue until the ultimate collapse of the comet still before it reaches perihelion. \n<!-- image --> \nearlier, it was about 21 March that the nucleus began to fragment profusely. Emission from a rapidly expanding surface area of short-lived activity could explain the stepped-up brightening, which continued over a period of 25 days (from 21 March to 15 April 2024) at heliocentric distances from 3.3 to 3.0 AU, ending up with the peak at 165 days before perihelion. Once the source of extra emission got exhausted, the remaining mass began to underperform, resulting in the light curve's downturn. Occasional flare-ups caused by further episodes of local fragmentation are likely to be the cause of large scatter among the data after 15 April. This scenario has now continued for more than two months and is expected to continue until the ultimate disintegration of the nucleus. The downward trend is described by somewhat uncertain parameters: H 0 = 11 . 8 ± 0 . 4 and n = -1 . 8 ± 0 . 4. \nThe evidence from the light curve is strongly supported by the data on a dust-content parameter Af ρ , introduced by A'Hearn et al. (1984) as a proxy to estimate the dust production rate. A website by Cazadores de Cometas 7 (Comet Hunters), which contains, among other information, an Af ρ database by mostly Spanish observers (including the Canary Islands), offers more than 150 data points for comet Tsuchinshan-ATLAS, starting in late February 2023. One of the plots shows that, reduced to a coma 10 000 km in radius, Af ρ gradually increased from a minimum of ∼ 500 cm in early March to ∼ 1800 cm in mid-August 2023. Between December 2023 and February 2024 Af ρ stayed nearly constant at 4000 cm, but in March it started increasing sharply, reaching a peak of ∼ 9000 cm by mid-April, followed by a steep drop. \nFigure 3. Variations in the dust production parameter Af ρ of comet Tsuchinshan-ATLAS between 15 May and 17 June 2024 (135 to 102 days before perihelion). The large scatter is not necessarily caused in its entirety by errors of observation, but could in part be a product of sudden dust release during episodes of fragmentation, as depicted for the two major features between 120 and 112 days before perihelion. The Af ρ data points near the bottom appear to decrease systematically with time at a crude average rate of ∼ 14 cm per day. \n<!-- image --> \nThe curve of Af ρ fairly closely parallels the light curve. Particularly interesting are the variations between midMay and mid-June (from 135 to 102 days before perihelion), which are reproduced in Figure 3. The considerable scatter, similar to the scatter in the light curve, is believed to be in part triggered by short-term release of dust during minor episodes of nuclear fragmentation. The rest of the scatter could be errors of observation. The baseline data in Figure 3 are shown to follow an approximately linear systematic decrease with time at a rate of about 14 cm day -1 , with an extrapolated value of more than 800 cm at perihelion. \nIt has been known for a fairly long time that some faint comets in orbits with small perihelion distance perish shortly before perihelion (Bortle 1991), although Jewitt & Luu (2019) did recently report a comet that disintegrated at 1.9 AU from the Sun. My investigation of the problem led to a conclusion that the objects that perish are always intrinsically faint Oort cloud comets depleted in dust (Sekanina 2019). To be able to predict the chance of survival from early observations, I introduced a synoptic index for perihelion survival , /Ifractur surv , given by a formula \n/Ifractur surv = H 0 -5 . 7 -35 6 log q -5 3 log( Afρ ) 0 , (1) \nwhere H 0 is the absolute magnitude, q is the perihelion distance in AU, and ( Af ρ ) 0 (in cm) is strictly to be taken at 1 AU from the Sun and at a zero phase angle. A comet is predicted to survive perihelion when /Ifractur surv < 0 and vice versa. \nIn the following section I will demonstrate that comet Tsuchinshan-ATLAS unquestionably has arrived from the Oort cloud, so that application of the synoptic index is relevant. Even though the quantities that determine the index are rather uncertain at present, it nonetheless is of inrerest to compute a preliminary value of /Ifractur surv . To \ndo so, I first of all adopt H 0 = 11 . 8 from the review of the light curve in Figure 2. Next, from its dependence on time proposed in Figure 3 I compute Af ρ at time t 0 when the comet reaches 1 AU from the Sun preperihelion, t 0 -t π = -38 days. The result is ( Af ρ ) 0 = 1360 cm. With the perihelion distance of 0.391 AU, the synoptic index amounts to \n/Ifractur surv = +3 . 3 (2) \nand the comet is predicted to have no chance of surviving perihelion. \nOf course, one may question whether the linear extrapolation of Af ρ is appropriate and the employed values of H 0 and ( Af ρ ) 0 compatible. As a check, I fit the relevant data from Figure 3 by a power law of heliocentric distance r , similar to the light curve: \nAfρ ( r ) = ( Afρ ) 0 r -z (3) \nand obtain by least squares ( Af ρ ) 0 = 1170 ± 40 cm and z = -0 . 90 ± 0 . 15. The synoptic index now equals \n/Ifractur surv = +3 . 4 , (4) \nvery close to the result (2). \nIn summary, from the standpoint of the perihelion survival of comet Tsuchinshan-ATLAS, both its systematic fading documented by the light curve between mid-April and mid-June 2024 and the continuing drop of Af ρ are extremely worrisome. The synoptic index predicts that the comet will perish before perihelion, as an instrumental correction of H 0 could not possibly exceed 1-2 mags. \nIndependent of this conclusion, I should note that from the spectra taken on 31 May 2024 when the comet was 2.33 AU from the Sun, Ahuja et al. (2024) obtained two major results. One, the production rate ratio Q (C 2 )/ Q (CN) < 0.32, suggesting a carbon depleted comet; and two, Q (H 2 O)=(1 . 50 ± 0 . 37) × 10 28 s -1 , an unexpectedly high rate of water production. At 2.33 AU from the Sun, a spatially averaged sublimation rate of water ice is 1 × 10 17 cm -2 s -1 , suggesting that the total sublimating surface was 15 km 2 ! As only a small fraction of a comet's surface is known to be active, a monolithic nucleus would have been close to 10 km across - an absurd result. An unresolved cloud of sublimating bouldersized fragments offers a more plausible solution. \nConsider ν fragments of an equal volume. Let the effective diameter of the parent nucleus before a fraction χ of its volume fragmented be D par and the effective diameter of each fragment D frg . Then \nχD 3 par = νD 3 frg . (5) \nLet the sublimating fraction of the surface of the parent nucleus be f par /lessmuch 1. Since the previously hidden icerich surface is, because of fragmentation, being exposed to sunlight for the first time, one can assume that for an overwhelming majority of fragments their entire surface facing the Sun is sublimating, so that f frg = 1 2 . The active surface area before fragmentation is \nS par = πf par D 2 par , (6) \nbut after fragmentation it is \nS frg = 1 2 πνD 2 frg = 1 2 πν 1 3 χ 2 3 D 2 par , (7) \nwhere I inserted for D frg from Equation (5). Here S frg is \nTable 1 \nSets of Orbital Elements for C/2023 A3 Computed by S. Nakano (Epoch of Osculation 2024/10/17) \nNote. \nknown from the observation. The ratio of S frg /S par is \nS frg S par = ν 1 3 χ 2 3 2 f par . (8) \nFrom Equation (7) the number of fragments needed to satisfy the observed large active surface S frg equals \nν = ( 2 S frg π ) 3 D -6 par χ -2 , (9) \nand the effective diameter of a fragment is \nD frg = πD 3 par χ 2 S frg . (10) \nThe counterintuitive inverse correlation between ν and χ is explained by the fragment size varying as χ . \nIt is enlightening to see the results after inserting some plausible numbers. The sublimation constraints dictate that S frg = 15 km 2 . From the orbital considerations in the next section one should rather prefer that D par be below 1 km. Then, for example, fracturing 75 percent of the volume of a nucleus 0.5 km in diameter will produce ∼ 100000 fragments, each 10 meters across. In reality, the fragments have of course some size distribution. The active-surface ratio S frg /S par becomes 190, if f par = 0 . 1, so that S par is less than 0.1 km 2 .", '3. THE ORBIT': "In nearly 17 months that have elapsed since its second discovery, the comet was observed for astrometry more than 4000 times. In addition, almost 30 pre-discovery images were subsequently located, obtained between 9 April and 22 December 2022 at three observatories: Panoramic Survey Telescope and Rapid Response System (Pan-STARRS 2), Haleakala Observatory (code F52); Dark Energy Camera Project (DECam), Cerro Tololo Inter-American Observatory (code W84); and Zwicky Transient Facility (ZTF), Palomar Observatory (code I41). As a result, at the time of this writing the observed arc of the orbit equals close to 27 months. One would expect that the orbital elements, including the semimajor axis, should by now be determined with very high accuracy. As described in some detail in the following, inspection of the published computation results suggests that this indeed is the case, even though there is a hitch. \nI start with the work by S. Nakano, who has been improving the orbital elements of this comet on a number of occasions. The selected data from his results, relevant to the issues of interest here, are summarized in Table 1. Presented in the individual columns are the reference; the reciprocal barycentric original semi-major axis, (1 /a b ) orig in AU -1 (the most important column); the mean residual; the number of astrometric observations used; the orbital arc covered by these observations (in days); the date of the last observation; and the difference between the reciprocal heliocentric osculating semi-major axis (whose value depends on the choice of the epoch of osculation) and the quantity in column 2, u b = (1 /a ) osc -(1 /a b ) orig . Since Nakano has always been using the standard epoch of osculation, u b should essentially be constant. \nThe tabulation of u b is intended to support the proposed explanation of a discrepancy with the results obtained at the Minor Planet Center (MPC), which are with two additional independent results listed in Table 2, whose format is similar to that of Table 1. Comparison of the orbital solutions that include recent observations shows an excellent agreement between the values of (1 /a b ) orig by JPL and by Williams, their difference amounting to less than 1 unit of 10 -6 AU -1 , within the errors of observation. Next comes Nakano's most recent solution, which gives a value that is more negative by 11-12 units of 10 -6 AU -1 . This is not surprising, as Kr'olikowska & Dones' (2023), who examined differences among the various methods of orbit determination, have found that Nakano's values of (1 /a b ) orig have systematically been lower by about 10 units. Whatever the source of this discrepancy, once it is accounted for, Nakano's result agrees with the other two to 1-2 units as well. \nThe huge problem appears to be with the MPC orbits, whose values of (1 /a b ) orig exhibit major deviations. The discrepancy is so large that its source must be a major oversight. I note that the first MPC value of (1 /a b ) orig , in particular, is very close to Nakano's value of u b . The possible confusion between these two quantities could explain the problem. On the other hand, this just may happen to be a coincidence, because the respective epochs of osculation are more than six months apart and u b may have changed substantially. While the source of the discrepancy remains unclear, I accept that the barycentric original orbit of the comet based on the observations available at present is essentially a parabola. \nTable 2 Additional Sets of Orbital Elements for C/2023 A3", 'Notes.': "- a See https://minorplanetcenter.net/db search .\n- b See https://ssd.jpl.nasa.gov . Automatically updated every few days; this entry dated 23 June 2024. \nThe orbit determination of comet Tsuchinshan-ATLAS involves more subtle issues related to the question of the object's survival. For example, no nongravitational orbital solution has as yet been published, apparently because the gravitational solutions have been entirely satisfactory. And this may have been so, at least in part, because the comet has so far been relatively far from the Earth and the amplitude of potential systematic residuals small. This will of course change in the future and so will other things. \nOne point of concern that I see from Nakano's results is a recent trend to more negative values of (1 /a b ) orig . The change is about 30 units or more than 7 σ of the error in the 5059 run and more than 20 σ of the error in the 5247 run. If this trend should continue, it could represent an independent confirmation of progressive fragmentation of the nucleus. This trend would necessitate to eventually incorporate the nongravitational terms. \nI call attention to this effect for two reasons. One is that the trend toward more negative values appears in the first orbital solutions that incorporate the observations made following the light-curve anomaly in midApril. The other reason is that the mean residual from these solutions has increased relative to the previous solutions by 0 '' .1 or about 30 percent. That is a lot and it indicates that for some reason the nuclear condensation has been measured with lesser accuracy. I will return to this issue and its implications in the following section. \nThe observers in the south have at most until the end of July to monitor the comet before it disappears in the solar glare; it will then still be more than 1 AU from the Sun. It should be seen no more afterwards.", '4. THE TAIL': "The tail completes the list of peculiarities that are being displayed by this unusual object. Plasma tails are seldom detected farther than 2 AU from the Sun, so it is not surprising that the tail of comet Tsuchinshan-ATLAS is definitely composed of dust. Except on special occasions that I address later, preperihelion dust tails are - because of dynamical constraints - rather narrow, but seldom as narrow and teardrop-shaped as displayed by this object. They typically exhibit two characteristics: have a tendency, however slight, to broaden with increasing distance from the comet's head and extend to a maximum distance along an axis that makes a small angle with the prolonged radius vector. \nThe tail of comet Tsuchinshan-ATLAS defies either of the two rules. Fortunately, this controversial behavior provides interesting information about the comet's history and activity. Tails of this appearance and outlines, especially the tendency of getting narrower farther from the head, have in the past been observed primarily among comets arriving from the Oort cloud in orbits with large perihelion distances. The tails of these comets deviated considerably from the radius vector and their investigation had a rather bizarre history (e.g., Osterbrock 1958, Brandt 1961, Roemer 1962, Belton 1965, Sekanina 1973, Meech et al. 2009). \nTo describe the physical significance of the odd-shaped tail of comet Tsuchinshan-ATLAS, I have selected one of many available images (which all look alike) from the recent past and I am going to explain the meaning of the displayed features below. \nThe selected picture, in Figure 4, has been cropped from a false-color computer-processed rendition of the image taken by R. Naves, of the Observatorio Montcabre, with a 30-cm f/9 reflector on 4 June 2024. It is a long, nearly two-hour exposure, which is important in that it shows the true extent, in excess of 8 ' , of the very delicate tail. The price paid is the apparent saturation of the central coma, whose projected radius of 13 '' is equivalent to 17 000 km. \nFigure 4 consists of two parts: the upper panel displays the image, while the lower panel offers a guide to its interpretation. Detailed information provided by Naves on the details of his observation, including the orientation and scale, has been most helpful for an accurate description of the properties of the tail. Besides its getting more narrow with increasing distance from the head, a rather stunning trait is its large angular separation of 16 · from the projected radius vector, the maximum dynamically allowed being 22 · .5 but plausible only 19 · . More than 100 days preperihelion in an orbit that comes to less than 0.4 AU from the Sun, this is by no means common. \nThe key to understanding the length, shape, and orientation of a dust tail is a computation of the motions of ejected particles relative to the nucleus as a function of their bulk properties and time of ejection. The properties (size, shape, bulk density, morphology, and composition) determine the magnitude of the radiation pressure acceleration that a particle is subjected to after ejection. The most widely varying parameter is the particle size and the acceleration varies inversely as the size. \n<!-- image --> \nFigure 4. Dust tail of comet Tsuchinshan-ATLAS. Upper panel: Cropped version of the computer processed image of the comet taken by R. Naves with a 30-cm f/9 reflector of the Observatorio Montcabre, Spain, on 2024 June 4.91744 UT, the exposure time of 103 minutes. North is up, east to the left. The image is 12 ' .6 on the diagonal. The tail extends more than 8 ' in a position angle of 98 · . Note that the tail gets more narrow as the distance from the head increases, having the appearance of a teardrop. Lower panel: Four synchrones that mimick the axial directions of dust grains ejected from the nucleus (the large open circle) at different times. The heavy line, fitting the direction of the observed tail, shows the ejecta about 400 days old at the time of observation, which left the nucleus at some 6.6 AU ( ∼ 500 days before perihelion); the other three synchrones (the thinner lines) show the predicted locations of the tail that would be occupied by dust ejected at 10.6 AU ( ∼ 1000 days before perihelion); at 2.8 AU (155 days before perihelion); and at 2.6 AU (135 days before perihelion), respectively. Each synchrone contains particles of different sizes, the greater ones nearer the nucleus, the smaller ones at larger distances from it. The locations of particles of specific sizes are marked by a dot, the magnitude of the radiation pressure acceleration β (in units of the solar gravitational acceleration) being indicated. With β essentially varying as an inverse effective diameter of a particle, the plot suggests that the observed tail consisted of submillimeter-sized and larger particles that could not be lifted from the nucleus by sublimating water ice at heliocentric distances greater than 6 AU. On the other hand, the image shows quite clearly that the comet did not eject microscopic dust (submicron- and micron-sized particles) in any nontrivial amounts in the weeks and months preceding the time of observation, as the ordinary tail is completely missing. This anomaly has important implications for the process of fragmentation that this comet has been subjected to. (Image courtesy of R. Naves, Observatorio Montcabre, Spain.) \n<!-- image --> \nTo describe the main features of the dust-emission history of comet Tsuchinshan-ATLAS from Figure 4, one needs to compute a number of synchrones, loci of particles of different sizes ejected at equal times. It turns out that the production of dust has been characterized by two anomalies, one of which is strong emission of submillimeter-sized and larger particles at heliocentric distances around 6.6 AU, some 500 days before perihelion. As the synchrones are crowded, one cannot determine the effective ejection time from the tail orientation accurately, but Figure 4 shows no apparent contribution from heliocentric distances beyond ∼ 10 AU or more than 1000 days before perihelion. Indeed, the dust ejected with a zero velocity lined up on 4 June 2024 in a position angle of 98 · .2 when released 500 days before perihelion (6.56 AU), in 99 · .5 when 400 days (5.61 AU), in 97 · .3 when 600 days (7.46 AU), and in 95 · .4 when 1000 days (10.63 AU). The position angle is probably measured to slightly better than ± 1 · in the image in Figure 4 and the error in ejection times, a little better than ± 100 days, is greater in the direction of early emissions. 8 \nAs 500 days before perihelion corresponds to mid-May 2023 ( ± a few months, because of the uncertainty of position angle measurements), it appears that the time of major activity (derived from the orientation of the tail) coincided crudely with the time of discovery of the comet in South Africa (583 days before perihelion). Interestingly, according to Yoshida (see footnote 2) the comet's light curve was much steeper before 2 March 2023 ( ∝ r -14 ) than after this date ( ∝ r -4 ). \nThe second anomaly is the absence of a tail consisting of freshly ejected microscopic (submicron- and micronsized) particles, whose axis is expected in position angles close to the radius vector. On 4 June 2024 the radius vector was in position angle 113 · .9, and Figure 4 shows that a 20 days old tail (ejected at 2.56 AU from the Sun) should be pointing in a position angle of 111 · and a 40 days old tail (ejected at 2.84 AU) in 109 · . Microscopic dust composed of silicate material would make the tail at least 5 ' and 18 ' long, respectively. If the emissions also contained organic or other absorbing material, the tail would be about four times longer. In any case, it should be long enough to clearly extend beyond the coma boundaries. \nThat obviously has not been happening, even though the Af ρ levels (addressed in Section 2) have not at all been low. Thus, the cross-sectional area of solid material in the comet's head is fairly large, but it does not translate into equivalent amounts of microscopic dust that would be blown away by radiation pressure into the tail. Instead, the fragmentation process appears to essentially cease when the sizes of fragments reach a certain lower limit. From the absence of an ordinary dust tail one cannot say whether the limit is the size of pebbles or boulders or much bigger objects. The relative speeds of these large fragments are low, which may explain why the central coma of not more than 17000 km in radius appears to be saturated in Naves' image. In any case, the absence of a tail containing freshly ejected microscopic dust ejecta is unusual in a comet that is not depleted in dust. I will return to this issue in the next section to address the implications. \nIn order to verify the conclusions based on the single image in Figure 4, I have next applied the same method to a set of 31 CCD tail-orientation observations made, along with his magnitude observations (Section 2), by A. Pearce between 13 February and 13 June 2024 (see footnote 4). Their comparison with the theoretical loci (synchrones) of dust ejected at 400, 500, and 600 days before perihelion is in Figure 5. The data between about 230 and 190 days before perihelion provide striking evidence in support of the results derived from Figure 4. Figure 5 also shows the crowding of the synchrones in June (June 4.9 UT = 114.8 days before perihelion) and illustrates the exceptional Sun-comet-Earth geometry that I referred to above. \nThe issue is as follows: dust ejected with a zero velocity and subjected to a radiation pressure acceleration from the Sun is always located between the radius vector and the reverse orbital-velocity vector (i.e., the direction behind the comet). The angle ψ between the two vectors increases as the comet proceeds along its path about the Sun; it is near 0 · long before perihelion, 90 · at perihelion, and approaches 180 · long after perihelion. For a parabolic orbit, an excellent approximation to the orbit of comet Tsuchinshan-ATLAS, the angle equals \ncos ψ = -1 2 √ r q sin u = ± √ 1 -q r , (11) \nwhere r and u are, respectively, the heliocentric distance and true anomaly at the position in the orbit, and q is the perihelion distance. The plus sign applies before perihelion, the minus sign afterwards. \nMy statement near the beginning of this section that the preperihelion dust tails of comets are narrow was based on Equation (11), as ψ provides an upper limit to the tail width. For example, in a case relevant to comet Tsuchinshan-ATLAS in April 2024, for q = 0 . 4 AU and r = 3 AU, one has ψ = 21 · .4. In reality, of course, the tails must be much more narrow than ψ . \nThe same applies to the tail width in projection onto the plane of the sky, unless it is distorted because of the position of the Earth relative to the comet and the Sun. This happens when the Earth lies in the comet's orbit plane inside the sector bounded by the two vectors. The radius vector and the vector opposite the orbit-velocity vector then project 180 · apart from one another. When ψ is small, the comet is seen near the Sun in the sky. But when the Earth lies inside the extended sector, between the comet and the Sun, the two vectors also project 180 · apart, with the comet near opposition with the Sun. And when the Earth is not in but near the comet's orbit plane, the projected sector gets only slightly less wide than 180 · , greatly enhancing the breadth of any feature inside the sector. This is what I meant by the exception. \nAs it turned out, comet Tsuchinshan-ATLAS offered the terrestrial observer exactly this opportunity. While the true width of the sector grew very gradually, from ∼ 19 · on 10 February to ∼ 21 · at the time of the Earth's transit across the orbit plane on 11 April, the projected width, which already equaled 90 · on 10 February, increased to 98 · on 1 March, to 124 · on 21 March, to 144 · on 31 March, and of course to 180 · at the transit time. Subsequently, the projected width dropped to 102 · by 20 April, to 48 · by 30 April, and to 27 · by 20 May, \nFigure 5. Tail orientation of comet Tsuchinshan-ATLAS from CCD observations by A. Pearce, who used his 35-cm f/5 Schmidt-Cassegrain. The observed position angles (solid circles; equinox J2000) are compared with the theoretical synchrones for dust ejected 500 days before perihelion (thick curve), as well as 100 days earlier and later (nearly parallel thin lines). The position angles of the radius vector are plotted as a dotted curve. Marked with a diamond is the time of the Earth's transit across the comet's orbital plane, 2024 April 11.3 UT or 169.44 days preperihelion, when the tail pointed at the Sun, in a position angle of 69 · .7, and technically became an antitail. The time extends from February 4.0 UT = 236.7 days before perihelion to June 22.0 UT = 97.7 days before perihelion, a total of 139 days. \n<!-- image --> \nwhen ψ = 23 · . As the projection conditions continued to worsen, by 9 June the projected width of the sector diminished to 22 · , below its true width, ψ = 25 · . \nBecause of the strong variations in the projected width of the sector between the radius vector and the reverse orbital-velocity vector, Pearce's tail orientation data, accurate to ± 2 · or so, contribute unequally to the needs of the applied method. It turns out that the degree of angular resolution of dust-emission times (i.e., the degree of crowding of synchrones), measured by the width of the occupied position-angle sector per unit range of relevant dust-emission times, must be neither too narrow nor too wide to be most useful. Between 13 February \nand 22 March, when Pearce made eight tail-orientation observations, the angular resolution was between about 2 · and 4 · per month of dust-emission activity. Then, even though he observed the comet nine times between 24 March and 8 April, he did not report the tail a single time. As Figure 5 shows, the tail was rotating fast in this period of time (187 through 172 days before perihelion) and the angular resolution reached 7 · per month of activity. This temporary 'disappearance' of the tail indicates that the emission of dust was continuing over a long period of time and was projecting as a wide fan, whose surface brightness was obviously low enough to escape detection. If instead the dust were emitted in a \nbrief, powerful outburst, the event would have been detected regardless of the angular resolution. \nOn the other hand, following the Earth's crossing the comet's orbit plane on 11 April, the angular resolution of dust-emission times remained very low, comparable to 0 · .33 per month of activity when the image in Figure 4 was taken. The most useful tail observations have obviously been those from February and March (230 through 190 days before perihelion), which show that the primary emission time was 500-550 days before perihelion, extending over an estimated period of at least several months. No observation by Pearce offers any evidence on a tail made up of freshly ejected microscopic dust, confirming the conclusion based on the image of 4 June. \nThe absence of micron-sized and smaller dust grains in the old emission is unquestionable, but only a very approximate lower limit on the size of the dust present in the tail could be esimated from the image in Figure 4. It was desirable to verify the veracity of this result. To do just that, I examined the angular tail lengths measured by Pearce in the CCD images that he obtained with his 35-cm f/5 Schmidt-Cassegrain. \nThe results are presented in Figure 6. The observations are compared with the expected angular lengths of a tail consisting of particles ejected 500 days before perihelion (mid-May 2023; 6.6 AU from the Sun) and subjected to radiation pressure accelerations not exceeding 0.01 the solar gravitational acceleration. At an assumed bulk density of 0.5 g cm -3 such particles are about 0.2 mm or more across. \nIt is well-known that tail lengths measured in CCD images depend on the exposure time, observing conditions (such as a degree of light pollution, air quality, etc.), and other circumstances. Furthermore, the surface brightness of tails has a tendency to decline with their age and it is rather common to see the reported length to get gradually shorter as the surface brightness drops below the detection threshhold. Given all these uncertainties, I deem the fit in Figure 6 rather satisfactory, confirming the conclusion from Figure 4 that the tail consisted of submillimeter-sized and larger dust particles.", '5. FRAGMENTATION': "In the preceding sections I have assembled extensive circumstantial evidence to support a notion that the comet's nucleus is currently, and has already been for some time, in the process of progressive fragmentation, which will continue until the point of complete deactivation and disintegration. Given the perihelion distance of 0.39 AU, I expect that the object will disappear and cease to exist as an active comet before perihelion. \nComet Tsuchinshan-ATLAS has been at the risk of perishing before perihelion by virtue of its arrival from the Oort cloud. The comets of this class have a tendency to disintegrate if they are intrinsically faint and depleted in dust by the time they are near 1 AU from the Sun. The first condition may be diagnostic of a small, subkilometer-sized nucleus. While the dimensions of the nucleus of comet Tsuchinshan-ATLAS are unknown, the object was neither intrinsically faint nor dust-poor prior to mid-April. However, both the observed light curve and measurements of Af ρ obtained more recently have suggested that from mid-April on the comet has been fading and the rate of its dust production dropping. \nFigure 6. Angular tail length of comet Tsuchinshan-ATLAS from CCD observations by A. Pearce, who used his 35-cm f/5 SchmidtCassegrain. Unlike for the tail orientation, the observations of tail length are affected by observing conditions and therefore less reliable. Yet, a fair degree of agreement is seen between the observations and a predicted length of a tail consisting of dust particles ejected from the nucleus 500 days before perihelion and subjected to radiation pressure accelerations of up to 0.01 the solar gravitational acceleration. Such particles are a fraction of 1 mm across and larger. \n<!-- image --> \nIn the following I first summarize the evidence presented in the preceding sections in the context of the proposed progressive fragmentation scenario. After that, I briefly address the expected developments in the near future, before the comet disappears in the Sun's glare. Obviously, the results of a search for the comet after perihelion will ultimately settle the issue of its whereabouts. \nThe process of fragmentation is proposed to have begun at the latest around 21 March 2024, 190 days before perihelion and 3.3 AU from the Sun, when the rate of brightening was abruptly increased from r -3 . 5 to r -7 . 2 . For an Oort cloud comet such a change is unusual, because typically the rate of brightening declines with decreasing heliocentric distance. \nThe likely cause of the surge or steep brightening (not an outburst!) was a rapid increase in the cross-sectional area of the sublimating surface of the nucleus, splitting in quick succession into a large number of sizable active fragments. On a short time scale of hours or days the outgassing fragments were breaking up time and again \ninto smaller subfragments as the sublimating area continued to grow. The cloud of these objects began to slowly expand, running at the same time out of ice. Its shortage became eventually kind of a firewall that brought the runaway activity to a stop only 25 days (on 15 April 2024; 165 days before perihelion and 3 AU from the Sun) after it had started. As a result, the light curve reached a peak because the cross-sectional area of fragments around the nucleus grew no more and their outgassing was coming to a halt. The value of Af ρ started to drop more rapidly than the brightness, as increasing numbers of fragments passed the boundary of the zone of 10 000 km in radius, where this parameter is measured. The activity of the comet, whose nucleus was now reduced in size and much of its ice-rich surface was gone, began to fade, until the depleted ice was replaced with fresh supplies from the interior. In late April, May, and June, this process appears to have worked haphazardly, on a local scale. As a result, the comet's brightness was subjected to shortterm fluctuations, which made the comet look as if its activity stalled. The light curve showed that the comet was struggling, unable to 'pick up steam.' \nThe observed appearance of the comet as a whole supports the fragmentation scenario in that starting in late June the false-color, computer-processed images showed the coma to be noticeably elongated tailward. The elongation did not change appreciably from day to day, suggesting that the flow of material was steady and/or that the material moved very slowly, as one would expect of fairly sizable relic objects left over from the original fragments after all ice sublimated away. A stunning piece of evidence that the fragments did not, on a large scale, disintegrate into microscopic dust is the absence of an ordinary dust tail adjoining the radius vector. Significantly, as of early July, no companion nuclei were detected. \nFigure 7. Distribution of residuals in right ascension and declination left by Nakano's recent set of orbital elements of comet Tsuchinshan-ATLAS (NK 5247) from 860 astrometric observations made between 1 May and 7 June 2024. Note the sub-arcsecond systematic trends in both right ascension and declination. Nakano rejected all observations whose residuals equaled or exceeded ± 2 '' . \n<!-- image --> \nMAY 2024 \nNakano's sequence of the sets of orbital elements provides evidence in support of the fragmentation scenario. The reciprocal original semimajor axis from his orbital runs that incorporated the astrometric observations made after the light-curve anomaly is ∼ 30 units of 10 -6 AU -1 more negative than from the runs that did not include those observations. The difference must at least in part be due to an increased sublimationdriven nongravitational acceleration on the nucleus that grew smaller because of fragmentation. The Oort cloud comets moving in orbits with small perihelion distances have been shown by Marsden et al. (1978) to possess a greater hyperbolic excess than those in orbits with larger perihelion distances. These authors presented a formula for the dependence of (1 /a b ) orig on the perihelion distance, which predicts -0.000014 AU -1 for comet Tsuchinshan-ATLAS, close to the current value. \nAnother relevant issue is that of a mean residual, which climbed from ± 0 '' .32-0 '' .33 to ± 0 '' .42 following inclusion of the post-anomaly data. This ∼ 30 percent increase in the error means that either the nuclear condensation became more difficult to measure accurately (because of its substantially larger dimensions, for example) or the fit provided by the purely gravitational orbits was less than entirely satisfactory. Which of the two reasons is correct could be decided by carefully inspecting the distribution of residuals: in the former case it should be broader than before but still essentially random, while one should see systematic trends in the latter case. \nBecause of time constraints, I have inspected the problem rather superficially. I have focused on the residuals from Nakano's recent set of orbital elements (NK 5247) left by 860 astrometric observations (a small fraction of more than 4000 data points used) made between 1 May and 7 June 2024. The results of my inspection are apparent from Figure 7 and Table 3, which consistently show sub-arcsecond deviations from a random distribution. \nTable 3 Daily Average Residuals from Orbit of Comet Tsuchinshan-ATLAS (Nakano NK 5247) in May-June 2024 \nThe distribution of residuals in right ascension in the upper half of the figure displays a wide letter V: most residuals in early May are positive trending down toward negative numbers, in late May and June back up toward positive ones. Four of the first six tabulated daily averages are positive and only one is negative. In the period of 15-20 May all averages are negative, while five of the last six averages are positive. In declination, the systematic trend in the figure is immediately obvious, as a great majority of the residuals is positive. The table shows that only six out of the 38 daily averages are negative. Whether or not these effects are products of a nongravitational acceleration remains to be seen. \nThe only measurement of the water production rate that I am aware of at this time implies a sublimation area of 15 km 2 , a rather bewildering piece of information. By itself it allows two very different interpretations, neither one of which looks reasonable to me: an enormous nucleus or a nucleus in a highly advanced stage of fragmentation. \nIn the short-period of time, over which the comet can still be observed from some locations, not much additional information can be gathered over what is currently known. It is hoped that the existing databases will further be expanded: the light curve, the curve of Af ρ , the orbital arc, and more data on the water production rate. Of special interest is evidence of the continuing absence of a tail made up of recent micron- and submicron-sized ejecta in images of the longest possible exposure. \nI expect that the light and Af ρ curves will continue to drop in an uneven fashion, even though the absolute magnitude of 11.8 from the data in Figure 2 may be too pessimistic. Whether the comet will undergo another surge of brightness of the kind experienced in late March and early April depends on the size of the nucleus; a chance is that there will be none. Essentially the same uncertainty applies to Af ρ , whose variations with time appear to be governed by the same processes. \nIn the field of orbit determination, I predict that the mean residual from gravitational orbital solutions will continue to grow and the reciprocal original semimajor axis will continue to get more hyperbolic. Sooner or later it will be necessary to introduce the nongravitational terms. Such orbital solutions should be attempted earlier rather than later and tested by how much the mean residual has been improved. \nNew data for the water production rate will be extremely useful not only for getting a more integrated picture of the comet's physical evolution, but also for verifying the rate from the end of May. \nOne dataset that unfortunately will not become available is a close-up view of the comet's nuclear region. One can only envision an image looking like superimposed multiple exposures of the nuclear region of comet C/2019 Y4, taken with the Hubble Space Telescope's camera (Ye et al. 2021) - a field with an enormous cloud of barely glimmering fragments, some brighter, some fainter. In the final phase of fragmentation, increasing numbers of devolatilized, fractured refractory solids stay assembled in dark, highly porous, and exotically shaped blobs that eventually become undetectable as they gradually disperse in space. One can only speculate that some (the largest?) of these blobs could possibly look like 1I/'Oumuamua.", '6. CONCLUSIONS': "The purpose of this paper is not to disappoint comet observers who have been looking forward to a new nakedeye object this coming October, but to present scientific arguments that do not appear to substantiate such hopes. Even though the prognostication of preperihelion disintegration of comets is admittedly a very risky undertaking, I believe that the time has come to go ahead with it. \nComet Tsuchinshan-ATLAS exhibits traits that are diagnostic of extensive fragmentation of the nucleus, even though no distinct companion has been observed as of early July. Some of the features resemble the performance of past members of the Oort cloud that disintegrated before reaching perihelion at a heliocentric distance substantially smaller than 1 AU. It is primarily the peculiar light curve, as pointed out by Ferrin, as well as the parallel changes in Af ρ , especially a surge in late March through early April, a sharp peak in mid-April, when the comet was at 3 AU from the Sun, and the subsequent fading with superimposed fluctuations, which are likely to testify to additional events of local fragmentation. Also worrisome is the apparent tailward elongation of the coma seen in June. The extremely high water production rate needs confirmation. \nIndependent support is provided by two effects in the comet's orbital motion: (i) a hyperbolic shift of 30 units of 10 -6 AU -1 in the reciprocal original semimajor axis after inclusion of astrometric observations made following the light-curve anomaly and (ii) parallel increase in the mean residual from ± 0 '' .32-0 '' .33 to ± 0 '' .42. The culprit appears to be a sublimation-driven nongravitational acceleration on the nucleus whose dimensions have been rapidly diminishing because of its continuing fragmentation. Even though detected systematic trends have as yet been in a sub-arcsecond range, they soon may grow and require the incorporation of the nongravitational terms in orbit determination efforts. \nMost unusual is the continuing absence of an ordinary dust tail, which means that large amounts of dry, fractured solid material do not disintegrate into microscopic dust, but stay assembled in dark and highly porous bizarre bodies that I refer to above as blobs . Once they disperse in space, they are nearly impossible to detect, yet they may be omnipresent though perhaps short lived. \nAdded after final editing: As observations continue and the data in the paper get increasingly incomplete, some of the numerical results may be affected. Apologies.", 'REFERENCES': "A'Hearn, M. F., Schleicher, D. G., Millis,R. L., et al. 1984, Astron. J., \n89, 579 Ahuja, G., Aravind, K., Sahu, D., et al. 2024, Astron. Tel., 16637 Belton, M. J. S. 1965, Astron. J., 70, 451 Bortle, J. E. 1991, Int. Comet Quart., 13, 89 Brandt, J. C. 1961, Astrophys. J., 133, 1091 Green, D. W. E., ed. 2023, Cent. Bur. Electr. Tel. 5228 Green, D. W. E., ed. 2024, Cent. Bur. Electr. Tel. 5404 Jewitt, D., & Luu, J. 2019, Astrophys. J., 883, L28 (6pp) Kr'olikowska, M., & Dones, L. 2023, Astron. Astrophys., 678, A113 Marsden,B.G., Sekanina, Z., & Everhart, E. 1978, Astron. J., 78, 64 Meech, K. J., Pittichov'a, J., Bar-Nun, A., et al. 2009, Icarus, 201, 719 Osterbrock, D. E. 1958, Astrophys. J., 128, 95 Roemer, E. 1962, Publ. Astron. Soc. Pacific, 74, 351 Sekanina, Z. 1973, Astrophys. Lett., 14, 175 Sekanina, Z. 2019, eprint arXiv:1903.06300 \nYe, Q., Jewitt, D., Hui, M.-T., et al. 2021, Astron. J., 162, 70 (13pp)"}

2024arXiv240403000D

We present the DESI 2024 galaxy and quasar baryon acoustic oscillations BAO measurements using over 5.7 million unique galaxy and quasar redshifts in the range 0.1ltzlt2.1. Divided by tracer type we utilize 300017 galaxies from the magnitudelimited Bright Galaxy Survey with 0.1ltzlt0.4 2138600 Luminous Red Galaxies with 0.4ltzlt1.1 2432022 Emission Line Galaxies with 0.8ltzlt1.6 and 856652 quasars with 0.8ltzlt2.1 over a 7500 square degree footprint. The analysis was blinded at the cataloglevel to avoid confirmation bias. All fiducial choices of the BAO fitting and reconstruction methodology as well as the size of the systematic errors were determined on the basis of the tests with mock catalogs and the blinded data catalogs. We present several improvements to the BAO analysis pipeline including enhancing the BAO fitting and reconstruction methods in a more physicallymotivated direction and also present results using combinations of tracers. We present a reanalysis of SDSS BOSS and eBOSS results applying the improved DESI methodology and find scatter consistent with the level of the quoted SDSS theoretical systematic uncertainties. With the total effective survey volume of 18 Gpc3 the combined precision of the BAO measurements across the six different redshift bins is 0.52 marking a 1.2fold improvement over the previous stateoftheart results using only firstyear data. We detect the BAO in all of these six redshift bins. The highest significance of BAO detection is 9.1sigma at the effective redshift of 0.93 with a constraint of 0.86 placed on the BAO scale. We find our measurements are systematically larger than the prediction of Planck2018 LCDM model at zlt0.8. We translate the results into transverse comoving distance and radial Hubble distance measurements which are used to constrain cosmological models in our companion paper abridged.

2024-04-01T00:00:00Z

['2024arXiv240403000D', 'arXiv:2404.03000', '10.48550/arXiv.2404.03000']

['Astrophysics - Cosmology and Nongalactic Astrophysics']

DESI 2024 III Baryon Acoustic Oscillations from Galaxies and Quasars

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2404.03000.pdf

{'DESI 2024 III: Baryon Acoustic Oscillations from Galaxies and Quasars': "- 78 42 29 35 \n```\nJ. Moustakas , N. Mudur, E. Mueller, A. Mu˜noz-Guti'errez, A. D. Myers, 79 S. Nadathur , 26 L. Napolitano , 79 R. Neveux, 17 J. A. Newman , 37 N. M. Nguyen , 7 J. Nie , 80 G. Niz , 57 , 11 H. E. Noriega , 12 , 35 N. Padmanabhan, 13 E. Paillas , 65 , 67 N. Palanque-Delabrouille , 9 , 2 J. Pan , 7 S. Penmetsa, 65 W. J. Percival , 65 , 66 , 67 M. Pieri, 81 M. Pinon, 9 C. Poppett, 2 , 39 , 40 A. Porredon , 17 , 82 , 44 F. Prada , 47 A. P'erez-Fern'andez , 35 , 77 I. P'erez-R'afols , 83 D. Rabinowitz, 13 A. Raichoor , 2 C. Ram'ırez-P'erez, 10 S. Ramirez-Solano, 35 M. Rashkovetskyi , 42 M. Rezaie , 15 J. Rich, 9 A. Rocher , 48 , 9 C. Rockosi , 70 , 71 , 84 N.A. Roe, 2 A. Rosado-Marin, 85 A. J. Ross , 32 , 63 , 44 G. Rossi, 86 R. Ruggeri , 19 , 34 V. Ruhlmann-Kleider , 9 L. Samushia , 87 , 15 , 88 E. Sanchez , 36 C. Saulder , 77 E. F. Schlafly , 89 D. Schlegel, 2 M. Schubnell, 7 H. Seo , 85 R. Sharples , 90 , 6 J. Silber , 2 A. Slosar, 91 A. Smith , 6 D. Sprayberry, 20 J. Swanson, 85 T. Tan , 9 G. Tarl'e , 7 S. Trusov, 69 R. Vaisakh , 64 D. Valcin , 85 F. Valdes , 20 M. Vargas-Maga˜na , 35 L. Verde , 76 , 55 M. Walther , 59 , 60 B. Wang , 92 , 93 M. S. Wang , 17 B. A. Weaver, 20 N. Weaverdyck , 2 R. H. Wechsler , 45 , 94 , 46 D. H. Weinberg , 63 , 44 M. White , 95 , 40 J. Yu, 48 Y. Yu , 38 S. Yuan , 46 C. Y'eche , 9 E. A. Zaborowski , 32 , 43 , 44 P. Zarrouk , 69 H. Zhang , 65 , 67 C. Zhao , 93 R. Zhao , 26 , 80 R. Zhou , 2 and H. Zou 80\n``` \nAffiliations are in Appendix B \nE-mail: spokespersons@desi.lbl.gov \nAbstract. We present the DESI 2024 galaxy and quasar baryon acoustic oscillations (BAO) measurements using over 5.7 million unique galaxy and quasar redshifts in the range 0 . 1 < z < 2 . 1. Divided by tracer type, we utilize 300,017 galaxies from the magnitude-limited Bright Galaxy Survey with 0 . 1 < z < 0 . 4, 2,138,600 Luminous Red Galaxies with 0 . 4 < z < 1 . 1, 2,432,022 Emission Line Galaxies with 0 . 8 < z < 1 . 6, and 856,652 quasars with 0 . 8 < z < 2 . 1, over a ∼ 7 , 500 square degree footprint. The analysis was blinded at the catalog-level to avoid confirmation bias. All fiducial choices of the BAO fitting and reconstruction methodology, as well as the size of the systematic errors, were determined on the basis of the tests with mock catalogs and the blinded data catalogs. We present several improvements to the BAO analysis pipeline, including enhancing the BAO fitting and reconstruction methods in a more physically-motivated direction, and also present results using combinations of tracers. We employ a unified BAO analysis method across all tracers. We present a re-analysis of SDSS BOSS and eBOSS results applying the improved DESI methodology and find scatter consistent with the level of the quoted SDSS theoretical systematic uncertainties. With the total effective survey volume of ∼ 18 Gpc 3 , the combined precision of the BAO measurements across the six different redshift bins is ∼ 0.52%, marking a 1.2-fold improvement over the previous state-of-the-art results using only first-year data. We detect the BAO in all of these \nsix redshift bins. The highest significance of BAO detection is 9 . 1 σ at the effective redshift of 0.93, with a constraint of 0.86% placed on the BAO scale. We find our measurements are systematically larger than the prediction of the Planck 2018-ΛCDM at z < 0 . 8. We translate the results into transverse comoving distance and radial Hubble distance measurements, which are used to constrain cosmological models in our companion paper.", '1 Introduction': "̸ \nCosmology today is characterized by a well-established, simple phenomenological model that explains observations over a broad range of scales and epochs. Of particular relevance to this paper is the background expansion rate of the Universe which is quantitatively described by the Hubble parameter H ( z ). Assuming a homogeneous and isotropically expanding Universe, H ( z ) is determined by the relative contributions of the various matter/energy components of the Universe, as well as its value at the present epoch, H 0 (the Hubble constant). One of the great successes in cosmology is that the parameters inferred from probes sensitive to the background expansion are (largely) consistent with probes sensitive to the growth of cosmic structure. Notwithstanding this concordance, modern cosmology faces two outstanding problems. The first is to understand the nature of the energy that is causing the accelerated expansion of the Universe. In particular, measurements of the expansion history aim to constrain whether this accelerated expansion is consistent with a cosmological constant, or if its source ('dark energy') evolves in time (most simply parametrized by an equation of state parameter w = -1) [1]. The second problem is the discrepancy in the measurement of the expansion rate today, the Hubble constant H 0 [2]. Local distance ladder measurements using Cepheids [3] favor a higher value ( H 0 = 73 . 04 ± 1 . 04 kms -1 Mpc -1 ), while measurements anchored at high redshift [4] favor a lower value ( H 0 = 68 . 18 ± 0 . 79 kms -1 Mpc -1 ). The current and next generation of background expansion measurements, using a combination of standard candles and standard rulers, aim to address these problems by increasing the precision (and accuracy) of the expansion measurements over a wide range of redshifts. This paper presents the measurement of this background expansion using the standard ruler provided by the imprint of baryon acoustic oscillations (BAO) measured in the first year of data from the Dark Energy Spectroscopic Instrument (DESI) survey. \nThe baryon acoustic oscillation (BAO) method is one of the principal methods to map the expansion history of the Universe. The physics of BAO has been discussed extensively in a number of papers [1, 5, 6], including a companion to this paper [7], and we refer the reader to these for a complete discussion. Very simply, acoustic oscillations in the baryon-photon fluid in the pre-recombination Universe imprint a characteristic scale in the clustering of matter [8, 9]. This is manifested as a bump in the two-point correlation function of matter, or equivalently as a series of oscillations in its Fourier transform, the power spectrum. The comoving scale of this feature, r d ≃ 100 h -1 Mpc, is given by the sound horizon at the end of baryon drag epoch and depends upon the photon and baryon content of the universe 1 and is precisely constrained by cosmic microwave background (CMB) measurements. While the 3D matter distribution is not directly measurable, the BAO feature is faithfully traced by galaxies, quasars, and the Lymanα forest. Measuring the apparent size of the BAO standard ruler perpendicular and parallel to the line of sight constrains the angular diameter distance D A ( z ) and the Hubble parameter H ( z ). Using a fiducial power spectrum template \nas a ruler, these measurements are frequently expressed in terms of Alcock-Paczynski-like dilation parameters [10, 11] \nα || = H fid ( z ) r fid d H ( z ) r d , α ⊥ = D A ( z ) r fid d D fid A ( z ) r d , (1.1) \nwhere the 'fid' denotes quantities measured in the fiducial cosmology. By simply comparing BAO measurements at different redshifts, we can constrain the relative evolution of these quantities with redshift. When combined with the CMB or BBN measurements that constrain the physical scale of the BAO ruler, these relative measurements become absolute measurements of D A ( z ) and H ( z ) 2 . We note that the measurements of D A ( z ) and H ( z ) obtained are not independent, but are correlated at ∼ 40%, where this correlation is determined by the distribution of modes perpendicular/parallel to the line of sight-motivated by this, Eq. (1.1) is often re-expressed in terms of the geometric mean α iso and ratio α AP of the dilation parameters: \nα iso = ( α ∥ α 2 ⊥ ) 1 / 3 , α AP = α ∥ /α ⊥ . (1.2) \nOne of the features of the BAO method as a standard ruler is its relative insensitivity to astrophysical and observational systematics. At a theoretical level, this derives from the fact that the scale of the BAO feature ( ∼ 100 h -1 Mpc) is much larger than the characteristic scales of nonlinear structure growth and galaxy formation ( ≲ 10 h -1 Mpc). Furthermore, these effects can be significantly reduced by the 'reconstruction' of the linear BAO signal. Reconstruction effectively reverses the flows on large scales using the observed density field, undoing the effects of gravitational evolution on the BAO feature [12-15] The large scale of the BAO feature also makes it amenable to perturbative treatments, allowing for a very accurate theoretical understanding of any possible systematic effects. These effects have been studied in detail in the literature [14, 16-18]; [7] provide a comprehensive review within the context of the precision of the DESI measurements. On the observational side, the BAO method relies on a well-localized feature in three dimensions, while most of the observational systematics are either two-dimensional (from the imaging surveys), or along the line of sight (from the selection function of galaxies). This allows for a robust extraction of the BAO signal even in the presence of significant observational systematics. This has been demonstrated previously in the literature [19-22], and we explicitly show this for our data as well [23, 24]. \nThe large scale and relatively low amplitude of the BAO feature required the advent of large volume galaxy surveys for it to be detected. The BAO feature was first detected in galaxy clustering by the SDSS [25] and 2dFGRS [26] surveys. The success of these measurements prompted the development of the next generation of spectroscopic surveys, most notably the 6dFGS [27], BOSS [28], eBOSS [4] and WiggleZ [29] surveys, that made distance measurements with increasing precision at redshifts 0 < z < 1. The BOSS and eBOSS [30] surveys also demonstrated that the BAO method using the Lymanα forest, both in the auto-correlation as well as cross-correlation with quasar samples, provide measurements at redshifts 2 < z < 4. These BAO constraints connect the low-redshift SN distance measurements with the distance to the CMB last scattering surface in an inverse distance ladder, yielding very precise constraints on the expansion history of the Universe, and in particular the curvature of the Universe [4, 31]. On the other hand, these BAO measurements also \nTable 1 . The list of the papers supporting this paper and the corresponding sections where their results are discussed. \nprovide a CMB-independent measurement of the Hubble constant using BBN to calibrate the BAO scale [32]. The Hubble constant measurements thus obtained are consistent with those from the CMB and provide an independent piece of evidence for the tension between the low and high redshift inferences of the Hubble constant. \nBAO measurements at sub-percent precision are considered the primary science targets of the Dark Energy Spectroscopic Instrument [DESI; 33], along with novel constraints on theories of modified gravity and inflation, and on neutrino masses. DESI, as a Stage-IV DE experiment, aims to provide multiple sub-percent distance measurements over a broad 0 < z < 3 . 5 redshift range. DESI is in the process of a five-year survey over 14,000 deg 2 , and will result in a spectroscopic sample that will be an order of magnitude larger than previous surveys, both in the volume surveyed and in the number of galaxies measured. It achieves this with a combination of new instrumentation, including a 5000-fiber multi-object spectrograph [34, 35] new imaging surveys and efficient target selection algorithms [36], and optimized data pipelines [37]. In addition, DESI builds in a number of internal systematics checks using multiple tracer populations to probe common volumes. The early data from DESI was presented in [38, 39]. These data were used to make an initial BAO measurements in [40] and [41] which presented initial BAO measurements with the DESI galaxy and Lymanα samples respectively. These measurements were used to validate the DESI BAO pipeline, to demonstrate the statistical power of the DESI data, and set the stage for future analyses. This paper presents the BAO measurements from the galaxy samples in the first year of DESI data, [42] presents the BAO measurements from the Lymanα forest, and [43] presents the cosmological implications of these measurements. \nThis paper is one of a series of Key Papers presenting measurements with the first year of data from the DESI survey, which is designated as DESI DR1. While BAO is now a mature cosmological probe, the improved statistical precision of the DESI project motivates reexamining all aspects of the pipeline. This paper and its accompanying supporting papers (see Table 1) are the result of this work. This paper serves both to present the actual DESI galaxy BAO measurements and to summarize the key findings of the supporting papers. \nGiven the length of this paper, we present both an outline and an executive summary to guide the reader through this paper. Section 2 describes the data and the construction \nof the large-scale structure (LSS) catalogs used. In particular, this includes discussing how these data were blinded to the underlying cosmology, a first for a BAO analysis. Section 3 discusses the construction of mock catalogs used to estimate the systematic and statistical errors and to validate the pipelines. Section 4 provides an overview of the entire BAO measurement pipeline - the measurement of the two-point functions, the reconstruction pipeline, the definition of the model for the two-point statistics, the fitting of the BAO scale, and the construction of the covariance matrices. There are a number of improvements over previous analyses discussed here - these include adopting the RecSym reconstruction convention (see below), revisiting the BAO fitting model including a new treatment of the broadband marginalization, and a comprehensive treatment of estimating the covariance matrices using both analytical and mock-based methods. Section 5 presents our systematic error budget. We approach this both with theoretical modeling as well as extensive tests on the data and mocks, clearly demonstrating that the systematic error level is significantly below the statistical errors of our sample. Since this is the first time that the analysis of BAO measurements have been blinded at the catalog level, we present the process and tests used to determine when to unblind the data in Section 6. Section 7 contains our results for the different samples, including a comparison/reanalysis of previous BOSS/eBOSS data, while Section 8 combines these results, and presents the measured expansion history. Section 9 concludes with a discussion of these results and a look to future DESI results. Readers interested only in the results might focus on Sections 7 to 9 and Table 18. \nThe analyses here use a fiducial cosmology to convert redshifts into distances; the BAO distance measurements are relative to this fiducial cosmology. Our fiducial cosmology matches the primary cosmology used in the AbacusSummit suite of simulations [55], which is a Planck 2018-ΛCDM cosmology [31]. 3 We refer the reader to the above references 4 for the complete specification of the cosmology, but the key parameters are ω b = 0 . 02237, ω CDM = 0 . 1200, h = 0 . 6736, N ur = 2 . 0328 and one massive neutrino with ω ν = 0 . 00064420.", '2.1 DESI DR1': "The DESI Data Release 1 (DR1; [56]) dataset includes observations using the DESI instrument [57] on the Mayall Telescope at Kitt Peak, Arizona during main survey operations starting from May 14, 2021, after a period of survey validation [58], through to June 14, 2022. DESI measures the spectra of 5,000 'targets' at once, using robotic positioners to place fibers in the focal plane at the celestial coordinates of the targets [59, 60]. The fibers are divided into ten 'petals' and carry the light to a corresponding ten climate-controlled spectrographs. The data was obtained via observations of 'tiles', using an observing strategy meant to prioritize completing observations in a given area of the sky [61]. Each tile represents a specific sky position and set of associated targets [36] assigned to each robotic fiber positioner. DESI dynamically divides its observing time into separate 'bright' time and 'dark' time programs, depending on observing conditions. 2744 tiles were observed in DR1 'dark' time and 2275 tiles were observed in 'bright' time. Observations of the bright galaxy sample [62] happen in bright time while the luminous red galaxies (LRGs [63]), quasars (QSO [64]), and emission \nFigure 1 . The comoving number density as a function of redshift for the samples used for the DESI DR1 galaxy/quasar BAO measurements. Here, we show simply the observed number of redshifts, per comoving volume element. The vertical lines represent the boundaries of the redshift bins used in this analysis, except for the QSO sample which is analyzed as a single redshift bin. \n<!-- image --> \nline galaxies (ELGs [65]) are observed in dark time. These data were first processed by the DESI spectroscopic pipeline [37] the morning following observations for immediate quality checks, and then in a homogeneous processing run (internally denoted as 'iron') to produce resulting redshift catalogs used in this paper and will be released in DR1.", '2.2 DR1 Large-scale structure catalogs': "The redshift and parent target catalogs were processed into large-scale structure (LSS) catalogs and two-point function measurements as described in [66, 67]. Table 2 presents the basic details on the tracer samples used in this paper; the catalogs used for the Lyα BAO measurements are presented separately in [42]. In total, over 5.7 million unique redshifts are used for galaxy and quasar BAO measurements in DR1, a factor of ∼ 3 increase compared to SDSS DR16 [4]. \nA key component of the LSS catalogs is the matched random sample ('randoms'), which accounts for the survey geometry. The randoms were first produced to match the footprint of DESI target samples, as described in [36]. These were then passed through the stage of the DESI fiberassign code that determines the 'potential assignments' for each input random target, using all of the properties of the observed DESI DR1 tiles. The potential assignments do not require that the randoms are allocated a fiber in a full assignment run, they simply determine whether a fiber could reach their angular positions. Thus, this selection is one based on the individual position, and no fiber assignment effects are imprinted on the randoms. This procedure works to the angular scale at which the DESI fiberassign software can predict the focal plan position of targets to be observed, which is better than 1 arcsecond. It is thus far more accurate than trying to sample a pixellated angular mask. The process is detailed in [66]. \nThese potential assignments are cut to the same combination of 'good' tiles and fibers \nTable 2 . Statistics for each of the DESI tracer types used for the DESI Y1 BAO measurements presented in this paper. Redshift bins are non-overlapping, except for the shot-noise dominated QSO sample and the 0 . 8 < z < 1 . 1 LRG and ELG. The effective volume calculation, V eff provides a rough estimate for the relative amount of cosmological information in each redshift bin. See text for further details on the calculation of values in this table and [67] for further details on the samples themselves. \n- \n∼ \n× \nas the DR1 data samples. Subsequently, veto masks are applied 5 following the same process as applied to DR1 data described in [67]. In DR1, the randoms are normalized such that the ratio of weighted data and random counts is the same in each of the distinct regions relevant to the photometry used to target the sample. Similarly, the redshift distributions are matched between data and randoms in each region separately. For all but the QSO sample, there are two distinct photometric regions: data targeted from BASS/MzLS photometry [68] in the North Galactic Cap (NGC) and declination greater than 32 . 375 · and those targeted from DECaLS photometry (in both Galactic caps). For QSOs, the DECaLS sample is further divided into DES and non-DES regions, as the target selection [64] is different in those regions. \nSupporting studies help define and correct for variations in the selection function due to the effects of imaging systematics on the input target samples [23, 69, 70] and variations in the DESI instrument's ability to successfully measure redshifts [24, 71]. Our ability to simulate and correct for incompleteness in target assignment is presented in [72-74]. The effects of all of those issues are combined into a weight column in the data and random catalogs, meant to be used for any subsequent calculations (i.e., both for reconstruction and two-point statistics). The number density of the DR1 DESI samples varies strongly with both redshift and the number of overlapping tiles (due to assignment completeness). Thus, 'FKP' 6 weights, inspired by [75], that are meant to maximize the signal to noise of clustering measurements at the BAO scale with respect to such number density variations are also included in all two-point calculations. Their calculation is fully detailed in [67]. \nIn what follows, we provide brief details on the properties of each sample, providing further context for the statistics of each sample that are presented in Table 2. The z eff values are calculated weighting by the square of the weighted number density of randoms (with the weights-including FKP-described above), n ran ( z ): \nz eff = ∫ zr 2 drn 2 ran ( z ) ∫ r 2 drn 2 ran ( z ) , (2.1) \nwhere r is the comoving distance to the redshift z . These represent the redshift at which the BAO fit parameters, α iso and α AP , can be converted into physical distances (see Section 8). The clustering amplitude is determined via a single parameter fit to the post-reconstruction power spectra at wavenumbers k < 0 . 07 h Mpc -1 , assuming the linear matter power spectrum given by our fiducial cosmology. The effective volume estimate is obtained in each redshift bin via \nV eff = ∫ [ ¯ n tracer ( z ) P 0 ( k = 0 . 14) 1 + ¯ n tracer ( z ) P 0 ( k = 0 . 14) ] 2 dV ( z ) (2.2) \nwhere for P 0 ( k = 0 . 14) we use the values listed in Table 2, which are rounded numbers taken from the DR1 P 0 ( k ) measurements. 7 The choice of k = 0 . 14 h Mpc -1 was chosen to be most representative for BAO measurements [76]. The comoving number density as a function of redshift for each sample is shown in Figure 1. \nThe Bright Galaxy Sample (BGS): The final BGS sample used for our DR1 BAO measurements comprises of 300 , 017 redshifts in the interval 0 . 1 < z < 0 . 4 over 7 , 473 deg 2 , with an assignment completeness of 61 . 6% (see [67] for full details). The completed DESI BGS sample is expected to have ∼ 80% assignment completeness over 14,000 deg 2 , implying the DR1 sample contains just over 2/5 of the final DESI BGS sample. The minimum bound of z = 0 . 1 is chosen to minimize any effects of bright limits ([77]) on the target sample. The maximum bound is chosen to separate the sample from the LRG sample (described next). \nThe nominal BGS sample is flux-limited [62] and thus has strong variation of its number density with redshift. In order to produce a more homogeneous sample a cut on the DR1 BGS sample was engineered to produce a sample of roughly constant number density of 5 × 10 -4 h 3 Mpc -3 , for a sample with 100% fiber assignment completeness. Thus, the DR1 number density is roughly constant at 3 × 10 -4 h 3 Mpc -3 , averaged over the DR1 footprint, as can be seen in the light green curve in Figure 1. The sample was defined using k -corrected r -band absolute magnitudes from [78, 79] and a correction for evolution that is linear in redshift (matching that used for [39]). The cuts make the sample a close match to both the LRG number density (for a complete sample) at z = 0 . 4 and the clustering amplitude, as can be seen by comparing to the red curve in Figure 1 and the P 0 entries in Table 2. The density is high enough to make shot-noise a minor contribution to the BGS statistical uncertainty in the DR1 two-point measurements at BAO scales. See [67] for more details. \nThe Luminous Red Galaxy Sample (LRG): The DESI DR1 LRG sample used for BAO measurements consists of 2 , 138 , 600 good redshifts over 5 , 840 deg 2 in the redshift interval 0 . 4 < z < 1 . 1. The DR1 assignment completeness is 69 . 2% in the Y1 sample; this is expected to increase to 90% over 14 , 000 deg 2 in the complete survey, so the DR1 LRG sample is thus approximately 30% of the full DESI LRG sample. \nThe lower redshift bound was chosen to separate the sample from BGS, as most lowredshift LRG targets are also BGS targets and we are able to match the densities of the two samples at z = 0 . 4 (when accounting for assignment completeness). The complete LRG sample has a nearly constant number density over 0 . 4 < z < 0 . 8 of 5 × 10 -4 h 3 Mpc -3 , which becomes just over 3 . 5 × 10 -4 h -3 Mpc 3 when averaged over the DR1 area, as shown in Figure 1. The upper bound of z < 1 . 1 was chosen to align with the ELG redshift bins. As for previous LRG samples, the clustering is significantly biased compared to the matter distribution, which enhances the BAO signal. The clustering amplitude is approximately \nconstant over the entire redshift range, though there is some evolution in both the clustering amplitude and composition of the sample for z > 0 . 95, as documented in [80], which causes the slight decrease in the bσ 8 clustering amplitude for the 0 . 8 < z < 1 . 1 bin. \nThis sample is further split for the clustering analysis into 3 disjoint redshift ranges, 0 . 4 < z < 0 . 6 ( LRG1 ), 0 . 6 < z < 0 . 8 ( LRG2 ), and 0 . 8 < z < 1 . 1 ( LRG3 ), with this higher redshift sample overlapping the low-redshift ELG sample. The redshift binning was chosen in part to align with the SDSS BAO measurements used in [4]. The redshift bin 0 . 4 < z < 0 . 6 matches one of the SDSS bins. Despite SDSS using 0 . 6 < z < 1 . 0 for one of their LRG bins, the SDSS sample is dominated by galaxies with z < 0 . 8 and their effective redshift is slightly lower than that of our 0 . 6 < z < 0 . 8 LRG sample ( z eff = 0 . 7 compared to z eff = 0 . 71). This allows us to present a direct comparison between the DESI DR1 measurements and the SDSS measurements (Section 7.6). The 0 . 8 < z < 1 . 1 redshift bin is aligned with the lower redshift bin chosen for ELGs (described next), which allows a combined LRG+ELG sample to be created and used in the 0 . 8 < z < 1 . 1 redshift range. \nThe Emission Line Galaxy Sample (ELG): The DR1 ELG sample defined for clustering analysis in [67] comprises 2 , 432 , 022 good redshifts in the interval 0 . 8 < z < 1 . 6 over a footprint 8 of 5 , 914 deg 2 . DESI ELGs are assigned fibers at the lowest priority of any DESI target class, and in DR1, the fiber assignment completeness is only 35 . 3%. This should increase to over 60% in the final dataset, with the footprint growing to 14,000 deg 2 . The DR1 sample is thus less than 1/4 of the final DESI ELG sample. The clustering amplitude is the lowest of any of the DESI target classes, consistent with the general knowledge that actively star-forming galaxies are generally less clustered than passive galaxies. These lower clustering amplitudes explain why the effective volume for the 0 . 8 < z < 1 . 1 ELG sample is significantly lower than the LRG sample in the same redshift bin, despite having more galaxies. One can see from Eq. (2.2) that there is a leading factor of P 2 0 . As the DESI survey progresses, the ELG completeness will increase and thus N tracer /V tot will grow larger and make the clustering amplitude less relevant to the effective volume calculation. \nThe sample is split into 2 disjoint redshift ranges for measuring two-point clustering: 0 . 8 < z < 1 . 1 ( ELG1 ) and 1 . 1 < z < 1 . 6 ( ELG2 ). The split at z = 1 . 1 is meant to align with the maximum redshift of the LRG sample. At z < 0 . 8, the ELG number density drops below that of the LRG sample, even after accounting for fiber assignment incompleteness (see [67]). Further trends with the imaging depth become severe, as shown in [23]. The upper limit of z < 1 . 6 is applied as the OII emission line doublet used to secure redshifts for the ELG sample is at a longer wavelength than covered by the DESI spectrographs for z > 1 . 6. \nCombined LRG and ELG ( LRG3 + ELG1 ): A combined LRG and ELG sample is used for 0 . 8 < z < 1 . 1. Ignoring any possibility of sample variance cancellation, the optimal BAO information extracted from data in this redshift range should come from combining the two samples into one, with an appropriate weighting given the difference in clustering amplitude. The construction of this combined sample and its validation are presented in [47] and results from it are denoted ' LRG3 + ELG1 '. Further details, including its motivation, are discussed in Section 7.5. \nThe Quasar Sample (QSO): The DR1 QSO sample used for BAO measurements consists of 856 , 652 good redshifts with 0 . 8 < z < 2 . 1. The sample covers an area of 7,249 deg 2 , \nwithin which the assignment completeness is 87 . 6%. The QSO assignment completeness in the completed DESI survey, which spans 14,000 deg 2 , is expected to be greater than 99%. Thus, the DESI DR1 QSO sample contains just under half of the number expected for the completed survey. \nThe upper bound of z < 2 . 1 is chosen as z > 2 . 1 quasars are used as sight-lines along which Lymanα forest absorption is measured and then used to obtain BAO measurements, as described in [42]. The z > 0 . 8 choice is somewhat arbitrary, but aligns both with the lower bound chosen by SDSS and with the redshift bin edge adopted for LRG and ELG. The clustering amplitude of the sample is intermediate between that of the LRG and ELG samples, which makes it the most highly biased sample, given the low σ 8 ( z eff ) at its effective redshift.", '2.3 Blinding': "In order to protect against confirmation bias in our DESI DR1 galaxy BAO measurements, the location of the BAO feature in our measured two-point functions for galaxies and quasars was intentionally shifted from its true values by an unknown amount by applying a blinding shift at the catalog level following the methods proposed in [81]. Full details of the specific methodology applied to DESI DR1 are presented in [54]. 9 We summarize some basic details here. First, w 0 , w a , f NL were randomly chosen from a pre-defined list of values, 10 which were bounded to keep α iso within 3% of its fiducial value within the redshift range 0 . 4 < z < 2 . 1. The primordial non-Gaussianity parameter was allowed to vary within -15 < f NL < 15. The choice of these bounds represented one of the final necessary choices by the DR1 cosmological analysis team prior to obtaining the first (blinded) DESI DR1 BAO measurements. The DESI redshifts were then coherently shifted based on the expected difference in redshift between the chosen w 0 , w a and w 0 = 1 , w a = 0, at the comoving coordinates initially determined from the true redshift and the fiducial cosmology. Redshifts were also shifted based on estimates of the local density in order to blind structure growth methods. Rather than being drawn randomly, a shift in the growth rate, f , was chosen so that it would compensate for the expected change in the monopole of the redshift-space clustering, up to 10% of the expected change using a linear redshift-space distortions model. The f NL shift was implemented via weights and demonstrated to have no effect on BAO measurements, beyond small random fluctuations from the effective re-weighting. Further description of the DESI DR1 blinding methodology and validation is in [54]. \nThe cosmology of the shift was kept constant with every catalog update and never revealed. When the DR1 analyses matured to the point where all choices were frozen (see Section 6 for the tests that were first performed on the shifted data), the LSS catalogs without any shifts applied were then used for the first time, to produce two-point clustering measurements (i.e., 'unblinded'). After the unblinded results were revealed, an error was found in the LSS catalogs related to the completeness weights that are assigned to random points. 11 The main effect of this on BAO measurements is to re-weight the contributions from different areas of the footprint. The greatest change to the BAO fits was a -0 . 7 σ shift in the α AP result for LRGs with 0 . 4 < z < 0 . 6. The magnitude of all other shifts were less than 0 . 5 σ , and most were less than 0.2 σ . Only two pieces of the BAO analysis were updated after \nTable 3 . The table summarizes the principal set of simulations used for each supporting task/paper and the sections of this paper that make use of them. Abacus-2 and EZmock DR1 duplicate the footprint of the DESI DR1, the variation of completeness with target number density, and an approximate fiber assignment effect. The tasks that need to simulate the DESI DR1 survey realism concluded their tests with the DR1 mocks. Abacus-1 Y5 approximately traces the expected DESI Y5 footprint. The tasks that needed to test the theoretical systematics with the least level of noise utilized the cubic simulations; in this case, we used Abacus-1 cubic instead of Abacus-2 , as the former was completed first and was sufficient for our analysis. CV denotes the control variate technique that was applied to the Abacus-1 cubic [82]. The CV technique allows to reduce sample variance noise from the two-point statistics by utilizing a highly correlated surrogate based on the Zeldovich approximation. \nunblinding: 1) the final choice on the covariance matrix and 2) the use of the LRG+ELG combined sample in the 0 . 8 < z < 1 . 1 redshift bin; however, both of these updates were decided/planned before unblinding.", '3 Mocks': 'Realistic and accurate mock simulations form the backbone of our analysis. They allow us to test the limitations of our theoretical models in the presence of non-linear evolution and galaxy-halo physics. In addition, they allow us to assess our ability to mitigate imperfections in our survey caused by effects such as atmospheric conditions, foreground astrophysical systems, and instrument limitations. Building a single set of mock simulations that can be used for all these tests is not feasible because the combination of volume, resolution and number of realizations needed is beyond our current computational resources. Therefore, we build a series of DESI mock simulations focusing on the different aspects of theoretical and observational systematics. A specific set of simulations used for each task is listed in Table 3. \nWe built two kinds of mock simulations. The first called Abacus used the high-resolution AbacusSummit simulation suite and hence produced highly accurate non-linear structure. The second called EZmock were computationally very cheap to produce over large volumes and resulted in highly accurate linear scales but non-linear scales that were not that well controlled. \nAll mocks are produced at the Planck 2018 ΛCDM cosmology, specifically the mean estimates of the Planck TT,TE,EE+lowE+lensing likelihood chains: Ω c h 2 = 0 . 1200, Ω b h 2 = 0 . 02237, σ 8 = 0 . 811355, n s = 0 . 9649, h = 0 . 6736, w 0 = -1, and w a = 0 [31].', '3.1 Abacus': "AbacusSummit is a large suite of high-resolution gravity-only N-body simulations using the Abacus N-body code [55, 83]. These simulations provide us with realizations of the density field and dark matter halos in cubic boxes with a range of cosmologies. The entire suite consists of over 150 simulations at 97 different cosmologies. This study makes use of the 25 'base' boxes of the Planck 2018-ΛCDM cosmology, each of which contains 6912 3 particles within a (2 h -1 Gpc) 3 volume corresponding to a particle mass of 2 . 1 × 10 9 M ⊙ /h . 12 \nThe dark matter halos are identified with the CompaSO halo finder [84]. We also run a post-processing 'cleaning' procedure to remove over-deblended halos in the spherical overdensity finder, and to intentionally merge physically-associated halos that have merged and then physically separated [85]. \nThe dark matter halo catalogs are then populated with galaxies using an extended halo occupation distribution (HOD) model with the AbacusHOD code [86]. These Abacus mocks were produced in two generations called Abacus-1 and Abacus-2 . The main difference between the two comes from the fact that they were produced at different times to be able to make early progress on testing the analysis pipeline while we collect more data and improve our model of the survey and instrument. The Abacus-1 used very early version of the DESI early data release (DESI-EDR)[39] to find the best fit halo occupation distribution model whereas Abacus-2 used the final DESI-EDR after correcting for all the systematics and including a detailed model for DESI focal plane effects. \nAbacus-1 These mocks were produced by fitting the galaxy two-point correlation function averaged in angular bins at small scales using Abacus halos and a flexible halo occupation distribution model (HOD) [87] to populate these halos with galaxies. We found best-fit HOD parameters at each available snapshot between redshift of 0 to 2 in the AbacusSummit suite. The details of HOD models used are described in [88]. We note that satellite galaxies were distributed using NFW profiles fit to the density profile of each halo in the simulation. For the QSO mocks we also included an additional velocity dispersion to account for the significant QSO redshift errors. Each tracer at each redshift is populated over all 25 base boxes, giving a total volume of 200 h -3 Gpc 3 . \nAbacus-2 The HOD parameters are tuned to the final DESI EDR redshift-space two-point correlation function measurements. We refer the readers to [80, 89, 90] for the exact HOD models and calibration. The final cubic mocks consists of BGS samples at z = 0 . 1 , 0 . 2 , 0 . 4, LRG samples at z = 0 . 5 , 0 . 8 , 1 . 1, ELG samples at z = 0 . 95 , 1 . 1 , 1 . 325, and QSO samples at z = 1 . 1 , 1 . 4 , 1 . 7. Each tracer at each redshift is populated over all 25 base boxes, for a total volume of 200 h -3 Gpc 3 . We also provide Zeldovich control variates (ZCV) mock simulations with suppressed sample variance for all AbacusSummit realizations (see [82] for description of the technique).", '3.2 EZmock': "In order to generate large simulation volumes for covariance matrices and pipeline validations, we use the EZmock code [91] which can be calibrated to accurately reproduce the two- and three-pt clustering on the scales relevant for this analysis without the cost of a full N-body simulation. It has been widely used in eBOSS [92] and DESI [93]. \nThe method comprises two steps: constructing a dark matter density field and populating galaxy catalogs. The dark matter density field is based on the the Zel'dovich approximation [94]. To populate the resulting density field with galaxies, EZmock uses an effective bias model to account for non-linear evolution and galaxy bias. The latest description of the effective bias model can be found in [92]. We produced two generations of DESI EZmock by fitting the two-point clustering of Abacus-1 and Abacus-2 in order to give equivalent covariance matrices. We produced 1000 realizations of each generation of EZmock with a box side of 2 h -1 Gpc. These provide the covariance matrix for equivalent 2 h -1 Gpc Abacus mocks. We also produced 1000 realizations of each generation of EZmock with a box side of 6 h -1 Gpc in order to fit the volume occupied by the DESI DR1 data without any repetition of structure to validate our covariance matrices for the full survey volume.", '3.3 Simulations of DR1': "Both the Abacus-2 and 6 h -1 Gpc EZmock have been used to simulate the DESI DR1 LSS dataset [95]. A brief outline of the steps to create such mocks is as follows: For both, the first step is to transform the box coordinates to angular sky coordinates and redshifts. Then, the data are sub-sampled as a function of redshift such that the total projected density matches that of the given target sample and the n ( z ) (after accounting for redshift failures) matches that of the observed DR1 sample [67]. This provides a simulated DESI target sample. The simulated target sample is then cut so that it covers the same sky area as the DESI target samples. Then, matching the process applied to randoms described in Section 2.2, it is passed through the stage of the DESI fiberassign code that determines the 'potential assignments' for each simulated target, using all of the properties of the observed DESI DR1 tiles. These potential assignments are cut to the same combination of 'good' tiles and fibers as the DR1 data samples. Subsequently, veto masks are applied following the same process as applied to DR1 data described in [96]. \nThe process described above reproduces the small-scale structure of the DESI DR1 footprint, but does not impart any incompleteness within it. For the Abacus-2 , mock LSS catalogs (hereafter mocks) were produced with three variations in the fiber assignment completeness. These are: \n- · The ' complete ' mocks that have no assignment incompleteness added and thus can be used as a baseline comparison for understanding the effect of the incompleteness.\n- · The ' altmtl ' mocks that represent our most realistic simulations of the DR1 data. They apply the process described in [72] to apply the DESI fiberassign code to tiles in the same ordering and cadence in a feedback loop to the target list as occurred for the observed data. The process was demonstrated to perfectly reproduce DESI fiber assignment on real DESI targets, with no approximations.\n- · The ' fast-fiberassign ' mocks that emulate the fiber assignment process by sampling from the average targeting probability of the galaxies multiple times, learned from the data as a function of number of overlapping tiles and local (angular) clustering. The final sample is obtained by recombining the multiple realisations in such a way that deliberately creates a small-scale exclusion effect, which approximately reproduces the fiber-collisions pairwise incompleteness. The process is much faster than the altmtl and is described and validated in [73]. \nThe computation time required for the altmtl mocks prohibits it from being run on all 1000 EZmocks . Thus, we apply only the fast-fiberassign process to the EZmocks . \nAll flavors of mocks go through the process of assigning redshifts and weights to randoms in the same way as for the real data samples and are normalized within the same regions, etc (e.g., all integral constraints effects, e.g., described in [97], are the same between data and mock LSS catalogs) following the prescription in [67]. More details about the creation and validation of the different mock flavours can be found in [95].", '4 Methods': 'This section summarizes the various methods used in the BAO analyses that follow. We refer the reader to the referenced supporting papers for more detail and validations.', '4.1 Two-point function codes': 'The BAO measurements derive from the two-point clustering statistics of the data, the correlation function in configuration-space and the power spectrum in Fourier space. The techniques for computing these are now well established. We use the Landy-Szalay estimator [98] for the correlation functions (modified as in [99] for the reconstructed data) and an FKP based estimator ([100, 101]) for the power spectrum. A more detailed discussion of our particular implementations can be found in [67]. The specific codes are pycorr 13 for correlation functions and pypower 14 for power spectra. We compress the angular dependence (to the line of sight) into Legendre multipoles; our analyses rely on the ℓ = 0 (monopole) and ℓ = 2 (quadrupole) components. The galaxies are weighted by terms to account for the selection function and to optimally measure two-point statistics (FKP weights), both summarized in Section 2 and fully defined in [96], unless otherwise noted. \nSince the clustering measurements are consistent for both Galactic caps, we combine these measurements when constructing our data vectors. In configuration-space, the combination is performed by summing the pair counts computed in each region independently. Similarly, the power spectrum estimates are obtained for each Galactic cap and the measurements are then combined by averaging the two power spectra, weighting by their respective normalizations [67]. The number of randoms used is more than 50 × the size of the data for all correlation functions measurements and more than 100 × for all power spectra 15 . \nThe DESI fiber assignment imprints structure into the two-point statistics, especially on small scales. This can be mitigated very effectively by removing small-angle/small-separation pairs in the two-point statistics [74]. While such an approach is necessary for analyses that use the full shape of the two-point function, we expect these fiber assignment issues to have no measurable impact on our extraction of the BAO distance scales. This is both a result of the fact that the BAO feature is at large scales, and that we marginalize out the overall shape of the two-point function in our analysis. We validate this with mocks with and without fiber assignment and find no impact at greater than 2 σ significance (Section 5.3). Note that this does not include accounting for the overall completeness, which we do in our galaxy weights. Given this insensitivity, we do not include any additional corrections for fiber assignment in our analysis.', '4.2 Density-field reconstruction': 'Density-field reconstruction [12] is now a well-established element of galaxy BAO analyses, as it robustly eliminates biases due to the nonlinear evolution of the density field and improves the statistical precision of the BAO method. Beyond the standard reconstruction method proposed in [12], which has been widely applied to observational datasets [19, 99, 102-105], there are several improved reconstruction algorithms that have been proposed in the literature [106-108]. Although these methods have significant promise for reconstructing the linear density field at small scales at the very low shot noise regime, the improvements in the BAO distance measurements are marginal at the galaxy number densities of DESI DR1. Considering this and the robustness and simplicity of the original method, we restrict ourselves to using it for this work, with the modifications described below. \n- · One of the main differences from previous applications of reconstruction in SDSS is the use of the RecSym convention [17]. RecSym shifts the tracers and randoms from the LSS catalogs in the same way using the redshift-space displacement, preserving RSD in the post-reconstruction clustering. The RecIso convention [99] used in BOSS and eBOSS approximately removes RSD, resulting in more isotropic clustering postreconstruction. We adopt the RecSym convention as our baseline since it is the choice that fully removes the nonlinear damping and shift of the BAO due to large-scale modes [7] and avoids artefacts in the correlation function on small scales. However, [46] shows that the DESI BAO constraints in practice are rather insensitive to this choice (a similar conclusion was found in [109]).\n- · We tested the sensitivity of the reconstruction method to the choice of scale that is used to smooth the density field. Using the blinded DESI data and mocks that match the expected clustering properties of DESI DR1, we determined the optimal smoothing scale to be used when reconstructing each DESI target sample, as described in [46] and presented in Table 4. \nOur reconstruction uses pyrecon , 16 a Python package developed by the DESI collaboration. This comprehensive toolkit offers a diverse range of reconstruction algorithms and accommodates various conventions, and provides the flexibility to process periodic-box simulations or survey data with non-uniform geometries. In terms of the numerical implementation of the reconstruction method, we adopt an efficient algorithm based on iterative Fast Fourier Transforms introduced in [110] as our baseline, which we find highly consistent with the output from MultiGrid . 17 The iterative Fast Fourier Transform we adopt is different from the method in eBOSS. In eBOSS [4, 111], the method iteratively updates the locations of the galaxy and random particles, which we will denote as IFFTP , while the method we adopt for DESI DR1 iteratively updates the density of given meshes (hereafter, simply IFFT ). This is the first time that the IFFT method (the latter) has been implemented in data analysis. An extensive comparison of reconstruction algorithms in the context of DESI is presented in [45]. \nTable 4 summarizes the different hyperparameters that we calibrated when reconstructing the DESI DR1 samples. The catalogs were reconstructed across the entire redshift range \nTable 4 . Redshift range, tracer linear bias, growth rate of structure, and smoothing scale assumed when reconstructing each DESI target sample. The smoothing scale was chosen after testing different values for each tracer, as detailed in [46]. \nof each tracer simultaneously, assuming a value of the growth rate of structure determined by our fiducial cosmology and the effective redshift of each sample.', '4.3 Defining the clustering model': 'We next describe the BAO fitting method used in the galaxy DR1 analysis. We design this method to fully isolate the BAO feature within the broader two-point clustering measurements by combining a physically motivated theory model from quasi-linear theory and a parameterised model to marginalise over non-linearities that may otherwise affect our measurements of the BAO scale. The design of our method is based on detailed past investigations, starting from the original works of [12] with improvements through the eras of BOSS and eBOSS [19, 20, 105, 111-115]. Although these references have demonstrated the BAO fitting methodology to be robust at a level beyond that required for these previous surveys, there remained some inconsistencies in the modelling of the power spectrum and correlation function, and some arbitrariness in the choice of free parameters dependent upon the specific configuration and signal to noise of the measurements. As such, the modelling choices motivated or adopted in previous works have all been revisited in [7] to ensure a robust fit for the DESI DR1 results, using a method based on the allowed physical degrees of freedom that can also be consistently used for future surveys without significant modification. We have taken particular care to develop a more consistent modelling in Fourier and configurationspace, and to better motivate (or remove the need for) different modelling choices. Where such choices remain, we have quantified their systematic differences in the BAO constraints (see Section 5). In this subsection, we provide an overview of the official DESI galaxy BAO modelling prescription and summarise those changes.', '4.3.1 Fourier-space fitting framework': "Our generic model for the galaxy power spectrum as a function of scale k and (cosine) angle µ in the 'true' cosmology can be written as \nP ( k, µ ) = B ( k, µ ) P nw ( k ) + C ( k, µ ) P w ( k ) + D ( k ) , (4.1) \nwhere P nw ( k ) and P w ( k ) denote the smooth (no-wiggle) and BAO (wiggle) components of the linear power spectrum, respectively, which are obtained using the peak average method from [116]. The linear matter power spectrum template is predicted from class 18 using our fiducial cosmology (see Section 1). Generally, the term C ( k, µ ) P w ( k ) encompasses the BAO component we are interested in, damped by non-linear galaxy motions [12]; while B ( k, µ ) P nw ( k ) uses quasi-linear theory to model the smooth component of the galaxy clustering, and D ( k ) \nis our parametric model to account for additional non-linearities and observational effects. This model generally matches that used in previous BAO analyses [19, 103, 104, 112], but the exact forms of each component differ and so will be discussed later in this section. \nIn order to make contact with the apparent size of the BAO seen in the fiducial cosmology, C ( k, µ ) P w ( k ) is evaluated at Eq. (1.1), \nk ' = α 1 / 3 AP α iso [ 1 + µ 2 obs ( 1 α 2 AP -1 )] 1 / 2 k obs (4.2) \nand \nµ ' = µ obs α AP [ 1 + µ 2 obs ( 1 α 2 AP -1 )] -1 / 2 , (4.3) \nwhere the subscript 'obs' is used to distinguish between observed coordinates and measurements (assuming a fiducial cosmology) and those in the (unknown, but to be constrained) true cosmology. Note that this transformation is not just a coordinate transformation between the true and fiducial cosmologies but also includes a rescaling of the BAO template from the template cosmology, reflecting that what is measured is the apparent size of the BAO relative to the template sound horizon. The true, fiducial and template cosmologies need not be the same ([51], see also Section 5.4), but the latter two are usually equated for simplicity. The rescaling in Eqs. (4.2) and (4.3) in principle should not apply to the non-BAO parts of the power spectrum. In order to prevent accidentally using broadband information in the smooth component in the DESI DR1 analysis, we hence evaluate the smooth components directly at the observed coordinates, without dilation. The model for the power spectrum multipoles is thus \nP ℓ, obs ( k obs ) = 2 ℓ +1 2 ∫ 1 -1 dµ obs L ℓ ( µ obs ) [ B ( k obs , µ obs ) P nw , obs ( k obs ) + C ( k ' ( k obs , µ obs ) , µ ' ( k obs , µ obs )) P w ( k ' ( k obs , µ obs )) ] + D ℓ ( k obs ) . (4.4) \nThis is similar to that used in [115], but with the key difference that the term B ( k, µ ) P nw ( k ) is not dilated. As a final technicality, we note that our rescaling is applied to the entire BAO component; strictly speaking the prefactor C ( k, µ ) is not subject to the exact same rescaling as P w , but the differences will be degenerate with the free parameters within C ( k, µ ) itself. We will now describe each remaining model component in turn. \nFollowing previous BAO studies [115, 117], we adopt the following parametric form for B ( k, µ ): \nB ( k, µ ) = ( b 1 + fµ 2 [1 -s ( k )] ) 2 F fog , (4.5) \nwhere F fog = ( 1 + 1 2 k 2 µ 2 Σ 2 s ) -2 accounts for the 'Fingers of God' effect due to halo virialization [118, 119] via a single free smoothing scale Σ s . The term ( b 1 + fµ 2 [1 -s ( k )] ) 2 is a generalised form of the Kaiser factor [120] that also accounts for impact of reconstruction; here b 1 is the linear galaxy bias for the particular sample we are fitting, while f is the linear growth rate of large scale structure, both of which are free parameters in our model. For pre-reconstruction and the RecSym convention, s ( k ) = 0, while for RecIso , s ( k ) = exp [ -( k Σ sm ) 2 / 2 ] and Σ sm is the smoothing scale we applied to the density field during the reconstruction process. \nThe function C ( k, µ ) captures the anisotropic non-linear damping of the BAO feature on top of linear theory. Similarly to the smooth component above we have that it takes the form ([7], see also ref. [121]) \nC ( k, µ ) = ( b 1 + fµ 2 [1 -s ( k )] ) 2 exp [ -1 2 k 2 ( µ 2 Σ 2 || +(1 -µ 2 )Σ 2 ⊥ )] (4.6) \nwhere Σ || and Σ ⊥ model the damping for modes along and perpendicular to the line of sight. The FoG factor in Equation 4.5 is dropped here due to its high correlation in fits with the damping parameters [7]. In RecIso two caveats apply: (a) the smoothing kernel s ( k ) is always evaluated in the observed coordinates k obs , since it is defined in the fiducial cosmology and (b) the simple exponential form here is approximate and only holds on small scales where the contribution from the randoms is negligible. At intermediate scales the damping due to long-wavelength modes takes on a more complex form since the randoms and galaxies are displaced by different amounts, which is one of the reasons we choose RecSym as our default convention, as it is the unique choice that removes the nonlinear damping and shift due to long-wavelength modes. However, we note that past BAO measurements have often empirically employed the exponential form along with the prefactor in Eq. (4.5) for RecIso , and we will continue to do so here in tests involving this scheme. \nFinally, the D ( k ) factor captures any deviation from linear theory in the broadband shape of the power spectrum multipoles. Past analyses have used a polynomial form for this [19, 20, 105, 111, 112, 115], although a single exact equation cannot be provided here for comparison due to different studies using polynomial equations with different numbers of terms. To improve on this, in DESI DR1 we instead parameterize it using a spline basis with bases separated by a single user defined scale ∆, \nD ℓ ( k ) = n max ∑ n = -1 a ℓ,n W 3 ( k ∆ -n ) (4.7) \nwhere W 3 is a piecewise cubic spline kernel [122, 123] and n max sets the number of broadband terms we consider. A suitable ∆ is chosen under the premise that the spline basis is able to match the broadband shape of the power spectrum without reproducing the BAO wiggles themselves. This sets a limit on the choice of ∆ to be larger than half the BAO wavelength ( π/r d where r d is the sound-horizon at the baryon-drag epoch). We hence use twice this minimum value, ∆ = 2 π/r d ≃ 0 . 06 h Mpc -1 . The number of broadband terms n max = 7 for our default fitting procedure then arises from considering how many spline terms of width ∆ are required to fully span the 0 . 02 h Mpc -1 < k < 0 . 30 h Mpc -1 range of our power spectrum measurements, i.e., there is no need to specify a choice for n max . Unlike previous analyses, our new method hence provides a more physically motivated and less arbitrary broadband model. \nFinally, the model multipoles need to be convolved by the data window function. This can be accomplished via matrix multiplication ˜ P ˜ ℓ ( ˜ k i ) = ∑ j W ˜ ℓℓ ( ˜ k i , k j ) P ℓ ( k j ) which can be compared directly to the data vector. This follows previous approaches [124, 125]. To ensure accuracy in this convolution, the unconvolved model is evaluated at k -points within 0 . 001 h Mpc -1 < k < 0 . 35 h Mpc -1 and separated by ∆ k = 0 . 001 h Mpc -1 . The computational binning is thus five times finer than the k -bins of the data vector ˜ k , while covering a larger k -range. For the theory input to P ℓ ( k j ) we include angular dependence up to the hexadecapole. This is because power from these higher order moments can still 'leak' into \nthe observed multipoles (monopole and quadrupole, or monopole only in case of 1D fits) due to the convolution with the window function W ˜ ℓℓ ( ˜ k, k ) - in principle higher order multipoles can also enter but their contributions are negligible on the scales we fit. \nIn summary, our model for the BAO can be evaluated for comparison to data using Eq. 4.4, with the functional forms and free model parameters described in Eqs. 4.5- 4.7. Fitting for these free parameters proceeds as detailed in Section 4.4.", '4.3.2 Configuration-space framework': 'Our approach for modelling the correlation function very closely follows that for the power spectrum, more so than in previous works [20, 112]. We start with the power spectrum multipoles in fiducial coordinates obtained from Eq. (4.4), before the window function convolution, and without the D ℓ ( k ) terms. These multipoles are Hankel-transformed to configurationspace to yield the correlation function multipoles in the same coordinates \nξ ℓ, obs ( s ) = i ℓ ∫ ∞ 0 dk k 2 2 π 2 j ℓ ( ks ) P ℓ, obs ( k ) , (4.8) \nwhere j ℓ are the spherical Bessel functions. As a direct transform of the power spectrum, the correlation function model hence also contains the same BAO dilation parameters, BAO and Fingers of God damping parameters, linear galaxy bias and growth of structure, with the same physical interpretation. We evaluate our theory model in narrow bins of ∆ s = 1 h -1 Mpc. We match our wider ∆ s = 4 h -1 Mpc measurement binning by averaging the theory, weighted by the number of random - random pair counts in each fine ∆ s = 1 h -1 Mpc bin. \nFor the remaining broadband modelling, we Hankel-transform the same spline basis functions D ℓ ( k ) as used for the power spectrum. However, all of these except for the n = [0 , 1] terms of Eq. (4.7) in the quadrupole quickly go to zero on large scales, and so do not need to be included given the choice of fitting scales we use in configuration-space model. These two terms are explicitly evaluated as \n∆ 3 B 2 ,n ( s ∆) = i 2 ∫ dk k 2 2 π 2 W 3 ( k ∆ -n ) j 2 ( ks ) (4.9) \nfor n = 0 and 1. However, we also introduce two additional nuisance terms for each multipole to account for the potential impact of uncontrolled large-scale data systematics. Such effects can be confined to k < k min in Fourier space and removed by truncating the range of scales used in our fit to the power spectrum data. However, in the configuration-space, these are modelled using \n˜ D ℓ ( s ) = b ℓ, 0 + b ℓ, 2 ( sk min 2 π ) 2 . (4.10) \nwith k min = 0 . 02 h Mpc -1 . In summary, the configuration-space broadband terms comprise of: \n˜ D 0 ( s ) = b 0 , 0 + b 0 , 2 ( sk min 2 π ) 2 , (4.11) \n˜ D 2 ( s ) = b 2 , 0 + b 2 , 2 ( sk min 2 π ) 2 +∆ 3 ( a 2 , 0 B 2 , 0 ( s ∆) + a 2 , 1 B 2 , 1 ( s ∆)) (4.12) \nwith B 2 , 0 ( s ∆) and B 2 , 1 ( s ∆) given by Eq. (4.9).', '4.3.3 Polynomial-based broadband modeling': 'For select comparisons with previous BOSS/eBOSS literature, we will occasionally compare our DESI results using the default spline-based broadband model with a more traditional polynomial-based model. In Fourier space, this latter choice consists of writing D ℓ ( k ) = ∑ 3 n = -1 a ℓ,n k n . In configuration-space, this is ˜ D ℓ ( s ) = ∑ 2 n =0 ˜ a ℓ,n s -n . Note that even when using this polynomial option, we still include all the other improvements made in DESI 2024, except for the different broadband functions.', '4.3.4 Main differences between DESI 2024 and previous BAO modelling': 'Here we provide a summary of the most important differences in our DESI 2024 BAO modeling compared to the previous methods used in BOSS and eBOSS [19, 20, 105, 111, 112, 115]. This covers only those differences we deem most relevant to the interpretation of our fitting results, or that contribute to our systematic error budget. A more comprehensive list including more minor changes can be found in [7]. Overall, we find that the sum total of the differences between our new methodology and that used in BOSS/eBOSS result in only a small systematic error, far below DESI DR1precision (See Section 5.1). \n- 1. As described in Section 4.2, our BAO fitting model is calibrated for RecSym as a fiducial reconstruction procedure. BOSS/eBOSS used the RecIso convention, with caveats as described around Eq. 4.6. The RecSym form of reconstruction is more physically motivated, and leads to a much simpler form for the BAO damping. We note however that [46] demonstrate consistent constraints on DESI DR1 blinded data even when using RecIso .\n- 2. BOSS/eBOSS most frequently used a polynomial-based broadband (Section 4.3.3, although with varying numbers of free-parameters between different studies). We prefer to use a spline-based model to marginalise over non-linear physics in the broadband as this leads to a less arbitrary choice in the number of free parameters, and greater consistency between the power spectrum and correlation function. Though we believe our new approach to be better, we detect small differences between the two approaches for α AP which we adopt into our systematic error budget.\n- 3. When including FoG damping, BOSS/eBOSS analyses usually applied this equally to both the wiggle and no-wiggle components (i.e., [20, 115]). Ref. [7] (figure 12 therein) demonstrated that this introduces a degeneracy between the FoG and BAO damping parameters. We hence include FoG damping only in the no-wiggle component, which removes this degeneracy. The freedom we give to the two BAO damping parameters is sufficient to account for non-linearities that can move them away from their theoretical values, and we find no systematic differences between the two methods.\n- 4. We treat the correlation function model purely as the Hankel transform of the power spectrum, applying all our modelling choices and BAO dilation to the power spectrum first. The exception is the broadband terms, which are applied separately to the power spectrum and correlation function; nonetheless, the form of these for the correlation function is still based purely on what one obtains from Hankel transforming the spline-based broadband model for the power spectrum. This is an improvement in the consistency of the modelling compared to [20, 112]. Nonetheless, we tested both methods on our mocks and data and conservatively adopt the small difference in α AP into our systematic error budget.', '4.4 Fitting the clustering data': "The galaxy DESI 2024 BAO results in this work are obtained using desilike , 19 a python package that provides a common framework for writing DESI likelihoods. The BAO theory and likelihood is implemented in JAX [126] 20 . Even though gradient-based sampling methods were implemented, we found that with analytic marginalization over broadband parameters that leaves a few sampled parameters, and using Jax just-in-time compilation and parallelization capabilities, the ensemble sampler emcee [127] 21 provided well-sampled posterior estimates in a just a few minutes. In addition to MCMC sampling, we also perform posterior profiling using desilike 's wrapping of Minuit [128]. \nDuring the course of this work, we also used/developed a fully independent galaxy BAO fitting pipeline ( Barry 22 [129]), for some of the supporting papers and with which we tested the consistency of our results prior to unblinding. This latter code was also used to demonstrate that our results are independent of the choice of MCMC/Nested sampling algorithm used [7]. We adopt desilike as our official pipeline, owing to the greater computational speed offered by its JAX implementation, and its better integration within the wider set of DESI pipelines used for producing the clustering measurements (Section 4.1 [67]), and for the cosmological interpretation of our BAO constraints [43]. \nFor our default fitting, we adopt Gaussian priors on the BAO and Finger-of-God damping parameters and flat priors for all other model parameters. We parameterize the linear RSD though a parameter dβ = f/f fid where f fid is the fiducial value for the growth rate at the effective redshift of the sample. Furthermore, a 0 ,n , a 2 ,n are scaled by the amplitude of the fiducial no-wiggle power spectrum at the pivot points k p = n ∆ before injecting them into Eq. (4.7). All of these are simple rescalings of the parameters described in Section 4.4 to improve the convergence of the fitting. \nAll our priors are listed in Table 5. The choices for the Gaussian priors, particularly their central values, were informed by a combination of theoretical calculations, measurements of the cross-correlation between the initial and post-reconstruction density fields in Abacus-2 DR1 simulations, and by running fits to mock catalogs for each tracer. By fitting the average over many mocks, we ensured that the signal-to-noise ratio is large enough to let the damping parameters vary freely during the fit. The resulting central values for the priors are given in Table 6. In Section 7, we show that the recovered BAO parameters from fits to DESI DR1 are largely insensitive to this choice of priors. \nWe perform two-dimensional fits to monopole and quadrupole data for LRG s and ELG2 . For the BGS , ELG1 , and QSO samples, we perform only one-dimensional fits to the monopole due to their relatively noisier clustering measurements based on the unblinding tests detailed in Section 6. In these 1-D cases, α AP and dβ are set to unity. \nFor Fourier-space fits, the theory model always includes all three multipoles for the product with the window matrix, but the broadband terms for the multipoles that are not fitted (e.g. quadrupole and hexadecapole for 1-D fits) are set to 0 as they are mostly unconstrained. \nIn the rest of this work we mainly report on the BAO scaling parameters α iso and α AP , but in doing so are fully marginalizing over the other parameters of our model, including the correlation function/power spectrum broadband parameters, the galaxy bias, and the damping parameters. \nTable 5 . The free parameters and their priors for Fourier-space (FS) and configuration-space analyses. N ( µ, σ ) refers to a normal distribution of mean µ and standard deviation σ , [ x 1 , x 2 ] to a flat distribution between x 1 and x 2 inclusive. For faster fitting and convergence, we re-parameterise our linear RSD parameter using dβ = f/f fid , where f fid is fiducial growth rate. a 0 ,n , a 2 ,n are also scaled by the amplitude of the fiducial no-wiggle power spectrum at the pivot points k p = n ∆ before injecting them into Eq. (4.7). Parameters with superscript ' ∗ ' are fixed to the following values when only a 1D fit is performed: α AP = 1, dβ = 1, a 2 ,n = 0, b 2 ,n = 0. The central values of the Gaussian BAO damping parameters, denoted by superscript 'fid' vary between tracers and so are provided separately in Table 6. Σ ⊥ , Σ ∥ and Σ s are additionally bounded between [0 , 20]. The total number of free parameters for our 1D and 2D fits are 15 and 25 respectively for P ( k ); and 7 and 13 respectively for ξ ( r ). The corresponding numbers of degrees-of-freedom for our 1D and 2D fits are 41 and 87 for P ( k ); and 19 and 39 for ξ ( r ). \nTable 6 . The central values of the BAO damping parameters used for our default Gaussian priors (see Table 5) when fitting each of the DESI DR1 tracers. These are obtained from a combination of theoretical calculations, measurements of the cross-correlation between the initial and post-reconstruction density fields in Abacus-2 DR1 mocks, and by running fits to data vectors averaged over many realizations of these mocks for each tracer.", '4.5 Covariance Matrices': 'Our analysis here has used both analytic and mock-based approaches to computing the covariance matrices for the two-point functions. We summarize the construction and validation of these covariance matrices below. \nThe analytic covariances assume Gaussian covariances based on the observed clustering \nFigure 2 . Comparison of fits using the RascalC semi-analytical and mock-based covariance matrices. The figure demonstrates the consistency of the inferred distance scales and errors using our two approaches for computing the covariance matrices. The panels on the left show the BAO distance scales α iso and α AP , while the panels on the right show the errors on these parameters. The x -axis show results using the mock-based covariances in the fits, while the y -axis show results using the RascalC semi-analytical covariances. The RascalC matrices have been tuned to the ensemble of mocks. Further tests were performed using matrices tuned to a single mock which yield consistent results. The ∆ values in the legends on the left show the fractional mean and standard deviation (in percent, relative to the analytic value) of the difference between the α values, while the values on the right correspond to the difference in the errors. \n<!-- image --> \nof the galaxies and account for the effects of the survey geometry and selection function. Note that while we largely limit to the disconnected terms, these are derived from the observed nonlinear clustering of the data or mocks. \nThe covariance matrices for the correlation functions are generated with the RascalC code[130-133], 23 a variant of which was used for the BAO analysis of the early DESI data [134]. In addition to the disconnected term, the code also includes an adjustable shot-noise parameter that serves as a proxy for the missing connected 4-point terms in the covariance matrix. This parameter can be calibrated with data jackknife resampling (or mocks). A complete discussion of this method is in [49], along with the validation of the data pipeline on mocks demonstrating high consistency with their sample covariance, in particular for errorbars of BAO scale parameters. The code used to generate the covariance matrices for this analysis is publicly available. 24 \nThe covariances for the power spectrum are based on the formalism by [135]. Our implementation is described in detail in [50]. 25 For the power spectrum we limit to the Gaussian terms and do not include any higher-order corrections. \nIn addition to the analytic covariances, we also use the EZmocks described previously to generate sample covariance matrices. The advantage of the EZmocks covariances is that we can directly assess the impact of observational systematics like fiber assignment. These mocks are used to test the approximations made in the generation of the analytic covariance matrices. A detailed comparison of the two approaches for generating covariance matrices is in [48]. In particular, we quantify how the fitting of the BAO scales and the corresponding errors depends on the choice of the covariance matrix used. Figure 2 compares the α values and their errors for the full sample of mocks obtained from fits in configuration space. We find that the analytic and mock-based covariances yield answers that are consistent to (much) better than 0.1% for α iso / AP and generally < 2% for the errors. Both these values sit comfortably within the respective standard deviations for all the galaxy samples. Note that, in addition to the consistent average BAO scales and average errors between the two types of covariance matrices, the individual measurements demonstrate high correlation, typically ranging between 0.965-0.980, as depicted by the distribution thickness in Figure 2. We stress that these tests are done using analytic covariances tuned on EZmocks , thus making the sample covariance baseline a fair estimate of the true covariance. While Figure 2 shows results for a matrix built using the mean of EZmock clustering and the shot-noise rescaled for an optimal match to the sample covariance, [48] shows that such a matrix is consistent with matrices built from a single EZmock realisation and the shot-noise rescaling based on a jackknife covariance estimate as is done using observational data. \nAcaveat is that we have not attained the desired level of agreement between the analytic and the EZmock covariance matrices in Fourier space as of the time of writing this paper, as will be presented in [48]. Hence, we consider our default results in configuration space the most robust measurements. \nGiven this level of agreement, particularly in configuration space, we adopt analytic covariance matrices built based on the unblinded DESI DR1 data and calibrated using jackknife resampling of this same data for our analyses in this paper. This choice allows us to tune our covariances to match the observed clustering of the data after we unblind the data, and to avoid the small discrepancies between the clustering seen in the data and the mocks. We note that we are focusing on the BAO observables here, although preliminary work seems to suggest that the analytic covariances may also work well for other observables, at least on large scales.', '5 Systematic Error Budget': 'This section provides an overview of the systematic errors on the BAO scales stemming from various components of our analysis pipeline, as determined through the tests performed in our supporting papers (Table 1). Although individual supporting papers may use a more stringent limit for investigating systematics, the overarching rule we use is to count systematics when we detect an effect that exceeds 3 σ , compared to the statistical error associated with the mock test. We will conclude this section by presenting the combined systematic errors on the BAO scales.', '5.1 Theoretical Systematics': 'Ref. [7] investigates the systematic error induced by various approximations made when modelling the BAO. These can potentially arise from either physical effects that are dropped from the model used in BAO fits, e.g., the shrinkage of the BAO feature due to non-linear \nTable 7 . Different contributions to the DESI galaxy BAO theory and modelling systematic error budget for α iso and α AP considered in this work, taken from [7]. The first two rows denote theory cases, where the exact contributions depend on the nature of the tracers (galaxy bias, redshift, etc.), we have estimated a conservative upper value considering all DESI tracers/redshift bins. The other rows are obtained from fitting the Abacus-1 cubic, control-variate (CV) measurements while varying our modelling assumptions/choices. Only two of the tests show that our results can potentially be subject to a small nonzero bias. The remainder return no difference within the statistical precision available from the simulations and so are reported as upper bounds. The combined theory and modelling systematic error is obtained by summing the two solid detections in quadrature and rounding up to account for the remaining upper limits. \nclustering/evolution in the galaxy distribution [136-140] or the imprint of relative baryondark matter perturbations. There can also be residual errors due to (small) imperfections in the numerical choices made in the broadband model and extraction of the BAO template. \nThe physical processes affecting the BAO feature are comprehensively computed in [7] within perturbation theory. The impact of the numerical choices made in the modeling is measured by fitting to our high-precision Abacus-1 cubic, control-variate (CV) measurements while changing the numerical prescriptions used in the modelling. The robustness of the BAO means that even with the large volume and sample-variance cancellation of the Abacus-1 cubic mocks most of the modelling choices we test give consistent constraints within the statistical uncertainties, and we report only upper bounds on the potential systematic error. We obtain positive detections of only two (small) systematic modelling errors. The first arises from comparing our preferred spline-based broadband model with the historically used polynomial-based model. The second arises from comparing our new approach of modelling the correlation function as the Hankel transform of the dilated power spectrum multipoles with free growth rate parameter, to the BOSS approach [20] of transforming the undilated multipoles and applying the dilation after the transformation, then varying two different bias parameters for the monopole and quadrupole rather than the growth rate parameter. In both cases, we argue that our new method is better physically motivated and preferred, but as these are nonetheless modelling choices that also passed the systematic checks used in previous studies, we absorb these differences into our systematic error budget. \nUsing these two detections, upper bounds on the other modelling considerations, and folding in the expected theoretical upper limits from nonlinearities, we obtain a (conservative) systematic error on α iso , ap of 0 . 1 and 0 . 2%, respectively. The different contributions to this total are in Table 7.', '5.2 HOD-dependent systematics': "Systematic errors can be introduced when we infer the underlying BAO scale in the matter density field from the BAO measurement using a specific choice of LSS tracer, which is typi- \nTable 8 . Summary table for the estimation of HOD-dependent systematics for individual tracers. The table is split into Fourier and configuration space results, and we include the systematics for both α iso and α AP . The results for LRGs and ELGs are presented in detail in [52] and [53]. The results for BGS and QSO are derived consistently to LRGs and ELGs. The bold-faced numbers indicate a detection of systematics; otherwise, the numbers reflect the statistical precision associated with no detection. As conservative upper limits, we adopt systematic error on α iso , ap of 0 . 1 and 0 . 2% for the HOD-dependent systematics. \ncally a biased tracer. Studies have shown that such systematics are subdominant compared to the shift in the BAO scale due to structure growth and redshift-space distortions; moreover, they effectively vanish after reconstruction [141]. Given the precision of the state-of-the-art DESI DR1 data and that the new major tracer, the ELGs, believed to exhibit distinct halo occupation features compared to LRGs, this test is revisited. Our supporting papers, [52] and [53], extensively test the systematics for the different assumptions of the halo occupation distribution (HOD) for LRGs and ELGs, respectively, using DESI mock catalogs constructed from Abacus simulations. [52] studied nine different variations of HODs that are within 3 σ of the best-fit HOD parameters to the One-Percent Survey [80]. [53] studied the impact of the ELG HODs using 22 different HOD models, including the standard models used for Abacus-1 and Abacus-2 as well as extended models that include galactic conformity, assembly bias, and modified satellite profile adopted from [89]. \nThe amplitude of HOD systematics is estimated using the same methodology in both of our supporting papers, [52] and [53]. We compute the differences in α iso and α AP between every pair of HOD models, averaging over the 25 realizations. We compute the significance of this difference to assess the systematic detection level. If we do not detect systematics at the level of 3 σ in terms of the typical dispersion of the average differences from the mocks, we use the distribution of the differences between all pairs of HOD models and quote the 68% region of the distribution as the systematic error. On the other hand, if any of the paired HODs shows a non-zero difference at 3 σ or above, we take the shift between that pair of HODs as the systematic error. If there is more than one pair of HODs with a non-zero difference above 3 σ , we quote the maximum of these shifts. Table 8 shows the summary of the systematics for each tracer. The systematics for BGS and QSO are derived consistently. \nNote that the quoted systematic error, in the case of no detection, often merely reflects the limited sample variance cancellation due to the different subsampling noise. Even for the detection cases, an analysis like this depends on how extreme the HOD models we decide to compare are; we compare the models that span the ± 3 σ contours around the best-fit HODs, which is a reasonable choice to incorporate viable HOD parameter values allowed by the data. In addition, this test includes the systematics in the process of reconstruction as well, as we fix the galaxy bias input to reconstruction, while the actual HOD model (and, therefore, the effective bias) is being varied, with variations with respect to the input bias scaling up to 10%. The theoretical systematics reported by [7] already include a contribution \nTable 9 . The mean and the error associated with the mean of the BAO measurements from the 25 Abacus-2 DR1 mocks using the DESI DR1 baseline BAO method, as presented in [46]. Results indicate that fits to the 'complete' mocks, i.e., before applying fiber assignment effects, align with unbiased BAO measurements within statistical error. Furthermore, the mean difference between the altmtl mocks and the ' complete' mocks demonstrates the recovery of unbiased BAO estimates despite the presence of the fiber assignment effect. LRG3 + ELG1 is omitted since it is the combination of LRG3 and ELG1 . \n± \n± \ndue to the reference HOD model assumed in the simulation they utilized, and therefore the HOD systematics reported in this section should be considered as additional systematics on the BAO scales that can be introduced by assuming different HODs than the reference HOD model tested in [7]. \nUsing the results presented in Table 8, we adopt a conservative approach and consider a common HOD-dependent systematic error for all tracers of 0.2% for both α iso and α AP .", '5.3 Observational Systematics': "Similar to results found in SDSS (e.g., [142]), we find that observational systematics have a negligible impact on our BAO measurements. We therefore do not add any additional observational systematic uncertainty. We briefly outline the studies justifying this conclusion. \nObservational systematics in DESI can be broadly divided into three classes: fiber assignment incompleteness, spectroscopic systematics, and imaging systematics. Any effects of fiber assignment incompleteness can be tested by comparing results on our DR1 mocks (described in Section 3) that have no fiber assignment incompleteness ('complete') and those that have had fiber assignment run in a way that mimics the DESI DR1 observing history (' altmtl '). Each have been run through the DESI DR1 BAO measurement pipeline and consistent results were found [46], as summarized in Table 9. The table first shows that the BAO fits from Abacus-2 DR1 'complete' mocks are consistent with no bias. The altmtl mocks are consistent with the 'complete' mocks within 1.5 σ , demonstrating that we can recover unbiased BAO measurements in the presence of the fiber assignment. The differences in α iso appear systematic. The weighted mean (assuming no correlation) is ⟨ α complete iso -α altmtl iso ⟩ all = 0 . 00149 ± 0 . 00077 and thus just under 2 σ . This is not significant enough for us to add as a systematic uncertainty, but does suggest a closer look is warranted in future DESI studies. \nSignificant trends between the observed DESI galaxy density and the observational properties of the imaging data used to define the target samples (e.g., the imaging depth) are found for all DESI tracers [96]. These trends are removed from the DR1 LSS catalogs via weights that are added to the sample based on regression analysis against maps of the observational properties. The overall impact of the weights on the DR1 clustering measurements is quite large in significance; the χ 2 difference between weighted and unweighted clustering for ELGs is greater than 1000 [23]. However, the impact on BAO measurements is negli- \nFigure 3 . The impact of imaging systematics on the BAO scales. In detail, we show the difference in the derived BAO scales when we use the mitigation method for imaging systematics and when we use the raw data ('No weight'). The two measurements for each tracer are slightly displaced horizontally for a better visualization. The BAO scales are very stable even when there was no mitigating and even after density field reconstruction using such raw data. Again, the BAO is robust against the imaging systematics. \n<!-- image --> \n<!-- image --> \nα \ngible, as shown in Figure 3. When comparing the post-reconstruction BAO measurements with and without weights to correct for imaging systematics, the variation is found to be at most 0 . 27 σ (for the QSO α iso ) and is typically less than 0 . 2 σ on any of the measured BAO parameters ( α iso and α AP ), without any coherent direction across tracers. The weights modulate the density field obtained from the LSS catalogs and thus there is expected to be some fluctuation in any measured parameter that is proportional to the original uncertainty, purely from the degree to which the weighting de-correlates the weighted and unweighted data. Thus, shifts as a fraction of the uncertainty on the measured BAO parameters are the most relevant thing to quote here. The 0.2 σ level of variation is equivalent to a correlation of 0.99 between the weighted and unweighted results. This is thus similar to that one would expect from randomly applying weights with the same variance as the imaging systematic weights to the sample. We stress that the weighted clustering results are expected to be much closer to the 'true' clustering than those without the imaging systematic weights. Thus, even if these shifts were not consistent with random variation, they would represent an extreme scenario. \nSpectroscopic systematics refer to both the effect of errors in the DESI redshift estimation and unphysical galaxy/quasar variations in the fraction of successful redshifts as a function of parameters relating to DESI hardware and observing conditions. Similar to the corrections for imaging systematics, trends with spectroscopic properties can be regressed against, and corrected with, weights. Such weights, w zfail , are determined as a function of effective observing time for all DESI tracers, as described in [24, 71]. Despite the significance of the observed trends, [24] find that difference in clustering between results including w zfail or not yields a maximum χ 2 difference of just 0.09 (for ELG samples at both redshift range) across all of the pre-reconstruction ξ 0 , 2 within the BAO fit range. Such a small χ 2 difference implies at most a 0.3 σ shift in any potential parameter; the differences are not localized and any effect on BAO measurements is thus expected to be negligible. Further corrections, such as extra weights to remove the remaining dependence of the success rate on \nthe focal plane location, were also studied in [24] and found to have similarly small effects on the measured clustering. The size of the effect of redshift success trends on the DESI DR1 two-point functions is thus negligible for BAO measurements. \nFinally, redshift errors have been characterized again in DR1. These are divided into two components: a typical uncertainty, characterized by a narrow Gaussian or Lorentizian profile, and a rate of catastrophic failure. The potential impact of the typical uncertainty on clustering measurements have been studied with galaxy mocks generated with the UNITSHAM method [143] and AbacusSummit-HOD models [80, 89]. In [143] the impact on ELGs was negligible at all scales and the effect for LRGs was ∼ 0 . 05 σ . In DR1, the ELG redshift uncertainty is even smaller [24], and LRGs show the same level of uncertainties [71]. The impact on measured QSO clustering was substantially greater, due to the larger uncertainty on their redshifts. However, the effect of redshift uncertainties as an additive component to the velocity dispersion is included for both LRG and QSO in the DESI Y1 mocks, where no detectable bias in BAO parameters is found. \nIn studying catastrophic failures in the redshift measurements of DESI ELGs (using, e.g., repeated observations of the same ELG), [24] find that the most prominent feature is confusion at z ∼ 1 . 32 between residuals from sky spectra and [O ii ] emission. The total failure rate from repeat spectra was found to be 0.27%, much smaller than the worst case of 1% [144]. However, the observed failure patterns with 0.27% rate show similar clustering impacts to the assumed 1% failures for ELG mocks at 1 . 1 < z < 1 . 6 due to the prominent confusion at z ∼ 1 . 32. Nevertheless, their impact on the measured configuration space clustering is < 0 . 01 σ at scales s > 60 h -1 Mpc and thus should not impact BAO measurements.", '5.4 Systematics due to the assumption of the fiducial cosmology': 'In this subsection, we summarize the impact of using a wrong fiducial cosmology in the BAO analysis, which were extensively studied in [51]. \nThe choice of reference cosmology comes into play at three different stages. First, we assume a set of cosmological parameters when converting redshift measurements into distances; we term this the grid cosmology . The difference between the grid and true cosmologies causes a distortion of the BAO scale along and across the line of sight [10], which is quantified by the parameters q ∥ = H grid ( z ) /H true ( z ) and q ⊥ = D true ( z ) /D grid ( z ) respectively. Without the three-dimensional standard ruler like BAO, this effect is somewhat degenerate with the redshift-space distortions, but with a sufficiently large data set, such as we have with DESI, and with the BAO feature it is possible to distinguish between the two [145, 146]. Second, a template cosmology is chosen in order to compute the linear power spectrum, which is then used to create the model power spectrum for the fitting ( P nw and P w in Eq. (4.1)). The effect of fixing the template is interpreted as an additional isotropic rescaling of the distances by a factor of r tem d /r true d . Lastly, the values for the linear bias b 1 and the growth rate f ( z ) input into the reconstruction algorithm are cosmology dependent, affecting the estimation of the displacement field. The separate effect of both the template and the values assumed for reconstruction have been comprehensibly studied in the past [147-150], as well as the joint effect of consistently changing the reference cosmology in the whole pipeline [114], while the separate effect of the grid is less explored [151]. Potential systematic shifts of the order of a few tenths of a percent in the alpha values have been reported in the most extreme scenarios. \nWe studied the effect of analysing mocks with DESI DR1 geometry with different fiducial cosmologies. Namely, the four secondary AbacusSummit [152] cosmologies (Table 10), which', 'LRG 0.8<z<1.1': 'Figure 4 . Contour plots of the differences of the (rescaled) alpha values measured with the different AbacusSummit cosmologies using DESI DR1 data. As an example, the LRG high redshift bin is shown. The quantities ∆ α c 000 are defined as follows. For a given cosmology c00x, the derived alpha values are rescaled to c000 units and the values measured directly with c000 are subtracted. \n<!-- image --> \ndeviate from the Planck 2018 cosmology in various aspects. For example, from Table 10, c002 introduces an offset on the distance to redshift relation at the level of ∼ 4 . 7 -7 . 5% and an offset in the sound-horizon scale at the level of 6.8%. Up to 2.9% of the anisotropic distortion (AP) is also tested with c002. \nOur baseline test consists of simultaneously changing the cosmology for grid, template and reconstruction (the separate contributions of grid and template are explored in the companion paper). The fiducial DESI cosmology c000 is used to create the Abacus-2 DR1 mock catalogs, but the different grid cosmologies c001-c004 are used to convert redshifts into distances to compute the pair counts and power spectrum. The change in template is implemented by simply changing the linear power spectrum computed with class and then following the fitting procedure with desilike as described in Section 4.3. The values for the linear bias and growth rate are derived assuming each fiducial cosmology and introduced in reconstruction. \nWe compare the expected BAO scales (the last two columns in Table 10) with the derived BAO scales for all tracers and for each of the wrong fiducial cosmologies using 25 Abacus-2 DR1 mocks, and regard a shift detection for values above a 3 σ based on the dispersion of the 25 mocks. Table 11 summarises our results in [51]. For cosmology c003 \nTable 10 . The level of scale dilation and anisotropy on the α parameters we introduce in the test of the fiducial cosmology, by adopting four fiducial cosmologies (c001-c004) when analyzing the mock catalogs generated using the true cosmology (c000). Each of c000-c004 is based on AbacusSummit [152]. The effect of the grid cosmology on the scale dilation depends on redshift and we therefore list the ranges for 0 . 1 < z < 2 . 1. Note that, with c002, we introduce an offset on the distance to redshift relation at the level of ∼ 4 . 7 -7 . 5% and an offset in the sound-horizon scale at the level of 6.8%. Up to 2.9% of the anisotropic distortion (AP) was also tested with c002. The last two columns are the expected BAO scales we should recover. A significant (3 σ ) offset of the derived BAO scale from the last two columns is considered as a bias. \nTable 11 . Estimation of systematics due to the assumption of an incorrect fiducial cosmology, based on the test by [51]. We show the residual bias in the BAO scales after accounting for the expected shift due to using incorrect grid and template cosmologies (c00X, where X = 1, 2, 4). Unless the residual bias in the BAO is detected at a significance greater than 3 σ (in terms of the standard deviation of the mean of the mocks), we consider there to be no net bias in the BAO scale due to the fiducial cosmology and put an ad-hoc value of 0.1%. We detected a significant bias in α iso for c003, which assumes an incorrect N eff in the BAO template. Systematics from an incorrect N eff are not included in our systematic budget, as this concerns the interpretation of the measured BAO scale. \nwe measure a systematic shift of the order of 0.2% for α iso that can be neatly explained by the phase shift in the BAO due to the difference in N eff (see for example [153]). Since this effect comes from the template and it is a matter of how the re-scaling by the sound horizon is interpreted in the BAO measurements, we opt to not include it as part of the systematic error budget. A detection at the level of 0.1% is reported for c002 from [51]; otherwise we do not see a detection of bias. Thus we adopt a conservative value of 0.1% for both α iso and α AP . Furthermore, we tested the consistency of the BAO fits against the choice of fiducial cosmology for the blinded Y1 catalogs, as part of the unblinding tests described in Section 6. Likewise, we conduct an analogous comparison with the unblinded data, for which consistent results are observed for the different cosmologies (Figure 4).', '5.5 Combining all systematics': 'In this subsection, we combine all the systematics in the basis of α iso and α AP , and then project these to the systematics in α ⊥ and α ∥ . It is imperative to include all pertinent systematics, while setting the upper limit of systematics conservative enough to avoid adding systematics after unblinding. \nTable 12 . Summary of our individual systematic errors. In some cases, we do not have any statistically significant detection of a systematic bias. \nTable 13 . The combined non-zero systematics. The total error in each space is obtained by summing contributions in quadrature and rounding to two significant digits. For the fiducial cosmology systematics, we do not include the bias that can be introduced due to an incorrect assumption of N eff in the fitting model. Although the tracer-by-tracer estimation of the systematics, such as the HODs, return smaller systematics for some tracers and in P ( k ), here we will adopt the worst case of the systematics across all tracers, 0 . 245% for α iso and 0.3% for α AP for both ξ ( r ) and P ( k ). \nThe summary of the systematic tests is in Table 12. From [45, 46], we conclude that the systematics due to the reconstruction choice, within a few comparably optimal options we narrowed down, is negligible. Moreover, the systematics due to incorrect estimation of bias and incorrect fiducial cosmology during reconstruction are already accounted for by the HOD systematics (Table 8) and the fiducial cosmology systematics budgets (Section 5.4). We also conclude that the observational effects (Section 5.3) and the covariance choice (Section 4.5 and [48]) are negligible. \nTable 13 shows the combination of all systematic budgets based on Table 12. We opt for the most conservative approach, namely, directly adding in quadrature individual systematic budgets. This approach only marginally increases the final error by ∼ 5% even for the best α iso measurement. \nAssuming the negligible correlation between the systematics in α iso and α AP , which was the reason we have been evaluating the systematics using this basis, we derive \nC Sys α iso ,α AP = ( 0 . 245 2 0 0 0 . 3 2 ) , (5.1) \nwhere the matrix elements are the fractional (%) covariances. \nWe projecting this to the systematics in α ⊥ and α ∥ following \nC -1 q i q j = d ln p a d ln q i C -1 p a ,p b d ln p b d ln q j (5.2) \nwhere q i , q j = α ⊥ , α ∥ and p a , p b = α iso , α AP . With C p a ,p b = C Sys α iso ,α AP and \nd ln p a d ln q i = ( 2 / 3 1 / 3 -1 1 ) , (5.3) \nwe find \nC Sys α ⊥ ,α ∥ = ( 0 . 265 2 0 . 478(0 . 265 × 0 . 316) 0 . 478(0 . 265 × 0 . 316) 0 . 316 2 ) . (5.4) \nThis again is the fractional (%) covariances. For the best BAO measurement, LRG3 + ELG1 , this increases an error on α ∥ by 1.5% and α ⊥ by 2.2%. \nNote that, unlike the statistical constraints in Table 15, this synthetic matrix for systematics (Eq. (5.4)) shows a positive cross-correlation, which is mainly because the variances in α iso and α AP are more comparable, unlike the statistical constraints that are subject to a particular combination of contribution of transverse and parallel wavemode. We add C Sys α iso ,α AP and C Sys α ⊥ ,α ∥ to the corresponding statistical BAO covariance to construct the final posteriors.', '6 The Unblinding Tests': 'One manifestation of confirmation bias is the tendency to attribute a higher level of systematic uncertainty to outcomes that are unexpected or perceived as outliers. To prevent this from affecting the DESI DR1 BAO measurements, we determined the systematic error budget detailed in Table 12 prior to unblinding. \nBefore unblinding, we also required the following set of the conditions to be met. \nTable 14 . Unblinding checklists. For each entry in the checklist, if the difference observed in the blinded data is within the full range covered by the 25 Abacus-2 DR1 mocks, we consider that the test was passed. For BGS , ELG1 , and QSO , 1-D BAO fits pass these tests while the other tracers pass these tests with 2-D as well as 1-D fits. \n- 1. The official DESI DR1 galaxy BAO analysis pipeline must be determined, including the density field reconstruction setup and the BAO fitting setup. We determined these configurations based on the tests on the blinded data and the mocks. These mocks were calibrated against the DESI One Percent sample rather than DESI DR1. While this was a limitation potentially affecting optimization for the unblinded data, it was a necessary trade-off due to the blinding process.\n- 2. The consistency tests in the checklist (Table 14) were to have been completed on the blinded data to the fullest extent possible. For each entry in the checklist, if the difference observed in the blinded data is within the entire range covered by the 25 Abacus-2 DR1 mocks, we consider that the test was passed. \nTable 14 presents the list of the tests we performed before unblinding. The tests are extensively described in [46]. Based on the tests, we decided the following. \n- 1. We decided the optimal reconstruction scheme and the smoothing scales for each tracer [46]. In particular, we decided to take the reconstructed QSO BAO fit as our fiducial choice regardless of its actual reconstruction efficiency when applied to the unblinded DESI DR1 data.\n- 2. We decided the prior ranges of the BAO damping scales, as detailed in [7].\n- 3. We decided to perform 1-D BAO fits for the lower signal-to-noise samples, i.e., BGS , ELG1 , and QSO . 2-D BAO fits on these samples showed either frequent outliers (i.e., failed to locate the BAO) in the mocks or failed some of the consistency tests in Table 14 with the blinded data.\n- 4. We decided to utilize the combined tracer of ELG1 and LRG3 for 0 . 8 < z < 1 . 1 bin to carry out the cosmology analysis. The information from the lowest signal-to-noise tracer, ELG1 , is propagated to cosmological constraints only through the combined tracer analysis. \nWe also decided on the aspects of the analysis that could be re-done after unblinding. \n- 1. The calibration of covariance matrices could be finalized after unblinding, based on EZmock and analytic covariance matrices. Consequently, the final decision on the covariance matrices between the numerical and analytic covariances could also be made at this stage. This was partly due to the unavailability of EZmocks for DR1 during the unblinding tests, and also because we could not determine beforehand how well the clustering in the EZmocks and Abacus-2 would align with that of the unblinded data.\n- 2. Given the bias expected with an incorrect assumption of N eff in the template (Section 5.4), we allowed a potential iteration of the BAO fits using a different assumption of N eff if we were to find the best fit N eff from the DESI DR1 cosmology analysis different to the value we assumed in the fitting.\n- 3. An update on the catalogs could be made if an error were to be found in checks unrelated to the BAO fitting. The error would then be fixed before assessing the impact of the correction on the BAO fits. As detailed in Section 2.3, an error was indeed identified in the LSS catalogs after unblinding and promptly rectified. The most significant shift on the BAO measurement observed was 0.7 σ shift in α AP for LRG1 , while other shifts are typically less than 0.2 σ .\n- 4. HOD systematics for QSO and BGS would be determined after unblinding, as a result of prioritizing the highest signal-to-noise tracers such as LRG s and ELG s. \nAfter unblinding, we discovered a minor discrepancy between the fitting model outlined in [7] (and Section 4.4 in this paper) and its implementation in the code. The correction of the code resulted in BAO measurement changes of less than a maximum of 0 . 3 σ , with most shifts under 0 . 1 σ , and no systematic correlation in the shifts across redshift bins.', '7 Results': 'In this section, we present the results of the BAO analysis using the unblinded DESI DR1 catalogs.', '7.1 Reconstructed two-point clustering of DESI DR1': 'We start with presenting the clustering measurements of the reconstructed catalogs using our default reconstruction method described in Section 4. While we refer to [67] for the full quantitative discussion comparing the mocks and the data, here we note the excellent consistency between the overall clustering amplitude of the Abacus-2 DR1 mocks and that of the data as a function of scale within the fitting ranges pre and post reconstruction, as shown in Figures 5 and 6. \nFigures 5 and 6 also compare the pre-reconstruction versus post-reconstruction clustering of DESI DR1 within the fitting ranges. As we adopted the RecSym convention, which preserves the linear RSD boost along the line of sight through the displacement of the random particles, the clustering before and after reconstruction is quite consistent on large scales, as expected. On smaller scales we see in Figure 6 that the post-reconstruction power spectrum is reduced relative to the pre-reconstruction one-this is not unexpected as in general we expect the reconstructed density field to have different nonlinear power due to the coordinate remapping, and there is some analytic evidence for the decrease of power specifically in the case of the post-reconstruction matter power spectrum [17, 106, 154, 155]. \n] \n2 \nc \nMp \n2 \n- \nh \n[ \n) \ns \n( \nξ \n2 \ns \n] \n2 \nc \nMp \n2 \n- \nh \n[ \n) \ns \n( \nξ \n2 \ns \n] \n2 \nc \nMp \n2 \n- \nh \n[ \n) \ns \n( \nξ \n2 \ns \n] \n2 \nc \nMp \n2 \n- \nh \n[ \n) \ns \n( \nξ \n2 \ns \n100 \n50 \n0 \n50 \n- \n- \n- \n100 \n150 \n100 \n50 \n0 \n50 \n- \n- \n100 \n60 \n40 \n20 \n0 \n20 \n40 \n60 \n20 \n0 \n20 \n40 \n- \n- \n- \n- \n- \n60 \n60 \n80 \n100 \n120 \n140 \n60 \n80 \n100 \n120 \n140 \n60 \n80 \n100 \n120 \n140 \n1 \ns [ \nh \n- \n60 \n80 \n100 \n120 \n140 \n1 \nMpc] \nFigure 5 . Two-point correlation functions of various tracers of the unblinded DESI DR1 (colored lines). We compare the overall clustering amplitude over the BAO fitting range (48 h -1 Mpc < r < 152 h -1 Mpc) before (dotted lines) and after reconstruction (solid lines), compared to the corresponding mean of the 25 Abacus-2 DR1 mocks (black lines). Gray shading represents the error associated with post-reconstruction DESI DR1 based on RascalC covariance. As a reminder, the data points of ξ are substantially correlated between different separation r . The upper set of the lines is for the monopole, and the lower set of the lines is for the quadrupole. The plots demonstrate excellent consistency in the overall clustering between the data and the mocks. \n<!-- image --> \ns [ \nh \n- \ns [ \nh \n- \ns [ \nh \n- \ns [ \nh \n- \nMpc] \n1 \nMpc] \nELG2 \nPre \nPost \n1 \nMpc] \nLRG3+ELG1 \nPre \nPost \n80 \n100 \n120 \n140 \n1 \nMpc] \nLRG2 \nPre \nPost \nBGS \nPre \nPost \n] \n2 \nc \nMp \n2 \n- \nh \n[ \n) \ns \n( \nξ \n2 \ns \n] \n2 \nc \nMp \n2 \n- \nh \n[ \n) \ns \n( \nξ \n2 \ns \n] \n2 \nc \nMp \n2 \n- \nh \n[ \n) \ns \n( \nξ \n2 \ns \n100 \n75 \n50 \n25 \n0 \n25 \n50 \n75 \n- \n- \n- \n- \n100 \n100 \n75 \n50 \n25 \n0 \n25 \n50 \n75 \n- \n- \n- \n- \n100 \n20 \n0 \n20 \n40 \n60 \n- \n- \n- \n60 \n80 \n100 \n120 \n140 \n60 \n80 \n100 \n120 \n140 \n1 \ns [ \nh \n- \ns [ \nh \n- \nMpc] \nELG1 \nPre \nPost \n1 \nMpc] \nLRG3 \nPre \nPost \nLRG1 \nPre \nPost \nFigure 6 . Power spectra multipoles of various tracers of the unblinded DESI DR1 (colored lines) in comparison to the Abacus-2 DR1 mocks (black lines). Similar to Figure 5. The gray shading is from the analytic covariance matrices for the DESI DR1 data in Section 4.5. Again, the plots demonstrate excellent consistency in the overall clustering between the data and the mocks. \n<!-- image --> \nFrom Figure 6, the change in clustering before and after reconstruction of the data (colored solid and dotted lines for monopole and quadrupole, respectively) appears highly consistent with the expectation based on Abacus-2 DR1 mocks within the 1 σ dispersion expected from \nFigure 7 . The isolated BAO feature in the correlation function of DESI DR1 data before (open circles) and after reconstruction (solid circles). A 1-D BAO fitting is performed for BGS , ELG1 , and QSO , while the rest is fitted for the 2-D BAO scales. The solid and dashsed lines are the best fit BAO models to the unblinded DESI DR1 before (open circles) and after reconstruction (solid circles), respectively. \n<!-- image --> \nthe RascalC covariance matrice (the black solid and dashed lines with the gray shade for the 1 σ dispersion). \nThe BAO feature appears moderately sharpened by reconstruction in the LRG s redshift bins, while the improvement is less obvious for other tracers. One can see that the Abacus-2 DR1 mocks replicate the observed level of the BAO sharpening in the DESI DR1 data. Hence, we qualitatively find that the reconstruction of the data is performing as expected \nFigure 8 . The isolated BAO feature in the monopole and quadrupole before (open circles) and after reconstruction (solid circles) in the power spectrum. The solid and dashsed lines are the best fit BAO models to the unblinded DESI DR1 before (open circles) and after reconstruction (solid circles), respectively. The unit in the y axis h -2 Mpc 2 is omitted due to the limited space. \n<!-- image --> \ngiven the survey configuration and the reconstruction method we chose. In the next section, where the results of the BAO fits are discussed, we make quantitative comparisons on the aforementioned aspects and show that all tracers had moderate gain from reconstruction at the level consistent with the mocks.', '7.2 BAO measurements from the DESI DR1 galaxies': "In this section, we present the BAO fitting of all galaxy and quasar tracers using the default fitting method defined in Section 4.3. We focus on the fitting results in configuration space for \nour discussion below, although we get consistent results in Fourier space Appendix A. 26 All results except for ELG1 and LRG3 will be directly used for the DESI DR1cosmology analysis of [43], with ELG1 and LRG3 incorporated into the analysis via LRG3 + ELG1 . Based on the unblinding tests and the decisions presented in Section 6, we performed two-dimensional BAO fits ( α iso and α AP ) for all tracers other than BGS , ELG1 , and QSO . For the latter, we derive α iso using only the monopole data. \nTable 15, Table 19, Figure 7 and Figure 8 summarize the resulting BAO constraints and visualize the isolated BAO features. Table 15 shows that the χ 2 values of the 2-D fits to the correlation functions fall between 33.1 for LRG3 + ELG1 and 58.6 for ELG2 , all for 39 degrees of freedom. The cases of 1-D fits show χ 2 = 17 . 3 for BGS , χ 2 = 20 . 4 for ELG1 , and χ 2 = 32 . 4 for QSO for 19 degrees of freedom. The goodness-of-fit is reasonable; the worst cases, ELG2 gives an associated p-value of 2.27% and the QSO gives an associated p-value of 2.82%. Table 19 in appendix summarises the corresponding fits using the power spectrum multiples and shows a reasonable level of the goodness-of-fit with P ( k ) as well (a smallest p-value of 4.35% for QSO ). The BAO measurement scales for the two conjugate spaces are very consistent (less than 0.3% in α iso and α AP except for QSO ). This level of consistency between the BAO fits between ξ ( r ) and P ( k ) agrees with the Abacus-2 DR1 mocks ([46]). \n̸ \nFigure 7 displays the BAO features in the correlation function selected for each tracer, compared to the best-fit ξ models for pre-reconstruction (dashed lines) and post-reconstruction data (solid lines). For the 2-D cases, LRG1 shows a distinct BAO feature in the postreconstruction quadrupole measurement. This is consistent with the 2 . 29 σ offset of α AP -1 measured in Table 15. The more the underlying α AP differs from the assumed α AP = 1 of the fiducial cosmology, the greater extent to which the feature in the quadrupole would be shifted from that of the monopole and also amplified, as the BAO feature leaks from the monopole to an extent proportional to α AP -1. 27 This behaviour of the BAO feature in the quadrupole is not evident in the mock data, as the mocks consistently assume α AP = 1 when the they were analyzed. That is, the discrepancy between the BAO signal in the mock and the data merely indicates a measurement of α AP = 1 with 2 . 29 σ significance. A similar quadrupole feature and the consistent α AP best fit is derived from the power spectrum fit. \nThe reconstruction applied to the QSO sample is less effective than the other tracers due to its high shot noise, as depicted in Figure 7 and detailed in Tables 15 and 19. The mock analysis in [46] suggests 3 -8% gain in the precision after reconstruction, though individual realizations will vary considerably due to high shot noise. Additionally, [7] and [138] find that on average the reconstruction will unbias the BAO location, even in this regime of high shot noise. Given these findings, in Section 6, we opted to use the reconstructed QSO result as our fiducial while recognizing its inefficiency from the unblinding test. This decision ensures consistency across cases rather than relying on the realization-dependent performance. In particular, with the future DESI data releases, we anticipate an improved reconstruction performance for the QSO sample, with a reduced fluctuation from realization to realization. \nP 0 → P 0 -2 ϵ 5 dP 2 d ln k -6 ϵ 5 P 2 , (7.1) \nP 2 → ( 1 -6 ϵ 7 ) P 2 -4 ϵ 7 dP 2 d ln k -2 ϵ dP 0 d ln k . (7.2) \nFigure 10 . Similar to Figure 9, but showing results for BGS , ELG1 , and QSO . The DESI DR1 results are, again, consistent with the range covered by Abacus-2 DR1, except for QSO ; the result of QSO shows the reconstruction of DESI DR1 is less efficient than the worst case of the mocks. \n<!-- image --> \n- \n<!-- image --> \n- \n<!-- image --> \n- \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 9 . Pre- and post-reconstruction BAO measurements, comparing DESI DR1 measurements (stars) with the 25 Abacus-2 DR1 (the open points with the error bars are the means and the standard deviations around the means of the 25 mocks). Due to the discrepancy between the BAO scales imprinted in the mocks and the BAO scales recovered from the data, we focus on the relative comparison between the pre and the post reconstruction. All DESI DR1 fits are consistent with what is expected from the mocks. \n<!-- image --> \n<!-- image --> \n- \n<!-- image --> \n- \n<!-- image --> \n- \n- \n- \n<!-- image --> \n<!-- image --> \nFigure 11 . Detection significance of the BAO features in various DESI DR1 galaxy tracers. Solid lines show ∆ χ 2 of the BAO fits as a function of α iso , i.e., χ 2 offset from the minimum χ 2 . The dashed lines correspond to ∆ χ 2 of the noBAO fits, i.e., using a model without the BAO wiggles, after subracting the minimum χ 2 of the BAO fit for each tracer. The detection level of the BAO feature for each tracer is shown in the legend. \n<!-- image --> \nFigures 9 and 10 compare the pre- and post-reconstruction BAO fits for the data and the 25 Abacus-2 DR1 mocks. Due to the discrepancy between the BAO location assumed in the mocks and its measurement in the data, we focus on the changes in the best fits and precision. Except for QSO , the measurements from the data (indicated by stars) lie well within the distribution of the 25 Abacus-2 DR1 mocks, which is the criteria we set as 'consistent' given the small number of the mocks. For QSO , the blinded data BAO fit was marginally consistent with the upper boundary (the least effective reconstruction limit) reached by the mocks, therefore passing the unblinding test ([46]). However, the reconstruction of the unblinded DESI DR1 is slightly worse than the blinded case and less efficient than in the worst case of the mocks. We believe that such a slight change before and after unblinding is acceptable. \nIn detail, we observe the change in the best fit α iso with reconstruction is the greatest for LRG2 , giving an 1.6% shift (0 . 87 σ compared to its pre-reconstruction precision) in ξ . In terms of the precision, LRG s show the biggest improvement from reconstruction, a factor of 1.3-1.6 higher precision in σ ( α iso ) and a factor of 1.5-1.8 in σ ( α AP ). For other tracers, we find an improvement of 1.4 for BGS , a factor of 3 for ELG1 28 , and 1.2 ( α iso ) and 1.4 ( α AP ) for ELG2 . Reconstruction of these tracers is not as effective as LRG s. In particular, the reconstruction for ELG2 is very likely inhibited due to its low completeness and irregular footprint for DESI DR1. Again, with the future DESI data releases, we anticipate an improved reconstruction performance for the ELG s sample. To summarize, the BAO reconstruction fits lead to a substantial gain for the LRG s, but only a moderate gain for the other tracers. \nFigure 11 shows the detection level of the BAO feature. For this plot, one-dimensional BAO fits were performed for all tracers including LRG s and ELG2 . Following the standard procedure [156], the fits with the template without the BAO feature were also made (i.e., setting the second term in Eq. (4.1) to zero, hereafter a 'noBAO fit'). The square root of \nTable 15 . Mean values and standard deviations from the marginalized posteriors of the BAO scaling parameters from fits to the unblinded DESI DR1 correlation functions in the α iso -α AP basis. These do not include the systematic effect. r off = C α iso ,α AP / √ C α iso ,α iso C α AP ,α AP . \n± \nthe difference in χ 2 between the pair of the BAO and noBAO fits at the best fit α iso of the former is quoted as the detection significance. 29 The flat likelihood of the noBAO fit is a trivial result of the improved separation between the BAO and noBAO modeling in Eq. (4.1) that the DESI DR1 analysis adopted, i.e., the noBAO modeling now has naturally no dependence on α iso , in contrast to previous configuration space analyses (see, e.g., the right-hand panel of figure 12 in [112]). The detection significance, in the increasing order, is 4.0 σ for BGS , 4.2 σ for QSO , 6.4 σ for LRG1 , 6.8 σ for LRG2 , 7.0 σ for ELG2 , and 8.7 σ for LRG3 , with the most significant case being 9 . 1 σ for LRG3 + ELG1 , closely correlated with the precision.", '7.3 BAO constraints projected onto α ⊥ and α ∥': 'The parameterisation of the BAO-scale fitting by α iso and α AP provides a natural separation of the isotropic and anisotropic dilation of the BAO that can be constrained somewhat independently. Table 15 (and Table 19 in Appendix for P ( k )) displays the cross-correlation coefficients, r off = C α iso ,α AP / √ C α iso ,α iso C α AP ,α AP . We find the absolute values of r off are less than ∼ 0 . 10 for the three LRG s after reconstruction, although it is larger for ELG2 ( ∼ 0 . 3). The low value of the cross-correlation allows us to incorporate systematic errors ignoring the covariance between the two dilation parameters, i.e., by adding the statistical covariance from Table 15 with the systematic covariance in Eq. (5.1). \nWe can also decompose the BAO locations into transverse and line-of-sight dilation parameters α ⊥ and α ∥ (Eq. (1.1)) that are more directly linked to the angular diameter distance and the Hubble parameter at the given redshift. Table 16 (and Table 20 for \nTable 16 . Mean values and standard deviations from the marginalized posteriors of the BAO scaling parameters from fits to the unblinded DESI DR1 correlation functions in the α ∥ -α ⊥ basis. These do not include the systematic effect. r off = C α ⊥ ,α ∥ / √ C α ⊥ ,α ⊥ C α ∥ ,α ∥ . \n± \n± \nP ( k )) shows the BAO constraints from re-fitting ξ ( r ) into this parameterization rather than transforming them using Eq. (5.2). 30 The cross-correlation coefficients in this case, r off = C α ⊥ ,α ∥ / √ C α ⊥ ,α ⊥ C α ∥ ,α ∥ , are in excellent agreement with the theoretical Fisher matrix prediction when the BAO feature is singled out [157]. The covariance based on Table 16 will be combined with the covariance for systematics (Eq. (5.4)).', '7.4 Test of the systematics using the unblinded data': 'We replicate the tests conducted on the blinded catalogs with the unblinded catalogs to investigate any indications of systematics. Figure 12 illustrates the BAO measurements when comparably optimal variations are introduced around our fiducial setup. The figure compares the baseline fit (using ξ ( r )) versus the P ( k ) fit, a different choice of the damping priors, a different choice of the broadband modeling, and a different choice of the reconstruction convention. These are the choices that we consider equally optimal (e.g., the power spectrum fit) or close to being optimal, compared to the default setup. In addition, we also include the pre-reconstruction results (second row). \nThe comparison between the baseline fits versus the fits using P ( k ) (third whisker) shows high consistency in the derived BAO scales across all cases, as discussed in Section 7.2. The BAO precision from the DESI DR1 ξ ( r ) fit reasonably agrees with these mock results, especially for α iso . In some cases, such as LRG2 and ELG2 , the P ( k )-BAO precision tends to be 20-30% better than the ξ ( r )-BAO error. The Abacus-2 DR1 mock test shows 5-10% larger error with ξ ; LRG2 lies somewhat beyond the range of the mocks. Thus, opting for the ξ -BAO fits is deemed a conservative choice. \nIn the fourth whisker, we display the constraints obtained by only fitting data from the NGC, which differs from the baseline case where both galactic caps (NGC+SGC) are combined at the clustering level. In most cases, we only see a small degradation in the parameter constraints compared to the baseline method. The largest deviations appear in α AP of LRG1 and LRG3 + ELG1 , but the comparison with mocks shows that this level of deviation can occur for DESI DR1 [46]. \nFigure 12 . Response of the constraints on the isotropic (top) or anisotropic (bottom) BAO scaling parameters to changes in the data vector or the model assumptions. The baseline configuration adopted for the BAO analysis is shown in the top row, while the remaining rows show single variations around that baseline (see the main text for a description of each case). The consistency between the ξ ( r ) and P ( k ) fits and the consistency between the spline broadband versus polynomial broadband are also very high. We identified the Gaussian prior on the sampling parameters as a more robust choice, but even the suboptimal flat priors return very consistent results. The difference between NGC alone and the default (NGC + SGC) shows mild discrepancies for α AP of LRG1 . \n<!-- image --> \nThe fifth whisker in Figure 12 shows the effect of using a flat, non-informative prior on the BAO non-linear damping parameters. Based on [7], we identified the Gaussian prior on the damping parameters as a more robust choice than a flat prior, but even the sub-optimal, flat priors return very consistent results. \nPrevious BAO analyses from BOSS [104] and eBOSS [4] adopted a different parameterization for the broadband component of the power spectrum and correlation function, using polynomials with varying degrees of freedom and functional forms. The comparison between our default spline broadband analysis and the polynomial-based broadband method (described in Section 4.3.3) demonstrates excellent consistency (sixth whisker). We note that we have performed this comparison in ξ ( r ) and have only modified the parameters that capture the broadband component in the modeling, while still integrating all the other theoretical improvements in our analysis. \nThe second to last whisker shows the constraints using the RecIso convention for reconstruction. This convention, which was the default in previous SDSS BAO analyses [104, 111, 158] supresses higher-order multipoles by removing linear RSD on large scales. To perform this test, we used the (e)BOSS polynomial broadband parametrization for consistency with previous analysis. Overall, we find excellent agreement in the BAO constraints compared to our baseline case using the RecSym convention, in agreement with what was found with mock galaxy catalogs [7] and the blinded DESI data [46]. \nFinally, in the last row, we show results using a smaller smoothing scale when reconstructing the density field. This corresponds to 10 h -1 Mpc for BGS , LRG s, and ELG s, and 20 h -1 Mpc for QSO . We find that the constraints on α iso are only slightly affected by this choice of scale. For α AP , the largest shift is of 1 σ for LRG3 . In [46], tests on the full ensemble of Abacus-2 DR1 show that no statistically significant shifts are observed when comparing these choices of smoothing scales with the baseline. However, variations in individual mocks are expected due to noise fluctuations and are consistent with what is observed in the data. \nIn summary, the systematic tests conducted directly using unblinded DESI DR1 data demonstrate high consistency across variations of the fiducial setups and data selections (e.g., NGC versus NGC+SGC). Hence, this test bolsters the robustness of our measurements. As a caveat, these consistency tests may not identify systematics that shift all the results together.', '7.5 BAO measurements from the combined tracers': "There are several redshift ranges in which more than one DESI target class substantially overlaps. Over 0 . 8 < z < 1 . 1, LRG s, ELG s, and QSO overlap, and over 1 . 1 < z < 1 . 6, ELG s, and QSO overlap. In particular, over 0 . 8 < z < 1 . 1, LRG3 and ELG1 are expected to have comparable densities for DESI Y5 and there is a compensatory crossing of the two populations within this redshift range. As both tracers probe the same volume, the two BAO constraints would be highly covariant on large scales, providing a test for consistency in the tracer-independent parameter, such as post-reconstructed BAO locations. Although we have estimated the tracer-dependent systematics on BAO in Section 5.2 within a reasonable (or limited, depending on a view) range of the HODs for each tracer, these overlapping tracers provide an utmost test of the tracer dependence. \nWhen combining the two BAO constraints we must account for this cross-covariance, either using a set of large mocks in which both tracers are present or with an analytical method. Another approach is to construct a combined catalog. A BAO constraint from such a catalog seamlessly combines the information from two BAO measurements from autoclustering statistics and one from the cross-clustering between ELG1 and LRG3 (hereafter ' LRG3 × ELG1 '). The combined catalog, which we defined as LRG3 + ELG1 in Section 2, may have an additional gain: due to the higher number density, the reconstruction can be more effective. [47] tested the optimal construction of DESI DR1 LRG3 and ELG1 . The work includes the test of systematics using auto-clustering measurements and the cross-clustering between the two tracers. \nFor the unblinded DESI DR1 LRG3 + ELG1 , we find 11% precision-improvement in α iso and 14% improvement in α AP relative to LRG3 , as shown in Figure 8 and Table 15. This is highly consistent with the ∼ 10% improvement predicted by [47] based on the mocks. \nWe find an offset of 0.4% in α iso between LRG3 and LRG3 × ELG1 ([47]) and 0.6% between LRG3 versus LRG3 + ELG1 (Table 15). In α AP , we find an offset of 0.4% between LRG3 and LRG3 × ELG1 ([47]) and 1.9% between LRG3 + ELG1 (Table 15). These differences are well within the range covered by the mocks. Therefore, we detect no tracer-dependent bias among the \nFigure 13 . A comparison of the impact of different pipelines on the derived distances, analyzing the published BOSS and eBOSS LRG data. There are three groups of points, one for each of the redshift bins, while the different panels show results for D V /r d , D M /r d , and D H /r d . In each group, the leftmost point is the published BOSS/eBOSS result. The second point from the left uses the published correlation functions and covariance matrices, but refits these using the methodology presented in this paper, while the last two points show the results of reprocessing the BOSS and eBOSS catalogs and randoms through the full DESI pipeline for different fiducial cosmologies. To make the comparison more direct (and to leave the covariance matrices unchanged), we use the RecIso reconstruction scheme here. The fiducial cosmology used to normalize the distance scales on the y -axis is the DESI fiducial cosmology used in this paper. \n<!-- image --> \nDESI DR1 BAO measurements from LRG3 , ELG1 , LRG3 × ELG1 , and LRG3 + ELG1 . With Y3 and Y5, as the completeness of the ELG s increases, we expect a greater improvement from the combined tracer analysis in terms of reconstruction as well as the systematic test. Note that we perform only the ξ ( r ) fit to the combined tracer, as we have only the RascalC covariance available for the combined tracer.", '7.6 Comparison to previous analyses': 'Given the changes described above in our pipeline, we revisit the SDSS (BOSS and eBOSS) data previously analyzed to quantify the impact of these changes on previously published results [159]. We do so in two stages. First, we refit the published correlation functions (using the published covariance matrices) to measure the distance scale. Second, we rerun the reconstruction pipeline using our new convention and then re-fit the updated correlation functions. Figure 13 shows the results of these comparisons for the three redshift bins used in the SDSS ([159]). \nThe red point of each group in Figure 13 (first from the left) is the published BOSS/eBOSS result for α , while the yellow point (second from the left) uses the published correlation functions and covariance matrices, but refits these using the baseline methodology described in this paper. For the fits performed here, we use damping parameters with Gaussian priors \nTable 17 . Comparison between the SDSS published results and the reanalysis using the DESI pipeline. Among several options we tested in Section 7.6, we present the DESI reanalysis with IFFT RecIso under the BOSS fiducial cosmology, to best match the clustering amplitude assumed in the published covariance matrices. We did not add the SDSS systematic errors in the middle group. We also did not add the DESI systematic errors from Eq. (5.4) to the bottom group, but the difference is negligible. \n± \n± \nto match this analysis, while the published analyses used a fixed damping scale. The differences between these points are due to two primary factors. The first are changes in how the BAO features are damped. When computing the theoretical correlation functions in the previous analyses, the power spectrum was damped by an additional smoothing of 1 h -1 Mpc to accelerate the convergence of the Hankel transform and reduce ringing. This effectively changes the damping scales. Furthermore, previous analyses applied the small-scale streaming velocities to both the BAO and the broadband shape, while our current analysis only applies it to the broadband shape (Section 4.3.4). Additionally, the current analysis has a new separation of the BAO and broadband shape. The top and middle group of Table 17 shows the net change from all of these effects. \nWe also explore the effect of different reconstruction conventions using the full DESI pipeline and setup. We reprocess the BOSS and eBOSS catalogs and randoms using pyrecon and pycorr under cosmodesi and fit using the fiducial setup of Section 4.3 implemented in desilike . The two rightmost points in each group of Figure 13 show the results of this reprocessing, and the bottom group of Table 17 shows their impact. When comparing these numbers, we emphasize that the exact details of how the data sets are assigned to grids and padded will also affect the derived displacement fields. These different implementations should be thought of as yielding different data sets, even though they are derived from the same initial sets of galaxy positions. Given that the remaining differences are well within the statistical precision of these samples, we therefore conclude that our new analysis does yield consistent results with published results. The companion cosmological interpretation paper [43] will present a detailed consistency check at the level of the underlying cosmological parameters.', '8 The final distance measurements and the Hubble Diagram': 'In this section, we present our final distance measurements, construct the Hubble diagram, compare with the previous BAO measurements, and discuss any implications. We adopt the \nFigure 14 . The final BAO-only distance posteriors, shown as measurements of D M /r d , D H /r d , D V /r d , and D H /D M , compared against the fiducial Planck 2018-ΛCDM distances. In the bottom panel, the prediction at the effective redshift of each sample is shown by the black dots along the dashed lines: z eff = 0 . 51 ( LRG1 ), 0.71 ( LRG2 ), 0.93 ( LRG3 ), and 1.32 ( ELG2 ). The uncertainty intervals on the plot do not take account of the systematic errors but doing so would make little visible difference. \n<!-- image --> \nFigure 15 . Hubble diagram of the BAO distance scales measured from the unblinded galaxy and quasar data, compared to those from earlier BAO measurements by the 6 degree Field Galaxy Survey (6dFGS, [27]), WiggleZ [103], the Sloan Digital Sky Survey (SDSS, [159]), and the Dark Energy Survey (DES Y6, [160]), as labelled. From top to bottom, the panels show D M /r d , D H /r d , D V /r d and D M /D H , all relative to the respective quantities evaluated in the DESI fiducial cosmology described in Section 1. For 6dFGS, WiggleZ and some redshift bins of SDSS and DESI, only D V /r d measurements were possible due to the low signal-to-noise ratio, so these points are only shown in the third panel. For the DESI and SDSS redshift bins where both D M /r d and D H /r d were measured, results for D V /r d and D M /D H in the third and fourth panels are displayed with open markers to indicate the repetition of information in the top two panels in a different parametrisation. Note that a slight offset has been applied to the effective redshifts of the SDSS results at z eff = 0 . 51 and 0 . 70 to avoid overlap and ensure visibility in this figure. \n<!-- image --> \nRedshift \nz \nconfiguration space measurements as our fiducial results. \nTable 18 . Final summary of the derived BAO distance scales from fits to the post-reconstruction correlation function multipoles for each DESI target sample, including the systematics from Eqs. (5.1) and (5.4). We display results in terms of the mean values and standard deviations from the marginalized posteriors of each parameter. r off is the correlation coefficient between D M /r d and D H /r d . The results in this table are used for the cosmology analysis in [43]. \n±', '8.1 The final distance measurements with systematics': 'In Table 18, we combine the best-fit BAO measurements, based on Tables 15 and 16, with the total systematic error budget from Eqs. (5.1) and (5.4), and present the final distance posteriors to be used for cosmology inference in [43]. With the knowledge of the fiducial distance-to-redshift relation and the sound horizon scale used in the analysis, we can convert the BAO measurements to the distance observables: comoving angular diameter distance D M ( z ), \nD M ( z ) r d ≡ D A ( z )[1 + z ] r d = α ⊥ D fid M ( z ) r fid d , (8.1) \nthe Hubble distance D H ( z ), \nD H ( z ) r d ≡ c H ( z ) r d = α ∥ D fid H ( z ) r fid d , (8.2) \nand the spherically-averaged distance D V ( z ), \nD V ( z ) r d ≡ [ zD 2 M ( z ) D H ( z )] 1 / 3 r d = α iso D fid V ( z ) r fid d . (8.3) \nFigure 14 graphically presents our distance constraints of all tracers relative to the fiducial Planck 2018-ΛCDM distances. The plot shows offsets from the Planck 2018-ΛCDM predictions for 0 . 4 < z < 0 . 6 ( LRG1 ( z eff = 0 . 51)) and 0 . 6 < z < 0 . 8. The deviation at LRG1 ( z eff = 0 . 51) is mainly for D H /r d . This means that the line-of-sight BAO scale appears bigger than the transverse BAO scale, when the observed space is mapped to the physical space using the metric of the Planck 2018-ΛCDM cosmology. \nOn the other hand, the deviation at LRG2 ( z eff = 0 . 71) is mainly in D V /r d in D M /r d . That is, the transverse BAO scale at this redshift appears much larger than the prediction of the Planck 2018-ΛCDM, giving a smaller D V /r d and D M /r d at this redshift. The cosmological implication of these deviations is extensively tested in [43]. At z > 0 . 8, our BAO measurements are consistent with the expansion history predicted by the Planck 2018ΛCDM. \nWe then construct a Hubble diagram using DESI DR1 BAO-only distance measurements and also with previous spectroscopic BAO measurements overlaid (Figure 15). Again, we \ndefer the thorough investigation of the cosmological implications of these measurements in combination with the DESI DR1 Lyα BAO measurement ([42]) to [43].', '8.2 Remarks on the discrepancy between SDSS and DESI over 0 . 6 < z < 0 . 8': 'Figure 15 shows noticeable discrepancy between the DESI DR1 measurement at z eff = 0 . 71 ( LRG2 ) and SDSS LRG (eBOSS DR16) at z eff = 0 . 7 (0 . 6 < z < 0 . 85). SDSS LRG measured D M (0 . 70) /r d = 17 . 86 ± 0 . 33 and D H (0 . 80) /r d = 19 . 33 ± 0 . 53, with r off = -0 . 32 [111], while this paper reports D M (0 . 71) /r d = 16 . 85 ± 0 . 32, D H (0 . 71) /r d = 20 . 08 ± 0 . 60 with r off = -0 . 420 for LRG2 . \nAn approximate estimate of the cross-correlation between the two tracers in the power spectrum space is 0.21. Assuming that this directly propagates to the cross-correlation of the BAO measurements, the discrepancy is close to ∼ 3 σ for D M /r d and D V /r d when we account for the difference between z eff = 0 . 7 and z eff = 0 . 71 using the Planck 2018-ΛCDM cosmology. 31 \nAn insight gleaned from Section 7.6 and numerous robustness tests conducted in this paper is that an estimator derived from a particular dataset undergoing different pipelines can be regarded as two distinct yet unbiased estimators of the truth, provided each is individually demonstrated to be unbiased. Hence, the cross-correlation coefficient could be much lower than the theoretical estimate, suggesting that the observed 3 σ discrepancy between SDSS and DESI is likely an upper limit. However, the lower limit, determined assuming no correlation, is still 2.7 σ . The DESI reanalysis in Table 17 yields a slightly lower BAO measurement compared to the published SDSS measurements, resulting in significances of 2.8 σ and 2.5 σ , assuming cross-correlations of 0.21 and 0, respectively. \nWhile this 2.5 σ -3 σ discrepancy warrants attention, we have not identified any sources of non-statistical discrepancy from both sets of the data. We reiterate the significant value of a blinded analysis of DESI DR1, conducted and unblinded without being affected by prior knowledge of any discrepancies compared to previous measurements. We anticipate gaining a deeper understanding of its nature through the analysis of DESI Y3 and Y5 data.', '9 Conclusion': 'In this paper, we presented the first BAO scale measurements 32 from the DESI galaxies and quasars since the start of its data collection in 2021. The measurements from DESI DR1 include over 5.7 million galaxy and quasar redshifts over the redshift range of 0 . 1 < z < 2 . 1, with the total effective survey volume of ∼ 18 Gpc 3 . 33 This marks the largest volume and the most redshifts used for any spectroscopic BAO measurements. We summarize the final distance constraints for six redshift bins, detailed in Table 18, as: \n- · Distance precision (i.e., D V /r d ) ranges of 1.9% ( BGS at z eff = 0 . 3), 1.2% ( LRG1 at z eff = 0 . 51), 1.2% ( LRG2 at z eff = 0 . 71), 0.8% ( LRG3 + ELG1 at z eff ∼ 0 . 95), 1.5% ( ELG2 at z eff = 1 . 32), and 2.6% ( QSO at z eff = 1 . 49). The aggregate precision of all measurements is 0.52% on the BAO scale. \n- · The anisotropic distortion of the BAO (i.e., D H /D M ) is measured with the precision of 4.1% ( LRG1 ), 4.1% ( LRG2 ), 2.7% ( LRG3 + ELG1 ), and 4.6% ( QSO ). \nOur measurement marks the highest precision on the cosmological distances for z > 0 . 8, compared to the previous survey, and also collectively marks the highest aggregate precision. Comparing this with the SDSS BAO measurements ([159, 161]), which report an aggregate precision of 0.64% on D V /r d for all galaxy and quasar tracers combined, the aggregate precision of DESI DR1 in this paper returns 0.52%. The detection levels range from the highest 9.1 σ for LRG3 + ELG1 to the lowest 4.0 σ for BGS . We produced the first high-significance BAO detection using galaxies z > 1, i.e., ELG2 at the significance of 7.0 σ . Proving the robust nature of the BAO for ELG s, despite its substantial imaging systematics, is an important result for upcoming Stage-V BAO surveys. \nAlongside the state-of-the-art dataset, the DESI DR1 BAO analysis incorporated several significant novel elements. Our analysis is one of the first to rigorously implement blinding in the BAO analysis (c.f. [160]) and the first catalog-level blinded BAO analysis (Section 2.3). This approach was adopted to mitigate confirmation bias. The true clustering was unveiled only once the blinded catalogs successfully passed a series of unblinding tests (refer to Section 6). Anticipating the unprecedented precision of DESI BAO beyond DESI DR1, our analysis incorporated physically motivated enhancements to the BAO fitting and reconstruction methods from previous surveys. This included the adoption of a new reconstruction method (Section 4.2) and improved treatment for separating BAO from broadband signals (Section 4.3). We introduced a spline-based broadband model to reduce parameterization dependence on data signal-to-noise. We constructed and analyzed a combined tracer catalog for LRG s and ELG s spanning 0 . 8 < z < 1 . 1, merging information from both tracers and facilitating an additional systematic test (refer to Section 7.5). This approach yielded an improvement of approximately 10%, with even greater improvements anticipated for future DESI analyses. \nWe employed a unified BAO analysis pipeline and methodology across all galaxy and quasar tracers, as well as between configuration space and Fourier space, marking a first in BAO surveys. The algorithms utilized in this study are incorporated into a publicly accessible repository developed by the collaboration, 34 with some currently available and others planned for future release. We meticulously coordinated a systematic study across all tracers and components. Our investigation covered theoretical modeling effects (Section 4.4), tracerspecific systematics (Section 5.2), assumptions regarding fiducial cosmology (Section 5.4), observational systemtaics (Section 5.3), reconstruction algorithm choices (Section 4.2), and covariance matrix choice (Section 4.5). Summary of these findings is provided in Table 13, yielding conservative systematics estimates of 0.245% for the isotropic BAO scale and 0.300% for the anisotropic distortion of the BAO feature (i.e., the Alcock-Paczynski effect). Further details are available in supporting papers listed in Table 1. \nTo quantify the impact of the changes we made in the DESI pipeline, we revisited the SDSS BAO measurements using the DESI pipelines. We conclude that the two measurement pipelines yield sufficiently similar results for these datasets (refer to Section 7.6), although we anticipate that the new method will be advantageous for the higher-precision dataset expected in the future. \nFrom the Hubble diagram constructed from our results, we found that our BAO measurements indicate systematically larger observed BAO scales than the prediction of Planck \n2018-ΛCDM at z < 0 . 8, and therefore lower distances. For z eff = 0 . 51, this discrepancy is sourced from the line-of-sight BAO: it appears bigger than the transverse BAO scale, when the observed space is mapped to the physical space using the metric of the Planck 2018ΛCDM cosmology. For z eff = 0 . 71, the discrepancy is sourced from the overall size of the BAO (and more transverse than the line-of-sight). The transverse BAO scale at this redshift appears much larger than the prediction of the Planck 2018-ΛCDM, implying a smaller D V /r d and D M /r d at this redshift. Accordingly, we found a large discrepancy between the DESI BAO measurement at z eff = 0 . 71 and the SDSS measurement at z eff = 0 . 7. The level of discrepancy is ∼ 3 σ on D M /r d . At z > 0 . 8, our BAO measurements are consistent with that of the Planck 2018-ΛCDM. We emphasize the value of our blinded analysis in achieving the DESI DR1 BAO measurements without prior knowledge of comparisons with previous measurements. An extensive investigation on the cosmological implication of these discrepancies from Planck 2018-ΛCDM, in combination with the BAO measurement from the Lymanα forest ([42]), is presented in the DESI DR1 cosmology paper, [43]. \nAssuming the same sample definitions (which are conservative for DESI DR1), the final DESI dataset will be more than a factor of 3 in terms of the effective volume (4-5 for ELG s, ∼ 3 for LRG s, and 2-3 for BGS and QSO ) compared to DESI DR1, using more than 20 million unique redshifts. Through the analysis of the DESI Y3 and Y5 data, we anticipate to gain a deeper understanding of the nature of the intriguing measurements we observed.', '10 Data Availability': 'The data used in this analysis will be made public along the Data Release 1 (details in https://data.desi.lbl.gov/doc/releases/).', 'Acknowledgments': "This material is based upon work supported by the U.S. Department of Energy (DOE), Office of Science, Office of High-Energy Physics, under Contract No. DE-AC02-05CH11231, and by the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility under the same contract. Additional support for DESI was provided by the U.S. National Science Foundation (NSF), Division of Astronomical Sciences under Contract No. AST-0950945 to the NSF's National Optical-Infrared Astronomy Research Laboratory; the Science and Technology Facilities Council of the United Kingdom; the Gordon and Betty Moore Foundation; the Heising-Simons Foundation; the French Alternative Energies and Atomic Energy Commission (CEA); the National Council of Humanities, Science and Technology of Mexico (CONAHCYT); the Ministry of Science and Innovation of Spain (MICINN), and by the DESI Member Institutions: https://www.desi.lbl.gov/ collaborating-institutions . Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the U. S. National Science Foundation, the U. S. 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As noted in the main section, our analytical and mock-based covariances have not fully converged in our Fourier space analysis, thus we consider the power spectrum results less robust. Nevertheless, the power spectrum results serve as a consistency check of our results based on the correlation function as the two are in good agreement, in line with expectations derived from mock tests.', 'B Author Affiliations': "- 1 Instituto de F'ısica Te'orica (IFT) UAM/CSIC, Universidad Aut'onoma de Madrid, Cantoblanco, E-28049, Madrid, Spain\n- 2 Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA \nTable 20 . Mean values and standard deviations from the marginalized posteriors of the BAO scaling parameters from fits to the unblinded DESI DR1 power spectra in the α ∥ -α ⊥ basis. Systematic effects are not included. r off = C α ⊥ ,α ∥ / √ C α ⊥ ,α ⊥ C α ∥ ,α ∥ . \n± \nTable 19 . Mean values and standard deviations from the marginalized posteriors of the BAO scaling parameters from fits to the unblinded DESI DR1 power spectra in the α iso -α AP basis. Systematic effects are not included. The P ( k ) fits do not include LRG3 + ELG1 , as we do not have a covariance available for the combined tracer in P ( k ). r off = C α iso ,α AP / √ C α iso ,α iso C α AP ,α AP . \n± \n± \n4 Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India \n- 5 Centre for Extragalactic Astronomy, Department of Physics, Durham University, South Road, Durham, DH1 3LE, UK\n- 6 Institute for Computational Cosmology, Department of Physics, Durham University, South Road, Durham DH1 3LE, UK\n- 7 Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA\n- 8 Leinweber Center for Theoretical Physics, University of Michigan, 450 Church Street, Ann Arbor, Michigan 48109-1040, USA\n- 9 IRFU, CEA, Universit'e Paris-Saclay, F-91191 Gif-sur-Yvette, France\n- 10 Institut de F'ısica d'Altes Energies (IFAE), The Barcelona Institute of Science and Technology, Campus UAB, 08193 Bellaterra Barcelona, Spain \n95 Department of Physics, University of California, Berkeley, 366 LeConte Hall MC 7300, Berkeley, CA 94720-7300, USA"}

2017arXiv170200786A

Following the selection of The Gravitational Universe by ESA and the successful flight of LISA Pathfinder the LISA Consortium now proposes a 4 year mission in response to ESAs call for missions for L3. The observatory will be based on three arms with six active laser links between three identical spacecraft in a triangular formation separated by 2.5 million km. LISA is an allsky monitor and will offer a wide view of a dynamic cosmos using Gravitational Waves as new and unique messengers to unveil The Gravitational Universe. It provides the closest ever view of the infant Universe at TeV energy scales has known sources in the form of verification binaries in the Milky Way and can probe the entire Universe from its smallest scales near the horizons of black holes all the way to cosmological scales. The LISA mission will scan the entire sky as it follows behind the Earth in its orbit obtaining both polarisations of the Gravitational Waves simultaneously and will measure source parameters with astrophysically relevant sensitivity in a band from below 104Hz to above 101Hz.

2017-02-01T00:00:00Z

['2017arXiv170200786A', 'arXiv:1702.00786', '10.48550/arXiv.1702.00786']

['Astrophysics - Instrumentation and Methods for Astrophysics']

Laser Interferometer Space Antenna

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/1702.00786.pdf

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2024A&A...690A.344M

Transitional millisecond pulsars tMSPs represent a dynamic category of celestial sources that establish a crucial connection between lowmass Xray binaries and millisecond radio pulsars. These systems exhibit transitions from rotationpowered states to accretionpowered ones and vice versa highlighting the tight evolutionary link expected by the socalled recycling scenario. In their active phase these sources manifest two distinct emission modes named high and low occasionally punctuated by sporadic flares. Here we present hightimeresolution spectroscopic observations of the binary tMSP J10230038 in the subluminous disc state. This is the first shorttimescale 1 min optical spectroscopic campaign ever conducted on a tMSP. The campaign was carried out over the night of June 10 2021 using the Gran Telescopio Canarias. The optical continuum shows erratic variability without clear evidence of high and low modes or of orbital modulation. Besides the analysis of these hightemporalcadence spectroscopic observations reveals for the first time evidence for a significant up to a factor of 2 variability in the emission line properties equivalent width and full width half maximum over a timescale of minutes. Intriguingly the variability episodes observed in the optical continuum and in the emission line properties seem uncorrelated making their origin unclear.

2024-10-01T00:00:00Z

['2024arXiv240912893M', '2024A&A...690A.344M', '10.1051/0004-6361/202449466', 'arXiv:2409.12893', '10.48550/arXiv.2409.12893']

['accretion', 'accretion disks', 'stars: neutron', 'X-rays: binaries', 'Astrophysics - High Energy Astrophysical Phenomena']

Hightemporalresolution optical spectroscopic observations of the transitional millisecond pulsar PSR J10230038

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']

https://arxiv.org/pdf/2409.12893.pdf

{'High-temporal-resolution optical spectroscopic observations of the transitional millisecond pulsar PSR J1023+0038': "M. M. Messa 1 , 2 , P. D'Avanzo 2 , F. Coti Zelati 3 , 4 , 2 , M. C. Baglio 2 , S. Campana 2 \n- 1 Università degli Studi di Milano, Dipartimento di Fisica, Via Celoria 16, 20133 Milano (MI), Italy; e-mail: marco.messa@unimi.it\n- 2 INAF, Osservatorio Astronomico di Brera, Via E. Bianchi 46, 23807 Merate (LC), Italy\n- 3 Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans s / n, E-08193, Barcelona, Spain\n- 4 Institut d'Estudis Espacials de Catalunya (IEEC), Carrer Gran Capità 2-4, E-08034 Barcelona, Spain \nReceived month day, year; accepted month day, year", 'ABSTRACT': 'Transitional millisecond pulsars (tMSPs) represent a dynamic category of celestial sources that establish a crucial connection between low-mass X-ray binaries and millisecond radio pulsars. These systems exhibit transitions from rotation-powered states to accretionpowered ones and vice versa, highlighting the tight evolutionary link expected by the so-called recycling scenario. In their active phase, these sources manifest two distinct emission modes named high and low, occasionally punctuated by sporadic flares. Here, we present high-time-resolution spectroscopic observations of the binary tMSP J1023 + 0038, in the sub-luminous disc state. This is the first short-timescale ( ∼ 1 min) optical spectroscopic campaign ever conducted on a tMSP. The campaign was carried out over the night of June 10, 2021 using the Gran Telescopio Canarias. The optical continuum shows erratic variability, without clear evidence of high and low modes or of orbital modulation. Besides, the analysis of these high-temporal-cadence spectroscopic observations reveals, for the first time, evidence for a significant (up to a factor of ∼ 2) variability in the emission line properties (equivalent width and full width half maximum) over a timescale of minutes. Intriguingly, the variability episodes observed in the optical continuum and in the emission line properties seem uncorrelated, making their origin unclear. \nKey words. stars: individual: PSR J1023 + 0038 - accretion, accretion discs - stars: neutron - pulsars: general - X-rays: binaries.', '1. Introduction': "Millisecond radio pulsars (MSPs) are rapidly rotating magnetic neutron stars (NSs) whose fast spin originates through the transfer of matter in a low-mass X-ray binary (LMXB, Alpar et al. 1982; Radhakrishnan & Srinivasan 1982). The discovery of the first 'transitional' MSP (tMSP), PSR J1023 + 0038 (hereafter J1023; Archibald et al. 2009), established a connection between rotation-powered MSPs and accretion-powered NSs in LMXBs. These systems are observed to switch back and forth between a rotation-powered radio pulsar state ('pulsar state') and a state characterised by X-ray pulsations and accretion disc features in the optical spectra ('disc state'). Besides J1023, three other tMSPs systems are known so far: J18245-2452 (M28I; Papitto et al. 2013), XSS J12270-4859 (Bassa et al. 2014; de Martino et al. 2013), plus a few candidates: RXS J154439.4112820 (Bogdanov & Halpern 2015; Bogdanov 2016; Britt et al. 2017), CXOU J110926.4-650224 (Coti Zelati et al. 2019), 4FGL J0407.7-5702 (Miller et al. 2020; Kennedy et al. 2020), 3FGL J0427.9-6704 (Strader et al. 2016; Li et al. 2020), Terzan 5 CX10 (Bahramian et al. 2018) and XMM J174457-2850.3 (Degenaar et al. 2014; Degenaar et al. 2015). \nJ1023 was discovered by Bond et al. (2003) as part of the Faint Images of the Radio Sky at Twenty Centimeters' (FIRST) survey. In 2001, it was categorised as a cataclysmic variable because the optical counterpart to the radio source exhibited rapid, shortduration flickering and a blue optical spectrum with doublepeaked emission lines, indicative of an accretion disc (Szkody et al. 2003). However, optical photometry conducted in 2003 \n(Woudt et al. 2004) and 2004 (Homer et al. 2006) did not show the intense, rapid flickering events observed in the 2001 light curve, indicating a change in the system's state. Only 4.75-hour single-humped modulation was detected (Woudt et al. 2004). Indeed, Thorstensen & Armstrong (2005) confirmed this state transition, as the optical spectrum taken in 2003 displayed strong absorption features and lacked the prominent emission lines seen in the 2001 spectrum. They conducted a time-resolved optical spectroscopic and photometric study of J1023. They identified the companion star as a late-type G5 star with an absorption-line radial velocity semi-amplitude of 268 ± 3 km s -1 modulated at the 4.75-hr orbital period. Optical light curves taken in 2004 revealed a single-humped modulation, attributed to the companion star strongly irradiated by the pulsar particle wind. The combination of photometric and radial velocity studies led to the conclusion that the system was not a cataclysmic variable but, rather, an X-ray binary hosting an NS. This X-ray binary scenario also explained the 2004 X-ray observations, which were characterised by a dominant hard X-ray power-law component (Homer et al. 2006). The discovery of a 1.69-ms radio pulsar in a 4.75-hr binary system (Archibald et al. 2009) made J1023 the first system showing the potential to alternate between an X-ray state and a radio pulsar phase powered by rotation. \nIn 2013, J1023 underwent a sudden and significant increase in the emission levels at both X-ray and gamma-ray frequencies, amplified by a factor of 5-10. Simultaneously, there was a noticeable enhancement in the emission at ultraviolet (UV) and optical frequencies, with a magnitude increase of 1-2. This notable shift in emissions coincided with the vanishing of the pulsed ra- \ndio signal (Stappers et al. 2014 and Patruno et al. 2014). Shortly after these events, double-peaked optical emission lines were detected, which provided further evidence for the formation of an accretion disc, as was documented by Coti Zelati et al. (2014). J1023 has since remained in this active state, maintaining an Xray luminosity of LX ∼ 7 × 10 33 erg s -1 in the energy range of 0.3-80 keV (Coti Zelati et al. 2018), based on a distance estimate of 1.37 kpc provided by Deller et al. (2012). This phase is also called sub-luminous because it di ff ers in brightness from the typical luminosity of NSs in their X-ray accretion state of the order of LX ∼ 10 36 -10 38 erg s -1 . During the disc state, the X-ray emission from J1023 exhibits a distinctive pattern of switching between two intensity modes, which have been labelled as high and low (Papitto et al. 2013). These transitions are occasionally punctuated by sporadic flares (see e.g., Linares 2014; Bogdanov et al. 2015). The high mode in the X-ray band is predominant, occurring roughly 70% of the time with a luminosity of LX ∼ 7 × 10 33 erg s -1 , while the low mode manifests 20% of the time with a luminosity of LX ∼ 3 × 10 33 erg s -1 . Typically, the low mode persists for durations ranging from a few tens of seconds to a few minutes. The rate of change in intensity during these mode switches happens on timescales of ≈ 10 s. In addition, in the X-ray high-intensity mode, there are concurrent pulsations at X-ray, UV, and optical frequencies, which cease during the low mode (Archibald et al. 2015; Papitto et al. 2019; Miraval Zanon et al. 2022, and Illiano et al. 2023). The presence of optical pulsations at the spin period of J1023 discovered by Ambrosino et al. (2017) in the disc state cannot be explained by an accretion of matter onto the NS. This finding is interpreted as a strong indication that a rotation-powered pulsar is active in the system, even when the accretion disc is present (Papitto et al. 2019). \nThe pulsar generates a wind of particles that interacts with the innermost accretion flow generating a region of shock very close to the pulsar (within two times the light cylinder radius; Papitto et al. 2019), maintaining coherence and giving rise to the pulsated emission observed from optical up to X-ray frequencies (Papitto et al. 2019; Veledina et al. 2019). During low modes, it has been proposed that the shock moves outwards (Papitto et al. 2019) or that the disc penetrates the light cylinder radius and the system enters the propeller regime (Veledina et al. 2019). More recently, Baglio & Coti Zelati et al., (2023) have shown that the high-to-low mode transition could be interpreted as being due to the evolution of the innermost accretion disc region into a compact radio jet, with the additional emission of discrete ejecta, and that the low-to-high mode switch is due to the re-enshrouding of the pulsar. \nThe emission at UV, optical, and near-infrared (NIR) frequencies is primarily influenced by the presence of the accretion disc and the irradiated companion star. In addition, occasional flaring and flickering are observed (Shahbaz et al. 2018; Kennedy et al. 2018; Papitto et al. 2018; Hakala & Kajava 2018, Baglio et al. 2019), which were interpreted as being due to a combination of reprocessing of the optical emission and direct NIR emission from plasmoids in the accretion flow that are channelled onto the NS and then expelled from the magnetosphere (Shahbaz et al. 2018). Time-resolved spectroscopic studies carried out in the optical band were presented by Hakala & Kajava (2018), who observed changes in the disc emission structure (specifically in the properties of the H α emission line) using a single exposure time of 400 s and a 2.5 m telescope. These observations, together with Doppler tomography, suggested that a propeller e ff ect might occur during flaring, as is suggested by Shahbaz et al. (2015). Additional minor contributions to the overall emission are provided \nby the shock between the relativistic pulsar wind and the surrounding material and by a jet (Coti Zelati et al. 2014; Baglio et al. 2019; Baglio & Coti Zelati et al., (2023)), seen at X-rays, NIR, and optical wavelengths, respectively. \nIn this paper, we report the results of a high-temporal-cadence spectroscopic study of J1023 carried out by obtaining a spectrum in the optical band when the source was in its disc state every ∼ 50 s with the 10 m Gran TeCan Telescope. The paper is organised as follows: in Sec. 2, we describe our observations and the data reductions; the results are presented and discussed in Sec. 3 and 4; and our conclusions are presented in Sec. 5. Throughout the paper, errors are at a 68% confidence level unless stated otherwise.", '2. Data reduction and analysis': 'J1023 spectra were taken on June 10, 2021 with the 10.4m Gran Telescopio Canarias (GTC) at La Palma. We used the OSIRIS (Optical System for Imaging and low-Intermediate-Resolution Integrated Spectroscopy, Cepa et al. 2000) instrument for longslit spectroscopy with the R1000B grism, achieving a resolution of ∼ 1000 and a dispersion value of 2.12 Å / pix, with spectral coverage from 3630 to 7500 Å. Observations were carried out over one night from 21:22 UT to 22:25 UT, covering 22% of the total orbital period of J1023; namely, from phase 0.13 to 0.35, based on the precise ephemerides of Miraval Zanon et al. (2022) 1 . Each spectrum had an exposure time of 20 s and overall we obtained 87 spectra, each every ∼ 50 s. This is the first time that such a high-cadence spectroscopic study has been performed on a tMSP (and on an NS LMXB in general). The data were reduced using the standard ESO-MIDAS procedures for bias subtraction, flat-field correction, and cosmic ray removal. All spectra were sky-subtracted and corrected for atmospheric extinction. Helium-argon lamp spectra were obtained for wavelength calibration during daytime and with the telescope vertically parked. The wavelength scale was then derived through third-order polynomial fits to 26 lines, resulting in an rms scatter of < 0.06 Å. Instrumental flexures during our observations were then accounted for using atmospheric emission lines in the sky spectra. Finally, each spectrum was flux-calibrated through the observation of a standard spectrophotometric star during the same night.', '3.1. Average spectrum': 'Fig. 1 shows the normalised and averaged disc-state spectrum of J1023. We clearly detect H α , H β , H γ , H δ , HeI ( λλ 4472, 4921, 5016-5048, 5876, 6678, 7065 Å), and HeII at 4686 Å. As has been shown in other work in the literature (Coti Zelati et al. 2014), these lines display a double-horned profile that indicates the presence of an accretion disc.', '3.2. Continuum emission': "The initial focus was to analyse how the various spectral characteristics evolved over time. We started by examining the trend of the optical continuum, which serves as a proxy for the average \nFig. 1: Average spectrum of J1023 normalised to the emission of the continuum. The emission lines studied in this work are highlighted. \n<!-- image --> \nbrightness of the source in the optical range. We used the molly 2 program to carry out this analysis, which allowed us to measure the variability of the continuum by choosing a wavelength range roughly centred in the V-band spectrum region, spanning from approximately 5200 to 5600 Å. As can be seen from Fig. 2, the optical continuum shows an erratic variability with possible flaring episodes at about phase 0.3. Despite the limited orbital coverage, no clear orbital modulation (sinusoidal or ellipsoidal) in the flux density seems to be present in our data. The analysis was also repeated for longer wavelengths (5900 to 6100 Å), obtaining the same kind of variability. \nFig. 2: Plot of the continuum (taken between 5200 and 5600 Å) V-band flux as a function of the orbital phase. For each spectrum, the uncertainty of the flux density is given by the square root of the number of counts. \n<!-- image --> \nPrevious research on J1023 has revealed that the short-term variations observed during the source's disc state, particularly the switches between the well-established low and high modes, demonstrate a notable bimodal distribution of the source counts, highlighted in particular in the X-ray light curve(Shahbaz et al. 2015; Linares 2014; Bogdanov et al. 2015; Coti Zelati et al. 2018; Baglio et al. 2023). Since the observation in the optical band was not covered by an X-ray observation, we do not know when the di ff erent high and low modes occurred. We tried anyway to search for evidence of the same behaviour from the observations in the optical band extracting a histogram of the flux densities. This histogram was constructed by segmenting the observed flux, taken from every single spectrum, into intervals of 0.025 mJy. The resulting histogram is presented in Fig. 3. This type of analysis did not reveal any clear sign of bi-modality in the optical flux. Searches for evidence of mode-switching in the optical / NIR light curve of J1023 have been carried out in the past, leading to both detections (Shahbaz et al. 2015) and nondetection (Baglio et al. 2019. Therefore, considering also both the limited orbital coverage of our data and the ∼ 50 s temporal sampling (which could prevent the detection of particularly short low mode episodes) the non-detection of mode switches is not unexpected. \nFig. 3: Count distribution for the continuum V-band flux of J1023. \n<!-- image -->", '3.3. Equivalent width': 'A first indication of the intensity variability in the observed source emission was obtained by studying the equivalent width (EW) 3 of the various emission lines. A plot showing the Nvariability of the H α line EW is shown in the top left of Fig. 4. As can be seen from the image, there are three clear relative maxima; namely, at orbital phases 0.14, 0.25, and 0.35. Interestingly, there is no evidence for flaring activity at the same time in the continuum emission (Fig. 2). This behaviour is also found in the various lines, albeit it is less obvious in the weaker ones (H δ or H γ , as is shown in Fig. 4). \nFig. 4: EW values over time for the H α , H β , H γ , H δ , HeI at 5876 Å, and HeI at 6678 Å emission lines. \n<!-- image -->', '3.4. Full width at half maximum': 'Another significant parameter for studying the line variability is the full width at half maximum (FWHM). For this study, we fitted the H α , H β , and HeI at 5876 Å lines (which are the most prominent emission lines for all 87 spectra), together with their underlying continuum normalised to the unity, with a Gaussian + constant model. The fit was performed on the wings of each emission line; that is, after masking the central part that is dominated by the double-horned profile, due to the presence of the accretion disc. In Fig. 5, the behaviour of the FWHM for the three emission lines is shown. As can be seen from this figure, the disc velocity behaviour does not seem to correlate with the trends seen from the EW values and the behaviour of the flux of the continuum emission (Fig. 4 and 2), although it does exhibit considerable variation (about 50 % on short timescales).', '3.5. Emission line variability': "Since the system is in its disc state, in order to better investigate the nature of the observed variations in line intensity, we tried to model the continuum and the expected double-horned line profile using a constant plus a double Gaussian. This analysis was specifically conducted on the most prominent line, the H α (Tab. A.1 in the appendix). This approach allows for the extraction of the characteristics associated with each component of the line and, potentially, the disentanglement of changes in their relative contributions to the overall line intensity over time. If the disc becomes homogeneously more luminous, it can reasonably be expected that the overall optical spectrum (continuum and emission lines) will increase in intensity. On the other hand, for a geometrically localised episode of disc variability (e.g. a flare originating from a specific disc region), one could expect a di ff erent behaviour. In such cases, the e ff ects of a geometrically localised flare on the shape of the peak might a ff ect only one of the two components; that is, the blue component if the flare occurs in the region where the matter in the disc is moving toward \nthe observer, and the red component if it occurs in the region where the matter in the disc is moving away from the observer. We first applied this model to the average spectrum. The fit was performed in the wavelength range (6510 - 6618 Å) corresponding to 51 data points. Two constraints were applied during the constant + two-Gaussian fitting process. First, the constant was frozen to 1 (since the continuum was normalised). The second constraint was to require equal FWHM values for the two different Gaussians. Every fit therefore had five free parameters (two normalisations, two centroids, and one FWHM) and 46 degrees of freedom (DOFs). These constraints are in line with expectations for a truncated disc emission and prove particularly e ff ective for spectra in which one component (either the blue or the red) dominates over the other. We obtained a robust fit ( ˜ χ 2 = 18.13, DOF = 46, P 4 = 0.99993). From an F-test we obtained that this model is preferred over the constant + single Gaussian one by more than 3 σ . Having successfully tested the model on the average spectrum, we then repeated the same analysis for all 87 available spectra. When carried out on the single spectra, we find that for the majority of the spectra (59 out of 87, ∼ 70%), the fit converged (Fig. 6) and the H α line profile could be adequately modelled by the procedure described above, with P -values > 1% in about 90% of cases. The temporal trends of these fits are presented in Fig. 7 (the first two plots). As is depicted in Fig. 7 (the first two plots), the behaviours of the blue and red components' values exhibit similarities at the start and end of the time series. However, during the mid-times, their behaviours appear markedly distinct. Specifically, the blue component exhibits significantly elevated values around phase 0.25, as was previously observed in the EW plot (see Fig. 4), while the red component maintains a relatively constant trend. The reliability of the double Gaussian analysis is confirmed by the trend of the combined areas of the blue and red components over time, as is illustrated in Fig. 7 (the last plot). This trend closely mirrors that of the EW, which was expected since the EW serves as a proxy for line in- \nFig. 5: Three plots showing, respectively, the trend for H α , H β , and HeI at 5876 Å emission line of the FWHM over the orbital phase. \n<!-- image --> \n. \nFor about 30% of the spectra, the applied fit did not converge; this may be related to the fact that the H α line displayed a profile \nwith three or more peaks (Fig. 6). The presence of a third component may be due to the presence of some anisotropy; for example, the so-called hotspot (the region where the gas stream from the companion star impacts the accretion disc) or the shock front between the incoming matter from the companion and the NS pulsar wind (if present), or the companion star. The behaviour of a third peak is challenging to comprehend due to the limited observational range. Investigating how this peak changes over time would be of interest, and a comprehensive understanding of its behaviour would require observations spanning at least an entire orbital period, particularly to discern the e ff ects related to the geometric position of the system. Another issue arises from the possible presence of multiple peaks, potentially more than three, the nature of which is di ffi cult to interpret. Multiple peaks could be attributed to system inhomogeneities of uncertain origin or simply to noisy data. Further examination of these unusual spectra will be pursued in future work. Additionally, conducting these observations over an entire orbital period will provide valuable insights.", '4. Discussion': "The results presented in the previous section indicate that the optical spectrum of J1023 exhibits rapid variability on timescales of the order of minutes. Although the limited coverage of the orbital period prevents us from drawing definitive conclusions, we can state that our data do not show clear evidence of 'high-low mode' variability (Fig. 3). On the other hand, this variability involves the continuum, the intensity, and the width of the spectral lines. Previous phase-resolved optical studies of PSR J1023 carried out since its transition to the disc state (e.g. Coti Zelati et al. 2014) provided evidence for an optical light curve clearly modulated at the orbital period and dominated by the emission of an irradiated companion star (i.e. well modelled with a sinusoidallike modulation with a single maximum at phase 0.5, when the companion shows its inner, irradiated face to the observer). As can be seen in Fig. 2, the flux of the continuum does not clearly present this modulation. However, more recent optical observations of PSR J1023 suggest that such a modulation is now less evident with respect to previous studies. As an example, Fig. 3 from Baglio & Coti Zelati et al., (2023) shows the optical light curve (REM data) of PSR J1023 observed on June 26-27, 2021 (i.e. just a few days after the GTC spectra presented in our paper). By comparing Fig. 3 of Baglio & Coti Zelati et al., (2023) with Fig. 3 of Coti Zelati et al. (2014), it is clear that the evidence for sinusoidal-like optical modulation is definitely less evident (if not missing). On the other hand, the optical variability observed by Baglio & Coti Zelati et al., (2023) is similar to what was observed for the optical continuum (Fig. 2), even over a shorter orbital range. This might indicate that the disc is increasing its relative contribution to the total emission of the system at optical frequencies, gradually becoming dominant with respect to the irradiated companion component. A similar behaviour has been observed before in the quiescent accreting millisecond Xray pulsar XTE J1814-338 (Baglio et al. 2013). To thoroughly study the possible correlation between the behaviours of continuum V-band spectra, EW, and FWHM, the values were compared through Pearson correlation coe ffi cient and Spearman's rank correlation coe ffi cient tests (Table 1). As can be seen from the values, there does not seem to be an obvious correlation between the trend of the two parameters for H α and H β emission lines and the continuum. In relation to the EW trend alongside the flux light curve, an inverse correlation is typically expected; in other words, as the continuum flux increases, the line be- \nFig. 7: Three plots showing the trend of the area (A) over the orbital phase. The first two plots, show the trend of the area of the blue Gaussian and red Gaussian for the H α emission line. The last one shows the sum of the areas of the blue and red Gaussians. \n<!-- image --> \nth the other quantities for any of the lines, although it does exhibit considerable variation in its values. \nFig. 6: Three normalised spectra, with the vertical line representing the typical error, centred on the H α emission line. For the first spectrum, the double Gaussian fit converges statistically within 3 σ . For the second one, the fit converges statistically just below 3 σ . For the last spectrum, the fit does not converge. \ncomes 'drowned out', resulting in a lower EW value. However, the peaks of EW are not matched by changes in the continuum. Similarly, the peak of the continuum does not correspond to any significant decrease in the EW value. As was described above, the behaviour of the FWHM does not appear to be correlated \nIn the context of emission line variability, the results derived from the fitting of the line profile using two Gaussian functions have unveiled intriguing patterns. As is illustrated in Fig. 7 (a) and (b), distinct behaviours are evident during mid-times at phase 0.25, while a consistent trend is observable at the beginning and towards the end of the time series. In general, these two parameters exhibit a positive correlation, with the exception of values corresponding to the EW at the time of the maximum observed around phase ∼ 0.25. We also derived the trailed spectrum for the H α line (Fig. 8), although the low resolution prevents us from deriving firm conclusions. The three peaks around 0.14, 0.25, and 0.35 are clearly evident, together with a hint of an overall lower contribution of the red component to the total line emission. \nWavelength (Angstroms) \n<!-- image --> \nFig. 8: Trailed spectrogram for the H α emission line. \nA possible interpretation of these findings suggests that the observed peaks in the overall line intensity are not solely attributable to uniform brightening of the disc. In particular, for the peak around phase 0.25, it appears that a single component, the blue one, is dominant, suggesting the presence of some anisotropy within the disc. It is evident that the blue component significantly influences the observed results. As proposed by Campana et al. (2019), it is possible that this short-term variability is due to magnetic reconnection within the disc. This model suggests that interactions between the strong magnetic \nfields, generated and amplified by the di ff erential rotation of the disc, and the accreting matter from the disc can lead to reconnections of the field lines, manifesting as intermittent 'flares' in the overall emission of the source. According to this scenario, these flares are anticipated to occur sporadically and without any discernible pattern within the inner disc region. However, alternative possibilities exist. Anisotropies in the disc structure may imply the presence of persistent bright spots that would align with the observer's line of sight, due to the orbital motion of the system, causing periodic increases in brightness. The observational data, which covers only around 22% of the orbital period, does not provide su ffi cient information to definitively distinguish between these two scenarios. \nTable 1: Pearson and Spearman test values and the respective Pvalue for correlations between continuum V-band flux, EW and FWHMof the H α and H β emission lines.", '5. Conclusions': "We have presented high-time-resolution spectroscopic observations of the binary tMSP J1023, in the sub-luminous disc state. The source shows, like other tMSPs, flux variability on short timescales (tens of seconds) in all bands. \nHere, we have shown the results of high-time-resolution spectroscopic observations of the binary tMSP PSR J1023 + 0038. We obtained a total of 87 optical spectra acquired over 1.1 hours of observation, covering 22% of the binary orbital period, making this the first optical spectroscopic study carried out on a tMSP (and an LMXB in general) with such a fast temporal cadence. Our main findings are: \n- 1. On average, each single spectrum looks rather similar to those reported in the literature for this source obtained over the same wavelength range and with longer time exposures: a blue continuum, indicative of a high-temperature disc, overlaid with intense Balmer and helium series emission lines. These lines show in most cases a double-horned emission profile indicative of the presence of an accretion disc, as was expected, the source being in its disc state at the time of its observations.\n- 2. We found evidence for variability in the main properties of the optical spectrum of J1023 (optical continuum, EW, and FWHM of the main emission lines) over timescales of minutes. This is the first time that variability in the spectral line properties of a tMSP has been observed over such short timescales. \n- 3. The episodes of variability observed in the continuum, EW, and FWHM seem to be erratic and not correlated with each other, which makes the origin of such episodes unclear. \nThe future development of this project is to repeat this study covering the full orbital period (4.75 hours) of the source, ideally in a multi-wavelength simultaneous observational campaign to also assess a possible correlation between the variability in the emission line properties and the mode-switching phenomenon. In the first stage, this study will verify whether the significant variability episodes are still observed, whether they have counterparts in the other bands, and whether they are sporadic or periodic. Besides, obtaining a series of optical spectra with a high time cadence over the entire orbital period would allow a Doppler map to be obtained through the technique of Doppler tomography; that is, a map in velocity coordinates displaying the geometry of the system optical emission. While a single map could not provide any direct insight into an intrinsically variable disc (or more generally, a variable emitting region), it should be possible to obtain di ff erent maps for di ff erent states of the optical spectra (e.g. flaring and not flaring; Hakala & Kajava 2018). \nAcknowledgements. We thank the referee for helpful comments. This research is based on observations made with the GTC telescope, in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias, under Director's Discretionary Time (Pr. ID: GTC2021-181). FCZ is supported by a Ramón y Cajal fellowship (grant agreement RYC2021- 030888-I), Catalan grant SGR-Cat 2021 (PI: Graber) and the program Unidad de Excelencia María de Maeztu CEX2020- 001058-M. FCZ, SCa and PD'A acknowledge financial support from INAF-Fundamental research astrophysics project 'Uncovering the optical beat of the fastest magnetised neutron stars' (FANS). MCB acknowledges support from the INAF-Astrofit fellowship.", 'References': "Alpar, M. A., Cheng, A. F., Ruderman, M. A., & Shaham, J. 1982, Nature, 300, 728 \n- Ambrosino, F., Papitto, A., Stella, L., et al. 2017, Nature Astronomy, 1, 854\n- Archibald, A. M., Bogdanov, S., Patruno, A., et al. 2015, ApJ, 807, 62\n- Archibald, A. M., Stairs, I. H., Ransom, S. M., et al. 2009, Science, 324, 1411\n- Baglio, M. C., Coti Zelati, F., Campana, S., et al. 2023, A&A, 677, A30\n- Baglio, M. C., D'Avanzo, P., Muñoz-Darias, T., Breton, R. P., & Campana, S. 2013, A&A, 559, A42 \nBaglio, M. C., Vincentelli, F., Campana, S., et al. 2019, A&A, 631, A104 \n- Bahramian, A., Strader, J., Chomiuk, L., et al. 2018, ApJ, 864, 28\n- Bassa, C. G., Patruno, A., Hessels, J. W. T., et al. 2014, MNRAS, 441, 1825 \nBogdanov, S. 2016, ApJ, 826, 28 \n- Bogdanov, S., Archibald, A. M., Bassa, C., et al. 2015, ApJ, 806, 148 \nBogdanov, S. & Halpern, J. P. 2015, ApJ, 803, L27 \n- Bond, H. E., Henden, A., Levay, Z. G., et al. 2003, Nature, 422, 405\n- Britt, C. T., Strader, J., Chomiuk, L., et al. 2017, ApJ, 849, 21 Campana, S., Miraval Zanon, A., Coti Zelati, F., et al. 2019, A&A, 629, L8 \nCepa, J., Aguiar, M., Escalera, V. G., et al. 2000, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 4008, Optical and IR Telescope Instrumentation and Detectors, ed. M. Iye & A. F. Moorwood, 623-631 \n- Coti Zelati, F., Baglio, M. C., Campana, S., et al. 2014, MNRAS, 444, 1783\n- Coti Zelati, F., Campana, S., Braito, V., et al. 2018, A&A, 611, A14\n- Coti Zelati, F., Papitto, A., de Martino, D., et al. 2019, A&A, 622, A211\n- de Martino, D., Belloni, T., Falanga, M., et al. 2013, A&A, 550, A89\n- Degenaar, N., Wijnands, R., Miller, J. M., et al. 2015, Journal of High Energy Astrophysics, 7, 137\n- Degenaar, N., Wijnands, R., Reynolds, M. T., et al. 2014, ApJ, 792, 109\n- Deller, A. T., Camilo, F., Reynolds, J. E., & Halpern, J. P. 2012, ApJ, 748, L1\n- Hakala, P. & Kajava, J. J. E. 2018, MNRAS, 474, 3297\n- Homer, L., Szkody, P., Chen, B., et al. 2006, AJ, 131, 562\n- Illiano, G., Papitto, A., Ambrosino, F., et al. 2023, A&A, 669, A26\n- Kennedy, M. R., Breton, R. P., Clark, C. J., et al. 2020, MNRAS, 494, 3912\n- Kennedy, M. R., Clark, C. J., Voisin, G., & Breton, R. P. 2018, MNRAS, 477, 1120\n- Li, K.-L., Strader, J., Miller-Jones, J. C. A., Heinke, C. O., & Chomiuk, L. 2020, ApJ, 895, 89 \nLinares, M. 2014, ApJ, 795, 72 \n- Miller, J. M., Swihart, S. J., Strader, J., et al. 2020, ApJ, 904, 49 Miraval Zanon, A., Ambrosino, F., Coti Zelati, F., et al. 2022, A&A, 660, A63\n- Papitto, A., Ambrosino, F., Stella, L., et al. 2019, ApJ, 882, 104\n- Papitto, A., Ferrigno, C., Bozzo, E., et al. 2013, Nature, 501, 517 Papitto, A., Rea, N., Coti Zelati, F., et al. 2018, ApJ, 858, L12 Patruno, A., Archibald, A. M., Hessels, J. W. T., et al. 2014, ApJ, 781, L3\n- Radhakrishnan, V. & Srinivasan, G. 1982, Current Science, 51, 1096\n- Shahbaz, T., Dallilar, Y., Garner, A., et al. 2018, MNRAS, 477, 566\n- Shahbaz, T., Linares, M., Nevado, S. P., et al. 2015, MNRAS, 453, 3461\n- Stappers, B. W., Archibald, A. M., Hessels, J. W. T., et al. 2014, ApJ, 790, 39\n- Strader, J., Li, K.-L., Chomiuk, L., et al. 2016, ApJ, 831, 89 Szkody, P., Fraser, O., Silvestri, N., et al. 2003, AJ, 126, 1499\n- Thorstensen, J. R. & Armstrong, E. 2005, AJ, 130, 759\n- Veledina, A., Nättilä, J., & Beloborodov, A. M. 2019, ApJ, 884, 144\n- Woudt, P. A., Warner, B., & Pretorius, M. L. 2004, MNRAS, 351, 1015", 'Appendix A: Line fitting details': 'Table A.1: Results of the fits of the H α line profile of J1023 with a constant + double Gaussian model. For each spectrum the exposure time is 20 s while the number of DOF is 46, corresponding to 51 points minus 5 free parameters (see Sect. 3.5 for details). \n. \n. \n0 \n9 \n1 \n2', 'A & A proofs: manuscript no. HTR\\_opt\\_spec\\_obs\\_J1023\\_Messa': 'Table A.1: continued. \n. \n.'}

2024A&A...690A.372L

We present a new constraint on the Galactic SUP12SUPCSUP13SUPC gradient with sensitive HCOSUPSUP absorption observations against strong continuum sources. The new measurements suffer less from beam dilution optical depths and chemical fractionation allowing us to derive the isotopic ratios precisely. The measured SUP12SUPCSUP13SUPC ratio in the solar neighborhood 665 is consistent with those obtained from CHSUPSUP. Two measurements toward the GC are 42.21.7 and 37.56.5. Though the values are a factor of two to three higher than those derived from dense gas tracers e.g. HSUB2SUBCO complex organic molecules toward Sagittarius Sgr B2 regions our results are consistent with the absorption measurements from cCSUB3SUBHSUB2SUB toward Sgr B2 40 and those from CHSUPSUP toward Sgr A and Sgr B2N gt30. We have calculated a new Galactic SUP12SUPCSUP13SUPC gradient of 6.41.9RSUBGCSUBkpc25.910.5 and found an increasing trend of the SUP12SUPCSUP13SUPC gradient obtained from highdensity to lowdensity gas tracers suggesting that opacity effects and chemical fractionation may have a strong impact on the isotopic ratios observed in highdensity regions.

2024-10-01T00:00:00Z

['2024arXiv240911821L', '2024A&A...690A.372L', 'arXiv:2409.11821', '10.1051/0004-6361/202451412', '10.48550/arXiv.2409.11821']

['astrochemistry', 'ISM: abundances', 'ISM: clouds', 'ISM: molecules', 'galaxy: evolution', 'Astrophysics - Astrophysics of Galaxies']

A new measurement of the Galactic SUP12SUPCSUP13SUPC gradient from sensitive HCOSUPSUP absorption observations

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']

https://arxiv.org/pdf/2409.11821.pdf

{'A new measurement of the Galactic 12 C/ 13 C gradient from sensitive HCO + absorption observations': "Gan Luo 1 , Laura Colzi 2 , Tie Liu 3 , Thomas G. Bisbas 4 , Di Li 5 , 6 , Yichen Sun 7 , and Ningyu Tang 8 \n- 1 Institut de Radioastronomie Millimetrique, 300 rue de la Piscine, 38400, Saint-Martin d'Hères, France e-mail: luo@iram.fr\n- 2 Centro de Astrobiología (CAB), CSIC-INTA, Ctra. de Ajalvir Km. 4, 28850 Torrejón de Ardoz, Madrid, Spain\n- 3 Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030, China\n- 4 Research Center for Astronomical Computing, Zhejiang Laboratory, Hangzhou 311100, China\n- 5 Department of Astronomy, Tsinghua University, Beijing 100084, China\n- 6 CAS Key Laboratory of FAST, National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, China\n- 7 School of Astronomy and Space Science, Nanjing University, Nanjing 210093, China\n- 8 Department of Physics, Anhui Normal University, Wuhu, Anhui 241002, China \nReceived xx; accepted xx", 'ABSTRACT': 'We present a new constraint on the Galactic 12 C / 13 C gradient with sensitive HCO + absorption observations against strong continuum sources. The new measurements su ff er less from beam dilution, optical depths, and chemical fractionation, allowing us to derive the isotopic ratios precisely. The measured 12 C / 13 C ratio in the solar neighborhood (66 ± 5) is consistent with those obtained from CH + . Two measurements toward the GC are 42.2 ± 1.7 and 37.5 ± 6.5. Though the values are a factor of two to three higher than those derived from dense gas tracers (e.g., H2CO, complex organic molecules) toward Sagittarius (Sgr) B2 regions, our results are consistent with the absorption measurements from c-C3H2 toward Sgr B2 ( ∼ 40) and those from CH + toward Sgr A ∗ and Sgr B2(N) ( > 30). We have calculated a new Galactic 12 C / 13 C gradient of (6.4 ± 1.9) R GC / kpc + (25.9 ± 10.5) and found an increasing trend of the 12 C / 13 C gradient obtained from high-density to low-density gas tracers, suggesting that opacity e ff ects and chemical fractionation may have a strong impact on the isotopic ratios observed in high-density regions. \nKey words. astrochemistry - ISM: abundances - ISM: molecules - ISM: clouds - Galaxy: evolution', '1. Introduction': 'The measurements of isotopic ratios in the interstellar medium (ISM) are important for understanding Galactic chemical evolution (GCE) due to di ff erent stellar yields with di ff erent masses (Guesten & Mezger 1982; Maeder 1983; Iben & Renzini 1983; Clayton 2003; Kobayashi et al. 2011; Romano et al. 2017; Kobayashi et al. 2020). One of the few isotopic ratios that can be easily measured is 12 C / 13 C . Notably, 12 C is mainly produced through He-burning as the primary element (e.g., core-collapse supernovae and He-burning shells in low-mass stars), while 13 C is an intermediate product of CNO cycling in intermediate-mass asymptotic giant branch (AGB) stars (Kobayashi et al. 2011; Romano et al. 2017). Since the timescale in which massive stars release the elements into the ISM is much shorter than for low-tointermediate mass stars, the 12 C / 13 C ratios predicted from GCE models would decrease as a function of time with a given initial mass function (IMF). In the Milky Way, a radial gradient in the 12 C / 13 C ratio inferred from 12 C 18 O and 13 C 18 O rises from approximately 24 at a galactocentric distance ( R GC) of ∼ 0.5 kiloparsec (kpc) to roughly 70 at R GC ∼ 12 kpc (Langer & Penzias 1990, 1993). \nWhile various methodologies and tracers have been employed in recent years to constrain the Galactic 12 C / 13 C gradient (Halfen et al. 2017; Yan et al. 2019; Jacob et al. 2020; Sun et al. 2024), the di ff erent methods have yielded inconsistent results, especially for the Galactic center (GC) regions. Measur- \ning the 12 C / 13 C isotopic ratio with molecular lines faces challenges from both observational factors (e.g., opacity, beam dilution) and chemical e ff ects (e.g., selective photodissociation and fractionation). The most abundant carbon-bearing isotopologs (e.g., 12 CO, H 12 CO + , H 12 CN) can easily become optically thick in dense clouds, making precise estimation of column densities di ffi cult. In di ff use and translucent clouds, the main isotopologs may not pose challenges for optical depths, but the rare isotopologs are often di ffi cult to detect due to their low abundance and sub-thermally excitation of even the ground state transition (Godard et al. 2010; Luo et al. 2020). Additionally, distant clouds have smaller beam-filling factors, which reduces the reliability of the measurements observed with a single dish. \nIsotope-selective photodissociation and isotopic exchange reactions can strongly alter the 12 C / 13 C isotopic ratios in molecules from the original element ratios. For abundant species such as CO, the self-shielding of 12 CO is significantly stronger than less abundant isotopologs (e.g., 13 CO and C 18 O). Thus, the 12 C / 13 C ratio in CO would be higher under the existence of external UV radiation (Bally & Langer 1982; Chu & Watson 1983; Liszt 2007). Additionally, due to the di ff erences in zero-point energy between di ff erent isotopologs, isotopic exchange reactions can further alter 12 C / 13 C ratios in various molecules. For instance, the isotopic exchange reaction between 13 C + and CO is exothermic, leading to an enrichment of 13 C in CO and other molecules primarily formed through CO in cold environments \n(Watson et al. 1976; Langer et al. 1984; Roue ff et al. 2015; Colzi et al. 2020; Sipilä et al. 2023). \nAbsorption lines against strong continuum sources can overcome such di ffi culties. The advantage of using absorption lines is to measure the optical depth as well as the column density precisely even if the transition is sub-thermally excited. Such a methodology has been widely used to quantify the physical properties of low-density gas (Lucas & Liszt 1996, 1998; Luo et al. 2020, 2023b; Rybarczyk et al. 2022b; Liszt & Gerin 2023; Gerin et al. 2024). Lucas & Liszt (1998) is one of the first works that measured the isotopic ratios in the solar neighborhood through millimeter molecular absorption lines. Liszt & Gerin (2018) measured the H 12 CO + / H 13 CO + ratios in four directions toward the Galactic bulge, and they inferred a much higher value ( ∼ 60) than the other tracers ( ∼ 20). Inspired by these pioneering works, systematic measurements at di ff erent R GC with such a methodology could be useful to constrain the 12 C / 13 C gradient. \nIn this work, we present a new measurement of the Galactic 12 C / 13 C gradient by using absorption lines of H 12 CO + and H 13 CO + with high-angular observations of the Atacama Large Millimeter / submillimeter Array (ALMA) and the Northern Extended Millimeter Array (NOEMA). We emphasize that such a methodology to constrain the isotopic gradient of the Milky Way with a large sample, especially for future searches in the anti-center direction (though rare and challenging), could provide valuable constraints for GCE models.', '2.1. Quasar sight lines': "We carried out H 12 CO + and H 13 CO + J = 1-0 observations toward six quasars (Fig. 1). Five were observed with ALMA (2022.1.01438.S, PI: Thomas G. Bisbas) during March 20-30, 2023, and one source (3C 111) was observed with NOEMA (W20BB, PI: Gan Luo) in December 2020 and January 2021. The angular resolution of ALMA is ∼ 0.6 '' at 89 GHz. The spectral resolution of ALMA is 122 kHz, corresponding to a velocity resolution of ∼ 0.4 km s -1 at 89 GHz. The raw data was calibrated using the standard pipeline with the Common Astronomy Software Applications (CASA, version 6.4.1; CASA Team et al. 2022). The imaging of the calibrated visibilities was performed using the tclean algorithm with Briggs weighting (robust = 0.5). The NOEMA observations have an angular resolution of ∼ 2.1 '' at 89 GHz and a velocity resolution of ∼ 0.21 km s -1 . The PolyFiX backend of NOEMA covers H 12 CN, H 13 CN, HN 12 C, and HN 13 C J = 1-0 transitions simultaneously. The calibration of the raw data was performed using clic software in the gildas . 1 The typical optical depth noise levels are 3 ∼ 7 × 10 -3 per 0.4 km s -1 velocity channel for ALMA observations and 2 × 10 -3 per 0.21 kms -1 velocity channel for NOEMA observations.", '2.2. UCH ii sight lines': "The ALMA observations toward ten UCH ii regions (red crosses in Fig. 1), which were selected from the ALMA Three-millimeter Observations of Massive Star-forming regions (ATOMS) survey (2019.1.00685.S, PI: Tie Liu), were observed from September to November 2019 with both 12-m and 7-m arrays at band 3. The baselines of ATOMS range from 15 to 783.5 m, corresponding to an angular resolution of ∼ 1.2 '' at 89 GHz. The spectral resolution of H 12 CO + and H 13 CO + J = 1- \nFig. 1. Distribution of the observed quasar sight lines (yellow lines) and ultra-compact H ii region (red crosses) projected onto a topdown schematic view of the Milky Way (artist's concept, R. Hurt: NASA / JPLCaltech / SSC). \n<!-- image --> \n0 transitions are 0.2 km s -1 and 0.422 km s -1 , respectively. The raw data were calibrated using CASA version 5.6, and the combination and imaging of 12-m and 7-m data were performed with CASA version 6.4.1. The sensitivity of the resultant spectra depends on the properties of each source, a detailed description of observations can be found in Liu et al. (2020).", '3.1. Absorption spectra and integrated optical depth': 'The absorption spectra of HCO + (hereafter, HCO + refers to H 12 CO + ) and H 13 CO + 2 were extracted from the continuum peak of each source. Figure 2 shows an example of normalized spectra of HCO + and H 13 CO + toward J1851 + 0035. Spectra toward all other sources can be found in Appendix A. \nSince most of the sources are located in the Galactic plane ( | b | < 2 · ), the foreground absorption components are mixed, and it is di ffi cult to decompose individual Gaussian components (except for the high-latitude source 3C111; see Appendix B). To obtain the column density ratios, we integrated the optical depths ( R τν d υ ) of HCO + and H 13 CO + in the same velocity range (shadowed region in Fig. 2 and A.1). We set four criteria when identifying the integrated range: 1) H 13 CO + absorption should not be contaminated with absorption features of other molecules (e.g., -25 km s -1 component of H 13 CO + in Fig. 2); 2) each of the integrated ranges of H 13 CO + should be distinguished from the others (no overlap); 3) the absorption profile of HCO + should not be saturated ( e -τ > rms); and 4) absorption spectra toward UCH ii regions should not be contaminated with emission features from the compact foreground envelopes. The uncertainty of R τν d υ \n( σ I) through an error propagation formula is thus defined by \nσ I = d υ 2 v u t N -1 X i = 1 ( σ 2 τ i + σ 2 τ i + 1 ) , (1) \nwhere d υ is the channel width, N is the length of the data array in the velocity range over which the opacity profile is integrated, and στ i = RMS / e -τ i is the uncertainty of τ at the i th channel. The integrated optical depths of HCO + and H 13 CO + toward all sources are listed in Table C.1. \nThe molecular column density can be written as a function of R τν d υ (Mangum & Shirley 2015): \nN tot = 3 h 8 π 3 | µ lu | 2 Q rot g u e E u kT ex e h ν kT ex -1 Z τν d υ, (2) \nwhere h is the Planck constant, k is the Boltzman constant, | µ lu | 2 is the dipole matrix element, Q rot is the rotational partition function, g u is the degeneracy of the upper energy level, E u is the upper energy level, and T ex is the excitation temperature. For each transition, | µ lu | 2 , E u, and the rest frequency ν were taken from the Cologne Database for Molecular Spectroscopy (CDMS; Müller et al. 2001, 2005), and they are listed in Table D.1. For linear molecules, the simplified partition function is given by McDowell (1987) \nQ tot = kT hB 0 e hB 0 3 kT , (3) \nwhere B 0 is the rigid rotor rotation constant. \nThe excitation temperature of HCO + in di ff use molecular clouds is usually close to the cosmic microwave background temperature ( T CMB = 2.73 K) (Godard et al. 2010; Luo et al. 2020). If we fill in all the constants in Eq. 2 and take T ex = 2 . 73 K for both HCO + and H 13 CO + , the column density ratios can be simplified as \nN HCO + N H 13 CO + = 0 . 974 R τ HCO + d υ R τ H 13 CO + d υ . (4) \nThough T ex of HCO + can rise above 2.73 K when the gas density is higher than 300 cm -3 (Rybarczyk et al. 2022a), a variance of T ex from 2.73 to 5 K only results in a variance of the column density ratio of less than 1%. The calculated ratios in our samples range from 12.7 ± 0.2 3 to 139.4 ± 38.0 (Table B.1).', '3.2. The 12 C/ 13 C gradient': 'Obtaining an accurate distance of a distant molecular cloud is di ffi cult, especially without parallax measurements from maser emissions. The most common estimation of such a distance is based on kinematic distance. We calculated the R GC for each velocity component with the Monte Carlo kinematic distance method, using the latest rotational curve and updated solar motion parameters (Wenger et al. 2018). This model is obtained through the trigonometric parallax results from the high-mass star-forming regions (HMSFRs) in the BeSSeL Survey (Reid et al. 2014, 2019). This method samples the local standard of rest velocity ( V lsr) and Galactic rotational curve parameters for HMSFRs and derives the probability density distribution (PDF) of kinematic distance, resulting in a median uncertainty of 13% \nJ1851+0035 \nFig. 2. Normalized absorption spectra of HCO + (black curve) and H 13 CO + (blue curve) toward J1851 + 0035. The spectrum of H 13 CO + has been scaled by a factor of five and shifted upward by 0.5 for better display. The gray shaded regions denote the velocity ranges used to calculate the column density ratios. The red vertical lines represent HCO absorption, where the -25kms -1 component of H 13 CO + absorption is contaminated with HCO J = 1 / 2-1 / 2, F = 1-1. Thus, these components were discarded since they can only obtain lower limits of 12 C / 13 C. \n<!-- image --> \nto the parallax distances. A detailed description of this method can be found in Wenger et al. (2018). \nThe optical depth-weighted V lsr in our samples is calculated by \nV lsr = R υτ HCO + d υ R τ HCO + d υ . (5) \nThe calculated V lsr and R GC as well as the uncertainties are listed in Table B.1. We note that such an estimation with pure circular motions may fail to predict the R GC near the GC (e.g., the highly negative V lsr toward J1720-3552), in which non-circular motions should be taken into account (Liszt & Gerin 2018). Therefore, we used the tilted-disk model presented by Burton & Liszt (1978) to estimate the velocity components of V lsr < -140 kms -1 toward J1720-3552, resulting in an R GC between 1.24 to 1.5 kpc. Thus, we adopted a value of 1.37 ± 0.04 kpc for this component. \nFigure 3 shows the measured HCO + / H 13 CO + ratios as a function of R GC. The weighted-mean 12 C / 13 C ratios increase from 42 ± 1 toward the GC to 66 ± 5 in the solar neighborhood. To avoid an underestimation of the HCO + optical depth when τ is large, we set stricter constraints ( τ < 3) when fitting a gradient. We used the Markov chain Monte Carlo (MCMC) method within the emcee code (Foreman-Mackey et al. 2013) to sample the free parameters and the posterior probability distribution. The maximum likelihood fit of the data is shown with a red solid curve in Fig. 3, which gives a H 12 CO + / H 13 CO + gradient as a function of R GC: \nH 12 CO + H 13 CO + = (6 . 4 ± 1 . 9) R GC kpc + (25 . 9 ± 10 . 5) . (6) \nFig. 3. Measurements of H 12 CO + / H 13 CO + as a function of R GC and comparison of the derived gradients between di ff erent works. Blue dots represent τ HCO + < 3, and black dots represent τ HCO + > 3. The two lower limits are shifted to R GC = 0 . 5 kpc for better visualization. The red curve denotes the maximum likelihood fit of the blue points only; orange curves denote the 3 σ deviation from the MCMC sampling. \n<!-- image -->', '4.1. The 12 C/ 13 C ratios in the solar neighborhood': 'While the 12 C / 13 C ratios in the nearby molecular clouds have been derived extensively with various molecules, the most reliable tracer thought to represent the true element 12 C / 13 C ratios is CH + (Ritchey et al. 2011). The measured 12 C / 13 ratios from our HCO + samples range from 62.0 ± 4.4 to 139.4 ± 38.0 within 500 pc of the solar circle (7 . 6 ≤ R GC ≤ 8 . 6 kpc), with a weighted mean value of 66 ± 5. This value is in good agreement with the mean value obtained through optical 12 CH + / 13 CH + measurements toward di ff use sight lines (74.4 ± 7.6, Ritchey et al. 2011) and the -0 . 9 kms -1 component of 3C 111 (Appendix B).', '4.2. The 12 C/ 13 C ratios in the Galactic center': 'The measurements of the 12 C / 13 C ratio within ∼ 2 kpc of the GC are rare. This is mainly due to the high gas column density (e.g., N H2 ≳ 10 24 cm -2 ), which makes the molecular transitions optically thick, even for the rare isotopologs (e.g., C 18 O). Halfen et al. (2017) measured the isotopic ratios using complex organic molecules (COMs) toward Sgr B2(N) and found an average value of 24 ± 7, which is similar to the value obtained from dense gas tracers (e.g., C 34 S, Humire et al. 2020). The measurements from absorption lines of CH toward Sgr B2(M) and H2CO toward Sgr B2 result in 12 C / 13 C ratios of 15.8 ± 2.4 (Jacob et al. 2020) and 11.48 ± 0.03 (Yan et al. 2019), respectively. \nPrevious absorption observations toward the Galactic bulge have revealed a significantly high 12 C / 13 C ratio ( ∼ 60, Liszt & Gerin 2018). However, since these observations integrated the full velocity range, which has HCO + absorptions (the H 13 CO + absorptions are much narrower), the absorption of HCO + could be contaminated with foreground di ff use gas, leading to a high value. Since the HCO + absorptions toward the dense regions (e.g., Sgr B2(N)) are completely saturated, the measured values at R GC < 2 kpc in our work are in fact the di ff use molecular components (a few hundred cm -3 , Gerin & Liszt 2017; Liszt & Gerin 2018). \nThe two measurements at R GC < 2 kpc from our observations have 12 C / 13 C ratios of 42.2 ± 1.7 and 37.5 ± 6.5, values which are consistent with absorption measurements of c-C3H2 ( ∼ 40 at R GC < 1 kpc, Corby et al. 2018) but still larger than the other tracers in high-density ( n H2 > 10 5 cm -3 ) regions. Furthermore, the two lower limits of 12 C / 13 C at -181 and -162 kms -1 of J1720-3552 are > 19 and > 31. These results are consistent with previous observations of CH + toward Sgr A ∗ and Sgr B2(N) (e.g., 12 CH / 13 CH > 30 at -168 ∼ -150 km s -1 of Sgr A ∗ , Godard et al. 2012), which is again distinct from the previously reported low values toward Sgr B2 but supports our results. Thus, either opacity e ff ects or fractionation may have an impact on the measured isotopic ratios in the high-density tracers (e.g., CN, and COMs), while our measurements at low density are less influenced.', '4.3. The 12 C/ 13 C gradient and comparison with GCE models': 'Di ff erent tracers have been used to calculate the Galactic 12 C / 13 C gradient. This gradient is mostly constrained by results from 12 C 18 O / 13 C 18 O (e.g., Langer & Penzias 1990, 1993; Wouterloot & Brand 1996; Giannetti et al. 2014); 12 CN / 13 CN (e.g., Savage et al. 2002; Milam et al. 2005; Sun et al. 2024); H 12 2 CO / H 13 2 CO (Henkel et al. 1982; Yan et al. 2019); C 34 S (Yan et al. 2023); and a combined analysis from various tracers (e.g., COMs, CH; Halfen et al. 2017; Jacob et al. 2020). The gradients from di ff erent methodologies are shown with di ff erent curves in Fig. 3. 4 \nBy comparing the gradients using di ff erent tracers, we found that the measurements in the low-density molecular clouds (HCO + , n H2 ∼ a few 10 2 cm -3 ) are higher than those from higher density environments ( n H2 ≥ a few 10 4 cm -3 ), such as C 18 O, H2CO, CS, and CN (Giannetti et al. 2014; Yan et al. 2019, 2023; Sun et al. 2024). The systematically increasing trend of the 12 C / 13 C gradient from high-density to low-density gas may again suggest that fractionation plays a crucial role in the measured 12 C / 13 Cratios in molecules. Predictions from the chemical models by Colzi et al. (2020) are in agreement with this observational trend, finding that 12 C / 13 C ratios of HCN, HNC, and HCO + tend to increase from the higher (10 6 cm -3 ) to the lower (10 3 cm -3 ) densities. \nThe comparison between the derived 12 C / 13 C gradient and the GCE models presented in Colzi et al. (2022) is shown in Fig. 4, in which the models consider di ff erent mass ranges for white dwarf progenitors and average ejected masses of 13 C and 15 N per nova outburst (Romano et al. 2019, 2021). Our result is reasonably consistent with the GCE models with a mass range of white dwarf progenitors 1-8 M ⊙ and higher average ejected masses of 13 C (models 3 and 4). Given the limited range of R GC in the current samples, future observations at R GC > 10 kpc would be exceptionally useful to constrain GCE models. \nHowever, one should always keep in mind that the formula of the isotopic gradient strongly depends on the accuracy of the R GC, which is mostly based on the kinematic distance estimation. For instance, the estimated R GC before and after the model by Reid et al. (2019) could di ff er by several kpc (Sun et al. 2024). The large scatter of 12 C / 13 C at R GC between 4 kpc to 8 kpc in both our work and the literature may suggest that our understanding of the Milky Way rotational model or inhomogeneous mixing of the elements may also contribute to the scatter. \nFig. 4. Comparison between observations and GCE models, in which the models are taken from Table 2 in Colzi et al. (2022) with the same numbers. \n<!-- image -->', '4.4. Chemical effects on the measured 12 C/ 13 C ratios': 'The isotopic exchange reactions could be one of the main processes that lead to di ff erent isotopic ratios in HCO + . In this section, we consider the isotopic exchange reaction that produces H 13 CO + (Langer et al. 1978; Roue ff et al. 2015; Colzi et al. 2020) 5 : \n13 C + + CO ⇌ 13 CO + C , (R1) \n13 CO + HCO + ⇌ H 13 CO + + CO . (R2) \nWe considered the main formation pathways of H 13 CO + in lowto-intermediate density gas (van Dishoeck & Black 1988; Luo et al. 2023b): \n13 CO + + H2 → H 13 CO + + H , (R3) \n13 C + + H2O → H 13 CO + + H , (R4) \n13 CH + O → H 13 CO + + e . (R5) \nThe destruction of HCO + (and H 13 CO + ) is dominated by electrons: \nHCO + + e -→ CO + H . (R6) \nThus, the isotopic exchange reaction would influence the HCO + / H 13 CO + ratio only if the right side reaction R2 is comparable to reactions R3 ∼ R5. The reaction rate of the above reaction is 2 . 6 × 10 -10 × ( T / 300) -0 . 4 cm 3 s -1 (Roue ff et al. 2015). The typical abundance of CO + is 10 -10 ∼ 10 -9 (Stäuber & Bruderer 2009; Treviño-Morales et al. 2016), the abundance of HCO + is a few 10 -9 (Lucas & Liszt 1996; Gerin et al. 2019; Luo et al. 2020), and the abundance of CH is 3 × 10 -8 (Liszt & Lucas 2002; She ff er et al. 2008; Tang et al. 2021; Luo et al. 2023a). Even if we only consider reaction R3 (the reaction rate k = 7 . 5 × 10 -10 cm 3 s -1 ; McElroy et al. (2013)) and scale the above abundance with 12 C / 13 C = 70, the formation of H 13 CO + through reaction R3 would overwhelm reaction R2 by over four orders of magnitude at the typical gas temperature of ∼ 50 K (Snow & McCall 2006). Therefore, fractionation would have little impact on the measured HCO + / H 13 CO + ratios in low-density gas. This is \nalso consistent with various chemical modelings of carbon fractionation, in which HCO + / H 13 CO + can overall best represent the original 12 C / 13 Celement ratio in low-to-intermediate density gas (Sz"ucs et al. 2014; Roue ff et al. 2015; Colzi et al. 2020; Sipilä et al. 2023). \nIn dense regions, CO becomes the precursor of HCO + (Dalgarno 2006; Indriolo & McCall 2012; Bisbas et al. 2015): \nCO + H + 3 → HCO + + H2 . (R7) \nSince reaction R6 is still the dominant formation channel of CO even in high-density clouds (Luo et al. 2023a), the fractionation of CO in the dense cloud could greatly impact the isotopic ratio in HCO + only if reaction R7 is comparable to reaction R6. However, the gas density that will significantly alter the 12 C / 13 C ratio measured in HCO + would be above 10 4 cm -3 (Sipilä et al. 2023). The highest column density of HCO + in Table B.1 is ∼ 3 . 4 × 10 13 cm -2 , corresponding to an H2 column density of 10 22 cm -2 , assuming a constant HCO + abundance (3 × 10 -9 , Lucas & Liszt 1996; Liszt & Gerin 2016) 6 Therefore, most components are not expected to trace such a high-density regime and will be less influenced by the fractionation e ff ect. Nevertheless, both detailed chemical modeling (Colzi et al. in prep) and the constraints of gas volume density and temperature through multiple lines in the future are necessary to determine where and how fractionation impacts the isotopic ratios.', '5. Conclusion': "In this work, we have performed high sensitivity absorption line observations toward strong continuum sources (including quasars and UCH ii regions). We derived 12 C / 13 C ratios from HCO + absorption and R GC according to the latest parallax-based distance calculation. Our main conclusions are as follows: \n- 1. The derived 12 C / 13 C gradient from HCO + absorption measurements is (6.4 ± 1.9) R GC / kpc + (25.9 ± 10.5), which is reasonably consistent with current GCE models.\n- 2. The derived weighted mean 12 C / 13 C ratio in the solar neighborhood is 66 ± 5, which is consistent with those measured from CH + (74.4 ± 7.6).\n- 3. Our measurements toward the GC are two to three times higher than those measured with dense gas tracers toward Sgr B2 (11 ∼ 24). Nevertheless, our results are supported by the CH + observations toward Sgr A ∗ and Sgr B2(N) ( > 30) as well as absorption measurements of c-C3H2 ( ∼ 40).\n- 4. The discrepancy between our method and those from dense gas tracers suggests that opacity e ff ects and fractionation may have a larger impact on the dense gas tracers in highdensity regions. \nWe highlight the use of absorption lines to measure the isotopic ratios with interferometry observations, which are less affected by optical depth, beam dilution, and chemical fractionation. Future large samples toward the GC and the anti-center directions may provide more constraints on the Galactic 12 C / 13 C gradient as well as the GCE models. \nAcknowledgements. We are grateful to the anonymous referee for the thoughtful comments and suggestions that greatly improved the clarity of our work, especially the suggestion of including the non-circular motion to estimate the distance near the GC. We thank Zhiyu Zhang, J'erˆome Pety, and Michel Gu'elin \nfor their useful comments, Donatella Romano for providing the GCE models, and the sta ff s at IRAM for carrying out the NOEMA observations and reducing the data. L. C acknowledges support from the grant No. PID2022136814NB-I00 by the Spanish Ministry of Science, Innovation and Universities / State Agency of Research MICIU / AEI / 10.13039 / 501100011033 and by ERDF, UE. Tie Liu acknowledges the supports by the National Key R&D Program of China (No. 2022YFA1603100), National Natural Science Foundation of China (NSFC) through grants No.12073061 and No.12122307, and the Tianchi Talent Program of Xinjiang Uygur Autonomous Region. D. L. is a New Cornerstone investigator. N.-Y. Tang is sponsored by the University Annual Scientific Research Plan of Anhui Province (No. 2023AH030052, No. 2022AH010013), the China Manned Space Program through its Space Application System, Zhejiang Lab Open Research Project (No. K2022PE0AB01). This paper makes use of the following ALMA data: ADS / JAO.ALMA#2022.1.01438.S and ADS / JAO.ALMA#2019.1.00685.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI / NRAO and NAOJ. This work is based on observations carried out under project number W20BB with the IRAM NOEMA Interferometer. IRAM is supported by INSU / CNRS (France), MPG (Germany) and IGN (Spain).", 'References': 'Bally, J. & Langer, W. D. 1982, ApJ, 255, 143 Bisbas, T. G., Papadopoulos, P. P., & Viti, S. 2015, ApJ, 803, 37 Burton, W. B. & Liszt, H. 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Note that for the UCH ii sight lines, the H40 α emissions are usually associated with UCH ii regions (Liu et al. 2020) (e.g., -150 ∼ -80 kms -1 toward I15254 + 5621), this is not the bad baseline. Furthermore, since the emission from the compact structures around UCH ii regions could have a high excitation temperature, the spectra may still show emission in conjunction with absorption. We avoid these velocity ranges in our calculation. Future high angular resolution and high sensitivity observations may help reconstruct both the emission and absorption profiles around UCH ii regions.', 'Appendix B: Gaussian decomposition of absorption profiles toward 3C 111': 'Absorption profiles and Gaussian decomposition of H 12 CN, H 13 CN, HN 12 C, and HN 13 C J = 1-0 transitions toward 3C 111 are shown in Fig. B.1, in which the spectra of H 12 CNand H 13 CNare hyperfine transitions. The Gaussian decomposition is performed with curve \\_ f it package in scipy using the Levenberg-Marquardt algorithm, and the resultant optical depths and linewidth ( ∆ V ) are shown in Table B.1. We set constraints when fitting hyperfine transitions that all hyperfine lines have 1) the same linewidth ( ∆ V ) and 2) the same velocity o ff set with respect to V lsr. \nThere are two velocity components ( -0 . 9 kms -1 and -2 . 5 kms -1 ) in front of 3C 111, the 3D extinction indicates that the cloud is within 300 pc from the Sun (Lucas & Liszt 1998). The gas volume density at -0 . 9 km s -1 is n H2 = 398 ± 22 cm -3 (Luo et al. in prep), and n H2 is supposed to be much lower at -2 . 5 km s -1 than at -0 . 9 kms -1 (Lucas & Liszt 1998). However, the HCO + absorption profile is almost saturated toward 3C 111 and we cannot get a good Gaussian decomposition. Thus, we use HCN and HNC to derive the 12 C / 13 C ratio. The optical depth ratios of the H 12 CN hyperfine transitions are 1:2.3:3.5 and 1:2.9:4.6 at -0.9 kms -1 and -2.5 km s -1 , respectively. While the latter represents the intrinsic line ratio (1:3:5), the former deviates from it by 30%, indicating that the optical depths of the H 12 CN (F = 1-1 and 2-1) at -0.9 km s -1 are underestimated. As an alternative, we can fix the optical depth ratios at 1:3:5 instead of treating them as free parameters to obtain a more accurate value of the column density. \nThe 12 C / 13 C ratio at -0 . 9 kms -1 derived from H 12 CN / H 13 CN is 74 ± 3, which is 30% higher than that previously obtained by Lucas & Liszt (1998). Measurement from HN 12 C / HN 13 C (62 ± 6) also show similar result. The 12 C / 13 C ratio at -2 . 5 kms -1 component derived from H 12 CN / H 13 CN is 138 ± 28, which is ∼ 2 times higher than that at -0 . 9 kms -1 . As suggested by Lucas & Liszt (1998), this could be a particular case, where the 13 CO has been enriched and fractionation leads to a deficit of 13 C in less abundant carbon carriers. Chemical models that consider the isotopic fractionation also predict a higher isotopic ratio of H 12 CN / H 13 CN than that from CO or HCO + (Roue ff et al. 2015; Colzi et al. 2020). Our detection of HN 13 C at -2 . 5 kms s -1 (S / N of R τ dv ∼ 4) is consistent with the hypothesis (HN 12 C / HN 13 C = 95 ± 39).', 'Appendix C: The source properties': 'The derived source properties from Sect. 3 are shown in Table C.1.', 'Appendix D: The Molecular transitions': 'The molecular transitions mentioned in this work are taken from the CDMS database (Müller et al. 2001, 2005), which are listed in Table D.1.', 'Appendix E: The unusual low H 12 CO + /H 13 CO + toward J1720-3552': 'The -22 kms -1 velocity component was recognized as an independent component in the H 13 CO + line profiles when we calculated the H 12 CO + / H 13 CO + ratios. The derived isotopic ratio is more than three times lower than the components toward the GC; however, the reason is still unclear. If this low value is caused by chemical fractionation, the gas component at -22 kms -1 must have a high density and compact size ( ≤ 0.01 pc at a distance of ∼ 3 kpc; otherwise, we would see emission lines around the continuum source). It should be surrounded by low-density envelopes since the H 12 CO + / H 13 CO + ratios toward the neighboring velocity components are much higher. Future highJ molecular line observations may help reveal the nature of this low value. \nFig. A.1. Same as Fig. 2 but for other sources. Red vertical lines denote HCO absorption. \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFig. B.1. Gaussian decomposition of normalized absorption profiles of HCN, H 13 CN, HNC, and H 13 NCtoward 3C 111. Red dashed curves denote the fitting results, and green dotted lines denote each Gaussian component. \n<!-- image --> \nTable B.1. Optical depths and line widths toward 3C 111. \nNotes. ( a ) Optical depth of J = 1-0, F = 0-1 transition. \nTable C.1. Source name, velocity, galactocentric distance, integrated optical depths, isotopic ratios, and RMS of the absorption spectra. \nNotes. ( a ) Estimation through tilted-disk model near the GC (see Sect. 3.2). ( b ) H 13 CO + is contaminated with HCO, thus, it is an upper limit. \nTable D.1. Molecular transitions mentioned in this work.'}

2024arXiv240912227M

Interferometric observations of gravitational microlensing events offer an opportunity for precise efficient and direct mass and distance measurements of lensing objects especially those of isolated neutron stars and black holes. However such observations were previously possible for only a handful of extremely bright events. The recent development of a dualfield interferometer GRAVITY Wide has made it possible to reach out to significantly fainter objects and increase the pool of microlensing events amenable to interferometric observations by two orders of magnitude. Here we present the first successful observation of a microlensing event with GRAVITY Wide and the resolution of microlensed images in the event OGLE2023BLG0061KMT2023BLG0496. We measure the angular Einstein radius of the lens with a subpercent precision thetarm E 1.280 pm 0.009 mas. Combined with the microlensing parallax detected from the event light curve the mass and distance to the lens are found to be 0.472 pm 0.012 Modot and 1.81 pm 0.05 kpc respectively. We present the procedure for the selection of targets for interferometric observations and discuss possible systematic effects affecting GRAVITY Wide data. This detection demonstrates the capabilities of the new instrument and it opens up completely new possibilities for the followup of microlensing events and future routine discoveries of isolated neutron stars and black holes.

2024-09-01T00:00:00Z

['arXiv:2409.12227', '2024arXiv240912227M', '10.48550/arXiv.2409.12227']

['Astrophysics - Instrumentation and Methods for Astrophysics', 'Astrophysics - Solar and Stellar Astrophysics']

Observations of microlensed images with dualfield interferometry onsky demonstration and prospects

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.12227.pdf

{'No Header': 'Draft version September 20, 2024', 'Observations of microlensed images with dual-field interferometry: on-sky demonstration and prospects': "Przemek Mróz, 1 Subo Dong, 2, 3 Antoine Mérand, 4 Jinyi Shangguan, 5 Julien Woillez, 4 Andrew Gould, 6, 7 Andrzej Udalski, 1 Frank Eisenhauer, 5 Yoon-Hyun Ryu, 8 Zexuan Wu, 2, 3 Zhuokai Liu 2, 3 And Hongjing Yang 9 \n- \nGuillaume Bourdarot, 5 Denis Defrère, 10 Antonia Drescher, 5 Maximilian Fabricius, 5 Paulo Garcia, 11, 12 Reinhard Genzel, 5 Stefan Gillessen, 5 Sebastian F. Hönig, 13 Laura Kreidberg, 7 Jean-Baptiste Le Bouquin, 14 Dieter Lutz, 5 Florentin Millour, 15 Thomas Ott, 5 Thibaut Paumard, 16 Jonas Sauter, 5, 7 T. Taro Shimizu, 5 Christian Straubmeier, 17 Matthias Subroweit 17 And Felix Widmann 5 \n(The GRAVITY+ Collaboration) \nMichał K. Szymański, 1 Igor Soszyński, 1 Paweł Pietrukowicz, 1 Szymon Kozłowski, 1 Radosław Poleski, 1 Jan Skowron, 1 Krzysztof Ulaczyk, 18, 1 Mariusz Gromadzki, 1 Krzysztof Rybicki, 19, 1 Patryk Iwanek, 1 Marcin Wrona 20, 1 And Mateusz J. Mróz 1 \n(The OGLE Collaboration) \nMichael D. Albrow, 21 Sun-Ju Chung, 8 Cheongho Han, 22 Kyu-Ha Hwang, 8 Youn Kil Jung, 8, 23 In-Gu Shin, 24 Yossi Shvartzvald, 25 Jennifer C. Yee, 24 Weicheng Zang, 24 Sang-Mok Cha, 8, 26 Dong-Jin Kim, 8 Seung-Lee Kim, 8 Chung-Uk Lee, 8 Dong-Joo Lee, 8 Yongseok Lee, 8, 26 Byeong-Gon Park, 8 Richard W. Pogge 27, 28 \n(The KMTNet Collaboration) \n1 Astronomical Observatory, University of Warsaw, Al. Ujazdowskie 4, 00-478 Warszawa, Poland \n2 Department of Astronomy, School of Physics, Peking University, Yiheyuan Rd. 5, Haidian District, Beijing, China, 100871 3 Kavli Institute for Astronomy and Astrophysics, Peking University, Yiheyuan Rd. 5, Haidian District, Beijing, China, 100871 4 European Southern Observatory, Karl-Schwarzschild-Straße 2, D-85748 Garching, Germany 5 Max Planck Institute for Extraterrestrial Physics, Giessenbachstraße 1, D-85748 Garching, Germany 6 Department of Astronomy, Ohio State University, 140 W. 18th Ave., Columbus, OH 43210, USA 7 Max Planck Institute for Astronomy, Königstuhl 17, 69117 Heidelberg, Germany 8 Korea Astronomy and Space Science Institute, Daejeon 34055, Republic of Korea 9 Department of Astronomy, Tsinghua University, Beijing 100084, China 10 Institute of Astronomy, KU Leuven, Celestijnenlaan 200D, 3001, Leuven, Belgium 11 CENTRA - Centro de Astrofísica e Gravitação, IST, Universidade de Lisboa, 1049-001 Lisboa, Portugal 12 Faculdade de Engenharia, Universidade do Porto, Rua Dr Roberto Frias, 4200-465 Porto, Portugal 13 School of Physics & Astronomy, University of Southampton, Southampton, SO17 1BJ, UK 14 Univ. Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France 15 Université Côte d'Azur, Observatoire de la Côte d'Azur, CNRS, Laboratoire Lagrange, Nice, France 16 LESIA, Observatoire de Paris, Université PSL, Sorbonne Université, Université Paris Cité, CNRS, 5 place Jules Janssen, 92195 Meudon, France 17 1st Institute of Physics, University of Cologne, Zülpicher Straße 77, 50937 Cologne, Germany 18 Department of Physics, University of Warwick, Coventry CV4 7 AL, UK 19 Department of Particle Physics and Astrophysics, Weizmann Institute of Science, Rehovot 76100, Israel 20 Department of Astrophysics and Planetary Sciences, Villanova University, 800 Lancaster Ave., Villanova, PA 19085, USA 21 University of Canterbury, School of Physical and Chemical Sciences, Private Bag 4800, Christchurch 8020, New Zealand 22 Department of Physics, Chungbuk National University, Cheongju 28644, Republic of Korea 23 National University of Science and Technology (UST), Daejeon 34113, Republic of Korea \n24 Center for Astrophysics | Harvard & Smithsonian, 60 Garden St.,Cambridge, MA 02138, USA \nCorresponding author: Przemek Mróz \npmroz@astrouw.edu.pl \n28 \n25 Department of Particle Physics and Astrophysics, Weizmann Institute of Science, Rehovot 7610001, Israel 26 School of Space Research, Kyung Hee University, Yongin, Kyeonggi 17104, Republic of Korea 27 Department of Astronomy, Ohio State University, 140 West 18th Ave., Columbus, OH 43210, USA Center for Cosmology and AstroParticle Physics, Ohio State University, 191 West Woodruff Ave., Columbus, OH 43210, USA \n(Received; Revised; Accepted)", 'ABSTRACT': 'Interferometric observations of gravitational microlensing events offer an opportunity for precise, efficient, and direct mass and distance measurements of lensing objects, especially those of isolated neutron stars and black holes. However, such observations were previously possible for only a handful of extremely bright events. The recent development of a dual-field interferometer, GRAVITY Wide, has made it possible to reach out to significantly fainter objects, and increase the pool of microlensing events amenable to interferometric observations by two orders of magnitude. Here, we present the first successful observation of a microlensing event with GRAVITY Wide and the resolution of microlensed images in the event OGLE-2023-BLG-0061/KMT-2023-BLG-0496. We measure the angular Einstein radius of the lens with a sub-percent precision, θ E = 1 . 280 ± 0 . 009 mas. Combined with the microlensing parallax detected from the event light curve, the mass and distance to the lens are found to be 0 . 472 ± 0 . 012 M ⊙ and 1 . 81 ± 0 . 05 kpc, respectively. We present the procedure for the selection of targets for interferometric observations, and discuss possible systematic effects affecting GRAVITY Wide data. This detection demonstrates the capabilities of the new instrument and it opens up completely new possibilities for the follow-up of microlensing events, and future routine discoveries of isolated neutron stars and black holes. \nKeywords: Gravitational microlensing (672), Optical interferometry (1168)', '1. INTRODUCTION': "In his seminal paper on gravitational microlensing, Einstein (1936) realized that the angular separation between the two images of the source star created by a Galactic gravitational lens would be well below the resolution limits of contemporary telescopes. That led him to believe that observing the microlensed images would be, in practice, impossible. This notion persisted for many decades until the first optical interferometry facilities were developed. \nIn point-source point-lens microlensing events, the gravitational lens creates two images of the source star (called the major and minor image). They are separated by approximately 2 θ E , where θ E = √ κMπ rel is the angular Einstein radius. Here, M is the lens mass, π rel is the lens-source relative parallax, and κ = 8 . 144 mas M -1 ⊙ is a constant. For typical configurations of stellar-mass microlensing events in the Milky Way, θ E ∼ 1 mas , and so the expected separation between the minor and major images is usually smaller than a few milliarcseconds. Interferometric observations of microlensing events, therefore, provide a direct way to precisely measure angular Einstein radii by resolving the images (and, as a consequence, measure the masses of lensing objects). \nAlthough a few tens of thousands of single-lens microlensing events have been discovered so far, only a \nfew had the angular Einstein radius measured. That was possible for only a small number of events exhibiting finite-source effects (Gould 1994; Nemiroff & Wickramasinghe 1994; Witt & Mao 1994), and the probability of such measurements is biased towards low-mass (planetary-mass) lenses. Another route for θ E measurements involved astrometric observations (Hog et al. 1995; Miyamoto & Yoshii 1995; Walker 1995), which led to the first detection of an isolated black hole (Sahu et al. 2022; Lam et al. 2022; Mróz et al. 2022; Lam & Lu 2023). These observations, however, required longterm astrometric monitoring, which is not feasible for a large sample of events. Conversely, interferometric observations are much more efficient as only one exposure is sufficient to resolve the microlensed images and measure the angular Einstein radius. When combined with the microlensing parallax ( π E ≡ π rel /θ E ) measurements from the light curve, they allow us to precisely determine the masses, distances, and transverse velocities of isolated objects, including neutron stars and black holes. \nDelplancke et al. (2001) were the first to discuss the prospects of using the European Southern Observatory (ESO) Very Large Telescope Interferometer (VLTI) to study gravitational microlensing events. They estimated that the first-generation VLTI instruments would be able to observe dozens of events every year in the mid- \n2000s. Yet, the first successful resolution of the microlensed images did not materialize until late 2017, when Dong et al. (2019) observed the microlensing event TCP J05074264+2447555 (a.k.a., Kojima-1) with the second-generation VLTI instrument GRAVITY (GRAVITY Collaboration et al. 2017). \nWhy did it take almost two decades to achieve this milestone? The early predictions on the VLTI performance underestimated the role and impact of many practical problems encountered during system operations, such as beam stability, vibrations, or air dispersion (e.g., Delplancke 2008). Moreover, even when these technical problems were solved or mitigated, the performance of a ground-based interferometer is ultimately limited by atmospheric turbulence, which breaks the coherence of wavefronts arriving at each telescope. Therefore, the observations must be carried out with exposure times ( ∼ 1 ms) shorter than the typical wavefront coherence timescale (on the order of 20-30 ms) and shorter than the mechanical vibrations timescale, which dominate in the 10-1000 Hz regime. Even for the largest modern 10-m-class telescopes, this requirement sets a limiting magnitude of K ≈ 10 for interferometric observations. Gravitational microlensing events brighter than this limit are exceedingly rare (Figure 1). They occur at most a few times a year, and they exceed the GRAVITY limiting magnitude for a short period of time. The three events with the published interferometric observations were unusually bright: TCP J05074264+2447555 (Dong et al. 2019; Zang et al. 2020), ASASSN-22av (Wu et al. 2024, submitted), and Gaia19bld (Cassan et al. 2022; Rybicki et al. 2022; Bachelet et al. 2022) reached V ≈ 11 . 5 , g ' ≈ 12 . 5 , and I ≈ 9 . 0 , respectively. The former two were observed using the standard on-axis mode of GRAVITY, whose fringe tracker allow minute-long science exposures (Lacour et al. 2019) that significantly enhance the VLTI sensitivity (Eisenhauer et al. 2023). \nThe dual-field interferometric observations were designed to alleviate this problem and reach fainter sources (e.g., Shao & Colavita 1992; Colavita et al. 1999; Delplancke 2008; Woillez et al. 2014). Still, a nearby bright ( K ≲ 10 ) reference (fringe-tracking) star is needed to stabilize the optical path difference between the telescopes and phase-reference the interferogram of the science target. The angular separation between the science object and the fringe-tracking star must be smaller than 20 -30 '' , which is set by the atmospheric conditions. In 2019-2022, the GRAVITY instrument was upgraded to enable such dual-field interferometric observations (GRAVITY+ Collaboration et al. 2022), hence the name of this observing mode, GRAVITY Wide. \nFigure 1. Cumulative distribution of the expected K -band peak magnitudes of events detected by the OGLE EWS in 2023 (blue line). For comparison, the red line shows the cumulative distribution of the K -band magnitudes of the brightest star within 30 '' of the event (fringe-tracking star). Only events with t E ≥ 50 day and an adaptive-optics guide star brighter than G = 14 are plotted. \n<!-- image --> \nThe upgraded instrument opened up an entirely new pathway for characterizing and studying microlensing events because, in the dense fields of the Galactic bulge, the probability of finding a suitable ( K ≲ 10 , within 30 '' ) fringe-tracking star is as large as 50-70%. According to the GRAVITY manual (Issue 110), 1 the GRAVITY Wide observations are possible for science targets as faint as K SC ≈ 16 -17 . However, the limiting magnitude strongly depends on the atmospheric conditions and the separation between the science object and the fringe-tracking star. In addition, closure phase observations lose the signal-to-noise ratio faster than the normal visibility data as the targets become fainter. Thus, in practice, we rarely considered observing targets fainter than K ≈ 13 . 5 -14 , which still left us with plenty of candidate events to scrutinize (Figure 1). \nAs soon as ESO offered regular GRAVITY Wide observations since October 2022 ('Period 110'), we initiated a program (PI: Mérand) of interferometric observations of microlensing events. Our primary science goal was to detect and measure precise masses, distances, and \ntransverse velocities of isolated stellar remnants - neutron stars and black holes. Our observations also served as a testbed for verifying the capabilities of the new instrument, and planning, executing, and analyzing the interferometric follow-up observations of a large number of microlensing events. \nThis paper presents the first resolution of microlensed images with GRAVITY Wide in the microlensing event OGLE-2023-BLG-0061/KMT-2023-BLG-0496.", '2. SELECTION OF THE TARGETS FOR THE INTERFEROMETRIC FOLLOW-UP': "The VLTI observations can be carried out with either four 8.2-m Unit Telescopes (UTs) or four 1.8-m Auxiliary Telescopes (ATs). The VLTI UTs observing runs are organized every month (around the full Moon) and typically last one week. During the rest of the month, observations are possible with ATs, which may be relocated to more than ten observing stations. GRAVITY Wide observations may be conducted with ATs, provided that the fringe-tracking star is brighter than K FT ≈ 9 -9 . 5 and the science target is brighter than K SC ≈ 13 -14 . For UTs, the limiting magnitudes are larger, K FT ≈ 10 -10 . 5 and K SC ≈ 16 -17 , respectively. As discussed above, the limiting magnitudes of science targets strongly depend on the atmospheric conditions (isoplanatic angle) and the separation of the fringetracking star. \nWe began the selection of microlensing targets about 7-10 days before the planned start of each VLTI UT run. The candidates were chosen from publicly available lists of microlensing alerts published by the Optical Gravitational Lensing Experiment (OGLE) Early Warning System 2 (EWS; Udalski 2003; Udalski et al. 2015), Korean Microlensing Telescope Network (KMTNet) Alert System 3 (Kim et al. 2018), and Microlensing Observations in Astrophysics (MOA) Transient Alerts 4 . In addition, we also checked transient alerts published by allsky surveys, such as Gaia (Hodgkin et al. 2013, 2021) or All Sky Automated Survey for SuperNovae (ASAS-SN) (Shappee et al. 2014). \nThe selection of targets was based on several scientific and technical criteria. First, we selected events near or past their maximum brightness so that parameters describing their light curves were reasonably well measured. However, the observations had to be secured before the event faded, when the contrast ratio between the microlensed images became too large. We required \nthe contrast ratio to be smaller than 10:1, which corresponds to the maximum lens-source separation (in Einstein radius units) of u max = 1 . 22 or, equivalently, the minimum amplitude of ∆ I min = 0 . 13 . \nBecause the primary scientific motivation of our project was searching for stellar remnants, which are expected to give rise to long-duration events, we selected events with Einstein timescales longer than t E = 50 day. However, the nature of the lens cannot be known at the time of selecting the targets. The mass of the lens can be determined only after the interferometric data are combined with the full light curve. That usually means waiting several weeks after the interferometric observations are taken because they are collected close to the maximum magnification. \nIn the next step, we checked if suitable fringe-tracking and adaptive-optics reference stars are located within 30 '' of the event. For possible fringe-tracking stars, we queried the Two Micron All Sky Survey (2MASS) Point Source Catalog (Skrutskie et al. 2006), while for adaptive-optics guide stars - the Gaia Data Release 3 (Gaia Collaboration et al. 2016, 2023). The guide star had to be brighter than G ≈ 14 for the Multi Application Curvature Adaptive Optics (Arsenault et al. 2003) system, or brighter than K = 8 for the Coudé Infrared Adaptive Optics (Kendrew et al. 2012) system. Finally, we estimated the expected K -band brightness of the event during the planned observations by assuming that the blending parameter in the K -band is identical to that in the I -band. The baseline K -band brightness of the event was taken from the VISTA Variables in the Via Lactea survey (VVV; Minniti et al. 2010), United Kingdom Infra-Red Telescope (UKIRT) Galactic Plane Survey (Lawrence et al. 2007; Lucas et al. 2008), or 2MASS (Skrutskie et al. 2006). We also examined the estimated K -band brightness of the source by using the best-fit I -band source fluxes as cross checks. For such estimates, we assumed de-reddened color ( I -K ) 0 , giant = 1 . 4 and ( I -K ) 0 , dwarf = 1 . 0 for sources roughly classified using the extinction-corrected source magnitudes as giants ( I 0 < 16 . 5 ) and dwarfs ( I 0 > 16 . 5 ), respectively. \nCandidate targets were selected independently by two teams (P.M. and S.D.) and subsequently investigated in more detail. In particular, we paid special attention to the microlensing parallax measurements, which are necessary for the lens mass determination. Since the value of the microlensing parallax is inversely proportional to the square root of the lens mass, we required it to be consistent with zero (or close to zero) during the trigger. We also run light curve simulations to ensure that the parallax will be precisely measured (or constrained) by the end of the observing season. We did not consider \nevents with variable source stars because the variability may affect the microlensing parallax measurements. \nThe blue solid line in Figure 1 shows the cumulative distribution of the expected K -band peak magnitudes of events detected by the OGLE's EWS in 2023. Only events with relatively long timescales ( t E ≥ 50 day) and suitable adaptive-optics guide stars ( G ≤ 14 ) within 30 '' are presented. Only a few events were bright enough ( K < 10 . 5 ) for standard GRAVITY on-axis observations. In contrast, the solid red line in Figure 1 shows the distribution of magnitudes of the brightest star within 30 '' of the event (which may serve as a fringe-tracking star). Nearly 100 events could have been considered for GRAVITY Wide observations.", '3. DATA': "The detection of the microlensing event OGLE-2023BLG-0061 was announced by the OGLE Early Warning System (Udalski 2003; Udalski et al. 2015) on 2023 Mar 13.60 UT ( HJD ' ≡ HJD -2460000 = 17 . 10 ). It was independently identified by the KMTNet Alert System (Kim et al. 2016, 2018) on 2023 Apr 20, and it was designated KMT-2023-BLG-0496. The event occurred on a bright red clump star ( I = 16 . 389 ± 0 . 001 , V -I = 2 . 26 ± 0 . 02 ) with equatorial coordinates (R.A., Decl.) J2000 = ( 17 h 43 m 04 . s 01 , -35 · 15 ' 32 . '' 3 ). According to data from the VVV survey, the event had K = 13 . 515 ± 0 . 011 in the baseline (Minniti et al. 2010).", '3.1. Photometric Data': "OGLE operates the 1.3-m Warsaw Telescope located at Las Campanas Observatory, Chile. The telescope is equipped with a mosaic camera covering a field of view of 1.4 deg 2 with the pixel scale of 0 . 26 arcsec pixel -1 . The event has been observed by the OGLE-IV survey since 2010. However, in this paper, we analyze the OGLE data collected from 2016 through 2023 because earlier observations do not contribute to constraining the parameters of the model. Observations were reduced using the OGLE-IV data reduction pipeline (Udalski et al. 2015), which employs a custom implementation of the Difference Image Analysis (DIA) method (Woźniak 2000). \nKMTNet uses three 1.6-m telescopes located at Cerro Tololo Inter-American Observatory (KMTC, Chile), South African Astronomical Observatory (KMTS, South Africa), and Siding Spring Observatory (KMTA, Australia). Each of the KMTNet telescopes is equipped with a camera with 4 deg 2 field of view and pixel scale of 0 . 40 arcsec pixel -1 . The analyzed KMTNet observations cover the years 2021-2023. However, because of saturation, we deleted data points near the peak of the \nevent. The KMTNet photometric data were reduced with the tender-loving-care (TLC) DIA-based pipeline (Yang et al. 2024), which was developed from pySIS (Albrow et al. 2009). The vast majority of OGLE and KMTNet images were taken in the I -band filter, with additional V -band observations to characterize the color of the source star. \nSome additional observations were taken in the R ' band with the 0.18-m Newtonian telescope (CHI-18; Wu et al. 2024) located at the El Sauce Observatory in Chile to cover the peak of the event. The CHI-18 images were reduced with the TLC pipeline. The original observations were taken at a 2-minute cadence. Because the event did not exhibit variability on such short timescales, we binned the CHI-18 data into 1-hour-long bins.", '3.2. VLTI Data': "OGLE-2023-BLG-0061/KMT-2023-BLG-0496 was considered a promising candidate for VLTI observations on 2023 July 20. Although the event was still before the peak, the available data predicted a relatively long timescale ( t E ≈ 100 d). At the same time, the optical brightness indicated that the microlens parallax should be robustly constrained. A nearby ( 13 . 2 '' ) bright ( G = 13 . 7 , K = 9 . 8 ) star 2MASS 17430426-3515195 could serve as a fringe-tracking and adaptive-optics guide star. The first GRAVITY Wide observations were secured with UTs on 2023 July 29 ('epoch 1'). We obtained four sets of medium-resolution observations ( R = λ/ ∆ λ ≈ 500 ) and one set of low-resolution data ( R ≈ 22 ), a new mode never tried before with GRAVITY Wide. The first three medium-resolution exposures were of low quality, and the resulting closure phases were very noisy. We, therefore, decided not to use them in the subsequent analysis. The preliminary modeling carried out at that time indicated that the medium- and low-resolution data were inconsistent with each other. \nWe thus attempted to collect additional VLTI data to study the magnitude of possible systematics affecting the low-resolution data. We triggered on-axis GRAVITY observations with ATs on 2023 August 23, when the event was approaching the peak (at magnification ∼ 90 ), but the data could not be collected. The next (successful) attempt to gather additional VLTI UTs observations took place on 2023 September 29 ('epoch 2'), when we secured six low-resolution exposures with GRAVITY Wide. Table 1 presents the log of VLTI observations. Each low-resolution observation consisted of twelve 30 s exposures; the medium-resolution observation consisted of four 100 s exposures. For each night, a bright star pair \nTable 1. Log of VLTI Observations \nwas observed to center the science fringe with GRAVITY differential delay line before the microlens observation. We did not adjust the science fringe when the telescope was moved to the science target. \nThe data were reduced with the standard GRAVITY pipeline (version 1.4.2). We first used the Python script run\\_gravi\\_reduced.py to reduce the raw data up to applying the pixel-to-visibility matrix (P2VM). The default options were used except that we adopted -gravity\\_vis.output-phase-sc=SELF\\_VISPHI to calculate the internal differential phase between each spectral channel and -gravity\\_vis.opd-pupil-stddev-maxsc=9999 to ignore the poor pupil measurements in the acquisition camera which do not affect our closure phase measurements. The pipeline performed the bias and sky subtraction, flat fielding, wavelength calibration, and spectral extraction. Application of the P2VM converts the pixel detector counts into complex visibilities, taking into account all instrumental effects, including relative throughput, coherence, phase shift, and cross-talk. The dark, bad pixel, flat field, wavelength calibration, and P2VM matrix data were reduced from the daily calibration data obtained close in time to our observations. We then used run\\_gravi\\_trend.py to calibrate the closure phase data. For epoch 1, we used the star pair to center the science fringe and calibrate the medium-resolution data. Unfortunately, we did not have calibrator data observed in the low-resolution data on the same night. For epoch 2, we used a bright star pair observed after the microlens observation for the calibration. The following analyses were based on the closure phase data from the calibrated medium-resolution and uncalibrated low-resolution data from epoch 1, and calibrated lowresolution data from epoch 2. In this way, we used all the data with good quality. Meanwhile, when modeling the low-resolution data from epoch 2, we found that the calibration makes little difference to the closure phase.", '4. LIGHT CURVE MODEL': "The light curve of the event can be well fitted by a standard point-source point-lens model with the annual parallax effect (which is caused by the orbital motion of the Earth). This model has five free parameters: time of the closest approach between the lens and the source t 0 , their minimum separation (in Einstein radius units) u 0 , the Einstein radius crossing timescale t E , and North and East components of the microlensing parallax vector π E = ( π E ,N , π E ,E ) . The latter is a vector quantity whose direction is parallel to the direction of the relative lens-source proper motion µ rel . The magnification is calculated using the formula: \nA ( t ) = u ( t ) 2 +2 u ( t ) √ u ( t ) 2 +4 , (1) \nwhere u ( t ) = √ τ ( t ) 2 + β ( t ) 2 . The latter quantity is evaluated in the geocentric frame that is moving with a velocity equal to the Earth's velocity at t 0 , par = 2460179 (Gould 2004) in which \nτ ( t ) = t -t 0 t E + δτ ( t ) , β ( t ) = u 0 + δβ ( t ) (2) \nand \n( δτ, δβ ) = ( π E · ∆ s, π E × ∆ s ) , (3) \nwhere ∆ s is the projected position of the Sun. Two models with different signs of u 0 are possible; both models are almost perfectly degenerate, with the u 0 < 0 being preferred by only ∆ χ 2 = 3 . 1 . The best-fit model (with u 0 > 0 ) is presented in Figure 2. The best-fit parameters and their uncertainties are reported in Table 2. In Table 2, we also report the best-fit source magnitude I s , baseline magnitude I 0 , and dimensionless blending parameter f s . \nFigure 3 shows the constraints on the microlensing parallax vector derived using data from different observatories. Blue, red, green, and orange contours mark \nFigure 2. Light curve of OGLE-2023-BLG-0061/KMT2023-BLG-0496. The black line marks the best-fit model with u 0 > 0 . Arrows mark the two epochs of VLTI observations. \n<!-- image --> \nTable 2. Best-fit Parameters of the Light Curve Model \nconstraints from OGLE, KMTC, KMTA, and KMTS data, respectively, whereas the solid black contours mark \nFigure 3. Constraints on the microlensing parallax derived from different photometric data sets (blue - OGLE, red KMTC, orange - KMTS, green - KMTA) and all light curve data (black contours). The dotted and dashed lines mark constraints from the two epochs of VLTI data, respectively. \n<!-- image --> \nthe best-fit model to all data. The East component of π E , which is parallel to the projected acceleration of the Sun, is relatively well measured in all data sets. However, the North component π E ,N (perpendicular to the projected acceleration of the Sun) has a considerably larger uncertainty. The North component is also more susceptible to noise in the data (Gould et al. 1994; Smith et al. 2003; Gould 2004); hence, slightly different values of π E ,N are determined using different data sets. The best-fit parameters for individual data sets are reported in Tables 3 and 4. \nThe light curve model of a microlensing event makes it possible to predict the brightness ratio η between the minor and major image at a given epoch of interferometric observations. Moreover, the detection of the microlensing parallax in the light curve allows us to predict the orientation of microlensed images in the sky. The position angle (north through east) of the microlensing parallax vector can be calculated using formula \nΦ π = arctan π E ,E π E ,N . (4) \nThe angle ϕ between the source-lens relative proper motion and source-lens relative position at a given time can also be calculated directly from the parameters of the light curve model: \nϕ ( t ) = arctan β ( t ) τ ( t ) . (5) \nTable 3. Best-fit Parameters of the Light Curve Model ( u 0 > 0) for Individual Data SetsTable 4. Best-fit Parameters of the Light Curve Model ( u 0 < 0) for Individual Data Sets \nNote that in the limit of no parallax, this equation simplifies to ϕ ( t ) = arctan( u 0 t E / ( t -t 0 )) , which is equivalent to Equation (8) derived by Dong et al. (2019). Then, following Dong et al. (2019), the position angle of the minor image relative to the major image (north through east) is simply PA = Φ π + ϕ if u 0 > 0 or PA = Φ π -ϕ if u 0 < 0 . Note that Dong et al. (2019) introduced a different definition of the position angle of the major image relative to the minor image ψ = PA+ π . \nTable 2 presents the predicted lens-source separation u (in Einstein radius units), the flux ratio between the minor and major image η , and the position angle of the images PA for both epochs of VLTI observations. Note that the expected lens-source separation and flux ratio of the images are virtually identical for u 0 > 0 and u 0 < 0 models. However, the expected position angles are different. For the positive u 0 solution, we can predict PA = -164 . 83 ± 0 . 95 deg and PA = 31 . 52 ± 0 . 58 deg for both VLTI epochs. For the negative u 0 solution, we predict PA = -156 . 60 ± 0 . 68 deg and PA = 14 . 29 ± 1 . 06 deg, respectively. \nWe also note that the flux ratio of the images is tightly constrained by the light curve model, with a precision better than 0.3%. Conversely, the uncertainty of the position angle is dominated by the uncertainty in the determination of the angle Φ π , which itself is dominated by the uncertainty of π E ,N . As we observed small systematic differences between π E ,N determined using different data sets (Figure 3), these differences propagate to systematic variations in Φ π and therefore PA . In particular, the angle Φ π can vary from 18 . 56 to 25 . 90 deg (Tables 3 and 4).", '5. CLOSURE PHASE MODELS': 'We first separately analyze the VLTI data collected during epochs 1a, 1b, and 2, because they were taken at different times and using different instrument configurations. That will allow us to study the consistency between the model parameters derived using different data sets and the consistency with the light curve model (Sections 5.1, 5.2, and 5.3). In Section 5.4, we combine all VLTI data sets to derive the final parameters of the system. The results presented in this section were independently checked using the PMOIRED software 5 (Mérand 2022) and we found virtually identical results.', '5.1. Binary-star Model': 'We start by fitting a simple binary-star model to the closure phase data. We assume that both images of the source star can be considered as pointlike. We place \nTable 5. Best-fit Parameters of the Binary-star Closure Phase Model \nthe major image in the origin of the coordinate system, whereas the position of the minor image is parameterized by a vector (∆ α, ∆ δ ) . If we denote the flux ratio between the minor and major image as η , then the complex visibility is \nVIS = 1 + η exp ( -2 πi λ ( U ∆ α + V ∆ δ ) ) 1 + η , (6) \nwhere λ is the wavelength of observations, and ( U, V ) is the separation of the telescopes in the UV plane. The visibility is calculated for each pair of telescopes, and then the closure phase T3 is calculated for each triangle. Four possible closure phases can be calculated for VLTI; in theory, one of them is not independent. However, since the measured closure phases may be affected by noise in the data, we employ all four closure phase sets in the fits. \nWe find the best-fit parameters by maximizing the log-likelihood function defined as: \nln L = -1 2 4 n exp ∑ i =1 Λ ∑ j =1 (T3 ij -T3 model ij ) 2 σ (T3) 2 ij + σ 2 0 + -1 2 4 n exp ∑ i =1 Λ ∑ j =1 ln ( σ (T3) 2 ij + σ 2 0 ) , (7) \nwhere n exp is the number of exposures in the epoch and Λ is the number of spectral channels. Since the error bars calculated from the pipeline may be underestimated, we add a constant error term σ 0 in quadrature. The best-fit parameters and their uncertainties are calculated using the Markov chain Monte Carlo (MCMC) algorithm coded by Foreman-Mackey et al. (2013). We assume flat (non-informative) priors on all parameters of the model. \nFigure 4. Closure phase data for the VLTI epoch 1. Color solid lines mark the best-fit binary-star model. \n<!-- image --> \nResults of the fits, separately for VLTI epochs 1a, 1b, and 2, are reported in Table 5. The best-fitting closure phase models are shown in Figures 4 and 5. In addition to four model parameters, we report four derived quantities: separation between the images s = √ ∆ α 2 +∆ δ 2 , their position angle (north through east) PA = arctan∆ α/ ∆ δ , source-lens separation u (in Einstein radius units), and the angular Einstein radius θ E . The latter quantity is calculated using the formula \nθ E = sη 1 / 4 √ η +1 (8) \nderived under the assumption that the flux ratio between the microlensed images determines their separation in Einstein radius units. Similarly, the source-lens separation is √ \nu = 1 -η η 1 / 4 . (9) \nResults of the fits to individual epoch 2 exposures are presented in Table 6. \nThere is some tension between the medium- and lowresolution interferograms obtained during epoch 1: the measured ∆ α position differs by 3 . 3 σ , ∆ δ position2 . 3 σ , position angle3 . 2 σ . On the other hand, the inferred angular Einstein radii are formally consistent between the three different data sets, although the differences between individual measurements can amount up to 1 . 5 -2 . 9 σ . These tensions indicate that at least one of the analyzed data sets may suffer from unaccounted, low-level systematic errors. Thus, error bars reported in Tables 5 and 6 may be underestimated. \nFurther checks are possible because the flux ratio η and position angle PA of the images can be independently measured using the light curve data (Section 4). In particular, there are two possible light curve models differing by the sign of u 0 , which predict the position angles of the microlensed images of ( -164 . 8 · , 31 . 5 · ) (positive u 0 ) or ( -156 . 6 · , 14 . 3 · ) (negative u 0 ), during the VLTI epochs 1 and 2, respectively. The measured angles ( -162 . 9 · , 26 . 7 · ) (Table 5) seem to favor the u 0 > 0 model. \nHowever, while the light curve and closure phase models agree well for epoch 1, the expected position angles differ by almost 4 . 8 · (that is, 7 . 5 σ ) during epoch 2. That is illustrated in Figure 6, which presents 1 , 2 , 3 σ confidence ellipses in the (PA , η ) plane. Gray contours mark the constraints from three different VLTI data sets, whereas black contours were calculated based on the light curve model ( u 0 > 0 ). Blue, red, green, and orange contours mark constraints on (PA , η ) from OGLE, KMTC, KMTA, and KMTS data, respectively. \nThe position angle of the images measured from interferometric observations can be projected onto the ( π E ,N , π E ,E ) plane (Figure 3). The dotted and dashed lines in this figure correspond to the best-fitting values of PA during epochs 1a and 2, respectively. The dashed line does not intersect color contours from the light curve model, which further exemplifies the tension discussed above. \nThe light curve model also predicts the flux ratios between the microlensed images: η = 0 . 5907 ± 0009 for epoch 1 and η = 0 . 4264 ± 0 . 0013 for epoch 2. The for- \nFigure 5. Closure phase data for the VLTI epoch 2. Color solid lines mark the best-fit binary-star model. \n<!-- image --> \nmer value is in 2 . 7 -3 . 5 σ tension with closure phase fits to epoch 1a and 1b data, respectively.', '5.2. Luminous-lens Model': 'Despite the slight tensions discussed above, the binary-star model overall describes the closure phase data well (Figures 4 and 5). We now modify the model to include a possibility of blended light coming from the lens itself or a luminous companion (either to the lens or source). We changed the primary parameters of the model to the lens-source separation u , the position angle of the lens relative to the source PA (which is equal to the position angle of the minor image relative to the major image), and the angular Einstein radius θ E , which provide a more natural description of a microlensing event than the binary-star model parameters. We keep σ 0 fixed at 6.990, 2.593, and 2.509 deg for epoch 1a, 1b, and 2, respectively. \nWe first consider models with a luminous lens. We place the lens in the origin of the coordinate system and calculate the positions of the minor and major image relative to it (Dong et al. 2019). We parameterize the ratio of the lens flux to the (unmagnified) source flux as η b . \nWe found that the closure phase data do not provide strong evidence for the light from the lens. Including the lens light in the fits improves the χ 2 by 1.3, 0.1, and 0.9 for epochs 1a, 1b, and 2, respectively. The corresponding 95% upper limits on η b are 0.74, 0.81, and 0.44, respectively. Moreover, we noticed that the lens flux is correlated with the projected lens-source separation (and so, the flux ratio between the microlensed images) and the angular Einstein radius. \nThus, we explored the possibility that the light from the lens may be a source of tensions between the photometric and interferometric data discussed above. We repeated modeling, taking into account priors on u from the light curve model. The best-fit parameters are reported in Table 7. While the tension between u determined from the light curve model and that from the closure phase model is removed, the position angles are still slightly different.', '5.3. Luminous-blend Model': 'We also consider the model with a luminous blend, which has two additional parameters compared to the luminous-lens model, the offset of the blend (∆ α b , ∆ δ b ) relative to the lens in the sky. We searched for possible blends on a grid of 201 × 201 positions uniformly spread over the range -20 ≤ ∆ α b , ∆ δ b ≤ 20 mas . We kept the position of the blend fixed but allowed the other parameters ( u , PA, θ E , and η b ) to vary. The best-fit \nparameters were found using a downhill approach using the Nelder-Nead algorithm. We adopted a prior on the lens-source separation u from the light curve model, and we required η b ≥ 0 . \nThe grid search results for the epoch 2 data are presented in Figure 7. There is only one local minimum around (∆ α b , ∆ δ b ) = ( -6 . 0 , 15 . 8) mas that may be statistically significant (the χ 2 statistics is improved by ∆ χ 2 = 26 . 6 ). We explored this local minimum using the MCMC approach and found u = 0 . 4295 ± 0 . 0016 , PA = 26 . 00 ± 0 . 24 deg, θ E = 1 . 2672 ± 0 . 0054 mas, ∆ α b = -5 . 91 ± 0 . 18 mas, ∆ δ b = 15 . 73 ± 0 . 32 mas, and η b = 0 . 0426 ± 0 . 0094 . Thus, even if we consider this solution statistically significant, the blend does not explain the tension between the position angle measured using the light curve and interferometric data.', '5.4. Final Closure Phase Models': 'Finally, we measure the angular Einstein radius of the event θ E using all VLTI data combined. The baseline model has five parameters: the source-lens separation u and position angle of the images PA during epochs 1 and 2, and θ E (the same during both epochs). To estimate the impact of systematic errors on the parameter uncertainties, we employ the bootstrapping method (e.g. Efron 1979; Kervella et al. 2004; Lachaume et al. 2019). It involves randomly selecting eight VLTI interferograms (with replacement), feeding them to our fitting routine and then repeating the procedure multiple times (in this case, 5000 times) to obtain the multivariate probability density function for the parameters of the model. Such a procedure enables us to retrieve more realistic uncertainties on the model parameters. The results of the bootstrapping method are reported in the second column of Table 8. \nWe then repeat the modeling, allowing an additional flux from the lens. We consider two models: with and without the prior on u from the light curve model (third and fourth columns of Table 8.) As in Section 5.2, we find that including the lens light does not significantly improve the fits, and we can only measure upper limits on η b (95% confidence).', '6. PHYSICAL PARAMETERS OF THE LENS': 'The mass and distance to the lens can be obtained from \nM = θ E κπ E , D l = au π E θ E +au /D s , (10) \nwhere D s ≈ 8 kpc is the source distance. As the interferometric data do not provide strong evidence for the light from the lens, we adopt θ E = 1 . 2802 ± 0 . 0085 mas (Table 8) as our final measurement. Using π E = \nFigure 6. Constraints on the flux ratio and position angle of the microlensed images measured from the VLTI data (gray contours). The color contours mark the constrains from the individual photometric data sets (blue - OGLE, red - KMTC, orange - KMTS, green - KMTA), the black contours are measured using all photometric data combined. \n<!-- image --> \nTable 6. Best-fit Parameters of the Binary-star Closure Phase Model for Individual Epoch 2 DataTable 7. Best-fit Parameters of the Luminous-Lens Closure Phase Model \nTable 8. Final Closure Phase Models \n0 . 3330 ± 0 . 0084 , we find M = 0 . 472 ± 0 . 012 M ⊙ , π rel = 0 . 426 ± 0 . 011 mas, and D l = 1 . 81 ± 0 . 05 kpc. The lens is, \ntherefore, most likely a main-sequence star located in the \nFigure 7. Detection map for possible luminous blends in the VLTI epoch 2 data. The color codes the χ 2 improvement over the model without the blend. The lens is located in the origin of the coordinate system. \n<!-- image --> \nEpoch 2 \nnearby Galactic disk. Alternatively, it may be a white dwarf, but because white dwarfs are much less frequent than main-sequence stars, this possibility is unlikely. \nIf the lens is a main-sequence star, its absolute magnitude is M K = 5 . 76 ± 0 . 04 (Pecaut & Mamajek 2013), and its apparent K -band magnitude is 17 . 20 ± 0 . 07 , assuming the K -band extinction of 0.15 mag toward the lens (Nataf et al. 2013). That corresponds to η b ≈ 0 . 03 in the K -band, consistent with the limits on the lens flux from the VLTI data. \nThe trajectory of the source in the sky is presented in Figure 8 by a gray solid line. Open symbols (red square and blue circle) mark the position of the source during the two epochs of VLTI observations, whereas the filled symbols mark the positions of the images of the source.', '7. DISCUSSION AND CONCLUSIONS': "The start of the scientific operations of GRAVITY Wide opens up completely new possibilities for the follow-up of microlensing events. Only very few of the brightest events could have been observed with the standard on-axis GRAVITY mode. In contrast, GRAVITY Wide observations provide the opportunity for studies of a large sample of events and measurements of masses, distances, and transverse velocities of isolated objects, including neutron stars and black holes. Interferometric observations are very efficient at the same time-only one-hour exposure is sufficient for determining the angular Einstein radius. Thus, interferometric observations \nFigure 8. Geometry of the event. The gray solid line marks the trajectory of the source in the sky. Open symbols (red square and blue circle) mark the position of the source during the two epochs of VLTI observations, whereas the filled symbols mark the positions of the images of the source. The dashed circle shows the Einstein radius. \n<!-- image --> \nwith GRAVITY Wide (and, its successor, GRAVITY+) hold promise for routine detections of isolated neutron stars and stellar remnants. \nIn this paper, we report the first successful observation of a microlensing event with GRAVITY Wide and the resolution of microlensed images in the event OGLE2023-BLG-0061/KMT-2023-BLG-0496. These observations serve as a testbed for verifying the capabilities of the new instrument. In particular, some observations were carried out using the low-resolution mode, which had never been tried before with GRAVITY Wide. The comparison of medium- and low-resolution interferograms (epochs 1a and 1b) reveals low-level systematic differences between the inferred parameters of the lensing system (Table 5). Similar systematic differences can be seen in models fitted to individual exposures taken during epoch 2 (Table 6). Judging from Figures 4 and 5, the systematic and correlated errors in the closure phase data can reach up to 5-10 deg. We defer a detailed investigation of systematic errors in the closure phase data to a separate study. \nWe use the bootstrapping method to deal with these systematic and correlated errors. Thanks to a large number of observations, we are still able to measure the angular Einstein radius with a sub percent precision, θ E = 1 . 2802 ± 0 . 0085 mas. By combining the informa- \non from the light curve and the closure phase data, we measure the mass of the lens with a precision of 2.6%, M = 0 . 472 ± 0 . 012 M ⊙ . \nIndependent tests of the accuracy of the closure phase model are possible with the light curve model. The model of the microlensing event light curve predicts the brightness ratio of the microlensed images and their position angle in the sky. While the predicted flux ratio matches that inferred from the closure phase data reasonably well (Figure 6), there is a tension in the position angles measured using the epoch 2 data. This tension cannot be explained by additional blended light in the closure phase data (Section 5.3). \nOne possible explanation for this systematic difference is the unaccounted systematic errors in the closure phase data. Alternatively, the problem may lie in the light curve model. However, we find this unlikely, because the residuals from the best-fit light curve model are smaller than ≈ 0 . 01 mag, and there is no strong evidence that the light curve model is inadequate. A binary-lens model can be ruled out because the binary lens would produce additional images of the source, which should have been detected in our grid search. Still, the unaccounted orbital motion of the source ('xallarap effect') could, in principle, partly explain the discrepancy. The xallarap effect would not create additional images but would deflect the path of the source in the sky. Our modeling, however, do not provide strong evidence for xallarap in the light curve data (Appendix A).", 'ACKNOWLEDGEMENTS': "Based on observations collected at the European Southern Observatory under ESO program 108.220D. This research was funded in part by National Science Centre, Poland, grants OPUS 2021/41/B/ST9/00252 and SONATA 2023/51/D/ST9/00187 awarded to P.M. This research is partly supported by the National Natu- \nral Science Foundation of China (Grant No. 12133005) and the science research grants from the China Manned Space Project with No. CMS-CSST-2021-B12. S.D. acknowledges the New Cornerstone Science Foundation through the XPLORER PRIZE. \nGRAVITY+ is developed by the Max Planck Institute for extraterrestrial Physics, the Institute National des Sciences de l'Univers du CNRS (INSU) with its institutes LESIA / Paris Observatory-PSL, IPAG / Grenoble Observatory, Lagrange / Côte d'Azur Observatory and CRAL / Lyon Observatory, the Max Planck Institute for Astronomy, the University of Cologne, the CENTRA - Centro de Astrofisica e Gravitação, the University of Southampton, the Katholieke Universiteit Leuven and the European Southern Observatory. D.D. has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement CoG - 866070). \nThis research has made use of the KMTNet system operated by the Korea Astronomy and Space Science Institute (KASI) at three host sites of CTIO in Chile, SAAO in South Africa, and SSO in Australia. Data transfer from the host site to KASI was supported by the Korea Research Environment Open NETwork (KREONET). This research was supported by KASI under the R&D program (project No. 2024-1-832-01) supervised by the Ministry of Science and ICT. W.Zang, H.Y., S.M., R.K., J.Z., and W.Zhu acknowledge support by the National Natural Science Foundation of China (Grant No. 12133005). W.Zang acknowledges the support from the Harvard-Smithsonian Center for Astrophysics through the CfA Fellowship. J.C.Y. and I.-G.S. acknowledge support from U.S. NSF Grant No. AST-2108414. Y.S. acknowledges support from BSF Grant No. 2020740. Work by C.H. was supported by the grants of National Research Foundation of Korea (2019R1A2C2085965 and 2020R1A4A2002885). J.C.Y. acknowledges support from a Scholarly Studies grant from the Smithsonian Institution.", 'REFERENCES': 'Albrow, M. D., Horne, K., Bramich, D. M., et al. 2009, \nMNRAS, 397, 2099 \nArsenault, R., Alonso, J., Bonnet, H., et al. 2003, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 4839, Adaptive Optical System Technologies II, ed. P. L. Wizinowich & D. Bonaccini, 174-185 \nBachelet, E., Zieliński, P., Gromadzki, M., et al. 2022, \nA&A, 657, A17 \nCassan, A., Ranc, C., Absil, O., et al. 2022, Nature \nAstronomy, 6, 121 Colavita, M. M., Wallace, J. K., Hines, B. E., et al. 1999, ApJ, 510, 505 Delplancke, F. 2008, NewAR, 52, 199 Delplancke, F., Górski, K. M., & Richichi, A. 2001, A&A, 375, 701 Dong, S., Mérand, A., Delplancke-Ströbele, F., et al. 2019, ApJ, 871, 70 Efron, B. 1979, The Annals of Statistics, 7, 1', 'A. XALLARAP MODELS': "In this section, we explore whether the orbital motion of the source (xallarap effect) can explain the discrepancy between the position angle of the microlensed images measured using VLTI data and calculated from the light curve model. Compared to the standard point-source point-lens model, models including the xallarap effect have seven additional parameters describing the shape of the source's orbit. These are P ξ - the orbital period, a ξ - the semimajor axis (expressed in Einstein radius units), i ξ - the inclination of the orbit, Ω ξ - the longitude of the ascending node of the orbit, u ξ - the argument of latitude at the reference epoch (which is taken to be t 0 ,ξ = t 0 , par ), e ξ - the eccentricity of the orbit, and ω ξ - the argument of periapsis of the orbit. \nFor simplicity, we consider only circular orbits, keeping e ξ = 0 and ω ξ = 0 . We also assume that the companion to the source is dark, given there is no strong evidence for the additional light in the VLTI data (Section 5.3). We employ the MulensModel package by Poleski & Yee (2019) to calculate the source trajectory and magnification. \nWe search for best-fit xallarap models on a grid of orbital periods P ξ spanning from 10 to 251 days. We keep the orbital period fixed, but the remaining parameters ( a ξ , i ξ , Ω ξ , u ξ , and the standard microlensing parameters) are allowed to vary. The best-fit parameters are found using the MCMC approach (Foreman-Mackey et al. 2013). Simultaneously, we calculate the position angle of the minor image relative to the major image during the two epochs of VLTI observations (Section 4). We assume uniform priors on all parameters. \nThe results of our calculations for the OGLE data are presented in Table 9, where we report the χ 2 statistic of the best-fit model and the predicted position angles. There is no strong evidence for xallarap in the data. The best-fit model, for the orbital period of P ξ = 79 . 4 days, is favored over the model without xallarap by only ∆ χ 2 = 11 . 6 , which is not statistically significant given the increased number of free parameters. The predicted position angles are all similar to those calculated using the standard point-source point-lens model. However, their uncertainties are typically larger, which quantifies the additional degrees of freedom on the position of the source due to orbital motion. \nTable 9. Results of Xallarap Fits to the OGLE Data"}

2024arXiv240911453B

We identify a pointsymmetric morphology of three pairs of earsclumps in the corecollapse supernova CCSN remnant CCSNR Puppis A supporting the jittering jets explosion mechanism JJEM. In the JJEM the three pairs of jets that shaped the three pairs of earsclumps in Puppis A are part of a large about 10 to 30 pairs of jets that exploded Puppis A. Some similarities in morphological features between CCSNR Puppis A and three multipolar planetary nebulae considered to have been shaped by jets solidify the claim for shaping by jets. Puppis A has a prominent dipole structure where one side is bright with a welldefined boundary while the other is faint and defused. The neutron star NS remnant of Puppis A has a proper velocity its natal kick velocity in the opposite direction to the denser part of the dipole structure. We propose a new mechanism in the frame of the JJEM that imparts a natal kick to the NS the kickbyearly asymmetrical pair kickBEAP mechanism. At the early phase of the explosion process the NS launches a pair of jets where one jet is much more energetic than the counter jet. The more energetic jet compresses a dense side to the CCSNR and by momentum conservation the NS recoils in the opposite direction. Our study supports the JJEM as the primary explosion mechanism of CCSNe and enriches this explosion mechanism by introducing the novel kickBEAP mechanism.

2024-09-01T00:00:00Z

['10.48550/arXiv.2409.11453', '2024arXiv240911453B', 'arXiv:2409.11453']

['Astrophysics - High Energy Astrophysical Phenomena']

The Puppis A supernova remnant an early jetdriven neutron star kick followed by jittering jets

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.11453.pdf

{'The Puppis A supernova remnant: an early jet-driven neutron star kick followed by jittering jets': 'Ealeal Bear, 1 Dmitry Shishkin, 1 and Noam Soker 1 \n1 Department of Physics, Technion, Haifa, 3200003, Israel; ealeal44@technion.ac.il; s.dmitry@campus.technion.ac.il; soker@physics.technion.ac.il \n(Dated: July 2024)', 'ABSTRACT': 'We identify a point-symmetric morphology of three pairs of ears/clumps in the core-collapse supernova (CCSN) remnant (CCSNR) Puppis A, supporting the jittering jets explosion mechanism (JJEM). In the JJEM, the three pairs of jets that shaped the three pairs of ears/clumps in Puppis A are part of a large, about 10 to 30 pairs of jets that exploded Puppis A. Some similarities in morphological features between CCSNR Puppis A and three multipolar planetary nebulae considered to have been shaped by jets solidify the claim for shaping by jets. Puppis A has a prominent dipole structure, where one side is bright with a well-defined boundary, while the other is faint and defused. The neutron star (NS) remnant of Puppis A has a proper velocity, its natal kick velocity, in the opposite direction to the denser part of the dipole structure. We propose a new mechanism in the frame of the JJEM that imparts a natal kick to the NS, the kick-by-early asymmetrical pair (kick-BEAP) mechanism. At the early phase of the explosion process, the NS launches a pair of jets where one jet is much more energetic than the counter jet. The more energetic jet compresses a dense side to the CCSNR, and, by momentum conservation, the NS recoils in the opposite direction. Our study supports the JJEM as the primary explosion mechanism of CCSNe and enriches this explosion mechanism by introducing the novel kick-BEAP mechanism. \nKeywords: supernovae: general - stars: jets - ISM: supernova remnants - stars: massive', '1. INTRODUCTION': "In the jittering jets explosion mechanism (JJEM; Papish & Soker 2011) of core-collapse supernovae (CCSNe), pairs of opposite jets with stochastically (fully or partially) varying directions explode the star. The newly born neutron star (NS; and the black hole, in cases where the NS collapses into a black hole) launches several to few tens of pairs of jets on a time scale of ≈ 0 . 5 -10 s (e.g., Soker 2024a) as it accretes mass from the collapsing core via intermittent accretion disks (later pairs of jets are possible). In the present study, we will propose that the first jets are launched already at t b ≃ 0 . 1 -0 . 2 s post-bounce (Section 5). The stochastic angular momentum of the gas that feeds the intermittent accretion disks results from instabilities above the NS that increase the angular momentum seed fluctuations of the collapsing gas. These instabilities above the NS and below the stalled shock at ≃ 150 km from the center include the modes of spiral standing accretion shock instability (e.g., Buellet et al. 2023 for a recent study of this instability). The seed fluctuations come from the \nconvective motion in the pre-collapse core (e.g., Papish & Soker 2014; Gilkis & Soker 2014, 2016; Shishkin & Soker 2023; Wang et al. 2024). Neutrino heating boosts the explosion process in the JJEM (Soker 2022a), but the jets play the primary role in the explosion process. \nIn the competing delayed neutrino explosion mechanism of CCSNe (e.g., Andresen et al. 2024; Burrows et al. 2024b; Janka & Kresse 2024; Muller 2024; Muller et al. 2024; Nakamura et al. 2024; van Baal et al. 2024; Wang & Burrows 2024; Laplace et al. 2024 for some very recent studies) neutrino heating revives the stalled shock and explode the star; jets play no roles in the explosion process. The jet-based magnetorotational explosion mechanism requires a rapidly rotating pre-collapse core to launch jets along a fixed axis (e.g., Shibagaki et al. 2024; Zha et al. 2024 for recent studies of this mechanism); therefore, it can power only a very small fraction of CCSNe. We considered the magnetorotational explosion mechanism part of the neutrino-driven mechanism despite the powering by jets because it still attributes \nmost CCSNe to the neutrino-driven mechanism. The JJEM attributes all CCSN explosions to jets. \nThe observational property of CCSN remnants (CCSNRs) that best differentiates between the two explosion mechanisms is point-symmetric morphological features. A point-symmetric morphology has two or more pairs of structural features on opposite sides of the center that do not share the same symmetry axis. The opposite structures include clumps, filaments, bubbles (faint structures closed and encircled by a brighter rim), lobes, which are bubbles with partial rims, and ears. An ear is a protrusion from the main CCSNR shell with decreasing cross sections away from the center. \nWhile the JJEM predicts that many, but not all, CCSNR morphologies are point symmetric, the neutrinodriven explosion mechanism has no explanation for most of these morphologies (for a detailed comparison between the two explosion models, see Soker & Shishkin 2024). The following is the list of CCSNRs with studies that attributed their point-symmetric morphologies to the JJEM: SNR 0540-69.3 (Soker 2022b), the Vela CCSNR (Soker 2023; Soker & Shishkin 2024), CTB 1 (Bear & Soker 2023), N63A (Soker 2024b), SN 1987A (Soker 2024c,a), G321.3-3.9 (Soker 2024d; Shishkin & Soker 2024), G107.7-5.1 (Soker 2024d), Cassiopeia A (Bear & Soker 2024), and the Cygnus Loop (Shishkin et al. 2024). Some other CCSNRs show only one pair of opposite ears (e.g., Grichener & Soker 2017). Some other CCSNRs are challenging in having a complicated structure lacking clear symmetry; the CCSNR Puppis A is such a CCSNR. \nEarlier studies have argued for the jet-shaping of Puppis A. Castelletti et al. (2006) consider jet shaping at very late times rather than exploding jets. Grichener & Soker (2017) consider shaping by a pair of jets that are part of the jittering jets that exploded the progenitor of Puppis A, and estimate the energy of each of the two jets that inflated the western and eastern ears to be ≃ 1% of the explosion energy. Clearly, these two jets could not have by themselves exploded the star. We expect more pairs of jets, as we discuss in this study. Other studies noticed the interaction of Puppis A with a circumstellar material (CSM) or interstellar medium (ISM; e.g., Dubner et al. 2013; Dubner & Giacani 2015; Reynoso & Walsh 2015; Meyer et al. 2022). However, Reynoso et al. (2017) find no strong indication for ejecta-cloud interaction in the east. In any case, although the ISM influences the morphology of old CCSNRs (e.g., Sofue 2024 for a recent paper), it cannot account for pointsymmetry and asymmetrical metal distribution as observed in many CCSNRs (e.g., Soker & Shishkin 2024). \nThis study considers the shaping jets part of the exploding jets. The similarities of pairs of ears, a main line of symmetry (main jet axis), and, above all, pointsymmetric morphologies of many CCSNRs to similar structures in planetary nebulae (e.g., Bear et al. 2017; Bear & Soker 2017, 2018a; Soker 2022c, 2024e) and cooling flow clusters (Soker 2024d) support the claim of shaping by energetic jets. Several properties, like the large volume of some jet-shaped structures, indicating energetic jets, and the high abundance of heavy elements in some jet-shaped structures show that the NS launched the shaping jets during the explosion process and that the jets have powered the explosion (Soker & Shishkin 2024). \nIn this study, we analyze the highly asymmetrical CCSNR Puppis A. We describe its structure as appears in observations of earlier studies but emphasize its point-symmetrical morphology and highly asymmetrical dipole structure (Section 2). In Section 3 we compare Puppis A's highly asymmetrical dipole structure to that of the SNR N49. In Section 4, we find some similarities with planetary nebulae that strengthen the claim for shaping by jets. In Section 5, we propose a novel mechanism for the kick velocity unique to the JJEM. We summarize this study in Section 6.", '2. THE POINT-SYMMETRIC STRUCTURE OF PUPPIS A': "Puppis A is a CCSN projected inside the Vela CCSNR. Its morphology and structure are highly nonspherical, and it has attracted attention over the years (e.g., Arendt et al. 1990; Hwang et al. 2005; Castelletti et al. 2006; Katsuda et al. 2010; Hewitt et al. 2012; Dubner et al. 2013; Reynoso et al. 2017, 2018; Aruga et al. 2022; Krivonos et al. 2022; Mayer et al. 2022; Giuffrida et al. 2023; Ghavamian et al. 2024). In this study, we concentrate on two aspects of its morphology. (1) We reveal a possible point-symmetric structure of three pairs of 'ears' (one pair of the three is more of two clumps). An 'ear' is a protrusion from the SNR's central nebula with a base smaller than the main SNR and a crosssection that decreases outward. The ear might appear differently from the main SNR, like being fainter. (2) The prominent dipole asymmetry (below and in more detail in Section 3), and in Section 5, we suggest a new mechanism to impart a kick velocity to the newly born NS that we suggest shaped the dipole asymmetry. \nIn Figure 1 we present an X-ray image of Puppis A, using data from the early data release (EDR) of eROSITA (Obs ID: 700195). We added three lines connecting three pairs of ears, A with A ' , B with B ' , and C with C ' (which can also be classified as a pair of clumps). Earlier stud- \nentified the pair AA ' . Grichener & Soker (2017) estimated that the combined energy of the two jets that inflated ears A and A ' is ≃ 2% of the explosion energy of Puppis A. They also connect each ear to the NS rather than to each other. This forms a bent in the direction of the two ears. Namely, as seen from the NS, the two ears are not exactly opposite at 180 · to each other. We prefer to connect the pair AA ' with a straight line. We also identify the pairs BB ' and CC ' . These two pairs' northern ears overlap on the plane of the sky but are not parts of a single structure. We take it to imply that the two axes (lines) of pairs BB ' and CC ' are highly inclined to each other. \nWe mark three central points on the image of Puppis A: the present location of the NS (red asterisk), the estimated location of the NS at the time of the explosion (red dot) determined from the proper velocity of the NS of v kick = 763 km s -1 (763 ± 73 km s -1 , Mayer et al. 2020; Holland-Ashford et al. 2017 give v kick = 437 km s -1 ), and an assumed SNR age of 4600 yr (Mayer et al. 2020; Aruga et al. 2022 give an age of ≃ 10 4 yr), and the average location of the three centers of the three symmetry lines (white asterisk). Although the three symmetry lines do not cross each other at the same point, they all cross between the initial (at the explosion time) and the present location of the NS. Considering also the highly non-spherical structure of Puppis A, namely, its prominent dipole structure, we believe the point-symmetric morphology is robust, although not easy to identify. \nThe dipole prominent structure of Puppis A includes a radio and X-ray bright emission region in the northeast with a more or less straight and sharp boundary. The bright zone forms a saddle-like shape; hence, we call it the 'Saddle.' It is seen as a yellow-colored zone on panel b of Figure 2 and a brighter zone in panel d. On the opposite side, the southwest side of Puppis A, on the other hand, the CCSNR does not have a sharp boundary, and it is much fainter. The southwest farthest structure is a faint potential ear, which we term 'P-ear(?)', where the question mark indicates that future studies should determine whether it is an ear. Southeast of the P-ear, we notice a zone that is brighter than the P-ear in radio (panel c) and in X-ray (panel d in Figure 2), which we term the 'Tongue'. The bright saddle on one side of Puppis A and the faint P-ear + the tongue on the other are the prominent components of the dipole morphology of Puppis A. \nIn addition to the point-symmetric structure and the large-scale dipole asymmetry, we point out a central band prominent in radio images of Puppis A (e.g., Castelletti et al. 2006; Reynoso et al. 2018) that we \nterm the 'corridor'; we mark it on panel c of Figure 2. Castelletti et al. (2006)'s processed radio images (figures 5 and 6 in their paper) show a central band from northeast to southwest, the 'corridor.' It is consistent with a flatter (than the surroundings) spectral index, which, they claim, might imply a region with emission being more thermal emission-dominated (than synchrotrondominated). The motion of the NS from explosion to present (proper motion) is inside and almost along the corridor. \nMayer et al. (2022) eROSITA observations show a prominent zone rich in ejecta material, namely, elements from the deep core of the progenitor, O, Ne, Mg, Si, S, and Fe. Mayer et al. (2022) mark in this zone the ejecta knot that was noticed by Katsuda et al. (2008) and the ejecta-rich region that was seen by Hwang et al. (2008). We mark this zone on panel a of Figure 2. Some studies attribute ejecta-rich clumps to jets and expect a symmetric structural feature on the other side of the center, as expected in the JJEM. A prominent example is the Vela SNR (Soker & Shishkin 2024). There is no ejectarich zone opposite the one northeast of Puppis A. However, we note that the tongue is on the opposite side of the ejecta-rich clumps, as we mark by the dashed-white line on panels a and d of Figure 2. At this time, we do not argue that the ejecta-rich zone and the tongue are opposite structural features formed by a pair of opposite jets. We only comment on the need for deeper observations to reveal the composition of the tongue and further exploit the structure of the ejecta-rich clump. \nSome studies attribute the shaping to the wind of the progenitor of Puppis A, e.g., Reynoso et al. (2018). There is also a dense ISM around Puppis A (e.g., Aruga et al. 2022). We do not question whether pre-explosion blown CSM and the ISM play a role in shaping Puppis A. However, we also attribute substantial shaping to pairs of jets that exploded the progenitor of Puppis A. We base our claim on the three pairs of ears that form the point-symmetric morphological structure and the 'corridor' that extends more or less along the kick direction of the NS to the southwest, beyond the present location of the NS. In Section 4, we find morphological similarities between Puppis A and a few planetary nebulae that are thought to be shaped by jets. This further supports the claim for jet-shaping of Puppis A. \nGhavamian et al. (2024) suggest the presence of a close companion to the progenitor of Puppis A that formed a funnel in the ejecta. They aim to explain the blueshifted series of nested, optically emitting rings near the center of Puppis A ('the Swirl'). We note that a binary companion might smear a point-symmetric morphology, \nFigure 1. X-ray log scaled counts image of Puppis A in the 0 . 2 -2 . 4keV range taken from the eROSITA EDR data. Colors indicate different energy bands, with levels balanced for display purposes (legend at bottom left). We mark with light-blue lines two point symmetrical pairs of ears (AA ' , BB ' ) and one pair of clumps (CC ' ). We denote the center of each symmetry line (axis) with a blue dot and the total average projected location of the three centers with a white asterisk. We mark the present NS location with a red asterisk. A dashed white arrow shows the direction of the NS's proper motion. A red dot indicates the location of the NS at explosion based on the proper motion (Mayer et al. 2020). In an inset on the bottom right, we display a zoom-in on the central region of Puppis A; the red arrow indicates the NS's proper motion direction. \n<!-- image --> \nRight Ascention \nas many other processes do, like interaction with a CSM and the ISM, the hot ejecta, and the NS natal kick.", '3. THE HIGHLY ASYMMETRICAL STRUCTURE OF N49': "We briefly mention the morphology of the CCSNR N49 in the Large Magellanic Cloud (other names are DEML190; LHA 120-N 49; MCSNRJ0526-6605). N49 shares some similarities with Puppis A, particularly a strong dipole asymmetry. However, there are two sig- \nnificant differences. The first is that there is no secure identification of an NS in that remnant; the magnetar in its north SGR0526-66 (PSR B0525-66) is likely not associated with N49 (e.g., Gaensler et al. 2001; Park et al. 2020; Ghavam et al. 2024). The second difference from Puppis A is that we could not identify clear signatures of jets despite some ear-like structures. For these differences, we only briefly mention N49 here. In Figure 3 we present images of N49. There is a bright and dense \nFigure 2. Puppis A morphological features. (a) X-ray counts image, with the Si/H abundances map from Mayer et al. (2022) (semi-transparent, ranging from 0 to 3). We denote the ejecta-rich clump, also enriched in O, Ne, Mg, S, and Fe (see Mayer et al. 2022 for details). We mark a southwest protrusion that we call the 'Tongue'. We draw a dashed white line between the ejecta-rich clump and the tongue, suggesting their potential formation in a pair of jets deserves further study. Two X-ray count contour lines outline the entire remnant and mark the higher-counts region in the northeastern part of Puppis A. (b) X-ray counts (green) and IR image (WISE, 22 µm , red). The yellow region at the NE part of the Puppis A SNR is where emission is strong in both X-ray and IR - which we nickname the 'Saddle'. We mark its northern and eastern edges. We also point at a potential ear (P-ear) that might be a counterpart to the saddle on the southwestern part of Puppis A. (c) Radio image of the spectral index distribution from Castelletti et al. (2006), in grayscale. With a dashed red line, we mark the darker central band in the image, corresponding to a flatter radio spectrum (see Castelletti et al. 2006 for details) and denote it the 'Corridor'. We also mark the NS kick direction (dashed white arrow) to note the small angle between the two. Contour lines are of X-ray emission from the original image by Castelletti et al. (2006) and are different from the contour lines in panel a. (d) X-ray counts image with all aforementioned relevant features we identified in Puppis A, as well as the point-symmetric wind-rose from Figure 1. A grid serves to guide the eye with the different features. All panels display the same region as Figure 1. In all panels, we note our proposed center (see Figure 1) with a white asterisk, and in panels a, c, and d, the NS location (red asterisk) and the calculated location of the NS at explosion (red dot). \n<!-- image --> \nzone in the southeast with a more or less straight outer boundary, resembling the bright-dense zone of Puppis A in the northeast, i.e., the 'Saddle' of Puppis A. \nYamane et al. (2018) observed a cloud in the southeast rim of the main shell of N49 that interacts with N49. This influences the morphology in the southeast, where the density is high. However, an interaction with a cloud in the southeast cannot explain some other morphological features of N49. (1) The western and northern sides should have a semi-circular boundary because, in those directions, interaction with CSM/ISM is much weaker. However, the boundary of N49 is structured, like having two protrusions in the northwest and southwest (marked by 'P' in the upper-left panel of Figure 3). This suggests a non-spherical explosion process. (2) Optical filaments extending from the southeast dense part to the two protrusions on the northwest and southwest. These filaments exist on the far and near side of N49 (Bilikova et al. 2007). An interaction with a cloud on one side cannot explain this large-scale structure of the filaments. (3) Some metals' distribution is along the morphology's dipole line (Zhou et al. 2019). This is most prominent for Ne and Mg, but O and Si also show this asymmetry; S is distributed in the south of N49, and Fe is scattered (Zhou et al. 2019). We present the density and Ne distribution from Zhou et al. (2019) in the lower panels of Figure 3. \nFrom this short discussion and the similarity to Puppis A, and assuming that the magnetar is not associated with N49, we speculate that the explosion was highly asymmetrical with a strong dipole morphology and that the compact object of N49 has a kick velocity to the northwest, as we mark by the double-line green arrow in the lower left panel of Figure 3 (see Section 5).", '4. HINTS FROM PLANETARY NEBULAE': "The tool of identifying point-symmetrical morphologies and attributing them to jets is prevalent among people studying planetary nebulae (e.g., Sahai et al. 2011, 2024; Clairmont et al. 2022; Danehkar 2022; Moraga Baez et al. 2023; Miranda et al. 2024). We use this tool here. \nIn Fig. 4, we present Puppis A alongside four planetary nebulae with some, but not full, morphological similarities to Puppis A. The two images of M 1-30 are from Hsia et al. (2014) with the marks from their figures beside the dashed-yellow line. M 1-30 has an enhanced density in the equatorial plane with two lobes on its sides and a prominent pair of ears, aa', and a less prominent pair, bb'. We find a similarity between the pair aa' in M 1-30 and AA ' in Puppis A. The upper right panel is an overexposed image from Hsia et al. (2014) where they \nidentify a pair of opposite jets (see their marks on the image). The axis of these two jets, the dashed-yellow line that we added on the upper right panel, is perpendicular to the dense strip that Hsia et al. (2014) mark with a dashed-black line (upper middle panel), which we take to be the equatorial plane. We find a similarity between this pair of jets in M 1-30 and the pair BB ' in Puppis A, which is also perpendicular to a dense stripe, more or less the saddle-tongue structure. We mark possible two 'equatorial planes' on the image of Puppis A by dotted-orange lines on the upper left panel of figure 4, one line ending at the P-ear(?) and the other ending at the tongue (see Figure 2 for definitions). A plane somewhere between these two might be considered an 'equatorial plane' only for morphological comparison with planetary nebulae. However, in the case of Puppis A, it is not an equatorial plane of a binary system. The planetary nebula M 1-30 belongs to a class of multipolar planetary nebulae that most researchers attribute to shaping by two or more pairs of jets. We claim the same for Puppis A. \nIn the lower panels of Figure 4, we present the planetary nebulae A 72, K 3-24, and NGC 650-1 originally from Manchado et al. (1996). The multipolar planetary nebulae K 3-24 and NGC 650-1 exhibit two pairs of ears, which we mark by pairs of light-blue lines and a dense equatorial plane between them (orange-dotted line). These features are also evident in SNR Puppis A, with a bright pair AA ' and two pairs more or less perpendicular to the dense strip, i.e., pairs BB ' and CC ' (for pair CC ' see Figure 1). The morphological similarities between Puppis A and these three planetary nebulae strengthen the claim that Puppis A was also shaped by jets. \nPuppis A texture is granular; tens of cells (or grains) appear on its surface. Several planetary nebulae have this texture. We present the planetary nebula A 72, which also exhibits granular texture. The similarity of the granular texture suggests that the granular texture is not the result of the explosion process or post-explosion radioactive heating (nickel bubbles). In planetary nebulae, the instability might result in the interaction of the main nebula with a previously ejected CSM or from a fast wind blown by the central star that accelerates the nebula. The same might hold in CCSNRs, where the ejecta interaction with a CSM might lead to granular texture, as the simulations by Orlando et al. (2022) suggest for Cassiopeia A. \n5. EARLY JET-DRIVEN NS KICK: THE KICK-BEAP MECHANISM \n<!-- image --> \n<!-- image --> \nFigure 3. Images of the SNR N49 emphasizing its large-scale dipole morphology resembling Puppis A. North is up and east to the left. Upper left: HST image (red: H α ; yellow: [S ii ]; blue: [O iii ]; from Bilikova et al. 2007), with our added marks. We point at three out of several filaments extending from the bright southeast side to the west. The letter P refers to the two protrusions. Upper right: A composite image from Ghavam et al. (2024): radio (5.5 GHz ATCA; red), optical (HST; green), and X-ray (Chandra; blue). Lower panels are from Zhou et al. (2019); the dashed circle is the outer boundary for their density calculation. The magnetar is not associated with N49 (see text). Note that the density and Ne distribution share the dipole axis; we mark the dipole axis with a dashed-orange line. The green double-line arrow is our speculation for the kick velocity direction of the NS (not found yet). \n<!-- image --> \n8 \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 4. Comparing the Puppis A's morphology to planetary nebulae. Upper left panel: X-ray image of SNR Puppis A (Credit # : NASA/CXC/IAFE/, Dubner et al. 2013; Arendt et al. 2010; for newer detailed SNR Puppis A morphology features, see Mayer et al. 2022). We added the cyan lines to mark the two pairs of ears AA ' and BB ' (see Figure 1). With dotted orange lines, we mark a plane for comparison with the equatorial plane of the planetary nebulae; one line ends at the P-ear, and the other ends at the tongue (see Figure 1). Upper middle and right panels: Two images of the multipolar planetary nebulae M 1-30 (PN G355 . 9 -04 . 2) and marks from Hsia et al. (2014); we added only the dotted-yellow line that connects the two jets which they marked. The position of the central star is marked with a black cross. We consider the plane that Hsia et al. (2014) marked by a black dotted line the equatorial plane. Hsia et al. (2014) marked the two pairs of ears by aa' and bb'. The lower panels are images of three different planetary nebulae from the IAC morphological catalog of northern Galactic planetary nebulae Manchado et al. 1996 & : PN A 72 (PN G059 . 7 -18 . 7), PN K 3-24 (PN G048 . 7 + 02 . 3), and NGC 650-1 (PN G130 . 9 -10 . 5). In the middle and right lower panels, we added cyan lines to mark pairs of ears and dotted-orange lines to mark a possible equatorial plane of each planetary nebula. \n<!-- image --> \n- # https://www.nasa.gov/image-article/unprecedented-x-ray-view-of-supernova-remains/\n- & Images taken from the PNIC catalog Balick 2006: https : // faculty . washington . edu / balick / PNIC / \nEarlier papers on the natal kick of the NS in the JJEM assumed that the tug-boat mechanism that operates in the delayed neutrino explosion mechanism (e.g., Scheck et al. 2004, 2006; Nordhaus et al. 2010, 2012; Wongwathanarat et al. 2010, 2013; Janka 2017) operates in the JJEM; there are other natal kick mechanisms, e.g., Yamasaki & Foglizzo 2008; Yao et al. 2021; Xu et al. 2022; for general discussion of natal kick velocities see, e.g., Igoshev 2020). We here also take the mechanism of the kick velocity to be hydrodynamical, as evident, for example, from the anti-correlation between the direction of motion of the primary X-ray emitting ejecta and the \nneutron star observed in Cassiopeia A (e.g., Hwang & Laming 2012). \nStudies of 14 CCSNe show that the distribution of angles between the jet-main axis in the frame of the JJEM and the NS natal kick avoids small angles (Bear & Soker 2018b, 2023; Soker 2022b). With the tug boat mechanism, this implies that the angle of the main-jet axis to the dense clump that accelerates the NS avoids small angles. The first possible explanation holds that the jets prevent the formation of dense clumps along their propagation directions, and therefore, no NS acceleration takes place in those directions. The second expla- \n<!-- image --> \nnation has several dense clumps that are falling towards the NS. Some clumps feed an accretion disk around the NS, and some escape and accelerate the NS by pulling it. This accretion disk launches jets along the angular momentum axis, perpendicular to the direction of the clumps' inflow, hence to the kick velocity. \nThe gravitational tug-boat mechanism operates for a relatively long time of several or more seconds. It converts the internal energy of the ejecta into kinetic energy of the ejecta, maintaining shell expansion (Wongwathanarat et al. 2013). Below, we suggest a very short mechanism to give the NS a kick velocity early in the explosion process. The tug-boat mechanism can operate later as well. \nThe characteristic lifetime of an intermittent accretion disk that launches a pair of jets in the JJEM is ≃ 0 . 01 -0 . 3 s (e.g., Soker 2024a), a time scale that is about equal or shorter than the viscous timescale of the accretion disk, which is its relaxation timescale (e.g., Soker 2024a). The implication is that the accretion disk has no time to fully relax. Considering that the gas that feeds the accretion disk has large fluctuations in its properties, the two sides of the accretion disk are born unequal and have no time to relax. Such a disk will likely launch two opposite jets unequal in power and opening angle (Soker 2024b). \nConsider the very early time after the shock bounces and stalls at r ≃ 100 -150 km s -1 . At the period of t b ≲ 0 . 2 s, where t b is the time measured from shock bounce, the mass accretion rate onto the very young central object, a bloated NS star, is ˙ M acc ≳ 1 M ⊙ s -1 (e.g., Muller et al. 2017; Burrows et al. 2024a). Due to its still lower mass and large radius, the escape velocity from the NS is lower than at the later phases of the explosion process, and we scale the jets' velocity by v j = 5 × 10 4 km s -1 ; this is an expected minimum velocity, as it might be somewhat larger, up to ≃ 10 5 km s -1 . Consider a short accretion period through an accretion disk that lives for ∆ t 1 ≃ 0 . 05 s. As discussed above, the disk doesn't have time to relax, and one jet can be much more powerful than the counter jet. We assume the more powerful jet carries much more momentum than the counter jet. This one jet carries a fraction f j1 ≃ 0 . 1 of the accreted mass, while the counter jet carries a fraction f j2 ≪ f j1 . The momentum and energy of the pair of jets are \np j12 = 250 ( f j1 -f j2 0 . 1 )( ∆ t 1 0 . 05 s ) ( ˙ M acc 1 M ⊙ s -1 ) × ( v j 5 × 10 4 km s -1 ) M ⊙ km s -1 , (1) \nand \nE j12 = 1 . 24 × 10 50 ( f j1 + f j2 0 . 1 )( ∆ t 1 0 . 05 s ) × ( ˙ M acc 1 M ⊙ s -1 ) ( v j 5 × 10 4 km s -1 ) 2 erg , (2) \nrespectively. \nThe observed NS kick velocity distribution ranges from very slow to v NS ≃ 1000 km s -1 , with two peaks at ≃ 80 km s -1 and ≃ 500 km s -1 (e.g., Igoshev 2020). For a typical NS mass of M NS = 1 . 4 M ⊙ the momenta of most NSs are in the range p NS ≃ 100 -700 M ⊙ km s -1 . The highly asymmetrical and powerful jet pair at a very early time can account for this natal kick; we term it the kick by early asymmetrical pairs (Kick-BEAP) mechanism. The tug-boat mechanism can act later and change the value of the kick velocity that the kick-BEAP mechanism imparted. \nOur suggested kick-BEAP mechanism imparts a large natal kick velocity to the NS at the beginning of the explosion process before the NS launches most jittering jets and therefore has the following properties and consequences. \n- 1. There might be two or even three pairs of jets that can impart large kick velocity at t b ≲ 0 . 2 s.\n- 2. Because the core is still intact, the strong early jets can lead to the nucleosynthesis of elements from silicon to iron group with high asymmetrical distribution in the opposite direction to the natal kick velocity. This is the focus of a future study.\n- 3. For the same reason, these early jets will all be choked inside the core and will not shape ears and bubbles in the CCSNR. Their marks are the kick velocity and the asymmetrical ejecta of some elements.\n- 4. The NS launches the later jets, including the last pairs of jets with large marks on the CCSNR morphology, while it is already moving relative to the center of the collapsing core. This can explain the avoidance of small angles between the main jet axis and the kick velocity. We elaborate on this below. \nConsider then that after the kick-BEAP phase, at t b ≳ 0 . 2 s, the NS moves at a velocity of v NS ≈ 50 -500 km s -1 relative to the center of mass of the pre-collapse core. We are interested in the final energetic pair of jets, the pair that is likely to shape the main jet axis (Soker 2024a). The mass that the NS accretes at the final accretion phases of the explosion \nprocess, about a second to a few after the shock bounce, originates from a radius of r f ≃ 3000 km (e.g., Shishkin & Soker 2021). The impact parameter, namely the distance of the accreted mass from the trajectory of the NS, is in the range 0 ≤ b ≤ r f . The specific angular moment that the kick velocity adds to an accreted parcel of gas with an impact parameter b is perpendicular to the kick velocity with a magnitude of \nj k = 3 . 75 × 10 15 ( b 1500 km ) ( v NS 250 km s -1 ) cm 2 s -1 . (3) \nThis value is of the order of magnitude of the specific angular momentum fluctuations in the core convective zones (e.g., Shishkin & Soker 2021, 2022). Therefore, ⃗ j k that is perpendicular to ⃗v NS prevents small angles between the kick velocity and the jet-main axis, and explains this finding (e.g., Bear & Soker 2023).", '6. SUMMARY': "We revealed a point-symmetric structure of three pairs of ears in the CCSNR Puppis A (Figure 1). The centers of the three symmetry lines that connect the two ears of each pair are within the general region between the present location of the NS remnant and its calculated location at the explosion time. Like with other CCSNRs with point-symmetric morphologies (Section 1), the point-symmetric morphology strongly suggests explosion by jittering jets, i.e., the JJEM; other shaping processes, like instabilities, interaction with a CSM and with the ISM, cannot account for the observed point-symmetric morphologies of CCSNRs (e.g., Soker & Shishkin 2024). In Section 4, we compared the pointsymmetric morphology of Puppis A with three multipolar planetary nebulae that researchers consider to have been shaped by two or more pairs of jets. The similarities between Puppis A's point-symmetric morphology and the three planetary nebulae (Figure 4) further solidify our claim that energetic jets shaped the pointsymmetric morphology of Puppis A. According to the JJEM, these jets are part of ≈ 10 -30 pairs of jets that exploded the progenitor of Puppis A. \nThe elongated morphological features of the 'corridor' and the line connecting the ejecta-rich clump with the 'tongue' (dashed lines on Figure 2) require deeper observations and analysis to explore their properties and whether jets also shaped their structures. \nIn this study, we also focused on the solid dipole structure of Puppis A. Its prominent components are the Xray and radio bright 'saddle' with a sharp edge on the northeast and the much fainter and diffuse other side. A possible ear, the P-ear(?), and the tongue (Figure 2) are the prominent components of the other dipole side of \nthe saddle. We notice a similar, but not identical, dipole structure in the SNR N49 (Section 3). In N49, filaments extend from the bright side of the dipole to its fainter opposite side (Figure 3). The filaments in N49 and the corridor in Puppis A that extend from one side to the other imply that an internal process shaped the dipole structures. A CSM or an ISM cloud on the dense side of an SNR can compress that side but cannot form filaments and structures extending to the other side. Also, the NS kick velocity in Puppis A is opposite to the side of the dense part, the saddle (Figure 2). \nBased on the dipole structure of Puppis A and its relation to the kick velocity direction, we proposed (Section 5) a mechanism in the frame of the JJEM to impart a natal kick to the NS. In this kick-BEAP (kick by early asymmetrical pair) mechanism, the very young NS, age of ≲ 0 . 2 s, launches a pair of jets where one jet is much more energetic than the counter jet. This takes place when the accretion rate is very high. Momentum conservation implies that the NS recoils in the opposite direction of the much more energetic jets. Such highly unequal jets in a pair in many, but not all, cases is one of the expectations of the JJEM (Soker 2024b). The more energetic jet in the pair compresses a dense side to the SNR while the NS acquires a natal kick velocity in the opposite direction. The kick-BEAP mechanism allows for the tug-boat mechanism, which occurs at a little later time in the explosion, to operate as well, but it is not necessary as the kick-BEAP might account for a large range of kick velocities (equation 1). \nOur study further supports the JJEM as the primary, or even sole, explosion mechanism of CCSNe and adds to the role of jets by introducing the novel kick-BEAP mechanism.", 'ACKNOWLEDGEMENTS': "A grant from the Pazy Foundation supported this research. \nWe thank Martin Mayer for advice regarding eROSITA data and useful comments. \nWe acknowledge the use of NASA's SkyView facility (https://skyview.gsfc.nasa.gov) located at NASA Goddard Space Flight Center. \nThis work made use of Astropy: 1 a communitydeveloped core Python package and an ecosystem of tools and resources for astronomy (Astropy Collaboration et al. 2013, 2018, 2022). \nThis work makes use of data from eROSITA, the soft X-ray instrument aboard SRG, a joint Russian-German \nscience mission supported by the Russian Space Agency (Roskosmos), in the interests of the Russian Academy of Sciences represented by its Space Research Institute (IKI), and the Deutsches Zentrum fur Luft- und Raumfahrt (DLR). The SRG spacecraft was built by Lavochkin Association (NPOL) and its subcontractors, and is operated by NPOL with support from the Max Planck Institute for Extraterrestrial Physics (MPE). The development and construction of the eROSITA X-ray instrument was led by MPE, with contributions from the Dr. Karl Remeis Observatory Bamberg & ECAP (FAU Erlangen-Nuernberg), the University of Hamburg Observatory, the Leibniz Institute for Astrophysics Potsdam (AIP), and the Institute for Astronomy and Astrophysics of the University of Tubingen, with the support of DLR and the Max Planck Society. The Argelander \nInstitute for Astronomy of the University of Bonn and the Ludwig Maximilians Universitat Munich also participated in the science preparation for eROSITA. \nThis publication makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. \nWe used the PNIC: Planetary Nebula Image Catalogue composed by Bruce Balick (http://faculty.washington.edu/balick/PNIC/) \nSoftware: Astropy (Astropy Collaboration et al. 2013, 2018, 2022), Matplotlib (Hunter 2007), NumPy (Harris et al. 2020)", 'REFERENCES': '- Wang, N. Y. N., Shishkin, D., & Soker, N. 2024, ApJ, 969, 163, doi: 10.3847/1538-4357/ad487f\n- Wang, T., & Burrows, A. 2024, ApJ, 969, 74, doi: 10.3847/1538-4357/ad5009\n- Wongwathanarat, A., Janka, H.-T., & Muller, E. 2010, ApJL, 725, L106, doi: 10.1088/2041-8205/725/1/L106\n- Wongwathanarat, A., Janka, H. T., & Muller, E. 2013, A&A, 552, A126, doi: 10.1051/0004-6361/201220636\n- Xu, F., Geng, J.-J., Wang, X., Li, L., & Huang, Y.-F. 2022, MNRAS, 509, 4916, doi: 10.1093/mnras/stab3342\n- Yamane, Y., Sano, H., van Loon, J. T., et al. 2018, ApJ, 863, 55, doi: 10.3847/1538-4357/aacfff\n- Yamasaki, T., & Foglizzo, T. 2008, ApJ, 679, 607, doi: 10.1086/587732\n- Yao, J., Zhu, W., Manchester, R. N., et al. 2021, Nature Astronomy, 5, 788, doi: 10.1038/s41550-021-01360-w\n- Zha, S., Muller, B., & Powell, J. 2024, arXiv e-prints, arXiv:2403.02072, doi: 10.48550/arXiv.2403.02072\n- Zhou, P., Vink, J., Safi-Harb, S., & Miceli, M. 2019, A&A, 629, A51, doi: 10.1051/0004-6361/201936002'}

2024A&A...690L...1O

Context. Magnetic wind braking drives the spindown of lowmass stars and the evolution of most interacting binary stars. A magnetic braking prescription that was claimed to reproduce both the period distribution of cataclysmic variables CVs and the evolution of the rotation rates of lowmass stars is based on a relation between the angular momentum loss rate and magnetic field complexity. Aims. The magnetic braking model based on field complexity has been claimed to predict a detached phase that could explain the observed period gap in the period distribution of CVs but has never been tested in detailed models of CV evolution. Here we fill this gap. Methods. We incorporated the suggested magnetic braking law in MESA and simulated the evolution of CVs for different initial stellar masses and initial orbital periods. Results. We find that the prescription for magnetic braking based on field complexity fails to reproduce observations of CVs. The predicted secondary star radii are smaller than measured and an extended detached phase that is required to explain the observed period gap a dearth of nonmagnetic CVs with periods between 2 and 3 hours is not predicted. Conclusions. Proposed magnetic braking prescriptions based on a relation between the angular momentum loss rate and field complexity are too weak to reproduce the bloating of donor stars in CVs derived from observations and in contrast to previous claims do not provide an explanation for the observed period gap. The suggested steep decrease in the angular momentum loss rate does not lead to detachment. Stronger magnetic braking prescriptions and a discontinuity at the fully convective boundary are needed to explain the evolution of close binary stars that contain compact objects. The tension between braking laws derived from the spindown of single stars and those required to explain CVs and other close binaries containing compact objects remains.

2024-10-01T00:00:00Z

['2024A&A...690L...1O', '10.48550/arXiv.2409.05673', '10.1051/0004-6361/202451829', '2024arXiv240905673O', 'arXiv:2409.05673']

['methods: numerical', 'binaries: close', 'stars: evolution', 'novae', 'cataclysmic variables', 'Astrophysics - Solar and Stellar Astrophysics']

Suggested magnetic braking prescription derived from field complexity fails to reproduce the cataclysmic variable orbital period gap

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']

https://arxiv.org/pdf/2409.05673.pdf

{'No Header': 'L etter to the E ditor', 'Suggested magnetic braking prescription derived from field complexity fails to reproduce the cataclysmic variable orbital period gap': 'Valentina Ortúzar-Garzón 1 , Matthias R. Schreiber 1 , and Diogo Belloni 1 \nDepartamento de Física, Universidad Técnica Federico Santa María, Avenida España 1680, Valparaíso, Chile \nReceived Accepted', 'ABSTRACT': 'Context. Magnetic wind braking drives the spin-down of low-mass stars and the evolution of most interacting binary stars. A magnetic braking prescription that was claimed to reproduce both the period distribution of cataclysmic variables (CVs) and the evolution of the rotation rates of low-mass stars is based on a relation between the angular momentum loss rate and magnetic field complexity. Aims. The magnetic braking model based on field complexity has been claimed to predict a detached phase that could explain the observed period gap in the period distribution of CVs but has never been tested in detailed models of CV evolution. Here we fill this gap. \nMethods. We incorporated the suggested magnetic braking law in MESA and simulated the evolution of CVs for di ff erent initial stellar masses and initial orbital periods. \nResults. We find that the prescription for magnetic braking based on field complexity fails to reproduce observations of CVs. The predicted secondary star radii are smaller than measured, and an extended detached phase that is required to explain the observed period gap (a dearth of non-magnetic CVs with periods between ∼ 2 and ∼ 3 hours) is not predicted. \nConclusions. Proposed magnetic braking prescriptions based on a relation between the angular momentum loss rate and field complexity are too weak to reproduce the bloating of donor stars in CVs derived from observations and, in contrast to previous claims, do not provide an explanation for the observed period gap. The suggested steep decrease in the angular momentum loss rate does not lead to detachment. Stronger magnetic braking prescriptions and a discontinuity at the fully convective boundary are needed to explain the evolution of close binary stars that contain compact objects. The tension between braking laws derived from the spin-down of single stars and those required to explain CVs and other close binaries containing compact objects remains. \nKey words. binaries: close - stars: evolution - stars: novae, cataclysmic variables - methods: numerical', '1. Introduction': 'Magnetic fields of low-mass stars force the mass lost in winds to co-rotate with the star up to the Alfvén radius. This causes the terminal specific angular momentum of the wind to be higher than the specific angular momentum of the stellar surface. The resulting angular momentum loss is called magnetic wind braking and represents a fundamental ingredient of stellar astrophysics. Magnetic braking drives the spin-down of single stars (Schatzman 1962; Mestel 1968) and the secular evolution of virtually all close binary stars (see Belloni & Schreiber 2023, for a recent review). Despite this importance, magnetic braking remains poorly understood. In particular, di ff erent prescriptions are used for single stars and stars in binaries. \nDecades ago, Skumanich (1972) found that the rotational periods ( P rot) of Sun-like stars are proportional to the square root of their age, which translates to an angular momentum loss of ˙ J ∝ MR 4 P -3 rot (with M and R representing the mass and the radius of the star). However, observations of chromospheric activity, coronal X-ray emission, flare activity, and magnetic field strengths in low-mass main-sequence stars reveal that these observables are all correlated and increase with rotation only up to a mass-dependent critical rotation rate. For shorter periods, the relation between activity and rotation saturates (e.g. Stau ff er et al. 1994; Reiners et al. 2009; Medina et al. 2020). The assump- \nn that these observables also relate with magnetic braking led to saturated magnetic braking prescriptions being postulated, in which the dependence of the magnetic braking torque on the spin period becomes shallower above a given rotation rate (e.g. Sills et al. 2000; Andronov et al. 2003). \nMore recent measurements of low-mass main-sequence stars in young clusters revealed a bimodal distribution of fast and slower rotation rates that is di ffi cult to explain with Skumanichlike or saturated magnetic braking prescriptions (e.g. Meibom et al. 2011; Newton et al. 2016). A relation between the strength of magnetic braking and the field complexity has been suggested by Garra ff o et al. (2016) to potentially solve this issue (Garra ff o et al. 2018, hereafter, CG18a). \nIn close binary stars with orbital periods shorter than ∼ 10 days, tidal forces cause the stellar rotation to be synchronised with the orbit (e.g. Levato 1974; Meibom et al. 2006; Fleming et al. 2019). A Skumanich-like magnetic braking prescription therefore predicts very strong orbital angular momentum loss in these close binaries (Rappaport et al. 1983). These high angular momentum loss rates for close binaries represent a key ingredient for the standard evolution theory of cataclysmic variables (CVs). According to this scenario, CVs with donor stars that still contain a radiative core experience strong angular momentum loss due to magnetic braking, which causes the donor stars to be bloated and the mass transfer rates to be high. \nWhen the secondary star becomes fully convective, at an orbital period of ∼ 3 hours, magnetic braking becomes much weaker, which allows the donor star to shrink and detach from its Roche lobe. The binary evolves as a detached system towards shorter periods until the mass transfer rate resumes at a much lower rate at a period of ∼ 2 hours. \nIn other words, to produce a detached phase that covers a period range from ∼ 2 to ∼ 3 hours, two conditions need to be fulfilled. First, above the gap, the donor star needs to be significantly oversized compared to its equilibrium radius. Second, at the fully convective boundary, the mass transfer timescale needs to become longer than the radius adjustment timescale of the donor star. In the standard scenario for CV evolution, the first condition requires strong magnetic braking above the gap, while the second condition is met by assuming a drastic decrease in magnetic braking at the fully convective boundary (see e.g. Belloni & Schreiber 2023, for more details). \nRecently, Schreiber et al. (2021b) showed that the late appearance of white dwarf magnetic fields (Bagnulo & Landstreet 2021), possibly related to a crystallisation-driven dynamo (Isern et al. 2017; Schreiber et al. 2021a, 2022; Ginzburg et al. 2022; Schreiber et al. 2023; Belloni et al. 2024a), a ff ects the evolution of CVs and should cause a reduction in magnetic braking if the field is strong enough. The evolution described above might thus fully apply only to CVs with weakly or non-magnetic white dwarfs. For CVs containing strongly magnetic white dwarfs, socalled polars, the white dwarf magnetic field can reduce the wind zones of the secondary star, thereby significantly reducing magnetic braking. This reduction has been predicted to cause a much less pronounced period gap, or none at all (see also Webbink & Wickramasinghe 2002; Belloni et al. 2020). \nThe evolution of CVs is observationally constrained by the measured orbital period distribution (Knigge et al. 2011; Inight et al. 2023a,b; Schreiber et al. 2024) and the mass transfer rate, which can be determined from the mass-radius relation of the donor stars (Knigge et al. 2011; McAllister et al. 2019) or the temperature of the white dwarf (Pala et al. 2017, 2022). The orbital period distribution of CVs containing weakly or nonmagnetic white dwarfs indeed shows a dearth of systems with periods between ∼ 2 and ∼ 3 hours. This period gap seems to be at least much less pronounced (maybe even absent) for CVs containing strongly magnetic white dwarfs (Schreiber et al. 2024). The mass transfer rates of CVs measured from donor star radii are significantly higher at periods longer than three hours (Knigge et al. 2011; McAllister et al. 2019). These observations are roughly consistent with the mass transfer rates determined from white dwarf temperatures (Pala et al. 2022). \nThese observational constraints support the idea that the field of a strongly magnetic white dwarf reduces magnetic braking. For non-magnetic CVs, the observations agree reasonably well with predictions made assuming a Skumanich-like magnetic braking for donor stars that still contain a radiative core, and significantly weaker magnetic braking for fully convective stars (Knigge et al. 2011; Belloni et al. 2018). \nHowever, CV evolution has turned out to be di ffi cult to explain when applying more recent prescriptions derived for single stars. In an attempt to unify magnetic braking prescription for single stars and close binaries, Garra ff o et al. (2018, hereafter, CG18b) applied their magnetic braking prescription based on field complexity to the evolution of CVs and claimed that it is possible to reproduce the observed orbital period distribution. If true, this result would greatly reduce the tension between magnetic braking prescriptions derived for single stars and those used in binary evolution studies. \nHere we use detailed binary calculations to show that the magnetic braking prescription suggested by CG18b can explain neither the existence of the period gap for CVs containing weakly or non-magnetic white dwarfs nor the large radii derived from observations of such CVs above the gap. We then discuss possible avenues towards a unified magnetic braking prescription.', '2. Binary star simulations': "We used the MESA code (Paxton et al. 2011, 2013, 2015, 2018, 2019; Jermyn et al. 2023), version r-23.03.1, to compute the evolution of CVs 1 . The MESA equation of state is a blend of the OPAL (Rogers & Nayfonov 2002), SCVH (Saumon et al. 1995), FreeEOS (Irwin 2004), HELM (Timmes & Swesty 2000), PC (Potekhin & Chabrier 2010), and Skye (Jermyn et al. 2021) equations of state. Radiative opacities are primarily from OPAL (Iglesias & Rogers 1993, 1996), with low-temperature data from Ferguson et al. (2005) and the high-temperature, Comptonscattering dominated regime from Poutanen (2017). Electron conduction opacities are from Cassisi et al. (2007) and Blouin et al. (2020). Nuclear reaction rates are from JINA REACLIB (Cyburt et al. 2010), NACRE (Angulo et al. 1999), and additional tabulated weak reaction rates (Fuller et al. 1985; Oda et al. 1994; Langanke & Martínez-Pinedo 2000). Screening is included via the prescription of Chugunov et al. (2007). Thermal neutrino loss rates are from Itoh et al. (1996). Roche lobe radii in binary systems are computed using the fit of Eggleton (1983). Mass transfer rates in Roche lobe overflowing binary systems are determined following the prescription of Ritter (1988). \nIn our simulation we furthermore assumed the white dwarf mass to be constant, that is, that the same amount of mass that is accreted during a nova cycle is expelled during the eruption, in rough agreement with model predictions (Yaron et al. 2005). The mass expelled during nova eruptions was assumed to carry the specific angular momentum of the white dwarf. While this likely underestimates the true value for CVs containing lower-mass white dwarfs (Schreiber et al. 2016), using this form of consequential angular momentum loss should not lead to significantly di ff erent predictions for the secular evolution of most CVs, which contain relatively massive white dwarfs ( > ∼ 0 . 8M ⊙ ). Our simulations take into account angular momentum loss through gravitational radiation according to Paczy'nski (1967). The orbital angular momentum loss through magnetic braking suggested by CG18a is summarised in what follows.", '3. Magnetic braking and field complexity relations revisited': 'The magnetic braking prescription we tested is based on a relation between angular momentum loss and the complexity of the magnetic field. This relation has been claimed to explain not only the spin-down rates of single stars (CG18a) but also the CV orbital period distribution (CG18b). This prescription for angular momentum loss through magnetic braking ( ˙ J MB), developed initially by Garra ff o et al. (2016), can be written as follows: \n˙ J MB = ˙ J Dip Q J( n ) , (1) \nwhere \nQ J( n ) = 4 . 05 e -1 . 4 n + ( n -1) / (60 Bn ) , (2) \nwith B representing the field strength. The dipole angular momentum loss rate is given by \n˙ J Dip = Ω 3 c τ, (3) \nwhere Ω = 2 π/ P orb is the angular velocity, and c is a constant related to the wind e ffi ciency and is assumed to be 1 × 10 41 g cm 2 (CG18a). Finally, the field complexity parameter, n , is defined as \nn = a τ P rot + b P rot τ + 1 . (4) \nThe parameters a and b were set to 0 . 02 and 2 in CG18a, respectively. However, when applying their model to CVs, slightly di ff erent values ( a = 0 . 01 and b = 1) were adopted (CG18b, their Sect. 2) with the justification that the parameter values were not well constrained in their previous study, which only dealt with stars with masses greater than 0 . 3 M ⊙ . As the aim of this Letter is to test the magnetic braking prescription that was suggested to reproduce the orbital period distribution of CVs, we used a = 0 . 01 and b = 1. \nFor close binary stars, the rotational period, P rot, is equal to the orbital period, P orb (i.e. the orbit is synchronised). The dependence of the convective turn-over time on the stellar mass is given by the empirical relation of Wright et al. (2011): \nlog( τ ) = 1 . 16 -1 . 49 log( M / M ⊙ ) -0 . 54 log 2 ( M / M ⊙ ) . (5) \nAdmittedly, this frequently used empirical dependence might need to be updated using larger, recently established datasets (Jao et al. 2022). However, given that we want to test the prescription calibrated by CG18b for CVs, we followed them in adopting the above relation. We furthermore note that this empirical convective turn-over time is di ff erent from the calculated global convective turn-over time used in alternative magnetic braking prescriptions (e.g. Van & Ivanova 2019, their Eq. 16). \nApplying their magnetic braking algorithm to CVs, CG18b dropped the term that depends on the field strength in Eq. 3 as it is negligible as long as n remains below 7 , which is the case for the parameter space relevant to the orbital period gap in CVs. Combining the above equations then results in a largely simplified angular momentum loss prescription through magnetic braking: \n˙ J MB = -4 . 05 e -1 . 4 n c Ω 3 τ. (6) \nTo simulate CV evolution, CG18b then assumed a massradius relation for the donor stars in CVs derived from observations (Knigge et al. 2011). This last step of their procedure is potentially problematic because it swaps cause and e ff ect. In CVs, magnetic braking causes angular momentum loss, which drives the mass transfer, and it is this mass loss of the donor star that determines the mass-radius relation of the donor star. In other words, angular momentum loss (largely through magnetic braking) determines the mass-radius relation of CV donors, which therefore cannot be assumed a priori.', '4. CV evolution driven by magnetic braking related to field complexity': 'We show in Fig. 1 the dependence of the field complexity parameter and the convective turn-over time on the orbital period assuming the CG18b magnetic braking prescription for a white dwarf mass of 0.83 M ⊙ and an initial donor mass of 0.65 M ⊙ . \nFig. 1. Comparison of the turn-over time, τ, evolution (upper panel) based on the empirical relation proposed by (Wright et al. 2011, solid orange line) and a constant value of 100 days (dashed black line) and the corresponding magnetic field complexities, n (lower panel). The shaded region corresponds to the period gap for non-polar CVs according to Schreiber et al. (2024). The dashed grey vertical line indicates the lower period gap edge according to Knigge et al. (2011). The assumption of a constant turn-over time is not justified, and it is incompatible with the condition n < 7. \n<!-- image --> \nThe bottom panel confirms that n indeed remains below ∼ 7 beyond the period gap if the formula from Wright et al. (2011) is used for the turn-over time. However, CG18b state that they obtained qualitatively identical results by assuming a constant convective turn-over time of 100 days. This is not the case in our simulations. The assumption that n remains below 7, and that the dropped term in Eq. 3 is thus indeed negligible, cannot be justified. \nIn Fig. 2 we show the relative angular momentum loss we obtain using the magnetic braking prescription of CG18b. The strength of angular momentum loss is similar to theirs if the dependence of the convective turn-over time on mass (Wright et al. 2011) is taken into account. For a constant value, a convective turn-over time of 100 days, the angular momentum loss rate becomes much lower, clearly inadequate to describe CV evolution. \nIn Fig. 3 we show the evolution of the donor star radius and mass for a white dwarf mass of 0.9 M ⊙ , which corresponds roughly to the mean white dwarf mass in CVs (Zorotovic et al. 2011; Pala et al. 2022), and an initial donor star mass of 0.85 M ⊙ , assuming the magnetic braking prescription based on field complexity (CG18b). The initial orbital period was assumed to be 12 hours. Also shown is the mass-radius relation derived by Knigge et al. (2011) from observations of CVs. The predicted radii are significantly smaller than those derived from observations, and the donor star does not su ffi ciently shrink to detach from its Roche lobe. Instead, the donor star mass decreases continuously. \nFigure 4 shows the evolution of the donor mass, the rate of angular momentum loss due to magnetic braking, and the mass transfer rate as a function of the orbital period for di ff erent white dwarf and donor star masses. The resulting accretion rates in the considered period range are roughly of the same order as the observed ones (Pala et al. 2022; Dubus et al. 2018), but a detached phase that could explain the orbital period gap (Knigge et al. 2011; Schreiber et al. 2024) is not predicted. In contrast to \nFig. 2. Relative angular momentum loss of CVs with M 1 = 0 . 83 M ⊙ and M 2 = 0 . 65 M ⊙ , with a constant τ = 100 days (dashed black line) calculated using the empirical relation from (Wright et al. 2011, solid orange line). The shaded region corresponds to the period gap for non-polar CVs according to Schreiber et al. (2024). The dashed grey vertical line indicates the lower period gap edge according to Knigge et al. (2011). \n<!-- image --> \nFig. 3. Donor star radius ( R 2) as a function of its mass ( M 2) according to Knigge et al. (2011) (dashed red line) compared to the evolution predicted by MESA for a CV with initial M 1 = 0 . 9 M ⊙ and M 2 = 0 . 85 M ⊙ and assuming magnetic wind braking as in CG18b (solid black line). The radii and masses of CV donors derived from observations (McAllister et al. 2019) are shown as blue dots with their respective error bars. It is clear that the prescription based on the complexity of the field fails to reproduce the observations. \n<!-- image --> \nthe claims by CG18b, our detailed simulations of CV evolution show that the steep decrease in angular momentum loss through magnetic braking predicted by their prescription does not cause the systems to detach, and the orbital period gap thus remains unexplained.', '5. Concluding discussion': 'Assuming that the e ffi ciency of magnetic braking is related to the magnetic field complexity is very reasonable because the rate of magnetic wind braking must be related to the number of open field lines. In particular, given that the field strength saturates in a similar fashion for fully convective stars and those that still contain a radiative core (Wright & Drake 2016), a dependence of angular momentum loss through magnetic braking on field complexity could represent an elegant way to reproduce the orbital gap. This idea goes back several decades (Taam & Spruit 1989). Finding a prescription for magnetic braking that depends on field complexity and that can reasonably well explain the spin-down of single low-mass stars and key observables \nFig. 4. Evolution of the donor mass (upper panel), angular momentum loss rate (central panel), and mass transfer rate (lower panel) for CVs with M 1 = 0 . 9 M ⊙ , M 2 = 0 . 42 M ⊙ (dashed blue line), M 1 = 0 . 83 M ⊙ , M 2 = 0 . 65 M ⊙ (solid orange line), and M 1 = 0 . 9 M ⊙ , M 2 = 0 . 85 M ⊙ (dash-dotted pink line). The angular momentum loss rate is calculated as in CG18b. The shaded region corresponds to the period gap according to Schreiber et al. (2024). The dashed grey vertical line indicates the lower period gap edge according to Knigge et al. (2011). For the lowestmass donor (blue track), the onset of mass transfer occurs close to the upper edge of the period gap, and the binary evolves through a wellknown loop (often called the period flag) at the onset of mass transfer (e.g. Stehle et al. 1996, their Fig. 4). As the donor stars do not detach from their Roche lobe, the considered magnetic braking model can not explain this crucial feature in the observed period distribution. \n<!-- image --> \nof CVs would therefore represent a significant step forwards in our understanding of magnetic braking. This is exactly what has been attempted by CG18b, and their results seemed promising. \nHowever, instead of modelling the detailed evolution of the donor star, CG18b assumed a mass-radius relation similar to those derived from observations of CVs (Knigge et al. 2011), which might not represent the mass-radius relation predicted by their angular momentum loss prescription. It is important to note that CG18b were well aware of the limitations of their simulations and stated that more detailed simulations were required. \nHere we filled this gap by implementing their prescription into the stellar evolution code MESA and simulating evolutionary tracks of CVs. We find that the magnetic braking model from CG18b does not allow fundamental observables of CVs to be re- \nproduced, such as the donor star radii and the famous orbital period gap. The di ff erences between our results and those of CG18b are most likely related to the assumed mass-radius relation. Detailed stellar evolution calculations are required to obtain the mass-radius relation for a given prescription of angular momentum loss through magnetic braking since the expansion of the donor star is driven by mass loss, which in turn is driven by angular momentum loss. In our detailed simulations, the magnetic braking prescription proposed by Garra ff o et al. (2015) and CG18a does not produce su ffi ciently bloated donor stars, and the steep decrease in their angular momentum loss rate is not sufficient to cause a detached phase. A more drastic reduction in magnetic braking is required to explain the observed orbital period distribution of CVs. \nThis result emphasises the previously mentioned tension (e.g. Knigge et al. 2011; Belloni et al. 2024b; Schreiber et al. 2024) between braking laws that describe the rotational evolution of single low-mass stars (Matt et al. 2015; Sills et al. 2000; Andronov et al. 2003; Garra ff o et al. 2018) and those used to reproduce the evolution of close compact binaries ranging from CVs to AMCVn stars (Belloni & Schreiber 2023), low-mass Xray binaries (Van & Ivanova 2019), or detached white dwarf plus M-dwarf post common envelope binaries (Schreiber et al. 2010). While relatively weak saturated magnetic braking seems to do relatively well in describing the evolution of single stars, much higher angular momentum loss rates are needed to explain the observations of close compact binaries. Apart from the di ff erent strengths of magnetic braking, a drastic decrease (often assumed to be a discontinuity) at the fully convective boundary is required to explain the CV orbital period gap, and such a discontinuity is usually not incorporated into saturated magnetic braking laws. \nGiven the recent evidence of saturated magnetic braking in close main-sequence binaries (El-Badry et al. 2022) and observations of detached white dwarf plus main-sequence binaries (Schreiber et al. 2010), which indicate a drastic change at the fully convective boundary similar to that required to explain the CV orbital period gap, Belloni et al. (2024b) developed a disrupted (at the fully convective boundary) and saturated prescription that could simultaneously explain observations of both types of detached binaries. A natural next step is to test the very same model for CVs. \nWhile this disrupted and saturated magnetic braking prescription represents a promising candidate to describe binary star evolution, one important piece of the puzzle, an explanation for the dramatically di ff erent angular momentum loss rates through magnetic braking in single stars and close binaries, is obviously missing. Perhaps the possibility of tidally enhanced mass loss, which is required to explain a number of observed binary star systems (Tout & Eggleton 1988), should be investigated in more detail. \nAcknowledgements. MRS and DB thank for support from FONDECYT (grant numbers 1221059 and 3220167). VOG and MRS thank C. Garra ff o and J. Drake for their insightful feedback. We also thank O. 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2024ApJ...968...38K

Observations with the James Webb Space Telescope JWST have uncovered numerous faint active galactic nuclei AGN at z 5 and beyond. These objects are key to our understanding of the formation of supermassive black holes SMBHs their coevolution with host galaxies as well as the role of AGN in cosmic reionization. Using photometric colors and size measurements we perform a search for compact red objects in an array of blank deep JWSTNIRCam fields totaling 640 arcminSUP2SUP. Our careful selection yields 260 reddened AGN candidates at 4 lt z SUBphotSUB lt 9 dominated by a pointsourcelike central component r SUBeffSUB lt 130 pc and displaying a dichotomy in their restframe colors blue UV and red optical slopes. Quasar model fitting reveals our objects to be moderately dustextincted A SUBVSUB 1.6 which is reflected in their inferred bolometric luminosities of L SUBbolSUB 10SUP4447SUP erg sSUP1SUP and fainter UV magnitudes M SUBUVSUB 17 to 22. Thanks to the large areas explored we extend the existing dusty AGN luminosity functions to both fainter and brighter magnitudes estimating their number densities to be 100 higher than for UVselected quasars of similar magnitudes. At the same time they constitute only a small fraction of all UVselected galaxies at similar redshifts but this percentage rises to 10 for M SUBUVSUB 22 at z 7. Finally assuming a conservative case of accretion at the Eddington rate we place a lower limit on the SMBH mass function at z 5 finding it to be consistent with both theory and previous JWST observations.

2024-06-01T00:00:00Z

['2024arXiv240109981K', '10.48550/arXiv.2401.09981', 'arXiv:2401.09981', '10.3847/1538-4357/ad4265', '2024ApJ...968...38K']

['Galaxies', 'High-redshift galaxies', 'Active galaxies', 'Active galactic nuclei', '573', '734', '17', '16', 'Astrophysics - Astrophysics of Galaxies']

A Census of Photometrically Selected Little Red Dots at 4 lt z lt 9 in JWST Blank Fields

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']

https://arxiv.org/pdf/2401.09981.pdf

{'No Header': '7', 'A Census of Photometrically Selected Little Red Dots at 4 < z < 9 in JWST Blank Fields': "Vasily Kokorev, 1 Karina I. Caputi, 1, 2 Jenny E. Greene, 3 Pratika Dayal, 1 Maxime Trebitsch, 1 Sam E. Cutler, 4 Seiji Fujimoto, 5, 2 Ivo Labb'e, 6 Tim B. Miller, 7 Edoardo Iani, 1 Rafael Navarro-Carrera, 1 and Pierluigi Rinaldi 1 \n1 \nKapteyn Astronomical Institute, University of Groningen, 9700 AV Groningen, The Netherlands 2 Cosmic Dawn Center (DAWN), Niels Bohr Institute, University of Copenhagen, Jagtvej 128, København N, DK-2200, Denmark 3 Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08544 4 Department of Astronomy, University of Massachusetts, Amherst, MA 01003, USA 5 Department of Astronomy, The University of Texas at Austin, Austin, TX 78712, USA 6 Centre for Astrophysics and Supercomputing, Swinburne University of Technology, Melbourne, VIC 3122, Australia Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA), Northwestern University, 1800 Sherman Ave, Evanston IL 60201, USA \n(Received n/a; Revised n/a; Accepted n/a) \nSubmitted to ApJ", 'ABSTRACT': 'Observations with the James Webb Space Telescope (JWST) have uncovered numerous faint active galactic nuclei (AGN) at z ∼ 5 and beyond. These objects are key to our understanding of the formation of supermassive black holes (SMBHs), their co-evolution with host galaxies, as well as the role of AGN in cosmic reionization. Using photometric colors and size measurements, we perform a search for compact red objects in an array of blank deep JWST/NIRCam fields totaling ∼ 640 arcmin 2 . Our careful selection yields 260 reddened AGN candidates at 4 < z phot < 9, dominated by a point-source like central component ( ⟨ r eff ⟩ < 130 pc) and displaying a dichotomy in their rest-frame colors (blue UV and red optical slopes). Quasar model fitting reveals our objects to be moderately dust extincted ( A V ∼ 1 . 6), which is reflected in their inferred bolometric luminosities of L bol = 10 44 -47 erg/s, and fainter UV magnitudes M UV ≃ -17 to -22. Thanks to the large areas explored, we extend the existing dusty AGN luminosity functions to both fainter and brighter magnitudes, estimating their number densities to be × 100 higher than for UV-selected quasars of similar magnitudes. At the same time they constitute only a small fraction of all UV-selected galaxies at similar redshifts, but this percentage rises to ∼ 10% for M UV ∼ -22 at z ∼ 7. Finally, assuming a conservative case of accretion at the Eddington rate, we place a lower limit on the SMBH mass function at z ∼ 5, finding it to be consistent with both theory and previous JWST observations. \nKeywords: Active galactic nuclei (16), High-redshift galaxies (734), Early universe (435)', '1. INTRODUCTION': "The remarkable sensitivity and angular resolution of the James Webb Space Telescope ( JWST ) at infrared wavelengths is enabling us to explore the distant Universe like never before. This allows for an exceptionally detailed examination of the characteristics of known highz sources (e.g. Bunker et al. 2023; Maiolino et al. \nCorresponding author: Vasily Kokorev \nkokorev@astro.rug.nl \n2023a) and, at the same time, reveals the presence of more and farther galaxies (e.g. Adams et al. 2023; Atek et al. 2023; Austin et al. 2023; Bradley et al. 2023; Casey et al. 2023; Finkelstein et al. 2023; Naidu et al. 2022; Robertson et al. 2023), some of them spectroscopically confirmed beyond z > 13 (Curtis-Lake et al. 2023; Wang et al. 2023). \nWhat truly tests our models and preconceived vision of galaxy evolution is not how early we can see these objects, but the questions they raise regarding the balance between their mass, UV luminosity and age. The excess of highz galaxies at the bright end ( M UV ≤ -20) of the \nUV luminosity function is in tension with current theoretical frameworks (Behroozi & Silk 2015; Dayal et al. 2017; Yung et al. 2019, 2020; Behroozi et al. 2019, 2020; Dav'e et al. 2019; Wilkins et al. 2022; Kannan et al. 2023; Mason et al. 2023; Mauerhofer & Dayal 2023), which suggests exotic initial mass functions, little to no dust attenuation, or a higher than anticipated density of galaxies undergoing active galactic nuclei (AGN) phenomena (e.g. Finkelstein & Bagley 2022; Pacucci et al. 2022; Boylan-Kolchin 2023; Ferrara et al. 2023; Fujimoto et al. 2023a; Lovell et al. 2023; Steinhardt et al. 2023; Sun et al. 2023). \nAlthough early hints also existed in prior works (Morishita et al. 2020; Fujimoto et al. 2022; Endsley et al. 2023), one of the most intriguing discoveries from early JWST imaging is that of compact red sources with a 'v-shaped' spectral energy distribution (SED), namely a blue UV continuum and a steep red slope in the restframe optical (Labb'e et al. 2023a,b; Furtak et al. 2023a). While the first photometric selections of these objects included spatially resolved targets that could be early massive compact galaxies (Barro et al. 2023), spectra revealed clear evidence for broad H α and/or H β emission indicative of actively accreting supermassive black holes (SMBH; Furtak et al. 2023b; Fujimoto et al. 2023a; Greene et al. 2023; Killi et al. 2023; Kocevski et al. 2023; Kokorev et al. 2023a; Matthee et al. 2023; Ubler et al. 2023). \nDubbed 'little red dots' (LRDs), these sources have SEDs characterized by a unique 'v-shaped' continuum combined with their point source morphology (Labb'e et al. 2023a,b; Furtak et al. 2023a). However, what truly makes the LRDs stand out is their high number densities. It appears that LRDs may account for a few percent of the galaxy population at z > 5, and are far more numerous than the lowest luminosity known UV-selected quasars. Likewise, they appear to account for ∼ 20% of broad-line selected active galactic nuclei (AGN) at z ∼ 5 -6 (Greene et al. 2023; Harikane et al. 2023; Labb'e et al. 2023b; Maiolino et al. 2023b), which is higher than the fraction of dusty red quasars at z < 2 (Banerji et al. 2015; Glikman et al. 2015). These red dots are generally observed at z ∼ 5 (Labb'e et al. 2023b), but can potentially be found even at z > 9 (Leung et al. 2023). However, these initial LRD studies were performed with limited spectroscopic samples and/or small areas of the sky, covering only ∼ 20 - 40 arcmin 2 . The numbers of compact red objects could therefore be further affected by cosmic variance, which makes it quite difficult to assess their real importance and diversity. \nExtending the selection of this compact red population of low-luminosity broad-line AGN candidates to \nlarger areas would thus be necessary to study their complete demographics, limiting the effects of cosmic variance. In addition, this would provide us with a sufficient level of detail toward a better understanding of the total number densities of obscured AGN at highz as well as the potential role that these sources play in cosmic reionization (e.g. see Grazian et al. 2018; Mitra et al. 2018; Dayal et al. 2020; Trebitsch et al. 2023; Dayal et al. 2024). \nIn this work we present a carefully selected sample of 260 reddened AGN candidates in the ∼ 640 arcmin 2 area covering some of the deepest blank extragalactic JWST fields. Examining such a large area will ensure that we are reducing the effects of cosmic variance to a minimum, while our focus on blank fields lessens the selection biases and avoids volume uncertainties arising from lensing magnification. \nThroughout this work we assume a flat ΛCDM cosmology (e.g. Planck Collaboration et al. 2020) with Ω m , 0 = 0 . 3, Ω Λ , 0 = 0 . 7 and H 0 = 70 km s -1 Mpc -1 , and a Chabrier (2003) initial mass function (IMF) between 0 . 1 -100 M ⊙ . All magnitudes are expressed in the AB system (Oke 1974).", '2. OBSERVATIONS AND DATA': 'In this work we use JWST data from the following programs/fields - CEERS (# 1345, PI: S. Finkelstein, Bagley et al. 2022) in the EGS, PRIMER (# 1837, PI: J. Dunlop) in COSMOS and UDS. For the GOODSS we combine the available data from multiple broad and medium band programs - FRESCO (# 1895, PI: P. Oesch, Oesch et al. 2023), JADES (# 1180, 1210, 1286, 1287 PIs: D. Eisenstein, N. Luetzgendorf, Eisenstein et al. 2023a,b) and JEMS (# 1963, PI: C. Williams, Williams et al. 2023b). We provide a general overview of these 4 fields in Table 1. More detailed information, including specific filters, depths and survey designs can be found in overview papers for each data release.', '2.1. JWST Imaging Data Reduction': "We homogeneously processed all the publicly available JWST imaging obtained with the NIRCam and MIRI in a variety of public JWST fields, presented in Table 1. The images have all been reduced with the grizli pipeline (Brammer 2023), using the jwst 1084.pmap , and follow the same methodology of (multiple) previous studies (e.g., Jin et al. 2023; Kokorev et al. 2023b; Valentino et al. 2023). Compared to the standard pipeline, we incorporate additional corrections to account for cosmic rays and stray light (see e.g., Bradley et al. 2023), 1/f noise, detector level artifacts ('wisps' and 'snowballs') and bias in individual exposures (see \nFigure 1. Selection and analysis of LRD candidates. Top: Sample selection criteria. The left and central panels show modified ' red 1 ' ( z ≲ 6) and ' red 2 ' ( z ≳ 6) color-color cuts from Labb'e et al. (2023b). The right panel shows the compactness cut of our sample. Selected objects are highlighted as maroon circles, while grayscale hexbins show the full catalog. The compact red sources are clear outliers in color-color-compactness space. Colorbar is shared between all plots. Bottom: An example of best-fit SEDs to the photometry of LRD candidates with the dust-free (blue) and dusty (red) AGN templates (Vanden Berk et al. 2001; Glikman et al. 2006) at representative redshifts of z ∼ 6 and z ∼ 8. The combined model is shown in black. Detections ( > 3 σ ) are shown as red circles, while upper limits (primarily from HST ) are shown as downward arrows. On the right of each SED we show 1 . '' 5 color composite cutouts in the short (F115W/F150W/F200W) and long (F277W/F356W/F444W) NIRCam filters. \n<!-- image --> \ne.g., Rigby et al. 2023). For the PRIMER data, we introduce an additional procedure that alleviates the detrimental effects of the diagonal striping seen in some exposures as was done in Valentino et al. (2023). Finally, our mosaics include the updated sky flats for all NIRCam filters. These reductions are publicly available as a part of the DAWN JWST Archive (DJA 1 ). \nThese data-sets are further complemented by including all available optical and near-infrared data from the Complete Hubble Archive for Galaxy Evolution (CHArGE, Kokorev et al. 2022). Individual JWST and HST exposures were aligned to the same astrometric \nreference frame by using the Gaia DR3 (Gaia Collaboration et al. 2021), then co-added and drizzled (Fruchter & Hook 2002) to a 0 . '' 04 pixel scale for all the JWST and HST filters. \nSome of the fields we examine in this work have also been observed with MIRI, in one or more filters, sampling mostly the rest-frame near-infrared (NIR) at z ≳ 4. These data are, however, not uniform in the wavelength coverage, depth and area. In fact, only about a third of the objects in areas we examine have public MIRI data and even fewer are actually detected. While the inclusion of the MIRI photometry can assist in further identifying the presence (or absence) of dusty, power law-like AGN component in galaxies (e.g. see Yang et al. 2023; Williams et al. 2023a), doing so appro- \niately within a context of a population study requires a degree of uniformity which the current MIRI data do not possess. Therefore, we have opted to exclude MIRI photometry from our current analysis to maintain consistency across various fields.", '2.2. Source Extraction': "The initial JWST catalog was constructed by utilizing a detection image combined from all noise weighted 'wide' (W) NIRCam Long Wavelength (LW) filters available, which includes F277W, F356W and F444W. A similar detection method was already successfully employed in several works (see e.g. Jin et al. 2022; Kokorev et al. 2023b; Valentino et al. 2023; Weaver et al. 2023a). To extract the sources and produce a segmentation map, we used sep (Barbary 2016), a Python version of SExtractor (Bertin & Arnouts 1996). Photometry was extracted in circular apertures of increasing size. Correction from the aperture to the 'total' values was performed by using the flux auto column output by sep , which is equivalent to the MAG AUTO from SExtractor , ensuring that for each source only flux belonging to its segment is taken into account. This method was shown to apply to both point-like and extended objects (Weaver et al. 2022, 2023b), so we believe it to be adequate for our sources. \nAdditionally, we introduce a correction to account for the missing flux outside the Kron aperture (Kron 1980), by utilizing a method similar to the one used in Whitaker et al. (2011) and Weaver et al. (2023a). In short, this procedure involves computing the fraction of the missing light outside the circularized Kron radius by analyzing curves of growth of the point spread functions (PSF), which were obtained empirically, by stacking stars in these various fields. This correction is then applied to the flux auto values for each source. However, since our work focuses on compact (sub NIRCam PSF size) AGN candidates this additional correction does not strongly influence the derived flux densities. For the same reason, we use the total fluxes, computed from D =0 . '' 36 apertures, unless specified otherwise.", '3. IDENTIFYING COMPACT RED OBJECTS': 'The data from CEERS, PRIMER and various programs covering GOODS-S are well-suited for a photometric search for compact obscured AGN candidates. The available HST and JWST photometry covers a complete wavelength range from 0.4 - 5 µ m, in at least 7 broad and medium bands, reaching a median 5 σ depth of 28.3 AB mag in F444W filter (Table 1). In our search, we explore blank fields covering a large area of ∼ 640 arcmin 2 , which are also completely independent. In return this will significantly limit the impact of cosmic \nFigure 2. Top: Distribution of the L bol assuming an AGN dominated rest-frame optical continuum, with best-fit dust correction applied. Our compact sources span a wide range of luminosities across the redshift range of interest. We contrast our results to the BL AGN from Greene et al. (2023, blue squares), Matthee et al. (2023, orange circles), z > 6 . 5 quasars from Yang et al. (2021, gray crosses), highz AGN from Maiolino et al. (2023b, magenta circles) and finally objects hosting Type 2 AGN from Scholtz et al. (2023, light blue circles). Assuming λ edd = 1, we show what L bol would correspond to M BH = 10 6 -8 M ⊙ (dashed lines). Bottom: Same as before, but we show the distribution of L bol without correcting for the best-fit dust-attenuation. Our final sample has a median A V ∼ 1 . 6 +1 . 1 -1 . 0 . On the side of each panel we also show histograms highlighting the redshift and L bol distributions. \n<!-- image --> \nvariance and enable us to avoid dealing with the cosmic volume uncertainties introduced by the lensing magnification.', '3.1. Color and Morphology Selection': "Recently, Labb'e et al. (2023b) published a large sample of photometrically identified compact red sources from the Cycle 1 JWST UNCOVER program (PIs: I. Labb'e, R. Bezanson; Bezanson et al. 2022). Subsequent follow-up of 17 such objects with NIRSPec/MSA \nTable 1. Properties of the observed fields with JWST /NIRCam observations. \nNote -NIRCam depths: expressed as 5 σ within the 0 . '' 36 apertures used for the photometric extraction in the area covered by F115W/F150W/F200W/F277W/ F356W/F444W. \nPRISM has resulted in a remarkable success rate of 83 %, with 14/17 photometrically selected targets confirmed as broad-line (BL) AGN at 4 < z < 8 . 5 (Greene et al. 2023), and 3/17 as brown dwarfs (Burgasser et al. 2023). In brief, the color cuts introduced in Labb'e et al. (2023b) are designed to catch the break between the red continuum slope in rest-frame optical, and the blue rest-UV emission ( λ rest ∼ 4000 ˚ A). This color selection requires that the red continuum slope is rising in more than one adjacent filter pair, to avoid selecting galaxies with strong emission lines. Indeed, currently available spectra of LRDs (e.g. Fujimoto et al. 2023a; Furtak et al. 2023b; Kocevski et al. 2023; Kokorev et al. 2023a; Greene et al. 2023; Matthee et al. 2023) display a remarkable dichotomy in their observed spectral shapes. In particular the SEDs at 1 -2 µ m (1000 - 2000 ˚ A rest) are blue ( f λ ∝ λ -2 ) and red ( f λ ∝ λ 0 -2 ) at 3 -5 µ m (3100 - 5200 ˚ A rest). As such we keep the Labb'e et al. color criteria largely unchanged, only introducing some further adjustments based on the UNCOVER spectra of LRDs, namely to limit the contamination of our sample by brown dwarfs as was suggested in Greene et al. (2023). \nColors alone would end up selecting both LRDs and extended red galaxies (see e.g. Labb'e et al. 2023b; Williams et al. 2023a,b), so we introduce a further 'compactness' cut to only select sources with high central flux concentration. To do that we use the ratio between the total flux in F444W between 0 . '' 4 and 0 . '' 2 apertures. Since roughly 17% of the LRD candidates followed up with NIRSpec turned out to be brown dwarfs (Burgasser et al. 2023), we would also like to minimize the incidence of these objects in our sample. To do that we adopt the brown dwarf removal criterion from Greene et al. (2023), based on the LRD spectra from NIRSPec/MSA. Finally, we also require our sources to be significantly ( > 14 σ ) detected in F444W, and be brighter than 27.7 AB mags, to be consistent with the UNCOVER selection. The im- \nr cuts are then: \nred1 = F115W -F150W < 0 . 8 & F200W -F277W > 0 . 7 & F200W -F356W > 1 . 0 \nor red2 = F150W -F200W < 0 . 8 & F277W -F356W > 0 . 6 & F277W -F444W > 0 . 7 , \nwhich are effectively selecting our low ( z < 6) and high ( z > 6) redshift samples, respectively. The compactness is given by: \ncompact = f f444w (0 . '' 4) /f f444w (0 . '' 2) < 1 . 7 . \nTo limit the number of brown dwarfs in the sample we also adopt: \nbd removal = F115W -F200W > -0 . 5 . \nThe final selection then becomes ( red1 | red2 ) & compact & bd removal . Applying the color criteria also means that every object has to be detected ( > 3 σ ) in at least one band per color to make the selection meaningful. In case of a non-detection we use the 2 σ upper limits, but only if the 'brighter' band in the color is detected. Out of ∼ 408 000 objects covering 4 fields of interest, we end up selecting 334. Most importantly, we note that no information about photometric redshifts and underlying galaxy/AGN SEDs is used at this stage to avoid being biased by models. We discuss our photometric redshift estimate and its agreement with specz for sub-samples in the next sub section.", '3.2. Size Measurements': "While the compactness cut alone already successfully manages to select PSF-dominated point sources, we would like to provide a further fine-tuning to provide a fully quantitative rather than qualitative assessment. To do that we fit our sources with pysersic (Pasha & \nFigure 3. Observed M UV compared to the M UV expected from the dust-corrected L bol . We derive the expected M UV values by following the relation from Shen et al. (2020). We also show the data for red dots from Greene et al. (2023) (blue squares) and Matthee et al. (2023) (orange circles), as well as broad-line AGN at z > 4 from Maiolino et al. (2023b) (gray triangles). The gray dotted line shows the 1:1 relation. The observed M UV we derive is fainter than the expected value, with A UV varying from ∼ 0 . 6 -4 . 2. While extreme, the UV attenuation is more than 5 magnitudes smaller when compared to values expected given our best-fit A V . \n<!-- image --> \nMiller 2023) in the F444W band. The primary goal of this is to ensure that the source is dominated by the PSF component in the reddest, least dust obscured, band as was done in Labb'e et al. (2023b). We focus on the F444W band for this analysis, as the galactic origin of the rest-UV can not be ruled out with current photometric (or even spectroscopic) observations. Moreover, if an object is dominated by a single star-forming region, it could appear compact in rest-UV bands, but still be extended in the redder filters, making the F444W band the most physically constraining for our type of study. \nTaking the PSF into account is imperative when measuring sizes of unresolved objects. We generate our F444W PSFs empirically for each field by following the methodology described in Skelton et al. (2014), Whitaker et al. (2019) and Weaver et al. (2023a). In brief, we identify non-saturated stars in every field by considering objects on the stellar locus, that are brighter than 24 AB mags and extract these candidates in stamps. These stamps are then centered and normalized to unity. The final PSFs are derived by averaging the weighted stamps, and are then normalized to match the enclosed energies of the expected JWST calibration \nlevels within 4' diameter apertures 2 . For more detail see the Appendix in Weaver et al. (2023a). \nThe light is modeled with a single S'ersic (S'ersic 1963) profile with the center, brightness, effective radius, S'ersic index, and axis ratio as free parameters. The prior for the index is uniform between 0 . 65 -6 and the effective radius uniform between 0 . 25 -5 pixels (0 . '' 01 - 0 . '' 2 ). For each source we create a 3' square cutout (75 pixels by 75 pixels) and mask any additional sources within the stamp. Parameter values and uncertainties are calculated using the Laplace approximation, assuming that the posterior is Gaussian. We exclude fits where the resulting χ 2 per pixel is greater than 2 or the best fit flux differs from the catalog value by more than 2 AB magnitudes. This excludes 15 sources from our sample which by visual inspection we find are untrustworthy due to contamination of bright nearby objects. \nA source can be considered to be point-like if its effective radius in F444W band is lower than the empirical PSF FWHM ( ∼ 0 . '' 15). It appears that our compact criterion is extremely effective at identifying PSF-like objects as none of the 319/334 sources with reliable fits exceed a diameter of 0 . '' 08, when considering the 95 % size upper limits, corroborating the effectiveness of the compactness criterion described in Section 3. After carefully considering both colors and morphology when selecting our sample of AGN candidates, we are now able to proceed directly to the SED fitting.", '3.3. Photometric Redshifts': "To calculate photometric redshifts ( z phot ) for our objects, we use the Python version of EAZY (Brammer et al. 2008). We choose the blue sfhz 13 model subset 3 that contains redshift-dependent SFHs, and dust attenuation values.More specifically, the linear combinations of log-normal SFHs included in the template set are not allowed to exceed redshifts that start earlier than the age of the Universe (for more detail see Blanton & Roweis 2007). These models are further complemented by a blue galaxy template, derived from a JWST spectrum of a z = 8 . 50 galaxy with extreme line equivalent widths (ID4590; Carnall et al. 2022). \nWhile it might seem counter-intuitive to use galaxy templates for what we believe to be AGN candidates, similar efforts presented in Labb'e et al. (2023b) report a good agreement between deriving z phot with stellar templates alone, as opposed to stellar+AGN models, finding \na very good agreement between the two. This is not surprising, as when it comes to photometric redshift fitting, the key deciding factors are the positions of the Lyman ( ∼ 912 ˚ A) and Balmer ( ∼ 4000 ˚ A) breaks. \nFor the LRDs, a general absence of significant stellar contribution in the rest-frame optical (e.g. Greene et al. 2023) would result in a lack of a noticeable Balmer break, however the trough of the 'v-shape' in the restframe SEDs of observed LRDs is also located at roughly 4000 ˚ A (e.g. Furtak et al. 2023b; Kokorev et al. 2023a). Indeed the existence of such a feature in LRDs has resulted in their misidentification as dusty star-forming galaxies, leading to stellar mass estimates which are in tension with ΛCDM, if all the light is attributed to starformation alone (e.g. see discussion in Boylan-Kolchin 2023; Kocevski et al. 2023; Labb'e et al. 2023a; Steinhardt et al. 2023). \nSpectroscopic follow-up of red compact objects hosting AGN BL emission has in fact shown a remarkable agreement between the z phot derived with EAZY (or similar routines) and z spec . For example, in GOODS-S, Matthee et al. (2023) report an average σ z = | ∆z | / (1 + z spec ) = 0 . 01, and UNCOVER LRDs presented in Greene et al. (2023) have shown σ z ∼ 0 . 04. Similar consistency was also found between the initial photometric source selection and final spectra in JADES and CEERS fields (Maiolino et al. 2023b; Kocevski et al. 2023; Andika et al. 2024). As such we consider that utilizing EAZY to derive redshifts is adequate for our sample. \nWe fit all the available photometry, and upper limits from HST/F435W ( λ obs ∼ 0 . 4 µ m) to JWST/F444W ( λ obs ∼ 4 . 4 µ m) filters for our sample of 319 LRDs, limiting the redshift grid between 0 . 01 < z < 20. From the best fit EAZY SEDs we only derive photometric redshifts, delegating the estimation of physical parameters to a different template set discussed in the next section. The uncertainties on the photometric redshift are computed from the 16th and 84th percentiles of the redshift probability distributions p ( z ). The availability of HST photometry allows us to securely constrain the presence of the Lyman break either through diminishing flux where a given filter overlaps with the break, or via upper limits for 40 % sources in our sample. Notably, however, the presence of the Lyman break is not required to securely constrain redshift for highz LRDs, as the break in the optical part of the SED already places these objects in a unique color-color space, as initially shown in Labb'e et al. (2023b) and then further confirmed in Greene et al. (2023). In addition, access to at least one NIRCam medium band further enhances redshift quality by allowing us to identify emission lines in \nbroad band photometry. As a result none of our objects have double peaked redshift solutions. Despite that, since our sample is still only identified photometrically, appropriately taking into account z phot uncertainties is crucial when deriving the physical parameters and luminosity functions in the upcoming sections.", '3.4. Quasar Template Fitting': "While the origin of the rest-frame UV light in LRDs remains elusive, growing samples of JWST spectra consistently show either a complete absence or a lack of a significant contribution from the host galaxy to the total flux in the rest-frame optical ( λ obs ≳ 2 µ m) (Furtak et al. 2023b; Greene et al. 2023; Kokorev et al. 2023a). This is generally evidenced by comparing the expected L 5100 from broad Balmer series lines (generally H β and/or H α ) to the observed values. For example Greene et al. (2023) find that H α -derived and observed L 5100 agree within a factor of two for the objects which have H α PRISM coverage. Supporting this, Furtak et al. (2023b) and Kokorev et al. (2023a) also find that black holes masses ( M BH ) derived via broad H β line luminosity and continuum are identical, given the scatter of the relations derived from AGN reverberation mapping (see e.g. Kaspi et al. 2000; Greene & Ho 2005), hinting at negligible stellar components. Furthermore none of the currently known spectroscopically confirmed LRDs in Abell 2744 are detected in ALMA at 1.2 mm down to < 70 µ Jy (2 σ ), which strongly limits the contribution of obscured star formation (see e.g. Labb'e et al. 2013, 2023b), unless the dust is either very cold, very hot or diffuse. Indeed, when both JWST data and ALMA upper limits (Fujimoto et al. 2023b,c) are considered in a joint AGN+galaxy template fitting for the objects described in Furtak et al. and Kokorev et al. the contribution of the galaxy model to the total rest-frame optical light is negligible. Finally, robust measurements of effective radii for all UNCOVER LRDs, while also taking into account the empirically derived PSFs (see Weaver et al. 2023a), find no strong evidence for extended emission associated with the host galaxy in the F444W band. \nUnfortunately, a lack of deep and uniform ALMA coverage for our objects prevents us from carrying out joint AGN+galaxy template fitting to ascertain the amount of AGN contribution to the optical SED. While it is possible to do it with only JWST photometry, such a fit would be too degenerate given the available number of bands and the number of models required. However, objects in our work were specifically selected with the color and compactness criteria largely mirroring those used to identify broad-line AGN in UNCOVER. It is reasonable therefore to assume that given similarly red \n( f λ ∝ λ 0 -2 at 3100 - 5200 ˚ A rest) slopes, the rest optical continuum in our sources is also dominated by AGN light. \nIn terms of luminosity, the dust-obscured component is dominating the light from LRDs, and must be substantially attenuated ( A V ∼ 1 -2) in order to fit the observed red slope. Given that, the rest-UV light should not be visible at all ( A UV > 10). From our photometry, however, we see that while the blue component is weak (only a few percent of red component), it is not reddened. This emission can be interpreted as either scattered light from the AGN itself, or the host galaxy (see discussion in Labb'e et al. 2023b; Greene et al. 2023). However, even when spectra are available (Greene et al. 2023), given the similarities between the UV slopes of quasars and young star-forming galaxies, these two models are equally good representations of the observed light. Our available data also do not allow us to make a clear distinction between these two possibilities, therefore to avoid over-interpreting the origins of the rest-UV emission, we will assume the scattered light (unreddened) only template in our modeling. We caution the reader that as a result of the unknown origin of the blue light, the rest-UV properties derived in this paper do not necessarily represent physical conditions of the potential AGN our LRDs might host. Due to the aforementioned similarity between UV slopes in quasars and SFGs, the M UV values derived from both galaxy and quasar fits are thus nearly identical. \nFollowing galaxy-only fits presented in Section 3.3 and keeping the above considerations in mind, we now would like to explore an AGN-only scenario where we model the observed light with a two component AGN model. The first one is the empirical model based on a composite of 2200 SDSS quasar spectra (Vanden Berk et al. 2001), and the second is derived from 27 near-infrared quasar spectra by Glikman et al. (2006). We then combine and renormalize both templates, allowing us to cover the full range from rest-UV to the near-infrared. \nThe same approach was already successfully employed in Labb'e et al. (2023b) for a photometrically selected sample of red dots, and then later for PRISM spectra of 14 such objects in Greene et al. (2023) and Kokorev et al. (2023a). We fit the unreddened AGN component together with the Small Magellanic Cloud (SMC) law (Gordon et al. 2003) attenuated ( A V = 0.1 - 4) version of the same composite template. With the photometric redshift being fixed, we are fitting for a total of three free parameters. \nWe find the AGN-only fits to be a marginally better representation of the observed photometry, when compared to galaxy-only EAZY fits, with ⟨ χ 2 ν ⟩ = 3 . 0 +3 . 7 -1 . 8 \nfor the former and ⟨ χ 2 ν ⟩ = 4 . 2 +6 . 6 -1 . 9 for the latter, with a difference of approximately ⟨ ∆ χ 2 ν ⟩ ∼ 1. Similar findings were also presented in Labb'e et al. (2023b), even without ALMA photometry, and Barro et al. (2023), where no significant χ 2 difference exists between dusty star-formation and reddened AGN models.", '3.5. Extreme Equivalent Width of Emission Lines': 'Before focusing on the final sample of reddened AGN candidates we would like to conduct one final test which concerns the potential presence of strong emission lines, particularly H α in the spectra of LRDs. Empirical quasar templates presented in Vanden Berk et al. (2001), which we used to fit our objects, generally contain bright AGN with a rest-frame H α EW ∼ 190 ˚ A. Conversely, the recent literature results which analyze LRD spectra (Killi et al. 2023; Matthee et al. 2023) have found that the EW of H α can reach and even exceed 500 ˚ A. Such strong emission lines can contribute to the flux observed in the medium and even broad-band JWST filters in a non-negligible way, making the observed colors redder. In return, if such strong emission lines are not present in the templates, the value of the A V , and subsequently other physical properties dependent on it (e.g. L bol ) can be overestimated. \nTo test the significance of this effect we do the following. Starting with the original combined Vanden Berk et al. (2001) and Glikman et al. (2006) template set, we isolate the regions that cover the H β +[OIII] and H α lines, and use a spline function to fit the continuum, while masking out the regions containing line complexes. While doing this we successfully verify that the measured rest-frame EW of these lines is exactly as the one reported in Vanden Berk et al. (2001). Finally, we uniformly boost the continuum subtracted spectrum to a point where the EW of the H α line measures at ∼ 500 ˚ A, and add back the continuum. We then re-fit all of our sources, following the same considerations as described in Section 3.4. \nUsing models with boosted emission line strengths, we indeed find the best-fit A V values to be systematically lower, albeit only by ∼ 0 . 1 mag, on average, compared to the original templates. This offset is well within our quoted uncertainty on the A V from the SED fitting. We thus conclude that even if some of our AGN candidates indeed contained very high EW H α emission lines, the physical properties derived with the original Vanden Berk et al. (2001) template set should still remain valid. Despite being small, this offset is systematic, so we still incorporate it into out uncertainties when computing number densities in the subsequent sections.', '3.6. Final Sample of Little Red Dots': "Following the initial object selection and SED fitting we are now in a position to define our final sample of 'little red dots'. The primary goal of this work is to explore the photometrically selected dusty AGN candidates in the highz Universe, compare these results to robust samples of spectroscopically identified BL AGN, and potentially extend these examinations to fainter UV magnitudes and bolometric luminosities. The accurate determination of these parameters is contingent upon good coverage of the spectral break between the blue and red components at ∼ 4000 ˚ A. This is crucial to confirm that the selected objects indeed exhibit the characteristic features of LRDs. Furthermore, a thorough sampling of the rest-frame UV around ∼ 1450 ˚ A is essential to accurately derive M UV , and the 5100 ˚ A rest-frame optical continuum is needed for determining the bolometric luminosity L bol . With the exception of CEERS, all of our fields benefit from full NIRCam filter coverage, spanning from F090W to F444W, which will cover the rest-frame UV at z ≳ 4. On the other hand CEERS has extremely deep ( ∼ 29 . 6 mag at 5 σ ) HST /ACS F814W coverage instead, which will also allow us to adequately compute M UV at 1450 ˚ A in the same redshift range. We thus limit our exploration only to objects which have z > 4. To do that we take into account the p ( z ) and ensure that the 16th percentile, rather than just the median of the p ( z ) lies above our redshift threshold (e.g. see Valentino et al. 2023). This final selection leaves us with a total of 260 red dots.", '3.7. Physical Parameters': "The physical sizes of objects in our final sample are extremely compact, with a median effective radius of r eff < 130 pc (95 % upper limit). This is much smaller when compared to the typical rest-optical sizes of starforming galaxies measured at z > 5 (e.g. see Kartaltepe et al. 2023; Ormerod et al. 2024), but is similar to the extremely compact red objects presented in Labb'e et al. (2023a,b); Baggen et al. (2023) and LRDs spectroscopically confirmed as BL AGN (Furtak et al. 2023b; Kokorev et al. 2023a). Curiously, dusty galaxies at z > 7 explored in Akins et al. (2023) also show a lack of extended bright component ( r eff < 200 pc), similar to LRDs. Although not as faint or centrally concentrated as our objects or other LRDs at these redshifts, some of these similarities might imply that these dusty objects can act as potential AGN hosts. \nUsing the standard relations, with the scatter, presented in Greene & Ho (2005) and taking into account our best-fit A V ( ∼ 0.6 - 3.7 mags), we derive the L bol from the 5100 ˚ A continuum, measured directly from \nbest-fit SEDs. While this is not ideal, and assumes that the red continuum is AGN dominated, the SED model-dependent values represent our best guess for the intrinsic AGN luminosities. The inferred bolometric luminosities for the compact red objects from our sample thus range from L bol ≃ 10 43 . 5 -10 46 . 5 erg/s. This range is slightly brighter than that derived in Labb'e et al. (2023b) as we are not including any lensed fields, and thus likely fail to detect intrinsically fainter LRDs. We show the dust-corrected and observed L bol values in Figure 2. \nIn Figure 3 we explore how the observed M UV values of our LRDs compare to the expectations derived from the dust corrected bolometric luminosity (Shen et al. 2020). Given our median ⟨ A V ⟩ ∼ 1 . 6, we expect the UV extinction to be large with A UV ∼ 9, however what we find is ⟨ A UV ⟩ ∼ 2 . 5 (similar to e.g. Greene et al. 2023; Maiolino et al. 2023b; Matthee et al. 2023), more than six magnitudes difference. Adding to this, the shape of the rest-UV spectrum, while faint, does not hint at any dust extinction. This suggests that a second component, different from a reddened AGN spectrum is present in LRDs, however with our current data its origin can not be determined. \nThe final table which contains photometry, sizes and the physical parameters we derive for our sample is available in full online 4 . We show an excerpt of the full table in the Appendix.", '4.1. Estimating Effective Volumes': "One of the key motivations for our work is to conduct an unbiased search for LRDs in some of the deepest blank fields observed with JWST . Our goal is to extend the existing luminosity functions that have been deduced from spectroscopic samples, for a larger sample covering a wider area. By applying the color and size criteria that have been adopted in recent LRD studies, such as those discussed in Greene et al. (2023) and Labb'e et al. (2023b), we aim to exploit the large area in our blank fields to get better statistics, particularly at the bright and faint ends. This approach should allow us to examine how much these objects contribute to the observed L bol and M UV number densities. However, as we are working with a photometrically selected sample our analysis will be focused on the aggregate characteristics of the LRDs, rather than on detailed examinations of individual objects. \nFigure 4. The UV luminosity function (UVLF) for the LRDs in our sample in the z ∈ [4 . 5 , 6 . 5] (left) and z ∈ [6 . 5 , 8 . 5] (right) bins (blue), derived from observed rest-UV light. Upper limits are shown is downward pointing arrows. The dashed red line and the shaded area correspond to our best-fit Schechter function and its 68 % confidence interval, respectively. Vertical maroon lines highlight the M UV completeness limit calculated from the average depth of F814W/F090W bands. We compare our derived number densities to the luminosity functions of Lyman break galaxies from Bouwens et al. (2021) (solid blue line), extrapolated quasar UVLF relations from Niida et al. (2020) at z ∼ 5, as well as an upper bound provided by Kulkarni et al. (2019) (green lines). At z ∼ 7, in green, we show the UVLF derived from bright quasars from Matsuoka et al. (2023). We highlight the spectroscopically identified LRDs from Greene et al. (2023) (blue squares), Matthee et al. (2023) (orange circles) and Kocevski et al. (2023) (blue pentagon). Furthermore, we show densities of BL AGN quasars from Maiolino et al. (2023b) (gray triangles) and Harikane et al. (2023) (open squares). Green stars show the UV number densities of the X-ray detected quasars at z ∼ 5 from Giallongo et al. (2019). Finally, light blue octagons represent the UVLF derived for galaxies hosting Type 2 AGN from Scholtz et al. (2023). We offset some of the literature points by ± 0 . 05 dex horizontally for visualization purposes. Note that our measured UV luminosities do not decompose the AGN emission from the potential galaxy light. \n<!-- image --> \nFocusing only on the blank fields allows us to estimate the effective volumes for our objects in a rather simple way. In order to measure the observed number densities of our sample, we follow the standard V max method (Schmidt 1968). The 1 /V max estimator has the advantage of simplicity and does not require prior assumptions on the functional form for the luminosity distribution, ideal for LRDs since their intrinsic luminosity/mass distributions are unknown. To compute the number density for some property x , we can then say: \nΦ( x ) = 1 ∆ x ∑ i V max , i ( A,z min , z max ) -1 , \nwhere ∆ x is the width of the bin and V max , i is the maximum volume over which a source can be detected. In return, V max , i depends on the effective survey area A , lower redshift bin boundary z min and maximum observable redshift z max . The latter is computed empirically from the detection limits of the survey, given the selection criteria, and cannot exceed the maximum redshift of the bin. \nWe obtain the total survey areas by adding up all the non-masked pixels in our detection images, as presented in Table 1. Given how bright we require our objects \nto be (F444W < 27 . 7 mags at SN > 14) it might seem that z max would always exceed the maximum redshift of the bin, however this does not take into account the fact that our objects have to be detected in at least four bands (at > 3 σ ) to make color selection robust. We choose to remain conservative with our volume corrections, by only requiring one band per color combination to be detected. The z max values for each object are then estimated by considering our color selection laid out in Section 3. Uncertainties on our number densities are then derived in the following way. We consider the standard errors arising from Poisson statistics and compute them as prescribed in Gehrels (1986). Given that we only consider a photometric sample in our work, the uncertainty on the photometric redshift has to be taken into account appropriately in order to derive realistic errors on the physical parameters and number densities. To do that we follow the approach described in Marchesini et al. (2009). Briefly, for each object we use Monte Carlo simulations to determine whether the objects fall into the redshift bin by considering their p ( z ). The final uncertainties are then a quadrature sum of the Poisson and p ( z ) errors. \nAccounting for magnitude incompleteness effects as it is normally done for galaxy luminosity functions is not possible in our case, since it relies on making assumptions regarding the intrinsic source distributions. However, as Labb'e et al. (2023b) already note, the requirement for objects to be bright in the detection band should lessen, but not eliminate altogether, the effect of magnitude incompleteness. Given that our sources are compact, we also expect that all of them will be detected above the brightness limit, diminishing the need to consider the incompleteness as a function of surface brightness. Despite the complex selection function, it is still possible to define a limit beyond which the derived number densities, for observed quantities, are expected to become incomplete. We will discuss this in the next sections.", '4.2. UV Luminosity Function': "In Figure 4 we present the UV luminosity functions in two redshift bins, at z ∼ 5 and z ∼ 7, derived from the continuum luminosity at rest frame 1450 ˚ A as normally done for blue quasars. We list the number counts alongside the uncertainties in Table 2. The widths of our redshift bins were chosen to best align with the current literature results, for ease of comparison, as well as to ensure that photometric redshift uncertainties have a minimal impact on the luminosity functions. \nAt z ∼ 5, we find that the number densities of our red color selected AGN are ∼ 2 dex higher compared to the UV-selected quasars at similar magnitudes, depending on the extrapolation (Niida et al. 2020). As an upper limit on number density of quasars at z ∼ 5, we also compare to the results presented in Kulkarni et al. (2019), which combine both UV-bright quasars ( M UV < -24) and UV-faint X-ray detected AGN (Giallongo et al. 2019) in their UVLF. \nBefore comparing to current observational results (Kocevski et al. 2023; Greene et al. 2023; Matthee et al. 2023), we note that it is difficult to accurately define the selection function for spectroscopically observed samples, and therefore derive the V max corrections. As such the number densities computed in these works should be treated as lower-limits. In our case the sample is selected via photometry and we derive our V max correction based on the selection criteria alone. This is done to avoid significantly over-estimating the number counts, thus misrepresenting the true abundance of red dots. It is also unlikely that this difference is a result of brown dwarfs contaminating our sample here, since we introduce an additional color cut from Greene et al. (2023) based on the spectra from Burgasser et al. (2023). \nTaking the uncertainties into account, we find that our UV number counts are consistent with JWST -selected red BL AGN samples (Greene et al. 2023; Labb'e et al. 2023b; Matthee et al. 2023), at least at M UV ∼ -19 and brighter. Confirming the initial findings for the UNCOVER red-dots presented in Greene et al. (2023) and Labb'e et al. (2023b), we also find that our sample accounts for ∼ 10 - 30 % of total BL AGN populations at highz (Harikane et al. 2023; Maiolino et al. 2023b) and is largely comparable to the X-ray selected quasars from Giallongo et al. (2019), although in the case of the latter we infer higher number densities at fainter UV magnitudes. However it is worth noting that differences between the resolution of Chandra X-ray data and optical light from HST can lead to uncertainties when associating X-ray emission to the galaxies being present in the same patch of the sky. Curiously enough, the recovered scarcity of compact red sources compared to galaxies is in stark contrast to the density of Type 2 AGN hosts inferred from the recent JADES spectra (Scholtz et al. 2023) which report as much as a 20% contribution to the galaxy luminosity functions at z ∼ 5. \nWhen moving to the z ∼ 7 bin, the results for the UVLF at both bright and faint luminosities are inconclusive, due to the limited number of objects and the uncertainty on the photometric redshifts. However, we are again consistent with the number densities of UNCOVER BL AGN from Greene et al.. Comparing to the luminosity functions of UV selected quasars from Matsuoka et al. (2023) at z ∼ 7, and extrapolating to fainter magnitudes, we find a 2 -3 dex offset between the number densities at M UV > -22, roughly a factor of ten larger than in the lower redshift bin. Alongside our UVLF we also highlight the median M UV 5 σ completeness limits. This is derived by considering the depths of filters covering the rest frame ∼ 1450 ˚ A at a given redshift, and whether a source of a given M UV would be detected at a S/N > 5. As such we should be complete down to M UV ∼ -18 . 5 at z = 5, and M UV ∼ -19 . 0 at z = 7. \nFollowing Bouwens et al. (2015), we fit our observed UV number densities with a Schechter (Schechter 1976) function, allowing all parameters to be free. We only fit data brighter than M UV = -18 . 5 at z ≃ 5 and M UV = -19 . 0 at z ≃ 7 as our number densities indicate that we are likely becoming incomplete at such faint magnitudes. The best-fit is shown in Figure 4 and the parameters are listed in Table 3. In both redshift bins we find that our red compact objects constitute roughly 3 -5% of the total star-forming galaxy populations (Bouwens et al. 2021), consistent with the spectroscopic samples of red-dots (Greene et al. 2023). We also \nreport shallower faint-end slopes compared to SF galaxies, however it is likely that the observed flattening of the UVLF for LRDs is induced by the incompleteness of our sample at fainter UV magnitudes, rather than any lack of compact red sources at fainter UV magnitudes. Deeper surveys would be required to robustly constrain the faint-end slope of LRDs. It appears that the LRD luminosities start to become comparable or even outshine galaxies at brighter ( M UV ∼ -23) magnitudes, which is particularly prominent in the z ∼ 7 bin. This might be an expected consequence of the assembly of increasingly massive black holes with cosmic time (e.g. see Piana et al. 2022), or selection effects (see Volonteri et al. 2017), however we note that our number counts for the brightest objects are uncertain due to a limited amount of detections available. \nProvided that our color and morphology selection is comparably successful at identifying reddened AGN as was previously shown (Labb'e et al. 2023b; Greene et al. 2023), it appears that the compact red sources identified in blank JWST fields are ∼ 1 -2 dex more numerous compared to the preJWST studies of known UVselected faint quasars ( M UV > -21). While this trend has been consistently re-emerging in the new JWST results (e.g. Furtak et al. 2023b; Kokorev et al. 2023a; Maiolino et al. 2023b; Pacucci et al. 2023), it is worth noting that earlier works have already hinted that the number density of UV faint, dusty active black holes could have been much higher than previously thought (Laporte et al. 2017; Morishita et al. 2020; Fujimoto et al. 2022). For example, both Fujimoto et al. (2022) and Morishita et al. (2020) find that the less-luminous red quasar population could be anywhere from 10 to 100 times more common at z ∼ 7 -8, compared to quasars luminosity functions at z ∼ 6, constructed from ground-based datasets (e.g. Matsuoka et al. 2018; Kato et al. 2020; Niida et al. 2020). The results of this work, together with the recent efforts to study compact red sources, therefore imply that these faint quasar populations, missed by previous surveys, are now being uncovered by the deep and rich multi-wavelength photometry and spectra from JWST . It is also important to highlight that if we extrapolate our UVLF to brighter magnitudes, the number density of LRDs becomes comparable to and then drops below the density of UV-selected quasars. Currently, however, it is not possible to speculate whether this is a real physical effect, or simply a consequence of insufficient volumes sampled.", '4.3. Bolometric Luminosity Function': "Our SED fitting results show that the fraction of the UV light contributing to the total luminosity is small as \na result of significant dust reddening ( A V = 0 . 6-2.7) in these objects. Even with spectra in hand (e.g. Greene et al. 2023), it is not easy to establish the origins of the rest UV light, which could be AGN light, either scattered or transmitted through patchy dust clouds, or unobscured light from star-formation in the host galaxy. As such, while we put our LRDs in the context of their observed UV luminosities, this does not explicitly describe the physics of potential AGN these compact objects host. Due to that, and also to carry out a comparison with existing spectroscopic bolometric luminosity functions of dusty BL AGN, we also present bolometric luminosity functions in Figure 5 and Table 2. While dust attenuation, estimated from SED fitting, can be an important source of uncertainty, we note that even if all our A V values were grossly overestimated, this would only change the dust-corrected L bol by a factor of × 5, given the average A V ∼ 1 . 6. This would thus only impact the number densities by ∼ √ 5 on average, which is insignificant when compared to the Poisson and z phot errors. Finally, to account for the potential presence of emission lines with high EW, we incorporate the additional ∼ 0 . 1 systematic shift in A V and apply it to the uncertainties on the L bol . \nUnderstanding where the bolometric luminosity functions start to become incomplete is less straightforward compared to the observed quantities like M UV , as the former also relies on dust correction derived via SED modeling, and the assumptions made regarding the AGN contribution to the rest-frame optical emission. For this reason we do not define a completeness cut, like we do for M UV , however for each bin of bolometric luminosity with a width of 1 dex, we also compute the V max correction as was described in Section 4.1. \nOur number densities again confirm that the redcompact AGN candidates are roughly 100 times more abundant compared to the UV-selected AGN at similar intrinsic luminosities (Shen et al. 2020) at z ∼ 5. The L bol number densities which we recover are comparable to the previous results for these objects derived in Greene et al. (2023), Labb'e et al. (2023b) and Matthee et al. (2023) for L bol -10 45 -46 erg/s. Curiously however, we find a factor of ten more LRDs compared to Greene et al. (2023) at L bol ∼ 10 44 erg/s. Nominally, the median NIRSpec depth at 4 µ m of the UNCOVER followup of Abell 2744 is shallower compared to the fields we examine, as such it is perhaps unsurprising that we can recover a large fraction of intrinsically faint objects. However since the L bol is not an observed quantity and depends on SED modeling to calculate the dust correction it is difficult to ascertain whether the higher number densities we recover are indeed caused by the depth dif- \nFigure 5. Bolometric luminosity functions in the z ∈ [4 . 5 , 6 . 5] (left) and z ∈ [6 . 5 , 8 . 5] (right) bins, derived from L 5100 , assuming rest-frame optical continuum is AGN dominated. The number densities have been V max and completeness corrected. Uncertainties are derived from Poisson noise (Gehrels 1986). Arrows show upper limits on the derived number densities. Blue squares show upper limits derived for spectroscopically confirmed 'little red dots' in the UNCOVER data from Greene et al. (2023). In addition in the lowest redshift bin we show the NIRCam grism result of Matthee et al. (2023) (open circle). Dashed lines show the preJWST L bol relation derived in Shen et al. (2020). Finally, the blue lines show the luminosity function from the Delphi semi-analytic models (Dayal et al. 2019) that grow SMBHs from seeds. \n<!-- image --> \ne, or simply the bias caused by the spectroscopiconly sample selection and lensed volumes in UNCOVER. Moreover, the mask design of the UNCOVER NIRSpec observations in the Abell 2744 field was also driven by optimizing the MSA coverage to include other targets of interest and was not just limited to LRDs. This, in return, induces selection effects which would not be possible to trace back and correct for. \nWe additionally compare our bolometric luminosity function to the latest version of the semi-analytic Delphi models (Dayal et al. 2019, 2024). In brief, these models follow the seeding and growth of BHs from z ∼ 40 down to z ∼ 4 . 5. Included are also all the key processes of merger- and accretion driven assembly of dark matter halos and their baryonic component (including black holes). The model also follows star formation and black hole growth and their respective feedbacks in determining the assembly of these early systems. Finally, Delphi models also include key dust processes to yield dust-to-stellar mass ratios, which with a baseline constructed against the latest ALMA observations (Dayal et al. 2022; Mauerhofer & Dayal 2023). All of this was specifically done to ensure Delphi can reproduce both the intrinsically faint and reddened sources in the recent literature, i.e. the LRDs. \nWe find that while our observations are comparable to Delphi results at L bol < 10 47 erg/s at z ∼ 7, these models fail to reproduce the high number density of bright \nobjects we report. At z ∼ 5 on the other hand, our densities consistently fall 1 dex below Delphi predictions. This in return could suggest that the fraction of dusty AGN is diminishing toward later times, as they potentially transition to unobscured quasars (Fu et al. 2017; Fujimoto et al. 2022). \nFinally, we also see a higher prevalence of intrinsically brighter objects at ∼ L bol -10 47 , which is likely consequence of larger volumes sampled in our analysis. As already mentioned in Greene et al. (2023) however, it is worth noting that the uncertainties on the L bol -L 5100 relation, dust correction and assuming that these objects are dominated by AGN light at rest-frame optical could cause objects to scatter upwards into the high luminosity bins. We only recover a single object above L bol = 10 47 erg/s at z ∼ 5 and below L bol = 10 44 erg/s at z ∼ 7, respectively. As this is insufficient to properly compute luminosity functions, these are shown as upper (lower) limits in Figure 5 derived by combining the Poisson (Gehrels 1986) and photoz (Marchesini et al. 2009) uncertainties.", '4.4. The z ∼ 5 SMBH Mass Function': 'With the data we have obtained, we will now derive and describe the measurement for the supermassive BH mass function which our compact objects potentially host. Computing the mass of the central black hole generally requires knowledge of the width of the broad \nFigure 6. The SMBH mass function, assuming λ edd = 1, of our sample in the 4 . 5 < z < 6 . 5 range. Red arrows show how our mass function would change, if we assumed a lower Eddington ratio of 10 % . We overlay the SMBH mass function from Matthee et al. (2023) at z ∼ 5 in orange and HSC+SDSS derived BH mass function from (He et al. 2023) in magenta. The maroon line shows the results from the EAGLE simulation at z ∼ 5 (Rosas-Guevara et al. 2016). The solid and dashed blue lines show the result from Delphi (Dayal et al. 2014, 2019, 2020) simulations for all and bright ( L bol ≳ 10 44 erg/s) black holes, respectively. Measured number densities of our LRDs agree well with the spectroscopic sample from (Matthee et al. 2023) and simulations given a λ edd ∼ 1. \n<!-- image --> \nlines (e.g. H α , H β in rest-optical or Mg II in rest-UV) coupled with the luminosity of their broad components or the luminosity derived from the AGN continuum at λ rest = 5100 ˚ A (see e.g. Kaspi et al. 2000; Greene & Ho 2005). \nWhile secure determination of the black hole mass in our compact objects is not possible due the photometric nature of the sample, we can still place a lower limit on black hole masses ( M BH ), by making a set of conservative assumptions. To do that, we adopt a scenario where all our AGN candidates accrete at Eddington rate (the physical limit at which outward radiation pressure balances inward gravitational force), such that L bol ∼ L edd , where L edd is directly proportional to M BH . While in the literature the Eddington rate ( λ edd ) for confirmed AGN in LRDs was found to vary between 10 -40 % (Furtak et al. 2023b; Greene et al. 2023; Kokorev et al. 2023a), we would like to remain conservative and compute a lower limit on the M BH . It is also worth noting that, given this range of λ edd in the literature, this is still small compared to other sources of uncertainty in our work. \nTable 2. Bolometric and UV ( λ rest =1450 ˚ A) luminosity functions, as well as a SMBH mass function for our sample of LRDs. \nWecalculate the M BH directly from the dust-corrected L bol and compute the SMBH mass function as described in the previous sections. We present our SMBH mass function in Figure 6 and Table 2, binned into 0.5 dex intervals to allow a direct comparison with the existing observational and theoretical results in this redshift \nrange. We limit this investigation to the z ∼ 5 range only. As before, we note that the effect of the A V uncertainty on our number densities is expected to be at most ∼ √ 5, largely overshadowed by the Poisson and redshift errors. \nWe are now in a position to compare our mass function to the existing samples of both bright and faint quasars at z ∼ 5. We start with the latest ground based examination of the quasar mass function at z ∼ 4 from He et al. (2023). The authors focus on a sample of ∼ 1500 faint broad-line AGN, from a combined Hyper Suprime Cam (HSC) and SDSS dataset, allowing them to extend their examination to a low mass range we are most interested in ( M BH ≃ 10 7 -8 M ⊙ ). We find that, while our result is consistent with the ground based mass function in the high mass regime M BH > 10 8 M ⊙ , our number densities diverge below that mass and continue to rise up to ∼ 10 -4 cMpc -3 at M BH ≃ 10 6 M ⊙ . Barring the color selection, it is possible that this effect is purely observational, as the SDSS/HSC detection limits in the rest-UV are much shallower compared to the JWST fields we explore. \nFurthermore, we contrast our result to the BH mass function at z ∼ 5 based on a sample of LRDs from the slitless JWST derived in Matthee et al. (2023). Given the uncertainties, we find our results to be consistent within 1 σ , although we do not find a sharp drop-off in number densities at M BH < 10 7 M ⊙ , likely driven by low mass incompleteness of grism data as mentioned in Matthee et al.. \nNaturally, the fact that both our result and Matthee et al. (2023) find more low mass black holes compared to He et al. (2023) is unsurprising, given the depth and wavelength coverage of JWST data. Quite curiously however, it appears that the mass function derived from dusty compact LRDs seems to nicely continue the rising trend of ground-based data, and extend the SMBH mass functions toward M BH ∼ 10 6 M ⊙ . In this redshift range, the maximum volume sampled by our multi-field investigation is roughly equal to ∼ 3 . 0 × 10 6 cMpc 3 . Therefore, taking into account the results of He et al. (2023), we should expect only one object with M BH ∼ 10 8 . 5 M ⊙ in our images, which indeed is the case. Detection of AGN hosting black hole with masses larger than that, would, however, require survey sizes ten to twenty times larger. \nBefore drawing conclusions, we would like to conduct a final check, and compare our result to the hydrodynamical simulation EAGLE (Rosas-Guevara et al. 2016) and semi-analytic Delphi (Dayal et al. 2014, 2019, 2020, 2024) simulations describing masses of SMBHs in the same redshift range. We limit our examination of the Delphi models to the bright ( L bol ≳ 10 44 \nTable 3. Best fit Schechter parameters for the rest-frame UVLF at λ rest = 1450 ˚ A, across blank JWST fields. \nerg/s) regime to match the same luminosity range covered by our objects. In the intermediate, to low mass end ( M BH < 10 7 . 5 M ⊙ ) our results are in broad ( ∼ 2 σ ) agreement with both EAGLE and Delphi, however in both cases we start to see a significant difference in number densities as we move to higher masses. Perhaps a worthwhile question to ask in this case, is whether more of these high-mass BHs would be found in larger areas, we will discuss this in the later section. \nExamining both the UV and bolometric luminosity functions we note that LRDs only represent ∼ 25 % of the total type I (broad-line) AGN population as inferred by Harikane et al. (2023); Maiolino et al. (2023b), even less so compared to the most recent examination of type II AGN hosts from JADES (Scholtz et al. 2023), where LRDs are 30-40 times less numerous. Taking this into account, we can conclude that, at least at z ∼ 5, LRDs appear to represent at most 1 % of the total accreting BHpopulation over the L bol ∼ 10 44 -47 erg/s range. The fact that LRDs are truly a distinct population of dusty broad-line AGN can therefore explain the observed ∼ 2 dex disparity between our results and simulations. Finally we would like to reiterate that our investigation of the BH mass function relies on assuming the most conservative case of accretion at exactly the Eddington rate, as we do not want to erroneously overestimate the number of high-mass black holes. Keeping that in mind, in Figure 6 we also show how our mass functions would change if we were to assume an Eddington ratio of 10 % instead. In this case we find that while our number densities compared to UV samples are still high, we now more closely match the abundance of high mass SMBHs predicted by Delphi . However, until broad emission line observations for all our sources are available, the value of λ edd will remain uncertain.', '5.1. Abundance of bright compact sources': "Previously limited to UV -selected samples at z ≲ 6 (Kashikawa et al. 2015; Ba˜nados et al. 2018; Matsuoka et al. 2018; Inayoshi et al. 2020; Wang et al. 2021; Fan et al. 2022) we are now able to use JWST to reveal the presence of AGN during (e.g. Kocevski et al. \n2023; Matthee et al. 2023; Ubler et al. 2023) and even beyond the epoch of reionization, (e.g. Bogdan et al. 2023; Furtak et al. 2023c; Goulding et al. 2023; Kokorev et al. 2023a; Larson et al. 2023; Lambrides et al. 2023; Maiolino et al. 2023a) only hundreds of millions of years after the Big Bang. Standing out among these early studies of active black holes, is the population of reddened type I AGN, the so called 'little red dots' (Greene et al. 2023; Labb'e et al. 2023b; Matthee et al. 2023). \nWhile the study of this unique population has been mostly limited to small spectroscopic samples, most recent efforts focused on the expansive Abell 2744 JWST data-set (Labb'e et al. 2023b) have shown great promise at using a combination of NIRCam colors and morphology to identify reddened AGN. This initial photometric selection was shown to be remarkably successful with ∼ 80 % of targets indeed confirmed as z > 5 dusty broad-line AGN (Fujimoto et al. 2023a; Furtak et al. 2023c; Greene et al. 2023; Kokorev et al. 2023a). It is clear that these objects play an important role in the story of black hole growth at early times, however so far a systematic review of these enigmatic AGN across multiple fields has not been undertaken. \nMotivated by the success of this photometric selection, we present a sample of 260 reddened BL AGN candidates in the 4 < z < 9 redshift range, covering 4 separate blank JWST fields with a total area of ∼ 640 arcmin 2 . We uniformly reduce the NIRCam JWST data from a variety of public programs, complementing our photometric coverage with archival HST observations. We perform a color and morphology selection to identify the most promising compact objects which display a dichotomy in their observed SED shapes, namely a blue rest-UV continuum, and a red power lawlike rest-optical. \nUsing model fitting, we derive photometric redshifts as well as a range of physical parameters including A V , M UV and dust corrected L bol . We split our objects into two redshifts bins at z ∼ 5 and z ∼ 7 and explore their contribution to the UV and bolometric luminosity functions of star-forming galaxies, as well as UV-selected quasars. Consistent with the previous works (Greene et al. 2023; Matthee et al. 2023; Maiolino et al. 2023b) exploring highz BL AGN, we find that number densities of these objects at z > 5 are surprisingly high, in excess of × 100 compared to faint UV selected quasars (e.g. Niida et al. 2020; He et al. 2023), while also accounting for ∼ 20% of the total BL AGN population (Harikane et al. 2023; Maiolino et al. 2023b), and ∼ 1 -2 % of UVselected star forming galaxies (e.g. Bouwens et al. 2021). Moreover, while some of these objects were potentially \npinned down as potential sources of reionization in their local environment (Fujimoto et al. 2023a), it appears that their UV luminosities are still insufficient to contribute to reionization in a significant way (Dayal et al. 2024). \nAssuming accretion at the Eddington rate, we also place a lower limit on the M BH of our objects, finding that some of these can already be very massive ( M BH > 10 7 M ⊙ ) only a few hundred of years after the Big Bang. Using these masses we were also able to construct our prediction for the SMBH mass function, and for the first time, extend it to the low-mass ( < 10 7 M ⊙ ) regime. We find that our mass function results are completely consistent with the number densities derived for faint dusty AGN from Matthee et al. (2023) at intermediate masses, and are comparable to those from UV-selected samples at high mass (He et al. 2023). We note however that while their number densities are similar, the sample presented in He et al. consists of unobscured quasars, and not LRDs, which are thought to be dust obscured AGN. We find that both hydrodynamical and semi-analytic predictions for the number of black holes at this redshift match our observations below ∼ 10 7 . 5 M ⊙ , however start to disagree by almost 2 dex at higher masses. These massive and bright black holes are likely to be heavily obscured in the rest-frame UV, and thus are not selected as LRDs due to a lack of a clear blue component.", '5.2. Final Remarks': "Using observed NIR colors to pick out active black holes in extragalactic fields is by no means a novel endeavor and has already been successfully done with the IRAC instrument, onboard the Spitzer Space Telescope (Lacy et al. 2004; Stern et al. 2005; Donley et al. 2012). This method, however, is still in its infancy when it comes to JWST (Labb'e et al. 2023b; Andika et al. 2024). As already pointed out by Matthee et al. (2023), very few JWST programs that detect dusty AGN, were actually designed with AGN in mind, implying that there is still more that we can to address a growing number of questions about this population. \nFirstly, the physical mechanisms that govern BH formation and growth in these systems are still poorly understood, however there exists already a growing body of works which try to decipher this enigmatic population (Greene et al. 2023; Silk et al. 2024). One such puzzle, is the origin of the blue light found in LRDs. The similarity between blue slopes of low-metallicity star-forming galaxies and quasars does not allow us to make a clear \nassessment of whether the rest-UV light originates from the AGN itself or the compact host galaxy surrounding it from the continuum alone. One way to solve this is to target the Mg II doublet ( λ rest ∼ 2800 ˚ A), the CIV, SiIV or HeII ( λ rest ∼ 1840 ˚ A) lines (e.g. see Maiolino et al. 2023a) and confirm whether these are broadened or not. This however would require longer integration times with NIRSpec as medium or even high resolution gratings would be required to achieve the necessary spectral fidelity. Moreover, while in principle detection of broad UV emission lines could hint at the AGN being responsible for some UV light, this would not necessarily mean that UV continuum also originates from the same source. \nSecondly, there are no models as of yet, which can adequately describe the light we see emerging from these objects. So far, we mainly have had to rely on combinations of dust-free and dust attenuated empirical models of local quasars, which might be adequately describing AGN at highz . Moreover, the lingering uncertainty on the A V correction can introduce some biases in our estimates of L bol and the M BH . One solution to alleviate this is to stack spectra of known LRDs, to define sets of reliable models describing these populations and aiding with further photometric selection. \nThirdly, it is crucial to note that a substantial proportion of massive SMBHs (with M BH > 10 8 M ⊙ ) at high redshifts (highz ) can be heavily obscured, as implied by Delphi . Similar conclusions were already reached from the X-ray luminosity functions at z > 5, both from simulations (Ni et al. 2020), and observations (Aird et al. 2015; Vito et al. 2018). In addition Trebitsch et al. (2019) have shown that accreting SMBHs in Lyman break galaxies are rarely UV-bright. With this in mind, selecting these massive AGN as LRDs would thus not be possible, as a combination of very deep rest-UV imaging and large areas are required. Despite that, these objects should still appear bright in near-infrared, which opens up the possibility of effectively identifying them through large or parallel surveys using MIRI. \nFinally, as was already shown recently by Williams et al. (2023a) and P'erez-Gonz'alez et al. (2024), MIRI can also assist with clarifying true numbers of AGN among LRDs as some of these could be dusty progenitors of compact ellipticals. \nEarly results from JWST have already provided us with quite unexpected and remarkable results regarding number densities of early AGN, leading to a shift in our understanding of their formation and growth in the early Universe. Our results highlight the potential of using NIRCam alone to select reddened AGN at highz in an effort to better understand their properties and \nabundance. While some limitations to this technique exist, as we already discuss in our work, this provides a crucial set of next steps in order to bridge the gap between UV bright quasars and faint SMBHs. However, it is already evident that the importance of faint, reddened AGN at early times can not be overlooked.", 'ACKNOWLEDGMENTS': "We thank the anonymous referee for a number of constructive suggestions, which helped to greatly improve the quality of this manuscript. We are grateful to Dale Kocevski and Kohei Inayoshi for their patience with helping us spot and correct minor inconsistencies in the manuscript. The authors would like to thank Sarah Bosman for insightful discussions about UV-bright quasars at high redshift. We also thank Mauro Giavalisco, Hollis Akins and Meghana Killi for useful discussion regarding the nature of dust obscured AGN. VK and KIC acknowledge funding from the Dutch Research Council (NWO) through the award of the Vici Grant VI.C.212.036. J.E.G. acknowledges support from NSF/AAG grant# 1007094, and also support from NSF/AAG grant # 1007052. PD & MT acknowledge support from the NWO grant 016.VIDI.189.162 ('ODIN'). PD also acknowledges support from the European Commission's and University of Groningen COFUND Rosalind Franklin program. This work is based on observations made with the NASA/ESA/CSA James Webb Space Telescope. The data were obtained from the Mikulski Archive for Space Telescopes at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-03127 for JWST. All the JWST and HST data used in this paper can be found on MAST: 10.17909/de9v-7893. Some of the data products presented herein were retrieved from the Dawn JWST Archive (DJA). DJA is an initiative of the Cosmic Dawn Center, which is funded by the Danish National Research Foundation under grant No. 140. TBM was supported by a CIERA Postdoctoral Fellowship. This work used computing resources provided by Northwestern University and the Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA). This research was supported in part through the computational resources and staff contributions provided for the Quest high performance computing facility at Northwestern University which is jointly supported by the Office of the Provost, the Office for Research, and Northwestern University Information Technology. \n```\nSoftware: EAZY(Brammer et al. 2008), FSPS (Conroy et al. 2009), pysersic (Pasha & Miller 2023), grizli (Brammer 2023), msaexp (Brammer 2022). Facilities: JWST , HST\n```", 'APPENDIX': "Table 4. An example of the table containing all properties of our sources. A full version of this table is available in the electronic format. \nNote -Sizes are measured in F444W band on the 0 . '' 04 images. The FWHM of the F444W PSF is 3.45 pixels.", 'REFERENCES': "Adams, N. J., Conselice, C. 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2024arXiv240905068M

The detection of gravitational waves GWs from binary neutron stars BNSs with possible telescope followups opens a window to groundbreaking discoveries in the field of multimessenger astronomy. With the improved sensitivity of current and future GW detectors more BNS detections are expected in the future. Therefore enhancing lowlatency GW search algorithms to achieve rapid speed high accuracy and low computational cost is essential. One innovative solution to reduce latency is the use of machine learning ML methods embedded in fieldprogrammable gate arrays FPGAs. In this work we present a novel textttWaveNetbased method leveraging the stateoftheart ML model to produce earlywarning alerts for BNS systems. Using simulated GW signals embedded in Gaussian noise from the Advanced LIGO and Advanced Virgo detectors third observing run O3 as a proofofconcept dataset we demonstrate significant performance improvements. Compared to the current leading MLbased earlywarning system our approach enhances detection accuracy from 66.81 to 76.22 at a 1 false alarm probability. Furthermore we evaluate the time energy and economical cost of our model across CPU GPU and FPGA platforms showcasing its potential for deployment in realtime gravitational wave detection pipelines.

2024-09-01T00:00:00Z

['arXiv:2409.05068', '2024arXiv240905068M', '10.48550/arXiv.2409.05068']

['Astrophysics - Instrumentation and Methods for Astrophysics']

Improving Early Detection of Gravitational Waves from Binary Neutron Stars Using CNNs and FPGAs

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.05068.pdf

{'Improving Early Detection of Gravitational Waves from Binary Neutron Stars Using CNNs and FPGAs': "Ana Martins Institute of Theoretical Astrophysics, University of Oslo Oslo, Norway Email: a.i.s.martins@astro.uio.no \nGregory Baltus STAR Institut, Universit'e de Li'ege Li'ege, Belgium Email: gbaltus@uliege.be \nMelissa Lopez, Quirijn Meijer, Marc van der Sluys ∗ and Chris Van Den Broeck Institute for Gravitational and Subatomic Physics (GRASP), Utrecht University. Nikhef, National Institute for Nuclear Physics and High-Energy Physics. \nNetherlands \nEmail: lopezm@nikhef.nl, r.h.a.j.meijer@uu.nl, sluys@nikhef.nl, vdbroeck@nikhef.nl", 'Sarah Caudill †': "Department of Physics, University of Massachusetts Dartmouth, Center for Scientific Computing and Data Science Research, University of Massachusetts Dartmouth, Dartmouth, USA Email: scaudill@umassd.edu \nAbstract -The detection of gravitational waves (GWs) from binary neutron stars (BNSs) with possible telescope follow-ups opens a window to ground-breaking discoveries in the field of multi-messenger astronomy. With the improved sensitivity of current and future GW detectors, more BNS detections are expected in the future. Therefore, enhancing low-latency GW search algorithms to achieve rapid speed, high accuracy, and low computational cost is essential. One innovative solution to reduce latency is the use of machine learning (ML) methods embedded in field-programmable gate arrays (FPGAs). \nIn this work, we present a novel WaveNet -based method, leveraging the state-of-the-art ML model, to produce earlywarning alerts for BNS systems. Using simulated GW signals embedded in Gaussian noise from the Advanced LIGO and Advanced Virgo detectors' third observing run (O3) as a proofof-concept dataset, we demonstrate significant performance improvements. Compared to the current leading ML-based earlywarning system, our approach enhances detection accuracy from 66.81% to 76.22% at a 1% false alarm probability. Furthermore, we evaluate the time, energy, and economical cost of our model across CPU, GPU, and FPGA platforms, showcasing its potential for deployment in real-time gravitational wave detection pipelines.", 'I. INTRODUCTION': "In recent years, Astrophysics and adjacent sciences have been expanding at an astronomical pace. A significant driver of this growth is the detection of gravitational waves (GWs), which are ripples in spacetime generated by astronomical cataclysms. On September 2015, the Laser Interferometer Gravitational-Wave Observatory (LIGO) [1] and Virgo [2] collaborations confirmed the existence of GWs by detecting a binary black hole (BBH) merger [3]. Since then, over \n∗ The research leading to these results has received funding from the European Union's Horizon 2020 Programme under the AHEAD2020 project (grant agreement No. 871158). This publication is part of the project Cortex with project number 00686766 of the research programme NWA which is (partly) financed by the Dutch Research Council (NWO). \n† S.C is supported by the National Science Foundation under Grant No. PHY-2309332. \n90 confident events have been detected in the past three observation runs by LIGO-Virgo collaboration [4]-[7]. Among these astronomical events, only two binary neutron star (BNS) mergers were detected [8]. \nAs gravitational wave (GW) detectors become increasingly sensitive with each upgrade, the enhanced capabilities of second-generation detectors and the future third-generation detectors, such as the Einstein Telescope (ET) [9] or Cosmic Explorer [10], will allow for more frequent detection of binary neutron star (BNS) mergers and other exotic systems. Nonetheless, this increase in sensitivity comes at increase in computational complexity, as more sources will be detected. In the era of ET, it is estimated that 8 × 10 4 yr -1 BBH [11] and 7 × 10 4 yr -1 BNS [12] will be detected, which could lead to over 400 daily GW merger events, as well as the detection of other exotic sources [13]. Moreover, while BNS GW signals are detectable for seconds with current detectors, they will be present for hours in the ET era. \nThe current state-of-the-art for detecting modelled gravitational wave (GW) signals, known as matched-filtering-a technique based on cross-correlating models, or templates, with GW detector data [14]-[18]-has led the LIGOVirgo-KAGRA collaboration to develop several low-latency matched-filtering pipelines for producing real-time GW alerts [19]-[23]. While these methods have been successful, matched-filtering techniques are known to be computationally intensive, posing challenges for future detectors and significantly contributing to increased pollution. \nIn recent years, machine learning (ML) techniques have gained significant interest due to their success in various tasks and domains. A key advantage of ML is its fast inference, with most computations occurring during the training stage. This is crucial for gravitational wave (GW) searches, particularly for early warning of BNS GW signals in multi-messenger astrophysics. The goal is to detect GW signals during the inspiraling phase, where both neutron stars are orbiting around \neach other, before they merge in a bigger astronomical object (known as merger phase), to enable electromagnetic telescope follow-ups. This task is challenging because GW signals are often buried in detector noise, with the signal being weaker during the inspiral phase and strengthening as it approaches the merger. Consequently, effective ML models need to be highly complex. \nGPUs, widely used for deploying ML algorithms, are wellknown for being costly, consuming large amounts of energy, having short lifetimes and potentially introducing high latencies within data transfer. In a low-latency framework, these characteristics could hamper the early detection of GW signals. An interesting alternative is to embed ML algorithms in fieldprogrammable gate arrays (FPGAs). FPGAs show promise by having longer lifetimes and being more energy-efficient than state-of-the-art hardware. Furthermore, high-end FPGAs are faster than CPUs and GPUs [24], and low-end FPGAs are more affordable than CPUs or GPUs in the same tier. We choose the latter to explore their applicability and limitations. \nIn this work, to build more sustainable and environmentallyfriendly alternatives, we explore the performance of FPGAs with respect to GPUs and CPUs to detect the early inspiral of BNS GW signals with ML-based algorithms 1 . This paper is structured as follows: in Section II, we address the state of the art in early warning with ML-based methods and motivate the need to improve the precision at a limited computational cost; in Section III, we describe the input data; in section IV, we describe the hardware employed and ML architectures, as well as learning and quantization strategies (reduction from floating-point to fixed-point representation); in Section V, we present the main findings of this work, comparing our model with the state of the art in terms of accuracy and computational cost; finally, Section VI presents the conclusions of this research with avenues for future research.", 'II. RELATED WORK': "The challenges in GW research require innovative solutions, and ML has emerged as a crucial tool for addressing them due to its adaptability and transversality. In the past few years, researchers have explored different ML applications to GW data analysis, such as detection of modelled [25], [26] and unmodelled GW signals [27]-[32], non-transient burst noise characterization [33], [34] and synthetic data generation [35][38], among others. Refer to [39] for a review. \nTraditional GW searches use the inspiral part of the waveform, as early efforts have shown that it is possible to detect a signal with only a fraction of the waveform model with matched-filtering techniques [19]. The first early warning ML-based algorithm implemented ResNet50 [40] training on spectrograms containing solely the inspiral part of the waveform [41]. However, as noted in [42], this added preprocessing step slows the inference by ∼ 0 . 5 s. To address this, [42] suggested using GW detector time series in design \nsensitivity, and employing 1-dimensional convolutional neural networks (CNNs), eliminating the need for this pre-processing step and enhancing efficiency. This proof of concept work led to subsequent efforts in this line of research [25], [26], [43][46]. \nIn particular, [26], building on the previous work of [42] and [25], extended this method to the simulated noise of third and fourth observing runs (O3 and O4), as well as real O3 data. While [26] is the current state of the art, some of the limitations of this study are described below: \n- · Large number of trainable parameters: The fixed input-signal of this work has a duration of 300 s with a sampling frequency of 512 Hz, resulting in 155 , 648 data points. Due to the long size of the input, large and complex models, that could lead to higher accuracy, would not fit into the device's memory.\n- · Catastrophic forgetting: This investigation successfully implements curriculum learning -where the model learns easier examples to then transition to harder ones- by lowering the maximum frequency of the signals or rather detecting the signal earlier. However, it is observed that the model forgets easier steps of curriculum learning due to its over-fitting.\n- · High false positive rate: The model has a large number of false positives, i.e. misclassifying detector noise as GW signal, regardless of its decision threshold. \nTo overcome the limitations of the previous work, hereafter referred to as FindCNN , we implemented an algorithm inspired by WaveNet [47] named GWaveNet . Similarly to [48] and [30], this choice was motivated by the similitude between GW time series data and audio or speech signals. For a fair comparison, we reproduced FindCNN and trained and tested both models on the same simulated GW detector data. \nFPGAs have been used for ML applications [49], also in high-energy physics [50], [51]. We highlight the pioneering work in [52] for the implementation of recurrent neural networks for anomaly detection in GW. As added value, in this work we evaluate the performance of the FindCNN and GWaveNet models in CPU, GPU and FPGA devices, as well as their time, energy consumption and economical cost.", 'III. DATA SET AND PRE-PROCESSING': "A GW search algorithm processes raw detector data, producing the probability of whether it contains a GW signal. We can find GWs using supervised learning, where we distinguish detector noise (called a negative class) from detector noise plus GW signal (called a positive class). \nIn a real scenario, the negative class is much larger than the positive class, as only ∼ 100 GW signals have been discovered at the present date. Thus, it is required to simulate GW signals, adding them to detector noise. This process is commonly referred to as 'injections', and they will be the positive class of this study. Hence, we construct a balanced data set of samples with only detector noise (negative class), and injections (positive class). Simulated Gaussian noise is \ngenerated with the average power spectral density of the third observing run (O3) for Hanford, Livingston and Virgo 2 . \nAs proposed in [26], to encompass all potential BNS systems, the component masses are uniformly distributed ∈ [1 , 3] M ⊙ , and the waveform approximant SpinTaylorT4 [55] (used to model inspirals) was chosen. Moreover, the sources are uniformly distributed over the sky, including spin effects. \nFig. 1. (Top) Representation of the inspiral phase, where both neutron stars are orbiting around each other, of a GW signal (blue) buried in simulated raw LIGO noise (grey). The vertical red line indicates the merger time. The GW waveform was modelled using SpinTaylorT4 -which only models the inspiral phase-with progenitor masses ( m 1 = 1 . 46 M ⊙ , m 2 = 1 . 27 M ⊙ ), similar to GW170817. The input to the ML model is contained within the yellow rectangle, i.e. 300 s of data sampled at 512 Hz. (Bottom) Frequency evolution of the full GW waveform (blue) and the partial waveform (yellow). \n<!-- image --> \nAs GWs have a weak amplitude, they are hard to distinguish from detector noise. For illustration, in Fig. 1 (top panel) we show the GW inspiral of a signal similar to GW170817 [8] in raw Advanced LIGO detector noise. To highlight its amplitude a common practice is to filter the data between 10 and 100 Hz, and whiten it, i.e. make the signal Gaussian-like with uniform variance by removing all the correlation of the noise [56]. Afterwards, we normalize the data ∈ [ -1 , 1] . As we can observe in Fig. 1, the inspiral phase lasts several minutes, so, as in [26], we pre-select 300 s of data. In the bottom panel, we show the relation between time and frequency of the source. Because of this relation, the difficulty of the input is governed by the maximum frequency f max within a time window. Samples with higher (lower) f max will be closer (further) to the merger, meaning that they will be detected later (earlier). \nAs the Hanford, Livingston and Virgo observatories measure GW signals independently, a signal that appears simultaneously in data from multiple detectors at the same time is more \n2 We use the PyCBC package [53]. In particular, the power spectral densities aLIGOaLIGO140MpcT1800545 and aLIGOAdVO3LowT1800545 , an average of O3, respectively for the Advanced LIGO and Advanced Virgo detectors [54]. Refer to https://gwosc.org/O3/o3 details/ for technical details of the observing run. \nlikely to be of astronomical origin. From an ML perspective, independent detector data is inputted as separate channels.", 'IV. METHODOLOGY': 'In this work, we do not only aim to detect GW signals as early as possible, going down in frequency f max , but also to minimize the number of false positives (FP), to avoid sending false alerts to electromagnetic telescopes. Thus, we have developed a WaveNet-like architecture, GWaveNet , that we compare to the current state of the art, FindCNN . In the following sections, we describe the hardware employed for the deployment of the models and provide an overview of FindCNN and GWaveNet .', 'A. Hardware specifications': "The manufacturing and use of state-of-the-art hardware are causing substantial damage to the environment and this is only predicted to increase, with the manufacturing of computer hardware predicted to take up more than 20% of global energy usage by 2030 [57] and computation energy predicted to hit the world's energy production capacity by 2040 [58]. In this context, it is of interest to understand the computational cost of low-latency algorithms of large physics experiments, such as Advanced LIGO-Virgo detectors, as well as their lifetime under heavy usage. In the following, we provide the specifications of the hardware employed in this work. \n- -CPU: We use the AMD EPYC 7551P 32-Core Processor [59], which has an expected lifetime of around 5 to 10 years. We used the full 32 cores for testing.\n- -GPU: We used the NVIDIA Tesla V100 [60] for training and validation, and NVIDIA GeForce GTX 108 [61] for testing and energy consumption calculation. Their expected lifetime is around 3 to 5 years.\n- -FPGA: We use the AMD Kria KV260 Vision AI Starter Kit [62]. Its expected lifetime is around 10 to 15 years. We use the most recent and fastest pre-made architecture for this system-on-chip, B4096, with 4096 peak operations per cycle. Furthermore, we use 10 threads for testing, which we found to be the optimal amount. \nAt the time of launch, the CPU and GPU used for inference were nine and three times more expensive than the FPGA used in this study. As of the current date of this investigation, the NVIDIA GeForce GTX 1080 GPU is eight years old, and the AMD EPYC 7551P is seven years old, with both considered obsolete. Typically, CPUs and GPUs become outdated even before their lifespan ends, as they struggle to compete with newer devices. In contrast, FPGAs are customizable, allowing modifications to keep them competitive throughout their lifetime. Consequently, the carbon footprint of using FPGAs is significantly lower than that of traditional processing units, not only because they consume less energy during operation but also due to their extended effective lifespan.", 'B. Basic components': '1) Common details: We implemented FindCNN and GWaveNet using the PyTorch package [64]. Both networks \nFig. 2. Overview of the modified WaveNet convolutional architecture for a single data stream, adapted from [63]. We present n = 7 1-dimensional WaveNet modules composed of dilated causal convolutions (blue) with the number of filters f , kernel size k , dilation d and padding p , batch normalization layers (orange), tanh and sigmoid σ activations (green). Furthermore, we show 1 × 1 convolutions (blue), dense layers (pink), and ReLU and Softmax activations (green). \n<!-- image --> \nminimize the binary cross-entropy (BCE) loss in PyTorch, the BCEWithLogitsLoss i.e., Sigmoid activation integrated with BCE loss [65]. We decided to employ this loss instead of BCELoss since it is more numerically stable. Note that the original FindCNN [26] uses BCELoss instead. \n2) FindCNN : It is a 1-dimensional CNN [26] that starts with a batch normalization layer [66], to stabilize and accelerate the training process, followed by 5 convolutional blocks and two dense layers. The convolutional blocks are composed of a convolutional layer with stride 4, a ReLU activation function [67], and a MaxPooling layer [68] with stride 1. Each convolutional layer has successively kernel sizes k [16 , 8 , 4 , 8 , 16] , and number of filters f [32 , 64 , 128 , 256 , 256] . This structure allows capturing high-level features at earlier layers while capturing finer details at intermediate layers. Furthermore, the combination of large kernel k and large filters allowed the last layers to capture high-level context with great expressiveness. Despite the compression of the input due to the usage of large kernel sizes k , the first dense layer is one of the most expensive ones, as it further compresses an input of 37 , 632 data points to only 128 . As expected, this model is highly complex, having 6 , 179 , 303 trainable parameters, mainly due to the length of the input and large dense layers. \n3) GWaveNet: WaveNet is an expressive CNN designed for the generation of high-fidelity speech audio [63]. The architecture is capable of handling long-range temporal dependencies, a characteristic desirable for GW data analysis, and in particular for BNS searches, due to their long inspirals. WaveNet-like models have been previously implemented in the field of GW searches [30], [48]. These implementations modify WaveNet for binary classification. This is a common point in the current investigation. However, previous WaveNet models were constructed as an ensemble with non-causal \ndilated convolutions, while we encode the information of all detectors in a single model with causal dilated convolutions. \nCausal dilated convolutions combine the power of dilation [63], which allows the NN to capture a larger context without increasing the computational load significantly, and causal convolutions, to effectively learn the time ordering of the data. Our experiments shown an improved performance when obeying the arrow of time with causal convolutions. These convolutions are encapsulated in WaveNet modules. \nWe present the full architecture of GWaveNet , including the WaveNet modules in Fig. 2. We can observe how an input x gets duplicated and each copy gets fed into a causal convolution block (blue), followed by a batch normalization layer (orange). Then, one output passes a tanh activation function (green), while the other passes a Sigmoid σ activation function (green). This is known as gated activation [69], designed for capturing complex relations in sequential data. Afterwards, the two outputs are multiplied (grey), generating the output of the module, y . At this point, the initial input x is added to the output y to form the residual input y res that passes to the next module. On the other hand, the output y is sent towards the final output of the network that will feed the BCEWithLogitsLoss . Before this step, all the y outputs of the modules are summed, passing a 1 × 1 convolution (blue) and two dense layers (pink) with ReLU activation functions (green) [67]. It is relevant to note that original WaveNet modules used 1 × 1 convolutions for further compression, but due to the fixed number of filters f , this was not needed. \nGWaveNet is composed of 7 WaveNet modules with kernels k = 16 , number of filters f = 64 and dilation d = 2 i for i ∈ [0 , . . . , 6] . Because of this, the modules are highly expressive, having large receptive fields, which allows the compression of the input from 155 , 648 to 2 , 496 data points \nand 64 filters. Using a 1 × 1 convolutional layer allows further compression of the first dense layer: now it needs to compress an input of 2 , 496 to 50 data points. Thus, the number of trainable parameters of GWaveNet is 522 , 598 , 12 times fewer parameters than FindCNN .', 'C. Learning strategy': 'As we mentioned in Sections II and III, [26] implemented a curriculum learning strategy going down in maximum frequency f max . For this aim, five data sets with different average f max are built (see Table I). In this way, the model first learns easier examples, closer to the merger, and slowly transitions to harder examples, further from the merger. Each data set is composed of 22 , 600 samples for both classes, such that for each curriculum learning step we use 71% for training, 9% for validation and 20% for testing. \nIn GW data analysis, it is common to measure the loudness of the signal in terms of signal-to-noise ratio (SNR) [18]. However, [25] and [26] introduced the concept of partial inspiral signal-to-noise ratio (PISNR) to describe the corresponding SNR of the partial waveform, seen by the CNN. Thus, we use this magnitude for comparison (Table I). \nTABLE I \nIN THIS CLASSIFICATION TASK THE POSITIVE CLASS IS A GW SIMULATED SIGNAL ADDED IN RAW DETECTOR NOISE. TO ENHANCE THE LEARNING WE CREATE FIVE CURRICULUM LEARNING DATA SETS D . IN THIS TABLE, WE PRESENT THE MOST RELEVANT MAGNITUDES OF THE GW SIMULATIONS FOR EACH CURRICULUM LEARNING DATA SET: AVERAGE MINIMUM FREQUENCY f min , MAXIMUM FREQUENCY f max , NETWORK (HANFORD, LIVINGSTON AND VIRGO) SNR AND PISNR. \nTo avoid catastrophic forgetting, we use a progressive curriculum learning approach, where the training set D T and validation sets D V are defined as \nD k ≡ k ⋃ i =1 D i , (1) \nHere, ∪ denotes the set union of the data, and the curriculum step is k = { 1 , 2 , 3 , 4 , 5 } . Each curriculum step in FindCNN lasts 6 epochs, minimizing a weighted BCEWithLogitsLoss to lower the number of FPs, i.e. it is more relevant to correctly classify the noise class (negative class) than the injection class (positive class) to avoid false alerts. Thus, as in the original work, we weigh the injection class by 0.4. Regarding the optimizer, we employ AdaMax [70] with a weight decay of 10 -5 and the learning rate is 8 × 10 -5 . \nWhile GWaveNet has less trainable parameters than FindCNN , it is quite a complex model. To control the overfitting of the model we employ an early stopping algorithm where the curriculum step stops early if \n|L e val -L e best val | < ϵ, given e -e best ≥ n, (2) \nwhere L e val represents the validation loss of the current epoch e , L e best val is the validation loss of the best epoch e best , ϵ the tolerance and n the patience. After several experiments we set ϵ = 10 -4 and n = 2 . Moreover, for each curriculum step, we select the weights of the best-performing epoch, i.e. the one with the lowest validation loss, as initialization for the next step. We also employ BCEWithLogitsLoss weighted at 0 . 1 . After several experiments, AdamW optimizer [70] showed the best performance. Furthermore, we use an adaptive learning rate to allow the network to learn more slowly at more difficult steps: given an initial learning rate lr 0 = × 10 5 , at each curriculum step c ∈ [1 , 5] , we set lr c = lr 0 /c .', 'D. Modified architectures and quantization': "In this work, we used the AMD Kria KV260 Vision AI Starter Kit [62], and the Vitis AI software [71] for the deployment of the models. As Kria KV260 is designed for vision applications, it is only prepared to deal with 2-dimensional operations. Hence, to run the models in the chosen FPGA, we transformed the 1-dimensional FindCNN and GWaveNet to 2-dimensional versions: FindCNN2D and GWaveNet2D . Nonetheless, in a low-latency context, this implies reshaping within the FPGA, but such operation is not supported to run on the data processing unit (DPU), needing to be run in the non-optimized CPU of the FPGA. \nFurther modifications were performed to maximize the number of layers that could run on the DPU. For FindCNN , we added a 1 × 1 convolution at the end of the convolution blocks to cut the number of filters in half. On the other hand, GWaveNet required more changes: we moved from constant to the supported replicate padding, we changed the traditional tanh and σ activations to linearized versions, we reduced the kernels size k from 16 to 2, to allocate the model in memory and we lowered the number of filters in the sixth module to 32. This last step forced us to add a 1 × 1 convolutional layer after each Wavenet module to reduce the number of filters of each block's output to 32. However, despite these changes, at the end of the day, no versions of GWaveNet could be compiled in the DPU IP because of software limitations. \nLast but not least, due to the limited memory of FPGAs, it is necessary to quantize our models, transforming the model's weights from floating point to lower bit-widths to save memory and accelerate inference. In this work, we use the power-of-two quantization, which is a robust logarithm-based quantization strategy implemented in Vitis AI [72].", 'A. Performance': "In Section III, we mentioned that the input data has 155 , 648 data points and three channels, corresponding to Hanford, Livingston and Virgo, that we use to train, validate and test both models. In Fig. 3, we present the losses (solid line) and accuracies (dashed line) for FindCNN (top panel) and GWaveNet (bottom panel) for training (blue) and validation (yellow). We use a decision threshold of 0 . 5 to calculate the accuracies, i.e. the cutoff value used to distinguish the \nclasses is 0.5. The progressive curriculum learning step is shown by vertical dotted lines, where at each step we add samples further away from the merger. Furthermore, we mark the best-performing epoch from each curriculum learning step in red, i.e. the epoch with the best validation loss. \nFig. 3. Training and validation of FindCNN (top) and GWaveNet (bottom). We present the average training (solid) and validation (yellow) losses (solid line) and accuracies (dashed line) at the default threshold of 0.5. The standard error is calculated at 3 standard deviations σ (shadowed region). The curriculum learning step is shown by vertical dotted lines, and we mark the best-performing epoch out of every step with a red star, i.e. the epoch with the best validation accuracy up to a given tolerance (see Eq. 2). \n<!-- image --> \nFor FindCNN (top panel) we can observe that at the beginning of the first curriculum step, both training and validation losses decrease rapidly, while the accuracy increases, with the best-performing epoch at the end of the step. A similar pattern is observed in the second step. However, by the third step ( f max = 30 Hz), signs of overfitting emerge, with the best performance occurring at the beginning of the step. Overfitting becomes more pronounced in the fourth and fifth steps, evidenced by the divergence between training and validation loss and accuracy. \nSuch overfitting is not observed in GWaveNet , as training and validation are close together throughout the progressive curriculum learning steps. Additionally, the best-performing epoch typically occurs in the middle or at the end of each step. Note that, due to the early stopping method (see Section IV-C), the curriculum steps are shorter for easier-to-learn datasets, and longer for harder ones. Another notable detail is the expected increase in loss at the beginning of each curriculum \nlearning step, followed by a decrease as learning stabilises, which is characteristic of curriculum learning strategies. \nFig. 4. Training losses over the number of floating point operations (FLOPs) in logarithmic scale for FindCNN (green) and GWaveNet (pink). We present the average loss per epoch for the cumulative amount of the number of FLOPs and 3 standard deviations σ . We mark with a blue star the beginning of each curriculum step. \n<!-- image --> \nBased on these observations, we can conclude that GWaveNet outperforms FindCNN with fewer trainable parameters, potentially implying lower computational cost. In Fig. 4, we present the training loss as a function of the number of floating point operations (FLOPs) for FindCNN (green) and GWaveNet (pink), and also mark in blue the begininning of each curriculum step. We can observe that for D 1 and D 2 curriculum steps the learning is smooth as the loss is minimized. However, for the hardest curriculum learning steps the loss becomes quite unstable, particularly D 5 for both models. From this plot, it is evident that GWaveNet performs more FLOPs than FindCNN , in part because it trains for more epochs, but also since the beginning of the training due to its architecture. Still, GWaveNet achieves a better loss reduction with a comparable number of FLOPs. \nWhile accuracy and loss functions are relevant magnitudes to measure performance, we are interested in minimizing the number of FPs. To assess the separability of both classes, in Fig. 5 we show the noise (negative class, in blue) and the noise + GW signal (positive class, in yellow) of the testing set as a function of the probability of containing a GW signal. We can observe that FindCNN is more decisive than GWaveNet , as it usually provides extreme probabilities, either 0 or 1. Nonetheless, this decisiveness is a symptom of being a 'yes-classifier', as many noise samples are classified with high probabilities. On the other hand, GWaveNet is more conservative and is less keen on providing extreme probability values, a desired behaviour for this particular application. Furthermore, for GWaveNet there is a notable separation between both classes for probabilities > 0 . 81 . Note that both models use a weighted loss function: FindCNN is weighted by 0.4, while GWaveNet is weighted by 0.1. \nIn a GW experiment, it is crucial assess the number of FP. \nFig. 5. Distribution of the probability of a GW signal being present, as determined by each model. We show the noise class in yellow, and noise + signal class in blue. \n<!-- image --> \n<!-- image --> \nFig. 6. (Top) False alarm probability (FAP)-or false positive rate- in logarithmic scale as a function of the decision threshold for FindCNN (green) and GWaveNet (pink), marking the threshold FAP = 1% in red. (Bottom) Receiver operator characteristic (ROC) curve of FindCNN (green) and GWaveNet (pink). \n<!-- image --> \nA common metric in GW is the false alarm probability (FAP), also known as false positive rate. In the top panel of Fig. 6 we \nplot FAP as a function of the decision threshold for FindCNN (green) and GWaveNet (pink). We mark the target FAP = 1% in red, as defined in [26]. We can observe that GWaveNet achieves FAP = 1% for a decision threshold of 0.46, while FindCNN needs a more aggressive decision threshold of 0.86. Furthermore, GWaveNet allows to set FAP ≤ 10 -4 . \nIn ML it is standard to use the receiver operator characteristic (ROC) curve, i.e. true positive rate as a function of false positive rate for different decision threshold steps. In Fig. 4 we show the ROC curve of FindCNN (green) and GWaveNet (pink), and we can observe how GWaveNet achieves a larger area under the curve, outperforming FindCNN . \nFig. 7. True alarm probability (TAP)-or true positive rate-as a function of PISNR for FindCNN (top) and GWaveNet (bottom) for different curriculum steps-stepping on average maximum frequency f max-at false alarm probability (FAP) of 1% . \n<!-- image --> \nOne of the the main limitations of FindCNN (see Section II), is that the network forgets the previous curriculum learning steps. To compare the 'memory' of FindCNN , we reproduce Fig. 5 of [26] in Fig. 7. Here, we represent the true alarm probability (TAP), also known as the true positive rate, as a function of the average PISNR for the final models after the progressive curriculum learning strategy. In the top panel of Fig. 7, we can see that the final FindCNN has a TAP > 0 . 5 for signals with PISNR > 30 and f max = 20 , 25 , 30 Hz, the most difficult samples. As expected, TAP decreases for lower PISNR, which corresponds to the most quiet signals. Nonetheless, its performance drops ∼ 20 percentage points (p.p.) for f max = 35 , 40 Hz at ≈ 25 PISNR, due to catastrophic forgetting. In a realistic GW search, this would be unreliable, as the algorithm would not alert about the signals closer to the merger. On the other hand, we can observe how the final GWaveNet can 'remember' easier data sets while \nmaintaining a comparable performance to FindCNN for the most difficult data set f max = 20 Hz. \nAs we present in Table II, GWaveNet outperforms FindCNN by 10 p.p., but these models are unable to run on an FPGA due to memory limitations (see Section IV-D). In Table, II we also show the accuracies of the modified models before and after quantization. It is relevant to note that FindCNN2D and GWaveNet2D outperform their 1dimensional versions by ∼ 3 p.p., which could imply that Pytorch [64] is better optimized for vision applications. This could also result in better numerical stability, leading to higher accuracies, but further investigation is needed. On the other hand, the modified versions perform worse than the original ones, due to a simplification in certain operations. This is particularly notorious for GWaveNet2DModified , which has ∼ 5 p.p. less accuracy than original GWaveNet , but its performance improves ∼ 1 p.p. after quantization, likely due to the smoothing of the learnt manifold. Nonetheless, the worstperforming GWaveNet still outperforms the best-performing FindCNN by ∼ 2 p.p. \nTABLE II \nTEST ACCURACIES AT FAP SET AT 1% BEFORE AND AFTER QUANTIZATION. ALL MODELS WERE TESTED ON 22000 SAMPLES, EXCEPT \nPOST-QUANTIZATION MODELS, WHICH WERE TESTED ON 5000 SAMPLES, DUE TO TIMEOUT CONSTRAINTS. WE HIGHLIGHT IN BOLD THE BEST \nFOR THE GWAVENET2D AND GWAVENET2DMODIFIED FINDCNN MODEL AND THE WORST GWAVENET MODEL. \n.", 'B. Time, energy and economic cost': "To compare the models' efficiencies, we measured the time taken to test 22,000 samples on both the CPU and GPU and 1,000 samples on the FPGA. The results of time (ms), energy consumption (mJ), and cost ( $ /year) to run each model for CPU, GPU and FPGA are presented in Table III. As mentioned in Section IV-D, we were unable to compile GWaveNet in the FPGA due to software limitations, potentially due to the lack of support of the padding needed for causal convolutions, but further investigation is required. \nRegarding the average inference time, we can observe how all FindCNN models take ∼ 25 ms to predict a sample in CPU and GPU. However, the time in an FPGA is 4 times longer. Further exploration revealed that the bottleneck was caused by the batch normalization layer in Vitis AI [73]. Ignoring this layer decreased the FPGA times and energy consumption in half. Nonetheless, the lack of speed of the FPGA could be due to our particular low-end FPGA, which was not designed for processing high-dimensional time series data. \nFor GWaveNet , the inference time on GPU is slightly higher than FindCNN , taking 1.5 times longer. Notably, on the CPU, original GWaveNet takes ∼ 2 times more than the 2dimensional versions, making GWaveNet2DModified the fastest, due to its simplified nature. It is also interesting to note that on the GPU the original models are slightly slower than their 2-dimensional equivalents. As we hypothesized in the previous section, this could be attributed to PyTorch's enhanced optimization for vision applications. However, it is worth noting that, given the input duration of 300 s, the time consumption of all the models is remarkably low. \nRegarding the energy consumption needed for inference (see Table III), when comparing the same model across various devices, we can see that while the GPU is one of the fastest, it is also the one that consumes the most energy. As before, if we ignore the batch normalization layers of FindCNN models on the FPGA, the energy consumption is halved, being comparable to the CPU consumption. Thus, we believe that a model fully optimized for an FPGA would have a better energy performance, and this will be explored in future works. \nIn a realistic situation, these models would run through the whole observing run. In Table III, we consider the economic cost of running the models for a year, taking the average industry electricity price in the USA, as most GW clusters are there. In 2022, the average electricity cost was 0 . 0845 $/kWh [74]. As expected, the GPU is the most expensive. On the other hand, the cost of running on an FPGA is ≈ 7% cheaper than running on a CPU. Note that we only take into account the electricity prices, but adding the upfront and upkeep prices of the devices, the FPGA is by far the least expensive. Assuming the price at launch date over the number of lifetime years, under heavy use, and the prices for running FindCNN2D for a year, running on a CPU ( ≈ 495$ ) or GPU ( ≈ 465$ ) have similar expenses, whereas running on an FPGA ( ≈ 65$ ) costs over seven times less. \nLastly, running FindCNN and GWaveNet incurs comparable economic costs. This highlights GWaveNet 's efficiency, achieving better performance without significantly increasing computational resources, making it a more cost-effective model for ML-based BNS search algorithms.", 'VI. CONCLUSIONS': 'In this work to develop a novel ML-based GW detection algorithm for the early inspiral of BNS mergers, GWaveNet , inspired by the audio generating WaveNet [63]. We compare its performance to the current state-of-the-art, FindCNN [26], reproducing this work in simulated detector noise from the third observing run. Moreover, we compare their time, energy consumption and cost in CPU, GPU and FPGA. \nAs in [26], we implement a progressive curriculum learning strategy, where easier (more difficult) samples have higher (lower) maximum frequency f max , and are closer (further) to the merger, meaning that they will be detected later (earlier). Both FindCNN and GWaveNet are binary classifiers, that differentiate between noise (negative class) and noise + signal (positive class) solely utilizing the inspiral of BNS mergers. \nTABLE III \nAVERAGE TIME (MS) AND ENERGY CONSUMPTION (MJ) TO PREDICT A SINGLE SAMPLE AND ECONOMIC COST OF RUNNING EACH MODEL IN A YEAR ( $ /YEAR) FOR DIFFERENT DEVICES (SEE DETAILS IN SECTION IV-A) AT 3 STANDARD DEVIATIONS GIVEN 10 EXPERIMENTS. NOTE THAT ONLY FINDCNN2D AND FINDCNN2DMODIFIED WAS COMPATIBLE WITH OUR QUANTIZATION FRAMEWORK. \nGWaveNet has less trainable parameters than FindCNN , but it is deeper and more complex, implementing key ideas to improve its performance at a limited computational cost. Furthermore, we implement a more flexible training strategy, accommodating GWaveNet to the difficulty of each curriculum step. In terms of accuracy at FAP=1%, FindCNN achieves 66.81%, while GWaveNet achieves 76.22%, outperforming FindCNN by ∼ 10 p.p. In terms of inference time, FindCNN is ∼ 2 ms faster on GPU, but we believe GWaveNet is the preferred model due to its enhanced performance. \nRegarding the FPGA, while we were able to test the performance of FindCNN , this was not possible with GWaveNet due to software limitations. In terms of costs, the FPGA is the most sustainable option. Although the GPU is the fastest device, it also has the highest energy consumption. \nThe present work demonstrated high performance in terms of accuracy at a limited computational cost. In future works, we will study more realistic scenarios, moving to real detector data. Furthermore, other FPGA software and/or hardware, better fitting for our application, could also be explored.', 'ACKNOWLEDGMENT': "The authors thank M. Hester and R. Aaij for the fruitful and inspiring discussions during this study. This project was supported by Nikhef Laboratory, and the authors extend their gratitude to the Nikhef computing group. M.L. is supported by the research program of the Netherlands Organisation for Scientific Research (NWO). This material is based upon work supported by NSF's LIGO Laboratory which is a major facility fully funded by the National Science Foundation.", 'REFERENCES': "- [1] J. Aasi et al. , 'Advanced LIGO,' Class. Quant. Grav. , vol. 32, p. 074001, 2015.\n- [2] F. 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2024JHEP...09..097C

The baryon acoustic oscillation BAO analysis from the first year of data from the Dark Energy Spectroscopic Instrument DESI when combined with data from the cosmic microwave background CMB has placed an upperlimit on the sum of neutrino masses mSUBSUBlt 70 meV 95. In addition to excluding the minimum sum associated with the inverted hierarchy the posterior is peaked at mSUBSUB 0 and is close to excluding even the minumum sum 58 meV at 2. In this paper we explore the implications of this data for cosmology and particle physics. The sum of neutrino mass is determined in cosmology from the suppression of clustering in the late universe. Allowing the clustering to be enhanced we extended the DESI analysis to mSUBSUBlt 0 and find mSUBSUB 16090 meV 68 and that the suppression of power from the minimum sum of neutrino masses is excluded at 99 confidence. We show this preference for negative masses makes it challenging to explain the result by a shift of cosmic parameters such as the optical depth or matter density. We then show how a result of mSUBSUB 0 could arise from new physics in the neutrino sector including decay cooling andor timedependent masses. These models are consistent with current observations but imply new physics that is accessible in a wide range of experiments. In addition we discuss how an apparent signal with mSUBSUBlt 0 can arise from new long range forces in the dark sector or from a primordial trispectrum that resembles the signal of CMB lensing.

2024-09-01T00:00:00Z

['2024arXiv240500836C', '2024JHEP...09..097C', '10.1007/JHEP09(2024)097', 'arXiv:2405.00836', '10.48550/arXiv.2405.00836']

['Cosmology of Theories BSM', 'Early Universe Particle Physics', 'Neutrino Interactions', 'Non-Standard Neutrino Properties', 'Astrophysics - Cosmology and Nongalactic Astrophysics', 'High Energy Physics - Phenomenology', 'High Energy Physics - Theory']

No s is Good News

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML']

https://arxiv.org/pdf/2405.00836.pdf

{'No ν s is Good News': 'Nathaniel Craig 1 , 2 , Daniel Green 3 , Joel Meyers 4 , and Surjeet Rajendran 5 \n- 1 \nDepartment of Physics, University of California, Santa Barbara, CA 93106, USA 2 Kavli Institute for Theoretical Physics, Santa Barbara, CA 93106, USA 3 Department of Physics, University of California, San Diego, La Jolla, CA 92093, USA 4 Department of Physics, Southern Methodist University, Dallas, TX 75275, USA 5 Department of Physics & Astronomy, The Johns Hopkins University, Baltimore, MD 21218, USA', 'Abstract': 'The baryon acoustic oscillation (BAO) analysis from the first year of data from the Dark Energy Spectroscopic Instrument (DESI), when combined with data from the cosmic microwave background (CMB), has placed an upper-limit on the sum of neutrino masses, ∑ m ν < 70 meV (95%). In addition to excluding the minimum sum associated with the inverted hierarchy, the posterior is peaked at ∑ m ν = 0 and is close to excluding even the minumum sum, 58 meV at 2 σ . In this paper, we explore the implications of this data for cosmology and particle physics. The sum of neutrino mass is determined in cosmology from the suppression of clustering in the late universe. Allowing the clustering to be enhanced, we extended the DESI analysis to ∑ m ν < 0 and find ∑ m ν = -160 ± 90 meV (68%), and that the suppression of power from the minimum sum of neutrino masses is excluded at 99% confidence. We show this preference for negative masses makes it challenging to explain the result by a shift of cosmic parameters, such as the optical depth or matter density. We then show how a result of ∑ m ν = 0 could arise from new physics in the neutrino sector, including decay, cooling, and/or time-dependent masses. These models are consistent with current observations but imply new physics that is accessible in a wide range of experiments. In addition, we discuss how an apparent signal with ∑ m ν < 0 can arise from new long range forces in the dark sector or from a primordial trispectrum that resembles the signal of CMB lensing.', '1 Introduction': 'The cosmological measurement of the sum of neutrino masses, ∑ m ν , is one of the most anticipated results from the coming generation of cosmic surveys [1-3]. From the measurement of neutrino flavor oscillations [4], which precisely determine the mass-squared splittings between neutrino mass eigenstates, it can be inferred that the sum of neutrino masses is necessarily greater than 58 meV. This provides a concrete prediction within the standard cosmological model that should be measurable (or detectable) with planned observations [5,6]. \nThe Dark Energy Spectroscopic Instrument (DESI) [7] is expected to provide the necessary increase in sensitivity to ∑ m ν to measure the minimum sum at 2 to 3 σ [5, 6]. Cosmological measurements of neutrino mass rely on the measurement of the clustering of matter on scales smaller than the free-streaming length of neutrinos [1]. A universe containing massive neutrinos will exhibit suppressed matter clustering compared to a universe with only massless neutrinos. This measurement can be achieved by combining observations of the cosmic microwave background (CMB) with the measurement of the baryon acoustic oscillations (BAO). The amplitude of clustering can be determined from the measurement of the CMB lensing power spectrum, and this amplitude is compared to what would be expected in a universe with only massless neutrinos [8]. In the absence of massive neutrinos, the amplitude of matter clustering is determined by the matter density and the primordial amplitude of scalar fluctuations. Measurements of the CMB angular power spectra allow for a determination of the primordial fluctuation amplitude. BAO measurements are needed to measure the abundance of non-relativistic matter to sufficient accuracy to isolate the effect neutrino mass [9]. \nThe release of the first year BAO analysis with DESI [10], combined with data from the CMB(Planck 2018 [11,12] and ACT DR6 lensing [13,14]), showed a remarkable upper-limit on ∑ m ν , reaching \n∑ m ν < 70 meV (95%) . (1.1) \nIn this paper, we will explore the current constraints on ∑ m ν and what an exclusion of ∑ m ν = 58 meV would mean for cosmology and particle physics. First, we will examine the current measurement and how it depends on different types of surveys. One particularly noteworthy aspect of the DESI measurement is that it appears to favor ∑ m ν < 0, though that region of parameter space was excluded from the DESI analysis by imposing a prior that ∑ m ν is positive. Although negative neutrino masses are unphysical, a preference in the data for ∑ m ν < 0 may simply reflect an excess of clustering in the late universe, \nThis is consistent with an earlier constraint from (e)BOSS of ∑ m ν < 82 meV [15] using CMB+BAO+Shape parameters (see also [16]). The DESI result is sufficient to exclude the minimum mass for an inverted neutrino mass hierarchy, 100 meV, at ∼ 3 σ . However, what is also noteworthy is that the posterior peaks at ∑ m ν = 0 and is very close to putting 58 meV in tension with observations. \nFigure 1: Posterior of neutrino mass in eV inferred from Planck + ACT Lensing + DESI data. The blue line shows constraints on a model with a physical neutrino mass, the orange line shows constraints where the neutrino mass is parametrized as an effect on the CMB lensing power spectrum and restricted to be positive, and the green line shows constraints on a parametrized neutrino mass that is allowed to be negative. The best fit for the parametrized neutrino mass is ∑ ˜ m ν = -160 meV, and the minimal neutrino mass of 58 meV is disfavored at 3 σ . For details about the parametrization of negative neutrino mass and the data sets used, see Section 2.2. \n<!-- image --> \n∑ \nrather than a deficit caused by free streaming neutrinos. We use this idea to define a neutrino mass, ∑ ˜ m ν , that is allowed to be negative and perform the same analysis as DESI without the positive mass prior. We find that that data does prefer negative mass, ∑ ˜ m ν = -160 ± 90 meV (68%), and corresponds to a 3 σ exclusion of the minimum neutrino mass. The full posterior is shown in Figure 1. \nAn absence of the neutrino mass signal, while forbidden in the Standard Model (plus neutrino masses), could be a natural consequence of a wide variety of beyond the Standard Model (BSM) scenarios. The most straightforward mechanisms to eliminate the signal would be to eliminate the SM neutrinos via decay (or annihilation), cool the neutrinos so that they behave like dark matter, or change their mass over cosmological history. Simple models for all three scenarios can be derived from new interactions in the neutrino sector \nThe preference of the current measurement for negative ∑ m ν is particularly important as it affects the bias in the measurement of cosmic parameters, particularly the optical depth, τ , that would be required to explain the current limits. For ℓ > 30, the CMB is only sensitive to the combination A s e -2 τ , where A s is the amplitude of primordial scalar fluctuations. The determination of τ is therefore essential for determining A s and suppression of power a late times, but requires (challenging) large angular scale measurements of the CMB. It is plausible that ∑ m ν = 0 could be explained by a statistical or systematic shift in τ , but it is far more challenging to explain ∑ m ν = -160 meV in this way. \nthat are weakly constrained by experiments. On the other hand, the CMB does provide stringent constraints on the parameter space of these models, as measurements of N eff are in good agreement with the expected temperature [17] and free-streaming [18-20] of the cosmic neutrino background (C ν B). Nevertheless, there have been hints of neutrino interactions [21-25] in cosmic data that may also point to new physics of this kind. \nThis paper is organized as follows: In Section 2, we review the measurement of ∑ m ν and extend the analysis to negative masses. We discuss what shifts in cosmic data would be required to make these measurements consistent with conventional neutrino physics. In Section 3, we present models that could explain ∑ m ν = 0 with new physics in the neutrino sector. In Section 4, we present models that could explain a cosmological inference of negative neutrino masses. We conclude in Section 5. Appendix A, we review the physics origin of the suppression of structure due to massive neutrinos. \nNegative neutrino masses, ∑ m ν < 0, are representative of enhanced clustering of matter, rather than any physical property of the neutrinos themselves. This kind of enhanced clustering can be achieved by changing the long range forces that act on matter. We discuss one simple mechanism, which is to introduce a new scalar force that acts only on the dark matter. Such forces are more weakly constrained than fifth forces acting on SM particles and thus could explain our signal without being in tension with other constraints. Alternatively, a CMB lensing measurement with ∑ m ν < 0 points to a larger than expected CMB trispectrum, which could result from a non-zero primordial trispectrum. These scenarios will all be testable with current and/or future cosmic data.', '2.1 How Neutrino Mass is Measured': "In order to understand what an apparent measurement of ∑ m ν = 0 would mean, we first need to review exactly what measurements allow us to infer ∑ m ν (see also [26, 27] for review). We will assume that ∑ m ν ≈ 60 meV, as this is the minimum sum consistent with neutrino oscillation experiments and is therefore the minimum value that would need to be excluded in order to favor ∑ m ν = 0. \nCosmic neutrinos are relativistic in the early universe, but become non-relativistic when their propagation speed, c ν , drops well below the speed of light. In a ΛCDM + m ν cosmology, the typical neutrino speed is given by \nc ν = ⟨ p ν ⟩ m ν = 3 T ν m ν ≈ 1 . 0 × 10 -2 ( 50 meV m ν ) (1 + z ) , (2.1) \nwhere we have set c = 1. As a result, the redshift where the heaviest neutrino becomes non-relativistic is z ν ≈ 100. For z < z ν , the energy density of neutrinos redshifts like \nnon-relativistic matter so that \nΩ m = Ω c +Ω b +Ω ν . (2.2) \nHowever, the neutrinos are still sufficiently hot that they do not cluster on scales below their effective Jeans scale. In terms of wavenumber, this free-streaming scale is given by \nk fs = √ 3 2 aH c ν = 0 . 04 h Mpc -1 × 1 1 + z ( ∑ m ν 58 meV ) . (2.3) \nBecause neutrinos don't cluster, the amplitude of clustering of matter, defined by the matter power spectrum \nP ( k ) = ⟨ δ m ( ⃗ k ) δ m ( ⃗ k ' ) ⟩ ' , (2.4) \nis suppressed on scales smaller than the neutrino free-streaming scale k ≫ k fs \nP ( ∑ m ν ) ( k ≫ k fs , z ) ≈ ( 1 -2 f ν -6 5 f ν log 1 + z ν 1 + z ) P ( ∑ m ν =0) ( k ≫ k fs , z ) , (2.5) \nwhere f ν = Ω ν / Ω m is the fraction of non-relativistic matter in the form of neutrinos, δ m ≡ δρ m / ¯ ρ m is the density contrast of non-relativistic matter, and the prime on the correlation function means that the delta function has been omitted. The suppression in this formula is the result of two distinct physical effects (see Appendix A for a derivation). The first term, -2 f ν , reflects the reduced fraction of matter that is actually clustering. The second, -6 5 f ν log 1+ z ν 1+ z , is due to reduced rate of growth of the dark matter perturbations in the presence of matter that doesn't cluster. Using \nΩ ν h 2 = 6 × 10 -4 ( ∑ m ν 58 meV ) → f ν ≈ 4 × 10 -3 , (2.6) \nthe suppression of the matter power spectrum at z = 1 is expected to be \nP ( ∑ m ν =58meV) ( k ≫ k fs , z ) ≈ (1 -0 . 02) P ( ∑ m ν =0) ( k ≫ k fs , z ) . (2.7) \nTherefore, the signal we are looking for is a 2% suppression of power on small scales around z = O (1). \nGalaxy surveys like DESI do not directly measure P ( k ) and instead primarily measure the clustering of galaxies. The power spectrum of galaxy overdensity has an overall amplitude that depends on the details of galaxy formation, and the baryonic physics inherent in galaxy formation is understood with insufficient precision to directly extract the amplitude of P ( k ) from these measurements. The best current measurements of the matter power spectrum come from gravitational lensing of the CMB. The CMB lensing convergence \npower spectrum C κκ ℓ is given in the Limber approximation by [28] \nC κκ ℓ ≈ 2 π 2 ℓ ∫ η 0 η ∗ η d η P Ψ ( ℓ/ ( η 0 -η ); η ) ( η ∗ -η ( η 0 -η ∗ )( η 0 -η ) ) 2 , (2.8) \nwhere η is the conformal time with η ⋆ and η 0 denoting the times of recombination and z = 0 respectively. We also defined P Ψ as power spectrum of the Weyl potential, Ψ, which can be written in terms of the matter power spectrum as \nP Ψ ( k ; η ) = 9Ω 2 m ( η ) H 4 ( η ) 8 π 2 P ( k ; η ) k . (2.9) \nUsing the fact that the matter power spectrum is proportional to the primordial scalar amplitude A s , we see that the amplitude of the CMB lensing power spectrum scales as \nC κκ ℓ ∝ (Ω m h 2 ) 2 A s ( 1 -0 . 02 f ν 4 × 10 -3 ) . (2.10) \nTherefore, in order to measure a three-percent suppression of the lensing power spectrum, we must determine the physical matter density Ω m h 2 (where h = H 0 / (100 km s -1 Mpc -1 ) is the dimensionless Hubble constant) and the primordial scalar amplitude A s to much better than three-percent accuracy. \nThe main impact of DESI on the cosmological neutrino mass constraint is to provide a precise measurement of ω m ≡ Ω m h 2 through the constraint on the expansion history from BAO. The impact of changing ω m on the CMB lensing power spectrum is shown in Figure 2 (for ∑ m ν = 0 and compared to the change from introducing ∑ m ν > 0). The reduction of ω m by 1 . 7% is roughly equivalent to introducing ∑ m ν = 58 meV, which implies that a 2 σ measurement of the minimum sum requires roughly 0.8% precision in the measurement of ω m .", '2.2 Negative Neutrino Mass': "The physical sum of neutrino masses is of course restricted to be positive. However, the combination of cosmological observables that we use to infer the mass of neutrinos are not restricted in this manner. We show in this subsection that the CMB+DESI data in fact prefer a negative neutrino mass (already hinted at in eBOSS [29]), corresponding to increased matter clustering compared to a model with only massless neutrinos. \nIn order to measure the preference of cosmological data for negative neutrino mass, we require an implementation of the effects of neutrino mass that is allowed to take either sign. The Boltzmann codes CAMB [30, 31] and CLASS [32] model neutrino mass in a way that is subject to the physicality constraint ∑ m ν > 0. We modified CAMB to include a new parameter, ∑ ˜ m ν , which is designed to mimic the effects of neutrino mass, but which \nFigure 2: Comparison of the fractional change to the CMB lensing power spectrum from changes to Ω m h 2 and the introduction of a non-zero neutrino mass. \n<!-- image --> \nis not restricted to be positive. Our new parameter simply scales the CMB lensing power spectrum in the same manner that would be expected from ∑ m ν . Specifically, we determine the fractional change A ℓ ( ∑ ˜ m ν ) ≡ C κκ ℓ [ ∑ m ν ] /C κκ ℓ [ ∑ m ν = 0] at fixed values of H 0 , ω m , and ω b . Once calibrated on positive values of neutrino mass, the effects of ∑ ˜ m ν can then be straightforwardly calculated for negative values as well. In the ΛCDM+ ∑ ˜ m ν cosmology, observables are computed with the physical ∑ m ν = 0 and the CMB lensing power spectrum is computed as C κκ ℓ = A ℓ ( ∑ ˜ m ν ) C κκ ℓ [ ∑ m ν = 0]. The temperature and polarization CMB power spectra are lensed using this modified CMB lensing power spectrum such that the set C TT ℓ , C TE ℓ , C EE ℓ , C κκ ℓ is calculated self-consistently for each point in parameter space. \nThis prescription is very similar, though not identical, to the effects of the physical neutrino mass in the regime ∑ m ν > 0. In particular, the physical neutrino mass in the ΛCDM+ ∑ m ν cosmology contributes to the non-relativistic matter density today Ω m = Ω b + Ω c + Ω ν . In our ΛCDM+ ∑ ˜ m ν cosmology, there is no neutrino contribution to Ω m . As a result, we anticipate that ∑ ˜ m ν should exhibit slightly weaker constraints than the physical ∑ m ν when measured using the same data combination. To check this, we derive constraints on three cosmological models: a model with a physical neutrino mass ΛCDM+ ∑ m ν , a model with our parametrized neutrino mass restricted to positive values ΛCDM+( ∑ ˜ m ν > 0), and finally a model with our parametrized neutrino mass \nwith no restriction on sign ΛCDM+ ∑ ˜ m ν . We analyze each model using the same data combination. \nThe results are presented in Table 1 and Figure 3. Notice that the parameter constraints in the ΛCDM+ ∑ m ν and ΛCDM+( ∑ ˜ m ν > 0) models are nearly identical, showing only slightly weaker constraints on ∑ ˜ m ν as compared to the physical ∑ m ν . This excellent agreement justifies our prescription for modeling the effects of neutrino mass, with the slightly weaker constraints on ∑ ˜ m ν expected from the differing treatment of Ω m in the two models. Notice that in the ΛCDM+ ∑ ˜ m ν model, the best-fit value for ∑ ˜ m ν is -160 meV, showing a preference for negative neutrino mass, and disfavoring even the minimal sum of neutrino masses inferred from flavor oscillation experiments at 3 σ . \nBoltzmann calculations were carried out using our modified version of CAMB [30, 31]. We utilized the likelihood for CMB temperature and polarization from Planck's 2018 data release [11], along with the combination of ACT DR6 [13,14] and Planck CMB lensing [12], and DESI BAO [10,33,34]. This combination of data is the same as that used by the DESI team to derive cosmological constraints [10]. Our analysis was performed with cobaya [35], using the Markov chain Monte Carlo sampler adapted from CosmoMC [36,37] using the fast-dragging procedure [38]. Analyses were run until the Gelman-Rubin statistic was R -1 < 0 . 01. \nWe also note in passing that in the ΛCDM+ ∑ ˜ m ν model, the best-fit value for S 8 ≡ σ 8 (Ω m / 0 . 3) 0 . 5 is lower than in ΛCDM+ ∑ m ν by about 1 . 5 σ and has 40% larger error bars (and is also smaller than the value inferred with Planck in the ΛCDM model, for which S 8 = 0 . 830 ± 0 . 013 [17]), representing a somewhat smaller S 8 tension [39] when neutrino mass is allowed to be negative.", 'Optical Depth': "The measurement of A s is limited by our understanding of the optical depth to reionization, τ . Thomson scattering of CMB photons into and out of the line of site by free electrons present after reionization suppresses the amplitude of CMB fluctuations. The observed amplitude of the CMB power spectrum is thereby reduced on small angular scales. CMB observations primarily constrain the combination [17] \nA s e -2 τ = (1 . 884 ± 0 . 011) × 10 -9 . (2.11) \nThis should be contrasted with the much less precise measurement of the primordial amplitude [17] \nA s = (2 . 100 ± 0 . 030) × 10 -9 . (2.12) \nTable 1: Parameter constraints from Planck + ACT lensing + DESI BAO in the three models described in the text. All constraints are given as 68% limits, except for the upper limits on the neutrino mass when it is restricted to be positive, which are reported as 95% CL. Values of neutrino mass are reported in eV and H 0 in kms -1 Mpc -1 . In the ΛCDM+ ∑ ˜ m ν model, the data favors a negative neutrino mass and disfavors the minimal physical neutrino mass of 58 meV at 3 σ . \n≡ \n± \n± \n± \nNoting that for these same analyses, \nτ = 0 . 0544 ± 0 . 0073 , (2.13) \nthe error on A s can be directly attributed to the error in τ and not the error in the measurement of A s e -2 τ . \nOf all the cosmological parameters defining ΛCDM, the optical depth is the most challenging to measure. For ℓ > 30, its effects on the CMB are completely degenerate with A s . It is only on large angular scales that the optical depth leaves a unique imprint, through the production of CMB polarization and the associated 'reionization bump' in the polarization power spectrum. The history of these measurements, shown in Figure 4 has involved significant changes in the central value with relatively small changes in sensitivity. \nIt is natural to wonder if the apparent measurement of ∑ m ν = -160 meV could also be attributed to an error in the measurement of τ . For this to be possible, we would need the true value of A s to be roughly 8.8% larger, so that the current measurement of the lensing includes the expected suppression of P ( k ) relative to A s . This would require a \nFigure 3: Triangle plot showing parameter constraints from Planck + ACT lensing + DESI BAO in three models described in the text and shown in Table 1. For the purposes of this plot, we treat the physical neutrino mass and our parametrized version on the same footing. Dashed lines show vanishing neutrino mass ∑ m ν = 0 and the minimal sum of neutrino mass ∑ m ν = 58 meV. Values of neutrino mass are reported in eV and H 0 in km s -1 Mpc -1 . \n<!-- image --> \n∑ \nvalue of the optical depth larger than that inferred from Planck τ = τ Planck + δτ , such that 2 δτ ≈ 0 . 088. Using τ Planck18 = 0 . 054 and σ Planck18 = 0 . 0073, this would require \nτ true ≥ 0 . 098 = τ Planck18 +6 . 0 σ Planck18 . (2.14) \nτ \nFigure 4: The historical measurement of the optical depth, τ , from WMAP data [40-44] and Planck [17, 45-48] by year of publication. The horizontal solid (dashed) blue line indicates the central value of τ that would be to move the peak of the DESI+CMB the ∑ m ν posterior from -160 meV (0 meV) to 58 meV. \n<!-- image --> \nSimilarily, if we take τ = 0 . 051 ± 0 . 006 or τ = 0 . 058 ± 0 . 006 from [47] and [48], we would require shifts of 7 . 8 σ or 6 . 7 σ respectively. For comparison, to shift ∑ m ν = 0 to 58 meV only requires A s to be 2.5% larger, which can be accomplished by a τ = 0 . 066 which is a 1.7 σ upward shift. Both lines are shown in Figure 4 and are consistent with some historical measurements; thus a systematic offset in the more recently inferred values of the optical depth is a plausible explanation for preference for negative neutrino mass. Yet, due to the magnitude of the difference it is unlikely to be the result of a statistical fluctuation. \nOne of the key challenges with the optical depth is that it is very difficult to measure with ground-based surveys (although it is currently being pursued, for example, by the Cosmology Large Angular Scale Surveyor (CLASS) collaboration [49,50]). The results of DESI alone point to the need for a confirmation of the Planck measurement of the optical depth, and in principle an improvement to the cosmic variance limit of σ ( τ ) = 0 . 002. This would be possible with another satellite, such as LiteBird [51]. However, there is the more immediate potential of balloon-based observations which could reach similar levels of sensitivity [52]. Other longer term possibilities include using measurements of crosscorrelations between the CMB and galaxy surveys to eliminate the need for an optical depth measurement [53,54] or to use measurement the patchy kinetic Sunyaev-Zeldovich effect to constrain the physical model of reionization [55-57], both of which might be possible with CMB-S4 [58]. \nFigure 5: The historical measurement of the matter density, Ω m h 2 , from WMAP data [40-43] and Planck [17, 45] by year of publication. The black Planck+DESI point is the result of our reanalysis of ΛCDM+ ∑ m ν using the same priors as [10]. The horizontal solid (dashed) blue line indicates the central value of Ω m h 2 that would be required to move the peak of the CMB+DESI ∑ m ν posterior from -160 meV (0 meV) to 58 meV. \n<!-- image -->", 'Matter Content': 'The measurement of the matter density ω m is equally important to the measurement of ∑ m ν as the optical depth. The primary CMB directly determines ω m through its influence on the height and locations of the acoustic peaks. This is, in part, why the CMB alone is capable of producing very stringent bounds on ∑ m ν , e.g. ∑ m ν < 240 meV (95%) from Planck TTTEEE + lensing [17]. \nImprovements in the measurement to ω m beyond the CMB has been driven by BAO measurements, most recently with DESI. As shown in Figure 5, the BAO has played a significant role in reducing uncertainty, but has been consistent with the measurements from the CMB data on which the BAO is calibrated. Like the measurement of the optical depth, there was a significant improvement from WMAP to Planck. However, unlike τ , the Planck measurements of ω m have been stable with the inclusion of more data, including from polarization and the BAO. \nThe measurement of ω m needs to be accurate to less than 0.8% in order to permit a reliable measurement of ∑ m ν . While this is a high standard, we have the benefit that ω m will be measured using a number of different CMB surveys that can be combined with several large-scale structure (LSS) surveys. Any large shifts in ω m due to systematic effects \nshould be different for different surveys and thus from planned measurements alone, we should be able to determine a robust value of ω m and/or identify systematic issues. This is in sharp contrast to the optical depth, of which Planck is currently the only measurement at the needed accuracy, and it is unclear if near term observations will reproduce or exceed their sensitivity. \n̸ \nIt is well known that introducing dynamical dark energy, e.g. in the form of w 0 = -1 and w a = 0, significantly weakens 1 the neutrino mass constraints [61]. This is for the simple reason that if we allow for more free parameters in the expression for H ( z ) at low redshifts, we cannot measure ω m at the accuracy needed to determine ∑ m ν . However, this will typically require fairly significant changes to the content and history of the universe. Leaving the content of the universe fixed, we will see that the neutrino mass signal can be explained with changes to the micro-physics in the neutrino and/or dark sector that otherwise leave the rest of cosmological history intact. \n̸', 'CMB Lensing': "Weak gravitational lensing of the CMB perturbs the path of photons, so that the apparently location on the sky is perturbed from the true direction ˆ n ' = ˆ n + ⃗ α (ˆ n ), where ⃗ α (ˆ n ) is deflection angle [28]. Since the gravitational lensing is time-independent on the scales of observations, the maps of the CMB temperature anistropies (for example) are also modified by the same effect, \nT lensed (ˆ n ) = T unlensed (ˆ n + ⃗ α (ˆ n )) . (2.15) \nThe deflection angle is related to the gravitational potential via the lensing potential ϕ (ˆ n ), via ⃗ α = ∇ ˆ n ϕ and \nϕ (ˆ n ) ≡ -2 ∫ η 0 η ⋆ d η η -η ⋆ ( η 0 -η ⋆ )( η 0 -η ) Ψ(( η 0 -η )ˆ n , η ) , (2.16) \nwhere Ψ is the Weyl potential and η ⋆ is the conformal time of CMB last scattering. \nWe can understand the main influence of lensing on the CMB by Taylor expanding \nT lensed (ˆ n ) ≈ T unlensed (ˆ n ) + ∇ ˆ n T · ∇ ˆ n ϕ + O ( ϕ 2 ) . (2.17) \nFor a small patch of sky, we can Fourier transform ˆ n → ⃗ ℓ so that the dot product is replaced with a convolution \nT lensed ( ⃗ ℓ ) ≈ T unlensed ( ⃗ ℓ ) -∫ d 2 ⃗ L 2 π ⃗ L · ( ⃗ ℓ -⃗ L ) T unlensed ( ⃗ ℓ -⃗ L ) ϕ ( ⃗ L ) + O ( ϕ 2 ) . (2.18) \nThis will induce a non-vanishing correlation between different Fourier modes, \n〈 T lensed ( ⃗ ℓ ) T lensed ( ⃗ L -⃗ ℓ ) 〉 T = δ ( ⃗ L ) C TT, unlensed ℓ + 1 2 π [ ( ⃗ L -⃗ ℓ ) · ⃗ LC TT, unlensed | ⃗ L -ℓ | + ⃗ ℓ · ⃗ LC TT, unlensed ℓ ] ϕ ( ⃗ L ) + O ( ϕ 2 ) , (2.19) \n̸ \nwhere C TT, unlensed ℓ is in the unlensed temperature power spectrum, and the subscript T on the left-hand side refers to an ensemble average over the unlensed CMB temperature realization. As the ⃗ L = 0 correlations would vanish without lensing, we can reconstruct ϕ ( ⃗ L ) from the presence of these correlations [62]. Estimating the CMB lensing power spectrum can therefore be achieved by measuring the temperature four-point function. Lensing also induces a measurable smoothing effect on the acoustic peaks of the CMB power spectrum, from convolving the unlensed power spectrum with the lensing power spectrum at second order. \nOnce the lensing potential is reconstructed, it can be used to calculate the power spectrum of lensing, remove lensing from the CMB maps [63-65], and/or cross-correlate with other data. For the neutrino mass, the only piece of information we need it the power spectrum of the lensing map C ϕϕ L . This is the same information that is contained in the connected trispectrum of the temperature, as ϕ ( ⃗ L ) was determined from a temperature twopoint function. As shown in Figure 2, ∑ m ν = 58 meV causes a roughly 2-3% suppression of the lensing power, while ∑ m ν = -160 meV is a 6-9% enhancement. \nThe reconstruction of the lensing map is a non-trivial process that could be influenced by other effects that correlate modes in the temperature maps. For example, it is known that the non-Gaussian statistics of unresolved foregrounds can induce biases in these maps [66]. Furthermore, these same correlations are relevant to the covariance of the primary CMB and thus are important for measurements of any other cosmological parameters. Yet, it is also noteworthy that the neutrino mass measurement not sensitive to non-linear effects in the matter power spectrum. Using current CMB data, the lensing map is too noisy to resolve modes that are strongly influenced by non-linear evolution. Yet, even with future data, such as from CMB-S4, these modes can be removed from the analysis with no loss of sensitivity to ∑ m ν [26].", '3 Vanishing Neutrino Mass': 'In this section, we will explore mechanisms for eliminating the signal of ∑ m ν ≥ 58 meV, while being consistent with ∑ m ν ≥ 0. The common element of all these models is that we will reduce or eliminate the suppression of power by directly altering the behavior of the neutrinos. In the next section, we will consider changes to the growth of structure beyond just the neutrinos, which could allow for an apparent enhancement of structure, which might be interpreted as ∑ m ν < 0.', '3.1 Decays': "Perhaps the most obvious way to reconcile a cosmological indication of ∑ m ν = 0 with the nonzero masses implied by neutrino oscillations is if massive neutrinos decay into massless degrees of freedom on cosmological timescales. While the two heaviest neutrino mass eigenstates are already unstable within the Standard Model, their lifetimes are far greater than the age of the universe ( τ ν ∝ ⟨ h ⟩ 4 /m 5 ν , where ⟨ h ⟩ ≃ 246 GeV is the vacuum expectation value of the Higgs field). Neutrino decays on cosmologically relevant timescales would therefore be unambiguous evidence of new physics, above and beyond the origin of neutrino masses. \nWhile decays involving photons are strongly constrained by CMB spectral distortions [67], decays into dark radiation (and either an active or sterile neutrino) are consistent with current limits over a wide range of lifetimes. A lower bound comes from the requirement that the decays and inverse decays of relativistic neutrinos do not prevent free streaming, τ ν ≳ 4 × 10 6 s ( m ν / 0 . 05 eV) 5 [68]. On the upper end, the maximum neutrino lifetime that can erase the cosmological signal of neutrino masses depends on the mass spectrum [69-73]. For the minimum masses implied by neutrino oscillations, the lifetime of the massive neutrinos should be roughly an order of magnitude shorter than the age of the universe, τ ν ≲ 4 × 10 16 s. For the sum of neutrino masses to be observable at KATRIN (sensitive to m ν e as small as 0.2 eV [74], which translates to ∑ m ν ∼ 0 . 6 eV), the maximum lifetime of all the active neutrinos should be around two orders of magnitude smaller, τ ν ≲ 4 × 10 14 s. \nThere are a variety of possible decay modes. Two-body decays of massive neutrinos necessarily proceed into a fermion and a boson, with the former either an active or sterile neutrino, and the latter a scalar ϕ or vector Z ' . As the masses of the bosons increase, the two-body decay channels close and the bosons instead mediate three-body decays into active and sterile neutrinos. As the viable parameter space for three-body decays is considerably more constrained, here we will restrict our attention to the two-body decays. \nIn the neutrino mass basis, decays into a (pseudo)scalar arise via couplings of the form \nL ϕ ⊃ λ ij 2 ¯ ν i ν j ϕ + ˜ λ ij 2 ¯ ν i γ 5 ν j ϕ +h . c . ( i, j = 1 , . . . 4) , (3.1) \nwhere i = 1 , 2 , 3(4) denote the primarily active (sterile) neutrino mass eigenstates; for definiteness we assume the neutrinos are Majorana. Assuming the lightest active or sterile neutrinos are much lighter than the heavy neutrinos, the corresponding lifetime for decay via the pseudoscalar coupling is [71] \nτ ( ν i → ν j ϕ ) ≃ 7 × 10 17 s × ( 0 . 05 eV m ν i ) ( 10 -15 ˜ λ 2 ij ) 2 . (3.2) \nFor two-body decays into active neutrinos to reconcile oscillation splittings with a cosmological measurement of ∑ m ν = 0, necessarily m ν 3 ≈ 0 . 05 eV. Erasing the energy density in massive neutrinos without spoiling free streaming then implies 4 × 10 -15 ≲ λ, ˜ λ ≲ 4 × 10 -10 [75-79]. The situation is analogous for decays into sterile neutrinos, although in this case the overall mass scale of active neutrinos may be significantly increased [72]. \nWhile the dimensionless couplings required to erase the cosmological neutrino mass signal are small, they are nicely compatible with expectations from UV-complete models. For example, models with spontaneously broken global horizontal lepton flavor symmetries [80] give rise to a goldstone mode coupling to neutrinos as in Eq. (3.1). In such models the off-diagonal pseudoscalar couplings ˜ λ ij are generated via mixing between heavy sterile and light active neutrinos of order ˜ λ ij ≃ √ m ν i m ν j /f , where f is the scale of spontaneous symmetry breaking. The desired size of ˜ λ corresponds to 50 MeV ≲ f ≲ 5 TeV, implying new physics associated with neutrino mass generation around the TeV scale. \nAlternately, decays into a vector arise via couplings of the form \nL Z ' ⊃ g L ij 2 Z ' µ ¯ ν i γ µ P L ν j + g R 44 2 Z ' µ ¯ ν 4 γ µ P R ν 4 +h . c . ( i, j = 1 , . . . 4) , (3.3) \nwhich set a lifetime via two-body decays of order \nτ ( ν i → ν j Z ' ) ≃ 7 × 10 17 s × ( 0 . 05 eV m ν i ) 3 ( m Z ' /g L ij 50 TeV ) 2 . (3.4) \nFor two-body decays into active neutrinos to erase the cosmological neutrino mass signal without spoiling free streaming requires 100 MeV ≲ m Z ' /g L ≲ 10 TeV, along with m Z ' ≪ m ν i . The situation is analogous for decays into sterile neutrinos, modulo the greater freedom in the active neutrino masses. \nAs in the scalar case, the dimensionless couplings required to erase the cosmological neutrino mass signal are nicely compatible with expectations from UV-complete models. For instance, in a model with a gauged lepton flavor symmetry such as U (1) L µ -L τ broken at a scale f , we have f = m Z ' /g L and the preferred range of decay couplings once again suggests new physics around the TeV scale. The preferred range of couplings and masses is also compatible with current limits, with the most stringent direct bounds m Z ' /g L > 1 . 3 GeV coming from monolepton + missing energy searches at the LHC [81].", '3.2 Annihilation': 'The cosmological neutrino mass signal may alternately be erased if the cosmological population of massive neutrinos annihilates away into light states at late times [82]. For simplicity, consider the case of a single light (pseudo)scalar coupling to neutrinos, as in Eq. (3.1). Whereas neutrino decays require off-diagonal couplings in the mass basis, an- \nnihilation is efficient even when the largest couplings are diagonal. For annihilations to effectively deplete the relic neutrino abundance, the couplings λ, ˜ λ should be large enough to keep ϕ in thermal equilibrium with neutrinos until after the neutrinos become nonrelativistic, at which point the neutrinos annihilate efficiently via νν → ϕϕ . The relic neutrino population is effectively erased provided λ, ˜ λ ≳ 10 -5 . However, such large couplings bring ϕ into thermal equilibrium before big bang nucleosynthesis (BBN), and the model is ruled out by a combination of free-streaming requirements and CMB bounds on N eff . \nHowever, mild variations on this scenario remain consistent with current cosmological bounds [83]. One natural possibility is for the active neutrinos to coannihilate into sterile neutrinos via a scalar or pseudoscalar ϕ through the couplings in Eq. (3.1). Avoiding efficient coannihilation while neutrinos are still in thermal equilibrium implies m ϕ ≲ MeV, while ending coannihilation before recombination implies m ϕ ≳ eV. Within this mass range, efficient conversion requires λ, ˜ λ ≳ 5 × 10 -11 × ( m ϕ keV ) 1 / 2 , while preserving free streaming at recombination requires λ, ˜ λ ≲ 5 × 10 -3 × ( m ϕ keV ) . \nThe above bounds are based on the direct coupling of active neutrinos to ϕ . These are significantly weakened if the active neutrino couples to ϕ via light right handed neutrinos. In this case, in the early universe, the mixing of relativistic active neutrinos to the right handed neutrino is suppressed by the small neutrino mass, suppressing annihilation at early times. At low redshift, the mixing of non-relativistic neutrinos is unsuppressed, leading to enhanced annihilation that can also explain this signal. We briefly comment on this possibility in Section 3.4.', '3.3 Cooling and Heating': "The origin of the neutrino mass signal in the matter power spectrum is that the neutrinos are cold enough to redshift like matter, but not cold enough to cluster like matter. Naturally, we could eliminate this signal by either heating or cooling the neutrinos. However, any large change to the temperature would have to come after recombination, as the measurement N eff = 2 . 99 ± 0 . 33 (95%) [17] is in precise agreement with the neutrino density predicted by the Standard Model [84-89] and inferred from BBN [90]. \nCooling the neutrinos can be an effective strategy if they can be cooled enough to reduce the free-streaming scale below the nonlinear scale, or equivalently k fs > k NL = O (1) h Mpc -1 . Recall that the free-streaming scale is defined by \nk fs = √ 3 2 aH c ν , (3.5) \nwhere the neutrino speed in the Standard Model is given by \nc ν = ⟨ p ν ⟩ m ν = 3 T ν m ν ≈ 1 . 0 × 10 -2 ( 50 meV m ν ) (1 + z ) , (3.6) \nAs a result, the free-streaming scale as a function of the neutrino temperature is \nk fs ( z ) = 0 . 04 h Mpc -1 ( ∑ m ν 58 meV ) × ( 1 . 95 K T ν ( z = 0) 1 1 + z ) . (3.7) \nThe role of the z -dependence puts a somewhat non-trivial requirement on T ν . At a minimum, if we have k fs ( z ≈ 100) > 0 . 1 h Mpc -1 , then we could expect the neutrinos to cluster on the scales in the linear regime of our late time observations. Less conservatively, we require k fs ( z = 0) > 0 . 1 h Mpc -1 . Together, these imply we need to cool the neutrinos by a factor of 10 to 1000 at redshifts z < 1000 to avoid the neutrino mass signal. \nSolving for the coupled linear evolution of the dark matter, baryon, and neutrinos numerically (see Appendix A), Figure 6 shows the suppression as a function of the neutrino temperature as z = 0, T ν for the minimum sum of neutrino masses, ∑ m ν = 58 meV. From these numerical results, we can conclude that T ν < O (1) × 10 -2 K at z = 0 is sufficient to move the free-streaming signal to the non-linear regime, assuming that neutrino cooling occurs near z = 100. \nAnatural mechanism for cooling the neutrinos is through interactions with dark matter. The dark matter is cold and therefore is a natural heat sink for the neutrinos. It is straightforward [91] to couple a right-handed neutrino, N , to dark matter, χ at lowredshifts through a light mediator ϕ , \nL ⊃ g N ϕNN + g χ ϕχχ + m 2 ϕ 2 + m N NN + λhLN + m χ χχ . (3.8) \nScattering between the dark matter and neutrinos scales as T -6 ν and thus avoids the constraints at earlier times (and higher temperatures) from BBN and the CMB [92]. \nOne could also consider the case where χ is a single particle sub-component of the dark matter with total energy fraction f χ . Without any additional light states, the scattering between ν and χ is purely elastic. In this scenario, the effect of the coupling is create a neutrino-dark matter fluid, much like the photon-baryon fluid that fills the universe before \nIn order to cool the neutrinos and reproduce the clustering in a ∑ m ν = 0 universe, it is important that the scattering between neutrinos and dark matter is ineslatic. This could be achieved through a number of mechanisms such a additional dark radiation coupled to χ or having nearly degenerate states associated with χ (like would occur with atomic dark matter, for example). This allows the dark matter to absorb energy from the neutrinos and allows for T ν to decrease. In the above model, g N ∼ g χ ≈ 10 -7 is sufficient to bring these two sectors into equilibrium at z ≲ 100 [91] and any efficient process for absorbing the neutrino's energy would lead to an effecive ∑ m ν = 0 signal. \nFigure 6: Suppression of P m ( k ) for ∑ m ν = 58 meV and various neutrino temperatures at redshift zero, T ν ( z = 0). As the suppression is a percent level effect, it will only be observable in the linear regime k < 0 . 1 h Mpc -1 . We see that cooling to T ν < 0 . 02 K, or cooling by factor of 100, is sufficient to eliminate the signal of free-streaming neutrinos. \n<!-- image --> \nrecombination, with a free-streaming scale: \nk fs ≈ 0 . 05 h Mpc -1 × ( f χ + f ν f ν ) 1 / 2 ( ∑ m ν 58 meV ) . (3.9) \nThe amplitude of the suppression on scales k ≫ k fs is proportional to f χ + f ν , the total energy fraction in this fluid. As a result, even if we could couple to all the dark matter so that k fs = 0 . 8 h Mpc -1 , the suppression is large enough to be constrained by the Lymanα forest [93-96] or counts of satellite galaxies [97,98]. \nHeating the neutrinos to avoid the suppression of matter clustering requires that the neutrino speed, shown in Eq. (3.6), remain near unity throughout cosmic history. This could be achieved by increasing T ν by a factor of ∼ 100 in the regime 1000 ≳ z ≳ 100; however, this would correspond to increasing the energy density of the cosmic neutrino background (C ν B) by at least the same factor (assuming no change to the number density of neutrinos). The extra energy density acquired by neutrinos needs to be transferred from another component, with the dark matter serving as the natural candidate during the matter-dominated era. A transfer of energy from the dark matter to the C ν B will have similar cosmological effects as models of dark matter decaying into dark radiation, which are subject to constraints from observations of the matter power spectrum and of the CMB that arise from a larger late-time integrated Sachs-Wolfe effect as compared to a standard cosmological history [99-102]. Current constraints set an upper limit of about 4% of dark matter decaying into radiation after recombination [102], comparable to the fraction of energy density that would need to be transferred from dark matter to heat the C ν B in order to keep neutrinos relativistic until the present time.", '3.4 Time Varying Mass': "The tension between the DESI data and the laboratory measurement of neutrino masses can also be alleviated if the mass of the neutrino is not a constant in either time or space. For example, it might be the case that neutrinos had a smaller mass in the early universe (until around z ∼ 10) but then subsequently had their mass change by O (1), as suggested in [103-105]. Alternately, it could be the case that the neutrino is a chameleon which acquires a larger mass near high density matter [106] but is otherwise lighter in the low density of the cosmos that is relevant to DESI and CMB lensing. For the purposes of illustration of this concept, in this paper, we study the possibility that the neutrino mass evolved in time and leave further exploration of potential chameleonic nature of neutrinos for future work. \nTo realize the phenomenology of lower mass neutrinos that become more massive around z ∼ 10, consider the following terms of the Lagrangian (3.8): \nL ⊃ yhLN + g N ϕNN + m 2 ϕ 2 . (3.10) \nWe take the Yukawa coupling y ∼ 10 meV ⟨ h ⟩ so that the neutrino's Dirac mass is comparable to the current neutrino mass ∼ 10 meV (per neutrino). Observe that when g N ϕ ≫ y ⟨ h ⟩ the phenomenology is identical to that of the conventional 'see-saw' mechanism and thus the neutrino mass will be light. When g N ϕ ≪ y ⟨ h ⟩ , the Dirac mass will dominate and equal the desired present day value. The cosmological evolution of ⟨ ϕ ⟩ naturally leads to such a change due to the fact that ⟨ ϕ ⟩ is sourced by the C ν B, whose number density drops as the universe expands. \nTo illustrate this dynamic, let us pick some example numbers. Suppose we assume that the neutrino mass was around ∼ 1 meV in the early universe. Such neutrinos would be relativistic until z ∼ 10. When they are relativistic, the C ν B sources ⟨ ϕ ⟩ ∼ g N meV 3 m 2 [91], independent of the temperature of the neutrinos. Once the neutrinos become nonrelativistic, ⟨ ϕ ⟩ scales with the number density of the C ν B and we get ⟨ ϕ ⟩ ∼ g N T 3 m 2 . When ⟨ ϕ ⟩ drops, the neutrinos become more massive, approaching their Dirac mass. The main constraint on this scenario is the bound g N ≲ 5 × 10 -8 in order to ensure that the neutrinos do not annihilate into ϕ when they are light (in fact, if they do, the situation reduces to the annihilation scenarios considered earlier). Setting g N ∼ 10 -8 and m ∼ 10 -12 eV, we see that at early times the neutrino mass is around ∼ 1 meV. These neutrinos become non-relativistic around z ∼ 10. The subsequent drop in ⟨ ϕ ⟩ raises the neutrino mass to around ∼ 20 meV today (per neutrino).", '3.5 Mirror Sectors and Relation to the Hubble Tension': 'The deviation from ΛCDM for ∑ m ν is roughly consistent with the suggestion that new physics might only impact dimensionful parameters [107,108]. The CMB and LSS directly \nmeasure dimensionless quantities (angles, redshifts) and thus are not directly related to dimensionful quantities like H 0 and ∑ m ν . This idea was put forward in Refs. [107, 108] to explain the Hubble tension. They realized this concept by introducing a mirror of the Standard Model in the dark sector, such that the gravitational signals remained unchanged but the Standard Model densities could be rescaled. \nNaturally, such a model could also easily explain the apparent ∑ m ν ≈ 0, by having massless neutrinos in the hidden sector. This would leave the other gravitational signals unchanged, but reduce the total gravitational influence of the Standard Model neutrinos. This dilutes ∑ m ν by the fraction of matter in the hidden sector to the mirror sector, and thus requires the Standard Model to be a small component of the total matter density. Unlike some of the other solutions to ∑ m ν , this requires an order one change to the universe and thus is difficult to make compatible with all observations. For example, BBN is sensitive to the physical baryon density and thus is not compatible with the simplest implementations of this idea. \nInterestingly, the suggestion that there could be multiple copies of the Standard Model with different mass parameters is a natural consequence of several recent mechanisms for solving the hierarchy problem [109-111]. However, these hidden sector typically increase N eff > 3 . 044 and ∑ m ν > 58 meV. Without fine tuning these models to take the form of those described in Refs. [107, 108], observations that favor ∑ m ν < 58 meV severely constrain these models.', "4 Negative 'Neutrino Mass'": 'The possibility of an apparent measurement with ∑ m ν < 0 would be most naturally explained by an increase in the amount of clustering in the late universe, or at least an apparent increase as measured through gravitational lensing of the CMB. Even if neutrinos were truly massless ∑ m ν = 0, this would require a change to the formation of structure or the statistical properties of the CMB. Such a mechanism could also erase the signal from conventional massive neutrinos and thus need not involve a change to the neutrino sector at all. In this section, we will explore representative examples of how this signal could arise. We will consider physically increasing the amount of clustering through a new long range force, and creating a apparent increase in lensing through changes to the statistics of the primordial density fluctuations. Both classes of ideas lead to observable consequences that may already be testable with existing cosmological data.', '4.1 Dark Matter with Long Range Forces': 'The most direct approach to enhancing the clustering of matter is to increase the strength of the long range force between dark matter particles. Such long range forces are very well constrained for ordinary matter, from tests of the equivalence principle [112]. However, \nif this new force is limited to the dark matter, it would evade most simple equivalence principle tests. It will nonetheless have observable implications for gravitational dynamics that impact structure on galactic [113-116] and cosmological scales [117, 118]. Interestingly, any such force would also violate the single-field consistency conditions for large-scale structure and thus would leave a measurable non-Gaussian imprint on cosmological correlators [119-121], in addition to the any change to the power spectrum. \nFollowing [118,121,122], suppose we introduce a massless field φ that couples only to the dark matter with a r -2 force similar to Newtonian gravity. This force will modify the momentum conservation equation for the dark matter, \n˙ u cdm + Hu cdm = -1 a (Φ + αφ ) , ∇ 2 φ = α 8 πG ¯ ρ cdm δ cdm . (4.1) \nThe resulting linear growth of the dark matter and baryons at k ≫ k fs where δ ν = 0, is described by \n¨ δ cdm + 4 3 t ˙ δ cdm = 2 3 t 2 [ (1 -f ν -f b )(1 + 2 α 2 ) δ cdm + f b δ b ] , (4.2) \n¨ δ b + 4 3 t ˙ δ b = 2 3 t 2 [(1 -f ν -f b ) δ cdm + f b δ b ] . (4.3) \nWe will define the new growth term as (1 -f ν -f b )(1 + 2 α 2 ) = 1 + ϵ , so that ϵ controls the change to the linear evolution. Taking δ cdm = t γ and δ b = ξδ cdm we find \nγ ( γ -1)+ 4 3 γ -2 3 (1+ ϵ + ξf b ) = 0 , ξγ ( γ -1)+ ξ 4 3 γ -2 3 ( 1 + ϵ 1 + 2 α 2 + ξf b ) = 0 . (4.4) \nTo linear order in ϵ and f b one finds the growing solution \nγ = 2 3 + 2 5 ( ϵ + f b ) , ξ = 1 -(2 α 2 ) . (4.5) \nIn the presense of this new long range force, the power spectrum is therefore modified \nP ( ϵ, ∑ m ν ) ( k ≫ k fs , z ) ≈ ( 1 -2 f ν + 6 5 ( ϵ + f b ) log 1 + z ⋆ 1 + z ) P ( ϵ =0 , ∑ m ν =0) ( k ≫ k fs , z ) . (4.6) \nHere z ⋆ is the redshift where the long-range force becomes important. In most simple models, z ⋆ is the redshift of horizon entry k = a ( z ⋆ ) H ( z ⋆ ). This would make the above signal scale dependent and thus would not mimic the neutrino signal. As a result, cosmological constraints already exclude α < 0 . 01 [117, 118]. Therefore, it is important that z ⋆ is a k -independent constant and that the field φ only becomes important at late times. In this case, if we assume the minimum ∑ m ν so that f ν = 4 × 10 -3 , as derived in Equation (2.6), we could explain an apparent ∑ m ν ≈ -160 meV with α 2 = 7 × 10 -3 . A phase transition, or some other time- or temperature-dependent physics could change \nthe mass of φ so that it became massless at z ⋆ ≈ 100. This would imply equivalence principle violation for the dark matter at later times. Current constraints [113-116] likely require α 2 ≪ 1 but have not been explored in detail. In addition, this type of equivalence principle violation leaves a number of cosmological [121] and astrophysical signals [123] that could be observed in near-term surveys and experiments. For example, the change to the evolution of matter also alters the galaxy bispectrum in a way that breaks the single-field consistency conditions. This effect is sufficient to measure α 2 ≳ 10 -3 [121] for a quasi-realistic survey.', '4.2 Primordial Trispectrum': "The trispectrum (four-point function) of the CMB plays two significant roles in the measurement of neutrino mass. First, gravitational lensing induces a connected four point function, and measuring the trispectrum allows us to reconstruct the lensing power spectrum. Secondly, the trispectrum is also what determines the variance of the primary CMB which sets the uncertainty in all our cosmic parameters [124,125]. \nA primordial trispectrum of the appropriate shape could mimic the effect of lensing and thus could lead to an apparent increase in the lensing amplitude. Both lensing and primordial trispectra can be measured using the same class of estimators defined in Ref. [126]. Concretely, we could couple the inflaton to an additional field, σ ( ⃗x ), that modulates the amplitude of the adiabatic fluctuations, ζ ( ⃗x ), by a term \nζ ( ⃗x ) = ζ G ( ⃗x ) + √ τ σ NL ζ G ( ⃗x ) σ ( ⃗x ) . (4.7) \nwhere ζ G ( ⃗x ) and σ ( ⃗x ) are Gaussian random fields. This modulation leads to a connected trispectrum \n〈 ζ ⃗ k 1 ζ ⃗ k 2 ζ ⃗ k 3 ζ ⃗ k 4 〉 ' = τ σ NL P ζ ( k 1 ) P ζ ( k 3 ) P σ ( | ⃗ k 1 + ⃗ k 2 | ) + permutations ≡ τ σ NL T ( ⃗ k 1 , ⃗ k 2 , ⃗ k 3 , ⃗ k 4 ) . (4.8) \nThis is not equivalent to the lensing signal because it is a three-dimensional correlation between the modes, rather than two dimensional. Bounds on this kind of non-Gaussianity for a scale invariant σ , P σ ≈ P ζ , have been derived from the CMB and yield τ local NL < 1700 (95%) [127]. However, if the power spectrum of σ were taken to be scale dependent to be degenerate with the lensing potential, ϕ ( ⃗ L ), it would be projected out of that analysis. Following Ref. [128] (see also Refs. [129,130]), we can estimate how correlated the proposed \ntrispectrum would be with the local model using the Fisher matrix, \nF ( T 1 , T 2 ) = V ∫ d 3 ⃗ k 1 d 3 ⃗ k 2 d 3 ⃗ k 3 d 3 ⃗ k 4 (2 π ) 12 〈 ζ ⃗ k 1 ζ ⃗ k 2 ζ ⃗ k 3 ζ ⃗ k 4 〉 ' 1 〈 ζ ⃗ k 1 ζ ⃗ k 2 ζ ⃗ k 3 ζ ⃗ k 4 〉 ' 2 P ζ ( k 1 ) P ζ ( k 2 ) P ζ ( k 3 ) P ζ ( k 4 ) (2 π ) 3 δ 3 ( ∑ ⃗ k i ) , (4.9) \nwhere V ∝ k -3 min is the survey volume. The ratio of the off-diagonal to diagonal terms defines the correlation coefficient between τ σ NL and τ local NL , C ( τ σ NL , τ local NL ), which is approximately \nC ( τ σ NL , τ local NL ) ≈ ∫ d 3 kP σ ( k ) P ζ ( k ) √ ∫ d 3 kP ζ ( k ) 2 √ ∫ d 3 kP σ ( k ) 2 . (4.10) \nAlthough this signal would be degenerate with lensing in the CMB, it would be introduce non-Gaussianity in the late universe that could be measured through the galaxy power spectrum [132] (via scale-dependent bias [133,134]) or cross-correlations between the CMB and LSS [135]. CMB lensing is currently measured at 40 σ [12-14,136] and therefore a trispectrum mimicking a 2 . 5%-7 . 5% shift in the lensing amplitude would visible at the 1-3 σ level. Given that the current constraints on primordial non-Gaussianity from related models are at least an order of magnitude weaker than Planck constraints [137], we do not expect 2 current galaxy survey data to be sensitive to such a trispectrum. However, data from DESI, Euclid [138], and particularly SPHEREx [139] are expected to be up to an order of magnitude more sensitive than Planck to this type of non-Gaussian signature. Concretely, SPHEREx is expected to be sensitive to τ NL = 130 at 2 σ [140] which is roughly 10 times the sensitivity of Planck [131]. \nTo match the CMB lensing power, we should choose P σ ( k ) ∝ P m ( k ) so that it takes a similar form to the lensing signal. We therefore require P σ ( k ) → 0 as k → 0, P σ ∝ k -3 as k →∞ , and have a maximum at some k = k ⋆ . We would then expect the correlation to be suppressed by C ( τ σ NL , τ local NL ) ≈ ( k min /k ⋆ ) 3 ≪ 1. In this regard, the shape of P σ ( k ) may not have to be finely tuned to contribute to the observed lensing trispectrum without violating other CMB trispectrum constraints. Other trispectrum shapes, like those considered in Refs. [128, 131] are usually scale invariant and peak in equilateral configurations where k ∼ k max . \nA second possibility is that additional contributions to the trispectrum could increase the true uncertainty in cosmic parameters. This could increase the probability that value of A s determined from the primary CMB is simply a statistical outlier. Specifically, a large primordial trispectrum increases the deviation of parameters from their mean values. One model that achieves such behavior is disorder in single field inflation [141]. In these models, random features in the inflationary potential introduce, on average, a trispectrum that is identical the Gaussian noise but with a larger or smaller amplitude. One can achieve a \nsimilar effect on ∑ m ν [142] from super-sample covariance [143], through a large amplitude of local-term non-Gaussianity (e.g. τ local NL ). To be consistent with CMB constraints, the effective amplitude τ NL would have to be scale-dependent to avoid the direct constraints from the CMB trispectrum.", '5 Conclusions': "The exclusion of the minimum sum of neutrino masses, from either the inverted or normal hierarchy, is a remarkable statement of the power of cosmological data. At these masses, neutrinos form only a fraction of a percent of the total energy density of the universe. The presence of cosmic neutrinos has been robustly established during the era of nucleosynthesis [90] (BBN) and recombination [17] (CMB), through the measurement of N eff and therefore their small but measurable impact on the late universe was to be expected. As we have no simple path to a direct measurement of cosmic neutrinos on earth, cosmological observations provide a novel window into the universe, capable of revealing new secrets. \nIt is important that the measurement of ∑ m ν from the CMB and DESI is incompatible with a wide range of proposals for BSM physics that are also otherwise unconstrained. Light but massive relics [144] are extremely common in models of BSM physics, including many approaches to the hierarchy problem, explanations of dark matter, models including light gravitinos [145, 146], etc. These necessarily contribute positively to N eff and ∑ m ν and thus would further exacerbate the tension with the minimum sum of neutrino masses. As a result, any such model would have to incorporate additional physics, of the kind discussed in this paper, in addition to the new physics relevant to these problems. It is interesting that our results from neutrino decay point to a possible origin from new physics at 10-100 TeV, which could provide a common origin for both effects. \nThe recent BAO measurements from DESI enrich this story. Allowing ∑ m ν < 0, as an indication of enhanced of clustering, we find data from CMB+DESI constrains ∑ m ν = -160 ± 90 meV (68%), excluding at about 3 σ even the minimum neutrino masses consistent with neutrino oscillation experiments. Yet, we showed that this measurement can be naturally explained by new physics in the neutrino and/or dark sectors that is otherwise weakly constrained by other experiments and observations. A measurement consistent with ∑ m ν = 0 could be naturally explained by neutrino decays, cooling, or time-dependent neutrino masses, pointing to new physics coupled to neutrinos and potentially dark matter (sectors). Achieving ∑ m ν < 0 requires physics beyond the neutrino sector but could be explained by new long range forces for dark matter or changes to the primordial statistics. Each class of models naturally suggests signals that could be present in existing data or testable with near term experiments or observations. \nAcknowledgements We are grateful to Kim Berghaus, Tim Cohen, Raphael Flauger, George Fuller, Peter Graham, Jiashu Han, Colin Hill, Mustapha Ishak, Thomas Kon- \nstandin, Tongyan Lin, and Ben Wallisch for helpful discussions. NC is supported by the US Department of Energy under grant DE-SC0011702. DG is supported by the US Department of Energy under grant DE-SC0009919. This work was supported by the U.S. Department of Energy (DOE), Office of Science, National Quantum Information Science Research Centers, Superconducting Quantum Materials and Systems Center (SQMS) under Contract No. DE-AC02-07CH11359. S.R. is also supported in part by the U.S. National Science Foundation (NSF) under Grant No. PHY-1818899, the Simons Investigator Grant No. 827042, and by the DOE under a QuantISED grant for MAGIS and Fermilab. JM is supported by the US Department of Energy under grant DE-SC0010129. Computational resources for this research were provided by SMU's Center for Research Computing. We acknowledge the use of CAMB [30], CLASS [32], IPython [147], and the Python packages Matplotlib [148], NumPy [149], and SciPy [150].", 'A The Suppression of Clustering': 'In this appendix, we review the calculation of the linear growth of structure in a universe with massive neutrinos. This calculation explains the suppression of small scale power due to neutrino free streaming, which is the dominant cosmological signal responsible for the constraints on ∑ m ν . \nFollowing [27], we define the density contrasts of the dark matter and baryons as δ cb = δρ cdm + δρ b ¯ ρ cdm +¯ ρ b , and the neutrinos, δ ν = δρ ν ¯ ρ ν . Energy and momentum conservation of these species after recombination is then described by the coupled equations \n˙ δ cb ( ⃗ k, t ) -a -1 k 2 u cb = 0 , ˙ δ ν ( ⃗ k, t ) -a -1 k 2 u ν = 0 , (A.1) \nand \n˙ u cb + Hu cb = -1 a Φ , ˙ u ν + Hu ν = -1 a Φ -c 2 ν a δ ν . (A.2) \nHere we have defined the scalar velocity potential u i for each species as ⃗v i = ⃗ ∇ u i . Finally, Φ is the Newtonian gravitational potential, which obeys \n∇ 2 Φ = 4 πG (¯ ρ cb δ cb + ¯ ρ ν δ ν ) . (A.3) \nIn a matter dominated universe, H 2 ∝ a -3 which implies that a ( t ) ∝ t 2 / 3 and ¯ ρ m ∝ t -2 . Differentiating these equations allows us to eliminate the velocity potential to find two second-order equations \n¨ δ cb + 4 3 t ˙ δ cb = 2 3 t 2 [ f ν δ ν +(1 -f ν ) δ cb ] , (A.4) \n¨ δ ν + 4 3 t ˙ δ ν = -2 α 3 t 2 δ ν + 2 3 t 2 [ f ν δ ν +(1 -f ν ) δ cb ] , (A.5) \nα ≡ 3 k 2 c 2 ν t 2 2 a 2 = k 2 k 2 fs , c ν ≡ ⟨ p ν ⟩ m ν , f ν ≡ Ω ν Ω m . (A.6) \nwhere \nFrom here, one can solve these equations numerically to understand the influence of the neutrinos on the matter fluctuations in the linear regime. \nIn the regime α ≫ 1, it easy to understand the solutions as follows: the homogeneous equation for δ ν (i.e. δ cb ≈ 0) can be solved to find that δ ν ∝ t -1 / 6 → 0 as t →∞ . We can also solve the inhomogeneous equation with δ ν = ξδ cb to find ξ ∝ 1 /α → 0. Therefore we can focus on δ cb with δ ν = 0. Taking the ansatz δ cb = t γ and δ ν = 0, we get \nγ ( γ -1) + 4 3 γ -2(1 -f ν ) 3 = 0 → γ = 2 3 -2 5 f ν + O ( f 2 ν ) (A.7) \nwhere we kept only the growing solution with γ > 0. In a matter-dominated universe, H 2 ∝ a -3 which implies that a ( t ) ∝ t 2 / 3 , and therefore \nδ cb ( ⃗ k, t ) ≈ δ cb ( ⃗ k, t ν ) a ( t ) 1 -3 f ν / 5 , (A.8) \nwhere 1 + z ν = a ( t ν ) -1 . Finally, since ¯ ρ m = ¯ ρ cb + ¯ ρ ν , the total matter density contrast δ m = δρ m / ¯ ρ m is given by \nδ m ( ⃗ k, t ) = δρ cb + δρ ν ¯ ρ m = δ cb ( ⃗ k, t ) ≈ (1 -f ν ) δ cb ( ⃗ k, t ν ) a ( t ) 1 -3 f ν / 5 ≈ δ cb ( 1 -f ν -3 5 f ν log 1 + z ν 1 + z ) . (A.9) \nThis gives rise to the suppression of the power spectrum \nP ( ∑ m ν ) ( k ≫ k fs , z ) ≈ ( 1 -2 f ν -6 5 f ν log 1 + z ν 1 + z ) P ( ∑ m ν =0) ( k ≫ k fs , z ) . (A.10) \nIn this regard, we see that the suppression is a straightforward consequence of the linear evolution.', 'References': "- [1] J. Lesgourgues and S. Pastor, 'Massive neutrinos and cosmology,' Phys. Rept. 429 (2006) 307-379, arXiv:astro-ph/0603494 .\n- [2] Topical Conveners: K.N. Abazajian, J.E. Carlstrom, A.T. Lee Collaboration, K. N. 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2024arXiv240902058L

In the age of gravitationalwave GW sources and newly discovered local black holes BH and neutron stars NS understanding the fate of stars is a key question. Not every massive star is expected to successfully explode as a supernova and leave behind a NS some stars form BHs. The remnant depends on explosion physics but also on the final core structure often summarized by the compactness parameter or iron core mass where high values have been linked to BH formation. Several groups have reported similar patterns in these parameters as a function of mass characterized by a prominent compactness peak followed by another peak at higher masses pointing to a common underlying physical mechanism. Here we investigate its origin by computing singlestar models from 17 to 50 solar masses with MESA. The first and second compactness increases originate from core carbon and neon burning respectively becoming neutrino dominated which enhances the core contraction and ultimately increases the ironcore mass and compactness. An early core neon ignition during carbon burning and an early silicon ignition during oxygen burning help counter the core contraction and decrease the final iron core mass and compactness. Shell mergers between CNe and Oburning shells further decrease the compactness and we show that they are due to an enhanced entropy production in these layers. We find that the final structure of massive stars is not random but already written in their cores at core helium exhaustion. The same mechanisms determine the final structure of any star in this core mass range including binary products though binary interactions systematical shift the range of expected BH formation. Finally we discuss the role of stellar physics uncertainties and how to apply these findings to studies of GW sources. Abridged

2024-09-01T00:00:00Z

['arXiv:2409.02058', '2024arXiv240902058L', '10.48550/arXiv.2409.02058']

['Astrophysics - Solar and Stellar Astrophysics', 'Astrophysics - High Energy Astrophysical Phenomena', 'General Relativity and Quantum Cosmology']

Its written in the massive stars The role of stellar physics in the formation of black holes

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.02058.pdf

{"It's written in the massive stars: The role of stellar physics in the formation of black holes": 'E. Laplace 1 ⋆ , F. R. N. Schneider 1 , 2 , and Ph. Podsiadlowski 3 , 1 \n- 1 Heidelberger Institut für Theoretische Studien, Schloss-Wolfsbrunnenweg 35, 69118 Heidelberg, Germany\n- 3 University of Oxford, St Edmund Hall, Oxford, OX1 4AR, United Kingdom\n- 2 Astronomisches Rechen-Institut, Zentrum für Astronomie der Universität Heidelberg, Mönchhofstr. 12-14, 69120 Heidelberg, Germany \nReceived ...; accepted ...', 'ABSTRACT': "In the age of gravitational-wave (GW) sources and newly discovered local black holes (BH) and neutron stars (NS), understanding the fate of stars is a key question. Not every massive star is expected to successfully explode as a supernova (SN) and leave behind a NS; some stars form BHs. The remnant left after core collapse depends on explosion physics but also on the final core structure, often summarized by the compactness parameter or iron core mass, where high values have been linked to BH formation. Several independent groups have reported similar patterns in these parameters as a function of mass, characterized by a prominent 'compactness peak' followed by another peak at higher masses, pointing to a common underlying physical mechanism. Here, we investigate its origin by computing detailed single star models from 17 to 50 solar masses with MESA. We show that the timing and energetics of the last nuclear burning phases determine whether stars will reach a high final compactness and iron-core mass and likely form BHs. The first and second compactness increases originate from core carbon and neon burning, respectively, becoming neutrino dominated, which enhances the core contraction and ultimately increases the iron-core mass and compactness. An early core neon ignition during carbon burning, and an early silicon ignition during oxygen burning both help counter the core contraction and decrease the final iron core mass and compactness. Shell mergers between C / Ne and O-burning shells further decrease the compactness and we show that they are due to an enhanced entropy production in these layers. We find that the final structure of massive stars is not random but already 'written' in their cores at core helium exhaustion when it is characterized by the central carbon mass fraction X C and the CO core mass. The same mechanisms determine the final structure of any star in this core mass range, including binary products, though binary interactions induce a systematical shift in the range of expected BH formation due to changes in X C. Finally, we discuss the role of stellar physics uncertainties and how to apply these findings to studies of GW sources. \nKey words. Stars: black holes - stars: massive - stars: interiors - stars: evolution - supernovae: general - gravitational waves", '1. Introduction': "Massive stars with masses larger than 10 M ⊙ are rare compared to their low-mass counterparts but their contribution to the evolution and chemical enrichment of galaxies is disproportionately important (Hopkins 2014). Their strong winds and supernova explosions create mechanical and energetic feedback and enrich their surroundings with heavier elements, determining the properties and evolution of galaxies and of the next generations of stars (Geen et al. 2023). However, not every massive star explodes. A fraction is expected to collapse directly and form black holes. Core-collapse supernovae are not always successful and failed explosions are another common path for black hole formation (see, e.g., the recent review by Heger et al. 2023). Even successful explosion can be associated with BH formation through fallback accretion (Burrows et al. 2023). Understanding and predicting the fate of stars remains one of the main unsolved problems in astrophysics. \nThe fate of stars is determined by the explosion mechanism, fallback dynamics, and their final core structure. In this work, we focus on the latter. Progress in stellar and supernova physics in recent years has shown that there is no simple initial mass threshold for a star to form a BH. In fact, almost 30 years ago, \nTimmes et al. (1996) pointed out that the final structure of a star, characterized for example by the final iron core mass, is not monotonic with mass. This picture was confirmed by multiple studies (e.g., Brown et al. 2001; O'Connor & Ott 2011; Sukhbold & Woosley 2014; Pejcha & Thompson 2015; Ertl et al. 2016; Sukhbold et al. 2018; Limongi & Chie ffi 2018; Schneider et al. 2021; Temaj et al. 2024). Moreover, it is now established that most massive stars live in close binary or multiple systems (e.g., Sana et al. 2012), further complicating this picture. Binary interactions have been shown to a ff ect the pre-collapse core structures of stars, both based on studies of pure He star models approximating stripped stars in binaries (Brown et al. 2001; Woosley 2019; Aguilera-Dena et al. 2022, 2023) and binary evolution models of stripped stars, accretors, and mergers (Laplace et al. 2021; Schneider et al. 2021, 2023; Schneider et al. 2024). In turn, this a ff ects their explodability (Müller et al. 2019; Vartanyan et al. 2021; Woosley et al. 2020; Antoniadis et al. 2022), nucleosynthesis (Farmer et al. 2021, 2023), significantly reduces the parameter space for the formation of compact object mergers observable with gravitational-wave (GW) observations (Schneider et al. 2021, 2023) and leads to features in the chirp-mass distribution of binary BH mergers (Schneider et al. 2023). However, to study the properties of BH populations and the formation of GW sources, current state-of-the-art studies necessarily have to \nmake simplifying assumptions regarding the formation of BHs, often using analytical prescriptions solely based on the core mass of single-star progenitors (e.g., Fryer et al. 2012). These can lead to substantially di ff erent outcomes compared to models that take the structure of stars into account (Patton et al. 2022). \nObservationally, only few direct hints of the link between the pre-supernova structure of massive stellar progenitors and the formation of BHs exist. A red supergiant (RSG), N6946-BH1, that was observed to suddenly vanish in the optical after a short outburst (Gerke et al. 2015; Adams et al. 2017; Sukhbold & Adams 2020; Basinger et al. 2021) could be the first direct progenitor of a BH ever observed. This event is compatible with model predictions for a single star with high compactness undergoing a failed supenova explosion and eventually forming a BH (Lovegrove & Woosley 2013; Sukhbold & Adams 2020; Temaj et al. 2024). Very recent observations with JWST potentially challenge this interpretation by identifying an infrared source at the location of this object, which may correspond to a surviving star enshrouded by dust (Beasor et al. 2024) or to the emission from a failed supernova (Kochanek et al. 2024). Future observations are needed to better understand if this event was indeed a RSG forming a black hole after a failed explosion. An additional, indirect observational clue on the link between RSGs and BH formation comes from observations of hydrogenrich (type II) core-collapse SNe. Archival data searches have unambiguously identified several RSG progenitors at the location of these SNe. These observed RSG SN progenitors tend to have low luminosities of log L / L ⊙ ≤ 5 . 1 (Smartt 2009). This is in tension with the observed maximum luminosity of about log L / L ⊙ ≊ 5 . 5 found for RSGs in the galaxy and in the Magellanic Clouds (Davies & Beasor 2020), and is known as the missing RSG problem. A possible interpretation for these 'missing' luminous RSG SN progenitors is that these are highmass stars that 'quietly' form black holes instead of exploding. The exact value of this maximum luminosity remains to be determined because of systematic uncertainties associated with photometric data of these objects (Davies et al. 2018; Davies & Beasor 2020). Nonetheless, these observations o ff er important insights into the structures of stars that may be the progenitors of BHs. In principle, the bolometric pre-SN luminosity of a RSG can be directly linked to its final core mass, independently of the uncertainties in convective boundary mixing or rotation (Temaj et al. 2024) 1 . However, the initial mass of these progenitors is very uncertain. This is because, even assuming single-star evolution, variations in internal mixing, mass loss history, and rotation all a ff ect the relation between the initial and final core mass of stars (Farrell et al. 2020). Taking binary evolution into account further complicates this relation (Zapartas et al. 2019; Zapartas et al. 2021). Generally, the connection between the final core mass and final observable properties of stars is much better constrained (Temaj et al. 2024). Finally, the lack of stars with high core masses exploding as supernovae is also supported by studies of their late-time supernova spectra (e.g., Jerkstrand et al. 2012) and by age-dating of supernova remnant environments (e.g, Jennings et al. 2014). \nBased on detailed stellar models, several summarizing quantities have been defined to evaluate the fate of massive stars. The compactness parameter ξ m (O'Connor & Ott 2011) is commonly \nused in recent literature and defined as \nξ m = m / M ⊙ R ( m ) / 1000km , (1) \nwhere m is the mass coordinate at which the compactness is evaluated, typically at a chosen value of 2 . 5 M ⊙ , and R is the radius at this mass coordinate. Essentially, it is a measure of the density (or mass-radius relation Chie ffi &Limongi 2020) outside the iron-rich core. This quantity, though arguably arbitrary in its definition, is known to correlate with other key properties, such as the iron-core mass, and the binding energy above the iron-rich core (e.g., Sukhbold & Woosley 2014; Schneider et al. 2021; Temaj et al. 2024, see also Fig. 1). Stars with large iron core masses tend to have a high binding energy outside this core, and are thus di ffi cult to explode by any explosion mechanism and tend to ultimately from BHs (Brown et al. 2001; Sukhbold & Woosley 2014; Heger et al. 2023; Temaj et al. 2024). In recent literature, the compactness parameter has been used as a predictor for the final remnant expected after core collapse, with high values indicating BH- and low values NS formation (O'Connor & Ott 2011; Ugliano et al. 2012; Sukhbold & Woosley 2014; Limongi & Chie ffi 2018; Schneider et al. 2021, 2023; Schneider et al. 2024; Heger et al. 2023). Multiple studies have pointed out that more sophisticated metrics are needed to accurately capture the explosion physics and understand the conditions for shock revival (Pejcha & Thompson 2015; Ertl et al. 2016; Müller et al. 2016; Sukhbold et al. 2016; Vartanyan et al. 2021; Burrows & Vartanyan 2021). The explodability of stars is a subject of active discussion in the community and several explodability criteria have been proposed and explored in recent years. These include the two-parameter criterion of Ertl et al. (2016), the presence of steep density profile with a density discontinuity around the Si / O interface (Vartanyan et al. 2021; Tsang et al. 2022; Wang et al. 2022; Boccioli et al. 2023), or a forced explosion condition (Murphy & Dolence 2017; Gogilashvili et al. 2023). \nPioneering work by Sukhbold & Woosley (2014) analyzed the pattern in the compactness parameter for the first time in great detail and linked it to the di ff erent nuclear burning conditions in the last evolutionary stages of massive stars post helium burning. Patton & Sukhbold (2020) demonstrated that variations in the final compactness are linked to the initial conditions for core carbon burning, i.e. the mass of the CO core and the initial central carbon abundance at core helium exhaustion. Sukhbold et al. (2018) found that the final compactness of a star is influenced by small variations in physical assumptions and resolution and interpreted this as a sign of intrinsic randomness in the core structure of stars. However, Chie ffi & Limongi (2020) argued that these apparently random variations can be traced back to their assumptions regarding the core helium-burning evolution, where in particular semi-convection can result in late inges- \nIndependently of the discussion about explodability criteria, several unrelated groups making di ff erent assumptions regarding the microphysics and using di ff erent methods have reported remarkably similar patterns in the final core structure of stars (often summarized by the compactness parameter) as a function of their core or initial mass (O'Connor & Ott 2011; Sukhbold & Woosley 2014; Sukhbold et al. 2018; Limongi & Chie ffi 2018; Chie ffi & Limongi 2020; Chie ffi et al. 2021; Schneider et al. 2021; Patton & Sukhbold 2020; Takahashi et al. 2023; Temaj et al. 2024). Typically, it consists of two prominent peaks in the compactness parameter, separated by ≈ 15 M ⊙ in initial mass and ≈ 7 M ⊙ in CO core mass. The robustness of this pattern points to a common underlying physical process determining the final core structure of stars, and with it, their fate. \ntion of helium in the core, generating 'breathing pulses' which change the central carbon abundance and lead to di ff erent initial conditions for core carbon burning and ultimately, to a di ff erent final core structure. In our work, which includes convective boundary mixing above the helium-burning core, we do not find these signs of intrinsic stochasticity either (Schneider et al. 2023; Temaj et al. 2024). \nSchneider et al. (2021) identified a connection between models with high compactness and the mass range for which carbon burning and neon burning become neutrino-dominated. Following these findings, in the present study, we investigate the origin of the observed patterns in the final structures of massive stars. Wecompute detailed simulations of massive single, non-rotating stars at solar metallicity that are the common progenitors of corecollapse events (17 - 50 M ⊙ ) and focus on the evolution of the innermost 6 M ⊙ . \nIndependent studies connected variations in the final core structure of stars to the number and size of carbon-burning shells and to the transition from convective to radiative carbon burning (Brown et al. 2001; Sukhbold & Woosley 2014; Sukhbold & Adams 2020; Chie ffi & Limongi 2020). However, this explanation appears incomplete. Even after the transition from convective to radiative core carbon burning, models of stars that undergo radiative core carbon burning can result in a low compactness (see, e.g., Fig. 2 of Sukhbold & Adams 2020). In addition, the cause of the prominent drop in compactness after the first peak remains unclear, though Sukhbold & Woosley (2014) identified a link between the base of the carbon shell exceeding the e ff ective Chandrasekhar mass and a smaller oxygen-burning core. \nWe present our computational setup in Sect. 2 and the overall properties of our models in Sect. 3. In Sect. 4, we conduct a simplified experiment to identify and understand the general physical mechanisms responsible for the observed trends in final core structure. These insights are then applied to our fiducial set of stellar models in Sect. 5. We summarize the physical mechanisms identified as being responsible for determining the main pattern in the final core structure of stars in Sect. 6. We discuss the implications and uncertainties of our results in Sect. 7 and present our conclusions in Sect. 8.", '2. Methods': "We compute the interior structure of massive single stars with initial masses between 17 and 50 M ⊙ with the MESA stellar evolution code (version 10398, Paxton et al. 2011, 2013, 2015, 2018). Our models build upon Schneider et al. (2021, 2023), with similar assumptions. Specifically, our models are computed at solar metallicity Z = 0.0142 (Asplund et al. 2009) and are non-rotating. We compute convective mixing with an mixing-length theory (Böhm-Vitense 1958) parameter of α MLT = 1 . 8 and assume step overshooting of 0.2 pressure-scale height, which is only applied over the H and He-burning convective cores. We adopt the Ledoux criterion for convection and assume a semi-convection e ffi ciency of α sc = 1 . 0 (Schootemeijer et al. 2019). We adopt the re-scaled 'Dutch' wind mass loss rates of Schneider et al. (2021) and enable the MLT ++ method of MESA that boosts the local energy transport for outer layers of our massive star models that locally exceed the Eddington limit. The models are computed with the MESA approx21\\_cr60\\_plus\\_co56.net nuclear network until the onset of core collapse, which is defined as the moment when the infall velocity of the iron-rich core exceeds 900 km s -1 . This network e ff ectively sets Y e in the entire iron-rich core by making \nthe approximation that deleptonizations in this core only occur through electron-captures onto 56 Fe. However, variations in Y e between di ff erent stellar models are generally small (Woosley et al. 2002) and do not a ff ect the formation of the main qualitative patterns in the final core structure that are the subject of this work, which already appear at the end of core Ne burning (see also Sec. 3.3). Using this nuclear reaction network is sufficient for our purpose, but we caution against employing these models as input for predictions of three-dimensional supernova simulations or nucleosynthesis yields, which require a larger nuclear network (Farmer et al. 2016; Renzo et al. 2024). To further verify that the final structure pattern is reproduced in models with larger networks, we compute three additional models with initial masses of 21 M ⊙ , 22 M ⊙ , and 23 M ⊙ , for which we employ a nuclear reaction network of 128 isotopes (as recommended by Farmer et al. 2016). The 21 M ⊙ and 22 M ⊙ models encounter numerical di ffi culties after the iron-core infall velocity exceeds 250 km s -1 . By this point the iron core mass and the central entropy change only slightly (by less than 2%), so we consider these to be comparable to our default collapse models. The reaction rates in our models are based on the JINA REACLIB database version 2.2 (Cyburt et al. 2010). \nWe ensure a high spatial resolution in our models, in particular in zones of high temperature and density (Farmer et al. 2016). More specifically, we adopt a minimum of 2000 grid points and a grid spacing option mesh\\_delta\\_coeff = 0.6 , which results in an average of 5000-6000 grid points for each model. We also ensure a high temporal resolution throughout the evolution with a maximum time step of 10 -4 years that is further limited based on changes in composition. Our final core collapse models and further information are available online 2 . \nTo better disentangle the e ff ects responsible for determining the final structure of massive stars, we perform a controlled experiment (see Sec. 4). These models have the same core masses but a di ff erent central carbon mass fraction at the moment of core helium depletion (when the central helium mass fraction is lower than 10 -4 ). This is achieved by computing additional models using the MESA relax\\_initial\\_composition method in which we artificially modify the central 12 C abundance and the central 16 O abundance while keeping the total mass fractions constant. Our base model for this experiment is our fiducial model with an initial mass of 22 M ⊙ , which corresponds to the first compactness peak, at the moment of core helium depletion. \nWe examine the e ff ect of changing the 12 C( α, γ ) 16 O reaction rate and discuss these in Appendix A. This notoriously uncertain nuclear reaction plays a crucial role for the evolution of stars, including their fate (Weaver & Woosley 1993; Austin et al. 2014; Sukhbold & Adams 2020; Farmer et al. 2020), as it determines the final central abundances at the end core helium depletion and the mass of the CO core (see also Section 3.3). Our fiducial set of models adopts the default MESA rate from Xu et al. (2013). The other two sets we compute adopt the approximately 15% lower rate from Kunz et al. (2002) which is often used in the literature, and a rate that is 10% higher than that of our default model. We compute additional sets of models for which we vary the semiconvection e ffi ciency, discussed in detail in Appendix B. Finally, we perform a resolution test in Appendix C to explore the e ff ect of numerical uncertainties on the occurrence of shell mergers. \nTo evaluate the final fate of our massive star models, we compute the expected explosion outcome of our models using the semi-analytical parametric neutrino-driven supernova explosion model of Müller et al. (2016), with the same assumptions \nFig. 1. Final (a) compactness, (b) specific central entropy, (c) iron core mass, and (d) binding energy above M 4 at the onset of core collapse as a function of the initial mass. The top axis shows the CO core mass at core helium exhaustion. Circles and crosses represent black hole formation and explosions, respectively according to the Ertl et al. (2016) criterion, while black and white colors indicate BH formation and explosions based on the Müller et al. (2016) supernova model. \n<!-- image --> \nas in Schneider et al. (2021, 2023); Schneider et al. (2024) and Temaj et al. (2024). For simplicity, we do not consider black holes formed by supernova fallback in this work. \nFor comparison, we also employ the two-parameter explodability criterion by Ertl et al. (2016). This criterion depends on the mass M 4, which is mass coordinate m where the entropy reaches a value of s / ( NAk B) = 4 and on µ 4 = ∆ m / M ⊙ ∆ r / 1000km GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> s = 4 , the radial mass gradient at M 4. Physically, this location M 4 typically corresponds to the Si / O interface (i.e. the transition point between the Si-rich and O-rich layers), which has been found to be a good predictor for successful multi-dimensional neutrinodriven supernova explosions (Ertl et al. 2016; Müller et al. 2016; Ertl et al. 2020). We use the s19.8 calibration of Ertl et al. (2016) to models by Woosley et al. (2002) to distinguish successful and failed explosions.", '3.1. Final stellar structure': "We characterize the final stellar structure of our models by several quantities, including the compactness parameter, final specific central entropy, iron-core mass, and final binding energy (Fig. 1). Here, the binding energy EB of a star of mass M above M 4 is defined as \nEB = -Z M M 4 Gm r dm . (2) \nWe identify specific mass ranges, labeled A, B, C, and D in Fig. 1, during which the compactness parameter, final specific central entropy, iron-core mass, and final binding energy follow specific trends (significant increase or decrease), whose physical origin we investigate in Sections 5.1, 5.2, 5.3, and 5.4, respectively. These mass ranges are indicated as a function of initial mass (bottom axis of Fig. 1) and of the CO core mass, based on the linear relation we derive between the two for our default assumptions (see also Fig. B.1): \nM CO = 0 . 532 Mi -5 . 31 . (3) \nFig. 1 demonstrates that all quantities summarizing the final stellar structure follow a very similar, non-linear trend as a function of mass, confirming earlier findings (e.g., Timmes et al. 1996; Brown et al. 2001; Sukhbold & Woosley 2014; Chie ffi &Limongi 2020; Schneider et al. 2021, 2023; Schneider et al. 2024; Takahashi et al. 2023; Temaj et al. 2024). For low masses, the values are approximately constant, with only small variations. A characteristic increase (region A) begins at initial (CO core) masses of about 19 (5.8) M ⊙ and reaches a maximum at CO core (initial) masses of 22 (6.5) M ⊙ , commonly referred to as the 'compactness peak'. It is followed by a decrease (region B) until a minimum is reached at about 25 (8) M ⊙ . After a mass range of about 3 M ⊙ with small variations, all quantities then experience a significant second increase from 30 (11) M ⊙ (region C) until a second peak is reached and the values generally decrease from masses of 40 (16) M ⊙ (region D). As shown in the lowest panel, at this point all models have a high binding energy above M 4. Even if a supernova shock were successfully propagating after core collapse in these models, potentially leading to an observable supernova, the high binding energy means that the formation of a black hole is likely for these mass ranges, independently of the explosion mechanism (Sukhbold & Woosley 2014; Heger et al. 2023). \nOur models show a lower intrinsic variability in the final structure than models by Sukhbold & Woosley (2014) and Sukhbold et al. (2016, 2018), just like the studies by Chie ffi & Limongi (2020); Chie ffi et al. (2021), and Takahashi et al. (2023). This is likely due to our choices of convective boundary mixing that prevent the occurrence of 'breathing pulses' during core helium burning (Chie ffi & Limongi 2020) that can change the relation between the CO core mass and the core carbon mass fraction X C at core helium exhaustion. Secondary peaks in the quantities shown in Fig. 1 are not observed for our default assumptions and mass sampling but can appear for di ff erent choices of physics. We argue that they are probably real (see Appendices A and B for more details). \nAll quantities shown in Fig. 1 follow similar patterns because they are intrinsically linked (Fryer 2014; Sukhbold & Woosley 2014; Takahashi et al. 2023; Schneider et al. 2021; Schneider et al. 2024; Temaj et al. 2024). For completeness, we repeat \nthese arguments below. The connection between the final central entropy (Fig. 1b) and the final iron-core mass (Fig. 1c) can be understood through the e ff ective Chandrasekhar mass (e.g., Timmes et al. 1996; Woosley et al. 2002; Sukhbold & Woosley 2014; Schneider et al. 2024). In massive stars, core collapse is triggered once the iron-rich core has reached a critical mass. However, in contrast to the cores of low-mass stars, the degenerate iron-rich cores of massive stars are hot, i.e. they have a finite temperature (entropy) that needs to be taken into account. This thermal structure leads to additional corrections compared to the classical Chandrasekhar mass M Ch , 0 = 1 . 457 GLYPH<16> Ye 0 . 5 GLYPH<17> 2 (e.g., Timmes et al. 1996, and references therein), and lead to the approximate e ff ective Chandrasekhar mass (ignoring special and relativistic corrections, and Coulomb corrections) \nM Ch ≈ M Ch , 0 se π Ye kBNA ! 2 + 1 , (4) \nwhere se is the average electronic entropy, which is roughly a third of the average central entropy (Baron & Cooperstein 1990; Timmes et al. 1996). This critical mass is mainly sensitive to changes in entropy because the electron fraction Ye tends to be very similar between di ff erent progenitors 3 (Sukhbold & Woosley 2014). Thus, the final iron-rich core mass that can form in a massive star before it exceeds M Ch and its core begins to dynamically collapse is directly linked to the central entropy. \nAs shown in Fig. 1, the compactness parameter also correlates well with the final entropy and iron-core mass, which may appear surprising at first. This connection can be understood by considering that the compactness is essentially a measure of the mass-radius relation outside the iron core (Chie ffi & Limongi 2020). By taking into account that the final iron-rich core is adiabatic, this mass radius relation is directly connected to entropy through polytrope relations (Schneider et al. 2021). Thus the final compactness can be understood as being equivalent to central entropy (see also Fryer 2014), itself linked to the final ironcore mass (and M Ch) as described above. The mass-radius relation and entropy both enter into the definition of binding energy above M 4 (see Eq. 2), which explains the observed correlation (Fig. 1d).", '3.2. Link to black hole formation': 'To estimate the range of stars for which BH formation is expected, beyond considering the binding energy, we apply the Ertl et al. (2016) explodability criterion to our models. This is shown by the di ff erent markers (crosses for explosions and circles for black holes) in Fig. 1. Based on this criterion, stars with initial (core) masses below 30 (12) M ⊙ are expected to preferentially explode. Models in the first compactness peak and beyond the second compactness increase are predicted to form black holes. In addition, we compute the expected explosion outcome of our models using the semi-analytical neutrino-driven supernova explosion model of Müller et al. (2016). Black hole formation is indicated with black and successful explosions with white markers in Fig. 1. We find that the Müller et al. (2016) model also predicts black hole formation for stars with a high compactness. The mass range for which successful explosions are expected is more extended compared to the outcome of the Ertl et al. (2020) criterion, reaching up to models with masses of 34 (12.5) M ⊙ . \nFig. 2. Evolution of the central specific entropy as a function of the initial mass for di ff erent evolutionary stages from core hydrogen depletion until core-collapse. Key di ff erence in the final entropy start to appear at the end of core carbon burning. Red markers indicate three models computed with a nuclear network comprising 128 isotopes. Taking into account more nuclear reactions generally leads to a slightly lower final central entropy, though the general qualitative pattern of the compactness peak is unchanged. The characteristic final landscape is already mostly determined at the end of core Ne burning and remains until core collapse. \n<!-- image --> \ni \n/circledot \nOverall, we find that based on these explosion criteria, BH formation is consistently expected for models with high compactness and high central entropy, including the compactness-peak models at a mass of 22-23 (6.5-7) M ⊙ and models beyond a mass of 34 (14) M ⊙ . We confirm that using a compactness threshold for models in which BH formation is expected (e.g., ξ f 2 . 5 > 0 . 4, see dashed line in Fig. 1a) is too simplified to fully reproduce the predictions by more sophisticated measures of explodability. Considering a larger set of models (Schneider et al. 2021; Temaj et al. 2024; Schneider et al. 2024) we note that there is no clear compactness threshold separating successful and unsuccessful explosions (Maltsev et al. in prep.). For ξ f 2 . 5 ≲ 0 . 3 successful explosions are found, and for ξ f 2 . 5 ≳ 0 . 45 explosions are unsuccessful, similarly to previous works (Müller et al. 2016; Takahashi et al. 2023; Zha et al. 2023). A compactness threshold can thus only be regarded as a rough first approximation of the core-collapse outcome. Approaches based on the entire final stellar structure, such as the Müller et al. (2016) model, can be applied more generally.', '3.3. Emergence of the final structure pattern': 'To understand at which point the observed pattern in the final stellar structure emerges, we trace the value of the central entropy at key evolutionary stages, shown in Fig. 2. At every stage, there is a general underlying trend of an increasing entropy as function of initial (or CO core) mass, which reflects the increasing core mass of stars as a function of their mass. As a function of time, the central entropy generally decreases. This can be understood as the e ff ect of a changing mass-radius relation of the core, as it becomes increasingly compact due to core contraction (Chie ffi &Limongi 2020; Schneider et al. 2021). Central nuclear burning episodes temporarily increase the central entropy, while \nthermal neutrino losses, which become important after core carbon burning, decrease it. At the end of core C burning (blue diamonds in Fig. 2), a first peak feature in the central entropy appears around 20 (5) M ⊙ initial (CO core) masses. It can be understood as a di ff erence in heat content between models before and after 21.5 (6) M ⊙ . Models below this mass experience convective core carbon burning while models above this mass burn carbon radiatively as the burning becomes more neutrinodominated (see also the Kippenhahn diagrams in the appendix Fig. D.1 and D.2). From the end of core neon burning on (orange crosses in Fig. 2), the characteristic feature corresponding to the compactness peak emerges around 22 (6.4) M ⊙ and remains until core collapse (black squares in Fig. 2). As we discuss later, this is because central entropy reflects the degree of contraction the core experiences in these models. Between the end of core O burning and core collapse, an additional feature emerges in the central entropy landscape for models above 40 (15) M ⊙ . The central entropy drops to lower values, though the general trend of increasing entropy as a function of mass remains. The key processes that determine the final entropy or compactness landscape thus already occur close to the time of core neon burning. For the models above 40 (15) M ⊙ additional processes play a role around the time of core silicon burning. \nThe emergence of the final structure landscape at the moment of core neon burning demonstrates that using a small nuclear reaction network is su ffi cient for characterizing the final structure landscape of stars. However, such models are not well-suited as input for multi-dimensional simulations of core collapse, which require larger reaction networks (Farmer et al. 2016; Renzo et al. 2024). We compare our default models to three additional models with initial masses of 21 M ⊙ , 22 M ⊙ , and 23 M ⊙ , for which we employ a nuclear reaction network of 128 isotopes (Farmer et al. 2016), shown with red markers in Fig 2. We find the same pattern in the formation of a compactness peak as in models computed with a smaller nuclear reaction network, though the final central entropy is 5% smaller. In addition, the patterns in final central entropy shown here are very similar to the ones found by Takahashi et al. (2023), who used a network comprising 300 isotopes and a finer grid (see their Fig. 11). The exact quantitative details of the final structure landscape, especially at the higher mass end, are thus still a ff ected by changes in Ye induced by taking into account more nuclear reactions, which ultimately change M Ch , e ff (see Eq. 4). This can slightly shift the mass range for which a high final central entropy is reached, but does not significantly a ff ect the main qualitative trends described here.', '4.1. Post core helium burning evolution': 'Beyond core helium burning, the evolution of massive stars proceeds di ff erently compared to earlier stages, as thermal neutrino losses become important. Because of their minuscule interaction cross sections, the vast majority of neutrinos escape the star, causing a tremendous energy loss and greatly accelerating the evolution (e.g., Woosley et al. 2002). At the end of core helium burning, the core contracts. For temperatures around log( T / K) ≈ 8 . 9 and densities around 10 5 g cm -3 , carbon burning ignites in the center (see Fig. 3). The nuclear energy generation rate ϵ nuc can be expressed as ϵ nuc ∼ ρ T 23 X C during carbon burning, where ρ and T are the density and temperature conditions close to the core, and X C the initial carbon abundance (Woosley et al. 2002). For high enough temperatures, log( T / K) ≳ 9, pair \nFig. 3. Central temperature as a function of the central density for models with the same core mass and di ff erent carbon mass fractions X C after core helium burning (indicated in the color bar). The previous evolution of the core (not shown here for clarity) is indistinguishable for all models, while the post core helium burning evolution is greatly a ff ected by the di ff erences in central carbon abundance. Symbols indicate key evolutionary phases. To the right of the gray dashed line electron degeneracy dominates the pressure. The inset axis indicates the final compactness of the models as a function of decreasing X C. \n<!-- image --> \nproduction, followed by electron-positron pair annihilation, can occur. In rare cases, a neutrino-antineutrino pair is produced, which escapes the star. This process is significant enough to be the dominant energy sink in our models with an energy loss rate ϵν ∼ ρ -1 T 12 (Sukhbold & Adams 2020). Photo-neutrinos from electron scattering processes also play a role, but the energy loss they cause is an order of magnitude lower than from pair-annihilation neutrinos. The balance between ϵν and ϵ nuc determines the final evolution of the core. As shown by the expressions for ϵ nuc and ϵν , it mainly depends on ρ and T , which are set by the core mass, and the initial central carbon abundance X C. \nTo disentangle the e ff ects of the core mass and of X C on the final structure (Patton & Sukhbold 2020) and understand the formation of the compactness peak identified between regions A and B in Fig. 1, we perform a controlled experiment. For a fixed CO core mass (6.62 M ⊙ , corresponding to an initial mass of M i = 22 M ⊙ for our assumption), we compute models with a modified central carbon abundance. Our fiducial model ( X C = 0 . 23) corresponds to the compactness peak model, while the models with carbon abundances of 0.28 and 0.17 have final compactness values before and after the peak, respectively. As shown in Fig. 3, small di ff erences in their central abundances cause large deviations in the post core helium evolution. Overall, the X C = 0 . 23 model (red line in Fig. 3) reaches higher central temperatures and lower densities after core Ne burning compared to the other models at similar evolutionary stages. \nFig. 4. (a) Evolution of key elements of the inner 6 M ⊙ core structure as a function of time from carbon ignition to core collapse for models with the same core mass and a di ff erent central carbon abundances XC . Full and dashed lines indicate the carbon and neon-burning fronts, respectively. Shaded (hatched) regions show convective carbon (neon) burning regions in the core. The iron-core masses are indicated with dash-dotted lines. The gray horizontal dashed line indicates the classical Chandrasekhar limit. The inset figure shows the final compactness of these models as a function of decreasing XC . (b) Total mass of 12 C in the convective carbon-burning shells for each of the models as a function of time. \n<!-- image --> \n- \n-', '4.2. Carbon burning and beyond': 'In Fig. 4, we summarize the final evolution of the stellar structure for our three models with the same core mass and a different initial core carbon mass fraction X C. As indicated in the inset figure in Fig. 4a, these models vary in final compactness and represent models before ( X C = 0 . 27), in ( X C = 0 . 23), and after ( X C = 0 . 18) the compactness peak, where the highest final central entropy and iron core mass are reached. Because they have the same core mass, their central temperature and density conditions at the end of core helium burning are the same (see also Fig. 3) and therefore the neutrino loss rate ϵν is the same. However, at lower X C, neutrinos increasingly dominate the energetic balance during core carbon burning because the nuclear energy generation rate ϵ nuc decreases. \nIn Fig. 4a, we show the progression of the C- and Ne-burning fronts in these models in a Kippenhahn-like diagram as a function of the time from core carbon ignition to core collapse. The location of these nuclear burning fronts are defined as the mass coordinate at which the maximum energy generation rate for a particular burning process (e.g., C or Ne), is reached. In other words, a burning front traces the location at which the maximum burning occurs. These burning fronts are important for the final core structure because their location determines the maximum potential growth of the underlying core mass. A model with carbon burning front that reaches a higher mass coordinate forms a larger carbon-free core, and can eventually form a larger ironrich core (dashed-dotted lines in Fig. 4). \nAfter core carbon burning ignites in the stellar cores, the Cburning front moves outward in mass for all models as it burns through the available carbon fuel. In the X C = 0 . 28 model (dark blue lines in Fig. 4), the high core carbon abundance leads to a nuclear energy generation rate that is large enough to overcome the neutrino energy loss rate, which triggers the development of a relatively small ( < 0.2 M ⊙ ) convective zone (Sukhbold & Adams 2020). It brings in 0.1 M ⊙ of fresh 12 C (see Fig. 4 b) and keeps the carbon-burning front in the center until all the carbon has been burned. Subsequently, the core contracts until the temperature in the carbon-rich layers above the former convective zone reach high enough temperatures for carbon burning to take place and the carbon-burning front moves outward in mass. The location of the burning front thus reflects the amount of core contraction that occurred below it. In this model ( X C = 0 . 28), this first convective episode is followed by two more successive convective regions that become more extended in mass as the temperature at the burning front increases. At the end of core carbon burning (blue cross in Fig. 4b), the core contracts even further, and core neon burning ignites in the center. The C-burning front moves further out in mass (though its progression is slowed by core neon burning, see also Fig. 5a) until the conditions for convection to occur are reached once again. At this point the carbonburning front has a high temperature and reaches a region with unburned carbon, which means it reaches a high energy generation rate that highly exceeds the neutrino losses, triggering the formation of a large convective zone with over 0.4 M ⊙ of carbon (see Fig. 4b). Because of the large amount of fuel remaining, \nthe C-burning front keeps producing energy at a high rate and stays at this mass coordinate of 3 M ⊙ until the end of the evolution. As a result, the carbon-free core below can only reach a relatively low mass. Hence, after silicon burning, a relatively low-mass iron core of 1.7 M ⊙ forms. \nFor the other models, which have lower central carbon abundances (red and yellow lines in Fig. 4), central carbon-burning proceeds radiatively. Even though these models have the same mass, and therefore initially the same temperature and density conditions as the X C = 0 . 28 model, the lower core carbon abundance means that less energy is generated from carbon burning. Neutrino losses dominate and during the initial central carbon burning, energy is transported solely through radiation (see also Fig. 5). The available carbon in the center is depleted quicker, as no fresh carbon is brought in, and thus the core contracts and the C-burning front moves further out in mass. \nThe contraction also increases the temperature at the burning front. Once the energy generation rate at the carbon-burning front reaches high enough values to exceed the neutrino losses (see also Fig. 5) two e ff ects occur. First, a convective region forms and the carbon-burning front stays at a constant mass coordinate. Second, this convective carbon-burning shell acts like a mirror between the layers above and the core below. As the core below the shell contracts, the layers above greatly expand (as seen in the summary of the gravothermal energy in the stellar structure shown in Fig. D.2). The convective region increases in mass, bringing in more fuel for the carbon-burning shell (see Fig. 4b) and prolonging the duration of this burning episode. \nFor X C = 0 . 23 and X C = 0 . 17 (red and yellow lines in Fig. 4), the carbon-burning front reaches further out in mass during core carbon burning (time coordinate t ≈ -2) than the X C = 0 . 28 model, signifying a larger core contraction. At this point, the mass coordinate of the C-burning front of the X C = 0 . 23 model, which corresponds to the compactness peak, is m C = 1 . 35 M ⊙ , close to the classical Chandrasekhar mass M Ch , 0 = 1 . 455 M ⊙ , while that of the X C = 0 . 17 model exceeds it ( m C = 1 . 83 M ⊙ ). As we discuss in Sect. 4.4, this is an important di ff erence because electron degeneracy pressure plays a role in these models, slowing down the core contraction during core carbon burning in the X C = 0 . 23 and instead speeding it up in the X C = 0 . 17 model. \nThe end of core carbon burning is the critical moment when the behavior of the compactness peak model ( X C = 0 . 23) and the model beyond the compactness peak start to di ff er significantly. In the X C = 0 . 17 model, core neon burning already ignites while the C-burning front is still experiencing the first convective episode. In contrast, in the compactness-peak model ( X C = 0 . 23) by the time neon ignites, convective carbon burning has mostly ended ( τ = -2 . 6). Because the core contraction is slower the model can burn a large fraction of the carbon in the convective region before core Ne burning ignites (see also Fig. 4b). Later on ( τ ≈ -3), it quickly burns through the remaining carbon in this region, growing a large C-free core. The C-burning front in the X C = 0 . 17 model grows more slowly and less far ( m / M ⊙ = 3 . 18 at τ = -5 . 15) than in the compactnesspeak model ( m / M ⊙ = 3 . 33 at τ = -4). As we show in Sect. 4.3, this is a direct consequence of the earlier core neon and oxygen ignition in the X C = 0 . 17 model, which suppresses carbon burning and slows down the progression of the C-burning front. \nMoreover, towards the end of the evolution ( τ = -5 . 2) the C-burning front of the X C = 0 . 17 model suddenly drops in mass and reaches the same level as the Ne-burning front. As we discuss in more detail in Sect. 4.5, this is caused by a shell merger between the C- and Ne-burning front. Eventually, a lower-mass \niron core forms in this X C = 0 . 17 model compared to the compactness-peak model. \nFrom this experiment, we have demonstrated that the final core structure of stars in this mass range is already determined at core helium depletion, where the initial conditions for core carbon burning are set. The same e ff ect of a change in the final structure pattern due to a change in X C can be observed for models with di ff erent initial or core masses. This is shown more generally in Appendix A, where we investigate changes induced by the 12 C( α, γ ) 16 O rate, which is largely responsible for determining X C at the end of core helium burning. \nWe have seen that the transition from a final stellar structure with low compactness and iron core mass to high compactness and iron core mass can be understood as the e ff ect of increasingly neutrino-dominated burning. It results in an overall larger core contraction and the formation of a larger carbon-free core, and eventually, to the formation of a larger iron-rich core. At the compactness peak, the neutrino-dominated carbon-burning conditions are such that most of the available carbon is burned before the end of core neon burning. Afterward, the carbon-burning front quickly moves out in mass and grows a particularly large core C-free core, which eventually forms a particularly large iron-rich core. For the transition between the compactness peak model and the model beyond, we identify three mechanisms that are responsible for the observed drop in compactness: (1) the e ff ect of an earlier central neon and oxygen ignition with decreasing X C, (2) an increased electron degeneracy, and (3) shell mergers.', '4.3. Effect of an early core neon ignition': 'We have observed that the timing of core neon ignition varies significantly between models in our experiment with the same core mass and a changing initial core carbon abundance X C (Fig. 4). The consequence of this early neon ignition is shown in Fig. 5, where the evolution of the ratio between the nuclear energy generation ϵ nuc and the neutrino loss rate ϵν at the location of the C, Ne, and O-burning fronts is plotted as a function of time until core collapse. Phases when ϵ nuc significantly exceeds ϵν correspond to convective episodes (Barkat & Marom 1990; Barkat 1994; Sukhbold & Adams 2020). In the X C = 0 . 28 model (Fig. 5a), core neon burning ignites immediately after core carbon depletion, when a core contraction occurs. Neon burning temporarily slows down the progression of the carbon-burning front and dominates the energy generation rate. However, it does not take long before the slowed down carbon-burning front reaches regions of unburned carbon, quickly becoming the main energy source again and forming a large convective region that remains at the same mass coordinate until the end of the evolution (see also Fig. 4b). \nIn the compactness-peak model ( X C = 0 . 23, Fig. 5b), core neon and oxygen burning ignite in rapid succession after the convective carbon-burning region has exhausted its fuel. They dominate the energy generation, slowing down the progression of the carbon burning front until the end of core oxygen burning is reached. The ensuing core contraction helps the carbonburning front reach layers where the nuclear energy generation rate dominates compared to the neutrino losses, triggering the formation of a large convective zone and marking the final location of this burning front. \nIn the X C = 0 . 17, model (Fig. 5c), while the convective carbon-burning episode is still ongoing, the conditions for neon burning are already reached in the center due to the stronger preceding contraction of the core aided by exceeding M Ch , 0 (see \nFig. 5. Time evolution of the ratio between the specific nuclear energy generation rate ϵ nuc and the neutrino loss rate ϵν at the location of the C, Ne, and O-burning fronts for models with the same core mass and di ff erent central carbon abundances. The dashed horizontal line indicates where ϵ nuc = ϵν . When the energy ratio significantly exceeds this line, convection occurs (Sukhbold & Adams 2020). With a decreasing core carbon abundance, the core becomes more neutrino dominated and neon burning, followed by oxygen burning, occur earlier. As highlighted in the red boxes, this temporarily suppresses carbon burning. \n<!-- image --> \n- \n- \n- \n- \n- \n- \nSect. 4.4). While core neon burning is more neutrino-dominated in this model and produces less energy, this simultaneous burning reduces the luminosity of the carbon-burning front, impacting the extend of the convective region above. Once oxygen burning ignites shortly after, it counters the contraction and helps drastically slow down the progression of the now very neutrinodominated carbon-burning front. In the subsequent phases, neon and oxygen burning dominate the energy generation, followed by core silicon burning. After silicon depletion, the neon-burning front, which moves out rapidly in mass, merges with the carbonburning front (see Sect. 4.5). The slowed C-burning implies a smaller carbon-free core and ultimately, a smaller iron core mass and compactness.', '4.4. Origin of the changing timing of core neon and oxygen ignition': 'Here, we investigate the origin of the systematically earlier ignition of neon and oxygen for lower X C. We find that it is mainly due to a systematic shortening of the duration of core carbon burning for a decreasing amount of fuel ( X C). This is shown for a large set of models with the same core mass and a varying core carbon abundance in Fig. 6a. For reference, we show the final compactness of these models in Fig. 6b. The contraction phase that follows core carbon depletion (when X C < 10 -4 ) before core neon burning ignites is significantly shorter than the duration of core carbon burning, on a neutrino-losses accelerated thermal time scale on the order of years (see Fig. 6c). It therefore plays a smaller role than the duration of core carbon burning in determining the timing of core neon ignition. \nBesides the duration of core carbon burning, a secondary effect related to electron degeneracy also appears to play a role in accelerating the evolution after core carbon burning, as first pointed out by Sukhbold & Woosley (2014). This is shown in Fig. 7, where the evolution of the electron degeneracy parameter η = µ/ k B T is displayed as colored contours in a Kippenhahnlike plot for each of our example models. η ≈ 0 indicates partial degeneracy, while η ≫ 0 signifies that electrons are strongly degenerate. We also show the evolution of the classical ( M Ch , 0) and e ff ective Chandrasekhar masses ( M Ch) (see Eq. 4). For a decreasing X C, the central region where electrons are degenerate during core carbon burning is more extended in mass. This can be un- \nFig. 6. (a) Duration of core carbon burning for models with the same mass and a varying initial core carbon abundance X C, indicated with colors. (b) Final compactness as a function of X C. Models that undergo convective core carbon burning are highlighted with black outlines. (c) Duration of the phase between core carbon depletion and core neon ignition as a function of mC , the maximum mass coordinate reached by the carbon-burning front before core neon burning. The classical ( M Ch , 0) and e ff ective ( M Ch) Chandrasekhar mass at core C depletion are shown as vertical dashed and full lines, respectively. \n<!-- image --> \nFig. 7. Kippenhahn diagram of the inner 6 M ⊙ core structure of models with the same core mass and di ff erent core carbon abundance at core C ignition. Colors indicate the dimensionless electron degeneracy parameter η = µ/ k B T (electrons are partially degenerate for η ≈ 0 and very degenerate for η ≫ 0). Convective zones are highlighted by the hatched regions and the dotted vertical lines indicate, from left to right, the moments when 12 C, 20 Ne, 16 O, and 28 Si are depleted at the center (central abundance lower than 10 -4 ). Grey full and dashed line indicates the classical ( M Ch , 0) and e ff ective ( M Ch) Chandrasekhar mass, respectively. Electron degeneracy during core carbon burning increases for models with lower central carbon abundances. \n<!-- image --> \nas a symptom of the overall stronger core contraction experienced by models that are more neutrino-dominated. For the model with X C = 0 . 28 (Fig. 7a), electron degeneracy plays a small role during central carbon burning. The C-burning front (at the base of the convective carbon burning shell) remains significantly below the values of the classical and e ff ective Chandrasekhar mass. For the compactness peak model ( X C = 0 . 23, Fig. 7b), degeneracy pressure helps support the core at the end of core carbon burning, reaching 15% of the total pressure in the center ( η c = 0 . 93) before it is lifted during core oxygen burning. The carbon burning front remains close to, but below M Ch , 0 and M Ch during core carbon burning. For the X C = 0 . 17 model (Fig. 7c) the region where electrons are at least partially degenerate extends up M Ch , 0. The carbon-burning front exceeds M Ch , 0 and even reaches the value of M Ch. This appears to help accelerate the core contraction, which is somewhat surprising since the Chandrasekhar mass is known to mainly play a role at full degeneracy, and deserves further study. \nTo understand if this e ff ect applies more generally, we investigate the behavior of a larger set of models with varying X C. In Fig. 6c, we show the duration of the phase between core carbon depletion and neon ignition as a function of mC , which is defined as the maximum mass coordinate of the carbon-burning front mC during core carbon burning (i.e., the base of the furthest convective carbon-burning shell) 4 . The changes in this timescale are very close to the compactness pattern shown in Fig. 6b as a function of decreasing core carbon abundance. A similar trend in this timescale was also noted by Chie ffi et al. (2021). At the end of core carbon burning, all models reach the same value of the classical Chandrasekhar M Ch , 0 = 1 . 455 M ⊙ , while the e ff ective Chandrasekhar mass varies slightly between models due to their \ndi ff erent central entropy, with values of M Ch ∼ 1 . 82 -1 . 84 M ⊙ . Only in the model with the lowest core carbon abundance ( X C = 0 . 17, shown in more detail in Fig. 7c), mC exceeds M Ch. \nFrom the variation of the timescale between core carbon depletion and core neon ignition, we derive the following interpretation. Models with X C < 0 . 25 ( mC > 1 . 2 M ⊙ ) are strongly neutrino-dominated and burn carbon radiatively in the core, experiencing a stronger contraction than those with larger X C that still burn carbon convectively in their cores. However, degeneracy pressure helps to counter the core contraction towards the end of core carbon burning and to delay the ignition of core neon burning. For models where mC > 1 . 5 M ⊙ , the entire region below M Ch , 0 is at least partially degenerate and degeneracy pressure no longer helps to counter the core contraction, leading to an earlier core neon ignition. Sukhbold & Woosley (2014); Sukhbold et al. (2018), and Sukhbold & Adams (2020) noted a smaller size of the oxygen-burning shell once the base of the carbon-burning shell mC exceeds M Ch and linked it to the role of degeneracy. It is noteworthy that we observe the same phenomenology here (see Fig. 7) despite the di ff erent assumptions and methods we use. As we discuss in Sect. 5.4, the same mechanism helps explain the second drop in compactness (region D in Fig.1) though in this case it involves the oxygen-burning front and an earlier core Si ignition.', '4.5. Shell mergers': 'In our experiment, the model with the lowest initial carbon abundance X C = 0 . 17 experiences a merger of the C- and Ne-burning layers shortly after core Si burning. This is highlighted within the red rectangle in the Kippenhahn diagram (Fig. 8a). The location of the C-burning and Ne-burning shells are indicated with markers. The origin of this shell merger can be understood by inspecting Fig. 8b, which shows the entropy profile in the highlighted region. The entropy in the layers corresponding to the location of the Ne-burning shell (2.5 to 2.9 M ⊙ ) rapidly increases. This is because after Si-burning ignites, a rapid contraction of the \nFig. 8. Top: Kippenhahn diagram showing the final evolution of the inner stellar structure of the model with XC = 0 . 17. Colors indicate the regions dominated by nuclear burning or neutrino losses and gray hatched regions indicate convective areas. Brown lines show contours of constant (logarithmic) density. Colored markers trace the burning front for C, Ne, and O burning. The red box highlights the mass and time range shown in the bottom panel. Small colored vertical lines indicate the times at which the entropy profile are shown in the bottom panel. Bottom panel: Entropy profiles in the mass and time range highlighted in the red box in the top panel. The entropy in the neon burning region (2 . 5 -2 . 9 M ⊙ ) gradually increases until it exceeds the entropy of the C-burning layers above, triggering a merger between these regions. \n<!-- image --> \n/circledot \ncore below the Ne-burning front leads to enhanced Ne burning, which generates entropy. Eventually, the entropy of these layers exceeds that of the carbon-burning layers above (see the entropy profile at a time of log 10 [( t cc -t ) / yr] = -3 . 33). This entropy contrast means that these layers are unstable against convection (Schwarzschild criterion). No convective boundary mixing is assumed in these models after core helium burning, which means that the mixing is solely caused by this entropy change. In Appendix C, we perform a resolution test and find that this shell merger is barely a ff ected by numerical uncertainties. The mixing leads to the merging of these shells and to a new configuration of the stellar structure.', '5. Final core structures of actual stellar models': 'In the previous section, we have identified key mechanisms that lead to a change in the final core structure through an experiment in which we varied the core carbon abundance for a fixed core mass. We now apply the insights gained to more realistic simulations of massive stars. \nThe initial conditions for central carbon burning are set by the core mass, which determines the central density and temperature, and by the initial carbon abundance. Hydrostatic equilibrium implies a characteristic trend in the central density and temperature of stars as a function of mass (e.g., Kippenhahn et al. 2013). While their central temperatures systematically increase with mass, their central densities decrease. The central carbon mass fraction at core helium depletion decreases systematically as a function of mass. This is a signature of the increased importance of the 12 C( α, γ ) 16 O reaction rate with respect to the triple-alpha process at the end of core helium burning for cores of higher mass and lower density, which e ff ectively leads to an increased destruction of carbon (e.g., Arnett 1972). For increasing masses, the higher central temperatures, lower central densities, and lower central carbon abundance imply stronger neutrino losses (see Sect. 4.2), with important consequences for the final \nAfter the merger, the Ne / C burning shell suddenly drops to a lower mass coordinate and a large convective zone develops above. These layers suddenly expand, leading to a dramatic density drop in these regions (see the density contours in Fig. 8a). This particular model additionally experiences a merger of the Ne-burning and O-burning shells (visible at a mass coordinate of 2 M ⊙ at log 10 [( t cc -t ) / yr] = -5 . 2). These mergers lead to a low final mass coordinate of the C / Ne / and O-burning fronts, eventually limiting the development of the Si-burning front, and thus setting a limit to the maximum mass of the iron-rich core. \n- \n- \n- \nFig. 9. Evolution of the C-burning front, the C-free core (which follows the progression of neon burning), and the iron-rich core as a function of time from carbon burning to core collapse for models before and up to the compactness peak (region A of Fig. 1, see inset). As carbon burning becomes increasingly neutrino-dominated, the C-burning front moves further out in mass, ultimately leading to the formation of a larger Cfree and iron-rich core. \n<!-- image --> \nstructure. For reference, a compilation of the changes in interior structure post core helium burning for all models is shown in the Appendix, Figs. D.1, D.2, D.3, D.4, D.5, and D.6.', '5.1. Origin of the increase in final compactness (region A)': 'Wefirst focus on stars at the lower-mass end before the compactness peak (region A in Fig. 1). The change in interior structure as a function of mass is summarized in Fig. 9, in which we show the progression of the C-burning front, the C-free core mass (which follows the progression of neon burning), and the iron core mass as a function of time. \nIn the lowest-mass model (20 M ⊙ ) core carbon burning is convective initially. The C-burning front remains at the center until the carbon fuel is exhausted (at τ ≈ -0 . 75) and the Cburning front moves further out in mass until the conditions for convective core carbon burning are reached once again. Because of the additional fuel from convective core carbon burning, the progression of the C-burning front proceeds somewhat di ff erently in this 20 M ⊙ model compared to the higher-mass models. The carbon-burning phase lasts longer and as a result, neon burning ignites later while the C-burning front is still moving out in mass (as shown by the growth of the C-free core at τ ≈ -0 . 75, dotted lines in Fig. 9). Eventually, the C-burning front reaches a higher value (2.3 M ⊙ ) than in the 21 M ⊙ model, which proceeds radiatively. At the end of the evolution, the model forms a relatively low-mass iron core of 1.59 M ⊙ . \nThe transition from convective to radiative core carbon burning has previously been invoked as the origin of patterns in compactness or iron-core mass (Brown et al. 1999; Sukhbold & Woosley 2014; Sukhbold & Adams 2020; Chie ffi & Limongi 2020; Takahashi et al. 2023). However, we can clearly see in Fig. 9 that the 21 M ⊙ model, which burns carbon radiatively, does not immediately develop a larger iron-core mass (1.56 M ⊙ ) \nFig. 10. Same as Fig. 9 for models beyond the compactness peak (region B in Fig. 1, see inset). As a function of initial mass, neon burning, ignites systematically earlier (as traced by the C-free core mass) and slows down the progression of the C-burning front. Shell mergers between the C / Ne / O-burning fronts are responsible for the sudden drops in the C-burning front at τ ≈ -5. Ultimately, for increasing masses, these models form systematically smaller iron-rich cores. \n<!-- image --> \n- \n- \nthan the 20 M ⊙ model (1.59 M ⊙ ). This transition in the energy transport mechanism from convection to radiation can be understood as a symptom for neutrino losses becoming increasingly dominant, but it is not the cause of the increase in compactness and iron core mass. \nFor increasing masses, the models experience larger neutrino losses during core carbon burning because they have higher central temperatures, lower densities, and a lower central carbon abundance (see Sect. 4.2). The neutrino losses imply that the layers above the burning front can contract more significantly than when neutrino losses are not dominant. This causes an overall more significant and earlier core contraction in these stars, which is reflected in the location of the burning fronts. The smaller amount of carbon fuel, together with the increased nuclear energy generation rate in higher-mass stars accelerate the outward progression of the C-burning front. For example, as shown in Fig. 9, at the end of the evolution, the C-burning front reaches 2.3 M ⊙ in the 21 M ⊙ model, while it reaches 3.35 M ⊙ in the 22 M ⊙ model (which corresponds to the compactness peak). This progression of the C-burning front halts when the energy generated by carbon burning greatly exceeds the neutrino losses and convection sets in. For the compactness-peak model (22 M ⊙ ) the C-burning front reaches a particularly high final mass coordinate. This is because most of the carbon in the large convective shell that forms during core carbon burning is already exhausted by the time of core neon ignition ( τ ≈ -2 . 1), and the carbonburning front quickly burns through the remaining fuel. A further progression of the C-burning front is a symptom of a larger core contraction which results in the growth of a systematically more massive C-free core as a function of initial mass (dotted lines in Fig. 9). In all models, the C-free core mass grows almost up to the final value of the C-burning front. Consequently, the Oand Si-burning fronts also move further out in mass and eventu- \nMi \nM \nally, a more massive iron-core grows in the higher-mass models (dash-dotted line in Fig. 9).', '5.2. Origin of the decrease in final compactness (region B)': 'For models in region B, the early behavior of the carbonburning front is similar to the neutrino-dominated models in region A (see Fig. 10). For higher-mass models, the C-burning front moves systematically further out in mass (e.g. reaching m ≈ 1 . 35 M ⊙ for the 22 M ⊙ model, compared to m ≈ 1 . 6 M ⊙ for the 24 M ⊙ model). However, around the time of core carbon depletion ( τ ≈ -2 Fig. 10), these models experience a clear reversal of the trends observed in region A. The convective carbon-burning episode at τ ≈ -2, where the C-burning front remains at a fixed mass coordinate, is systematically shorter. Afterward, the C-burning front moves out systematically slower in mass for increasing masses and reaches lower final values (e.g., m ≈ 3 . 35 M ⊙ for the compactness-peak model with an initial mass of 22 M ⊙ compared to m ≈ 2 . 28 M ⊙ for the 25 M ⊙ model). Consequently, the final iron-core mass is smaller in the highermass models compared to the lower-mass ones. \nThe observed change in final compactness and iron core mass can be traced back to the same mechanisms as outlined in Sect. 4. Higher neutrino losses in higher-mass models and the ensuing contraction of the core, together with the low initial core carbon abundance and shorter duration of core carbon burning, mean that the conditions for core neon burning to occur are met systematically earlier than in lower-mass models. This can be observed by the timing of the growth of the C-free core (dotted lines in Fig. 10), which occurs systematically earlier in highermass models. Additionally, as shown in Fig. 11, the extent of the degenerate region within the core increases and helps accelerate the core contraction in models where the C-burning front (at the base of the C-burning shell) exceeds the classical Chandrasekhar mass. This earlier central neon ignition increases the core luminosity which slows down the core contraction above the neon-burning front. This causes the observed slower increase in the C-burning front which implies a slower growth of the Cfree core. \nIn addition, central neon burning is more neutrino-dominated for higher mass-models, and leads to a smaller increase in entropy during these burning phases. After core Silicon ignition (as traced by the growth of the iron-rich core, dash-dotted lines in Fig. 10), we observe a phase during which neon and oxygen shell burning become more energetic for higher masses, leading to a higher entropy, which ultimately causes shell mergers just like in our previous experiment (see Sect. 4.5). These shell mergers are found for all models in region B beyond the compact peak and can be identified by a sudden drop in the C-burning front shortly after core silicon burning sets in and the iron-core mass starts to increase (e.g. at τ ≈ 5 in the 24 M ⊙ model). In most cases, this involves a merger of the neon-burning and carbon-burning shells. In the highest mass models ( Mi > 23 M ⊙ ), the oxygen and Ne-burning shell merge additionally. The shell merger implies a reconfiguration of the stellar structure and a large expansion of the layers above the C / O burning shells (see Fig. D.2). The lower final location of the C and O-burning fronts after the shell mergers (e.g., m ≈ 2 . 28 M ⊙ for the 25 M ⊙ model) limit the progression of the Si-burning front. Consequently, the iron-core mass of these higher-mass models remains small.', '5.3. Origin of the second compactness increase (region C)': 'For models between 25 and 28 M ⊙ , neutrino losses become increasingly important as the central temperature increases and the initial central carbon abundance decreases further. The core contracts even further during core carbon burning and the carbonburning front moves further out in mass. Core neon burning ignites even earlier and becomes more neutrino-dominated, which means neon burning proceeds in smaller convective zones (see Fig. D.3). These models are characterized by a large convective oxygen-burning shell that temporarily stops the progression of the carbon-burning front after it reaches its initial location. The final compactness and iron-core mass of these models remains low (see Fig.1). \nAfter an initial mass of about 27 M ⊙ , the final compactness increases again (see Fig.1). This is due to core neon burning becoming fully neutrino dominated. This is best seen in the interior structure evolution of the models with initial masses from 27 M ⊙ to 32 M ⊙ (Fig. D.3). For higher initial masses, neon burning changes from a succession of multiple small convection zones (four in the 27 M ⊙ model), to one or two tiny ( < 0.2 M ⊙ ) convective zones (27.5-30 M ⊙ ) followed by radiative burning, before eventually transitioning to fully radiative, neutrino-dominated neon burning (from models of 31 M ⊙ , where convective core neon burning regions completely disappear, see Fig. D.4). Once again, this transition from convective to radiative burning does not immediately lead to a change in the final compactness. Rather, the compactness gradually increases for fully neutrinodominated models as the core contracts further to compensate the energy loss due to neutrinos. \nIn Fig. 12 we summarize the changes in the stellar structure for models in region C. All models show a very similar trend as in the neutrino-dominated models in region A. As the neutrino losses increase and the core contracts further, the C-burning front moves further out in mass until it ignites a convective-burning shell. In these models, neon burning ignites even earlier than in region B (as shown by the progression of the C-free core at τ ≈ -2, dotted line in Fig. 12). However, for these masses, it becomes neutrino-dominated, which means that the contraction of the layers above the Ne-burning front cannot be slowed significantly by this early ignition of neon. Furthermore, as the initial carbon abundance decreases, so does the neon abundance after core carbon burning, which shortens the duration and impact of core neon burning. Thus core neon ignition barely a ff ects the progression of the C-burning front and neon shell burning no longer leads to a high enough increase in entropy to cause shell mergers with the C-burning layers. The Ne-burning front moves systematically further out in mass for higher-mass models as the core contracts and the C-free core almost reaches the C-burning front (e.g., 2.09 M ⊙ in the 28.0 M ⊙ model, and 3.9 M ⊙ in the 40.0 M ⊙ model, see dotted lines in Fig. 12). Eventually, for increasing masses, the Si-burning front can also move further out in mass and a systematically larger iron-core mass forms (e.g., 1.7 M ⊙ in the 28.0 M ⊙ model compared to 2.35 M ⊙ in the 40 M ⊙ model, see dash-dotted lines in Fig. 12).', '5.4. Origin of the second compactness decrease (region D)': 'For models beyond 40 M ⊙ , we once again observe a drop in the final compactness (see Fig. 1). As best observed in Fig. D.5, in these models, both carbon and neon are neutrino-dominated and the respective nuclear burning fronts move out so far in mass from the central regions that they no longer play a large role in setting the final iron core mass (see Fig. 13). Instead, \nFig. 13. Same as Fig. 9 for models in the mass range of the second decrease in compactness (region D in Fig. 1, see inset). The dashed line traces the O-burning fronts. For increasing masses, the progression of the O-burning front is slowed by a systematically earlier ignition of core silicon burning (as traced by the iron core mass). \n<!-- image --> \n-3.2 \n-0.3 \n0.0 \n-3.2 \n-0.3 \n0.0 \n0.3 \n3.2 \n(b \n23.0 \n24.0 \nlogo(time until collapse yr) \nMi \nM \n-2 \n4 \nuntil \ncollapse \nyr) \nFig. 11. Same as Fig. 7 for three stellar models of our grid after the first compactness peak. \nMi \nM \n<!-- image --> \n- \n- \nFig. 12. Same as Fig. 9 for models in the mass range of the second compactness increase (region C in Fig. 1, see inset). In these models, both carbon and neon burning are fully neutrino-dominated. \nthe core oxygen and silicon-burning phase increase in importance, as shown in Fig. 14. Oxygen burning produces a large amount of energy per gram and occurs convectively (Woosley et al. 2002). In analogy to the e ff ects observed for the C- and Neburning front in lower-mass models, after core oxygen burning the oxygen-burning front initially moves further out in mass the more neutrinos dominate in the center (as indicated in Fig. 13)). The timescale between core oxygen and core silicon ignition shortens for increasing masses (as shown by the timing of the growth of the iron-rich core, dash-dotted lines in Fig. 13). The (at least partially) degenerate core region becomes more extended, as it generally reaches up to the base of the oxygen-burning front (see Fig. 14). The core contraction accelerates for models in which the oxygen-burning front reaches a mass coordinate higher than the (classical) Chandrasekhar mass. For example, in \n<!-- image --> \n- \n- \nthe 42 M ⊙ model, the oxygen-burning front (at the base of the oxygen-burning shell) even reaches the e ff ective Chandrasekhar mass (indicated by a dotted line in Fig. 14c) with m ∼ 1 . 85 M ⊙ at log( t / yr) = -2 . 5. When silicon burning occurs during the oxygen shell-burning phase, it slows down the contraction of the layers above, and with it the progression of the oxygen-burning front. Eventually, this limits the maximum progression of the Si-burning front, which forms a lower iron-core mass than in models in which core silicon burning occurs later. For example, as shown in Fig. 14, in the 43.0 M ⊙ model the oxygen-burning front (dashed lines) remains at a mass coordinate of 2.1 M ⊙ after core silicon burning ignites (as traced by the growth of the iron-rich core, dash-dotted lines) . It only moves further out in \nFig. 14. Same as Fig. 7 for three stellar models of our grid after the second compactness increase. Here, neon burning is fully neutrino-dominated and radiative, and not visible in the figure. After convective core oxygen burning (hatched region in the center) the degenerate core region reaches up to the oxygen burning front (at the base of the convective oxygen-burning shell) for all models. Panel c: When this front exceeds the (e ff ective) Chandrasekhar mass, the degenerate core contracts and core silicon burning ignites earlier, slowing the contraction of the layers above and a ff ecting the extent of the convective region. \n<!-- image --> \nmass shortly before core collapse ( τ = -7), which does not leave enough time for the iron-core mass to grow further in mass than 1.85 M ⊙ in this model. \nBeyond region D, for models with higher masses the final compactness remains lower than during the second peak (see Fig. 1a) but above a value of 0.4. The 44 M ⊙ model has an unusually high compactness (0.7) compared to the other models with initial masses larger than 40 M ⊙ . We link this back to the oxygenburning front remaining at a rather low mass coordinate (close to 1.45 M ⊙ , not exceeding the e ff ective Chandrasekhar mass) after core oxygen burning, triggering a small (0.1 M ⊙ ) convective episode (see Fig. D.5). The core contracts less during core oxygen shell burning than for the adjacent models, which means silicon burning ignites later and does not slow down the progression of the oxygen-burning front as e ff ectively, leading to a larger final compactness and iron core mass. Small changes in the energy generation rate or mixing could influence the existence of this small oxygen-burning episode and we therefore consider this model not to be representative of the more general patterns described here. We note that the emergence and disappearance of such small secondary peaks in the final structure landscape can be observed when physical assumptions are varied, such as the e ffi ciency of semiconvection (see Appendix B) and deserve further study. Despite the observed variations, it should be noted that for all these models, the binding energy above the inner core is so high (see Fig. 1 d) that the formation of a BH is expected even if a successful explosion were triggered in the core.', '6. Global picture': "Based on our findings, we can derive a global picture of the physical processes that determine the final core structures of massive stars. After core helium burning, thermal neutrino emission becomes important and complicates the final evolution by taking away energy from the interior, leading to an acceleration of the evolution. The transition from convective to of radiative carbon and neon burning is a symptom of neutrino losses becoming dominant, but not the cause of the observed changes in \nthe final structure patterns. For higher masses (lower densities, higher temperatures, and lower central carbon abundance), neutrino losses become increasingly important and a ff ect the e ffi -ciency and timing of nuclear burning episodes and with it the associated core growth. \nWe identified physical mechanisms explaining the emergence and decline of the two prominent 'compactness peaks' (see Fig. 1) found in several independent studies, i.e. mass ranges for which stars develop a high final compactness, iron core mass, central entropy, and binding energy. These are summarized schematically in Fig. 15, where we show how neutrinodominated nuclear burning in the cores of massive stars post core helium depletion leads to di ff erent stellar structures. These outcomes are 'written' in the stellar cores at core helium depletion, when the initial burning conditions of the subsequent burning phases are set.", '6.1. Increase in final iron core mass (Region A and C):': 'For increasing stellar masses, the central nuclear burning source (C or Ne) becomes increasingly neutrino-dominated. As observed in multiple studies (e.g. Brown et al. 2001; Sukhbold & Woosley 2014; Sukhbold et al. 2018; Sukhbold & Adams 2020; Chie ffi &Limongi 2020; Takahashi et al. 2023), energy transport at the center transitions from convection in a large central region followed by shell burning, to ever smaller and more numerous successive convective burning episodes, until it becomes fully radiative for stars in which nuclear burning is fully neutrinodominated. In Fig. 15 I, the structure of a star with a neutrinodominated, radiative core is shown schematically mid burning (left wedge) and at the onset of core collapse (right wedge). The decreasing amount of fuel together with the neutrino dominance lead to an acceleration of the core contraction and to a faster and further outward progression of the burning front (the point where the maximum nuclear energy generation rate is reached, red lines in Fig. 15 I) and associated growth of the C / Ne-free core below. When the burning front reaches layers with a high C / Ne abundance, the generated nuclear energy becomes large enough \nFig. 15. Schematic representation of the mechanisms leading to the emergence and decline of peaks in final iron core mass and compactness. Horizontal gray dotted guidelines are included to aid the comparison. Star I: Before peak Schematic stellar structures of a star with a low final iron core mass at two points in the evolution represented by two adjacent wedges. The initial mass fraction of the fuel is still high (see red area in the rectangle) but the main burning stage is neutrino-dominated (arrows). Because neutrinos take away energy, the core contracts and the burning front (red line) moves out in mass as the burning progresses. The large amount of fuel prevents it from moving far out before a convective zone forms above the burning front once the energy generated is high (shaded red area). Ultimately (right wedge), it grows a small fuel-free core and the silicon-burning front (yellow line) cannot move far out in mass, forming a small iron core (black area). Star II: Peak With a higher mass and lower initial fuel mass fraction, the burning is even more neutrino dominated and a strong contraction occurs. Hence the burning front moves further out in mass than in I, but stays just below the e ff ective Chandrasekhar mass. Aided by degeneracy pressure support, the star burns through almost all available fuel before the next burning stage ignites, after which it quickly moves out in mass, growing a large fuel-free core and eventually a large iron core. Star III: Beyond peak: For an even lower initial fuel abundance and higher mass, the burning is even more neutrino dominated and the core contraction accelerates. The burning front moves further out and exceeds the e ff ective Chandrasekhar mass. This triggers a fast contraction of the partially-degenerate core and the next burning stage (blue area) ignites early. This next stage suppresses nuclear burning at the front above. As a result, the burning front moves out in mass slowly and eventually (right wedge) the star grows a low-mass iron core. \n<!-- image --> \nto significantly exceed the neutrino losses, and convection sets in above the front (shaded red area in Fig. 15 I). The star with the structure I still contains a large amount of fuel, which means (a) that a small core contraction is su ffi cient for convection to set in and (b) that the newly formed convective region contains enough fuel to remain until core collapse without the need of a large core contraction. This limits the growth of the fuel-free core below as the burning front moves only slowly out in mass. Ultimately (right wedge in Fig. 15 I), this structure grows a relatively small fuel-free core, which in turn limits the growth of the Si-rich core, which then forms a relatively low-mass iron-rich core. \nThe final location of the burning front reflects the contraction of the inner core and sets the maximum core growth for the subsequent nuclear-burning episodes. Through this process, for increasing masses, stars in which the central burning source becomes neutrino-dominated grow increasingly massive C / Nefree cores, which in turn form massive oxygen-free cores and eventually, more massive iron-rich cores. For higher masses, as the cores contract further and their density increases, electron degeneracy also starts to play a role. Degeneracy pressure contributes to supporting the core during the main burning phase (carbon or oxygen, depending on the mass range), delaying the ignition of the next burning phase as long as the burning front does not exceed the e ff ective Chandrasekhar mass. \nAt the compactness peak (structure II in Fig. 15), nuclear burning at the center is strongly neutrino-dominated due to a higher mass (lower initial density) and low initial fuel mass fraction. The strong core contraction leads to an outward progression of the burning front until it nearly reaches the e ff ective Chandrasekhar mass. Support from degeneracy pressure helps slow the contraction during the main burning phase such that nearly all the fuel in the convective region is burned before the next burning stage ignites in the center. Afterward, the burning front quickly burns through the former convective region as the core contracts further. It grows a particularly massive fuel-free core and eventually, a massive iron core.', '6.2. Decrease in final iron core mass (Region B and D)': 'For even higher masses, stars develop structures that are even more neutrino-dominated (star III in Fig. 15). Two mechanisms, (a) a lower initial fuel abundance decreasing the duration of the main fuel burning phase (core C-burning or Ne-burning), and (b) electron degeneracy no longer helping support the stellar structure once the main burning front and the fuel-free core it produces exceed the (e ff ective) Chandrasekhar mass, both lead to an acceleration of the core contraction. What follows is an early ignition of the next burning fuel while the main nuclear burning phase is still in progress (e.g. Ne ignition during C burning or Si ignition during Ne / O burning, indicated by the blue central \nregion in Fig. 15 III). The luminosity of this new nuclear energy source contributes to slowing down the contraction of the layers above and prevents the further growth of the C / O-free core. Consequently, the Si-burning front cannot move far out in mass and grows a lower-mass iron core than in star II. \nAdditionally, in particular in region B, mergers of the Neand C-burning shells, and, in some cases, of the Ne and Oburning shells, can occur due to the high energy generation rate and entropy at these fronts. These lead to a reconfiguration of the stellar structure in which the burning fronts all merge together at the lowest mass coordinate. If these mergers occur before the end of core silicon burning, they limit the progression of the silicon-burning front, and, with it, the growth of the ironrich core. Eventually, stars that experience such shell mergers form iron cores of even lower mass. Shell mergers before the end of core Si burning are thus an additional mechanism which produces final stellar structures with low iron core masses. \nBecause of these physical mechanisms, we expect a nonmonotonic landscape of final compactness, iron core mass, central entropy, and binding energy as a function of the initial and core masses of massive stars. Based on a neutrino-driven supernova model we have shown that stars with high compactness, iron core masses and binding energy are predicted to form black holes. Hence, the existence of these robust final structure patterns of massive stars observed by multiple independent studies imply the formation of black holes at characteristic mass ranges.', '7.1. Uncertainties in stellar physics': 'Our stellar evolution models are subject to uncertainties concerning the physics of massive stars that can a ff ect the lanscape of the final structure of stars, and with it, their fate. \nNuclear reaction rates: As discussed previously, the final core structures of stars are sensitive to the conditions under which core helium and carbon burning take place. In particular, the 12 C( α, γ ) 16 O is a key helium-burning process which is still very uncertain and has a tremendous impact on the final fate of stars (e.g. Austin et al. 2014; Farmer et al. 2020). This nuclear reaction rate sets the amount of carbon left after core helium burning ( XC ) by fusing carbon into oxygen. As shown in Appendix A, varying this reaction rate changes the core mass range at which stellar structures reach a high final compactness and iron-core mass and are expected to form black holes. A higher nuclear reaction rate resulting in a lower core carbon abundance means that neutrinodominated burning occurs at lower CO core masses, systematically shifting the compactness landscape (Sukhbold & Adams 2020; Schneider et al. 2023). For example, a 10% higher carbon alpha-capture rate leads to a compactness peak occurring at 0.4 M ⊙ lower CO core masses (see Appendix A). In turn, this increases the number of stars expected to form black holes, as stars with lower (CO core) masses are more common. It will also affect the final masses of black holes, which is expected to be proportional to the final mass of a star and is generally lower for progenitors with lower masses (see also Schneider et al. 2023). \nSimilarly, the value of the 12 C + 12 C cross section also systematically shifts the final core structures of stars (Chie ffi et al. 2021). The newly determined, higher rate found by Tumino et al. (2018) due to low-lying resonances shifts the compactness landscape to slightly lower core and initial masses compared to the classical rate we use in the present work (Caughlan & Fowler \n1988) by allowing core carbon burning to take place at lower temperatures and densities. Carbon thus ignites under conditions where neutrino losses are less severe, which means that the overall core contraction is slightly less significant. This results in a systematic shift in the final compactness to slightly lower values (0.05 in ξ 2 . 5, see Fig. 9 in Chie ffi et al. 2021). Finally, the triplealpha reaction rate also plays an important role, though it is less uncertain (Austin et al. 2014). \nMass loss: Wind mass loss (and equivalently, metallicity) has a large impact on the properties of stars and is still very uncertain. Latest observational constraints suggest that wind mass loss at solar metallicity may be less strong than we assume here. When mass loss is strong enough to remove a large fraction of the hydrogen-rich envelope, it can have a large impact on the final structure of stars as it changes the evolution of the helium core, and with it the initial conditions for core carbon burning. In particular, strong wind mass loss can lead to a less e ffi -cient hydrogen-burning shell and to a recession of the convective helium-burning core, which results in a large core carbon abundance XC at core helium depletion. This in turn shifts the parameter space for neutrino-dominated carbon burning to higher CO core masses. Stars that experience strong wind mass loss are thus generally more explodable (just like binary-stripped stars, see Sect. 7.3). In our models with high enough initial masses ( ≳ 35 M ⊙ ), stars may also enter a Luminous Blue Variable phase in which eruptive mass loss has been observed to occur (Smith et al. 2004; Davies et al. 2018). The mechanism behind these eruptions has been suggested to be related to helium opacity in the outer layers (Jiang et al. 2018; Grassitelli et al. 2021). Unlike line-driven winds, this mechanism would also be important at low metallicity. This highly uncertain mass loss is not taken into account in our models but can lead to the removal of the entire hydrogen-rich envelope, which would significantly a ff ect the final core properties, and lead to a final core structure that explodes more easily. \nMixing: How mixing proceeds in stellar interiors remains an importance uncertainty. Convective core boundary mixing, computed in our models through step overshooting, influences the size of convective cores. During core hydrogen burning, it helps determine the initial mass to helium core mass relation. Similarly, core overshooting during core helium burning influences the resulting CO core mass and thus has a particularly large influence of their final fate. Temaj et al. (2024) showed that core overshooting systematically shifts the compactness landscape of stars as a function of their initial masses, as a larger overshooting allows a star to behave as if it had the core of an initially higher-mass star. This also implies a larger number of black holes formed, as more stars can reach the mass range for black hole formation (see also Fig 6. of Schneider et al. 2023). As a function of the CO core mass, the compactness landscape is similar to that of single stars, though the compactness peak shifts towards higher CO core masses (1.3 M ⊙ by varying the step overshooting scaling parameter α OV from 0.05 to 0.5, Temaj et al. 2024). This shift is because, as convective overshooting a ff ects the core mass of stars, it also changes their luminosity and therefore mass loss rate. Especially for high initial masses, when the total mass loss is large, XC increases and thus the compactness landscape shifts to higher CO core masses (Temaj et al. 2024). Overshooting during core helium burning also influences the ratio of the total mass to the CO core mass, thus the final black \nhole mass. In stripped helium stars, large overshooting implies a lower black hole mass (see also Fig. 6 of Schneider et al. 2023). \nAs we explore in more detail Appendix B, semiconvection is another mixing process that has a large e ff ect on the final core structure of stars. It changes the compactness landscape as a function of the initial mass but only slightly a ff ects the compactness landscape as a function of the CO core mass (see Fig. B.3). Semiconvection a ff ects the stellar structure after core hydrogen exhaustion through the development of intermediate convection zones (Sibony et al. 2023). Furthermore, it can lead to late ingestion of helium during core helium burning depending on the chemical gradient at the edge of the helium core (Langer et al. 1985; Langer 1991) and thus not only shifts the CO core mass but also XC (see Fig. B.2). This can explain the "noise" or "randomness" reported by Sukhbold et al. (2018) in the compactness landscape (see also Chie ffi & Limongi 2020). Schootemeijer et al. (2019) found through a comparison with observations that semi-convection may be an e ffi cient process, i.e. favoring high values of α sc ≥ 1. However, it remains a major uncertainty in our models.', '7.2. Shell mergers': 'In our models beyond the compactness peak, we identify several models that experience mergers of their C-, Ne- and in some cases also O-burning shells. As we show in Sect. 4.5 these are the result of an increasing entropy at a nuclear burning front (e.g. Ne or O) that exceeds the entropy of the layers above, implying that these layers are unstable against convection and must mix. Several studies have reported the occurrence of such shell mergers in their one-dimensional models for a similar parameter space (i.e models before and just beyond the compactness peak, typically in the CO core mass range from 3 to 7 M ⊙ , Sukhbold & Woosley 2014; Sukhbold et al. 2018; Sukhbold & Adams 2020; Collins et al. 2018; Laplace et al. 2021). We argue that these shell mergers are not artifacts from our treatment of convective boundary mixing since we do not include overshooting above shells post helium burning. In Appendix C, we verify that these shell mergers are barely a ff ected by our choices of spatial and temporal resolution. The exact predictions for the outcomes of shell mergers will still depend on a more accurate treatment of convective mixing than the mixing-length approximation used here, i.e. informed by multi-dimensional simulations (Rizzuti et al. 2023). Several three-dimensional calculations have also observed the occurrence of shell mergers, strengthening the finding from one-dimensional simulations (e.g., Couch & Ott 2013; Collins et al. 2018; Yoshida et al. 2019; Andrassy et al. 2020; Yadav et al. 2020; McNeill & Müller 2020). Collins et al. (2018) shows that shell mergers help develop stellar structures that have extended oxygen shells with high convective Mach numbers. In such stellar structure, perturbations can be induced by this convective oxygen shell burning at the time of core collapse. 3D supernova simulations suggest that such perturbations are crucial for increasing the e ffi ciency of neutrino heating in neutrinodriven supernova explosions and for a successful shock revival (e.g., Couch & Ott 2013). Thus, stellar structures experiencing shell mergers may be particularly prone to successful explosions. In addition, due to the mixing process and ensuing nuclear burning episodes, stars experiencing shell mergers are expected to produce peculiar abundance patterns. Typically, these are enhanced alpha-capture products in the oxygen-shell region, which in turn a ff ect the nucleosynthesis, in particular the Ca / O ratio of core-collapse supernovae (e.g., Ritter et al. 2018; Dessart & John Hillier 2020; Laplace et al. 2021). In our work, we find \nthat shell mergers are expected in stellar structures with strong neon shell burning. We observe that this systematically occurs in models just beyond the compactness peak (23 < Mi / M ⊙ < 27, 6 . 3 < M CO / M ⊙ < 8 . 5), in which carbon burning is neutrinodominated and core neon burning is becoming neutrino dominated.', '7.3. Final structures of stars that experience binary interactions': 'In this work, we have focused on the core structures of single massive stars. However, it is well established that the majority of massive stars live in binary systems and are expected to interact during their lifetime (Sana et al. 2012). As a result of binary interactions, the core structure of stars can be greatly modified, and lead to a di ff erent fate even for stars with the same core mass (Laplace et al. 2021; Schneider et al. 2021, 2023; Schneider et al. 2024). In addition, the vast majority of black holes are found in binaries observationally. In these systems their properties, in particular their masses, can be measured most accurately (Casares & Jonker 2014). It is therefore important to discuss the e ff ect of binary interactions on the final structures of stars. \nThe mechanisms we identified here that set the final structure of stars (i.e. the impact of neutrino-dominated burning) apply generally for any stellar structure, as shown by our experiment in Sect. 4 and are mainly influenced by the initial conditions for core carbon burning (i.e. the CO core mass and central carbon abundance, see also Patton & Sukhbold 2020). These conditions can be influenced by binary interactions or strong wind mass loss, in particular if the structure of a star is significantly affected after core H-burning, which will ultimately change the initial conditions for core carbon burning (Brown et al. 1999, 2001; Woosley 2019; Schneider et al. 2021; Laplace et al. 2021). This implies a shift in the compactness landscape. \nWhen it comes to black hole formation, binary-stripped stars are particularly important binary products. In isolated massive binary systems, they are the natural progenitors of both black holes that eventually form binary black hole mergers that are now commonly detected as GW sources (Schneider et al. 2023). After mass transfer or strong wind mass loss removes the outer envelope of these stars, the conditions under which core helium burning occur can change drastically compared to single stars. A weaker or even absent hydrogen-burning shell, and strong wind mass loss leads to a shrinking of the convective helium-burning core. This results in a higher final core carbon abundance compared to single stars for the same CO core mass (Brown et al. 2001; Schneider et al. 2021; Laplace et al. 2021). The higher core carbon abundance means that the compactness peak, and the corresponding expected parameter space for BH formation is shifted to higher core masses in these stars ( M CO ≈ 7 . 5 M ⊙ , see Fig. 2 in Schneider et al. 2023). Therefore, binary-stripped stars are expected to be common progenitors of supernovae and to only form black holes for high initial masses (of about 30 M ⊙ ), which significantly reduces the parameter space for BH formation (Schneider et al. 2021). Moreover, the core structures of binary-stripped stars have a weak metallicity dependence, implying the formation of universal BH masses at the compactness peaks, with strong observational signatures (Schneider et al. 2023). \nThe final core structures of accretors and stellar mergers can also be a ff ected by binary interactions. In our recent work (Schneider et al. 2024) we have shown that mass accretion, in particular onto stars that have completed core hydrogen burning, can lead to systematically di ff erent final core structures. \nIn particular, the compactness peak in case B and C accretors shifts systematic to lower core masses (Schneider et al. 2024). Contrary to binary-stripped stars, this means that accretors form black holes in stars with lower initial masses than in single stars.', '7.4. Compactness peaks and black hole formation': 'In this work, we estimate the outcome of core-collapse for our models based on the Ertl et al. (2016) criterion and on the 1D semi-analytical neutrino-driven model of Müller et al. (2016). We find a correspondence between the formation of black holes and models with high final compactness, which all have a high final binding energy, iron-core mass, and central entropy (see Fig. 1). To first order, compactness (or equivalently, any of these quantities) captures the information that stars with high binding energy are di ffi cult to explode. Thus, independently of the uncertainties regarding the explosion mechanism of core-collapse supernovae, a signature of the characteristic final structure of stars described here is expected in their final remnants and in the properties of their explosions. \nHowever, while compactness captures the general final structure of the star, it is not a good predictor of the detailed explosion outcome (see Sect. 3.1). Explodability criteria and the remnants expected after the core collapse are the subject of active discussion in the supernova modeling community, including within groups studying the neutrino-driven explosion mechanism. For example, some 3D neutrino-driven simulations (e.g., Chan et al. 2018; Kuroda et al. 2018; Ott et al. 2018; Burrows et al. 2023) have recently found successful shock revival in models with high compactness. However, the revival of the supernova shock does not necessarily imply a full supernova explosion with complete ejection of the stellar layers, or that these will all form neutron stars. Due to the high accretion rate typical for models with high compactness, these are still likely to form black holes through fallback accretion, especially in stars with high core masses (Heger et al. 2023). For example, Chan et al. (2018) found that a 40 M ⊙ progenitor star with high compactness forms a black hole early on through fallback accretion. In their simulation, the supernova shock succeeds in unbinding parts of the envelope despite the explosion energy being lower than the binding energy of the envelope. More recently, Burrows et al. (2023) found that a 40 M ⊙ model with high compactness results in a successful explosion, though the remnant eventually forms a black hole due to the high accretion rate onto the proto neutron star. Black holes formed through fallback accretion have systematically lower masses than those formed through direct collapse because parts of the star can be expelled. They therefore have distinct properties, which may be constrained through observations (Chan et al. 2020). However, consistently finding explosion of models with high masses and high final compactness would be at odds with observational constraints, which seem to point to a lack of core-collapse supernovae from stars with high core masses (e.g., Smartt 2009; Davies & Beasor 2020).', '7.5. Observational constraints on the pre-supernova structure of stars': 'The robustness of the patterns found for the final core structure of massive stars, i.e. the \'compactness peaks\' based on the neutrino-dominated burning mechanism identified here, suggests that signatures of this pattern could be identifiable in observations. \nSukhbold & Adams (2020) showed that the luminosity of immediate BH progenitors could be used to test the existence of characteristic core structures corresponding to the compactness peaks observationally. The observation of the disappearing redsupergiant N6946-BH1 could be a piece of evidence (Sukhbold & Adams 2020), though more observations are needed to understand the latest constraints on this object (Beasor et al. 2024; Kochanek et al. 2024). Finding more disappearing stars or additional direct observational link between black holes and supernovae, and their progenitors, will provide crucial constraints. For example, Temaj et al. (2024) showed that the luminosity of supernova progenitors in hydrostatic equilibrium can be directly linked to their core mass, independently of uncertainties in interior mixing or wind mass loss. \nIf there is a direct link between massive star structures in the "compactness peak" and BH formation, i.e. if it is not fully obscured by supernova physics and fallback dynamics, then black hole masses should also encode information about the final structure of stars in the compactness peak. We expect that such a link should be readily identified in binary-stripped stars, which have characteristic final structures with a weak metallicity dependence (see also Sec. 7.3). We predicted that this would lead to the formation of BHs of universal masses (Schneider et al. 2023). Alternatively, if supernova and fallback physics \'wash out\' these signals, then identifying broad features in the BH mass distribution and significant numbers of low-mass black holes may provide further constraints into these processes and their relative importance. The complete absence of features in the BH mass distribution of binary-stripped stars (e.g., a flat black hole mass distribution) would also give particularly strong constraints on stellar, binary, and supernova physics, as this would require narrow ranges of physical ingredients. \nUsing GW observations to find features in the chirp-mass distribution of binary BH mergers could provide evidence for the existence of the characteristic final stellar structures of binarystripped stars (Schneider et al. 2023; Disberg & Nelemans 2023). Considering a population of merging binary BHs resulting from isolated binary evolution, we predicted peaks in the chirp-mass distribution of binary BH mergers at ∼ 8 M ⊙ and ∼ 14 M ⊙ , and a dearth between ∼ 10 -12 M ⊙ (Schneider et al. 2023). Based on the published data from LIGO-Virgo-KAGRA (LVK) observations, a peak at a chirp mass of ∼ 8 M ⊙ is found to be statistically significant, while the existence of a feature at ∼ 14 M ⊙ is less robust (Talbot & Thrane 2018; Tiwari & Fairhurst 2021; Tiwari 2022; Abbott et al. 2023; Edelman et al. 2023; Farah et al. 2023). Tentative evidence for the predicted dearth in the distribution between 10 and 12 M ⊙ has recently been reported, though more data, even beyond the upcoming O4a data release by the LVK collaboration, is required to further test its existence (Adamcewicz et al. 2024; Galaudage & Lamberts 2024). \nSignatures of the characteristic core structures of massive stars that form black holes may also be identified in the mass distribution of BHs from electromagnetic observations (Nambena et al. in prep.). Studies of black holes in X-ray binaries point to the existence of a characteristic peak in black hole masses of about ∼ 8 M ⊙ (e.g., Kreidberg et al. 2012; Casares et al. 2017), which is similar to the mass we predict for black holes in binary-stripped stars originating from the compactness peak (Schneider et al. 2023). The recent discovery of new X-ray quiet black holes in wide binaries (Shenar et al. 2022; El-Badry et al. 2023; El-Badry et al. 2023; Gaia Collaboration et al. 2024) provides a promising avenue to discover more black holes and better \nunderstand their properties, and ultimately, their formation and progenitors.', '8. Conclusions': "Understanding the fate of massive stars, i.e. whether stars form black holes at the end of their lives or successfully explode in a supernova is a major goal of astrophysics. In this work, we have explored how stellar physics determines the pre-supernova core structure and the role it plays in the fate of massive stars by studying the final evolution of single star in the range from 17 to 50 M ⊙ at solar metallicity. Our findings can be summarized as follows: \n- -We confirm that the final structure landscape of massive stars found in multiple independent one-dimensional stellar evolution studies (Timmes et al. 1996; Brown et al. 1999; Sukhbold & Woosley 2014; Sukhbold et al. 2018; Müller et al. 2016; Chie ffi & Limongi 2020; Chie ffi et al. 2021; Schneider et al. 2021, 2023; Schneider et al. 2024; Takahashi et al. 2023; Temaj et al. 2024) is robust. The prominent features in the final compactness and iron core mass can be traced back to the conditions under which late nuclear burning occurs in the cores of stars at this mass range.\n- -Summarizing quantities such as the final compactness, central entropy, iron core mass, and binding energy are all equivalent and intrinsically linked. Stars with high final compactness all have a high binding energy, and we verify that the formation of a black hole is likely in these stars based on a neutrino-driven supernova model. Independently from uncertainties in the supernova mechanism and explodability of stars, stellar physics thus plays a large role in determining the final remnants of stars, in particular the formation of black holes. We confirm that the final structure of massive stars is already mostly 'written' in their cores at the end of core helium burning.\n- -We find that the final compactness increases when carbon burning becomes neutrino-dominated (region A in Fig. 1). The transition from convective to radiative core carbon burning and the change in the extent and number of carbonburning shells identified in previous studies is a symptom of this e ff ect, but not its cause. Strong neutrino losses force a larger core contraction and the development of a larger carbon-free core. Eventually, this enables the formation of a larger iron core. The same mechanism leads to the formation of a second compactness increase once neon burning becomes neutrino dominated (region C in Fig. 1).\n- -Wetrace back the drop in compactness after the compactness peaks (regions B and D in Fig. 1) to the e ff ect of an earlier ignition of the next nuclear fuel (neon or silicon) in these stars, which slows down the core contraction and eventually leads to the formation of smaller iron core masses. We find that the next burning phase ignites earlier in these stars because of two main mechanisms (1) a shorter duration of the main burning stage due to a decrease in available fuel and increase in temperature and (2) the role of electron degeneracy, which further accelerates the contraction of the core when the fuel-free core exceeds the e ff ective Chandrasekhar mass.\n- -Shell mergers between the C and Ne- burning shells, and in some cases also between the Ne and O-burning shells, occur in our models after the first compactness peak, just like other studies have observed (Sukhbold & Woosley 2014; Sukhbold et al. 2018; Collins et al. 2018). These contribute to the drop \nin compactness after the compactness peak, lead to smaller iron-core masses, and make these stars more explodable. We find that shell mergers are not numerical artifacts but take place because of energetic neon and oxygen burning which generates enough entropy to mix with the layers above, independently of convective boundary mixing. Shell mergers not only a ff ect the pre-supernova structure but also change the pre-supernova composition and may have observable signatures which call for further investigation. \n- -The exact final core structure landscape of stars is subject to uncertainties in stellar evolution, such as semi-convective mixing, mass loss, and uncertain nuclear reactions, in particular the 12 C( α, γ ) 16 O reaction rate. The origin of small variations and secondary peaks in compactness and central entropy observed when nuclear burning conditions change remains to be explored further. However, the robustness of the main features identified in the final structures of massive stars as a function of their CO core masses provides a clear theoretical prediction and an opportunity to constrain stellar physics from the observation of black holes and their immediate progenitors. \nThe final structure landscape discussed here applies for any massive star, including those that interact in binary systems. However, binary interactions shift the compactness landscape to di ff erent core masses, which a ff ects the parameter space for expected black hole formation and successful supernovae explosions (Schneider et al. 2021, 2023; Schneider et al. 2024). For accurate predictions of supernova and black hole formation, in particular for understanding the population of gravitational-wave sources, it is crucial to take the di ff erent final core structures of single and binary stars into account. \nAcknowledgements. We thank the referee for constructive comments that helped improve the manuscript. This work was funded by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (Grant agreement No. 945806) and supported by the Klaus Tschira Stiftung. It was also supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy EXC 2181 / 1-390900948 (the Heidelberg STRUCTURES Excellence Cluster).", 'References': "- Abbott, R., Abbott, T. D., Acernese, F., et al. 2023, Physical Review X, 13, 011048\n- Adamcewicz, C., Lasky, P. D., Thrane, E., & Mandel, I. 2024, arXiv e-prints, arXiv:2406.11111\n- Adams, S. M., Kochanek, C. S., Gerke, J. R., Stanek, K. Z., & Dai, X. 2017, MNRAS, 468, 4968 \nAguilera-Dena, D. R., Langer, N., Antoniadis, J., et al. 2022, A&A, 661, A60 Aguilera-Dena, D. 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E., Justham, S., et al. 2021, A&A, 645, A6 \n- Zha, S., Müller, B., Weir, A., & Heger, A. 2023, ApJ, 952, 155 \nAppendix A: The effect of the 12 C( α, γ ) 16 O reaction rate \n<!-- image --> \n/circledot \nFig. A.1. Central carbon mass fraction at core helium depletion as a function of the CO core mass for varying 12 C( α, γ ) 16 O reaction rate. \nAmong all nuclear reactions taking place in the interiors of stars, the 12 C( α, γ ) 16 O reaction rate is particularly important for determining the fate of stars (e.g., Weaver & Woosley 1993). However, it remains notoriously uncertain (Farmer et al. 2020). Towards the end of core helium burning, this reaction becomes the main helium-burning process. It determines the fraction of remaining carbon and oxygen in the core after core helium burning, as shown in Fig. A.1. For a higher reaction rate, the core carbon mass fraction X C reaches systematically lower values. As this sets the initial conditions for the subsequent core carbonburning stage, this reaction thus plays a crucial role in determining the final structure and fate of stars. In Fig. A.2, we show the e ff ect of varying this reaction rate, from a ∼ 15% lower (Kunz et al. 2002) to a 10% higher rate than our fiducial assumptions on the final central entropy and compactness. Both as a function of initial and CO core mass, varying this reaction rate systematically shifts the location of the compactness peak to lower masses for a higher reaction rate. This is because a higher rate leads to a lower core carbon abundance and thus to carbon burning becoming neutrino-dominated at lower masses. This is thus a similar e ff ect as described in our experiment in which we vary the core carbon abundances for models with the same core mass (Sect. 4). A di ff erence comes from the role of the 12 C( α, γ ) 16 O reaction rate in setting the mass of the CO core (see the small shift in CO core mass in Fig. A.1). Secondary compactness and central entropy peaks appear at CO core masses between 5 and 6 M ⊙ and 7 to 8 M ⊙ for a varying 12 C( α, γ ) 16 Orate. These are discussed in more detail in Appendix B. Finally, shifts in the maximum and minimum compactness values can be observed. These can be understood as a varying degree of maximum or minimum contraction in the stellar structure, induced by the varying e ff ect of neutrino losses for di ff erent nuclear burning conditions during core carbon burning.", 'Appendix B: The impact of semi-convective mixing on the final structure': "Semi-convection is a particularly uncertain mixing process that occurs in stellar layers that are unstable against convection because of their superadiabaticity (Schwarzschild criterion for convection) but supported by a stabilizing molecular weight gradient (Ledoux criterion). It occurs for example at the end \nof core hydrogen burning when a H / He composition gradient is created by the receding convective core. This can form a transient intermediate convection zone (ICZ) and hence change the structure of the core outside the inner convective region (Sibony et al. 2023). Semi-convection remains di ffi cult to constrain observationally, though a recent study suggest that it may be an e ffi cient process, with an e ffi ciency parameter of α sc ≥ 1 (Schootemeijer et al. 2019). It has a large influence on the final core structure of stars because it a ff ects the development of the convective helium-burning core (Langer et al. 1985; Langer 1991), generally leading to smaller core masses for a higher semi-convection e ffi ciency (see Fig. B.1). In addition, even small semi-convective layers above a convective burning core can drastically change the initial conditions for the next burning phase. For example, late ingestion of helium due to semiconvection during core helium burning can strongly a ff ect the core carbon abundance at core helium depletion (Langer 1991). This is shown in Fig. B.2, where the core carbon abundance X C at the end of core helium depletion is plotted as a function of the CO core mass for varying degrees of semi-convective mixing. Between the models with no semi-convection mixing and the α sc = 0 . 1 and α sc = 10 models, there appears to be a systematic shift to a higher X C. This can be attributed to a systematically smaller helium core mass after core hydrogen burning with respect to the initial mass, which creates a systematically smaller CO core and generates more carbon overall (see also Fig. B.1). Generally, for lower masses and higher densities, the triple alpha reaction (which depends on the density cubed) is favored during core helium burning and alpha captures onto carbon (which linearly depends on density) play less of a role (Brown et al. 2001), which means less carbon is destroyed at lower masses, leading to a higher X C. However, this trends is no longer reproduced for the α sc = 1 models, where X C decreases for CO core masses larger than 5 M ⊙ compared to the α sc = 0 . 1 models. This can be understood as the e ff ect of a late ingestion of helium due to semi-convection during core helium burning in these models, which boost alpha captures onto carbon, which reduce X C. \nFor the α sc = 10 models, we observe a significant jump of ∼ 0 . 03 in X C for models with CO core masses larger than 4.5 M ⊙ in Fig. B.2. This is because the lowest-mass models ( Mi < 19 M ⊙ , M CO < 4 . 5 M ⊙ ) have a di ff erent structure compared to higher- models, which all burn helium as red supergiants. Due to the e ffi cient semiconvection, these low-mass models develop a large ICZ after core hydrogen burning and experience a blue loop, igniting helium as blue supergiants (BSGs). The characteristic strong hydrogen-burning shell of the BSG grows a more massive helium core compared to their initial mass than is the case in RSG structures (Fig. B.1). For example, the 18.5 M ⊙ model, which develops a BSG structure, has the same He core mass of 6.2 M ⊙ as the 19 M ⊙ model which has a RSG structure. This larger helium core mass and the ensuing di ff erent burning conditions during core helium burning favor carbon destruction through alpha captures and result in a systematically lower X C in these models. \nChanges in semi-convection lead to systematic variations in the final stellar structure, as shown in Fig. B.3. The shift of the compactness and entropy peaks to higher initial masses (2 M ⊙ for a shift of α sc from 0 to 10) reflects the change in the initial mass to helium core mass relation (see Fig. B.1). However, the CO core mass at which the compactness peak occurs is remarkably similar and varies by only 0.2 M ⊙ between models with di ff erent semi-convective mixing. This is because the onset \nof neutrino-dominated burning happens at a similar core mass ( ∼ 6 . 5 M ⊙ ) and similar X C ∼ 0 . 23. Small variations in the compactness landscape, i.e. the existence of small secondary peaks next to the main compactness peak are observed for di ff erent values of semi-convection in Fig. B.3. For example, the small peak in central entropy and compactness at an initial (CO core mass) of 20 (5.6) M ⊙ has been identified out in multiple other studies (Sukhbold & Woosley 2014; Sukhbold et al. 2018; Chie ffi & Limongi 2020; Chie ffi et al. 2021). This appears to be a recurrent feature in this narrow CO core mass and XC range. Models with initial (CO core) masses between 24 and 28 M ⊙ (8 and 10 M ⊙ ) also show variations in compactness and final entropy for a changing semi-convection e ffi ciency. This is the region at the transition when neon burning becomes neutrino-dominated and develops through several convective shells. We observed that for a varying semi-convection e ffi ciency, the development of this neon-burning front is a ff ected, which leads to small variations in compactness. However, the exact mechanisms behind these secondary peaks remains to be understood. \nUncertainties in semi-convective mixing can thus lead to variations in the final stellar structure by influencing the initial conditions for core helium and carbon burning. This creates in part the'noise' observed by Sukhbold et al. 2018 in their study of the compactness of massive stars. However, it is noteworthy that semi-convection does not a ff ect the existence of the main features observed in the compactness landscape and does not significantly change the CO core mass range at which the main compactness peak occurs.", 'Appendix C: Resolution test for the occurrence of shell mergers': 'To test how the occurrence of shell mergers is a ff ected by numerical uncertainties, we perform a resolution test for the model presented in Sect. 4.5, i.e. our default 22 M ⊙ model with a modified central carbon abundance X C = 0 . 17, starting from core oxygen exhaustion. We double the spatial and temporal resolution, and in pariticular, double the number of grid points in zones containing composition gradients and nuclear burning regions of C, O, and Ne (MESA settings mesh\\_logX\\_species and mesh\\_dlog\\_burn\\_c\\_dlogP\\_extra and equivalent). In Fig. C.1 we show the comparison of the Kippenhahn diagrams of the default model and the model with double the resolution. We find no di ff erence in the occurrence of the shell merger between the C and Ne shell (highlighted by the red box). The timing and extent is nearly identical, and can be traced back to the same mechanism, i.e. the increase in entropy in the Ne-burning shell. We therefore verify that these shell mergers between the C and Ne shell do not originate from poor resolution. The second shell merger between the C / Ne and the O-burning shells (highlighted by the blue box in Fig. C.1) occurs 5 min earlier for the model with higher resolution. Since this shell merger occurs towards the end of Si shell-burning, after most of the iron-rich core has already formed, it barely a ff ects the final core structure. Numerical uncertainties mainly influence the timing of this shell merger event. \nIn Fig. C.2 we compare the final density profiles at the onset of core collapse. The density profiles are almost identical, except for small variations of order ρ ∼ 10g cm -3 in the He-rich layer. The final iron core mass is a little larger for the higher-resolution model ( M Fe core = 1 . 80 compared to M Fe core = 1 . 89). Similarly, the final compactness values is a little larger ( ξ 2 . 5 = 0 . 357 for the default model and ξ 2 . 5 = 0 . 364 for the models with double the resolution). We conclude that the shell mergers identified in \nthis work are barely a ff ected by numerics. The C and Ne shell mergers are not numerical artifacts, but instead originate from the high entropy contrast between the Ne-burning shell and the C shell in this model (see Sec. 4.5).', 'Appendix D: Interior structure evolution diagrams': "We provide an overview of the evolution of the interior 6 M ⊙ structure of all our models from core carbon ignition to core collapse, shown in Fig. D.1, D.2, D.3, D.4, D.5, and D.6. \nFig. A.2. Final central entropy (top) and compactness (bottom) as a function of the core mass (left) and initial mass (right) for a varying 12 C( α, γ ) 16 O rate. \n<!-- image --> \n/circledot \n/circledot \nFig. B.2. Central carbon mass fraction at core helium depletion as a function of the CO core mass for varying semi-convection e ffi ciency. \n<!-- image --> \n/circledot \nFig. B.1. He (full lines) and CO (dashed lines) core mass at core helium depletion for models with a varying semiconvection e ffi ciency. A higher e ffi ciency generally leads to a smaller core mass. \n<!-- image --> \n/circledot \n<!-- image --> \n/circledot \n/circledot \nFig. B.3. Final central entropy and compactness as a function of the initial mass for varying semi-convection e ffi ciency.Fig. C.1. (a) Same as Fig. 8 but from core oxygen depletion to core collapse. (b) The same model in which we double the spatial and temporal resolution. The red and blue boxes highlight the shell merger between the C and Ne-burning shells, and the C / Ne and O-burning shells, respectively. \n<!-- image --> \nFig. C.2. Final density profile of the models in Fig. C.1 with our default assumptions compared to a model with double the resolution at the onset of core collapse. From left to right, vertical lines indicate the Fe, Si, CO, and He core masses in black and orange for the default and higher resolution model, respectively. \n<!-- image --> \n/circledot \nFig. D.1. Final interior structure evolution of massive stars from core C ignition to the onset of core collapse. Colors indicate the specific gravothermal energy released or used by contraction (red) and expansion (blue). Brown contours show lines of constant (logarithmic) density. Convective regions are shown by hatches and the evolution of the central specific entropy is shown with a black line. From left to right, vertical white dashed lines mark the moments of core C, Ne, O, and Si depletion, respectively. The dashed horizontal line represents the value of the classical Chandrasekhar mass before the end of core oxygen burning. Here we show models from 17 M ⊙ to 20.5 M ⊙ , where central carbon burning is slowly becoming neutrino dominated and the core structures are expected to lead to a successful supernova. \n<!-- image --> \nE. Laplace et al.: It's written in the massive stars: The role of stellar physics in the formation of black holesFig. D.2. Same as Fig. D.1 with initial masses between 21.0 M ⊙ and 24.5 M ⊙ (region A and B), where central carbon burning is neutrino dominated. The final core structure favors black hole formation for the 22 M ⊙ - 23 M ⊙ models and successful supernova explosions in the other cases. \n<!-- image --> \nFig. D.3. Same as Fig. D.1 for models with initial masses between 25 M ⊙ and 29 M ⊙ , where central carbon burning is neutrino dominated and central neon burning becomes increasingly neutrino-dominated. All models are expected to lead to a successful supernova explosion. \n<!-- image --> \nE. Laplace et al.: It's written in the massive stars: The role of stellar physics in the formation of black holesFig. D.4. Same as Fig. D.1 for models with initial masses between 30.0 M ⊙ and 37.0 M ⊙ (region C), where central carbon burning and central neon burning are both neutrino-dominated. Above an initial mass of 34.0 M ⊙ , Because of the high final binding energy of the layers above the central core, these models are expected to lead to black hole formation. \n<!-- image --> \nFig. D.5. Same as Fig. D.1 for models with initial masses between 38.0 M ⊙ and 45.0 M ⊙ (region D and beyond), where central carbon burning and central neon burning are both neutrino-dominated, and core oxygen burning is increasingly neutrino dominated. Because of the high final binding energy of the layers above the central core, these models are expected to lead to black hole formation. \n<!-- image --> \nE. Laplace et al.: It's written in the massive stars: The role of stellar physics in the formation of black holesFig. D.6. Same as Fig. D.1 for models with initial masses between 46.0 M ⊙ and 50.0 M ⊙ . Because of the high final binding energy of the layers above the central core, these models are expected to lead to black hole formation. \n<!-- image -->"}

2024arXiv240911708D

Accretion disc outbursts are reoccurring events observed in various astrophysical systems including Xray binaries and cataclysmic variables. These outbursts are characterized by a sudden increase in luminosity due to various instabilities in the accretion disc. We need to investigate the timedependent accretion flow models to understand the mechanisms driving these outbursts. Timedependent models incorporate the discs time evolution and can capture the buildup of instabilities. This review aims to give a basic overview of accretion disc outburst and stability analysis. The paper highlights the necessity of considering the hierarchy of different timescales dynamical viscous and thermal when investigating the instabilities occurring in the accretion disc. The importance and observational implications of studying these accretion disc outbursts are also discussed.

2024-09-01T00:00:00Z

['arXiv:2409.11708', '2024arXiv240911708D', '10.48550/arXiv.2409.11708']

['Astrophysics - High Energy Astrophysical Phenomena']

Accretion Disc Outbursts and Stability Analysis

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.11708.pdf

{'Accretion Disc Outbursts and Stability Analysis': "Liza Devi *; Asish Jyoti Boruah; Biplob Sarkar \n* Corresponding author Department of Applied Sciences, Tezpur University, Napaam, Assam-784028, India E-mail of corresponding author: app23110@tezu.ac.in \nAbstract - Accretion disc outbursts are re-occurring events observed in various astrophysical systems, including X-ray binaries and cataclysmic variables. These outbursts are characterized by a sudden increase in luminosity due to various instabilities in the accretion disc. We need to investigate the time-dependent accretion flow models to understand the mechanisms driving these outbursts. Time-dependent models incorporate the disc's time evolution and can capture the build-up of instabilities. This review aims to give a basic overview of accretion disc outburst and stability analysis. The paper highlights the necessity of considering the hierarchy of different timescales, dynamical, viscous, and thermal, when investigating the instabilities occurring in the accretion disc. The importance and observational implications of studying these accretion disc outbursts are also discussed. \nKeywords-accretion disc outburst, instabilities, dwarf novae, cataclysmic variable.", 'I. ACCRETION DISC OUTBURSTS: AN OVERVIEW': 'Accretion discs are swirling discs of gases rotating differentially around a compact object, like a neutron star or a black hole. These discs are an important component of various binary systems. These systems are consist of a compact object and a companion star, including X-ray binaries and cataclysmic variables [1,2,5,6,9]. The accretion disc is fuelled by matter drawn from the companion star and accreted onto the compact object. Sometimes, accretion discs experience a remarkable characteristic by a quick increase in brightness, followed by a slower decline. This is known as an accretion disc outburst [1,2,8]. Due to the sudden increase in the accretion rate, outbursts occur and result in a dramatic increase in luminosity. This phenomenon is often observed in X-ray binaries and cataclysmic variables. After an outburst, the system typically enters to an extended period of inactivity, known as quiescence [1]. This state can last for a significant amount of time before occurring another outburst. The period and recurrence time of outbursts vary from system to system ranging from minutes to months and even years. \nThis review aims to provide a basic overview of accretion disc outbursts and stability analysis along with highlighting the importance of considering the hierarchy of timescales involved.', 'II. TIME-DEPENDENT ACCRETION FLOW MODELS': "Time-dependent accretion flow models consider the disc's time evolution, which helps us to study the generation of instabilities that lead to outbursts. The magnitude of the viscosity governs the disc flow's time dependence [8,11,40]. Thus, one of the few sources of \nquantitative data regarding disc viscosity is provided by observations of time-dependent disc models. These models helps us to pinpoint the origins of various instabilities, monitor their growth rates and determine their eventual impacts [2]. The time dependence study mainly deals with the phenomenon of dwarf nova outbursts and is a field of great interest on its own. \nIn traditional steady-state optically thick discs, the viscosity has little effect on the observable properties, which is fortunate for confirming their existence, but it also implies that it is unlikely that observations of steady discs will provide much information about the viscosity [2,5,43]. Moreover these models failed to explain the time evolution phenomenon and instabilities occurring in outbursting candidates in binary systems. This is where the need of time-dependent accretion flow models comes into picture. Instabilities in accretion discs are caused by a variety of physical mechanisms.", 'III. THE HIERARCHY OF TIMESCALES': "When studying instabilities in accretion discs, it's important to examine the hierarchy of timescales. The three main timescales are:", 'A. Dynamical Timescale': "The shortest timescale in an accretion disc. This timescale characterizes the disc material's orbital period around the central object. In other words, it is the amount of time, the disc particles require to move around the accreting object in a single Kepler orbit. In addition, it is the characteristic period for restoring hydrostatic equilibrium perpendicular to the disc plane [1,2,9,28].", 'B. Thermal Timescale': 'The timescale on which the disc can adjust its temperature in response to external perturbations. The thermal time scale indicates the duration needed for viscous forces to produce the thermal energy of a specific disc annulus. It is calculated by dividing the total thermal energy present by the local energy dissipation rate [1,2,9,28].', 'C. Viscous Timescale': "The viscous time or radial drift timescale refers to the period required for the fluid in a disc to move substantially in a radial direction. We can also say that it is the period needed for matter to slowly move inward due to viscosity. The viscous timescale is substantially longer than the thermal timescale and the dynamical timescale is shorter than the later [1,2,9,28]. \nThe accretion disc's stability depends on all three timescales. Radiative processes on the thermal timescale alter the disc's vertical structure, while viscous forces on the viscous timescale drive angular momentum transfer [7,10,40]. A comprehensive stability analysis must \nencompass all three timescales in a coherent framework to accurately understand the origin of possible instabilities resulting in outbursts [1,2,39].", 'IV. INSTABILITIES IN THE ACCRETION DISC': 'Various instabilities can arise in accretion discs due to the interaction between these timescales. Some common instabilities include:', 'A. Thermal instability': 'It is caused by temperature fluctuations within the disc. In a disc annulus, when rate of heating is out of step with the rate of cooling, then a non-equilibrium situation arises. In this case it is said that the disc is subjected to thermal instability. This instability grows in thermal timescale [1,2,7,30].', 'B. Viscous instability': 'Viscous instabilities occur due to the buildup of viscous stress in the disc (resulting from changes in viscosity or angular momentum transfer). We can explain viscous instability by applying a perturbation in the form of extra mass to a disc ring. If the extra mass we added as a disturbance, diffuses or drifts away from the disc ring and and the disc ring gradually returns to its original surface density, the disc ring is said to be viscously stable. Otherwise, it is considered viscously unstable [1,2,30].', 'C. Thermal-Viscous instability': "Examining how an accretion disc reacts to perturbations, which might be local or global, is necessary to determine the stability of the disc. The most well-known and researched instability is the thermal-viscous instability, which is responsible for some X-ray binary outbursts as well as dwarf nova outbursts [29]. When angular momentum is transported viscously and there is a non-linear feedback between the heating and cooling processes, thermal-viscous instability results. Under some circumstances, a local region in the disc may change from a cool, low-viscosity state to a hot, high-viscosity one. The mass accretion rate significantly increases because of the change in the viscous regime, which in turn impacts the disc's heating and cooling processes. The hot state's temperature increase causes a higher equilibrium mass accretion rate, which enhances the outburst [26]. \nThe basic principles of thermal-viscous instability can be explained by the S-shaped equilibrium curve [7] (Fig.1.). In an accretion disc, hydrogen is the most abundant element. The degree of ionization of hydrogen determines the stability of the disc [38,39,40]. The hydrogen fully ionized state is known as the high state or HII state. However, in low state or HI state, hydrogen is neutral. Both these branches are stable (thermally and viscously) [30,31]. Partially ionized hydrogen occurs in the intermediate state. In the accretion disc's outer regions, when the temperatures vary between log10T = 3.5-4 K, the partial ionization of hydrogen occurs [27,28]. During partial ionization of hydrogen, the disc may become unstable due to changes in opacity. In the context of viscous and thermal instabilities, this state is unstable [30]. Three solutions are found for a specific range of surface density Σ (ΣB < Σ <ΣA) for an \naccretion disc annulus [30]. Let us consider a disc ring where Ṁout and Ṁin are mass outflow and inflow rate from and into the ring respectively. In the HI state, where Ṁin > Ṁout, surface density profile as well as disc mass will increase, until a maximum density ΣA is reached and jumps to HII state [30]. In HII state, Ṁin < Ṁout satisfies. This as a result decreases the disc's mass there. This continues until a minimum surface density ΣB is reached and the disc annulus jumps to HI state. The outbursts phenomena result from the constant oscillations between these two states. \nFig.1. Schematic diagram of S-shaped equilibrium curve. \n<!-- image -->", 'D. Radiation Pressure Instability': "When the pressure of radiation surpasses the pressure of gas, the black hole accretion disc, which has a classical heating component proportional to the pressure and viscosity parameter α (a parameter that describes the efficiency of angular momentum transport in an accretion disc) [20,42], can experience both thermal and viscous instabilities. This happens within the accretion disc's innermost radii that encircle a compact object (in case of black holes and neutron stars) [20,21]. Shakura and Sunyaev (1973) identified radiation pressure instability in their traditional α-models early on [22,43], which they thoroughly examined in 1976. According to Taam et al. [24] and Deegan et al. [23], the microquasar GRS 1915+105's recurrent outbursts lasting hundreds of seconds could be explained by radiation pressure instability. \nThe magnetorotational instability (MRI), which is caused by the interplay between weak magnetic fields and the disc's differential rotation, is another significant instability [39]. Viscosity can be efficiently enhanced, and angular momentum can be transported within the disc via an MRI. On the other hand, stability analysis of discs under MRI influence is a more complex field that is still being investigated.", 'V. STABILITY ANALYSIS': 'To analyze the stability of an accretion disc, we require knowledge on how the perturbations or disruption to its \nstructure grows and evolves over time. Linear stability analysis is commonly used to examine how a system responds to small perturbations [1,42]. After the addition of perturbation, if the perturbation grows, it may indicate an instability that could lead to sudden and intense outburst of energy.', 'A. Linear Stability Analysis': "Linear stability analysis involves simplifying the basic governing equations of the flow adding small perturbations to the disc's equilibrium state to examine their stability [1,19,34]. By solving the simplified equations, we can determine the conditions which give rise to instabilities. Fujimoto and Arai [18], investigated the stability of an optically thick, slim accretion disc around a black hole. They obtained a dispersion relation of the fourth order within the context of the study of linear stability. Also, for the cataclysmic variable disc models and their boundary layer (BL), Collins et al. [19] reported on the findings of a linearized perturbation analysis using local, linear stability analysis.", 'B. Non-Linear stability analysis': 'Although linear stability analysis helps us to know how perturbations first develop, nonlinear analysis and numerical simulations are mostly required to fully understand occurrence of instabilities. Non-linear effects can lead to a range of outcomes, which includes the formation of turbulent patterns [16,17], interactions between various unstable modes [14] and saturation of large-scale instabilities [15]. Balbus et al. [16] used a mix of analytical and numerical approach to explore the hydrodynamical nonlinear stability of fluid flows that rotate differentially at different distances from the center (differential rotation) and flows that slide past each other (pure shear flows) in threedimensions.', 'C. Numerical Simulations': 'Numerical simulations work as the most effective way to investigate the broad range dynamics of accretion discs [10,12,13,35]. Simulations can help us to understand how the interplay between thermal, viscous, and magnetorotational instabilities lead to disc outbursts and fluctuations [36,37,38]. Bergaulinger et al. [13] used semiglobal simulations to explore the growth and saturation of the MRI in core collapse supernovae by studying its evolution. The accretion-ejection instability in magnetized accretion discs is simulated numerically by Caunt and Tagger [12].', 'VI. OBSERVATIONAL IMPLICATIONS OF OUTBURSTS': 'Accretion disc outbursts can be observed in a variety of astrophysical systems, including cataclysmic variables (dwarf novae) and X-ray binaries.', 'A. Cataclysmic variables': 'This kind of binary system consists of a white dwarf which draws matter from a companion star [1,4,29]. Primarily, the companion star is a main sequence star or red dwarf [27]. Of all known cataclysmic variables, nearly fifty percent belong to the dwarf novae class, which has a known orbital period [1,41]. The outbursts last a few days, then happen again in a few weeks or months, and have an \namplitude ranging from two to five magnitudes [1,25,43]. The recurrence time and the form of the outburst light curves are not exactly periodic. The most studied example is the SS Cygni system [6]. The spectral characteristics of dwarf novae in outbursts are very similar to those of novalike variables that are continuously bright, or cataclysmic variables without dwarf nova outbursts. It is assumed that dwarf novae also feature a quasi-steady state disc during outbursts since nova-likes are believed to have bright, steady-state accretion discs [1].', 'B. X-ray Binaries': 'A compact object (a neutron star or black hole) that is accumulating mass from a companion star is what makes up an X-ray binary [1,2,4,5]. These systems are highly variable, exhibiting outbursts capable of several orders of magnitude increase in X-ray luminosity. Soft X-ray transients are a kind of low mass X-ray binaries that exhibit outbursts that are similar to dwarf novae outbursts, but with a considerably longer timescale and amplitude [1,3,32]. Compared to dwarf novae, the form of the light curve is significantly more variable. They have a typical shape which rises quickly over a few days and then slowly decays exponentially over several months. While there is concurrent brightening in other wavebands, particularly in the optical range, the outburst is most noticeable in the X-ray regime. There has only been one observation of several soft X-ray transient outbursts [1]. This suggests that the recurrence period is extremely long (decades, centuries, or even longer). Certain systems are seen repeating every year or every few years. \nStudying accretion disc outbursts may enhance our understanding of other astrophysical systems, including active galactic nuclei (AGN) [20,21] and gamma-ray bursts (GRB) [33]. These systems show rapid and extreme brightness increases, possibly due to similar instabilities in their accretion discs.', 'VII. CONCLUSION': 'This paper presents a summary of current state of knowledge about stability analysis and accretion disc outbursts. It is necessary to use time-dependent models which use hierarchy of dynamical, thermal, and viscous timescales to explain accretion disc outbursts. The various instabilities which lead to outbursts are also reviewed here. We have also discussed linear, non-linear and numerical stability analysis which are essential for determining the factors that contribute to outbursts. Investigation of accretion disc outbursts can help us to understand the importance of different complex binary systems along with other astrophysical systems which display similar behaviors. Further research in this area is necessary for fully understanding the process that drives outbursts in accretion discs.', 'ACKNOWLEDGMENT': 'The authors LD, AJB and BS acknowledge the reviewer for thoroughly reviewing the manuscript and providing beneficial comments. The authors also acknowledge the use of Meta AI and Google Gemini AI to improve the readability of the text.', 'REFERENCES': "- [1] Kolb, U., Extreme Environment Astrophysics . 2010.\n- [2] Frank, J., King, A., and Raine, D. J., Accretion Power in Astrophysics: Third Edition . 2002, p. 398.\n- [3] Mineshige, S., Kim, S.-W., and Wheeler, J. C., 'Time-dependent XRay Emission from Unstable Accretion Disks around Black Holes', The Astrophysical Journal , vol. 358, IOP, p. L5, 1990. doi:10.1086/185766.\n- [4] Seward, F. D. and Charles, P. A., Exploring the X-ray Universe . 2010.\n- [5] Courvoisier, T. J.-L., High Energy Astrophysics: An Introduction . 2013. doi:10.1007/978-3-642-30970-0.\n- [6] Shapiro, S. 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A., 'The outburst duration and duty cycle of GRS1915+105', Monthly Notices of the Royal\n- Astronomical Society , vol. 400, no. 3, OUP, pp. 1337-1346, 2009. doi:10.1111/j.1365-2966.2009.15573.x.\n- [24] Taam, R. E., Chen, X., and Swank, J. H., 'Rapid Bursts from GRS 1915+105 with RXTE', The Astrophysical Journal , vol. 485, no. 2, IOP, pp. L83-L86, 1997. doi:10.1086/310812.\n- [25] Cannizzo, J. K., Ghosh, P., and Wheeler, J. C., 'Convective accretion disks and the onset of dwarf nova outbursts.', The Astrophysical Journal , vol. 260, IOP, pp. L83-L86, 1982. doi:10.1086/183875.\n- [26] Faulkner, J., Lin, D. N. C., and Papaloizou, J., 'On the evolution of accretion disc flow in cataclysmic variables- I.The prospect of a limit cycle in dwarf nova systems.', Monthly Notices of the Royal Astronomical Society , vol. 205, OUP, pp. 359-375, 1983. doi:10.1093/mnras/205.2.359.\n- [27] Hōshi, R., 'Accretion Model for Outbursts of Dwarf Nova', Progress of Theoretical Physics , vol. 61, no. 5, pp. 1307-1319, 1979. doi:10.1143/PTP.61.1307.\n- [28] Livio, M., 'Accretion Discs: Limit Cycles and Instabilities', in Astrophysical Discs - an EC Summer School , 1999, vol. 160, p. 33. doi:10.48550/arXiv.astro-ph/9810035.\n- [29] Meyer, F. and Meyer-Hofmeister, E., 'On the elusive cause of cataclysmic variable outbursts.', Astronomy and Astrophysics , vol. 104, pp. L10-L12, 1981.\n- [30] Mineshige, S., 'Accretion Disk Instabilities', Astrophysics and Space Science , vol. 210, no. 1-2, Springer, pp. 83-103, 1993. doi:10.1007/BF00657876.\n- [31] Mineshige, S. and Osaki, Y., 'Disk-instability model for outbursts of dwarf novae Time-dependent formulation and one-zone model', Publications of the Astronomical Society of Japan , vol. 35, no. 3, OUP, pp. 377-396, 1983.\n- [32] Mineshige, S. and Wheeler, J. C., 'Disk-Instability Model for Soft XRay Transients Containing Black Holes', The Astrophysical Journal vol. 343, IOP, p. 241, 1989. doi:10.1086/167701.\n- [33] Janiuk, A., Yuan, Y., Perna, R., and Di Matteo, T., 'Instabilities in the Time-Dependent Neutrino Disk in Gamma-Ray Bursts', The Astrophysical Journal , vol. 664, no. 2, IOP, pp. 1011-1025, 2007. doi:10.1086/518761.\n- [34] Latter, H. N., Fromang, S., and Faure, J., 'Local and global aspects of the linear MRI in accretion discs', Monthly Notices of the Royal Astronomical Society , vol. 453, no. 3, OUP, pp. 3257-3268, 2015. doi:10.1093/mnras/stv1890.\n- [35] Hawley, J. F., 'Numerical Simulations of MHD Accretion Disks', Highlights of Astronomy , vol. 15, pp. 237-238, 2010. doi:10.1017/S1743921310009014.\n- [36] Kadam, K., Vorobyov, E., Regály, Z., Kóspál, Á., and Ábrahám, P., 'Outbursts in Global Protoplanetary Disk Simulations', The Astrophysical Journal , vol. 895, no. 1, IOP, 2020. doi:10.3847/15384357/ab8bd8.\n- [37] Ross, J., Latter, H. N., and Tehranchi, M., 'MRI turbulence and thermal instability in accretion discs', Monthly Notices of the Royal Astronomical Society , vol. 468, no. 2, OUP, pp. 2401-2415, 2017. doi:10.1093/mnras/stx564.\n- [38] Habibi, A. and Abbassi, S., 'Thermal Instability of Thin Accretion Disks in the Presence of Wind and a Toroidal Magnetic Field', The Astrophysical Journal , vol. 887, no. 2, IOP, 2019. doi:10.3847/15384357/ab5793.\n- [39] Balbus, S. A. and Hawley, J. F., 'A Powerful Local Shear Instability in Weakly Magnetized Disks. I. Linear Analysis', The Astrophysical Journal , vol. 376, IOP, p. 214, 1991. doi:10.1086/170270.\n- [40] Pringle, J. E., 'Accretion discs in astrophysics', Annual Review of Astronomy and Astrophysics , vol. 19, pp. 137-162, 1981. doi:10.1146/annurev.aa.19.090181.001033.\n- [41] Osaki, Y., 'An Accretion Model for the Outbursts of U Geminorum Stars', Publications of the Astronomical Society of Japan , vol. 26, OUP, p. 429, 1974.\n- [42] Piran, T., 'The role of viscosity and cooling mechanisms in the stability of accretion disks.', The Astrophysical Journal vol. 221, IOP, pp. 652-660, 1978. doi:10.1086/156069.\n- [43] Shakura, N. I. and Sunyaev, R. A., 'Black holes in binary systems. Observational appearance.', Astronomy and Astrophysics , vol. 24, pp. 337-355, 1973."}

2024MNRAS.533.3222D

We present a new determination of the evolving galaxy ultraviolet UV luminosity function LF over the redshift range inlineformulatexmath idTM0002 notationLaTeX8.5lt zlt 15.5texmathinlineformula using a combination of several major Cycle1 JWST imaging programmes Public Release IMaging for Extragalactic Research JWST Advanced Deep Extragalactic Survey and Next Generation Deep Extragalactic Exploratory Public Survey. This multifield approach yields a total of inlineformulatexmath idTM0003 notationLaTeXsimeq 370texmathinlineformula arcminSUP2SUP of JWSTNIRCam imaging reaching 5inlineformulatexmath idTM0004 notationLaTeXsigmatexmathinlineformula depths of inlineformulatexmath idTM0005 notationLaTeXsimeq 30texmathinlineformula AB mag in the deepest regions. We select a sample of 2548 galaxies with a significant probability of lying at high redshift inlineformulatexmath idTM0006 notationLaTeXpzgt 8.5gt 0.05texmathinlineformula to undertake a statistical calculation of the UV LF. Our new measurements span inlineformulatexmath idTM0007 notationLaTeXsimeq 4texmathinlineformula mag in UV luminosity at inlineformulatexmath idTM0008 notationLaTeXz912.5texmathinlineformula placing new constraints on both the shape and evolution of the LF at early times. Our measurements yield a new estimate of the early evolution of cosmic starformation rate density inlineformulatexmath idTM0009 notationLaTeXrho rm SFRtexmathinlineformula confirming the gradual decline deduced from early JWST studies at least out to inlineformulatexmath idTM0010 notationLaTeXz simeq 12texmathinlineformula. Finally we show that the observed early evolution of the galaxy UV LF and inlineformulatexmath idTM0011 notationLaTeXrho rm SFRtexmathinlineformula can be reproduced in a inlineformulatexmath idTM0012 notationLaTeXrm Lambda texmathinlineformulacold dark matter Universe with no change in dust properties or starformation efficiency required out to inlineformulatexmath idTM0013 notationLaTeXz simeq 12texmathinlineformula. Instead a progressive trend towards younger stellar population ages can reproduce the observations and the typical ages required at inlineformulatexmath idTM0014 notationLaTeXz simeqtexmathinlineformula 8 9 10 and 11 all converge on inlineformulatexmath idTM0015 notationLaTeXsimeq 380330texmathinlineformula Myr after the big bang indicative of a rapid emergence of early galaxies at inlineformulatexmath idTM0016 notationLaTeXz simeq 12 13texmathinlineformula. This is consistent with the first indications of a steeper dropoff in inlineformulatexmath idTM0017 notationLaTeXrho rm SFRtexmathinlineformula we find beyond inlineformulatexmath idTM0018 notationLaTeXz simeq 13texmathinlineformula possibly reflecting the rapid evolution of the halo mass function at earlier times.

2024-09-01T00:00:00Z

['2024arXiv240303171D', '2024MNRAS.tmp.1993D', '10.48550/arXiv.2403.03171', 'arXiv:2403.03171', '2024MNRAS.533.3222D', '10.1093/mnras/stae2037']

['Astrophysics - Astrophysics of Galaxies']

JWST PRIMER a new multifield determination of the evolving galaxy UV luminosity function at redshifts z 9 15

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']

https://arxiv.org/pdf/2403.03171.pdf

{'JWST PRIMER: A new multi-field determination of the evolving galaxy UV luminosity function at redshifts z ≃ 9 -15': '- C. T. Donnan 1 ★ , R. J. McLure 1 , J. S. Dunlop 1 , D. J. McLeod 1 , D. Magee 2 , K. Z. Arellano-Córdova 1 , \nL. Barrufet 1 , R. Begley 1 , R. A. A, Bowler 3 , A. C. Carnall 1 , F. Cullen 1 , R. S. Ellis 4 , A. Fontana 5 , G. D. Illingworth 2 , N. A. Grogin 6 , M. L. Hamadouche 1 , A. M. Koekemoer 6 , F.-Y. Liu 1 , C. Mason 7 , 8', 'P. Santini 5 , T. M. Stanton 1': '- 1 Institute for Astronomy, University of Edinburgh, Royal Observatory, Edinburgh, EH9 3HJ, UK\n- 2 Department of Astronomy and Astrophysics, UCO/Lick Observatory, University of California, Santa Cruz, CA 95064, USA\n- 3 Jodrell Bank Centre for Astrophysics, Department of Physics and Astronomy, School of Natural Sciences, The University of Manchester, Manchester, M13 9PL, UK\n- 4 Department of Physics & Astronomy, University College London. Gower St., London WC1E 6BT, UK\n- 5 INAF - Osservatorio Astronomico di Roma, via di Frascati 33, 00078 Monte Porzio Catone, Italy\n- 6 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA\n- 7 Cosmic Dawn Center (DAWN), Jagtvej 128, DK-2200, Copenhagen N, Denmark\n- 8 Niels Bohr Institute, University of Copenhagen, Jagtvej 128, DK-2200, Copenhagen N, Denmark \nAccepted XXX. Received YYY; in original form ZZZ', 'ABSTRACT': 'We present a new determination of the evolving galaxy UV luminosity function (LF) over the redshift range 8 . 5 < 𝑧 < 15 . 5 using a combination of several major Cycle-1 JWST imaging programmes - PRIMER, JADES and NGDEEP. This multi-field approach yields a total of ≃ 370 sq. arcmin of JWST /NIRCam imaging, reaching (5𝜎 ) depths of ≃ 30 AB mag in the deepest regions. We select a sample of 2548 galaxies with a significant probability of lying at high redshift ( 𝑝 ( 𝑧 > 8 . 5 ) > 0 . 05) to undertake a statistical calculation of the UV LF. Our new measurements span ≃ 4 magnitudes in UV luminosity at 𝑧 = 9 -12 . 5, placing new constraints on both the shape and evolution of the LF at early times. Our measurements yield a new estimate of the early evolution of cosmic star-formation rate density ( 𝜌 SFR ) confirming the gradual decline deduced from early JWST studies, at least out to 𝑧 ≃ 12. Finally we show that the observed early evolution of the galaxy UV LF (and 𝜌 SFR ) can be reproduced in a Λ CDM Universe, with no change in dust properties or star-formation efficiency required out to 𝑧 ≃ 12. Instead, a progressive trend towards younger stellar population ages can reproduce the observations, and the typical ages required at 𝑧 ≃ 8, 9, 10, and 11 all converge on ≃ 380 -330 Myr after the Big Bang, indicative of a rapid emergence of early galaxies at 𝑧 ≃ 12 -13. This is consistent with the first indications of a steeper drop-off in 𝜌 SFR we find beyond 𝑧 ≃ 13, possibly reflecting the rapid evolution of the halo mass function at earlier times. \nKey words: galaxies:high-redshift - galaxies:evolution - galaxies:formation', '1 INTRODUCTION': 'The discovery and study of the earliest galaxies is critical for refining our understanding of cosmology and structure formation, and in particular for clarifying the first stages of galaxy formation/evolution and the progress/drivers of cosmic hydrogen reionization (Dunlop 2013; Stark 2016). Over the last decade, deep near-infrared surveys with the Hubble Space Telescope ( HST ) and Spitzer have enabled the evolution of galaxies to be mapped out to redshifts 𝑧 ≃ 9 (e.g. Ellis et al. 2013; McLure et al. 2013; Finkelstein et al. 2015; McLeod et al. 2015, 2016; Oesch et al. 2018; Bouwens et al. 2021, 2022). Moreover, there is now growing evidence that the UV radiation emitted from these early star-forming galaxies powered cosmic hydrogen \nreionization, with this phase transition concluding by 𝑧 ≃ 5 . 5 -6 . 0 (Robertson et al. 2015; Aird et al. 2015; Bosman et al. 2021). \nThe reliable detection of galaxies at 𝑧 ≥ 7 was first made possible by the installation of the near-infrared camera WFC3/IR on HST . The detection of galaxies at 𝑧 ≥ 6 relies on the robust identification of the Lyman-break due to the attenuation of UV photons short-ward of 𝜆 rest = 1216 Å produced by the increasingly neutral intergalactic medium (IGM) at these redshifts. However, as a result of the longwavelength limit of HST at 1 . 7 𝜇 m, galaxies could only be robustly selected up to 𝑧 ∼ 10 with only a small number of more tentative detections reported at higher redshifts (e.g. Ellis et al. 2013; Oesch et al. 2016) which relied on single filter detections. In parallel with these space-based efforts, wider area (degree-scale) ground-based near-infrared imaging surveys (e.g. UltraVISTA; McCracken et al. 2012) have revealed an excess of bright galaxies ( 𝑀 UV ≤ -20) \nat high redshifts compared to what is expected from an evolving Schechter function (Bowler et al. 2020; Varadaraj et al. 2023; Donnan et al. 2023a). When the ground-based and space-based data are combined it is now clear that a double-power law is a more appropriate functional form for describing the UV luminosity function (LF) at 𝑧 > 6. \nNow, since summer 2022, the study of early galaxies has been revolutionised by the advent of JWST . The NIRCam instrument on JWST provides deep multi-band near-infrared imaging out to 𝜆 ≃ 5 𝜇 m, and this has enabled the robust detection of galaxies at extreme redshifts ( 𝑧 ≥ 10) for the first time. Indeed, in just the first year of JWST operations, several early NIRCam imaging surveys have already been completed, revealing a significant number of galaxy candidates at 𝑧 > 10 (e.g. Donnan et al. 2023a,b; McLeod et al. 2024; Adams et al. 2023; Naidu et al. 2022; Castellano et al. 2022; Harikane et al. 2023; Finkelstein et al. 2022b; Austin et al. 2023; Hainline et al. 2024). JWST has also been successfully used to spectroscopically confirm the redshift of many of these candidates using the NIRSpec instrument (up to 𝑧 spec = 13 . 2) (e.g. Curtis-Lake et al. 2023; Arrabal Haro et al. 2023a; Harikane et al. 2024; Wang et al. 2023; Castellano et al. 2024). These early spectroscopic results have shown very good agreement with the inferred photometric redshifts for the vast majority of robust targets (Arrabal Haro et al. 2023b; Bunker et al. 2023a). \nOut of the galaxy candidates revealed by JWST at 𝑧 > 10 there have been a number that are particularly (arguably surprisingly) bright (e.g. Castellano et al. 2022, 2023; McLeod et al. 2024). This has led to some suggestions that the early JWST results in this field present a challenge theoretical models of galaxy evolution, with lower dust attenuation and/or higher star-formation efficiencies, or even primeval (PopIII) stellar populations being proposed as necessary to explain the observations (e.g. Tacchella et al. 2018; Yung et al. 2019; Mason et al. 2023; Harikane et al. 2023). \nPrior to the launch of JWST , and due primarily to the aforementioned wavelength limitations of HST , there was uncertainty over the evolution of the cosmic star-formation rate density ( 𝜌 SFR ) at 𝑧 ≥ 9. This issue now been largely resolved by the first year of JWST observations, with early measurements of the UV LF with JWST showing only gradual evolution over the redshift range 𝑧 = 8 -13 (Donnan et al. 2023a,b; Harikane et al. 2023; Finkelstein et al. 2023; Bouwens et al. 2023b,a; Adams et al. 2024; McLeod et al. 2024; Leung et al. 2023). This is consistent with the smoother more gradual decline in 𝜌 SFR , consistent with the results/predictions of McLeod et al. (2016). \nHowever, the early high-redshift studies undertaken with JWST suffer from several limitations. One limitation is the modest total area coverage and limited availability of deep imaging in the Early Release Science (ERS) and the smaller-scale early Cycle-1 JWST NIRCam programmes. As a result there has remained significant uncertainty over the exact shape of the UV LF at 𝑧 > 10. In particular, the constraints on the faint-end slope of the LF are relatively poor, an uncertainty which in turn limits the accuracy with which the cosmic star-formation rate density can be inferred. Moreover, a free fit of the functional form to the UV LF at 𝑧 > 10 has remained challenging due to the limited dynamic range in the UV luminosity ( 𝑀 UV ) of the galaxies uncovered at these redshifts in the early JWST samples. The limited area of these early studies, as well as inevitably yielding rather small samples, also makes them vulnerable to cosmic variance. In particular, the Abell2744 field imaged by the UNCOVER (Bezanson et al. 2022) and GLASS (Treu et al. 2022) programmes is now known to be highly over-dense at 𝑧 ∼ 10 (Castellano et al. 2023). \nAnother limitation of the early JWST studies of high-redshift galaxy evolution has been the methodology used to determine the \nUV LF. Typically, the 1 / 𝑉 max method (Schmidt 1968) has been used, which involves selecting galaxies at high redshift using the Lyman break technique and simply adopting the best-fitting photometric redshift (photo-z). However, the redshift of a galaxy is often not very well defined by the multi-band photometry and therefore the photo-z can be unreliable, particularly for galaxies close to the detection limit. This is one possible explanation for the discrepant galaxy samples produced in a number of the early JWST studies (Bouwens et al. 2023a). \nIn this paper we aim to substantially advance our knowledge of the evolving galaxy UV LF and hence cosmic star-formation rate density at 𝑧 = 9 -15. First, we are now able to utilise a number of the more major extragalactic NIRCam imaging surveys (both public and GTO) which have been largely completed in JWST Cycle-1. Second, to do this expanded dataset justice, we have implemented a more statistically robust method of calculating the LF. \nOur primary imaging dataset is the Public Release IMaging for Extragalactic Research (PRIMER; Dunlop et al., in preparation) survey which provides imaging over ≃ 380 arcmin 2 in 8 NIRCam filters, an order-of-magnitude larger area than covered in the ERS NIRCam programmes. We also include the ultra-deep imaging from the JWST AdvancedDeepExtragalactic Survey (JADES; Eisenstein et al. 2023) and the Next Generation Deep Extragalactic Exploratory Public survey (NGDEEP; Bagley et al. 2024). By taking this multi-field approach, combining surveys with different (and largely complementary) depths and areas (covering a total of ∼ 400 sq. arcmin while reaching depths of 𝑚 AB ≃ 30) we now have substantially improved dynamic range in both UV luminosity and redshift compared to previous JWST -based studies. As a result of this, and our improved methodology, we are now able to significantly improve the accuracy with which the UV LF can be constrained at extreme redshifts, 𝑧 = 8 -15. \nThe paper is structured as follows. In Section 2 we describe the imaging data and the creation of our source catalogues. In Section 3 we explain the sample selection and the spectral energy distribution (SED) fitting to the galaxy photometry. In Section 4 we present our derived galaxy UV luminosity function as a function of redshift, and the resulting inferred early evolution of the cosmic star-formation rate density. In Section 5 we then discuss our results in the context of other recent observational studies and the predictions of various theoretical/numerical models of galaxy formation and evolution. Finally, in Section 6 we summarise our conclusions. Throughout we use magnitudes in the AB system (Oke 1974; Oke & Gunn 1983), and assume a standard cosmological model with 𝐻 0 = 70 km s -1 Mpc -1 , Ω 𝑚 = 0 . 3 and Ω Λ = 0 . 7.', '2.1 Survey Fields': 'We utilise a number of major JWST Cycle-1 imaging surveys covering 4 separate fields. Due to the uncertainties and limited cosmological volumes associated with gravitational lensing, we choose to focus on blank/unbiased fields to ensure a robust determination of the UV luminosity function. The largest programme used is the Public Release IMaging for Extragalactic Research (PRIMER, PI: J. Dunlop) survey which images the COSMOS and UDS fields using NIRCam through the F090W, F115W, F150W, F200W, F277W, F356W, F410M and F444W filters. We also use the first epoch of imaging from the Next Generation Deep Extragalactic Exploratory Public (NGDEEP, PI: S. Finkelstein) survey which provides ultradeep imaging in the GOODS-South field, specifically targeting the \nHubble Ultra Deep Field parallel 2 field (HUDF par. 2). This programme uses the NIRCam F115W, F150W, F200W, F277W, F356W and F444W filters. Finally we use the JWST Advanced Deep Extragalactic Survey (JADES; Eisenstein et al. 2023; Rieke et al. 2023) NIRCam data release 2. This programme targets a region within the GOODS-South field centred on the Hubble Ultra Deep Field (HUDF), and uses the same filter set as the PRIMER survey. \nThesesurvey fields also have the advantage of deep optical imaging from the HST /ACS instrument. The PRIMER UDS and COSMOS fields have been imaged in in the HST F435W,F606WandF814Wfilters from the Cosmic Assembly Near-IR Deep Extragalactic Legacy Survey (CANDELS; Grogin et al. 2011; Koekemoer et al. 2011). The JADESandNGDEEPfieldsalsohaveimaginginthesefilterswiththe addition of F775W and F850LP. This imaging was also taken as part of CANDELS as well as through the Great Observatories Origins Deep Survey (GOODS; Giavalisco et al. 2004). This includes the deepest HST /ACS imaging ever taken in the HUDF. The existence of deep optical data is critical for obtaining robust measurements of the redshift probability distributions ( 𝑝 ( 𝑧 ) ) of the sources detected in each field. \nThe PRIMER and NGDEEP imaging data was reduced using the PRIMER Enhanced NIRCam Image Processing Library (PENCIL; Mageeetal., in preparation) software. For the JADES imaging we use the reductions described in Rieke et al. (2023). The astrometry of all the reduced images was aligned to GAIA DR3 (Gaia Collaboration et al. 2023) and stacked to the same pixel scale of 0.03 arcsec.', '2.2 PSF homogenisation': 'To create a catalogue with accurate photometry, the differences in the point spread function (PSF) need to be accounted for. This is corrected by homogenising the PSFs produced by the imaging through the different filters to one common PSF, using a similar technique to that which was utilised for the ground-based imaging in Donnan et al. (2023a). For all of the survey fields used in this study, the F444W imaging is available and has the broadest PSF. The F444W PSF was therefore chosen as the natural target for the homogenisation of the imaging resolution at all other wavelengths. We measured the PSF by selecting ≃ 20 bright but unsaturated stars across the PRIMER COSMOSimaging. Each star was then centroided and stacked to generate a measurement of the PSF in each filter. Our measured PSFs are comparable to those from WebbPSF (Perrin et al. 2014) to within ≃ 1% -3%. Using a combination of Moffat and Gaussian profiles a series of kernels were produced which, when convolved with the imaging, result in the homogenisation of the imaging through each NIRCam filter to the PSF at F444W, to within an accuracy of 3 per cent.', '2.3 Photometric Depths': 'The global depths for each field were calculated by determining the distributions of fluxes measured in 0.3-arcsec diameter apertures placed within source-free regions of the PSF-homogenised imaging. The 1 𝜎 depth is then given by 1 . 4826 × MAD where MAD is the median absolute deviation of the flux in the source-free apertures. This gives a global depth, because it is calculated over the full (sourcefree) area of the imaging. In Fig. 1 we show the spatial distribution of the 5 𝜎 depth across each field. This illustrates the different depths achieved by the different NIRCam surveys, as well as revealing the extent of any depth variations across the images. In particular, the \'wedding-cake\' structure of the PRIMER and JADES surveys can be \nidentified by the distinct regions of differing depth. The depth maps indicate that the NGDEEP and UDS fields are relatively flat whereas the COSMOS and JADES fields have (by design) sub-regions that differ significantly in depth. In COSMOS we found it useful to define 2 individual sub-regions corresponding to the deeper core and the wider/shallower outer region which we name \'COSMOS Deep\' and \'COSMOS Wide\' respectively. In JADES we define 3 distinct subregions. \'JADES Deep\' comprises the deeper stripes in the north of the imaging as well as the deep pointing in the South often labelled as the JADES Origins Field (JOF; Robertson et al. 2023a). \'JADES Medium\' and \'JADES Shallow\' define the remaining area. In Table 1 we list the 5 𝜎 global depths for each field where COSMOS and JADESare divided into their respective sub-regions. The depths have been corrected to total assuming a point source correction. The areas of each region are also stated. \nTo ensure consistent and robust high redshift galaxy selection, we restricted the analysis (i.e. the source selection) area in each field to those regions in which deep HST /ACS F814W and JWST /NIRCam F090W imaging is available; this guarantees that there are always a sufficient number of "short"-wavelength filters to confirm the anticipated non-detections for 𝑧 ≥ 8 . 5 galaxy candidates. In practice this requirement only has a significant effect on the NGDEEP and UDS fields, with the useful area of the JWST imaging in both these fields being reduced by approximately 1/3. This reduction in usable area has been accounted for in the areas noted in Table 1. \nDue to the varying depths across much of the imaging, the global depth is a rather poor representation of the uncertainty in the flux of individual objects. To improve on this we have adopted the 1 𝜎 local depth as the uncertainty of the measured fluxes in each filter for each object. To establish the local depths we use the same technique as utilised by Donnan et al. (2023a,b) in which the depth is calculated using the 200 empty apertures that lie closest to each detected source.', '3 SAMPLE SELECTION': "For each field we constructed three catalogues using SE/x.pc/t.pc/r.pc/a.pc/c.pc/t.pc/o.pc/r.pc (Bertin & Arnouts 1996) in dual-image mode. We performed a restframe UV selection by using the F150W, F200W and F277W images as the detection images, as these filters sample the flux long-ward of the Lyman break at 9 ≲ 𝑧 ≲ 20. Master catalogues were constructed from the three single-filter catalogues, removing duplicates by retaining the duplicate with the highest signal-to-noise in its respective detection image. Photometry was measured within 0.3 '' -diameter circular apertures on the PSF homogenised images, corresponding to ∼ 70 per cent of the total flux for a point source. Based on the individual curves of growth, further corrections of the order ∼ 1 -2 per cent were made to correct the imaging through every filter to exactly 70 per cent of the total flux in order to improve the photometric homogenisation.", '3.1 SED fitting': 'We used /e.pc/a.pc/z.pc/y.pc (Brammer et al. 2008) to perform our intial SED fitting, exploring the redshift range 0 < 𝑧 < 20 with the P/e.pc/g.pc/a.pc/s.pc/e.pc (Fioc & Rocca-Volmerange 1999) template set that includes nebular emission lines. \nWhen computing the UV LF, the standard approach is to adopt the maximum likelihood (i.e. minimum 𝜒 2 ) photometric redshift solution for each source. However, this approach makes the implicit assumption that the integrated 𝑝 ( 𝑧 ) within the adopted redshift bin is unity. This assumption can hold true when the signal-to-noise of the \n<!-- image --> \nPRIMER COSMOS ( ∼ 136 sq. arcmin) 31 \n<!-- image --> \n<!-- image --> \nFigure 1. The 5 𝜎 depth maps in the F277W filter imaging of each NIRCam survey field used in this analysis, demonstrating the variation in depths between the different survey fields, and in some cases within a given field. All images are shown with bins of 200 pixels where the original images are on a 0.03-arcsec pixel scale. The colour-bar shows the 5 𝜎 depth in AB mag on the same scale for each field. The grayed out region shows where there is a lack of deep HST /ACS F814W imaging, which only affects the NGDEEP and UDS fields. \n<!-- image --> \nphotometry is sufficient to exclude any alternative redshift solutions as being statistically unacceptable fits to the data. However, this is not the case for a significant number of the sources initially selected, particularly if they are close to the detection limit of the imaging. Consequently, a more robust approach to calculating the UV LF is to consider the full 𝑝 ( 𝑧 ) of each source in the initial catalogue over the entire redshift range 0 < 𝑧 < 20.', '3.2 Posterior probability distributions': 'For each galaxy, we assume that the posterior probability distribution of the redshift, given the observed fluxes ( 𝐹 ), is given by \n𝑝 ( 𝑧 | 𝐹 ) = L( 𝐹 | 𝑧 ) 𝑝 ( 𝑧 | M UV ) , (1) \nwhere L( 𝐹 | 𝑧 ) is the likelihood of the observed fluxes given the redshift, taken to be \nL( 𝐹 | 𝑧 ) ∝ 𝑒 -𝜒 2 ( 𝑧 )/ 2 , (2) \nand 𝑝 ( 𝑧 | M UV ) is the prior probability of the redshift based on the implied absolute UV magnitude. This prior is based on an evolving model of the UV LF from 𝑧 = 0 -20. At redshifts 𝑧 < 7 we adopt the evolving Schechter function parameterisation from Bouwens et al. (2021). At 𝑧 ≥ 7 we adopt a new parameterisation, designed to reproduce the double power-law (DPL) fits to the 𝑧 ≃ 7 -11 UV LF from Bowler et al. (2017, 2020); Donnan et al. (2023a); McLeod et al. (2024). The evolution of the DPL parameters is described as \nTable 1. The derived 5 𝜎 global depths for all the images used in this analysis. All depths (given in AB mag) have been measured in 0.3-arcsec diameter apertures on the PSF-homogenised images and then corrected to total assuming a point-source correction. \nFigure 2. Acomparisonofourevolvingdoublepower-lawparameterisation of the 𝑧 = 11 UV LF (green solid line) with the observational data from McLeod et al. (2024) at the same redshift. The prediction of the evolving Schechter function parameterisation of the UV LF from Bouwens et al. (2021) is shown as the dashed blue line. \n<!-- image --> \nfollows: \n𝑀 ∗ = -20 . 95 + 0 . 11 𝑧 (3) 𝜙 ∗ = 10 (-0 . 14 𝑧 -2 . 36 ) (4) 𝛼 = -2 . 04 × 10 -4 𝑧 -2 . 1 (5) (6) \n𝛽 = 0 . 138 𝑧 -5 . 13 . \nFig. 2 demonstrates that our model of the evolving UV LF is a good fit to the observational data at 𝑧 ≃ 11 from McLeod et al. (2024). In contrast, it can be seen that an extrapolation of the Schechter model from Bouwens et al. (2021) under predicts the latest JWST observational data at these redshifts. \nThe impact of the UV LF prior is particularly important for very bright and/or extreme redshift sources, for which the posterior redshift probability distribution becomes weighted towards lower redshift if the photometric data are not deep enough to robustly exclude such solutions . In Fig. 3 we demonstrate the the effect of the UV LF \nFigure 3. The posterior redshift probability distribution for NGDEEP-3594 (red) and JADES-92420 (green) using a flat redshift prior (dashed-lines) and our UV LF prior (solid-lines). The first source is a 𝑧 ∼ 16 candidate reported by Austin et al. (2023) and Leung et al. (2023), while the second is a spectroscopically-confirmed galaxy at 𝑧 = 11 . 6 (Curtis-Lake et al. 2023). The UV LF prior strongly weights the posterior redshift probability distribution to lower redshift for the less robust source, whereas the redshift solution of the robust high-redshift galaxy is unaffected. The top panel shows cut-out images of the two sources (indicated by their respective border colours) in the HST /ACS F606W, F814W filters and the JWST /NIRCam F115W, F150W, F200W, F277W, F356W and F444W filters. \n<!-- image --> \nprior on the posterior redshift distribution for two different sources: a galaxy uncovered in the HUDF which now has a spectroscopicallyconfirmed redshift of 𝑧 = 11 . 6, and an object with a photometric redshift of 𝑧 ∼ 16 selected from the NGDEEP survey. We also show their images in a number of filters. \nThe first source is a galaxy that was initially identified using HST data in the HUDF (Bouwens et al. 2011; Ellis et al. 2013; McLure et al. 2013) with only a single-band detection implying an extreme photometric redshift of 𝑧 ≃ 12. This redshift solution has since been refined first with photometry from JWST /NIRCam (Bouwens et al. 2023b; Donnan et al. 2023b; Robertson et al. 2023b) and subsequently with spectroscopy from JWST /NIRSpec (Curtis-Lake et al. 2023) which yielded a robust redshift of 𝑧 = 11 . 6. Fig. 3 demonstrates that the 𝑝 ( 𝑧 ) we derive for this source (JADES-92420) fromfitting our JWST photometry is actually unaffected by the UV LF redshift prior due to the robust nature of its photometric redshift. This source is robustly at 𝑧 > 8 . 5 with an integrated 𝑝 ( 𝑧 ≥ 8 . 5 ) = 0 . 995. It most strongly contributes to the 𝑧 = 12 . 5 UV LF bin as described in Section 4 with 𝑝 ( 11 . 5 < 𝑧 < 13 . 5 ) = 0 . 97. \nThe second source was identified by Austin et al. (2023) to be at 𝑧 phot ≃ 15 . 6 (NGD-z15a) which, if correct, would make it one of the most distant galaxy candidates discovered to date. This source was also reported in Leung et al. (2023), with a very similar photometric redshift estimate of 𝑧 phot ≃ 15 . 8 (NGDEEP 1369). It can be seen from Fig. 3 that, with a flat redshift prior, we would also identify this source (here listed as NGDEEP-3594) as a robust 𝑧 ≥ 15 candidate based on our photometry. However, the application of our adopted UV LF prior changes the posterior probability distribution in a way which strongly favours the lower-redshift solution at 𝑧 ∼ 4, indicating that, given the available data, this candidate is unlikely to be at ultrahigh redshift, with a integrated 𝑝 ( 𝑧 ≥ 8 . 5 ) = 0 . 054. Although this candidate does still contribute to the tentative 𝑧 = 14 . 5 bin described in Section 4, it only contributes the equivalent of 0.025 galaxies. This situation may appear similar to that of the very luminous 𝑧 ≃ 16 candidate detected in CEERS (Donnan et al. 2023a; Finkelstein et al. 2023) where subsequent spectroscopy with JWST /NIRSpec conclusively revealed that the redshift was in fact 𝑧 = 4 . 9(Arrabal Haro et al. 2023a) where the photometry was dominated by strong rest-frame optical emission lines (Harikane et al. 2024). However, the 𝑝 ( 𝑧 ) for this highly-unusual candidate was in fact unaffected by the application of a UV LF prior, due to the relatively high signal-to-noise of the JWST photometry. Instead, the introduction of an extreme emissionline template SED was required to reveal the correct photometric redshift for this particular source.', '3.3 Selection of galaxy candidates': 'Based on the master photometric catalogues for each field, we produced initial samples of galaxy candidates at 8 . 5 < 𝑧 phot ≲ 20. To be included in our initial samples, objects were required be < 2 𝜎 detections in all ACS filters and the F090W NIRCam filter. Objects were also required to be a ≥ 5 𝜎 detection in one of the three detection filters: F150W, F200W and F277W as well as a ≥ 3 𝜎 detection in any one of the other available NIRCam filters. To minimise contamination from artefacts, we masked the low SNR regions around the edges of each field, together with bright stars and their associated diffraction spikes. These further corrections/reductions are accounted for in the areas noted in Table 1. \nThe next step in the selection process was to calculate the posterior redshift distribution for each source (see Eqn. 1), using the UV LF prior. The implementation of the UV LF prior required knowledge of the best-fitting value of 𝑀 UV at each redshift, which was calculated using a top-hat filter centred on 1500 Å in the rest-frame of the best-fitting SED. In order to calculate the total 𝑀 UV , the aperturebased fluxes were scale to a Kron aperture flux (Kron 1980), with an additional correction of 10 per cent to account for extended flux not accounted for by the Kron aperture (McLeod et al. 2024). \nFigure 4. The distribution of apparent magnitude in the F277W filter for the galaxies in the final sample, split by the field in which they reside. The histogram is presented in bins with a width of 0.2 mag and the number of galaxies selected from each field is noted in the legend. The final total combined sample contains 2548 galaxies. \n<!-- image --> \nAll objects with an integrated posterior redshift distribution of 𝑝 ( 𝑧 | 𝐹 ) ≥ 0 . 05 at 𝑧 ≥ 8 . 5 were kept as viable high-redshift candidates. The final stage in the selection process was a visual inspection of the sample in order to remove the small number of remaining artefacts and diffraction spikes. \nThe apparent magnitude distribution of the final sample in the F277W filter is shown in Fig. 4 across the six survey areas employed in this study.', '4 UV LUMINOSITY FUNCTION': 'The galaxy selection process produced a final sample of 2548 galaxies selected over a total area of ≃ 369 square arcminutes of JWST /NIRCam imaging. In this section we use this sample to determine the UV luminosity function at 8 . 5 < 𝑧 < 15 . 5.', '4.1 Completeness': 'The accurate determination of the UV LF requires that the incompleteness in the final galaxy sample is accounted for. To achieve this we employed a simulation designed to mimic as closely as possible to real galaxy selection process. In order to calculate the fraction of recovered sources as a function of 𝑀 UV and 𝑧 , artificial sources were injected into the real imaging for each of the fields over the range -22 < 𝑀 UV < -17 and 8 . 5 < 𝑧 < 19 . 5. This included separating COSMOS and JADES into their respective sub-fields as noted in Section 2.3. In steps of Δ 𝑚 = 0 . 5 and Δ 𝑧 = 0 . 5, sources were injected into all of the available images for each field as point sources, based on an SED template with a UV-slope typical of the 𝑧 ≥ 8 . 5 population ( 𝛽 ∼ -2 . 2; Cullen et al. 2024; Morales et al. 2024; Topping et al. 2024). \nBy performing the completeness simulation in this way, we are able to test every step of our selection process, ensuring that the recovered completeness is as accurate as possible. Sources were injected as point sources for simplicity, as detailed simulations have shown that modelling the physical size distribution of the sources has \na negligible impact on the recovered completeness at these redshifts (McLeod et al. 2024). After injecting the sources, catalogues were extracted from the injected images in the same manner as for the main galaxy sample and then passed through the same selection process. The fractional completeness at each point on the 𝑀 UV -𝑧 plane was then calculated for each field.', '4.2 Calculating the UV luminosity function': 'In order to calculate the UV luminosity function, we populate the 𝑀 UV -𝑧 plane based on the normalized posterior redshift probability distribution of all galaxies in the final sample. In practice, we split the 𝑀 UV -𝑧 plane into bins of dimension Δ 𝑚 = 0 . 5 and Δ 𝑧 = 0 . 5 within the range -22 < 𝑀 UV < -17 and 8 . 5 < 𝑧 < 19 . 5, in order to match the resolution of the completeness simulation. The population of the 𝑀 UV -𝑧 plane was calculated for each sub-field individually, before the fields were combined to mimic a single survey. The total combined number density at each 𝑀 UV and 𝑧 is therefore given by, \nΦ ( 𝑀 UV , 𝑧 ) = 𝑁 ∑︁ 𝑖 = 1 𝑝 𝑖 ( 𝑀 UV , 𝑧 ) 𝑉 𝑖 𝐶 𝑖 ( 𝑀 UV , 𝑧 ) (7) \nwhere 𝑁 = 7 (the total number of sub-fields); 𝑝 𝑖 ( 𝑀 UV , 𝑧 ) is the total probability in a given ( 𝑀 UV , 𝑧 ) bin for a given field; 𝐶 𝑖 ( 𝑀 UV , 𝑧 ) is the corresponding completeness and 𝑉 𝑖 is the cosmological volume provided by that field. When computing the combined UV LF, we conservatively restrict the contribution of each sub-field to the UV magnitude range where it is ≥ 50 per cent complete. The final result provides a continuous expression of the UV LF in two dimensions, from which the one-dimensional UV LF can be extracted over any chosen range in redshift. As a result, and unlike many literature studies, it is not necessary to calculate the the UV LF over wide redshift bins in order to counteract the impact of photometric-redshift uncertainties. \nWe extract and plot (in Fig. 5) the one-dimensional UV LF centered on 𝑧 = 9 , 10 , 11 and 𝑧 = 12 . 5 using redshift bins spanning 8 . 5 < 𝑧 < 9 . 5, 9 . 5 < 𝑧 < 10 . 5, 10 . 5 < 𝑧 < 11 . 5 and 11 . 5 < 𝑧 < 13 . 5. The galaxy number densities are tabulated in Table 2 along with their corresponding uncertainties. The uncertainties were calculated using Poisson confidence intervals from Gehrels (1986) combined in quadrature with the cosmic variance uncertainty. The cosmic variance was estimated using the calculator from Trenti &Stiavelli (2008) for each of the survey fields using the default cosmological parameters with a 𝜎 8 = 0 . 9 and a halo-filling factor of unity. This was then combined using the prescription from (equation (9) in Moster et al. 2011). However, the cosmic variance uncertainty (which ranges from 15% to 25%) in fact has minimal impact on the final uncertainties and including them does not significantly alter our results. For comparison, in Fig. 5 we also plot a number of other measurements of the UV LF from the recent literature. \nAt 𝑧 = 9 our new measurement of the UV LF is consistent with the early JWST -based results of Donnan et al. (2023a), as well as with other recent JWST and preJWST studies (e.g. McLeod et al. 2016; Bouwens et al. 2021). We also find good consistency with the faint-end slope measurements from Bouwens et al. (2022) which were based on an analysis of the available lensing fields from HST which enabled a measurement of the UV LF down to 𝑀 UV ≃ -16 at 𝑧 = 9. At 𝑧 = 10 there is good consistency between our new measurement and the preJWST measurement from McLeod et al. (2016), as well as an early analysis of JWST imaging from Bouwens et al. (2023a). At 𝑧 = 11 we can compare to a significant number of recent JWST studies and find that our new measurement of the UV \nLF is consistent with many of them, in particular the recent widearea study of McLeod et al. (2024). Our determination of the UV LF at 𝑧 = 11 is also consistent with measurements of the faint-end of the 𝑧 = 11 UV LF by Leung et al. (2023) and Pérez-González et al. (2023), who studied a relatively small area ( ≃ 8 sq. arcmin) covered by the NGDEEP and MIDIS fields, respectively. However, it can be seen that our new measurement of the 𝑧 = 11 UV LF is consistently higher than the recent determination by Willott et al. (2024). \nAt 𝑧 = 9 , 10 and 𝑧 = 11 we are able to achieve a dynamic range of ≃ 4 magnitudes ( -21 < 𝑀 UV < -17) due to our multi-field approach, without relying on gravitational lensing. This is transformative compared to the early measurements of the UV LF from JWST and, in combination with our large sample size, allows the UV LF to be measured with significantly lower uncertainties. Overall we find a particular lack of evolution at 𝑧 = 9 -11 which is consistent with previous JWST studies (e.g. Finkelstein et al. 2024). At 𝑧 = 12 . 5 we are still able to measure the UV LF over a dynamic range of ≃ 3 magnitudes ( -21 < 𝑀 UV < -18), significantly increasing the number of LF bins at this redshift. Our new determination of the 𝑧 = 12 . 5 UV LF is in reasonable agreement with recent JWST -based measurements from the literature (e.g. Adams et al. 2024; Robertson et al. 2023a). \nWith this sample we are able to robustly measure the evolution of the UV LF from 𝑧 = 9 -13. However, beyond 𝑧 = 13 . 5 there is a noticeable dearth in the total 𝑝 ( 𝑧 ) and therefore we are unable to precisely measure the UV LF at this redshift. That said, although limited by small-number statistics (the equivalent of ≃ 1 . 3 galaxies), we are able to plot a single bin at 𝑧 ≃ 14 . 5, spanning the range 13 . 5 < 𝑧 < 15 . 5. This single bin is shown in Fig. 6, where we compare to measurements from other recent studies centred on redshifts from 𝑧 ≃ 13 -15.', '4.2.1 Luminosity function fitting': 'It is now typical to fit the UV LF at 𝑧 ≥ 7 with a double-power law (DPL) rather than a Schechter function, due to the excess of very bright galaxies that have been detected from large area groundbased surveys (e.g. Bowler et al. 2017, 2020; Donnan et al. 2023a; Varadaraj et al. 2023). We therefore fit our new observational data points at 𝑧 = 9 , 10 , 11 and 𝑧 = 12 . 5 with a DPL parameterisation. We perform the fitting with the S/c.pc/i.pc/p.pc/y.pc curve\\_fit function (Virtanen et al. 2020) using a least-squares method to fit the data. To enhance the dynamic range in UV luminosity, we include in the fits the bright-end data points from Bowler et al. (2020) and Donnan et al. (2023a) at 𝑧 = 9 and from McLeod et al. (2024) at 𝑧 = 11. \nGiven the limited constraints on the bright-end of the LF at 𝑧 = 10, we fix the bright-end slope to 𝛽 = -4 . 05, as this is the midpoint between the best-fitting values at 𝑧 = 9 and 𝑧 = 11. Due to the more limited number of data points available at 𝑧 = 12 . 5, there is insufficient dynamic range in UV luminosity to perform a free-fit to the data. Consequently, we fix the faint-end and bright-end slopes to their best-fitting values at 𝑧 = 11, but keep the LF normalisation ( 𝜙 ∗ ) and the characteristic magnitude ( 𝑀 ∗ ) as free parameters. \nThe results of the LF fits are shown as the solid black lines in Fig. 5 and the best-fitting DPL parameters are reported in Table 3. We also fit a DPL to the single bin at 𝑧 = 14 . 5, fixing 𝑀 ∗ , 𝛼 and 𝛽 to their best-fitting values at 𝑧 = 12 . 5, and fitting 𝜙 ∗ alone. This fit is shown as the black solid line in Fig. 6. \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 5. Our new measurements of the rest-frame UV LF at 𝑧 = 9 , 10 , 11 and 𝑧 = 12 . 5 are shown as the black data points. For comparison we also show data points from McLeod et al. (2016); Oesch et al. (2018); Bowler et al. (2020); Bouwens et al. (2021, 2022); Harikane et al. (2023); Adams et al. (2024); Leung et al. (2023); Donnan et al. (2023a,b); McLeod et al. (2024); Pérez-González et al. (2023); Finkelstein et al. (2024); Casey et al. (2024); Bouwens et al. (2023a); Willott et al. (2024); Robertson et al. (2023a) with colours indicated in the figure legend. The best fitting double-power law functions are shown as the solid black lines at each respective redshift. \n<!-- image -->', '4.3 The cosmic star-formation rate density': 'The UV luminosity density ( 𝜌 UV ) was calculated by performing a luminosity-weighted integral of our best DPL fits to the data. We integrated down to a limit of 𝑀 UV = -17 which is consistent with many other JWST studies (e.g. Finkelstein et al. 2024; Harikane et al. 2023; Adams et al. 2024). The UV luminosity density was then converted to the cosmic star-formation rate density ( 𝜌 SFR ) using the conversion factor K UV = 1 . 15 × 10 -28 M ⊙ yr -1 /erg s -1 Hz -1 (Madau & Dickinson 2014) which assumes a Salpeter (1955) initial mass function (IMF). The results are shown in Fig. 8. \nThe improved statistics and dynamic range in UV luminosity of this study provide measurements of the UV luminosity density at 𝑧 = 9 to 𝑧 = 12 . 5 with much lower uncertainties than early JWSTbased measurements. \nIn Donnan et al. (2023a) we performed a log-linear fit to the evolution of 𝜌 UV with redshift, motivated in part by the analytical expression from Hernquist & Springel (2003). An updated log-linear fit to our new data is shown as the solid black line in Fig. 8 and has the functional form \nlog ( 𝜌 UV ) = (-0 . 140 ± 0 . 068 ) 𝑧 + ( 26 . 5 ± 0 . 6 ) . (8) \nIt can be seen that this expression provides an excellent description of the evolution of 𝜌 UV over the redshift range 8 < 𝑧 < 12 . 5. Interestingly, it is noticeable that an extrapolation of this relation (dashed black line) sits somewhat higher than our tentative measurement of 𝜌 UV at 𝑧 = 14 . 5.', '5.1 A multi-field approach to measuring the galaxy LF': 'By combining the data from the PRIMER, JADES and NGDEEP JWST NIRCam surveys we have been able to conduct a relatively wide-area survey which also samples ≃ 4 mag of dynamic range in UV luminosity at 𝑧 = 9 to 𝑧 = 12 . 5. This, coupled with a rigorous, statistically robust methodology, has enabled us to substantially improve our knowledge of the form and evolution of the UV galaxy LF at 𝑧 ≥ 9 compared to what was achieved by previous studies. \nFirstly, by combining four separate survey fields with a total combined area of ≃ 370 sq. arcmin, we have been able to mitigate the impact of cosmic variance on our measurements. Primarily due to the impact of PRIMER, our survey area is an order-of-magnitude \nTable 2. Computed UV LF data points shown in Fig. 5 at 𝑧 = 9 , 10 , 11 , 12 . 5 and in Fig. 6 at 𝑧 = 14 . 5. The columns show the central redshift and central UV absolute magnitude of each bin, and then the source number densities within each bin, along with their corresponding uncertainties. \nlarger than that which was available in the initial studies of the highredshift UV LF undertaken with JWST data. For example, the first measurements of the UV LF with JWST were derived from early imaging which covered only ∼ 40 sq. arcmin (Donnan et al. 2023a; Finkelstein et al. 2022a; Bouwens et al. 2023a). As JWST Cycle 1 progressed, the area of available imaging expanded somewhat, to ∼ 100 sq. arcmin (Harikane et al. 2023; Adams et al. 2024; Finkelstein et al. 2024). However, all of these early NIRCam-based studies essentially utilised the same survey fields, notably the CEERS (Bagley et al. 2024) ERS imaging of the EGS field (initially covering ≃ 35 sq. arcmin, subsequently expanding to ≃ 90 sq. arcmin after programme completion) and the Abell 2744 cluster field which was imaged by the GLASS (Treu et al. 2022) and UNCOVER (Bezanson et al. 2022) programmes. As noted in Castellano et al. (2023), as well as being relatively small (and complicated by gravitational lensing), this latter \nFigure 6. Our tentative measurement of the rest-frame UV LF at 𝑧 = 14 . 5 is shown as the black data point. We include data points from Harikane et al. (2023); Donnan et al. (2023a); McLeod et al. (2024); Finkelstein et al. (2024); Casey et al. (2024); Robertson et al. (2023a) for comparison. The best-fitting double power-law function is shown as the solid black line, where only the normalisation was allowed to evolve from 𝑧 = 12 . 5. The dashed grey line shows the prediction from the evolving UV LF model described in Section 3.2. The solid coloured lines show our best fitting double power-law fits at 𝑧 = 9 , 10 , 11 and 𝑧 = 12 . 5, with colours as indicated in the figure legend. \n<!-- image --> \n<!-- image --> \n<!-- image --> \na \n<!-- image --> \nFigure 7. The evolution of the best-fitting double-power law parameters at 𝑧 = 9 , 10 , 11 , 12 . 5 from our UV LF fitting. The solid black line shows a linear fit to the evolving parameters. Where a parameter was fixed, its value is indicated by a data point with no error bar. \n<!-- image --> \nfield has a significant over-density at 𝑧 ≃ 10, severely exacerbating the impact of cosmic variance on LF determinations at these redshifts. \nThese limitations motivated the work of McLeod et al. (2024), which represents the most extensive study of the UV LF at 𝑧 = 11 completed prior to the present work. This study was based on a combined survey area of ≃ 210 sq. arcmin, and the results solidified the earlier measurements of the extreme-redshift UV LF, with the data again supporting a slow, gradual evolution of the LF between 𝑧 = 9 and 𝑧 = 11. The results presented here are in excellent agreement \nTable 3. The derived parameter values for the best-fitting double power-law (DPL) models fitted to our data over the redshift range 9 < 𝑧 < 15 . 5. The LF fits derived at 𝑧 = 9 and 𝑧 = 11 utilised the new data presented here along with the data-points presented by Bowler et al. (2020) and McLeod et al. (2024). At 𝑧 = 10 and 𝑧 = 12 . 5 the fits are based purely on the new analysis and galaxy samples presented in this work. The first column gives the central redshift of the binned LF. This is followed by the values of the best-fitting characteristic density 𝜙 ∗ , the best-fitting or fixed characteristic absolute magnitude 𝑀 ∗ , the fitted or assumed faint-end slope 𝛼 , and the fitted or adopted bright-end slope 𝛽 (see text for details). In the cases where a parameter was fixed, the value is denoted with an asterisk. The final column states the resulting UV luminosity density derived at each redshift. \n<!-- image --> \nFigure 8. The redshift evolution of the UV luminosity density, 𝜌 UV , and hence the inferred cosmic star-formation rate density, 𝜌 SFR , at 𝑧 > 8 with our new measurements at 𝑧 ∼ 9 , 10 , 11 , 12 . 5 (solid black circular data points) and our tentative measurement at 𝑧 = 14 . 5 (open black circular data point). Estimates at 𝑧 ≃ 9 -10 from Oesch et al. (2013, 2018) and McLeod et al. (2016) are shown by the purple and blue data points respectively. The green, grey and light red points show 𝜌 UV derived from the LFs in Willott et al. (2024), Harikane et al. (2023) and Pérez-González et al. (2023) respectively. All values were determined using a limit of 𝑀 UV = -17 in the luminosity-weighted integral. The orange data points show the results from Donnan et al. (2023a,b). The solid black line shows the best-fitting linear relation at 𝑧 = 9 -12 . 5 with an extrapolation shown by the dashed black line. The left-hand panel shows a comparison to theoretical models, again setting the LF integration limit to 𝑀 UV = -17. The green, cyan, blue and yellow lines show semi-analytic models from Tacchella et al. (2018), Mason et al. (2023), Behroozi & Silk (2015) and Yung et al. (2019) respectively. The red line shows the dust-free model from Ferrara et al. (2023). The green, brown and pink lines show the results of hydrodynamical simulations from FiBY (Johnson et al. 2013; Paardekooper et al. 2015), FLARES (Wilkins et al. 2023), DELPHI (Mauerhofer & Dayal 2023) and MilleniumTNG (Kannan et al. 2023). The dashed purple line shows an extrapolation at 𝑧 ≥ 8 of the 𝜌 UV ∝ ( 1 + 𝑧 ) -2 . 9 relation from Madau & Dickinson (2014). The dashed lines in the right panel show the predictions from our halo mass function (HMF) models. We show the HMF model where log ( 𝑀 ∗/ M ⊙) = 9 corresponds to 𝑀 UV = -22 . 47 assuming a constant stellar mass to halo mass ratio (light blue), the same mass-to-light conversion but with a halo-mass dependent stellar mass to halo mass ratio consistent with that at 𝑧 = 0 (green) and then where dust is introduced (orange). The dark blue curve shows the same as the model in green but with an alternative scaling where log ( 𝑀 ∗/ M ⊙) = 10 corresponds to 𝑀 UV = -22 . 4. \n<!-- image --> \nwith those derived by McLeod et al. (2024), which is all the more significant because the present study is based on a different set of survey fields, and utilises a different method for calculating the evolving LF. These consistent results, now based on effectively ≃ 600 sq. arcmin of NIRCam imaging, contrast with the evolution of the LF recently presented by Willott et al. (2024). This study used imaging from the CANUCS survey targeting five cluster fields, covering a relatively small total area of ≃ 50 sq. arcmin, with the medium-band imaging (a strength of the CANUCS dataset) only covering ≃ 35 sq. arcmin. Therefore, as they note, they may have targeted one or more relatively underdense fields, which most likely explains why their results differ somewhat from the higher number densities found here and in other wider area studies (e.g. McLeod et al. 2024; Finkelstein et al. 2024). \nSecondly, our new study benefits from vastly improved statistics, \nsimply because our combined multi-tiered survey has yielded a very large sample of galaxies from which to compute the UV LF over a wide range in UV luminosity and redshift. This has led to a significant reduction in the statistical uncertainties in our measurements as well as enabling us to produce LF measurements in an increased number of redshift bins over the redshift range 𝑧 = 9 -12 . 5. By increasing the dynamic range in UV luminosity compared to previous JWST studies, we are better able to constrain the shape of the LF and hence explore how the parameters of the adopted DPL function evolve with redshift. In performing the DPL fits, we found that the data were of sufficient quality to enable us to allow the characteristic magnitude, 𝑀 ∗ , and the characteristic density, 𝜙 ∗ , to be fitted as free parameters at 𝑧 = 9 -12 . 5. The results as a function of redshift are shown in the upper two panels of Fig. 7, where it can be seen that the derived \nevolution in 𝜙 ∗ is stronger than that inferred for 𝑀 ∗ . This indicates that there is more density evolution than luminosity evolution in the UV LF over the redshift range 𝑧 = 9 -12 . 5, consistent with the persistence of relatively bright galaxies out to extreme redshifts (e.g. Castellano et al. 2023; McLeod et al. 2024). \nWe can also explore the evolution of the faint- and bright-end slopes of the DPL fit. Although the bright-end slope, 𝛽 , is (by necessity) fixed for the fits at 𝑧 = 10 and 𝑧 = 12 . 5, we see a lack of evolution between the free fits achieved at 𝑧 = 9 and 𝑧 = 11. The same is also true of the faint-end slope, 𝛼 , where we find no significant evolution between 𝑧 = 9 (where we obtain 𝛼 = -2 . 00 ± 0 . 47) and 𝑧 = 11 (where we find 𝛼 = -2 . 19 ± 0 . 69). Our results are consistent with other recent studies of the faint-end slope, which also observe no significant change over this redshift range (Leung et al. 2023; Pérez-González et al. 2023). \nUsing wide-area ground-based surveys, such as UltraVISTA, it has been demonstrated that there is little if any evolution in the bright end of the UV LF from 𝑧 ≃ 7 to 𝑧 ≃ 10 (e.g. Stefanon et al. 2019; Bowler et al. 2020; Donnan et al. 2023a). However, due to the wavelength restriction of ground-based telescopes, the bright-end of the UV LF at 𝑧 > 10 can only be measured with wide-area surveys from JWST . Our results re-affirm the findings of McLeod et al. (2024) and now extend the evidence for the lack of evolution in the brightend of the LF out to to 𝑧 ≃ 12 . 5. Several previous studies have discussed the potential physical mechanisms that might allow/explain this (arguably unexpected) lack of evolution in the bright galaxy population at early times (e.g. Bowler et al. 2017, 2020; Finkelstein & Bagley 2022). A decrease in dust attenuation or a lack of AGN feedback have been suggested as contributing factors, while others have inferred that the data imply increased star-formation efficiency in the very young Universe (Harikane et al. 2023). However, as we discuss further below, while these proposed astrophysical changes may be important, and can certainly not be excluded at present, they are not in fact required to explain the results of the present study. \nIn the left-hand panel of Fig. 8 we compare our new measurements of 𝜌 SFR to a number of theoretical models and cosmological simulations. It can be seen that the model predictions diverge widely beyond 𝑧 ≃ 9, and so the diagnostic power of the new measurements is clear. In particular, the observational data now clearly lie above the constant star-formation efficiency models of Tacchella et al. (2018), Yung et al. (2019) and Mason et al. (2023). However, there is good agreement between the observations and the predictions of the FLARES (Vijayan et al. 2020; Lovell et al. 2020; Wilkins et al. 2023) and DELPHI (Mauerhofer & Dayal 2023) cosmological hydrodynamical simulations. Our results are also consistent with the predictions of the semi-empirical dust-free model presented by Ferrara et al. (2023), which is discussed further in Section 5.3. It should be noted that the DELPHI and Ferrara et al. (2023) dust-free models were both published after initial JWST studies suggested a high abundance of luminous galaxies at 𝑧 ≥ 10.', '5.2 The star-formation rate density at z ≥ 13': 'The wealth of new deep NIRCam imaging delivered by JWST has already enabled a large number of galaxy candidates to be detected up to 𝑧 ≃ 12, and several of these extreme redshift candidates have already been spectroscopically confirmed with NIRSpec (e.g. Harikane et al. 2024; Arrabal Haro et al. 2023a). Indeed, as shown in the present work, the galaxy samples that can now be assembled at 𝑧 ≃ 12 (by combining the major Cycle-1 programmes) are large and robust enough to enable the basic form and amplitude of the UV LF to be well constrained at these redshifts. However, the nature of \ngalaxy evolution at still higher redshifts, 𝑧 ≥ 13, remains much more uncertain. Despite the abundance of JWST imaging now available, and the absence of any wavelength limitation, there remain very few robust galaxy candidates at 𝑧 ≥ 13, with only one spectroscopic confirmation of a faint galaxy at 𝑧 = 13 . 2 (Curtis-Lake et al. 2023). This has inevitably led to uncertainty in constraining the UV galaxy LF and hence cosmic star-formation rate density at 𝑧 ≥ 13. \nDonnan et al. (2023a) proposed a log-linear relationship between comoving cosmic star-formation rate density, 𝜌 SFR , and redshift, 𝑧 , at 𝑧 ≥ 8, motivated at least in part by the theoretical expectations articulated by Hernquist & Springel (2003). Indeed, the early JWST results did appear consistent with the inferred smooth, gradual decline in 𝜌 SFR to higher redshift, and the results of the present study re-affirm this conclusion, showing that the log-linear relation remains a good fit to our new robust constraints over the redshift range 𝑧 = 9 -12 . 5, as shown by the solid black line in Fig. 8. However, at some point a departure from this relationship is inevitable as we enter the epoch of the very first galaxies. To explore to yet higher redshifts, as described in Section 4, we have attempted to calculate a basic estimate of the galaxy number density at 𝑧 = 14 . 5 from our data. This measurement provides some tentative evidence for the onset of a steeper decline in the UV LF between 𝑧 = 12 . 5 and 𝑧 = 14 . 5, with the inferred 𝜌 SFR at 𝑧 = 14 . 5 lying below the extrapolation of the log-linear relation fitted at 𝑧 = 9 -12 . 5. However, this measurement is highly uncertain, as indicated by the error bars, and we caution against any strong interpretation of this result given that our UV LF measurement at 𝑧 = 14 . 5 is based on a total 𝑝 ( 𝑧 ) equivalent to ≃ 1 . 3 galaxies. \nRobertson et al. (2023a) also recently explored this very early epoch within the ≃ 8 sq. arcmin of the JADES Origins Field (JOF). Theyreport 3 candidates at 𝑧 > 13 . 5 but note that none of them can be regarded as robust. Therefore they discuss two alternative scenarios in which they calculate 𝜌 SFR with and without the 2 highest-redshift candidates (proposed to lie at 𝑧 > 14), yielding the expected result that there is a more rapid decline in 𝜌 SFR with the extreme redshift candidates removed. Although, as noted above, our own measurement of 𝜌 SFR at 𝑧 = 14 . 5 is somewhat poorly constrained, it is nonetheless more closely aligned with this more rapid decline scenario at 𝑧 > 13. Indeed, consistent with this, our own investigation of the JADES data supports the removal of all 3 of the 𝑧 > 13 . 5 candidates tentatively reported by Robertson et al. (2023a). We recover one of these galaxies in our sample (JADES+53.02868 -27.89301) but find a 𝑝 ( 𝑧 ) which peaks at 𝑧 ∼ 3 . 5 (albeit still with a non-neglible probability of lying at 𝑧 ∼ 13). The 2 candidates at 𝑧 > 14 tentatively reported by Robertson et al. (2023a) do not contribute significantly to our high-redshift LFs because in both cases the bulk of their redshift probability distribution lies at much lower redshifts in our analysis (which we emphasize, however, involves the use of LF priors). This therefore adds to the tentative but growing evidence that there is a change from the gradual evolution in the LF observed at 𝑧 = 9 -13 to a more rapid decline at 𝑧 > 13. However, still better constraints on the UV LF (and hence 𝜌 SFR ) are needed at these extreme redshifts to confirm the existence and/or severity of this transition.', '5.3 Modelling the growth of the galaxy population at z > 8': 'As discussed above, the initial observations from JWST revealed a high abundance of (UV) bright galaxies at 𝑧 ≥ 10 which stimulated a number of theoretical attempts to explain their abundance. One natural point of astrophysical interest is the way in which the dust attenuation of galaxies might change ( i.e. reduce) with increasing redshift. Indeed, Ferrara et al. (2023) have recently proposed a model in which galaxies are essentially dust-free at very early times, \nfinding good agreement with the first measurements of the UV LF from JWST . In this specific model they propose that the dust could have been ejected from the galaxies as a by-product of intense starformation activity, leading to (temporarily) dust-free galaxies that might populate the bright end of the LF at 𝑧 > 10. This model does indeed provide a good fit to our new observational constraints on 𝜌 UV , as shown by the red solid line in the left-hand panel of Fig. 8. There is now also independent observational evidence for a lack of significant dust in galaxies at 𝑧 > 10 as inferred from analyses of the UV continuum slopes, 𝛽 , displayed by early galaxies. In particular, Cullen et al. (2024) report that, at 𝑧 = 11 . 5, the average UV slope plateaus at 𝛽 ≃ -2 . 6, consistent with dust-free stellar populations. This is also consistent with the UV-slope measurements of Morales et al. (2024) and Topping et al. (2024). It has also been suggested that increasingly stochastic star formation, with spells of enhanced star-formation efficiency, can help to explain the high number densities of galaxies observed in the rest-frame UV at 𝑧 ≥ 10 (Mason et al. 2023). \nTo explore the physical processes which might explain our observational measurements we have constructed a simple model of galaxy evolution based on the evolving dark matter halo mass function (HMF). We first calculated the evolving HMF at 𝑧 = 8 -15 using HMFcalc (Murray et al. 2013) with the model from Reed et al. (2007). This was then converted to an evolving galaxy stellar mass function (GSMF) simply using a form of the mass-dependent stellar-mass to halo-mass ratio consistent with that at 𝑧 ≃ 0 (Behroozi et al. 2010). This step essentially applies the impact of the inferred feedback processes at both high and low halo masses which regulate the shape of the GSMF and, correctly or incorrectly, assumes this is unchanged with redshift (see Appendix A). The derived evolving GSMF was then converted to a UV LF at each redshift by a scaling equivalent to assuming that a galaxy with a stellar mass of log ( 𝑀 ∗ / M ⊙ ) = 9 has a UVluminosity equivalent to 𝑀 UV = -22 . 47. This mass to UV magnitude conversion was determined from a BC03 stellar population model (Bruzual & Charlot 2003) with a metallicity of 𝑍 / Z ⊙ = 0 . 2, a Chabrier (2003) IMF and an assumed constant star-formation history with an age of 30 Myr. We compare the predictions of this simple model to the observed evolution of 𝜌 UV in the right-hand panel of Fig. 8, and show a detailed comparison of the LF predicted by this model with our observed UV LF at 𝑧 = 11 in the left-hand panel of Fig. 9. \nWe show our primary model by the dashed green line which is consistent with the data at 𝑧 ≥ 10 but overshoots the observations at 𝑧 < 10. To attempt to correct this we introduce a dust component to the model by adding mass-dependent UV dust attenuation, 𝐴 1500 , given by the dust to stellar-mass relation at 𝑧 = 2 derived by McLure et al. (2018). This model is shown by the orange line in the righthand panel of Fig. 8 which lies slightly below the green dashed line, but still overshoots the data at 𝑧 < 10. This is perhaps unsurprising as galaxies at 𝑧 < 10 are likely to have average ages older than the assumed value of 30 Myr. Moreover, additional feedback at low stellar masses in the reionization era, or reduced halo occupation could also play a role in reducing the predicted 𝜌 UV at 𝑧 < 10. \nWe also show two secondary models. The first is created by altering the step where we apply feedback at the low- and high-mass ends of the HMF. Instead we simply use 𝑀 ∗ / 𝑀 𝐻 = 1 / 35, which is the peak value found in the present-day Universe, corresponding to the observed maximum in historical star-formation efficiency ( 𝜖 ≃ 0 . 17). This effectively provides an upper limit on what is physically permitted by the assumed Λ CDM cosmology while assuming that star-formation efficiency has never been more efficient at any mass than for present-day Milky-Way mass galaxies. This simple \nremoval of both high- and low-mass feedback predicts the evolution of 𝜌 UV shown by the dashed light-blue line in the right-hand panel of Fig. 8 and the form of the 𝑧 = 11 UV LF shown by the solid light-blue line in the left-hand panel of Fig. 9. \nClearly the predictions of this modified model vastly exceed the observational results, demonstrating that our new measurements of the UV galaxy LF certainly do not threaten the viability of the standard Λ CDM cosmological model. We also show another modified model illustrated by the dashed dark-blue line in the right-hand panel of Fig. 8. This is the same as the primary model except that it assumes a different mapping from galaxy stellar mass to UV luminosity. Here we assume that a galaxy of stellar mass of log ( 𝑀 ∗ / M ⊙ ) = 10 has a UV luminosity equivalent to 𝑀 UV = -22 . 4 as assumed at 𝑧 = 7 in Bowler et al. (2014). This demonstrates that a much steeper decline is inevitably predicted when fixing the stellar mass to UV luminosity mapping at 𝑧 = 7 assuming no dust obscuration.', '5.3.1 The effect of dust at 𝑧 > 10': 'As mentioned above, Ferrara et al. (2023) have presented a dust-free model of the UV LF and suggest physical mechanisms by which the dust could be expelled from galaxies at 𝑧 > 10. The galaxy UV LF predicted by this model at 𝑧 = 11 is shown by the dashed red line in the left-hand panel of Fig. 9, where it can be seen to be in excellent agreement with our observational data points. However, as shown in the right-hand panel of Fig. 8, there is no significant difference in the inferred evolution of 𝜌 UV predicted by models which do or do not include dust obscuration at 𝑧 > 10. This is because the dust-mass relation deduced by McLure et al. (2018) at 𝑧 = 2 indicates that there is essentially no significant dust attenuation ( i.e. 𝐴 1500 ≃ 0) for stellar masses below log ( 𝑀 ∗ / M ⊙ ) ≃ 8 . 4, and, based on our primary model, the vast majority of the galaxies currently observed at 𝑧 > 10 have stellar masses smaller than this. Indeed, our brightest bin at 𝑧 = 11, corresponding to 𝑀 UV = -21 . 25, equates to a stellar mass of log ( 𝑀 ∗ / M ⊙ ) = 8 . 5 according to our adopted mass to UV luminosity conversion. In other words, even assuming a Universe as dusty as observed at 𝑧 ≃ 2 ( i.e. at "cosmic noon", where dustobscured star formation dominates 𝜌 SFR ) we would not expect any of the galaxies that contribute to our observed UV LF at 𝑧 ≃ 11 to be significantly attenuated by dust in the rest-frame UV. Or, in other words, the stellar-mass range (and hence 𝑀 UV regime) in which dust, if present at such early times, would have an observable impact has not yet been accessed in this study. Thus, within the range of UV luminosities probed by the JWST imaging surveys analysed here, we do not need to invoke any physical mechanism to remove or destroy dust to explain the data, as the observed galaxies are not expected to be massive enough to contain significant quantities of dust at any redshift. Further work, including the exploitation of still largerarea surveys, is thus required to measure the very bright end of the galaxy UV LF ( 𝑀 UV ≲ -21 . 5) at 𝑧 = 11, and hence to determine whether dust removal processes are required to explain the properties of higher-mass galaxies at these early times. \nDespite this current lack of robust statistical constraints on the bright-end form of the 𝑧 ≃ 11 galaxy UV LF at 𝑀 UV < -21 . 5, a few \'bright\' galaxy candidates have been uncovered in this luminosity regime at 𝑧 > 10. These include the spectroscopicallyconfirmed 𝑀 UV ≃ -21 . 8 galaxy at 𝑧 = 10 . 6 reported by Bunker et al. (2023b) and Tacchella et al. (2023). Interestingly, these authors report a modest dust attenuation of 𝐴 V = 0 . 17 for a stellar mass of log ( 𝑀 ∗ / M ⊙ ) = 8 . 73, consistent with the stellar mass-to-UV magnitude mapping and dust-mass relation used in our primary model. In addition we note that Casey et al. (2024) have reported an initial \n<!-- image --> \nFigure 9. Left: The observed UV LF at 𝑧 = 11 compared to our DPL fit (black), our HMF model assuming a constant 𝑀 ∗/ 𝑀 𝐻 = 1 / 35 ( 𝜖 ≃ 0 . 17, light blue), our HMF model with 𝑀 ∗/ 𝑀 𝐻 = 𝑓 ( 𝑀 𝐻 ) (solid green), and the dust-free model from Ferrara et al. (2023) (dashed red). Right: The observed evolving UV luminosity density, 𝜌 UV , and cosmic star-formation rate density, 𝜌 SFR , at 𝑧 > 8 compared to the predictions of our model of the HMF in which the typical stellar population age is allowed to be redshift-dependent (solid green line). The dashed green line shows the extreme-redshift prediction of our HMF model if the stellar population age is simply fixed to 10 Myr at 𝑧 ≥ 12 . 5. The two inset panels show the age (left) and inferred formation redshift (right) of the stellar populations required to best fit our evolving HMF model to the UV LF at each redshift. \n<!-- image --> \nsample of more luminous ( 𝑀 UV ≲ -21) galaxy candidates from the first half of the COSMOS-Web programme (Casey et al. 2023), derived from imaging covering ∼ 0 . 28 deg 2 . This yields an estimate of the LF at 𝑧 = 11 indicated by the dark-green data-points in the bottom-left panel of Fig. 5. These inferred number densities are consistent with our best-fitting DPL and suggests that there may indeed be galaxies massive enough to require dust-removal mechanisms at these extreme redshifts. However, the lack of contiguous filter coverage in the COSMOS-Web programme may result in more significant numbers of low-redshift contaminants in the high-redshift galaxy samples, in turn leading to more uncertain and potentially biased estimates of the bright end of the galaxy UV LF at extreme redshifts. Spectroscopic verification of these luminous high-redshift candidates is therefore required to accurately constrain their abundance.', '5.3.2 An age-dependent model': 'Finally, we alter our primary model to introduce a stellar population age-dependence which creates a redshift-dependent mapping of stellar mass to UV luminosity. This was determined by adjusting the typical stellar age at a given redshift to best map the GSMF onto the UVLFat 𝑧 = 8 -14 . 5. This model is shown by the green solid line in the right-hand panel of Fig. 9. Unsurprisingly, and largely by design, this provides an excellent representation of the evolution of 𝜌 UV out to 𝑧 ≃ 13. Also unsurprising, and physically sensible, are the relatively young inferred ages of the stellar populations at each redshift which dominate the rest-frame UV light, as tabulated in Table 4. \nThe required stellar ages are consistent with those obtained from fitting SED models to the JWST /NIRCam photometry. For example, Robertson et al. (2023b,a) derive stellar ages of 𝑡 ∗ ≃ 10 -70 Myr for galaxies at 𝑧 ≥ 10. This trend of younger stellar ages at increasing redshift is also consistent with theoretical predictions, where increased gas accretion rates at high redshift lead to increased star-formation rates for fixed stellar mass (Mason et al. 2015, 2023). Therefore at fixed stellar mass, galaxies at higher redshifts have greater UV luminosities and younger stellar ages, consistent with the results of \nTable 4. The age-dependent UV magnitude mapping to stellar mass as a function of redshift for our age-dependent model. This model is based on a BC03 stellar population model (Bruzual & Charlot 2003) with a metallicity of 𝑍 / Z ⊙ = 0 . 2. The first column is the redshift. The second column is the UV magnitude, 𝑀 UV , that is mapped to a stellar mass of log ( 𝑀 ∗/ M ⊙) = 9 determined by the age given in the third column. The fourth column is the formation time after the Big Bang associated with this age and the final column is the formation redshift. \nour model. What is more surprising, and potentially very interesting, is the extent to which, for 𝑧 = 8, 9, 10, and 11, the required stellar population ages converge on a common formation time corresponding to ≃ 380 Myr after the Big Bang (equivalent to a formation redshift 𝑧 𝑓 ≃ 12). This is again tabulated in Table 4, with the results of this analysis shown in the two inset panels in the right-hand panel of Fig. 9. We note that our fit quality and overall conclusion is unchanged with different choices of the HMF in the model, with only a modest increase in the required stellar ages resulting from the adoption of a Sheth & Tormen (1999) HMF (pushing the galaxy emergence epoch back slightly to 𝑧 ≃ 12 . 5) or decrease for a Tinker et al. (2008) HMF. Our adopted model from Reed et al. (2007) is positioned in the middle of the scatter between different models of the HMF at 𝑧 > 8 . 5. \nObviously the small number of galaxies discovered at 𝑧 ≥ 13 require a still higher formation redshift (albeit with now very young stellar ages, corresponding to a formation redshift 𝑧 𝑓 ≤ 14), but there is nothing in our analysis which would a priori have required \nan inferred common epoch of formation for the galaxy populations observed over the redshift range 𝑧 ≃ 8 -11. These results indicate not only that the observed high-redshift evolution of the UV galaxy LF (and hence 𝜌 SFR ) can be explained without requiring any changes to cosmology, star-formation efficiency, or indeed dust, but intriguingly they also point towards the rapid emergence of early galaxies at 𝑧 ≃ 12 -13, consistent with the first suggestions of a steeper decline in galaxy number density at 𝑧 ≥ 13 seen here in Fig. 8. \nIn this context, to illustrate what is expected to happen once everyounger stellar populations can no longer offset the rapid decline of the halo mass function back to earlier times, we also plot a green dashed line in the right-hand panel of Fig. 9 which shows the extremeredshift prediction of our model when the stellar population age is simply fixed to 10 Myr at 𝑧 ≥ 12 . 5. The prediction is that, unless the inevitable extreme-redshift decline in the halo mass function is offset by more extreme/exotic stellar populations or enhanced starformation efficiency, UV luminosity density is expected to decline more rapidly, by roughly two orders-of-magnitude from 𝑧 ≃ 13 to 𝑧 ≃ 16.', '6 CONCLUSIONS': 'Wehave completed an analysis of the major Cycle-1 JWST /NIRCam imaging surveys PRIMER, JADES and NGDEEP, covering a total area of ≃ 370 sq. arcmin and reaching a 5𝜎 depth of ≃ 30 AB mag in the deepest regions. Rather than simply selecting galaxy candidates at high redshift by their "best" photometric redshift, we selected all galaxies that have at least a 5 per cent probability of lying at 𝑧 ≥ 8 . 5 andconsider their 𝑝 ( 𝑧 ) . Through careful selection of galaxies we have derived the 𝑝 ( 𝑧 ) for 2548 galaxies using a UV LF prior and hence calculated the evolution of the galaxy UV LF at 8 . 5 < 𝑧 < 15 . 5. Our multi-field approach has allowed new constraints to be placed on the form of the UV LF spanning a UV luminosity range corresponding to ≃ 4 AB mag over the redshift range 𝑧 = 9 -12 . 5. This has allowed us to reach a number of conclusions. \nFirst, the large dynamic range in UV luminosity resulting from our multi-field approach has enabled us to define the shape of the UV LF at 𝑧 = 9 -12 . 5 from 𝑀 UV ≃ -21 to 𝑀 UV ≃ -17. We have fitted our new measurements with a double-power law (DPL) functional form and explored how the parameters evolve with redshift. We find a lack of evolution in the bright- and faint-end slopes as well as at most only modest evolution in the characteristic magnitude, 𝑀 ∗ . Much stronger evolution is seen in the LF normalisation suggesting that the evolution of the LF at 𝑧 = 9 -12 . 5 is dominated by density evolution rather than luminosity evolution. \nSecond, our new measurements of the UV LF have yielded improved constraints on the evolution of cosmic star-formation rate density, 𝜌 SFR , at 𝑧 ≥ 9. We find good agreement with prior work showing a smooth, slow evolution over the redshift range 𝑧 = 9 -12 . 5 which can be described with a log-linear relationship. Indeed, in the present study we see very little evolution between 𝑧 = 9 and 𝑧 = 11. We have also presented a tentative measurement of the galaxy number density at 𝑧 = 14 . 5 which suggests the onset of a steeper decline in 𝜌 SFR , below the extrapolation of the log-linear relation to these extreme redshifts. This, albeit still necessarily uncertain result, hints at a more rapid build-up of galaxies at very early times corresponding to 𝑧 > 13. \nFinally, we explore the evolution in the UV LF and 𝜌 SFR at 𝑧 ≥ 8 through simple modelling based on the halo mass function. We demonstrate that our measurements are fully consistent with a Λ CDMcosmology, and moreover currently require no change \nto the star-formation efficiency or the dust properties of galaxies as observed in the low-redshift Universe. Rather, we show that a simple evolution in the ages of the stellar populations can explain our measurements, with (unsurprisingly) the typical age of the galaxies which populate the UV LF decreasing with increasing redshift. Intriguingly we find that the typical ages required at 𝑧 ≃ 8, 9, 10, and 11 all converge on a time ≃ 380 -330 Myr after the Big Bang, equivalent to a formation redshift 𝑧 ≃ 12 -13. This is consistent with the aforementioned first signs of a steeper drop-off in the galaxy population we find beyond 𝑧 ≃ 13, as expected given the very rapid evolution of the halo mass function at earlier times.', 'ACKNOWLEDGEMENTS': "We thank Andrea Ferrara for providing his model data. C. T. Donnan, D. J. McLeod, R. J. McLure, J. S. Dunlop, R. Begley, M. L. Hamadouche, F. Liu acknowledge the support of the Science and Technology Facilities Council. J. S. Dunlop also acknowledges the support of the Royal Society through a Royal Society Research Professorship. A.C. Carnall thanks the Leverhulme Trust for their support via the Leverhulme Early Career Fellowship scheme. F. Cullen, K. Z. Arellano-Córdova and T. M. Stanton acknowledge support from a UKRI Frontier Research Guarantee Grant [grant reference EP/X021025/1]. P. Santini acknowledges INAF Mini Grant 2022 'The evolution of passive galaxies through cosmic time.' R. A. A. Bowler acknowledges support from an STFC Ernest Rutherford Fellowship [grant number ST/T003596/1]. This work is based [in part] on observations made with the NASA/ESA/CSA James Webb Space Telescope. The data were obtained from the Mikulski Archive for Space Telescopes at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-03127 for JWST. These observations are associated with program 1837, 1180, 1210, 2079. CM acknowledges support by the VILLUM FONDEN under grant 37459 and the Carlsberg Foundation under grant CF22-1322. The Cosmic Dawn Center (DAWN) is funded by the Danish National Research Foundation under grant DNRF140.", 'DATA AVAILABILITY': 'All JWST and HST data products are available via the Mikulski Archive for Space Telescopes ( https://mast.stsci.edu ). Additional data products are available from the authors upon reasonable request.', 'REFERENCES': '```\nAdams N. J., et al., 2023, MNRAS, 518, 4755 Adams N. J., et al., 2024, ApJ, 965, 169 Aird J., Coil A. L., Georgakakis A., Nandra K., Barro G., Pérez-González P. G., 2015, MNRAS, 451, 1892 Arrabal Haro P., et al., 2023a, Nature, 622, 707 Arrabal Haro P., et al., 2023b, ApJ, 951, L22 Austin D., et al., 2023, ApJ, 952, L7 Bagley M. B., et al., 2024, ApJ, 965, L6 Behroozi P. S., Silk J., 2015, ApJ, 799, 32 Behroozi P. S., Conroy C., Wechsler R. H., 2010, ApJ, 717, 379 Bertin E., Arnouts S., 1996, A&AS, 117, 393 Bezanson R., et al., 2022, arXiv e-prints, p. arXiv:2212.04026\n``` \nBosman S. E. I., Ďurovčíková D., Davies F. B., Eilers A.-C., 2021, MNRAS,', 'APPENDIX A: THE STAR-FORMATION EFFICIENCY MODEL': 'In this short Appendix we detail and explore the role of star-formation efficiency within our simple theoretical model of the evolving galaxy UV LF. As described in Section 5.3, we model the star-formation efficiency as a function of halo-mass, 𝜖 ( 𝑀 h ) through the stellar-tohalo mass relation (SHMR), where the stellar and halo masses are \nsimply related by the universal baryon fraction, 𝑓 b = 0 . 167, through the relation \n𝑀 ∗ 𝑀 ℎ = 𝜖 ( 𝑀 h ) 𝑓 b (A1) \nwhich assumes that the efficiency is solely dependent on the mass of a galaxy\'s host dark-matter halo. We adopt a functional form for 𝜖 ( 𝑀 h ) given by the double power-law relationship described in Tacchella et al. (2018), namely: \n𝜖 ( 𝑀 h ) = 2 𝜖 0 " GLYPH<18> 𝑀 h 𝑀 c GLYPH<19> -𝛽 + GLYPH<18> 𝑀 h 𝑀 c GLYPH<19> 𝛾 # -1 (A2) \nwhere 𝜖 0 is the peak efficiency, 𝑀 c is the characteristic mass (the mass at 𝜖 0 ), 𝛽 is the low-mass slope and 𝛾 is the high-mass slope. We use ( 𝜖 0 , 𝑀 c , 𝛽 , 𝛾 ) = ( 0 . 16 , 10 11 . 7 , 0 . 9 , 0 . 65 ) , and our resulting model relation is shown by the solid red line in Fig. A1. Also shown in Fig. A1 are the observational constraints on the 𝑧 ≃ 0 SHMR as compiled by Wechsler & Tinker (2018) (with the uncertainty captured by the gray shaded region), as well as the 𝑧 = 0 . 1 SHMR from Behroozi et al. (2010). At high masses our model relation is essentially identical to that adopted by Behroozi et al. (2010), but the low-mass form of our model relation is chosen to better track the observational constraints on the SHMR at 𝑧 = 0. \nGiven that there exist a number of theoretical models of the high𝑧 UV LF, we also compare our adopted relation to the predictions of 𝜖 ( 𝑀 h ) from a number of alternative models. It can be seen that these cover quite a large range, with some adopted relations clearly inconsistent with the low-redshift observations. For example, Tacchella et al. (2018) assume a higher efficiency ( ∼ 1 dex across the mass ranges relevant for the 𝑧 ≥ 9 UV LF) whereas the efficiencies from Mason et al. (2015) and Harikane et al. (2022) are substantially lower. The semi-analytical model from Yung et al. (2019) has a redshift-dependent 𝜖 ( 𝑀 h ) which, for simplicity, we plot here assuming 𝑧 = 10, where it transpires to be very similar to our (redshift independent) model at low halo masses. However, one common feature of the models presented by Yung et al. (2019); Harikane et al. (2022) is that they assume a significantly lower star-formation efficiency at the high-mass end ( ∼ 1 -2 dex at log ( 𝑀 h ) > 11) compared either to our own model or to the observationally-defined relationship at 𝑧 = 0. \nIt should be noted that there is an obvious degeneracy between the star-formation efficiency and the age of the stellar population when converting between stellar mass and 𝑀 UV . If a higher efficiency is assumed then the same UV LF can be achieved by an older stellar population. This may explain why although Tacchella et al. (2018) assume a significantly higher efficiency than that at 𝑧 = 0, they predict lower number densities in the UV LF at 𝑧 ≥ 9 compared to our model. \nTo break this degeneracy requires direct tests of the evolving galaxy stellar mass function predicted by theoretical models. Using the 𝜖 ( 𝑀 h ) the HMF can be converted to a stellar mass function (SMF) by \n𝑀 ∗ = 𝜖 ( 𝑀 h ) 𝑓 b 𝑀 h (A3) \nwhere 𝑀 ∗ is stellar mass. Therefore if one assumes a form for the HMF, the SMF allows a direct measurement of the star-formation efficiency as a function of both stellar and halo mass which is now, thanks to JWST robustly measurable up to 𝑧 ≃ 8. Therefore, as a check on the viability of our model, in Fig. A2 we compare our model prediction for the evolving SMF at 𝑧 = 6 -8 to the latest observational \nFigure A1. The adopted (redshift independent) star-formation efficiency ( 𝜖 ) as a function of halo mass in our model (solid red line) compared to the corresponding relations in the models from Tacchella et al. (2018), Harikane et al. (2022), Yung et al. (2019), Mason et al. (2015) and Sun & Furlanetto (2016). The dashed grey line shows the 𝑧 = 0 . 1 Behroozi et al. (2010) stellarto-halo mass relation (SHMR) which is identical to the relation adopted here at high masses. The low-mass form of our adopted relation is chosen to better track the observational constraints on the SHMR at 𝑧 = 0 (Wechsler & Tinker 2018), the uncertainty in which is shown here by the gray shaded region. \n<!-- image --> \n/circledot \nmeasurements from Weibel et al. (2024). This comparison shows that our model for 𝜖 ( 𝑀 h ) is able to closely match the measurements of the SMF at 𝑧 = 6 -8, thus in the process confirming that the starformation efficiency as a function of halo mass found at 𝑧 = 0 is still able to reproduce the SMF out to the highest redshifts yet probed. Fig. A2 also reveals that several of the other theoretical models discussed above, while potentially able to reproduce the high-redshift UV LF, clearly fail this crucial test. \nThis paper has been typeset from a T E X/L A T E X file prepared by the author. \n/circledot \n/circledot \n/circledot \nFigure A2. Our model prediction for the evolving galaxy stellar mass function (SMF) at 𝑧 = 6 -8 (using the 𝜖 ( 𝑀 ℎ ) from Fig. A1) is shown in each panel by the solid red line where it is compared with the latest JWST PRIMER-based observational measurements of the SMF from Weibel et al. (2024) (black data points). The predictions of the SMF using the alternative 𝜖 ( 𝑀 ℎ ) relations shown in Fig. A1 from Tacchella et al. (2018), Harikane et al. (2022), Yung et al. (2019), Mason et al. (2015), Sun & Furlanetto (2016) and Behroozi et al. (2010) are shown by the dashed lines for comparison. \n<!-- image -->'}

2020JHEP...03..149A

We consider a gravity theory coupled to matter where the matter has a higherdimensional holographic dual. In such a theory finding quantum extremal surfaces becomes equivalent to finding the RTHRT surfaces in the higherdimensional theory. Using this we compute the entropy of Hawking radiation and argue that it follows the Page curve as suggested by recent computations of the entropy and entanglement wedges for old black holes. The higherdimensional geometry connects the radiation to the black hole interior in the spirit of EREPR. The black hole interior then becomes part of the entanglement wedge of the radiation. Inspired by this we propose a new rule for computing the entropy of quantum systems entangled with gravitational systems which involves searching for islands in determining the entanglement wedge.

2020-03-01T00:00:00Z

['2020JHEP...03..149A', '10.48550/arXiv.1908.10996', '10.1007/JHEP03(2020)149', 'arXiv:1908.10996', '2019arXiv190810996A']

['2D Gravity', 'Black Holes', 'Gauge-gravity correspondence', 'High Energy Physics - Theory', 'General Relativity and Quantum Cosmology']

The Page curve of Hawking radiation from semiclassical geometry

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML']

https://arxiv.org/pdf/1908.10996.pdf

{'The Page curve of Hawking radiation from semiclassical geometry': 'Ahmed Almheiri, 1 Raghu Mahajan, 1 , 2 Juan Maldacena, 1 and Ying Zhao 1 \n1 Institute for Advanced Study, Princeton, NJ 08540, USA \n2 Jadwin Hall, Princeton University, Princeton, NJ 08540, USA \nalmheiri@ias.edu, raghu m@princeton.edu, malda@ias.edu, zhaoying@ias.edu', 'Abstract': "We consider a gravity theory coupled to matter, where the matter has a higher-dimensional holographic dual. In such a theory, finding quantum extremal surfaces becomes equivalent to finding the RT/HRT surfaces in the higher-dimensional theory. Using this we compute the entropy of Hawking radiation and argue that it follows the Page curve, as suggested by recent computations of the entropy and entanglement wedges for old black holes. The higher-dimensional geometry connects the radiation to the black hole interior in the spirit of ER=EPR. The black hole interior then becomes part of the entanglement wedge of the radiation. Inspired by this, we propose a new rule for computing the entropy of quantum systems entangled with gravitational systems which involves searching for 'islands' in determining the entanglement wedge. \nDedicated to the memory of Steven S. Gubser", '1 Introduction': "The central question of the information paradox [1, 2] is whether the process of formation and evaporation of a black hole can be described in a unitary fashion. In particular, unitarity implies that the von Neumann entropy of the Hawking radiation should initially rise but then fall back down, following the so called 'Page curve' [3,4]. If we replace a black hole by its dual quantum mechanical description, via AdS/CFT, we know that this happens. However, one would like to understand how it happens from the gravity point of view. In gravitational theories we now have formulas that compute von Neumann entropies: the Ryu-Takayanagi formula and its extentions [5-7]. Indeed, interesting recent work [8, 9] addressed the closely related problem of studying the evolution of the von Neumann entropy of an evaporating black hole via the minimal quantum extremal surface (QES) prescription of [7]. (A QES is a surface that extremizes the generalized entropy functional.) This amounted to locating and tracking the minimal QES in the evaporating black hole spacetime as a function of boundary time. \nThe main result of [8,9] was that, for an old black hole, past the Page time, the QES is located just behind the event horizon, which thereby excludes most of the interior from the entanglement wedge of the boundary. This is in contrast to the situation at early times, where the minimal QES is the trivial surface, and hence the entanglement wedge for a black hole that forms in a pure state includes all of the interior. For such a black hole, the von Neumann (or fine-grained) entropy increases due to the early Hawking radiation, giving the initial rise of the Page curve, and then decreases once the entanglement wedge moves out to the near horizon region. The early growth of the boundary entropy is related to the growth of the entropy of the quantum fields inside the black hole, which is manifest using the nice slices picture of [10,11]. At the Page time, once this region is removed form the entanglement wedge, the entropy can start decreasing. \nIf we assume that the combined state of the black hole and Hawking radiation is pure, then a Page curve for one implies a Page curve for the other. However, this would amount to assuming away \nthe information paradox, and therefore a more direct argument for the Page curve of the radiation is desirable. In [9] the entropy of the Hawking radiation was computed in the semiclassical limit and was found to not have a Page curve, reproducing Hawking's original result of information loss. (See also [12] for similar calculations in the CGHS model.) As discussed in [8] and briefly alluded to in [9,13], there is perhaps a way to argue that the QES of the Hawking radiation coincides with that of the black hole. \nIn this paper, we argue that this is the case by considering an evaporating black hole in a gravitational theory with holographic matter. Namely, we consider a gravity theory with matter described by a quantum field theory that itself has a higher dimensional gravity dual. This allows us to compute the entropy of Hawking radiation holographically. In this case, the prescription of extremizing the generalized entropy in [7] is equivalent, at leading order, to the standard RT/HRT prescription [5,6] of extremizing the area. \nThe upshot is that the minimization condition in the RT/HRT formula in the higher-dimensional holographic dual ensures that the minimal surfaces of the evaporating black hole and the Hawking radiation must coincide. The entropy of the radiation computed in this way follows the Page curve, rising initially at early times and then decreasing after the Page time due to the phase transition between two surfaces. This is the same transition discussed in the computations of the von Neumann entropy of the black hole in [8,9]. \nA crucial point is that the region deep inside the black hole interior is connected to the radiation via the extra dimension. Therefore, when we search for the minimal QES, we can end up with an entanglement wedge for the radiation that reaches all the way into the interior of the black hole, which is in fact what happens after the Page time. This geometric connection is related to the entanglement between the Hawking radiation and the interior modes of the quantum matter. We can view this extra dimension as an example of a geometric connection between the radiation and the black hole interior, a realization of ER=EPR [14,15] idea (see also [16,17]). Our analysis in this paper is restricted to the case of a two-dimensional theory of gravity coupled to a two-dimensional conformal field theory, since in this case the three-dimensional gravity dual is very simple, but we expect that the results should generalize without much change to higher dimensions. \nThis holographic example suggests a new rule for computing von Neumann entropies of quantum systems entangled with quantum fields in a gravitational theory. This new rule allows for the inclusion of new 'quantum extremal islands' which are regions in the gravitational theory that contain matter entangled with the external quantum system. Including these islands can result in a penalty due to their areas, but they can also give larger 'savings' by reducing the bulk entropy piece of the generalized entropy. See also [8,13] for motivation of this new rule. \nThis paper is organized as follows. In Section 2, we discuss theories of gravity coupled to holographic matter and their bulk interpretation. We focus on a two-dimensional gravity theory coupled to conformal matter, which itself has a three-dimensional dual. In Section 3, we discuss quantum extremal surfaces and entanglement wedges for an evaporating two-dimensional black hole. We discuss the computation of the entropy of the radiation and the black hole, and explain how the Page curve arises. Finally, in Section 4 we discuss a new rule for computing entropies and entanglement wedges for systems entangled with a gravity theory. Conclusions are presented in Section 5.", '2 Two-dimensional gravity with holographic matter': 'Consider a general two-dimensional theory of gravity. The Einstein-Hilbert term in two dimensions is purely topological, but it contributes a constant term to the total entropy of the system. Nontrivial \ngravitational dynamics arise when we add an extra \'dilaton\' 1 field φ , and consider the general action \nI grav [ g (2) ij , φ ] = ∫ d 2 y √ -g ( 1 16 πG (2) N φR (2) + U ( φ ) ) , (1) \nwhere we have absorbed a possible purely Einstein-Hilbert term by a shift of φ . Adding matter to this system, which is taken to be a CFT 2 with some fields collectively denoted by χ , we consider the total action \nI [ g (2) ij , φ, χ ] = I grav [ g (2) ij , φ ] + I CFT [ g (2) ij , χ ] . (2) \nWe take this CFT 2 to have a three-dimensional holographic dual. To justify working in the semiclassical limit in the 2d theory, and to ensure that we have a large-radius dual in 3d, we require that the central charge of the CFT 2 satisfies 1 glyph[lessmuch] c glyph[lessmuch] φ 4 G (2) N . 2 In addition, to have an Einstein gravity dual we need that the CFT is suitably strongly coupled. \nFirst, let us think about this CFT 2 on a fixed background metric g (2) ij . Its three-dimensional dual has a two-dimensional boundary, where the metric obeys the boundary condition \ng (3) ij ∣ ∣ ∣ bdy = 1 glyph[epsilon1] 2 g (2) ij . (3) \nHere i, j are indices along the boundary (see figure 1), and glyph[epsilon1] is a short-distance cutoff. According to the usual rules of AdS/CFT [18-20], a 3d theory with the boundary metric fixed to be g (2) ij is dual to a CFT 2 described by the action I CFT [ χ ; g (2) ij ]. The extrinsic curvature of the two-dimensional boundary is related to the stress tensor of the CFT [21]. \nNext, in order to find the three-dimensional dual to the full action (2), we start from the geometry of the previous paragraph, add a scalar field φ that lives on the 2d boundary with the action (1), and integrate over φ and g (2) ij . The three-dimensional bulk metric is locally AdS 3 , with a boundary at a finite location where the 2d theory with with action (1) lives. We emphasize that unlike usual AdS/CFT, g (2) ij is also integrated over. This is essentially identical to the RandallSundrum model [22], and the dynamical boundary brane is called the \'Planck\' brane in that paper (for the relationship of the Randall-Sundrum model to holography, see e.g. [23]). \nThe embedding of the Planck brane in AdS 3 is determined by using the two-dimensional metric and stress tensor profile from the solution of (2). Let us describe this in detail. Imagine that we have a 2d gravity solution with some profile for the 2d metric and stress tensor \nds 2 = -e 2 ρ ( y ) dy + dy -, T y + y + ( y + ) and T y -y -( y -) . (4) \nHere the stress tensor is measured in the flat metric ds 2 = -dy + dy -. The full stress tensor in the original metric (4) also contains terms coming from derivatives of ρ ( y ) which can easily be obtained from the conformal anomaly. \nIt is useful to introduce the coordinate transformations w + ( y + ) and w -( y -) that make the stress tensor vanish locally. These are obtained by solving the equations \nT y + y + = -c 24 π { w + , y + } , T y -y -= -c 24 π { w -, y -} , (5) \n<latexit sha1\\_base64="PbtnKSmKw6Dl9/kt05Y9t+udOhg=">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</latexit> \nI grav [ g (2) ij , φ ]+ I CFT [ g (2) ij , χ ] \n<latexit sha1\\_base64="7YkrqCnz2Vbu7H+evH/9lV0sPI8=">AAACCnicbVDJSgNBEO2JW4xb1KOX1iBEkDATBT0GvegtglkgGYeeTmfSpmehuyYYhjl78Ve8eFDEq1/gzb+xsxw08UHB470qquq5keAKTPPbyCwsLi2vZFdza+sbm1v57Z26CmNJWY2GIpRNlygmeMBqwEGwZiQZ8V3BGm7/cuQ3BkwqHga3MIyY7RMv4F1OCWjJye9fO21gD5B4kgzSluck/D69S4rlo/QYt6Met518wSyZY+B5Yk1JAU1RdfJf7U5IY58FQAVRqmWZEdgJkcCpYGmuHSsWEdonHmtpGhCfKTsZv5LiQ610cDeUugLAY/X3REJ8pYa+qzt9Aj01643E/7xWDN1zO+FBFAML6GRRNxYYQjzKBXe4ZBTEUBNCJde3YtojklDQ6eV0CNbsy/OkXi5ZJ6XyzWmhcjGNI4v20AEqIgudoQq6QlVUQxQ9omf0it6MJ+PFeDc+Jq0ZYzqzi/7A+PwBCqyadQ==</latexit> \n<latexit sha1\\_base64="U0OBaYPKAAXD1kmKTpBIWAwwIcw=">AAACAHicbVA9SwNBEN3zM8avqIWFzWIiWIW7RNDCImJjGdF8QHIce3ubZMne3rE7J4YjjX/FxkIRW3+Gnf/GTXKFJj4YeLw3w8w8PxZcg21/W0vLK6tr67mN/ObW9s5uYW+/qaNEUdagkYhU2yeaCS5ZAzgI1o4VI6EvWMsfXk/81gNTmkfyHkYxc0PSl7zHKQEjeYXDLrBHSK+Cu5JXLWE/SmRA1GjsFYp22Z4CLxInI0WUoe4VvrpBRJOQSaCCaN1x7BjclCjgVLBxvptoFhM6JH3WMVSSkGk3nT4wxidGCXAvUqYk4Kn6eyIlodaj0DedIYGBnvcm4n9eJ4HehZtyGSfAJJ0t6iUCQ4QnaeCAK0ZBjAwhVHFzK6YDoggFk1nehODMv7xImpWyUy1Xbs+Ktcssjhw6QsfoFDnoHNXQDaqjBqJojJ7RK3qznqwX6936mLUuWdnMAfoD6/MH+ieV/g==</latexit> \nFigure 1: On the left, we have a 2d dilaton-gravity theory coupled to a matter CFT 2 . The fields of the matter CFT 2 are denoted collectively by χ , and this CFT 2 is assumed to be holographic. On the right, we display a 3d geometry obtained by replacing the matter CFT 2 with its 3d dual. This is a version of the Randall-Sundrum setup [22,23]. On the 2d boundary of this 3d geometry, we have the dilaton-gravity action. The boundary fields φ and g (2) ij are also integrated over in the functional integral. \n<!-- image --> \nwhere { f ( y ) , y } = f \'\'\' f \' -3 2 ( f \'\' f \' ) 2 is the usual Schwarzian derivative. 3 The w ± coordinates have the property that the stress tensor vanishes in the corresponding flat metric, obtained after a Weyl transformation from (4): \nds 2 = -dw + dw -, T w + w + = T w -w -= 0 . (6) \nWe therefore observe that the solution (4) is related to the vacuum solution on flat space (6) by a combination of Weyl and coordinate transformations. \nThis determines the location of the Planck brane in the w ± coordinates in following way [25]. The vacuum of the holographic CFT 2 has pure AdS 3 as its associated bulk dual \nds 2 = -dw + dw -+ dz 2 w z 2 w . (7) \nThe stress tensor components T w + w + and T w -w -vanish for a suface of constant z w . Therefore, we expect that the geometry near the Planck brane looks similar to (7). The condition we need to impose is that the induced metric on the brane is fixed by (4) and (3), which gives \n-dw + dw -z 2 w = -1 glyph[epsilon1] 2 e 2 ρ ( y ) dy + dy -. (8) \nThis locates the Planck brane at \nz w = glyph[epsilon1] e -ρ ( y ) √ dw + dy + dw -dy -. (9) \nAfter obtaining this we can check that the usual formula for the stress tensor in terms of the extrinsic curvature [21] gives the one we started in (4). These formulas show that once we know the two-dimensional dynamics, given by (4), we can easily find the embedding of the 2d geometry into the 3d one. \n<latexit sha1\\_base64="+BvkGntS0QbvYt/BCMhNxRLC070=">AAAB83icbVBNT8JAEJ3iF+IX6tFLIzHxRFow0SPqxSNGQRLakO12gQ3bbbM7NZKGv+HFg8Z49c9489+4QA8KvmSSl/dmMjMvSATX6DjfVmFldW19o7hZ2tre2d0r7x+0dZwqylo0FrHqBEQzwSVrIUfBOoliJAoEewhG11P/4ZEpzWN5j+OE+REZSN7nlKCRPA/ZE2aX4d2kV++VK07VmcFeJm5OKpCj2St/eWFM04hJpIJo3XWdBP2MKORUsEnJSzVLCB2RAesaKknEtJ/Nbp7YJ0YJ7X6sTEm0Z+rviYxEWo+jwHRGBId60ZuK/3ndFPsXfsZlkiKTdL6onwobY3sagB1yxSiKsSGEKm5utemQKELRxFQyIbiLLy+Tdq3q1qu127NK4yqPowhHcAyn4MI5NOAGmtACCgk8wyu8Wan1Yr1bH/PWgpXPHMIfWJ8/6mORmQ==</latexit> \n<latexit 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sha1\\_base64="Wqi3e7MNuxI8JekpeglN8Ow79AU=">AAAB9XicbVDLSgNBEOz1GeMr6tHLYCJ4CrtRMMdAQDxGyAuSNcxOJsmQ2QczvWpY8h9ePCji1X/x5t84SfagiQUNRVU33V1eJIVG2/621tY3Nre2MzvZ3b39g8Pc0XFTh7FivMFCGaq2RzWXIuANFCh5O1Kc+p7kLW9cnfmtB660CIM6TiLu+nQYiIFgFI1030X+hEn1pl7olQrTXi5vF+05yCpxUpKHFLVe7qvbD1ns8wCZpFp3HDtCN6EKBZN8mu3GmkeUjemQdwwNqM+1m8yvnpJzo/TJIFSmAiRz9fdEQn2tJ75nOn2KI73szcT/vE6Mg7KbiCCKkQdssWgQS4IhmUVA+kJxhnJiCGVKmFsJG1FFGZqgsiYEZ/nlVdIsFZ3LYunuKl8pp3Fk4BTO4AIcuIYK3EINGsBAwTO8wpv1aL1Y79bHonXNSmdO4A+szx97eZHP</latexit> \nFigure 2: We sketch three different pictures of the same system. The first is a 2d dilaton-gravity theory, plus a matter CFT 2 , coupled to a bath consisting of the same CFT 2 . This CFT 2 is assumed to have a holographic dual. The second is 3d gravity, where we replace the CFT 2 by its holographic dual. It contains a dynamical boundary metric on the Planck brane. More details about the state of the CFT are encoded deeper inside the 3d geometry. The third is the fully quantum mechanical description, where we replace the 2d gravity+matter theory by its quantum mechanical dual. This quantum mechanical system lives at the boundary of the bath CFT. In all cases, the thick dot represents the point σ y = 0 . \n<!-- image --> \nThe computation of the RT/HRT surfaces is particularly simple in the ( w + , w -, z w ) coordinates. We should emphasize that more details about the state of the conformal field theory are encoded deeper into the three-dimensional geometry. In other words, the RT/HRT surfaces used to compute the various entropies live in a geometry that is not necessarily the same as (7) deeper in the interior, and their areas can depend on the detailed geometry that we encounter in the interior.', '2.1 Two-dimensional black hole coupled to a bath': "Consider a black hole in the two-dimensional theory (2), which we allow to evaporate into an external 'bath'. For simplicity, we take the bath to be the same CFT 2 as the matter sector in (2), but now living on a rigid flat space; see figure 2. We can think of this setup as a toy model for the case where the dilaton becomes extremely large in some region of the geometry, so that we can neglect backreaction effects and think of the matter as living on a fixed, non-dynamical background. \nWe will mostly be interested in the case where the 2d gravitational theory has AdS 2 asymptotics. Prior to coupling in the bath, the matter CFT on this AdS 2 spacetime is defined with a conformal boundary condition at the asymptotic AdS 2 boundary. Prior to the coupling, the bath CFT is also defined on the half line. Coupling the two systems amounts to joining them at their boundaries, allowing them to freely exchange stress energy. Defining σ y = ( y + -y -) / 2, positive and negative values of σ y correspond to points in the bath and the AdS 2 systems, respectively. \nThis combined system has three alternative descriptions that are useful, see figure 2. \n2d-Gravity: A two-dimensional gravity-plus-matter theory living on σ y < 0 coupled to a twodimensional field theory living on σ y > 0. \n3d-Gravity: A three-dimensional gravity theory in AdS 3 with a dynamical boundary (Planck brane) on part of the space ( σ y < 0), and with a rigid boundary on the rest ( σ y > 0). \nQM: A two-dimensional CFT on the half-line σ y > 0 with some non-conformal boundary degrees of freedom at σ y = 0. \nThe first description is the one we have already described in detail. In the second description, which involves three-dimensional gravity, we replace the CFT 2 by its three-dimensional dual. This 3d dual has a Planck brane with dynamical gravity on part of the space ( σ y < 0) and the usual UV boundary on the rest of the space ( σ y > 0). \nIn the third, fully quantum-mechanical description with no gravity, we replace the 2d black hole by its quantum-mechanical dual (assuming that it has one). Then we have a CFT 2 living on a half-line coupled to a quantum mechanical system living at σ y = 0. In other words, in the case where the 2d gravity has AdS 2 asymptotics, we want to imagine that the entire 2d theory (2) arises as the holographic dual of a (0+1)-dimensional (nonconformal) quantum-mechanical system. After coupling the nearly AdS 2 gravity theory to the bath CFT, we get a CFT on the half-line coupled to a holographic quantum-mechanical system on its boundary, as shown in figure 2. A higher dimensional version of this set up and its gravity interpretation was discussed in [26]. \nThe story so far has been for an evaporating black hole in a general 2d gravity theory with AdS 2 asymptotics, coupled to a non-gravitational bath. This formalism can be directly applied to the case considered in [9] by specializing to Jackiw-Teitelboim gravity, with the only difference being that we consider matter composed of a holographic CFT 2 , rather than general matter. Using their solution for the dynamics of the 2d model, we can follow the simple steps outlined above to find the embedding of the Planck brane and the bath UV brane in the 3d geometry.", '2.2 Quantum extremal surfaces become ordinary RT/HRT surfaces': "In the two-dimensional gravity theory, we can compute the fine-grained entropy of its quantummechanical boundary theory using the prescription of extremizing the generalized entropy [7]. This involves first constructing a quantity similar to the generalized entropy \nS gen ( y ) = φ ( y ) 4 G (2) N + S Bulk-2 d [ I y ] . (10) \nHere y is a point in the two-dimensional bulk and I y is an interval from the point y to the boundary of the two-dimensional space (or to some region far away where the dilaton is very large and the theory is very weakly coupled). The quantity S Bulk-2 d [ I y ] is the bulk von Neumann entropy of this interval. This bulk entropy includes the entropy coming from the bulk matter fields χ , and also the entropy due to quantum fluctuations of φ and g (2) ij . Note also that φ ( y ) = Area (2) ; in two dimensions, the area of a point is the coefficient of the curvature term in (1). Figure 3(a) shows a slice of the 2d theory including its boundary dual system and indicates where S gen is evaluated. \nOnce we construct S gen ( y ) as in (10), we are instructed to extremize it over the choice of the point y . And finally, we take the minimum over all such extrema. The point ( y + e , y -e ) that results from this is called the quantum extremal 'surface.' \nThe contribution of the CFT 2 fields χ to S Bulk-2 d [ I y ] dominates over contributions from the quantum fluctuations of φ and g (2) ij , since we are assuming that the CFT 2 has a large number of degrees of freedom. Furthermore, since the CFT 2 has a holographic dual, the contribution of the χ fields to the entropy can be computed to leading order using the RT/HRT formula [5, 6]. This involves finding a minimal or an extremal surface Σ y in the three-dimensional geometry. \nThe extremal surface Σ y is here just an interval and is shown in blue in figure 3(b). We therefore \nFigure 3: (a) We have shown, from the 2d perspective, the two contributions φ ( y ) 4 G (2) N and S Bulk-2d [ I y ] to S gen ( y ) from equation (10). (b) Since the matter CFT 2 is holographic, the quantity S Bulk -2 d [ I y ] can be computed using a 3d RT formula. \n<!-- image --> \nhave \nS gen ( y ) = φ ( y ) 4 G (2) N + S Bulk-2 d [ I y ] ≈ φ ( y ) 4 G (2) N + Area (3) [Σ y ] 4 G (3) N . (11) \nWe used a ≈ sign because we are neglecting the contributions from the quantum fluctuations of φ and g (2) ij , and also dropping the subleading 3d bulk entanglement entropy terms. \nThe extremization of generalized entropy in 2d is equivalent to the standard RT/HRT area extremization in the 3d with a dynamical brane. In other words, we look for an area-extremizing surface in 3d with an endpoint on the Planck brane. This 'area' has a contribution coming from the length of the line Σ y as well as a contribution from the dilaton φ at the Planck brane, see figure 3. We are instructed to extremize the whole thing, which involves also the position of the point ( y + , y -) on the Planck brane. Of course, this observation lends further support to the notion that quantum extremal surfaces are computing von Neuman entropies [7], since in this setup it reduces to the simpler RT/HRT proposal in three dimensions. This discussion generalizes naturally to higher dimensions.", '3 Entanglement wedges for evaporating black holes and Hawking radiation': "In this section we review the quantum extremal surfaces found in [9] for black holes in JT gravity that evaporate into a non-gravitational bath, and present their corresponding three-dimensional picture. Our discussion will be out of time order, we will first discuss the late time picture, past the so called 'Page time' and then discuss the picture for early times. We do this because the late time picture is less dependent on the detailed formation history of the black hole. It is also the more surprising one, because the entanglement wedge of the radiation will be found to contain the region inside the black hole. In two dimensions, this region is manifestly disconnected from the radiation, but we show that it is connected to the radiation through the third dimension.", '3.1.1 Entanglement wedge of the black hole at late times': 'We start by recalling the position of the quantum extremal surface of the black hole at late times found in [8,9]. The idea is to consider a subsystem that includes the black hole, or more precisely, the whole gravity region. From the point of view of the QM description, we imagine that we take a small interval [0 , σ 0 ] where σ 0 is very small, but it includes the quantum-mechanical degrees of freedom at the boundary of the CFT. We do this at some late time t = ( y + + y -) / 2. See figure 4. \nFigure 4: The entanglement wedge for the black hole at late times. We show a spatial slice Σ Late at some late time that passes through the quantum extremal surface, (see also figure 5). In the leftmost picture, we have drawn Σ Late in the 2d geometry. The middle picture is a spatial slice of the three dimensional geometry that ends on Σ Late and contains the RT/HRT surface, the pink region being the entanglement wedge. In the rightmost picture, we have an interval that contains the left boundary and whose entropy we are trying to compute. \n<!-- image --> \n<latexit sha1\\_base64="DXcV7Ft6cd75OCMFKoj+hApzYzc=">AAAB+nicbVBNS8NAEN3Ur1q/Uj16CRbBiyWpgh6LHvRYwX5AG8pms2mXbjZhd1INsT/FiwdFvPpLvPlv3LY5aOuDgcd7M8zM82LOFNj2t1FYWV1b3yhulra2d3b3zPJ+S0WJJLRJIh7JjocV5UzQJjDgtBNLikOP07Y3up767TGVikXiHtKYuiEeCBYwgkFLfbPcA/oIWc0/vZF4zCCd9M2KXbVnsJaJk5MKytHom189PyJJSAUQjpXqOnYMboYlMMLppNRLFI0xGeEB7WoqcEiVm81On1jHWvGtIJK6BFgz9fdEhkOl0tDTnSGGoVr0puJ/XjeB4NLNmIgToILMFwUJtyCypjlYPpOUAE81wUQyfatFhlhiAjqtkg7BWXx5mbRqVeesWrs7r9Sv8jiK6BAdoRPkoAtUR7eogZqIoAf0jF7Rm/FkvBjvxse8tWDkMwfoD4zPH2xOlBs=</latexit> \n<latexit 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sha1\\_base64="Wqi3e7MNuxI8JekpeglN8Ow79AU=">AAAB9XicbVDLSgNBEOz1GeMr6tHLYCJ4CrtRMMdAQDxGyAuSNcxOJsmQ2QczvWpY8h9ePCji1X/x5t84SfagiQUNRVU33V1eJIVG2/621tY3Nre2MzvZ3b39g8Pc0XFTh7FivMFCGaq2RzWXIuANFCh5O1Kc+p7kLW9cnfmtB660CIM6TiLu+nQYiIFgFI1030X+hEn1pl7olQrTXi5vF+05yCpxUpKHFLVe7qvbD1ns8wCZpFp3HDtCN6EKBZN8mu3GmkeUjemQdwwNqM+1m8yvnpJzo/TJIFSmAiRz9fdEQn2tJ75nOn2KI73szcT/vE6Mg7KbiCCKkQdssWgQS4IhmUVA+kJxhnJiCGVKmFsJG1FFGZqgsiYEZ/nlVdIsFZ3LYunuKl8pp3Fk4BTO4AIcuIYK3EINGsBAwTO8wpv1aL1Y79bHonXNSmdO4A+szx97eZHP</latexit> \nThe final conclusion of [8,9] is that the quantum extremal surface is at a point ( y + e , y -e ) that lies behind the horizon and is such that a past directed light ray from it would reach the AdS 2 boundary at about a scrambling time earlier than the time t at which we are computing the entropy, \ny + e = t -1 2 πT ( t ) log S Bek ( T ( t )) -S 0 c + . . . . (12) \nSee figure 5. Here T ( t ) is the temperature of the black hole at time t , S 0 is the extremal entropy (which is assumed to be small), and S Bek ( T ) is the Bekenstein-Hawking entropy for a black hole of temperature T . The entanglement wedge is just the causal domain of a spacelike slice going from ( t, σ 0 ) to ( y + e , y -e ). This implies that the computation of the bulk entanglement entropy is just that of an interval. The answer is slightly nontrivial because the stress tensor is nonzero once we take into account the effects of Hawking radiation, 4 and hence the map w ± ( y ± ) determined from (5) is nontrivial. In the w ± coordinates, we are just considering an interval in ordinary flat space, and the nontrivial part of the entropy comes from the dependence of the cutoff z w on the length of the interval (9). \nAs shown in [8,9], this leads to an entropy for the black hole of the from \nS Black Hole ( t ) = S Bek ( T ( t )) + logs , (13) \n<latexit sha1\\_base64="3gOzB1SL6yFgCnFxkhUqpEVtVCk=">AAAB9HicbVBNS8NAEJ34WetX1aOXYBE8laQKCl4KXvRWwX5AG8pms2mXbjZxd1Isob/DiwdFvPpjvPlv3LY5aOuDgcd7M8zM8xPBNTrOt7Wyura+sVnYKm7v7O7tlw4OmzpOFWUNGotYtX2imeCSNZCjYO1EMRL5grX84c3Ub42Y0jyWDzhOmBeRvuQhpwSN5HWRPWF2pwWRwaRXKjsVZwZ7mbg5KUOOeq/01Q1imkZMIhVE647rJOhlRCGngk2K3VSzhNAh6bOOoZJETHvZ7OiJfWqUwA5jZUqiPVN/T2Qk0noc+aYzIjjQi95U/M/rpBheeRmXSYpM0vmiMBU2xvY0ATvgilEUY0MIVdzcatMBUYSiyaloQnAXX14mzWrFPa9U7y/Ktes8jgIcwwmcgQuXUINbqEMDKDzCM7zCmzWyXqx362PeumLlM0fwB9bnD0+hkm0=</latexit> \n<latexit sha1\\_base64="9xemsbOBlHUAEMvgfjc6uYKtW1g=">AAAB8HicbVDLSgNBEJyNrxhfUY9eBoPgKexGQcFLQASPEcxDkiXMTjrJkNnZZaZXDEu+wosHRbz6Od78GyfJHjSxoKGo6qa7K4ilMOi6305uZXVtfSO/Wdja3tndK+4fNEyUaA51HslItwJmQAoFdRQooRVrYGEgoRmMrqd+8xG0EZG6x3EMfsgGSvQFZ2ilhw7CE6Y3zUm3WHLL7gx0mXgZKZEMtW7xq9OLeBKCQi6ZMW3PjdFPmUbBJUwKncRAzPiIDaBtqWIhGD+dHTyhJ1bp0X6kbSmkM/X3RMpCY8ZhYDtDhkOz6E3F/7x2gv1LPxUqThAUny/qJ5JiRKff057QwFGOLWFcC3sr5UOmGUebUcGG4C2+vEwalbJ3Vq7cnZeqV1kceXJEjskp8cgFqZJbUiN1wklInskreXO08+K8Ox/z1pyTzRySP3A+fwAEhZCG</latexit> \n<latexit sha1\\_base64="DXcV7Ft6cd75OCMFKoj+hApzYzc=">AAAB+nicbVBNS8NAEN3Ur1q/Uj16CRbBiyWpgh6LHvRYwX5AG8pms2mXbjZhd1INsT/FiwdFvPpLvPlv3LY5aOuDgcd7M8zM82LOFNj2t1FYWV1b3yhulra2d3b3zPJ+S0WJJLRJIh7JjocV5UzQJjDgtBNLikOP07Y3up767TGVikXiHtKYuiEeCBYwgkFLfbPcA/oIWc0/vZF4zCCd9M2KXbVnsJaJk5MKytHom189PyJJSAUQjpXqOnYMboYlMMLppNRLFI0xGeEB7WoqcEiVm81On1jHWvGtIJK6BFgz9fdEhkOl0tDTnSGGoVr0puJ/XjeB4NLNmIgToILMFwUJtyCypjlYPpOUAE81wUQyfatFhlhiAjqtkg7BWXx5mbRqVeesWrs7r9Sv8jiK6BAdoRPkoAtUR7eogZqIoAf0jF7Rm/FkvBjvxse8tWDkMwfoD4zPH2xOlBs=</latexit> \n<latexit sha1\\_base64="lkiJUjd/niOd9Y2z3PItjaUDGa0=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoCcpePFY0X5AG8pmO2mXbjZhdyOE0p/gxYMiXv1F3vw3btsctPXBwOO9GWbmBYng2rjut1NYW9/Y3Cpul3Z29/YPyodHLR2nimGTxSJWnYBqFFxi03AjsJMopFEgsB2Mb2d++wmV5rF8NFmCfkSHkoecUWOlh6yP/XLFrbpzkFXi5aQCORr98ldvELM0QmmYoFp3PTcx/oQqw5nAaamXakwoG9Mhdi2VNELtT+anTsmZVQYkjJUtachc/T0xoZHWWRTYzoiakV72ZuJ/Xjc14bU/4TJJDUq2WBSmgpiYzP4mA66QGZFZQpni9lbCRlRRZmw6JRuCt/zyKmnVqt5FtXZ/Wanf5HEU4QRO4Rw8uII63EEDmsBgCM/wCm+OcF6cd+dj0Vpw8plj+APn8wdcII3V</latexit> \n<latexit sha1\\_base64="7FJzzRD0Vgnl3PBiCzraOo1za08=">AAAB73icbVBNSwMxEJ3Ur1q/qh69BIvgqexWQU9S8OKxgv2AdinZNNuGJtk1yQpl6Z/w4kERr/4db/4b03YP2vpg4PHeDDPzwkRwYz3vGxXW1jc2t4rbpZ3dvf2D8uFRy8SppqxJYxHrTkgME1yxpuVWsE6iGZGhYO1wfDvz209MGx6rBztJWCDJUPGIU2Kd1OkZPpSk7/XLFa/qzYFXiZ+TCuRo9MtfvUFMU8mUpYIY0/W9xAYZ0ZZTwaalXmpYQuiYDFnXUUUkM0E2v3eKz5wywFGsXSmL5+rviYxIYyYydJ2S2JFZ9mbif143tdF1kHGVpJYpulgUpQLbGM+exwOuGbVi4gihmrtbMR0RTah1EZVcCP7yy6ukVav6F9Xa/WWlfpPHUYQTOIVz8OEK6nAHDWgCBQHP8Apv6BG9oHf0sWgtoHzmGP4Aff4AxFuPxg==</latexit> \n<latexit sha1\\_base64="9xemsbOBlHUAEMvgfjc6uYKtW1g=">AAAB8HicbVDLSgNBEJyNrxhfUY9eBoPgKexGQcFLQASPEcxDkiXMTjrJkNnZZaZXDEu+wosHRbz6Od78GyfJHjSxoKGo6qa7K4ilMOi6305uZXVtfSO/Wdja3tndK+4fNEyUaA51HslItwJmQAoFdRQooRVrYGEgoRmMrqd+8xG0EZG6x3EMfsgGSvQFZ2ilhw7CE6Y3zUm3WHLL7gx0mXgZKZEMtW7xq9OLeBKCQi6ZMW3PjdFPmUbBJUwKncRAzPiIDaBtqWIhGD+dHTyhJ1bp0X6kbSmkM/X3RMpCY8ZhYDtDhkOz6E3F/7x2gv1LPxUqThAUny/qJ5JiRKff057QwFGOLWFcC3sr5UOmGUebUcGG4C2+vEwalbJ3Vq7cnZeqV1kceXJEjskp8cgFqZJbUiN1wklInskreXO08+K8Ox/z1pyTzRySP3A+fwAEhZCG</latexit> \nFigure 5: The spacetime diagram describing the coupling of the black hole to the bath, the energy pulse coming from the moment they are coupled, the formation of the black hole and its subsequent evaporation. We pick some late time nice slice Σ Late and we compute the entanglement wedge for what is to the left of σ 0 . This contains only a portion of the time slice in the interior. We have also displayed the Wheeler de Witt patch, or causal domain of dependence that describes the full entanglement wedge. \n<!-- image --> \nwhere T ( t ) is the approximate black hole temperature at time t . The \'logs\' denote terms that are logarithmic in the black hole entropy of the initial state. Note that this entropy is decreasing with time because the temperature is decreasing.', '3.1.2 Entanglement wedge of the radiation at late times': 'Now we consider the entropy of the radiation. More precisely, we compute the entropy of the state in the bath CFT outside some point σ 0 , the complement of the interval that we considered for the black hole in the previous subsection, see figure 6 right. \nFigure 6: The analog of figure 4 for the late time entanglement wedge of the radiation . The rightmost picture depicts the interval whose entropy is being considered. The leftmost picture depicts Σ Late in the 2d gravity picture. There is an entanglement island, disconnected from the region where the radiation lives. 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sha1\\_base64="dxTK8f4jLn9e/KfneO9jxQLO3hw=">AAAB+XicbVBNS8NAEN3Ur1q/oh69LBbBU0mqoOCl4MVjFfsBbSibzaZdutmE3UmxhP4TLx4U8eo/8ea/cZvmoK0PBh7vzTAzz08E1+A431ZpbX1jc6u8XdnZ3ds/sA+P2jpOFWUtGotYdX2imeCStYCDYN1EMRL5gnX88e3c70yY0jyWjzBNmBeRoeQhpwSMNLDtPrAnyB5IwHNlNrCrTs3JgVeJW5AqKtAc2F/9IKZpxCRQQbTuuU4CXkYUcCrYrNJPNUsIHZMh6xkqScS0l+WXz/CZUQIcxsqUBJyrvycyEmk9jXzTGREY6WVvLv7n9VIIr72MyyQFJuliUZgKDDGex4ADrhgFMTWEUMXNrZiOiCIUTFgVE4K7/PIqaddr7kWtfn9ZbdwUcZTRCTpF58hFV6iB7lATtRBFE/SMXtGblVkv1rv1sWgtWcXMMfoD6/MHLMCT/A==</latexit> \n<latexit sha1\\_base64="7FJzzRD0Vgnl3PBiCzraOo1za08=">AAAB73icbVBNSwMxEJ3Ur1q/qh69BIvgqexWQU9S8OKxgv2AdinZNNuGJtk1yQpl6Z/w4kERr/4db/4b03YP2vpg4PHeDDPzwkRwYz3vGxXW1jc2t4rbpZ3dvf2D8uFRy8SppqxJYxHrTkgME1yxpuVWsE6iGZGhYO1wfDvz209MGx6rBztJWCDJUPGIU2Kd1OkZPpSk7/XLFa/qzYFXiZ+TCuRo9MtfvUFMU8mUpYIY0/W9xAYZ0ZZTwaalXmpYQuiYDFnXUUUkM0E2v3eKz5wywFGsXSmL5+rviYxIYyYydJ2S2JFZ9mbif143tdF1kHGVpJYpulgUpQLbGM+exwOuGbVi4gihmrtbMR0RTah1EZVcCP7yy6ukVav6F9Xa/WWlfpPHUYQTOIVz8OEK6nAHDWgCBQHP8Apv6BG9oHf0sWgtoHzmGP4Aff4AxFuPxg==</latexit> \n<latexit sha1\\_base64="HyCMH0FSarNArmpEPpSpe79/5Sw=">AAAB8HicbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxYuQgHlIsoTZySQZMju7zPSKYclXePGgiFc/x5t/4yTZgyYWNBRV3XR3BbEUBl3321lZXVvf2Mxt5bd3dvf2CweHDRMlmvE6i2SkWwE1XArF6yhQ8lasOQ0DyZvB6GbqNx+5NiJS9ziOuR/SgRJ9wSha6aGD/AnT2t2kWyi6JXcGsky8jBQhQ7Vb+Or0IpaEXCGT1Ji258bop1SjYJJP8p3E8JiyER3wtqWKhtz46ezgCTm1So/0I21LIZmpvydSGhozDgPbGVIcmkVvKv7ntRPsX/mpUHGCXLH5on4iCUZk+j3pCc0ZyrEllGlhbyVsSDVlaDPK2xC8xZeXSaNc8s5L5dpFsXKdxZGDYziBM/DgEipwC1WoA4MQnuEV3hztvDjvzse8dcXJZo7gD5zPHwm2kI8=</latexit> \nAt first sight this seems straightforward to compute. We start from the entire slice and simply trace out everything that is outside the region under consideration. This would reproduce the \nFigure 7: Spacetime diagram for the formation and evaporation of the black hole. We consider the entanglement wedges of the radiation. The \'island\' is the left wedge, which is disconnected from the right one. \n<!-- image --> \ncomputation in [2] which results in an entropy that continues to grow past the Page time. On the other hand, since the entanglement wedge of the black hole covers only a portion of the interior, it is tempting to think that the rest of the interior should belong to to the entanglement wedge of the radiation [8,9,13]. This seems surprising from the two-dimensional point of view where the interior is disconnected from the outside, forming a disconnected \'island\', see figure 6 left. \nWhen we think about this problem from the three-dimensional point of view we find that the region in the interior is actually connected to the exterior, see figure 6 middle. If we compute the entropy in the outside CFT by using the standard RT/HRT formula, which instructs us to find the minimal extremal surface, candidate surfaces can end on the Planck brane. In fact, the extremal surface is essentially the same one as the one we found for the black hole, up to possible IR contributions depending on the precise initial state which not important conceptually, and will be discussed later. \nThis connection through the extra dimensions can be viewed as a realization of the ER=EPR idea that the radiation would be connected to the interior of the black hole. 5 The extra dimension provides a \'bridge\' that connects the \'island\' to the \'mainland\', the mainland being the CFT interval whose entropy we are computing. \nThe crucial point is that the interior region, which in the two-dimensional picture is disconnected from the outside radiation, is actually connected through the extra dimension. We expect that this should be a feature of any situation where we have a black hole coupled to holographic matter, even in higher dimensions, d > 2. 6 \nThis was already suggested as the right prescription on the basis of unitarity in [8, 13]. The argument we gave using the holographic example allows us to derive this fact from the usual rules of RT/HRT surfaces and entanglement wedges. Of course this derivation relies on the assumed correctness of the RT/HRT formula for computing von Neumann entropies.', '3.2 Early time entanglement wedges': 'We consider the setup in [9] that starts from a low temperature black hole, initially decoupled from the bath. The coupling between the black hole and the bath is turned on at t = 0. In order to describe the three-dimensional geometry, we will need some preliminaries. Many of these points are not essential for the main point of our paper, so the reader who finds them confusing can jump to the next subsection.', '3.2.1 The decoupled state': 'Suppose that we have a two-dimensional conformal field theory with a simple, Cardy conformal boundary condition [31]. This type of boundary does not carry any energy, so that T ++ = T --at the boundary. The holographic dual of such a boundary condition corresponds to a brane in AdS 3 with extrinsic curvature proportional to the metric. The metric will be (7) with boundary at σ w = ( w + -w -) / 2 = 0. We will call such a brane a \'Cardy\' brane. For simplicity, we consider a brane that goes straight down at σ w = 0 in the bulk, see figure 8. This brane contains no dilaton or any Ricci curvature term on its surface. 7 \nConsider an initial state where the bath and black hole are decoupled. This means that the CFT will have a boundary condition, which is taken to be the conformal boundary condition described above. The holographic dual consists of two disconnected geometries, see figure 8. \nFigure 8: The decoupled black hole (on the left) and the bath CFT (on the right). We have shown a constant time slice in the 3d geometry. The thick vertical lines are the \'Cardy\' branes, which are the holographic dual of conformal boundary conditions. \n<!-- image --> \n<latexit sha1\\_base64="LyaGTLeRZqI6FIWmYsPJhZDWBpU=">AAAB/HicbVDLSgNBEJyNrxhf0Ry9DAbBU9iNgoKXgBePEcwDkiXMTjrJkNnZZaZXDEv8FS8eFPHqh3jzb5wke9DEgoaiqpvuriCWwqDrfju5tfWNza38dmFnd2//oHh41DRRojk0eCQj3Q6YASkUNFCghHasgYWBhFYwvpn5rQfQRkTqHicx+CEbKjEQnKGVesVSF+ER07pkio9poJmCaa9YdivuHHSVeBkpkwz1XvGr2494EoJCLpkxHc+N0U+ZRsElTAvdxEDM+JgNoWOpYiEYP50fP6WnVunTQaRtKaRz9fdEykJjJmFgO0OGI7PszcT/vE6Cgys/FSpOEBRfLBokkmJEZ0nQvtDAUU4sYVwLeyvlI6YZR5tXwYbgLb+8SprVindeqd5dlGvXWRx5ckxOyBnxyCWpkVtSJw3CyYQ8k1fy5jw5L86787FozTnZTIn8gfP5AwxClQA=</latexit> \n<latexit sha1\\_base64="GMvGpDNJpn1dCV5yyO37eSkqTxQ=">AAAB+nicbVDLSgMxFM34rPU11aWbYBFclZkqKLgpuumygn1AO5RMmrahmWRI7qhl7Ke4caGIW7/EnX9j2s5CWw8EDufcV04YC27A876dldW19Y3N3FZ+e2d3b98tHDSMSjRldaqE0q2QGCa4ZHXgIFgr1oxEoWDNcHQz9Zv3TBuu5B2MYxZEZCB5n1MCVuq6hQ6wR0ivBaEjXFWCTbpu0St5M+Bl4mekiDLUuu5Xp6doEjEJVBBj2r4XQ5ASDZzaeflOYlhsx5MBa1sqScRMkM5On+ATq/RwX2n7JOCZ+rsjJZEx4yi0lRGBoVn0puJ/XjuB/mWQchknwCSdL+onAoPC0xxwj2tGQYwtIVRzeyumQ6IJBZtW3obgL355mTTKJf+sVL49L1ausjhy6Agdo1PkowtUQVVUQ3VE0QN6Rq/ozXlyXpx352NeuuJkPYfoD5zPHzS2k/A=</latexit> \n<latexit sha1\\_base64="OwBtsfyj2d2WyMBtpndIigX0izY=">AAAB/HicbVDLSgNBEJz1GeNrNUcvg0HwFHajoOAlkIvHCOYBSQizk04yZHZ2mekVlyX+ihcPinj1Q7z5N04eB00saCiquunuCmIpDHret7O2vrG5tZ3bye/u7R8cukfHDRMlmkOdRzLSrYAZkEJBHQVKaMUaWBhIaAbj6tRvPoA2IlL3mMbQDdlQiYHgDK3UcwsdhEfMqkz3UxpopsBMem7RK3kz0FXiL0iRLFDruV+dfsSTEBRyyYxp+16M3YxpFFzCJN9JDMSMj9kQ2pYqFoLpZrPjJ/TMKn06iLQthXSm/p7IWGhMGga2M2Q4MsveVPzPayc4uO5mQsUJguLzRYNEUozoNAnaFxo4ytQSxrWwt1I+YppxtHnlbQj+8surpFEu+Rel8t1lsXKziCNHTsgpOSc+uSIVcktqpE44SckzeSVvzpPz4rw7H/PWNWcxUyB/4Hz+AB+PlQ0=</latexit> \n<latexit sha1\\_base64="Kz+YroUHTUeesP/6mnKWBRgT+6g=">AAAB83icbVBNSwMxEJ2tX7V+VT16CRbBU8mKoBeh4MVjBfsB3aVk02wbmmSXJKuUpX/DiwdFvPpnvPlvTNs9aOuDgcd7M8zMi1LBjcX42yutrW9sbpW3Kzu7e/sH1cOjtkkyTVmLJiLR3YgYJrhiLcutYN1UMyIjwTrR+Hbmdx6ZNjxRD3aSslCSoeIxp8Q6KQgMH0rSf0I3CPerNVzHc6BV4hekBgWa/epXMEhoJpmyVBBjej5ObZgTbTkVbFoJMsNSQsdkyHqOKiKZCfP5zVN05pQBihPtSlk0V39P5EQaM5GR65TEjsyyNxP/83qZja/DnKs0s0zRxaI4E8gmaBYAGnDNqBUTRwjV3N2K6IhoQq2LqeJC8JdfXiXti7qP6/79Za2BizjKcAKncA4+XEED7qAJLaCQwjO8wpuXeS/eu/exaC15xcwx/IH3+QPSn5DQ</latexit> \n<latexit sha1\\_base64="Kz+YroUHTUeesP/6mnKWBRgT+6g=">AAAB83icbVBNSwMxEJ2tX7V+VT16CRbBU8mKoBeh4MVjBfsB3aVk02wbmmSXJKuUpX/DiwdFvPpnvPlvTNs9aOuDgcd7M8zMi1LBjcX42yutrW9sbpW3Kzu7e/sH1cOjtkkyTVmLJiLR3YgYJrhiLcutYN1UMyIjwTrR+Hbmdx6ZNjxRD3aSslCSoeIxp8Q6KQgMH0rSf0I3CPerNVzHc6BV4hekBgWa/epXMEhoJpmyVBBjej5ObZgTbTkVbFoJMsNSQsdkyHqOKiKZCfP5zVN05pQBihPtSlk0V39P5EQaM5GR65TEjsyyNxP/83qZja/DnKs0s0zRxaI4E8gmaBYAGnDNqBUTRwjV3N2K6IhoQq2LqeJC8JdfXiXti7qP6/79Za2BizjKcAKncA4+XEED7qAJLaCQwjO8wpuXeS/eu/exaC15xcwx/IH3+QPSn5DQ</latexit> \n<latexit sha1\\_base64="Kz+YroUHTUeesP/6mnKWBRgT+6g=">AAAB83icbVBNSwMxEJ2tX7V+VT16CRbBU8mKoBeh4MVjBfsB3aVk02wbmmSXJKuUpX/DiwdFvPpnvPlvTNs9aOuDgcd7M8zMi1LBjcX42yutrW9sbpW3Kzu7e/sH1cOjtkkyTVmLJiLR3YgYJrhiLcutYN1UMyIjwTrR+Hbmdx6ZNjxRD3aSslCSoeIxp8Q6KQgMH0rSf0I3CPerNVzHc6BV4hekBgWa/epXMEhoJpmyVBBjej5ObZgTbTkVbFoJMsNSQsdkyHqOKiKZCfP5zVN05pQBihPtSlk0V39P5EQaM5GR65TEjsyyNxP/83qZja/DnKs0s0zRxaI4E8gmaBYAGnDNqBUTRwjV3N2K6IhoQq2LqeJC8JdfXiXti7qP6/79Za2BizjKcAKncA4+XEED7qAJLaCQwjO8wpuXeS/eu/exaC15xcwx/IH3+QPSn5DQ</latexit> \n<latexit sha1\\_base64="Kz+YroUHTUeesP/6mnKWBRgT+6g=">AAAB83icbVBNSwMxEJ2tX7V+VT16CRbBU8mKoBeh4MVjBfsB3aVk02wbmmSXJKuUpX/DiwdFvPpnvPlvTNs9aOuDgcd7M8zMi1LBjcX42yutrW9sbpW3Kzu7e/sH1cOjtkkyTVmLJiLR3YgYJrhiLcutYN1UMyIjwTrR+Hbmdx6ZNjxRD3aSslCSoeIxp8Q6KQgMH0rSf0I3CPerNVzHc6BV4hekBgWa/epXMEhoJpmyVBBjej5ObZgTbTkVbFoJMsNSQsdkyHqOKiKZCfP5zVN05pQBihPtSlk0V39P5EQaM5GR65TEjsyyNxP/83qZja/DnKs0s0zRxaI4E8gmaBYAGnDNqBUTRwjV3N2K6IhoQq2LqeJC8JdfXiXti7qP6/79Za2BizjKcAKncA4+XEED7qAJLaCQwjO8wpuXeS/eu/exaC15xcwx/IH3+QPSn5DQ</latexit> \n<latexit sha1\\_base64="YK/Q7vEqvH930yeB7BJLd8zMRaM=">AAAB+HicbVDLSgMxFM3UV62Pjrp0EyyCqzJTBQU3xYK4rNAXtEPJpJk2NJMZkjtiHfolblwo4tZPceffmLaz0NYDgcM593JPjh8LrsFxvq3c2vrG5lZ+u7Czu7dftA8OWzpKFGVNGolIdXyimeCSNYGDYJ1YMRL6grX9cW3mtx+Y0jySDZjEzAvJUPKAUwJG6tvFHrBHSG8IjHDttjHt2yWn7MyBV4mbkRLKUO/bX71BRJOQSaCCaN11nRi8lCjgVLBpoZdoFhM6JkPWNVSSkGkvnQef4lOjDHAQKfMk4Ln6eyMlodaT0DeToUmol72Z+J/XTSC48lIu4wSYpItDQSIwRHjWAh5wxSiIiSGEKm6yYjoiilAwXRVMCe7yl1dJq1J2z8uV+4tS9TqrI4+O0Qk6Qy66RFV0h+qoiShK0DN6RW/Wk/VivVsfi9Gcle0coT+wPn8ATDOS0w==</latexit>', '3.2.2 Coupling the black hole and the bath CFT': 'We now consider coupling the two systems. (See [34-39] for studies of local quenches in twodimensional CFTs.) If we were to suddenly couple them at t = 0, we would get an infinite pulse of energy. Instead we imagine that we couple them over some time ∆ t . The state that we get after this will have a pulse of energy of the order E ∝ c ∆ t . An approximate form for the state can be \nobtained by considering the Euclidean problem, joining it suddenly and then evolving in Euclidean time for an amount ∆ t . \nThis leads to a Lorentzian 3d geometry such that, at t = 0, the Cardy brane is situated somewhere in the bulk, displaced away from the physical boundary towards the AdS 3 interior. Its precise shape depends on the coordinates used, but a sketch can be seen in figure 9(a). As time progresses, this brane falls deeper towards the interior of AdS 3 and becomes more distant from the physical boundary. \nFigure 9: We show the profile of the simple boundary brane in the bulk at t = 0 . At later times it falls in, in the sense that the physical distance to the boundary increases. We have also shown the black hole entanglement wedges. Where we assumed we started from a black hole at at low temperature where the initial horizon is located where we indicated. The pink region is the entanglement wedge of the black hole, while the complementary blue region is the entanglement wedge of the bath. \n<!-- image --> \n<latexit sha1\\_base64="/OImNnW21bro2sPUKthtuh1Kz4k=">AAAB+HicbVBNS8NAEN3Ur1o/WvXoZbEInkoigh4LXrxZwX5AG8pmO2mXbjZhdyLW0F/ixYMiXv0p3vw3btsctPXBwOO9GWbmBYkUBl332ymsrW9sbhW3Szu7e/vlysFhy8Sp5tDksYx1J2AGpFDQRIESOokGFgUS2sH4eua3H0AbEat7nCTgR2yoRCg4Qyv1K+UewiNmt1oMhWJy2q9U3Zo7B10lXk6qJEejX/nqDWKeRqCQS2ZM13MT9DOmUXAJ01IvNZAwPmZD6FqqWATGz+aHT+mpVQY0jLUthXSu/p7IWGTMJApsZ8RwZJa9mfif100xvPIzoZIUQfHFojCVFGM6S4EOhAaOcmIJ41rYWykfMc042qxKNgRv+eVV0jqveW7Nu7uo1t08jiI5JifkjHjkktTJDWmQJuEkJc/klbw5T86L8+58LFoLTj5zRP7A+fwBYpSTfQ==</latexit> \n<latexit 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sha1\\_base64="4ryyd/uDJf29UVtL1ucW6gsFliI=">AAAB8HicbVBNSwMxEJ2tX7V+VT16CRbBU9kVQY8FETxWsK3SLiWbpm1okl2SWbEs/RVePCji1Z/jzX9j2u5BWx8MPN6bYWZelEhh0fe/vcLK6tr6RnGztLW9s7tX3j9o2jg1jDdYLGNzH1HLpdC8gQIlv08MpyqSvBWNrqZ+65EbK2J9h+OEh4oOtOgLRtFJDx3kT5hdtybdcsWv+jOQZRLkpAI56t3yV6cXs1RxjUxSa9uBn2CYUYOCST4pdVLLE8pGdMDbjmqquA2z2cETcuKUHunHxpVGMlN/T2RUWTtWketUFId20ZuK/3ntFPuXYSZ0kiLXbL6on0qCMZl+T3rCcIZy7AhlRrhbCRtSQxm6jEouhGDx5WXSPKsGfjW4Pa/U/DyOIhzBMZxCABdQgxuoQwMYKHiGV3jzjPfivXsf89aCl88cwh94nz//2ZB3</latexit> \n<latexit sha1\\_base64="CaXfPbJVl2g25QcxjF+xCWGCmf8=">AAAB73icbVBNSwMxEJ3Ur1q/qh69BIvgqexWwR4LXjxWsB/QLiWbZtvQJLsmWaEs/RNePCji1b/jzX9j2u5BWx8MPN6bYWZemAhurOd9o8LG5tb2TnG3tLd/cHhUPj5pmzjVlLVoLGLdDYlhgivWstwK1k00IzIUrBNObud+54lpw2P1YKcJCyQZKR5xSqyTun3DR5IMvEG54lW9BfA68XNSgRzNQfmrP4xpKpmyVBBjer6X2CAj2nIq2KzUTw1LCJ2QEes5qohkJsgW987whVOGOIq1K2XxQv09kRFpzFSGrlMSOzar3lz8z+ulNqoHGVdJapmiy0VRKrCN8fx5POSaUSumjhCqubsV0zHRhFoXUcmF4K++vE7atap/Va3dX1ca9TyOIpzBOVyCDzfQgDtoQgsoCHiGV3hDj+gFvaOPZWsB5TOn8Afo8wfCjY/A</latexit> \nWe consider an initial black hole which is at a very low (but nonzero) temperature, much lower than the temperature of the black hole that results after the energy pulse falls in. The entanglement wedge goes very close to the original horizon (the horizon before the energy pulse comes in), and the entanglement wedge of the black hole contains most of the region associated to the black hole, see figure 9(a). \nAs time progresses, the topology of the figure 9(a) stays similar, but as the brane falls deeper into the bulk, its distance from the boundary increases, leading to a growing entropy, as shown in figure 9(b). This growing entropy can be physically interpreted as the building up of entanglement between the Hawking modes escaping into the bath and their partners trapped behind the event horizon. This entropy is equal to the entropy of the radiation and goes as \nS Black Hole ( t ) ∼ S Rad ( t ) = πc 6 ∫ t 0 dt \' T ( t \' ) = 2 S i Bek (1 -e -κ 2 t ) , (14) \nwhere T ( t \' ) is the temperature at time t \' . Here S i Bek is the coarse-grained Bekenstein-Hawking entropy of the black hole that forms after the pulse falls in. Denoting its temperature by T i , we find that T ( t ) ∼ T i e -κ 2 t due to black hole evaporation [40]. Here κ is proportional to c and also to the effective gravitational coupling of the 2d gravity theory, see [9] for details. \nThe important point about (14) is that it rises continuously and it saturates at twice the initial black hole entropy. In the context of figure (9) this means that we can neglect the contribution of the leftmost RT/HRT surface that ends at the original horizon. The factor of two in (14) arises because the Hawking radiation is not adiabatic and it generates coarse grained entropy [4] (see also [8]).', '3.2.3 A slightly more precise picture for the entropy of radiation': 'Having specified the initial state in more detail, we can be slightly more precise about the entropy of radiation at late times. There are a couple of new contributions. \nFirst, the initial state had some entropy, S 0 , associated to the horizon of the original lowtemperature black hole. We are imagining that S 0 is much smaller than the Bekenstein-Hawking entropy S i Bek of the black hole created by the pulse of energy. Nevertheless, this implies that, to the left of the \'island\', there is also a piece of the RT/HRT surface associated with this original horizon, see figure 10 middle. \nσ \nσ \nFigure 10: An improved version of figure 6 showing the entanglement wedge of the radiation including a contribution from the initial black hole horizon, and a contribution from the IR cutoff in the CFT. Both these contributions are smaller than S i Bek . \n<!-- image --> \n<latexit sha1\\_base64="/OImNnW21bro2sPUKthtuh1Kz4k=">AAAB+HicbVBNS8NAEN3Ur1o/WvXoZbEInkoigh4LXrxZwX5AG8pmO2mXbjZhdyLW0F/ixYMiXv0p3vw3btsctPXBwOO9GWbmBYkUBl332ymsrW9sbhW3Szu7e/vlysFhy8Sp5tDksYx1J2AGpFDQRIESOokGFgUS2sH4eua3H0AbEat7nCTgR2yoRCg4Qyv1K+UewiNmt1oMhWJy2q9U3Zo7B10lXk6qJEejX/nqDWKeRqCQS2ZM13MT9DOmUXAJ01IvNZAwPmZD6FqqWATGz+aHT+mpVQY0jLUthXSu/p7IWGTMJApsZ8RwZJa9mfif100xvPIzoZIUQfHFojCVFGM6S4EOhAaOcmIJ41rYWykfMc042qxKNgRv+eVV0jqveW7Nu7uo1t08jiI5JifkjHjkktTJDWmQJuEkJc/klbw5T86L8+58LFoLTj5zRP7A+fwBYpSTfQ==</latexit> \n<latexit 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Then there is a second surface that ends on this endpoint and goes to the \'Cardy\' brane. See figure 10. This surface gives rise to a relatively small entropy proportional to c 6 log σ IR glyph[lessmuch] S i Bek .', '3.3 The Page curve': 'The Page time [3] is defined to be the time where the increasing early-time form of the entropy (14) is equal to the decreasing late-time one (13), and the entanglement wedge undergoes a transition. The minimality condition in the RT/HRT prescription leads to the entropy of the radiation reaching a maximum and then decreasing, see figure 11. (Recall that we are assuming that S 0 glyph[lessmuch] S i Bek .) Both surfaces exist on either side of the transition, it is just that they may not be the minimal ones. \nNote that in the case that the extremal entropy S 0 is not negligible, then the minimization process could be different. For example, if we had S 0 glyph[greatermuch] S i Bek , then the entropy of the radiation will take the short time form for all times. The reason is that we need to pay a 2 S 0 price to create the \'island\' (becasue of the two endpoints of the island), and this is not favorable when S 0 is large. 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sha1\\_base64="YuC4AolqjnFCIxMWNT0jEKqgFKo=">AAAB73icbVDLSgNBEOz1GeMr6tHLYBA8hV0R9Bjw4jGCeUCyhN7JbDJkdnacmRXCkp/w4kERr/6ON//GSbIHTSxoKKq66e6KlODG+v63t7a+sbm1Xdop7+7tHxxWjo5bJs00ZU2ailR3IjRMcMmallvBOkozTCLB2tH4dua3n5g2PJUPdqJYmOBQ8phTtE7q9MaoFBLbr1T9mj8HWSVBQapQoNGvfPUGKc0SJi0VaEw38JUNc9SWU8Gm5V5mmEI6xiHrOioxYSbM5/dOyblTBiROtStpyVz9PZFjYswkiVxngnZklr2Z+J/XzWx8E+ZcqswySReL4kwQm5LZ82TANaNWTBxBqrm7ldARaqTWRVR2IQTLL6+S1mUt8GvB/VW17hdxlOAUzuACAriGOtxBA5pAQcAzvMKb9+i9eO/ex6J1zStmTuAPvM8fv/GPtQ==</latexit> \nκ \nt \nκt \nFigure 11: Sketch of the entropy of the radiation, computed using the early time surface and the late time one. We are assuming that S 0 glyph[lessmuch] S i Bek . There is a transition between the two at the Page time. Here κ is proportional to c and also to the effective gravitational coupling of the 2d gravity theory. \n<!-- image -->', '4 A new rule for computing the entropy when gravitational systems are involved': "The calculation of the Page curve using the three-dimensional geometry that we discussed above looks fairly natural from the three-dimensional point of view. It does not look very different from other examples where the entanglement wedge extends beyond the causal wedge. \nHowever it looks surprising from the purely two-dimensional point of view. There, we have a hybrid system consisting of a black hole coupled to a CFT on a half line. If we want to compute the entropy of a region in the CFT, then we are strongly tempted to do what Hawking did [2] and restrict attention to the region of the CFT, without including anything else. However, the holographic computation suggests that we should also include the interior region. \nThis suggests that we should introduce a new rule when we compute entropies in effective theories involving gravity. If we consider a state in a quantum field theory that is entangled with some other system that lives in a gravity theory, then we should use the RT/HRT/EW method and include all possible 'islands' that could extremize the entropy. When we include such an island we will often have to pay a price due to the boundary area of the island, the area term in the gravity theory. However, there can be situations where the quantum system is entangled with fields inside a closed universe, or the interior of a black hole that has evaporated completely. In such cases we do not have to pay an area price because we could, in principle, take the whole space. \nThe prescription is that the actual entropy of some region A in a quantum field theory is given by extremizing a generalized entropy-like functional over islands I g followed by a minimization over all extrema: \nS ( A ) = Min I g Ext I g [ S eff ( A ∪ I g ) + Area[ ∂ I g ] 4 G N ] , (15) \nwhere Area[ ∂ I g ] is the area of boundary of the island. The subscript in S eff reminds us that we are computing the entropy of the state in semi-classical gravity. We call the islands I g that extremize this functional quantum extremal islands . The area contribution form the boundary of the island can be zero if it includes a whole closed universe. We imagine minimizing over all possible regions \nI g that could reduce the bulk term for the entropy and also include the areas of such regions, see figure 12. The subscript in I g reminds us that this is an island in a gravity theory.Figure 12: On the left side we have a quantum system, A , entangled with quantum fields living in a dynamical geometry, which we take to have one spatial dimension. The rule is that we can consider islands, labelled here as blue regions I g . In (a) the region I g is a portion of the whole geometry which has a boundary with area Area ( ∂ I g ) . In (b) region I g is the whole universe and it has zero area. \n<!-- image --> \nWe should emphasize that this is a rule in the effective field theory. If we have the complete and exact state in the quantum system A , then we simply use the usual formula for computing the entropy, namely S ( A ) = -Tr( ρ A log ρ A ), where ρ A is the density matrix of A in the exact quantum theory. On the other hand, in the new rule (15) we have a state ρ eff A ∪I g which is a semiclassical gravity state. \nNote, in particular, that we cannot 'forget' how we obtained the state. When we obtained it using a gravity theory, we should also compute its entropy using the gravity formula (15). The system A could be a simple spin chain. But if we obtained the state in this spin chain by collecting Hawking radiation from a black hole, we should still include the interior in computing the entropy. If we solved the black hole evolution exactly and we give the exact state in the spin chain, then we can use the standard von Neumann formula for the entropy of this spin chain. We can also view the geometry as part of the specification of the state and therefore we need to use the proper RT/HRT/EW prescription to compute its entropy. \nAnother perspective is the following. If we have a configuration that has a quantum system in one region of space and has dynamical gravity in some other region of space, even to compute the entropy of the quantum system, we should still use the rules for gravity theories. In other words, \nif we transfer the Hawking radiation to a set of spins, using the gravity theory, then the combined configuration of spins and black hole spacetime is the full state. When we compute the density matrix we need to follow the rules of gravity to find the entanglement wedge of the spins and its corresponding entropy. The full configuration can be thought of as a tensor network preparing the state of the spins [41] [42]. So, when we compute the entropy we need to take into account the full network. More precisely, the entanglement displayed in figure 12 should be viewed as internal links in that tensor network. \nIf the whole computation occurs in a holographic theory, such as in the usual EngenhardtWall [7] quantum extremal surface prescription, then islands should also be allowed in the bulk when we perform the search for a minimal extremal surface. \nNotice that the size of the island can be very large, even if the entanglement is relatively small. So, this new rule says, for example, that if we have a single spin-1 / 2 particle entangled with another spin-1 / 2 particle in a very large closed universe which is otherwise in a pure state (its total bulk entropy arises only due to the spin half particle), then the entanglement wedge of the original spin half particle includes the whole closed universe, see figure 12(b). Now, this is surprising because the whole closed universe could be in different pure states. However, if we do not know which pure state we have, and we want to include those alternatives, we generate a mixed state and therefore the entropy inside this universe increases and we would not get that the entanglement wedge of the outside spin includes the whole universe. In other words, this formula is not saying that we can learn about the state of the whole closed universe by just looking at a single spin-1/2 particle. Note that by this rule, the entanglement wedge could potentially also include a whole set of additional closed universes in pure states, but which would not contribute to the generalized entropy at all, and we continue to not learn anything about them. \nIt is possible that there is also an exact description of the gravity theory where (15) is exact. (As discussed in [8], such a description might involve something similar to a final state projection [43].) We are calling (15) as an effective prescription, because in general, we can only view the gravity theory as an effective field theory. It is of course very interesting that gravity still knows about the fine-grained von Neumann entropy of the state. We can view this as evidence that there is some bulk theory that contains the precise information about the state. \nNotice that this set of ideas is also connected to the Bekenstein area bound in an interesting way. The Bekenstein bound says that the entropy of a region bounded by some surface S should be smaller than the area of S . This bound is clearly violated if we consider S to be the horizon of an old evaporating black hole and use semiclassical reasoning. However, in such situations when the bound is violated, we expect that there is a nontrivial quantum extremal surface E inside S , and now the S bulk term should only contain the entropy between E and S , instead of the entropy of the entire region bounded by S . The entanglement wedge thus might be smaller than the entire region inside S , and the large entropy remains outside the entanglement wedge. This suggests a modified version of the Bekenstein bound stating that: \nThe generalized entropy of the entanglement wedge inside a region with boundary S should be less than Area( S ) / (4 G N ) . \nLet us discuss this in more detail. This statement presumes that we can consider an arbitrary surface S and view it as being a holographic-type boundary where a quantum system lives. Then we consider a candidate quantum extremal surface E inside the region bounded by S . This modified bound would then follow simply from the minimization prescription, if we extremize over the choice of E . This is due to the fact that a very 'thin' entanglement wedge, where E almost coincides with S , would be an example of the surfaces over which we are extremizing. Since it is very thin, it does not \ncapture any bulk entropy, and thus the generalized entropy functional evaluates to Area( S ) / (4 G N ) on this very thin entanglement wedge. Therefore, the true minimal extremal surface should have smaller generalized entropy. \nWe should emphasize how surprising the RT/HRT/EW formulas are [5-7]. They claim that we can compute the fine-grained entropy just by looking at the effective gravity theory. This is surprising because one might expect that we need some knowledge that goes beyond the effective field theory (the details of the UV completion of the theory, for example) in order to compute the fine-grained entropy. It is expected that if one wanted to compute detailed matrix elements of the density matrix, then we would need to have an accuracy of order e -S , where S is the black hole entropy. Such computations are expected to be very sensitive to the microscopic details of the theory. They are also likely to involve other topologies, as in the long time discussion in [44, 45]. However just the effective field theory is smart enough to know about the correct entropy of Hawking radiation. One simply has to apply the correct prescription for its entropy! 8", '5 Conclusions': "To summarize, we have considered the computation of the entropy of Hawking radiation for an evaporating black hole. We studied a two-dimensional black hole coupled to a two-dimensional matter CFT, where this matter CFT has a holographic dual. In this case, the quantum extremal surface prescription of [7] reduces to the usual RT/HRT [5, 6] prescription in the bulk geometry. When interpreted in terms of the two-dimensional theory, the entanglement wedge has an 'island' in the black hole interior. The appearance of this island was previously noted in the computation of the entanglement wedge of the black hole in [8, 9]. This island is connected to the exterior, where radiation lives, by the extra dimension of the holographic theory. One can view the resulting geometry as a particular realization of the ER=EPR [14,15] idea (see also [16,17]). \nThe radiation is described by a density matrix living in an ordinary quantum field theory without gravity. So in principle, we can compute its entropy using the standard formula S = -Tr( ρ log ρ ). This presumes that we know the state ρ precisely enough. If the final state was obtained via the evaporation of a black hole, or some other process involving gravity, then, when we trace out the rest in the semiclassical approximation, we do not get a precise enough approximation for ρ . In fact, we would get the standard Hawking answer [2]. However, the full geometry does contain enough information to compute the fine-grained entropy, but perhaps not enough to compute the precise individual matrix elements of the density matrix. For this, one needs to make up a new rule for how to compute entropies for quantum systems that are entangled with gravitational systems. We need to consider the addition of 'islands' and use the quantum extremal surface prescription [5-7] to find the precise shape for the island. \nThe case with holographic matter that we have considered here makes this new rule plausible. But we expect it to hold even in cases where matter is given by free fields which do not have a standard Einstein gravity dual. \nGiven the existence of these islands, it would be interesting to see whether there exists a method to extract the information contained within them that has a clear bulk interpretation, in the spirit of what was discussed for the Hayden-Preskill problem [48] in [49,50]. \nNotice that these islands, together with a suitable statement about entanglement wedge reconstruction, suggests a degree of non-locality for the theory. So it would be interesting to understand better how it is compatible with ordinary local gravitational bulk physics. \nNote that for the CGHS model [12,51-54], which is asymptotically flat, we expect a very similar story. We need to compute the entropy of radiation in the asymptotic region, where the dilaton is very large. Again we expect the development of an island in the black hole interior. In the case that the matter is a CFT with a holographic dual, the island is connected to the radiation through the extra dimension. \nFinally, even though we restricted our analysis to the simple case of two dimensions, we expect that the results should be similar in higher dimensions when we have a gravity theory in d dimensions which contains matter that is holographically dual to a ( d +1)-dimensional geometry.", 'Acknowledgments': 'We would like to thank Netta Engelhardt, Daniel Harlow, Andreas Karch, Donald Marolf, Henry Maxfield, Geoffrey Penington, Steve Shenker, Douglas Stanford, Sandip Trivedi, and Edward Witten for useful conversations. A.A. is supported by funds from the Ministry of Presidential Affairs, UAE. R.M. is supported by US Department of Energy grant No. DE-SC0016244. The work of R.M. was performed in part at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611. J.M. is supported in part by U.S. Department of Energy grant DE-SC0009988 and by the Simons Foundation grant 385600. Y.Z. is supported by the Simons foundation through the It from Qubit Collaboration.', 'References': "- [1] S. W. Hawking, 'Particle Creation by Black Holes,' Commun. Math. Phys. 43 , 199-220 (1975)\n- [2] S. W. Hawking, 'Breakdown of Predictability in Gravitational Collapse,' Phys. Rev. D14 , 2460-2473 (1976)\n- [3] Don N. Page, 'Information in black hole radiation,' Phys. Rev. Lett. 71 , 3743-3746 (1993), arXiv:hep-th/9306083 [hep-th]\n- [4] Don N. Page, 'Time Dependence of Hawking Radiation Entropy,' JCAP 1309 , 028 (2013), arXiv:1301.4995 [hep-th]\n- [5] Shinsei Ryu and Tadashi Takayanagi, 'Holographic derivation of entanglement entropy from AdS/CFT,' Phys. Rev. Lett. 96 , 181602 (2006), arXiv:hep-th/0603001 [hep-th]\n- [6] Veronika E. 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2024ApJS..271...55K

A complete accounting of nearby objectsfrom the highestmass white dwarf progenitors down to lowmass brown dwarfsis now possible thanks to an almost complete set of trigonometric parallax determinations from Gaia groundbased surveys and Spitzer followup. We create a census of objects within a Suncentered sphere of 20 pc radius and check published literature to decompose each binary or higherorder system into its separate components. The result is a volumelimited census of 3600 individual star formation products useful in measuring the initial mass function across the stellar lt8M SUBSUB and substellar 5M SUBJupSUB regimes. Comparing our resulting initial mass function to previous measurements shows good agreement above 0.8M SUBSUB and a divergence at lower masses. Our 20 pc space densities are best fit with a quadripartite power law xi MdNdMpropto Malpha with longestablished values of 2.3 at high masses 0.55 lt M lt 8.00M SUBSUB and 1.3 at intermediate masses 0.22 lt M lt 0.55M SUBSUB but at lower masses we find 0.25 for 0.05 lt M lt 0.22M SUBSUB and 0.6 for 0.01 lt M lt 0.05M SUBSUB. This implies that the rate of production as a function of decreasing mass diminishes in the lowmass starhighmass brown dwarf regime before increasing again in the lowmass brown dwarf regime. Correcting for completeness we find a star to brown dwarf number ratio of currently 41 and an average mass per object of 0.41 M SUBSUB.

2024-04-01T00:00:00Z

['arXiv:2312.03639', '10.48550/arXiv.2312.03639', '2024ApJS..271...55K', '2023arXiv231203639K', '10.3847/1538-4365/ad24e2']

['Initial mass function', 'Stellar mass functions', 'Brown dwarfs', 'Trigonometric parallax', 'Solar neighborhood', 'Binary stars', '796', '1612', '185', '1713', '1509', '154', 'Astrophysics - Solar and Stellar Astrophysics', 'Astrophysics - Earth and Planetary Astrophysics', 'Astrophysics - Astrophysics of Galaxies']

The Initial Mass Function Based on the Fullsky 20 pc Census of 3600 Stars and Brown Dwarfs

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https://arxiv.org/pdf/2312.03639.pdf

{'The Initial Mass Function Based on the Full-sky 20-pc Census of ∼ 3,600 Stars and Brown Dwarfs': "J. DAVY KIRKPATRICK, 1, 2 FEDERICO MAROCCO, 1, 2 CHRISTOPHER R. GELINO, 3 YADUKRISHNA RAGHU, 4, 2 JACQUELINE K. FAHERTY, 5, 2 DANIELLA C. BARDALEZ GAGLIUFFI, 5, 2 STEVEN D. SCHURR, 1 KEVIN APPS, 6 ADAM C. SCHNEIDER, 7, 2 AARON M. MEISNER, 8, 2 MARC J. KUCHNER, 9, 2 DAN CASELDEN, 5, 2 R. L. SMART, 10 S. L. CASEWELL, 11 ROBERTO RADDI, 12 AURORA KESSELI, 1 NIKOLAJ STEVNBAK ANDERSEN, 13, 2 EDOARDO ANTONINI, 2 PAUL BEAULIEU, 2 THOMAS P. BICKLE, 14, 2 MARTIN BILSING, 2 RAYMOND CHIENG, 2 GUILLAUME COLIN, 2 SAM DEEN, 2 ALEXANDRU DEREVEANCO, 2 KATHARINA DOLL, 2 HUGO A. DURANTINI LUCA, 2 ANYA FRAZER, 2 JEAN MARC GANTIER, 2 LÉOPOLD GRAMAIZE, 2 KRISTIN GRANT, 2 LESLIE K. HAMLET, 2 HIRO HIGASHIMURA ( 東 村 滉 ) , 15 MICHIHARU HYOGO, 16, 2 PETER A. JAŁOWICZOR, 2 ALEXANDER JONKEREN, 2 MARTIN KABATNIK, 2 FRANK KIWY, 2 DAVID W. MARTIN, 2 MARIANNE N. MICHAELS, 2 WILLIAM PENDRILL, 2 CELSO PESSANHA MACHADO, 2 BENJAMIN PUMPHREY, 2 AUSTIN ROTHERMICH, 17, 2 REBEKAH RUSSWURM, 2 ARTTU SAINIO, 2 JOHN SANCHEZ, 2 FYODOR THEO SAPELKIN-TAMBLING, 2 JÖRG SCHÜMANN, 2 KARL SELG-MANN, 2 HARSHDEEP SINGH, 2 ANDRES STENNER, 2 GUOYOU SUN ( 孙 国 佑 ) , 18, 2 CHRISTOPHER TANNER, 2 MELINA THÉVENOT, 2 MAURIZIO VENTURA, 2 NIKITA V. VOLOSHIN, 2 JIM WALLA, 2 ZBIGNIEW W ˛EDRACKI, 2 JOSE I. ADORNO, 19, 5 CHRISTIAN AGANZE, 20 KATELYN N. ALLERS, 21 HUNTER BROOKS, 22, 2 ADAM J. BURGASSER, 23 EMILY CALAMARI, 5, 24 THOMAS CONNOR, 25, 26 EDGARDO COSTA, 27 PETER R. EISENHARDT, 28 JONATHAN GAGNÉ, 29 ROMAN GERASIMOV, 30 EILEEN C. GONZALES, 31, 32, ∗ CHIH-CHUN HSU, 33, 34 ROCIO KIMAN, 35 GUODONG LI, 36, 37 RYAN LOW, 38 ERIC MAMAJEK, 39 BLAKE M. PANTOJA, 40 MARK POPINCHALK, 41, 42, 24 JON M. REES, 43 DANIEL STERN, 28 GENARO SUÁREZ, 5 CHRISTOPHER THEISSEN, 23 CHAO-WEI TSAI, 36 JOHANNA M. VOS, 44, 5 DAVID ZUREK, 5 AND THE BACKYARD WORLDS: PLANET 9 COLLABORATION \n1 IPAC, Mail Code 100-22, California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125, USA \n2 Backyard Worlds: Planet 9 \n3 NASA Exoplanet Science Institute, Mail Code 100-22, California Institute of Technology, 770 S. Wilson Avenue, Pasadena, CA 91125, USA 4 Washington High School, 38442 Fremont Blvd., Fremont, CA 94536, USA \n5 Department of Astrophysics, American Museum of Natural History, Central Park West at 79th Street, New York, NY 10024, USA 6 Independent scholar, UK \n7 \nUnited States Naval Observatory, Flagstaff Station, 10391 West Naval Observatory Road, Flagstaff, AZ 86005, USA \n8 NSF's National Optical-Infrared Astronomy Research Laboratory, 950 N. Cherry Ave., Tucson, AZ 85719, USA \nNASA Goddard Space Flight Center, Exoplanets and Stellar Astrophysics Laboratory, Code 667, Greenbelt, MD 20771, USA \n10 \nIstituto Nazionale di Astrofisica, Osservatorio Astrofisico di Torino, Strada Osservatorio 20, I-10025 Pino Torinese, Italy \n11 \nSchool of Physics and Astronomy, University of Leicester, University Road, Leicester LE1 7RH, UK \nUniversitat Politècnica de Catalunya, Departament de Física, c/ Esteve Terrades 5, 08860 Castelldefels, Spain \n13 Sygehus Lillebalt, Department of Cardiology, Kolding, Denmark \n14 School of Physical Sciences, The Open University, Milton Keynes, MK7 6AA, UK \n15 Earl of March Intermediate School, 4 The Pkwy, Kanata, ON K2K 1Y4, Canada \n16 Meisei University, 2-1-1 Hodokubo, Hino, Tokyo 191-0042, Japan \nPhysics Department, University of Central Florida, 4000 Central Florida Boulevard, Orlando, FL 32816, USA \n18 Xingming Observatory, Mt. Nanshan, Urumqi, 830011, Xinjiang, PR China \n19 Department of Physics, University of Miami, Coral Gables, FL 33124, USA \n20 Department of Physics, Stanford University, Stanford CA 94305, USA \n21 Department of Physics and Astronomy, Bucknell University, Lewisburg, PA 17837, USA \n22 Department of Astronomy and Planetary Science, Northern Arizona University, Flagstaff, AZ 86011, USA \nDepartment of Astronomy & Astrophysics, University of California San Diego, 9500 Gilman Drive, La Jolla, CA 92093, USA \n24 The Graduate Center, City University of New York, New York, NY 10016, USA \n25 \nJet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA \n26 \nCenter for Astrophysics \n| \nHarvard & Smithsonian, 60 Garden St., Cambridge, MA 02138, USA \n27 Universidad de Chile, Casilla 36-D, Santiago, Chile \nJet Propulsion Laboratory, California Institute of Technology, MS 169-237, 4800 Oak Grove Drive, Pasadena, CA 91109, USA \n29 Institute for Research on Exoplanets, Université de Montréal, Montréal, Canada \n30 Department of Physics & Astronomy, University of Notre Dame, Notre Dame, IN 46556, USA \nDepartment of Physics and Astronomy, San Francisco State University, 1600 Holloway Avenue, San Francisco, CA 94132, USA \n31 \n32 Department of Astronomy and Carl Sagan Institute, Cornell University, 122 Sciences Drive, Ithaca, NY 14853, USA \n9 \n23 \n28 \n12 \n17", 'KIRKPATRICK ET AL.': 'Table A1 (continued) \n- a The bands refer to 1 = ch1 = 3.6 µ mand 2 = ch2 = 4.5 µ m. \n- c Source identification is uncertain or blended in both bands.\n- d Source is undetected in both bands, as these data are very shallow.\n- e Source is badly blended and not extracted in ch1.', 'ABSTRACT': 'Acomplete accounting of nearby objects - from the highest-mass white dwarf progenitors down to low-mass brown dwarfs - is now possible, thanks to an almost complete set of trigonometric parallax determinations from Gaia, ground-based surveys, and Spitzer follow-up. We create a census of objects within a Sun-centered sphere of 20-pc radius and check published literature to decompose each binary or higher-order system into its separate components. The result is a volume-limited census of ∼ 3,600 individual star formation products useful in measuring the initial mass function across the stellar ( < 8 M ⊙ ) and substellar ( ≳ 5 MJup ) regimes. Comparing our resulting initial mass function to previous measurements shows good agreement above 0.8 M ⊙ and a divergence at lower masses. Our 20-pc space densities are best fit with a quadripartite power law, ξ ( M ) = dN / dM ∝ M -α with long-established values of α = 2 . 3 at high masses (0 . 55 < M < 8 . 00 M ⊙ ) and α = 1 . 3 at intermediate masses (0 . 22 < M < 0 . 55 M ⊙ ), but at lower masses we find α = 0 . 25 for 0 . 05 < M < 0 . 22 M ⊙ and α = 0 . 6 for 0 . 01 < M < 0 . 05 M ⊙ . This implies that the rate of production as a function of decreasing mass diminishes in the low-mass star/high-mass brown dwarf regime before increasing again in the low-mass brown dwarf regime. Correcting for completeness, we find a star to brown dwarf number ratio of, currently, 4:1, and an average mass per object of 0.41 M ⊙ . \nKeywords: stars: mass function - brown dwarfs - parallaxes - stars: distances - solar neighborhood - binaries: close', '1. INTRODUCTION': 'The concept of the initial mass function is one of the most fundamental paradigms in astronomy. It embodies the observational evidence for how the universe turns gas into stars and provides an empirical framework on which to test and inform the underlying theory. The initial mass function has far-reaching influence, from providing the cornerstone for galaxy formation scenarios across all cosmic epochs to determining which stellar and substellar populations we see in our own solar neighborhood. \nDebate continues on whether the initial mass function is variable with time or dependent on environment, but its description over most of the range of stellar masses in the Milky Way is well determined. Bastian et al. (2010) conclude that the initial mass function is universal for hydrogen-burning stars, at least within the measurement errors of most cur- \nrent observations, and Andersen et al. (2008) specifically conclude that there is no strong evidence for environmentspecific effects at masses above ∼ 30 MJup . However, far less is known about the mass function at the low-mass end. Knowledge in this area tells us the creation ratio between stars and brown dwarfs and enlightens us on whether planetary mass objects formed via star formation are common compared to those formed via protoplanetary disks. \nIn this paper, we use recent advances in our knowledge of the nearby stellar census to explore in unprecedented detail the field initial mass function. Gaia has helped refine the nearby census down to spectral types of mid-/late-L out to 20 pc (Gaia Collaboration et al. 2020). For colder spectral types, the WISE mission, together with follow-up parallaxes measured by Spitzer, has filled out this census down to earlyY dwarfs (Kirkpatrick et al. 2019, 2021a), with the help of many other ground-based endeavors (e.g., Best et al. 2021). Our understanding of the low-mass end is dominated by solivagant L, T, and Y dwarfs, but much less is known about the \n42 \nfrequency with which these low-mass objects exist as companions to hotter objects in the census. We rectify that gap in our understanding by building a complete census of all objects within 20 pc of the Sun and splitting those systems into their individual components. \nIn Section 2 we use previous nearby star lists, additions from Gaia, and published or newly discovered objects lacking Gaia astrometry to construct the census of objects in the 20-pc volume. In Section 3 we discuss the format of the compiled census, which includes data on nomenclature, astrometry, spectral types, photometry, radial velocities, multiplicity, masses, and effective temperatures. In Section 4 we discuss the methods used to directly measure masses. In Section 5 we discuss the fact that some objects in our sample have strong evidence for multiplicity but generally lack sufficient evidence to characterize the mass of the subcomponents, which is a source of uncertainty in our final analysis. In Section 6 we discuss mass estimation for white dwarf progenitors, giants/subgiants, brown dwarfs, young stars, low metallicity stars (subdwarfs), and normal main sequence stars and discuss what role objects labeled as exoplanets play in our analysis. In Section 7 we perform analysis of the brown dwarf initial mass function, and then we mate that to the stellar initial mass function. In Section 8 we discuss the resulting initial mass function over the entire mass range by compar- \ning our fit of the functional form to other estimates in the literature, and in Section 9 we summarize our conclusions. Auxiliary data and analyses are found in the Appendices. In Appendix A we present photometric, spectroscopic, and astrometric follow-up used to further characterize 20-pc census members and candidates, and in Appendix B we present a list of the "proximal" systems for each constellation.', '2.1. Building the list of 20-pc systems': 'Our starter list for compiling the census of 20-pc systems was the Preliminary Version of the Third Catalog of Nearby Stars (CNS3; Gliese & Jahreiß 1991), which represents the sum knowledge, prior to large-area digital surveys, of stars believed to lie within 25 pc of the Sun. We took all objects in CNS3 and cross-identified them with the Gaia Early Data Release 3 (eDR3; Gaia Collaboration et al. 2020) to provide updated parallaxes. Objects with parallax values < 50 mas were removed from further consideration, and those with values ≥ 50 mas or lacking a Gaia eDR3 parallax were retained. Two objects listed in the CNS3 as possibly being within 20 pc had no parallax in Gaia DR2, Gaia eDR3, or the literature. These were added to a list, shown in Table 1, of potential 20pc members to consider further. Other additions to this list are discussed in Section 2.1.1. \nTable 1 . Stars Lacking Trigonometric Parallaxes but Possibly within 20 pc \nTable 1 continued', '2.1.1. Other published M dwarfs': 'To better complete the M dwarf list, we first consulted the all-sky compilation of Finch et al. (2014), who used the US Naval Observatory fourth CCD Astrograph Catalog (UCAC4; Zacharias et al. 2013) in concert with the American Association of Variable Star Observers (AAVSO) Photometric All-Sky Survey (APASS 5 ) and Two Micron All-Sky Survey (2MASS; Skrutskie et al. 2006) to identify objects within 25 pc of the Sun. The methodology used a suite of color to absolute magnitude relations to provide distance estimates for detections, although this was supplemented with proper motion detection in order to further distinguish nearby stars from background sources. We took this list (their tables 5 and 6) and selected those candidates having Finch et al. (2014) estimated distances ≤ 20 pc and, if available, other published distance estimates ≤ 20 pc from their table 6. This resulted in 267 objects not already in our master census created above. Of these, 251 had parallaxes in Gaia eDR3 (or Gaia DR2, if parallaxes were lacking in eDR3) placing them outside of 20 pc. Of the remaining 16 objects, seven were found to have other published parallaxes or additional distance estimates placing them beyond 20 pc. The final nine possible additions are listed in Table 1 for further scrutiny. \nSecond, we cross-checked our master table against a volume-complete subsample of 0.1-0.3 M ⊙ Mdwarfs within 15 pc of the Sun (Winters et al. 2021) whose parallax data were pulled from both Gaia DR2 and the literature. We found that all of the host stars in those systems were already included in our master list. \nThird, we combed through The Solar Neighborhood series of papers by RECONS - specifically papers I (Henry et al. 1994) through XLIX (Vrijmoet et al. 2022) - to identify all objects verified or suspected to fall within 20 pc of the Sun. Our earlier checks had identified all of the confirmed 20-pc objects, but there were, however, a small number of nearby candidates from Winters et al. (2015) that still lack a trigonometric parallax from any source. These were also added to Table 1. \nAs discussed in the footnotes to Table 1, we have used available photometry and spectroscopy to update the distance estimates for these objects. After additional scrutiny, we find that only one of these - LP 822-37 AB - likely falls within 20 pc, so it has been added to our master census.', '2.1.2. M, L, T, and Y dwarf discoveries from Backyard Worlds': 'Since the recent publication of our 20-pc L, T, and Y dwarf census (Kirkpatrick et al. 2021a), new nearby low-mass stars and brown dwarfs have continued to be recognized via discovery and/or additional follow-up. Examples are a new parallax confirming the nearby nature of the extreme T subdwarf WISEA J181006.18 -101000.5 (Lodieu et al. 2022), the discovery and confirming parallax of the late-T dwarf VVV J165507.19 -421755.5 (Schapera et al. 2022), and the discovery and established physical companionship of the possible Y dwarf companion to Ross 19 (Schneider et al. 2021). Some other 20-pc suspects, such as CWISE J061741.79+194512.8 AB (Humphreys et al. 2023), have also been shown to fall outside the 20-pc volume after additional follow-up. Still other candidates - other isolated field brown dwarfs identified by the Backyard Worlds team - may yet prove to be new members of the 20-pc census. \nTo assess the status of each of these, we list in Table 2 all newer M, L, T, and Y dwarf discoveries that had initial distance estimates of < 25 pc. To obtain more informed distance estimates of these candidates, in addition to providing additional data on other objects previously believed to be in the 20-pc census, we have searched photometric archives for additional data longward of 1 µ m (along with Gaia magnitudes in the case of brighter sources) and have performed other photometric 6 , spectroscopic, or astrometric follow-up on selected targets. Our own 1.25 µ m and 1.65 µ m followup and reductions, along with our reduction of archival data at 3.6 µ m 4.5 µ m, are described in section A.1. Our optical and near-infrared spectroscopic follow-up is discussed in Section A.2. Additional parallactic measurements are described in Section A.3. \nUsing this set of compiled data, we have recomputed distance estimates, as listed in Table 2. Column d J is the distance estimate derived by comparing the measured JMKO magnitude to the predicted MJMKO magnitude derived from the M JMKO vs. JMKO -W2 relation of Kirkpatrick et al. (2021a) 7 . This estimate is valid only for objects with JMKO -W2 ≥ 4 . 0 mag, as smaller values may lead to nonunique solutions for M JMKO (figure 20a of Kirkpatrick et al. \n2021a). Column d H is the distance estimate derived by comparing the measured H magnitude to the predicted MH magnitude derived from the M H vs. spectral type relation of Kirkpatrick et al. (2021a) 8 . The published relation is restricted to types of L0 and later. Column d ch 2 is the distance estimate derived by comparing the measured ch2 magnitude to the predicted Mch 2 magnitude derived from the M ch 2 vs. ch1 -ch2 relation of Kirkpatrick et al. (2021a). This estimate is valid only for objects with 0 . 3 ≤ ch1 -ch2 ≤ 3 . 7 mag, as shown in Figure 18c of Kirkpatrick et al. (2021a). Method 1 (M = 1 in the table) takes the average of these three independent distance measurements - or as many of these as can be derived - as the adopted distance. \nColumn d W 2 is the distance estimate derived by comparing the measured W2 magnitude to the predicted MW 2 magnitude derived from the M W 2 (M ch 2) vs. W1 -W2 relation of Kirkpatrick et al. (2021a). This estimate is valid only for objects with 1 . 0 ≤ W1 -W2 ≤ 4 . 5 mag, as shown in figure 19c of Kirkpatrick et al. (2021a). Method 2 (M = 2 in the table) takes this estimate as the adopted distance. \nColumn d G is the distance estimate derived by comparing the measured Gaia G magnitude to the predicted MG magnitude derived from an M G vs. G -J relation derived specifically for this paper. This estimate is valid only for objects with 1 . 5 ≤ G -J ≤ 5 . 0 mag. Method 3 (M = 3 in the table) uses this as the adopted distance. Method 4 (M = 4 in the table) is exactly the same as Method 3 except that its distance estimate, d GRP uses the Gaia G RP magnitude instead of G and uses an M GRP vs. G RP -J relation also derived specifically for this paper. Method 5 (M = 5 in the table) uses the Gaia DR3 parallax, if available, to establish the distance. \nWhen none of the five estimation methods above apply, we use combinations of colors to solve for degeneracies among possible spectral type or absolute magnitude solutions, as discussed in the notes to the table. For a very small number of objects, the adopted distance is left blank, as no estimate will be possible until additional follow-up is acquired. \nFinally, as another arbiter of proximity to the Sun, Table 2 lists the measured CatWISE2020 proper motions (Marocco et al. 2021; Eisenhardt et al. 2020) and how significantly those measurements differ from zero. Also, using the color vs. spectral type relations given in Kirkpatrick et al. (2021a), we have estimated spectral type based on the W1 -W2 color (SpW), ch1 -ch2 color (SpS), and J -W2color (SpJW), where the valid color ranges are 0 . 4 ≤ W1 -W2 ≤ 4 . 0 mag, 0 . 3 ≤ ch1 -ch2 ≤ 3 . 0 mag, and 4 . 0 ≤ J -W2 ≤ 8 . 5 mag. \nOf the 211 candidate objects in the table, 44 have adopted distance estimates placing them closer than 20 pc (res = in, as listed in the table). Although we have tentatively added these 44 objects to the 20-pc census, obtaining parallaxes of all objects in Table 2 still believed to be within 25 pc would', '20-PC MASS FUNCTION': 'Table A1 . Ancillary Spitzer Data \nTable A1 continued', '2.2. 20-pc stars with newly discovered companions': "While assembling our nearby census, we discovered a small number of objects that fall in close proximity to other, higher mass stars in the list. These known stars and their possible companions are discussed further below and are illustrated in Figure 1. \nHD 13579 (0215+6740), a K2 dwarf (Bidelman 1985) at 18.6 pc (Gaia DR3) : The motion object CWISER J021550.96+674017.2 from Table 2 was discovered by D. Caselden during a targeted search for companions to known 20-pc stars using multi-epoch imaging data from WISE (Figure 1a). Follow-up JMKO -band photometry from Keck/MOSFIRE (see Section A.1) shows that current location of the companion coincides with a background source, rendering the J -W2 = 0.78 ± 0.02 mag color useless as a gauge of spectral type. The measurement of W1 -W2 = 0.49 ± 0.02 mag from the CatWISE2020 Reject Catalog (Marocco et al. 2021) is also contaminated, as the WISE imaging sequence shown in WiseView 9 (Caselden et al. 2018) indicates that this is a much redder source. The motion measurement from the CatWISE2020 Reject Catalog, 392 ± 11 mas yr -1 in RA and -102 ± 10 mas yr -1 in Dec, is also contaminated by background sources but shows a magnitude and direction roughly similar to the values for the 41 '' -separated K2 star HD 13579 (518.178 ± 0.012 mas yr -1 in RA and -305.636 ± 0.014 mas yr -1 in Dec; Gaia DR3). Using the WISE W2 epochal positions from the unTimely Catalog (Meisner et al. 2023a), a linear least-squares fit results in motions of 698 ± 35 mas yr -1 in RA and -500 ± 71 mas yr -1 in Dec, which are discrepant from the primary's motion values by 5.1 σ and 2.7 σ in RA and Dec, respectively. Curiously, the CatWISE2020 and unTimely motions bracket the Gaia motion values of the primary despite the fact that both the CatWISE2020 and unTimely measurements are WISE-based and are affected by the same background contaminants. Implanting a fake source into the WiseView image sequence with the same W2 magnitude as CWISER J021550.96+674017.2 but with the Gaia-measured motions of HD 13579 provides an excellent match to observed motion of CWISER J021550.96+674017.2 itself, but suggests that the CatWISE2020 value of W2 = 13.84 ± 0.01 mag may be too bright. Given the close separation between the CWISER source and HD 13579 and motions that appear similar, we tentatively denote these as a physical pair with an apparent separation of 760 AU. If associated, the distance to HD 13579 implies a spectral type of > T4.5 for CWISER J021550.96+674017.2 based on the CatWISE W2 magnitude's possibly being biased too bright. \nHD 17230 (0246+1146), a K6 dwarf (Gray et al. 2003) at 16.2 pc (Gaia DR3) : K. Apps (see Section 3.6.3) notes that there is a fainter star, Gaia DR3 25488745411919488 (G = \n15.62 mag, ∆ G = 7.51 mag), 3 . '' 6 south of HD 17230 that has no parallax or proper motion solution in Gaia DR3. A search of the Keck Observatory Archive 10 by C. Gelino reveals two epochs of observations of HD 17230 with Keck/NIRC2 behind the adaptive optics system (Wizinowich et al. 2000). Raw images in the Kp and J filters with HD 17230 under a coronagraph (PI: J. Crepp; Program ID: C182N2) clearly show a star located ∼ 3.7 '' from HD 17230 at a position angle of ∼ 195 · . This observation, taken on 2011 Aug 30 UT, can be compared to another taken on 2014 Oct 13 UT (PI: J. Crepp; Program ID: N100N2) in the narrow-band K -continuum. Given the substantial proper motion of HD 17230 of 263.88 ± 0.03 mas yr -1 in RA and -211 . 58 ± 0 . 03 mas yr -1 in Dec (Table 4), the fainter star should fall at a separation of 3 . '' 4 and position angle of 211 · if it were a background source. However, the second epoch shows the secondary at nearly the same separation and position angle as the first epoch, proving that the two stars are a common motion pair. This conclusion is further bolstered by the 2016-epoch Gaia DR3 positions, that place the fainter star at a separation of 3 . '' 64 and position angle of 195 . 3 · from HD 17230. (Figure 1b shows a first-epoch coronagraphic image in which the A component is seen only via its scattered light.) Using the Gaia DR3 parallax for the primary, this implies MG = 14 . 58 mag (spectral type ∼ M8) for the secondary. However, the true type might be somewhat later than this, as the lack of an astrometric solution in Gaia DR3 may mean that this companion is itself a multiple system. This companion, at an apparent physical separation of 59 AU, may be responsible for the radial velocity acceleration seen for HD 17230 over a decades-long timespan by Rosenthal et al. (2021). \nG 43-23 (1002+1149), an M4 dwarf (Reid et al. 1995) at 17.9 pc (Gaia DR3) : The motion object WISEU J100241.49+145914.9 from Table 2 was discovered by D. Caselden during a targeted search for companions to known 20-pc stars using multi-epoch imaging data from WISE (Figure 1c). This object lies only 15 . '' 6 away from G 43-23, which has astrometry from Gaia DR3 of ϖ abs = 56 . 01 ± 0 . 11 mas, µ α = 157 . 02 ± 0 . 11 mas yr -1 , and µ δ = -235 . 65 ± 0 . 10 mas yr -1 . WISEU J100241.49+145914.9 itself is not listed in either the CatWISE2020 Catalog or the CatWISE2020 Reject Table, but a linear least-squares fit to its epochal unTimely positions (Meisner et al. 2023a) in W2 gives motions of 164 ± 56 mas yr -1 in RA and -233 ± 43 mas yr -1 in Dec, nearly identical to the Gaia motions for the primary. Implanting a W2 = 14.55 mag source with the motions of G 43-23 into the WISE image sequence of WiseView (Caselden et al. 2018) makes for a convincing doppelgänger to WISEU J100241.49+145914.9 itself. The WISEU source's UHS detection at JMKO = 18 . 18 ± 0 . 05 mag results in a color of J -W2 = 3.63 ± 0.11 mag, suggesting a type of ∼ T8.5 and a distance of ∼ 14.3 pc, which is slightly closer than the 17.9 pc distance measured for G 43-23. Nonetheless, given the proximity of the two objects to each other and their nearly identical mo- \nFigure 1. Images illustrating the six new multiple systems discussed in the text. Each image has north up and east to the left, and the size of each (in arcsec) is noted in the legend. (a) HD 13579 and CWISER J021550.96+674017.2, (b) HD 17230 and Gaia DR3 25488745411919488, (c) G 43-23 and WISEU J100241.49+145914.9, (d) HD 170573 and CWISE J183207.94 -540943.3, (e) G 155-42 and CWISE J184803.45 -143232.3, and (f) 2MASS J19253089+0938235 A and B. All panels show WISE W1+W2 images from WiseView, except for panel (b), which shows a Keck/NIRC2 Kp -band image, and panel (f), which shows a Keck/NIRC2 K -band image. \n<!-- image --> \nis to be a physical pair at an apparent physical separation of 280 AU. \nHD 170573 (1833 -5415), a K4.5 dwarf (Gray et al. 2006) at 19.1 pc (Gaia DR3) : The T7 dwarf CWISE J183207.94 -540943.3 was discovered by G. Colin and B. Pumphrey and first published in Kirkpatrick et al. (2021a), where Spitzer astrometric monitoring gave ϖ abs = 57 . 0 ± 4 . 3 mas, µ α = -129 . 1 ± 11 . 6 mas yr -1 , and µ δ = -172 . 1 ± 9 . 7 mas yr -1 . In assembling the full 20-pc census for this paper, it was noted that this object lies 10 . ' 3 from the K4.5 dwarf HD 170573 (Figure 1d), which has Gaia DR3 astrometric values of ϖ abs = 52 . 29 ± 0 . 02 mas, µ α = -121 . 05 ± 0 . 02 mas yr -1 , and µ δ = -142 . 04 ± 0 . 02 mas yr -1 . Until more accurate astrometry for the T dwarf becomes available, we will consider this pair to be physically associated because these values are only 1.1 σ , 0.7 σ , and 3.1 σ different for ϖ abs , µ α , and µ δ , respectively. If a true binary, the projected separation is 11,800 AU. \nG 155-42 (1848 -1434), an M3 dwarf (Gaidos et al. 2014) at 17.1 pc (Gaia DR3) : The motion object CWISE J184803.45 -143232.3 from Table 2 was discovered by S. \nGoodman while searching for unpublished motion objects in WISE imaging data. While assembling the 20-pc census for this paper, it was noted that this source falls 2 . ' 45 away from G 155-42 (Figure 1e). The CatWISE2020 Catalog (Marocco et al. 2021) lists motions for CWISE J184803.45 -143232.3 of µ α = -145 ± 33 mas yr -1 and µ δ = -104 ± 37 mas yr -1 . A linear least-squares fit to the WISE W2 epochal positions from the unTimely Catalog (Meisner et al. 2022) results in motions of µ α = -181 ± 22 mas yr -1 and µ δ = -158 ± 31 mas yr -1 . The Gaia DR3 astrometry for G 155-42 is ϖ abs = 58 . 60 ± 0 . 02 mas, µ α = -236 . 45 ± 0 . 02 mas yr -1 , and µ δ = -237 . 26 ± 0 . 02 mas yr -1 . The measured motion values between the two sources differ by 2.8 σ and 3.6 σ in RA and Dec, respectively, for the CatWISE2020 motion of the potential secondary and by 2.5 σ and 2.6 σ for the unTimely motion. The J -W2color of CWISE J184803.45 -143232.3 from Table 2 suggests a ∼ T7.5 dwarf at a distance of ∼ 15.7 pc, which is sufficiently close to the 17.1 pc distance of G 15542 that we tentatively consider them to be a physical pair with apparent physical separation of 2500 AU, pending improved astrometry for the secondary. \n2MASS J19253089+0938235, an M8 dwarf (West et al. 2015) at 17.0 pc (Gaia DR3) : C. Gelino finds two epochs of Keck/NIRC2 data for this object in the Keck Observatory Archive. The first epoch (2019 May 22 UT; PI: Bond; Program ID: H299) shows two objects separated by 194 mas at a position angle of 146 · and magnitude difference of ∆ K =0.29 mag. Two objects are still present in the second epoch (2020 Jun 2 UT; PI: Mawet; Program ID: C249) but with a separation of 199 mas and position angle of 137 · (Figure 1f). We conclude that 2MASS J19253089+0938235 is a closelyseparated binary showing orbital motion because the pair shows measurably different separations and position angles but the astrometry of the second object is inconsistent with the motion of a background star, which would have exhibited a relative motion of approximately -80 mas in RA and +240 mas in Dec. Using a UKIDSS Galactic Plane Survey DR11PLUS star visible in the field and located at J2000 RA = 291.3801844 deg and Dec= +9.6377532 deg, we find K =10.53 ± 0.03 mag for 2MASS J19253089+0938235A (the northwest component) and K =10.82 ± 0.03 mag for 2MASS J19253089+0938235B (the southeast component). This object has been flagged as a possible member of the AB Dor Moving Group (Gagné & Faherty 2018).", '2.3. Checks against the Fifth Catalog of Nearby Stars': 'After we had completed our accounting of the 20-pc census, we were presented with an additional opportunity to further check for omissions or subtractions. Golovin et al. (2022) recently published the Fifth Catalog of Nearby Stars (CNS5), a compilation of all stars and brown dwarfs within 25 pc of the Sun. Within the CNS5, there are 3,002 objects with parallaxes of 50 mas or greater, whereas our list has 3,588 individual objects that meet this criterion 11 . For the purposes of checking the completeness of our own census, we find that only twenty-two of these CNS5 objects were not included in our list. These are given in Table 3. Five of these are Gaia discoveries with relatively large Gaia parallax uncertainties. We show in section A.2 that three of these are background objects based on their spectra, and we assume that the other two, given their even larger parallactic errors, are also background objects. Another fifteen have preferred parallaxes that place them beyond 20 pc, and these preferred parallaxes are either revised values in Gaia DR3 or published parallaxes (or new parallaxes discussed in Section A.3) with \nsmaller uncertainties than those quoted in CNS5 12 . The remaining two objects in Table 3 deserve special note. The first, 2MASSI J0639559 -741844, has a CNS5 parallax with a 16% uncertainty, so we consider our spectrophotometric distance estimate, which places the object beyond 20 pc, to be preferable. (See Kirkpatrick et al. 2021a for a discussion on the credibility of parallaxes when the uncertainties exceed 12.5%.) The second, APMPM J2330 -4737 B, is a bit of a mystery, as we can find no corroborating evidence in the literature that it exists, and this is why it is not included in our census. In conclusion, our comparison to the CNS5 results in no new additions to our list.', '3. THE 20-PC CENSUS': 'Our final 20-pc census is presented in Table 4. The content of this table is described in more detail in the subsections that follow.', '3.1. Nomenclature': 'Not all researchers refer to the same star by the same name, so having a list of aliases is needed. As we entered each object into the census, we searched SIMBAD for alternative names. The name listed under the heading "DefaultName" in Table 4 is the one that appeared as the default name in SIMBAD 13 when our initial search was performed. For all of these objects, a deep dive into the literature is required to establish the current knowledge of multiplicity, spectral type, etc., so we also list alternative names to aid the literature search. Table 4 therefore lists common names (e.g., Sirius), Bayer and Flamsteed designations, and designations from the HR, HD, BD, CD, and CPD catalogs. Table 4 also lists designations from proper motion catalogs (Wolf, Ross, L, LP, G, LHS, LFT, NLTT, LTT, LSPM, SCR, UPM, APMPM, LEHPM, WT, SIPS, PM, and PM J), white dwarf catalogs (WD, LAWD, EGGR), all-sky photometric catalogs (2MASS, WISE), all-sky astrometric catalogs (Gaia, HIC, HIP, TYC, UCAC4, TIC), along with a few other miscellaneous catalogs that also have high usage (GJ, V*, Karmn, **). The field "VarType" is filled with the type of variability seen, if the object is a known variable star; this informa-', 'Table 4 (continued)': 'Table 4 continued', '3.2. Astrometry': 'For each object in the 20-pc census, we list approximate sexagesimal Right Ascension (RA) and Declination (Dec) coordinates at equinox J2000, given under "SexagesimalRA" and "SexagesimalDec" in Table 4. For close multiple systems, the positions of the two objects may be identical, as these coordinates are meant to provide only a crude position for matching the system across catalogs. For more precise coordinates, we also provide RA and Dec in decimal degrees ("RA" and "Dec") at the yearly epoch provided in the "Epoch" column, along with the coordinate uncertainties ("RA\\_unc" and "Dec\\_unc"). Also listed are the absolute parallax ("Parallax") and its uncertainty ("Parallax\\_unc") and the (usually) absolute proper motions and their uncertainties in RA and Dec ("PMRA", "PMDec", "PMRA\\_unc", and "PMDEC\\_unc"). The reference for these decimal coordinates, parallax, and motion measurements is also given ("PlxPMRef"). Note that for some multiple systems, this more precise astrometry may exist only for the composite system or primary and not for each individual component. (As asterisk in the "PlxPMRef" column indicates that the parallax and motion values for another object in the system are used in lieu of actual measurements for this component.) Furthermore, for some recent brown dwarf discoveries, only positions and proper motions are given, as parallaxes have not yet been measured. \nAs a final note on positions, we provide the constellation in which each object is located ("Constellation"), based on the VizieR tool 17 that uses the constellation boundaries provided by Roman (1987). This column can be used to determine the nearest object in each constellation, as further explored in Appendix B.', '3.3. Spectral types': 'For higher mass stars - typically those with types earlier than mid-M - our primary sources for spectral types were the NStars papers by Gray et al. (2003) and Gray et al. (2006). This was done to assure that as many of our referenced types as possible were classified against a homogeneous system of standards, in this case, the MKK System of Morgan et al. (1943). This system was subsequently updated to the MK system of Johnson & Morgan (1953), which itself was expanded and updated by Morgan & Keenan (1973) (the revised MK system), Keenan & McNeil (1976), and Morgan et al. (1978). (See Hearnshaw 2014 more a more detailed history.) \nClassification for objects of later type has followed the precepts of the MK System, thereby pushing this homogeneity into the late-M (Boeshaar 1976, Boeshaar & Tyson 1985, Kirkpatrick et al. 1991, Kirkpatrick et al. 2010), L (Kirkpatrick et al. 1999, Kirkpatrick et al. 2010), T (Burgasser et al. 2006, Kirkpatrick et al. 2010), and Y (Cushing et al. 2011, Kirkpatrick et al. 2012) dwarf sequences. Classification is \ndependent upon the wavelength range over which the typing is done, so Table 4 specifies whether the spectral type was obtained in the visible to photographic near-infrared region ( < 1 µ m; "SpecTypeOpt") or the classical near-infrared region (12.5 µ m; "SpecTypeNIR"). References for the spectral types can be found under "SpTOpt\\_ref" and "SpTNIR\\_ref". For ease of plotting, the spectral types have been converted into a numerical code, with the luminosity type (if listed) ignored. The scale 18 is set so that 0=A0, 10=F0, 20=G0, 30=K0, 40=M0, 50=L0, 60=T0, and 70=Y0; a type of L8.5 would thus be encoded as 58.5. These codes can be found under "SpTOpt\\_indx" and "SpTNIR\\_indx". Note that the original MK classification system\'s standards jump from K5 to K7 to M0 in the late-K sequence, although K6 standards were eventually added in the late 1980s (Keenan & Yorka 1988; Keenan & McNeil 1989). As a result, there are very few objects with codes of ∼ 36 or ∼ 38-39. \nFor white dwarfs, Table 4 uses types primarily taken from the compilations of Sion et al. (2014) and McCook & Sion (2016), with post-2016 discoveries taken from more recent literature or from Section A.2. The use of these references assures that all white dwarfs are on the spectroscopic classification system proposed by Liebert & Sion (1994). All white dwarf classifications have been assigned based on optical spectra, and the corresponding optical spectral index, "SpTOpt\\_indx", is coded to be the Liebert & Sion (1994) temperature index + 100. That is, our index is set so that DA2=102, DAZ5.8=105.8, DA9.2=109.2, DZ12.6 =112.6, etc. For any white dwarf lacking a temperature index, our spectral index has been arbitrarily assigned a code of 100, as a temperature index of 0.0 cannot exist (and no white dwarf in Table 4 has a temperature index lower than 2.0).', '3.4. Photometry': 'Table 4 provides photometry in several systems that have hemispheric or all-sky coverage. As discussed in section 5.3, objects in the 20-pc census span a vast dynamic range in absolute luminosity, amounting to over twenty-nine magnitudes (a difference of 5 × 10 11 in brightness) in J -band alone. Thus, special care must be taken when choosing photometry for Table 4. \nFor the traditional "visible" wavelength regime, Gaia eDR3 magnitudes and uncertainties at G , GBP , and GRP are listed ("G", "G\\_BP", "G\\_RP", "G\\_unc", "G\\_BP\\_unc", and "G\\_RP\\_unc"). The brightest reported G -band magnitude is ∼ 2 mag (Gaia Collaboration et al. 2021), and objects with G < 8 mag, GBP ≲ 4 mag, and GRP ≲ 4 mag have residual saturation effects, as detailed in Gaia Collaboration et al. (2021) and Riello et al. (2021). At the faint end, Gaia is complete to G ≈ 20 mag (depending upon source crowding and galactic latitude; Gaia Collaboration et al. 2021), which means \nthat the more distant late-L dwarfs in the 20-pc census, along with most of the T dwarfs and all of the Y dwarfs, are too faint for Gaia photometric measurements (figure 26 of Gaia Collaboration et al. 2020; see also figure 2 of Theissen 2018). \nFor the traditional near-infrared wavelength regime, J , H , and K magnitudes are provided, with the caveat that there are two main filter systems in use: the 2MASS filter system 19 and the MKO filter system (Tokunaga et al. 2002). The H -band filter is almost identical between the two, but the J and K filters are quite different. As a result, we provide five separate entries to cover the possibilities JMKO , J 2 MASS , H , K , and Ks - along with their uncertainties ("JMKO", "J2MASS", "H", "K", "Ks", "JMKOerr", "J2MASSerr", "Herr", "Kerr", and "Kserr"). The references for this photometry are given in the "JHK\\_ref" column. The J 2 MASS entries mostly come from 2MASS, whereas the JMKO entries come mostly from surveys based at the United Kingdom Infrared Telescope (UKIRT; e.g., the UKIRT Hemisphere Survey, UHS - McMahon et al. 2013) and the Visible and Infrared Survey Telescope for Astronomy (VISTA; e.g. the VISTA Hemisphere Survey, VHS - Dye et al. 2018). The KMKO entries come primarily from UKIRT-based surveys, whereas the Ks entries come from both 2MASS and VISTA-based surveys. The H entries are pulled from all three sets of surveys. 2MASS provides the only reliable photometry at the bright end of our sample, albeit with large uncertainties, and extends to a S/N = 10 limit of J = 15 . 8 mag, H = 15 . 1 mag, and Ks = 14 . 3 mag at its faint end 20 . UKIRT and VISTA provide reliable photometry between their bright limit ( J ≈ 12 mag for UHS and ∼ 11.512.5 mag in J , H , and Ks for VHS; Dye et al. 2018, GonzálezFernández et al. 2018) and their detection limit ( J ≈ 19 mag for UHS and J ≈ 20 mag, H ≈ 19 mag and Ks ≈ 18 mag for VHS; Dye et al. 2018, González-Fernández et al. 2018) and provide higher angular resolution than 2MASS. We have therefore favored 2MASS photometry for near-infrared magnitudes brighter than ∼ 12 mag and UKIRT/VISTA for fainter magnitudes. For objects even fainter than the UKIRT/VISTA limits, or for objects in areas not yet covered by the public UKIRT and VISTA releases, we have pulled objects from the literature or from Appendix A.1. \nAt longer near-infrared wavelengths and extending into the near mid-infrared, we also provide WISE-based W1 (3.4 µ m), W2 (4.6 µ m), W3 (12 µ m), and W4 (22 µ m) magnitudes and their uncertainties ("W1", "W2", "W3", "W4", "W1err", "W2err", "W3err", and "W4err"). The reference for the WISE photometry is given in the "WISEphot\\_ref" column. For W1 and W2, magnitudes brighter than W1 ≈ 8 mag and W2 ≈ 7 mag were pulled from the WISE All-sky Source Catalog and fainter magnitudes were pulled from the AllWISE Source Catalog, in accordance with the suggestion \nmade in the AllWISE Explanatory Supplement 21 . Photometry at W3 and W4 was pulled from the WISE All-sky Release. The only exceptions to the above are objects that were not detected in either of these releases and are instead found only in the CatWISE2020 Catalog. For these sources, only W1 and W2 photometry is presented, as CatWISE2020 has no W3 or W4 photometry. \nIn the case of 2MASS and WISE photometry, we further provide columns "2MASS\\_contam?" and "WISE\\_contam?". A "yes" in these columns indicates that the associated photometry is likely compromised by another nearby object or artifact, as judged via our by-eye assessments of the multiepoch WiseView image blinks (Caselden et al. 2018), as the poorer image scales of 2MASS (pixel scale of 1 \'\' ) and WISE (pixel size of 1 . \'\' 375) translate to a higher likelihood of source blending.', '3.5. Radial velocities': 'Gaia DR3 provides all-sky radial velocities for stars with GRVS ≲ 14 mag (Katz et al. 2022) and effective temperatures as high as 14,500 K (Blomme et al. 2022). These radial velocities and their uncertainties are also listed in Table 4 ("GaiaRV" and "GaiaRV\\_unc").', '3.6. Multiplicity': 'Even after all systems within the 20-pc volume have been noted, one difficult step remains: correctly determining, based on current knowledge, the multiplicity of each system so that each individual component can be correctly accounted for in the mass distribution. We took a multi-pronged approach at tackling this problem, as described below.', '3.6.1. The "Stellar Ambassadors" program': 'The first approach was to crowdsource the initial reconnaissance of the literature. With the help of the citizen scientist super users of the Backyard Worlds: Planet 9 project, we set up a program whereby volunteers could sign up to investigate the multiplicity of randomly selected 20-pc systems. To make this more enjoyable, the following mission statement was provided: \n"Our science-fictional Earth Coalition is currently laying the groundwork to explore all of the \'worlds\' within 20 parsecs (65 light years) of the Sun. Scientists on the Earth Coalition\'s Board of Advisors have a list of host \'suns\' within this volume of space, but the details in that list are a bit spotty. The Coalition is seeking to flesh out these details using our Stellar Ambassadors program. \n"If you choose to become a Stellar Ambassador, your role will be to represent planet Earth to a small number of stellar systems within 20 parsecs. As we reach out for the first time to each of \nthese stellar neighbors, you will be Earth\'s representative to them. But you need to be knowledgeable of the systems for which you\'re responsible, and that will involve your gaining knowledge of each system you\'re assigned. (By \'system\', we\'re referring to a host star and any of its companions - other stars, brown dwarfs, or exoplanets - in orbit around it.)" \nEach volunteer was tasked with determining (a) the number of stars, brown dwarfs, and exoplanets in each system, (b) the spectral types of the (sub)stellar components, and (c) the masses of each component, if the masses have been measured. Each Stellar Ambassador was initially assigned a set of ∼ 12 systems, and additional sets would be assigned if the Ambassador wished to analyze more. Importantly, participants were asked to track the reference material that they used for their data collection, regardless of whether they started with SIMBAD, VizieR, Wikipedia, or some other encyclopedic compendium. In total, twenty-one super users participated in the program, which allowed us to cover 56% of the systems with primaries earlier than L0. (All 20-pc objects with primaries later than this had already been scrutinized in Kirkpatrick et al. 2021a.) These efforts were coordinated in weekly and bi-weekly calls with the volunteers. \nThe product of this exercise was, as expected, an inhomogeneous set of results, as individual Ambassadors concentrated on different portions of the exercise or used entirely different methodologies in their workflows. Nonetheless, it was these varied approaches that enabled us to determine the references on which it would be the most lucrative to focus our early attention. For example, despite the varied approaches, many of the same references kept appearing again and again in the Ambassadors\' reports. These repeating references underscored the vast groundwork laid by exoplanetfinding searches in characterizing potential host stars, as well as the breadth of methods used to measure masses of stars within the solar neighborhood, a topic explored more fully in Section 4. The references that arose from the Stellar Ambassador program were the first resources we used to populate Table 4 with information on multiplicity, mass measurements, and mass estimates.', '3.6.2. In-depth literature checks': 'After this first reconnaissance of the oft-referenced literature, our second approach was the inevitable deep-dive into the literature for each individual object. For this, we used the extensive per-object references compiled by SIMBAD. We concentrated on literature with high-resolution imaging and radial velocity monitoring, in order to judge the multiplicity of each system. We also looked for paper titles that referenced mass measurements and variability (such as eclipsing binaries, RS CVn variables, etc.). Because of time constraints, we were not able to review each reference in detail, but a paper well stocked with results would often allow us to populate Table 4 with information for many systems at once, which sped up the process for objects further down the list. \nWe also relied heavily on the Washington Double Star Catalog 22 and the Ninth Catalog of Spectroscopic Binaries (Pourbaix et al. 2009), although the former reference lists both confirmed and possible companions that themselves must be studied individually to gauge true companionship.', '3.6.3. The Apps catalog': 'After our in-depth literature checks were completed, we became aware of an unpublished list of objects within 30 pc of the Sun that (now co-author) K. Apps has been carefully curating since 2009. A comparison of the Apps catalog to our list revealed twenty-eight objects, mainly companions, that have been disproved via published literature but that our list still included. These have now been removed from Table 4. The comparison to the Apps list also revealed another twenty-three objects, almost all of which are the second components in spectroscopic binaries or companions revealed by high-resolution imaging, whose discovery literature we had missed. These objects have now been added to Table 4.', '3.6.4. Multiplicity parameters and exoplanets': 'To encapsulate knowledge from the multiplicity checks above, we include several additional columns in Table 4 and split the components of each system into separate rows. An example for one system is illustrated in the mobile diagram (see Evans 1968) of Figure 2. In the column "DefaultName", the entry for the system as a whole appears as " xi UMa AB & WISE 111838.70+312537.9 ". The names of the first subdivision in the mobile diagram of this multiple are denoted by a double hyphen at the beginning of the name, which in this case are " --xi UMa AB " and " --WISE 111838.70+321537.9 ". Further hierarchical branches are denoted by four hyphens (e.g., " ----xi UMa A "), six hyphens (e.g., " ------xi UMa Aa "), etc. The column "#CompsOnThisRow" refers to the number of known components on that row of the table. To select only individual objects in the census, for example, one can downselect only those rows for which "#CompsOnThisRow" equals 1. There is also a "SystemHierarchy" column, giving a code for each division within the system. This is comprised of a four-digit integer (e.g., " 1297 ") that uniquely identifies the system, followed by decimal subdivisions (e.g., " 1297.1 " and " 1297.2 ") to identify subcomponents. For subcomponents that are themselves binaries, further decimal subdivisions (e.g., " 1297.1.1 " and " 1297.1.2 ") are assigned, etc. Table 4 also lists a column called "#CompsInThisSystem" that gives the total number of components in the system. This field is populated only for the top level of each system (those rows having no decimal subdivisions in the "SystemHierarchy" column) and can be summed to find the total number of individual components in the table. Additionally, Table 4 includes a column called "SystemCode" that collapses the "SystemHierarchy" format into an eight-digit integer comprised of the four-digit system identifier followed by \n<!-- image --> \nFigure 2. Mobile diagram for the ξ UMa system along with sample columns from Table 4. The mobile diagram at top shows a stylized representation of this quintuple system, illustrating the pair of close doubles ( ξ UMa A and ξ UMa B) and their distant common proper motion companion (WISE J111838.70+312537.9). The table at bottom shows the nine rows for this system, representing the nine vertices (with labels) in the mobile diagram. Table 4 entries for DefaultName, #CompsOnThisRow, SystemHierarchy, #CompsInThisSystem, and SystemCode are shown for illustration. \nfour additional digits representing any other subdivisions of the "SystemHierarchy" code, but with the decimals removed (e.g., " 12971210 "). Note that when lower subdivisions are lacking, those digits are filled with zeroes. This "SystemCode" column is useful if the user prefers to sort the systems in Table 4 into their mobile diagrams rather than keeping the table\'s default ordering, which sorts by RA. \nNote that our accounting of components above includes only those stellar and brown dwarf members of the system, but not any of the known exoplanets. For the latter, we also include a column in Table 4 named "#Planets" that reports the number of exoplanets listed in the NASA Exoplanet \nArchive 23 as of 01 Sep 2022. To match objects from Table 4 to objects in this archive, we used the Transiting Exoplanet Survey Satellite (TESS; Ricker et al. 2015) Input Catalog (TIC; Stassun et al. 2019) designations. It should be noted that, whereas we use a formation-based definition for brown dwarfs in this paper, the NASA Exoplanet Archive uses a mass-based definition for exoplanet vs. brown dwarf and sets the dividing line, somewhat arbitrarily, at 30 M Jup 24 . As a result, there will be some double counting of objects, as these \nmay appear in both the substellar and exoplanet lists. We will return to this point in Section 6.2.3.', '3.7. Mass parameters and effective temperature': 'The final parameters in Table 4 relate to our need to assign masses to all individual objects within the 20-pc census. In Section 4, we discuss the various methods for which masses can be directly measured. For objects whose masses must, instead, be estimated, Section 6 provides additional discussion. Stars with measured accelerations (see Section 4.1.4) are further discussed in Section 5.1, and objects whose Gaia astrometry suggests hidden companions are discussed in Section 5.2. \nMass estimation techniques work well for hydrogenburning stars because there is a direct mapping from color, temperature, and spectral type to mass on the main sequence. These same techniques fail for brown dwarfs because color, temperature, and spectral type vary with age, and the age of a brown dwarf is generally an unmeasurable quantity. Estimating the masses for brown dwarfs, therefore, requires a different tack, one that we approach statistically through their distribution of effective temperatures, as further discussed in Section 6.1.3.', '4. MASSES FROM DIRECT MEASUREMENT': 'There are many ways of measuring stellar masses. Some methods (1) measure mass directly using only observational data, (2) lean lightly on theoretical assumptions when a full suite of needed observational data is not available, (3) derive masses by comparing available data to an empirical data grid for stars with directly measured masses, and (4) compare observables to theoretical models. Examples of these third and fourth groups are methodologies such as The Cannon (Ness et al. 2016) and StarHorse (Queiroz et al. 2018). However, the aim of this section is to establish nearby fiducial objects for which masses have been (semi-)directly measured, in order to establish our own empirical grid (method 3) to estimate masses for the remainder of the 20-pc census. Toward this goal, we use the next two subsections to discuss methods 1 and 2 as they have been applied to nearby objects. Table 4 includes directly measured masses for objects that have such values ("Mass" and "Mass err") along with the technique used for the measurement ("Mass method") and its citation ("Mass reference").', '4.1. Multiple systems': 'Mass measurements can be made for objects in binary or multiple systems, once sufficient information has been collected to define the orbits. For compact objects, mass can also be deduced from the gravitational redshift; observationally, this can only be done in multiple systems, as it requires at least one additional, non-compact, co-moving object with which to disentangle the part of the redshift due to radial velocity. More specifics are given below. \n4.1.1. Visual binaries \nFor a visual binary whose orbit can be observed, the ratio of the masses is just \nM 1 M 2 = a 2 a 1 , (1) \nwhere M 1 and M 2 are the masses of the two objects and a 1 and a 2 are the (physical, not apparent) semi-major axes of their respective orbits. The total mass of the system, M 1 + M 2, can be derived from the equation \nM 1 + M 2 = 4 π 2 ( a 1 + a 2) 3 GP 2 cos 3 i , (2) \nwhere G is the gravitational constant, P is the orbital period, and i is the inclination of the orbit on the plane of the sky. The distance to the system must also be measured so that a 1 and a 2 are in physical, not angular, units, and the inclination can be deduced from the difference between the offset of the center of mass and the focus of the projected ellipse (Carroll &Ostlie 1996). Individual masses can be measured by combining equations 1 and 2. A list of visual (and other) multiple systems can be found in the Washington Double Star Catalog 25 .', '4.1.2. Spectroscopic binaries with eclipses': 'For spectroscopic binaries in which the radial velocities of both stars can be measured (SB2s), the ratio of the masses is just \nM 1 M 2 = v 2 v 1 , (3) \nwhere v 1 and v 2 are the maximum velocity amplitudes with respect to the mean radial velocity curves of the system. The sum of the masses can be obtained via the equation \nM 1 + M 2 = P ( v 1 + v 2) 3 2 π G sin 3 i . (4) \nThe inclination cannot be determined unless the SB2 is also an eclipsing system, in which case the nearly edge-on orientation means that i ≈ 90 · , allowing for a mass determination for both components. \nThere is a class of eclipsing single-lined spectroscopic binary (SB1) systems for which masses can also be derived (Stassun et al. 2017; Stevens et al. 2018). These are systems with a single stellar host and a transiting exoplanet. Because the combined light of the system is almost entirely that of the host star, available all-sky data sets can provide photometry across a wide swath of the electromagnetic spectrum - from the ultraviolet to the near mid-infrared - so that the star\'s apparent bolometric luminosity can be measured. Accurate parallaxes from Gaia provide the distances needed to convert this to absolute bolometric luminosity. These photometric points span either side of the flux peak in these objects, so they also provide a semi-empirical measurement of effective \ntemperature, as well. The radius of the host star, R , can then be derived from the Stefan-Boltzmann Law \nR = √ L bol 4 πσ T eff 4 , (5) \nwhere L bol is its bolometric luminosity, T eff is its effective temperature, and σ is the Stefan-Boltzmann constant. In the limit where the mass and radius of the exoplanet are far smaller than those of the host star, the density of the host star, ρ , can be calculated directly from observable quantities using the equation \nρ = 3 π GP 2 a n 3 , (6) \nwhere a n is the "normalized" semi-major axis (see Sandford & Kipping 2017 for details) and P is the orbital period, both of which can be measured from the transit light curve. (This simplified form assumes a circular orbit. More generalized forms of this equation can be found in Seager & MallénOrnelas 2003.) The stellar mass, M , then follows from \nM = 4 3 π R 3 ρ. (7) \nA list of SB1 and SB2 systems (see Pourbaix et al. 2009) can be found at the Centre de Données astronomiques de Strasbourg 26 . A list of 158 detached eclipsing binaries with well measured stellar properties is given in Stassun & Torres (2016).', '4.1.3. Astrometric binaries': 'Astrometric binaries are those systems in which the presence of an unseen companion can be inferred from the nonlinear motion of the primary, once its parallactic motion is accounted for. A careful mapping of the astrometric orbit results in the following measurement \nM 2 ( M 1 + M 2) 2 / 3 = rap (1 + e ) ( 2 π P √ G ) 2 / 3 (8) \nwhere M 1 is the mass of the luminous component, M 2 is the mass of the invisible component, rap is the orbital separation of the luminous component at apastron, e is the eccentricity of the orbit, P is the orbital period, and G is the gravitational constant (Andrews et al. 2019). \nIt is possible to measure individual masses in astrometric binaries if the right conditions are met. We consider here an astrometric binary in which the secondary contributes little or no light to the system, as would be the case in a system comprised of a main sequence star and a black hole, neutron star, cold brown dwarf, or exoplanet companion. In this case, the light of the system comes almost entirely from the primary star, so an analysis of its broad-wavelength spectrum or spectral energy distribution built from broad-wavelength \nFigure 3. Schematic diagram demonstrating the concept of proper motion anomaly. A binary star system, comprised of an A component (solid black orbit and black points) and a lower-mass B component (dotted pink orbit and pink points) is shown at four separate times corresponding to approximate start and end dates of Hipparcos (left pair) and Gaia DR2 (right pair). The center of mass (grey squares) moves from left to right over time, and the true proper motion of the system over the Hipparcos and Gaia timeframes is represented by the two grey arrows. Assuming that the A component dominates the light of the system, neither Hipparcos nor Gaia will measure this true motion because the center of light will move with component A as the stars orbit their barycenter. The black arrow at left thus shows the proper motion that would be measured by Hipparcos, and the black arrow at right shows the motion measured by Gaia DR2. The disagreement between these two independent measurements is termed "proper motion anomaly" and provides evidence that the system has an unseen component. (For simplicity, we have removed parallactic motion by showing only those points at the same parallax factor, as depicted by the time stamps at the bottom of the figure.) \n<!-- image --> \nphotometry can be used to deduce, with the help of empirical relations, its mass, M 1. Then the mass of the companion, M 2, can be measured using Equation 8. Gaia will produce orbits of hundreds of thousands of such astrometric binaries over its anticipated lifetime (Halbwachs et al. 2023).', '4.1.4. Binaries with acceleration (aka proper motion anomaly)': 'Proper motion measurements at two different epochs have the capability of identifying hidden companions if those two motion values differ significantly from one another. (This would be labeled as an astrometric binary, see Section 4.1.3, once additional astrometric epochs are obtained.) The reason is that the proper motion of the system\'s photocenter will deviate from a straight line unless both components contribute equally to the light output. This methodology was first used by Bessel (1844) to deduce hidden companions to Sirius and Procyon. An illustration of the effect, which is known both as "proper motion anomaly" and as "acceleration", is shown in Figure 3. This procedure has seen a recent revival now that high quality Hipparcos motions from the early 1990s and high quality Gaia DR2 motions from the mid-2010s can be compared. \nThe lack of agreement between the motion measurements is sufficient to identify a hidden companion, and only a few other measurements are needed to derive the companion\'s mass. This can be computed from the following equation \nfrom Brandt et al. (2019) \nM = s 2 ( aPM 2 + aRV 2 ) 3 2 G ( ϖ aPM ) 2 , (9) \nwhere s is the projected separation between the companion and host star, aPM is the host star\'s acceleration on the plane of the sky, aRV is the host star\'s acceleration along the line of sight, and ϖ is the parallax of the system. (See Equation 17 for a different treatment.) This equation holds only if all measurements can be approximated to refer to the same orbital epoch. Otherwise, as detailed in Brandt et al. (2019), more complex orbital fitting is required.', '4.1.5. Compact objects with gravitational redshifts': "Finally, gravitational redshift can be used to measure the mass if the surface gravity can also be determined. Within the 20-pc sample, this is realistically measurable only in relatively massive compact objects like white dwarfs 27 . The observed velocity shift, vgr , due to gravitational redshift is given by \nvgr = GM Rc , (10) \nwhere c is the speed of light (e.g., Chandra et al. 2020). Because the star's mass is related to its surface gravity, g , via the equation \ng = GM R 2 , (11) \nthe mass can be computed from \nM = c 2 vgr 2 Gg . (12) \nThe surface gravity can be measured from the white dwarf's spectrum by comparing to model atmospheres. In practice, though, this method cannot be applied to single white dwarfs because the gravitational redshift is not separable from the radial velocity. If the star is part of a co-moving multiple system or is a member of a young cluster or association, however, then the degeneracy between the radial velocity component and gravitational redshift component can be broken.", '4.2. Single objects': 'Researchers have employed several methods that are capable of measuring the masses of individual objects. These techniques - gravitational lensing, asteroseismology, and surface convection monitoring (aka "flickering") - are described below.', '4.2.1. Gravitational lensing': 'Gravitational lensing occurs when a mass moves very close to the line of sight between an observer and a background object. The mass of the intervening object acts as a lens that alters the apparent position of the background source as seen by the observer (Gaudi 2012) and is potentially measurable for any object. The two temporarily generated images of the background source have a morphology that is azimuthally asymmetric, and this manifests itself observationally as a shift in the centroid. The astrometric shift of the photocenter is given by \nδ ( t ) = u ( t ) θ E u ( t ) 2 + 2 (13) \nwhere θ E is the angular Einstein radius, which can be expressed as \nθ E 2 = 4 GMl c 2 ( Dl -1 -Ds -1 ) (14) \n(Walker 1995, Lu et al. 2016). Here, u and u represent the scalar and vector time-dependent lens-source separation in the plane of the sky normalized to θ E , Ml is the mass of the lens, and Dl and Ds are the distances to the lens and source, respectively. When the distances to the lens and source are known, the monitoring of the astrometric shift as a function of time enables a measurement of the mass of the lens. These equations show that closer lenses produce larger astrometric signals, which makes this a valuable technique for measuring the masses of nearby objects, the main limitation being that such encounters of a lens and a background source happen only rarely and very accurate astrometry is needed to predict such encounters a priori. This technique has so far been successfully applied to only two objects in the 20-pc sample (Sahu et al. 2017, Zurlo et al. 2018) but promises to become more valuable as more accurate Gaia parallaxes and proper motions become available for stars all across the Milky Way.', '4.2.2. Asteroseismology': "Asteroseismology is the study of oscillations in stellar atmospheres, and the characteristics of these oscillations can be used to deduce a star's physical parameters. Any star having a mechanism that can drive oscillations - such as surface convection, pulsations, tidal effects in a close binary, or opacity effects (the κ -mechanism) - can potentially have its mass measured. Equation 52 in Aerts (2021) can be rewritten to show that the stellar mass, M , can be determined from these oscillations using the relation \nM ∼ ν max 3 T eff 3 2 ∆ ν 4 . (15) \n(In the absence of a definitive theoretical model for convection, the scaling of this relation is based on observations of the Sun, as described in Kjeldsen & Bedding 1995.) Here, ∆ ν is the large frequency separation, ν max is the frequency of maximum power, and T eff is the effective temperature. The quantity ν can also be thought of as the inverse of twice the \nsound travel time between the stellar center and the stellar surface (Eq. 39 of Aerts 2021). Figures 4 and 10 of Aerts (2021) graphically demonstrate how ν and ν max are measured in practice. The effective temperature, T eff, is obtained by comparing broad-wavelength spectroscopy of the star to model atmospheres.", '4.2.3. Surface convection monitoring ("flickering")': "The full asteroseismology treatment above requires highquality data over a sufficient time baseline with which to resolve the individual oscillation modes. However, variations in surface convection alone require less exquisite data and can be used to measure the mass, if certain ancillary quantities have also been well measured (Stassun et al. 2018). The needed quantity is ν max from above, which has been shown to depend on the star's gravity, g , and effective temperature, T eff, through the relation \ng = ν max √ T eff C (16) \n(Brown et al. 1991), where C is a normalization constant obtained by calibrating to stars with gravity measurements independently determined via asteroseismology (Kallinger et al. 2016). The effective temperature is, as above, obtained by comparing broad-wavelength spectroscopy to model atmospheres. The mass can then be measured via Equation 11, where the star's radius can be measured directly via interferometry or through the Stefan-Boltzmann Law in Equation 5. The bolometric luminosity can be measured from the aforementioned broad-wavelength spectrum along with an accurate trigonometric parallax.", '5. MULTIPLES LACKING SUFFICIENT DATA FOR MASS DETERMINATION': 'There are some systems for which acceleration has been measured or whose astrometry indicates the presence of multiple components but for which insufficient data exist to compute the masses of the individual objects. Such systems are important to note because their mass accounting is still incomplete. This serves as an additional source of uncertainty in our mass function analysis.', '5.1. Multiples known only through limited acceleration data': 'Currently, there are many accelerating objects within the 20-pc census that lack the additional data needed for companion mass computations via Equation 9. We nonetheless still note these as binaries in Table 4, and we split out those cases here for individual discussion. \nKhovritchev & Kulikova (2015) have identified likely accelerators by comparing the proper motion measured between the first and second Palomar Observatory Sky Surveys (POSS-I and POSS-II; Minkowski & Abell 1963; Reid et al. 1991; Lasker & STSCI Sky-Survey Team 1998) to a motion derived using first-epoch data from other sky surveys (2MASS, SDSS, WISE) and their own second-epoch follow-up astrometry. With these two independent measurements, they can compare a long-baseline motion over 50 yr to \nTable 5. New 20-pc Accelerators from the Khovritchev & Kulikova (2015) Sample \none derived more instantaneously, over only ∼ 10 yr. Brandt (2021) have similarly intercompared the near-instantaneous Hipparcos-measured proper motion from the early 1990\'s, the near-instantaneous Gaia-measured motion from the midto late-2010\'s, and a long-baseline motion constructed from the Hipparcos-to-Gaia baseline. Both sets of authors have identified objects with significant motion discrepancies and labeled these as likely binaries. These objects are noted in Table 4 using the column labeled "Accelerator?".', '5.1.1. Accelerators from POSS vs. recent motion comparison': 'The Khovritchev & Kulikova (2015) list of ∼ 2400 objects covers only a portion of the northern sky (30 · < Dec < 70 · ) for bright ( V < 17 mag), high motion ( µ > 300 mas yr -1 ) stars. Within 20 pc of the Sun, nine such accelerators are identified, only two of which - BD+66 34 and G 96-29 (Capella HL) - were already identified as known multiples in Table 4. The other seven are listed in Table 5. \nWe note that none of these seven objects is identified as a high-significance accelerator in the Brandt (2021) reference discussed in the following subsection. This is because the Brandt (2021) Hipparcos-to-Gaia accelerations could not be computed for these seven stars, as none are in the Hipparcos Catalog 28 . To further explore the underlying data for these Khovritchev & Kulikova (2015) accelerators, we have produced finder charts that show all seven in the POSS-I, POSSII, 2MASS, and WISE images. A few of these appear to be blended with a background object at one of the POSS epochs. The most notably affected are G 172-30, which is blended at POSS-I with an object fainter by ∆ G = 5.8 mag; Wolf 47 29 , which is blended at POSS-I with an object fainter by ∆ G = 6.4 mag; and G 192-13, which is blended at POSS-II with an object fainter by ∆ G = 6.2 mag. (Ross 10 moves past a star \nof near-equal magnitude in all of the images, the possible blending being worst at the POSS-II and 2MASS epochs.) This having been noted, whether or not objects with these magnitude differences could perturb the POSS measurements enough to affect the 50-yr proper motion measurements is not clear. Future releases from a longer baseline Gaia data set should determine whether the accelerations seen for these seven objects are real.', '5.1.2. Accelerators from Hipparcos vs. Gaia comparisons': 'The Brandt (2021) list of ∼ 115,000 objects covers the entire sky for objects in common to Hipparcos and Gaia eDR3 ( G ≲ 11 mag). This list also gives the computed χ 2 value between the two proper motions measured with the best precision, which is usually the Gaia-specific and Hipparcos-toGaia measurements. We conservatively set a false alarm rate of Q = e -χ 2 / 2 < 0 . 1%, corresponding to χ 2 > 13 . 8, to select high-confidence accelerators for analysis here. Using this criterion produces ∼ 33,750 objects, of which 194 fall within the 20-pc census. These 194 are denoted in Table 4 with a "yes" in the "Accelerator?" column. \nKervella et al. (2022) have also produced a catalog of possible accelerators based on a comparison of the short-baseline Gaia-specific motions and the long-baseline Hipparcos-to-Gaia motions. As this list is based on the same underlying data as the list produced by Brandt (2021), many of the same accelerators are flagged by both teams. Under the \nassumption that the companion mass is much less than that of the primary and that the (circular) orbit is perpendicular to the line of sight, Kervella et al. (2022) have further used the proper motion measures to estimate the mass of the hidden companion using the equation \nm = ( 4740 . 470 ∆ µ ϖ ) √ rM G , (17) \nwhere m is the companion mass, M is the primary mass, G is the gravitational constant, r is the orbital radius, ∆ µ (the difference in motion measurements) is in units of mas yr -1 , and ϖ is in units of mas. The constant of 4740.470 is used to convert ∆ µ/ϖ into units of m s -1 (Kervella et al. 2019). Companion masses are estimated using estimated primary masses generally from isochrone fitting for the brightest stars or from an absolute K -band relation for the fainter stars, as described further in Kervella et al. (2022). Companion masses are dependent upon the unknown value of the separation between components, so Kervella et al. (2022) constructed estimates for assumed separations of 3, 5, 10, and 30 AU. In Table 4 we include the extrema of these mass estimates in columns labeled "EstMassAt3AU" and "EstMassAt30AU" for all objects tagged as accelerators. (In a small number of cases, a Brandt 2021 accelerator was not deemed to be an accelerator by Kervella et al. 2022, so these estimates are not given.) \nTable 6 . 20-pc Accelerators in Known Close Binary/Multiple Systems \nTable 6 continued \nTable 6 (continued) \nTable 6 continued \nTable 6 (continued) \nWe divide the resulting list of 20-pc accelerators into three subgroups. The first, listed in Table 6, comprises eightythree objects in known close binary and multiple systems. For all of these, the host star is known to have a close-in companion that Gaia eDR3 fails to detect or provide a full astrometric solution for, and these companions range in mass from the substellar regime into the planetary regime. For a host star at a distance of 10 pc, its Hipparcos-to-Gaia acceleration can be detected if the companion has a separation below a few × 100 AU (Figure 12 from Kervella et al. 2022). Companions at this separation range can also be detected with high-resolution imaging techniques or via radial velocity monitoring, and some have independently measured masses. As one example, the companion in the 19.5-yr spectroscopic binary HD 10307 AB has a measured dynamical mass from Torres (2022) of 0.254 ± 0.019 M ⊙ , and that system has a = 7 . 7 AU, i = 100 · and e = 0 . 44. The Kervella et al. (2022) companion mass estimates of 0.20 M ⊙ at 3 AU and 0.63 M ⊙ at 30 AU bracket the dynamically measured values well, as the assumptions used were reasonable for this system. As another example, the companion to the 1.35-yr spectroscopic binary HD 184467 AB has a measured dynamical mass of 0.868 ± 0.025 M ⊙ (Piccotti et al. 2020), and the system has a = 0 . 7 AU, i = 145 · and e = 0 . 34 (Arenou et al. 2000). The Kervella et al. (2022) companion mass estimate of 0.03 M ⊙ at 3 AU compares unfavorably to the measured \nvalue possibly because of the Kervella et al. (2022) assumption that the secondary mass is much less than that of the primary. This demonstrates that, although the Kervella et al. (2022) companion mass estimates listed in Table 6 provide a guide as to whether the companion causing the acceleration is already known or is a still hidden member, additional astrometric data is needed before the masses can be reliably measured. As can be seen from the full entries in Table 4, many objects in Table 6 are triples, so it is also unclear how many objects are contributing to the measured acceleration. \nThe second list, shown in Table 7, gives fourteen objects known to host exoplanets but lacking any "close" stellar or substellar companions. Here we define "close" to mean within ∼ 50 AU. Six of these objects, as listed in the final column of the table, have more widely separated stellar companions at apparent separations of ≳ 70 AU. The Kervella et al. (2022) mass estimates for all fourteen of these objects are quite low and, for assumed separations of a few AU, correspond to masses traditionally thought of as being in the planetary range. Thus, the accelerations for these objects are likely caused by the known exoplanet(s) in the system. Kervella et al. (2022) provides additional analysis on the stars ϵ Eri, Kapteyn\'s Star, ϵ Ind A, and π Men, while noting that Kapteyn\'s Star has no significant proper motion anomaly as measured by them. \nThe final list, shown in Table 8, has ninety-seven objects whose closest known companions are resolved by Gaia eDR3 or have no known companions at all. For many of these, the nearest known companion falls close enough to the accelerator star ( ≲ 100 AU; Figure 12 of Kervella et al. 2022) \nTable 7. 20-pc Accelerators whose only Close Companions are Known Exoplanets \nthat it may be the object causing the acceleration. Examples are CD -44 3045 A, VV Lyn Aa, CD -36 6589 A, Ross 52 A, BD+45 2247 A, and Wolf 1225 A. Objects for which the nearest known companion lies beyond this separation or for which no companions are currently known are the hosts most likely to harbor new additions to the 20-pc census. Examples of stars with likely hidden companions are G 32-7, CD -22 526, HD 13579, LP 837-53, HD 43162 A, HD 52698, G 25034, BD -17 3088, µ Vir, β TrA, and θ Cyg. \nTables 6-8 highlight that the accounting of all components within the 20-pc census is still incomplete, as there is overwhelming evidence of additional, tightly separated companions. As only < 200 of the ∼ 3,000 Gaia-detected primaries show such evidence, it is tempting to conclude that our tally of higher mass (non-exoplanet) companions is nearing com- \netion. We caution, however, that our criteria for selecting accelerators was set very conservatively and that many real accelerators likely exist with a measured significance below our cutoff value. As the time baseline of Gaia observations is extended, accelerations will be increasingly sensitive to longer-period companions that, for higher (non-exoplanet) masses, are potentially verifiable with direct imaging techniques. Furthermore, Gaia observations over this same extended time baseline will remove the need to compare to the shallower Hipparcos data, enabling acceleration data for lower-mass primaries between the Hipparcos and Gaia limits (11 ≲ G ≲ 21 mag). Finally, less than a third of all systems in the 20-pc census of Table 4 have both a Hipparcos entry and a Gaia DR3 astrometric solution, so many objects within our sample volume are unavailable for similar acceleration analysis. \nTable 8 . 20-pc Accelerators with More Distant (or No Known) Companions \nTable 8 continued \nTable 8 (continued) \nTable 8 continued \nTable 8 (continued)', '5.2. Multiples with large RUWE values': 'The Gaia Renormalized Unit Weight Error (RUWE) is a measure of the goodness of fit of the single-star astrometric model to the observed astrometry and is expected to be ∼ 1.0 if the fit is a good representation (Lindegren et al. 2021b). This parameter is pulled from Gaia DR3 and is listed in the "RUWE" column of Table 4. Values significantly higher than unity can indicate either that the object is an unresolved, physical multiple (Penoyre et al. 2020) or that some other effect is causing the photocenter to deviate from expectations. Two examples of the latter are a chance alignment with a marginally resolved background star or single-star variability that confounds the RUWE renormalization itself (Belokurov et al. 2020). The typically quoted value for selecting likely binaries using this statistic is RUWE > 1.4 (e.g., Fabricius et al. 2021), although Stassun & Torres (2021) have shown that values of 1.0 < RUWE < 1.4 are also highly predictive of unresolved multiplicity. While the RUWE normalization works well across the full population of Gaia-measured stars, Penoyre et al. (2022b) note that it does a somewhat less adequate job when a selection of nearby (d < 100 pc; the GCNS of Gaia Collaboration et al. 2020) stars alone is analyzed. For that reason, they define a new statistic, which they term the Local Unit Weighted Error (LUWE), that improves upon RUWE for these nearer objects. \nValues of RUWE and LUWE change with each subsequent release of Gaia data, and there is valuable information contained within the differences. Later Gaia releases have data (and astrometric solutions) covering a longer timespan, so for unresolved multiple systems with periods roughly equal to or longer than the timespans of the data release, the RUWE (or LUWE) values may continue to run high or even become larger between Gaia DR2 and Gaia eDR3 simply because the photocentric displacement caused by orbital motion in an unresolved binary makes the single-object astrometric solution fit less well with an extended data set. (See Penoyre et al. 2022a for additional discussion.) Conversely, unresolved binaries with shorter periods should improve and eventually get full astrometric solutions in the Gaia non-single star lists. \nWith these observations in mind, Penoyre et al. (2022b) devised a two-part criterion to select the most likely hidden multiples in the 100-pc sample: (1) LUWE eDR 3 > 2 and (2) ∆ LUWE ≡ LUWE eDR 3 -LUWE DR 2 > -LUWE eDR 3 / 3. Within our 20-pc census, 104 objects meet these criteria, and these are the ones labeled with a "yes" in the Table 4 column named "LUWE\\_binary?". Of these, 73 are known from previous literature to be binary and were already labeled as such in our census. The other 31, listed in Table 9, are newly identified multiples. Nine of these are part of higher-order multiples, as indicated by the notes in the table. For eight of these systems, Gaia has detections of both the new high-LUWE object and the other component (sometimes a double itself) with which it has physical companionship. The ninth system, however, is a new triple system for which Gaia detects only the new high-LUWE binary G 43-23 but not the common-proper-motion T dwarf compan- \nWISEU J100241.49+145914.9, discussed in Section 2.2. Note that the LUWE criteria from Penoyre et al. (2022b) are meant to be conservative, so other hidden binaries will exist with LUWE or ∆ LUWE values outside of the bounds noted above. \nTable 9 . New 20-pc Multiple Systems Identified Through LUWE \nTable 9 continued \nTable 9 (continued)', 'NOTE-': "- · (1) 0146 -5339: This is the 35 . ' 9-distant companion to the F9 dwarf q01 Eri.\n- · (2) 0749 -0320: This is the 3 . ' 9-distant companion to the M3.5 binary PM J07498 -0317 AB.\n- · (3) 1002+1459: This object also has a 15 . '' 6-distant companion, WISEU J100241.49+145914.9, announced in this paper (Section 2.2).\n- · (4) 1145 -2021: This is a physical system with the M5e star LP 793-34, 15 . '' 2 distant.\n- · (5) 1417+4525: This is the 59 . '' 2-distant companion of the M0 star BD+46 1951.\n- · (6) 1428+0518: This is a physical system with the M4 star G 65-54, 1 . ' 0 distant.\n- · (7) 1843 -3322: This is a physical system with the M6 star CE 507, 15 . '' 0 distant.\n- · (8) 1936+5013: This is the 1 . ' 9-distant companion to the F3+ dwarf θ Cyg, which is listed amongst the accelerators in Table 8.\n- · (9) 2217 -0848: This is a physical system with the M5 dwarf binary Wolf 1561 BaBb, 7 . '' 9 distant.", '5.3. Multiplicity (and oddities) identified through color-magnitude diagrams': 'In Figures 4-9, we show several color-type, color-color, and color-magnitude diagrams as a final method for identifying unresolved binaries. These diagrams also illustrate the rich diversity of colors and absolute magnitudes that objects within the 20-pc census possess. \nEach plot shows photometry only for those objects believed to be single components ("# Components this Row" = 1 in Table 4) and whose photometry is uncontaminated ("2MASS\\_contam?" and/or "WISE\\_contam?" not equal to "yes" in Table 4). Each object is color coded by its spectral type, as shown by the color bar in each figure 30 . Preference is given to the near-infrared spectral type if listed; otherwise, the optical spectral type is used. (It should be noted that for stars of type A through M, near-infrared classifications are given in Table 4 only when no optical type is available, so this criterion is only relevant for the L, T, and Y dwarfs.) Each object is plotted as a solid black dot, the center of which is colored if the spectral type is known; that is, objects lacking a spectral type appear only as black dots. Furthermore, for plots that involve J or K bands, preference is given to MKO magnitudes; otherwise 2MASS magnitudes are used. \nIn Figure 4, only those objects with absolute magnitude uncertainties below 1.0 mag are shown, to keep the plots more legible. In Figure 5, objects are shown only if their uncertainties in MG are below 1.0 mag and their color uncertainties are below 0.5 mag. In Figure 6 (or 7), objects are shown only if their absolute magnitude uncertainties are below 1.0 mag and their J -W2(or H -W2) uncertainties are also below 1.0 mag. In Figure 8, objects are shown only if the color uncertainty is less than 0.5 mag for Gaia-based color plots or less than 1.0 mag for all other colors. In Figure 9, points are shown only if their color uncertainties are generally less than 0.10-0.20 mag. \nWe have examined each of these diagrams in detail and have identified objects that fall significantly far from the common loci of main sequence stars or white dwarfs to warrant special attention. There are several classes of objects, however, that we do not discuss in this section but address elsewhere: (1) Stars with bright magnitudes may be problematic and have quoted uncertainties insufficiently small to capture these problems. Given that these bright stars are generally well characterized already, we concentrate only on those not believed to be main sequence stars (category #4 below). (2) L, T, and Y dwarfs have already been examined in detail via color-type, color-magnitude, and color-color diagrams in Kirkpatrick et al. (2021a). (3) Subgiant, giant, and bright giant stars are discussed in Section 6.1.2. (4) Low-metallicity (subdwarf) stars are discussed in Section 6.2.2. (5) Young objects are discussed in Section 6.2.1. \nWe begin with objects whose placement on these diagrams could potentially highlight a problem with their measured parallaxes. These are all objects that the Gaia survey is placing within the 20-pc volume for the first time. With the exception of the object at 11 h 59 m -36 · 34 \' , all of these objects have higher than normal Gaia parallax uncertainties as compared to objects of similar G magnitude. We discuss each of these individually below: \n- · Gaia EDR3 4966072879648455296 (0229 -3606): This object, whose spectral type has yet to be determined but whose Gaia eDR3/DR3 parallax is 50 . 66 ± 0 . 61 mas, falls near or just above the main sequence on most color-magnitude diagrams. Its apparent magnitudes are similar to 20-pc objects of the same color, so there is no reason to question its inclusion in our census. Its location on color-magnitude diagrams along with its high DR3 RUWE value of 5.438 indicate possible unresolved binarity.\n- · Gaia EDR3 3330473222213987072 (0623+1018): This M3 dwarf (see Section A.2) has a Gaia eDR3/DR3 parallax of 50 . 80 ± 1 . 55 mas. The derived MG value is ∼ 8 magnitudes fainter than that expected for an average M3 dwarf, and the MW 2 value is ∼ 9 magnitudes fainter. The Gaia parallax value for this object is clearly in error, so it has been removed from the 20-pc census. \nFigure 4. Various absolute magnitudes plotted against spectral type for the 20-pc census. See text for details. \n<!-- image --> \nFigure 5. Absolute G -band magnitude plotted against various colors for the 20-pc census. See text for details. The spray of mostly black points (i.e., objects with no measured spectral types) to the left of the main sequence in the GBP -G vs. MG diagram and to the right of the main sequence in the G -GRP vs. MG diagram represents components in close binaries near the Gaia resolution limit. The GBP and GRP magnitudes are calculated from the fluxes in a 3.5 × 2.1 arcsec 2 field, whereas the G magnitudes are calculated from a profile fit to a much higher-resolution image (section 8 of Evans et al. 2018). For binaries just above the Gaia resolution limit, this means that per-component BP and RP fluxes will often include light from the other object, whereas the G flux will not (Halbwachs et al. 2022). This effect pushes such objects blueward in GBP -G color and redward in G -GRP color, as these diagrams show. \n<!-- image --> \nFigure 6. Various absolute magnitudes plotted against J -W2 color for the 20-pc census. See text for details. \n<!-- image --> \nW2 (mag) \nFigure 7. Various absolute magnitudes plotted against H -W2 color for the 20-pc census. See text for details. \n<!-- image --> \nFigure 8. Various colors plotted against spectral type for the 20-pc census. See text for details. \n<!-- image --> \n- · Gaia EDR3 3460907947316392704 (1159 -3634): This is an M9.5 dwarf with a Gaia eDR3/DR3 parallax of 50 . 10 ± 0 . 18 mas. Its apparent magnitudes fall within the range of other M9.5 dwarfs within the 20pc census. In absolute magnitude, it falls above the main sequence by as much as a magnitude for objects of similar color, and its Gaia RUWE value is 1.482. This position on color-magnitude diagrams cannot be explained by binarity alone, but a slightly larger parallax in tandem would solve the discrepancy. In any event, there is no reason to exclude this object from Table 4.\n- · Gaia EDR3 6025146733201615616 (1624 -3212): This object, of unknown type, has a Gaia eDR3/DR3 parallax of 59 . 01 ± 0 . 12 mas. Its apparent magnitude falls in the range expected for objects of similar color within 20 pc. On plots of absolute magnitude vs. color, however, it appears anomalous. On the MG vs. GBP -GRP plot, it falls 0.4 mag more luminous than objects of similar color; on the MG vs. G -J plot, it is also more luminous, but by 2.4 mag. Whether these issues indicate a problem with the measured photometry, the measured astrometry, or both - or whether the object has an unusual spectrum - is currently unknown. This object is retained in Table 4.\n- · Gaia DR2 4062191480304598656 (AB) (1736 -2515): This object, also of unknown type, has a Gaia DR2 parallax of 60 . 24 ± 0 . 83 mas and is a known double (Vrijmoet et al. 2022). Gaia DR3 lists two sources \nnear this position, but neither have a parallax or proper motion measurement. The apparent magnitude of the DR2 source is at odds with the range expected for objects of similar color within 20 pc for many combinations of apparent magnitude vs. color, such as G vs. G -W2, J vs. J -W2, and H vs. J -Ks . The object also has a very small Gaia-measured proper motion of 25 . 2 ± 1 . 1 mas yr -1 and lies near the Galactic Center at l , b = (2 . · 1 , + 3 . · 1). This object is most likely a background object with a faulty parallax, so we have removed it from the 20-pc census. \n- · Gaia DR2 1795813379365971072 (2151+2328): No spectrum has been acquired of this object, and it appears to be a very close double in both Gaia DR2 and eDR3/DR3. However, only one of these components has a parallax measurement in DR2, and neither one does in DR3. This object lies well below and blueward of the main sequence on many apparent magnitude vs. color plots such as J vs. G -J , W2 vs. GBP -GRP , and H vs. J -W2. The object also has a small Gaia DR2 motion of only 45 . 7 ± 0 . 7 mas yr -1 . This is likely a background source with a bogus parallax, so we have removed it from the 20-pc census. \nThe rest of our analysis deals with objects that are outliers for various other reasons. As discussed below, these reasons include possible unresolved binarity, unusual atmospheric composition, variability corrupting pan-epoch colors, and suspected typographical errors in published literature values. \nFigure 9. Various color-color diagrams for the 20-pc census. See text for details. \n<!-- image --> \n- · HD 1237 B (0016 -7951): For its spectral type of M4, this object has GBP -G and GBP -GRP colors much bluer than expected, while its G -GRP color is much redder than expected. No other separate photometry of the B component is given in Table 4. Given that the A component is eight magnitudes brighter in G than the B component and lies only 4 . \'\' 0 away, we suspect a problem with the measured photometry of B that is not adequately reflected in its quoted uncertainties.\n- · EGGR 246 (0041 -2221): This is an oddly blue white dwarf in colors that use W1 or W2 magnitudes. The object is also blue relative to other white dwarfs in J -H color, though normal in G -J . This carbonbearing object has a peculiar spectral type, DQpec9.3, and is believed to have a mixed hydrogen-helium atmosphere. The known infrared flux deficit is thought to be caused by absorption by H2 via collisions with neutral He (Giammichele et al. 2012; Bergeron et al. 1994, 2022).\n- · LP 941-19 (0213 -3345): Although this DA4.5 white dwarf has contaminated WISE photometry, it falls in an odd position on plots based only on Gaia photometry. Specifically, at its value of MG it falls ∼ 0.5 mag blueward of the white dwarf locus in GBP -G and ∼ 0.3 mag redward in G -GRP . There is very little literature on this source, and our spectrum of it (Section A.2) is the first published. It is not yet clear if this spectrum differs markedly enough from other DA white dwarfs to account for the color discrepancies or whether the Gaia magnitudes themselves are at fault.\n- · HD 21209 A (0323 -4959): The only oddity with this K dwarf is its anomalously blue W1 -W2 color. The value in Table 4, which is from the WISE All-Sky Source Catalog, is W1 -W2 = -0 . 17 ± 0 . 06 mag. Although this is the preferred WISE catalog for sources of this brightness (W1 = 5.56 mag), the AllWISE Source Catalog gives a very similar color of W1 -W2 = = -0 . 15 ± 0 . 13 mag. This color may be due to the slightly subsolar metallicity of the object ([Fe/H]= -0 . 44 ± 0 . 19, Soto & Jenkins 2018; -0 . 41 ± 0 . 04, Sousa et al. 2008; -0 . 39 ± 0 . 02, Tsantaki et al. 2013).\n- · HD 23189 (0348+6840): This early-K dwarf is underluminous for its type at MG , MJ , MH , MKs , and MW 2. When colors formed from Gaia-based magnitudes are compared to the mean colors of objects of the same type, it appears normal, whereas the W1 -W2 color is slightly bluer than normal. We suspect that the Gray et al. (2003) type of K2 V is a typographical error, as independent assessments of the type from spectra, colors, and luminosity considerations (Adams et al. 1935; Bidelman 1985; Mermilliod 1987; Stassun et al. 2019) suggest a spectral type closer to K7.\n- · 2MASS J05053461+4648017 (0505+4648): This is an M8 dwarf (see Section A.2) with a Gaia DR3 parallax value of 56 . 84 ± 0 . 60 mas. The absolute values calculated with this parallax are similar to those of other known M8 dwarfs in the census, and a previous parallax of 69 . 5 ± 4 . 7 mas (Dittmann et al. 2014) also places it within 20 pc. Its location on color-magnitude diagrams such as MG vs. G -W2 along with its high DR3 RUWEvalue of 4.823 indicate possible unresolved binarity.\n- · DENIS J071807.3 -350220 (0718 -3502): On the GBP -GRP and G -J vs. various absolute magnitude diagrams, this object appears to be ∼ 0.7 mag above the locus of other objects of the same color. This is, therefore, likely a near equal-magnitude double. This object is also flagged as a possible binary in the Apps Catalog, again based on its position in color-magnitude diagrams.\n- · SCR J0818 -3110 (0818 -3110): This DZ white dwarf is an outlier on the G -J vs. W1 -W2 diagram and, in fact, any diagram involving W1 -W2 color. This issue has been hinted at previously in Figure 4 of Kawka et al. (2021), which shows that the best model fit to existing spectra and photometry fails to match the observed W1 -W2 color. Although the effect is known, its reason has apparently not yet been established and may be caused by variability or missing opacity sources in the models.\n- · UPM J0901 -6526 (0901 -6526): This K5 dwarf is an outlier on all plots showing spectral type but appears normal on color-color and color-magnitude plots. We suggest that the published type of K5 (Riaz et al. 2006) results from a transcription error in the data for this star and that the actual spectral type is closer to M5.\n- · APMPM J1251 -2121 (1250 -2121): This M6/6.5 dwarf has a Gaia DR3 parallax of 56 . 79 ± 0 . 19 mas. Its apparent magnitudes fall within the range expected for an M6 dwarf within the 20-pc volume, and a previous parallax measurement of 57 . 7 ± 1 . 7 mas (Winters et al. 2015) is in agreement with the Gaia one. On the MG vs. GBP -GRP diagram, it falls ∼ 0.5 mag above the main sequence, and on the MG vs. G -W2 diagram, it falls ∼ 0.7 mag above. This and the Gaia DR3 RUWE value of 2.888 suggest unresolved binarity.\n- · HD 113194 (1302 -2647): Although this K5 dwarf has a Gaia DR3 parallax with a relatively large uncertainty (56 . 94 ± 0 . 19 mas), the earlier Hipparcos parallax (56 . 87 ± 1 . 11 mas) is in agreement. This object also has a high Gaia DR3 RUWE value, is listed as a high-LUWE object (see Section 5.2), and shows acceleration (see Section 4.1.4), in addition to falling ∼ 0.5 mag above the main sequence on the G vs. GBP -GRP diagram. This object is almost certainly an unresolved binary. It is also considered to be binary in the Apps \nCatalog, based on its position on color-magnitude diagrams. The Gaia DR3 main catalog reports a radial velocity of -17 . 56 ± 7 . 24 km s -1 using seventeen observations over 920 days, along with an amplitude of radial velocity variations of 60.25 km s -1 , further supporting the hypothesis of binarity. The P-value for radial velocity constancy ( rv\\_chisq\\_pvalue ) is also 0.0. \n- · 2MASSW J1421314+182740 (1421+1827): This M9.5 dwarf has WISE photometry contaminated by a background source, but it appears unusual in nonWISE colors as well. Specifically, it has oddly blue GBP -G and GBP -GRP colors compared to other objects of similar MG . However, it appears normal for its MG in G -GRP , G -J , G -H , and J -Ks . This may simply indicate an issue with the GBP magnitude that the formal uncertainty fails to adequately capture.\n- · LP 222-65 (1516+3910): This mid-M dwarf lies consistently ∼ 0.6 mag above the main sequence relative to objects of the same color and spectral type on colormagnitude diagrams. This is an isolated object with no obvious problems with its photometry, so we believe this is an unresolved near-equal magnitude binary.\n- · UCAC4 554-051865 (1518+2036): This mid-M dwarf is ∼ 0.6 mag more luminous than objects of similar color on the MG vs. GBP -GRP and MG vs. GBP -G diagrams and has a large RUWE and LUWE value (see Section 5.2). It is likely an unresolved binary.\n- · L 339-19 (1640 -4559): This M3 dwarf shows anomalously red G -W3 and J -W3 colors for its absolute magnitude, and even more anomalously red G -W4and J -W4 colors. A more careful look at the WISE images shows that the W3 detection is likely real, but the W4 detection likely is not. The W3 photometry from AllWISE (reported in Table 4) is 7.10 ± 0.05 mag and that from WISE All-Sky is 6.57 ± 0.04 mag. In G -W3 and using the AllWISE value, the object lies 1.0 mag redward of objects of the same absolute G magnitude; using the WISE All-Sky value shows the object to lie 0.5 mag redward. Archival Spitzer/IRAC and Spitzer/MIPS photometry of this object exists in the GLIMPSE I Spring \'07 Catalog (Benjamin et al. 2003) and MIPSGAL Archive (Carey et al. 2009) at IRSA: ch1 = 7.830 ± 0.038, ch2 = 7.781 ± 0.045, ch3 = 7.724 ± 0.037 (5.8 µ m), ch4 = 7.705 ± 0.026 (8.0 µ m), and [24 µ m] = 7.14 ± 0.24 mag. Running these new data points and the tabulated Table 4 photometry through the Virtual Observatory Spectral energy distribution Analyzer (VOSA 31 ; Bayo et al. 2008) suggests not only that the W4 magnitude is in error but that the W3 magnitude is spuriously bright relative to \nthe bracketing IRAC and MIPS data points. The spectral energy distribution is otherwise typical of that of an M3 dwarf. We therefore conclude that there is no infrared excess in this object. \n- · UCAC4 317-104829 (1706 -2643): This DAH white dwarf is normal in Gaia-only colors, colors formed using Gaia minus near-infrared magnitudes, and colors formed from J , H , and Ks magnitudes. It is, however, oddly blue in W1 -W2. We assume that this anomalous color may be intrinsic to the star and a result of its strong magnetic field, although it should be cautioned that this white dwarf is located against a busy region of the Galactic Plane and may suffer from contamination in its WISE photometry.\n- · DENIS-P J1733423 -165449 (1733 -1654): This L1 dwarf has WISE photometry that is contaminated by background sources, but it also shows unusual colors in Gaia-only measurements. Gaia DR3 lists two other point sources within 2 . \'\' 1 of this object, so its Gaia photometry may be adversely affected in a way that the formal uncertainties fail to capture.\n- · LSPM J1733+1655 (1733+1655): The Gaia DR3 parallax of 60 . 91 ± 0 . 48 mas has a relatively large uncertainly for its magnitude and is in disagreement with an earlier published value of 85 . 40 ± 3 . 30 mas by Dittmann et al. (2014). This mid-M dwarf is more luminous than objects of similar color by ∼ 1.6 mag on the G vs. GBP -G , GBP -GRP , G -GRP , G -J , and G -W2 diagrams, if the Gaia DR3 parallax is used. This overluminosity decreases to ∼ 0.9 mag if the Dittmann et al. (2014) parallax is used instead. This is a high-RUWE/LUWE object as well (Section 5.2), and so is likely an unresolved multiple system with problematic Gaia astrometry. Clark et al. (2022) identify a candidate companion at separation 0 . \'\' 14 and position angle 101 · at epoch 2017.3 and again at separation 0 . \'\' 36 and position angle 63 · at epoch 2019.7. C. Gelino also finds a single epoch of Keck/NIRC2 data in the Keck Observatory Archive for LSPM J1733+1655. These are Br γ and J -continuum observations taken on 2015 Jul 10 UT (PI: Hansen; Program ID: U050N2), from which we measure a separation of 0 . \'\' 11 at position angle 248 · . If we assume all three of these measurements refer to the same star and it is a stationary background object, we derive motions of LSPM J1733+1655 of -0 . 100 \'\' yr -1 in RA and -0 . 051 \'\' yr -1 in Dec, which can be compared to the measured Gaia DR3 values of -0 . 135 \'\' yr -1 in RA and -0 . 130 \'\' yr -1 in Dec. The derived magnitude and direction of motion lead us to conclude that the background star hypothesis is sound.\n- · LP 388-55 A (1735+2634): This late-M dwarf is anomalously red, by 0.25 mag, in G -GRP color but looks normal in the GBP -G color compared to objects \nof similar MG magnitude. Curiously, all Gaia-based absolute magnitudes ( MG , MGBP , and MGRP ) are consistent with the reported spectral type. The B component is an early-L that is not directly imaged by Gaia but may nonetheless be subtly affecting the Gaia magnitudes of the A component. \n- · LP 44-334 A (1840+7240): This primary in a M6.5 dwarf system has a Gaia DR3 parallax (52 . 78 ± 0 . 09 mas) with a relatively large uncertainty for its magnitude, but this value compares favorably to the earlier published value of 59 . 3 ± 2 . 2 mas by Lépine et al. (2009). The GBP -G color is too blue for its MG value, the GBP -GRP color is normal, and the G -GRP color is too red. These issues are likely caused by the nearness of the B component, only 0 . \'\' 8 away, which is likely corrupting the photometry of the A component.\n- · LP 867-15 (1842 -2328): The colors for this M0 dwarf are more consistent with an M4 dwarf than with an M0. Pending spectroscopic verification, we assume that this object has been misclassified.\n- · SCR J2012 -5956 (2012 -5956): This object, a DC9.9 white dwarf, falls below the white dwarf locus for most colors. It is very blue relative to other white dwarfs in J -Ks , J -H , and H -Ks but looks like other white dwarfs in colors made with Gaia-only magnitudes. It is somewhat blue in G -J , G -H , and G -Ks colors. As with EGGR 246 above, the infrared flux deficit is believed to be caused by H2-He collision-induced absorption (Giammichele et al. 2012).\n- · LEHPM 2-783 (2019 -5816): This M6.5 dwarf is overluminous in all Gaia-based colors. (Many other colors are nearly degenerate with absolute magnitude or type in this spectral type range.) On both the GBP -GRP vs. MH and the G -W2 vs. MW 2 plots, the overluminosity is ∼ 0.7 mag. Ujjwal et al. (2020) mark this as a possible member of the β Pic Moving Group, and Riaz et al. (2006) note that it is a strong X-ray emitter with strong H α emission.\n- · LP 12-90 (2322+7847): This mid-M dwarf lies above the main sequence by ∼ 0.75 mag on the MG vs. GBP -G plot. On many other plots of absolute magnitude vs. color, it lies similarly above (and redward of) the main sequence. This could be another unresolved binary - if confirmed, this would make its system with HD 220140 AB a quadruple - but the primary in this system is a young, naked T Tauri star (Makarov et al. 2007), meaning that its position may be solely due to its youth.\n- · ZZ Psc (2328+0514): This white dwarf is anomalously red in colors involving WISE magnitudes - so much so that it falls far from the white dwarf sequence itself. On a plot of MG vs. W1 -W2, for example, it \nlies substantially redward of both the white dwarf locus and the main sequence. This object, also known as G 29-38, is known to have a debris disk around it, the first evidence of which was uncovered by Zuckerman & Becklin (1987). For an update on this object, see Cunningham et al. (2022).', '6. MASSES FROM ESTIMATION': 'Only a small fraction of objects within the 20-pc census has masses measurable by methods 1 or 2 described in the introduction of Section 4. For the rest, we must rely on methods 3 and 4 of that section, which depend on comparison to empirical trends or to theoretical models. \nIn the first subsection below, we discuss mass measurements for objects not on the main sequence - namely, white dwarfs, giants/subgiants, and brown dwarfs. In the second subsection, we summarize mass estimation for main sequence stars. In the third subsection, we discuss other complications - youth, subsolar metallicity, and formation scenario - that may need to be considered when assigning accurate mass estimates for special objects.', '6.1.1. White dwarfs': "Masses have been measured via one of the methods described in Section 4 for a handful of white dwarfs in the 20-pc census, but these represent the end-state masses of the stellar remnants and are not suitable for analysis of the initial mass function. Rather, what is needed are the initial masses before evolution off the main sequence. Techniques have been established that use the final mass of the remnant to estimate the initial mass of the progenitor. \nFor white dwarfs lacking a direct mass measurement, one can estimate the final mass of the white dwarf using one of the following two semi-empirical methods. The first is to use spectroscopic observations of the depth and width of the hydrogen Balmer, He I, or He II lines to establish, after comparison to atmospheric models, the log( g ) and T eff for each object. Further comparison of these two parameters to cooling models provides the remnant mass (e.g., Tremblay et al. 2011; Genest-Beaulieu & Bergeron 2019; Bergeron et al. 1992; Finley et al. 1997). Whereas this first method is applicable only to DA (hydrogen atmosphere) or DB (helium atmosphere) white dwarfs, an alternate method can be used both for objects lacking hydrogen lines as well as for objects lacking spectroscopic observations. In this second method, masses can still be estimated if an accurate parallax has been measured. Here, absolute fluxes across as wide a swath of wavelength space as possible are compared to model atmospheres to provide log( g ) and R , from which the mass can be derived from equation 11 (e.g., Bergeron et al. 2019; Tremblay et al. 2019; Giammichele et al. 2012; Bergeron et al. 2001; Koester et al. 1979). \nThe next step is to convert this final mass into an initial mass using an initial-to-final mass relation (e.g., Weidemann 2000). The empirical form of this relationship has been es- \ntablished using white dwarfs that are members of open clusters of known age. As described above, spectroscopic observations of the Balmer lines in these stars can be compared to atmospheric models to derive log( g ) and T eff for each object. A comparison of these parameters to cooling models provides both the remnant mass as well as the cooling time since the object left the tip of the asymptotic giant branch. The known cluster age minus this cooling time gives the main sequence lifetime of the object, which can then be related back to an initial mass using theoretical evolutionary isochrones. This same technique can also be applied to white dwarfs in globular clusters. Due to their much older ages, these clusters can provide white dwarfs of lower final mass than those available in young open clusters. Because these globular clusters are much more distant, their white dwarfs are faint and more difficult to study, so old low-mass white dwarfs are still not well represented by cluster methods. \nThis lack of low-mass examples can be partly mitigated by the use of old, wide binaries for which the second component can be age dated and the separation between components is large enough that no mass transfer has occurred during the system's evolution. Examples are wide subgiant + white dwarf binaries in which the system can be dated from its more recently evolved member (Barrientos & Chanamé 2021), wide F/G/K dwarf + white dwarf binaries in which the age of the main sequence star can be estimated from activity diagnostics (Catalán et al. 2008, Zhao et al. 2012), and white dwarf pairs in which comparison of the higher-mass white dwarf to known cluster white dwarfs can provide an age for the binary, and the difference in the cooling times for the white dwarf pair gives the main sequence lifetime of the lower-mass white dwarf (Andrews et al. 2015). Using the results of these methods, the trend of final mass with initial mass can be fit. As can be seen from figure 9 of Bar- \nentos & Chanamé (2021), the relation shows considerable scatter at lower masses, as the age dating methods for individual systems are generally less robust than those from clusters. The relations we adopt here are the cluster-based tripartite parameterization found in equations 4-6 of Cummings et al. (2018) and the quadripartite parameterization found in table 1 of El-Badry et al. (2018), The former relation is applicable to white dwarfs with 0 . 56 < M final < 1 . 24 M ⊙ (0 . 83 < M initial < 7 . 20 M ⊙ ), and the latter relation, which is based on nearby white dwarfs with accurate Gaia parallaxes, is applicable to white dwarfs with 0 . 50 < M final < 1 . 37 M ⊙ (0 . 95 < M initial < 8 . 00 M ⊙ ). We further note that neither the cluster nor field methods have yet extended the initial-to-final mass relation below final masses of 0 . 50 M ⊙ . (As discussed further below, white dwarfs with final masses below 0 . 45 M ⊙ require binary interactions, as a single progenitor would imply an age older than the Universe; Marsh et al. 1995). \nSpecifically, we apply the following methodology to assign final masses to white dwarfs in the 20-pc census. First, we use directly measured masses, whenever such measurements are available. For others, we use final mass estimates that are based on accurate parallaxes, high S/N spectra, and/or broad-wavelength data spanning the white dwarf's spectral energy distribution. For all other objects, we resort to the Gaia-centric estimates of Gentile Fusillo et al. (2021) and Gentile Fusillo et al. (2019). These estimates use only a small fraction of the white dwarf's spectral energy distribution spanning the Gaia optical bandpasses - and thus lead to separate solutions for hydrogen- vs. helium-atmosphere objects. When our own follow-up has determined the spectral type of the object, we use this information to break the degeneracy; otherwise, a hydrogen-atmosphere object is assumed, as noted in Table 10. \nTable 10 . Mass Measurements and Estimates for White Dwarfs in the 20-pc Census", '6.1.2. Giants and subgiants': 'There are a number of objects in the 20-parsec census that have evolved off the main sequence but have not yet become white dwarfs. Table 11 includes all objects in Table 4 that have a luminosity class more luminous than V and/or fall in a locus on the absolute magnitude vs. color diagrams that identifies them as post-main sequence stars. \nSeveral of these have direct mass measurements either from orbital dynamics or asteroseismology. The rest have had their masses estimated from other methods, primarily via comparison of their placement on the HR diagram in relation to modeled evolutionary tracks or via fits of their spectra to atmospheric models. For some objects with IV-V or IV luminosity classes, other published spectral types indicate a V luminosity class or their placement on the HR diagram suggests a main sequence star. This is reflected in the mass estimates given in Table 11. \nTable 11 . Giants and Subgiants in the 20-pc Census \nTable 11 continued', '6.1.3. Brown dwarfs': 'Brown dwarfs follow no mass-luminosity relation because they constantly cool over time. If the age of the brown dwarf is known, this can be used to estimate the mass from evolutionary models, but age is a difficult parameter to measure for non-youthful disk objects. We therefore must resort to simulations to tease out information regarding the mass function. In Kirkpatrick et al. (2019, 2021a), we took the empirical distribution of brown dwarf effective temperatures and compared that to various predicted temperature distributions modeled by taking the shape of the brown dwarf mass function, the value of its low-mass cutoff, and the underlying evolutionary model suite as free parameters. For the analysis of this paper, we will employ those same methods, using an updated suite of predictions by Raghu et al. (submitted). \nHere, we compare the Kirkpatrick et al. (2021a) accounting of all 525 known 20-pc L, T, and Y dwarfs to that given \nin Table 4. Additions and subtractions to this tally are listed in Table 12. We find that eight objects have fallen out of the 20-pc sample, all because of new parallax measurements or revised distance estimates that place them outside of 20 pc. On the other hand, sixty-five objects are newly added. These additions include thirty-eight new discoveries (thirtyseven by the Backyard Worlds citizen science group, four of which are new companions), nine new companions recently announced in the literature, one new companion announced here but found in Gaia, three new published parallaxes with \nd < 20 pc, twelve previously overlooked companions, and two previously overlooked objects (DENIS J065219.7-253450, presumably due to a transcription error, and SSSPM J14442019, whose subdwarf type had earlier been updated from late-M to early-L). To facilitate analysis on the revised T eff distribution, we have listed in Table 12 the estimated temperatures of each of the additions and subtractions. Further analysis can be found in Section 7. \nTable 12 . Additions to and Subtractions from the 20-pc L, T, and Y Census of Kirkpatrick et al. (2021a) \nTable 12 continued', '6.2. Other complications': "6.2.1. Youth \nWill the estimation of masses for young objects be biased if those estimates use a relation based on much older stars? Evolutionary models suggest that below a mass of ∼ 0 . 4 M ⊙ , the contraction of a star down to the main sequence follows a Hayashi track along which the star's effective temperature remains approximately fixed (section 16.2.5 of Stahler & Palla 2004). If a temperature-based metric is used for estimating the masses of such stars, then such estimates will be accurate. At higher masses, however, the descent along the Hayashi track will be interrupted when a radiative zone develops. The star then moves via a Henyey track along which the temperature slowly increases until the star reaches the main sequence. For stars with masses above ∼ 0 . 4 M ⊙ , this evolution to the main sequence occurs within the first 100 Myr. \nThis means that objects in the 20-pc census that have masses above ∼ 0 . 4 M ⊙ and ages less than 100 Myr should have their mass estimates more carefully considered. The \nMontreal Open Clusters and Associations database (https: //mocadb.ca/, Gagné et al., in prep., Gagné et al. 2018) is a compilation of known stellar associations, stellar streams, moving groups, and open clusters within 500 pc of the Sun. A search of this database on 2023 May 18 for objects within 20 pc of the Sun and likely belonging to one of these young groups yielded 217 systems. The only objects in this list with ages below 100 Myr are those believed to be members 37 of the β Pic Moving Group ( ∼ 26 Myr), the Columba Association ( ∼ 42 Myr), the Argus Association ( ∼ 45 Myr), the Carina Assocation ( ∼ 45 Myr), and the Octans-Near Association ( ∼ 55 Myr) 38 . These are listed in Table 13. \nBecause a mass of 0 . 4 M ⊙ corresponds to a spectral type of M2.5-M3 (table 7 of Mann et al. 2019), we can use spectral type to identify which of the young 20-pc objects are the ones whose mass estimations may need special handling. \nTable 13. 20-pc Members of Young Associations and Moving Groups with Ages < 100 Myr \nThe only objects in Table 13 with spectral types earlier than this are BD -21 1074 ABC, V2689 Ori, β Pic, α Cir AB, HD 182488 A, AU Mic, and HD 220140 A. Three of these are early M dwarfs for which the brief jog along the Henyey track before reaching the main sequence covers such a small range in temperature that their mass estimates should not be unduly affected. \nThis leaves only four individual objects to consider, and two of these have dynamical mass measurements already. For β Pic, Lacour et al. (2021) used astrometry of the exoplanet system to derive the mass of the host star (1.75 ± 0.03 M ⊙ ) using only a uniform mass prior between 1.4 and 2.0 M ⊙ on β Pic itself. HD 182488 A has a loosely constrained dynamical mass measurement from Brandt et al. (2019) of 0 . 94 + 0 . 17 -0 . 27 M ⊙ . This leaves only two young systems with possibly skewed mass estimates, and this represents such a small percentage of 20-pc stars with types earlier than M0 ( < 1%) that no bias will be imparted on the overall derived mass distribution. \nWe acknowledge that our understanding of young moving groups near the Sun is still evolving. Our Sun is currently moving through three groups - the β Pic Moving Group, the AB Dor Moving Group, and the recently identified (but older) Oceanus Group (Gagné et al. 2023) - but it remains unlikely that many new early-M and hotter dwarfs within 20 pc will be associated with any newly recognized groups. Such young objects would have already revealed themselves through, for example, high chromospheric activity. \nYoung brown dwarfs , on the other hand, require their own special handling. For brown dwarfs, we deduce the form of the mass function via the empirical temperature distribu- \ntion. It has been well established, however, that young brown dwarfs follow a different spectral type (or color) to T eff relation than their older counterparts (Faherty et al. 2016). Corrections to the temperature estimates for these objects were already established for the brown dwarf portion of the 20pc census in Kirkpatrick et al. (2021a), and none of the new brown dwarfs discussed in Table 12 are known to be youthful themselves. Therefore, no additional work is required here.", '6.2.2. Non-solar metallicity': 'Objects with non-solar metallicity raise two concerns. The first is that metal-poor objects may belong to the Galactic halo population and could skew our calculation of the nearby mass function, which concentrates on the Galactic disk. The second is that these objects, even if true disk members, may be sufficiently metal poor that standard solar-metallicity relations will not adequately predict their masses. Are either of these concerns justified? \nA number of objects in Table 4 have spectroscopic classifications indicating subsolar metallicity. For objects earlier than early-M, these classifications can generally be identified via the iron index, "Fe#", which attempts to encode the abundance of metals relative to hydrogen in the spectrum if the spectrum does not match the standards of solar-metallicity (Gray & Corbally 2009). Underabundances are encoded as negative numbers. For objects of spectral type late-K and later, metal poor spectral types (Gizis 1997; Lépine et al. 2003; Kirkpatrick 2005; Lépine et al. 2007; Burgasser et al. 2007c; Zhang et al. 2017) are usually denoted with prefixes of sd (subdwarf), esd (extreme subdwarf), or usd (ultra subdwarf). Table 14 lists all objects in the 20-pc census that have one of these low-metallicity classifications. \nTable 14 . 20-pc Objects with Low-metallicity and/or Halo Kinematics \nTable 14 continued \nTable 14 (continued) \nTable 14 continued \nTable 14 (continued) \nReferences -The references for radial velocity are - (1) Gaia Collaboration et al. 2018, (2) Fouqué et al. 2018, (3) Gaia Collaboration et al. 2022, (4) Holmberg et al. 2007, (5) Pourbaix et al. 2004, (6) Maldonado et al. 2010, (7) Abazajian et al. 2009, (8) Gizis 1997. \nothers most likely belong to the thin or thick disk populations. \nTo answer the first concern, we use the sky positions, parallaxes, and proper motions in Table 4 along with published radial velocities in Table 14 to calculate the U , V , W space velocities with respect to the Local Standard of Rest (LSR). We also calculate the U , V , W values for all objects in Table 4 with Gaia-based radial velocity measurements to see if any objects lacking low-metallicity spectral classifications are found to be halo members merely from their kinematics 39 . Figure 10 shows the Toomre diagram for both sets of objects. Also shown for comparison are stars having radial velocity measurements in Gaia DR2 and lying within 100 pc of the Sun, color coded as thin disk ( V tot ≤ 85 km s -1 ), thick disk (85 < V tot ≤ 180 km s -1 ), or halo ( V tot > 180 km s -1 ) in accordance with the kinematic criteria of Nissen (2004). This comparison demonstrates that only six objects - LP 651-7, Ross 578, HD 25329, Kapteyn\'s Star, HD 103095, and Ross 769 - appear to belong to the kinematic halo population. All \nAs stated in Section 3.5, Gaia contains radial velocities only for those objects having GRVS ≲ 14 mag, which omits many of the M dwarfs and all of the L, T, and Y dwarfs within 20 pc. For these objects, we leverage spectroscopic indications of low metallicity to build a list of potential halo members, then we scour the literature for other published radial velocities. These objects are also listed in Table 14. Many of these lack any radial velocity measurements, so assumed values from -200 to +200 km s -1 , in increments of 50 km s -1 , were used to calculate a range of possible U , V , W velocities. These results, shown in Figure 11, suggest that only two of these colder objects - SSSPM J1444 -2019 and WISEA J153429.75 -104303.3 (aka "The Accident") - are likely to be true halo members. \nFigures 10 and 11 taken together suggest that only eight objects (all of them believed to be single) out of 3,589 total in the 20-pc census, or 0 . 22%, are halo interlopers. Although this is slightly higher than the percentage of 0 . 15% used in Table A of Bensby et al. (2014) based on F and G stars alone, it nevertheless confirms that contamination by halo objects in the 20-pc census is extremely small. Although these objects \nFigure 10. Toomre diagram of UVW space motions corrected to the Local Standard of Rest (LSR) for 74,066 Gaia DR2 stars within 100 pc of the Sun and having parallax errors < 10% (Kirkpatrick et al. 2021b). Thin disk (light grey dots), thick disk (medium grey crosses), and halo (dark grey pluses) objects are marked, with halo stars falling outside the outer dashed circle (red) and thin disk objects falling inside the inner dashed circle (navy). Objects with measured radial velocities in Table 4 or Table 14 are shown in navy if lying in the thin disk velocity zone, yellow for the thick disk zone, and red for the halo zone. The six halo members are highlighted with black labels. \n<!-- image --> \nwill still be included in our mass function, any systematic offset imprinted upon their mass estimates can be ignored in subsequent analyses. \nThe second concern is difficult to address, as very few lowmetallicity objects have had their masses measured via direct methods. The coldest subdwarfs, for instance, have a multiplicity fraction of only ∼ 1% (González-Payo et al. 2021); therefore, few such objects exist for dynamical analyses (e.g., Rebassa-Mansergas et al. 2019). Single subdwarfs are obvious targets for lensing-based mass measurements, as their high velocities increase the likelihood of "encounters" with background objects, but accurate whole-sky astrometry is just now advancing to the stage at which such measurements can be predicted and planned for (e.g. Sahu et al. 2020). So, to address this concern, we instead note that only forty-two \nFigure 11. Toomre diagram of the 100-pc sample from Figure 10, now overplotted with the seven objects (various colors and symbols) from Table 14 that lack radial velocity measurements. For these, results are shown for nine assumed radial velocities ranging from -200 to +200 km s -1 , in steps of 50 km s -1 . As in Figure 10, the demarcation of the thin disk, thick disk, and halo populations are shown by the dashed circles in red and navy. \n<!-- image --> \nlow-metallicity systems are known within the 20-pc census 40 (Table 14), which represents only 1 . 5% of the total. Thus, if small biases are present in converting a subdwarf\'s spectral type, colors, or absolute magnitudes to masses, the bias in the overall 20-pc mass distribution will be negligible. \nThe above logic on the scarcity of objects also holds for systems with higher metallicity than the Sun. This set of objects has a much smaller range in metallicity than the metalpoor objects above, and there are just a handful of examples. Only the higher-mass stars ι Hor AB (Fe+0.3), ν Phe (Fe+0.4), HD 176051 AB (slightly metal strong), and HD 207129 (Fe+0.4) have spectroscopic classifications that fall into this class. Another object, 14 Her, has a supersolar metallicity ([Fe/H] ≈ 0 . 4; Rosenthal et al. 2021) although its listed spectral type in Table 4 gives no indication of this. Curiously, even though members of the Hyades Cluster have metallicities that are slightly supersolar ([Fe/H] = 0 . 14 ± 0 . 05; \nPerryman et al. 1998) and lie, on average, only 47.0 ± 0.2 pc from the Sun (Lodieu et al. 2019), there are no confirmed Hyads within the 20-pc volume (Gaia Collaboration et al. 2020; Schneider et al. 2022).', '6.2.3. Formation process': 'Because we are interested in objects formed via the star formation process, we need a criterion to distinguish objects that may have formed via alternative formation mechanisms at the lowest masses. When brown dwarfs were first theorized (Kumar 1963; Hayashi & Nakano 1963), they were regarded as direct products of the star formation process - ones that had insufficient mass to sustain prolonged thermonuclear fusion in their cores - and as such represented the lowermass extension of hydrogen-burning stars themselves. These could be contrasted with another low-mass formation product, planets, that were believed to be formed via a secondary process - from a protoplanetary disk created around a newly formed protostar or brown dwarf. In the early 1960s, there were no known examples of brown dwarfs, and our own Solar System provided the only known examples of planets. \nAs brown dwarf and exoplanet discoveries began in earnest (see reviews by Kirkpatrick 2005; Winn & Fabrycky 2015), it became clear that nature produces some low-mass products that are difficult to classify as either brown dwarf or exoplanet (e.g., 2MASSWJ 1207334 -393254b, Chauvin et al. 2004). The earlier definition based on formation was cumbersome to use in practice; unless an object was still in its infancy, its exact formation process would be difficult, if not impossible, to ascertain from observations. As an alternative, Burrows et al. (1997) proposed another theoretically based definition. This alternative uses mass to distinguish between a brown dwarf and an (exo)planet, the dividing line being the somewhat arbitrarily chosen deuterium burning limit, which is ∼ 13 MJup for solar metallicity. Somewhat surprisingly, this definition was thereafter widely (though not universally) adopted, in no small part because lower-mass discoveries that earlier would have been called "brown dwarfs" could now be referred to by a more attention-grabbing label of "exoplanet." \nThis alternative definition, however, came three and a half decades after the original brown dwarf definition, and the concept of planets having being born from a protoplanetary disk (the "nebular hypothesis") had been in the astronomical lexicon for over two centuries (Kant 1755; Laplace 1796). Thus, labeling an object below 13 MJup as a planet often leads to confusion, as some readers - and even researchers - unwittingly apply both definitions in tandem. That is, they assume that a so-named "planet" (by the new, mass-based definition) must have formed via a protoplanetary disk (by the former, formation-based definition). It is difficult to divorce the term "planet" from its formation scenario. \nIn this paper, one of our goals is to define, or place limits on, the low-mass terminus of star formation. If we were to use the newer definition to include/exclude objects for the mass function analysis, our results would return a terminus of 13 MJup , which merely reflects the dividing line chosen by the arbitrary definition. We must, therefore, more carefully \nconsider whether the lowest mass objects in the 20-pc census should be counted as star-formation products or planetaryformation products. \nAs stated earlier, this definition also lacks easy observational verification. Nonetheless, some methods have been proposed to distinguish formation mechanisms. Öberg et al. (2011) postulated that the carbon to oxygen ratio could be used as one tracer. Planets that formed close to a star would have a solar-like C / O value, like brown dwarfs formed via gravitational collapse, whereas planets formed via accretion of ices beyond the water snowline would have a supersolar C / O value. Those authors acknowledged, however, that measuring an accurate value of C / O is fraught with difficulties (even within our own Solar System), and Calamari et al. (2022) made a similar conclusion based on their analysis of the spectrum of the brown dwarf Gliese 229B. Mollière et al. (2022) show that this simplified picture of the C / O ratio is somewhat more complicated when disk chemical evolution and pebble accretion are taken into account, as well. \nSimilarly, Morley et al. (2019) showed that the deuterium to hydrogen ratio could be used to distinguish between planets with solar D / H values like Jupiter and Saturn, that formed directly from accretion of gas in the protostellar nebula, and planets with enhanced D / H values like Neptune and Uranus, that presumably formed from accretion of ices. Both C / O and D / H thus have limitations: some objects formed via a protoplanetary disk have values indistinguishable from those of objects born via star formation. \nAnother promising avenue is the overall metallicity. The giant planets of our Solar System have metal enhancements well above solar values (Wong et al. 2004, Fletcher et al. 2009), and exoplanets are preferentially found around metalrich host stars (Fischer & Valenti 2005, Wang & Fischer 2015). These facts led Fortney et al. (2008) to propose metallicity-based diagnostics that could distinguish between formation scenarios. Specifically for objects with T eff < 1400 K , a strong 4.5 µ m CO absorption band along with enhanced H - and K -band fluxes (from a relative lack of collision-induced absorption by H 2) are proposed as fingerprints of planet-like formation. However, these diagnostics are likely only useful when comparing populations of objects and not when establishing the formation pathway of individual objects. Metal enrichment is not unique to planet formation, as a collapsing metal-rich cloud can also produce low-mass objects. \nBowler et al. (2023) note that the orientation between the spin axis of the star and the orbital plane of the companion shows promise as another marker of formation, as starlike formation shows a wide range of orientations, whereas planet-like formation prefers values near 90 · . This is, however, another marker that can distinguish between populations but cannot be used on an individual object basis. \nSchlaufman (2018) demonstrates that companions above ∼ 10 MJup lack the tendency to fall primarily around metalrich hosts that companions below ∼ 4 MJup exhibit, which is taken as evidence of core accretion in the lower-mass set. Hoch et al. (2023) likewise find a tentative difference in the \ntrend of C / O values at ∼ 4 MJup , which is taken as further evidence that those objects are primarily formed via core accretion, although, as stated above, C / O ratios can be difficult to interpret. Similarly, Ribas & Miralda-Escudé (2007) find differing radial velocity distributions above and below M sin( i ) values of ∼ 4 MJup . Schlaufman (2018) states that planet-like formation appears to cease above ∼ 4 MJup , but not necessarily that star-like formation ceases below ∼ 10 MJup . There might still be a range in mass, below ∼ 4 MJup , where both processes contribute. \nThe methods addressed above require data that are so far lacking for most exoplanets or can be used only in comparing populations. Instead, for this paper, we propose a simple scheme whose purpose is merely to exclude objects with a high likelihood of having been formed via a protoplanetary disk while including all others as possible products of star formation. For our scheme, we require at least three bodies in a system because the only parameters available for two-body systems - mass ratio, separation, etc. - can lead to ambiguities when trying to distinguish between formation scenarios. \nAs an example, Bowler et al. (2020) have used twentyseven long-period companions labeled as giant planets and brown dwarfs to search for differences in parameters. They find that the population of brown dwarfs has an eccentricity distribution peaking in the range 0 . 6 < e < 0 . 9, whereas binaries with mass ratios significantly different from one have an eccentricity distribution peaking closer to e ≈ 0. These results indicate that the star formation process tends to create binaries with large eccentricity, and the protoplanetary process tends to form binaries with near-zero eccentricity. To reiterate a point from above, while such trends may be indicative of a population of objects, eccentricity alone cannot be used on an object-by-object basis to distinguish between formation scenarios. The same is true for mass ratio, as doing so can bias our list of potential companions to only the higher mass ones, which could impact our ability to determine star formation\'s low-mass cutoff. (Similarly, not se- \ncting on mass ratio can bias our results in the opposite direction, a point we address further in the next section.) \nIn triple (and higher-order) systems, however, we have other parameters available. Specifically, we note that the hierarchy of empirically observed triple star systems is such that the period of the outer component must be at least five times that of the inner pair (Tokovinin 2004). This is in good agreement with dynamical stability expectations for objects in circular orbits, and the period of the outer component must be even larger than five times the inner one when elliptical orbits are considered (Mardling & Aarseth 2001). Planets that have formed from a protoplanetary disk, on the other hand, can often arrange themselves in stable orbital configurations (e.g., in resonances with one another) that violate the above law. A multi-star system that formed via a collapsing cloud could, presumably, arrange itself in a similar manner if conditions were ideal, but such examples must be exceedingly rare. Therefore, we will use the ratios of orbital periods to identify "exoplanet" systems in the 20-pc census that most likely formed via a protoplanetary disk, and we retain all others for consideration as possible products of star formation. \nTo this end, Table 15 lists all of the host objects from Table 4 that were labeled as having one or more confirmed exoplanets in the NASA Exoplanet Archive as of 2022 Sep 01 41 . For systems in which any pair of "exoplanets" violate the Pouter < 5 Pinner , we indicate the innermost pair that violates the rule and exclude all of the planets, thus including only the host star in later analysis. For all others, we have used the NASA Exoplanet Archive to compile their mass measurements. For objects identified only through radial velocity monitoring, we list the M sin( i ) values, since the inclination of the system is not known. For other objects - transiting systems, radial velocity systems with astrometric imaging, etc. - we list the actual measured masses. Incorporating these objects in to the stellar mass function analysis will be discussed further in Section 7. \nTable 15 . 20-pc Objects Hosting "Planets" \nTable 15 continued \nTable 15 (continued) \nTable 15 continued \nTable 15 (continued) \nTable 15 continued \nTable 15 (continued)', '6.3. Objects on the main sequence': 'Main sequence objects with directly measured masses can be used to calibrate relations of mass vs. absolute magnitude or mass vs. spectral type. Studies have shown that the relation with the smallest intrinsic scatter for K and M dwarfs is the one using absolute K -band magnitude (Delfosse et al. 2000). The fact that the K -band relation shows the least scatter across the optical to near-infrared range is also predicted by model atmospheres, as this is the wavelength regime where competing physical effects modulated by metallicity variations largely cancel one another (Delfosse et al. 2000; Mann et al. 2019). More (and improved) dynamical mass measurements of binary stars 42 along with improved Gaia parallaxes have enabled Mann et al. (2019) to construct a mass vs. MKs relation that results in estimated masses with only 2-3% uncertainty. Specifically, Ks is used because 2MASS provides all-sky coverage at this band. We use the Mann et al. (2019) massMKs relation (their equation 2) over the range 5 . 0 ≤ MKs ≤ 11 . 0, roughly corresponding to spectral types from early-M to late-M 43 . These estimates and their propagated uncertainties are listed in columns "EstMassMKs" and "EstMassMKsErr" of Table 4. \nFor other main sequence stars, we can use the methodology employed by Stassun et al. (2019). Using ∼ 20,000 (non-reddened) stars within 100 pc of the Sun with spectroscopically determined effective temperatures, they established a relation between T eff and GBP -GRP color. This is then mated with the results of Torres et al. (2010) that relate T eff to mass for stars with dynamically measured masses (Stassun et al. 2018b). This gives mass estimates with uncertainties of ∼ 6 . 4% (Stassun et al. 2019). We take mass estimates and their uncertainties directly from the revised TESS Input Catalog (TIC; Stassun et al. 2019) for stars in our Table 4. These values are listed in columns "EstMassTIC" and "EstMassTICErr". We note, however, that the Stassun et al. (2019) prescription for stars with T eff ≲ 4000K (see their appendix A.1 along with Muirhead et al. 2018) followed a different methodology. For these objects, masses were estimated using Ks magnitudes, Gaia DR2 parallaxes, and the Mann et al. (2019) mass-vs.MKs relation. \nSome main sequence stars lack both Ks magnitudes and an entry in the TESS Input Catalog. For these, we resort to two other estimation methods. The first is the mass vs. MG relation. Chontos et al. (2021) took a list of well-studied late-K and M dwarfs (tables 5-7 from Mann et al. 2015) and \nFigure 12. Absolute G-band magnitude plotted against estimated mass for 180 well studied late-K and M dwarfs from Mann et al. (2015). The solid blue line shows our fitted relation from Equation 18. See text for details. \n<!-- image --> \nrefined their mass estimates using more precise Gaia DR3 parallaxes and the Mann et al. (2019) mass-vs.MKs relation from above. They derived a relation between this estimated mass and the absolute G-band magnitude. However, the coefficients in Chontos et al. (2021) are published with insufficient accuracy to re-create the relation show in their figure 7, so we have re-derived them here. Our methodology is identical to theirs except that we exclude the sdM3 object L 750-42 (Gizis 1997) and do not incorporate a dependence on metallicity because the metallicity has not been measured for most of the M dwarfs within 20 pc. Using a functional form of \nMass = 4 ∑ i =0 ci ( MG -10 . 5) i , (18) \nwhere MG is the G-band absolute magnitude in magnitudes and Mass is in units of M ⊙ , we find best-fit coefficients of c 0 = 0 . 30548, c 1 = -0 . 10588, c 2 = 0 . 011471, c 3 = 0 . 0021352, and c 4 = -0 . 00041023. Our fit is illustrated in Figure 12. The relation is valid from 7 . 5 ≤ MG ≤ 15 . 0 (spectral types from ∼ K7 to ∼ M8). For uncertainty propagation, we adopt the Chontos et al. (2021) practice of a 2.2% uncertainty added in quadrature to the ∼ 3% uncertainty inherent to the Mann et al. (2019) relation. This massMG relation is particularly useful for estimating masses of individual components of close double systems that are currently resolved only by Gaia. In Table 4 we provide columns labeled "EstMassMG" and "EstMassMGErr" listing mass estimates for all objects for which these MG -based estimates can be computed. \nThe second alternative estimation method is StarHorse (Anders et al. 2022), which uses Gaia EDR3 data crossmatched to photometry from Pan-STARRS1, SkyMapper, 2MASS, and AllWISE to estimate stellar parameters from stellar isochrones (from PARSEC 1.2S; Marigo et al. 2017) providing the closest match. When Anders et al. (2022) \nmass estimates are available, these are listed in columns "EstMassSH" and "EstMassSHErr" of Table 4. These published mass uncertainties can be anomalously low compared to the other estimates discussed in this section because they pertain only to the internal model errors and do not include the systematic component coming from a model-to-truth comparison. \nFigure 13 shows the four estimation techniques compared to each other. The top three panels show the intercomparisons between the TESS Input Catalog estimates, the MKs estimates, and the MG estimates. As these are all based on the same underlying mass vs. MKs relation of Mann et al. (2019), the correspondence is generally excellent. (In fact, the correspondence between the TESS Input Catalog estimates and estimates from our MKs technique are nearly perfect, differing only in the Gaia data release from which the parallax values were obtained.) The only deviation is for masses greater than ∼ 0 . 65 M ⊙ in the comparison between the TESS Input Catalog values and those derived from MKs , where the difference can be as large as 13%. \nThe bottom three panels of Figure 13 show small systematics between the three estimation techniques above and StarHorse. As stated above, masses from StarHorse are based on theoretical models, so such systematics might be expected between theory and observation. At masses of ∼ 0 . 3 M ⊙ , StarHorse tends to overpredict (by ∼ 10%) the mass relative to the other techniques, and at smaller masses may significantly underpredict (by ∼ 35%). At masses near 0 . 8 M ⊙ , a small underprediction (by < 5%) relative to the TESS Input Catalog becomes an overprediction (by ∼ 5%) relative to masses from the MKs relation. At masses closer to 1 . 0 M ⊙ , StarHorse leads to underpredictions (by ∼ 10%) relative to estimates from the TESS Input Catalog. \nGiven that systematic offsets of up to 15% are seen even between the sets with empirical underpinnings, we are reluctant to apply corrections to offsets smaller than this value. The only exception to this is the ∼ 35% offset seen for StarHorse estimates below StarHorse values of ∼ 0 . 275 M ⊙ . In this case, rather than applying an offset, we will simply not use any StarHorse estimates below 0 . 275 M ⊙ .', '7. FURTHER ANALYSIS': 'For each individual object ("#CompsOnThisRow" = 1) in Table 4, we have adopted a mass and its uncertainty. These are listed in columns "AdoptedInitialMass" and "AdoptedInitialMassErr" along with an additional column, "AdoptedInitialMassNote", indicating the origin of the data from elsewhere in the table. These are labeled with the term "Initial" as a reminder that for white dwarfs, we need their initial masses on the main sequence; for all other objects, their current masses are assumed identical to their initial masses. The codes for "AdoptedInitialMassNote" are as follows, listed in their order of selection: \n- · wd IFMR, wd low, wd ultra-low , or wd conjecture : The initial mass and its uncertainty have been computed via \nthe initial-to-final mass relation or other means (see Table 10), if this object is a white dwarf. \n- · measured : Directly measured mass values from "Mass" and "MassErr" are used. The methodology used and its reference are listed in columns "MassMethod" and "MassRef". (For L, T, and Y dwarfs, directly measured masses are not retained because these are estimated in bulk through statistical means; see the Teff bullet, below.)\n- · M\\_Ks : The mass and its uncertainty from the Mann et al. (2019) MKs relation ("EstMassMKs" and "EstMassMKsErr") are used.\n- · TIC : The mass and its uncertainty from the TESS Input Catalog (Stassun et al. 2019; "EstMassTIC" and "EstMassTICErr") are used.\n- · M\\_G : The mass and its uncertainty from the MG relation of Equation 18 ("EstMassMG" and "EstMassMGErr") are used.\n- · SH : The mass and its uncertainty from StarHorse (Anders et al. 2022; "EstMassSH" and "EstMassSHErr") are used, unless that estimate falls below 0 . 275 M ⊙ (see section 6.3).\n- · literature : The mass and its uncertainty are taken from columns "EstMassLit" and "EstMassLitErr", the mass estimation method and reference for which are listed in "EstMassLitMethod" and "EstMassLitRef". (Literature values can supersede other values above if the object is listed as a giant or subgiant in Table 11.)\n- · see GeneralNotes : For objects with this code, the mass and its uncertainty were computed by us, as detailed in the "GeneralNotes" column of the table.\n- · Teff : For objects of type L, T, or Y, individual masses are not computed. These are handled statistically via the distribution of T eff values and their uncertainties ("Teff" and "Teff\\_unc"), as described in detail below. \nFor cases in which literature values did not list a mass uncertainty, a value of 10% is arbitrarily assumed. The quoted StarHorse uncertainty is also replaced with a 10% uncertainty, based on the under- and over-predictions noted when comparing StarHorse values to other estimates (see discussion at end of Section 6.3), unless the quoted StarHorse internal uncertainty is already larger, in which case we retain the published value. \nFor cases where only a miscellaneous magnitude or delta magnitude of a companion were available, it is instructive to estimate a spectral type for the object in order to estimate its mass. Figure 14 shows a comparison between masses and measured spectral types for those Table 4 objects having mass estimates (or direct measures) from one of the other methods. The piecewise fit shown in the figure is the one we \nFigure 13. Intercomparisons of results from our four mass estimation techniques. The line of one-to-one correspondence is shown by the blue dashes. See text for details. \n<!-- image --> \nTable 16. Piecewise Fit to Mass vs. Dwarf Spectral Type Relation \nNOTE-Each row in this table represents an inflection point in the red, piecewise fit of Figure 14. \nuse to translate a dwarf spectral type estimate into a mass estimate. Other per-object details can be found in the General Notes column of Table 4.', '7.1. Analysis of brown dwarfs': 'We use the methodology adopted by Kirkpatrick et al. (2019, 2021a) and Raghu et al. (submitted) to determine the mass function for L, T, and Y dwarfs, most of which are brown dwarfs lacking any color (or spectral type or absolute magnitude) to mass correlation. Specifically, the mass function for these objects is determined by comparing the distribution of present-day temperatures to predicted temperature distributions. Predictions are drawn from a grid of models with varying mass functions, birthrates, and low-mass cutoffs. For each point in the grid, we build a predicted mass/age distribution that is then passed through a set of evolutionary models to predict the current-day T eff distribution. Using this grid of predictions allows us to find the combination of mass function, birthrate, and cutoff mass that best fits the observed temperature distribution. \nFor the empirical distribution, we estimate the T eff value for each L, T, or Y dwarf (see Table 12 in this paper and table 11 of Kirkpatrick et al. 2021a) and then calculate space densities as a function of T eff. To compute space densities, we need to determine the distances at which our brown dwarf subsamples are truly complete, as the coldest Y dwarfs are so intrinsically dim that we are unable to push their completeness to the 20-pc limit targeted in this paper. As described in Kirkpatrick et al. (2021a), we determine completeness via the V / Vmax test (Schmidt 1968) using 150K bins and computing ⟨ V / Vmax ⟩ at half-parsec steps within each bin. The \nFigure 14. Mass as a function of spectral type for 20-pc objects with measured (black points) or estimated (grey points) masses and optical spectral types in Table 4. The adopted initial mass (see text) is used for each object. For objects with estimated (not measured) masses, a random value between -0.25 and +0.25 has been added to the spectral type to better visualize otherwise overlapping data points. Our piecewise fit to the relation for dwarf stars is shown by the solid red line and is quantified in Table 16. For comparison, we show the average mass per spectral type as tabulated in the 2022.04.16 version of https://www.pas.rochester.edu/~emamajek/ EEM\\_dwarf\\_UBVIJHK\\_colors\\_Teff.txt (Pecaut & Mamajek 2013; magenta dashed line). \n<!-- image --> \ncomputation starts with the first half-parsec step falling just larger than the distance of the bin\'s nearest object and advances in distance out to d = 20 pc. These results are shown in Figure 15. \nA comparison of this figure to figure 23 of Kirkpatrick et al. (2021a) shows that, despite the many new discoveries (and many fewer retractions) noted in Table 12, each 150K bin has the same completeness limit as before. As one example, consider the bin with the largest change, 600-750K. In both Kirkpatrick et al. (2021a) and here, this bin is complete out to 20 pc, but the number of objects has nonetheless increased from eighty-three in Kirkpatrick et al. (2021a) to ninety-eight in this paper; see also Table 17. (The V / Vmax test is only as robust as the Poisson statistics allow, which is why both \nsets of numbers were deemed to be complete.) As another example, the number of objects interior to the completeness limit of 15.0 pc in the 450-600K bin has increased from fiftythree to fifty-six. \nAs noted in Kirkpatrick et al. (2021a), the V / Vmax test does not check for inhomogeneities in surface area, the most likely cause of which would be confusion along the Galactic plane that hinders our ability to find nearby brown dwarfs. Do the increased densities now reported in this paper indicate that these corrections can be reduced or dropped altogether? \nFigure 16 shows the positions in Galactic coordinates of all 583 L, T and Y dwarfs in the 20-pc census. As was done in Kirkpatrick et al. (2021a), we divide the sky into two zones: a zone along the Galactic plane ( | glat | < 14 . · 48) and another ( | glat | ≥ 14 . · 48) well outside of the plane. This value of | glat | was chosen so that the non-plane zone contains exactly three times the area of the plane zone. If there is no incompleteness along the Galactic plane, then the ratio of non-plane to plane objects should be three. For the volumecomplete portions of our 20-pc census, we find that this ratio is 138 / 44 = 3 . 1 for L dwarfs, 257 / 65 = 4 . 0 for T dwarfs, and 31 / 4 = 7 . 8 for Y dwarfs, suggesting that the Galactic plane does not introduce any significant incompleteness ( < 1%) for L dwarfs but does still impede the discovery of fainter T and Y dwarfs. In contrast, Kirkpatrick et al. (2021a) derived ratios of 137 / 34 = 4 . 0, 234 / 34 = 6 . 9, and 24 / 4 = 6 . 0 for the L, T and Y dwarf samples, respectively. \nIncompleteness along the Galactic plane has improved in the current 20-pc census for the L and T dwarfs. For the Y dwarfs, the view is complicated by smaller number statistics. Taking the non-plane numbers of Y dwarfs as truth, then the number of plane Y dwarfs in the current sample should be 31 / 3 ± ( √ 31) / 3 = 10 . 3 ± 1 . 9, which is 3 . 3 σ different from the value of 4 actually found. The same computation for the Kirkpatrick et al. (2021a) numbers gives a number of plane Y dwarfs that was only 2 . 5 σ different. Hence, the underdensity of Y dwarfs in the plane is now significantly worse, due to the fact that all new discoveries of Y dwarfs within the volume have been found outside of the plane zone. \nL dwarfs no longer show an underdensity in the plane, so no correction is needed for our derived L dwarf space densities. T dwarf space densities should, however, be multiplied by 1.06 to account for the observed incompleteness. The Y dwarf incompleteness is harder to assess given the small number of Y dwarfs in the plane, but the raw numbers suggest a conservative correction factor of 1.15, slightly larger than the 1.13 factor adopted by Kirkpatrick et al. (2021a). These factors are listed in Table 17. \nThe final step in measuring the space densities of L, T, and Y dwarfs is assessing their measurement uncertainties. For this we adopt the same methodology used in Kirkpatrick et al. (2021a). To summarize, our confidence in assigning an object to a T eff bin is directly related to the measurement uncertainty on T eff, which is often comparable to the bin size itself. To estimate our confidence in the numbers of objects in each bin, we have run simulations with 10,000 Monte \nFigure 15. The average V / V max value in 0.5-pc intervals across fourteen 150-K bins encompassing L, T, and Y dwarfs. Blue dots show the empirical sample, and red labels denote the number of objects at each 0.5-pc computation. The black dashed line shows the ⟨ V / Vmax ⟩ = 0 . 5 level indicative of a complete sample. The grey error bars show the approximate 1 σ range that a sample of the size shown in red would exhibit, given random statistics. The brown error bars, offset by +0.05 pc from the grey error bars for clarity, show the 1 σ variation obtained by simulations using 10,000 Monte Carlo realizations having the number of objects and completeness limit listed in Table 17. See section 8.2 of Kirkpatrick et al. (2021a) for more details. \n<!-- image --> \nFigure 16. Plots of the 20-pc L, T, and Y dwarf census in Galactic coordinates. The four panels display (a) the sample in its entirety (black), (b) only the L dwarfs (blue), (c) only the T dwarfs (green), and (d) only the Y dwarfs (red). New additions to the sample since Kirkpatrick et al. (2021a) are plotted with grey haloes in panels (b) through (d). \n<!-- image --> \nTable 17. Space Densities for Early-L through Early-Y Dwarfs \nb This value is computed via the equations \ndens = ( raw )( corr ) / ( 4 3 π dmax 3 ) \nand \nFigure 17. Our measured L, T, and Y dwarf space densities from Table 17 (black dots) as a function of effective temperature overplotted on different simulations from Raghu et al. (submitted). In all panels, simulations assuming a constant birthrate are shown, along with the results for three different low-mass cutoffs: 10 MJup (light blue), 5 MJup (green), and 1 MJup (red). Panels in the left column use the Saumon & Marley (2008) evolutionary models, and panels in right column use Baraffe et al. (2003). The top row shows simulations with a power law of α = 0 . 8, the middle row shows α = 0 . 6, and the bottom row shows α = 0 . 4. The best overall fits are those shown in the left panel in the middle row, using α = 0 . 6 and the Saumon & Marley (2008) models. \n<!-- image --> \noutside the completeness limit of the colder bin. This last loss is one-sided, however, as any colder objects scattering into the warmer bin would be necessarily retained. Hence, we compute our adopted space densities using the raw number counts, but including the uncertainties derived from our simulations, as shown in the footnote of Table 17. These densities are graphically illustrated in Figure 17. \nThe measured space densities can now be compared to the simulated T eff distributions (Raghu et al., submitted) to infer the form of the mass function at this low-mass end. Following on the results of Kirkpatrick et al. (2021a), which showed the best match to be a power law, dN / dM ∝ M -α , with α ≈ 0 . 6, Raghu et al. (submitted) assume power-law functional forms with α values between 0.3 and 0.8 and, like \nσ dens = √ ( σ raw 2 + σ ad j 2 ) ( corr ) / ( 4 3 π dmax 3 ) \nwhere σ raw = √ raw . \nCarlo realizations wherein we take the uncertainty in T eff and multiply it by a random value generated from a normal distribution having a mean of 0 and a standard deviation of 1. For each simulation, this uncertainty is added onto the measured value and the object (re-)assigned to the appropriate T eff bin. The computed means and standard deviations across all 10,000 realizations are given in column 5 of Table 17. We use only these computed standard deviations in our adopted space densities, but not the adjusted means. As further explained in Kirkpatrick et al. (2021a), the reason for this is that the number of objects is not preserved across the Monte Carlo simulations because some objects scatter into the hotter, incomplete bin at 2100-2250K and are lost, while objects at the other temperature extreme may be lost because they fall \nKirkpatrick et al. (2021a), choose low-mass cutoffs of ∼ 1, 5, and 10 MJup . Unlike Kirkpatrick et al. (2021a), however, they vary the birthrate to include not only a constant birthrate over the lifetime of the Milky Way, but also consider two other birthrates - called inside-out and late-burst - from Johnson et al. (2021) that are constrained by new results from Gaia. The inside-out birthrate represents a declining birthrate over the 10 Gyr lifetime of the Galactic disk, and the late-burst birthrate is identical to the inside-out form, except with an abrupt increase (by a factor of ∼ 3) in star formation ∼ 3-5 Gyr ago. \nEvolutionary models are used to infer the current T eff value of each simulated object (using its mass and age). Raghu et al. (submitted) expand the model set used in Kirkpatrick et al. (2021a) by including the newer Marley et al. (2021) predictions and show (again) that the only evolutionary models able to fit the bump in the L/T transition in the T eff distribution are those of Saumon & Marley (2008). \nIt has been shown in Kirkpatrick et al. (2021a) and Raghu et al. (submitted) that the low-mass cutoff has little effect on the shape of the mass function at T eff values above 450K, where our fitting is taking place. Therefore, we consider each α + birthrate pair and compute the median of the least squared values for the simulations across all three cutoff masses. The minimum is achieved for α = 0 . 6 and a constant birthrate, identical to the findings in Kirkpatrick et al. (2021a). The second best fit is achieved for α = 0 . 5 and a constant birthrate. The third best fit is a tie among the α = 0 . 7 + constant, the α = 0 . 4 + late-burst, and α = 0 . 5 + late-burst models. Use of either the late-burst or inside-out birthrates results in a slightly reduced α because those birthrates create a small overabundance, relative to the constant birthrate models, of older brown dwarfs that have already cooled to cooler temperatures. \nIn Figure 17, we show the fits for three values of α (0.4, 0.6, and 0.8) all paired with a constant birthrate. The panels in the left column of the figure show that the α = 0 . 6 model with a constant birthrate and using the Saumon & Marley (2008) evolutionary models is an excellent representation of the empirical data. Can any new conclusions be gleaned regarding the low-mass cutoff? As figure 4 of Raghu et al. (submitted) illustrates, the Saumon & Marley (2008) models are incomplete below masses of ∼ 0.015 M ⊙ ( ∼ 16 MJup ), so they are a poor choice for determining what the low-mass cutoff might be. Instead, we revert back to the Baraffe et al. (2003) models, that are complete down to ∼ 5 MJup . As the rightmost panels in Figure 17 illustrate, our ability to distinguish between low-mass cutoffs depends on measuring accurate space densities below 450K. Using the 20-pc census to say confidently that star formation\'s terminus is below 10 MJup or even 5 MJup depends on surveying the sky more deeply at the wavelengths of these objects\' peak emission and obtaining the necessary astrometry to measure accurate distances. As the simulations using the Baraffe et al. (2003) models show, measuring an accurate space density for the 300-450K bin will allow us to distinguish between the cutoff masses, and even a few more objects discovered in the \nFigure 18. Plots of the implied space densities of brown dwarfs (brown) in 0.001 M ⊙ bins compared to the measured space densities of other possible low-mass star formation products from Table 15 (blue). The brown dwarf space densities are divided into three mass zones M > 10 MJup (solid brown), 5 MJup < M < 10 MJup (dashed brown), and 1 MJup < M < 5 MJup (dotted brown). Note that the densities of the possible pseudo-exoplanets do not affect our measurement of the brown dwarf space densities, as their numbers only become appreciable at masses well below 5 MJup . \n<!-- image --> \n150-300K bin, which currently has only the 250K Y dwarf WISE J085510.83 -071442.5 in it, will provide even tighter constraints. \nHave some of these ultra-low mass products of star formation already been identified, and are they masquerading in the literature as exoplanet discoveries to higher mass objects? We use our analysis in Table 15 to see first if the omission of these objects has biased our derivation of the brown dwarf mass function above. With the exception of the two objects (the companions to UCAC4 211-005570 and L 119213) lacking mass estimates, we take all objects labeled as "consider" in column 4 of Table 15 and estimated their contribution to the overall mass function. For objects with M sin( i ) measurements only, we pull a random number from a distribution of values uniformly distributed between 0 and 1 and multiply that number by 90 degrees to assign each an inclination, which we then use to assign an actual mass value. For all masses, whether or not they are true masses or adjusted M sin( i ) measurements, we then pull a random number from a normal distribution with a mean of 0 and standard deviation of 1 and multiply that number by the uncertainty, which we then add back to the mass value. We perform this methodology over 10,000 Monte Carlo iterations and find the mean and standard deviation of the resulting space density, binned over 0.001M ⊙ mass intervals, as illustrated in Figure 18. \nThis figure shows that our derived space density of brown dwarfs, (which we find to be ξ ( M ) = dN / dM = 0 . 0469 × M -0 . 6 in units of # pc -3 [1 M ⊙ ] -1 , with M in units of M ⊙ ; see \nSection 8), overwhelms the space density above 5 MJup where our fitting took place. So, the omission of these objects has no impact on our derivation. However, the second question is whether any of these objects could be products of star formation itself rather than the secondary by-products of a protoplanetary disk. That question cannot be answered from this diagram, but it is a statistical certainty that at least a few of the objects on the high-mass tail of this distribution are star formation products. One striking result from Figure 18, however, is the high space density of objects in the lowest mass bin, given that the census of such low-mass objects, whether resulting from star formation or proto-planetary disks, is still woefully incomplete. There is clearly no shortage of ultralow mass objects in the Milky Way.', '7.2. Combined stellar and brown dwarf space densities': "With the brown dwarf portion of the mass function now fitted, we can combine the stellar and brown dwarf portions to determine the shape of the overall mass function. \nFirst we take the number counts across the stellar regime and perform a similar Monte Carlo analysis as was done on the brown dwarfs. Specifically, for each object we pull a random number from a normal distribution with a mean of 0 and a standard deviation of 1. We then multiply the mass measurement uncertainty by the random number and add that back to the mass value to get a true mass. We do this for each of the stars in our sample, and repeat the process 10,000 times to simulate 10,000 possible histograms. We then compute the mean value in each histogram bin along with its standard deviation. Because the 20-pc volume around the Sun is just one of many such volumes that can be taken as a sample of the Milky Way, we add the Poisson uncertainty and the standard deviation from above in quadrature to provide the final uncertainty per bin. (This parallels the brown dwarf space density analysis of Table 17.) \nWe can now append the substellar contribution onto this stellar distribution. To do this, we look at the predictions from the best fit Raghu et al. (submitted) model to the brown dwarf T eff distribution from above, which is the α =0 . 6 power law with a constant birthrate function and passed through the Saumon & Marley (2008) evolutionary models. We also choose a 0.005 M ⊙ ( ∼ 5 MJup ) cutoff to parallel the more detailed cutoff analysis from Kirkpatrick et al. (2021a). This simulation gives the predicted mass distributions shown in Figure 19. Each histogram is scaled so that the total number of objects in each histogram matches the raw numbers of objects per bin listed in Table 17. As one example, the 27 objects in the 1950-2100K bin are predicted to fall almost exclusively in the 0.075-0.080 M ⊙ bin, and these predictions suggest that our 27 objects be apportioned as 20.5 objects in the 0.075-0.080 M ⊙ bin, 2.0 objects in the 0.0700.075 M ⊙ bin, 1.0 object in the 0.065-0.070 M ⊙ bin, and fractional numbers of objects in bins of lower mass. As another example, the 63 objects in the 750-900K bin are spread over a wide range of masses from 0.005-0.060 M ⊙ and are apportioned as 2.6 objects in the 0.055-0.060 M ⊙ bin, 10.8 objects in the 0.050-0.055 M ⊙ bin, 12.3 objects in the 0.045- \n0.050 M ⊙ bin, 10.7 objects in the 0.040-0.045 M ⊙ bin, 8.3 objects in the 0.035-0.040 M ⊙ bin, etc. \nWe take the apportionment across all thirteen temperature bins and tally the results in each of the 0.005 M ⊙ -wide mass bins, after also applying the factor ( corr in Table 17) to correct for losses of objects along the Galactic plane and extrapolating the numbers to the full 20-pc volume if that temperature bin was not complete to 20 pc. For example, the raw number counts in the 450-600K bin shown in both Table 17 and Figure 19 were multiplied by the 1.06 correction factor then multiplied by (20 / 15) 3 to extrapolate to the full volume. In Table 17, we find that our lowest temperature bin with a space density measurement, 300-450K, is considered to be incomplete, and the mass distribution for that bin in Figure 19 suggests that that bin's objects fall exclusively below 0.025 M ⊙ . Therefore we consider any space density measurements below this mass value to be lower limits only. \nWe now add these brown dwarf masses to the results of our Monte Carlo analysis of stellar masses above to produce a mass function across the entire mass range. This initial mass function is illustrated in Figure 20. Panel (a) shows the mass function across the full mass range from 0 to 8 M ⊙ , binned in 0.1 M ⊙ increments. The mass function rises with decreasing mass, and it continues to rise beyond our 0.025 M ⊙ ( ∼ 26 MJup ) completeness limit. Subsequent panels show details. Panel (b) shows the high-mass end of the initial mass function from 1.5 to 8.0 M ⊙ , again with 0.1 M ⊙ binning. The statistics above 3 M ⊙ are poor but nonetheless show a steady increase from there down to 1.5 M ⊙ . Panel (c) shows the mid-mass range (0 . 4 < M < 1 . 5 M ⊙ ), now binned into smaller 0.02 M ⊙ increments because the statistics here are richer. Panel (d) zooms in on the smallest mass portion, below 0.4 M ⊙ , and chooses yet a smaller mass binning of 0.005 M ⊙ . With the exception of a few small features (discussed below), the mass function is seen to rise monotonically from 1.5 to 0.025 M ⊙ . Mostly within the measurement errors (see more discussion below), the initial mass function is seen to continue rising well below our 0.025 M ⊙ completeness limit and at least down to 0.015 M ⊙ . \nThe numbers on which Figure 20 is based are given in Table 18. For ease of reference, both the number of stars and the space density is given for each mass bin. Three mass binnings are tabulated, roughly paralleling what is shown in Figure 20: 0.1 M ⊙ binning across the entire 0.0-8.0 M ⊙ range (80 bins), 0.02 M ⊙ binning across the range 0.0-1.6 M ⊙ (80 bins), and 0.005 M ⊙ binning across the range 0.0-0.4 M ⊙ (80 bins). Mass bins with incomplete statistics are labeled as lower limits in the final column of the table. \nThere are a few features in Figure 20(c) and (d) that warrant special attention. The first is the bump in the object counts near 0.55 M ⊙ in panel (c). This falls near the point at which our mass estimation switches from that of the TIC relations of Stassun et al. (2019) at higher masses to that of the MKs relation of Mann et al. (2019) at lower masses. Currently, we switch between these two relations at MKs = 5 . 0 mag, corresponding to a mass of ∼ 0 . 6 M ⊙ . As a test, if we \nSun \nFigure 19. The predicted distributions of brown dwarf masses in each of our 150K effective temperature bins based on the best-fit Raghu et al. (submitted) simulation to our measured L, T, and Y dwarf space densities (see text for details). Each histogram is scaled to match the total number of objects listed for that T eff bin in Table 17. \n<!-- image --> \nTable 18. Number of Objects and Space Densities per Mass Bin for the 20-pc Census \nNOTE-(This table is available in its entirety in machine-readable form.) \n- a The [ M bin] -1 portion of the units should be replaced with the bin size for that row. For example, for the first row of the table, the units will be pc -3 [0 . 10 M ⊙ ] -1 because that bin is 0 . 10 M ⊙ wide. \nchange the switchover point to be at MKs = 4 . 0 mag ( M ≈ 0 . 7 M ⊙ ) instead, we find that this bump in the space densities moves to higher masses, with a deficit around 0 . 8 M ⊙ , as shown in Figure 21. We also note that the uncertainties in the masses resulting from the Mann et al. (2019) relation are three to four times smaller than those derived from the Stassun et al. (2019) relation. As another test, we can artificially inflate the mass uncertainties on the Mann et al. (2019)derived masses while keeping the current switchover point at MKs = 5 . 0 mag. That result is also shown in Figure 21. In this case, the bump is greatly diminished in the number counts, but an inflection point is still seen near 0.6 M ⊙ . Given that this feature in the number counts moves in response to the mass estimation used, we believe it is an artificial effect. Furthermore, given that the Stassun et al. (2019) mass estimation relies on dynamically measured individual masses whereas the Mann et al. (2019) relation uses Bayesian statistics to ferret out individual masses from binaries in which only the total system mass is measured, this likely indicates a small systematic offset that slightly deflates the Mann et al. (2019)-derived masses relative to truth. In fact, an effect in this direction and representing a systematic offset of ∼ 2% is seen when comparing results of the MKs relation to individually derived masses (figure 15 of Mann et al. 2019). Obtaining more directly measured individual masses in this regime, corresponding to late-K and early-M dwarfs, would help to put this issue to rest. \nOther features are seen in Figure 20(d). There is a small drop in the number counts near 0.13 M ⊙ followed by a sudden rise near 0.11 M ⊙ . This feature is likely artificial, as the bump at 0.11 M ⊙ is due primarily to components in multiple systems about which little information is known, and these were arbitrarily assigned masses of 0.11 M ⊙ based on an an- \n0.00 \nFigure 20. The 20-pc initial mass function across all stellar and substellar masses. Our measured values and their uncertainties are shown in black. The raw number counts for stars of type M9.5 and earlier are shown by the blue histogram. (a) The full mass range, 0.0-8.0 M ⊙ , with 0.1 M ⊙ binning; (b) A zoom-in of the high-mass end, from 1.5 to 8.0 M ⊙ , with the same binning; (c) A zoom-in of the mid-range, from 0.4 to 1.5 M ⊙ , with 0.02 M ⊙ binning. (d) A zoom-in of the low-mass portion, 0.0-0.4 M ⊙ , with 0.005 M ⊙ binning. Our fit to the mass function is shown by the orange line. \n<!-- image --> \nticipated spectral type of M5.5. This type lies at a very sharp inflection point (see Table 16) in our mass vs. spectral type relation (Figure 14). \nThe other feature is the rapidly changing number count between masses of 0.06 and 0.08 M ⊙ , a mass range that straddles the stellar/substellar break. Some of the early- to mid-L dwarfs that we have included in our brown dwarf mass function analysis are likely very low-mass stars and not brown dwarfs themselves. As a consequence, these are assigned masses that are a bit too high (the extraneous high point in the 0.075-0.080 M ⊙ bin), which likely leads to concomitant deficits in the next higher mass bins. In fact, Table 4 lists three early-L dwarfs in the 20-pc census that have dynamical mass measurements, and one of these (LP 388-55 B; Dupuy \n& Liu 2017) has a mass just above the 0.075-0.080 M ⊙ mass bin (0.083 ± 0.03 M ⊙ ).", '8. DISCUSSION': 'How do our 20-pc results compare to other attempts in the literature to measure the initial mass function? Pioneering work by Salpeter (1955) found that a power law form ξ ( M ) = dN / dM ∝ M -α with α =2 . 35 best fit the initial mass function over the range 0 . 3 ≤ M ≤ 10 M ⊙ . Subsequent work by Miller &Scalo (1979) found α =1 . 4 for 0 . 1 ≤ M ≤ 1 M ⊙ and α =2 . 5 for 1 ≤ M ≤ 10 M ⊙ . Scalo (1986) determined α = 2 . 7 for 2 ≤ M ≤ 10 M ⊙ , and Reid et al. (2002) found α = 1 . 3 for 0 . 1 ≤ M ≤ 0 . 7 M ⊙ . \nAs more accurate measurements of the initial mass function became possible, researchers realized the importance of \n<!-- image --> \n<!-- image --> \nFigure 21. Tests of the bump in the initial mass function seen near 0.55 M ⊙ in Figure 20. (a) A zoom-in showing the bump in Figure 20(c). (b) The number counts over the same mass range but where we have moved the switchover in the mass estimate from MKs = 5 . 0 mag to MKs = 4 . 0 mag. (c) The number counts over the same mass range but where we have kept the MKs = 5 . 0 mag switchover point and inflated the uncertainties on the MKs -derived masses by a factor of four. For other details, see the caption to Figure 20. \n<!-- image --> \ndistinguishing whether the masses used for the computation were that of the stellar system or of its individual components. For example, in an analysis of data from the Sloan Digital Sky Survey, Bochanski et al. (2010) found α = 2 . 38 for 0 . 32 < M < 0 . 80 M ⊙ and α = 0 . 35 for 0 . 10 < M < 0 . 32 M ⊙ for the mass function of systems but α = 2 . 66 and α = 0 . 98 for the mass function of single stars over the same two mass regimes, respectively. Earlier, Kroupa (2001) had found that single-star initial mass functions resulting from the analysis of young star clusters generally gave values of α that were higher by ∼ 0.5 (for 0 . 1 < M < 1 . 0 M ⊙ ) than the field initial mass function, for which systems were not resolved. Reid (2005) cautions that unresolved multiplicity complicates interpretation of the initial mass function; the initial mass function of systems is more directly tied to the fragmentation of the original molecular cloud, but the initial mass function of individual objects give the mass distribution of the (sub)stellar bodies formed. \nOur work on the 20-pc census has concentrated on analysis of the individual products of star formation, as we are curious to know how frequently this process produces, for example, very low-mass brown dwarfs compared to higher-mass objects. We will therefore restrict subsequent analysis here to the single-object initial mass function and leave analysis of the 20-pc census regarding multiplicity and the system mass function to those investigating how the formation of systems relates back to cloud fragmentation.', '8.1. Comparison of Initial Mass Functions': "Here we compare two very well established forms of the initial mass function and compare their predictions to our results based on the 20-pc census. \nChabrier (2001, 2003a,b) developed several functional forms for the initial mass function. The latest one relating to single objects is given by Chabrier (2003b) and is comprised of a power law at high masses and a log-normal form \nat lower masses: \nξ ( M ) = C 1 ln10 M -α , for M > 1 . 0 M ⊙ = C 2 M ln10 e -(log M -log Mc ) 2 2 σ 2 , for M ≤ 1 . 0 M ⊙ (19) \nwhere C 1 = 0 . 0443, α = 2 . 3, C 2 = 0 . 158, Mc = 0 . 079 M ⊙ , and σ = 0 . 69 M ⊙ . The value of ξ ( M ) is in units of # of objects per pc 3 per M ⊙ . See equation 2 and table 1 of Chabrier (2003b) for the derivation shown above. We note that the values of C 1 and C 2 are set by Chabrier (2003b) to match an empirical space density measurement (at 1 M ⊙ from Scalo 1986) of the initial mass function in the Milky Way's disk population. \nLikewise, Kroupa et al. (2013) found that a tripartite power-law form best describes the single-object initial mass function: \nξ ( M ) = C ( 0 . 5 0 . 07 ) -α 2 ( M 0 . 5 ) -α 1 , for 0 . 5 < M < 150 M ⊙ = C ( M 0 . 07 ) -α 2 , for 0 . 07 < M < 0 . 5 M ⊙ = C 3 ( M 0 . 07 ) -α 3 , for 0 . 01 < M < 0 . 15 M ⊙ (20) \nwhere α 1 = 2 . 3, α 2 = 1 . 3, and α 3 = 0 . 3. As above, the value of ξ ( M ) is in units of # of objects per pc 3 per M ⊙ . Note that there are two components of this mass function that contribute to the mass range 0 . 15 < M < 0 . 07 M ⊙ . The value of C is not specified by Kroupa et al. (2013). However, figure 4-24 of Kroupa et al. (2013) provides a comparison of this initial mass function with the Chabrier (2003b) version in Equation 19, showing that they are identical at ∼ 0.85 M ⊙ , resulting in a value of C ≈ 1 . 15. \nAs shown in Figure 22, neither of these parameterizations adequately describes the 20-pc initial mass function derived here. The main reason for this is that prior determinations were done pre-WISE, pre-Spitzer, and pre-Gaia before the L, \nT, and Y dwarf complement of the mass function was fully characterized and before exquisite parallax determinations became available for almost all objects in the 20-pc volume. As such, these prior works relied on poorer statistics with fits done in logarithmic scaling on both the dN and dM axes. Our careful accounting of objects within the 20-pc volume now allows for a more precise determination of the single-object initial mass function. \nWe thus provide a new multi-part power law parameterization that is bounded by the following caveats: (1) We assume α = 2 . 3 at the high mass end, as has been determined from earlier studies, and we do this because our 20-pc census has few stars with M > 2 M ⊙ to better constrain this. (2) We assume α = 1 . 3 at intermediate masses, as this has also been established by earlier studies. (3) We take α = 0 . 6 in the brown dwarf regime, as was determined in Section 7.1. We do not constrain the stitch points in mass between the power law pieces nor do we limit the number of power law pieces to only three. We perform these fits by eye, keeping in mind the caveats from Section 7.2 concerning the non-physical bumps and troughs in the number counts as a function of mass. Given the constraints above, we find that a three-piece power law does not adequately describe the number counts in the mid- to late-M dwarf regime (0 . 08 ≲ M ≲ 0 . 20 M ⊙ ), but that a four-piece power law can. This best fit is given below and illustrated by the orange line in Figure 20: \nξ ( M ) = C 1( M ) -α 1 , for 0 . 55 < M < 8 . 0 M ⊙ = C 2( M ) -α 2 , for 0 . 22 < M < 0 . 55 M ⊙ = C 3( M ) -α 3 , for 0 . 05 < M < 0 . 22 M ⊙ = C 4( M ) -α 4 , for 0 . 01 < M < 0 . 05 M ⊙ (21) \nwhere C 1 = 0 . 0150, α 1 = 2 . 3, C 2 = 0 . 0273, α 2 = 1 . 3, C 3 = 0 . 134, α 3 = 0 . 25, C 4 = 0 . 0469, α 4 = 0 . 6, As above, the value of ξ ( M ) is in units of # of objects per pc 3 per M ⊙ . \nIf we integrate under our best fit from 0.075 to 8.0 M ⊙ , we find 3002 stars, which can be compared to the 3000 individual objects in the 20-pc census that have (measured or implied) types of M9.5 or earlier. The integration under our fit for 0.020 to 0.075 M ⊙ gives 789 brown dwarfs, compared to the 582 individual L, T, and Y dwarfs in Table 4. Most of the missing ∼ 200 brown dwarfs are ones with T eff values 450-600K and distances of 15-20 pc or ones with 300-450K temperatures and 11-20 pc distances, where our current accounting is known to be incomplete (see Table 17). These results show that the number of stars relative to brown dwarfs is 3002/789, or ∼ 4. However, we believe that the brown dwarf mass function extends to at least 0.010 M ⊙ , which would give a ratio of 3002/986 ( ∼ 3) if the α = 0 . 6 functional form continues to that mass. In the limiting case in which it continues to zero mass, we find a star-to-brown-dwarf ratio of 3002/1602, or ∼ 2. We note that as recently as a decade ago, this ratio in the Solar neighborhood was believed to be as high as 6:1 (Kirkpatrick et al. 2012). \nThis decrease in the ratio of stars to brown dwarfs is not in tension with microlensing results, as analysis of OGLE data found that an even steeper power law in the brown dwarf \nregime ( α = 0 . 8 for 0 . 01 < M < 0 . 08 M ⊙ ; Mróz et al. 2017) best fits the observed distribution of short-timescale (lowmass) events. Furthermore, the possibility that the mass function extends into a regime significantly below 0.010 M ⊙ is bolstered by recent JWST observations of the Orion Nebula Complex that show a significant population of objects, down to at least 0.001 M ⊙ , that are apparently direct products of star formation (McCaughrean & Pearson 2023; Pearson & McCaughrean 2023). \nFinally, we note that our accounting of the mass of hydrogen-burning stars in the 20-pc census along with our best fit to the mass function of brown dwarfs allows us to calculate the average mass of an object in this sample. Integrating our mass function down to a mass of 0.020 M ⊙ , we find that value to be 0.41 M ⊙ . There are likely many undiscovered brown dwarfs in the solar neighborhood too faint to be currently detected, so this average value could be pushed lower. Assuming there is no low-mass cutoff of star formation and the brown dwarf mass function continues to zero mass with a power law slope of α = 0 . 6, we find that the average mass of objects in the 20-pc census would drop to 0.34 M ⊙ . This can be considered as the limiting case unless the functional form at the lowest masses has an α value considerably greater than 0.6.", '9. CONCLUSIONS': 'In this paper our aim was to study the initial mass distribution of star formation\'s by-products, from the highestmass progenitors of present-day white dwarfs to the lowestmass brown dwarfs. For this, we have produced a volumecomplete sample of stellar and substellar objects within 20 pc of the Sun. We have split multiple systems into their separate components and characterized each individual object to provide an accurate mass assignment. \nOur main conclusions can be summarized as follows: \n1) The initial mass function steadily increases as a function of descending mass. Its peak in (linear) mass space is not yet defined but is located below 0.020 M ⊙ ( ∼ 20 MJup ; Figure 20). We find that a quadripartite power-law ( ξ ( M ) = dN / dM ∝ M -α ) fits the observed space densities well (Equation 21). Going from high mass to low mass, we find exponents of α = 2.3, 1.3, 0.25, and 0.6, with stitch points between segments of 0.55, 0.22, and 0.05 M ⊙ , respectively. Although the rate of ascent of the mass function is slowly retarded as a function of descending mass through the stellar and high-mass brown dwarf regimes, its ascent increases again for the lower-mass brown dwarfs. \n2) This initial mass function agrees well with previous determinations in the high-mass regime (by design) but differs markedly from other established formalisms in the M, L, T, and Y dwarf regimes (Section 8). Functional forms proposed by Chabrier (2003b) and Kroupa et al. (2013) overpredict the number of these lower-mass dwarfs relative to their more massive counterparts. \n3) The 20-pc census currently consists of ∼ 3000 stars and ∼ 600 brown dwarfs (Table 4). At face value, this implies a stellar-to-substellar ratio of ∼ 5, but corrections for in- \n<!-- image --> \n<!-- image --> \nFigure 22. A comparison of our 20-pc number counts (black points with error bars) and our fit of the initial mass function (solid orange line) to the functional forms of Kroupa et al. (2013) (dotted blue line) and Chabrier (2003b) (dashed green line). Each panel shows a zoom-in of a different mass segment: (a) 1 . 5 < M < 8 . 0 M ⊙ , (b) 0 . 4 < M < 1 . 5 M ⊙ , (c) 0 . 0 < M < 0 . 4 M ⊙ . \n<!-- image --> \nteness for brown dwarfs with temperatures from 300600K shows that the ratio is currently measured at ∼ 4. Incompletenesses at lower temperatures may yet bring this ratio as low as ∼ 3 (Section 8). The average mass of objects in the 20-pc census is currently measured as 0.41 M ⊙ but could drop as low as 0.34 M ⊙ if many colder brown dwarfs, yet to be discovered, actually exist (Section 8). \n4) The 20-pc census of objects colder than 600K, corresponding to spectral type ∼ T8.5, is still incomplete beyond 15 pc, and the completeness volume shrinks to 11 pc at 450K, corresponding to spectral type ∼ Y0 (Table 17). Moreover, an additional source of incompleteness for objects as warm as 1350K is high backgrounds and confusion along the Galactic plane (Section 7.1). \n5) There are direct indications that many unrecognized companions still exist to already identified members of the 20-pc census. Acceleration (aka proper motion anomaly) has been used to flag many such systems (Section 5.1.1), and large Gaia RUWE/LUWE values significantly higher than 1.0 flag many others (Section 5.2). Additional follow-up of these systems would help to better flesh out the 20-pc census itself while also providing much firmer statistics on multiplicity and the prevalence of hierarchical systems. \n6) Our "complete" (see caveats #4 and #5, above) 20-pc census produced for this paper is available for additional uses (Table 4). As one example, this nearby sample is particularly useful as the hunting grounds for the closest habitable worlds to our own Solar System and is thus also available via the NASA Exoplanet Archive 44 . \n7) Except for white dwarfs (Section 6.1.1) and brown dwarfs (Section 6.1.3), masses are used directly when they have been measured (Section 4). Most objects, though, lack actual mass determinations. For these we use a variety of \nmass estimation methods and select the ones that provide the most reliable results, when comparison to truth is available (Section 6.3). Nonetheless, our resulting space density computations binned by mass show some spurious features that appear to be caused by shortcomings in the estimation method. These are most obvious in the early-M dwarf region and in the regime from late-M to early-L dwarfs (Section 7.2). Dedicated programs, such as those by Vrijmoet et al. (2022) and Dupuy et al. (2022) directly determining masses of objects in these zones are clearly needed. \n8) Dueling definitions in the literature of "brown dwarf" and "exoplanet" could bias our results if objects tagged as exoplanets are omitted from the initial mass function. We account for this and find that most of the objects labeled exoplanets (via the 13 MJup -based definition) fall in a small-mass regime separate from the objects that have been more traditionally labelled as brown dwarfs (via the formation-based definition) and do not affect our conclusions regarding the mass function (Section 6.2.3). This having been stated, future studies of the initial mass function might wisely consider no such division, as planet formation can be thought of merely as a (delayed) secondary process resulting from star formation itself. Including planetary formation products as another branch of the initial mass function will, however, not be feasible until a statistically robust, volume-complete set of exoplanets can be reliably measured. \n9) Gaia DR3 detections comprise only ∼ 75% of the volume-complete 20-pc census. Objects within 20 pc of the Sun can be missed by Gaia because they are too bright for Gaia observations, too faint for Gaia to detect, or are companions to Gaia-detected host stars that are (presently) inadequately characterized astrometrically (Section 2.1). \n10) Citizen science continues to produce new discoveries within 20 pc (Section 2.1.2), even including a possible new Y dwarf with a "bonus" Spitzer parallax (Section A.3). Such discoveries are coming largely from WISE data sets, but these sets are being pushed to their sensitivity limits. Com- \npleting the 20-pc census in the 300-600K temperature range will require a deeper survey at ∼ 5 um, the best prospect for which is the NASA mission NEO Surveyor (Kirkpatrick et al. 2019b; Mainzer et al. 2023). Results from that mission, along with searches for cooler companions to known 20-pc hosts with JWST, represent our best short-term prospects for determining the occurrence rate of objects such as WISE J085510.83 -071442.5 that reside below 300K. \n11) The 20-pc census enables us to identify the nearest star or brown dwarf in each constellation (Table B6). Interestingly, six of the eighty-eight constellations have a Y dwarf as their nearest member despite the fact that Y dwarfs have not yet been fully mapped within this 20-pc volume. \nNOTE ADDED IN PROOF: The nearby brown dwarf candidate CWISE J165909.91 -351108.5 from Table 2 has been confirmed as a late-L dwarf by Robbins et al. (2023a), but it likely falls outside the 20-pc census. Additionally, Robbins et al. (2023b) find that CWISE J105512.11+544328.3, from Table 14, is not a T subdwarf but rather an anomalously blue Y dwarf. \nACKNOWLEDGMENTS: Davy Kirkpatrick, Federico Marocco, and Chris Gelino acknowledge support from grant #80NSSC20K0452 awarded for proposal 18-2ADAP180175 under the NASA Astrophysics Data Analysis Program. Data presented here are based on observations obtained at the Hale Telescope, Palomar Observatory as part of a continuing collaboration between the California Institute of Technology, NASA/JPL, Yale University, and the National Astronomical Observatories of China. Some of the data presented herein were obtained at the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W. M. Keck Foundation. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Maunakea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain. This research has made use of the Keck Observatory Archive (KOA), which is operated by the W. M. Keck Observatory and the NASA Exoplanet Science Institute (NExScI), under contract with the National Aeronautics and Space Administration. The first author would like to thank Patrick Shopbell at Caltech for resurrecting an Exabyte drive that successfully read raw CTIO data from 1997. Part of this research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration (80NM0018D0004). He would also like to thank Mike Cushing for discussion of evolved star loci in the 20-pc colormagnitude diagrams. \nOne observation reported in this paper was obtained with the Southern African Large Telescope (SALT) under program 2021-2-SCI-027 (PI: Faherty). Roberto Raddi acknowledges support from Grant RYC2021-030837-I funded by MCIN/AEI/ 10.13039/501100011033 and by \'European Union NextGenerationEU/PRTR\', and partial support by the AGAUR/Generalitat de Catalunya grant SGR-386/2021 and by the Spanish MINECO grant PID2020-117252GBI00. Eileen Gonzales acknowledges support from the Heising-Simons Foundation through a 51 Pegasi b Fellowship. This publication makes use of VOSA, developed under the Spanish Virtual Observatory (https://svo.cab.inta-csic. es) project funded by MCIN/AEI/10.13039/501100011033/ through grant PID2020-112949GB-I00. This research made use of the Montreal Open Clusters and Associations (MOCA) database, operated at the Montréal Planétarium (J. Gagné et al., in preparation). Backyard Worlds research was supported by NASA grant 2017-ADAP17-0067 and by the NSF under grants AST- 2007068, AST-2009177, and AST2009136. Johanna Vos acknowledges support from a Royal Society - Science Foundation Ireland University Research Fellowship (URF \\ 1 \\ 221932). \nFacilities: WISE, Gaia, Spitzer (IRAC), Hale (WIRC, DBSP, TSpec), Keck:I (MOSFIRE, NIRES), IRTF (SpeX), Gemini:South (FLAMINGOS-2), Blanco (RCSPec, ARCoIRIS), NTT (SOFI), Magellan:Baade (FIRE), Shane (Kast), SALT (RSS), SOAR (OSIRIS), ARC (TSpec). \nSoftware: WiseView (Caselden et al. 2018), Spextool (Cushing et al. 2004; Vacca et al. 2003), MOPEX/APEX (Makovoz & Khan 2005; Makovoz & Marleau 2005), IRAF (Tody 1986, 1993), kastredux (Burgasser et al., in prep.), FIREHOSE/MASE (Bochanski et al. 2009), crowdsource (Schlafly et al. 2018).', 'REFERENCES': "Abazajian, K. N., Adelman-McCarthy, J. K., Agüeros, M. A., et al. 2009, ApJS, 182, 543. doi:10.1088/0067-0049/182/2/543 \nAberasturi, M., Burgasser, A. J., Mora, A., et al. 2014, AJ, 148, 129. doi:10.1088/0004-6256/148/6/129 Aberasturi, M., Caballero, J. 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R., et al. 2016, ApJS, 224, 2. doi:10.3847/0067-0049/224/1/2\n- Huber, D., Matthews, J. M., Croll, B., et al. 2009, A&A, 505, 715. doi:10.1051/0004-6361/200912139 \nHumphreys, A., Meisner, A. M., Burgasser, A. J., et al. 2023, Research Notes of the American Astronomical Society, 7, 184. doi:10.3847/2515-5172/acf4a0", 'APPENDIX': '(FLAMINGOS-2; Eikenberry et al. 2006, Elston et al. 2003, Jannuzi & Bechtold 2004) at the 8.1-m Gemini-South Observatory on Cerro Pachón, Chile, on the nights of 2014 December 01, 2015 June 30, 2019 April 14, 2019 April 29, 2021 February 22, 2021 July 17, 2021 July 21, and 2021 October 18 UT. Photometric acquisition and reductions followed the procedures described in section 9.1 of Meisner et al. (2020a). \nA.1.4. SOAR/OSIRIS \nOne object was observed with the Ohio State Infra-Red Imager/Spectrometer (OSIRIS) at the 4.1-m SOuthern Astrophysics Research (SOAR) Telescope on Cerro Pachón, Chile, on the night of 2012 March 12 UT. Photometric acquisition and reduction of this lone H -band data point followed the methodology outlined in section 2.2.5 of Kirkpatrick et al. (2012).', 'A.1.5. Spitzer/IRAC': 'The Spitzer Heritage Archive was queried for directed or serendipitous observations of objects in our 20-pc candidate list (Table 2). The locations of thirty-one of these candidates were found to have Spitzer/IRAC observations in ch1 and/or ch2. The full list is shown in Table A1. Although some of the Spitzer data were too shallow to detect our objects or were obtained at an epoch when our candidate was blended with a background source, most objects had measurable photometry. We used the MOPEX/APEX software (Makovoz & Khan 2005, Makovoz & Marleau 2005) on each Astronomical Observation Request (AOR) to create mosaics, perform source detection, and then measure the photometry using the stack of individual frames. The output of the APEX code is the flux, in µ Jy, for each detection using both aperture and PRFfit techniques. The Spitzer photometry reported in Table 2 is the PRF-fit photometry after converting to magnitudes using the correction factors listed in Table C.1 of the IRAC Handbook 46 along with the flux zeropoints in each band, as given in Table 4.1 of the Handbook. For objects having multiple AORs for a band, the reported photometry in that band is a weighted mean of the individual measurements in each AOR.', 'A.1. Photometry': 'For possible M, L, T, and Y dwarf additions to the 20pc census, we have searched for published near- and midinfrared photometry using online surveys such as the Two Micron All Sky Survey (2MASS; Skrutskie et al. 2006), the VISTA Hemisphere Survey (VHS; McMahon et al. 2013), the UKIRT Hemisphere Survey (UHS; Dye et al. 2018), and the Wide-field Infrared Survey Explorer (WISE; Wright et al. 2010). These photometric measurements are listed in Table 2. In other cases, we have obtained our own ground-based follow-up or have searched the Spitzer Heritage Archive 45 for images with which to measure photometry. These results are also presented in Table 2 but discussed further in the subsections below.', 'A.1.1. Palomar/WIRC': "Eighteen objects were observed with the Wide-field Infrared Camera (WIRC; Wilson et al. 2003b) at the Hale 5-m telescope on Palomar Mountain during the nights 2014 July 03, 2014 September 14, 2016 February 26, 2018 September 01, 2019 July 14, 2020 February 05, 2020 June 03, 2020 July 03, 2020 September 03, 2020 October 09, 2021 July 01, and 2021 August 10 UT. Data were acquired in the Maunakea Observatory filter system's J and H bands. Our standard observing technique, calibration strategy, and reduction methodology have been discussed in section 3.1.5 of Kirkpatrick et al. (2011) with updates as discussed in section 9.2 of Meisner et al. (2020a).", 'A.1.2. Keck/MOSFIRE': "Thirteen objects were observed with the Multi-Object Spectrometer For Infra-Red Exploration (MOSFIRE; McLean et al. 2012) at the 10-m W. M. Keck I telescope on Maunakea, Hawai'i, on the nights of 2021 August 27 and 2022 January 21 UT. Photometric acquisition and reductions followed the procedures described in section 3.1.1 of Schneider et al. (2021).", 'A.1.3. Gemini-South/FLAMINGOS-2': 'Nine objects were observed with the FLoridA Multi-object Imaging Near-infrared Grism Observational Spectrometer 2', 'A.2. Spectroscopy': 'To aid in the characterization of objects, spectroscopy was acquired of 20-pc members discovered by Gaia, 20-pc suspects discovered by the Backyard Worlds: Planet 9 citizen science group, or 20-pc members lacking published spectral types. These reduced spectra are illustrated in Figures A1 through A5. Ten different instruments were used for this follow-up, as detailed below and summarized in Table A2.', 'A.2.1. CTIO/RCSpec': 'Optical follow-up of five objects was obtained on the UT dates of 1995 August 13-14, 1997 July 14, and 1997 July 16 at the Cerro Tololo Interamerican Observatory (CTIO) 4m telescope and on 1996 May 20 at the CTIO 1.5m telescope using the R-C Spectrograph with Folded Schmidt Camera (RCSpec). For the 1995 and 1996 observations, a 300 line mm -1 grating with a GG 495 order-blocking filter was used with the 1024 × 1024 CCD to cover a wavelength range from 6050 to 9550 Å. For the 1997 observations, a 316 line mm -1 grating with an OG 515 order-blocking filter was used with the Loral 3K CCD to cover a useable wavelength range from 5200 to 10000 Å. In addition to the targets, standard calibrations - biases, dome flats, arcs, and flux calibration standards - were also obtained. Reductions were accomplished using the Image Reduction and Analysis Facility ( IRAF ; Tody 1986, 1993), as described in Kirkpatrick et al. (1997).', 'A.2.2. Lick/Kast': 'The Kast Double Spectrograph at the Lick 3m Shane Telescope was used for optical follow-up of six objects on UT dates 2019 September 20, 2020 March 06, 2020 August 1516, 2020 December 14, and 2022 July 02. The only data used were from the red arm, which employed a 600 line mm -1 grating blazed at 7500 Å to cover the wavelength range from ∼ 6000 to ∼ 9000 Å. In addition to standard wavelength and flux calibrations, G2V and A0V stars were obtained near in time and on sky to the targets to correct for telluric absorption. The data were reduced using the kastredux 47 package (Burgasser et al., in prep.), as further described in Schneider et al. (2021).', 'A.2.3. Palomar/DBSP': 'The Double Spectrograph (DBSP; Oke & Gunn 1982) at the Hale 5m telescope on Palomar Mountain was used for sixty additional follow-up spectra. The UT nights of observation were 1995 December 02, 2021 December 06, 2022 January 02, 2022 February 03, 2022 February 07, 2022 May 24, 2022 May 30, 2022 June 07, and 2022 August 27. For the 1995 run, the D68 dichroic was used to split the light at ∼ 6800 Å between the two arms. Gratings with 300 line mm -1 blazed at 3990 Å and with 316 line mm -1 blazed at 7150 Å were used in the blue and red arms, respectively, producing continuous wavelength coverage from ∼ 5100 to \n9200 Å. For the 2021 and 2022 runs, the D55 dichroic was used instead to split the light near 5500 Å. A 600 line mm -1 grating blazed at 3780 Å was used in the blue arm and a 316 line mm -1 blazed at 7150 Å was used in the red arm, producing coverage from ∼ 3300 to 5500 Å on the blue side and from ∼ 5700 to 10000 Å on red side. (For the 2022 June run, fringing in the blue arm caused data shortward of 4500 Å to be unusable.) Standard calibrations and IRAF data reductions were employed, as further described in Kirkpatrick et al. (1991).', 'A.2.4. SALT/RSS': "An additional optical spectrum was acquired with the Robert Stobie Spectrograph (RSS; Burgh et al. 2003; Kobulnicky et al. 2003) on the 11.1 × 9.8-m Southern African Large Telescope (SALT; Buckley et al. 2006) on 2021 December 26 UT. The spectrograph was used in long slit mode using the PG0900 grating at an angle of 20 · , which produces coverage over the ranges 6033-7028, 7079-8045, and 8091-9023 Å across the 3 × 1 mini-mosaic. Our reductions began with the observatory-provided pre-processed data, for which gain correction, correction for cross-talk, and overscan subtraction had been applied. We then wavelength calibrated using neon arc lines obtained immediately after the target's spectroscopic data and flux calibrated using the Hamuy et al. (1994) standard EG21 acquired with the same spectroscopic setup on 2023 January 24 UT.", 'A.2.5. APO/TSpec': 'TripleSpec (Wilson et al. 2004) on the ARC 3.5m telescope at Apache Point Observatory was used for near-infrared follow-up of two objects on the nights of 2018 September 23 and 2019 October 08 UT. The spectrograph provides spectral coverage from 0.95 to 2.46 µ m across five spectral orders. Data were taken with the conventional near-infrared technique of nodded pairs to perform background/bias subtraction, and standard calibrations were also acquired, including quartz lamps for flat fielding and A0 stars for telluric correction and flux calibration. Wavelength calibration was accomplished using night sky lines. Data reduction used Tspectool 48 , a modified version of Spextool (Cushing et al. 2004) rewritten specifically for APO/TripleSpec.', 'A.2.6. CTIO/ARCoIRIS': "Three objects were observed on the nights of 2018 April 02-03 UT using the Astronomy Research with the Cornell Infra Red Imaging Spectrograph (ARCoIRIS) at the 4m Victor Blanco telescope at CTIO. Spectra are acquired across six cross-dispersed orders covering 0.8 to 2.4 µ mat a resolving power of ∼ 3500. Science exposures were taken with AB nod positions along the slit, which has a fixed width of 1 . '' 1. Standard calibrations were acquired as discussed in Greco et al. (2019), and data were reduced using a modified version of \nFigure A1. Optical spectroscopic follow-up for objects classified as early-M through mid-M. Each target object (black) is normalized to one at 7500 Å and overplotted (in other colors) with the spectral standard nearest the same spectral type. Integral offsets have been added to separate the spectra vertically. Target objects are labeled with brief RA/Dec (hhmm ± ddmm) identifiers. The two target objects at upper left 1839+0901 and 0623+1018 - have been smoothed to improve the signal-to-noise in each wavelength bin. Most spectra have not been corrected for earth's atmospheric absorption, so the contaminating B- and A-bands of O2 at ∼ 6850-6900 and ∼ 7600-7700 Å and telluric bands of H2O at ∼ 7150-7300, 8150-8350, and 9000-9600 Å remain. \n<!-- image --> \nthe Spextool package (Cushing et al. 2004), which utilizes A0 stars for telluric correction and flux calibration following the methodology of Vacca et al. (2003).", 'A.2.7. IRTF/SpeX': 'The SpeX instrument on the NASA Infrared Telescope Facility (IRTF) was used for near-infrared spectroscopy of twenty-seven objects over the nights of 2018 June 16, 2018 November 25, 2019 January 23, 2019 March 16-17, 2020 October 30, 2020 November 25, 2021 May 31, 2021 June 30, 2021 September 11, 2021 October 23, 2022 January 09, 2022 January 19, 2022 February 12, 2022 February 21, 2022 March 07, and 2022 March 11 UT. Two different setups were employed. The first, used mainly for brighter targets, was a cross-dispersed mode that provides spectra over the range 0.9-2.4 µ mat a resolving power of R ≡ λ/ ∆ λ ≈ 1200. The second, used primarily for fainter targets, was the prism mode that provides spectra over the range 0.8-2.5 µ mat a resolving power of R ≡ λ/ ∆ λ ≈ 100 -150. As discussed in the subsections above, standard near-infrared calibrations were \nobtained, and data were reduced using Spextool (Cushing et al. 2004, Vacca et al. 2003).', 'A.2.8. Keck/NIRES': 'Three objects were observed over the nights of 2019 February 14, 2020 July 07, and 2021 February 24 UT using the Near-Infrared Echellette Spectrometer (NIRES; e.g., Wilson et al. 2004) at the 10m W. M. Keck II telescope. These data provided spectral coverage from 0.94 to 2.45 µ m. Setup and calibrations were identical to those described in Meisner et al. (2020b), and reductions used a modified version of Spextool (Cushing et al. 2004, Vacca et al. 2003).', 'A.2.9. Magellan/FIRE': 'The Folded-port Infrared Echellette spectrograph (FIRE; Simcoe et al. 2013) at the 6.5 m Walter Baade (Magellan I) telescope at Las Campanas Observatory was used to observe three objects over the nights of 2016 January 23, 2019 December 11, and 2020 February 18 UT. Observations were done in prism mode, which covers the range from 0.80 to \nFigure A2. Optical spectroscopic follow-up for objects classified as mid-M through early-L. Each target object (black) in the two left panels is normalized to one at 7500 Å and overplotted (in other colors) with the spectral standard nearest the same spectral type. In the far right panel, this normalization is done instead at 8250 Å. Offsets in steps of 1.5 have been added to separate the spectra vertically. A few target objects - 1921 -2915 (M7), 1906+4011 (L1), 0033+4340 (L2), and 0617+1945A (L2) - have been smoothed to improve the signal-to-noise in each wavelength bin. The spectrum of 1921 -2915 (M7) also suffers from residual cosmic ray hits. See the caption to Figure A1 for more details. \n<!-- image --> \n2.45 µ m. Standard calibrations were acquired, and data were reduced using the FIREHOSE pipeline, which is based on the MASE (Bochanski et al. 2009) and Spextool (Cushing et al. 2004, Vacca et al. 2003) packages.', 'A.2.10. Palomar/TSpec': "Finally, seven near-infrared spectra were acquired with the Triple Spectrograph (TSpec; Herter et al. 2008) at Palomar Mountain's 5m Hale telescope on the nights of 2018 April 28-29, 2018 October 17, and 2019 September 18 UT. Setup and calibrations were identical to those described in Kirkpatrick et al. 2011. As with many of the other near-infrared spectroscopic data sets discussed above, TSpec data were also reduced with a modified version of Spextool (Cushing et al. 2004, Vacca et al. 2003).", 'A.2.11. Analysis': 'Spectral classification was accomplished by comparing spectra of the target objects to established on-sky anchors for each integral spectral type. For the optical spectra, these anchor points were taken from Kirkpatrick et al. (1991) for objects of type mid-K through late-M and from Kirkpatrick et \nal. (1999) for L dwarfs (Figures A1-A2). Optical classifications of objects earlier than type K (Figure A3) used spectral anchors taken from Gray & Corbally (2009). Near-infrared classification (Figures A4-A5) for M dwarfs, L dwarfs, and early-T dwarfs used the anchors described in Kirkpatrick et al. (2010), with the rest of the T dwarf anchors coming from Burgasser et al. (2006). For more on the methodology employed for both optical and near-infrared classifications, see Kirkpatrick et al. (2010). \nThese classification anchors are generally old disk objects with metallicities similar to the Sun. In a few cases, described below, the target object failed to match an anchor spectrum because of anomalous features attributable to extreme youth, lower metallicity, or other reasons including unresolved binarity. These special cases are addressed further below: \nCWISE J045334.34+203350.2: At J -band, this object best matches the L5 standard, but there are clear discrepancies with the L5 standard across all wavelengths (Figure A4). The continuum of 0453+2033 is much flatter between 1.1 and 1.3 µ m, the FeH band at 0.99 µ m is much stronger, and the H - and K -band portions emit less flux rel- \nFigure A3. Optical spectroscopic follow-up of objects not classified as M dwarfs or L dwarfs. Each target object is normalized to one at its peak flux. Objects in the far left panel are hot stars, and objects in the two right panels are colder stars or other background objects. Integral offsets have been added to separate the spectra vertically. A few spectra - 0213 -3345 (wd), 1955+2224 (mid-F?), 1318+3810 (early-G), and all of those in the rightmost panel - have been smoothed to improve the signal-to-noise in each wavelength bin. See the caption to Figure A1 for more details. \n<!-- image --> \nive to J -band than does the standard. We find that the J -band spectrum of 0453+2033 is a better match to 2MASS J17561080+2815238, which is typed in both the optical and the near-infrared as an sdL1 (Kirkpatrick et al. 2010, Zhang et al. 2018), in both the continuum shape and the strength of the FeH band. However, 0453+2033 has more flux at H and K relative to J than does 2MASS J1756+2815, possibly indicating that the former is a slightly later subdwarf. Given that the set of anchors for the L subdwarf spectral sequence is still incomplete (Zhang et al. 2018), we tentatively classify this object as an early- to mid-sdL. \nCWISE J055942.94 -012002.4: Of the spectra in Figures A4-A5 that have a "red" or "slightly red" classification, only 0559 -0120 has the triangular-shaped H -band peak indicative of low gravity. Such low-gravity objects are necessarily young, as they have not yet contracted to their final radii. Using just the sky position and proper motion values (Table 2), as its parallax and radial velocity have not yet been measured, BANYAN Σ (Bayesian Analysis for Nearby Young Associations; Gagné et al. 2018) gives the object an \n80% chance of belonging to a known, young moving group - either the AB Doradus group or, less likely, the β Pictoris group. If an AB Dor member, BANYAN Σ predicts 46 ± 3 pc with a radial velocity of 22 ± 2 km s -1 ; if a β Pic member, the predictions are d = 21 ± 3 pc and RV = 19 ± 2 km s -1 . Using solely an MKs vs. spectral type relation (Dupuy & Liu 2012), as advocated for young L dwarfs in Schneider et al. (2023), we estimate a distance of ∼ 28.8 pc for this L5.5 dwarf, based on a value of Ks =14 . 34 ± 0 . 09 mag from the 2MASS All-Sky Point Source Catalog. \nCWISE J075227.38+053802.6: We classify this object as L9 pec (Figure A5). The peculiarities stem from the two unusual absorption troughs at 1.63 and 1.67 µ m within the H -band plateau and the unusual inflection near 2.21 µ m at K -band. Such features are indicative of methane absorption, which should not be present shortward of 2.5 µ m in an L9 dwarf. As previous papers such as Burgasser (2007b), Burgasser et al. (2010b), and Bardalez Gagliuffi et al. (2014) have noted, such spectra may indicate the presence of an unresolved binary comprised of two morphologically distinct \nFigure A4. Near-infrared spectroscopic follow-up of objects classified as mid-M through mid-L. Each target object (black) is normalized to one at 1.28 µ m and overplotted (in other colors) with the spectral standard nearest the same spectral type. Integral offsets have been added to separate the spectra vertically. Target objects are labeled with brief RA/Dec (hhmm ± ddmm) identifiers. One spectrum - 0907 -4308 (L5) - has been smoothed to improve the signal-to-noise in each wavelength bin. Data deep within the telluric water bands near ∼ 1.4 and ∼ 1.75 µ m are not displayed for some targets because of the poor signal-to-noise in those regions. \n<!-- image --> \nspectra - a non-methane M or L dwarf and a methane-rich T dwarf. If 0752+0538 represents such an unresolved binary, modeling (see section 4.5 of Kirkpatrick et al. 2016) suggests it is likely a late-L plus early-T composite system. \nCWISE J075853.12 -232645.8: We classify this object as T2.5 pec (Figure A5). Although its J -band peak matches both the T2 and T3 standards equally well, the H -band flux is suppressed and the K -band flux elevated relative to the standards. We find that a synthetic spectrum made up of components of types L8-L9 and T5-T6 fits the overall spectral shape slightly better than the single standards, suggesting perhaps that this object is an unresolved binary. \nCWISE J132403.81 -052631.4: The width of the J -band peak in this object best matches that of the T4 standard (Figure A5), but the fits at both shorter and longer wavelengths are much poorer. Specifically, the H - and K -band portions of 1324 -0526 emit less flux relative to J -band than does the standard, and the K -band portion is notably flattened, an effect often ascribed to increased collision-induced absorption by H2. Moreover, the Y -band portion emits more flux relative to J -band than does the standard. Elevated Y -band flux \nand suppressed K -band flux are seen in a comparison of the sdT5.5 dwarf HIP 73786B (Figure 1 of Zhang et al. 2019) to standards of type T5 and T6, although the discrepancies are stronger in 1324 -0526, and the latter also shows suppressed H -band flux. In the case of HIP 73786, the system has a measured subsolar metallicity of [Fe/H] = -0 . 3 ± 0 . 1 (Murray et al. 2011) from the K5 V primary, suggesting that the metallicity of 1324 -0526 is somewhat lower still. We classify 1324 -0526 as sdT4. \nCWISE J221859.41+114642.7: The width of this object\'s J -band peak is most similar to the T7 standard, but its H - and K -band flux peaks are suppressed, with the latter being noticeably flattened. As with 1324 -0526 above, such features are typical of subdwarfs, although the suppression of the Y -band peak in 2218+1146 runs contrary to the trend seen in T subdwarfs of slightly earlier type. In the sdT8 WISE J200520.38+542433.9, a wide companion in the low-metallicity ([Fe/H] = -0 . 64 ± 0 . 17) Wolf 1130 system (Mace et al. 2013b), the Y -band peak is shifted notably to the blue - from 1.09 to 1.03 µ m - relative to the standards, an effect also seen in the isolated sdT6.5 dwarf ULAS \nFigure A5. Near-infrared spectroscopic follow-up of objects classified as mid-L through late-T. One spectrum - 0133+8031 (T4) - has been smoothed to improve the signal-to-noise in each wavelength bin. Data deep within the telluric water bands near ∼ 1.15, ∼ 1.4, and ∼ 1.75 µ m are not displayed for some targets because of the poor signal-to-noise in those regions. See the caption to Figure A4 for other details. \n<!-- image --> \nJ131610.28+075553.0 (Burningham et al. 2014). Our spectrum is too noisy in this region to determine whether the same \neffect is present in 2218+1146, so we classify this object as T7 pec pending further confirmation of its subdwarf status. \nTable A2 . Spectroscopic Follow-upTable A2 continued \nTable A2 (continued) \nTable A2 continued \nTable A2 (continued) \nTable A2 (continued)', 'A.3. Astrometry': 'Additional parallaxes have been measured as part of an ongoing ground-based program and through serendipitous imaging data found in the Spitzer Heritage Archive. These results are discussed further below.', 'A.3.1. NPARSEC results': 'Nearby objects continue to be targeted as part of the NTT PARallaxes of Southern Extremely Cool objects (NPARSEC) project, a long-term program (186.C-0756 with R. Smart, PI; 105.C-0781, 108.21XQ.0001, and 108.21XQ.002 with E. Costa, PI) using the infrared spectrograph and imaging camera Son OF ISAAC (SOFI; Moorwood et al. 1998) on the New Technology Telescope (NTT). The observational methodology and reduction procedures are identical to those \ndiscussed in Smart et al. (2013). For the eleven objects listed in Table A3, the new NPARSEC preliminary values have smaller uncertainties than previously published parallaxes. The table gives the object names, J2000 coordinates, mean epoch of observation, the absolute parallax, the correction applied to the relative parallax to convert to absolute, the proper motion values in Right Ascension and Declination, the total time baseline of the NTT observations, the number of reference stars used, and the total number of separate observational epochs. \nNine of the targets have absolute parallaxes greater than 50 mas, but for SDSS J163022.92+081822.0 and 2MASS J23312378 -4718274, these better determined parallaxes have values below 50 mas, so we now exclude these three from the 20-pc census. We note, however, that these results are still considered preliminary and will be finalized once the NPARSEC program draws to a close. \nTable A3 . Preliminary Parallax and Motion Fits for Objects on the NPARSEC Parallax Programs \nTable A3 (continued)', 'A.3.2. Spitzer results': "CWISE J181125.34+665806.4 (hereafter 1811+6658; see Figure A6) is located only 1.2 degrees from the north ecliptic pole (NEP), and the area around the NEP was routinely observed by the Spitzer Space Telescope. As shown in Table A1, the location of 1811+6658 was observed repeatedly in post-cryogenic programs 10147 (PI: Bock) and 13153 (PI: Capak) in an attempt to explore the genesis of fluctuations in the extragalactic background light and to provide IRAC/ch1 and IRAC/ch2 data on touchstone fields that will be used by Euclid, Roman, and JWST to study galaxy growth during the epoch of reionization. The data in program 10147 cover the timeframe from May 2014 to Sep 2014, and those in program 13153 cover Feb 2017 to Feb 2019. \nTo extract astrometry from these data sets, we searched for blocks of ch2 coverage that had sufficient depth and redundancy to provide a similar per-epoch astrometric accuracy to that obtained in our own Spitzer parallax programs (Kirkpatrick et al. 2019, 2021a). (This cold brown dwarf is much brighter at ch2, 15.95 mag, than at ch1, 18.23 mag, so only the longer-wavelength band would provide sufficient signal-to-noise for our astrometric needs.) Program 10147 used 30s exposures per frame, and the position of 1811+6658 was observed at four or fewer epochs. The data from program 13153, on the other hand, used 100s exposures per frame and had more coverage at each sky position. \nWe pared this data set down to include only those frames for which the location of 1811+6658 was far enough from the frame edge to provide a reasonable number of Gaia DR3 reference stars surrounding the target's location. Specifically, we retained only those ch2 frames that imaged all six of our pre-selected Gaia DR3 astrometric reference objects encircling a 60 '' zone centered on the location of 1811+6658. \nFor program 10147, this left only two or three frames per epoch; this lack of redundancy combined with the short exposure time means that these data are unsuitable for astrometric analysis. For program 13153, however, we are left with four to six redundant, longer exposure frames per epoch (defined here to be per AOR), which is suitable for our reduction techniques. Of those program 13153 AORs listed in Table A1, only the ch2 data in 62377728, 65133312, 68615680, and 68631296 lacked sufficient redundancy. The time span covered by the remaining data sets is Jul 2017 to Jan 2019. These data were extracted and astrometrically calibrated to the Gaia DR3 reference frame as described in (Kirkpatrick et al. 2021a). \nGiven that the usable Spitzer data only cover a year and a half, we turned to WISE astrometry to provide the additional baseline needed to disentangle parallax from proper motion. The NEP is within the boresight of the WISE spacecraft on every orbit, but given the 47 ' -wide field of view, 1811+6658 is not within the continuous viewing zone. However, that location is viewed by WISE during a span of 50+ days every six months as the scan pattern rotates around the ecliptic pole. As such, there are several weeks of coverage twice per year covering its location. \nWe ran the crowdsource detection software (Schlafly et al. 2018) on time-resolved unWISE coadds (Meisner et al. 2018) for all ten-day epochal mosaics covering the position of 1811+6658. We retained those source lists for which the frame coverage depth at the location of 1811+6658 was 40 or greater. This was done in an effort to assure that the area surrounding the target's location also had sufficient coverage, as this area is needed for the astrometric calibrators. The measured positions of these surrounding astrometric standards were used to place the measured position of 1811+6658 onto the same Gaia DR3 reference frame used for the Spitzer observations. \nFigure A6. Cutout images, 120 '' on a side with north up and east to left, of 1811+6658. (Left) WISE data at epoch 2021.6. The separate W1 and W2 bands have been mapped into a color scheme in which objects appearing at roughly equal brightness in each will appear black, and those appearing primarily in W2 will appear orange (Caselden et al. 2018). The brown dwarf 1811+6658 is the orange object at the center of the field. (Center) PanSTARRS data. Bands y / i / g have been mapped into red/green/blue. Note the two blue background sources lying near the center of the field, which is shown in the zoomed inset. (Right) Keck/MOSFIRE data at epoch 2021.7. The detection of 1811+6658 is marked with a red circle (matched to the size of the aperture used in our photometric reductions) and is sandwiched between the two blue background sources seen in the PanSTARRS view. The inset shows a zoom of the field center. \n<!-- image --> \nThis astrometry from Spitzer and WISE is listed in Table A4. A fit to the parallax and proper motion was performed on the combined astrometry using the methodology outlined in Kirkpatrick et al. (2021a), resulting in the values shown in Table A5. The results of this fit are shown graphically in Figure A7. We find that the object has a distance of 14.3 + 1 . 6 -1 . 2 pc and a value of Mch 2 = 15 . 16 ± 0 . 21 mag. A comparison to Figure 16d of Kirkpatrick et al. (2021a) suggests a spectral type of early-Y for this absolute magnitude. We further note that the measured colors JMKO -ch2 = 5.66 ± 0.04 mag, W1 -W2 = 3.04 ± 0.09 mag, and ch1 -ch2 = 2.28 ± 0.02 mag - suggest a slightly earlier spectral type of around T9T9.5 based on Figures 16e, 16g, and 16h of Kirkpatrick et al. (2021a). A comparison of our Keck/MOSFIRE JMKO , and WISE W1+W2 images with data from PanSTARRS (Figure A6) shows that 1811+6658 is passing near two blue PanSTARRS sources. Given the low spatial resolution of the WISE (and Spitzer) data, we believe that our measurements of W1 (and ch1) are contaminated by these background objects. The higher resolution of the Keck/MOSFIRE data allows us to separate all three components, but our aperture photometry at JMKO is likely still compromised given our aperture radius of 6 pixels (1 . '' 1). Therefore, our measured JMKO -ch2, W1 -W2, and ch1 -ch2 color are all likely bluer than their true values, supporting our assertion of a Y dwarf spectral type.", 'B. THE LIST OF PROXIMAL SYSTEMS': 'Despite recent WISE-based discoveries of the L+T dwarf binary system WISE J104915.57 -531906.1 AB (Luhman 2013; 1.99 pc distant) and the Y dwarf WISE J085510.83 -071442.5 (Luhman 2014; 2.28 pc distant) adding to our knowledge of the Sun\'s immediate neighbors, Proxima Centauri (Innes 1915; 1.30 pc distant) remains the closest. It is often just referred to as "Proxima", Latin for "nearest", because it is the nearest to the Sun. Yet, its full name translates to "Nearest of Centaurus". This has led some curious individuals to wonder what are the nearest stars - i.e., the other proximal objects - of each of the other eighty-seven official constellations. \nTable 4 allows us to answer this question, given our current knowledge of the 20-pc census. Proxima Centauri and its primary, α Centauri AB, represent a rare multi-object system for which the parallaxes of the individual components are so accurate that we can determine the far-flung companion to be closer to us than its host binary. For other multiobject systems, discerning the closest component may be far more difficult. Using a short-period binary as an example, determining the closest object in the system depends upon the orbital period and orientation with respect to the Sun, as one component may be the closer one at some epochs and the more distant one at others. Hence, we will identify only the proximal systems in each constellation when a multi-object system arises. \nTable B6 lists these proximal systems. As examples, the closest system in Canis Major is the binary Sirius AB, and the closest in Delphinus is the Y dwarf WISEPC J205628.90+145953.3. For several constellations, the proximal system is still ambiguous, given current uncertainties in the measured trigonometric parallaxes of the closest can-', 'CWISE J181125.34+665806.4 (B = 88.809)': '<!-- image --> \n<!-- image --> \nFigure A7. Best fit to the parallax and proper motion of 1811+6658. (Upper left) Sky plot showing the track of the object along the sky. Black points with large uncertainties are the 57 individual unWISE time-resolved measurements. The orange curve shows the best fit as seen from the vantage point of WISE, and the blue curve shows this same fit from the vantage point of Spitzer. Red lines connect each observation to its predicted point along the best-fit curve. (Upper right) The parallax solution (green) with the proper motion component subtracted out. For clarity, only the 10 Spitzer data points are shown. Red lines connect the times of the Spitzer observations to their predicted points on the curve. (Lower left) The parallactic fit (green) as a function of time in RA and Dec, along with the measured Spitzer astrometry. (Lower right) Residuals around the parallactic fit as function of time in both RA and Dec. Blue lines mark residuals of zero. For additional info on this plot, see Figure 2 of Kirkpatrick et al. (2021a). \n<!-- image --> \n<!-- image --> \nTable A4. Astrometry on the Gaia DR3 Reference Frame for 1811+6658 \nNOTE-(This table is available in its entirety in a machine-readable form in the online journal. A portion is shown here for guidance regarding its form and content.) \nTable B6 (continued) \nTable B6 continued \nTable A5. Parallax and Motion Fit for 1811+6658 \nNOTE-The RA and Dec values are listed on the ICRS coordinate system. The last three rows represent the number of Spitzer ch2 epochs (# Spitzer ) and the number of unWISE W2 epochs (# WISE ) used in the fits, along with the number of five-parameter Gaia DR3 stars used for the astrometric reregistration (# Gaia ). \ndidates. Constellations having two objects within 1 σ of the same closest value are indicated by footnotes. Note that all constellations have a proximal system within the 20-pc limit of Table 4, the most distant being the K dwarf HD 200779 at 15.05 pc, the closest known object in Equuleus. \nTable B6 . The Proximal Systems for Each Constellation \nTable B6 continued \nTable B6 (continued) \nIt is worth noting the prevalence of brown dwarfs in Table B6. There are four T dwarf companions residing in proximal systems with K or M dwarf primaries, one L+T binary as its own proximal system, and eleven solivagant L, T, or Y dwarfs that are proximal objects in their own right. In this latter group, over half (six) of the these are Y dwarfs, a spectral type that is not yet fully sampled near the Sun.'}

2024arXiv240911317S

The detection of gravitational waves GWs from coalescing compact binaries has become routine with groundbased detectors like LIGO and Virgo. However beyond standard sources such as binary black holes and neutron stars and neutron star black holes no exotic sources revealing new physics have been discovered. Detecting ultracompact objects such as subsolar mass SSM compact objects offers a promising opportunity to explore diverse astrophysical populations. However searching for these objects using standard matchedfiltering techniques is computationally intensive due to the dense parameter space involved. This increasing computational demand not only challenges current search methodologies but also poses significant obstacles for thirdgeneration 3G groundbased GW detectors. In the 3G era signals may last tens of minutes and detection rates could reach one per minute requiring efficient search strategies to manage the computational load of longduration signals. In this paper we demonstrate a hierarchical search strategy designed to address the challenges of searching for longduration signals such as those from SSM compact binaries and the anticipated issues with 3G detectors. We show that by adopting optimization techniques in a twostage hierarchical approach we can efficiently search for the SSM compact object in the current LIGO detectors. Our preliminary results show that conducting matched filtering at a lower frequency of 35 Hz improves the signaltonoise ratio by 6 and enhances the detection volume by 1020 compared to the standard twodetector PyCBC search. This improvement is achieved while reducing computational costs by a factor of 2.5.

2024-09-01T00:00:00Z

['arXiv:2409.11317', '2024arXiv240911317S', '10.48550/arXiv.2409.11317']

['General Relativity and Quantum Cosmology', 'Astrophysics - Instrumentation and Methods for Astrophysics']

Hierarchical searches for subsolarmass binaries and the thirdgeneration gravitational wave detector era

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.11317.pdf

{'Hierarchical searches for subsolar-mass binaries and the third-generation gravitational wave detector era': '<!-- image --> \n1 \nDepartment of Physics, Syracuse University, Syracuse, New York 13244, USA \n2 Inter-University Centre for Astronomy and Astrophysics, Pune 411007, India', 'ABSTRACT': 'The detection of gravitational waves (GWs) from coalescing compact binaries has become routine with ground-based detectors like LIGO and Virgo. However, beyond standard sources such as binary black holes and neutron stars and neutron star black holes, no exotic sources revealing new physics have been discovered. Detecting ultra-compact objects, such as subsolar mass (SSM) compact objects, offers a promising opportunity to explore diverse astrophysical populations. However, searching for these objects using standard matched-filtering techniques is computationally intensive due to the dense parameter space involved. This increasing computational demand not only challenges current search methodologies but also poses significant obstacles for third-generation (3G) ground-based GW detectors. In the 3G era, signals may last tens of minutes, and detection rates could reach one per minute, requiring efficient search strategies to manage the computational load of long-duration signals. In this paper, we demonstrate a hierarchical search strategy designed to address the challenges of searching for long-duration signals, such as those from SSM compact binaries, and the anticipated issues with 3G detectors. We show that by adopting optimization techniques in a two-stage hierarchical approach, we can efficiently search for the SSM compact object in the current LIGO detectors. Our preliminary results show that conducting matched filtering at a lower frequency of 35 Hz improves the signal-to-noise ratio by 6% and enhances the detection volume by 10-20%, compared to the standard two-detector PyCBC search. This improvement is achieved while reducing computational costs by a factor of 2.5. \nKeywords: Primordial black holes (1292); Gravitational waves (678); Gravitational wave detectors (676)', '1. INTRODUCTION': "The field of gravitational-wave astronomy has been rapidly expanding ever since the detection of the first binary black hole merger GW150914 (Abbott et al. 2016a). To date, nearly 90 gravitational wave (GW) sources are cataloged by LIGO-Virgo-KAGRA (LVK) Collaboration, including dozens of binary black holes, two binary neutron stars, and three neutron star-black hole mergers (Abbott et al. 2023a). Additional GW sources were independently cataloged (Nitz et al. 2021; Olsen et al. 2022; Nitz et al. 2023; Mehta et al. 2024; Wadekar et al. 2023) using publicly available data (Abbott et al. 2023b, 2021a). The third observation (O3) \nksoni01@syr.edu \nrun of Advanced LIGO (Aasi et al. 2015) and Advanced Virgo (Acernese et al. 2014) lead to the detection of compact objects within sub-3 M ⊙ range with GW190814 (Abbott et al. 2020) where the secondary compact object had a mass of ∼ 2 . 59 M ⊙ and a low spin ( ≤ 0 . 07). Several other events like GW190425, GW191219, GW200105, GW200115, and GW200210 identified during this run also had one of the component masses less than 3 M ⊙ . The ongoing fourth observation run detected GW230529 (Abac & others 2024). This event's primary object had a mass ranging between 2.5 and 4 . 5 M ⊙ , making it an additional compact object, likely a black hole (BH), existing within the 'lower mass gap' (Bailyn et al. 1998; Ozel et al. 2010; Farr et al. 2011). Although several studies have provided insights into the mass and spin distributions of compact sources detected through current GW detectors (Abbott et al. 2021b; Roulet & Zaldarriaga 2019), the possibility of \ndiscovering ultra-compact objects with masses less than a solar mass range remains an open question (Abbott et al. 2022a, 2019, 2022b; Nitz & Wang 2022, 2021a; Collaboration et al. 2023; Miller 2024). \nSubsolar mass (SSM) compact objects do not follow the standard stellar evolution pathway. These objects, if BHs, are expected to form through nonstellar evolution models and could be primordial black holes (PBHs) (Carr et al. 2010). If they are neutron stars (Doroshenko et al. 2022), they might result from non-standard supernova explosion models (Muller et al. 2024). Although the search for SSM black holes began quite early (Nakamura et al. 1997; Alcock et al. 2000), no candidates have been found yet. Since then, numerous models have proposed various formation pathways for these sources. The most common formation mechanism of PBHs suggests their origin from the direct collapse of early, small-scale fluctuations (Zel'dovich & Novikov 1967; Hawking 1971) due to certain features of the inflationary potential. Additionally, there are alternative formation channels where PBHs emerge from phase transitions (Byrnes et al. 2018) in the early universe or through the collapse of topological defects like cosmic strings (Hawking 1989; Polnarev & Zembowicz 1991; Hong-bo & Xin-zhou 1996). \nStudies show that a small fraction of dark matter could be due to PBHs (Carr et al. 2010, 2021). While many cosmological investigations have ruled out their existence at extremely low masses (Sasaki et al. 2018), exploration continues in a mass range spanning several orders of magnitude. If these black holes appear in a binary system, the emitted GWs can be detected through ground-based interferometers. Several studies have investigated the search for SSM black holes (Abbott et al. 2019; Nitz & Wang 2021a,b; Abbott et al. 2022a,b), but no significant detections have been made to date. A confirmed detection within the LIGO-Virgo frequency band would provide critical insights into the formation mechanisms of PBHs and contribute to constraining the fraction of dark matter in the universe. \nThe offline search for GWs from the inspiralling and merging binaries uses the matched filtering technique (Sathyaprakash & Dhurandhar 1991; Dhurandhar & Sathyaprakash 1994; Dhurandhar & Schutz 1994; Owen & Sathyaprakash 1999; Allen et al. 2012; Usman et al. 2016; Davies et al. 2020). In this method, a bank of modeled signals, or templates, is correlated with well-calibrated interferometer data (Siemens et al. 2004; Abadie et al. 2010). However, this approach becomes computationally demanding, particularly for lowmass binaries, as the cost increases with the number of templates and the length of the signal model used as \na matched filter template. To mitigate the computational challenges, suboptimal choices are often made by limiting the search parameters. For instance, searches may only filter data above 45 Hz (Abbott et al. 2016b), or limit the duration of the templates to nearly 512 seconds (Nitz & Wang 2021a, 2022). While these restrictions help reduce computational costs, they also reduce the sensitive volume by approximately 24%, within which PBHs might be detected. \nObserving long-duration GW signals poses significant challenges with existing search methods. The main difficulties arise from the necessity of using a very dense template bank ( O (10 7 )), which significantly increases the computational cost of the matched filtering search. Furthermore, the search sensitivity can be compromised by non-stationary data, which may contain long-duration correlations that hinder current signal-vetoing techniques and statistical analyses. This non-stationarity can also impact the statistics and signal-vetoing methods used in current search pipelines. These issues are expected to become considerably more severe in the era of third-generation (3G) ground-based detectors. 3G detectors such as the Cosmic Explorer (Abbott et al. 2017; Reitze et al. 2019) and the Einstein Telescope (Hild et al. 2009; Punturo et al. 2010; Grado 2023; Maggiore et al. 2020; Di Pace et al. 2022), are anticipated to detect binary mergers at rates two to three orders of magnitude higher than current detectors (Evans et al. 2021). These detectors, designed to operate from very low frequencies (starting from 2 Hz), will observe signals for several minutes or hours. Due to their high sensitivity in the lower frequency band, the likelihood of detecting eccentric or precessing binaries will be higher, which will indirectly expand the template bank's parameter space-both in dimensionality and parameter ranges-thereby increasing the computational cost of the search. Furthermore, since signals would remain in the sensitivity band for longer periods, the Earth's rotation will reduce search sensitivity by altering the detector's response functions and affecting matched filter statistics. Therefore, developing an efficient, cost-effective matched filtering strategy for long-duration GW signals is essential to advance the current state-of-the-art search techniques. This approach will not only mitigate computational burdens but also prepare us to effectively search for GW signals from compact binary coalescences (CBCs) with 3G detectors. \nOne approach to efficiently search for long-duration signals, such as those from SSM binaries, is implementing a hierarchical search strategy (Soni et al. 2022; Dhurkunde et al. 2022; Soni et al. 2024). In this method, a two-stage matched filtering search is performed using multiple template banks of varying densities. In the \nfirst stage, the search is conducted over coarsely sampled data using a coarse bank to identify coincident triggers that could represent true GW events. These triggers are followed up in the second stage with a finer search, focusing on the neighborhood of the parameter space identified in the first stage. This two-stage approach effectively reduces the number of matched filtering operations required for the search, significantly reducing computational time. \nIn this paper, we present a comprehensive analysis of real data to demonstrate how hierarchical search strategies can be effectively applied to SSM binary searches in Advanced LIGO data. We also discuss the necessary modifications to extend these techniques to generic CBC searches in the upcoming 3G detectors. We implement a two-stage hierarchical search method, as detailed in Soni et al. (2022, 2024), specifically targeting SSM compact objects. This method allows us to optimize the search by employing different sampling rates and varying template bank densities across the two stages. \nOur search focuses on SSM binaries with primary masses m 1 ∈ [0 . 2 , 10] M ⊙ and secondary masses m 2 ∈ [0 . 2 , 1] M ⊙ , defined in the detector frame. We also consider component spins of up to 0.9 for each compact object. The parameter space for our template bank is similar to that used by the LVK collaboration (Abbott et al. 2022a), but we operate with different frequency settings. Specifically, our search is tuned to detect SSM candidates beginning at 35 Hz within the Advanced LIGO frequency band. By lowering the operational frequency, we aim to reduce the loss in astrophysical volume by approximately 10%. Although this adjustment increases the density of the coarse template bank by a factor of 1.5 compared to the bank used in the PyCBC search in Abbott et al. (2022a), we achieve a reduction in the computational cost of matched filtering by utilizing two stages with varying data sampling rates in our search pipeline.", '2. METHOD': "The matched filtering search for long-duration signals as in the case of binaries containing SSM compact objects is expensive as it requires a very dense template bank. To optimize the search, often the length of the template is reduced to a manageable duration ( ∼ 512 seconds) so that search analysis can be performed. This could be enabled by performing matched filtering from 45 Hz rather than 15 Hz (Abbott et al. 2023a). However, such adjustments affect the horizon distance of the binary and the expected signal-to-noise (SNR) ratio. \nThe horizon distance (Thorne 1987; Allen et al. 2012) for an inspiraling binary is given by \nD max ∝ M 5 / 6 ρ √ ∫ f max f min f -7 / 3 S n ( f ) df , (1) \nwhere f min and f max are the minimum and maximum frequencies of the LIGO's sensitivity range. S n ( f ) is the power spectral density (PSD) of the noise in the detector and ρ is expected matched filter SNR for an inspiraling binary with chirp mass M observed in the detector's frame. \nFor a particular source of chirp mass of a few that have a fixed SNR in the detector's frame, changing the operating frequency band can affect the detectability of a signal. This means that the fractional SNR loss, as also shown in Magee et al. (2018), due to a change in the operating frequency band would be \nF SNR -loss = 1 -D max ( f min , f max ) D max (15Hz , 2048Hz) , (2) \nwith respect to the standard operational band of Advanced LIGO for matched filtering for generic CBC search (Abbott et al. 2023a). \nIf we perform a search from 35 Hz, assuming the other source parameters do not change, the percentage loss in SNR is about 3.1% for a frequency band of 35 -2048 Hz. This loss is relatively small compared to the SSM searches conducted in the 45 -2048 Hz band (Abbott et al. 2022a), which experience an SNR loss of about 8 -9%. This comparison can also be seen in Fig. 1. Ideally, the lower frequency limit can be further lowered to match those used for a generic CBC search. However, this step will increase the computation demand and require very long-duration templates in the bank. Therefore, in our work, we select a lower frequency cutoff of 35 Hz. By making this choice, we expect the loss in astrophysical volumetric sensitivity to the inspiral stage by ∼ 10% of binary coalescence, which is lower than 24% as observed in Abbott et al. (2022a). \nGenerating templates at lower frequencies enhances the sensitivity of the search, but it can also lead to a higher density of the template bank. This increase in density may raise the computational cost of matched filtering in traditional offline PyCBC or flat search (Usman et al. 2016; Davies et al. 2020). However, this added computational burden can be mitigated by adopting hierarchical search strategies.", '2.1. Review of hierarchical search': "The two-detector flat search performs matched filtering over a discretely sampled data segment using a dense bank of templates and generates triggers with SNRs ( ρ ) (Usman et al. 2016). In contrast, a hierarchical \nFigure 1. The percentage loss in SNR as a function of lower frequency limit ( f min ) used for matched filtering data, as described in Eq. (2). The SNR loss increases with larger f min values in matched filtering. For comparison, the black dotted line represents the lower frequency limit of 45 Hz (Abbott et al. 2022a), while the gray line indicates the 35 Hz limit used in this work. \n<!-- image --> \nsearch strategy offers a more efficient approach by enabling a multi-stage matched filtering process, where the number of templates is progressively reduced at each stage. As described in Soni et al. (2022), this search divides a flat search into two stages: coarse and fine . During the coarse stage, the data is matched filtered using a coarse template bank, which consists of sparsely placed templates. The sparseness of these templates is determined by the minimal match (Owen 1996) at which the bank is constructed, typically set lower (below 0.97) than the value used for constructing a flat bank. \nPerforming a coarse search reduces the number of matched filtering operations. This reduction is further enhanced when the data is sampled at a lower frequency (512 Hz). However, this approach comes with the tradeoff of potentially lower matched filter SNR values for the resulting triggers. To address this, the SNR thresholds are lowered ( ρ = 3 . 5) compared to those used in a flat search ( ρ = 4). Given these triggers could be generated by non-Gaussian features or glitches (Abbott et al. 2021c) present in the data, the SNRs are further down-weighted using chi-square vetoes (Allen 2005; Nitz 2018). Only those triggers that pass these vetoes are then subjected to a coincidence test (Usman et al. 2016), during which a ranking statistic is computed to assess their significance (Nitz et al. 2017). \nThe coincident triggers obtained from the coarse search, with ranking statistics above a certain threshold (approximately 7, as used in Soni et al. (2022)), are followed up in the second stage for a finer search. In this stage, a focused search is conducted within the vicinity or neighborhood ( nhbd ) of the coarse template's \nparameter space. This nbhd is a region around a coarse template where the minimal match between templates within the nbhd ( ∼ 10-100) and the coarse template ranges from 0.75 to 0.99. \nTo avoid the computational burden of calculating the nbhd for each coarse template on the fly, a pre-computed nbhd bank is used. This bank contains all the nbhd regions and the corresponding templates for each coarse template. During the second stage search, a union of all the nbhds corresponding to the coarse triggers in each data segment is used for matched filtering. To further improve the SNR, the data sampling rate is increased to 2408 Hz, which is higher than that used in the coarse search. Triggers with SNRs above 4 that pass all chisquare tests are then subjected to a final coincidence search, compiling a list of foreground candidates. \nTo assess the significance of potential GW events, the false alarm rate (FAR) is estimated based on the background (Usman et al. 2016). Unlike the flat search, which estimates the background by applying millions of time shifts to triggers from a single detector, the hierarchical search employs a hybrid approach in its second stage (Soni et al. 2024). At first, a few time shifts are applied to generate coincident background triggers. Then, an exponential fit is applied to the cumulative distribution of these background triggers, and the fitted curve is used to calculate the FAR for the foreground triggers obtained in the second stage. This method of estimating the background is particularly effective for long-duration signals, as the expected background distribution tends to follow the tail of a Poisson distribution (Usman et al. 2016).", '2.2. Template bank': 'We construct two aligned-spin banks- a coarse bank at a minimal match of 0.92 and a fine bank of 0.97, using a geometric placement algorithm (Harry et al. 2014) for the hierarchical search. Both banks are designed to cover parameters where m 1 ranges between 0 . 2 -10 M ⊙ and m 2 between 0 . 2 -1 . 0 M ⊙ in the detector frame. The dimensionless spins span up to 0 . 9 for both compact objects. These bank parameter ranges are consistent with the bank used for the LVK search (Abbott et al. 2022a; Collaboration et al. 2023). From here, we refer to this bank as flat bank . \nIn contrast to the flat bank, where templates commence at a frequency of 45 Hz, the templates in our banks start at a frequency of 35 Hz. This choice reduces the fractional loss in the matched filter SNR, as shown in Sec. 2. As a result, even though our coarse bank is constructed at a lower minimal match, it is approximately \n1.6 times the size of the flat bank. These distinctions are summarized in Table 1. \nTable 1. Summary of the coarse and fine template banks constructed for the hierarchical search, with a comparison to the flat bank used in the LVK SSM search (Abbott et al. 2022a). The banks are characterized by different minimal match values, which denote the minimum match between neighboring templates, and different starting frequencies f 0 . A lower f 0 results in increased template density, as demonstrated by the fine bank, even though the fine and flat banks have similar minimal match values. The coarse bank is approximately 1.6 times denser than the flat bank, primarily due to its lower starting frequency. \nThe hierarchical search requires the construction of nbhd bank for the second stage search. Therefore, we use the generated fine bank to construct the nbhd bank using the method described in Sec. IV of Soni et al. (2022). Figure 2 shows the parameter space covered by the coarse templates in the chirp mass and effective spin plane. This plot shows that the distribution of templates within a nbhd is not uniform across the parameter space, primarily due to boundary effects. These effects occur because the match between neighboring templates gradually decreases as the mismatch in the τ 3 mass parameter increases relative to τ 0 , thereby significantly extending the nbhd region along this coordinate. \nFigure 2. Figure depicting the distribution of coarse templates in the logarithm of chirp mass ( M ) - effective spin ( χ eff ) plane. The color bar represents the total number of templates in the nbhd of each coarse template parameter. \n<!-- image --> \nAs shown in Fig. 2, the number of templates in a nbhd typically ranges from a few tens to hundreds. This can significantly affect the final background in the second stage. To improve background estimation, more noise coincidences from the coarse search need to be followed up. However, since the number of templates in the nbhds is relatively small, the search cost is not expected to be significantly higher than that of the flat search.', '3. APPLICATION TO SUBSOLAR MASS SEARCH': 'We perform SSM search on publicly available datasets using a two-stage hierarchical approach, as outlined in Sec. 2. The data consists of approximately five days of coincident observations from the O3 run of the two LIGO detectors-LIGO Hanford and LIGO Livingston, covering the period from April 1 to April 8, 2019. \nTo begin, we first conduct a coarse search over the data sampled at 512 Hz using the coarse bank described in Sec. 2.2. The templates are generated at 35 Hz using the TaylorF2 (Sturani et al. 2010) waveform model with phase corrections up to 3.5 post-Newtonian order. The lengths of these templates range between 10 2 and 10 3 at 35 Hz. To prevent the templates from wrapping around during the Fast Fourier Transform operation in the matched filtering step, we ensure that the data segment length exceeds that of the longest template in the bank. Consequently, we set the data segment length to 2048 seconds for our analysis. \nWe identify triggers with a matched filter SNR and reweighted SNR (Allen 2005; Usman et al. 2016) of 3.5 or greater. This threshold is chosen to increase the likelihood of detecting true signals that might be missed due to lower data sampling and the use of a coarse bank. To reduce the impact of short-duration glitches in the data, the triggers are further weighted using a chi-square and sine-Gaussian vetos. The surviving triggers then undergo a coincidence test, where they are shifted in time by 5,000 seconds, and a ranking statistic (Λ) (Nitz et al. 2017; Davies et al. 2020) is computed. \nIn the next step, we perform a finer search in the second stage on coincident triggers with Λ ≥ 7. During this stage, matched filtering is conducted again on data sampled at 2048 Hz, using a union of nbhds around the identified coarse templates. The data sampling rate is increased by a factor of 4 to improve the matched filter SNR. Triggers from this stage are collected if their SNRs and re-weighted SNRs exceed a threshold of 4. These triggers are then re-weighted using chi-square and sineGaussian vetoes before undergoing a coincidence test, over the same time-shift interval as in the first stage. Finally, a list of foreground and background triggers is \nTable 2. Results from a two-detector analysis over data duration from April 1 to April 8, 2019, using flat and hierarchical search pipelines. Listed candidates are arranged in descending order of false alarm rate (FAR). The table also compares chirp mass ( M ) and network SNR (ˆ ρ T = √ ρ 2 H + ρ 2 L ) for each identified candidates. The FARs of the detected events in the flat search were determined using the time-shift method, whereas those for the hierarchical search were determined by the method described in Soni et al. (2024). \ncompiled, and the FAR for the foreground triggers is calculated following the procedure described in Soni et al. (2024). \nThe primary differences between the SSM search using the hierarchical method and the search adopted by the LVK collaboration (Abbott et al. 2022a) lie in two key aspects: the parameter space covered by the template banks and the lower frequency cutoff for matched filtering. In this work, we use a coarse bank and a nbhd bank specifically designed to optimize the detection of SSM compact objects by covering a more targeted and dense parameter space. Additionally, while the flat search in Abbott et al. (2022a) typically starts matched filtering at 45 Hz, our approach begins at a lower frequency of 35 Hz. This choice enhances the sensitivity of our search to potential GW signals, particularly those that might be present at lower frequencies. By adjusting the matched filtering start frequency, we aim to improve the detection capabilities for signals that might be overlooked in the flat search.', '3.1. Results': 'The hierarchical search yielded a list of GW candidates, many of which were statistically insignificant due to their FAR values exceeding 1 per year. These candidates along with the ones identified through a flat search using the same dataset are summarized in Table 2. While a few candidates are common to both search pipelines, none are statistically significant. As shown in Fig. 3, the foreground events overlap with the background distributions for both searches, indicating that the candidates are primarily noise coincidences. \nFigure 3 also highlights the disparity in backgrounds between hierarchical and flat searches, with the hierar- \nFigure 3. False alarm rate (FAR) versus ranking statistics for foreground and background computed from our reference flat and hierarchical searches. The flat search background, represented by the black curve, is computed with a time shift interval of 0.1 s using a template bank with f 0 = 45 Hz. The hierarchical search background, shown by the gray curve, uses the method proposed in Soni et al. (2024) and a bank from a union of nbhds where templates are generated at f 0 = 35 Hz. \n<!-- image --> \nchical search producing a larger background. This difference is due to the usage of different numbers of templates within their respective search pipelines. As shown in Table 1, the coarse bank is approximately 1.6 times and the fine bank is 4.8 times denser than the flat bank owing to the template generation at 35 Hz. If the flat search had employed the fine bank, its background distribution would likely resemble that of the hierarchical search, especially in the tail. Although the number of templates used in the second stage search is low ( ∼ 10 -1000 per \ndata segment), the background generated in the second stage is expected to increase leading to more instances of noise coincidence. However, this also improves the chances of detecting GW sources that might be missed in a flat search.', '3.2. Sensitivity and Search Efficiency': 'The sensitivity of a search is determined by how many signals it can detect at a particular significance level within a given observation time (T). This can be quantified by estimating the observable volume-time (VT) product (Tiwari 2018). For a constant merger rate of the population of binaries, the average VT sensitivity product is given by \n⟨ V T ⟩ = V 0 N det N inj T , (3) \nwhere N det is the number of detected sources in the search, and N inj is the total number of injected sources. V 0 is the volume defined as \nV 0 = ∫ z max 0 dV c dz 1 (1 + z ) dz , (4) \ndV c dz is the differential comoving volume in an expanding universe with redshift z. \nTo test the sensitivity of our search method, we conducted a comparative analysis through an injection campaign on a simulated binary population. In this population, we assumed that one of the compact objects has a mass below a solar mass, while the other ranges from 1 to 10 solar masses. We created three distinct sets of injections, each defined by different spin conditions: high spin, low spin, and a mixed case where only one compact object has low spin. The distributions and ranges of the component masses and spins for these three scenarios are detailed in Table 3. \nFor the given parameter space, we generated GW signals using the waveform approximants listed in Table 3, starting at a frequency of 35 Hz. Each signal was injected into the data with a minimum interval of 100 seconds between injections, assuming an isotropic distribution for the sky locations of the sources. Following this procedure, approximately 6,500 injections were made across the three sets, and a search was performed using both the traditional flat method and our hierarchical approach. \nFigure 4 shows the VT ratio computed for the two search methods. The hierarchical search outperforms the flat search, showing an improvement in the VT ratio by approximately 1.1 to 1.2. This improvement is primarily due to the SNR enhancement when the search is performed at a lower frequency of 35 Hz instead of 45 \nTable 3. Overview of three injection sets focusing on one of the component masses in SSM ranges. These ranges are specifically chosen to align with the parameters explored in the SSM search conducted by PyCBC (Abbott et al. 2022a). Each injection set has component masses (in the detector frame) and spin parameters uniformly distributed. \nHz. When the search is performed at 35 Hz, we expect the gain in the SNR to be approximately 6% and an astrophysical volumetric gain of ∼ 20%. However, the injection study results indicate that the detection volume improves by only 10 -20%. This discrepancy likely arises from an increase in the noise background when the lower bound of the matched filtering frequency is reduced. \nTable 4. Comparison of CPU core hours required for the flat search and the coarse and fine stages of the hierarchical search. The values outside the parentheses represent the CPU core hours for the Hanford detector, while the values inside the parentheses correspond to the Livingston detector. \nWe compared the matched filtering cost by comparing the number of CPU core hours required by each detector in use in the two searches. As shown in Table 4 the number of CPU core hours required by flat search is more than coarse and fine searches owing to the different number of templates used in each search. The numbers show that the overall cost of a hierarchical search is approximately 2.5 times less than a flat search. This is a huge advantage of hierarchical search given that the search sensitivity is also more than that of the flat search. \nFigure 4. Plot showing the sensitive volume-time (VT) ratio for the hierarchical and flat searches, averaged across all three injection sets (see Table 3). The VT ratio, binned over inverse false alarm rates (IFARs), demonstrates an improvement in search sensitivity across all chirp mass ( M chirp ) bins. \n<!-- image -->', '4. CONCLUSION AND DISCUSSION': "The search for long-duration GW signals from compact binary mergers, such as SSM binaries, low-mass BNSs, precessing binaries, and binaries with moderate mass ratios ( m 1 /m 2 , m 1 ≥ m 2 ) with eccentricity in their orbits, presents significant challenges in the current LIGO-Virgo frequency band. These challenges primarily arise from the requirement of large, densely populated template banks necessary for matched filtering. To mitigate the computational burden, suboptimal choices are often made, which inevitably limit the sensitivity of such searches. With the advent of 3G detectors, these challenges are expected to become more pronounced, as GW signals will be observed over longer durations, ranging from tens to hundreds of minutes. This extended observation window will substantially increase the computational demands due to the rapid expansion of the parameter space. Consequently, the development of efficient hierarchical search strategies is critical, not only to enhance current detection capabilities but also to ensure readiness for the vastly more computationally expensive searches required by next-generation detectors. \nIn this paper, we demonstrated that a hierarchical search strategy can be effectively implemented for the search of SSM binaries without restricting the parameter space. In Sec. 2, we presented a preliminary calculation indicating that the SNR improves by approximately 6% when the matched filtering is conducted starting at a frequency of 35 Hz. This means that a ∼ 20% increase in the sensitive volume could be expected in the search. Building on the results from Sec. 2, we constructed the necessary template banks (see Sec. 2.2) and \nconducted a hierarchical search on a small dataset from the O3 run (see Sec. 3). As described in Sec. 3.1, our search did not yield any significant candidates, which is consistent with previous searches (Collaboration et al. 2023; Abbott et al. 2019; Nitz & Wang 2021a,b; Abbott et al. 2022a,b). Through injection studies, we demonstrated that the hierarchical search gives a sensitive VT ratio improvement of about 10 -20% compared to the flat search method employed by the LVK Collaboration. This volumetric improvement is important as this can increase the possibility of finding sources in the upcoming LIGO and Virgo observation runs. Moreover, this improvement can provide better constraints on the fraction of dark matter, potentially ruling out several models proposing SSM black holes as dark matter candidates. \nOur findings highlight that near-optimal sensitivity can be achieved using a hierarchical search, even when compared to a direct flat search. By optimizing different aspects of the search process at two stages, such as adjusting the frequency of operation, data sampling rates, and the density of the template banks, we achieved computational savings of up to a factor of 2.5 while simultaneously enhancing the sensitivity of the SSM search. This represents a significant advancement in search optimization as we prepare for 3G detectors. \nLooking ahead, 3G detectors are expected to introduce several significant challenges for CBC searches due to their enhanced low-frequency sensitivity, as discussed in Sec. 1. With the ability to detect GW signals from a broader range of CBC sources-including eccentric and precessing binaries over extended durations-the search space will expand dramatically. This increase in the \nsearch space will, in turn, raise computational demands to potentially unmanageable levels. Therefore, a hierarchical approach may be proposed to address these challenges effectively. \nIn the 3G era, the hierarchical search could be structured into stages, with the first stage focused on efficiently identifying potential candidates by maximizing the likelihood of detecting GW signals based on all intrinsic parameters, while the second stage aims to improve the SNR and address any losses from the first stage. Since the primary goal of the first stage is to locate regions where signals are likely to be present, general optimizations related to reducing the template bank size, adjusting data sampling, and defining the operating frequency range for matched filtering can be applied, as shown in this work. The density of the template bank could be reduced by coarsening the bank and adjusting the frequency at which templates are generated. However, this step should concentrate on regions of the parameter space where the SNR due to features in the binary's orbit is expected to be higher. For instance, in the case of eccentric binaries, the effects of eccentricity are most prominent in the lower frequency region, while at higher frequencies, the binary's orbit is expected to circularize. Therefore, the template bank, matched filtering frequency band, and data sampling rate could be adjusted to focus only on the higher frequencies to reduce \nthe search cost. To enhance the detection probability, a nbhd search could be performed with templates having eccentricities at lower frequencies in the second stage. A similar strategy could be applied to binaries with moderate precession. For these binaries, the first stage can be adjusted to search for signals with variable starting frequencies for matched filtering, excluding less significant merger-ringdown phases. However, these regions could be relaxed in the second stage. Since we expect the Earth's rotation to impact the antenna response functions, potentially reducing search sensitivity, approximate response functions, which would include the effect of the source's orientation with respect to the detector, could be introduced only in the second stage of the hierarchical search, thereby improving overall sensitivity. The second stage could also be optimized to recover any SNR loss from the first stage resulting from the various optimizations applied earlier. \n- KS and AHN acknowledge support from the National Science Foundation grant (PHY-2309240). KS acknowledges the support for computational resources provided by the IUCAA LDG cluster Sarathi and Syracuse University through the OrangeGrid High Throughput Computing (HTC) cluster. KS expresses sincere gratitude to 1 2 3 4 5 6\n- Sanjit Mitra for discussions in the very early stages of 7\n- this work. 8", 'REFERENCES': 'Aasi, J., Abbott, B. P., Abbott, R., et al. 2015, Classical Quantum Gravity, 32, 074001, \ndoi: 10.1088/0264-9381/32/7/074001 \nAbac, A. G., et al. 2024, The American Astronomical \nSociety, doi: 10.3847/2041-8213/ad5beb \nAbadie, J., Abbott, B., et al. 2010, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 624, 223, doi: https://doi.org/10.1016/j.nima.2010.07.089 \nAbbott, B. P., Abbott, R., Abbott, T. D., et al. 2016a, Phys. Rev. Lett., 116, 061102, \ndoi: 10.1103/PhysRevLett.116.061102 \nAbbott, B. P., et al. 2016b, Phys. Rev. D, 93, 112004. \nhttps://journals.aps.org/prl/abstract/10.1103/ PhysRevLett.123.161102 \nAbbott, B. P., Abbott, R., Abbott, T. D., et al. 2017, \nClassical and Quantum Gravity, 34, 044001, \ndoi: 10.1088/1361-6382/aa51f4 \nAbbott, B. P., et al. 2019, Phys. Rev. Lett., 123, 161102, doi: 10.1103/PhysRevLett.123.161102 \nAbbott, R., Abbott, T. D., Abraham, S., et al. 2020, The Astrophysical Journal Letters, 896, L44, \ndoi: 10.3847/2041-8213/ab960f \nAbbott, R., et al. 2021a, SoftwareX, 13, 100658, \ndoi: 10.1016/j.softx.2021.100658 \nAbbott, R., Abbott, T. D., Abraham, S., et al. 2021b, The Astrophysical Journal Letters, 913, L7, \ndoi: 10.3847/2041-8213/abe949 \nAbbott, R., et al. 2021c, Phys. Rev. D, 104, 122004, \ndoi: 10.1103/PhysRevD.104.122004 \n- -. 2022a, Phys. Rev. Lett., 129, 061104, \ndoi: 10.1103/PhysRevLett.129.061104 \n-. 2022b, Phys. Rev. Lett., 129, 061104, \ndoi: 10.1103/PhysRevLett.129.061104 \n-. 2023a, Phys. Rev. X, 13, 041039, \ndoi: 10.1103/PhysRevX.13.041039 \n- -. 2023b, Astrophys. J. Suppl., 267, 29, \ndoi: 10.3847/1538-4365/acdc9f \nAcernese, F., Agathos, M., Agatsuma, K., et al. 2014, Classical Quantum Gravity, 32, 024001, doi: 10.1088/0264-9381/32/2/024001 \nPunturo, M., Abernathy, M., Acernese, F., et al. 2010, Classical and Quantum Gravity, 27, 084007, \ndoi: 10.1088/0264-9381/27/8/084007 \nReitze, D., Adhikari, R. X., Ballmer, S., et al. 2019, Cosmic Explorer: The U.S. Contribution to Gravitational-Wave Astronomy beyond LIGO. \nhttps://arxiv.org/abs/1907.04833 \nRoulet, J., & Zaldarriaga, M. 2019, Monthly Notices of the Royal Astronomical Society, 484, 4216-4229, doi: 10.1093/mnras/stz226 \n- Sasaki, M., Suyama, T., Tanaka, T., & Yokoyama, S. 2018, Classical and Quantum Gravity, 35, 063001, \ndoi: 10.1088/1361-6382/aaa7b4 \n- Sathyaprakash, B. S., & Dhurandhar, S. V. 1991, Phys. Rev. D, 44, 3819, doi: 10.1103/PhysRevD.44.3819\n- Siemens, X., Allen, B., Creighton, J., Hewitson, M., & Landry, M. 2004, Classical and Quantum Gravity, 21, S1723, doi: 10.1088/0264-9381/21/20/015 \nSoni, K., Dhurandhar, S., & Mitra, S. 2024, Phys. Rev. D, 109, 024046, doi: 10.1103/PhysRevD.109.024046'}

2024MNRAS.534.1257B

The radial velocity RV method of exoplanet detection requires mitigation of nuisance signals arising from stellar activity. Using analytic cool and facular spot models we explore the use of central line moments CLMs for recovering and monitoring rotation induced RV variability. Different spot distribution patterns photospherespot contrast ratios and the presence or absence of the convective blueshift lead to differences in CLM signals between M and G dwarfs. Harmonics of the rotation period are often recovered with the highest power in standard periodogram analyses. By contrast we show the true stellar rotation may be more reliably recovered with string length minimization. For solar minimum activity levels recovery of the stellar rotation signal from CLMs is found to require unfeasibly high signaltonoise observations. The stellar rotation period can be recovered at solar maximum activity levels from CLMs for reasonable crosscorrelation function CCF signaltonoise ratios gt10005000. The CLMs can be used to recover and monitor stellar activity through their mutual correlations and correlations with RV and bisector inverse span. The skewness of a CCF a measure of asymmetry is described by the third CLM inlineformulatexmath idTM0001 notationLaTeXM3texmathinlineformula. Our noisefree simulations indicate the linear RV versus inlineformulatexmath idTM0002 notationLaTeXM3texmathinlineformula correlation is up to 10 per cent higher than the RV versus bisector inverse span correlation. We find a corresponding 5 per cent increase in linear correlation for CARMENES observations of the M star AU Mic. We also assess the effectiveness of the time derivative of the second CLM inlineformulatexmath idTM0003 notationLaTeXM2texmathinlineformula for monitoring stellar activity.

2024-10-01T00:00:00Z

['arXiv:2409.10306', '2024arXiv240910306B', '10.48550/arXiv.2409.10306', '2024MNRAS.tmp.2084B', '10.1093/mnras/stae2125', '2024MNRAS.534.1257B']

['Astrophysics - Solar and Stellar Astrophysics', 'Astrophysics - Earth and Planetary Astrophysics', 'Astrophysics - Instrumentation and Methods for Astrophysics']

Identifying activity induced RV periodicities and correlations using central line moments

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']

https://arxiv.org/pdf/2409.10306.pdf

{'J. R. Barnes 1 , S. V. Jeffers 2 , C. A. Haswell 1 , M. Damasso 3 , F. Del Sordo 4 , F. Liebing 2 , M. Perger 4 , 5 , G. Anglada-Escudé 4 , 5': "- 1 School of Physical Sciences, The Open University, Walton Hall, Milton Keynes. MK7 6AA. UK\n- 2 Max Planck Institute for Solar System Research, Justus-von-Liebig-Weg 3, 37077 Göttingen. Germany 3 INAF - Osservatorio Astrofisico di Torino Strada Osservatorio 20, I-10025 Pino Torinese (TO), Italy 4 Institut de Ciències de l'Espai (ICE, CSIC), Campus UAB, Carrer de Can Magrans s/n, 08193 Bellaterra, Spain\n- 5 Institut d'Estudis Espacials de Catalunya (IEEC), c/ Gran Capità 2-4, 08034 Barcelona, Spain \nAccepted for publication in MNRAS - 2024 September 10.", 'ABSTRACT': 'The radial velocity (RV) method of exoplanet detection requires mitigation of nuisance signals arising from stellar activity. Using analytic cool and facular spot models, we explore the use of central line moments (CLMs) for recovering and monitoring rotation induced RV variability. Different spot distribution patterns, photosphere-spot contrast ratios and the presence or absence of the convective blueshift lead to differences in CLM signals between M dwarfs and G dwarfs. Harmonics of the rotation period are often recovered with the highest power in standard periodogram analyses. By contrast, we show the true stellar rotation may be more reliably recovered with string length minimisation. For solar minimum activity levels, recovery of the stellar rotation signal from CLMs is found to require unfeasibly high signal-to-noise observations. The stellar rotation period can be recovered at solar maximum activity levels from CLMs for reasonable cross-correlation function (CCF) signal-to-noise ratios > 1000 - 5000. The CLMs can be used to recover and monitor stellar activity through their mutual correlations and correlations with RV and bisector inverse span. The skewness of a CCF, a measure of asymmetry, is described by the third CLM, 𝑀 3 . Our noise-free simulations indicate the linear RV vs 𝑀 3 correlation is up to 10 per cent higher than the RV vs bisector inverse span correlation. We find a corresponding ∼ 5 per cent increase in linear correlation for CARMENES observations of the M star, AU Mic. We also assess the effectiveness of the time derivative of the second CLM, 𝑀 2 , for monitoring stellar activity. \nKey words: stars: activity - stars: atmospheres - techniques: spectroscopic - methods: observational', '1 INTRODUCTION': "of planets, we must study both younger and older populations of planets that orbit stars with significantly different activity levels. \nThe recovery of dynamically induced radial velocity signals due to orbiting exoplanets can be severely hampered by activity signals from the host star. As a consequence, the majority of radial velocity surveys have targeted low-activity stars that are well into their main sequence lifetime. Although large scale photometric surveys searching for transiting planets do not necessarily pre-select low-activity stars, the youngest stellar clusters with ages of a few 10 -100 Myr are generally avoided due difficulties posed by activity-related variability. In addition, moderately active stars that have reached the main sequence can also still exhibit activity signatures that are comparable or greater in amplitude than the dynamical signatures induced by orbiting low-mass planets. Demographic features found in older exoplanets depend on orbital evolution or interaction with the host star at younger ages. To fully understand the Galaxy's population \nThe study of stellar activity and identification of spectroscopic activity signals has been established since the first exoplanet detections were made (Saar et al. 1998). Techniques to correct for long term magnetic variability arising from changes in absorption line shapes soon followed (Saar & Fischer 2000). Subsequent studies have provided detailed assessments of the stellar line bisector variability due to starspots (Desort et al. 2007). The signatures of more complex cool spot and hot facular regions have also been investigated by Meunier et al. (2010a) in detail, using the Sun as a laboratory. Extrapolation of the solar analogue has enabled stars with different spectral types, activity features and activity levels to be modelled and studied in detail (Barnes et al. 2011a; Meunier & Lagrange 2013; Dumusque et al. 2014; Jeffers et al. 2014; Borgniet et al. 2015; Herrero et al. 2016; Barnes et al. 2017a; Jef- \nfers et al. 2022). While direct physical modelling plays a crucial and important role in characterising and correcting activity signatures, non-parametric posterior modelling and machine learning techniques have also been successfully employed to account for variable stellar activity signals combined with planetary induced radial velocity variations (Haywood et al. 2014; Rajpaul et al. 2015; Cretignier et al. 2022; de Beurs et al. 2022; Perger et al. 2023; Liang et al. 2024; Zhao et al. 2024). These techniques rely on either prior estimates of activity related effects and timescales or on activity indicators simultaneously observed and derived from the same data used to make the radial velocity measurements. \nIn this paper we systematically investigate the use of central line moments (hereafter, CLMs) as a simple and consistent standardised measure of absorption line shape changes due to stellar activity. CLMs have been employed in the study of M dwarf stars, in the characterisation of instrumental and activity effects (Berdinas et al. 2016), and for the characterisation of activity signatures in the discovery of Proxima Centauri b (Anglada-Escudé et al. 2016). We demonstrate the expected line moment signatures from simple single spots and scaled solar spot models and simulate the expected sensitivities of the method for an M dwarf and a G dwarf star.", '2 CENTRAL LINE MOMENTS': 'The CLMs are a basic tool for measuring a statistical distribution. Similarly, by analogy they can describe the shape of a stellar emission or absorption line. Factors such as cool or hot spots, magnetic sensitivity due to the Zeeman effect and convective blueshift affect individual lines by different degrees. Because activity induced absorption line changes are often small effects, data with good signal-to-noise ratios (SNRs) are generally required for effective CLM measurement. In radial velocity studies, the cross correlation function (CCF) is commonly calculated by considering thousands of photospheric absorption lines. The CCF, can thus be thought of as a high SNR distribution that describes the effective mean shape of all the absorption lines considered. \nFor a perfectly stabilised spectrograph the CLMs arising from a single observation can be associated with different physical phenomena. From a planet hunting perspective, we are interested in the velocity shift of the star due to the reflex motion induced by an orbiting unseen exoplanet. The radial velocity measurement can alternatively be measured using the first CLM, 𝑀 1 , which measures the centre weighted velocity of the line. Because so many absorption lines are used to calculate a typical CCF or LSD line profile, the signal-to-noise ratio can be very high at typically SNR ∼ 1000 or more. This is a necessity for precise RV measurements with m s -1 precision, when a spectrum is recorded with a typical detector pixel velocity increment that is ∼ 1000 × greater. \nIf cool starspots or hot facular regions are present, a time series of observations may show periodic CLM variability in 𝑀 1 as a result of line-distorting active regions combined with stellar rotation. In other words, 𝑀 1 , behaves in the same manner as the RV measured from fitting a symmetric function to the CCF in that it is sensitive to both dynamically induced translations in velocity due to an orbiting exoplanet and to CCF asymmetries. In an attempt to resolve this ambiguity, a very simple measure of the shape of the line CCF, in the form of the bisector inverse span (BIS), is commonly made (Toner &Gray 1988; Martínez Fiorenzano et al. 2005; Desort et al. 2007). This measure of line asymmetry measures the difference in the mean velocity of two regions of a CCF that are representative of a deep and shallow region of the line profile. It can be used to assess whether \nparticular radial velocity periodicities arise from stellar activity. The BIS is defined such that a cool spot signature induces a radial velocity that anti-correlates with the measured centroid of the CCF (i.e. a negative correlation). The use of BIS calculates the difference in mean bisector in two regions of the CCF, simplifying the activity induced distortions, and thus running the risk of losing information. (Figueira et al. 2013) already demonstrated that alternative measures of the CCF shape can lead to more efficient recovery of activity induced RV variability. The skewness of the CCF, defined by the third CLM, 𝑀 3 , can be used to measure asymmetry. One advantage of measuring skewness rather than the BIS is that more of the CCF can be used, potentially offering increased sensitivity to activity induced distortions. \nWhile RV and BIS measurement, or similarly, 𝑀 1 and 𝑀 3 , are metrics used for investigating planetary and activity indices (e.g. see simulations in Desort et al. 2007; Dumusque et al. 2014 and Herrero et al. 2016), 𝑀 2 is also commonly monitored as the directly related line full width at half maximum (FWHM) or differential line width (Zechmeister et al. 2018). The CCF FWHM is also routinely modelled and examined in precision RV analyses (e.g. Herrero et al. 2016; Haswell et al. 2020; Jeffers et al. 2022; Zicher et al. 2022). \nThe use of CLMs thus standardises and extends common RV metrics that enable the signatures of activity to be studied. We consider the first 5 line moments, 𝑀 0 to 𝑀 4 , of a continuum normalised CCF. The CLMs in velocity space, 𝑣 , are then \n𝑀 0 = " 𝑖 F 𝑖 𝑛 Mean of CCF area 𝑀 1 = ¯ 𝑣 = " 𝑖 F 𝑖 𝑣 𝑖 " 𝑖 F 𝑖 Mean weighted velocity 𝑀 2 = 𝜎 𝑣 = √︂ " 𝑖 F 𝑖 ( 𝑣 𝑖 -𝑀 1 ) 2 " 𝑖 F 𝑖 Standard deviation 𝑀 3 = " 𝑖 F 𝑖 GLYPH<16> 𝑣 𝑖 -𝑀 1 𝑀 2 GLYPH<17> 3 " 𝑖 F 𝑖 Skewness 𝑀 4 = " 𝑖 F 𝑖 GLYPH<16> 𝑣 𝑖 -𝑀 1 𝑀 2 GLYPH<17> 4 " 𝑖 F 𝑖 -3 Kurtosis (1) \nwhere F 𝑖 is simply the flux of CCF velocity bin 𝑣 𝑖 (each with width Δ 𝑣 𝑖 ) in the normalised CCF (i.e. the continuum level is zero and the maximum flux, F max ≤ 1, is at or close to the line centre). Hence, 𝑀 0 is simply the mean of the CCF summed area, where 𝑛 is the number of data points. Dividing by 𝑛 preserves the value of 𝑀 0 for changes in instrument resolution and detector binning. The first even moment, 𝑀 2 , is the flux weighted standard deviation. Finally, the fourth CLM, 𝑀 4 , measures the Kurtosis or tailedness of the line CCF. The CLMs, 𝑀 0 , 𝑀 3 and 𝑀 4 are dimensionless, while 𝑀 1 and 𝑀 2 have units of velocity. \nWhile the CLMs are easily obtained from CCFs that are typically calculated by precision RV pipelines, they can alternatively be calculated via Least Squares Deconvolution (LSD) (Donati et al. 1997) of the stellar spectra. This procedure may provide less biased absolute reference estimates of higher order moments since LSD also minimises the effects of line shape distortion due to blending of absorption lines. In the ideal case, where 𝑁 lines are considered, we can expect an effective gain in SNR by a factor of up to √ 𝑁 lines . For an échelle spectrograph, several thousand absorption lines are generally considered. For a typical G star spectrum observed with mean SNR ∼ 100 -200, we can thus expect SNRs of order 5000 - \nFigure 1. Model local intensity profiles representing a spot, the photosphere and a facula. Local intensities at a spectral resolution of 𝑅 = 115 , 000 are shown normalised to the G dwarf photospheric continuum intensity level for 8 limb angles from 𝜇 = 1 . 0 at disc centre to 𝜇 = 0 . 13 near the limb. \n<!-- image --> \n10000 to be routinely achieved in either the CCF or least squares deconvolved profile Donati et al. (1997); Barnes et al. (1998, 2001).', '3 MODELS': 'We investigate the behaviour of CLMs by modelling absorption lines obtained from 3D stellar models. We use the Doppler imaging code, DoTS (Collier Cameron 2001), which enables line profiles to be calculated from synthetic spot models with any user-defined spot pattern (Jeffers et al. 2014). A simulated line profile is generated for each desired rotation angle with a specified SNR, emulating the CCF obtained via cross correlation. The CLM is then calculated directly from each simulated line profile. We have modified DoTS to generate absorption lines from a three temperature model and user-specified spot-filling factors. It has similar functionality to the code of Dumusque et al. (2014). Use of DoTS for spot modelling is well documented in a number of related publications (Barnes et al. 2011b; Jeffers et al. 2014; Barnes et al. 2017a).', '3.1.1 Cool Spots': 'Spot models of cool dark spots were simulated for the case of a G dwarf and a cooler M dwarf. For the G dwarf, we assumed 𝑇 phot = 5800 K and 𝑇 spot = 4000 K, while for the M dwarf, 𝑇 phot = 3500 K and 𝑇 spot = 3000 K (Berdyugina 2005; Panja et al. 2020; Jeffers et al. 2023). We adopted typical observation wavelengths in each case, performing simulations with resolution 𝑅 = 115 , 000 at V band wavelengths for the G dwarf (e.g. assuming HARPS-like observations) and at R band wavelengths for the M dwarf (e.g. the visible arm of CARMENES, but also assuming 𝑅 = 115 , 000). \nThe cool spot and photospheric intensities as a function of limb-angle, 𝜇 = cos ( 𝜃 ) , were obtained from the limb-dependent \nmodel intensity spectra provided by the Göttingen Spectral Library 1 (Husser et al. 2013) at 0 . 5 -0 . 6 𝜇 m(Gdwarf) and 0 . 6 -0 . 7 𝜇 m(M dwarf) wavelengths. The corresponding photospheric intensity to cool spot intensity ratios at disc centre for the chosen temperatures and wavelengths are 𝐼 phot / 𝐼 spot = 7 . 66 for the G dwarf and 𝐼 phot / 𝐼 spot = 3 . 40 for the M dwarf. A cool spot is thus expected to induce a larger amplitude signature on a G dwarf compared with an M dwarf. Simulated limb-dependent local intensity line profiles for cool spots and photosphere are shown in Fig. 1 (left and middle panels) for 8 limb angles. Further details are given below in §3.1.2 - 3.1.5.', '3.1.2 Facular contrasts': 'Simple models have been used to describe the observed hot facular intensity as a function of limb angle. Lawrence & Chapman (1988) and Lawrence (1988) investigated facular contrast using solar images. Lawrence (1988) proposed the contrast between quiet photosphere and faculae as a function of limb angle, 𝜇 , takes the form ( 𝐼 fac -𝐼 phot )/ 𝐼 phot = 𝑏 ( 1 / 𝜇 -𝑎 ) . Ahern & Chapman (2000) found the variation in facular contrast at 6723 Å shows very similar behaviour to bluer wavelengths (4706 Å), but with reduced contrast at the limb. Unruh et al. (1999) also computed models of facular contrasts as a function of limb angle and wavelength, finding general agreement with observations, but higher contrasts. More recent studies have examined the dependence of this trend on magnetic field strength and spectral type (e.g. Yeo et al. 2013; Norris et al. 2017). Simulations using the MURaM code (Vögler et al. 2005) were used by Johnson et al. (2021) to obtain limb-dependent variation of facular features of different magnetic field strengths for a range of spectral type. These were adopted by Zhao & Dumusque (2023) in their modelling of stellar activity.', '3.1.3 Mdwarf Facular contrasts': 'The simulations of Beeck et al. (2015) showed that for active M dwarfs, faculae are not significant. We adopted the M2 dwarf model with 500G vertical magnetic field strength, simulated for Kepler band wavelengths (Fig. 3 of Johnson et al. 2021). For this model, faculae remain dark (like cool spots) with a very small maximum contrast of ( 𝐼 fac -𝐼 phot )/ 𝐼 phot = -0 . 03) until they become indistinguishable (i.e. zero contrast) from the photosphere at limb angle, 𝜇 = 0 . 34. As faculae approach the limb further, they exhibit very small positive contrasts of around 0 . 015 at the maximum simulated limb angle of 𝜇 = 0 . 2. We fit the quadratic law found by Borgniet et al. (2015) for the Sun (see §3.1.4 below) to obtain contrast at any limb angle. The model tends to a contrast of 0 . 04 at 𝜇 = 0 . 0. The subtle change from negative to positive ratios in the photospheric vs facular intensity contrast can be discerned by close examination of the corresponding continuum intensities at 𝜇 = 0 . 12 and 𝜇 = 1 . 00 in Fig. 1 (bottom right panel). \nWealso investigated the effect of M dwarf faculae by assuming that the solar law can be applied to the M dwarf case at R band wavelengths, following the observations and modelling of Lawrence & Chapman (1988), Ahern & Chapman (2000) and Unruh et al. (1999). In this case, we used the same Borgniet et al. (2015) solar facular contrast law as for the G2 dwarf model described below.', '3.1.4 G dwarf Facular contrasts': 'For simplicity, for the G2 dwarf model, we adopted the law found by Borgniet et al. (2015), based on the solar observation modelling work of (Meunier et al. 2010a), where ( 𝐼 fac -𝐼 phot )/ 𝐼 phot = 0 . 131618 -0 . 218744 𝜇 + 0 . 104757 𝜇 2 . Again, although faculae have a relatively low contrast with the quiet photospheric level at disc centre (see Fig. 1, top left panel), the contrast rises significantly near the limb: for the V band, at limb angles of 𝜇 = 1 . 0 , 0 . 5 and 0 . 1 (limb angles 𝜃 = 0 · , 60 · and 84 · ), ( 𝐼 fac -𝐼 phot )/ 𝐼 phot = 0 . 018 , 0 . 048 and 0 . 11. The contrast tends to 0 . 13 at 𝜇 = 0 . 0.', '3.1.5 Convective Blueshift': 'At the solar photospheric level, the balance between the hotter upwelling central regions of granular cells and the sinking cooler intergranular regions results in a net effective blueshift of spectral lines. This results in skewed absorption line bisectors with a distinctive C-shape (Dravins et al. 1981). Inside a sunspot or facular region, where convection is suppressed, the degree of blueshift is reduced leading to both an apparent relative redshift of spectral lines and a change in line bisector shape. Moreover, Gray (2005) showed that changes in convective motions can be probed as a function of spectral type and class by measuring the bisector shape changes. \nThe convective blueshift (CB) effect contributes the highest degree of activity induced solar RV variability as the average number of active regions fluctuates on solar magnetic activity cycle timescales (Meunier et al. 2010a). Consequently, it can have an important systematic effect on precision RV studies aiming to detect exoplanets and precisely measure their masses. We investigated CB in our modelling of RV signatures in active stars (Jeffers et al. 2014). The effects of CB have also been reviewed, modelled and examined in detail by Meunier et al. (2010a), Meunier et al. (2010b), Borgniet et al. (2015), Dumusque et al. (2014) and Meunier et al. (2017). \nMeunier et al. (2017), examined the convective blueshift as a function of spectral type using a sample of late-F, G and K stars. Liebing et al. (2021) extended the sample size and included M stars and found that there is no convective blueshift or redshift for stars with 𝑇 eff < 4000 K. In the following sections, we consider models without CB for all the M dwarf simulations. \nDumusque et al. (2014) showed that Δ 𝑣 ∼ 330 ms -1 on average for G dwarf photospheric regions compared with active regions where convection is suppressed. Moreover the different line bisector shapes in the photosphere and spot regions have subtle and important effects on the mean line profile of an active star (Dumusque et al. 2014). For our G dwarf model, we have followed a more elaborate prescription by incorporating limb-dependent line shape changes and shifts for both active regions and the quiet photosphere. These are informed by the high resolution limb-dependent solar observations of Fe/i.pc lines by Cavallini et al. (1985) and Löhner-Böttcher et al. (2019). Fig. 1 shows the local intensity profiles with the bisectors and shifts that were derived and used by Zhao & Dumusque (2023) (see their Fig. 8).', '3.2 Individual spot construction': 'Line profiles can be simulated from our 3D stellar model for any spot distribution pattern. Each visible 0.5 · pixel was assigned a local intensity profile appropriate for either a cool spot, photosphere or facular region. For cool spots, we simulated both umbral regions at the spot intensity and penumbral regions with an intensity that is \nhalf of the difference of the photospheric and spot continuum intensities. The continuum intensity of each visible pixel is obtained by interpolating from the appropriate limb-dependent spectra (Husser et al. 2013), or in the case of the faculae, scaled relative to the photospheric continuum as described above in §3.1. We investigated cases for a cool spot, facular spot and cool + facular spots. The latter comprises a cool spot surrounded by a facular annulus. The ratio of facular coverage to cool spot area is known to be a function of cool spot coverage: we estimated facular areas from equations 1 & 2 of (Shapiro et al. 2014), which give combined cool spot umbra and penumbra areas ( 𝐴 u + 𝐴 p ), and facular areas ( 𝐴 f ). The three single spot cases are: \n- (1) A single cool spot with umbral radius, 𝑟 u = 2 · . A penumbral and outer umbral annulus is modelled for each spot, with 𝐴 p / 𝐴 u = 4, following Solanki & Unruh (2004). The area ratio typically varies between 3 and 5 (Brandt et al. 1990; Steinegger et al. 1990; Beck & Chapman 1993). The corresponding penumbral/umbral radius ratio for 𝐴 p / 𝐴 u = 4 is 𝑟 p = √︁ ( 𝐴 p / 𝐴 u + 1 ) 𝑟 u yielding √ 5 × 2 · = 4 . 47 · .\n- (2) A facular spot. For the purposes of illustration, we simulated a spot with the same total area as a cool spot. i.e. 𝐴 f = 𝐴 p = 4 . 47 · .\n- (3) A cool spot surrounded by a facular ring. For illustration, we simulated two cases: (a) 𝐴 f = 𝐴 u + 𝐴 p and (b) facular rings with 𝐴 f /( 𝐴 u + 𝐴 p ) = 26.9, 10.3 and 5.29 appropriate for solar minimum, maximum and high activity cases following the relationships of Shapiro et al. (2014). i.e. the facular contribution is greatest for the solar minimum case and lowest for the high activity case. \nOur 𝑟 u = 2 · spots have penumbral radius, 𝑟 p = 4 . 47 · and facular outer radii of 𝑟 f = √ 5 √︁ 𝐴 f /( 𝐴 u + 𝐴 p ) + 1 × 2 · = 6 . 32 · for case (a) with 𝐴 f = 𝐴 u + 𝐴 p . For case (b), 𝑟 f = 11 . 2 · , 15 . 1 · and 23 . 6 · respectively for the high activity, solar maximum and solar minimum cases.', '4 CENTRAL LINE MOMENTS FROM SINGLE SPOT MODELS': 'Wecharacterise the expected behaviour of CLMs by first simulating equatorial single starspots during a complete stellar rotation. We then investigate CLM signatures for single spots located at different latitudes before attempting to recover periodicities from the CLMs. Figure 2 shows the noise-free CLM signatures (calculated from absorption lines with SNR = ∞ ) for the M dwarf and G dwarf models for each of the three cases described above in §3.2 with an axial inclination of 𝑖 = 90 · and a spot located on the equator.', '4.1 CLMsignatures for an M dwarf': 'The CLM signatures for the cool, facular and cool + facular single spots are shown respectively in Figure 2 columns 1,2 and 3. Apart from differences in amplitude, a given CLM shows very similar behaviour in each of the three spot cases.', '4.1.1 Mdwarf with single cool spot': 'For approximately half of the rotation cycle, when spots are not visible, all CLM signatures remain flat. A reduction in local intensity due to a cool spot results in a localised apparent bump in the line profile at the velocity of the spot. The total stellar continuum flux is also reduced since the intensity of a cool spot is lower than the photospheric intensity. This leads to a reduction in line contrast relative \nFigure 2. Central Line Moments as a function of rotation phase for an M dwarf star (columns 1-3) and a G dwarf star (columns 4-6) with 𝑣 sin 𝑖 = 2 km s -1 and axial inclination, 𝑖 = 90 · . The central line moments as defined in Equation 1 are shown for an equatorial (latitude, 𝑙 = 0 · ) single cool dark spot with respective umbral and penumbral radii of 𝑟 u = 2 · and 𝑟 p = 4 . 47 · (columns 1 and 4), a bright facular spot with 𝑟 f = 4 . 47 · (columns 2 and 5). For the cool + facular models in columns 3 and 6, 𝑟 u and 𝑟 p are the same as in columns 1 and 4, while the ratios of facular area to combined umbral and penumbral areas are 𝐴 f /( 𝐴 u + 𝐴 p ) = 1 , 5 . 29 , 10 . 3 and 26 . 9 (solid, dotted, short-dashed and long-dashed curves). The solid line in column 2 shows the CLMs using the Johnson et al. (2021) facular limb-dependent contrasts, while the long dashed curve shows the corresponding CLM using the Borgniet et al. (2015) limb-dependent contrast used for the G dwarf model. The vertical dashed lines indicate the phases of minimum and maximum deviation for each CLM. N.B. the y-axis extent in columns 1, 3, 4 and 5 are the same (with the exception of 𝑀 1 for the G dwarf facular spot) and 1/10th this range in column 2 while the column 6 y-axis has a much larger extent. \n<!-- image --> \nto the normalised continuum. A minimum 𝑀 0 (Fig. 2, column 1, row 1), occurs when the spot is on the meridian, at the centre of the line profile. At this phase, the spot is travelling transverse to the observer with no Doppler shift. Relative limb darkening and foreshortening, as the cool spot is seen at different limb angles, are also important and lead to maxima in 𝑀 0 when the phase angle subtended by the spot relative to the stellar meridian is close, but not necessarily exactly, 45 · . The phases of maximum and minimum 𝑀 0 are shown in Fig. 2 by the vertical dashed lines for each CLM. The amplitude of variability in line area measured by 𝑀 0 is over an order of magnitude smaller than the velocity amplitudes measured by 𝑀 1 and 𝑀 2 and the dimensionless morphology changes measured by 𝑀 3 and 𝑀 4 (Fig. 2, column 1, rows 1 -5). \nThe line centroid, 𝑀 1 , is analogous to measuring a flux weighted RV signature due to an active region. With no CB in the M dwarf model, a cool equatorial spot traces a symmetric sinewave signature for the half of the rotation period over which it is visible. This signature results primarily as a consequence of the localised relative local intensity deficit due to the presence of the cool spot. The maximum and minimum values of 𝑀 1 occur when the line distortion due to the starspot is at a maximum. As with 𝑀 0 , the exact phases are related to the foreshortening limb angle of the spot, the relative limb darkening effect between spot and photosphere and \nthe spot size. When the spot signature is in the blueshifted portion of the line profile, the profile appears deeper in the unperturbed redshifted side, leading to a positive shift of the line centroid. The reverse signature occurs when the spot signature is in the redshifted portion of the line profile. \nFor the M dwarf behaviour of 𝑀 2 , the presence of a cool spot located at a phase angle between the limb and the meridian acts to slightly reduce the width of the profile when the maximum spot signature is present. The maximum occurs when the spot is on the meridian because the profile is effectively more U-shaped compared with the spot-free profile, thereby giving relatively more weight to regions away from the profile centre. The cool spot 𝑀 3 signature behaves in a roughly inverse sense to 𝑀 1 . It is defined such that a spot in the blueshifted portion of the line effectively creates a longer, skewed, tail on the redshifted side of the profile (negative skewness). This behaviour is exactly as seen with the anticorrelation behaviour between RV and BIS. Finally, 𝑀 4 measures the kurtosis or tailedness of the line profile. Here the maximum spot signature amplitude in 𝑀 1 will not only skew the profile (as seen in 𝑀 3 ), but will lead to a more tailed profile (positive Kurtosis) on average. Negative Kurtosis, corresponding to the minimum in 𝑀 4 , is seen when the profile appears more U-shaped due to a spot at the centre of the line. \nFigure 3. Central Line Moment signatures as a function of rotation phase for an M dwarf and G dwarf star with axial inclination 𝑖 = 60 · and spots at respective equatorial, low, high and circumpolar latitudes of 𝑙 = 0 , 30 , 60 & 80 · . The cool, facular and cool + facular models are plotted in the same columns as Fig. 2. The cool + facular models are plotted for 𝐴 f /( 𝐴 u + 𝐴 p ) = 5 . 29. \n<!-- image --> \no \no \no \no \nThe overall behaviour of the even numbered moments for a single spot is similar, with 𝑀 0 and 𝑀 4 showing the same general phase-dependent behaviour, while 𝑀 2 behaves in an inverse manner, but with the same general morphology. This is similar to the inverse behaviour of the odd numbered moments, 𝑀 1 and 𝑀 3 . However, close inspection of the CLMs in Fig. 2 reveals the relative phases of the maxima and minima (vertical dashed lines) are different for each CLM. This feature is important when considering the degree of linearity in the correlation between 𝑀 1 and 𝑀 3 , which we examine in detail in §6.', '4.1.2 Mdwarf with single facular spot': 'Afacular spot with the same total area as a cool spot ( 𝐴 f = 𝐴 u + 𝐴 p ) yields a signal that is more than an order of magnitude lower owing to the much lower contrast with the photosphere (Fig. 2, column 2, with 1 / 10 th the scale in Fig. 2 column 1). Because the adopted limb-dependent facular contrast with the photosphere is small and negative for limb angles in the range 0 . 34 < 𝜇 < 1 (Johnson et al. 2021; see §3.1.3), the general morphology of M dwarf faculae is the same as for a cool spot as shown by the solid curves in Fig. 2, column 2. Once spots approach the stellar limb, with 𝜇 < 0 . 34, the facular contrast becomes positive, though foreshortening results in smaller intensity contributions near the stellar limb, leading to relatively low amplitude effects. \nIn the solar-like case, where the facular contrast is always positive and increases towards the limb (Borgniet et al. 2015; see \n§3.1.2 - §3.1.4), behaviour that is broadly inverse to the cool spot CLM behaviour is seen (Fig. 2, column 2, long-dashed curve). However, since the facular intensity contrast with the photosphere is greatest at the limb, the maximum deviation for a facular spot occurs ∼ 0 . 05 earlier in phase (before phase 0.5), or later (after phase 0.5) than for a cool spot for our model setup (as shown by the vertical dashed lines in columns 1 and 2 of Fig. 2). For subsequent simulations, we use the facular contrast law described by Johnson et al. (2021) and assume that CB effects are not present.', '4.1.3 Mdwarf with single cool + facular spot': 'The third column in Fig. 2 demonstrates how a facular region around a cool spot reinforces the cool spot signature, leading to an increased amplitude. This is a consequence of the behaviour noted in §4.1.2 for the Johnson et al. (2021) facular contrast law (the Borgniet et al. (2015) solar law would lead to the opposite effect: a partial damping of the dominant cool spot component). The solid curve represents case (a) where the facular region has the same area as the cool spot. For case (b) with the highest facular area of 𝐴 f /( 𝐴 u + 𝐴 p ) = 26 . 9 (see §3.2), the signature is only augmented by a factor of up to around ∼ 2 in CLMs 𝑀 1 to 𝑀 4 , because of the low facular contrast.', '4.2 CLMsignatures for a G dwarf': 'The G dwarf CLM signatures for the cool, facular and cool + facular \n<!-- image --> \nFigure 4. Period recovery matrices showing the dominant recovered period, 𝑃 rec , as a fraction of the simulated period, 𝑃 rot , for M dwarf (left) and G dwarf (right) models with 𝑖 = 60 · . 𝑃 rec is colour coded and shown for CLM vs simulated 𝑣 sin 𝑖 value in each sub-panel. Sub-panel rows 1-4 show respective results for spot latitudes 𝑙 = 0 · , 30 · , 60 · , 80 · . Sub-panel columns 1-3 for each stellar model show results for the cool spot, facular spot and combined cool + facular spot models. \n<!-- image --> \nsingle spots are shown respectively in Figure 2 columns 4, 5 and 6. Here, with a fixed spot size, the 340 ms -1 convective blueshift yields a facular signature for a given CLM that is different from the corresponding cool spot signature in terms of morphology, but similar in terms of amplitude. As a consequence, the facular signature dominates the morphology of the cool + facular CLMs for the simulated 𝐴 f /( 𝐴 u + 𝐴 p ) ratios, leading to relatively large CLM amplitudes.', '4.2.1 G dwarf with single cool spot': 'The cool spot CLMs, 𝑀 0 -𝑀 2 , in the G dwarf model (Fig. 2, column 4) have slightly larger amplitudes compared with the M dwarf model owing to the higher contrast between photosphere and spots. The cool spot distortion in an absorption line, and consequently, the CLM signature, is dominated by the relatively large local intensity continuum contrast between the spot and photosphere. At this contrast, the local intensity line shape, or any shift in the local intensity line profile (i.e. from CB effects) have only secondary effects. However, the relative shift between the local intensity line profiles of the photosphere and spot can be seen as slight asymmetries. For example, in 𝑀 1 , the positive deviation before phase 0.5 is ∼ 5% greater than the negative deviation after phase 0.5. Similarly, the pre-phase 0.5 minimum in 𝑀 2 is ∼ 6% greater than the post-phase 0.5 minimum. The additional complexity induced by adopting limbdependent local intensity line shapes and CB blueshift velocity offset between photosphere and active regions (see Zhao & Dumusque 2023, Fig. 8) dilutes the amplitude of the higher order moments with \nthe effect that the 𝑀 3 and 𝑀 4 amplitudes are less than those seen for the M dwarf cool spot model. 𝑀 0 shows higher amplitude than for the M dwarf model, with a lower degree of complexity (i.e. it passes through a minimum when the spot is on the stellar meridian) but is still an order of magnitude lower than the other CLMs.', '4.2.2 G dwarf with single facular spot': 'The presence of a facular region results in a net apparent redshift of the line profile, illustrated by the behaviour of 𝑀 1 (Fig. 2, column 5). A very small, almost negligible, blueshift (negative velocity) at phase 0.28 is seen. To first order, the shift is symmetric. The maximum redshift values occur when the facular region is viewed at phases ∼ 0 . 45 and ∼ 0 . 55 (the peak at ∼ 0 . 55 is slightly higher than the peak at ∼ 0 . 45) as a result of the limb angle dependency of the line bisector shapes and the relative convective blueshift values as a function of limb angle (see Zhao & Dumusque (2023), Fig. 8). Compared with the other CLMs, 𝑀 1 is unique in its morphology when a facular spot or dominant facular region (see §4.2.3 below) is present. This behaviour also contrasts with the M dwarf model where clear correlations or anti-correlations are seen between all odd moments or even moments.', '4.2.3 G dwarf with single cool + facular spot': 'The cool + facular case (Fig. 2, column 6) shows how the convective blueshift of the facular region dominates the morphology for all 𝐴 f /( 𝐴 u + 𝐴 p ) > 1 models (i.e. case (b) in §3.2); that is, the cool +', '8 J. Barnes': 'facular CLM morphologies more closely resemble the facular morphologies than the cool morphologies. Only for 𝐴 f /( 𝐴 u + 𝐴 p ) = 1 (i.e. case (a) in §3.2), are more equal contributions seen from individual cool and facular components, as would be expected from Fig. 2 columns 4 and 5. For the solar minimum case, when 𝐴 f /( 𝐴 u + 𝐴 p ) = 26 . 9 (long-dashed lines), the convective blueshift effect due to the large facular area results in an order of magnitude increase in the CLMs. With an average spot radius of 1 · , as seen on the Sun (Takalo 2020), the cool + facular variability from our models is of order 20 m s -1 , which is larger than the solar RVs derived by Meunier et al. (2010a). In reality facular regions are not confined to an annulus around a cool spot, but are more distributed, which would lead to lower amplitude signatures.', '4.3 The effect of instrumental resolution': 'We investigated the effect of increasing the resolution from the standard 𝑅 = 115 , 000 for the single spot G dwarf models with 𝑣 sin 𝑖 = 2 kms -1 discussed in the preceding sections. The CLM signatures for a 2 · equatorial spot, assuming 𝑖 = 90 · are shown in columns 4-6 of Fig. A1. Specifically, we looked at the higher resolutions of 𝑅 = 140 , 000 and 𝑅 = 190 , 000 of ESPRESSO (Pepe et al. 2021) in addition to a very high resolution of 𝑅 = 500 , 000. Because the rotational broadening is a significant fraction of the line broadening at 𝑣 sin 𝑖 = 2 kms -1 , we find that the effects on lower order moments is fairly subtle. The large convective blueshift that dominates the 𝑀 1 signature in a G dwarf (see equivalent RV signature contributions in Zhao & Dumusque (2023)) is already well resolved at 𝑅 = 115 , 000 so that further increasing the resolution only increases the amplitude of 𝑀 1 by 1 . 8%, 1 . 3% and 0 . 5% respectively for the cool, facular and cool + facular (solar maximum ratio with 𝐴 f /( 𝐴 u + 𝐴 p ) = 10 . 3) models at 𝑅 = 500 , 000. By contrast, the equivalent 𝑀 3 amplitude increases for the cool, facular and cool + facular models are respectively 36%, 63% and 60%. However, for the G dwarf model, the degree of correlation, which we investigate below, decreases , with increasing resolution. Although the amplitude changes are more pronounced in the higher order moments, we note that 𝑀 4 notably resolves more structure, which arises from the convective blueshift and limb-dependent bisector changes. For completeness, we also show CLM signatures for a very low 𝑣 sin 𝑖 = 1 kms -1 in Fig. A1 (columns 1-3). Although the CLM amplitudes are predominantly lower compared with 𝑣 sin 𝑖 = 2 kms -1 , there is a visibly more dramatic change in amplitude in 𝑀 3 , while the cool + facular 𝑀 4 signature at 𝑅 = 500 , 000 shows a very large amplitude signature.', '4.4 Dependence of CLM signature and spot latitude': "Figure 2 demonstrates that single spots located on the stellar equator do not produce purely sinusoidal signatures in the CLMs. This was noted by Boisse et al. (2011) (see their Fig. 7), who also showed that fractions of the true stellar rotation period, typically 𝑃 rot / 2 or 𝑃 rot / 3, may be recovered with greater power than 𝑃 rot when measuring CCF RVs. The recovered period with dominant power, 𝑃 rec , thus depends on the form of the CLM when it is visible, but also on the duration of visibility of a spot throughout a complete rotation, which is determined by the stellar axial inclination, 𝑖 , and the latitude, 𝑙 , of the spot. We investigated the behaviour of the CLMs for a star with 𝑖 = 60 · , with spots at latitudes, 𝑙 = 0 · , 30 · , 60 · and 80 · . We again considered cool spot, facular spot and cool + facular spot models. For the cool + facular spot case, we used 𝐴 f /( 𝐴 u + 𝐴 p ) = 5 . 29. \nTable 1. Sensitivity simulation 𝑣 sin 𝑖 and 𝑃 rot combinations for the G and Mdwarf models with respective 𝑅 ∗ = 1R ⊙ and 0 . 5R ⊙ . \nThe CLM signatures for a single spot at different latitudes are shown in Figure 3. Equatorial and low latitude (i.e. 𝑙 = 0 · and 30 · ) spot CLM signatures replicate the CLM signatures seen in the 𝑖 = 90 · simulation. In contrast, the circumpolar spots at 𝑙 = 80 · induce signals that more closely resemble pure sinusoids in almost all CLMs because they are always visible, but with varying intensity contribution modulated on the 𝑃 rot timescale to first order. As expected, the CLM amplitudes are largest for spots at the subobserver's latitude (90 o -𝑖 = 30 · ). The signatures for 𝑙 = 60 · are intermediate between those of low and circumpolar spots in both amplitude and morphology.", '4.5 Recovering periodicities from single spot models': 'Using the single spot models, with 𝑖 = 60 · and 𝑙 = 0 · , 30 · , 60 · and 80 · , we investigated the recovered periodicities of CLM signatures for an expanded parameter space by including the effect of rotation velocity. We considered 𝑣 sin 𝑖 = 1, 2, 5, 10 and 20 km s -1 for both the M dwarf and G dwarf models. Corresponding periods, based on the assumed stellar radii of 𝑅 ∗ = 0 . 5 R ⊙ and 1 R ⊙ were estimated and modified slightly for the shorter periods to minimise integer day-aliasing effects. The 𝑣 sin 𝑖 and rotation period combinations are shown in Table 1, with periods chosen to further minimise integer day rotation period aliasing. For each spot latitude, 𝑙 , and 𝑣 sin 𝑖 vs 𝑃 rot combination, 120 daily noise free (SNR = ∞ ) absorption profiles spanning 120 days were synthesised with observation time uncertainties of 3 hrs to further minimise 1 d period aliasing. The effect of adding finite noise is explored with more complex models in §5.', '4.5.1 Periodicities in Generalised Lomb-Scargle periodograms': 'Period searches were performed on each derived CLM timeseries using the generalised Lomb-Scargle (GLS) periodogram analysis(Zechmeister & Kürster 2018). In each case, the dominant recovered period, 𝑃 rec (i.e. the periodogram peak with the highest power) was recorded by searching for the most significant peak power at 𝑃 rot and its fundamental harmonics at 𝑃 rot / 2, 𝑃 rot / 3, 𝑃 rot / 4, 𝑃 rot / 5 and 𝑃 rot / 6. A tolerance of 10 per cent in 𝑃 rec was required for identification as either 𝑃 rot or one of its harmonics. Fig. 4 shows the recovered periodicities for the M dwarf and G dwarf models with an axial inclination of 𝑖 = 60 · for cases with a single spot placed at latitudes 0 · , 30 · , 60 · and 80 · . The recovered period, 𝑃 rec , for combinations of SNR and 𝑣 sin 𝑖 are colour coded according to the dominant GLS peak and plotted as solid circles. Broadly, the results can be summarised as follows: \n- (i) Single spots induce dominant CLM periodicities related to, but not necessarily at 𝑃 rot \n- (ii) Dominant periodicities at 𝑃 rot / 2 , 𝑃 rot / 3 or 𝑃 rot / 4 are found, depending on spot latitude and type\n- (iii) High latitude spots ( 𝑙 = 80 · ): the spots are always visible, predominantly resulting in sinusoidal behaviour at 𝑃 rot\n- (iv) As a single spot is simulated at successively lower latitudes, progressively higher harmonics of 𝑃 rot tend to be recovered\n- (v) For the G dwarf and M dwarf cool spot models, 𝑃 rec , is very similar, since spot contrast is the predominant contributor to CLM variation \n(vi) The G dwarf facular and cool + facular models recover 𝑃 rec at lower harmonics of 𝑃 rot compared with the M dwarf in 𝑀 0 , 𝑀 1 and 𝑀 2 because the convective blueshift effect dominates and simplifies the phase dependent signature \nThe simulations demonstrate that recovery of the true rotation period for even single spot configurations with a GLS periodogram should not always be expected, particularly for spots that appear at low latitudes. When facular regions are present and more sinusoidal behaviour is seen in the CLMs (e.g. G dwarf, 𝑀 1 models), 𝑃 rot is recovered. The lack of convective blueshift in M dwarfs means that the CLM signatures are more prone to showing higher harmonics of 𝑃 rot . Nevertheless, the contribution from a facular region is much smaller on an M dwarf compared with a G dwarf because the facular contrasts are small. They would thus be relatively difficult to detect in data with finite SNR. \nA single spot (or dominant compact spot group) will only potentially yield 𝑃 rot as the dominant periodicity in all CLMs if the spot is always visible at high latitude. This is most likely for stars with significant axial tilt relative to the observer (i.e. 𝑖 << 90 · ), and is dependent on the expected spot distribution. Under the solar paradigm, where spots generally only emerge at low-intermediate latitudes, with 𝑙 < 40 · , a star would need to be significantly inclined to yield this behaviour. Nevertheless, there is a large body of evidence that suggests active stars possess polar or circumpolar spots of significant size (e.g. see Strassmeier (2009) for a review of stars with Doppler images and Almenara et al. (2022) who have recently invoked a polar spot to explain unusual planet transit lightcurves).', '4.5.2 Periodicities from String Length Minimisation': "Alternative approaches have been used for recovering periodicities in data, particularly where signals are not strictly sinuosoidal. McQuillan et al. (2013) demonstrated that the auto-correlation function (ACF) is able to recover the correct rotation period for simulated cases where there are amplitude changes, discontinuities in the data, or where the variability isn't strictly sinusoidal. This method works well for data that are well sampled, and was used by McQuillan et al. (2013) to recover rotation periods from high cadence Kepler photometry. However, the ACF method is not effective when there are fewer data with more phase gaps. Instead, we opted to use the string length minimisation method (Burke et al. 1970, Dworetsky 1983). This method was applied successfully to the planet hosting DMPP-3 system (Barnes et al. 2020), which shows highly eccentric RV variability from the DMPP-3AB binary orbit at 507 d, which standard periodogram methods were not initially able to identify. A String Length (SL) periodogram is obtained simply by phase-folding the data using a sequence of trial periods. The phase-folded data for each trial period are ordered by phase (i.e. from 0 to 1) and the string length, defined by the total summed distance between neighbouring points, is calculated. An SL periodogram is thus constructed, in which the period with the minimum string length indicates the op- \ntimal phase-fold. This nonparametric approach is difficult to assess when data become noisy. For the purposes of our period recovery, we have chosen to identify 𝑃 rec by searching for the minimum peak that lies at 𝑃 rot or its harmonics (as described above in §4.5.1) by searching for SL minima that lie at ≥ 99% or 2 . 5𝜎 below the SL periodogram mean. \nFig. 4 shows 𝑃 rec identified from the SL minimum for for CLM vs 𝑣 sin 𝑖 at each spot latitude. The SL recovered 𝑃 rec is indicated by the open circle symbol, which surrounds the GLS recovered 𝑃 rec indicated by the closed circles . Importantly, the SL method recovers the simulated rotation in all cases: all open circles are grey, indicating 𝑃 rec = 𝑃 rot , unlike the recovered GLS periodicities. In §5, we next investigate CLM signature morphology and recovery of rotation periods in the CLMs with GLS and SL searches using more realistic spot distributions of stars with solar-like activity.", '5 CENTRAL LINE MOMENTS FOR SCALED MULTI-SPOT DISTRIBUTIONS': "Based on solar observations, we don't expect a single spot to be present at most observation epochs, except during periods of lower activity. Although simple quasi-sinusoidal variability is possible over a single stellar rotation, CLM variability is expected to be more complex when more than one spot or spot group is present at different stellar longitudes and latitudes. To investigate this further, we generated spot distributions for an M dwarf and G dwarf model.", '5.1 Multi-spot model construction': 'We simulated three models with different activity levels based on solar observations. Spot coverage fractions and other statistics are given in Table 2. Spot sizes were drawn from the log-normal distributions described by Solanki & Unruh (2004). We assumed the same spot size distribution models for both G dwarfs and M dwarfs, with spot sizes defined via their angular radii (i.e. the absolute spot sizes scale with stellar radius). \nWe modelled spots in groups, with the first spot group as the largest, dominant group. To achieve this, the number of spots in each successive group was divided by an ad hoc factor of 1.5 until a minimum spot group size of 2 spots. For the solar minimum, solar maximum and high activity models, we assumed the first group contained respectively 3, 24 and 36 spots (see Table 2 for further statistics). For each given spot group, a centroidal longitude and latitude was randomly allocated. Spots were then added to the group at Gaussian distributed random offsets with 1-sigma distance of 2 · (solar minimum and maximum) and 3 · (high activity). Spot sizes were randomly selected without replacement from the appropriate spot size distribution. We did not attempt to model individual spots as spot pairs. \nTo account for any overlapping spots, the total spot count for each model was scaled to ensure the total cool spot area coverages of 0.03%, 0.3% and 1.4% for the solar minimum, solar maximum and high activity models, as defined in (Solanki & Unruh 2004). The chances of spot overlap are higher for the larger facular regions, which were again painted as annuli around each cool spot. All facular regions were also scaled to maintain the appropriate 𝐴 f /( 𝐴 u + 𝐴 p ) ratios used in the one spot simulations (see §3.2). A minimum spot size equivalent to the 0 . 5 · pixel resolution of our models was assumed.', '10 J. Barnes': 'Figure 5. Top: Starspot distribution Mollweide projection maps (showing rotation phase and latitude) and corresponding snapshot images at arbitrary phase of the M dwarf model (columns 1 and 2, orange) and G dwarf model (column 3 and 4, yellow). For the G dwarf, three spot models are shown: solar min, solar max and high activity models with respective cool spot coverage of 0.03%, 0.3% and 0.14%. For the M dwarf high activity models, scenario i uses the same spot coverage as the G dwarf models, but with spots distributed at all latitudes. In scenario ii, for the high activity case only (0.14% spot fraction), all spots are located within a single group at high latitude. Umbral (darkest), penumbral, photospheric (yellow/orange) and facular (white) regions are indicated. \n<!-- image -->', '5.1.1 Mdwarf spot distributions': 'The simulated M dwarf spot maps with cool spots and facular regions are shown on the left of Fig. 5 with snapshots of the corresponding stellar models at random phases (by definition, longitude on the models runs in the opposite sense to phase on the maps). There is evidence that supports both distributed dynamo activity in M dwarfs and also large-scale dynamo activity. While Zeeman Doppler imaging studies of early M dwarfs tend to favour toroidal and non-axisymmetric large scale field geometry, strong axisymmetric poloidal structure is more common in stars later than M3.5V, when stars are thought to become fully convective (Donati et al. 2008; Morin et al. 2008, 2010). Brightness Doppler maps of rapidly rotating M0 - M9 M dwarfs reveal cool spot patterns that show spots both distributed on the stellar surface, but also in some cases, concentrated at higher or circumpolar latitudes (Barnes & Collier Cameron 2001; Barnes et al. 2004, 2015, 2017b,a), in broad agreement with the magnetic field reconstructions. Discussion of dynamo models, numerical simulations and their various predictions can be found in Morin et al. (2010). \nAlthough we simulated M2 dwarf spot contrasts, we investi- \ngated two M dwarf spot distribution patterns, which we subsequently refer to as scenario i and scenario ii, as follows: \n- i) random longitude and latitude spot group placement with group sizes defined as for the G dwarf (Fig. 5)\n- ii) for the high activity model only, a single large group at latitude 60 · . The standard deviation of the spots within the group is 13 ·', '5.1.2 G dwarf spot distributions': "The G dwarf model approximates Spörer's law for the Sun by restricting spot group centroids to latitudes 25 · < 𝑙 < 40 · (solar min) and 5 · < 𝑙 < 15 · (solar max and high activity. The simulated G dwarf spot maps are shown on the right of Fig. 5.", '5.2 Multi-spot model CLM signatures': "Fig. 6 illustrates the CLM signatures for the three activity models of solar minimum, solar maximum and high activity with the three spot type models with cool, facular and cool + facular spot distributions for 𝑣 sin 𝑖 = 2 km s -1 . For each activity model, the facular spot \nFigure 6. Central Line Moments as a function of phase for spot solar min, solar max and high activity models for a star with 𝑣 sin 𝑖 = 2 kms -1 . Bottom panels: M dwarf model CLMs for scenario i with uniformly distributed spot groups (solid lines) and for scenario ii, for the high activity model only with a single high latitude spot group (dashed lines). Top panels: G dwarf models with spot group centroids restricted to latitudes 5 · < 𝑙 < 15 · (solar min) and 25 · < 𝑙 < 40 · (solar max and high activity) and the two largest spot groups located 180 · of longitude apart. \n<!-- image --> \nmodels use the same annular regions as the cool + facular spot models, but with the cool spot umbral and penumbral regions replaced with photosphere. This more readily enables a comparison between the cool and facular signatures and their relative contributions to the cool + facular signatures. Overlap of randomly located spots can lead to modification of the facular to cool spot ratio. The mean ratio of 𝐴 f /( 𝐴 u + 𝐴 p ) per spot was modified in each multi-spot model to ensure the global 𝐴 f /( 𝐴 u + 𝐴 p ) values described in §3.2. \nThe morphology of the CLMs vs phase in the multi-spot models is more complex than for a single spot. For multi-spot models, \nsince spots are more distributed, we expect continuous variation of CLMs to be common throughout a single rotation, unlike single spot models, where the CLMs may be constant for half the rotation phase. It is thus useful to describe a CLM's variability via its harmonic complexity, Ψ , which simply describes the number of complete cycles during a single rotation spanning phases 0 < 𝜙 < 1. This relates directly to the previous simulations and discussion for single spot models: a harmonic complexity of Ψ might reasonably be expected to lead to recovery of stellar rotation at 𝑃 rot / 𝜓 . \nFig. 6 shows that the degree of harmonic complexity in the \nTable 2. Spot model statistics. The factional spot coverage of umbral regions, number of spots and maximum and minimum umbral radii are given. The high activity model for the single high latitude spot group (scenario ii) are given in parentheses. \nCLMs is often higher than for a single spot, as expected for more complex spot patterns. Nevertheless, despite the random longitude placement in both models, low-order harmonic complexities still prevail in the CLM signatures. This is likely due to most of the spots appearing in just a few spot groups.", '5.2.1 Mdwarf multi-spot CLM signatures': "The M dwarf CLMs for spots distributed at all latitudes (scenario i) are shown in the lower panels of Figure 6. The solar minimum model, comprising 2 small spot groups, produces CLM signatures with relatively low Ψ , but the semi-amplitudes are very small and do not exceed ± 0 . 3 m s -1 . It is also worth noting that because the two spot groups in the solar minimum model are separated by ∼ 90 · , no spots are visible for a significant portion of a rotation, resulting in little or no modulation of the CLMs. Since the cool and facular regions appear dark (at least for most facular limb angles), the CLM morphologies are similar. The much lower contrast of faculae on the solar minimum model is balanced by the high relative area ratio with cool spots, 𝐴 f /( 𝐴 u + 𝐴 p ) = 26 . 9. These factors yield cool and facular signatures of similar amplitude that reinforce each other in the cool + facular spot model. The facular contribution introduces greater harmonic complexity at 𝑀 4 , but at an extremely low amplitude. \nThere is an order of magnitude increase in CLM amplitude in the solar maximum models compared with the solar minimum models. The harmonic complexity doesn't appear to change substantially, but shows greater amplitude in the signatures for phases in the approximate range 0 < 𝜙 < 0 . 5, where the dominant spot groups are located. As more spots are added in the high activity model, Ψ appears to increase in the higher order moments, as a consequence of the more even distribution of spot groups (see Fig. 6). For 𝑀 1 , on the other hand, Ψ appears to show a decrease. As the relative ratio of 𝐴 f /( 𝐴 u + 𝐴 p ) decreases with increasing activity, the facular contribution can be seen to decrease relative to the cool spot contribution. The cool + facular, high activity, model CLMs are dominated by the cool spots. with the consequence that they have a smaller or smoothed-out contribution relative to the larger spot groups. In general, the amplitudes of the CLMs is relatively low for scenario i, suggesting that the signatures may be difficult to discern with realistic data and SNRs. \nScenario ii, simulated for the high activity model only with one large spot cluster, resembles a single spot at high latitude. In Fig. 6, the dashed curves in columns 7 -9 show the CLM amplitudes for scenario ii with amplitudes divided by a factor of 5 to aid visual comparison. Clearly, Ψ is much lower and exhibits behaviour that is closer to the single spot simulations with 𝑙 = 60 · .", '5.2.2 G dwarf multi-spot CLM signatures': "The cool spot CLM amplitudes increase by an order of magnitude between the solar minimum and maximum models. This trend continues to a lesser degree between the solar maximum and high activity models, but is greater than was seen for the M dwarf. This is in part due to the spots being more clustered in lower and narrower latitude bands in the G dwarf model. \nThe facular CLM amplitudes are somewhat larger when compared with the corresponding cool spot CLMs. The relative decrease in 𝐴 f /( 𝐴 u + 𝐴 p ) with increasing activity means that the facular 𝑀 1 doesn't change amplitude between solar minimum and maximum models despite an increased total area. Further, the faculae are found in more groups and at relatively low contrast, so that their signatures are resolved at more longitudes or phases. In addition, since the facular signatures are greatest near the limb, when features are also foreshortened, the change in 𝑀 1 amplitude is fairly small as facular area increases. In these noise-free cases, the higher order moment amplitudes, 𝑀 2 -𝑀 4 , show comparatively larger changes. \nThe morphology of the cool + facular model signature is dominated by the facular contribution, contrasting with the M dwarf models, where cool and facular CLM amplitudes were more closely matched. Even the high activity facular + cool G dwarf model with the lowest 𝐴 f /( 𝐴 u + 𝐴 p ) = 5 . 29 more closely resembles the facularonly model for all the CLMs. The primary cause of facular dominance stems form a combination of the larger area, albeit at lower contrast than a cool spot, and the CB of 340 m s -1 , as was illustrated for the single spot models in Fig. 2. We note that in the high activity model, the CLM amplitude is greatest in the phase 0 . 0 < 𝜙 < 0 . 5 interval owing to the two spot groups at closely separated longitudes. The overall effect is reminiscent of the single spot simulations. Generally, the harmonic complexity, Ψ does not appear to change considerably across the three activity models. Arguably, it appears to decrease in the high activity model owing to the two dominant spot groups. \nSince 𝑀 1 is effectively equivalent to measuring spot induced RV (i.e. in our simulations with no dynamic Keplerian signals present), our findings are in good agreement with those of Meunier et al. (2010a) for solar minimum and maximum models (e.g. see their Fig. 7) when comparing the cool spot and facular components with convective blueshift included. Because 𝑀 1 is effectively centre-weighted, the total amplitude is lower than for RV, for which it can be considered a proxy.", '5.2.3 CLM signatures summary': 'Our simulations do not lead us to expect significant differences between cool spot and cool + facular model CLMs for an M dwarf because our models adopt the findings of Beeck et al. (2015) and Johnson et al. (2021), who do not predict strong facular contrasts on M dwarfs, and also Liebing et al. (2021), who empirically find that convective blueshift is absent. Because the low contrast faculae may appear dark on an M dwarf (except at the smallest 𝜇 ), the primary effect of adding them to the model is to fractionally increase the amplitude of cool spot signatures. There are however some further subtle changes to morphology of the CLMs when faculae are modelled in conjunction with cool spots. The more pronounced facular and convective contributions for earlier spectral types are the key ingredients that differentiate G dwarf CLM signatures from those of the M dwarf models. \nCLM signatures are lower in the M dwarf scenario i models compared with G dwarfs because (a) spots have lower contrast (b) \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 7. Period recovery matrices for the simulated spot activity models using Central Line Moments. The matrices indicate the combinations of SNR and 𝑣 sin 𝑖 for which 𝑃 rec , the simulated rotation period or a harmonic of the rotation is recovered. For GLS periodogram searches, finite SNR recoveries are indicated with filled squares, and SNR = ∞ (noise-free) recoveries are shown with filled circles. The key shows the colours that represent the recovered period harmonics. Symbols with grey borders and open symbols plotted in grey, indicate SNR and 𝑣 sin 𝑖 combinations where the SL period analysis returns 𝑃 rec = 𝑃 rot . A stellar inclination of 𝑖 = 90 · was simulated for all models. Bottom panels: Period recovery matrices for an M dwarf. The panels show results for scenario i with randomly placed cool + facular spot groups at solar minimum, solar maximum and high activity levels and scenario ii, with cool + facular spots for the high activity high latitude spot group. Top panels: Period recovery matrices for a G dwarf. The simulations are grouped into models with cool spots, facular spots and cool + facular spots, each showing the three simulated activity models: solar minimum, solar maximum and high activity. \n<!-- image --> \nspots may appear at higher latitudes, which means they are more foreshortened and do not traverse the full velocity extent of photospheric absorption lines and (c) facular contrasts are low and there is no convective blueshift effect. However, the relative amplitude of the CLMs increases considerably for the high activity M dwarf spot distribution scenario ii with all spots in a single group, where the CLM amplitudes are similar to those of the high activity G dwarf model.', '5.3 Recovered periodicities from the spot distribution models': 'We investigated period recovery for various combinations of SNR and 𝑣 sin 𝑖 using our spot distribution models. Although evolution of activity features, including spot growth, decay and differential rotation will modify CLM signatures, for the purposes of this study we assumed fixed spot patterns with no spot evolution. Based on our findings in §5.2, we consider M dwarf models with cool + facular spots only in this section and focus instead on differences between the M dwarf scenario i and scenario ii models. \nDetermination of stellar rotation directly from CLMs is dependent on several factors apart from activity level. Activity is likely to be greater for more rapidly rotating stars with larger 𝑣 sin 𝑖 , making 𝑃 rot easier to determine. For slow rotators, rotational modulation becomes difficult to resolve. The balance between activity signature amplitude, 𝑣 sin 𝑖 and data SNR is thus an important consideration, especially when we also consider that line equivalent width is conserved as 𝑣 sin 𝑖 increases (i.e. lines are shallower relative to the continuum). \nIn general, empirical data sets are obtained with unique sampling that is governed by various observing constraints, impacting on the ability to recover periodicities. For simplicity, we simulated 60 observations and assumed alternate-day sampling for 120 days with 3 hr uncertainty on the time of each observation to reduce aliasing. As in §4.5, the three spot activity models, solar minimum, solar maximum and high activity, were simulated with a range of 𝑣 sin 𝑖 and rotation periods appropriate for our G and M dwarf models. We assumed stellar axial rotation, 𝑖 = 90 · for these simulations. Trial values for SNR = 500, 1000, 2000, 5000, 10000, 20000 and 50000, representative of typical CCF profiles obtained in precision RV studies spanning several thousand absorption lines, were simulated along with the noise-free (SNR = ∞ ) case. We performed GLS and SL period searches to identify and compare the recovered period, 𝑃 rec , with the simulated rotation period, 𝑃 rot and its harmonics as outlined in §§4.5 - 4.5.2. Fig. 7 shows the recovered GLS and SL periodicities for combinations of SNR and 𝑣 sin 𝑖 , colour coded according to the period harmonic of the dominant GLS peak.', '5.3.1 Mdwarf multi-spot CLM periodicities': 'The lower panels of Fig. 7 show the periodicities recovered from CLMsforthesimulatedcool + facular M dwarf spot model scenarios described in §5.2.1. Scenario i, with spots randomly located in longitude and latitude shows changes in GLS 𝑃 rec , as the spot coverage increases. For both SNR = ∞ (filled circles) and finite SNRcases (filled squares), recovered GLS 𝑃 rec are found at a range of 𝑃 rot harmonics, though there is a preponderance of 𝑃 rot / 3 in the solar maximum model and 𝑃 rot for 𝑀 1 in the high activity model. It is also clear that recovery of periodicity in the solar minimum model, with only 5 small spots in 2 spot groups (Table 2), will not be possible for feasible SNRs. For the high activity model, with many scattered spot groups, recovery sensitivities are similar to the solar \nmaximum models, but the GLS 𝑃 rec is more unpredictable in terms of which harmonic of 𝑃 rot is recovered. Higher harmonics, such as 𝑃 rot / 5 and 𝑃 rot / 6 are recovered for 𝑣 sin 𝑖 = 10 and 20 km s -1 , but the presence at low activity or detection borderlines is likely less secure. \nFig. 7 also shows the recovered string length periodicity, SL 𝑃 rec , for instances where the true rotation period, 𝑃 rec = 𝑃 rot , is recovered. As in §4.5.2, we show the SL 𝑃 rec as an open symbol where there is no GLS 𝑃 rec at 0.1% FAP and as a grey open symbol enclosing the filled GLS 𝑃 rec symbols. For almost all cases, where werecover 𝑃 rot or a harmonic of 𝑃 rot with GLS, we recover precisely the simulated 𝑃 rot with SL. There are isolated borderline cases where the SL method was not able to recover a period or obtain 𝑃 rec = 𝑃 rot . In some cases, with the 2 . 5 -𝜎 requirement for identifying SL minima (§4.5.2), we are in fact more sensitive than the GLS method. \nFor the M dwarf with scenario i spot distribution patterns, it is clear that very high SNRs ≥ 5000 - 10000 are required to recover the stellar rotation signature. But even then, this may only prove reliable when spots are clustered in groups with solar maximum activity numbers and not too distributed as in the high activity model. While 𝑃 rot or a harmonic may be recovered with 𝑀 1 and a GLS period search in the scenario i high activity model, higher order moments that enable stellar activity and dynamically induced Keplerian signals to be distinguished do not confidently enable period recovery. The SL period method appears to be slightly more sensitive here and has the benefit of unambiguously recovering the true 𝑃 rot . \nIn scenario ii, CLMs from 𝑀 1 to 𝑀 3 allow rotation signatures to be recovered at lower SNRs of 1000 -2000. But here, period recovery for the higher order moments requires 𝑣 sin 𝑖 ≥ 5 km s -1 . Although scenario ii resembles the single spot simulation with a spot at high latitude, we have used 𝑖 = 90 · for the multi-spot simulation. Consequently, for the GLS 𝑃 rec , 𝑃 rot / 2 is common in 𝑀 1 and 𝑀 3 since the high latitude spot group is not always visible. The string length minimisation method again effectively recovers the true rotation period, 𝑃 rot , in contrast with GLS 𝑃 rec .', '5.3.2 G dwarf multi-spot CLM periodicities': "For the G dwarf model, GLS 𝑃 rec is predominantly 𝑃 rot and the lower harmonics from 𝑃 rot / 2 to 𝑃 rot / 4. Again, 𝑀 3 and 𝑀 4 are most likely to return GLS 𝑃 rec with higher harmonics of 𝑃 rot , especially for higher 𝑣 sin 𝑖 values. Inspection of the CLMs shows that considerable harmonic complexity may indeed be present within a single rotation, so GLS 𝑃 rec = 𝑃 rot / 5 and 𝑃 rot / 6 should not be surprising. Given the additional higher order structure that we find for higher spectral resolutions in §4.3 and Fig. A1, it isn't unreasonable to expect that higher order harmonics of 𝑃 rot might be recovered at lower 𝑣 sin 𝑖 values when observing with instruments such as ESPRESSO. \nThe presence of faculae significantly improves the sensitivity at all activity levels, but particularly, makes recovering periodicity in the solar minimum model more feasible. Despite this, it is clear that for the solar minimum model, high CCF SNR ≥ 10 , 000 is required to recover higher order moments ≥ 𝑀 2 that are typically useful as activity diagnostics. For solar maximum and high activity models, periodicities are clearly recovered at lower SNRs than for the solar minim models. Here, SNR ∼ 2000 -5000 is sufficient to recover periodicities in 𝑀 2 and 𝑀 3 . The increase in ability to recover a GLS 𝑃 rec between solar minimum and solar maximum activities is more pronounced when compared with the M dwarf. The further slight increase in sensitivity seen in the G dwarf high \nactivity model again contrasts with the M dwarf models, where the scattered nature of spots did not favour recovery of higher order moments. \nAs with the M dwarf, the SL periodograms return 𝑃 rec = 𝑃 rot when the GLS returns harmonics of 𝑃 rot . There are again some borderline sensitivity instances where the SL periodogram doesn't identify any peak. However, there are also instances where an SL 𝑃 rec is found, but no GLS 𝑃 rec is returned. \n5.3.3 CLM signatures and periodicities for a G dwarf with active longitudes \nWe also investigated the CLM signatures and periodicities for a G dwarf for spots that appear predominantly at active longitudes. For the Sun, Berdyugina & Usoskin (2003) showed that spots tend to appear in active regions separated by 180 · . We simulated the two largest spot groups for each activity model to be separated by 180 · in longitude in our simulations, rather than allocating their longitudes randomly. The CLM signature and period recovery matrices are shown in Figs A2 and A3. The CLM signatures in A2 show a notable difference when compared with the spots placed randomly in longitude. Visually, there is an obvious preponderance of signatures that appear with harmonic complexity Ψ = 2. Higher order harmonic structure starts to assert itself for 𝑀 3 and 𝑀 4 , but the amplitudes of the additional structure often appear smaller at the plotted 𝑣 sin 𝑖 = 2 km s -1 . These observations are born out in the period recovery matrices in Fig. A3, where there is a very clear dominance of GLS 𝑃 rec = 𝑃 rot / 2. The SL periodograms are again able to correctly recover the injected periods, so that SL 𝑃 rec = 𝑃 rot . The higher amplitude variability seen in the CLMs also increases the sensitivity slightly compared with the random spot group allocation.", '6 CENTRAL LINE MOMENTS AS ACTIVITY INDICATORS': 'When stellar activity signatures are present in absorption lines, care is needed to correctly and efficiently disentangle Keplerian signals from stellar astrophysical periodicities. It is common practice to use the FWHM or BIS derived from the CCF since they are able to describe line distortion periodicities due to stellar activity (e.g. Barragán et al. 2022, Zicher et al. 2022, Barnes et al. 2023). In the following sections, we look at the most significant CLM correlation signatures of single spot models before investigating the signatures arising from our M dwarf and G dwarf spot models.', '6.1 Single spot CLM correlations': "The amplitudes and slopes of RV vs BIS variations for stars with various spectral types were first investigated by Saar & Donahue (1997) and in detail by Desort et al. (2007). For a cool spot, the RV vs BIS shape resembles an inclined lemniscate (i.e. figure-of-eight or ∞ shape) with negative slope correlation (Desort et al. 2007; Boisse et al. 2011). This behaviour is also seen when the RV correlation with chromatic index (CRX) (Zechmeister et al. 2018) is measured for active M stars (Baroch et al. 2020; Jeffers et al. 2022). The degeneracy of the BIS vs RV relationship implied by the lemniscate arises as a result of foreshortening and limb-darkening effects due to the velocity-resolved cool spot changing its instantaneous limbangle on the stellar disc as the star rotates. As a result, the absolute value of the peak of the RV and the BIS deviations, and equivalently, \n𝑀 1 and 𝑀 3 deviations, occur at different limb angles (§4.1.1 and Fig. 2). It has already been noted that for BIS vs RV, the correlation length becomes shorter and the lemniscate collapses to a (near)linear correlation as spot latitude increases (Boisse et al. 2011). The degree of linear correlation thus increases for high latitude cool spots, but in the presence of noise, the decreased amplitude of the negative correlation will likely hamper the measurement of a linear correlation. Rapidly rotating stars that only possess higher latitude cool spots might thus be expected to enable more efficient activity modelling via a linear correlation compared with lower activity stars, provided that good SNR data can be obtained. \nFig. 8 illustrates the correlations of line moments for cases with a single 𝑟 u = 2 · , 𝑟 p = 4 . 47 · spot. For purely facular spots, we used 𝑟 f = 4 . 47 · and for the combined cool + facular spot cases, we assumed the high activity area ratio, 𝐴 f /( 𝐴 u + 𝐴 p ) = 5 . 29 (see §3.2). The spot was located on the equator at 𝑙 = 0 · of an M dwarf and G dwarf rotating with 𝑣 sin 𝑖 = 5 kms -1 . CLMs were derived from absorption lines with 3 · rotation resolution. \nThe lower left panel for the cool spot cases illustrates the behaviour of RV and 𝑀 1 with BIS and 𝑀 3 . The lemniscate with negative slope is clear and is comparable for both M dwarf and G dwarf models. In all sub-panels, the extent of 𝑀 1 is scaled by a constant to optimally match RV. Similarly, 𝑀 3 is scaled up to match BIS to enable a visual comparison of the correlations. Because RV and BIS have respectively higher corresponding amplitudes than 𝑀 1 and 𝑀 3 , we might expect a higher degree of correlation, suggesting that BIS is a more sensitive indicator of activity. However, as Fig. 8 columns 1 and 3 reveal, RV vs 𝑀 3 marginally shows the highest degree of (negative) linear correlation. The degree of correlation of RVvs 𝑀 3 compared with RV vs BIS is around ∼ 3%greater in both the M dwarf and G dwarf cool and cool + facular models for spots at low-mid latitudes. Corresponding improvements of respectively 6% and 10% are found for facular spots for the M dwarf and G dwarf models. \nAlthough inspection of Fig. 2 leads us to expect negative linear correlations of 𝑀 1 and 𝑀 3 in many cases, it can also be seen that a correlation between 𝑀 1 and 𝑀 2 is less obvious. The FF ' method for predicting RVs from photometric data (Aigrain et al. 2012) involves taking the (time) derivative of the photometric signal. In terms of morphology, this has the equivalent effect of transforming the even CLM signatures ( 𝑀 0 , 𝑀 2 and 𝑀 4 ) for a cool single spot into a signature more closely resembling the odd ( 𝑀 1 and 𝑀 3 ) CLM signatures. Fig. 8 columns 2 and 4 show the correlation of RV and 𝑀 1 with the time derivative of 𝑀 2 , which we denote 𝑀 ' 2 . While lemniscate figures with negative correlation are seen for all M dwarf 𝑀 3 signatures as well as the cool single spot G dwarf signature, the corresponding 𝑀 ' 2 signatures show lemniscates with positive correlation, though visual inspection suggests the degree of correlation is lower compared with 𝑀 3 . For models with both facular and cool + facular spots, the signatures for the G dwarf model are somewhat different in both morphology and amplitude where a skewed figure is seen. Facular spots have higher intensity nearer to the limb, when the velocity amplitude is greatest, leading to a more linear correlation between RV and 𝑀 ' 2 for the G dwarf where the asymmetry of the correlation over the rotation cycle is a consequence of the CB effect. The lower spot contrasts and lack of CBeffect in the M dwarf leads to low-amplitude facular signals. The Mdwarf facular spot signatures are an order of magnitude smaller than the cool spot signatures. For the G dwarf, the cool + facular signatures appear to be much higher than in the individual cool and facular spots cases because the area of the facular annulus (i.e. we used the high activity ratio 𝐴 f /( 𝐴 u + 𝐴 p ) = 5 . 29) is considerably \nFigure 8. The correlated behaviour of CLMs for M dwarf (left) and G dwarf (right) models with a cool spot (bottom), facular spot (middle) and cool + facular spot (top). The behaviour of BIS, 𝑀 3 , and 𝑀 2 time derivative, 𝑀 ' 2 (see key) as a function of RV or 𝑀 1 is shown for 𝑣 sin 𝑖 = 5 km s -1 for a spot located at latitude 𝑙 = 0 · . The spot radii ( 𝑟 u = 2 · ) are defined in §3.2. \n<!-- image --> \ngreater than for a single 𝑟 f = 4 . 47 · facular spot. Obviously, for solar minimum and solar maximum with 𝐴 f /( 𝐴 u + 𝐴 p ) = 26 . 9 and 10 . 3, the facular contribution will dominate the G dwarf signatures even further. \nPractically, taking time derivatives is only possible with dense time-sampling over a stellar rotation cycle. Instead, Barragán et al. (2022) incorporated the time derivative of the GP when performing a multidimensional approach to analysing spectroscopic timeseries to account for the difference in behaviour of RV with FWHM as compared with RV and BIS. A better correspondence between RV and FWHM variability due to cool spots is obtained this way, though good time sampling will still be desired to reliably inform the modelling process. In the following sections we investigate the effectiveness of using 𝑀 3 and 𝑀 ' 2 for our M dwarf and G dwarf single spot models via the Pearson's 𝑟 linear correlation statistic.", '6.2 CLMs single spot correlations as a function of spot latitude': "To further investigate the effective correlations between CLMs, we first performed simulations with single spot models for a 2 · spot located in the latitude range 0 · ≤ 𝑙 ≤ 80 · at 10 · intervals. We obtained RV and BIS in addition to the CLMs and their time derivatives. We again simulated 120 measurements over a single rotation with no noise for the case with axial inclination 𝑖 = 90 · . Fig. 9 shows Pearson's 𝑟 for single cool spots, facular spots and cool + facular spots for both the M dwarf and G dwarf models with rotational broadening of 𝑣 sin 𝑖 = 5 km s -1 . The most significant correlations are plotted for RV vs BIS, RV vs 𝑀 3 , 𝑀 1 vs BIS and 𝑀 1 vs 𝑀 3 (purple and cyan) and RV vs 𝑀 ' 2 and 𝑀 1 vs 𝑀 ' 2 (green and yellow). \nThe Pearson's 𝑟 correlations can be summarised as follows: \nMdwarf and G dwarf - cool spots: \n- (i) 𝑀 3 and BIS show negative linear correlation while 𝑀 ' 2 shows positive correlation with RV or 𝑀 1 \nFigure 9. Illustrative Pearson's 𝑟 correlations between absorption line (or CCF) metrics for single spots with 𝑟 u = 2 · and 𝑣 sin 𝑖 = 5 km s -1 located at latitudes 0 o to 80 o . The lower plots show Pearson's 𝑟 linear correlation for an M dwarf, while the upper panels show the same correlations for a G dwarf. The key indicates the line metric pairs that are considered in each case. \n<!-- image --> \n- (ii) 𝑀 3 shows a greater degree of linear correlation than 𝑀 ' 2\n- (iii) 𝑀 3 and 𝑀 ' 2 show a greater degree of correlation with RV than BIS\n- (iv) The degree of correlation is greatest for high latitude spots\n- Mdwarf - facular and cool + facular spots:\n- (i) The 𝑀 3 and BIS correlations, follow the same trend as cool spots, but with improved correlation at low-intermediate latitudes\n- (ii) 𝑀 ' 2 is moderately correlated with RV or 𝑀 1 with greatest correlation for a spot at intermediate latitudes and a fall in correlation for high latitude spots\n- G dwarf - facular and cool + facular spots:\n- (i) 𝑀 1 vs 𝑀 3 and 𝑀 1 vs BIS correlations for low latitude spots are significantly lower compared with cool spots\n- (ii) 𝑀 ' 2 shows fairly uniform moderate-high correlation at all latitudes \nIf only cool spots are found on M dwarfs, we can expect both 𝑀 3 and 𝑀 ' 2 to act as good linear activity correlators with RV. For the M dwarf model, the drop in correlation of RV with 𝑀 ' 2 for high latitude spots with a facular contribution is likely due to the switch in contrast behaviour of our models near to the limb. For the G dwarf models, cool spots also induce high linear correlations in 𝑀 3 and 𝑀 ' 2 . Here, the negative correlation between RV and 𝑀 3 is \nhigh, with 𝑟 ≤ -0 . 75 for low-intermediate latitude spots (Fig. 9, solid purple line in lower left panel); the range over which spots are typically found on the Sun. The inclusion of a facular contribution in the cool + facular model reduces the RV vs 𝑀 3 correlation for intermediate latitudes to 𝑟 ≤ -0 . 60. RV vs 𝑀 ' 2 also performs well for G dwarf model cool spots, with 𝑟 = 0 . 66 for an equatorial 𝑙 = 0 · spot and 𝑟 = 0 . 83 for 𝑙 = 80 · . RV vs 𝑀 ' 2 exhibits less variation when cool + facular regions are present on the G dwarf model, with 𝑟 = 0 . 76 ( 𝑙 = 0 · ) and 𝑟 = 0 . 61 ( 𝑙 = 80 · ). \nIncreasing the spectral resolution leads to very small improvements in correlation between RV and 𝑀 3 . At 𝑣 sin 𝑖 = 5 km s -1 , for resolutions of 𝑅 = 140 , 000 and 190 , 000, we find respective (negative) linearity increases in Pearson's r of 1 . 3% and 2 . 3% for an equatorial spot. These improvements in linearity decrease with spot latitude with essentially no improvement for a spot with 𝑙 = 80 · . The gains are smaller for RV and 𝑀 ' 2 , with 0 . 7%and1 . 2%increases in Pearson's r for low latitude spots. \nFor rotational broadening of 𝑣 sin 𝑖 = 2 km s -1 , Fig. A4 shows that correlations are reduced. In particular, high latitude spots become harder to discern, leading to poorer correlations compared with 𝑣 sin 𝑖 = 5 km s -1 . The correlations with BIS and 𝑀 3 in Fig. A4 are plotted for -1 < 𝑟 < 1since the correlations clearly either switch sign, or become positive when faculae are present. This is particularly pronounced for the G dwarf model and likely arises because the 340m s -1 convective blueshift is a significant fraction of the stellar 𝑣 sin 𝑖 and because the shape of the bisectors is a function of limb angle. Spots at low latitude traverse wider ranges of limb an- \nat higher latitudes. The exact correlation coefficient is therefore dependent on combinations of these factors. \nFor RV vs 𝑀 ' 2 , moderate correlations persist for spots at most latitudes. In addition, Fig. A4 shows that despite a reduction in Pearson's r at the lower 𝑣 sin 𝑖 = 2 km s -1 , a moderate correlation remains for spots at lower latitudes. For solar-like activity levels, with solar-like facular levels, it may thus be more useful to use the time derivative of 𝑀 2 or a similar metric. \nWe find that 𝑀 1 generally correlates less well than RV, but since regular precision radial velocity work uses RVs from templatematched spectra of cross-correlation profiles (as in our simulations), 𝑀 1 is potentially most useful as an additional monitoring diagnostic parameter only. Unlike the BIS, which measures the skewness of the line or CCF via the difference of the mean of two sub-regions of the profile, 𝑀 3 can potentially make use of the entire profile, leading to increased correlation sensitivity of 𝑀 3 over BIS. We note that in practice, it is necessary to reject the unstable profile wing regions when measuring line shapes. This is routine procedure when measuring BIS in observed data and should also be applied to CLM analysis.", '6.3 CLMcorrelations for spot models': "We computed the RV and CLM correlations using the solar minimum, solar maximum and high activity spot models. For the M dwarf, we show correlations for the cool + facular spot models only using scenario i and also scenario ii for the high activity case only. For the G dwarf, correlations are shown for cool, facular and cool + facular spot models. Pearson's 𝑟 values are plotted in Fig. 10 for RV vs 𝑀 3 and RV vs 𝑀 ' 2 for 𝑣 sin 𝑖 = 1 , 2 , 5 , 10 and 20 km s -1 and are colour coded for 500 ≤ SNR ≤ ∞ . The SNR = ∞ correlations (black curves in Fig. 10) give an indication of the maximum attainable correlation and indicate the true sense of a correlation (i.e. either negative, with 𝑟 < 0, or positive, with 𝑟 > 0). Typically, a lower SNR reduces the degree of negative or positive correlation, and in some cases, results in a misleading outcome, where a true negative correlation may appear as a positive correlation or vice-versa. \nEven rotation velocities of 𝑣 sin 𝑖 = 1 km s -1 enable significant solar minimum RV vs 𝑀 3 activity correlations to be detected, but only for very high SNR or SNR = ∞ . For RV vs 𝑀 3 , when faculae are present, the correlations change sign at 𝑣 sin 𝑖 = 1 and 2 km s -1 , as we found for single spots in §6.2; hence, for the appropriate panels in Fig. 10, we again plot -1 < 𝑟 < 1. This suggests that odd moments such as 𝑀 3 alone can not reliably distinguish between activity induced RV shits and dynamically induced shifts. Although the corresponding RV vs 𝑀 ' 2 correlations may be lower at a given finite SNR, they are more stable in the presence of facular regions. 𝑀 ' 2 may thus be a more reliable activity indicator at low 𝑣 sin 𝑖 values, though the necessary SNR and low correlations may prove challenging for solar minimum activity levels. Further details for the correlations with RV from the three activity models are discussed briefly below.", '6.3.1 Solar minimum': "Reliable correlations are not discernible for low 𝑣 sin 𝑖 and activity levels for the M dwarf model with SNRs that could feasibly be obtained with empirical data. A positive RV vs 𝑀 3 correlation of 𝑟 = 0 . 5 is found for the G dwarf cool + facular model at 𝑣 sin 𝑖 =1kms -1 and SNR = 10,000. Here, it would likely be more advisable to use \n𝑀 ' 2 , but clearly, the degree of correlation is much lower and very high CCF SNR ≥ 20 , 000 would be needed. For the G dwarf models, 𝑣 sin 𝑖 = 5 km s -1 and 10 km s -1 , SNR ≥ 5000 yields RV vs 𝑀 3 with Pearson's 𝑟 ≤ -0 . 5. Low-moderate degrees of correlation are recovered for RV vs 𝑀 ' 2 for SNR > 10000 when facular regions are present.", '6.3.2 Solar maximum': "The degree of correlation at a given SNR improves notably for both M dwarf and G dwarf models compared with the solar minimum models. For the M dwarf, SNR ∼ 5000 -10000 and 𝑣 sin 𝑖 ≥ 5 kms -1 yields Pearson's 𝑟 < -0 . 5 for RV vs 𝑀 3 , but SNR ∼ 20000 is needed to yield similar correlations for 𝑀 ' 2 . For the cool + facular G dwarf model; a star with rotation 𝑣 sin 𝑖 ≥ 5 km s -1 , requires SNR ∼ 1000 for strong RV vs 𝑀 3 correlation to be recovered. Here, moderate to strong correlations are also found in RV vs 𝑀 ' 2 for SNR ≥ 5000. The degree of linear correlation decreases significantly by 𝑣 sin 𝑖 = 20 km s -1 for models with facular spots. \n6.3.3 High activity \nFor the M dwarf, the degree of correlation in the high activity scenario i models decreases compared with the solar maximum model. This is a consequence of the uniform distribution of spots. As expected, for scenario ii with all spots are located in a single high latitude spot group, more significant correlations are found. For scenario ii, SNR ≥ 5000 enables 𝑟 < -0 . 5 to be recovered at 𝑣 sin 𝑖 ≥ 2 km s -1 for RV vs 𝑀 3 , though lower SNRs are needed when 𝑣 sin 𝑖 ≥ 5 kms -1 . Correlations increase significantly in the G dwarf high activity cool spot models. Here, despite a larger number of spots, the decline in relative facular contribution in the high activity models yields very similar correlation levels to the solar maximum models. Again, strong correlations are found for 𝑣 sin 𝑖 = 5 kms -1 and 10 km s -1 with high SNRs generally required for 𝑀 ' 2 when compared with the 𝑀 3 at a given 𝑣 sin 𝑖 . \nWe also investigated the effect of increasing the instrumental resolution. Figure A5 shows Pearson's 𝑟 correlations for the G dwarf models at instrumental resolutions of 𝑅 = 140 , 000 and 𝑅 = 190 , 000. Subtle increases in correlation are seen at a given SNR as resolution increases. While low activity and low 𝑣 sin 𝑖 values appear to preclude effective recovery of activity signatures at all simulated resolutions, it is also important to realise that, within the specified bounds of the models, the simulated spot distributions are just a single realisation. As such, the spot model correlations described above should only be used as a basic guideline for discerning trends and correlation amplitudes.", '7 SUMMARY AND DISCUSSION': "Our primary goal has been to demonstrate the use of CLMs derived from simulated high SNR absorption lines or cross-correlation functions to identify and recover activity related periodicities and trends. Wehavefocused on stars with activity levels similar to the Sun to enable us to determine the effective limits at which CLMs can provide useful information. \n<!-- image --> \n<!-- image --> \nCool + Facular spots - Scenario ii \n<!-- image --> \nFigure 10. Pearson's 𝑟 correlations for M dwarf (left panels) and G dwarf (right panels) for solar minimum, solar maximum and high activity models. The correlations for RV vs 𝑀 3 and RV vs 𝑀 ' 2 are shown for a range of 𝑣 sin 𝑖 values and SNR combinations in each sub-panel (see key). For the M dwarf model, the two spot distributions, scenario i and scenario ii, are shown for cool spots. For the G dwarf model, the cool, facular and cool + facular spot model correlations are shown. \n<!-- image -->", '7.1 Summary of main results': "For a single spot, we find the following behaviour of CLM signatures \n- (i) Cool spot CLM signatures are similar for both M dwarfs and G dwarfs; higher intrinsic spot-photosphere contrasts in G dwarfs (and also V band simulations vs R band for M dwarfs) yield larger signatures\n- (ii) Facular regions show a negative correlation with cool spots when convective blueshift is not present - i.e. in M dwarfs\n- (iii) Convective blueshift modifies the signature of CLMs in facular regions (and hence ii above) since a large asymmetric velocity shift affects the CCF wings where facular signatures have highest amplitude (i.e. for G dwarfs)\n- (iv) A consequence of (iii) is that the facular component dominates combined cool + facular signatures (i.e. for G dwarfs) \nThe recovered GLS periodicities, 𝑃 rec , that appear in the CLMs for a single spot are as follows: \n- (i) Low and intermediate latitude spot: a mixture of 𝑃 rot and harmonics of 𝑃 rot \n- (ii) High latitude spot: predominantly, 𝑃 rot\n- (iii) Higher harmonics of 𝑃 rot appear in stars with higher 𝑣 sin 𝑖 where more phase dependent CLM structure is resolved \nPeriod searches that assume sinusoidal variability, such as the (Generalised) Lomb-Scargle method, are not optimally matched to recover stellar activity signals and often lead to recovery of a harmonic of the true stellar rotation period. Instead, string length minimisation is particularly effective at recovering the true 𝑃 rot for spots at any latitude for the CCF SNRs that are readily achieved with precision radial velocity observations. \nFor the three simulated activity models with multiple spots: solar minimum, solar maximum and high activity, we similarly find \n- (i) For G dwarfs, the GLS period or harmonic of the period can be recovered for moderate to high SNRs and is generally easier at moderate and high 𝑣 sin 𝑖 values\n- (ii) Recovery of periods with GLS analysis for M dwarfs is challenging, often requiring SNR ≥ 10000 and 𝑣 sin 𝑖 ≥ 10 kms -1 and is most successful for high activity when cool spots are located at high latitudes in a single group \n- (iii) String length period searches reliably recover the true rotation period in most cases for the simulated range of 𝑣 sin 𝑖 and SNRs where GLS periodogram searches recover harmonics of the true rotation period \nWhen CLM periodicities are present, correlations between RV and higher order CLMs or their time derivatives show moderate-high degrees of linear correlation. \n- (i) line moments 𝑀 3 and the time derivative, 𝑀 ' 2 , show the strongest correlations with activity induced RV variability.\n- (ii) RV vs 𝑀 3 shows a stronger linear trend than the traditionally adopted RV vs BIS\n- (iii) RV vs 𝑀 ' 2 shows a moderate but more consistent linear trend across the range of simulated 𝑣 sin 𝑖 values, particularly for the G dwarf models with a facular component\n- (iv) Spot groups confined to high latitudes yield much higher degrees of linear correlation \nOur simulations have restricted spot latitude distributions for the G dwarf model to those seen on the Sun and expected from more distributed dynamo activity in M dwarfs (scenario i). However, based on the simulations of Granzer et al. (2000), we might expect spots to appear at intermediate latitudes for a G dwarf and high latitudes for an M dwarf (as per our M dwarf scenario ii model) for the fastest rotation rates we considered. While this may improve the degree of linear correlation, restricting spot latitudes in this way will also to lead to smaller CLM signatures for a fixed spot size or ensemble of spots, making recovery more challenging at any fixed SNR. Similarly, we expect a preference for periodicities recovered as lower harmonics of 𝑃 rot with GLS periodogram analysis, as evidenced by the M dwarf scenario ii model we considered. However, simulating changes in spot distributions with longitude and spot clustering within spot groups is generally more difficult.", '7.2 Application of CLM analysis to observations: AU Mic': "To illustrate the recovery of periodicities from CLMs, we used observations of the young active M1Ve star, AU Mic (HD 197481 / GJ 803), with photometric 𝑃 rot = 4 . 863 ± 0 . 010 d (Plavchan et al. 2020). 𝑃 rot (or very close to 𝑃 rot ) has been recovered by Doppler imaging methods, RVs and activity indicators, including chromospheric indicators, FWHM and BIS (Klein et al. 2021, Klein et al. 2022). Most importantly, the Zeeman Doppler map presented in Klein et al. (2021) reveals a decentered polar region of radial magnetic field suggesting a highly dipolar global magnetic field. Thebrightness Doppler images however suggest very weak cool spot structure close to the equator. In the HARPS images presented in Klein et al. (2022) at the later epoch, a weak high latitude cool spot at latitude 60 · is seen in the 2020 image. Two weak spots at equatorial latitudes are also seen in the same rotation hemisphere. The images also map weak, warm structure, at high and equatorial latitudes. The 2021 image reveals very weak, cool and warm, structure. It is likely that AU Mic is more active than our high activity spot simulations and the recovered radial magnetic field structure is probably more akin to a hybrid of scenarios i and ii for the high activity model. We note that the 𝑣 sin 𝑖 = 7 . 8 kms -1 is very low for brightness Doppler imaging and that reliable spot recovery, and particularly, latitude recovery of features, is difficult in this regime, especially considering the observations were made with a spectral resolution equivalent to ∼ 4 kms -1 . There is also a degeneracy between the signatures of facular and cool regions in Doppler images, which \nFigure 11. AU Mic line moments, 𝑀 0 -𝑀 4 , phased on the rotation period. The data are colour coded according to rotation number and phase relative to the first observation for the 18 rotations of AU Mic in the time interval of the observations. The dashed lines show a sinusoidal fit to the phased data. \n<!-- image --> \nmakes it difficult to recover hot spots in addition to cool spots. Further, Klein et al. (2021) note that the variability seen in the line profiles may well be the result of magnetic field effects (i.e. from Zeeman broadening) rather than true brightness variations. \nWe posit that cool high latitude spots are the most likely cause of measurable line profile and CCF distortions, particularly since the facular contribution decreases with active stars. Given the simulations of Beeck et al. (2015), we do not expect bright facular signatures to be discerned on M dwarfs. We note that purely high latitude cool spots do lead us to expect recovery of 𝑃 rot in all CLMs as shown in Fig. 4 for a single spot, exactly as the GLS periodogram of Klein et al. (2022) suggests. The distributed spot group in the high activity scenario ii model predicts a mixture of 𝑃 rot and harmonics of 𝑃 rot , likely because our simulated spots occur in a group with a larger effective radius. In addition, our scenario ii was simulated for an 𝑖 = 90 · case with mean latitude of 𝑙 = 60 · . The inclination of AU Mic's rotation axis is very close to 𝑖 = 90 · , so a decentered or asymmetric polar spot would be sufficient to yield periodicities at 𝑃 rot in the RVs and CLMs. \nObservations of AU Mic were made with CARMENES in 2019 and 2020 (Ribas et al. 2023). We used the more densely sampled observations in July to October 2019 to search for the signature of periodicity in the CLMs. We applied our implementation (Barnes et al. 1998) of least squares deconvolution (Donati et al. 1997) to derive line profiles with SNR = 3700 with the effects of line blending removed. The mean deconvolved profile velocity increment of \nFig. 12 shows the GLS and SL periodograms for each line moment. The most significant periods, at the maximum GLS power and minimum string length, are tabulated in Table 3 for each CLM. While the GLS method preferentially ( 𝑃 1 column) recovers a harmonic of 𝑃 rot , the second strongest peak ( 𝑃 2 column) generally recovers 𝑃 rot with almost the same significance as indicated by the fractional 𝑃 1 -𝑃 2 change in power (Table 3, column 4). For 𝑀 4 , we note that 𝑃 rot is only just recovered above the 10% false alarm level. The SL periodograms on the other hand recover 𝑃 rot for all CLMs, except for 𝑀 3 , where 𝑃 rot yields the 4th shortest string length after harmonics close to 3 𝑃 rot / 4, 𝑃 rot / 2 and 𝑃 rot / 4. Given the simulations in the previous sections, it should be pretty clear that taken together, the GLS and SL periodograms complement each other and are able to convincingly identify 𝑃 rot . In brief, the folded CLM signatures show evidence for higher order harmonics, which is supported by the presence of periodogram peaks at harmonics of 𝑃 rot . This appears to confirm the findings of Klein et al. (2021) that high latitude spot structure is the most satisfactory explanation for the signatures. \n<!-- image --> \nFigure 12. Periodograms of the Central Line Moments for AU Mic. Left: The Generalised Lomb Scargle periodograms with false alarm probabilities at FAP = 0.1, 1 and 10%. Right: normalised SL periodograms. The known period of 𝑃 rot = 9 . 863 d is indicated with the vertical dashed line. The fundamental harmonics at 𝑃 rot / 2 to 𝑃 rot / 6 are indicated by the dotted vertical lines. See Table 3 for further details of recovered periodicities. \n<!-- image --> \n1 . 25 kms -1 is large compared with the 𝑣 sin 𝑖 = 9 . 6 kms -1 (Han et al. 2023). Since the line distortions due to active regions can be small, before calculating the CLMs, we re-sample to a finer velocity increment via bicubic spline interpolation to minimise systematics. This also enables the degree of linear correlation between RV and the CLMs to be fine-tuned empirically by rejecting the wing regions of the profiles that tend to degrade the correlation. We found the greatest degree of anti-correlation in Pearson's 𝑟 by rejecting profile pixels at a fixed level corresponding to the 28.7 per cent of the mean deconvolved profile in the wings (i.e. the pixels nearest the effective continuum level). Pixels within the velocity range of ± 10 . 6 km s -1 were thus used to calculate the CLMs. This range is likely determined by the 𝑣 sin 𝑖 = 9 . 6 kms -1 combined with the stellar line broadening and instrumental broadening. Determination of CLM uncertainties is important if they are to be used for subsequent analysis. The least squares deconvolved profile uncertainties are fully propagated for each CLM according to Eqn. 1 and account for the re-interpolation described above. \nTheCLMsderivedfrom75spectra are shown in Fig. 11 phased on the known rotation period, 𝑃 rot = 4 . 863 d. The relative amplitude of the weighted moments 𝑀 1 to 𝑀 4 decreases in accordance with the simulations shown in Figs 2 and 6. 𝑀 0 shows much lower amplitude variability as expected for activity features covering a small fraction of the stellar surface. To first order, the CLMs show sinusoidal variability on the rotation timescale, though there are indications of either higher order harmonic structure or low-level evolution over the 88 d timespan of the observations. \nWe investigated the correlations between RV and BIS, 𝑀 1 and 𝑀 3 and RV and 𝑀 3 . Figure 13 shows the results for these three correlations. The points and connecting lines are colour-coded in time. We find exactly the same behaviour as our simulations predicted, with the tightest negative correlation between RV vs 𝑀 3 , where Pearson's 𝑟 = -0 . 775, a modest improvement over the more usual RV vs BIS of 4.6%. \nFinally, we also applied the same analysis to the young F8V \nFigure 13. The trends of RV vs BIS, 𝑀 1 vs 𝑀 3 and RV vs 𝑀 3 over 18 rotations of AU Mic. Error bars are plotted on the points. The lines connect the points in observing sequence with the colour coding used in Fig. 11. \n<!-- image --> \nTable 3. Recovered AU Mic GLS and SL periodicities showing peaks with the highest and second highest power (Note 1: 𝑃 rot is recovered as the 4th strongest peak in the String Length 𝑀 3 case). \nHyadescluster star, HD 30589. Ross (2024) have presented evidence for both Keplerian RV variability and have used BIS to simultaneously model significant stellar activity RV modulation. Ross (2024) also presented an analysis of the CLMs, finding that the RV vs 𝑀 3 correlation of 𝑟 = -0 . 659 yields an 8.0% improvement over the RV vs BIS correlation of 𝑟 = -0 . 610.", '8 CONCLUSIONS AND FUTURE WORK': 'As we attempt to retrieve low amplitude dynamical RV signals, it becomes increasingly important to be able to discern and correct for the effects of purely stellar signals in RV timeseries. We have explored the first few central line moments and have shown that distinct signatures of 𝑃 rot and its harmonics are to be expected. The String Length minimisation method is particularly effective at recovering 𝑃 rot , where more widely adopted periodogram methods tend to return harmonics of 𝑃 rot . We advocate the use of both methods in tandem to obtain a more robust assessment of activity induced modulations. The behaviour of the CLMs differs between Mdwarfs and solar type stars due to the differing character of stellar activity in the two cases. For low activity levels, we have shown that recovering activity signatures is challenging. Sensitivity improvements may be obtained by considering the most activity-sensitive lines (Dumusque 2018), though a balance between loss of SNR in the resulting CCF against the potential gain in activity induced amplitude, would need careful consideration. The calculation of CLMs is straightforward to implement, with 𝑀 3 offering a potential improvement over BIS as an activity correlator. We similarly advocate the use of methods that exploit the time derivative of second order line moment indicators, such as 𝑀 2 . \nIn this work, we have limited simulations to a single realisation (e.g. random spot pattern) within the bounds of our models: with further runs, uncertainties could be determined on each detection periodicity, though the CPU time to achieve this would be considerable. Similarly, we have assumed static spot patterns and a fixed number of observations to estimate recovery of signal periodicities at fixed SNRs. We have not considered the effects of differential rotation, which complicates the recovery of periodicities and long term stability of active features, particularly for earlier spectral types (Reiners & Schmitt 2003a,b; Barnes et al. 2005, 2017b). Active regions are known to be relatively stable on both M dwarfs and G dwarfs, which may exhibit active longitudes on timescales of months to many years. Further exploration of the latitude dependence of activity as a function of rotation rate could provide another refinement of the simulations. Ultimately, customised simulations, informed by observations and tailored to individual stars that include time evolution of features would likely shed more light on further factors that could impact on recovery of periodicities and activity signals. With the increasing number of facilities pushing to 10 cm s -1 stability, simulations that include stellar variability arising from lower activity levels in addition to more stochastic processes may provide further useful insights.', 'ACKNOWLEDGEMENTS': 'JRB and CAH were funded by STFC under consolidated grant ST/T000295/1 and ST/X001164/1. SVJ and FL acknowledge the support of the DFG priority program SPP 1992 "Exploring the Diversity of Extrasolar Planets (FL, SVJ: JE 701/5-1). \nMP and GAE acknowledge support by spanish grants PID2021125627OB-C31 funded by MCIU/AEI/10.13039/501100011033 and by \'ERDF A way of making Europe\', PID2020-120375GBI00 funded by MCIU/AEI, by the programme Unidad de Excelencia María de Maeztu CEX2020-001058-M, and by the Generalitat de Catalunya/CERCA programme. This work makes use of data from the CARMENES data archive at CAB (CSIC-INTA). The CARMENES archive is part of the Spanish Virtual Observatory project (http://svo.cab.inta-csic.es), funded by MCIN/AEI/10.13039/501100011033/ through grant PID2020112949GB-I00. We thank the anonymous referee for providing constructive suggestions that led to an improved manuscipt.', 'DATA AVAILABILITY': 'The data underlying this article will be shared on reasonable request to the corresponding author.', '24 J. Barnes': '```\nZechmeister M., Kürster M., 2018, GLS: Generalized Lomb-Scargle periodogram (ascl:1807.019) Zechmeister M., et al., 2018, A&A, 609, A12 Zhao Y., Dumusque X., 2023, A&A, 671, A11 Zhao Y., et al., 2024, A&A, 687, A281 Zicher N., et al., 2022, MNRAS, 512, 3060 de Beurs Z. L., et al., 2022, AJ, 164, 49\n```', 'APPENDIX A: ADDITIONAL FIGURES': 'This paper has been typeset from a T E X/L A T E X file prepared by the author. \nFigure A1. The effect on CLMs of increasing resolution from 𝑅 = 115 , 000 (lightest colour curves with smallest amplitudes), through resolutions of 𝑅 = 140 , 000 and 𝑅 = 190 , 000, to 𝑅 = 500 , 000 (darkest curves with highest amplitude or degree of complexity). \n<!-- image --> \nFigure A2. Central Line Moments as a function of phase for a G dwarf model with the two largest spot groups separated by 180 · . For further details, see main text and Fig. 6. \n<!-- image -->', '26 J. Barnes': "<!-- image --> \n<!-- image --> \nFigure A3. Period recovery matrices for a G dwarf with the two largest spot groups separated by 180 · in longitude. For further details, see main text and Fig. 7. \n<!-- image --> \nFigure A4. Companion to Fig. 9 in the main text for 𝑣 sin 𝑖 = 2 km s -1 . Pearson's 𝑟 correlations between absorption line (or CCF) metrics for single spots with 𝑟 u = 2 · and 𝑣 sin 𝑖 = 2 km s -1 located at latitudes 0 o to 80 o . The upper plots show Pearson's 𝑟 linear correlation for an M dwarf, while the lower panels show the same correlations for a G dwarf. The key indicates the line metric pairs that are plotted in each case. \n<!-- image --> \nFigure A5. Pearson's 𝑟 correlations for G dwarf models at spectral resolutions of 𝑅 = 140 , 000 (left panels) and 𝑅 = 190 , 000 (right panels). The can be compared directly with the G dwarf correlations for 𝑅 = 115 , 000 in Fig. 10 (right panels). \n<!-- image --> \n<!-- image --> \n<!-- image --> \n1 \n2 \n5 \n10 20 \n1 \n2 \n5 \n10 20 \n-1 \nvsini [kms \n] \n1 \n2 \n5 \n10 20 \n1 \n2 \n5 \n10 20 \n-1 \nvsini [kms \n] \n1 \n2 \n5 \n10 20 \n1 \n2 \n5 \n10 20 \n-1 \nvsini [kms \n] \n-1 \nvsini [kms \n] \n-1 \nvsini [kms \n] \nCool spots \n-1 \nvsini [kms \n] \nPearson's r \nPearson's r \n<!-- image --> \nFacular spots \n' \n2 \nRV vs M \n3 \nRV vs M \n' \n2 \nRV vs M \n3 \nRV vs M \n1 \n0.5 \n0 \n1 \n0.5 \n0 \n-0.5 \n-1 \n1 \n0.5 \n0 \n1 \n0.5 \n0 \n-0.5 \n-1"}

2024ApJ...976...39M

We address the critical need for accurate Rosseland mean gas opacities in highpressure environments spanning temperatures from 100 K to 32000 K. Current opacity tables from Wichita State University and SOPUS 2.0 are limited to inlineformula mmlmath overflowscrollmmlmilogmmlmimmlmo stretchyfalsemmlmommlmiRmmlmimmlmo stretchyfalsemmlmommlmommlmommlmn1mmlmnmmlmath inlineformula where inlineformula mmlmath overflowscrollmmlmiRmmlmimmlmommlmommlmimmlmimmlmspace width0.25emmmlmspacemmlmsubsupmmlmrowmmlmiTmmlmimmlmrowmmlmrowmmlmn6mmlmnmmlmrowmmlmrowmmlmommlmommlmn3mmlmnmmlmrowmmlmsubsupmmlmath inlineformula in units of inlineformula mmlmath overflowscrollmmlmi mathvariantnormalgmmlmimmlmspace width0.25emmmlmspacemmlmsupmmlmrowmmlmicmmmlmimmlmrowmmlmrowmmlmommlmommlmn3mmlmnmmlmrowmmlmsupmmlmsupmmlmrowmmlmo stretchyfalsemmlmommlmsupmmlmrowmmlmn10mmlmnmmlmrowmmlmrowmmlmn6mmlmnmmlmrowmmlmsupmmlmi mathvariantnormalKmmlmimmlmo stretchyfalsemmlmommlmrowmmlmrowmmlmommlmommlmn3mmlmnmmlmrowmmlmsupmmlmath inlineformula. This is insufficient for modeling very lowmass stars brown dwarfs and planets with atmospheres exhibiting higher densities and pressures inlineformula mmlmath overflowscrollmmlmilogmmlmimmlmo stretchyfalsemmlmommlmiRmmlmimmlmo stretchyfalsemmlmommlmogtmmlmommlmn1mmlmnmmlmath inlineformula. Leveraging extensive databases such as ExoMol ExoMolOP MoLLIST and HITEMP we focus on expanding the SOPUS opacity calculations to cover a broad range of pressure and density conditions inlineformula mmlmath overflowscrollmmlmommlmommlmn8mmlmnmmlmommlmommlmilogmmlmimmlmo stretchyfalsemmlmommlmiRmmlmimmlmo stretchyfalsemmlmommlmommlmommlmommlmommlmn6mmlmnmmlmath inlineformula. We incorporate the thermal Doppler mechanism and microturbulence velocity. Pressurebroadening effects on molecular transitions leading to Lorentzian or Voigt profiles are explored in the context of atmospheric profiles for exoplanets brown dwarfs and lowmass stars. We also delve into the impact of electron degeneracy and nonideal effects such as ionization potential depression under highdensity conditions emphasizing its notable influence on Rosseland mean opacities at temperatures exceeding 10000 K. As a result this study expands the SOPUS public web interface for customized gas chemical mixtures promoting flexibility in opacity calculations based on specific research needs. Additionally precomputed opacity tables inclusive of condensates are provided. We present a preliminary application to evolutionary models for very lowmass stars.

2024-11-01T00:00:00Z

['arXiv:2409.10905', '10.3847/1538-4357/ad7b27', '2024arXiv240910905M', '10.48550/arXiv.2409.10905', '2024ApJ...976...39M']

['Stellar atmospheric opacity', 'Astrochemistry', 'Low mass stars', 'Brown dwarfs', 'Exoplanets', 'Collisional broadening', '1585', '75', '2050', '185', '498', '2083', 'Astrophysics - Solar and Stellar Astrophysics', 'Astrophysics - Earth and Planetary Astrophysics', 'Astrophysics - Astrophysics of Galaxies', 'Astrophysics - Instrumentation and Methods for Astrophysics']

SOPUS 2.1 Lowtemperature Opacities Extended to High Pressure

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']

https://arxiv.org/pdf/2409.10905.pdf

{'ÆSOPUS 2.1: Low-Temperature Opacities Extended to High Pressure': "Paola Marigo , 1 Francesco Addari , 2 Diego Bossini , 1 Alessandro Bressan , 2 Guglielmo Costa , 3, 4 L'eo Girardi , 4 Michele Trabucchi , 1, 4 and Guglielmo Volpato 1, 4 \n1 Department of Physics and Astronomy G. Galilei, University of Padova, Vicolo dell'Osservatorio 3, I-35122, Padova, Italy 2 Scuola Internazionale Superiore di Studi Avanzati, Via Bonomea, 265, I-34136, Trieste, Italy \n3 \nUniv Lyon, Univ Lyon1, Ens de Lyon, CNRS, Centre de Recherche Astrophysique de Lyon UMR5574, \nF-69230 Saint-Genis-Laval, France \n4 INAF-Osservatorio Astronomico di Padova, Vicolo dell'Osservatorio 5, I-35122 Padova, Italy \n(Received May 16, 2024; Accepted September 14, 2024)", 'ABSTRACT': 'We address the critical need for accurate Rosseland mean gas opacities in high-pressure environments, spanning temperatures from 100 K to 32000 K. Current opacity tables from Wichita State University and ÆSOPUS 2.0 are limited to log( R ) ≤ 1, where R = ρT -3 6 in units of g cm -3 (10 6 K) -3 . This is insufficient for modeling very low-mass stars, brown dwarfs, and planets with atmospheres exhibiting higher densities and pressures (log( R ) > 1). Leveraging extensive databases such as ExoMol , ExoMolOP , MoLLIST , and HITEMP , we focus on expanding the ÆSOPUS opacity calculations to cover a broad range of pressure and density conditions ( -8 ≤ log( R ) ≤ +6). We incorporate the thermal Doppler mechanism and micro-turbulence velocity. Pressure broadening effects on molecular transitions, leading to Lorentzian or Voigt profiles, are explored in the context of atmospheric profiles for exoplanets, brown dwarfs, and low-mass stars. We also delve into the impact of electron degeneracy and non-ideal effects such as ionization potential depression under high-density conditions, emphasizing its notable influence on Rosseland mean opacities at temperatures exceeding 10 , 000 K. As a result, this study expands ÆSOPUS public web interface for customized gas chemical mixtures, promoting flexibility in opacity calculations based on specific research needs. Additionally, pre-computed opacity tables, inclusive of condensates, are provided. We present a preliminary application to evolutionary models for very low-mass stars. \nKeywords: Stellar atmospheric opacity(1585) - Astrochemistry(75) - Low mass stars(2050) - Brown dwarfs(185) - Exoplanets(498) - Collisional broadening(2083)', '1. INTRODUCTION': "Thanks to extensive databases such as ExoMol (Tennyson & Yurchenko 2012; Tennyson et al. 2016), ExoMolOP (Chubb et al. 2021a), MoLLIST (Bernath 2020) and HITEMP (Rothman et al. 2010), we can now rely on the availability of a substantial amount of molecular line list data, which is a significant asset for modeling the atmospheres of hot exoplanets, as well as cool stellar and sub-stellar atmospheres. \nRosseland mean opacities are critical components in modeling stars and sub-stellar objects. To our knowledge, the most widely distributed suppliers of Rosseland mean opacities for T ≲ 10000 K are the Wichita State University group (Alexander 1975; Alexander & Ferguson 1994; Ferguson et al. 2005), and the ÆSOPUS team (Marigo & Aringer 2009; Marigo et al. 2022). Two standard variables are used to build Rosseland mean opacity tables: \n-3 \nT and R = ρT 6 , (1) \nCorresponding author: Diego Bossini \ndiego.bossini@unipd.it \nFigure 1. Atmospheric pressure-temperature profiles for exoplanets, brown dwarfs, and very low mass stars. The two hot Jupiter exoplanets, WASP-17b with mass of 0 . 51 M J , and WASP-19b with mass of 1 . 14 M J , were analyzed by Sing et al. (2016) through HST/Spitzer observations. Calamari et al. (2022) used an atmospheric retrieval analysis to derive the profile for the brown dwarf Gliese 229B with an estimated mass of 50 M J . We also add two theoretical profiles for brown dwarfs with a surface gravity of log( g ) = 4 . 5 (Baraffe et al. 2002), and a very low mass star model from PARSEC (Bressan et al. 2012). The dashed black line denotes the limit below which pressure broadening for molecular spectral lines should be included. Refer to Section 3.2 for more details. \n<!-- image --> \nwhere T is the temperature (in K) and the R parameter (in g cm -3 (10 6 K) -3 ) includes both the temperature ( T 6 = T/ (10 6 K)) and the gas mass density ρ (in g cm -3 ). The fact that smooth opacity interpolations are feasible is the primary motivation behind the selection of these two parameters. \nSo far, low-temperature opacities have been computed encompassing the range -8 ≤ log( R ) ≤ +1 (Ferguson et al. 2005; Marigo et al. 2022). This range, however, is insufficiently broad to cover the structural characteristics of very low-mass stars, brown dwarfs, and planets (see Section 2). In fact, these objects have atmospheres with high density and pressure, so log( R ) can easily exceed 1. Very low-mass stars (0 . 08 ≲ M/M ⊙ ≲ 0 . 6) and sub-stellar objects (10 -3 ≲ M/M J ≲ 13 for exoplanets, 13 ≲ M/M J ≲ 90 for brown dwarfs, with M J ≡ 1 Jupiter mass) have atmospheres that typically cover temperature ranges of 100 ≲ T/ K ≲ 3500 and gas pressure ranges of 10 -4 ≲ P gas / bar ≲ 10 3 (Burrows et al. 2001; Spiegel et al. 2011; Wilson et al. 2016; Mulders et al. 2021). This demonstrates the need of extending the Rosseland mean opacity tables at higher densities and pressures (with log( R ) > 1). \nThe thermal Doppler mechanism is used by both the Wichita State University and ÆSOPUS teams for molecular line broadening. Micro-turbulence velocity is also included in ÆSOPUS . As we will see in Section 3.2, this approximation is mostly valid over the standard range -8 ≤ log( R ) ≤ +1, but it becomes inadequate when higher R values, hence larger densities and pressures are considered. When moving into the high density regime, pressure effects broaden molecular transitions, resulting in either Lorentzian or Voigt profiles (Burrows et al. 2001; Sharp & Burrows 2007). Figure 1 depicts the pressure-temperature atmospheric profiles of two exoplanets, one brown dwarf, and very low mass stars. Pressure broadening of spectral molecular lines affects all of these objects' atmospheres (below the dashed black line). Few works in the past literature, to our knowledge, computed Rosseland mean gas opacities at high densities, namely Kurucz (1993), Freedman et al. (2008), Malygin et al. (2014), Freedman et al. (2014), adopting scaled-solar abundances according to Grevesse & Sauval (1998) or Lodders (2003). For zero-metallicity gas, we also refer to the work of Lenzuni et al. (1991). \nUnder high density conditions, additional important phenomena, namely electron degeneracy and Coulomb interactions between charged particles, must be accounted for in the equation of state (Cox & Giuli 1968; Hansen & Vieillefosse 1976; Potekhin et al. 2009). Furthermore, all charges present in a radiating gas, electrons and ions, contribute to reduce the energy required to free an electron in the fundamental state. This process is designated as ionization potential depression (IPD; Ecker & Kroll 1963; Stewart & Pyatt 1966). We will show that the IPD effect will have a notable impact on Rosseland mean opacities, especially for T > 10000 K. As a result, appropriate physics must be included to describe the gas at high pressure and its opacity interaction with radiation. \nThis study aims to increase the accessibility of Rosseland mean gas opacities in high-pressure environments, across a broad temperature range, from 100 K to 32000 K. Expanding the opacities to cover a wide range of pressure and density conditions ( -8 ≤ log( R ) ≤ +6) is essential for accurate modeling of sub-stellar objects and very low mass stars. We provide our results in a public web interface that allows users to customize the chemical mixture according to their specific research requirements. This flexibility is crucial because different research projects may focus on different chemical compositions and environmental conditions. In addition we produce pre-computed opacity tables with the inclusion of condensates, similarly to Marigo et al. (2023). \nThis paper is organized as follows. Section 2 recaps the basic ingredients of the equation of state in ÆSOPUS and GGchem codes, as well the the method for calculating the Rosseland mean opacity. We additionally present some physical structure of very low mass stars to demonstrate the need for the pressure and density ranges of the opacity tables to be expanded to higher values. Section 3 describes how we implement the IPD effect in ÆSOPUS partition functions and abundance differential equations, as well as how we treat line pressure broadening for atomic and molecular transitions. In Section 4 we analyze and discuss the findings, focusing on the impact of the IPD effect and pressure broadening on Rosseland mean opacities and comparing our results to others found in the literature. Section 5 presents a few examples of Rosseland mean opacities with solid grains included, while Section 6 briefly discusses the impact of these opacities in low mass stellar models.", '2. EQUATION OF STATE AND ROSSELAND MEAN OPACITIES': 'The ÆSOPUS code solves the equation of state encompassing more than 800 species, including about 300 atoms and ions and 500 molecules, in the gas phase under conditions of thermodynamic and instantaneous chemical equilibrium (see Appendix A). In order to expand the opacity computations in the low-temperature regime where solid grains condense, we (Marigo et al. 2023) recently coupled ÆSOPUS (Marigo & Aringer 2009; Marigo et al. 2022) and GGchem codes (Woitke et al. 2018). In practice, we use ÆSOPUS for temperatures between 30000 K and 3000 K, and then we switch to GGchem for temperatures between 3000 K and 400 K. We recall that GGchem computes the abundances of approximately 568 gas molecules, 55 liquid species, and nearly 200 types of solid particles. \nThere is an extensive description of how to calculate the Rosseland mean opacity in Marigo et al. (2022) and Marigo & Aringer (2009), so it will not be repeated here. To summarize, we compute the total monochromatic opacity cross section per unit mass (in cm 2 g -1 ) for any chosen ( ρ, T ) pair by incorporating all the contributions from true absorption and scattering, for both gas and solid particles. \nAs previously stated in Section 1, the standard range of Rosseland mean opacities, -8 ≤ log( R ) ≤ 1, is not enough to describe the physical properties of very low-mass stars, brown dwarfs and planets, which reach much higher densities and pressures. To demonstrate this fact, we display a few stellar structures in Figure 2 that correspond to brown dwarfs and very low-mass stars with initial masses M i between 0 . 05 M ⊙ and 0 . 7 M ⊙ . \nLooking at the left panel of Figure 2, it is evident that a sizable portion of the structures of stars with M i ≤ 0 . 7 M ⊙ exceeds the upper limit of log( R ) = 1, reaching values up to log( R ) ≃ 4. The brown dwarf model with M i = 0 . 05 M ⊙ extends up to log( R ) ≃ 6. As we will discuss in Section 6, opacity extrapolation outside of the validity range of tables may result in incorrect structural properties. \nFrom stellar evolution theory, we know that main-sequence stars with 0 . 08 ≲ M i /M ⊙ ≲ 0 . 35 are completely convective and adiabatic, regardless of the convection model used. However, convection becomes less effective for transporting energy closer to the surface, particularly in layers where hydrogen and helium are partially ionized, and the true temperature gradient, ∇ = d log( T ) /d log( P ), becomes super-adiabatic, that is ∇-∇ ad > 0. In these regions, micro-physics plays a crucial role, as factors such as the equation of state and opacities become essential for determining the solution of the stellar structure. The right panel of Figure 2 highlights the regions where the temperature gradient becomes super-adiabatic. The mixing-length theory (Bohm-Vitense 1958) is used to estimate ∇-∇ ad . It is clear that in very low-mass stars, low-temperature opacities extend in stellar layers where log( R ) > 1 and convection is super-adiabatic, and thus accurate opacity estimation is critical. \nWhat are the chemical species that have to be considered in the calculation of Rosseland mean opacities at high values of R ? To answer this question, Figure 3 depicts the concentrations of several molecules in the gas phase as a function of temperature for log( R ) = 6. At high density the most abundant molecules are molecular hydrogen (H 2 ) and water vapor (H 2 O), carbon monoxide (CO), silicon monoxide (SiO), followed by methane (CH 4 ), ammonia (NH 3 ) and silane (SiH 4 ) towards lower temperatures. \nFigure 4 shows a number of condensed species, including metal oxides such as SiO, ZrO, VO, silicates, Cr, Fe. It is worth noting that corundum (Al 2 O 3 ) does not condense in appreciable amounts at high densities (hence it is not shown in Fig. 4). The condensation sequence at low temperatures is closed by iron sulfide (FeS), water ice (H 2 O) and ammonia ice (NH 3 ). \n<!-- image --> \nFigure 2. Structures of very low-mass stars and brown dwarfs computed with the PARSEC code (Bressan et al. 2012), from the center to the atmosphere, in a stage close to the zero-age main sequence. The initial mass changes in increments of 0 . 05 M ⊙ , from 0 . 05 M ⊙ to 0 . 7 M ⊙ . The initial chemical composition is scaled-solar according to Caffau et al. (2011), with a metallicity Z = 0 . 014, and helium abundance Y = 0 . 273. The horizontal line marks the maximum value, log( R ) = 1, currently available in the low-temperature Rosseland mean opacities (Ferguson et al. 2005; Marigo et al. 2022). Left panel: structures in the log( T ) -log( R ) plane, color-coded according to the initial stellar mass. Right panel: structures in the log( T ) -log( R ) plane, with convective regions color-coded according to the degree of super-adiabaticity, ∇-∇ ad . The gray sections correspond to convective regions treated as adiabatic, while black sections refer to the radiative regions. See text for more details. \n<!-- image -->', '3. OPACITIES AT HIGH DENSITY': "When entering a high density regime, opacities become increasingly complex and must be handled with great care due to a variety of physical processes, including electron degeneracy, which increases gas pressure; non-ideal effects due to Coulomb interactions among charged particles, which can lower atom and molecule ionization potentials; and line pressure broadening of molecular bands and atomic transitions. In the following sections, we will detail the necessary improvements we made in ÆSOPUS to deal with these high-density conditions. \nFigure 3. Gaseous molecular concentrations, in number density, for some selected species as a function of temperature, without condensation. We assume log( R ) = 6, with a chemical composition solar-scaled according to Magg et al. (2022), with metallicity Z = 0 . 01 and hydrogen abundance X = 0 . 7. \n<!-- image --> \nFigure 4. Distribution of several condensed species as a function of temperature at log( R ) = 6, for solar abundances in phase equilibrium, computed with GGchem . The abundances are evaluated with respect to the number density of hydrogen nuclei. The assumed log( R ) value and chemical composition are the same as in Fig. 4. \n<!-- image --> \n<!-- image --> \nFigure 5. Left panel: The equation of state for a gas of free particles in the log( T ) -log( ρ ) plane. The dashed blue lines are approximate boundaries between regions where radiation pressure, classical ideal gas pressure, and non-relativistic electron degeneracy dominate. We assume Magg et al. (2022)'s solar composition with X = 0 . 7 and Z = 0 . 01. The gray area denotes the upper extension of our opacity tables. The red and black lines mark the locus of points where the ionization degree of hydrogen is 0.5 and 0.2, as indicated. Right panel: Ionization degree of hydrogen as a function of temperature for log( R ) = 6 (black line) and log( R ) = 3 (magenta line). In both panels the plasma IPD effect is represented by solid lines, while dashed lines are the result of the classical Saha ionization equation. See text for more details. \n<!-- image --> \nThe ÆSOPUS opacity calculations are extended up to log( R ) = 6 in a high density regime. At the highest temperature, log( T/ K) = 4 . 5, this corresponds to a mass density of ρ ≃ 32 g cm -3 . As shown in Figure 5 (left panel), electrons are partially degenerate at this density. We recall that ÆSOPUS incorporates an accurate treatment of electron degeneracy, based on generalized Fermi-Dirac integrals (Cox & Giuli 1968). At increasing densities some non-ideal effects appear, such as the lowering of the ionization potentials of atoms and molecules. \nLet us briefly discuss this aspect. We know that the ionization potential U of an ion embedded in a plasma is reduced due to the interaction of all charged particles (ions and electrons) with that ion. The IPD effect is accounted for using the method developed by Ecker & Kroll (1963), who formulated a generalized Saha equation as a function of the chemical potential of the plasma. \nThe IPD is calculated with \n∆ U ( z ) = z e 2 λ D , (2) \nwhere z is the charge of the ion after the ionization occurrence, e is the electron's charge, and λ D is a generalized Debye length, which is computed through \nλ D = √ kT 4 π ( n e + n ion ) , (3) \nwhere k is the Boltzmann constant, n e and n ion are the number densities of free electron and ions in the plasma. \nThe implementation of the IPD effect in ÆSOPUS is carried out as follows. The ÆSOPUS code uses the NewtwonRaphson technique to solve the equation of state assuming instantaneous chemical equilibrium, employing a set of dissociation-recombination and/or ionization equilibrium constants for each of the 800 particles. Concerning the Saha equation for ionization, at each iteration we correct the ionization potential using the IPD effect inside the equilibrium constant, which depends on U ( z ) -∆ U ( z ), T , n e and n ion , until convergence is reached. \nThis is illustrated in the right panel of Figure 5. It compares the hydrogen ionization degree x (H) 1 with and without the IPD effect as a function of temperature for two R parameter values. Particularly for log( R ) = 6, with the classical Saha equation, x (H) remains extremely low, whereas with the IPD plasma effect, x (H) increases significantly at the highest temperatures. \nThe locus of points in the log( T ) -log( ρ ) diagram where the ionization degree for hydrogen x (H) is equal to 0.5 and 0.2 is plotted in Figure 5 (left panel). We observe that, when the IPD effect is considered, the same ionization \ndegree is obtained at higher densities than in the ideal case. This is especially noticeable at the highest temperatures for x (H) = 0 . 2, while at lower densities the differences become less and less significant. \nThe reduction in the ionization potential of atoms and molecules has a sizable impact on the Rosseland mean opacities at high density, as discussed in Section 4.1.", '3.2. Line pressure broadening for atomic and molecular transitions': "Various processes in planet, substellar, and stellar atmospheres naturally broaden spectral lines. Doppler and pressure broadening are the most common types of line broadening. Doppler broadening is caused by the thermal velocities of each atom and molecule and is normally described by a Gaussian line profile (Yurchenko et al. 2018). The width of the Doppler line core is directly related to the temperature. As the temperature increases, the thermal motion of particles becomes more significant, causing broader line profiles. \nFor instance, Ferguson et al. (2005) uses a pure thermal Doppler profile in their opacity computations. In ÆSOPUS we do similarly but also account for micro-turbulence velocity by producing a normalized broadening profile, ϕ ( ν ), according to the equation: \nϕ ( ν ) = 1 ∆ ν √ π e -( ν -ν 0 ∆ ν ) 2 , (4) \nwhere ν 0 is the line center position in frequency, and ∆ ν is the line width, obtained with \n∆ ν = ν 0 c √ 2 k B T m + ξ 2 . (5) \nIn this Equation, c denotes the speed of light, k B the Boltzmann constant, m the molecule's mass, and ξ the microturbolent velocity, which is set to 2 . 5 km/s (see Marigo & Aringer 2009; Marigo et al. 2022, for more details). \nTable 1 . Spectral Line Data for Molecular Absorption with Pressure Broadening taken from the ExoMolOP Database \nTable 1 continued on next page", 'Opacities at High Pressure': "Table 1 (continued) \nTable 1 continued on next page \nMarigo et al. \nTable 1 (continued) \nNote -For each species λ l , λ u denote the minimum and maximum wavelength of the corresponding line list; T max is the highest temperature available. \nPressure broadening, which varies depending on the perturbing species (such as H, He, H 2 ) in addition to pressure, results in a Lorentzian or Voigt profile. While a Lorentzian profile is typically pressure-dependent, a Voigt profile is a convolution of the Doppler and Lorentzian profiles. It accounts for both thermal motion and collisional broadening, making it suitable for modeling line shapes in a broader range of conditions. \nIn addition to temperature and pressure, the line width in Lorentzian and Voigt profiles is also influenced by various other quantities, the so-called broadening parameters, that can be challenging to determine accurately. Broadening parameters, if available, are present in ExoMol database (Barton et al. 2017; Yurchenko et al. 2017). Based on ExoMol broadening data, Chubb et al. (2021a) recently created a publicly accessible database ( ExoMolOP 2 ) of opacities for over 80 molecules of astrophysical interest computed at various pressures (10 -5 to 10 2 bar) and temperatures (the range depends on the line list). Atomic data for the alkali neutral metals, Na and K, is additionally provided, based on NIST database (Kramida et al. 2022) and the most recent profiles for the resonance lines (Allard et al. 2016, 2019). The data can be recovered in a variety of formats that are compatible with different exoplanet atmosphere retrieval codes. \nFor this work we use the cross section data for the retrieval code TauREx (Waldmann et al. 2015b,a; Al-Refaie et al. 2021), in HDF5 format, with a spectral resolution of R = λ/ ∆ λ = 15000, wavenumber coverage of 200 -33333 cm -1 . The TauREx table format is compatible with another retrieval code petitRADTRANS (Molli'ere et al. 2019), so we could benefit from its publicly available software. Utilizing the Exo k 3 code (Leconte 2021), we can create cross section tables appropriate for ÆSOPUS that are interpolated in pressure for each temperature value. The grids of temperature and pressure typically have 18 and 15 nodes, respectively, distributed throughout the corresponding \nranges (the number of temperature nodes as well as the wavenumber grid may vary depending on the line list). Table 1 contains a complete list of molecules, alkali neutral metals, and other atoms for which we have pressure broadening line profiles. For atoms with temperatures above 4000 K, we use Opacity Project cross sections (Seaton et al. 1994), which are expressed as a function of temperature and electron density. Line broadening includes effects produced by thermal Doppler, radiation damping and pressure. \nThe molecular data consist of 65 species, which is slightly less than the 80 species included in Marigo et al. (2022, see their Table 2). The molecules that currently lack pressure broadening are: O 3 , ClO, HI, CS 2 , OCS, NaO, N 2 , KOH, H 2 , HCl, ZrO, C 3 , CH 3 Cl, SO. We will incorporate new data into ÆSOPUS as soon as it becomes available. To avoid opacity gaps, we include monochromatic cross sections for these molecules using thermal Doppler plus microturbulent velocity profiles for any ( T , R ) combinations. We interpolate the TauREx tables as a function of wavenumber, temperature, and pressure to compute monochromatic cross sections for molecules, and alkali atoms Na and K. The same procedure is applied for all molecules at T values above the T max limits listed in Table 1. \nIt is important to know where in the pressure-temperature ( P -T ) space each of the two broadening mechanisms contributes most significantly. Hedges & Madhusudhan (2016) compared Doppler and Lorentzian broadening profiles over the P -T diagram in terms of half-width at half-maximum to gain a picture of where each profile impacts substantially. Based on their findings, we depict the two broadening regimes in Figure 6. According to Chubb et al. (2021a) analysis, we also add a lower pressure limit (10 6 bar, black line) above which molecular lines are treated with a Voigt profile. The data can be found in the ExoMolOP database. As expected, thermal Doppler broadening \nFigure 6. Comparison of the widths of Doppler line cores versus Lorentzian profiles in the pressure-temperature diagram according to the analysis of Hedges & Madhusudhan (2016). The boundary between the two regimes is denoted by the white line. Above the horizontal black line molecular transitions are treated with a Voigt profile extracted from the ExoMolOP database (Chubb et al. 2021a). \n<!-- image --> \ncontributes significantly to the final profile core at low pressures, whereas pressure (Lorentzian) broadening is more effective at high pressures. Both broadening mechanisms are likely to contribute considerably to the core of the line profiles closer to the border between these two regimes. In this work, we use the Voigt profile for molecular lines, which is a convolution of Lorentzian and Doppler broadening mechanisms. This convolution takes into account both pressure broadening (Lorentzian) and temperature-induced broadening (Doppler), making it a versatile tool for accurately modeling spectral lines in a variety of physical conditions. Below the black horizontal line in Figure 6 we assume that pressure effects become insignificant, and we use the thermal Doppler plus micro-turbulence velocity broadening. \nHigh pressure can have a sizable impact on the monochromatic absorption cross sections σ . To illustrate the effect Figure 7 compares σ with applied Doppler plus micro-turbulence broadening to pressure broadening, for two molecules, water vapor (H 2 O) and methane (CH 4 ). It is evident that pressure broadening reduces the excursion of σ to higher and lower values. In the case of water vapor, this is particularly clear at 2000 K. When compared to the thermal Doppler Gaussian profiles, the Lorentzian line profiles produce a sigma that is most concentrated at higher values at a pressure of 1 bar. At 100 bar of pressure and 1000 K of temperature, methane experiences a similar effect, with a cross section that does not exhibit large fluctuations when compared to the Doppler line profiles. The tendency of σ \n<!-- image --> \nFigure 7. Comparison of two broadening profiles used for monochromatic absorption cross sections: the Lorentzian profile produced by pressure and perturbing species (blue line) and the Gaussian thermal Doppler plus micro-turbulent velocity profile (magenta line; see Equation 5). Left panel: water vapor; right panel: methane. See Table 11 for details. Values for pressure and temperature are labeled. \n<!-- image --> \ntowards higher values at increasing pressure will have a noticeable impact on Rosseland mean gas opacities, which will tend to increase.", '4. RESULTS': '4.1. Ionization potential depression effects \n<!-- image --> \nFigure 8. Properties of the Rosseland mean opacity towards the highest temperature of our tables, at high density for log( R ) = 6. The chemical composition is defined by X = 0 . 7 , Z = 0 . 01, with a scaled-solar chemical mixture following Magg et al. (2022). Each curve corresponds to log( κ R ) -log( κ R i, off ), where κ R is the full opacity including all opacity sources considered here, and κ R i, off denotes the reduced opacity obtained by excluding the specific absorbing species. This displays the temperature window to which a given opacity source contributes the most. The IPD effect is ignored in the computations of the left panel, whereas it is incorporated in the results of the right panel. In both cases, we apply thermal Doppler plus micro-turbulent velocity profiles for molecular transitions. \n<!-- image --> \nTo illustrate the impact of the IPD on the opacities, we compare in Figure 8 the different contributions to the opacity for a scaled-solar mixture, assuming log( R ) = 6. Moving up in temperature, we notice the significant contribution of collision-induced absorption (CIA) at low temperatures, which is primarily caused mainly by H 2 -H 2 collisions. The inclusion of the IPD for ions does not produce discernible effects in log( κ R ), with differences < 0 . 02 dex up to log( T ) = 3 . 75. Beyond this temperature, the contribution of different opacity sources can vary significantly as the temperature and density increase (we assume log( R ) = 6). When IPD is considered, the most striking facts are: a \nFigure 9. Map of the differences in Rosseland mean opacities when the IPD effect is included or ignored at high density. Contour levels are distributed every 0 . 025 dex in ∆ log( κ R ). \n<!-- image --> \nsignificant decrease in H -opacity as the number density of neutral hydrogen decreases (see right panel of Figure 5), a shift to lower temperatures of the bound-bound hydrogen line opacity, and a remarkable increase in H + 2 opacity given its higher abundance. We also notice that at high density, metals contribute significantly to opacity in the temperature range 3 . 5 ≲ log( T/ K) ≲ 4 . 0. There is a noticeable opacity bump at 3 . 6 ≲ log( T/ K) ≲ 3 . 7, and we verify that the major absorption contributions come from Fe, Al, Na, and Ca. Such a bump will be visible in the Rosseland mean opacity as well (see Section 4.2). \nFinally, Figure 9 shows a map of the differences in log κ R when the IPD effect is taken into account or neglected. The major consequence of IPD is of increasing H and H + 2 ionization which results in higher Rosseland mean opacity for log( T ) > 4 . 1 and log( R ) > 1 (red area).', '4.2. Pressure broadening effects on mean gas opacity': "As thoroughly discussed in several studies (e.g., Freedman et al. 2008; Helling & Lucas 2009; Malygin et al. 2014) mean gas opacities without a dust continuum contribution have several astrophysical applications. They are important, for example, in a dust-depleted low-metallicity medium or when the equilibrium temperature exceeds the local dust sublimation temperature. As a result, gas opacities can be relevant in describing the inner regions of accretion disks (Muzerolle et al. 2004), calculating the energy balance of Type Ia supernovae (Dessart et al. 2014), estimating the cooling of non-accreting hot white dwarfs (Rohrmann et al. 2012), quantifying stellar feedback processes in the interstellar medium (Pelupessy & Papadopoulos 2009), and simulating star and planet formation (Helling & Lucas 2009). \nTo assess the impact of pressure broadening on Rosseland mean gas opacities we performed two independent runs of ÆSOPUS , one adopting the thermal Doppler plus micro-turbulent velocity molecular line profiles, and the other assuming the Lorentzian molecular line profiles that depend on both temperature and pressure. \nThe results are illustrated in Figure 10. We note that the dynamical range of κ R is extremely broad, spanning ∼ 14 orders of magnitude, making eye-comparison somewhat difficult. Nonetheless, we reckon it is useful to show the opacity trends in the two cases. To quantify the differences in κ R , we create a map that spans the whole area of the table, as shown in the right panel of Figure 11. In Section 3.1, the IPD effect has already been discussed. To help the discussion, in the left panel we also draw a map of the gas pressure, indicating the contour level (white line) above which we begin to consider pressure broadening of molecular transitions. \n<!-- image --> \nFigure 10. Rosseland mean gas opacities computed in the temperature range 100 ≤ T/ K ≲ 30000 and encompassing the R interval -8 ≤ log( R ) ≤ 6 in steps of 0.5 dex. The chemical composition is defined by X = 0 . 7 , Z = 0 . 01, with a scaled-solar chemical mixture following Magg et al. (2022). Left panel: opacities assuming Doppler plus micro-turbulent velocity molecular line profiles and ignoring the IPD effect. Right panel: opacities assuming pressure broadening for molecular line profiles and accounting for the IPD effect. \n<!-- image --> \n<!-- image --> \nFigure 11. Left panel: map of the gas pressure as a function of T and R . Contour levels are distributed every 1 dex in log( P gas ). The white contour draws the locus of points where the logarithmic difference in Rosseland mean opacity treated with a thermal Doppler profile and pressure broadening equals 0.001 dex in the regime where molecular transitions are important, for T ≤ 4000 K. Right panel: map of the differences in Rosseland mean opacities between computations that include the IPD effect and pressure broadening for molecular transitions and those that ignore the IPD and assume Doppler broadening regardless of pressure. Contour levels are distributed every 0 . 05 dex in ∆ log( κ R ). The chemical composition is the same as in Figure 10. \n<!-- image --> \nThe contribution of alkali atoms in atmospheric opacity of cool sub-stellar objects was initially established by studying the far red spectra of T dwarfs (Burrows et al. 2000). Atomic pressure-broadened lines, especially those of Na and K, are major opacity sources over certain spectral ranges, temperatures, and densities (Freedman et al. 2008). With their large absorption cross sections at near infra-red and far-red wavelengths, sodium and potassium fill what would otherwise be a spectral region of relatively low opacity. \nThe left panel of Figure 12 compares the total molecular opacity (magenta line) as a function of wave-number for a gas temperature of 1585 K and a gas pressure of 211 bar, with a calculation that does not account for alkali opacity (blue line). Above about a wave-number of 10000 cm -1 , the alkali opacity plays a significant role in determining the \n<!-- image --> \nFigure 12. Left panel: Molecular gas opacity as a function of wavenumber at T = 1585 K, gas pressure P gas = 211 bar, and density ρ = 0 . 004 g cm -3 . The reference solar chemical composition is taken from Magg et al. (2022), with metallicity Z = 0 . 02 and hydrogen abundance X = 0 . 7. The blue curve represents the monochromatic opacity without the contribution of alkali atoms Na and K, whereas the magenta curve includes the two atoms' opacity contributions. Right panel: Rosseland mean opacity as a function of temperature at three densities (g cm -3 ). Magenta lines include the opacity from alkali atoms Na and K, whereas dashed blue lines do not. The chemical composition is scaled-solar according to Magg et al. (2022), with metallicity Z = 0 . 01 and hydrogen abundance X = 0 . 7. \n<!-- image --> \ntotal summed opacity. The resonance Na doublet at ≃ 17000 cm -1 stands out as a prominent source of absorption. The right panel of Figure 12 compares the Rosseland mean opacity with and without the contribution of alkali metals at various densities. It is evident that the alkali opacity fills in the opacity minimum from about 1000 to 3200 K at higher densities. Na and K lines play a much smaller role at lower densities, so the differences are minor. \n4.2.2. Comparison with other authors \n<!-- image --> \nFigure 13. Comparison of Rosseland mean gas opacities computed in this work and by other studies. Left panel: Comparison with Malygin et al. (2014). The reference solar chemical composition is taken from Grevesse & Sauval (1998), with metallicity Z = 0 . 01696 and hydrogen abundance X = 0 . 7347. Right panel: Comparison with Freedman et al. (2014). The reference solar chemical composition is taken from Lodders (2003), with metallicity Z = 0 . 0133 and hydrogen abundance X = 0 . 7491. Values for log( R ) are labeled. \n<!-- image --> \nTo test our results we consider the Rosseland mean gas opacities computed by Malygin et al. (2014) and Freedman et al. (2014). In Malygin et al. (2014) work Kurucz's CD-ROMs were used to extract the line and continuum opacity data (Kurucz 1993), and the mean opacities were calculated using the publicly available DFSYNTHE code (Castelli 2005). The results are presented in the left panel of Figure 13 for a temperature range of 700 K to 10000 K. The \nagreement between the two opacity sets is good for log( T ) > 3 . 6. At lower temperatures, some deviations start to appear, most likely due to different line lists for atoms and molecules, as well as different line pressure broadening treatments. The most prominent discrepancy shows up at log( T ) < 3 . 3. While our opacity results remain either relatively flat or even increase for the highest R values, Malygin et al. (2014) opacities do, in fact, significantly decrease at lower temperatures. For example, our opacity curve for log( R ) = -7 only includes thermal Doppler profiles for the molecular transitions, as pressure is irrelevant in this case. Despite this, the decrease at lower temperatures is much less pronounced than in Malygin et al. (2014). One plausible explanation is that we rely on molecular line lists that extend down to 100 K in temperature, whereas in Kurucz opacity the contributions of molecules cease for T < 1995 K, and therefore extrapolations below that limit may be inaccurate. \nIn comparison to Freedman et al. (2014) results shown in the right panel of Figure 13, there are significant differences in Rosseland mean opacities. We found no explicit information about pressure broadening for molecular transitions, and molecular absorption is limited to 12 species. Conversely, we consider 80 absorbing molecules in our study. Both facts could explain the disparity in outcomes.", '5. ROSSELAND MEAN OPACITIES WITH SOLID GRAINS': 'We present a few examples of Rosseland mean opacities with solid grains included. The dust prescriptions are the same as in Marigo et al. (2023). We are aware that dust clouds in brown dwarfs and planets are crucial for understanding their atmospheric properties, formation mechanisms, and overall behavior, providing valuable insights into the broader field of planetary science. Dust clouds can significantly impact their atmospheric composition. They often consist of various particles, including silicates, iron, and other compounds, contributing to the chemical makeup of the atmosphere. In this context, important contributions were provided by Tsuji et al. (1996), Burrows et al. (2000), Ackerman & Marley (2001), Tsuji (2002), Sharp & Burrows (2007), Helling et al. (2008), Witte et al. (2009), Allard et al. (2012), Juncher et al. (2017), Woitke et al. (2020). Because our primary interest is in very low mass stars, in this work we do not take into account the formation of dust clouds and we postpone the effort to a future study. \nFigure 14 compares major solid species at low and high densities that contribute most to the opacity. There are significant differences between the two regimes. \nFirst, we notice that the corundum opacity bump that appears for log( R ) = -3 at temperatures ranging from 1500 to 1200 K is missing for log( R ) = 6. Al 2 O 3 does not condense at high pressure conditions. Even at log( R ) = -3, where corundum contributes to opacity, molecular band absorption by water molecules continues to play a significant role in opacity. This extends down to around 400 K. In contrast, for log( R ) = 6, the opacity contribution of water vapor is significantly reduced. Another major distinction is that at high density, solid iron is the dominant opacity source from ≃ 1870 K to 650 K, whereas at lower temperatures, silicates begin to prevail. Furthermore, we see that amorphous carbon does not condense, whereas Troilite (FeS) has a discernible contribution, from about 700 K to 380 K. \nFigure 15 shows the behavior of Rosseland mean opacities over the range -8 ≤ log( R ) ≤ 6. As previously discussed, we see that the corundum opacity bump (at 3 . 08 ≲ log( T ) ≲ 3 . 17) is present for log( R ) < 2, while for log( R ) > 2 the species does not condense and solid iron makes the most important opacity contribution for 650 ≲ T/ K ≲ 1870 (2 . 81 ≲ log( T ) ≲ 3 . 27).', '6. IMPACT ON STELLAR MODELS': "Figure 2 suggests that the updated Rosseland mean opacities could have a significant impact on stellar models of masses below ∼ 0 . 7 M ⊙ , particularly influencing the temperature gradient in their superadiabatic regions. This conjecture holds true, as illustrated in Fig. 16, depicting the consequences of employing the revised opacities on the fundamental properties of PARSEC models within the mass range of 0.1 to 0.85 M ⊙ . These results are compared with models utilizing the prior version of ÆSOPUS opacities (specifically, ÆSOPUS v2.0 from Marigo et al. 2022). In this latter case, the opacities were not readily available for log( R ) values exceeding 1, prompting PARSEC to use the opacity values taken at the border of the available tables. This assumption was grounded in the expectation that, in the extensive convective regions of such stars, the temperature gradient would approach adiabatic conditions, rendering it largely insensitive to variations in the opacity tables. \nThe models in Figure 16 are computed with PARSEC code version 2.0 (Costa et al. 2019; Nguyen et al. 2022), and adopting T -τ relations interpolated from the PHOENIX stellar atmosphere models (Allard et al. 2012), implemented and discussed in Chen et al. (2014). Here, suffice it to recall that we use the solar composition from Caffau et al. \n<!-- image --> \nFigure 14. Properties of the Rosseland mean opacity for temperatures where solid grains dominate, at low density (log( R ) = -3, left panel) and high density (log( R ) = 6, right panel). The chemical composition is defined by X = 0 . 735 , Z = 0 . 0165, with scaled-solar elemental abundances following Magg et al. (2022). Each curve represents the contribution of major solid species to the total Rosseland mean opacity, and it is calculated as log( κ R ) -log( κ R i, off ), where κ R is the full opacity including all opacity sources considered here, and κ R i, off is the reduced opacity computed by excluding the specific absorbing species. \n<!-- image --> \nFigure 15. Rosseland mean opacities with the inclusion of condensed dust grains. The chemical composition is the same as in Figure 14. The curves are distributed every 1 dex in log( R ), in the interval -8 ≤ log( R ) ≤ 6. \n<!-- image --> \n(2011) 4 and the mixing length theory with a parameter α = 1 . 74 derived from the calibration of the Solar model (Bressan et al. 2012). We present models either using (labelled with S) or not using the shift in the T -τ relation advocated by Chen et al. (2014). Further details will be discussed in a subsequent paper dedicated to very-low mass star models. \nSince low-mass models evolve minimally after settling into their main sequences, only their properties at the age of 5 Gyr are presented in Figure 16. From the Hertzsprung-Russell (HR) diagram, it is evident that the use of new opacities consistently produces cooler and fainter models. Furthermore, the impact of the new opacities diminishes as \nFigure 16. The impact of our new Rosseland mean opacities on low-mass stellar models computed with the PARSEC v2.0 code. The left panel shows the HR diagram for models in the mass range from 0.1 to 0.85 M ⊙ and computed at intervals of 0.05 M ⊙ (from bottom-right to top-left), and plotted at ages of 5 Gyr. The blue symbols indicate models computed with the previous version of the opacity tables from Marigo et al. (2022), while the red and green symbols are for models computed with present opacity tables. The green symbols correspond to models with a shift in their T -τ relation (see text for details). The right panel shows the same models in the mass-radius plane, now compared with the empirical data of low-mass stars in double-lined eclipsing binary catalogs from S'egransan et al. (2003); Demory et al. (2009); Torres et al. (2010); Carter et al. (2011); Doyle et al. (2011); Kraus et al. (2011); Parsons et al. (2012a,b); Southworth (2015). \n<!-- image --> \nwe approach stellar models with a mass of 0.6 M ⊙ , as expected for stellar structures that predominantly evolve in the range of log( R ) < 1 (refer to Figure 2). \nThe right panel of Figure 16 displays the same models in the mass-radius plane. In this instance, the empirical data derived from double-lined eclipsing binary catalogs is superimposed. As observed, our PARSEC models computed with ÆSOPUS v2.0 opacities generally align with the lower limit of the empirical mass-radius relation. This once again highlights the ongoing manifestation of the mass-radius discrepancy, extensively documented in various studies (see, for example, Chen et al. 2014; Torres et al. 2014; Somers et al. 2020). It is evident that the use of the new ÆSOPUS v2.1 opacities somewhat mitigates this discrepancy by slightly inflating all models in the 0.1-0.6 M ⊙ range. Therefore, we advocate the use of proper opacity tables extended to high densities and pressures, to describe very-low mass models. \nFigure 17 shows the relative differences between the observed and model radii, for the same models and data as in Fig. 16, so as to allow a better visualisation of the small improvements reached by using ÆSOPUS v2.1 instead of v2.0 opacities. An additional panel presented additional models, labelled ÆSOPUS 2.1 S, in which the T -τ relation is shifted by just half the amount calibrated in Chen et al. (2014) - that is, these models adopt a shift of ∆ log( T/T eff ) = 0 . 03 dex for log( T eff / K) < 3 . 5, and gradually reduce this shift to 0 for log( T eff / K) between 3.5 and 3.765. These latter models practically cancel out the discrepancy in the mass-radius relation at low masses. \nTherefore, while our new ÆSOPUS v2.1 opacities reduce the systematic discrepancies between the predicted and observed radii of very low mass stars, additional model assumptions - like the shift in the T -τ relation by Chen et al. (2014) or the stellar spots by Somers et al. (2020) - seem to be still necessary to eliminate them completely.", '7. CONCLUDING REMARKS': "We compute the equation of state and provide Rosseland mean opacity tables for temperatures ranging from 32,000 K to 100 K. These tables are expected to be useful for a series of applications, which we leave to the readers to explore. Tables for different chemical compositions can be retrieved with the updated ÆSOPUS web interface in \nFigure 17. The relative difference between the observed and model radii for the 3 sets of models and the data presented in the right panel of Fig. 16, as a function of stellar mass. It can be seen that while our new opacities reduce the systematic discrepancies between the predicted and observed radii for stars in the M < 0 . 6 M ⊙ range, a shift in the T -τ relation (or other alternatives as discussed in the text) seems to be still necessary to eliminate it completely. \n<!-- image --> \nhttp://stev.oapd.inaf.it/aesopus 5 , where one will also find links to a set of pre-computed tables for 10 popular chemical mixtures (i.e.Anders & Grevesse 1989, Grevesse & Noels 1993, Grevesse & Sauval 1998, Holweger 2001, Lodders 2003, Grevesse et al. 2007, Asplund et al. 2009, Caffau et al. 2011, Asplund et al. 2021, and Magg et al. 2022), and in each case spanning wide ranges in Z and X values. We have so far verified that these new opacity tables lead to important changes in the modeling of very low mass stars at near-solar metallicities, causing shifts in their masseffective temperature and mass-luminosity relations. More modest shifts are present in the mass-radius relation, and they go in the right direction to alleviate the discrepancies in the radii of very low mass stars that have been widely reported in the literature. \nWe acknowledge support from Padova University through the research project PRD 2021. PM and AB acknowledge the Italian Ministerial grant PRIN2022, 'Radiative opacities for astrophysical applications', no. 2022NEXMP8. GC acknowledges support from the Agence Nationale de la Recherche grant POPSYCLE number ANR-19-CE31-0022. LG acknowledges partial support from an INAF Theory Grant 2022. \nSoftware: ÆSOPUS (Marigo et al. 2022; Marigo & Aringer 2009), EXOCROSS (Yurchenko et al. 2018), DIANA (Woitke et al. 2016), GGchem (Woitke et al. 2018)", 'REFERENCES': "- Stewart, J. C., & Pyatt, Kedar D., J. 1966, ApJ, 144, 1203, doi: 10.1086/148714 \nSyme, A.-M., & McKemmish, L. K. 2021, Monthly Notices \nof the Royal Astronomical Society, 505, 4383, \ndoi: 10.1093/mnras/stab1551 \n- Tennyson, J., & Yurchenko, S. N. 2012, MNRAS, 425, 21, doi: 10.1111/j.1365-2966.2012.21440.x\n- Tennyson, J., Yurchenko, S. N., Al-Refaie, A. F., et al. 2016, Journal of Molecular Spectroscopy, 327, 73, doi: 10.1016/j.jms.2016.05.002\n- Torres, G., Andersen, J., & Gim'enez, A. 2010, A&A Rv, 18, 67, doi: 10.1007/s00159-009-0025-1\n- Torres, G., Sandberg Lacy, C. H., Pavlovski, K., et al. 2014, ApJ, 797, 31, doi: 10.1088/0004-637X/797/1/31 Tsuji, T. 2002, ApJ, 575, 264, doi: 10.1086/341262\n- Tsuji, T., Ohnaka, K., & Aoki, W. 1996, A&A, 305, L1\n- Upadhyay, A., Conway, E. K., Tennyson, J., & Yurchenko, S. N. 2018, Monthly Notices of the Royal Astronomical Society, 477, 1520, doi: 10.1093/mnras/sty998\n- Waldmann, I. P., Rocchetto, M., Tinetti, G., et al. 2015a, ApJ, 813, 13, doi: 10.1088/0004-637X/813/1/13\n- Waldmann, I. P., Tinetti, G., Rocchetto, M., et al. 2015b, ApJ, 802, 107, doi: 10.1088/0004-637X/802/2/107 \nWilson, P. A., H'ebrard, G., Santos, N. C., et al. 2016, \nA&A, 588, A144, doi: 10.1051/0004-6361/201527581 \n- Witte, S., Helling, C., & Hauschildt, P. H. 2009, A&A, 506, 1367, doi: 10.1051/0004-6361/200811501\n- Woitke, P., Helling, C., & Gunn, O. 2020, A&A, 634, A23, doi: 10.1051/0004-6361/201936281\n- Woitke, P., Helling, C., Hunter, G. H., et al. 2018, A&A, 614, A1, doi: 10.1051/0004-6361/201732193\n- Woitke, P., Min, M., Pinte, C., et al. 2016, A&A, 586, A103, doi: 10.1051/0004-6361/201526538", 'A. CHEMICAL SPECIES CONSIDERED IN ÆSOPUS': 'For completeness, Table 2 presents the complete list of chemical species considered at the moment in the equation of state of ÆSOPUS 2.1 . \nTable 2. Chemical species considered in the EOS of ÆSOPUS .'}

2024arXiv240913657S

Based on the recently solidified notion that the jittering jets explosion mechanism JJEM is the primary explosion mechanism of corecollapse supernovae CCSNe I estimate some typical properties of the jittering jets. From the imprints of jittering jets in the outskirts of some CCSN remnants I estimate the halfopening angles of jittering jets that shape CCSN remnants to be 110 degrees. I also estimate that intermittent accretion disks around the newly born neutron star NS can launch jets after they live for only several times their orbital period around the NS. To operate the JJEM requires the intermittent accretion disks that launch the jets to amplify the magnetic fields in a dynamo and the magnetic fields to reconnect to release their energy rapidly. I estimate the width of magnetic field reconnection zones to be 0.005r0.1km near the surface of the NS. This width requires a numerical resolution several times smaller than the resolution of present CCSN simulations. I argue therefore that existing simulations of the CCSN explosion mechanism are still far from correctly simulating CCSN explosions.

2024-09-01T00:00:00Z

['arXiv:2409.13657', '10.48550/arXiv.2409.13657', '2024arXiv240913657S']

['Astrophysics - High Energy Astrophysical Phenomena']

Learning from corecollapse supernova remnants on the explosion mechanism

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.13657.pdf

{'Learning from core-collapse supernova remnants on the explosion mechanism': 'Noam Soker 1 \n1 Department of Physics, Technion, Haifa, 3200003, Israel; soker@physics.technion.ac.il \n(Dated: September 2024)', 'ABSTRACT': 'Based on the recently solidified notion that the jittering jets explosion mechanism (JJEM) is the primary explosion mechanism of core-collapse supernovae (CCSNe), I estimate some typical properties of the jittering jets. From the imprints of jittering jets in the outskirts of some CCSN remnants, I estimate the half-opening angles of jittering jets that shape CCSN remnants to be α j /similarequal 1 · -10 · . I also estimate that intermittent accretion disks around the newly born neutron star (NS) can launch jets after they live for only several times their orbital period around the NS. To operate, the JJEM requires the intermittent accretion disks that launch the jets to amplify the magnetic fields in a dynamo and the magnetic fields to reconnect to release their energy rapidly. I estimate the width of magnetic field reconnection zones to be D rec ≈ 0 . 005 r ≈ 0 . 1 km near the surface of the NS. This width requires a numerical resolution several times smaller than the resolution of present CCSN simulations. I argue, therefore, that existing simulations of the CCSN explosion mechanism are still far from correctly simulating CCSN explosions. \nKeywords: supernovae: general - stars: jets - ISM: supernova remnants - stars: massive', '1. INTRODUCTION': "Two competing theoretical explosion mechanisms of massive stars utilize the gravitational energy the collapsing core releases as it forms a new neutron star (NS) to power core-collapse supernovae (CCSNe). In the delayed-neutrino explosion mechanism, neutrinos mediate the gravitational energy to explode the star (e.g., Bethe & Wilson 1985; Heger et al. 2003; Nordhaus et al. 2010, 2012; Janka 2012; Burrows 2013; Muller et al. 2019, 2024; Fujibayashi et al. 2021; Fryer et al. 2022; Boccioli et al. 2022, 2023; Boccioli & Roberti 2024; Nakamura et al. 2022, 2024; Olejak et al. 2022; Burrows et al. 2023; Andresen et al. 2024; Burrows et al. 2024b; Janka & Kresse 2024; Muller 2024; van Baal et al. 2024; Wang & Burrows 2024; Laplace et al. 2024). The magnetorotational explosion mechanism, where a pair of energetic opposite jets with a fixed axis explode the star, requires a rapidly rotating pre-collapse core (for recent studies see, e.g., Shibagaki et al. 2024; Zha et al. 2024a); therefore, it can power only a very small fraction of CCSNe. Despite being powered by jets, I take the magnetorotational explosion mechanism part of the neutrino-driven mechanism because it attributes most CCSNe, those with \nno pre-collapse rapidly rotating cores, to the delayedneutrino explosion mechanism. \nIn the jittering jets explosion mechanism (JJEM), jets that the newly born NS launches mediate the gravitational energy and explode the star in all CCSNe (e.g., Soker 2010; Papish & Soker 2011, 2014a; Gilkis & Soker 2014, 2016; Soker 2020, 2022a,b, 2023a; Shishkin & Soker 2021, 2022, 2023; Wang et al. 2024). Neutrino heating plays a role in the JJEM in boosting the effect of jets, but it does not play the primary role (Soker 2022c). \nThe morphological signatures left by pairs of jets in many CCSN remnants (CCSNRs) are compatible with the JJEM, which considers such jets part of the explosion mechanism. Such signatures include opposite pairs of ears, nozzles, clumps, and filaments in CCSNRs, (e.g., Grichener & Soker 2017 who estimated the energies of the pairs of jets in several CCSNRs). The similarities of jet-shaped morphological features in CCSNRs with jet-shaped morphologies in planetary nebulae (e.g., Bear et al. 2017; Bear & Soker 2017, 2018; Soker 2022b, 2024a; Bear et al. 2024) and in cooling flow clusters (Soker 2024b) solidify the claim for CCSNR shaping by energetic jets. The claim for shaping by energetic jets that are part of the pairs of jets that explode progenitors \nof the CCSNRs is particularly strong on CCSNRs that possess point-symmetric morphology (Soker & Shishkin 2024). A point-symmetric morphology has two or more pairs of opposite structural features (ears, clumps, filaments, nozzles) that do not share the same symmetry axis. The present list of point-symmetric CCSNRs and the studies that attributed their morphologies to jittering jets are: SNR 0540-69.3 (Soker 2022a), the Vela CCSNR (Soker 2023b; Soker & Shishkin 2024), CTB 1 (Bear & Soker 2023), N63A (Soker 2024c), SN 1987A (Soker 2024d,e), G321.3-3.9 (Soker 2024b; Shishkin & Soker 2024), G107.7-5.1 (Soker 2024b), Cassiopeia A (Bear & Soker 2024), the Cygnus Loop (Shishkin et al. 2024), and Puppis A (Bear et al. 2024). \nBased on this list of point-symmetric CCSNRs and other CCSNRs with signatures of one pair of jets, I consider the JJEM to be CCSNe's primary or even sole explosion mechanism. This implies that the newly born NS manages to launch jets within several seconds from the formation of the proto-NS and the bounce of the shock that the proto-NS forms as the collapse stops due to nuclear forces. I use this implication to deduce some properties of the jet-launching process from young NSs (Section 2). I then show (Section 3) that to simulate the relevant processes for the JJEM, three-dimensional (3D) magneto-hydrodynamical (MHD) numerical codes must have very high spatial resolution beyond the reach of most present computers available to the community. I summarize this short study and conclude that the community is far from simulating the correct CCSN explosion process.", '2.1. Previously derived JJEM parameters': 'According to the JJEM, the basic explosion process starts when the newly born NS (and later the black hole if the NS collapses into a black hole) accretes mass with stochastic angular momentum from the collapsing core via intermittent accretion disks. The stochastic angular momentum results from instabilities above the NS, including the spiral standing accretion shock instability (e.g., Buellet et al. 2023 for a recent study of this instability), that amplify angular momentum seed fluctuations in the collapsing core material. The seed fluctuations come from the convective motion in the pre-collapse core (e.g., Papish & Soker 2014a; Gilkis & Soker 2014, 2016; Shishkin & Soker 2023; Wang et al. 2024), and, in some cases, possibly from the envelope (e.g., Quataert et al. 2019; Antoni & Quataert 2022, 2023). In cases of a rapid precollapse core rotation, the stochastic angular momentum variations are around this rotation axis (e.g., Soker \n2023a). Namely, it is not entirely stochastic at a whole solid angle. \nEarlier studies have derived some of the basic properties of the JJEM, which I summarize in Table 1. The first 11 properties are in a table from an earlier paper (Soker 2024e). In this study, I added the last three rows (12-14). Table 1 presents the range of typical values of the JJEM and the justifications for these values. The values the table lists are for iron CCSNe, not electroncapture CCSNe. In electron-capture CCSNe, the primary explosion phase (jet-launching period) might be much longer, up to several minutes to even a few hours rather than seconds (Wang et al. 2024).', '2.2. Half-opening angle of the jets': "The ears on the outskirts of many CCSNRs (e.g., Grichener & Soker 2017; Bear et al. 2017; see Section 1) show that the shaping jets could penetrate the ejecta. For that, the momentum per unit area of the jets cannot be much smaller than the average of the CCSN ejecta and might be even larger. The typical momentum of a jet that carries a mass of m 1j = 0 . 5 m 2j , by values from Table 1, is \np 1j = m 1j v 1j /similarequal 5 -100 M /circledot km s -1 . (1) \nThe radial momentum of a typical CCSN is \np ej , r = M ej v ej ≈ 3 × 10 4 M /circledot km s -1 , (2) \nfor a typical ejecta mass of M ej /similarequal 8 M /circledot and ejecta velocity of v ej /similarequal 4000 km s -1 , such that the kinetic (explosion) energy is E k /similarequal 1 . 3 × 10 51 erg. The value of p ej , r can vary greatly from one CCSN to another. I use equation (2) only to scale the next equation. For the momentum of the jets per unit area to be as large as that of the ejecta or more, the half-opening angle of the jets should be \nα j /lessorsimilar 2 ( p 1j p ej , r ) 1 / 2 = 3 . 6 · ( p 1j 10 -3 p ej , r ) 1 / 2 . (3) \nConsidering the extensive range in the possible jet and ejecta momenta values, I take the half-opening angle's demand to be α j /lessorsimilar 1 · -10 · . \nIn an earlier paper (Soker 2022c), I suggested the collimation of the jets by the pressure of the ambient gas close to the NS, below the gain region, because the pressure in that region is much larger than in the gain region (e.g., Janka 2001). Further out, the material that the jet shocks, the so-called cocoon, maintains the collimation of the jet or might even compress it somewhat (Soker 2022c). Here, I quantify the collimation and find that these collimation processes should be efficient in bringing the opening angle of the jets to be small. \nTable 1. Typical parameters of jittering jets in iron-core CCSNe \nNotes: The first 11 properties are from an earlier table (Soker 2024e); the last three rows are quantities I estimate in this study. # The values in the table are for CCSNe of collapsing iron-rich core. Jets of electron capture supernovae are less energetic, and the explosion might last for minutes to a few hours (Wang et al. 2024). \n& Izzo et al. (2019) reported the possible indication for jets at /similarequal 10 5 km s -1 in SN 2017iuk associated with GRB 171205A. Guetta et al. (2020) claim that most CCSNe have no signatures of relativistic jets, supporting the non-relativistic jet velocity. References: [B+25]: Bear et al. (2024); [BGS17]: Bear et al. (2017); [GiSo15]: Gilkis & Soker (2015);[GrSo17]: Grichener & Soker (2017); [Li04]: Livio (2004); [Ni18]: Nisini et al. (2018); [PaSo11]: Papish & Soker (2011); [PaSo14a]: Papish & Soker (2014a); [PaSo14b]: Papish & Soker (2014b); [ShSo14]: Shishkin & Soker (2021); [So24e]: Soker (2024e); [So24f]: Soker (2024f). \nClumps are also observed in point-symmetric CCSNRs. Clumps can be the tip of jets. Another possibility is that jet-inflated bubbles in the core compress clumps, as Papish & Soker (2014a) have shown, and the clumps penetrate the ejecta and imprint the morphology. Similarities with cooling flow clusters support this process of forming clumps in some, but not all, cases (Soker 2024b).", '2.3. Minimum time for jet launching': 'Bear et al. (2024) proposed a new mechanism to impart a natal kick velocity to the NS, the kick by early asymmetrical pair (kick-BEAP) mechanism. In the kickBEAP mechanism, the NS launches a pair (or two) \nof two opposite jets where one is much more powerful. The momentum asymmetry between the two jets is significant enough to impart a large velocity to the NS (Bear et al. 2024). The kick-BEAP operates within t b /lessorsimilar 0 . 2 s after shock bounce when the accretion of core material onto the NS is very high. At that early phase, the radius of the hot NS is much larger than its final radius, R ( < 0 . 2 s) /similarequal 40 km (e.g., Raynaud et al. 2020) instead of R Ns /similarequal 12 km; it is a proto-NS. \nThe jets of the kick-BEAP mechanism are launched from a radius of /similarequal 40 km. The orbital period at this radius, for a mass of still M /similarequal 1 . 2 M /circledot , is τ Kep /similarequal 0 . 004 s. The high-mass accretion rate lasts for /similarequal 0 . 2 s (e.g., \nMuller et al. 2017; Burrows et al. 2024a), and a jetlaunching episode for a shorter time; Bear et al. (2024) scale with a jet launching period of ∆ t 1 = 0 . 05 s. But the accretion disk starts to launch jets before it ceases to exist. Namely when its age is < 0 . 05 s /similarequal 10 τ Kep . \nThe conclusion from this discussion, namely, under the assumption of the kick-BEAP mechanism in the frame of the JJEM, is that an accretion disk can launch jets even if it exists for only a short time, as brief as /lessorsimilar 10 τ Kep , namely, for only several times the orbital period at the jet-launching radius. This time is shorter than the relaxation time of the accretion disk, which implies unequal opposite jets (Soker 2024c).', '2.4. The rotational energy of the accreted material': "In the JJEM, there are jet-launching episodes with varying angular momentum directions. This implies that the orbital velocity of an accretion disk that launches jets is inclined to the orbital velocity of the accretion disk from the previous jet-launching episode. One effect is the amplification of magnetic fields. The differential rotation within an accretion disk amplifies the azimuthal magnetic field. The accretion disk of the next jet-launching episode is not in the same plane, i.e., it is inclined to that plane. Therefore, it stretches the magnetic field lines in its orbital motion direction, which is inclined to the previous one. This further amplifies the magnetic fields in the stochasticω (S ω ) effect that I suggested in an earlier paper (Soker 2020). Here, I consider the possible dynamic implications of the inclined consecutive accretion disks. \nThe orbital velocity near the surface of the NS during the explosion process when the NS is not yet settled to its final radius is v Kep /similarequal 10 5 km s -1 . This is about the typical jets' velocity in the JJEM, v j /similarequal v Kep (Table 1). The two jets in each jet-launching episode carry a fraction η 2j /similarequal 0 . 1 of the mass in the accretion disk so that a fraction of 1 -η 2j is accreted. The jets carry angular momentum, and the accreted mass settles onto the NS with lower rotational velocity than in the accretion disk. As with young stellar objects that rotate at tens of percent of the break-up velocity (e.g., Huang et al. 2010), I take here the rotational velocity of the accreted mass to be v rot = δ K v Kep , with δ rot /similarequal 0 . 3 -0 . 5. Since v j /similarequal v Kep , I take v rot = δ j v j , with δ j /similarequal 0 . 3 -0 . 5. The ratio of the rotational energy of the accreted mass in an episode to the energy of the jets of that episode is then \nE rot E 2j /similarequal 1 . 4 ( δ j 0 . 4 ) 2 ( 1 -η j 9 η j ) . (4) \nThe kinetic energy of the accreted mass is about equal to the kinetic energy of the pair of jets from the same jet-launching episode. \nThe implication of the relation E rot /similarequal E 2j is as follows. Because the next accretion phase will have a different angular momentum, the velocity in the new accretion disk will not be parallel to that in the previous one that has just been accreted. There will be a large friction in the boundary between the accreted material of the two accretion episodes. A large fraction of the kinetic energy will be dissipated, much more than in an accretion flow with a fixed angular momentum axis. The dissipated kinetic rotational energy ends in heat, high pressure that accelerates material out, and magnetic fields. Any heat carried by neutrinos will not affect the launching of the jets. On the other hand, the hot gas can accelerate material outwards before it cools, adding energy directly to the outflow, namely, jets. A fraction of the dissipated energy ends in magnetic fields, which play a crucial role in jet launching. \nFuture magnetohydrodynamic simulations of the CCSN explosion process should include this friction between accreted material from consecutive jet-launching episodes. This demands a high, but not very high, numerical resolution. The zone between consecutive accretion episodes behaves like a boundary layer where the velocity has a large gradient. A boundary layer in fixedaxis accretion disks can boost mass ejection if shocks occur within it (e.g., Soker & Regev 2003). The width of the boundary layer is of the order of the disk's halfthickness. Allowing for a disk half-thickness of /similarequal 0 . 1 r and demanding resolving the boundary layer by at least four cells requires the numerical resolution near the NS to have a cell size of ∆ /lessorsimilar 0 . 025 r . This is about the resolution of present simulations of CCSNe, as I discuss in Section 3, where I also show that magnetic processes require even finer resolution.", '3. THE REQUIRED NUMERICAL RESOLUTION': "The conversion of the gravitational energy of the collapsing stellar core to the kinetic energy of the jets that explode the star involves intermediate energy forms of rotational energy of the accretion disk (or of a sub-Keplerian accretion flow through a belt; Schreier & Soker 2016) and magnetic energy. Magnetic energy is crucial in the JJEM (Soker 2018, 2019a,b, 2020). However, as I show next, accurate simulations of jittering jets demand numerical resolutions beyond those in present three-dimensional hydrodynamical and magnetohydrodynamical simulations. \nThe sheared velocity of the accreted gas, particularly differential rotation in the intermittent accretion disks, and the convection amplify the magnetic energy, i.e., the dynamo process. The rapid release of magnetic energy requires rapid reconnection of magnetic field lines. \nI turn to estimate the necessary resolution that allows for the correct simulation of the rapid reconnection. If reconnection occurs on a too large scale, then the dynamo is inefficient, and reconnection is not as violent as required. \nConsider a reconnection process where two magnetic flux tubes with opposite field lines approach each other. Let λ tube be the typical size of a magnetic flux tube. After reconnection, the material is ejected to opposite sides at the Alfven velocity v A , while the two magnetic flux tubes approach each other at a velocity of v rec ≈ 0 . 1 v A (e.g., Parker 1979). The width of the reconnection zone is very small \nD rec ≈ λ tube v rec v A ≈ 0 . 1 λ tube . (5) \nFor a violent magnetic reconnection event, the reconnection should occur on a time scale τ rec that is shorter than the material's escape time from the considered radius, /similarequal r/v Kep . I crudely demand, therefore, \nτ rec ≈ λ tube 0 . 1 v a /lessorsimilar r v Kep . (6) \nTo resolve the reconnection process, the size of the grid cells should resolve the two halves of the reconnection zone, each half of an approaching flux tube (of opposite field lines). Namely, ∆ < 0 . 5 D rec . With equations (5) and (6), the constraint on the grid cells becomes \n∆ /lessorsimilar 0 . 05 λ tube /lessorsimilar 0 . 005 ( v a v Kep ) r. (7) \nIn a powerful magnetic activity v a /similarequal v Kep , the Alfven speed will not be larger than the rotation speed. Considering then this ratio of v a /v Kep and that the reconnection zone should be resolved by more than two cells, better at least four cells, I conclude that for simulating the JJEM, the grid cells should be ∆ /lessorsimilar 0 . 002 r . \nIn the recent magnetohydrodynamical simulations of CCSNe that Varma et al. (2023), Powell & Muller (2024), and Zha et al. (2024b), conducted, the numbers of grid cells in the three coordinates were ( N r , N θ , N φ ) = (550 , 128 , 256), corresponding to an angular resolution of 1 . 4 · , or ∆ θ,φ = 0 . 0245 r . They have near the NS ∆ r /similarequal 0 . 015 r in the radial direction. This resolution is several times, and up to an order of magnitude, too coarse for the demand of the JJEM. Raynaud et al. (2020) performed magnetohydrodynamic simulations of magnetar formation. They have high resolution but simulated only the inner /similarequal 40 km, namely, the proto-NS but not its surroundings. With the grid of ( N r , N θ , N φ ) = (257 , 512 , 1024), the resolution of ∆ θ,φ = 0 . 006 r is almost as required for simulating the JJEM, but not yet as needed. \nThe main finding of this discussion is that the present three-dimensional magnetohydrodynamical simulations do not yet have the resolution and ingredients to simulate the JJEM.", '4. DISCUSSION AND SUMMARY': 'Last year, studies of the JJEM analyzed in depth the point-symmetrical morphologies of about ten CCSNRs (see list in Section 1). The main conclusion of the indepth analysis is that jittering jets shaped all these CCSNRs, and the shaping jets were part of the tens of jets that exploded the star (e.g., Soker & Shishkin 2024). The conclusion of jet-shaping of CCSNRs is based in part on carefully studying many morphological similarities between CCSNRs and jet-shaped planetary nebulae and hot gas in clusters of galaxies where pairs of jets shape pairs of bubbles, clumps, rims, and nozzles (e.g., Soker 2024b; Bear et al. 2024; see a talk from 2024: https://www.youtube.com/watch?v=hJYc6EfgxJU). \nAlthough the tool of identifying the morphological imprints of jets is not familiar in the CCSN community, it is standard in analyzing structures of planetary nebulae and hot gas in clusters of galaxies, making the claim for jet-shaped CCSNRs robust. The analysis of these point-symmetrical CCSNRs motivated the present paper. \nBased on the solidification of the JJEM in 2024 and the understanding that jets shape many CCSNRs, I estimated more parameters of the JJEM (rows 12-14 in Table 1). I estimated the typical half opening of the shaping jittering jets to be α j /similarequal 1 · -10 · . Namely, these are not wide jets. I also estimated that an accretion disk around the newly born NS can launch jets even when its lifetime is only several times the orbital period in the radius where it launches the jets. Table 1 lists the estimated parameters of the JJEM. \nWhile studies of the JJEM in 2024 focused on point-symmetrical morphologies of CCSNRs, studies of the competing delayed-neutrino explosion mechanism have focused on three-dimensional simulations to revive the stalled shock (e.g., Burrows et al. 2024a; Janka & Kresse 2024; Muller 2024; Muller et al. 2024; Nakamura et al. 2024; van Baal et al. 2024; Wang & Burrows 2024). As far as I know, when I wrote this paper, none of the studies of the delayedneutrino explosion mechanism suggested any explanation for point-symmetrical CCSNRs. Some researchers in private communications mentioned shaping by postexplosion jets and/or interaction with the interstellar medium or the circumstellar material. However, in a detailed analysis of several CCSNRs, Soker & Shishkin (2024) find that these mechanisms encounter severe dif-fi \nlties. Post-explosion jets and ambient gas cannot explain the point-symmetrical structures of some heavy elements that come from the deep core of the CCSN progenitor (e.g., O, Ne, Mg, Si, S, Fe) and as observed in some CCSNRs, and nor the point-symmetrical structures in the inner zones of some CCSNRs. \nThe strongest argument against the JJEM made by supporters of the delayed neutrino explosion mechanism is that their simulations do not obtain jittering jets. In this study, I dismissed this argument. In Section 3, I argued that the numerical resolution in magnetohydro- \ndynamical simulations appropriate to the JJEM should be much higher than in current simulations. Specifically, the typical size of grid cells in radius r should be ∆ /lessorsimilar 0 . 002 r . Current CCSN simulations have grid cells larger by /greaterorsimilar 3 -5 times from this constraint. I conclude that the simulations of the CCSN explosion mechanism are still a long way from simulating the correct CCSN explosion mechanism.', 'ACKNOWLEDGEMENTS': 'A grant from the Pazy Foundation supported this research.', 'REFERENCES': '- -. 2018, arXiv e-prints, arXiv:1805.03447,\n- doi: 10.48550/arXiv.1805.03447\n- -. 2019a, arXiv e-prints, arXiv:1907.13312, \ndoi: 10.48550/arXiv.1907.13312 \n- -. 2019b, Research in Astronomy and Astrophysics, 19, 095, doi: 10.1088/1674-4527/19/7/95\n- -. 2020, Research in Astronomy and Astrophysics, 20, 024, doi: 10.1088/1674-4527/20/2/24\n- -. 2022a, Research in Astronomy and Astrophysics, 22, 035019, doi: 10.1088/1674-4527/ac49e6\n- -. 2022b, Research in Astronomy and Astrophysics, 22, 122003, doi: 10.1088/1674-4527/ac9782\n- -. 2022c, Research in Astronomy and Astrophysics, 22, 095007, doi: 10.1088/1674-4527/ac7cbc\n- -. 2023a, Research in Astronomy and Astrophysics, 23, 095020, doi: 10.1088/1674-4527/ace9b3\n- -. 2023b, Research in Astronomy and Astrophysics, 23, 115017, doi: 10.1088/1674-4527/acf446\n- -. 2024a, Galaxies, 12, 29, doi: 10.3390/galaxies12030029\n- -. 2024b, The Open Journal of Astrophysics, 7, 49, doi: 10.33232/001c.120279\n- -. 2024c, The Open Journal of Astrophysics, 7, 12, doi: 10.21105/astro.2311.03286\n- -. 2024d, NewA, 107, 102154,\n- doi: 10.1016/j.newast.2023.102154\n- -. 2024e, Research in Astronomy and Astrophysics, 24, 075006, doi: 10.1088/1674-4527/ad4fc2\n- -. 2024f, The Open Journal of Astrophysics, 7, 12, \ndoi: 10.21105/astro.2311.03286 \n- Soker, N., & Regev, O. 2003, A&A, 406, 603,\n- doi: 10.1051/0004-6361:20030809\n- Soker, N., & Shishkin, D. 2024, arXiv e-prints, arXiv:2409.02626, doi: 10.48550/arXiv.2409.02626\n- van Baal, B. F. A., Jerkstrand, A., Wongwathanarat, A., & Janka, H.-T. 2024, MNRAS, \ndoi: 10.1093/mnras/stae1603 \n- Varma, V., Muller, B., & Schneider, F. R. N. 2023, MNRAS, 518, 3622, doi: 10.1093/mnras/stac3247\n- Wang, N. Y. N., Shishkin, D., & Soker, N. 2024, arXiv e-prints, arXiv:2401.06652, \ndoi: 10.48550/arXiv.2401.06652 \nWang, T., & Burrows, A. 2024, ApJ, 969, 74, \ndoi: 10.3847/1538-4357/ad5009 \nZha, S., Muller, B., & Powell, J. 2024a, arXiv e-prints, \narXiv:2403.02072, doi: 10.48550/arXiv.2403.02072 \n- -. 2024b, ApJ, 969, 141, doi: 10.3847/1538-4357/ad4ae7'}

2024arXiv240913562D

Highcadence surveys of the sky are revealing that a large fraction of redsupergiant RSG stars which are progenitors of Type IIPlateau IIP supernovae SNe explode within circumstellar material CSM. Such SNe IIPCSM exhibit considerable diversity with interaction signatures lasting from hours to days potentially merging with the Type IIn subclass for which longerduration interaction typically occurs. To tackle this growing sample of transients and to understand the preSN mass loss histories of RSGs we train on the highest quality spectropolarimetric observations of a young Type IIn SN taken to date Those of SN1998S at 5d after explosion. We design an approach based on a combination of radiation hydrodynamics with HERACLES and polarized radiative transfer with CMFGEN and LONGPOL. The adopted asymmetries are based on a latitudinal depth and timeindependent scaling of the density of 1D models of SNe IIPCSM e.g. model r1w6b with a wind massloss rate of 0.01Msunyr used for SN2023ixf. For a poletoequator density ratio of five we find that the polarization reaches and then remains for days at a maximum value of 1.0 1.4 and 1.8 as the CSM extent is changed from 6 to 8 and 10x1014cm. The polarization is independent of wavelength away from funnelshaped depolarizations within emission lines. Our models implicate a significant depolarization at line cores which we use to constrain the interstellar polarization of SN1998S. Our 2D prolate ejecta models with moderate asymmetry match well the spectropolarimetric observations of SN1998S at 5d supporting a polarization level of about 2. This study provides a framework for interpreting future spectropolarimetric observations of SNe IIPCSM and SNe IIn and fostering a better understanding of the origin of their preSN mass loss.

2024-09-01T00:00:00Z

['arXiv:2409.13562', '2024arXiv240913562D', '10.48550/arXiv.2409.13562']

['Astrophysics - High Energy Astrophysical Phenomena', 'Astrophysics - Astrophysics of Galaxies', 'Astrophysics - Solar and Stellar Astrophysics']

Spectropolarimetric modeling of interacting Type II supernovae. Application to earlytime observations of SN1998S

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.13562.pdf

{'Spectropolarimetric modeling of interacting Type II supernovae. Application to early-time observations of SN 1998S.': "Luc Dessart 1 , Douglas C. Leonard 2 , Sergiy S. Vasylyev 3 , 4 , and D. John Hillier 5 \n- 1 Institut d'Astrophysique de Paris, CNRS-Sorbonne Université, 98 bis boulevard Arago, F-75014 Paris, France\n- 2 Department of Astronomy, San Diego State University, San Diego, CA 92182-1221, USA\n- 3 Department of Astronomy, University of California, Berkeley, CA 94720-3411, USA\n- 4 Steven Nelson Graduate Fellow in Astronomy\n- 5 Department of Physics and Astronomy & Pittsburgh Particle Physics, Astrophysics, and Cosmology Center (PITT PACC), University of Pittsburgh, 3941 O'Hara Street, Pittsburgh, PA 15260, USA", 'ABSTRACT': "High-cadence surveys of the sky are revealing that a large fraction of red-supergiant (RSG) stars, which are progenitors of Type II-Plateau (II-P) supernovae (SNe), explode within circumstellar material (CSM). Such SNe II-P / CSM exhibit considerable diversity, with interaction signatures lasting from hours to days, potentially merging with the Type IIn subclass for which longer-duration interaction typically occurs. To tackle this growing sample of transients and to understand the pre-SN mass loss histories of RSGs, we train on the highest quality, spectropolarimetric observations of a young Type IIn SN taken to date: Those of SN 1998S at ∼ 5 d after explosion. We design an approach based on a combination of radiation hydrodynamics with HERACLES and polarized radiative transfer with CMFGEN and LONG\\_POL . The adopted asymmetries are based on a latitudinal, depth- and time-independent, scaling of the density of 1D models of SNe II-P / CSM (e.g., model r1w6b with a 'wind' mass-loss rate of 0.01 M ⊙ yr -1 used for SN 2023ixf). For a pole-to-equator density ratio of five, we find that the polarization reaches, and then remains for days, at a maximum value of 1.0, 1.4, and 1.8 % as the CSM extent is changed from 6, to 8 and 10 × 10 14 cm. The polarization is independent of wavelength away from funnel-shaped depolarizations within emission lines. Our models implicate a significant depolarization at line cores, which we use to constrain the interstellar polarization of SN 1998S. Our 2D, prolate ejecta models with moderate asymmetry match well the spectropolarimetric observations of SN 1998S at 5 d, supporting a polarization level of about ∼ 2 %. This study provides a framework for interpreting future spectropolarimetric observations of SNe II-P / CSM and SNe IIn and fostering a better understanding of the origin of their pre-SN mass loss. \nKey words. radiative transfer - polarization - supernovae: general - supernova: individual: SN 1998S", '1. Introduction': 'There is much interest today about the origin of pre-explosion mass loss and subsequent circumstellar material (CSM) that surrounds the massive star progenitors that lead to Type II supernovae (SNe). Some rare and extraordinary super-luminous SNe II exhibit signatures of ejecta interaction with CSM for weeks or months (e.g., SN 2010jl; Zhang et al. 2012; Fransson et al. 2014; Dessart et al. 2015), suggesting extreme CSM properties of several M ⊙ that extend out to many 10 15 cm, and earning these events a classification as Type IIn SN (Niemela et al. 1985; Schlegel 1990). In contrast standard-luminosity Type II SNe, with more typical Plateau or fast-declining light curves, exhibit signatures of interaction for only a few days after the emergence of the shock at the progenitor surface before showing the more standard spectral properties of noninteracting Type II SNe with Doppler-broadened lines (Bruch et al. 2021, 2023; JacobsonGalán et al. 2024) - we may refer to this second class of events as SNe II-P / CSM. From this growing sample of events emerges a finer diversity, with objects like SN 2013fs exhibiting interaction, IIn-like signatures for about a day (Yaron et al. 2017), and others such as SN 1998S for one or two weeks (Leonard et al. 2000, hereafter L00). \nThere are likely multiple origins for this diversity of CSM and SN properties, which may involve wave excitation from the stellar core (Quataert & Shiode 2012; Fuller 2017), nuclear \nflashes (Woosley & Heger 2015), surface pulsations (Yoon & Cantiello 2010), or binary e ff ects (Wu & Fuller 2022). Redsupergiant (RSG) star atmospheres may also be more massive and extended than typically envisioned, serving as a wasteland for the numerous instabilities taking place in the stellar envelope or at its surface and acting over the entire RSG-phase duration (see, e.g., Dessart et al. 2017; Soker 2021; Fuller & Tsuna 2024). Some of these phenomena may occur in unison. \nCharacterizing this massive-star mass loss is a prerequisite for understanding its nature, both in terms of CSM density and extent. New insights about the dynamical properties of the CSM are starting to be revealed with high-cadence high-resolution observations at the earliest post-breakout times (Shivvers et al. 2015; Smith et al. 2023; Pessi et al. 2024). Constraining the geometry of the CSM may also provide clues on the mechanism at the origin of the CSM. For unresolved sources, polarization is a powerful means to find evidence for asymmetry and constrain its nature (Wang & Wheeler 2008). \nSpectropolarimetric observations of interacting SNe have been secured for only a few objects such as SN 2010jl (Patat et al. 2011) or SN 2009ip (Mauerhan et al. 2014). In Type II SNe with early-time signatures of interaction present for only a few days, the spectropolarimetric observations must be conducted at the earliest times after shock breakout when a large part of the CSM remains unshocked. This has been achieved recently for \nFig. 1. Comparison of the optical luminosity of model r1w6b at 5 d after explosion and computed either with the nonmonotonic solver or assuming homologous expansion and using the blanketed mode in CMFGEN . [See discussion in Section 2.] \n<!-- image --> \nSN2023ixf (Vasylyev et al. 2023; Singh et al. 2024), but undoubtedly the best quality, early-time, spectropolarimetric observations of a SN with signatures of interaction are those obtained for SN 1998S by L00 at an estimated post-explosion time of 5 d. Being of relatively high spectral resolution (i.e., 6 Å), these data reveal a polarization that varies significantly from continuum to emission line regions, as well as across line profiles between the broad wings and the narrow cores. While much uncertainty and debate surround the correction for interstellar polarization (ISP; see. e.g., L00 or Wang et al. 2001), SN 1998S clearly shows percent level intrinsic polarization that indicates significant departures from spherical symmetry. \nConverting this intrinsic polarization into a specific CSM asymmetry is di ffi cult. The presence of electron-scattering wings on all emission lines indicates that the continuum and lines form under optically-thick conditions (Chugai 2001; Dessart et al. 2009) - estimates requiring optically-thin conditions (see., e.g., Brown & McLean 1977) cannot be used. Interpretations connecting an overall polarization level to a specific degree of asphericity have been formulated (Hoflich 1991) but applied to noninteracting SNe like SN 1987A where the ejecta structures are vastly di ff erent from those present in interacting SNe (see also Dessart & Hillier 2011). In such phenomena, modeling of both the radiation-hydrodynamics and the radiative transfer are necessary, as done in a first attempt by Dessart et al. (2015) and applied to the spectropolarimetric observations of the Type IIn SN2010jl (Patat et al. 2011). \nIn this letter, we extend our previous work on the polarization modeling of noninteracting Type II SNe during the photospheric (Dessart et al. 2021b) and nebular phases (Dessart et al. 2021a; Leonard et al. 2021) by considering the cases of interacting Type II SNe. The essence of the approach (our \'ansatz"), which is simple, is to enforce homologous expansion throughout the interaction region in order to use the 2D polarized radiativetransfer code LONG\\_POL (Hillier 1994, 1996; Dessart & Hillier 2011; Dessart et al. 2021b) in which homologous expansion is currently assumed. As we show in the next section, our ansatz is well motivated at early times, but could also be used at all times if only the continuum polarization is sought. In the next section, we present our modeling procedure, followed by our results in Section 3 and a comparison of our models to the spectropolarimetric observations of SN 1998S in Section 4. While the presentation is kept concise and focused in the main text, we present the code LONG\\_POL in Appendix B. All simulations in this work will be uploaded on zenodo. 1', '2. Numerical procedure': 'In this work, we use o ff shoots of the model r1w6, which itself was one instance in the grid of interacting, SNe II-P / CSM models from Dessart et al. (2017). It corresponds to a solarmetallicity progenitor star of 15 M ⊙ on the zero-age main sequence, whose explosion following core collapse is designed to produce an ejecta with a kinetic energy of 1 . 2 × 10 51 erg. This explosion occurs in a CSM corresponding to a wind with a mass loss rate of 0.01 M ⊙ yr -1 , a terminal velocity V ∞ of 50 km s -1 and a velocity profile versus radius R given by V ( R ) = V ∞ (1 -R ⋆/ R ) β with β = 2 ( R ⋆ is the progenitor surface radius). The dense part of the CSM extends to R CSM, beyond which the mass-loss rate drops smoothly and within a few 10 14 cm to 10 -6 M ⊙ yr -1 . This configuration is modeled in 1D with the radiation-hydrodynamics code HERACLES (González et al. 2007) as discussed in Dessart et al. (2017). For a selection of epochs, the HERACLES calculations are post-processed with the 1D radiative transfer code CMFGEN (Hillier & Dessart 2012; Dessart et al. 2015) in order to generate for each model the variation of the electron density, opacity, and emissivities versus radius. During the CMFGEN calculation, the gas temperature is kept fixed and equal to the value from the HERACLES snapshot. \nIn our modeling of the polarization, the asymmetry of the interaction is considered only at this last stage, by introducing a latitudinal scaling of the densities, opacities and emissivities, which are then used by the 2D, polarized radiative transfer code LONG\\_POL (Hillier 1994, 1996; Dessart & Hillier 2011; Dessart et al. 2021b). This approach is not fully-consistent hydrodynamically but it is flexible and allows for explorations of di ff erent asymmetries. In practice, we use a simple latitudinal scaling of the density that goes as X ( µ ) = a (1 + A µ 2 ), where µ = cos θ , θ is the polar angle, A takes values typically of a few, and a is chosen to preserve the density at a given depth. We solve the 2D polarized radiative transfer for such 2D ejecta from 3800 to 9500 Å. In the 2D ejecta, the electron density is scaled by X , whereas the opacities and the emissivities are scaled by X 2 (for full details of the method and an application to the modeling of spectropolarimetric observations of Type II-P SN 2012aw, see Dessart et al. 2021b; see also Appendix B). \nIn Dessart et al. (2017), all radiative-transfer calculations were done with the 1D steady-state, nonmonotonic solver in CMFGEN in order to capture the complex interaction structure and model the spectral evolution in detail. Unfortunately, LONG\\_POL does not currently handle nonmonotonic flows - homologous expansion is required. To circumvent this current limitation of LONG\\_POL , we modify the velocity from the HERACLES snapshot that is read in by CMFGEN , and enforce homologous expansion by setting V = R / t and t = R phot / V phot. We then run CMFGEN with a di ff erent solver, in steady-state and 1D, but in blanketed mode (Hillier & Miller 1998). A similar blanketed mode in CMFGEN was also used by Shivvers et al. (2015) in their modeling of highresolution spectra of SN 1998S, although without any coupling to radiation hydrodynamics. \nFigure 1 compares the emergent optical spectra for model r1w6b, for which R CSM is 8 × 10 14 cm, at 5 d using either the original nonmonotonic velocity (and nonmonotonic solver) or assuming homologous expansion (and using the blanketed mode in CMFGEN ; see also Appendix A). The di ff erence in the optical luminosity is small and the morphology of emission lines is largely preserved - this confirms that the main broadening mechanism is electron scattering. However, the blanketed mode resolves a discrepancy in the strength of the blend of N iii and C iii multiplets around 4640 Å, which the nonmonotonic solver \nFig. 2. Evolution of the total flux FI and the polarized flux FQ computed with LONG\\_POL assuming a 2D, prolate ejecta based on model r1w6b ( ρ pole /ρ eq = 5). Epochs cover from 2.5 to 10.0 d after explosion. Both fluxes are scaled to the inferred distance of SN 1998S, with an additional scaling for the total flux (see label) to match the magnitude of the polarized flux at 6300 Å ( FQ flips sign at the last epoch). All fluxes have been rebinned at 6 Å. [See discussion in Section 3.] \n<!-- image --> \nsystematically underestimates relative to observations (see, e.g., Jacobson-Galán et al. 2023). These changes in line strength arise in part from the greater UV luminosity obtained with the blanketed mode. Other optical emission lines are of similar strength in both CMFGEN calculations and in good agreement with observations of SN 1998S (see Section 4) or SN 2023ixf (JacobsonGalán et al. 2023). This close correspondance between the two CMFGEN calculations suggests that our ansatz is acceptable. \nIn this work, we focus mostly on model r1w6b but also consider 2D ejecta built from variants in which R CSM is 6 and 10 × 10 14 cm (models r1w6[a,c]). Assuming homologous expansion, we performed CMFGEN calculations for the r1w6[a,b,c] models at times between 1.67-2.5 d up to 5-10 d after explosion. These epochs straddle the phase during which the photosphere is located in the unshocked CSM and eventually into the cold-dense shell (for a general overview of this evolution, see Dessart 2024). In all cases presented here, the 2D ejecta built from these interaction models are prolate and have a moderate pole-to-equator density ratio of five. Furthermore, the variation with latitude as µ 2 is progressive and thus excludes jets and disks.', '3. Polarization modeling results': "Figure 2 shows the results from the 2D polarized radiativetransfer code LONG\\_POL for a 2D, prolate ( ρ pole /ρ eq = 5) ejecta based on model r1w6b, for a 90-deg inclination, and from 2.5 to 10.0 d after explosion. The total flux FI shows emission lines \nwith narrow cores and broad, symmetric, electron-scattering broadened wings for the first four epochs whereas at the last epoch at 10.0 d, the total flux exhibits weak, blueshifted, Doppler broadened lines that are starting to show blueshifted, P-Cygni absorptions. With the normalization of the total and polarized fluxes (i.e., total flux multiplied by the percent polarization, here denoted as FQ ) at 6300 Å, one sees that the slope of each is essentially identical. In other words, the overall continuum polarization is constant with wavelength. Across most lines, the polarized flux decreases, with a maximum reduction at the line cores; this indicates substantial depolarization of both line and continuum photons at these wavelengths. \nThe lower polarization across lines arises in part from the fact that all lines form exterior to the continuum-formation region and thus at lower electron-scattering optical depth (see second panel from top in Fig. A.1). The linear polarization arising from scattering with free electrons is therefore greater for continuum than for line photons. In the narrow line cores, we see photons that have undergone no frequency redistribution in wavelength (or in velocity space) as they travelled outwards from their original point of emission (many of these 'core' photons are emitted beyond the photosphere, at low electron-scattering optical depth, and thus could not exhibit much polarization). These line core photons thus experienced little or no scattering with free electrons and are thus unpolarized. Furthermore, because the optical depth in lines such as H α is huge, the (polarized) continuum photons overlapping with the line cores su ff er from absorption, leading to a narrow polarization dip in emission line cores. \nThe funnel-shaped profile observed in polarized flux across lines indicates that the associated photons are depolarized (i.e., the polarized flux is below the level obtained by interpolating FQ from the adjacent continuum regions). This funnel shape arises because continuum photons originally emitted at a λ init not too far from a line's rest wavelength λ c will be absorbed by that line if they ever come within a few Doppler widths (say 10 km s -1 ) of λ c. This eventuality may occur as photons random-walk and scatter with free electrons in the CSM. The probability for this is greater the smaller is | λ initλ c | . Line cores act as a sink for neighboring (in λ -space), scattered continuum photons. \nSome subtleties are also visible. For example, the C iii / N iii blend exhibits a weaker depolarization. These lines form deeper than H α and thus closer to the region of continuum formation (see also Fig. A.1, and L00), but also through di ff erent processes (i.e., continuum fluorescence and dielectronic recombination). The polarization is found to be positive for the first four epochs, thus aligned with the axis of symmetry. This results from an optical depth e ff ect, suggesting that polarization is controlled primarily by the lower-density, equatorial regions where the bulk of the radiation emerges rather than the higher density regions where scattering is enhanced (see discussion by Dessart & Hillier 2011). The polarization flips in sign at the last epoch (equivalent to a 90-deg rotation of the polarization angle), when the unshocked CSM has been fully swept-up and the entire spectrum forms in the dense shell. \nIn this 2D, prolate ejecta model with ρ pole /ρ eq = 5, the continuum polarization (which is essentially constant throughout the optical range) has a maximum value of 1.4 % until 4.17 d, dropping to 1.3 % at 5.0 d and 0.5 % at 10.0 d. Hence, the presence of an extended unshocked CSM with a modest asymmetry can produce percent-level polarization at early times (as inferred in SNe 1998S or 2010jl), without invoking extreme explosion asymmetries (e.g., jets or disks), and thus comparable to peak values obtained at the transition to the nebular phase of \n- \nFig. 3. Polarization data in the q -u plane for SN 1998S at ∼ 5 d after explosion, from L00. Photons with wavelengths corresponding to the broad wings and narrow-line core of the H α flux profile are indicated with red and green circles, respectively. Each point represents a bin 10 Å wide, except for the green circles, which are binned at 2 Å for enhanced resolution. The blue line spans the range of potential ISP choices bounded by the ISP derived from di ff erent assumptions about the extent of depolarization in the narrow-line core region of H α . The blue squares indicate four specific choices of ISP discussed in the text, which yield the inferred intrinsic polarizations displayed in the bottom panels of Fig. 4. [See Section 4 for discussion.] \n<!-- image --> \n2 \nnon-interacting, Type II SNe (Leonard et al. 2015; Nagao et al. 2021). Modulations of R CSM do not a ff ect the qualitative results obtained for r1w6b but change the values of the maximum polarization and its evolution in time (Appendix C). The equivalent 2D prolate ejecta based on model r1w6a (r1w6c) exhibit a maximum polarization of 1.0 % (1.8 %) and a decline at ∼ 5 d ( ∼ 10 d). Consequently, SNe II-P / CSMwith longer-lived IIn-like signatures should on average exhibit a greater level of polarization. Other parameters that may impact the polarization behavior are the CSM density or associated wind mass loss.", '4. Comparison to spectropolarimetric observations of SN1998S': "We now compare our modeling results with the single-epoch spectropolarimetry of SN 1998S at ∼ 5 d post-explosion presented and thoroughly discussed by L00. We choose to compare our models with these data since they remain (to our knowledge) the only optical spectropolarimetry obtained both early enough to capture the strong ejecta-CSM interaction and with su ffi cient resolution ( ∼ 6 Å) to clearly reveal the spectropolarimetric behaviors of both the narrow- and broad-line features in an SN IIn. \nThe observed data of SN 1998S from L00 present a high degree ( ∼ 2 %) of wavelength-independent continuum polarization across the observed spectral range (4314-6850 Å) and sharp depolarizations across all strong emission lines (see Fig. 3, as well as the 'ISP2' panel of Fig. 4 here). To convert these observed data into intrinsic polarization that can be directly compared with our models, we must first determine a plausible value of the ISP. \nWepresent the observed data of SN 1998S in the Stokes q -u plane in Fig. 3. In this representation, the black circles clustering around [ q , u ] ≈ [-0.9%,-1.7%] arise from the continuum regions; the red and green circles correspond specifically to the broad wings and narrow core spectral regions of H α . \nThis presentation enables immediate identification of an interesting fact: The broad wings and narrow core photons 'point' along di ff erent directions in the Stokes q -u plane. Indeed, L00 used this to derive two di ff erent ISP possibilities, one with the \nFig. 4. Comparison of the spectropolarimetric observations of SN1998S at 5 d after explosion with the counterparts obtained with LONG\\_POL assuming a 2D, prolate ejecta ( ρ pole /ρ eq = 5) based on model r1w6b at 5 d. The observer's inclination relative to the axis of symmetry is 90 deg. Four ISP choices are shown, corresponding to ISP[1,2,3,4] indicated in Fig. 3 ; note that 'ISP2' represents the observed data, with no ISP removed. Observations have been corrected for a recession velocity of 840 km s -1 and a reddening E ( B -V ) of 0.11 mag. Model spectra have been smoothed and rebinned to 6 Å to match the resolution of the observations. [See Section 4 for discussion.] \n<!-- image --> \nassumption of unpolarized photons in the broad wings of strong lines, and one under the assumption of unpolarized photons contributed solely by the narrow emission lines; these choices are indicated by the points identified in Fig. 3 as 'ISP-L00' and 'ISP4', respectively. \nHere we shall be guided by the modeling results of the previous section, which predict that the spectral region corresponding specifically to the narrow-line core of strong emission features (i.e., H α ) should consist not only of intrinsically unpolarized narrow-line photons, but also a significantly - if not completely -depolarized underlying continuum at these wavelengths. This presents to us a new range of ISP choices, for which the bounds can be easily established. First, if the narrow-line core of H α is assumed to have zero intrinsic polarization (i.e., unpolarized line photons and completely depolarized underlying continuum), then the ISP must lie simply at the location of the tip of the narrow H α line in the q -u plane; this is indicated by 'ISP1' in Fig. 3. At the other extreme lies the result that obtains if we assume only unpolarized narrow-line photons and no depolarization of the underlying continuum photons; this is indicated by 'ISP4' in Fig. 3. The line connecting these two points then spans the range of allowable ISP values under di ff erent assumptions about the degree of depolarization of the underlying continuum. 'ISP3', located at [ q , u ] = [0.55%, 0.40%] is one such example, and was chosen specifically to best fit the depolarization at H α found by model r1w6b at 5 d. Curiously, the origin is also an al- \nble ISP choice, and so we also consider that possibility and label that point 'ISP2'. \nWhen the four ISP choices labeled in Fig. 3 are removed from the SN 1998S data, they yield the intrinsic polarizations shown in the bottom panels of Fig. 4. While the inferred overall level of continuum polarization changes drastically with ISP choice, it is notable that the basic features of the resulting spectropolarimetry do not: A high level of wavelength-independent continuum polarization with 'funnel-shaped' depolarizations across the strong line features. The level of depolarization across lines is best matched by choices ISP[1,2,3], which imply an intrinsic continuum polarization in SN 1998S of ∼ 2 %, in rough agreement with our 2D prolate ejecta model with ρ pole /ρ eq = 5. Choice ISP4 implies a high intrinsic polarization but a very weak depolarization within lines, which is not as well matched by our models; as noted by L00, such a high degree of ISP also strains the allowable values from reddening considerations. We thus arrive at the conclusion that the SN 1998S spectropolarimetry likely su ff ered little ISP contamination; in fact, assuming zero ISP produces results in as good agreement with our model predictions as any other. \nAs the data in the q -u plane do not lie perfectly along a straight line (Fig. 3), the ejecta associated with SN1998S must exhibit departures from axial symmetry. The small degree of polarization angle rotation across the broad emission lines is not presently captured by our models and is a limitation of the imposed axisymmetry in LONG\\_POL . Modeling of such features requires 3D polarized radiative-transfer and is left to future work.", '5. Conclusion': "We have presented an end-to-end modeling of RSG stars exploding inside an asymmetric CSM. The models are based on 1D radiation-hydrodynamics calculations with HERACLES of the ejecta interaction with the CSM, the post-treatment of multiepoch snapshots with the 1D radiative transfer code CMFGEN , and finally the 2D polarized radiative transfer with LONG\\_POL . The main limitation of our work is the incomplete physical consistency since asymmetry is only introduced at the last modeling stage with LONG\\_POL . Another adjustment is the enforcement of homologous expansion, as currently required by LONG\\_POL , although we show that this modification is acceptable (see Section 2 and Appendix A). We select the interaction model r1w6b (and o ff shoots r1w6[a,c]), presented and confronted to SN2023ixf by Jacobson-Galán et al. (2023). \nAdopting a simple, depth-independent, latitudinal scaling of the density and assuming suitable scaling relations for the opacities and emissivities computed with CMFGEN for the 1D model r1w6b, we model the spectropolarimetry of 2D, prolate ejecta with a pole-to-equator density ratio of five. In those H-rich, ionized environments where the electron-scattering opacity dominates everywhere away from line cores, we find that the polarization is essentially constant with wavelength apart from funnel-shaped depolarizations across lines. The lower polarization across lines follows from the formation of lines exterior to the continuum, and thus at lower electron-scattering optical depth, but the additional depolarization is caused by the disappearance of continuum photons that wander in λ -space too close to highly-absorbing line cores. Our simulations with a fixed, depth-independent asymmetry, suggest the overall polarization should remain constant for several days in SNe II-P interacting with CSM before dropping, and flipping sign, as the shock emerges from the CSM. For a fixed CSM density, models with \nmore compact / extended CSM exhibit a lower / greater maximum polarization with a more / less rapid evolution. \nOur finding that the narrow line cores should exhibit some level of depolarization is used to set constraints on the ISP of SN1998S, leading to the conclusion that the intrinsic polarization of SN 1998S is of ∼ 2 %. Our models replicate the wavelength independence of the continuum polarization, with the funnel-shaped depolarization across lines. This similarity lends support to the essential ansatz of our models and the numerical approach. As discussed in Section 3 (see also results for models r1w6[a,c] in Appendix C), tweaks to a few input parameters (e.g., R CSM, ρ pole /ρ eq, etc.) could in principle be made in order to provide better fits to the actual levels of (even temporally changing) continuum polarization for objects where the ISP is more tightly constrained than is the case for SN 1998S. Overall, our methodology extended here to include spectropolarimetry, o ff ers a means to model and constrain the properties of the growing sample of SNe II-P / CSM like 2023ixf and better understand the origin of pre-SN mass loss. \nAcknowledgements. This work was supported by the 'Programme National Hautes Energies' of CNRS / INSU co-funded by CEA and CNES. This work was granted access to the HPC resources of TGCC under the allocation 2023 - A0150410554 on Irene-Rome made by GENCI, France. D.C.L. acknowledges support from NSF grant AST-2010001, under which part of this research was carried out. D.J.H. gratefully acknowledges support through NASA astro-physical theory grant 80NSSC20K0524.", 'References': '- Brown, J. C. & McLean, I. S. 1977, A&A, 57, 141\n- Bruch, R. J., Gal-Yam, A., Schulze, S., et al. 2021, ApJ, 912, 46\n- Bruch, R. J., Gal-Yam, A., Yaron, O., et al. 2023, ApJ, 952, 119\n- Chandrasekhar, S. 1960, Radiative transfer (New York: Dover)\n- Chugai, N. N. 2001, MNRAS, 326, 1448\n- Chugai, N. N., Blinnikov, S. I., Fassia, A., et al. 2002, MNRAS, 330, 473\n- Dessart, L. 2024, arXiv:2405.04259\n- Dessart, L., Audit, E., & Hillier, D. J. 2015, MNRAS, 449, 4304\n- Dessart, L. & Hillier, D. J. 2011, MNRAS, 415, 3497\n- Dessart, L., Hillier, D. J., & Audit, E. 2017, A&A, 605, A83\n- Dessart, L., Hillier, D. J., Gezari, S., Basa, S., & Matheson, T. 2009, MNRAS, 394, 21\n- Dessart, L., Hillier, D. J., & Leonard, D. C. 2021a, A&A, 651, A10\n- Dessart, L., Leonard, D. C., Hillier, D. J., & Pignata, G. 2021b, A&A, 651, A19\n- Fransson, C., Ergon, M., Challis, P. J., et al. 2014, ApJ, 797, 118\n- Fuller, J. 2017, MNRAS, 470, 1642\n- Fuller, J. & Tsuna, D. 2024, The Open Journal of Astrophysics, 7, 47\n- González, M., Audit, E., & Huynh, P. 2007, A&A, 464, 429\n- Hillier, D. J. 1994, A&A, 289, 492\n- Hillier, D. J. 1996, A&A, 308, 521\n- Hillier, D. J. & Dessart, L. 2012, MNRAS, 424, 252\n- Hillier, D. J. & Miller, D. L. 1998, ApJ, 496, 407\n- Hoflich, P. 1991, A&A, 246, 481\n- Jacobson-Galán, W. V., Dessart, L., Davis, K. W., et al. 2024, ApJ, 970, 189\n- Jacobson-Galán, W. V., Dessart, L., Margutti, R., et al. 2023, ApJL, 954, L42\n- Leonard, D. C., Dessart, L., Hillier, D. J., et al. 2021, ApJL, 921, L35\n- Leonard, D. C., Dessart, L., Pignata, G., et al. 2015, in IAU General Assembly, Vol. 29, 2255774\n- Leonard, D. C., Filippenko, A. V., Barth, A. J., & Matheson, T. 2000, ApJ, 536, 239\n- Mauerhan, J., Williams, G. G., Smith, N., et al. 2014, MNRAS, 442, 1166 \nNagao, T., Patat, F., Taubenberger, S., et al. 2021, MNRAS, 505, 3664 \n- Niemela, V. S., Ruiz, M. T., & Phillips, M. M. 1985, ApJ, 289, 52\n- Patat, F., Taubenberger, S., Benetti, S., Pastorello, A., & Harutyunyan, A. 2011, A&A, 527, L6\n- Pessi, T., Cartier, R., Hueichapan, E., et al. 2024, A&A, 688, L28\n- Quataert, E. & Shiode, J. 2012, MNRAS, 423, L92\n- Schlegel, E. M. 1990, MNRAS, 244, 269\n- Shivvers, I., Groh, J. H., Mauerhan, J. C., et al. 2015, ApJ, 806, 213\n- Singh, A., Teja, R. S., Moriya, T. J., et al. 2024, arXiv:2405.20989\n- Smith, N., Pearson, J., Sand, D. J., et al. 2023, ApJ, 956, 46\n- Soker, N. 2021, ApJ, 906, 1\n- Vasylyev, S. S., Yang, Y., Filippenko, A. V., et al. 2023, ApJL, 955, L37\n- Wang, L., Howell, D. A., Höflich, P., & Wheeler, J. C. 2001, ApJ, 550, 1030\n- Wang, L. & Wheeler, J. C. 2008, ARA&A, 46, 433\n- Woosley, S. E. & Heger, A. 2015, ApJ, 810, 34\n- Wu, S. C. & Fuller, J. 2022, ApJL, 940, L27\n- Yaron, O., Perley, D. A., Gal-Yam, A., et al. 2017, Nature Physics, 13, 510\n- Yoon, S.-C. & Cantiello, M. 2010, ApJL, 717, L62\n- Zhang, T., Wang, X., Wu, C., et al. 2012, AJ, 144, 131', 'Appendix A: The assumption of homologous expansion in CMFGEN simulations of interacting supernovae': "We show some results from the CMFGEN calculation assuming homologous expansion in Fig. A.1 and using the model r1w6b ( R CSM = 8 × 10 14 cm) - this model was found by Jacobson-Galán et al. (2023) to yield a satisfactory match to the photometric and spectroscopic evolution of SN 2023ixf, a close analog of SN1998S (model r1w6b derives from model r1w6 of Dessart et al. (2017), which has been extensively used in the SN community). The time is 5 d after first detection. \nLooking at the velocity structure first, we see that the original, nonmonotonic velocity and the new ('fudged'), homologous velocity (bottom panel of Fig. A.1) di ff er significantly. By assuming homologous expansion, the fast inner regions are now the slowest while the originally slow outer CSM regions are now the fastest. The unadulterated velocities di ff er from the initial velocities because of the radiative acceleration of the unshocked CSM (see simulation results for this phenomenon in Dessart et al. 2017; see also Chugai et al. 2002). Here, in the HERACLES calculation, the entire CSM is predicted to move at a velocity greater than 200 km s -1 , and as fast as 300 km s -1 at the photosphere location at 5 d. Such relatively large velocities suggest that a resolution of 300 km s -1 is not so bad for spectropolarimetric observations of interacting SNe (as obtained by L00 for SN1998S). \nThese o ff sets have, however, a moderate impact for what concerns us. First, the inner, fast moving layers are at high optical depth and thus contribute negligibly to the emergent flux (both are shown in the second panel from top in Fig. A.1). The outer CSM regions are of low density and low optical depth and will thus have a weak impact, essentially limited to the narrow emission line cores. Because these velocities are still small, they are typically unresolved in spectropolarimetric observations our models will simply overestimate the width of this narrow, line core emission by a factor of about two. The bulk of the spectrum forms between optical depth 0.1 and 10 (i.e., around the photosphere; see quantity P δ ¯ L in Fig. A.1, second panel from the top), where the original and the modified velocities are similar by design. Electron scattering being the dominant line broadening mechanism at such times and in those regions, and since we are not concerned here with information at the 100 km s -1 scale only available in high-resolution spectra, this slight change in material velocity is unimportant. \nThis adjustment of the original nonmonotonic velocity into a homologous flow leads to di ff erent velocities at di ff erent epochs or in di ff erent models. While we set V = R / t and t = R phot / V phot in all cases, the quantities R phot and V phot are specific of each HERACLES snapshot for each model studied. Obviously, this 'homologous' time does not correspond to any physical time for either the ejecta, the CSM, or the interacting SN, but this time plays no role in the steady-state radiative transfer to be performed.", 'Appendix B: Polarized radiative transfer with LONG\\_POL': "For completeness, we summarize the nomenclature and sign conventions adopted in LONG\\_POL and also presented by Dessart & Hillier (2011). We assume that the polarization is produced by electron scattering. The scattering of electromagnetic radiation by electrons is described by the dipole or Rayleigh scattering phase matrix. To describe the 'observed' model polarization \n3 \nFig. A.1. Illustration of ejecta and radiative properties of model r1w6b at 5 d after explosion, as computed by HERACLES and CMFGEN . From top to bottom, we show the variation with radius of the mass density (dots indicate the location of the hundred grid points in the CMFGEN simulation), the electron-scattering optical together with the regions of formation of H α , He ii 4685.7 Å, and N iii 4640.6 Å, the temperature together with the H, He, and N ionization (a value of one means a species once ionized), and the original, nonmonotonic velocity from the HERACLES simulation (dashed line) together with the fudged, homologous velocity adopted for the CMFGEN calculation (solid line). [See discussion in Section 2.] \n<!-- image --> \nwe adopt the Stokes parameters I , Q , U , and V (Chandrasekhar 1960). Since we are dealing with electron scattering, the polarization is linear and the V Stokes parameter is identically zero. For clarity, IQ and IU refer to the polarization of the specific intensity, and FQ and FU refer to the polarization of the observed flux. \nFor consistency with the earlier work of Hillier (1994, 1996) we choose a right-handed set of unit vectors ( ζ X , ζ Y , ζ W ). Without loss of generality the axisymmetric source is centered at the origin of the coordinate system with its symmetry axis lying along ζ W , ζ Y is in the plane of the sky, and the observer is located in the XW plane. \nWe take FQ to be positive when the polarization is parallel to the symmetry axis (or more correctly parallel to the projection of the symmetry axis on the sky), and negative when it is perpendicular to it. With our choice of coordinate system, and since the SN ejecta are left-right symmetric about the XW plane, FU is zero by construction. This must be the case since symmetry requires that the polarization can only be parallel, or perpendicular to, the axis of symmetry. For a spherical source, FQ is also identically zero. \nI ( ρ, δ ), IQ ( ρ, δ ) and IU ( ρ, δ ) refer to the observed intensities on the plane of the sky. IQ is positive when the polarization is parallel to the radius vector, and negative when it is perpendicular. In the plane of the sky we define a set of polar coordinates ( ρ, δ ) with the angle δ measured anti-clockwise from ζ Y . The polar coordinate, ρ , can also be thought of as the impact parameter of an observer's ray. We also use the axes defined by the po- \nFig. C.1. Same as Fig. 2 but now for model r1w6a and from 1.67 to 5.00 d. \n<!-- image --> \nate system to describe the polarization. FI is obtained from I ( ρ, δ ) using \nFI = 2 d 2 Z ρ max 0 Z π/ 2 -π/ 2 I ( ρ, δ ) dA , (B.1) \nwhere dA = ρ d δ d ρ . Since ζρ is rotated by an angle δ anticlockwise from ζ Y , FQ is given by \nFQ = -2 2 Z ρ max Z π/ 2 GLYPH<2> IQ ( ρ, δ ) cos 2 δ + IU ( ρ, δ ) sin 2 δ GLYPH<3> dA . \nd 0 -π/ 2 (B.2) \nIn a spherical system, IQ is independent of δ , and IU is identically zero. \nThe percentage polarization P λ is defined as 100 | FQ GLYPH<14> FI | , where we have dropped the λ subscript of the fluxes for clarity. \nIn this paper and for brevity, we report mostly on the maximum polarization obtained for a 90-deg inclination angle i . The variation with i has been discussed in previous studies and may deviate from a simple sin 2 i dependence if optical-depth e ff ects are present (Dessart et al. 2021a,b).", 'Appendix C: Additional models and illustrations': 'Fig. C.2. Same as Fig. 2 but now for model r1w6c \n<!-- image -->'}

2024ApJ...977...44I

The perturbations of the hyperbolic motion of a test particle due to the general relativistic gravitoelectromagnetic Schwarzschild and LenseThirring components of the gravitational field of a rotating massive body are analytically worked out to the first postNewtonian level in terms of the osculating Keplerian orbital elements. To the Newtonian order the impact of the quadrupole mass moment of the source is calculated as well. The resulting analytical expressions are valid for a generic orientation in space of both the orbital plane of the probe and the spin axis of the primary and for arbitrary values of the eccentricity. They are applied to Oumuamua an interstellar asteroid which recently visited our solar system along an unbound heliocentric orbit and to the Near Earth Asteroid Rendezvous spacecraft during its flyby of the Earth. The calculational approach developed can be straightforwardly extended to any alternative models of gravity as well.

2024-12-01T00:00:00Z

['10.3847/1538-4357/ad8dc6', 'arXiv:2409.12063', '10.48550/arXiv.2409.12063', '2024arXiv240912063I', '2024ApJ...977...44I']

['General relativity', 'Celestial mechanics', 'Planetary probes', '641', '211', '1252', 'General Relativity and Quantum Cosmology', 'Astrophysics - Earth and Planetary Astrophysics', 'Physics - Space Physics']

PostKeplerian Perturbations of the Hyperbolic Motion in the Field of a Rotating Massive Object Analysis in Terms of Osculating and Nonosculating Contact Elements

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']

https://arxiv.org/pdf/2409.12063.pdf

{'Post-Keplerian perturbations of the hyperbolic motion in the field of a rotating massive object. Analysis in terms of osculating and nonosculating (contact) elements': "Lorenzo Iorio 1 \n1 Ministero dell' Istruzione e del Merito. Viale Unit'a di Italia 68, I-70125, Bari (BA), Italy", 'Abstract': "The perturbations of the hyperbolic motion of a test particle due to the general relativistic gravitoelectromagnetic Schwarzschild and Lense-Thirring components of the gravitational field of a rotating massive body are analytically worked out to the first post-Newtonian level in terms of the osculating Keplerian orbital elements. To the Newtonian order, the impact of the quadrupole mass moment of the source is calculated as well. The resulting analytical expressions are valid for a generic orientation in space of both the orbital plane of the probe and the spin axis of the primary, and for arbitrary values of the eccentricity. They are applied to ' Oumuamua, an interstellar asteroid which recently visited our solar system along an unbound heliocentric orbit, and to the Near Earth Asteroid Rendezvous (NEAR) spacecraft during its flyby of the Earth. The calculational approach developed can be straightforwardly extended to any alternative models of gravity as well. \nKeywords: General relativity (641); Celestial mechanics (211); Planetary probes (1252)", '1. Introduction': "Let a localized gravitational source like, e.g., a planet, a natural satellite, a main sequence star or any astrophysical compact object endowed with mass M , equatorial radius R e, quadrupole mass moment J 2 and angular momentum J be considered. Let its external gravitational field be calculated in points far enough so that it is weak and the speeds of any moving test particles are small with respect to the speed of light in vacuum c . Then, in addition to the dominant Newtonian inverse-square mass monopole, also further post-Keplerian (pK) terms of both Newtonian and post-Newtonian (pN) origin come into play. The most relevant ones are the classical contribution of J 2 and, to the first post-newtonian (1pN) order, the so-called gravitoelectromagnetic Schwarzschild and Lense-Thirring (LT) components induced by M and J , respectively. \nUntil now, their orbital e GLYPH<11> ects have been studied mainly in the case of bound, otherwise Keplerian elliptical trajectories (Brumberg 1991; So GLYPH<11> el 1989; Kopeikin et al. 2011; Gurfil and Seidelmann 2016; So GLYPH<11> el and Han 2019; O'Leary 2021; Iorio 2024), used as tools to perform tests of gravitational theories. The most famous case is represented by the then anomalous perihelion precession of Mercury of 42 : 98 arcseconds per century (arcsec cty GLYPH<0> 1 ) (Nobili and Will 1986) in the field of the Sun, known since the second half of the nineteenth century (Le Verrier 1859b,a), and its successive explanation by Einstein (Einstein 1915) in terms of his newborn General Theory of Relativity (GTR). For a historical overview, see, e.g., Roseveare (1982). \nInstead, studies of pK perturbations of hyperbolic trajectories are comparatively much more rare, being mainly focussed on the e GLYPH<11> ects of the primary's oblateness for a particular orientation 1 of its spin axis ˆ J (Sauer 1963; Anderson and Giampieri 1999; Rappaport et al. 2001; Martinusi and Gurfil 2013; Kim and Park 2015). Other works investigated the hyperbolic motions of test particles and photons in the Schwarzschild spacetime at various levels of completeness (Morton 1921; Hagihara 1930; Leavitt 1939; Darwin 1959, 1961; Mielnik and Pleba'nski 1962; Davidson 1980; Hioe and Kuebel 2010; Chowdhuri et al. 2024). The case of the hyperbolic motion of a spinning particle in the Schwarzschild metric was treated by Bini and Geralico (2017), while Battista and Esposito (2022) dealt with geodesic motion in Euclidean Schwarzschild geometry. To the author's knowledge, the gravitomagnetic e GLYPH<11> ects of the rotation of the primary on hyperbolic trajectories have never been treated so far, apart from the study by Mummery and Balbus (2023) in the Kerr metric. \nlorenzo.iorio@libero.it \nFlybys of planets and natural satellites by artificial spacecraft traveling along patched hyperbolic conical sections are commonplace in current astrodynamics and planetary sciences (Flandro 1966; Anderson 1997; van Allen 2003; Anderson et al. 2007). Furthermore, they are often repeated several times within the same missions; su GLYPH<14> ce it to think about the grand tour of the Cassini probe in the Kronian system (Wolf and Smith 1995). Finally, also the Galactic Centre and the cluster of stars surrounding the supermassive black hole at Sgr A GLYPH<3> (Genzel et al. 2010) may be considered. Indeed, the star S111 is following a hyperbolic path (Trippe et al. 2008; Gillessen et al. 2009, 2017); other stars like that might be discovered in the future. Eventually, such kind of trajectories may represent, in principle, further opportunities to test gravitational theories in addition to the traditional bound ones. \nHere, in order to make closer contact with observations in actually accessible astronomical scenarios, a perturbative approach is followed. It allows to analytically calculate the variations experienced by all the usual Keplerian orbital elements of a hyperbolic trajectory perturbed by the aforementioned pK components of the gravitational field of the primary. In this respect, the present work follows a similar strategy as that adopted in Sauer (1963); Anderson and Giampieri (1999); Rappaport et al. (2001); Kim and Park (2015). Nonetheless, the e GLYPH<11> ects of the Newtonian quadrupole mass moment J 2 and of the 1pN gravitomagnetic LT field are worked out in full generality, without any a priori simplifying assumptions about the orientations of both ˆ J and the orbit in space. Furthermore, all the formulas obtained are valid for any values of the eccentricity. \nThe paper is organized as follows. In Section 2, the basics of the Keplerian hyperbolic motion is reviewed and the perturbative equations for the rates of change of the Keplerian orbital elements in the form of Lagrange are presented for such kind of unperturbed, reference trajectory. Furthermore, the way of calculating the disturbing function, to be used with the aforementioned equations, for the pK e GLYPH<11> ects considered is discussed as well. The 1pN gravitoelectric shifts induced solely by M are calculated in Section 3. The 1pN gravitomagnetic LT perturbations due to J is the subject of Section 4, while the impact of J 2 is worked out, to the Newtonian order, in Section 5; both e GLYPH<11> ects are calculated without any a priori assumptions on both ˆ J and the orientation of the orbital plane. Certain subtleties concerning the proper use of the Lagrange planetary equations in presence of velocitydependent disturbing functions are dealt with in Section 6. The results of the previous Sections are used for numerical calculation in Section 7 for two astronomical scenarios in our solar system: the interstellar asteroid ' Oumuamua and the Sun in Section 7.1, and the spacecraft Near Earth Asteroid Rendezvous (NEAR) approaching the Earth in Section 7.2. Section 8 summarizes the findings and o GLYPH<11> ers conclusions.", '2. Calculational overview': 'In the Keplerian hyperbolic motion (Rappaport et al. 2001; Roy 2005; Gurfil and Seidelmann 2016), a is the semimajor axis, e is the eccentricity, I is the inclination, GLYPH<10> is the longitude of the ascending node, ! is the argument of pericentre, and GLYPH<17> is the mean anomaly at epoch. The semimajor axis measures the distance between the vertex, namely the point Q of closest approach to the primary, and the centre O of the hyperbola; it is a < 0. For the eccentricity, which is at any time the constant ratio of the distance of the test particle at the point P( t ) on the hyperbola to the focus F where the primary resides to the distance of P( t ) itself to the directrix, it always holds e > 1; the larger it is, the straightest the hyperbola, while its asymptotes tend to get closer for e & 1. The inclination is the tilt of the orbital plane to the reference f x ; y g plane of the body-fixed reference frame adopted. The longitude of the ascending node is the angle, counted in the reference plane, from the reference x direction to the point N on the line of nodes crossed by the test particle from below; the line of nodes is the intersection between the orbital and the fundamental planes. The argument of pericentre is the angle, reckoned in the orbital plane, from N to Q. The mean anomaly at epoch is proportional to the time of closest approach t p; indeed, from the definition of the mean anomaly \nM ( t ) = n K GLYPH<16> t GLYPH<0> t p GLYPH<17> = n K t + GLYPH<17>; (1) \nit follows \nIn Equations (1)-(2), \nn K = r GLYPH<0> m a 3 (3) \nis the Keplerian mean motion which, of course, has not the same meaning as for the elliptic orbits. Moreover, \nm : = GM (4) \nis the standard gravitational parameter of the source of the gravitational field given by the product of its mass by the Newtonian gravitational constant G . Instead, I ; GLYPH<10> and ! determine the orientation of the orbit in space and of the orbit itself within its orbital plane also for the hyperbolic motion. \nGLYPH<17> : = GLYPH<0> n K t p : (2) \nwhere \np : = GLYPH<0> a GLYPH<16> e 2 GLYPH<0> 1 GLYPH<17> (14) \nis the semilatus rectum. The hyperbolic excess velocity is defined as (Rappaport et al. 2001) \nv 1 = GLYPH<0> n K a = r GLYPH<0> m a : (15) \nEquations (6)-(7) allow to express Equations (11)-(13) in terms of f . \nThe components of r and v can be referred to the primary-fixed reference frame by means of the rotation matrix (Montenbruck and Gill 2000) \nR ( GLYPH<10> ; I ; ! ) = R z ( GLYPH<0> GLYPH<10> ) R x ( GLYPH<0> I ) R z ( GLYPH<0> ! ) ; (16) \nwhere, for a generic angle GLYPH<30> , it is \nR ( GLYPH<0> GLYPH<30> ) z = 0 B B B B B B B B B B B @ cos GLYPH<30> GLYPH<0> sin GLYPH<30> 0 sin GLYPH<30> cos GLYPH<30> 0 0 0 1 1 C C C C C C C C C C C A ; (17) \nIn view of the forthcoming calculation, it is convenient to express the mean anomaly in terms of the hyperbolic eccentric anomaly H ( t ) as (Rappaport et al. 2001) \nM = e sinh H GLYPH<0> H : (5) \nFurthermore, it is (Rappaport et al. 2001) \nsinh H = sin f p e 2 GLYPH<0> 1 1 + e cos f ; (6) \ncosh H = e + cos f 1 + e cos f : (7) \nIn Equations (6)-(7), f ( t ) is the true anomaly, counted from Q to P( t ) in such a way that f = 0 at the pericentre and \nGLYPH<0> f 1 GLYPH<20> f GLYPH<20> f 1 ; (8) \nwhere \nf 1 = arccos GLYPH<0> 1 e ! : (9) \nFrom Equation (1) and by using Equations (5)-(7), one gets \nd t d f = GLYPH<16> e 2 GLYPH<0> 1 GLYPH<17> 3 = 2 n K (1 + e cos f ) 2 : (10) \nThe instantaneous distance of the test particle from the primary can be expressed as (Rappaport et al. 2001) \nr = a (1 GLYPH<0> e cos H ) : (11) \nThe position and velocity vectors, referred to the orbital plane 2 f X ; Y g , are (Rappaport et al. 2001) \nr = n a (cos H GLYPH<0> e ) ; p GLYPH<0> ap sin H ; 0 o ; (12) \nv = ( p GLYPH<0> m a r sin H ; p m p r cos H ; 0 ) ; (13) \nR ( GLYPH<0> GLYPH<30> ) x = 0 B B B B B B B B B B B @ 1 0 0 0 cos GLYPH<30> GLYPH<0> sin GLYPH<30> 0 sin GLYPH<30> cos GLYPH<30> 1 C C C C C C C C C C C A : (18) \nEquation (16) allows to determine the orientation of the orbit in space and of the orbit itself within the orbital plane in full generality. \nFor calculational purposes, it is convenient to introduce the following mutually orthogonal unit vectors (So GLYPH<11> el 1989; Brumberg 1991; So GLYPH<11> el and Han 2019) \nˆ l : = f cos GLYPH<10> ; sin GLYPH<10> ; 0 g ; (19) \nˆ m : = fGLYPH<0> cos I sin GLYPH<10> ; cos I cos GLYPH<10> ; sin I g ; (20) \nˆ h : = f sin I sin GLYPH<10> ; GLYPH<0> sin I cos GLYPH<10> ; cos I g ; (21) \n(22) \nˆ l is directed along the line of nodes towards the ascending node, ˆ h is perpendicular to the orbital plane, being aligned with the orbital angular momentum, and ˆ m lies in the orbital plane so that \nˆ l GLYPH<2> ˆ m = ˆ h (23) \nholds. \nThe planetary equations in the form of Lagrange which allow to calculate the perturbations of the Keplerian orbital elements in the case of the hyperbolic motion are (Rappaport et al. 2001) \nd a d t = GLYPH<0> 2 n K a @ R @GLYPH<17> ; (24) \nd e d t = p e 2 GLYPH<0> 1 n K a 2 e @ R @! + GLYPH<16> e 2 GLYPH<0> 1 GLYPH<17> n K a 2 e @ R @GLYPH<17> ; (25) \nd I d t = GLYPH<0> 1 n K a 2 p e 2 GLYPH<0> 1 sin I @ R @ GLYPH<10> + cos I n K a 2 p e 2 GLYPH<0> 1 sin I @ R @! ; (26) \nd GLYPH<10> d t = 1 n K a 2 p e 2 GLYPH<0> 1 sin I @ R @ I ; (27) \nd ! d t = GLYPH<0> p e 2 GLYPH<0> 1 n K a 2 e @ R @ e GLYPH<0> cos I n K a 2 p e 2 GLYPH<0> 1 sin I @ R @ I ; (28) \nd GLYPH<17> d t = 2 n K a @ R @ a GLYPH<0> GLYPH<16> e 2 GLYPH<0> 1 GLYPH<17> n K a 2 e @ R @ e ; (29) \nwhere R is the disturbing function. R is given by the pK part L pK of the Lagrangian per unit mass L of the test particle which can be obtained from the spacetime metric tensor g GLYPH<22>GLYPH<23> ; GLYPH<22>; GLYPH<23> = 0 ; 1 ; 2 ; 3 as follows. \nWritten in spatially isotropic or harmonic coordinates, the latter can be expressed, to the pN order, as \ng 00 \' 1 + h 00 = 1 + 2 U ( r ) c 2 + 2 U 2 ( r ) c 4 + O GLYPH<16> 1 = c 6 GLYPH<17> ; (30) \ng 0 i \' h 0 i = O GLYPH<16> 1 = c 3 GLYPH<17> ; i = 1 ; 2 ; 3 ; (31) \ngij \' GLYPH<0> 1 + hij = GLYPH<0> " 1 GLYPH<0> 2 U ( r ) c 2 # GLYPH<14> i j + O GLYPH<16> 1 = c 4 GLYPH<17> ; i ; j = 1 ; 2 ; 3 : (32) \nIn Equations (30)-(32), the coe GLYPH<14> cients h GLYPH<22>GLYPH<23> ; GLYPH<22>; GLYPH<23> = 0 ; 1 ; 2 ; 3 are the pN corrections to the constant components GLYPH<17>GLYPH<22>GLYPH<23> ; GLYPH<22>; GLYPH<23> = 0 ; 1 ; 2 ; 3 of the \'flat\' Minkowskian spacetime metric tensor, \nGLYPH<14> i j : = 8 > > < > > : 1 for i = j 0 for i , j ; i ; j = 1 ; 2 ; 3 ; (33) \nis the 3-dimensional Kronecker delta (Olver et al. 2010), and U ( r ) is the Newtonian potential of the source including also J 2 \nU ( r ) = GLYPH<0> m r " 1 GLYPH<0> J 2 GLYPH<18> R e r GLYPH<19> 2 P 2 GLYPH<16> ˆ J GLYPH<1> ˆ r GLYPH<17> # ; (34) \nin which \nP 2 ( GLYPH<24> ) = 3 GLYPH<24> 2 GLYPH<0> 1 2 (35) \nis the Legendre polynomial of degree \' = 2 in the generic dimensionless argument GLYPH<24> . Furthermore, \nh 0 i = 2 GJ GLYPH<15> i jk ˆ J j x k c 3 r 3 ; i = 1 ; 2 ; 3 (36) \nwhere \nGLYPH<15> i jk : == 8 > > > > > < > > > > > : + 1 if ( i ; j ; k ) is (1 ; 2 ; 3), (2 ; 3 ; 1), or (3 ; 1 ; 2) GLYPH<0> 1 if ( i ; j ; k ) is (3 ; 2 ; 1), (1 ; 3 ; 2), or (2 ; 1 ; 3) 0 if i = j , or j = k , or k = i (37) \nis the 3-dimensional Levi-Civita symbol (Olver et al. 2010), are the components of the gravitomagnetic LT potential. In Equation (36), ˆ J i ; i = 1 ; 2 ; 3 are the components of the spin unit vector ˆ J , and x k ; k = 1 ; 2 ; 3 are the Cartesian coordinates x ; y ; z of the test particle. In Equation (36), the Einstein summation convention (Olver et al. 2010) is applied to the dummy summation indexes j and k . \nTo the 1pN order, the Lagrangian per unit mass turns out to be (Brumberg 1991, p. 56, Equation (2.2.53)) \nL = L N + L 1pN ; (38) \nwhere 3 \nL N = 1 2 v 2 GLYPH<0> 1 2 c 2 h ( 1 = c 2 ) 00 ; (39) \nL 1pN = GLYPH<0> 1 2 c 2 h ( 1 = c 4 ) 00 + v 4 8 c 2 GLYPH<0> 1 4 h 00 v 2 + c 2 8 h 2 00 GLYPH<0> 1 2 hijv i v j GLYPH<0> ch 0 jv j ; (40) \nwhere h GLYPH<11>GLYPH<12> ; GLYPH<11>; GLYPH<12> = 0 ; 1 ; 2 ; 3 are given by Equations (30)-(32); h ( 1 = c 2 ) 00 and h ( 1 = c 4 ) 00 denote the 1pN and second post-Newtonian (2pN) parts of h 00, respectively; both of them are needed to keep the Lagrangian to the 1pN level. To this aim, it is meant that only h ( 1 = c 2 ) 00 enters the third and fourth terms of Equation (40). \nStrictly speaking, the Lagrange equations in the form of Equations (24)-(29) return either the osculating elements if R depends only on the position r of the test particle, or the nonosculating, contact elements, to the first 4 order in the perturbation, if R does depend also on the velocity v (Brumberg 1991; Kopeikin et al. 2011). The consequences of this fact, often overlooked, will be treated in detail in Section 6. \nwhere \nmin \nGLYPH<0> f 1 < f min < 0 ; (44) \n0 < f max < f 1 ; (45) \none gets \nGLYPH<1> a GE ( f min ; f max) = 0 ; (46) \nGLYPH<1> e GE ( f min ; f max) = 0 ; (47) \nGLYPH<1> \nI \n( \nf \n; \nf \nmax) \n= \n0 \n; \n(48) \nGLYPH<1>GLYPH<10> GE ( f min ; f max) = 0 ; (49) \nGLYPH<1> ! GE ( f min ; f max) = 2 m c 2 ae 2 0 B B B B B B B @ GLYPH<0> 3 GLYPH<16> 1 + e 2 GLYPH<17> GLYPH<1> f e 2 GLYPH<0> 1 GLYPH<0> 2 GLYPH<16> 3 + e 2 GLYPH<17> p e 2 GLYPH<0> 1 8 > > > < > > > : arctanh 2 6 6 6 6 6 6 4 ( e GLYPH<0> 1) tan GLYPH<16> f max 2 GLYPH<17> p e 2 GLYPH<0> 1 3 7 7 7 7 7 7 5 GLYPH<0> arctanh 2 6 6 6 6 6 6 4 ( e GLYPH<0> 1) tan GLYPH<16> f min 2 GLYPH<17> p e 2 GLYPH<0> 1 3 7 7 7 7 7 7 5 9 > > > = > > > ; \nGLYPH<0> e sin f max 1 + e cos f max + e sin f min 1 + e cos f min ! ; (50) \nGLYPH<1> GLYPH<17> GE ( f min ; f max) = m 2 c 2 ae 2 0 B B B B B B B @ 2 GLYPH<16> 12 GLYPH<0> 11 e 2 GLYPH<17> 8 > > > < > > > : arctanh 2 6 6 6 6 6 6 4 ( e GLYPH<0> 1) tan GLYPH<16> f max 2 GLYPH<17> p e 2 GLYPH<0> 1 3 7 7 7 7 7 7 5 GLYPH<0> arctanh 2 6 6 6 6 6 6 4 ( e GLYPH<0> 1) tan GLYPH<16> f min 2 GLYPH<17> p e 2 GLYPH<0> 1 3 7 7 7 7 7 7 5 9 > > > = > > > ; \nGLYPH<0> p e 2 GLYPH<0> 1 ( 12 GLYPH<1> f + e GLYPH<16> e 2 GLYPH<0> 4 GLYPH<17> " sin f max 1 + e cos f max GLYPH<0> sin f min 1 + e cos f min #)! ; (51) \nGLYPH<1> f : = f max GLYPH<0> f min : (52) \nGE \nThe orbital shifts experienced by the test particle during the flyby can be explicitly worked out by integrating the right hand sides of Equations (24)-(29), calculated onto the unperturbed Keplerian hyperbola, by means of Equations (6)-(7) and Equations (10)(14) from f min to f max. As it will be shown, while for the 1pN LT and the Newtonian J 2 terms one can analytically work out shifts covering the whole motion by assuming \nj f min j = f max = f 1 ; (41) \nit is not possible for the 1pN gravitoelectric perturbations since they diverge when calculated with Equation (41); however, analytical expressions valid for restricted ranges of values of f including the passage at the pericentre can be obtained.', '3. The 1pN gravitoelectric shifts': 'The 1pN gravitoelectric disturbing function, due solely to M , can be extracted from Equation (40) by neglecting the last o GLYPH<11> -diagonal term and using Equation (30) and Equation (32) calculated for J 2 ! 0. It turns out to be \nR GE = r 2 v 4 + 12 m rv 2 GLYPH<0> 4 m 2 8 c 2 r 2 : (42) \nBy inserting Equation (42) in Equations (24)-(29) and integrating their right hand sides by means of Equation (10) within the range \nf min GLYPH<20> f GLYPH<20> f max ; (43) \nwith \nand \nAs anticipated in Section 2, Equations (50)-(51) turn out to be singular for \nf min = GLYPH<0> f 1 ; (53) \nf max = f 1 : (54) \nIt may happen that observations are collected during a larger time interval before the passage at the point of closest approach than after it, or vice versa; thus, the condition \nj f min j , j f max j (55) \nshould be generally allowed. If, instead, data are taken during identical finite time spans before and after the flyby, it is \nf min = GLYPH<0> f max ; (56) \nso that Equations (50)-(51) become \nGLYPH<1> ! GE ( f max) = GLYPH<0> 4 m c 2 ae 2 8 > > > > > > < > > > > > > : 3 GLYPH<16> 1 + e 2 GLYPH<17> f max e 2 GLYPH<0> 1 + 2 GLYPH<16> 3 + e 2 GLYPH<17> arctanh " ( e GLYPH<0> 1) tan GLYPH<16> f max 2 GLYPH<17> p e 2 GLYPH<0> 1 # p e 2 GLYPH<0> 1 + e sin f max 1 + e cos f max 9 > > > > > > = > > > > > > ; ; (57) \nGLYPH<1> GLYPH<17> GE ( f max) = m c 2 ae 2 8 > > > < > > > : 2 GLYPH<16> 12 GLYPH<0> 11 e 2 GLYPH<17> arctanh 2 6 6 6 6 6 6 4 ( e GLYPH<0> 1) tan GLYPH<16> f max 2 GLYPH<17> p e 2 GLYPH<0> 1 3 7 7 7 7 7 7 5 GLYPH<0> p e 2 GLYPH<0> 1 2 6 6 6 6 6 6 4 12 f max + e GLYPH<16> e 2 GLYPH<0> 4 GLYPH<17> sin f max 1 + e cos f max 3 7 7 7 7 7 7 5 9 > > > = > > > ; (58) \nExpressions valid for short time intervals symmetric with respect to the flyby can be obtained by expanding Equations (57)-(58) in powers of f max, assumed close to zero; indeed, f = 0 corresponds just to the passage at pericentre. Thus, one obtains \nGLYPH<1> ! GE p ( f max) \' GLYPH<0> 4 m (2 + e ) c 2 ae ( e GLYPH<0> 1) f max + O GLYPH<16> f 2 max GLYPH<17> ; (59) \nGLYPH<1> GLYPH<17> GE p ( f max) \' m GLYPH<16> 8 + 3 e GLYPH<0> 10 e 2 GLYPH<0> e 3 GLYPH<17> c 2 ae p e 2 GLYPH<0> 1 f max + O GLYPH<16> f 2 max GLYPH<17> : (60) \nFrom Equation (15) and Equation (46), it can be straightforwardly inferred that v 1 is not changed by the 1pN gravitoelectric acceleration.', '4. The 1pN gravitomagnetic Lense-Thirring shifts': "The 1pN gravitomagnetic disturbing function, arising from the last term in Equation (40) calculated with Equation (36), turns out to be \nR LT = GLYPH<0> 2 GJ c 2 r 3 GLYPH<16> ˆ J GLYPH<2> r GLYPH<17> GLYPH<1> v : (61) \nIntegrating Equations (24)-(29), calculated with Equation (61), by means of Equations (8)-(10) finally yields \nGLYPH<1> a LT 1 = 0 ; (62) \nGLYPH<1> e LT 1 = 0 ; (63) \nGLYPH<1> I LT 1 = GLYPH<0> 4 GJ h arcsec ( GLYPH<0> e ) + p e 2 GLYPH<0> 1 i Jl c 2 n K a 3 GLYPH<0> e 2 GLYPH<0> 1 GLYPH<1> 3 = 2 ; (64) \nwhere \nGLYPH<1>GLYPH<10> LT 1 = GLYPH<0> 4 GJ csc I h arcsec ( GLYPH<0> e ) + p e 2 GLYPH<0> 1 i Jm c 2 a 3 n K GLYPH<0> e 2 GLYPH<0> 1 GLYPH<1> 3 = 2 ; (65) \nGLYPH<1> ! LT 1 = 4 GJ n e 2 cot I h arcsec ( GLYPH<0> e ) + p e 2 GLYPH<0> 1 i Jm + h 5 e 2 arcsec ( GLYPH<0> e ) + GLYPH<16> 3 + 2 e 2 GLYPH<17> p e 2 GLYPH<0> 1 i Jh o c 2 a 3 n K e 2 GLYPH<0> e 2 GLYPH<0> 1 GLYPH<1> 3 = 2 ; (66) \nGLYPH<1> GLYPH<17> LT 1 = GLYPH<0> 12 GJ p e 2 GLYPH<0> 1 Jh c 2 a 3 n K e 2 ; (67) \nJl : = ˆ J GLYPH<1> ˆ l = ˆ Jx cos GLYPH<10> + ˆ Jy sin GLYPH<10> ; (68) \nJm : = ˆ J GLYPH<1> ˆ m = cos I GLYPH<16> GLYPH<0> ˆ Jx sin GLYPH<10> + ˆ Jy cos GLYPH<10> GLYPH<17> + ˆ Jz sin I ; (69) \nJl : = ˆ J GLYPH<1> ˆ h = sin I GLYPH<16> ˆ Jx sin GLYPH<10> GLYPH<0> ˆ Jy cos GLYPH<10> GLYPH<17> + ˆ Jz cos I : (70) \nEquations (62)-(67), which cover the full motion of the test particle, retain a general validity since they hold for arbitrary orientations in space of both the orbit and the primary's spin axis. \nFrom Equations (62)-(67) and Equations (68)-(70) it turns out that the inclination and the node stay constant for equatorial orbits, characterized by \nJh = GLYPH<6> 1 ; (71) \nJl = Jm = 0 ; (72) \nwhile the pericentre and the mean anomaly at epoch undergo nonvanishing net shifts. Instead, for polar orbits ( Jh = 0), the inclination, the node and the pericentre are, in general, shifted. \nFrom Equation (15) and Equation (62), it can be straightforwardly inferred that v 1 is not changed by the LT acceleration.", '5. The Newtonian J 2 shifts': "The Newtonian disturbing function due to the primary's oblateness, obtained from the J 2-driven pK component of Equation (39) calculated with Equation (34), turns out to be \nR J 2 = m J 2 R 2 e GLYPH<20> 1 GLYPH<0> 3 GLYPH<16> ˆ J GLYPH<1> ˆ r GLYPH<17> 2 GLYPH<21> 2 r 3 : (73) \nThe resulting orbital shifts, integrated according to Equations (24)-(29) and Equations (8)-(10), are \nGLYPH<1> a J 2 1 = 0 ; (74) \nGLYPH<1> e J 2 1 = J 2 R 2 e p e 2 GLYPH<0> 1 a 2 e 3 6 X i = 1 E J 2 1 ; i b Ti ; (75) \nGLYPH<1> I J 2 1 = J 2 R 2 e a 2 e 2 GLYPH<0> e 2 GLYPH<0> 1 GLYPH<1> 2 6 X i = 1 I J 2 1 ; i b Ti ; (76) \nwhere \nand \n; \nGLYPH<1>GLYPH<10> J 2 1 = J 2 R 2 e csc I a 2 e 2 GLYPH<0> e 2 GLYPH<0> 1 GLYPH<1> 2 6 X i = 1 N J 2 1 ; i b Ti ; (77) \nGLYPH<1> ! J 2 1 = J 2 R 2 e 2 a 2 e 4 GLYPH<0> e 2 GLYPH<0> 1 GLYPH<1> 2 6 X i = 1 G J 2 1 ; i b Ti ; (78) \nGLYPH<1> GLYPH<17> J 2 1 = 3 J 2 R 2 e 2 a 2 e 4 6 X i = 1 H J 2 1 ; i b Ti ; (79) \nb T 1 : = 1 ; (80) \nb T 2 : = Jl 2 + Jm 2 ; (81) \nb T 3 : = Jl 2 GLYPH<0> Jm 2 ; (82) \nb T 4 : = JhJl ; (83) \nb T 5 : = JhJm ; (84) \nb T 6 : = JlJm ; (85) \nE J 2 1 ; 1 : = 0 ; (86) \nE J 2 1 ; 2 : = 0 ; (87) \nE \nJ \n2 \n1 \n3 \nE J 2 1 ; 4 : = 0 ; (89) \nE J 2 1 ; 5 : = 0 ; (90) \nE J 2 1 ; 6 : = GLYPH<0> 2 E J 2 1 ; 3 cot 2 !; (91) \nI J 2 1 ; 1 : = 0 ; (92) \nI J 2 1 ; 2 : = 0 ; (93) \nI J 2 1 ; 3 : = 0 ; (94) \n: \n= \nsin 2 \n(88) \n!; \n1 \n2 \nI J 2 1 ; 4 : = GLYPH<0> 3 e 2 arcsec ( GLYPH<0> e ) + p e 2 GLYPH<0> 1 h GLYPH<0> 3 e 2 GLYPH<0> GLYPH<16> e 2 GLYPH<0> 1 GLYPH<17> cos 2 ! i ; (95) \nI J 2 1 ; 5 : = GLYPH<0> GLYPH<16> e 2 GLYPH<0> 1 GLYPH<17> 3 = 2 sin 2 !; (96) \nI J 2 1 ; 6 : = 0 ; (97) \nN \nJ \n2 \n1 \nN \nJ \n; \n2 \n1 \n; \nN J 2 1 ; 3 : = 0 ; (100) \nN \nJ \n2 \n1 \n; \n4 \nJ \n2 \n1 \n; \n5 \nN J 2 1 ; 5 : = GLYPH<0> 3 e 2 arcsec ( GLYPH<0> e ) + p e 2 GLYPH<0> 1 h GLYPH<0> 3 e 2 + GLYPH<16> e 2 GLYPH<0> 1 GLYPH<17> cos 2 ! i ; (102) \nN J 2 1 ; 6 : = 0 ; (103) \nG J 2 1 ; 1 : = 6 e 2 h p e 2 GLYPH<0> 1 GLYPH<16> 1 + e 2 GLYPH<17> + 2 e 2 arcsec ( GLYPH<0> e ) i ; (104) \nG J 2 1 ; 2 : = GLYPH<0> 3 2 G J 2 1 ; 1 ; (105) \nG J 2 1 ; 3 : = GLYPH<0> 3 p e 2 GLYPH<0> 1 GLYPH<16> 2 GLYPH<0> 3 e 2 + e 4 GLYPH<17> cos 2 !; (106) \nG J 2 1 ; 4 : = 2 e 2 GLYPH<16> e 2 GLYPH<0> 1 GLYPH<17> 3 = 2 cot I sin 2 !; (107) \nG J 2 1 ; 5 : = 2 e 2 n 3 e 2 arcsec ( GLYPH<0> e ) + p e 2 GLYPH<0> 1 h 3 e 2 GLYPH<0> GLYPH<16> e 2 GLYPH<0> 1 GLYPH<17> cos 2 ! io cot I ; (108) \nG J 2 1 ; 6 : = GLYPH<0> 6 p e 2 GLYPH<0> 1 GLYPH<16> 2 GLYPH<0> 3 e 2 + e 4 GLYPH<17> sin 2 !; (109) \nH J 2 1 ; 1 : = GLYPH<0> 2 e 2 ; (110) \nH J 2 1 ; 2 : = GLYPH<0> 3 2 H J 2 1 ; 1 ; (111) \nH J 2 1 ; 3 : = GLYPH<16> 2 + e 2 GLYPH<17> cos 2 !; (112) \n: \n= \nI \n(101) \n: \n= \n0 \n: \n= \n0 \n; \n; \n(98) \n(99) \n; \nH J 2 1 ; 4 : = 0 ; (113) \nH J 2 1 ; 5 : = 0 ; (114) \nH J 2 1 ; 6 : = 2 H J 2 1 ; 3 tan 2 !: (115) \nAlso Equations (74)-(79), covering the whole motion, are valid for any spatial orientations of the primary's spin axis and the orbital plane. \nFrom Equations (74)-(115) and Equations (68)-(70) it turns out that, for equatorial orbits, the eccentricity, the inclination and the node stay constant, while the pericentre and the mean anomaly at epoch do generally vary. Instead, for polar orbits, only the inclination and the node remain una GLYPH<11> ected. \nFrom Equation (15) and Equation (74), it can be straightforwardly inferred that v 1 is not changed by the primary's oblateness.", '6. A subtlety about the results obtained: choosing the gauge of the Lagrange planetary equations': 'If the disturbing function depends also on the velocity v , as in the case of the general relativistic Equation (42) and Equation (61), the Lagrange planetary equations as given by Equations (24)-(29) actually provide, to the first order in the perturbation, the instantaneous rates of change of the so-called contact Keplerian orbital elements. They are not osculating, and parameterize confocal instantaneous conics which may be generally neither tangent nor coplanar to the actual trajectory. Thus, the correct position of the test particle is returned at each instant of time, but not its velocity since v , v K. Non-osculating Keplerian orbital elements were used also for studying the impact of, e.g., J 2 (Gurfil 2004) and the 1pN gravitoelectric field (Gurfil and Efroimsky 2022) on bound, quasi-elliptical orbits. For a thorough analysis of the subtle technicalities involved, see Efroimsky and Goldreich (2004); Efroimsky (2005b,a); Kopeikin et al. (2011). The physical and geometrical meaning of the contact elements is not as straightforward as for the osculating ones, and wrong interpretations 5 of (mathematically correct) results actually obtained in terms of nonosculating orbital elements can be found in the literature, as pointed out in Efroimsky and Goldreich (2004). Thus, it may be preferable to have expressions for the orbital shifts written in terms of the osculating Keplerian elements also for velocity-dependent disturbing functions. Indeed, in general, it is not guaranteed that, in the case of a hyperbolic motion, any variations of the osculating and contact elements coincide when they are integrated over some range for f . \nIt turns out that the di GLYPH<11> erence between the instantaneous values of the contact elements C ct i and the osculating ones C os i is given by (Kopeikin et al. 2011, p. 74, Equation (1.323)) \nC ct i ( f ) GLYPH<0> C os i ( f ) = GLYPH<0> 6 X j = 1 n C os i ; C os j o @ r @ C os j GLYPH<1> @ R @ v : = Z i ( f ) ; i = a ; e ; I ; GLYPH<10> ; !; GLYPH<17>; (116) \nwhere n C os i ; C os j o are the Poisson brackets 6 \nA straightforward calculation yields \nfor the ith and jth elements. \nZ a ( f ) = Z x a @ R @ vx + Z y a @ R @ vy + Z z a @ R @ vz ; (117) \nZ e ( f ) = Z x e @ R @ vx + Z y e @ R @ vy + Z z e @ R @ vz ; (118) \nZ I ( f ) = Z x I @ R @ vx + Z y I @ R @ vy + Z z I @ R @ vz ; (119) \nZ GLYPH<10> ( f ) = Z x GLYPH<10> @ R @ vx + Z y GLYPH<10> @ R @ vy + Z z GLYPH<10> @ R @ vz ; (120) \nwhere \n; \nL. Iorio \nZ ! ( f ) = Z x ! @ R @ vx + Z y ! @ R @ vy + Z z ! @ R @ vz ; (121) \nZ GLYPH<17> ( f ) = Z x GLYPH<17> @ R @ vx + Z y GLYPH<17> @ R @ vy + Z z GLYPH<17> @ R @ vz ; (122) \n(123) \nZ x a ( f ) = 0 ; (124) \nZ y a ( f ) = 0 ; (125) \nz \na \nZ \n( \nf \n) \n= \n0 \n(126) \nZ x e ( f ) = GLYPH<16> e 2 GLYPH<0> 1 GLYPH<17> 3 = 2 (cos GLYPH<10> sin u + cos I sin GLYPH<10> cos u ) aen K (1 + e cos f ) ; (127) \nZ y e ( f ) = GLYPH<16> e 2 GLYPH<0> 1 GLYPH<17> 3 = 2 ( GLYPH<0> cos I cos GLYPH<10> cos u + sin GLYPH<10> sin u ) aen K (1 + e cos f ) ; (128) \nZ z e ( f ) = GLYPH<0> GLYPH<16> e 2 GLYPH<0> 1 GLYPH<17> 3 = 2 sin I cos u aen K (1 + e cos f ) ; (129) \nZ x I ( f ) = GLYPH<0> p e 2 GLYPH<0> 1 sin I sin GLYPH<10> cos u an K (1 + e cos f ) ; (130) \nZ y I ( f ) = p e 2 GLYPH<0> 1 sin I cos GLYPH<10> cos u an K (1 + e cos f ) ; (131) \nZ z I ( f ) = GLYPH<0> p e 2 GLYPH<0> 1 cos I cos u an K (1 + e cos f ) ; (132) \nZ x GLYPH<10> ( f ) = GLYPH<0> p e 2 GLYPH<0> 1 sin GLYPH<10> sin u an K (1 + e cos f ) ; (133) \nZ y GLYPH<10> ( f ) = p e 2 GLYPH<0> 1 cos GLYPH<10> sin u an K (1 + e cos f ) ; (134) \nZ z GLYPH<10> ( f ) = GLYPH<0> p e 2 GLYPH<0> 1 cot I sin u an K (1 + e cos f ) ; (135) \nZ x ! ( f ) = p e 2 GLYPH<0> 1 n cos GLYPH<10> h 2 e + GLYPH<16> 1 + e 2 GLYPH<17> cos f i cos u GLYPH<0> cos I sin GLYPH<10> ( e + cos f ) sin u o aen K (1 + e cos f ) ; (136) \nZ y ! ( f ) = p e 2 GLYPH<0> 1 n cos I cos GLYPH<10> ( e + cos f ) sin u + sin GLYPH<10> h 2 e + GLYPH<16> 1 + e 2 GLYPH<17> cos f i cos u o aen K (1 + e cos f ) ; (137) \nZ z ! ( f ) = p e 2 GLYPH<0> 1 csc I h 3 e + cos f + 2 e 2 cos f GLYPH<0> cos 2 I ( e + cos f ) i sin u aen K (1 + e cos f ) ; (138) \nZ x GLYPH<17> ( f ) = GLYPH<0> GLYPH<16> e 2 GLYPH<0> 1 GLYPH<17> 2 (cos GLYPH<10> cos u GLYPH<0> cos I sin GLYPH<10> sin u ) cos f aen K (1 + e cos f ) ; (139) \nZ y GLYPH<17> ( f ) = GLYPH<0> GLYPH<16> e 2 GLYPH<0> 1 GLYPH<17> 2 (cos I cos GLYPH<10> sin u + sin GLYPH<10> cos u ) cos f aen K (1 + e cos f ) ; (140) \nZ z GLYPH<17> ( f ) = GLYPH<0> GLYPH<16> e 2 GLYPH<0> 1 GLYPH<17> 2 sin I cos f sin u aen K (1 + e cos f ) : (141) \nIn Equations (124)-(141), \nu : = ! + f (142) \nis the argument of latitude. In general, also the components of the gradient of R with respect to v are time-dependent through f \n. \nThe next step is calculating the di GLYPH<11> erence of the integrated shifts of the contact and osculating orbital elements. To this aim, by taking the time derivative of both members of Equation (116), one gets \nd C ct i ( f ) d t GLYPH<0> d C os i ( f ) d t = @ Z i ( f ) @ f d f d t ; i = a ; e ; I ; GLYPH<10> ; !; GLYPH<17>: (143) \nThe analytical expressions for @ Z i ( f ) =@ f ; i = a ; e ; I ; GLYPH<10> ; !; GLYPH<17> , obtainable straightforwardly from Equations (117)-(141), are too cumbersome to be explicitly displayed. Integrating both members of Equation (143) from f min to f max finally yields \nGLYPH<1> C os i ( f min ; f max) = GLYPH<1> C ct i ( f min ; f max) + GLYPH<4> i ( f min ; f max) ; i = a ; e ; I ; GLYPH<10> ; !; GLYPH<17>; (144) \nwhere \nGLYPH<4> i ( f min ; f max) : = GLYPH<0> Z f max f min @ Z i ( f ) @ f d f ; i = a ; e ; I ; GLYPH<10> ; !; GLYPH<17>: (145) \nIn the case of the general relativistic disturbing functions of Equation (42) and Equation (61), the integrated shifts of the contact elements GLYPH<1> C ct i ( f min ; f max) ; i = a ; e ; I ; GLYPH<10> ; !; GLYPH<17> are given, to the 1pN order, by Equations (46)-(51) and Equations (62)-(67), respectively. \nFrom the above considerations, it turns out that the oblateness-driven classical orbital shifts of Equations (74)-(79) are to be intended as written in terms of the osculating elements. Indeed, the disturbing function of Equation (73) depends only on the position vector r . In this case, the contact and the osculating elements coincide, as per Equation (116). \nFor elliptical motions perturbed by Equation (42) and Equation (61), it can be shown that GLYPH<4> i (0 ; 2 p ) = 0 ; i = a ; e ; I ; GLYPH<10> ; !; GLYPH<17>; when Equation (145) is integrated from 0 to 2 p .', '6.1. The 1pN gravitoelectric corrections to the shifts of the contact Keplerian orbital elements': "The corrections GLYPH<4> GE i ( f min ; f max) ; i = a ; e ; I ; GLYPH<10> ; !; GLYPH<17> to the 1pN gravitoelectric integrated shifts of the contact Keplerian orbital elements are obtained in the following way. \nThe calculation of the gradient of Equation (42) with respect to v returns \n@ R GE @ vx = m n K GLYPH<16> 7 + e 2 + 8 e cos f GLYPH<17> [cos GLYPH<10> ( e sin ! + sin u ) + cos I sin GLYPH<10> ( e cos ! + cos u )] 2 c 2 GLYPH<0> e 2 GLYPH<0> 1 GLYPH<1> 3 = 2 ; (146) \n@ R GE @ vy = GLYPH<0> m n K GLYPH<16> 7 + e 2 + 8 e cos f GLYPH<17> [cos I cos GLYPH<10> ( e cos ! + cos u ) GLYPH<0> sin GLYPH<10> ( e sin ! + sin u )] 2 c 2 GLYPH<0> e 2 GLYPH<0> 1 GLYPH<1> 3 = 2 ; (147) \n@ R GE @ vz = GLYPH<0> m n K sin I GLYPH<16> 7 + e 2 + 8 e cos f GLYPH<17> ( e cos ! + cos u ) 2 c 2 GLYPH<0> e 2 GLYPH<0> 1 GLYPH<1> 3 = 2 : (148) \nThe functions Z GE i ( f ) ; i = a ; e ; I ; GLYPH<10> ; !; GLYPH<17> entering Equation (116) can be calculated by inserting Equations (146)-(148) in Equations (124)-(141). One gets \nZ GE a ( f ) = 0 ; (149) \nZ GE e ( f ) = GLYPH<0> m GLYPH<16> 7 + e 2 + 8 e cos f GLYPH<17> 2 c 2 ae ; (150) \nZ GE I ( f ) = 0 ; (151) \nZ GE GLYPH<10> ( f ) = 0 ; (152) \nZ GE ! ( f ) = m h 6 e GLYPH<16> 3 + e 2 GLYPH<17> + GLYPH<16> 7 + 24 e 2 + e 4 GLYPH<17> cos f + 4 e GLYPH<16> 1 + e 2 GLYPH<17> cos 2 f i sin f 2 c 2 a GLYPH<0> e 2 GLYPH<0> 1 GLYPH<1> (1 + e cos f ) 2 ; (153) \nZ GE GLYPH<17> ( f ) = GLYPH<0> m p e 2 GLYPH<0> 1 GLYPH<16> 7 + e 2 + 8 e cos f GLYPH<17> sin 2 f 4 c 2 a (1 + e cos f ) 2 : (154) \nAccording to Equations (149)-(154), the instantaneous osculating eccentricity, argument of pericenter and mean anomaly at epoch are generally di GLYPH<11> erent from their contact counterparts, while the osculating and contact semimajor axis, inclination and longitude of the ascending node always coincide. \nBy plugging Equations (149)-(154) in Equation (145), the explicit expressions of GLYPH<4> GE i ( f min ; f max) ; i = a ; e ; I ; GLYPH<10> ; !; GLYPH<17> can be obtained. They are \nGLYPH<4> GE a ( f min ; f max) = 0 ; (155) \nGLYPH<4> GE e ( f min ; f max) = 4 m (cos f min GLYPH<0> cos f max) c 2 a ; (156) \nGLYPH<4> GE I ( f min ; f max) = 0 ; (157) \nGLYPH<4> GE GLYPH<10> ( f min ; f max) = 0 ; (158) \nGLYPH<4> GE ! ( f min ; f max) = m 2 c 2 a GLYPH<0> e 2 GLYPH<0> 1 GLYPH<1> 8 > > > < > > > : h 6 e GLYPH<16> 3 + e 2 GLYPH<17> + GLYPH<16> 7 + 24 e 2 + e 4 GLYPH<17> cos f min + 4 e GLYPH<16> 1 + e 2 GLYPH<17> cos 2 f min i sin fmin (1 + e cos f min) 2 \nGLYPH<0> h 6 e GLYPH<16> 3 + e 2 GLYPH<17> + GLYPH<16> 7 + 24 e 2 + e 4 GLYPH<17> cos f max + 4 e GLYPH<16> 1 + e 2 GLYPH<17> cos 2 f max i sin fmax (1 + e cos f max) 2 9 > > > = > > > ; ; (159) \nGLYPH<4> GE GLYPH<17> ( f min ; f max) = m p e 2 GLYPH<0> 1 2 c 2 a 2 6 6 6 6 6 6 4 GLYPH<16> 7 + e 2 + 8 e cos f max GLYPH<17> sin 2 f max 2 (1 + e cos f max) 2 GLYPH<0> GLYPH<16> 7 + e 2 + 8 e cos f min GLYPH<17> sin 2 f min 2 (1 + e cos f min) 2 3 7 7 7 7 7 7 5 : (160) \nAlso Equations (155)-(160), as Equations (46)-(51), hold only for the range of values of f given by Equations (43)-(45). \nBy using Equation (56), Equations (156)-(160) reduce to \nGLYPH<4> GE a ( f max) = 0 ; (161) \nGLYPH<4> GE e ( f max) = 0 ; (162) \nGLYPH<4> GE I ( f max) = 0 ; (163) \nGLYPH<4> GE GLYPH<10> ( f max) = 0 ; (164) \nGLYPH<4> GE ! ( f max) = GLYPH<0> m h 6 e GLYPH<16> 3 + e 2 GLYPH<17> + GLYPH<16> 7 + 24 e 2 + e 4 GLYPH<17> cos f max + 4 e GLYPH<16> 1 + e 2 GLYPH<17> cos 2 f max i sin fmax c 2 a GLYPH<0> e 2 GLYPH<0> 1 GLYPH<1> (1 + e cos f max) 2 ; (165) \nGLYPH<4> GE GLYPH<17> ( f max) = m p e 2 GLYPH<0> 1 GLYPH<16> 7 + e 2 + 8 e cos f max GLYPH<17> sin 2 f max 2 c 2 a (1 + e cos f max) 2 : (166) \nBy expanding Equations (165)-(166) in powers of f max around 0, as done for Equations (59)-(60), yields \nGLYPH<4> GE ! ( f max) ' m (7 + e ) c 2 a (1 GLYPH<0> e ) f max + O GLYPH<16> f 2 max GLYPH<17> ; (167) \nGLYPH<4> GE GLYPH<17> ( f max) ' m (7 + e ) p e 2 GLYPH<0> 1 c 2 a ( e + 1) f max + O GLYPH<16> f 2 max GLYPH<17> : (168) \n6.2. The Lense-Thirring corrections to the shifts of the contact Keplerian orbital elements \nExplicit expressions for the corrections GLYPH<4> LT i ( f min ; f max) ; i = a ; e ; I ; GLYPH<10> ; !; GLYPH<17> to the LT integrated shifts of the contact Keplerian orbital elements can be obtained as follows. \nCalculating the gradient of Equation (61) with respect to v yields \n@ R LT @ vx = 2 GJ (1 + e cos f ) 2 hGLYPH<16> ˆ Jz cos I cos GLYPH<10> GLYPH<0> ˆ Jy sin I GLYPH<17> sin u + ˆ Jz sin GLYPH<10> cos u i c 2 a 2 GLYPH<0> e 2 GLYPH<0> 1 GLYPH<1> 2 ; (169) \n@ R LT @ vy = GLYPH<0> 2 GJ (1 + e cos f ) 2 h ˆ Jz cos GLYPH<10> cos u GLYPH<0> GLYPH<16> ˆ Jx sin I + ˆ Jz cos I sin GLYPH<10> GLYPH<17> sin u i c 2 a 2 GLYPH<0> e 2 GLYPH<0> 1 GLYPH<1> 2 ; (170) \n@ R LT @ vz = 2 GJ (1 + e cos f ) 2 hGLYPH<16> ˆ Jy cos GLYPH<10> GLYPH<0> ˆ Jx sin GLYPH<10> GLYPH<17> cos u GLYPH<0> cos I GLYPH<16> ˆ Jx cos GLYPH<10> + ˆ Jy sin GLYPH<10> GLYPH<17> sin u i c 2 a 2 GLYPH<0> e 2 GLYPH<0> 1 GLYPH<1> 2 : (171) \nEquations (169)-(171), inserted in Equations (124)-(141), allow to calculate the functions Z LT i ( f ) ; i = a ; e ; I ; GLYPH<10> ; !; GLYPH<17> entering Equation (116). They turn out to be \nZ LT a ( f ) = 0 ; (172) \nZ LT e ( f ) = 2 GJ (1 + e cos f ) Jh c 2 a 3 n K e p e 2 GLYPH<0> 1 ; (173) \nZ LT I ( f ) = GLYPH<0> 2 GJ (1 + e cos f ) cos u ( Jm cos u GLYPH<0> Jl sin u ) c 2 a 3 n K GLYPH<0> e 2 GLYPH<0> 1 GLYPH<1> 3 = 2 ; (174) \nZ LT GLYPH<10> ( f ) = GLYPH<0> 2 GJ csc I (1 + e cos f ) sin u ( Jm cos u GLYPH<0> Jl sin u ) c 2 a 3 n K GLYPH<0> e 2 GLYPH<0> 1 GLYPH<1> 3 = 2 ; (175) \nZ LT ! ( f ) = 2 GJ cot I (1 + e cos f ) sin u ( Jm cos u GLYPH<0> Jl sin u ) c 2 a 3 n K GLYPH<0> e 2 GLYPH<0> 1 GLYPH<1> 3 = 2 ; (176) \nZ LT GLYPH<17> ( f ) = 0 : (177) \nAccording to Equations (172)-(177), the instantaneous osculating eccentricity, inclination, longitude of ascending node and argument of pericenter are generally di GLYPH<11> erent from their contact counterparts, while the osculating and contact semimajor axis and mean anomaly at epoch always coincide. \nBy means of Equation (145), calculated with Equations (172)-(177), one straightforwardly obtains the explicit expressions of GLYPH<4> LT i ( f min ; f max) ; i = a ; e ; I ; GLYPH<10> ; !; GLYPH<17> which, apart from the semimajor axis and the mean anomaly at epoch, turn out to be generally nonvanishing functions valid for any values of f min and f max. They are too cumbersome to be explicitly displayed for the general case of Equation (55). Instead, it is possible to write down manageable formulas if the condition of Equation (56) holds. The resulting nonvanishing shifts are \nGLYPH<4> LT I ( f max) = 2 GJ (1 + e cos f max) sin 2 f max ( Jl cos 2 ! + Jm sin 2 ! ) c 2 a 3 n K GLYPH<0> e 2 GLYPH<0> 1 GLYPH<1> 3 = 2 ; (178) \nGLYPH<4> LT GLYPH<10> ( f max) = 2 GJ csc I (1 + e cos f max) sin 2 f max ( Jl sin 2 ! GLYPH<0> Jm cos 2 ! ) c 2 a 3 n K GLYPH<0> e 2 GLYPH<0> 1 GLYPH<1> 3 = 2 ; (179) \nGLYPH<4> LT ! ( f max) = 2 GJ cot I (1 + e cos f max) sin 2 f max ( GLYPH<0> Jl sin 2 ! + Jm cos 2 ! ) c 2 a 3 n K GLYPH<0> e 2 GLYPH<0> 1 GLYPH<1> 3 = 2 : (180) \n(181) \nEquations (178)-(180) vanish if \nf max = f 1 : (182) \nThus, the LT total shifts of the osculating orbital elements accumulated over the entire path are equal just to those given by Equations (62)-(67).", '7. Numerical evaluations for some natural and artificial bodies': "Here, the results obtained in the previous Sections are applied to some flybys occurred in our solar system. \n7.1. ' Oumuamua in the field of the Sun \nThe case of the interstellar, cigar-shaped asteroid 7 1I / 2017 U1 ( ' Oumuamua) (Meech et al. 2017), which briefly visited the inner regions of out solar system in 2017 along an unbound trajectory, is considered here. Its orbital parameters are listed in Table 1. It may be interesting to calculate the size of the pK e GLYPH<11> ects of gravitational origin treated in the previous Sections if only \nTable 1. Orbital parameters of the heliocentric trajectory of the interstellar asteroid ' Oumuamua referred to the International Celestial Reference Frame (ICRF) at epoch J2000.0. retrieved from the HORIZONS WEB interface maintained by the Jet Propulsion Laboratory (JPL) for the epoch 23th November 2017. The distance of closest approach turns out to be r p = 0 : 38 au = 82 R GLYPH<12> e , while f 1 = 146 : 4 deg = 2 : 55 rad. \nto get an idea of the potential o GLYPH<11> ered by this type of unusual objects (Jewitt 2024) whose number may increase in the future 8 . However, it should be made clear that using them as probes for tests of gravitational theories would be quite challenging non only because of the observational accuracy needed but also because of the heavy non-gravitational accelerations perturbing their motion (Micheli et al. 2018). \nIn Table 2, the relevant physical parameters of the Sun are listed. The components of the Sun's spin axis, parameterized in \nTable 2. Relevant physical parameters of the Sun (Pijpers 1998; Seidelmann et al. 2007; Emilio et al. 2012; Park et al. 2017; Mecheri and Meftah 2021; Park et al. 2021). R.A. GLYPH<11> GLYPH<12> and decl. GLYPH<14> GLYPH<12> of the north pole of rotation are equatorial coordinates referred to the International Celestial Reference Frame (ICRF) at epoch J2000.0. \nterms of the right ascension (R.A.) GLYPH<11> GLYPH<12> and declination (decl.) GLYPH<14> GLYPH<12> of its north pole of rotation, are \nˆ J GLYPH<12> x = cos GLYPH<11> GLYPH<12> cos GLYPH<14> GLYPH<12> ; (183) \nˆ J GLYPH<12> y = sin GLYPH<11> GLYPH<12> cos GLYPH<14> GLYPH<12> ; (184) \nˆ J GLYPH<12> z = sin GLYPH<14> GLYPH<12> ; (185) \nthey are needed to calculate Equations (62)-(67) and Equations (74)-(79). \nThe nominal values of the gravitational pK orbital shifts of ' Oumuamua are displayed in Table 3. They are exceedingly small. Su GLYPH<14> ce it to say that the e GLYPH<11> ects of the Sun's oblateness and angular momentum are at the microarcseconds ( GLYPH<22> as) level over the whole trajectory, while the 1pN gravitoelectric shifts, to be rescaled by f max & 0 since they are valid just around the flyby, are of the order of less than a hundred milliarcseconds (mas). \nTable 3. Nominal values of the pK orbital shifts of ' Oumuamua calculated with the values of Table 2 and Table 1; f max entering the 1pN gravitoelectric shifts is assumed to be close to 0, and its value has to be given in rad. Here, mas and GLYPH<22> as stand for milliarcseconds and microarcseconds, respectively.", '7.2. NEAR in the field of the Earth': "Here, the case of the spacecraft NEAR 9 (Prockter et al. 2002) when it approached the Earth is treated. \nTable 4 lists the orbital parameters of such a spacecraft for the flyby of the Earth occurred on 23th January 1998. Table 5 \nTable 4. Orbital parameters of the geocentric trajectory of the probe NEAR referred to the International Celestial Reference Frame (ICRF) at epoch J2000.0. retrieved from the HORIZONS WEB interface maintained by the Jet Propulsion Laboratory (JPL) for the epoch 23th January 1998. The distance of closest approach turns out to be r p = 6 : 90 GLYPH<2> 10 3 km = 1 : 08 R GLYPH<8> e , while f 1 = 123 : 4 deg = 2 : 15 rad. \ndisplays the nominal values for the pK orbital shifts experienced by NEAR; the relevant physical parameters of the Earth needed to calculate them were retrieved from Petit and Luzum (2010). It turns out that the nominal orbital shifts due to the Earth's first even zonal harmonic, being as large as ' 10 6 GLYPH<0> 10 8 GLYPH<22> as, neatly overwhelm the pN ones; su GLYPH<14> ce it to say that the LT displacements are as little as . 10 GLYPH<22> as, while the gravitoelectric ones are less than a few mas. However, the present-day relative uncertainty in determining J GLYPH<8> 2 from several dedicated satellite missions is 10 \nGLYPH<27> J GLYPH<8> 2 J GLYPH<8> 2 ' 10 GLYPH<0> 8 ; (186) \nas it can be inferred by inspecting the latest Earth's gravity models retrievable from, e.g., the webpage 11 of the International Centre for Global Earth Models (ICGEM) maintained by the GeoForschungsZentrum (GFZ). Thus, the mismodelled classical shifts would be smaller than the nominal pN ones by about one order of magnitude (LT), or more (Schwarzschild). Also in the case of artificial probes like NEAR, the impact of the non-gravitational accelerations during flybys should be carefully investigated. \nGiven the low altitude \nh p ' 532 km (187) \n2 \nTable 5. Nominal values of the pK orbital shifts of the probe NEAR calculated with the values of Table 4; f max entering the 1pN gravitoelectric shifts is assumed to be close to 0, and its value has to be given in rad. Here, mas and GLYPH<22> as stand for milliarcseconds and microarcseconds, respectively. \nreached by NEAR at the perigee of its Earth's flyby, one may wonder what could be the impact of, say, the mismodeling GLYPH<27> J GLYPH<8> 4 of the second even zonal harmonic J GLYPH<8> 4 of the geopotential. It can be naively evaluated by multiplying the nominal values of the shifts due to J 2 listed in Table 5 by the following scaling factor \na J GLYPH<8> 4 ' GLYPH<18> R a GLYPH<19> 2 GLYPH<27> J GLYPH<8> 4 J GLYPH<8> 2 : (188) \nThe square of the ratio of the Earth's equatorial radius to the semimajor axis of NEAR, listed in Table 4, is of the order of \nGLYPH<18> R a GLYPH<19> 2 ' 0 : 5 : (189) \nAccording to the most recent Earth's gravity models listed by ICGEM, the formal uncertainty in J GLYPH<8> 4 is as little as \nGLYPH<27> J GLYPH<8> 4 ' 10 GLYPH<0> 13 : (190) \nThe nominal value of the first even zonal harmonic of our planet is \nJ GLYPH<8> 2 ' 10 GLYPH<0> 4 : (191) \nThus, the bias due to the imperfect knowledge of the Earth's octupole mass moment seems somewhat negligible with respect to the pN e GLYPH<11> ects of interest, shown in Table 5, since \na J GLYPH<8> 4 ' 10 GLYPH<0> 10 : (192) \nFinally, as a further source of potential systematic bias, it may be mentioned the coe GLYPH<14> cient of degree ' = 2 and order m = 2 of the geopotential accounting for the Earth's dynamical triaxiality. Its impact on spacecraft's flybys was analytically investigated by, e.g., Rappaport et al. (2001).", '8. Summary and conclusions': "Analytical expressions of the variations of all the Keplerian orbital elements of an otherwise unperturbed hyperbolic trajectory are obtained in full generality for some known post-Keplerian perturbing accelerations of both Newtonian and postNewtonian origin: that due to the primary's quadrupole mass moment, and, to the first post-Newtonian level, the general relativistic Schwarzschild and Lense-Thirring ones. In the case of the classical perturbation, the resulting formulas describe the shifts of the osculating Keplerian orbital elements. Instead, the general relativistic e GLYPH<11> ects are to be intended as written in terms of the nonosculating, contact elements, to the first post-Newtonian order, because their disturbing functions depend on the velocity of the test particle. The corrections required to have final expressions in terms of the osculating elements are explicitly calculated. It turns out that, for a generally asymmetric range of values of the true anomaly smaller than the maximum allowed one, \nthe gravitoelectric variations of the contact eccentricity, argument of pericentre and mean anomaly at epoch are di GLYPH<11> erent from their osculating counterparts. Instead, the gravitomagnetic orbital shifts, written in terms of either the contact or the osculating elements, coincide if they are calculated over the entire unbound trajectory. \nThe resulting formulas for the Newtonian and post-Newtonian shifts, all expressed with the osculating elements and valid for any spatial orientations of both the orbital plane and the spin axis of the source, are applied to ' Oumuamua, the first asteroid of interstellar origin which recently entered the inner regions of our solar system. While the quadrupole mass moment of the Sun and its angular momentum induce shifts as little as a few microarcseconds throughout the whole path, the post-Newtonian gravitoelectric e GLYPH<11> ect due to the solar mass amounts to less than a hundred milliarcseconds around the passage at perihelion. Actual usage of such kind of natural bodies as possible probes to perform gravitational experiments is likely prevented from the usually large non-conservative accelerations heavily perturbing their motions. As far as the Earth's flyby by the NEAR spacecraft is concerned, the size of its post-Newtonian disturbances is close to those of ' Oumuamua within about one order of magnitude. Instead, the perturbations due to the terrestrial oblateness are nominally much larger. However, the present-day relative uncertainty in our knowledge of the first even zonal harmonic of the geopotential is small enough to make the mismodeled part of such classical orbital shifts smaller than the corresponding relativistic ones. \nThe situation may become more favorable in the case of numerous planetary or satellite flybys by the many artificial probes currently travelling through the solar system. Furthermore, dedicated gravity experiments may be suitably designed relying upon the analytical results obtained. \nThe calculational approach adopted can straightforwardly be extended to any modified model of gravity as well.", 'Data availability': 'No new data were generated or analysed in support of this research.', 'Conflict of interest statement': 'I declare no conflicts of interest.', 'References': "- J. D. Anderson. Gravity-Assist Navigation. In J. H. Shirley and R. W. Fairbridge, editors, Encyclopedia of Planetary Sciences , pages 287-289. Chapman & Hall, 1997.\n- J. 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2024arXiv240907038H

The compact object with a mass of 2.502.67Modot observed by LIGO Scientific and Virgo collaborations in GW190814 as well as the recent report of a light compact object with a mass and radius of M0.770.200.17Modot and R10.40.860.78 km within the supernova remnant HESS J1731347 have posed a great challenge to the investigations into the supranuclear matter. In the inner core region of the neutron star the strangeness degrees of freedom such as the hyperons can be present which is also named as a hyperonic star. In this work the neutron star consisting of nucleons and leptons and the hyperonic star including the hyperons will be studied in the framework of the densitydependent relativistic meanfield DDRMF model. Some popular DDRMF parameterizations will be adopted to investigate the properties of nuclear matter and the mass radius tidal deformability and other properties of neutron star and hyperonic stars. We find that the maximum masses of neutron star calculated by DDMEX DDMEX1 DDMEX2 DDMEXY and DDLZ1 sets can be around 2.52.6Modot with quite stiff equations of state EOSs generated by their strong repulsive contributions from vector potentials at high densities. Moreover by investigating the influence of the crust EOS and core EOS on the neutron stars we find that the observational data from HESS J1731347 suggest the requirement of a crust EOS with a higher L parameter and a core EOS with a lower L parameter and the MR relations from the constructed EOSs can also be consistent with the observables of PSR J07406620 PSR J00300451 from NICER and the GW170817 event. With the inclusion of hyperons the hyperonic star matter becomes softer compared to the neutron star matter. But the massive hyperonic star can also be obtained with DDRMF parameter sets if the vector coupling constants are strong.

2024-09-01T00:00:00Z

['arXiv:2409.07038', '10.48550/arXiv.2409.07038', '2024arXiv240907038H']

['Astrophysics - High Energy Astrophysical Phenomena', 'Nuclear Theory']

Investigations on the equation of state of neutron star matter with densitydependent relativistic meanfield model

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.07038.pdf

{'Investigations on the equation of state of neutron star matter with density-dependent relativistic mean-field model': 'Kaixuan Huang \nSchool of Physics, Nankai University, Tianjin 300071, China', 'Jinniu Hu ∗': 'School of Physics, Nankai University, Tianjin 300071, China and Shenzhen Research Institute of Nankai University, Shenzhen 518083, China (Dated: September 12, 2024) \nThe neutron star as a supernova remnant is attracting high attention recently due to the gravitation wave detection and precise measurements about its mass and radius. In particular, the compact object with a mass of 2 . 50 -2 . 67 M /circledot observed by LIGO Scientific and Virgo collaborations in GW190814, as well as the recent report of a light compact object with a mass and radius of M = 0 . 77 +0 . 20 -0 . 17 M /circledot and R = 10 . 4 +0 . 86 -0 . 78 km within the supernova remnant HESS J1731-347, have posed a great challenge to the investigations into the supranuclear matter. In the inner core region of the neutron star, the strangeness degrees of freedom, such as the hyperons, can be present, which is also named as a hyperonic star. In this work, the neutron star consisting of nucleons and leptons, and the hyperonic star including the hyperons will be studied in the framework of the densitydependent relativistic mean-field (DDRMF) model. Some popular DDRMF parameterizations will be adopted to investigate the properties of nuclear matter and the mass, radius, tidal deformability, and other properties of neutron star and hyperonic stars. We find that the maximum masses of neutron star calculated by DD-MEX, DD-MEX1, DD-MEX2, DD-MEXY and DD-LZ1 sets can be around 2 . 5 -2 . 6 M /circledot with quite stiff equations of state (EOSs) generated by their strong repulsive contributions from vector potentials at high densities. Moreover, by investigating the influence of the crust EOS and core EOS on the neutron stars, we find that the observational data from HESS J1731-347 suggest the requirement of a crust EOS with a higher L parameter and a core EOS with a lower L parameter, and the M -R relations from the constructed EOSs can also be consistent with the observables of PSR J0740+6620, PSR J0030+0451 from NICER and the GW170817 event. With the inclusion of hyperons, the hyperonic star matter becomes softer compared to the neutron star matter. But the massive hyperonic star can also be obtained with DDRMF parameter sets if the vector coupling constants are strong. \nPACS numbers: 21.10.Dr, 21.60.Jz, 21.80.+a', 'I. INTRODUCTION': "With the rapid developments of astronomical technology, many significant achievements have been made in neutron star observation in the past decade. The massive neutron stars, whose masses are around 2 M /circledot , PSR J1614-2230 ( 1 . 928 ± 0 . 017 M /circledot ) [1], PSR J0348+0432 ( 2 . 01 ± 0 . 04 M /circledot ) [2], PSR J2215+5135 ( 2 . 27 +0 . 17 -0 . 15 M /circledot ) [3] and PSR J0740+6620 ( 2 . 08 ± 0 . 07 M /circledot ) [4, 5], were measured within the relativistic Shapiro delay effect. The simultaneous measurements of mass-radius observations of the pulsars, PSR J0030+0451 [6, 7] and PSR J0740+6620 [5, 8], using the pulse profile modeling of X-ray emissions from hot spots by NICER. What's more, the gravitational wave and the electromagnetic counterpart of a binary neutron-star merger were firstly detected by LIGO Scientific and Virgo Collaborations (LVC) as event GW170817 [9-11] and the dimensionless tidal deformability of 1 . 4 M /circledot neutron star can be extracted from GW170817 as 190 +390 -120 [12-14]. In 2019, a new event of a compact binary merger with a 22 . 2 -24 . 3 M /circledot black hole and a compact component with a mass of 2 . 50 -2 . 67 M /circledot was reported by LVC as GW190814 [15]. The secondary object of GW190814 attracts a lot of attentions, since it may be either the heaviest neutron star [16] or the lightest black hole ever discovered [15]. In addition to the massive neutron stars, there has been a growing focus on low-mass neutron stars. Recently, a central compact object (CCO) was reported within the supernova remnant HESS J1731-347. The estimated mass and radius of this object are M = 0 . 77 +0 . 20 -0 . 17 M /circledot and R = 10 . 4 +0 . 86 -0 . 78 km, respectively [17]. There are various speculations regarding its internal constitution. Some studies have focused on investigating it as a quark star [18], a hybrid star [19], or a neutron star [17, 20]. \nThe structure of a static neutron star is separated into five regions. The outer layer is the atmosphere consisting of the atom and is very thin. In the next layer as the outer crust, the electrons of an atom are ionized and form the uniform Fermi gas, where the \nnuclei are immersed. With the density increasing, neutrons drip out from the neutron-rich nuclei and generate the neutron gas, which is called the inner crust. When the nucleon density approaches ρ 0 / 2 , heavy nuclei dissolve and the neutron star matter becomes homogeneous, which corresponds to the outer core of neutron star. It plays an essential role in determining the mass and radius of the neutron star [21-25]. In the inner core region, the baryons including the strangeness degree of freedom, such as Λ , Σ , and Ξ hyperons, may appear, when the Fermi energies of nucleons are larger than their chemical potentials, which is also called as a hyperonic star. \nThe mass, radius, and tidal deformabilities of neutron stars and hyperonic stars can be determined by solving the Tolman-Oppenheimer-Volkoff (TOV) equation [26, 27] with the equation of state (EOS) of neutron star matter as input. Many attempts have been made to obtain the EOS of supranuclear matter in neutron stars from different models. The relativistic meanfield (RMF) model has been widely successful in describing the properties of finite nuclei and naturally extends to high-density regimes with the ρ meson, nonlinear terms of σ and ω mesons, the coupling terms with ρ meson to σ or ω meson introduced step by step [28-33]. Furthermore, the nonlinear terms of various mesons could be replaced by the density-dependent mesonnucleon coupling constants in the density-dependent RMF (DDRMF), which consider the nuclear medium effect originated by the relativistic Brueckner-Hartree-Fock model [34]. \nThe symmetry energy ( E sym ) and its density dependence play a crucial role in the EOS of neutron star matter because of its highly isospin-asymmetric nature. The behavior of E sym near the saturation nucleon density ( ρ 0 ) affects the structure of the neutron star crust [35, 36] as well as the radius of neutron stars in the intermediate mass range [37]. In addition to their impact on neutron stars, E sym and its slope ( L ) can be constrained by terrestrial experiments. Recent measurements of the neutron skin thickness ( R skin ) of 208 Pb by PREX-I and PREX-II resulted in R 208 skin = 0 . 283 ± 0 . 071 fm [38-40], leading to derived values of E sym = 38 . 1 ± 4 . 7 MeV and L = 106 ± 37 MeV based on the linear relation between L and R 208 skin [41-43]. Similarly, the CREX collaboration reported the neutron skin thickness of 48 Ca as R 48 skin = 0 . 121 ± 0 . 026 fm [44], using the same method as PREX-II. Notably, the value of L obtained from 48 Ca is much smaller compared to that from PREX-II [45-47] and the big difference between the two measurements brings significant challenges to the understanding of nuclear many-body methods. \nThe hyperons were investigated from the 1980s by Glendenning in the framework of RMF model, where the coupling strengths between the mesons and baryons were simply generated by the quark power counting rules [48, 49]. Actually, the appearances of hyperons are strongly dependent on the hyperon-nucleon and hyperon-hyperon potentials, which can be extracted from the properties of various hypernuclei. In the past 40 years, the RMF parameterizations about nucleon-nucleon and nucleon-hyperon interactions were largely improved through reproducing the ground-state properties of finite nuclei and above hypernuclei experimental data, that were adopted to investigate the properties of neutron stars and hyperonic stars[50-64]. With the discoveries of two-times-solar-mass neutron stars, a 'hyperon puzzle' was proposed, since the neutron star maximum mass will be reduced by about 20% once the hyperons are self-consistently introduced in the nuclear many-body methods. Therefore, it is difficult to explain the existence of massive neutron stars with hyperons at the beginning and many schemes were raised to solve such a puzzle [59, 65-68]. Recently, the role of vector meson including the strange quark, φ was also discussed in the hyperonic star with various hyperons [63, 69]. \nIn this work, the neutron star and hyperonic star in the framework of DDRMF models will be systematically calculated with the most popular parameterizations, where the exchange mesons with strangeness quarks will be completely included and coupling strengths between the mesons and baryons will be constrained with the latest hypernuclei experimental data. When calculating the properties of neutron stars, the EOS for light neutron stars will be constructed with DDVT model, incorporating the tensor coupling of vector mesons, to study the compact object in HESS J1731-347. This paper is arranged as follows. In Section II, the formulas about the neutron star and hyperonic star with the DDRMF model are shown in detail. In Section III, the parameterizations of the DDRMF model are listed and the properties of massive neutron stars, light neutron stars and hyperonic stars are calculated and discussed. In Section IV, the summary and conclusion will be given.", 'II. THE DENSITY-DEPENDENT RELATIVISTIC MEAN-FIELD MODEL FOR NEUTRON STAR': "In the DDRMF model, the baryons interact with each other via exchanging various light mesons, including scalar-isoscalar meson ( σ ) with mass m σ , vector-isoscalar meson ( ω ) with mass m ω , vector-isovector meson ( ρ ) with mass m ρ , scalar-isoscalar meson ( δ ) with mass m δ , and strange scalar and vector mesons ( σ ∗ and φ ) with mass m σ ∗ and m φ , respectively [33, 70-74]. The baryons considered in the present calculation are nucleons ( n and p ) and hyperons ( Λ , Σ , Ξ ). The Lagrangian density of \nDDRMF model is written as [75, 76] \nL = ∑ B ψ B [ γ µ ( i∂ µ -Γ ωB ( ρ B ) ω µ -Γ φB ( ρ B ) φ µ -Γ ρB ( ρ B ) 2 /vector ρ µ /vectorτ ) -( M B -Γ σB ( ρ B ) σ -Γ σ ∗ B ( ρ B ) σ ∗ -Γ δB ( ρ B ) /vector δ/vectorτ )] ψ B + 1 2 ( ∂ µ σ∂ µ σ -m 2 σ σ 2 ) + 1 2 ( ∂ µ σ ∗ ∂ µ σ ∗ -m 2 σ ∗ σ ∗ 2 ) + 1 2 ( ∂ µ /vector δ∂ µ /vector δ -m 2 δ /vector δ 2 ) -1 4 W µν W µν + 1 2 m 2 ω ω µ ω µ -1 4 Φ µν Φ µν + 1 2 m 2 φ φ µ φ µ -1 4 /vector R µν /vector R µν + 1 2 m 2 ρ /vector ρ µ /vector ρ µ , (1) \nwhere ψ B represents the wave function of baryons with mass M B . σ, σ ∗ , ω µ , φ µ /vector ρ µ denotes the fields of σ, σ ∗ , ω, φ , and ρ mesons, respectively. W µν , Φ µν , and /vector R µν are the anti-symmetry tensor fields of ω , φ and ρ . In nuclear matter, the tensor coupling between the vector meson and nucleon does not provide any contributions. Therefore, it is neglected in the present Lagrangian. The coupling constants of σ and ω mesons are expressed as a fraction function of the total vector density, ρ B = ∑ B ρ v B . In most of DDRMF parameterizations, such as DD2 [77], DD-ME1 [78], DD-ME2 [79], DD-MEX [80], PKDD [81], TW99 [75], and DDV, DDVT, DDVTD [82], they are assumed as, \nΓ iN ( ρ B ) = Γ iN ( ρ B 0 ) f i ( x ) (2) \nwith \nf i ( x ) = a i 1 + b i ( x + d i ) 2 1 + c i ( x + d i ) 2 , x = ρ B /ρ B 0 , (3) \nfor i = σ, ω . ρ B 0 is the saturation density of symmetric nuclear matter. The couplings Γ iN ( ρ B 0 ) , and the coefficients a i , b i , c i , and d i are obtained by fitting the properties of finite nuclei . Five constraints on the coupling constants f i (1) = 1 , f '' i (0) = 0 , f '' σ (1) = f '' ω (1) can reduce the numbers of independent parameters to three in Eq. (2). For the isovector mesons ρ and δ , their density-dependent coupling constants are assumed to be, \nΓ iN ( ρ B ) = Γ iN ( ρ B 0 )exp[ -a i ( x -1)] . (4) \nWhile in other parameterizations, such as DD-LZ1 [83], the coefficient in front of fraction function, Γ i is fixed at ρ B = 0 for i = σ, ω : \nΓ iN ( ρ B ) = Γ iN (0) f i ( x ) . (5) \nThere are only four constraint conditions as f i (0) = 1 and f '' i (0) = 0 for σ and ω coupling constants in DD-LZ1 set. The constraint f '' σ (1) = f '' ω (1) in previous parameter sets was removed in DD-LZ1 parameterization. For ρ meson, its coupling constant is also changed accordingly as \nΓ ρN ( ρ B ) = Γ ρN (0)exp( -a ρ x ) . (6) \nTo solve the nuclear many-body system in the DDRMF model, the mean-field approximation should be adopted, in which various mesons are treated as classical fields, 〈 σ 〉 = σ, 〈 σ ∗ 〉 = σ ∗ , 〈 ω µ 〉 = ω, 〈 φ µ 〉 = φ, 〈 /vector ρ µ 〉 = ρ, 〈 δ 〉 = δ. The space components of the vector mesons are removed in the parity conservation system. In addition, the spatial derivatives of baryons and mesons are neglected in the infinite nuclear matter due to the transformation invariance. Finally, with the Euler-Lagrange \nequation, the equations of motion for baryons and mesons are obtained, \n[ iγ µ ∂ µ -γ 0 (Γ ωB ( ρ B ) ω +Γ φB ( ρ B ) φ + Γ ρB ( ρ B ) 2 ρτ 3 +Σ R ( ρ B ) ) -M ∗ B ] ψ B = 0 . m 2 σ σ = ∑ B Γ σB ( ρ B ) ρ s B , m 2 σ ∗ σ ∗ = ∑ B Γ σ ∗ B ( ρ B ) ρ s B , m 2 ω ω = ∑ B Γ ωB ( ρ B ) ρ B , m 2 φ φ = ∑ B Γ φB ( ρ B ) ρ B , m 2 ρ ρ = ∑ B Γ ρB ( ρ B ) 2 ρ 3 B , m 2 δ δ = Γ δB ( ρ B ) ρ s 3 B . (7) \n∑ B \nThe rearrangement contribution, Σ R , is expressed as \nΣ R ( ρ B ) = -∂ Γ σB ( ρ B ) ∂ρ B σρ s B -∂ Γ σ ∗ B ( ρ B ) ∂ρ B σ ∗ ρ s B -∂ Γ δB ( ρ B ) ∂ρ B δρ s 3 B + 1 2 ∂ Γ ρB ( ρ B ) ∂ρ B ρρ 3 B + [ ∂ Γ ωB ( ρ B ) ∂ρ B ω + ∂ Γ φB ( ρ B ) ∂ρ B φ ] ρ B , (8) \nwhere the scalar, vector densities, and their isospin components are \nρ s B = 〈 ψ B ψ B 〉 , ρ s 3 B = 〈 ψ B τ 3 ψ B 〉 , ρ B = 〈 ψ † B ψ B 〉 , ρ 3 B = 〈 ψ † B τ 3 ψ B 〉 . (9) \nτ 3 in the above equation is the isospin third component of the baryon species B . The effective masses of baryons are dependent on the scalar mesons σ , σ ∗ and δ , \nM ∗ B = M B -Γ σB ( ρ B ) σ -Γ σ ∗ B ( ρ B ) σ ∗ -Γ δB ( ρ B ) δτ 3 . (10) \nBecause of the mass-energy relation, the corresponding effective energies of baryons are \nwhere k FB is the Fermi momentum of baryons. \nE ∗ FB = √ k 2 FB +( M ∗ B ) 2 , (11) \nWith the energy-momentum tensor in a uniform system, the energy density, E and pressure, P of nuclear matter in DDRMF model are obtained respectively as \nE = 1 2 m 2 σ σ 2 + 1 2 m 2 σ ∗ σ ∗ 2 -1 2 m 2 ω ω 2 -1 2 m 2 φ φ 2 -1 2 m 2 ρ ρ 2 + 1 2 m 2 δ δ 2 +Γ ωB ( ρ B ) ωρ B +Γ φB ( ρ B ) φρ B + Γ ρ ( ρ B ) 2 ρρ 3 + E B kin , P = ρ B Σ R ( ρ B ) -1 2 m 2 σ σ 2 -1 2 m 2 σ ∗ σ ∗ 2 + 1 2 m 2 ω ω 2 + 1 2 m 2 φ φ 2 + 1 2 m 2 ρ ρ 2 -1 2 m 2 δ δ 2 + P B kin , (12) \nwhere the contributions from kinetic energy are \nE B kin = γ 2 π 2 ∫ k FB 0 k 2 √ k 2 + M ∗ B 2 dk = γ 16 π 2 k FBi E ∗ FB 2 k 2 FB + M ∗ B 2 + M ∗ B 4 ln M ∗ B k FB + E ∗ , (13) \n= γ 48 π 2 k FB 2 k 2 FB -3 M ∗ B 2 E ∗ FB +3 M ∗ B 4 ln k FB + E FB M ∗ . (14) \n[ ( ) FB ] P B kin = γ 6 π 2 ∫ k FB 0 k 4 dk √ k 2 + M ∗ B 2 [ ( ) ∗ B ] \nγ = 2 is the spin degeneracy factor. \nThe binding energy per nucleon for the symmetric nuclear matter can be defined by \nE A = E ρ B -M. (15) \nThe symmetry energy is calculated by \nE sym = 1 2 ∂ 2 E/A ∂α 2 = k 2 F 6 E ∗ F + Γ 2 ρN 8 m 2 ρ ρ N , (16) \nwhere α is the asymmetry factor, defined as α = ( ρ Bn -ρ Bp ) / ( ρ Bn + ρ Bp ) and its slope at the saturation point, L is given by \nL = 3 ρ B ∂E sym ∂ρ B ∣ ∣ ∣ ∣ ρ B = ρ B 0 . (17) \n∣ In the uniform neutron star matter, the compositions of baryons and leptons are determined by the requirements of charge neutrality and β -equilibrium conditions. All baryon octets ( n, p, Λ , Σ -, Σ 0 , Σ + , Ξ -, Ξ 0 ) and leptons ( e, µ ) will be included in this work. The β -equilibrium conditions can be expressed by [49, 52] \nµ p = µ Σ + = µ n -µ e , µ Λ = µ Σ 0 = µ Ξ 0 = µ n , µ Σ -= µ Ξ -= µ n + µ e , µ µ = µ e , (18) \nwhere µ i can be derived from the thermodynamics equations at zero temperature, \nµ B = √ k 2 FB + M ∗ 2 B +[Γ ωB ( ρ B ) ω +Γ φB ( ρ B ) φ + Γ ρB ( ρ B ) 2 ρ 3 B +Σ R ( ρ B ) ] , µ l = √ k 2 Fl + m 2 l . (19) \nThe charge neutrality condition has the following form, \nρ p + ρ Σ + = ρ e + ρ µ + ρ Σ -+ ρ Ξ -. (20) \nThe total energy density and pressure of neutron star matter will be obtained as a function of baryon density within the constraints of Eqs. (II) and (20). The Tolman-Oppenheimer-Volkoff (TOV) equation describes a spherically symmetric star in the gravitational equilibrium from general relativity [26, 27], \ndP dr = -GM ( r ) E ( r ) r 2 [ 1 + P ( r ) E ( r ) ] [ 1 + 4 πr 3 P ( r ) M ( r ) ] 1 -2 GM ( r ) r , dM ( r ) dr = 4 πr 2 E ( r ) , (21) \nwhere P and M are the pressure and mass of a neutron star at the position r and G is the gravitational constant. Furthermore, the tidal deformability becomes a typical property of a neutron star after the observation of the gravitational wave from a binary neutron-star (BNS) merger, which characterizes the deformation of a compact object in an external gravity field generated by another star. The tidal deformability of a neutron star is reduced as a dimensionless form [84, 85], \nΛ = 2 3 k 2 C -5 . (22) \nwhere C = GM/R is the compactness parameter. The second order Love number k 2 is given by \nk 2 = 8 C 5 5 (1 -2 C ) 2 [2 + 2 C ( Y R -1) -Y R ] { 2 C [6 -3 Y R +3 C (5 Y R -8)] +4 C 3 [ 13 -11 Y R + C (3 Y R -2) + 2 C 2 (1 + Y R ) ] +3(1 -2 C ) 2 [2 -Y R +2 C ( Y R -1)ln(1 -2 C )] } -1 . (23) \nHere, Y R = y ( R ) . y ( r ) satisfies the following first-order differential equation, \nr dy ( r ) dr + y 2 ( r ) + y ( r ) F ( r ) + r 2 Q ( r ) = 0 , (24) \nwhere F ( r ) and Q ( r ) are functions related to the pressure and energy density \nF ( r ) = [ 1 -2 M ( r ) r ] -1 { 1 -4 πr 2 [ E ( r ) -P ( r )] } , (25) r 2 Q ( r ) = { 4 πr 2 [ 5 E ( r ) + 9 P ( r ) + E ( r ) + P ( r ) ∂P ∂ E ( r ) ] -6 } × [ 1 -2 M ( r ) r ] -1 -[ 2 M ( r ) r +2 × 4 πr 2 P ( r ) ] 2 × [ 1 -2 M ( r ) r ] -2 . \nThe second Love number corresponds to the initial condition y (0) = 2 . It is also related to the speed of sound in compact matter, c s , \nc 2 s = ∂P ( ε ) ∂ E . (26)", 'A. The DDRMF Parameterizations': 'We list some of the DDRMF parameterizations in Table I, where the DD2 [77], DD-ME1 [78], DD-ME2 [79], DD-MEX [80], PKDD [81], TW99 [75], DDV, DDVT, DDVTD [82], and DD-LZ1 [83] have been applied to study the properties of nuclear matter and the neutron stars in our previous works [16, 76]. DD-MEX1, DD-MEX2, and DD-MEXY [86] are the new DDRMF parameter sets based on the DD-MEX set. In DDVT and DDVTD sets, the tensor coupling between the vector meson and nucleon was included. The scalar-isovector meson, δ was taken into account in DDVTD set. \nThe density-dependent behaviors of the coupling constants as functions of the vector density are shown in Fig. 1. It can be found that all of these coupling constants decrease when the nuclear density becomes larger due to the nuclear medium effect. For the ρ meson coupling constants in panel (c), all parameter sets have very similar density-dependent behaviors in the whole density region. In DDVT and DDVTD, the tensor coupling constants play obvious roles in finite nuclei due to their derivative forms, while they do not provide any contribution in nuclear matter. Their coupling constants of σ and ω mesons in panel (a) and panel (b) are dramatically smaller than other sets. Furthermore, the coupling constants from several typical nonlinear RMF models, NL3 [87], TM1 [31], IUFSU [88], and BigApple [89] are also shown to compare their differences with those in \nD \nT \nV \nD \nD \nT \nV \nD \nD \nV \nD \nD \n9 \n9 \n7 \ncoefficients of meson coupling constants, \nΓ \nat nuclear saturation densities. \n3 \n1 \n4 \n5 \n6 \n5 \n. \n9 \n3 \n9 \n3 \n1 \n4 \n5 \n6 \n5 \n. \n9 \n3 \n9 \n3 \n1 \n4 \n5 \n6 \n5 \n. \n9 \n3 \n9 \n0 \n0 \n0 \nin DD-LZ1 are the values at zero density, while other parameter sets are dependent on the values \n1 \n8 \n0 \n2 \n7 \n2 \n. \n8 \n3 \n9 \n1 \n8 \n0 \n2 \n7 \n2 \n. \n8 \n3 \n9 \n1 \n8 \n0 \n2 \n7 \n2 \n. \n8 \n3 \n9 \n0 \n0 \n0 \n3 \n4 \n8 \n9 \n1 \n6 \n. \n2 \n0 \n5 \n2 \n0 \n6 \n8 \n9 \n5 \n. \n2 \n0 \n5 \n8 \n9 \n0 \n0 \n0 \n6 \n. \n7 \n3 \n5 \n0 \n0 \n0 \n0 \n0 \n0 \n0 \n. \n3 \n8 \n7 \n0 \n0 \n0 \n0 \n. \n3 \n8 \n7 \n0 \n0 \n0 \n0 \n. \n3 \n8 \n7 \n0 \n0 \n0 \n0 \n0 \n0 \n0 \n. \n3 \n6 \n7 \n0 \n0 \n0 \n0 \n. \n3 \n6 \n7 \n0 \n0 \n0 \n0 \n. \n3 \n6 \n7 \n0 \n0 \n0 \n9 \n6 \n2 \n9 \n7 \n3 \n. \n8 \n3 \n6 \n8 \n2 \n8 \n3 \n. \n8 \n0 \n6 \n9 \n6 \n3 \n1 \n. \n0 \n1 \n4 \n5 \n8 \n3 \n3 \n4 \n0 \n8 \n9 \n. \n0 \n1 \n6 \n0 \n1 \n7 \n8 \n9 \n. \n0 \n1 \n0 \n5 \n4 \n0 \n7 \n7 \n. \n2 \n1 \n5 \n1 \n0 \n8 \n3 \n0 \n6 \n0 \n. \n8 \n2 \n1 \n1 \n7 \n9 \n6 \n. \n7 \n3 \n3 \n8 \n4 \n8 \n. \n7 \n6 \n9 \n3 \n5 \n1 \n. \n0 \n6 \n3 \n5 \n1 \n. \n0 \n1 \n1 \n5 \n1 \n. \n0 \n0 \n3 \n3 \n4 \n6 \n9 \n1 \n. \n1 \n7 \n9 \n3 \n0 \n2 \n. \n1 \n3 \n9 \n9 \n0 \n2 \n. \n1 \n9 \n6 \n4 \n3 \n. \n1 \n0 \n7 \n9 \n3 \n. \n1 \nσ \na \n8 \n4 \n7 \n2 \n6 \n0 \n. \n1 \nσ \na \n3 \n6 \n2 \n1 \n7 \n1 \n9 \n1 \n. \n0 \n4 \n1 \n3 \n0 \n1 \n2 \n9 \n1 \n. \n0 \n4 \n4 \n8 \n6 \n8 \n2 \n1 \n2 \n. \n0 \n1 \n6 \n0 \n1 \n. \n2 \n0 \n5 \n3 \n3 \n. \n1 \nσ \nb \n7 \n2 \n6 \n3 \n6 \n7 \n. \n1 \nσ \nb \n9 \n5 \n8 \n6 \n7 \n3 \n7 \n2 \n. \n0 \n6 \n6 \n5 \n3 \n7 \n7 \n7 \n2 \n. \n0 \n7 \n9 \n1 \n8 \n9 \n7 \n0 \n3 \n. \n0 \n4 \n0 \n7 \n1 \n. \n3 \n1 \n7 \n6 \n0 \n. \n2 \nσ \nc \n8 \n2 \n9 \n8 \n0 \n3 \n. \n2 \nσ \nc \n5 \n0 \n7 \n3 \n4 \n3 \n0 \n1 \n. \n1 \n7 \n1 \n8 \n2 \n5 \n5 \n9 \n0 \n. \n1 \n2 \n4 \n3 \n4 \n3 \n0 \n4 \n0 \n. \n1 \n5 \n9 \n9 \n3 \n. \n0 \n6 \n1 \n0 \n4 \n. \n0 \nσ \nd \n7 \n5 \n9 \n9 \n7 \n3 \n. \n0 \nσ \nd \n3 \n9 \n6 \n6 \n1 \n. \n1 \n4 \n8 \n0 \n6 \n1 \n. \n1 \n6 \n4 \n7 \n3 \n2 \n. \n1 \n8 \n8 \n4 \n3 \n. \n1 \n6 \n3 \n9 \n3 \n. \n1 \nω \na \n1 \n8 \n1 \n9 \n5 \n0 \n. \n1 \nω \na \n6 \n1 \n0 \n0 \n4 \n6 \n2 \n0 \n. \n0 \n0 \n5 \n8 \n9 \n5 \n4 \n4 \n0 \n. \n0 \n2 \n2 \n4 \n1 \n1 \n9 \n3 \n0 \n. \n0 \n7 \n7 \n5 \n8 \n. \n1 \n1 \n9 \n1 \n0 \n. \n1 \nω \nb \n3 \n7 \n2 \n8 \n1 \n4 \n. \n0 \nω \nb \n0 \n1 \n0 \n3 \n3 \n2 \n4 \n0 \n. \n0 \n9 \n5 \n7 \n1 \n2 \n7 \n6 \n0 \n. \n0 \n9 \n3 \n9 \n9 \n3 \n2 \n7 \n0 \n. \n0 \n3 \n9 \n2 \n8 \n. \n2 \n0 \n6 \n0 \n6 \n. \n1 \nω \nc \n3 \n6 \n6 \n8 \n3 \n5 \n. \n0 \nω \nc \n3 \n8 \n4 \n7 \n1 \n6 \n0 \n8 \n. \n2 \n8 \n5 \n5 \n8 \n8 \n6 \n2 \n2 \n. \n2 \n2 \n4 \n4 \n1 \n7 \n5 \n4 \n1 \n. \n2 \n5 \n5 \n9 \n3 \n. \n0 \n6 \n5 \n5 \n4 \n. \n0 \nω \nd \n9 \n4 \n6 \n6 \n8 \n7 \n. \n0 \nω \nd \n2 \n0 \n9 \n5 \n9 \n7 \n5 \n5 \n. \n0 \n0 \n0 \n2 \n0 \n7 \n8 \n4 \n5 \n. \n0 \n9 \n9 \n8 \n5 \n6 \n2 \n5 \n3 \n. \n0 \n0 \n5 \n5 \n. \n0 \n2 \n0 \n2 \n6 \n. \n0 \nρ \na \n5 \n9 \n0 \n6 \n7 \n7 \n. \n0 \nρ \na \nTABLE I: Masses of nucleons and mesons, meson coupling constants, and the nuclear saturation densities in various DDRMF models. The \nD \nD \nX \nE \nM \n- \nD \nD \n1 \nZ \nL \n- \nD \nD \n9 \n3 \n9 \n0 \n0 \n0 \n0 \n. \n9 \n3 \n9 \nn \nm \n0 \n0 \n0 \n0 \n0 \n9 \n. \n8 \n3 \n9 \n] \nV \ne \nM \n[ \nn \nm \n9 \n3 \n9 \n0 \n0 \n0 \n0 \n. \n9 \n3 \n9 \np \nm \n0 \n0 \n0 \n0 \n0 \n9 \n. \n8 \n3 \n9 \n] \nV \ne \nM \n[ \np \nm \n3 \n5 \n5 \n7 \n2 \n3 \n3 \n. \n7 \n4 \n5 \nσ \nm \n6 \n1 \n2 \n9 \n1 \n6 \n. \n8 \n3 \n5 \n] \nV \ne \nM \n[ \nσ \nm \n3 \n8 \n7 \n0 \n0 \n0 \n0 \n. \n3 \n8 \n7 \nω \nm \n0 \n0 \n0 \n0 \n. \n3 \n8 \n7 \n] \nV \ne \nM \n[ \nω \nm \n3 \n6 \n7 \n0 \n0 \n0 \n0 \n. \n3 \n6 \n7 \nρ \nm \n0 \n0 \n0 \n0 \n. \n9 \n6 \n7 \n] \nV \ne \nM \n[ \nρ \nm \n. \n0 \n1 \n7 \n6 \n0 \n7 \n. \n0 \n1 \n) \n0 \nB \nρ \n( \nσ \nΓ \n9 \n2 \n4 \n1 \n0 \n0 \n. \n2 \n1 \n) \n0 \n( \nσ \nΓ \n. \n3 \n1 \n8 \n8 \n3 \n3 \n. \n3 \n1 \n) \n0 \nB \nρ \n( \nω \nΓ \n5 \n2 \n5 \n2 \n9 \n2 \n. \n4 \n1 \n) \n0 \n( \nω \nΓ \n0 \n0 \n2 \n5 \n1 \n. \n0 \n0 \nB \nρ \n1 \n8 \n5 \n1 \n. \n0 \n] \n3 \n- \nm \nf \n[ \n0 \nB \nρ \n2 \n. \n7 \n0 \n8 \n3 \n2 \n. \n7 \n) \n0 \nB \nρ \n( \nρ \nΓ \n4 \n3 \n9 \n0 \n5 \n1 \n. \n5 \n1 \n) \n0 \n( \nρ \nΓ \n0 \n2 \n4 \n7 \n8 \n4 \n8 \n. \n0 \n- \n- \n2 \n0 \n9 \n5 \n9 \n7 \n5 \n5 \n. \n0 \n- \n- \ni \n0 \n0 \n0 \n0 \n. \n0 \n8 \n9 \n- \n- \n- \nδ \nm \n- \n] \nV \ne \nM \n[ \nδ \nm \n- \n) \n0 \nB \nρ \n( \nδ \nΓ \n- \n) \n0 \n( \nδ \nΓ \n- \nδ \na \n- \nδ \na \nFIG. 1: The coupling constants of ω, σ , and ρ mesons as functions of vector density in various DDRMF models and several nonlinear RMF models. \n<!-- image --> \nB \nB \nTABLE II: Nuclear matter properties at saturation density generated by DDRMF parameterizations. \nDDRMF model. At low density region, the coupling constants in DDRMF models are usually stronger than those in nonlinear RMF modes, while weaker at higher density. \nThe saturation properties of symmetric nuclear matter calculated with different DDRMF effective interactions are listed in Table II, i.e. the saturation density, ρ 0 , the binding energy per nucleon, E/A , incompressibility, K 0 , symmetry energy, E sym, the slope of symmetry energy, L , and the effective neutron and proton masses, M ∗ n and M ∗ p . Notably, DD-MEX2 and PKDD exhibit a distinct L compared to other DDRMF sets, while DDVT demonstrates a notable difference in effective nucleon mass ( M ∗ N ). The fitting process of DD-MEX2 did not account for constraints from pure neutron matter and neutron skin, whereas the tensor coupling terms in DDVT suppress the magnitude of the σ field, resulting in a larger effective nucleon mass. To examine the influence of effective mass on neutron star properties, we also considered three non-relativistic density-functional theory parameterizations, namely BSk19, BSk20, and BSk21 [90, 91], based on Skyrme-type effective interactions. These parameterizations have an relatively large effective mass of 0 . 8 M N at saturation density and relatively small L values of 31 . 90 , 34 . 70 , and 46 . 60 MeV, respectively. \nThe binding energies per nucleon as functions of vector density for symmetric nuclear matter are plotted in panel (a) of Fig .2 with the present DDRMF parameterizations. These EOSs of nuclear matter below 0 . 2 fm -3 are almost identical since all the parameters were determined by properties of finite nuclei, whose central density is around nuclear saturation density ρ B 0 ∼ 0 . 15 fm -3 . Their differences increase from 0 . 30 fm -3 and they are separated into the softer group with DDV, DDVT, DDVTD and TW99, and the stiffer group with DD2, DD-ME1, DD-ME2, DD-MEX, DD-MEX1, DD-MEX2, DD-MEXY and DD-LZ1 since the scalar and vector coupling strengths in softer group sets are obviously weaker than those in stiffer group sets [16]. \nIn panel (b) of Fig. 2, the pressures in nuclear matter as functions of density are shown and compared to the constraints from heavy-ion collisions at 2 -4 ρ B 0 by Danielewicz et al. [92]. We can find that the EOSs from the softer group sets are completely consistent with the experiment data, while the other group is indeed stiffer than the heavy-ion collisions constraints. \n) \nB \nρ \n( \nωN \nΓ \n17 \n16 \n15 \n14 \n13 \n12 \n11 \n10 \n9 \n0.0 \n0.2 \n0.4 \n0.6 \n0.8 \n1.0 \n] \n(b) \nDD \n-ME1 \nDD-ME2 \nDD-MEX \nρ \n[fm \n-3 \nDD-MEX1 \nDD-MEX2 \nDD-MEXY \n) \nB \nρ \n( \nρN \nΓ \n16 \n14 \n12 \n10 \n8 \n6 \n4 \n2 \n0 \n0.0 \n0.2 \n0.4 \n0.6 \n0.8 \n1.0 \n] \n(c) \nρ \n[fm \nBigAppl( \nIUFSU \nTM1 \nNL3 \n-3 \nHowever, we want to emphasize here that the constraints from the heavy-ion collisions are strongly model-dependent, which is determined by many inputs, such as the NN interaction. To our knowledge, there were few investigations about heavy-ion collisions, which adopted the RMF model as NN interaction. It cannot certainly claim that the EOSs generated by DD2, DD-ME1, DD-ME2, DD-MEX, DD-MEX1, DD-MEX2, DD-MEXY and DD-LZ1 parameterizations are clearly excluded by the constraints of heavy-ion collisions. \nFIG. 2: The binding energies per nucleon the pressures as functions of vector density for symmetric nuclear matter with various DDRMF models. \n<!-- image --> \nFor the hyperonic star matter with strangeness degree of freedom, the hyperon masses are chosen as m Λ = 1115 . 68 MeV, m Σ + = 1189 . 37 MeV, m Σ 0 = 1192 . 64 MeV, m Σ -= 1197 . 45 MeV, m Ξ 0 = 1314 . 86 MeV, and m Ξ -= 1321 . 71 MeV [93], while the masses of strange mesons, φ and σ ∗ are taken as m φ = 1020 MeV and m φ = 980 MeV. We adopt the values from the SU(6) symmetry for the coupling constants between hyperons and vector mesons [94] both in nonlinear RMF and DDRMF models. Here, the coupling constants of the DDRMF model are taken as an example, \nΓ ω Λ = Γ ω Σ = 2Γ ω Ξ = 2 3 Γ ωN , 2Γ φ Σ = Γ φ Ξ = -2 √ 2 3 Γ ωN , Γ φN = 0 , Γ ρ Λ = 0 , Γ ρ Σ = 2Γ ρ Ξ = 2Γ ρN , Γ δ Λ = 0 , Γ δ Σ = 2Γ δ Ξ = 2Γ δN , (27) \nwhere Γ iN has been defined in Eq. (2)-Eq. (3) for DDRMF models. The coupling constants of hyperon and scalar mesons are constrained by the hyperon-nucleon potentials in symmetric nuclear matter, U N Y , which are defined by [76] \nU N Y ( ρ B 0 ) = -R σY Γ σN ( ρ B 0 ) σ 0 + R ωY Γ ωN ( ρ B 0 ) ω 0 , (28) \nwhere Γ σN ( ρ B 0 ) , Γ ωN ( ρ B 0 ) , σ 0 , ω 0 are the values of coupling strengths and σ, ω meson fields at the saturation density. R σY and R ωY are defined as R σY = Γ σY / Γ σN and R ωY = Γ ωY / Γ ωN . We choose the hyperon-nucleon potentials of Λ , Σ and Ξ as U N Λ = -30 MeV, U N Σ = +30 MeV and U N Ξ = -14 MeV, respectively from the recent hypernuclei experimental observables [58, 95, 96]. \nThe coupling constants between Λ and σ ∗ is generated by the value of the ΛΛ potential in pure Λ matter, U Λ Λ at nuclear saturation density, which is given as \nU Λ Λ ( ρ B 0 ) = -R σ Λ Γ σN ( ρ B 0 ) σ 0 -R σ ∗ Λ Γ σN ( ρ B 0 ) σ ∗ 0 + R ωY Γ ωN ( ρ B 0 ) ω 0 + R φ Λ Γ ωN ( ρ B 0 ) φ 0 , (29) \nWe similarly define that R σ ∗ Λ = Γ σ ∗ Λ / Γ σN and R φ Λ = Γ φ Λ / Γ ωN . R σ ∗ Λ is obtained from the Λ -Λ potential as U Λ Λ ( ρ B 0 ) = -10 MeV, which was extracted from the Λ bonding energies of doubleΛ hypernuclei. R φ Λ = -√ 2 / 2 is corresponding to the SU(6) symmetry broken case [58]. Here, the coupling between the Σ , Ξ hyperons and σ ∗ mesons are set as R σ ∗ Ξ = 0 , R σ ∗ Σ = 0 , since the information about their interaction is absent until now. The values of R σY and R σ ∗ Λ with above constraints in different DDRMF effective interactions are shown in Table III. Here, for DD-MEX parameter series, we only select the DDMEXset for investigation. \nTABLE III: The Coupling constants between hyperons and σ meson, Γ σY and Λ -σ ∗ , Γ σ ∗ Λ in different DDRMF effective interactions.', 'B. Neutron Star from DDRMF model': "FIG. 3: The pressure P versus energy density ε and the pressure P versus baryon density ρ B of neutron star matter from DDRMF models. Panels (a) and panel (b) for the neutron star matter. Panels (c) and panel (d) for the hyperonic star matter. The joint constraints in panel (a) and panel (c) are from GW170817 and GW190814. The corresponding speeds of sound in units of the speed of light shown in subfigure of panel (b) and panel (c). \n<!-- image --> \nTogether with the conditions of beta equilibrium and charge neutrality in Eq. (II) and Eq. (20), the EOSs of neutron star matter with DDRMF models can be obtained in panel (a) and panel (b) in Fig. 3, which shows the pressures of neutron star matter as a function of energy density and the pressures as functions of density, respectively. The crust EOS of the non-uniform matter is generated by TM1 parameterization with Thomas-Fermi approximation [22]. In the core of neutron star, EOSs of the uniform matter are calculated with various DDRMF sets discussed above. Their density-dependent behaviors are very similar with those in symmetric nuclear matter. At high density region, the stiffer group sets provide higher pressures due to the stronger vector potentials. The joint constraints on EOS extracted from the GW170817 and GW190814 are shown as a shaded band \nhere. When the energy density is smaller than 600 MeV/fm 3 , the EOSs from stiffer group sets satisfy the constraints from the gravitational wave detection, while the pressures obtained from softer group sets start to be lower than the constraint band from ε = 300 MeV/fm 3 . The speeds of sound in neutron star matter, c s from softer group sets in the insert of panel (b) are around 0 . 6 at ρ B = 1 . 0 fm -3 . They are much lower than those from stiffer group EOSs, which rapidly increase from ρ B = 0 . 2 fm -3 and can reach around 0 . 8 at high density. \nFIG. 4: Mass-radius relations and the tidal deformabilities as functions of star mass using the EOSs from different DDRMF sets. Panel (a) and panel (b) from the neutron star matter with different DDRMF models. Panels (c) and panel (d) from the hyperonic star matter. The dotted contours show the 68.3% and 95.4% credibility mass-radius constraints from PSR J0740+6620 [5] and PSR J0030+0451 [6]. The solid contours represent the central compact objects within HESS J1731-347 [17]. The horizontal error bar at 1.4 M /circledot is from GW170817 [10]. The horizontal error bar represents the tidal deformability constraint with a range of 70 < Λ 1 . 4 < 580 from GW170817. \n<!-- image --> \nThe mass-radius ( M -R ) relation of a static neutron star can be obtained by solving TOV equation in Eq. (II) with the EOS of neutron star matter as input. The M -R relations from various DDRMF models are shown in panel (a) and panel (b) of Fig. 4. The M -R relations from BSk series [90, 91] are also shown for comparison. Additionally, we include massradius observations from measurements of PSR J0740+6620 and PSR J0030+0451 by NICER, the mass-radius constraints on the compact central object (CCO) from HESS J1731-347 [17], as well as the radius constraint, R 1 . 4 = 11 . 9 ± 1 . 4 km, from gravitational wave event GW170817 [10]. As depicted in the figure, nearly all these DDRMF parameter sets yield mass-radius relations that satisfy the 95 . 4% confidence constraints of PSR J0740+6620 and PSR J0030+0451 while only the parameter sets in the stiffer group can satisfy the 68 . 3% confidence constraints of PSR J0740+6620. The maximum masses calculated from the EOSs of the stiffer group are all above 2 . 3 M /circledot , where DD-LZ1, DD-MEX, DD-MEX1, DD-MEX2, and DD-MEXY sets can even support the neutron star above 2 . 5 -2 . 6 M /circledot because of their strongest repulsive contributions from ω meson, which are in accord with the observed mass of the secondary compact object in GW190814, 2 . 50 -2 . 67 M /circledot . Among these parameter sets, PKDD and DD-MEX2 produces the largest radius of 1 . 4 M /circledot due to its highest values of L from Table (II), which exceeds the radius constraint from GW170817. What's more, these DDRMF parameter sets from the stiffer group fail to describe the measurement data from HESS J1731-347, while the EOSs from TW99, DDVT, DDVTD in the softer group, as well as the BSk series, which have larger effective masses, are capable of generating smaller radii in the lower mass region and can satisfy the 95 . 4% confidence constraint from HESS J1731-347. In particular, the M -R relation from BSk19 with smaller L aligns with the 68 . 3% uncertainties of HESS J1731-347 compared to BSk20 and BSk21. The GW170817 event also provides a valuable \nconstraint on the tidal deformability, which corresponds to a radius of a 1 . 4 M /circledot neutron star within the range of 70 < Λ 1 . 4 < 580 [10], which is represented in panel (b) of Fig 4 by a horizontal error bar, favoring the soft EOSs from the softer group of the DDRMF parameter sets, as well as the EOSs from BSk19, BSk20, and BSk21 sets. \nTABLE IV: Parameter Γ ρN ( ρ 0 ) and a ρ generated from the DDVT model for different L at the saturation point from E sym fixed at ρ B = 0 . 11 fm -3 . \nNext we want to further construct an EOS which can produce the M -R relation that complies with the HESS J1731-347 observation constraints using the DDRMF model. We choose the DDVT set to calculate the core EOS since the DDVT set, which include the tensor coupling and can produce rather smaller radii at the low-mass region due to its larger effective mass from Fig. (4) (a). The crust EOSs were derived from the IUFSU model at various values of L , with the symmetry energy fixed at ρ N = 0 . 11 fm -3 using the self-consistent Thomas-Fermi approximation [97] and this choice based on the similarity of their nuclear saturation properties, which are listed in the last line in Table (II). We choose two extreme crust EOSs generated by the IUFSU family of models with L = 47 , 110 MeV, while the core EOSs will be obtained from the current family of DDVT parameterizations with L = 26 , 30 , 40 MeV because a smaller L for the core EOS can produce a smaller radius in the intermediate mass region of neutron stars [20]. By manipulating the coupling strength of the isovector meson in the DDVT parameter set, the hadronic EOSs with L = 26 , 30 , 40 MeV can been obtained. It should be noted that when the slope of the symmetry energy falls below L = 26 MeV, the speed of sound in nuclear matter becomes less than zero. Table IV displays the ρ meson coupling constants for different L and the couplings of the original DDVT model with L = 42 . 35 MeV is also shown for comparison. It's obvious that there is a decrease in the ρ meson coupling strengths at nuclear saturation density, Γ ρN ( ρ 0 ) , as L is reduced. \nFIG. 5: Symmetry energy E sym as a function of the nucleon density for the DDVT sets with different L . The behaviors of the slope of symmetry energy L are shown in the insert. \n<!-- image --> \nThe density dependence of the symmetry energy E sym is plotted in Fig. 5 for the various L parameter sets presented in Table IV, which plays a crucial role in determining the properties of neutron stars. Smaller values of the L parameter yield larger symmetry energy values below the sub-saturation density, while exhibiting smaller values in the high-density region. Unlike the behavior of E sym obtained from nonlinear RMF parameter sets like the TM1 model in Ref. [98, 99], the density-dependent model's E sym converges above a density of 0.8 fm -3 due to the last term in Eq. (16) tends to approach zero exponentially and the symmetry energy at high densities is primarily determined by the contribution of the first term. \nThe corresponding M -R relations for neutron stars with the constructed EOSs are shown in Fig. 6. It is observed that the L parameter has minimal impact on the maximum mass and the corresponding radius of the neutron star. The mass and radius of a CCO within the supernova remnant HESS J1731-347 have been estimated as M = 0 . 77 +0 . 20 -0 . 17 M /circledot and R = 10 . 4 +0 . 86 -0 . 78 km, respectively [17], making the radius at 0 . 77 M /circledot an important factor to consider. As L increases from 47 MeV to 110 MeV, the radii at 0 . 77 M /circledot have a reduction of approximately about 0.35 km, and as the L of the core EOS decreases to 26 -40 MeV from the original value of 42 . 35 MeV, the radii corresponding to the low-mass neutron stars obtained from the EOSs become \nFIG. 6: Mass-radius relations of neutron stars obtained using the sets from Table (IV). \n<!-- image --> \nsufficiently small to meet the 68 . 3% credibility constraint, except for the combination of L = 47 MeV for the crust EOS and L = 40 MeV for the core EOS. Therefore, the observational data from HESS J1731-347 suggest the requirement of a crust EOS with a higher L parameter and a core EOS with a lower L parameter, representing an extremely soft EOS in both segments which is also consistent with the observables of PSR J0740+6620, PSR J0030+0451 from NICER, the GW170817 event, and the PREXII. This explains why the BSk19 EOS can also accurately describe HESS J1731-347. The properties of the neutron stars, i.e. maximum mass ( M max ), the corresponding radius ( R max ), the central density ( ρ c ), the radius ( R 1 . 4 ), and the dimensionless tidal deformability ( Λ 1 . 4 ) at 1 . 4 M /circledot , as well as the radius ( R 0 . 77 ) and the corresponding density at 0 . 77 M /circledot , can be seen in Ref. [20].", 'C. Hyperonic Star from DDRMF Model': "Similarly, EOSs from DDRMF models for the hyperonic star matter are obtained in the panel (c) and panel (d) of Fig. (3). The EOS of the inner crust is also chosen from the TM1 parameterization with self-consistent Thomas-Fermi approximation as before [97] and the EOSs of the core region are calculated with various DDRMF parameter sets. They almost become softer from ε ∼ 300 MeV/fm 3 compared to the neutron star matter in Fig. (3) (a) due to the appearances of hyperons. In high-density region, all of them are below the joint constraints on the EOS from GW170817 and GW190814 events. The onset densities of the first hyperon are marked by the full discretized symbols, which are around 0 . 28 -0 . 45 fm -3 . c 2 s of hyperonic star matter in panel (d) of Fig. (3) is not smooth anymore since the appearance of hyperon can sharply reduce the magnitude of c 2 s , especially at first onset density. For the hard EoS, the c 2 s becomes 0 . 6 in hyperonic star matter from 0 . 8 in neutron star matter at high-density region. \nThe M -R relations of hyperonic star from DDRMF parameter sets are shown in panel (c) of Fig. (4). The onset positions of the first hyperon in the relations are shown as the discretized symbols. After considering the strangeness degree of freedom, the maximum masses of the hyperonic star will reduce about 15% ∼ 20% . Among these parameter sets, the DD-LZ1, DD-MEX, DD-ME2, DD-ME1, DD2, and PKDD sets generate the hyperonic star heavier than 2 M /circledot . Furthermore, the central densities of the hyperonic star become higher compared to the neutron star, all of which are above 5 ρ 0 . The role of hyperons in the lower mass hyperonic star is strongly dependent on the threshold density of the first onset hyperon. The properties of a hyperonic star whose central density is below this threshold are identical to those of a neutron star. When the central density of the hyperonic star is larger than the threshold, the properties of hyperonic star will be influenced. For the softer EOSs, the lower mass neutron stars are more easily affected. This is because the central densities at low-mass region of the neutron stars from the softer EOSs are much higher than the onset densities of the first hyperons. However, the central densities of the harder EOSs at low-mass region are almost lower than the onset densities of the first hyperons. For example, the central densities from the softer EOSs at 1 . 4 M /circledot are about 0 . 50 -0 . 58 fm -3 and the onset densities of the first hyperons are about 0 . 35 -0 . 45 fm -3 , so the appearance of hyperons has a big impact on the low mass region. But for the harder EOSs, the central densities at 1 . 4 M /circledot are about 0 . 32 -0 . 35 fm -3 and the onset densities of the first hyperons are about 0 . 33 -0 . 35 fm -3 , the appearance of hyperons have very little effect. Therefore, the radii of the hyperonic stars at 1 . 4 M /circledot from DDV, DDVT, and DDVTD decrease from 12 . 3060 , 11 . 6058 , 11 . 4615 km to 10 . 8990 , 11 . 4515 , 10 . 9880 km, a reduction of about 5% compared to those of the neutron stars. \nThe dimensionless tidal deformabilities of hyperonic star are plotted in panel (d) of Fig. (4). For the harder EOSs, the hyperons only can influence the magnitudes of Λ at the maximum star mass region, while they can reduce the dimensionless \ntidal deformability at 1 . 4 M /circledot for the softer EOSs, such as TW99, DDV, DDVT, and DDVTD set, whose Λ 1 . 4 are closer to the constraints from GW170817. Therefore, the compact stars in the GW170817 events may be the hyperonic stars. \nThe properties of neutron star and hyperonic star, i.e., the maximum mass ( M max /M /circledot ), the corresponding radius ( R max ), the central density ( ρ c ), the radius ( R 1 . 4 ) and dimensionless tidal deformability ( Λ 1 . 4 ) at 1 . 4 M /circledot from present DDRMF models can be found in our previous works [16, 20, 76]. \nNow, the maximum masses of hyperonic stars are 1 . 50 ∼ 2 . 20 M /circledot , and the corresponding radii are in the range of 9 . 30 ∼ 11 . 86 km, which are reduced compared to the cases without considering the strangeness degree of freedom. The central densities become larger compared to those of neutron stars. The smallest radius of the hyperonic star at 1 . 4 M /circledot is 10 . 90 km from DDV, whose dimensionless tidal deformability is just 136 . In general, the maximum mass of a hyperonic star can exceed 2 M /circledot if the EOS is a hard type, whose maximum mass approaches 2 . 2 M /circledot with DDRMF model for a neutron star. Therefore, one solution to the 'hyperon puzzle' is to adopt the stiff EOS. The softer EOS only can describe the hyperonic star whose mass is around 1 . 5 M /circledot .", 'IV. SUMMARY': 'The neutron star consisting of nucleons and leptons, and the hyperonic star considering additional contributions from hyperons were reviewed in the DDRMF model due to recent rapid achievements about astronomical observations on the compact star. The DDRMF parameter sets (DD-LZ1, DD-MEX, DD-MEX1, DD-MEX2, DD-MEXY, DD-ME2, DD-ME1, DD2, PKDD, TW99, DDV, DDVT, DDVTD) were adopted to calculate the properties of the neutron star and hyperonic star, which were created by reproducing the ground-state properties of several finite nuclei with different considerations. \nThe EOSs of symmetric nuclear matter and neutron star matter at high-density region are separated into the softer type (DDV, DDVT, DDVTD , TW99) and stiffer one (DD-MEX, DD-MEX1, DD-MEX2, DD-MEXY, DD-ME2, DD-ME1, DD2, PKDD). The maximum masses of neutron stars generated by the softer EOSs cannot approach 2 . 0 M /circledot , which can hardly satisfy the 68.3% confidence constraints PSR J0740+6620. However, the radii of the corresponding neutron star are relatively small that can comply with the 95 . 4% confidence constraint from HESS J1731-347 and their dimensionless tidal deformability at 1 . 4 M /circledot are around 275 ∼ 510 which are in accord with the value extracted from the GW170817 event. Meanwhile, the harder EOS can lead to a very massive neutron star because of their strongest repulsive contributions from ω meson. In particular, the DD-MEX, DD-MEX1, DD-MEX2, DD-MEXY and DD-LZ1 parameter sets even can produce neutron stars with masses of 2 . 5 -2 . 6 M /circledot , which can explain the secondary object in GW190814 with a mass of 2 . 50 -2 . 67 M /circledot . \nTo further study the light compact object in HESS J1731-347, a reasonable hadronic EOS has been obtained by manipulating the coupling strength of the isovector meson of the DDVT parameter set for the core EOS and the crust EOS is obtained by the IUFSU parameterization set [97]. The observational data from HESS J1731-347 suggest the requirement of a crust EOS with a higher L parameter and a core EOS with a lower L parameter, representing an extremely soft EOS in both segments. The corresponding M -R relations can also be consistent with the observables of PSR J0740+6620, PSR J0030+0451 from NICER, the GW170817 event. \nFor the hyperonic stars, the meson-hyperon coupling strengths in DDRMF parameter sets were generated by the empirical hyperon-nucleon potential in symmetric nuclear matter at nuclear saturation density. The strangeness scalar and vector mesons were introduced to consider the Λ -Λ potential in hyperonic star with the bond energies of double Λ hypernuclei. The hyperonic star matter becomes softer compared to the neutron star matter. The onset densities of the first hyperon were around 2 ρ 0 ∼ 3 ρ 0 in present DDRMF models. 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2024ApJ...977..108M

In the solar atmosphere flux ropes are subject to currentdriven instabilities that are crucial in driving plasma eruptions ejections and heating. A typical ideal magnetohydrodynamics instability developing in flux ropes is the helical kink which twists the flux rope axis. The growth of this instability can trigger magnetic reconnection which can explain the formation of chromospheric jets and spicules but its development has never been investigated in a partially ionized plasma PIP. Here we study the kink instability in PIP to understand how it develops in the solar chromosphere where it is affected by chargeneutral interactions. Partial ionization speeds up the onset of the nonlinear phase of the instability as the plasma of the isolated plasma is smaller than the total plasma of the bulk. The distribution of the released magnetic energy changes in fully ionized plasma and PIP with a larger increase in internal energy associated with the PIP cases. The temperature in PIP increases faster also due to heating terms from the twofluid dynamics. PIP effects trigger kink instability on shorter time scales which is reflected in more explosive chromospheric flux rope dynamics. These results are crucial to understanding the dynamics of smallscale chromospheric structuresminifilament eruptionsthat thus far have been largely neglected but could significantly contribute to chromospheric heating and jet formation.

2024-12-01T00:00:00Z

['2024arXiv240906901M', 'arXiv:2409.06901', '10.48550/arXiv.2409.06901', '10.3847/1538-4357/ad79f6', '2024ApJ...977..108M']

['Solar chromosphere', 'Solar magnetic reconnection', '1479', '1504', 'Astrophysics - Solar and Stellar Astrophysics', 'Physics - Plasma Physics']

Kink Instability of Flux Ropes in Partially Ionized Plasmas

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']

https://arxiv.org/pdf/2409.06901.pdf

{'Kink instability of flux ropes in partially-ionised plasmas': '<!-- image --> \nG/i.pc/u.pc/l.pc/i.pc/a.pc \nM/u.pc/r.pc/t.pc/a.pc/s.pc \n, \n1, 2 \nA/n.pc/d.pc/r.pc/e.pc/w.pc \nH/i.pc/l.pc/l.pc/i.pc/e.pc/r.pc \n, \n2 \n/a.pc/n.pc/d.pc \nB/e.pc/n.pc \nS/n.pc/o.pc/w.pc \n2 \n1 Los Alamos National Laboratory, Los Alamos NM, 87545, USA \n2 Department of Mathematics and Statistics, University of Exeter, Exeter EX4 4QF, UK', 'ABSTRACT': 'In the solar atmosphere, flux ropes are subject to current driven instabilities that are crucial in driving plasma eruptions, ejections and heating. A typical ideal magnetohydrodynamics (MHD) instability developing in flux ropes is the helical kink, which twists the flux rope axis. The growth of this instability can trigger magnetic reconnection, which can explain the formation of chromospheric jets and spicules, but its development has never been investigated in a partially-ionised plasma (PIP). Here we study the kink instability in PIP to understand how it develops in the solar chromosphere, where it is affected by charge-neutral interactions. Partial ionisation speeds up the onset of the non-linear phase of the instability, as the plasma 𝛽 of the isolated plasma is smaller than the total plasma 𝛽 of the bulk. The distribution of the released magnetic energy changes in fully and partially-ionised plasmas, with a larger increase of internal energy associated to the PIP cases. The temperature in PIP increases faster also due to heating terms from the two-fluid dynamics. PIP effects trigger the kink instability on shorter time scales, which is reflected in a more explosive chromospheric flux rope dynamics. These results are crucial to understand the dynamics of small-scale chromospheric structures - mini-filament eruptions - that this far have been largely neglected but could significantly contribute to chromospheric heating and jet formation. \nKeywords: Solar chromosphere (1479) - Magnetohydrodynamics (1964) - Solar magnetic reconnection (1504)', '1. INTRODUCTION': 'In the solar chromosphere, the formation of highly dynamic features such as spicules (Roberts 1945) and jets (Mohler 1951) is strongly bound to changes in connectivity of the magnetic field. The energy fuelling such explosive events can be released through the onset of magnetohydrodynamics (MHD) instabilities, that promote the conversion of stored magnetic energy through processes such as magnetic reconnection (Yamada et al. 2010; De Pontieu et al. 2012). The energy release associated with these processes promotes the formation of several types of dynamical structures, and can provide an important contribution to the chromospheric heating (Roberts 1945; Mohler 1951; Osterbrock 1961). \nFlux ropes, which are ubiquitous in the solar atmosphere, are twisted by convective motions, reaching an equilibrium that can be unstable to the kink modes (Hood & Priest 1979; Hood 1992; Priest & Forbes 2000). Kink modes define a class of ideal, current-driven MHD instabilities that can become particularly important in triggering explosive events. When subjected to the kink instability, a plasma column bends and develops displacements that are transverse to the column axis and driven by an unbalance in magnetic pressure (Hood 1992). The instability depends both on the length of the flux tube and the twist of magnetic field lines wrapped around it (Velli et al. 1990): if the field lines wrap multiple times along the length of the flux tube, then the system becomes unstable to kink modes (Biskamp 2000). \nWhen a plasma column is traversed by an axial magnetic field, the 𝑚 = 1 kink mode develops a helical deformation of the plasma column. Such deformation is magnified by the compression of the magnetic field lines at the internal side of the bends, with a consequent increase in magnetic pressure, and the expansion of field lines at the external side of the bends, that results in a decrease of magnetic pressure (Goedbloed & Poedts 2004). The magnetic pressure imbalance leads to an increase in the distortion \ngiuliamurtas31994@gmail.com \nof the plasma column, which consequently results in a larger magnetic pressure gradient and a faster growth of the instability. The onset of the helical kink instability reduces the magnetic energy within the flux rope by converting the twist of the magnetic field lines in an helical deformation of the rope axis itself, namely the writhe (Torok et al. 2014). Frequent observations of writhing in erupting filaments and prominences in the solar corona led to the suggestion, supported by many works, that the helical kink instability could be responsible for triggering filament eruptions and CMEs (Sakurai 1976; Sturrock et al. 2001; Torok & Kliem 2005; Fan 2005). \nIn systems where magnetic diffusivity is significant the kink instability develops in flux ropes, with an initial growth phase followed by a nonlinear resistive phase, characterised by current sheets formation and the onset of magnetic reconnection (Parker 1988; Hood et al. 2009; Bareford & Hood 2015; Snow et al. 2018). The compression and expansion of magnetic field lines triggers the relaxation process (Taylor 1974, 1986), where reconnection straightens the magnetic field lines and lead to the release of the energy stored in the highly stressed and twisted field configuration. The magnetic field tends to relax toward a state of minimum magnetic energy defined by a force-free field, called the Taylor state (Taylor 1974), where the magnetic helicity is preserved. This phase results in a less twisted configuration of the flux rope. During relaxation, magnetic energy is converted into kinetic energy and heat (Heyvaerts & Priest 1984; Browning et al. 2008). The role of relaxation has been examined through several studies involving numerical simulations of cylindrical coronal loops (Browning et al. 2008; Botha et al. 2011; Gordovskyy et al. 2016; Pinto et al. 2016). \nFlux ropes have been well studied in the fully-ionised coronal environment, where the kink instability has been identified in several observations (Srivastava et al. 2010; Liu et al. 2016), but it is unclear how these structures evolve in the lower atmosphere. Below the corona, twisted magnetic fields reconnect in regions where the plasma composition is heterogeneous and the plasma is only partially-ionised: this could be the case of micro-filament and mini-filament eruptions (Sterling et al. 2015, 2016; Sterling & Moore 2016; Samanta et al. 2019; Sterling et al. 2020), taking place in small-scale loop systems observed in extreme ultraviolet and X-rays (Harden et al. 2021; Madjarska et al. 2022; Sterling et al. 2022). The kink instability has been investigated as an explanation for jet formation through several numerical studies between the upper chromosphere (Nishizuka et al. 2008) and the lower corona (Yokoyama & Shibata 1995; Moreno-Insertis & Galsgaard 2013; Fang et al. 2014). However, numerical models that study unstable flux ropes in partially-ionised chromospheric plasma are still absent. Therefore, it is essential to investigate how the helical kink instability grows in a partially-ionised plasma, and the differences with the development of the same instability in a fully-ionised plasma. \nThe goal of this study is to examine the kink instability developing in single loops as a potential explanation for mini-filament eruptions in the solar chromosphere, and compare a fully-ionised plasma case (MHD) with two partially-ionised plasma cases (PIP). In Section 2, the models of our simulations are presented. In Section 3 results are discussed, with a focus on the energy release during the onset of magnetic reconnection within the nonlinear phase of the kink instability. Conclusions are presented in Section 4.', '2. METHODS': 'The simulations presented in this work are performed through the (PIP) code (Hillier et al. 2016; Snow & Hillier 2021). The equations are solved throughout the domain through a fourth-order central difference scheme, and the time updates are computed by a four-step Runge-Kutta scheme for time integration. \nThe partially-ionised plasma is modelled through a two-fluid ion-neutral hydrogen plasma. Ions and neutrals are described by two separate sets of equations and coupled though elastic collisions, charge-exchange, collisional ionisation and recombination. The neutral fluid is governed by: \n𝜕𝜌 𝑛 𝜕𝑡 + ∇ · ( 𝜌 𝑛 v 𝑛 ) = 𝐷, (1) \n𝜕 𝜕𝑡 ( 𝜌 𝑛 v 𝑛 ) + ∇ · ( 𝜌 𝑛 v 𝑛 v 𝑛 + 𝑝 𝑛 I ) = R , (2) \n𝜕𝑒 𝑛 𝜕𝑡 + ∇ · [ v 𝑛 ( 𝑒 𝑛 + 𝑝 𝑛 )] = 𝐸, (3) \n𝑒 𝑛 = 𝑝 𝑛 𝛾 -1 + 1 2 𝜌 𝑛 𝑣 2 𝑛 , (4) \n𝑇 𝑛 = 𝛾 𝑝 𝑛 𝜌 𝑛 , (5) \n𝜕𝜌 𝑝 𝜕𝑡 + ∇ · ( 𝜌 𝑝 v 𝑝 ) = -𝐷, (6) \nwhile the ionised fluid is governed by: \n𝜕 𝜕𝑡 ( 𝜌 𝑝 v 𝑝 ) + ∇ · 𝜌 𝑝 v 𝑝 v 𝑝 + 𝑝 𝑝 I -BB + B 2 2 I ! = -R , (7) \n𝜕 𝜕𝑡 𝑒 𝑝 + 𝐵 2 2 ! + ∇ · [ v 𝑝 ( 𝑒 𝑝 + 𝑝 𝑝 ) - ( v 𝑝 × B ) × B + 𝜂 (∇ × B ) × B ] = -𝐸 -Φ 𝐼 + 𝐴 heat , (8) \n𝜕 B 𝜕𝑡 - ∇ × ( v 𝑝 × B -𝜂 ∇ × B ) = 0 , (9) \n𝑒 𝑝 = 𝑝 𝑝 𝛾 -1 + 1 2 𝜌 𝑝 𝑣 2 𝑝 , (10) \n∇ · B = 0 , (11) \n𝑇 𝑝 = 𝛾 𝑝 𝑝 2 𝜌 𝑝 . (12) \nThe subscripts 𝑝 and 𝑛 identify physical quantities of the ion-electron plasma and of the neutral fluid respectively. The variables v , 𝑝 , 𝜌 , 𝑇 and 𝑒 are the fluids velocity, gas pressure, density, temperature and internal energy, 𝛾 = 5 / 3 is the adiabatic index and B is the magnetic field. \nThe terms 𝐷 , R and 𝐸 are respectively the source terms for mass, momentum and energy transfer between the two species: \n𝐷 = Γ rec 𝜌 𝑝 -Γ ion 𝜌 𝑛 , (13) \nR = -𝛼 𝑐 𝜌 𝑛 𝜌 𝑝 ( v 𝑛 -v 𝑝 ) + Γ rec 𝜌 𝑝 v 𝑝 -Γ ion 𝜌 𝑛 v 𝑛 , (14) \n𝐸 = -𝛼 𝑐 𝜌 𝑛 𝜌 𝑝 " 1 2 ( v 2 𝑛 -v 2 𝑝 ) + 𝑇 𝑛 -𝑇 𝑝 𝛾 ( 𝛾 -1 ) # + 1 2 ( Γ rec 𝜌 𝑝 v 2 𝑝 -Γ ion 𝜌 𝑛 v 2 𝑛 ) + Γ rec 𝜌 𝑝 𝑇 𝑝 -Γ ion 𝜌 𝑛 𝑇 𝑛 𝛾 ( 𝛾 -1 ) . (15) \nBoth fluids are subject to the ideal gas law. The factor of 2 in equation (12) is included to account for the electron pressure in the plasma fluid. \nThe two-fluid collisional coupling is mediated by 𝛼 𝑐 ( 𝑇 𝑛 , 𝑇 𝑝 , 𝑣 𝐷 ) (Draine 1986; Zank et al. 2018), that is defined as: \n𝛼 𝑐 = 𝛼 𝑐 ( 0 ) √︂ 𝑇 𝑛 + 𝑇 𝑝 2 √︄ 1 + 9 𝜋 64 𝛾 2 ( 𝑇 𝑛 + 𝑇 𝑝 ) 𝑣 2 𝐷 , (16) \nwhere 𝛼 𝑐 ( 0 ) is the initial coupling and 𝑣 𝐷 = | v 𝑛 -v 𝑝 | is the magnitude of the drift velocity between the neutral components and the hydrogen plasma. The collisional frequencies are determined by the product of 𝛼 𝑐 and the fluids density: the ion-neutral collisional frequency is defined as 𝛼 𝑐 𝜌 𝑛 = 𝜏 -1 𝑖𝑛 , and the neutral-ion collisional frequency is 𝛼 𝑐 𝜌 𝑝 = 𝜏 -1 𝑛𝑖 , where 𝜏 𝑖𝑛 and 𝜏 𝑛𝑖 are the ion-neutral and neutral-ion coupling time scales, respectively. While the coupling between ions and neutrals is governed by 𝛼 𝑐 , the collisional coupling between ions and electrons is modelled by imposing a diffusivity 𝜂 in the system. The distribution for 𝜂 chosen in this work is spatially irregular and time-dependent: more details can be found in Section 2.1. Initially, the two fluids are in thermal and ionisation equilibrium. \nThe terms Γ rec and Γ ion are the recombination and collisional ionisation rates for a hydrogen atom. The normalised empirical forms of the rates (Voronov 1997; Smirnov 2003) are: \nΓ rec = 𝜌 𝑝 √︁ 𝑇 𝑝 √︁ 𝑇 𝑓 𝜉 𝑝 ( 0 ) 𝜏 IR , (17) \nΓ ion = 𝜌 𝑝 𝑒 -𝜒 𝜒 0 . 39 0 . 232 + 𝜒 ! ˆ 𝑅 𝜉 𝑝 ( 0 ) 𝜏 IR , (18) \n𝜒 = 13 . 6 𝑇 𝑓 𝑇 𝑒 0 𝑇 𝑝 , (19) \nˆ 𝑅 = 2 . 91 · 10 -14 2 . 6 · 10 -19 √︁ 𝑇 𝑒 0 . (20) \nTwo characteristic temperatures appear in Equations (17)-(20), and are based on a physical reference electron temperature 𝑇 0, expressed in Kelvin. 𝑇 𝑒 0 is the value of 𝑇 0 converted in electron volts. In our model we assume that protons and electrons are at the same temperature. Therefore, the choice of 𝑇 0 at the beginning of calculations determines the initial ion fraction 𝜉 𝑝 ( 0 ) . 𝑇 𝑓 is a normalisation factor defined as \n𝑇 𝑓 = 𝛽𝛾 4 2 𝜉 𝑝 ( 0 ) 𝜉 𝑛 ( 0 ) + 2 𝜉 𝑝 ( 0 ) , (21) \nand ensures that the reference temperature 𝑇 0 becomes the initial non-dimensional temperature of the simulation. The free parameter 𝜏 IR determines the ratio between recombination time scale and dynamic time: following our normalisation, 𝜏 IR states the initial recombination rate. \nThe terms Φ 𝐼 and 𝐴 heat in equation (8) are associated to the ionisation potential and account for radiative losses. Φ 𝐼 approximates the energy removed by the system through ionisation, while 𝐴 heat is an arbitrary heating term included to obtain an initial equilibrium. Their non-dimensional forms are: \nΦ 𝐼 = Γ ion 𝜌 𝑛 ˆ Φ , (22) \n𝐴 heat = Γ ion ( 𝑡 = 0 ) 𝜌 𝑛 ( 𝑡 = 0 ) ˆ Φ , (23) \nwhere ˆ Φ = 13 . 6 𝛽 2 𝐾 𝐵 𝑇 0 ensures consistency between the normalisation of ionisation potential and the equations modelling the system, which are normalised to the total Alfv\'en speed 𝑣 𝐴 = 1 and the total density 𝜌 0. As we include these heating/cooling terms associated to ionisation and recombination processes, the total energy of the system is not conserved in PIP. It is however conserved in fully-ionised MHD, as the solver is known to conserve energy in the absence of losses. More details on the atomic internal structure used to estimate the ionisation potential can be found in Snow & Hillier (2021). \nThe physical variables in our system are non-dimensionalised as follows: \nr → 𝑟 \' ˜ r , B → 𝐵 \' ˜ B , v → 𝑣 \' ˜ v , 𝑡 → 𝑡 \' ˜ 𝑡, 𝜌 → 𝜌 \' ˜ 𝜌, 𝑃 → 𝑃 \' ˜ 𝑃, (24) \nwhere 𝑟 \' is the radius of the flux tube, 𝐵 \' = 1 is the initial axial field at 𝑟 = 0, 𝑣 \' = 𝑣 𝐴 = 𝐵 \' / √ 𝜌 \' as the permeability is set as 𝜇 0 = 1, the characteristic time scale 𝑡 \' = 𝑡 𝐴 = 𝑟 \' / 𝑣 𝐴 is the Alfv\'en transit time across the flux rope, 𝜌 \' = 1 is the total density and the total gas pressure 𝑃 \' = 𝛽 / 2. This non-dimensionalisation is consistent with the work in Hood et al. (2009).', '2.1. Initial conditions': "The initial setup is provided by a twisted magnetic flux tube in force-free equilibrium that is unstable to an ideal MHD kink instability. The initial non-potential, force-free magnetic field satisfies the condition: \n∇ × B = 𝛼 ( r ) B , (25) \nwhere 𝛼 = 𝜇 0 J · B / 𝐵 2 is the initial stress of the flux tube, and can relax through reconnection and reach a lower energy state, while preserving magnetic helicity. The field relaxes towards the minimum energy state, which must have a linear or constant 𝛼 -field profile: for this mechanism to work, the magnetic field must be sufficiently stressed beyond the relaxed state, before an instability releases the stored magnetic energy. \nIn our simulations we model a cylindrical flux rope with a radius 𝑟 = 1. The magnetic field components, consistent with the case studied in Hood et al. (2009) for an initial smooth 𝛼 -profile, are reported in the following equations in cylindrical coordinates. For 𝑟 < 1 we set: \n𝐵 𝜃 ( 𝑟 ) = 𝜆𝑟 ( 1 -𝑟 2 ) 3 , (26) \n𝐵 𝑧 ( 𝑟 ) = √︂ 1 -𝜆 2 7 + 𝜆 2 7 ( 1 -𝑟 2 ) 7 -𝜆 2 𝑟 2 ( 1 -𝑟 2 ) 6 , (27) \n𝛼 = 2 𝜆 ( 1 -𝑟 2 ) 2 ( 1 -4 𝑟 2 ) 𝐵 𝑧 ( 𝑟 ) , (28) \nwhile for 𝑟 ≥ 1 these components become: \n𝐵 𝜃 = 0 , 𝐵 𝑧 = √︂ 1 -𝜆 2 7 , 𝛼 = 0 . (29) \nFigure 1. Left: initial 2D profile of 𝐵 𝑧 at 𝑧 = 0 (top panel) and 𝐽 𝑧 at 𝑧 = 0 (bottom panel) in the 𝑥𝑦 -plane for all simulations. The black contour lines in the bottom panel represent the field lines of the 𝐵 𝑥 𝐵 𝑦 vector field. Right: initial 3D setup of the twisted magnetic field lines of case M1. The slices at 𝑧 = -10 , 0 , 10 represent the magnitude of the plasma temperature. \n<!-- image --> \nThis choice of the initial magnetic field components leads to a configuration with zero net-current. In the equations above, 𝜆 is a constant parameter corresponding to a measure of the twist in the field. Due to the requirement from equation (27) that 𝐵 2 𝑧 must be positive, the value of 𝜆 is constrained to be 𝜆 < 64 / 965 √ 1351 = 2 . 438. In this work we impose 𝜆 = 1 . 8, consistent with Hood et al. (2009): the choice of these initial conditions also leads to a change of sign for 𝛼 at 𝑟 = 0 . 5. Figure 1 shows the initial profiles of 𝐵 𝑧 and the current density component 𝐽 𝑧 in the 𝑥𝑦 -plane ( 𝑧 = 0) set for all simulations. The current density profile is continuous across the domain, and a smooth transition occurs from 𝐽 ≠ 0 to 𝐽 = 0 at 𝑟 = 1. The three-dimensional form of the magnetic field can be observed in Figure 1, compared to the initial plasma temperature of case M1. \nThe plasma 𝛽 associated with the bulk pressure is 0.1, consistent with the upper chromosphere (Gary 2001). Chromospheric neutrals play an important role in the decay of the non force-free components of the photospheric field, as it transitions to the force-free coronal field (Goodman 2000; Khodachenko et al. 2004; Arber et al. 2009). In a two-fluid system, the balance with the Lorentz force can only be achieved by varying the plasma pressure: as the charges have an effective 𝛽 of 0.02, as such we expect the system to be approximately force-free. Therefore, a force-free equilibrium is a suitable starting point for this work. \nThe calculations are run with a non-uniform, current-dependent resistivity, that accounts for electron-ion collisions and is determined as follows: \n𝜂 = 𝜂 0 + 0 , | 𝐽 | < 𝐽 crit 𝜂 1 , | 𝐽 | > 𝐽 crit (30) \nwhere 𝜂 0 is the background uniform resistivity, set equal to 𝜂 0 = 10 -4 , and 𝜂 1 = 10 -3 is the anomalous resistivity component that is only switched on when the magnitude of the current exceeds a critical value, chosen to be 𝐽 crit = 5 to be above the initial maximum current magnitude determined by the initial conditions. This choice of diffusivity, resolved by the grid, is similar to the work of Arber et al. (1999), also used by Browning et al. (2008) and Hood et al. (2009). As ambipolar diffusion represents strongly coupled two-fluid effects in a single-fluid regime (Khomenko & Mart'ınez-G'omez 2024), its role is already covered by the more complex two-fluid physics of our setup. As such, no separate resistive term (i.e. ambipolar diffusion, Brandenburg & Zweibel 1994, 1995; Chae & Litvinenko 2021) is included to account for the effects of neutral-charge interactions. \nIn order to trigger the instability in the system, a small velocity perturbation is included. The initial velocity perturbation, imposed in both the ion and neutral fluids to break the initial equilibrium, is a 𝑥𝑦 -plane white noise of the form: \n𝑣 𝑥, 𝑝 , 𝑣 𝑦, 𝑝 = 0 . 05 𝑒 -( 𝑥 2 + 𝑦 2 ) · ( random noise ) . (31) \nThe perturbation results in a larger velocity magnitude at the centre of the flux rope, perpendicular to the flux rope axis, and a smaller velocity magnitude at the extremities of the flux rope. As this perturbation is set to be the same for all cases, its effects \nTable 1. List of the simulation parameters. \nmight vary following the different coupling between ions and neutrals, resulting in the onset of different modes. Evidence of this process is discussed in Section 3. \nThe simulations are run in a Cartesian computational domain 𝑥 = [-2 , 2 ] , 𝑦 = [-2 , 2 ] and 𝑧 = [-10 , 10 ] with a uniform grid. Although the analytical equilibrium is presented in cylindrical coordinates, the field components have been mapped onto the Cartesian grid to provide the initial states for the simulations. The domain is resolved by 496 × 496 × 868 grid points, corresponding to a cell size of Δ 𝑥 = 8 · 10 -3 , Δ 𝑦 = 8 · 10 -3 and Δ 𝑧 = 2 . 3 · 10 -2 . All boundaries are periodic, mimicking previous studies of instabilities in flux ropes (Shafranov 1957; Kruskal et al. 1958). Following this choice, the system simulates a potentially infinite number of infinitely long flux ropes, but the boundaries are set sufficiently far that the ropes never interact with each other, and the kink instability can be studied for a single structure.", '3. RESULTS': "We analyse the kink instability in three simulations, corresponding to a fully-ionised plasma case (M1) and two PIP cases (P1 and P2), run at different collisional coupling. M1 represents the limit for a completely coupled system ( 𝛼 𝑐 →∞ ), where neutrals and charges act as a single fluid. P1 corresponds to a weakly coupled ion-neutral plasma, where collisions occur on timescales ∼ 𝜌 -1 for a reference density 𝜌 , while P2 is run with a stronger collisional coupling between the fluids, with collisions occurring on timescales ∼ 0 . 1 𝜌 -1 . The parameter 𝜏 IR is varied in the PIP cases to maintain the same proportionality between collision frequencies and ionisation/recombination rates. The full array of parameters is shown in Table 1. \nThe development of the kink instability is shown in Figure 2 for three evolution stages in both fully-ionised plasmas (M1) and partially-ionised plasmas (P1 and P2). At the onset of the instability, the current density is larger at the centre of the flux rope, as shown by panels ( 𝑎 ), ( 𝑏 ) and ( 𝑐 ). Under the distortion generated by the kink, the plasma flow pushes the field lines to form current sheets, that appear in panels ( 𝑑 ), ( 𝑒 ) and ( 𝑓 ) of Figure 2 as dark, thin and elongated regions where 𝐽 > 5. As the magnetic tension reduces with the straightening of the field lines, and the gas pressure increases with the thermal energy released from reconnection, the flux rope expands. During the expansion phase, shown at early times in panels ( 𝑑 ), ( 𝑒 ) and ( 𝑓 ), current sheets form at different locations and the reconnection process moves to regions further away from the centre of the flux rope. At later stages, shown in panels ( 𝑔 ), ( ℎ ) and ( 𝑖 ) reconnection occurs mostly near the external boundary of the flux rope for P1 and P2, while in case M1 several structures of high concentrations of current density are still present around the centre of the flux rope. \nAs shown by Figure 2, where the same instability phase appears in the bottom panels at 𝑡 = 120 for M1, 𝑡 = 105 for P2 and 𝑡 = 65 for P1, respectively, the kink instability evolves much faster in partially-ionised plasmas than in fully-ionised plasmas. By comparing the two PIP cases, the flux rope expansion is faster when the collisional coupling 𝛼 𝑐 is smaller. The same development stage of the instability occurs earlier ( 𝑡 = 65) for case P1 (where 𝛼 𝑐 ( 0 ) = 1) than case P2 ( 𝑡 = 105, where 𝛼 𝑐 ( 0 ) is set to be equal to 10). This occurs as the growth of the kink instability scales with the Alfv'en speed, which is inversely proportional to the density. In case M1 the fluid is completely ionised, and the ion density coincides with the bulk density. In both PIP cases, run with the same bulk density as M1, the effective Alfv'en speed is larger as it depends on a combination of the smaller ion density and the density of the neutrals that are collisionally coupled to the charges: for this reason, the kink instability grows faster. The lower coupling in case P1 speeds up the reconnection rate and pushes the flux rope to expand faster than in case P2, where the stronger coupling between ions and neutrals results in the evolution of the system more closely resembling that of a single, completely ionised fluid. \nThe charges temperature 𝑇 𝑝 and the current density magnitude at the three instability phases in Figure 2 are shown in Figure 3 in the 𝑥𝑧 -plane ( 𝑦 = 0) laying along the flux rope axis for cases P1 (top six panels), P2 (central six panels) and M1 (bottom six panels). The writhe of the flux rope axis is observed in the top two panels of each set, corresponding to 𝑡 P1 = 45, 𝑡 P2 = 75 and 𝑡 M1 = 90. In case P1, where the two fluids are almost decoupled, four crests located at regular distances from each other are observed along the 𝑧 -axis, which suggests the onset of a single, dominant mode of the instability. Similarly, four crests are \nFigure 2. Contour plot of the current density magnitude 𝐽 at 𝑧 = 0, showing the development of the kink instability in M1 (left), P2 (centre) and P1 (right). Each row displays the same stage of development of the kink instability. Panels ( 𝑎 ), ( 𝑏 ) and ( 𝑐 ) show the onset of the kink instability. Panels ( 𝑑 ), ( 𝑒 ) and ( 𝑓 ) display the evolution at a later stage, and the formation on current sheets inside the flux rope. Panels ( 𝑔 ), ( ℎ ) and ( 𝑖 ) show the flux rope expanding and the formation of further current sheets. The arrows show the direction of the plasma flow v 𝑝 . Times are given in the same non-dimensional unit for all three cases. 𝐽 is saturated by fixing the maximum value of the colour scale to the critical value 𝐽 crit = 5, in order to enhance the structures characterised by strong currents. \n<!-- image --> \nobtained in case M1. In the intermediate case, however, it is possible to identify five crests, with a new, central peak appearing around 𝑧 = 0. While a dominant mode is observed both in the PIP case with the lowest coupling and in the full coupling limit (M1), the very presence of oscillations with crests located at irregular distances in P2 suggests the overlapping of two modes of the instability. Therefore, the partially coupled neutrals leads to the selection of different modes of the kink instability. \nFigure 4 shows the time variation of the energy components integrated over the volume 𝑉 of the simulation box, and the maximum current density of cases M1, P1 and P2. In order to better show the variation of the energy terms, the net variation of internal and magnetic energy are displayed. The total kinetic energy is shown fully, and it differs from zero at 𝑡 = 0 due to the initial velocity perturbation. \nThe passage from the linear growth phase to the nonlinear reconnection phase of the kink instability is identified by the decrease of magnetic energy Δ 𝑀 (panel 𝑏 of Figure 4), defined as: \nΔ 𝑀 = ∫ 𝑉 𝐵 2 2 𝑑𝑉 -∫ 𝑉 𝐵 2 ( 𝑡 = 0 ) 2 𝑑𝑉, (32) \nand the increase of kinetic and internal energy components (panels 𝑎 and 𝑐 of Figure 4) due to the energy conversion from reconnection, which for case M1 starts at 𝑡 ∼ 90, for case P1 is at 𝑡 ∼ 50 and for case P2 is at 𝑡 ∼ 80. The maximum current density reached in all three cases is approximately similar, as suggested by the trend of 𝐽 displayed in panel ( 𝑑 ). \nt = 55 \nt = 105 \nt = 105 \nt = 65 \nt = 65 \nt = 65 \nt = 65 \nt = 90 \nt = 105 \nt = 105 \nt = 75 \nt = 75 \nt = 45 \nt = 45 \nt = 90 \nt = 90 \nt = 65 \nt = 105 \nt = 105 \nt = 90 \nt = 55 \nt = 45 \nt = 45 \nt = 75 \nt = 75 \nt = 90 \nt = 90 \nt = 105 \nt = 105 \nt = 105 \nt = 45 \nt = 55 \nt = 65 \nt = 55 \nt = 90 \nt = 75 \nt = 90 \nt = 90 \nt = 105 \nt = 120 \nt = 105 \nt = 120 \nt = 55 \nt = 65 \nt = 65 \nt = 90 \nt = 105 \nt = 105 \nt = 120 \nt = 90 \nt = 75 \nt = 105 \nt = 65 \nt = 120 \nt = 90 \nt = 75 \nt = 90 \nt = 75 \nt = 90 \nt = 105 \nt = 90 \nt = 75 \nt = 90 \nt = 105 \nt = 90 \nt = 105 \nt = 105 \nt = 120 \nt = 120 \nt = 90 \nt = 105 \nt = 105 \nt = 105 \nt = 120 \nt = 90 \nt = 105 \nt = 120 \nt = 90 \nt = 105 \nt = 90 \nt = 90 \nt = 105 \nt = 120 \nt = 105 \nt = 120 \nt = 105 \nt = 120 \nt = 120 \nt = 105 \nt = 105 \nt = 65 \nt = 65 \nt = 55 \nt = 55 \nt = 65 \nt = 65 \nt = 90 \nt = 90 \nt = 105 \nt = 105 \nt = 75 \nt = 75 \nt = 45 \nt = 45 \nt = 90 \nt = 90 \nt = 65 \nt = 65 \nt = 65 \nt = 105 \nt = 105 \nFigure 3. Contour plot of the plasma temperature 𝑇 𝑝 (left) and current density magnitude 𝐽 (right) in the 𝑥𝑧 -plane ( 𝑦 = 0), showing the development of the kink instability in case P1 (top panels), P2 (central panels) and M1 (bottom panels). Each set of panels displays the same stage of development of the kink instability shown in Figure 2. 𝑇 𝑝 is saturated by fixing the maximum value of the colour scale to the value 0.1, to enhance the plasma structures where the largest temperature increase occurs. Similarly, 𝐽 is saturated by fixing the maximum value of the colour scale to the critical value 𝐽 crit = 5 in order to enhance the structures characterised by strong currents. \n<!-- image --> \nRegarding the distribution of the converted magnetic energy, the individual contributions of plasma (blue) and neutrals (red) to the kinetic energy in panel ( 𝑎 ) and the total kinetic energy, defined as: \n𝐾 TOT = ∫ 𝑉 1 2 𝜌 𝑝 v 2 𝑝 + 1 2 𝜌 𝑛 v 2 𝑛 ! 𝑑𝑉, (33) \nare smaller than the MHD case, as shown by the dashed black line (P1) and the dotted black line (P2), while the variation of the total internal energy 𝐼 TOT , shown in panel ( 𝑐 ) and calculated as: \nΔ 𝐼 TOT = ∫ 𝑉 𝑝 𝑝 + 𝑝 𝑛 𝛾 -1 -𝑝 𝑝 ( 𝑡 = 0 ) + 𝑝 𝑛 ( 𝑡 = 0 ) 𝛾 -1 ! 𝑑𝑉, (34) \nis larger in the PIP cases than in the MHD case (solid black line) when compared at the same growth phase of the instability, despite smaller single contribution from charges and neutrals. \nt = 55 \nt = 90 \nFigure 4. Kinetic energy ( 𝑎 ), time variation of ( 𝑏 ) magnetic energy Δ 𝑀 and ( 𝑐 ) internal energy Δ 𝐼 , and maximum current density ( 𝑑 ) for M1, P1 and P2. In all four panels, the solid lines are associated to variables of the MHD case (M1), the dashed lines refer to case P1 and the dotted lines refer to case P2. In panels ( 𝑎 ) and ( 𝑐 ) blue lines are associated to the plasma energy components, red lines are associated to the neutral energy components and black lines are associated to the bulk energy component (only plasma for case M1, plasma and neutrals for cases P1 and P2). The kinetic energy is the only energy component that is perturbed initially. \n<!-- image --> \nThe difference in the increase of the bulk internal energy between PIP and MHD cases means that the temperature of cases P1 and P2 grows faster. In general, it must be expected that the temperature is higher in the PIP cases than in the MHD case: this comes from the inclusion of further heating terms that result from the interaction of charges and neutrals. This will be discussed in more detail in Section 3.2. \nFollowing the work of Browning et al. (2008), the instability growth rate can be calculated as: \n𝜎 = 1 2 𝑑 log ( 𝐾𝐸 ) 𝑑𝑡 . (35) \nTherefore, in this work, the linear growth rate of the kink instability is estimated as the angular coefficient of the slope given by the initial increase of the natural logarithm of the total kinetic energy (black lines in panel 𝑎 Figure 4), divided by a factor of 2. The dimensionless growth rate of the three cases is, respectively, \n- · 𝜎 (P1) = 0 . 18 ± 0 . 02, in the interval 𝑡 = 20 -60,\n- · 𝜎 (P2) = 0 . 118 ± 0 . 006, in the interval 𝑡 = 30 -75,\n- · 𝜎 (M1) = 0 . 082 ± 0 . 004, in the interval 𝑡 = 55 -115, \nwhere the error is determined as the standard deviation from the linear fit of the kinetic energy logarithm. \nThe estimated growth rate decreases at the increase of 𝛼 𝑐 . The variation in growth rate between the fully coupled case (M1) and the least collisionally coupled case (P1) is of a factor of 2.2. The larger growth rate comes as a consequence of the variation in effective plasma density: in case of a fully decoupled case ( 𝛼 = 0) run at the PIP ion fraction = 0 . 1, the growth of the instability is expected to be a factor ∼ √ 10 (∼ 3 . 16 ) larger than the MHD case presented here. The value of the growth rate for our intermediate coupling cases prove that there is a direct response of the coupling in affecting the instability growth rate.", '3.1. Two-fluid effects': "In a partially-ionised plasma, the interaction of ions and neutrals can lead to further dynamics that might affect the physics of the kink instability. Here case P1 is examined at 𝑡 = 90, a late stage of the nonlinear evolution of the kink instability. Figure 5 shows the temperature difference 𝑇 𝑛 -𝑇 𝑝 , the magnitude of the drift velocity between fluids 𝑣 𝐷 = | v 𝑛 -v 𝑝 | , the ionisation rate \nFigure 5. Contour plot of ( 𝑎 ) the temperature difference 𝑇 𝑛 -𝑇 𝑝 , ( 𝑏 ) the drift velocity magnitude | 𝑣 𝐷 | , ( 𝑐 ) the current density magnitude 𝐽 , ( 𝑑 ) the ionisation rate Γ ion and ( 𝑒 ) the recombination rate Γ rec for the PIP case P1 at 𝑡 = 90 (top panels), and details of the same variables in the area around a current sheet (bottom panels). The variables are shown at the centre of the flux rope ( 𝑧 = 0). The magnitude of ionisation and recombination rates is presented with a logarithmic color scale. The area covered by the bottom panels corresponds to the region identified in the black boxes in the top panels. \n<!-- image --> \nΓ ion, the recombination rate Γ rec and the current density magnitude 𝐽 at the centre of the flux rope ( 𝑧 = 0). The whole section of the flux rope is shown in the top panels, while the detail of a smaller area surrounding a current sheet is displayed in the bottom panels. \nThe temperature distribution differs between neutral and plasma, as shown by the panels in column ( 𝑎 ) of Figure 5 where golden areas represent the regions where the plasma is hotter than the neutrals and purple areas identify the regions where the neutral fluid is hotter than the plasma. The plasma temperature exceeds the neutral temperature mostly in regions of high current density magnitude, as shown by the comparison between the panels in column ( 𝑎 ) and ( 𝑐 ), while patches of higher neutral temperature than the plasma are observed outside the reconnecting regions. The maximum temperature difference reached in these areas ( ∼ 0 . 17) is very large when compared to the background temperature of 0.075 of both species, and dimensionally is equivalent to a difference of about 3 · 10 4 K. The magnitude of the drift velocity is also very high, reaching values of about a fifth of the bulk Alfv'en speed, mostly in correspondence of high current density structures as shown by the comparison with 𝐽 . This means that in high current density regions the two fluids are mostly decoupled, a feature that is also shown by the large difference in temperature in these areas, with peaks in plasma temperature within the current sheets and peaks in neutral temperature immediately outside. \nIn P1, ionisation and recombination processes occur with a comparable increase inside the flux rope. Initially Γ ion = 10 -6 and Γ rec = 10 -5 , values that remain constant in the background plasma. Spikes of ionisation rate are observed within the areas of larger current density and 𝑇 𝑝 , with a maximum value Γ ion = 2 . 4 · 10 -3 at 𝑡 = 90: inside the current sheet, the plasma compression and high temperature allow to ionise the neutrals dragged by collisions with the ions. The larger ionisation rate leads to a thickening of the current sheet, as noted by Murtas et al. (2022). Outside the longer current sheets, recombination rates increase proportionally to the plasma temperature, as shown by the comparison with the temperature difference map in column ( 𝑎 ), with peaks of Γ rec = 1 . 2 · 10 -5 at 𝑡 = 90. A similar trend is found for the case with a larger ion-neutral coupling (P2). In case P2, the background Γ ion = 10 -5 and Γ rec = 10 -4 , with peaks of Γ ion = 4 . 4 · 10 -2 and Γ rec = 1 . 4 · 10 -4 at 𝑡 = 105. \nAs shown by the zoom-in of the area around a current sheet, displayed in the bottom panels of Figure 5, peaks of drift velocity (column b ) occur in the outflow regions of the current sheet, while in the inflow both neutral and plasma velocity are comparable. This has previously been observed in the results presented in Murtas et al. (2021) and Murtas et al. (2022), where an evident \ndecoupling of the species occurred in the outflow regions of current sheets. The expulsion of neutrals from the current sheet lead to a more turbulent mixing in the regions around, and to local peaks of the neutral temperature. \nFigure 6 shows the time evolution of the kink instability and the variation of the drift velocity at three locations of the flux rope. At all times the drift velocity produces similar structures of comparable magnitude along the flux rope. In the early stages of the simulation ( 𝑡 = 0) characterised by the largest writhe, the drift velocity is negligible. The first structures in 𝑣 𝐷 appear when the nonlinear phase of the instability begins ( 𝑡 = 30), and their distribution along a spiral structures follows the relaxation of the magnetic field lines. At 𝑡 = 60 the drift velocity increases are concentrated primarily at the external surface of the flux rope, where the first current sheets are formed, as expected from the distribution of 𝑣 𝐷 magnitude shown in Figure 5 and discussed above. Finally, the velocity structures created by the drift increase in complexity with the increasing turbulent motions occurring within the flux rope at later times ( 𝑡 = 90).", '3.2. Temperature and heating terms': "Beyond the term of Ohmic heating, the high velocity drifts observed in partially-ionised plasmas and discussed in Section 3.1 can lead to heating generated by collisions taking place between ions and neutrals at these high speeds. In this Section we examine the evolution of the fluids temperature and of the heating terms with the growth of the kink instability. \nThe time variation of both neutral and plasma temperatures are shown in Figure 7 for case P1 and Figure 8 for case P2 at the same evolution phases displayed in Figure 2. In a partially-ionised plasma, fluids can be heated through Ohmic heating 𝜂𝐽 2 , generated by the current dissipation in regions at high current densities, and by frictional heating, which is generated by the drift between ions and neutrals. Frictional heating is characterised by two different components (Snow et al. 2023). The first component is a collisional term related to the drift velocity between ions and neutrals. The non-dimensional form of the collisional frictional heating is: \n𝐹 heat , 1 = 1 2 𝛼 𝑐 ( 𝑇 𝑛 , 𝑇 𝑝 , 𝑣 𝐷 ) 𝜌 𝑛 𝜌 𝑝 𝑣 2 𝐷 , (36) \nand depends on both the collisional frequencies and the drift velocity between charges and neutrals. The second term adding to frictional heating is linked to the work done through ionisation-recombination processes, and takes the form: \n𝐹 heat , 2 = Γ rec 𝜌 𝑝 v 2 𝑝 - ( Γ rec 𝜌 𝑝 + Γ ion 𝜌 𝑛 ) v 𝑛 · v 𝑝 + Γ ion 𝜌 𝑛 v 2 𝑛 . (37) \nThe contour maps of the Ohmic heating and the collisional frictional heating terms are shown in Figure 7 for case P1 and 8 for case P2. In both Figures, the colour scales are set equal for an easier comparison between the two PIP cases. At the beginning of the kink instability both 𝑇 𝑝 and 𝑇 𝑛 of cases P1 and P2 present very small variations with respect to the initial conditions, as shown by the panels in row ( 𝑎 ) of Figure 7-8. The plasma temperature starts increasing when the first current sheets form inside the flux rope, as shown in rows ( 𝑏 ) and ( 𝑐 ) of Figure 7-8, and as a consequence of the thermal coupling between the two fluids the neutral temperature starts to increase in the same areas of high 𝑇 𝑝 . The Ohmic term is larger in magnitude than frictional heating, as shown by the comparison of the heating terms magnitude in both Figure 7-8, and its action is localised to the current sheets structures. Outside these areas, collisional frictional heating provides a comparable contribution to heating both charges and neutrals. The frictional heating produced by ionisation-recombination processes is a few orders of magnitude smaller in both PIP cases, reaching a peak of 1 . 78 · 10 -5 for P1 at 𝑡 = 65 and 4 . 13 · 10 -5 for P2 at 𝑡 = 105, hence its contribution can be considered negligible. \nThe temperatures and heating terms of the two PIP cases can be compared at similar growth stages of the instability, and across the central section of the flux rope. In case P1, where the initial collisional coupling is 𝛼 𝑐 ( 0 ) = 1, the plasma temperature at the later stages displays the largest increase, reaching peaks of 𝑇 𝑝,𝑃 1 ∼ 0 . 25, 3.3 times the background temperature in the reconnecting regions (corresponding to peaks of ∼ 4 . 2 · 10 4 K), against 𝑇 𝑝,𝑃 2 ∼ 0 . 21 for case P2 where 𝛼 𝑐 ( 0 ) = 10, which is about 2.7 times the background temperature (corresponding to ∼ 3 . 4 · 10 4 K). From the underlying empirical ionisation and recombination equations 17-18, the equilibrium ionisation fraction can be defined as a function of temperature only (Snow & Hillier 2021), i.e., \n𝜉 𝑝 = 1 𝐹 ( 𝑇 ) 𝐺 ( 𝑇 ) + 1 , (38) \nwhere 𝐹 ( 𝑇 ) , 𝐺 ( 𝑇 ) are the temperature dependent components of the recombination and ionisation rates respectively. The temperature maxima in both P1 and P2 cases is high enough that the plasma should be, locally, completely ionised ( 𝜉 𝑝 ≈ 1) based on Equation 38. However, Equation 38 is based on steady-state equilibrium, whereas the simulation is time-dependent and allows departures from equilibrium to exist. The ionisation and recombination rates are relatively slow, with Γ ion ≈ 10 -3 in \nFigure 6. Time series of the kink instability of case P1 showing magnetic field lines (grey) and slices of the drift velocity squared ( v 𝑛 -v 𝑝 ) 2 . The times showed are, from top to bottom, 𝑡 = 0, 𝑡 = 30, 𝑡 = 60 and 𝑡 = 90. \n<!-- image --> \nFigure 7. Left to right: neutral temperature 𝑇 𝑛, norm and plasma temperature 𝑇 𝑝, norm normalized by the respective background temperatures, collisional frictional heating 𝐹 heat , 1 and Ohmic heating 𝜂𝐽 2 of case P1 at 𝑡 = 45 (row 𝑎 ), 𝑡 = 55 (row 𝑏 ) and 𝑡 = 65 (row 𝑐 ). \n<!-- image --> \nFigure 8. Left to right: neutral temperature 𝑇 𝑛, norm and plasma temperature 𝑇 𝑝, norm normalized by the respective background temperatures, collisional frictional heating 𝐹 heat , 1 and Ohmic heating 𝜂𝐽 2 of case P2 at 𝑡 = 75 (row 𝑎 ), 𝑡 = 90 (row 𝑏 ) and 𝑡 = 105 (row 𝑐 ). \n<!-- image --> \nthe reconnection region, as shown in Figure 5 for case P1 where Γ ion peaks at 2 . 4 · 10 -3 at 𝑡 = 90. As such, the timescale for ionisation-recombination equilibrium to be obtained is long compared to the life of the feature. Within the reconnection region, the ionisation fraction is roughly 0.1 and as such, the local neutral density is far larger than would be predicted by ionisationrecombination equilibrium. The larger temperature increase in P1 (the case with the smallest ion-neutral coupling) depends on the difference in the effective plasma 𝛽 , which is smaller in the most decoupled case as a lower coupling results in a lower gas density. The largest increases in neutral temperatures for the two PIP cases are closer, being 𝑇 𝑛,𝑃 1 ∼ 0 . 11 and 𝑇 𝑛,𝑃 2 ∼ 0 . 13, 1.5 and 1.74 times the background temperature respectively, despite the difference in the magnitude of the plasma temperature. \nFigure 9. Global Ohmic heating (black) and collisional frictional heating (red) as a function of time for case M1 (solid line), P1 (dashed line) and P2 (dotted line). \n<!-- image --> \nAsecond difference is observed in the magnitude of the heating components between case P1 and P2. While the Ohmic heating 𝜂𝐽 2 is similar in both cases, the maximum being ∼ 0 . 25 for case P1 and ∼ 0 . 26 for case P2 in the last phase of the instability (row 𝑐 of Figure 7-8), the contribution of the collisional frictional heating is larger in case P2, being ∼ 0 . 04 against a maximum value of ∼ 0 . 014 for case P1. The difference in frictional heating comes from the difference in the collision coupling between the two cases. Collisional frictional heating is directly proportional to 𝛼 𝑐 as shown by the definition in equation (36), and directly affects the increase in temperature of both charges and neutrals. \nWhile the collisional frictional heating is smaller locally than the Ohmic heating, its total contribution is larger than Ohmic heating in both PIP cases. This is shown in Figure 9, where Ohmic heating and collisional frictional heating integrated over the domain are displayed as a function of time. This occurs because the Ohmic heating, despite being larger in magnitude, is localised in the few current sheet structures at higher current densities, while the frictional heating produced by the drift between charges and neutrals is produced more uniformly inside the whole flux rope. As already noticed from the magnitude of the collisional frictional heating in Figure 7-8, P2 develops larger amounts of collisional frictional heating than P1 due to the stronger coupling between fluids. \nFigure 10 shows the changes of the mean plasma and neutral temperatures, averaged over the whole domain, of all three simulations. Due to the normalization based on the Alfv'en speed, the initial mean plasma temperature is slightly higher in the MHD case, where 𝑇 𝑝 ( 𝑡 = 0 ) = 0 . 0833, than in the PIP cases, where at 𝑡 = 0 𝑇 𝑝 = 𝑇 𝑛 = 0 . 0758. The time variation of the mean temperature is very small, due to averaging across a large portion of the domain where the plasma is unperturbed. Comparing similar stages of the instability, the plasma temperature of M1 increases up to 𝑇 𝑝 = 0 . 0842 with a rise of 1% at 𝑡 = 120, and a similar increase is seen for P2, where 𝑇 𝑝 = 0 . 0768 (1.3%) at 𝑡 = 105, while for case P1 𝑇 𝑝 = 0 . 077 (1.6%) at 𝑡 = 65. Due to the original difference in background temperature, the mean temperature of case M1 is higher than P1 and P2, but the PIP cases show a temperature increase that is inversely proportional to the two-fluid coupling.", '4. SUMMARY AND DISCUSSION': "The kink instability of flux ropes may take place in several layers of the solar atmosphere, but has been well modelled only in coronal plasmas. Due to the lack of numerical models, it is still unknown of how this instability develops in a plasma that is only partially-ionised, such as the chromospheric plasma. In this work we present a study of the kink mode developing in PIP, as the growth of the instability is observed in presence of partial ionisation and compared to the evolution in a fully-ionised plasma. Our results can be summarised as follows. \n- · The growth time scale of the helical kink instability increases with the collisional coupling. The nonlinear phase begins earlier in PIP, and is faster for the most decoupled case (P1). This depends on the scaling with respect to the Alfv'en speed, which is larger the lower the ion fraction and the coupling with neutrals. Consequently, the growth rate is larger the lower the ion-neutral collisional coupling. In all three cases, the estimated growth rate of the instability, normalised by the Alfv'en time, matches the values found in Browning et al. (2008) ( 𝜎 ∼ 0 . 05 -0 . 20) and Hood et al. (2009) ( 𝜎 ∼ 0 . 075). In the limit of a completely decoupled system ( 𝛼 𝑐 = 0), as pointed out by Murtas et al. (2021, 2022), we expect the instability of \nFigure 10. Evolution of the global \n<!-- image --> \nmean plasma temperature (blue) and mean neutral temperature (red) of cases M1 (solid line), P1 (dashed line) and P2 (dotted line). \nthe growth rate to be the fastest: this can be considered as a fully-ionised plasma, whose total density is equal to the density of the partially-ionised plasma. \n- · The loss of magnetic energy during reconnection results in a larger increase of kinetic energy for the MHD case and of internal energy for the PIP cases. One of the reasons for the faster temperature increase of the PIP cases is that there are more heating terms, such as frictional heating, contributing.\n- · The maximum Ohmic heating is similar in both PIP cases and locally bigger than frictional heating, but frictional heating is more distributed and its total value in the system is larger. In PIP cases, collisional frictional heating is the major contribution to heating. In the weakly coupled regimes explored in this work, collisional frictional heating is bigger in case P2 than P1, exhibiting an increase that is directly proportional to the collisional coupling. In more strongly coupled systems, collisional frictional heating is expected to reduce, following the increasingly smaller drift velocity between charges and neutral. \nThese results suggest that flux ropes forming in partially-ionised plasmas are more unstable to the kink mode, compared to fully-ionised plasmas of equal total density. This could lead to a faster release of energy following the shorter time scale of the instability, and contribute to the onset of chromospheric jets, where plasmoid formation has been already observed as a product of reconnection (Singh et al. 2011) and chromospheric micro- and mini-filament eruptions (Sterling et al. 2015, 2016; Sterling & Moore 2016; Samanta et al. 2019; Sterling et al. 2020). Using the values of temperature and plasma 𝛽 imposed as initial conditions, the Alfv'en speed is ∼ 40 km s -1 . The typical length of a mini-filament is ∼ 8 · 10 3 km (Sterling et al. 2015), therefore we estimate the initial flux rope radius to be ∼ 400 km. This leads to a time scale 𝜏 𝐴 ∼ 10 s. For typical chromospheric parameters, including a total number density 𝑛 = 10 20 m -3 , the ion-neutral collisional frequency is calculated as \n𝜈 𝑖𝑛 = 𝑛 𝑛 √︂ 8 𝐾 𝐵 𝑇 𝜋𝑚 𝑖𝑛 𝜎 𝑖𝑛 , (39) \nwhere 𝑛 𝑛 is the neutral number density is a fraction of the total density, 𝑚 𝑖𝑛 = 𝑚 𝑖 𝑚 𝑛 /( 𝑚 𝑖 + 𝑚 𝑛 ) ∼ 𝑚 𝑖 and 𝜎 𝑖𝑛 = 5 · 10 -19 m 2 is the cross section, and varies in the interval 10 3 -10 6 s -1 (Leake et al. 2005; Khomenko & Collados 2012). In non-dimensional form, this would correspond to an 𝛼 𝑐 ∼ 10 4 -10 7 , which is a strongly coupled regime. The effects of weak ionisation and coupling are enhanced in much smaller structures: for flux ropes with radius on the order of 10s of meters the dimensional ion-neutral collisional frequency 𝜈 𝑖𝑛 = 0 . 09 -0 . 9 s -1 . This however has to be considered as a lower limit for which strong two-fluid effects can be observed: both ion fraction and plasma 𝛽 can be orders of magnitude lower than the values imposed here, which would lead to stronger local two-fluid effects. It has to be mentioned that in this work we haven't explored the regime between weakly and strongly coupled plasmas ( 𝛼 𝑐 = 10 2 -10 3 ), which would lead to the enhancement of partial ionisation effects in larger structures. We defer this to future studies. \nPartial ionisation is also responsible for more heating than what produced during the instability growth in fully-ionised plasmas. The new heating term is generated by the drifts between charges and neutrals (frictional heating) that is dominant over the Ohmic heating. Flux ropes are therefore more heated in partially-ionised plasmas, as more energy is converted into heat than kinetic energy, unlike the fully-ionised coronal processes. This result identifies an important contribution to the heating of the chromosphere, and can be connected to the works of Gudiksen & Nordlund (2005) and Hansteen et al. (2007), whose simulations suggest that most of the heating in the solar atmosphere occurs at chromospheric heights. The localised heating, produced by the small-scales explosive events that are triggered by the instability of chromospheric flux ropes (e.g. chromospheric jets), provides an important contribution to the chromospheric heating, and could also significantly fuel the coronal heating (De Pontieu et al. 2009). \nThe study of mini-filaments of finite length would require a different type of boundary conditions at the footpoints (i.e. conductive walls, Hood et al. 2009, line-tying conditions, Hood & Priest 1979). While periodic boundaries are still appropriate to examine how partial ionisation changes the evolution from the standard MHD case. This allows us to infer that even with more realistic boundaries we will see faster growth of the instability and more heating. We defer the use of a more realistic setup to future studies.", 'ACKNOWLEDGEMENTS': "The authors thank Prof. Rony Keppens and Dr. Claire Foullon for the precious comments and suggestions that led to the completion of this study. GM is supported by the NASA Grant No. 80HQTR21T0087. AH and BS are supported by STFC Research Grant No. ST/R000891/1 and ST/V000659/1. For the purpose of open access, the authors have applied a 'Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript version arising. This work used the DiRAC@Durham facility managed by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). The equipment was funded by BEIS capital funding via STFC capital grants ST/P002293/1, ST/R002371/1 and ST/S002502/1, Durham University and STFC operations grant ST/R000832/1. DiRAC is part of the National e-Infrastructure. This research also used resources of the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility located at Lawrence Berkeley National Laboratory, operated under Contract No. DE-AC02-05CH11231.", 'DATA AVAILABILITY': 'The data that support the findings of this study are available from the corresponding author upon reasonable request. The (PIP) code is available at the following url: https://github.com/AstroSnow/PIP. Details of the code and equations are available in Hillier et al. (2016).', 'REFERENCES': 'Arber, T. D., Botha, G. J. J., & Brady, C. S. 2009, ApJ, 705, 1183, doi: 10.1088/0004-637X/705/2/1183 \nArber, T. D., Longbottom, A. W., & Van der Linden, R. A. M. \n1999, ApJ, 517, 990, doi: 10.1086/307222 \nBareford, M. R., & Hood, A. W. 2015, Philosophical Transactions of the Royal Society of London Series A, 373, 20140266, doi: 10.1098/rsta.2014.0266 \nBiskamp, D. 2000, Magnetic Reconnection in Plasmas \nBotha, G. J. J., Arber, T. D., & Hood, A. W. 2011, A&A, 525, A96, \ndoi: 10.1051/0004-6361/201015534 \nBrandenburg, A., & Zweibel, E. G. 1994, The Astrophysical \nJournal Letters, 427, L91, doi: 10.1086/187372 \n-. 1995, The Astrophysical Journal, 448, 734, \ndoi: 10.1086/176001 \nBrowning, P. K., Gerrard, C., Hood, A. W., Kevis, R., & van der \nLinden, R. A. M. 2008, A&A, 485, 837, \ndoi: 10.1051/0004-6361:20079192 \nChae, J., & Litvinenko, Y. E. 2021, Research in Astronomy and Astrophysics, 21, 232, doi: 10.1088/1674-4527/21/9/232 \nDe Pontieu, B., Carlsson, M., Rouppe van der Voort, L. H. M., et al. 2012, ApJL, 752, L12, doi: 10.1088/2041-8205/752/1/L12 \nDe Pontieu, B., McIntosh, S. W., Hansteen, V. H., & Schrijver, C. J. 2009, ApJL, 701, L1, doi: 10.1088/0004-637X/701/1/L1 \nDraine, B. T. 1986, MNRAS, 220, 133, \ndoi: 10.1093/mnras/220.1.133 \nFan, Y. 2005, ApJ, 630, 543, doi: 10.1086/431733 \nFang, F., Fan, Y., & McIntosh, S. W. 2014, ApJL, 789, L19, \ndoi: 10.1088/2041-8205/789/1/L19 \nGary, G. A. 2001, SoPh, 203, 71, doi: 10.1023/A:1012722021820 \nGoedbloed, J. P. H., & Poedts, S. 2004, Principles of \nMagnetohydrodynamics \nGoodman, M. L. 2000, ApJ, 533, 501, doi: 10.1086/308635 \nGordovskyy, M., Kontar, E. P., & Browning, P. K. 2016, A&A, \n589, A104, doi: 10.1051/0004-6361/201527249'}

2024ApJ...974..208S

We develop a theoretical framework and use 2D hydrodynamical simulations to study the repulsive effect between two close orbiters embedded in an accretion disk. We consider orbiters on fixed Keplerian orbits with masses low enough to open shallow gaps. The simulations indicate that the repulsion is larger for more massive orbiters and decreases with the orbital separation and the disks viscosity. We use two different assumptions to derive theoretical scaling relations for the repulsion. A first scenario assumes that each orbiter absorbs the angular momentum deposited in its horseshoe region by the companions wake. A second scenario assumes that the corotation torques of the orbiters are modified because the companion changes the underlying radial gradient of the disk surface density. We find a substantial difference between the predictions of these two scenarios. The first one fails to reproduce the scaling of the repulsion with the disk viscosity and generally overestimates the strength of the repulsion. The second scenario however gives results that are broadly consistent with those obtained in the simulations.

2024-10-01T00:00:00Z

['2024ApJ...974..208S', '2024arXiv240910751S', 'arXiv:2409.10751', '10.48550/arXiv.2409.10751', '10.3847/1538-4357/ad737e']

['Active galactic nuclei', 'Exoplanet dynamics', 'Hydrodynamical simulations', 'Planetary-disk interactions', 'Planetary migration', 'Protoplanetary disks', '16', '490', '767', '2204', '2206', '1300', 'Astrophysics - Earth and Planetary Astrophysics', 'Astrophysics - Astrophysics of Galaxies', 'Astrophysics - Solar and Stellar Astrophysics']

A Close Pair of Orbiters Embedded in a Gaseous Disk The Repulsive Effect

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https://arxiv.org/pdf/2409.10751.pdf

{'A close pair of orbiters embedded in a gaseous disk: the repulsive e ff ect': "<!-- image --> \n1 \nInstituto de Ciencias F'ısicas, Universidad Nacional Aut'onoma de M'exico, Cuernavaca, Morelos, Mexico and Max Planck Institute for Solar System Research, Justus-von-Liebig-Weg 3, D-37077 Gottingen, Germany \nInstituto de Astronom'ıa, Universidad Nacional Aut'onoma de M'exico, AP 70-264, Mexico City 04510, Mexico 2 Instituto de Ciencias F'ısicas, Universidad Nacional Aut'onoma de M'exico, Cuernavaca, Morelos, Mexico 3", 'ABSTRACT': "We develop a theoretical framework and use two-dimensional hydrodynamical simulations to study the repulsive e ff ect between two close orbiters embedded in an accretion disk. We consider orbiters on fixed Keplerian orbits with masses low enough to open shallow gaps. The simulations indicate that the repulsion is larger for more massive orbiters and decreases with the orbital separation and the disk's viscosity. We use two di ff erent assumptions to derive theoretical scaling relations for the repulsion. A first scenario assumes that each orbiter absorbs the angular momentum deposited in its horseshoe region by the companion's wake. A second scenario assumes that the corotation torques of the orbiters are modified because the companion changes the underlying radial gradient of the disk surface density. We find a substantial di ff erence between the predictions of these two scenarios. The first one fails to reproduce the scaling of the repulsion with the disk viscosity and generally overestimates the strength of the repulsion. The second scenario, however, gives results that are broadly consistent with those obtained in the simulations. \nKeywords: Active galactic nuclei (16); Exoplanet dynamics (490); Hydrodynamical simulations (767); Planetary-disk interactions (2204); Planetary migration (2206); Protoplanetary disks (1300)", '1. INTRODUCTION': "A massive body embedded in an accretion disk can migrate radially due to the disk torques. For instance, in the accretion disks in the center of active galactic nuclei (AGNs), disk torques may lead to the inward migration of stellarmass ( ∼ 10 M ⊙ ) and intermediate-mass (defined as those with masses between 60 and 10 5 M ⊙ ) black holes (BHs) (e.g., Kocsis, Yunes & Loeb 2011; McKernan et al. 2011). It has been suggested that 10 M ⊙ BHs embedded in the accretion disks of AGNs can accumulate, scatter, and merge in migration traps (Bellovary et al. 2016; Secunda et al. 2019; Yang et al. 2019; McKernan et al. 2020). \nIn protoplanetary disks, embryos and protoplanets can also migrate inwards or outwards (e.g., Baruteau & Masset 2013; Nelson 2018). During their migration, planets can be trapped in mean-motion resonance (MMR) (e.g., Izidoro et al. 2017). Nevertheless, observations indicate that planetary pairs in compact multi-planet systems are generally not found in MMR. There is, however, a small population of pairs that are \nCorresponding author: F. J. S'anchez-Salcedo \nin near-resonance but with a tendency to have orbital period ratios larger than required for exact first-order MMR (Lissauer et al. 2011; Fabrycky et al. 2014). Understanding the processes that move systems just out of MMR has been the subject of numerous works (e.g. Charalambous et al. 2022, and references therein). \nBaruteau & Papaloizou (2013) suggest that the interactions between a planet and the wake of its companion can cause a 'repulsion' of the orbits, which could account for the observed shifts from the nominal commensurated period ratios. In this scenario, the disk mediates an exchange of angular momentum between the planets. Unlike resonant repulsion e ff ects (e.g., Lithwick & Wu 2012; Choksi & Chiang 2020), the interaction with the wake of the companion can induce orbital repulsion between the planets without involving the direct gravitational coupling between them. The orbital repulsion due to the 'wake-planet' interaction appears more e ff ective when the planets open partial gaps in the disk. Interestingly, the wake-planet repulsion between an inner Jovian planet and an outer super-Earth can halt convergent migration even before they are captured in first-order MMR(Podlewska-Gaca et al. 2012). Cui et al. (2021) study the migration of two super-Earths (planet-to-star mass ratios \n∼ 10 -5 ) in the inner parts ( ∼ 1 au) of protoplanetary disks (see also Ataiee & Kley 2021). They include a central cavity in the disk surface density, producing a migration trap for the inner planet. The migration changes from convergent to divergent in the models explored by Cui et al. (2021). \nThe repulsive e ff ect between pairs is relevant not only in interpreting the architectures of planetary systems but it could also help understand the role of the gas in the evolution of a closely-packed pair of BHs in the accretion disks of AGNs and to quantify the chance for the formation of a bound BH binary system (Rowan et al. 2023). \nIt has been suggested that the repulsion e ff ect seen in the simulations is a consequence of the modification of the corotation torques acting on each orbiter because a fraction of the angular momentum carried by the wake excited by the inner orbiter can be deposited in the coorbital region of the outer orbiter and vice versa, altering the corotation torques (Baruteau & Papaloizou 2013; Cui et al. 2021). It seems that this interpretation can explain qualitatively the repulsion effect, but a quantitative analysis of this hypothesis would be useful for full satisfaction. \nIn fact, Kanagawa & Szuszkiewicz (2020) study the migration of two gap-opening planets and show that the transition from convergent to divergent migration can be accounted for without invoking any repulsion mechanism. In the first stage, the planets undergo convergent migration because they have insu ffi cient time to clear their gaps. After being captured in resonance, they open their gaps, and the migration can become divergent. In this scenario, the migration will switch from convergent to divergent if the migration rate of the inner planet in a steady state, calculated as if it were at isolation in the disk, is larger than that of the outer planet. \nTo gain a deeper insight into the nature and magnitude of the disk-mediated repulsion e ff ect, in Section 2, we estimate the strength of the orbital repulsion adopting two di ff erent hypothesis. In Section 3, we compare the predicted scalings with those obtained from two-dimensional simulations of a pair of orbiters to determine which hypothesis is more appropriate. A discussion and summary of the results are given in Sections 4 and 5, respectively.", '2. THE REPULSION EFFECT: THEORETICAL ASPECTS': "Weconsider a pair of massive bodies embedded in the midplane of an accretion disk around a central object with mass M · . These orbiters can be either two planets in a protoplanetary disk or a pair of BHs in the AGN accretion disk. The orbiters have masses M 1 and M 2, semi-major axes a 1 and a 2 and eccentricities e 1 and e 2, respectively. We will assume that a 1 < a 2 so that orbiter 1 is the inner body. In this paper, our aim is to shed light on the disk-orbiters interaction taking into account that the disk is disturbed by the companion. \nFor this purpose, we will ignore the gravitational forces between the orbiters and consider only the disk force on each perturber arising due to the density perturbations induced in the disk by both bodies. \nEach body excites density waves that carry angular momentum. These waves can modify the structure of the disk. Due to the wake of the companion, the torque acting on orbiter j , denoted by Tj , will change by an amount δ Tj . Throughout this paper, we will use the convention that Tj is positive (negative) if the body gains (loses) angular momentum. \nIt is expected that a fraction λ 1 of the one-sided torque excited in the disk by the orbiter 2, denoted by T 1 s , 2, is absorbed by orbiter 1 and vice versa. Hence, δ T 1 = -λ 1 | T 1 s , 2 | , whereas T 2 will change by an amount δ T 2 = λ 2 | T 1 s , 1 | . \nTwo di ff erent cases with di ff erent assumptions are used to derive λ 1 and λ 2. In the first scenario, we assume that each orbiter absorbs the angular momentum flux, excited by its companion, that is deposited in its horseshoe region (e.g., Baruteau & Papaloizou 2013; Cui et al. 2021). For brevity, we will refer to this case as 'angular-momentum modified torques' (hereafter AMMT). In Appendix A, we provide the formulae to evaluate λ 1 and λ 2 in AMMT. \nThe second scenario assumes that the corotation torque on each orbiter is modified because the companion changes the radial profile of the disk (or, more specifically, the vortensity gradient). In this scenario, the outer orbiter will feel a larger (positive) corotation torque than when it is at isolation in the disk, because it lies at the outer edge of the gap opened by the inner orbiter. On the contrary, the corotation torque on the inner orbiter decreases because it lies on the inner edge of the gap opened by the outer orbiter. We will refer to this case as 'density-gradient modified torques' (hereafter DGMT). In Appendix B, we give λ 1 and λ 2 in DGMT. In Section 2.2, we will see that AMMT and DGMT provide di ff erent predictions. \nFor simplicity, in the derivation of λ 1 and λ 2, we will assume that the orbital eccentricity of both orbiters remains small. This is generally seen in the simulations. For instance, the orbital eccentricity of the planets reaches values up to e ∼ 0 . 01 in the simulations of Cui et al. (2021), and up to e ∼ 0 . 03 in the simulations of Baruteau & Papaloizou (2013). \nWe warn that the theoretical estimates for λ 1 and λ 2 in the Appendices A and B are accurate for perturbers that, when they are alone in the disk, open a partial gap, that is, Σ gap , j / Σ un , j ≥ 0 . 65. Here Σ gap , j is the disk surface density at the bottom of the gap created by orbiter j , and Σ un , j is the unperturbed disk surface density at R = aj . Using the empirical formula of Du ff ell (2015) for the gap depth created by a single perturber, we find that this condition implies \nq 2 j ≲ 10 α h 5 , where qj ≡ Mj / M · , h is the disk aspect ratio and α the Shakura-Sunyaev viscosity parameter.", '2.1. Convergent and divergent migration': 'The evolution of the orbital parameters of the orbiter j is determined by the power Pj = v j · F d , j and the torque Γ j = r j × F d , j acting on perturber j , where r j and v j are the position and velocity vectors, respectively. In particular, the evolution of the semi-major axis aj is \ndaj dt = 2 Pj ω 2 j ajMj , (1) \nwhere ω j is the orbital frequency of orbiter j . \nThe ratio of orbital radii ξ ≡ a 2 / a 1 would evolve according to \n1 ξ d ξ dt = 2 D ω 0( a 3 0 a 2) 1 / 2 , (2) \nwith \nD≡ P 2 ω 2 M 2 -ξ 1 / 2 P 1 ω 1 M 1 . (3) \nIn Equation (2), we have written ω j = ω 0( a 0 / aj ) 3 / 2 , where ω 0 is the orbital frequency at a reference radius a 0. Note that D has dimensions of specific torque. We will say that the migration is convergent if d ξ/ dt < 0, i.e. if D < 0, and divergent if D > 0. \nWe may express D as the sum of two terms \nD = D 0 + D I , (4) \nwhere D 0 is the value of D under the assumption that the torques the orbiters experience are not a ff ected by the presence of their companion, and D I represents the change of D caused by the presence of the companion. We will refer to D I as the interaction o ff set. \nIf the orbiters move on quasi-circular orbits, then Pj /ω j = Tj and \nD I = λ 2 M -1 2 | T 1 s , 1 | + λ 1 ξ 1 / 2 M -1 1 | T 1 s , 2 | . (5) \nIn both AMMT and DGMT scenarios, it holds that λ 1 and λ 2 are positive and hence D I > 0 (repulsive e ff ect). The oneside torque excited in the disk by orbiter j is given by \n| T 1 s , j | = f 0 γ jh -3 q 2 j a 4 j ω 2 j Σ un , j , (6) \nwhere \nγ j = 1 + f 0 3 π q 2 j α h 5 -1 , (7) \nwith f 0 ≃ 0 . 45 (e.g., Du ff ell 2015). \nFigure 1. ( ξ -1) 2 D η versus η ≡ q 2 / q 1 assuming AMMT (top panel) and DGMT (bottom panel), for α = 0 . 005 (black curves) and for α = 0 . 05 (red curves). The ratio of semi-major axes ξ varies from curve to curve according to the line style given in the upper-left corner of the top panel. For all curves q 1 = 1 . 5 × 10 -5 and h = 0 . 028. \n<!-- image -->', '2.2. Theoretical predictions': "We first explore how D I depends on the orbiters' separation. For simplicity, we will assume that Σ un( R ) = Σ 0( a 0 / R ) p , where Σ 0 is the surface density at the reference radius a 0. \nCombining Eqs. (5) and (6), the interaction o ff set can be written as \nD I = f 0 h -3 ω 2 0 a 3 + p 0 a 1 -p 1 Σ 0 M -1 · q 1 D η ( ξ ) , (8) \nwhere \nD η ( ξ ) = η 2 λ 1 γ 2 ξ 3 / 2 -p + η -1 λ 2 γ 1 (9) \nand η ≡ q 2 / q 1. We found empirically that the combination ( ξ -1) 2 D η , with D η given in Equation (9), varies little to changes in ξ , implying that, in the range of ξ under consideration, D η depends on ξ as D η ∼ ( ξ -1) -2 , in both AMMT \nand DGMT. This is shown in Figure 1, where we compare D η for three di ff erent values of ξ . We took q 1 = 1 . 5 × 10 -5 , h = 0 . 028 and p = 0 . 5. Indeed, throughout this paper, we will take p = 0 . 5 unless otherwise stated. \nAclear di ff erence between the models is that, for any value of η , D η is significantly smaller in DGMT than in AMMT. Another di ff erence is that, at fixed ξ and α , D η increases monotonically with η in DGMT, whereas it presents a minimum at η ≃ 0 . 5 in AMMT. \nThe dependence of D η with α in AMMT also di ff ers from that in DGMT. In AMMT, D η for α = 0 . 05 is larger than it is for α = 0 . 005, especially for η > 2. The reason is that the angular momentum deposited by the outer perturber decreases as its mass increases because it opens a deeper gap. However, DGMT predicts a reduction of the repulsive e ff ect because radial gradients in the disk surface density induced by the companion become smaller as viscosity increases. \nThe repulsion mechanism is significant when D I ≳ |D 0 | . If we define R as the ratio D I / |D 0 | , and using the equation for D 0 derived in the Appendix C, and D I from Equation (8), the above condition implies: \nR ≡ f 0 χ h ξ 1 / 2 ! D η | ηγ 2 ξ 1 / 2 -p -γ 1 | ≳ 1 . (10) \nFigure 2 shows R versus ξ for h = 0 . 028 and di ff erent combinations of q 1, q 2 and α . For the brevity of the notation, we use the quantities ˜ qj ≡ qj / 10 -5 . We consider values of ξ that satisfy the following three conditions. First, the Hill condition (see Appendix D). The second condition is that the horseshoe regions are separated, i.e. a 2 -a 1 > x hs , 1 + x hs , 2, where x hs , j is the half-width of the horseshoe region of orbiter j . Finally, to justify the two-dimensional approximation, we demand that a 2 -a 1 > 2 H 12, where H 12 is the vertical scale height of the disk at R = a 12, where a 12 ≡ ( a 1 + a 2) / 2. \nAs expected, R decreases as the separation between the orbiters increases. For the parameters under consideration in Figure 2, R ≲ 1 in DGMT. In AMMT, the condition R > 1 is fulfilled at ξ ≲ 1 . 3. \nIt is interesting to note that, for the case ˜ q 1 = 0 . 25 (corresponding to the bottom row of Fig. 2), the repulsive e ff ect is a bit larger for ˜ q 2 = 2 than for ˜ q 2 = 0 . 5, in both AMMT and DGMT. This result implies that although the inward migration rate of the outer perturber is larger for ˜ q 2 = 2 than for ˜ q 2 = 0 . 5, this increase is correspondingly lower than the increase in the inward migration rate of the inner perturber. \nAMMTpredicts that when ˜ q 1 and ˜ q 2 are su ffi ciently small, R is almost independent of α (see lower left panel of Figure 2). The trend is di ff erent in DGMT; R decreases as α increases, even if ˜ q 1 and ˜ q 2 are small. \nIn the case that the inner body does not migrate because it is in a migration trap, the corresponding interaction o ff set is \nD I = λ 2 M -1 2 | T 1 s , 1 | , (11) \nFigure 2. Repulsive ratio R versus the ratio of semi-major axes ξ for di ff erent combinations of ˜ q 1, ˜ q 2 and α parameter. The horizontal lines indicate the value R = 1, above which the repulsion e ff ect counteracts convergent migration. The left column corresponds to the AMMT scenario and the right-hand column is for the DGMT scenario. The disk has h = 0 . 028. \n<!-- image --> \nand the condition D I ≳ |D 0 | is simplified to \nR ≡ f 0 λ 2 γ 1 ξ p -1 χ h η 2 γ 2 ≳ 1 . (12) \nWe have checked that if all the parameters (˜ q 1 , ˜ q 2 , h , α, ξ ) are fixed, R is significantly larger when both orbiters can migrate.", '3. THE SIMULATIONS': 'In this section, we compute the disk forces acting on a pair of massive bodies embedded in a two-dimensional disk, using the publicly available code FARGO (Masset 2000). We aim to explore the separations between the pair at which the repulsion e ff ect is important and to test whether AMMT or DGMT can correctly predict the scaling and magnitude of D I. In order to isolate how the disk torques change in a disk already disturbed by its companion, the bodies are forced to move on fixed Keplerian orbits with orbital radii a 1 and a 2, and eccentricities e 1 and e 2 (constant over time). \nIn all our models, the initial (unperturbed) radial profile of the surface density of the disk, Σ un( R ), follows the powerlaw assumed in Section 2.2. For simplicity, h is taken to be constant with R and over time (locally isothermal disk) and \nFigure 3. Ratio between the azimuthally-averaged surface density ⟨ Σ ⟩ ( R ) at t = 250 orbits and its initial value Σ un( R ). A single object on a fixed circular orbit with ˜ q = 5 was inserted in the disk. Di ff erent curves correspond to di ff erent values of h and α . \n<!-- image --> \ncan take one of two values: h = 0 . 028 and h = 0 . 05. We will consider two values for α (0 . 005 and 0 . 05) in our simulations. We limit ourselves to mass ratios ˜ qj ≤ 5. For this range of masses and the values of h under consideration, the horseshoe regions of the orbiters do not overlap if ξ > 1 . 09. \nTo model the gravitational potential of the orbiters, we introduced a gravitational softening length Rs = 0 . 6 H . The forces on each perturber are calculated by summing the gravitational force in each grid cell. We use two tapering Gaussian functions to reduce the contribution of material bound to the orbiters \nf j ( sj ) = 1 -exp[ -s 2 j / R 2 H , j ] , (13) \nwhere s j = r -r j and RH , j = ( qj / 3) 1 / 3 rj is the individual Hill radius of orbiter j . Accretion onto the orbiters is not included in our simulations. \nIn code units, the inner boundary is located at R = 0 . 35 and the outer boundary at R = 3, with wave-killing boundary conditions (de Val-Borro et al. 2006). In all the simulations with two perturbers, the initial semi-major axis of the inner orbiter is a 1 = 0 . 93. The surface density of the disk is scaled so that the disk mass contained within R = 0 . 93 is 2 . 4 × 10 -3 M · . In code units, M · = 1 and the orbital period is 2 π at R = 1. We measure the time in terms of the orbits revolved by a body at R = 1. \nIn all our simulations, the grid has NR = 950 (logarithmically spaced) and N ϕ = 2400 zones in the radial and azimuthal directions, respectively. \nA certain model is specified by seven dimensionless parameters ˜ q 1, ˜ q 2, ξ , e 1, e 2, h and α . Throughout this section, however, we use either ξ or ∆ / H 12, with ∆ ≡ a 2 -a 1, \nand remind that H 12 is the vertical scale-height of the disk at R = a 12. The relation between ξ and ∆ / H 12 is given by \n∆ H 12 = 2( ξ -1) ( ξ + 1) h . (14)', '3.1. Individual orbiter': 'We start by considering the case where a single object with mass ratio ˜ q = 5 is on a fixed circular orbit with semi-major axis a = 1. Figure 3 shows the ratio between the azimuthallyaveraged surface density ⟨ Σ ⟩ at t = 250 orbits and the initial surface density, for di ff erent combinations of h and α . We see that the gap depth is small for h = 0 . 05 and α = 0 . 005 and h = 0 . 028 and α = 0 . 05. However, for h = 0 . 028 and α = 0 . 005, a partial gap ( Σ gap / Σ 0 ≃ 0 . 66) is carved in the disk. Since we restrict ourselves to models with ˜ qj ≤ 5, our orbiters open shallow or partial gaps when they are alone in the disk. \nFigure 4 shows the steady-state magnitude of the specific power, more specifically P / ( ω M ), versus ˜ q , for two orbital eccentricities. In the cases with h = 0 . 05 and α = 0 . 005 (left panel), P / ( ω M ) is a monotonic function of ˜ q . For h = 0 . 028 and α = 0 . 05, P / ( ω M ) presents a gap. The position of the gap in P depends on the orbital eccentricity; it is located at ˜ q ≃ 3 for e = 0 and at ˜ q ≃ 8 for e = 0 . 014. The origin of this gap in the torque in high-viscosity disks was already discussed in Masset et al. (2006).', '3.2. Torques on a pair: Fixed circular orbits': 'In this section, we will focus on the particular case where the two components of the pair move on fixed circular orbits ( e 1 = e 2 = 0, at any time) around the central object.', '3.2.1. Temporal behaviour of D': "Whentwo orbiters are immersed in the disk, the disk forces onto one of them display large variations in time because of the interaction with the perturbed density field created by the other one. Thus, we compute the torques using time averages over N av τ syn where N av is an integer and τ syn is the time between two successive passages of the orbiters through opposition (or synodic period): \nτ syn = 2 π ω 1 -ω 2 . (15) \nWe will use a bar over a quantity (e.g. ¯ Tj and ¯ D ) to denote its time-average over N av τ syn. \nFigures 5 and 6 show representative examples of the temporal evolution of ¯ Tj / Mj and ¯ D , for h = 0 . 05 and h = 0 . 028, respectively, in the circular case ( e 1 = e 2 = 0). After su ffi -ciently long times, ¯ Tj and ¯ D reach an almost constant value. We will denote these 'steady-state' values with the subscript ss, e.g. ¯ T 1 , ss, ¯ T 2 , ss and ¯ D ss. \nFigure 4. Magnitude of P / ( ω M ) when only one perturber with mass ratio ˜ q is inserted in the disk. The object is in circular orbit (solid lines) or quasi-circular orbit ( e = 0 . 014; dashed lines) with semi-major axis a = 1. The dotted lines represent the formula -P / ( ω M ) = χγ h -2 q ω 2 a 4 Σ 0 M -1 · , which corresponds to the value for a perturber in a circular orbit in an inviscid disk (Tanaka et al. 2002), including a factor γ as suggested by Kanagawa et al. (2018) (see Appendix C). The values of h and α of the disk vary from panel to panel. \n<!-- image --> \nFigure 5. Specific torques acting on the inner orbiter (solid red line), which has ˜ q 1 = 3, and on the outer perturber (solid blue line), which has ˜ q 2 = 5. The solid black line represents ¯ D ≡ ¯ D 0 + ¯ D I, whereas the dashed lines labeled with ¯ T 0 / M 1 and ¯ T 0 / M 2 indicate the torques when the orbiters are isolated in the disk. The arrows indicate the change in the torque due to the presence of the companion. The disk has h = 0 . 05 and α = 0 . 005. The orbits are circular with a separation ∆ = 3 H 12 (or, equivalently, ξ = 1 . 16). \n<!-- image --> \nFor orbiters in circular orbits with separations ∆ = 3 . 1 H 12, or in disks with α = 0 . 05, our simulations are long enough to reach a state where ¯ D (and also the torques) is almost constant. Therefore, ¯ D ss is computed as the value at those steady states. For the simulations with α = 0 . 005 and ∆ = 5 H 12, we ran the simulations to at least 800 orbits. If the torques had not reached a steady-state value, we continued the simulations until ¯ D varied less than 4% for the last 100 orbits. Then, we took ¯ D ss as the value of ¯ D at the end of the simulation.", '3.2.2. ¯ D I , ss versus ∆': "It is worthwhile to look again at Figures 5 and 6 to gain insight into the repulsion e ff ect. In all cases, ¯ T 1 , ss is more negative (the magnitude of the torque is larger) than it is when the orbiter is isolated (red horizontal lines). On the other hand, ¯ T 2 , ss is always more positive than the torque when it is treated as a single body (blue horizontal lines). Therefore, the disk-mediated interaction between orbiters leads to a repulsive e ff ect. Interestingly, in all cases presented in Figures 5 and 6, ¯ D ss > 0. \nThe interaction o ff set in the steady state, ¯ D I , ss, can be computed in our simulations as ¯ D I , ss = ¯ D ss -¯ D 0 , ss, with ¯ D 0 , ss calculated from Equation (3) in simulations where each orbiter is alone in the disk (as those computed in Section 3.1). \nFigure 7 shows the dependence of ¯ D I , ss on the orbital separation ∆ / H 12 for di ff erent combinations of ˜ q 1 and ˜ q 2. We have taken h = 0 . 028, α = 0 . 005 and e 1 = e 2 = 0. In all cases shown in Figure 7, the maximum of ¯ D I , ss occurs at the smallest value of ∆ considered. For these lowest values of ∆ , ¯ D ss > 0, implying that the migration would be divergent. In general, there is a trend for ¯ D I , ss to decrease with the separation (see the left and middle-upper panels). However, it becomes almost constant between ∆ = 6 H 12 and ∆ = 13 H 12 in the third panel (for ˜ q 1 = 3 and ˜ q 2 = 5). \nIn Figure 8, we present the radial profile of the azimuthally-averaged surface density after 1000 orbits at R = 1, for two representative values of the orbital separation ∆ . We see that the orbiters share a common gap for ∆ ≲ 6 H 12. For the largest orbital separations explored, it is possible to distinguish the 'individual' gaps (solid lines in the third and fourth panels). Note that the gap depths in the first and second panels are similar. \n5 \nFigure 6. Similar as Fig. 5 but for h = 0 . 028 and ∆ = 3 . 1 H 12 (which corresponds to ξ = 1 . 09 in this case), and di ff erent combinations of q 1 and q 2. In the left column, we take α = 0 . 005, whereas α = 0 . 05 in the right-hand column. The dashed lines indicate the values when the orbiters are isolated in the disk. Recall that the solid black lines represent ¯ D≡ ¯ D 0 + ¯ D I. Note that the blue arrows are always pointing upward (the outer torque is less negative), whereas the red arrows are always pointing downwards (the inner torque becomes more negative). \n<!-- image --> \nFigure 7. ¯ D I , ss versus ∆ / H 12 as obtained in the simulations (symbols), together with the predicted values in AMMT (orange lines) and DGMT (magenta lines), in a disk with h = 0 . 028 and α = 0 . 005. Black dots indicate divergent migration ( ¯ D ss > 0), whereas green diamonds indicate that the migration is convergent ( ¯ D ss < 0). \n<!-- image --> \nFigure 8. Azimuthally-averaged surface density ⟨ Σ ⟩ after 1000 orbits at R = 1. The position of the inner and outer orbiters are marked with red and blue dots, respectively. In all the cases h = 0 . 028 and α = 0 . 005. \n<!-- image --> \nFigure 9. Similar to Fig. 7 but for α = 0 . 05 (again h = 0 . 028). We do not show, however, the corresponding panel for ˜ q 1 = 0 . 5 and ˜ q 2 = 3 because D I , ss becomes very small, almost compatible with zero. In all the simulations in this figure, it holds ¯ D ss > 0. \n<!-- image --> \nFigure 10. ¯ D ss (dots) and ¯ D 0 , ss (squares) along curves of constant ˜ q 1 as a function of ˜ q 2, in a disk with h = 0 . 05 and α = 0 . 005. We take ∆ = 3 H 12, which corresponds to ξ = 1 . 16. \n<!-- image --> \nIn Figure 7, we have also shown ¯ D I , ss predicted in AMMT and DGMT described in Section 2 and Appendices A and B. We see that AMMT overestimates ¯ D I , ss up to one order of magnitude in some cases. DGMT, on the other hand, can predict the value of ¯ D I , ss within a factor of 3. It can also reproduce the slight ascend (from left to right-hand panels) of the curves from the simulations. \nFigure 9 shows ¯ D I , ss versus ∆ , as in Figure 7, but now for α = 0 . 05. We see that ¯ D I , ss for ˜ q 1 = 1 . 5 and ˜ q 2 = 3 is generally smaller for α = 0 . 05 than for α = 0 . 005. However, in the two models with ˜ q 1 ≥ 3 and ˜ q 2 = 5, ¯ D I , ss for α = 0 . 05 is quite similar to that for α = 0 . 005. \nIn the cases shown in Figure 9, AMMT overestimates ¯ D I , ss and predicts that ¯ D I , ss should be larger as viscosity increases, but the simulations do not show that trend. On the other hand, DGMTreasonably predicts the magnitude of the repulsive effect. A more comprehensive comparison between the results of simulations and the models will be presented in Section 3.2.4.", '3.2.3. Dependence of the repulsion e ff ect on ˜ q 1 and ˜ q 2': 'To see whether the orbiters can reverse the direction of migration from converging to diverging, it is worthwhile to compare ¯ D ss with ¯ D 0 , ss. Figure 10 shows ¯ D 0 , ss and ¯ D ss, both obtained from the simulations, for h = 0 . 05 and α = 0 . 005, when ∆ = 3 H 12 (implying ξ = 1 . 16). Along the curves, ˜ q 2 is varied, keeping ˜ q 1 constant. We see that for ˜ q 1 = 0 . 5 -1 . 5 and for the values of ˜ q 2 under consideration, ¯ D ss is rather similar to ¯ D 0 , ss, meaning that the repulsion is small. The curves for ¯ D ss and for ¯ D 0 , ss essentially overlap for ˜ q 2 = 0 . 5 and separate from each other as q 2 increases. We find that for ˜ q 1 ≥ 3 \nFigure 11. ¯ D ss (dots) and ¯ D 0 , ss (squares), as in Figure 10, but now in a disk with h = 0 . 028, and two di ff erent values of α : 0 . 005 (left column) and 0 . 05 (right-hand column). The orbital separation is ∆ = 5 H 12 (or, equivalently, ξ = 1 . 15). The numbers below the dots at ˜ q 2 = 0 . 5 and at ˜ q 2 = 5 in the left panels indicate the mutual Hill separation ≡ ( a 2 -a 1) / R mH; all the experiments satisfy the Hill stability condition. \n<!-- image --> \nand ˜ q 2 ≥ 3, the disk-mediated interaction between orbiters should be taken into account, for separations ξ = 1 . 16. Yet for these values of ˜ q 1 and ˜ q 2, the depth of the common gap is still very shallow. For instance, for ˜ q 1 = ˜ q 2 = 5, the depth of the gap is only Σ gap / Σ un , gap = 0 . 935. \nFigure 11 shows ¯ D ss and ¯ D 0 , ss for h = 0 . 028 and ∆ = 5 H 12. In this case, the semi-major axis ratio ξ = 1 . 15 is very similar to the value adopted in Figure 10. We find that ¯ D ss > ¯ D 0 , ss, implying that the e ff ect is repulsive. ¯ D I , ss increases as ˜ q 2 \n5 \nFigure 12. ¯ D ss (dots) and ¯ D 0 , ss (squares) for di ff erent combinations of ˜ q 1 and ˜ q 2, separated by ∆ = 3 . 1 H 12 in a disk with h = 0 . 028 (i.e. ξ = 1 . 09). The viscosity parameter α is 0 . 005 in the left panels and 0 . 05 in the right panels. For reference, we give the mutual Hill separation for the first and last points on the left panels; all the experiments satisfy the Hill stability condition. \n<!-- image --> \nincreases (fixed ˜ q 1) and as ˜ q 1 increases (fixed ˜ q 2). In some cases, ¯ D 0 , ss < 0, but ¯ D ss > 0. \nConsider first the low viscosity disk ( α = 0 . 005; left panels in Fig. 11). For ˜ q 2 = 2, ¯ D ss > 0 for any value of ˜ q 1 ∈ [0 . 5 , 5]. In addition, for ˜ q 1 ≥ 1 . 5 and ˜ q 2 ≤ 3, we have ¯ D ss > 0. In particular, if we take ˜ q 1 = 1 . 5 and ˜ q 2 = 2 . 4, similar to those in Kepler-36 for illustration, we find that the migration is divergent if ∆ = 5 H 12, provided that h = 0 . 028, α = 0 . 005 and the orbits are circular. \nIn a disk with h = 0 . 028 and α = 0 . 05 (right panels in Fig. 11), the curves for ¯ D 0 , ss exhibit a peak at ˜ q 2 = 3; this \nis reminiscent of the reduction of the torque for this value of the mass ratio (see Figure 4). ¯ D ss also shows this peak but only when ˜ q 1 ≤ 1 . 5. For ˜ q 1 ≤ 1 . 5 and ˜ q 2 ≤ 3, ¯ D ss ≃ ¯ D 0 , ss. Interestingly, for ˜ q 1 = 1 . 5 -3 and ˜ q 2 = 5, we have that ¯ D 0 , ss < 0 (convergent migration), whereas ¯ D ss > 0 (divergent migration). \nFigure 12 shows the results for ∆ = 3 . 1 H 12, again for h = 0 . 028 (for these parameters, ξ = 1 . 09). Quite remarkably, even for the smallest inner mass considered in this figure, the repulsive e ff ect is su ffi ciently strong to yield a divergent migration for ˜ q 2 = 2, in the low viscosity case ( ¯ D ss > 0). The migration is also divergent in all the models explored in that figure with α = 0 . 005 and ˜ q 1 ≥ 0 . 5. For α = 0 . 005, ˜ q 1 = 0 . 5 and ˜ q 2 ≳ 0 . 5, migration switches from convergent to divergent because ¯ D 0 , ss < 0 and ¯ D ss > 0. \nFinally, we focus on models with α = 0 . 05 (right panels in Figure 12). For ˜ q 1 ≤ 1 . 5, the shape of the curves for ¯ D ss is rather similar to the shape of ¯ D 0 , ss curves. However, for ˜ q 1 = 3, the large discrepancy between ¯ D 0 , ss and ¯ D ss indicates that the disk-mediated interaction plays an important role.', '3.2.4. Comparison with the predictions of AMMT and DGMT': "In Section 3.2.2, we already mentioned that, for the cases considered in that section, DGMT is more consistent with the results of simulations than AMMT, as AMMT generally overestimates the magnitude of ¯ D I , ss. This is confirmed in Figure 13 where we show a one-to-one comparison between the predicted values of ¯ D I , ss and those obtained from the simulations. We see that AMMT fails to predict the scalings of ¯ D I , ss and, in addition, overestimates ¯ D I , ss for orbital separations ≤ 5 H 12. However, DGMT correctly captures the scaling of ¯ D I , ss with h , α , ˜ q 1, ˜ q 2 and radial separation ∆ . When ¯ D I , ss ≲ 0 . 4 × 10 -5 , the simulations show a mildly weaker e ff ect than predicted by DGMT. This has a simple explanation. The time scale to reach a steady state is ∼ 200 orbits for α = 0 . 05, and ∼ 2000 orbits for α ∼ 0 . 005 (e.g., Ataiee et al. 2018). For α = 0 . 005, the time is a bit longer than the running times of our simulations. This is particularly true for low-mass perturbers (˜ q 1 + ˜ q 2 < 2) in a disk with h = 0 . 05 and α = 0 . 005 (the three star symbols at the bottom left corner of Figure 13), for which our simulations were not long enough to establish a steady state. As a consequence, the variations of the slope of surface density have not reached their steady state values, hence numerical simulations show a repulsion weaker than predicted. On the other hand, the scatter in the upper part of the diagram (at values ¯ D I , ss ≳ 0 . 4 × 10 -5 ) may reflect the fact that the analytical model for the gap profile given by Equation (B10), as well as the expression for the corotation torque in Equation (B7) used to compute ¯ D I , ss in DGMT become less accurate for masses larger than the thermal mass ( q ≳ h 3 ). \n3.3. Eccentric orbits \nFigure 13. The interaction o ff set ¯ D I , ss in the numerical simulations shown in Figs. 10, 11 and 12, is compared with the results of AMMT (upper panel) and DGMT (lower panel). The attributes of the symbols are given in the legend in the lower panel. The color bar indicates ˜ q 1 + ˜ q 2. The solid lines represent the identity function. \n<!-- image --> \nIn the previous section, we assumed that the orbiters have null eccentricities. However, the gravitational interaction between the pair's components will excite their eccentricities. Therefore, it is worthwhile to see how sensitive ¯ D I , ss is to the orbiters' eccentricities. To do so, we have conducted the following test. For a subset of the simulations shown in Figs. 11 and 12, we have redone the simulations with the same parameters but adopting e 1 = e 2 = h / 2 = 0 . 014 for the orbital eccentricities of the perturbers, which were kept constant throughout the duration of the runs. We ran the models for the same number of orbits as we did in their circular counterparts and computed ¯ D ss as the average of ¯ D over the last 3 complete oscillations (typically ≃ 100 orbits). Since \nFigure 14. A comparison of ¯ D I , ss in simulations with e 1 = e 2 = 0 . 014 and in simulations with e 1 = e 2 = 0. In all the experiments h = 0 . 028. The solid line represents the identity function. \n<!-- image --> \nFigure 15. Specific torque exerted by the disk on an orbiter in two di ff erent cases: (1) without any companion (green curve), and (2) when the orbiter is alone in the disk but an external positive torque Λ imp is applied to the disk (magenta curve). \n<!-- image --> \nthe shape of the radial profile of the gaps open by individual orbiters with e = 0 . 014 is rather similar to that for individual orbiters in a circular orbit, we expect that DGMT will be for these eccentric pairs as accurate as for circular pairs. Figure 14 compares ˜ D I , ss as obtained in the simulations with e 1 = e 2 = 0 with the values in simulations having e 1 = e 2 = 0 . 014. We see that ˜ D I , ss may di ff er by up to 50% for the models under consideration. In summary, as long as the orbital eccentricities are moderate ( e ≤ h / 2), ¯ D I , ss hardly \nchanges by a factor larger than 2 as compared to the case where the eccentricities are fixed to zero.", '4. DISCUSSION: WHY DOES AMMT YIELD WRONG RESULTS?': "In Section 3.2.4 we have found that AMMT generally overestimates the e ff ect of repulsion, implying that the orbiters absorb less angular momentum than the amount deposited in their horseshoe regions. Here, we present a simple simulation that clearly shows that the excess torque on the orbiter is not that deposited in the disk within its horseshoe region. More specifically, we conducted a simulation of the outer orbiter alone in the disk but applying an imposed (external) positive torque density, Λ imp( R ), to the disk. This external torque imitates the torque deposited by the inner orbiter. To avoid spurious changes in the surface density due to the imposed torque, we require that the specific torque Λ imp / Σ scales with R as ∝ B ( R ) / Σ un( R ), where B ( R ) = Ω ( R ) / 4 is the second Oort constant, with Ω ( R ) the disk orbital frequency (Masset &Papaloizou 2003). In the absence of the orbiter, this torque induces a constant outward drift in a disk with a stationary axisymmetric density profile Σ un( R ). In this particular choice of Λ imp( R ), no waves are excited by the external torque. \nWe have computed the torque on an orbiter with ˜ q 2 = 5 at a 2 = 1 . 34, in a simulation where it is isolated in the disk (with h = 0 . 028 and α = 0 . 005), and in a simulation where the e ff ect of the inner orbiter has been replaced by Λ imp( R ), with Λ imp = 0 . 8 × 10 -8 (in code units) at R = a 2. This magnitude of the external torque density corresponds to the torque density deposited at R = 1 . 34 by the damping of the wake excited by an inner orbiter with ˜ q 1 = 5 at a 1 = 0 . 93, when alone in the disk. It was measured in the simulations of a single orbiter, as Λ dep , 1( R ) = -d ˙ J w / dR , where ˙ J w is the angular momentum flux carried by the waves, which is given by \n˙ J w( R ) = R 2 Z 2 π 0 ( v ϕ -¯ v ϕ )( vR -¯ vR ) Σ d ϕ, (16) \nwhere vR and v ϕ are the radial and azimuthal components of the gas velocity, respectively, and the bar indicates the azimuthal average. We comment that this procedure neglects a potential excitation term (by the inner perturber) at the orbit of the outer perturber. The ratio of orbital radii between the outer and inner perturber is 1 . 44 and the ratio of orbital periods is 1 . 72, which indicates that all the outer Lindblad resonances of the inner perturber are located inside of the outer perturber's orbit, except that of the m = 1 Fourier component of the inner planets' potential, which lies outside. The residual excitation that may take place over the width of the horseshoe region should therefore represent a tiny fraction of incident flux of angular momentum, especially in a disc as thin as the one considered in this experiment. Furthermore, these considerations about the magnitude of the deposition \nof angular momentum within the horseshoe region will soon appear futile, as the torque excess on the outer orbiter does not even match the sign of the torque deposited. \nFigure 15 shows the torques as a function of time. According to AMMT, the magnitude of the torque acting on the outer orbiter should be augmented by an amount of 2 Λ dep , 1 R hs , 2 (evaluated at R = a 2) with respect to the isolated case. This implies an o ff set in the specific torque T 2 / M 2 of ∼ 2 × 10 -5 if ˜ q 2 = 5. This result is in sharp contrast with the outcomes of the simulation with Λ imp, where we see that the torque presents an o ff set of ∼ -0 . 3 × 10 -5 after 800 orbits (the torque is more negative than it is when the orbiter is isolated in the disk; see magenta and green curves in Fig. 15). However, this result is in agreement with studies of type III migration, where the relative planet-disk drift is achieved by inducing a disk drift under the action of external torque. The torque applied to the disk entails an outward drift. This is equivalent to a situation where the disk does not drift, and the planet migrates inwards. If the horseshoe region is partially depleted, as is the case here, type III torques arise and exert a positive feedback on migration (Masset & Papaloizou 2003), i.e. they tend to make the negative torque larger in absolute value. This is precisely what we see in Fig. 15. This goes against the expectation that the torque on the orbiter due to the action of the external torque can be evaluated by calculating the angular momentum given to the disk within its horseshoe region. While it is true that fluid elements trapped in the horseshoe region ultimately transmit the positive torque deposited to the orbiter, fluid elements flowing (outwards) through the horseshoe region extract angular momentum from the orbiter during their unique horseshoe U-turn with the latter. The net e ff ect is an increase in the absolute value of the torque. \nThe standard AMMT scenario contemplates only the contribution from within the horseshoe region and arbitrarily discards the e ff ect of material immediately interior to that region that flows towards the outer disk. By construction, this scenario yields a torque excess that has the correct sign, but it cannot have the correct value, as it considers only half of the problem.", '5. SUMMARY': "In this work, we have re-examined the repulsion e ff ect in packed pairs in cases where the pair components carve partial gaps in the disk ( Σ gap / Σ un , gap ≳ 0 . 5). We focus on cases where the total mass of the pair is ≲ 10 -4 M · . In the case of protoplanetary disks, this corresponds to sub-Neptune-mass planets. In the case of AGN disks, this corresponds to the so-called extreme mass ratio inspirals (EMRIs). It includes an inspiral formed by (10 -300) M ⊙ BHs in an accretion disk around a central 10 6 -7 M ⊙ massive BH. \nWe have used ¯ D I , ss as a measure of the repulsion effect. We have developed a semi-analytical framework to pro- \nide a quantitative view of the scalings of ¯ D I , ss with the orbiters' mass, orbital separation, disk aspect ratio, and viscosity. We applied two di ff erent assumptions: AMMT and DGMT. AMMT assumes that the torques are modified because each orbiter absorbs the angular momentum deposited on its horseshoe region by the wakes excited by its respective companion. DGMT assumes that the corotation torques are modified because the companion modifies the surface density profile of the disk. The formulation can also be applied when the inner orbiter does not migrate if it is in a migration trap. \nWe have measured ¯ D I , ss in two-dimensional simulations of a pair of orbiters in fixed orbits for a wide range of orbiters' mass ratios (0 . 1 ≲ q 2 / q 1 ≲ 10), and disk viscosity (a variation of a factor of 10). The magnitude of ¯ D I , ss spans nearly three orders of magnitude. We have confirmed, in agreement with previous results, that (1) high-mass orbiters present a larger repulsion e ff ect than low-mass orbiters, (2) the repulsive e ff ect may be large enough to stall convergent migration, regardless of whether the pair is close or not to resonance, and (3) the repulsion e ff ect decreases as the disk viscosity is increased. In most of our simulations, we forced the perturbers to be on circular orbits. Nevertheless, we have also \nexplored orbits with eccentricities of 0 . 5 h and found similar repulsion. \nWe have compared the predictions of AMMT and DGMT with the results of the hydrodynamical simulations. AMMT is incapable of reliably predicting the scaling of ¯ D I , ss with q 1, q 2, h and α . In particular, for our adopted range of parameters, AMMT generally overestimates the magnitude of the wake-orbiter interaction by up to two orders of magnitude, especially in disks with high viscosity. \nWe find that a more robust approach is DGMT, which can successfully account for the scaling of ¯ D I , ss, at least for the range of parameters explored in this paper. Note that this work focuses exclusively on the repulsion e ff ect due to the forces mediated by the disk, ignoring the mutual gravitational interaction between the pair components, which can be strong, especially when the planets are near first-order commensurabilities. \nWe would like to thank the referee for a thorough reading of the manuscript and constructive comments that improved the quality of the paper. \nSoftware: FARGO (Masset 2000).", 'REFERENCES': "Kanagawa, K. D., Tanaka, H., Szuszkiewicz, E. 2018, ApJ, 861, 140 \nKanagawa, K. 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R. 2002, ApJ, 572, 566 \nRowan, C., Boekholt, T., Kocsis, B., Haiman, Z. 2023, MNRAS, 524, 2770 \nSecunda, A., Bellovary, J., Mac Low, M.-M., Ford, K. E. S., McKernan, B., Leigh, N. W. C., Lyra, W., S'andor, Z. 2019, ApJ, \n878, 85 Tanaka, H., Takeuchi, T., Ward, W. R. 2002, ApJ, 565, 1257 Ward, W. R. 1991, Lunar Planet. Sci. Conf., 22, 1463 \nWard, W. R. 1992, Lunar Planet. Sci. Conf., 23, 1491 \nYang, Y., Bartos, I., Haiman, Z., Kocsis, B., M'arka, Z., Stone, N. C., M'arka, S. 2019, ApJ, 876, 122", 'A. λ 1 AND λ 2 IN THE AMMT ASSUMPTION': "In this Appendix, we derive the angular momentum deposited in the horseshoe regions. The angular momentum flux excited by orbiter j can be written as Φ j ( R ) T 1s , j , with Φ j ( R ) the dimensionless angular momentum flux. For isolated perturbers that do not open deep gaps, Φ ( R ) can be found in Rafikov (2002) (see also Goodman & Rafikov 2001). It can be approximated by: \nΦ j = 1 if τ j ( R ) < τ sh , j p τ sh /τ j if τ j ( R ) > τ sh , j , (A1) \nwhere τ sh , j is the dimensionless wave-to-shock timescale (see Rafikov (2002) for details), given by \nτ sh , j = 1 . 89 + 0 . 53 h 3 / qj , (A2) \nand \nτ j ( R ) = 3 (2 h 2 ) 5 / 4 GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> Z R / aj 1 | s 3 / 2 -1 | 3 / 2 s ( p -3) / 2 ds GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> . (A3) \nWe recall that p is the power-law exponent of the unperturbed surface density of the disk. The fraction of the angular momentum excited by the perturber 3 -j that is deposited in the horseshoe region of perturber j is \nλ ' j = | Φ 3 -j ( b + j ) -Φ 3 -j ( b -j ) | (A4) \nwhere b ± j = aj ± x hs , j , with x hs , j ≃ aj p qj / h the half-width of the horseshoe region. \nIn principle, the deposition torque depends on the surface density profile of the disk. Equations (A1)-(A4) assume that the radial profile of the surface density follows a power-law and ignore the fact that the companion could open a gap that modifies the underlying structure of the disk. For perturbers that open partial gaps, this non-linear e ff ect is likely small (e.g., Ginzburg & Sari 2018). \nFollowing Cui et al. (2021), we will assume that the perturber can absorb the angular momentum deposited in its horseshoe region if it is less than the one-sided horseshoe drag T hs. Using that | T hs , j | = 0 . 5 γ j Σ un , j ω 2 a 4 j ( qj / h ) 3 / 2 = 0 . 5 h -3 / 2 γ j ω 2 0 a 3 + p 0 a 1 -p j Σ 0 q 3 / 2 j , the condition | λ 3 -jT 1s , j | ≤ | T hs , 3 -j | implies: \nλ 1 = min λ ' 1 , h f 0 γ 1 γ 2 h q 1 ! 1 / 2 η -2 ξ p -1 , (A5) \nand \nλ 2 = min λ ' 2 , h f 0 γ 2 γ 1 h q 2 ! 1 / 2 η 2 ξ 1 -p . (A6)", 'B. λ 1 AND λ 2 IN THE DGMT ASSUMPTION': 'In this Appendix, we compute the change in the corotation torque as the companion modifies the disk surface density. \nThe unsaturated corotation torque on perturber 1 in the steady state case, when it is alone in the disk on a fixed circular orbit, \nis given by \nT (uns) CR , 1 = 3 4 Σ un , 1 ω 2 1 x 4 hs , 1 d ln( Σ un / B ) d ln R GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> R = a 1 , (B7) \nwhere B is the second Oort\'s constant (Goldreich & Tremaine 1979; Ward 1991, 1992). This formula is valid if the perturber does not open a deep gap in the disk. In order to take into account that the corotation torque may di ff er from the unsaturated value because it can be partially saturated, we include a correction factor C , so that T CR , 1 = C · T (uns) CR , 1 . The correction factor C depends on z ν ≡ a ν/ ( ω x 3 hs ), the ratio between the libration timescale and di ff usion timescale across the horseshoe region. For z ν ≤ 1, we take C from Masset & Casoli (2010): \nC = 8 π 3 z ν F ( z ν ) , (B8) \nwhere F is defined by Eq. (120) of Masset & Casoli (2010). For z ν > 1, we take the values reported in the Figure 12 of the same paper. \nIf, in the presence of the companion, the surface density profile changes from Σ un( R ) to Σ un( R ) + δ Σ ( R ), the change in the corotation torque will be \n∆ T CR , 1 = 3 4 C ω 2 1 x 4 hs , 1 a 1 d ( Σ un + δ Σ ) dR GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> R = a 1 + 3 2 δ Σ 1 + p Σ un , 1 = 3 4 C ω 2 1 x 4 hs , 1 " a 1 d ( δ Σ ) dR + 3 2 δ Σ # GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> R = a 1 , (B9) \nwhere we have assumed that the rotation of the disk is Keplerian (i.e. d ln B / d ln R = -3 / 2). From Du ff ell (2015), we know that \nδ Σ ( R ) = -Σ un( R ) " f 0 Φ 2 K 2 / (3 π ) 1 + f 0 K 2 / (3 π ) r a 2 R # , (B10) \nwith K 2 = q 2 2 / ( α h 5 ) and Φ 2( R ) as given in Equation (A1). Using our definition of λ 1 ≡ -∆ T CR , 1 / | T 1s , 2 | and recalling that | T 1s , 2 | = f 0 γ 2 h -3 q 2 2 a 4 2 ω 2 2 Σ un , 2, we obtain \nλ 1 = -3 4 Ch 3 f 0 ξ p γ 2 q 2 2 x 4 hs , 1 a 3 1 a 2 " a 1 Σ un , 1 d ( δ Σ ) dR + 3 2 δ Σ Σ un , 1 # GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> R = a 1 . (B11) \nA similar procedure can be followed to derive λ 2 ≡ ∆ T CR , 2 / | T 1s , 1 | . We get \nλ 2 = 3 4 Ch 3 f 0 ξ -p γ 1 q 2 1 x 4 hs , 2 a 1 a 3 2 " a 2 Σ un , 2 d ( δ Σ ) dR + 3 2 δ Σ Σ un , 2 # GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> R = a 2 . (B12) \nHere δ Σ is given by Equation (B10) but replacing the subscript 2 with 1. In the cases of interest, λ 1 and λ 2 are both positive, so the e ff ect is generally repulsive. As expected, λ 1 and λ 2 are proportional to the fourth power of x hs , 1 and x hs , 2, respectively. Thus, to predict correctly the magnitude of λ 1 and λ 2, it is important to have a precise value for x hs , j . The procedure to compute x hs in our simulations is described in the Appendix E. \nIn principle, the di ff erential Lindblad torque on each orbiter will also be modified if the structure of the disk changes due to the presence of the companion. We note, however, that the dependence of the Lindblad torque on the slope of surface density is much weaker than that of the corotation torque, not only in a realistic, 3D disk (Tanaka et al. 2002), but also in the 2D disks considered in this paper, even for the low-mass planets for which the width of the horseshoe region is not enhanced (see Appendix E). In fact, Paardekooper et al. (2010) showed that the di ff erential Lindblad torque, T L, on perturber j (alone in the disk) in circular orbit in a 2D disk with constant h is given by \nγ T L , j T 0 , j = -4 . 2 -0 . 1 d ln Σ un d ln R GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> R = aj 0 . 4 Rs / H ! 0 . 71 , (B13) \nwith T 0 , j = q 2 j Σ un , ja 4 j ω 2 j / h 2 . From the above equation and for the smoothing length Rs used in this work, the change in the di ff erential Lindblad torque when the background density Σ un is modified to Σ un + δ Σ is \n∆ T L , j = T 0 , j -0 . 075 aj Σ un , j d ( δ Σ ) dR -3 . 1 δ Σ Σ un , j ! GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> R = aj . (B14) \nWe may compare ∆ T CR , 1 given in Equation (B9) with ∆ T L , 1, assuming, for simplicity, that the outer perturber opens a Gaussian gap so that δ Σ = -β Σ un exp( -( R -a 2) 2 / W 2 ), where β is the depth of the gap and W its width. By combining Eqs. (B9) and (B14), it is simple to show that \n| ∆ T L , 1 | | ∆ T CR , 1 | ≃ 3 C 1 + 2( ξ -1) a 2 1 W 2 -1 . (B15) \nTo derive this equation, we have taken conservatively that x hs , 1 ≃ 1 . 1 p q 1 / h in Equation (B9). For W ≃ 3 ha 2 and the values of ξ considered in this paper, we find that | ∆ T L , 1 | ≲ 0 . 15 | ∆ T CR , 1 | . A similar argument leads to | ∆ T L , 2 | ≲ 0 . 15 | ∆ T CR , 2 | . It is therefore legitimate to neglect the Lindblad torque in the DGMT scenario.', 'C. TWO INDEPENDENT MIGRATORS': 'In this section, we estimate D 0 defined as D , given in Equation (3), when the bodies are so far apart that the torques they experience are not a ff ected by the presence of their companion. The torque exerted on an isolated migrator in a steady state, i.e. when the gap has become a steady-state structure, was computed by Kanagawa et al. (2018), using two-dimensional hydrodynamical simulations. For α ≥ 5 × 10 -3 , they find that the torque exerted on a migrator that opens a partial gap is approximately given by the Type I migration formula but replacing the disk surface density by the surface density at the bottom of the gap Σ gap , j (see Figure 7 in Kanagawa et al. 2018 1 ). From Du ff ell (2015), we have Σ gap , j = γ j Σ un , j , where γ j is given in Equation (7). On the other hand, we may use the formula for the Type-I torque from Tanaka et al. (2002) to obtain Tj ≃ -χγ jh -2 q 2 j ω 2 j a 4 j Σ un , j , with χ = 1 . 16 + 2 . 83 p in a two-dimensional disk. Combining these equations, we find from Equation (3): \nD 0 = χ h -2 ξ 1 / 2 ω 2 0 a 3 + p 0 a 1 -p 1 Σ 0 M -1 · GLYPH<16> γ 1 q 1 -γ 2 q 2 ξ 1 / 2 -p GLYPH<17> . (C16) \nTherefore, if h and α are constant along the disk and γ 2 q 2 / ( γ 1 q 1) = ξ p -1 / 2 then D 0 = 0 and, consequently, ξ remains constant over time (see Eq. 2). On the other hand, the migration is divergent if γ 2 q 2 / ( γ 1 q 1) < ξ p -1 / 2 , while it is convergent if γ 2 q 2 / ( γ 1 q 1) > ξ p -1 / 2 .', 'D. HILL STABILITY CONDITION': "Assume that initially, the orbiters are on circular orbits and focus on pairs satisfying the condition for Hill stability. In the absence of the accretion disk, the pair is said to be Hill stable if the gravitational interactions between the two orbiters remain moderate. The Hill stability condition is commonly written in terms of the mutual Hill radius, defined as: \nR mH = GLYPH<18> qt 3 GLYPH<19> 1 / 3 a 12 , (D17) \nwhere qt ≡ q 1 + q 2 and a 12 ≡ ( a 1 + a 2) / 2. In the case of initially circular orbits, Gladman (1993) showed that the condition ∆ > 2 √ 3 R mH, where ∆ ≡ a 2 -a 1, ensures Hill stability because close encounters between the orbiters are forbidden. Even though the interaction between the orbiters may lead to episodes where their eccentricities can increase, conservation of angular momentum (if the disk is not present) implies that the orbital separation increases, impeding close encounters. In terms of ξ ≡ a 2 / a 1, Gladman's condition implies \nξ > 1 + 3 1 / 6 q 1 / 3 t 1 -3 1 / 6 q 1 / 3 t . (D18) \nFor instance, for qt = 10 -4 , it implies ξ > 1 . 12.", 'E. THE HALF-WIDTH OF THE HORSESHOE REGION IN OUR SIMULATIONS': "Since some planets in our samples have a mass near or slightly above the thermal mass h 3 M · , the standard, low-mass estimate for the width of the horseshoe region may lead to an underestimation of the variation of the corotation torque. It is, therefore, desirable to use a formula for the width of the horseshoe region that captures the transition from the low- to the high-mass regime. Jim'enez & Masset (2017) have studied this transition, but their results apply to planets in three-dimensional disks. Here, we have planets in two-dimensional disks and a smoothing length of the potential of 0 . 6 H , for which no result exists in the literature. We have therefore undertaken a dedicated study similar to that of Jim'enez & Masset (2017), but in 2D disks with the smoothing length quoted above. We performed 20 runs over 10 orbital periods, with planet masses ranging from 10 -6 M · to 10 -4 M · in a geometric sequence, in an inviscid disk with aspect ratio h = 0 . 028. The resolution adopted in these runs was N ϕ = 1000 and NR = 400, the radial bins having a constant spacing covering the range 0 . 5 to 1 . 5 (the planet's orbital radius being one). The half-width x hs of the horseshoe region is then obtained as the mean of the rear and front upstream half-widths, measured at ± 60 · from the azimuth of the planet. Following Jim'enez & Masset (2017), expressing the planet's mass Q in units of the thermal mass ( Q = q / h 3 ) and the half-width X hs of the horseshoe region in units of the pressure lengthscale H ( X hs = x hs / H ), we obtain the following regimes: \nX low hs = 1 . 11 Q 1 / 2 in the low mass limit , (E19) \nX high hs = 1 . 6 Q 1 / 3 in the high mass limit . (E20) \n(E21) \nAt any given mass, we find that the horseshoe half width is within 2 %, at most, of the linear combination of these two extreme regimes given by: \nX = ε X low + (1 -ε ) X high , (E22) \nhs hs hs \nwhere ε = 1 / (1 + 0 . 7 Q 3 ). From this we infer the value x hs = HX hs used in Appendices A and B."}

2023arXiv230914815H

The paper analyses a spectral approach to reconstructing a scalar field on the sphere given only information about a masked version of the field together with precise information about the smooth mask. The theory is developed for a general mask and later specialised to the case of an axially symmetric mask. Numerical experiments are given for the case of an axial mask motivated by the cosmic microwave background assuming that the underlying field is a realisation of a Gaussian random field with an artificial angular power spectrum of moderate degree ell le 100. The recovery is highly satisfactory in the absence of noise and even in the presence of moderate noise.

2023-09-01T00:00:00Z

['2023arXiv230914815H', 'arXiv:2309.14815', '10.48550/arXiv.2309.14815']

['Mathematics - Numerical Analysis', 'General Relativity and Quantum Cosmology', '15A23', '15A29', '60G60']

Removing the mask reconstructing a scalar field on the sphere from a masked field

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2309.14815.pdf

{'REMOVING THE MASK - RECONSTRUCTING A SCALAR FIELD ON THE SPHERE FROM A MASKED FIELD': 'JAN HAMANN, QUOC T. LE GIA, IAN H. SLOAN, AND ROBERT S. WOMERSLEY \nABSTRACT. The paper analyses a spectral approach to reconstructing a scalar field on the sphere, given only information about a masked version of the field together with precise information about the (smooth) mask. The theory is developed for a general mask, and later specialised to the case of an axially symmetric mask. Numerical experiments are given for the case of an axial mask motivated by the cosmic microwave background, assuming that the underlying field is a realisation of a Gaussian random field with an artificial angular power spectrum of moderate degree ( ℓ ≤ 100 ). The recovery is highly satisfactory in the absence of noise and even in the presence of moderate noise.', '1. INTRODUCTION': "In this paper, we study the reconstruction of a scalar field (for example temperature or pressure) on the unit sphere, given (possibly noisy) data on a masked version of the field, together with precise knowledge of the mask. The underlying motivation is the cosmic microwave background (CMB) for which the temperature observations, so important for the modern understanding of the early universe, are obscured over substantial portions of the sky by our own Milky Way, creating the need for masking some portions before attempting reconstruction. \nThe paper aims at a proof-of-concept for a new spectral approach to such problems. While the theory is general, the numerical experiments are restricted to the case of an axially symmetric mask, and limit the field's angular power spectrum to polynomial degree ℓ ≤ 100 , corresponding to an angular resolution of approximately 2 · . Within these limitations, the recovery is shown to be highly satisfactory in the no-noise case, and also in the case of moderate noise. \nThere is a rich literature on the inpainting problem for the particular case of the CMB [1, 20], with techniques based on harmonic methods [4, 9], iterative methods [15, 8], constrained Gaussian realisations [10, 5], group sparse optimisation methods [11] or neural networks [25, 17, 18, 13, 24]. \nClosest to the present approach is the work of Alonso et al. [2], which however differs in aim (which was to recover the angular power spectrum from a knowledge of the 'pseudo C ℓ ', through pooling together Fourier coefficients of different degrees). \nThe problem is formulated in the next section, by reducing the problem to that of solving a large ill-posed linear system. Section 3 establishes properties of the matrix in that linear system. Section 4 outlines our stochastic approach to the solution of the linear system. Because the full linear system is currently beyond our resources, in Section 5 we obtain a large reduction in \ndifficulty by specialising to the case of an axially symmetric mask. The final section is devoted to numerical experiments in both the no-noise and noisy cases.", '2. THE PROBLEM SETTING': "Taking r to be any point in the unit sphere S 2 in R 3 (i.e. r is a unit vector in R 3 ), it can be represented in spherical coordinates as \nr = (sin θ cos ϕ, sin θ sin ϕ, cos θ ) , θ ∈ [0 , π ] , ϕ ∈ [0 , 2 π ) . \nThe underlying real scalar field a ( r ) is assumed to be partially obscured by a known mask v = v ( r ) to give a masked field a v = a v ( r ) := a ( r ) v ( r ) . \nIt is well-known [12] that the space of square-integrable functions L 2 ( S 2 ) admits an orthonormal basis formed by spherical harmonics \n{ Y ℓ,m : ℓ = 0 , . . . ; m = -ℓ, . . . , ℓ } , \nwhere the Y ℓ,m is given explicitly by \n(2.1) Y ℓ,m ( θ, ϕ ) = √ 2 ℓ +1 4 π ( ℓ -m )! ( ℓ + m )! P m ℓ (cos θ ) e imϕ , m ≥ 0 , Y ℓ,m ( θ, ϕ ) = ( -1) m Y ℓ, -m ( θ, ϕ ) , m< 0 , \nwhere { P m ℓ } denote the associated Legendre function, which is defined in terms of Legendre polynomials { P ℓ : ℓ -0 , 1 , . . . } by \nP m ℓ ( t ) = ( -1) m (1 -t 2 ) m/ 2 d m d t m P ℓ ( t ) , m = 0 , 1 , . . . , ℓ ; ℓ = 0 , 1 , 2 , . . . \nWe assume that the incompletely known field a is a spherical polynomial of degree at most L , and hence expressible as an expansion in terms of orthonormal (complex) spherical harmonics Y ℓ,m ( r ) of degree ℓ ≤ L , \n(2.2) a ( r ) = L ∑ ℓ =0 ℓ ∑ m = -ℓ a ℓ,m Y ℓ,m ( r ) , r ∈ S 2 , \nwhere \na ℓ,m := ∫ S 2 a ( r ) Y ℓ,m ( r )d S ( r ) , \nwhere d S ( r ) = sin θ d θ d ϕ , following from the orthonormality of the spherical harmonics, \n∫ S 2 Y ℓ,m ( r ) Y ℓ ' ,m ' ( r )d S ( r ) = δ ℓ,ℓ ' δ m,m ' , ℓ, ℓ ' ≥ 0 , m = -ℓ, . . . , ℓ, m ' = -ℓ ' , . . . , ℓ ' . \nWe assume that the mask can be adequately approximated by a partial sum of spherical harmonics series as 1 \n(2.3) v ( r ) = K ∑ k =0 k ∑ ν = -k v k,ν Y k,ν ( r ) , \nwhere \nv k,ν := ∫ S 2 v ( r ) Y k,ν ( r )d S ( r ) . \nThus the masked signal a v ( r ) = a ( r ) v ( r ) is expressible as a spherical harmonic expansion of degree L + K , \n(2.4) a v ( r ) = a ( r ) v ( r ) = L + K ∑ j =0 j ∑ µ = -j a v j,µ Y j,µ ( r ) , r ∈ S 2 , \nwhere \na v j,µ := ∫ S 2 a v J ( r ) Y j,µ ( r )d S ( r ) = ∫ S 2 a ( r ) v ( r ) Y j,µ ( r )d S ( r ) (2.5) = ∫ S 2 ( L ∑ ℓ =0 ℓ ∑ m = -ℓ a ℓ,m Y ℓ,m ( r ) )( K ∑ k =0 k ∑ ν = -k v k,ν Y k,ν ( r ) ) Y j,µ ( r )d S ( r ) = L ∑ ℓ =0 ℓ ∑ m = -ℓ K ∑ k =0 k ∑ ν = -k a ℓ,m v k,ν ∫ S 2 Y ℓ,m ( r ) Y k,ν ( r ) Y j,µ ( r )d S ( r ) = L ∑ ℓ =0 ℓ ∑ m = -ℓ E j,µ ; ℓ,m a ℓ,m , \nwhere, for j = 0 , . . . , L + K , µ = -j, . . . , j and ℓ = 0 , . . . , L , m = -ℓ, . . . , ℓ , \n(2.6) E j,µ ; ℓ,m = K ∑ k =0 k ∑ ν = -k ∫ S 2 Y ℓ,m ( r ) Y k,ν ( r ) Y j,µ ( r )d S ( r ) v k,ν . \nThus the essential task in the reconstruction is to solve as accurately as possible the large linear system \n(2.7) L ∑ ℓ =0 ℓ ∑ m = -ℓ E j,µ ; ℓ,m a ℓ,m = a v j,µ , \nwhere E is defined by (2.6). \nEquation (2.7) can be considered as an overdetermined (but possibly not full-rank) set of linear equations for the a ℓ,m . We will find it convenient to replace the upper limit L + K in (2.7) by a more flexible upper limit J with L ≤ J ≤ L + K . Then the equation can be written as \n(2.8) E a = a v , \nwhere E is a ( J +1) 2 × ( L +1) 2 matrix, \n(2.9) a = ( a ℓ,m , ℓ = 0 , . . . , L, m = -ℓ, . . . , ℓ ) ∈ C ( L +1) 2 . \nand \n(2.10) a v = ( a v j,µ , ℓ = 0 , . . . , J, m = -ℓ, . . . , ℓ ) ∈ C ( J +1) 2 . \nIn Section 4 we come to the most challenging part of the paper, which is the approximate solution of the ill-posed linear system, and before it the computation of the matrix E . Before then, however, it is useful to establish properties of the matrix E .", '3. PROPERTIES OF E': 'This section summarizes useful properties of the matrix E defined in (2.6). We first note that the product of three spherical harmonics (known as a Gaunt coefficient) can be evaluated in terms of Wigner 3j symbols, see eg. [6, Eq. 34.4.22] or [2]. Explicitly, \nD ℓ,m ; k,ν ; j,µ := ∫ S 2 Y ℓ,m ( r ) Y k,ν ( r ) Y j,µ ( r )d S ( r ) (3.1) = ( -1) µ √ (2 ℓ +1)(2 k +1)(2 j +1) 4 π × ( ℓ k j 0 0 0 )( ℓ k j m ν -µ ) . \nAs important special cases, \n(3.2) D ℓ,m ; k,ν ; j,µ = 0 if j + ℓ + k is odd, or k < | j -ℓ | , or k > j + ℓ, or m + ν = µ. \n̸ \nThe following lemma gives several elementary properties of the matrix E , beginning with an explicit integral expression in terms of the mask function v .', 'Lemma 3.1. The elements of the matrix E satisfy': "(3.3) E j,µ ; ℓ,m = ∫ S 2 Y j,µ ( r ) Y ℓ,m ( r ) v ( r ) d S ( r ) , \nand, for a real mask v , \nE ℓ,m ; j,µ = E j,µ ; ℓ,m , (3.4) \nE j,µ ; ℓ,m = K ∑ k =0 [ D ℓ,m ; k, 0; j,µ v k, 0 +2 k ∑ ν =1 ℜ ( D ℓ,m ; k,ν ; j,µ v k,ν ) ] , (3.5) \nE j,µ ; ℓ, -m = ( -1) m -µ E j, -µ ; ℓ,m . (3.6) \nProof. Firstly, from the definition of E in (2.6) we have \nE j,µ ; ℓ,m = K ∑ k =0 k ∑ ν = -k ∫ S 2 Y ℓ,m ( r ) Y k,ν ( r ) Y j,µ ( r ) d S ( r ) v k,ν = ∫ S 2 Y j,µ ( r ) Y ℓ,m ( r ) K ∑ k =0 k ∑ ν = -k v k,ν Y k,ν ( r ) d S ( r ) = ∫ S 2 Y j,µ ( r ) Y ℓ,m ( r ) v ( r ) d S ( r ) , \nestablishing (3.3). From the definition of D ℓ,m ; k,ν ; j,µ in (3.1) as an integral, together with the spherical harmonic property \n(3.7) \nit follows that \nD j,µ ; k,ν ; ℓ,m ; = ( -1) ν D ℓ,m ; k, -ν ; j,µ . \nBecause both the mask v and the field a are real we have \n(3.8) a ℓ,m = ( -1) m a ℓ, -m , v k,µ = ( -1) µ v k, -µ , \nfor all relevant values of ℓ, m, k and µ , and (3.4) then follows from (2.6). Also, from (3.1) and (3.7), \nD ℓ, -m ; k,ν ; j,µ = ( -1) m D ℓ,m ; k,ν ; j,µ , D ℓ,m ; k, -ν ; j,µ = ( -1) ν D ℓ,m ; k,ν ; j,µ , \nso (3.8) gives \nD ℓ,m ; k, -ν ; j,µ v k, -ν = D ℓ,m ; k,ν ; j,µ v k,ν , \nand (2.6) then yields (3.5). Finally, for a real mask (3.6) follows easily from (3.3). \n□ \n- 3.1. Singular values of E . This subsection gives upper bounds on the singular values of the rectangular matrix E in terms of the real mask v . We use E ∗ to denote the complex conjugate transpose of the matrix E . \nTheorem 3.2. Assume that the mask v ( r ) is approximated by the partial sum of its spherical Fourier series of degree K ≥ 1 and that L ≤ J ≤ L + K is the degree of the approximation a v J ( r ) to the masked field a v ( r ) , so that E is a ( J +1) 2 by ( L +1) 2 matrix. The singular values σ of E satisfy \n(3.9) 0 ≤ σ ≤ v max \n, \nY ℓ, -m ( r ) = ( -1) m Y ℓ,m ( r ) , \nwhere \nv max := max r ∈ S 2 | v ( r ) | . \n̸ \nProof. Let u = 0 be an eigenvector of the positive semi-definite Hermitian matrix E ∗ E corresponding to the non-negative real eigenvalue σ 2 , so E ∗ E u = σ 2 u . Then, using (3.3), and writing the elements of u as u ℓ,m , ℓ = 0 , . . . , L, m = -ℓ, . . . , ℓ , we have \n( E u ) j,µ = L ∑ ℓ =0 ℓ ∑ m = -ℓ u ℓ,m ∫ S 2 Y j,µ ( r ) Y ℓ,m ( r ) v ( r )d S ( r ) = ∫ S 2 Y j,µ ( r ) u ( r ) v ( r )d S ( r ) , \nfor j = 0 , . . . , J , µ = -j, . . . , j , where \nu ( r ) := L ∑ ℓ =0 ℓ ∑ m = -ℓ u ℓ,m Y ℓ,m ( r ) , \ngiving \nu ∗ E ∗ E u = ∥ E u ∥ 2 ℓ 2 = J ∑ j =0 j ∑ µ = -j ∣ ∣ ∣ ∣ ∫ S 2 u ( r ) v ( r ) Y j,µ ( r )d S ( r ) ∣ ∣ ∣ ∣ 2 . \nThus, using Parseval's identity for uv and then u , \nσ 2 ∥ u ∥ 2 ℓ 2 = u ∗ ( σ 2 u ) = u ∗ ( E ∗ E u ) = J ∑ j =0 j ∑ µ = -j ∣ ∣ ∣ ∣ ∫ S 2 u ( r ) v ( r ) Y j,µ ( r )d S ( r ) ∣ ∣ ∣ ∣ 2 (3.10) = ∥ u v ∥ 2 L 2 ≤ ( v max ) 2 ∥ u ∥ 2 L 2 = ( v max ) 2 ∥ u ∥ 2 ℓ 2 . \nThis gives the upper bound (3.9) on the singular values σ of E . \n□ \n- 3.2. Eigenvalues of E . In this subsection we take J = L , making the matrix E square. From the second statement in Lemma 3.1, E is Hermitian thus its eigenvalues are real, and eigenvectors belonging to distinct eigenvalues are orthogonal. \n̸ \nLet λ ∈ R be an eigenvalue of E and let q = 0 be a corresponding eigenvector, thus \nE q = λ q . \nThe following result provides both lower and upper bounds on λ , in terms of the minimum and maximum values of the mask v . \nTheorem 3.3. Assume that the mask v is approximated by the partial sum of its spherical Fourier series of degree K ≥ 1 . Assume also that J = L , so that the matrix E is square. Then the eigenvalues of E lie in the interval ( v min , v max ) , where \nv min := min r ∈ S 2 v ( r ) , v max := max r ∈ S 2 v ( r ) . \nProof. Let λ ∈ R be an eigenvalue of E , with corresponding eigenvector q ∈ C ( L +1) 2 . Then from (3.3), \nq ∗ E q = L ∑ j =0 j ∑ µ = -j L ∑ ℓ =0 ℓ ∑ m = -ℓ q j,µ E j,µ ; ℓ,m q ℓ,m (3.11) = L ∑ j =0 j ∑ µ = -j L ∑ ℓ =0 ℓ ∑ m = -ℓ q j,µ q ℓ,m ∫ S 2 Y j,µ ( r ) Y ℓ,m ( r ) v ( r ) d S ( r ) = ∫ S 2 q ( r ) q ( r ) v ( r )d S ( r ) , \nwhere \nq ( r ) := L ∑ ℓ ∑ q ℓ,m Y ℓ,m ( r ) \n(3.12) ℓ =0 m = -ℓ . \nIt follows that \n∫ (3.13) \n∥ q ∥ 2 L 2 v max = J ∑ j =0 j ∑ µ = -j | q j,µ | 2 v max = ∥ q ∥ 2 ℓ 2 v max , \nλ ∥ q ∥ 2 ℓ 2 = q ∗ ( λ q ) = q ∗ E q = S 2 | q ( r ) | 2 v ( r ) d S ( r ) < \nwhere the inequality is strict because v , being a spherical polynomial of non-zero degree, cannot be identically equal to either its maximum or minimum value. Similarly, we have a lower bound \nλ ∥ q ∥ 2 ℓ 2 > ∥ q ∥ 2 ℓ 2 v min , \ntogether proving λ ∈ ( v min , v max ) . \n□ \nNote that even if the true (non-polynomial) mask lies in [0 , 1] for all r ∈ S 2 , the Gibbs phenomenon will typically produce oscillations in v , making v min < 0 and v max > 1 . \nWe also note in passing that q is an eigenvector belonging to λ for the integral equation \n∫ S 2 K L ( r , r ' ) q ( r ' ) d r ' = λq ( r ) , \nwhere K L ( r , r ' ) is the integral kernel given by \nK L ( r , r ' ) = L ∑ ℓ =0 ℓ ∑ m = -ℓ Y ℓ,m ( r ) Y ℓ,m ( r ' ) v ( r ' ) (3.14) = L ∑ ℓ =0 2 ℓ +1 4 π P ℓ ( r · r ' ) v ( r ' ) , \nand in the last step we used the addition theorem for spherical harmonics. Here P ℓ is the Legendre polynomial of degree ℓ , normalised so that P ℓ (1) = 1 .", '4. SOLVING E a = a v': "As with any ill-posed system, it is essential to build in a priori knowledge of the solution. Neumaier [14, Section 8], knowing that the true solution of an ill-posed problem is generally smooth, controls the smoothness through a smoothing operator S . However, in this problem a smoothing operator would not be appropriate because the solution is the opposite of smooth, since for each ℓ, m the unknown quantity a ℓ,m is a realisation of an independent random variable. That is a property we must build into the solution. Accordingly, we assume, in accordance with the usual assumptions for the CMB, that the a ℓ,m are mean-zero uncorrelated random variables with covariance ( C ℓ ) L ℓ =0 , where C ℓ is real. Details on using Gaussian random fields to model the CMB can be found in the book by Marinucci and Peccati [12]. \nWe allow general J in the range L ≤ J ≤ L + K , giving a linear system with ( J + 1) 2 equations, so typically an over-determined linear system with more equations than unknowns, implying that an exact solution does not in general exist. \nMoreover, we assume that the original field coefficients a ℓ,m are corrupted by noise, so the actual model is \n(4.1) E a ε = a v + ε v = ( a + ε ) v , \nwhere ε is a vector of independent mean-zero random variables ε ℓ,m with a diagonal covariance matrix Υ , and a ε is an approximation to a . We also assume that the ε ℓ,m and the a ℓ,m are all statistically independent, so that in terms of expected values we have \n⟨ ε ℓ,m ⟩ = 0 , ⟨ a ℓ,m ⟩ = 0 , ⟨ ε ℓ,m a ℓ ' ,m ' ⟩ = 0 , ⟨ a ℓ,m a ℓ ' ,m ' ⟩ = C ℓ δ ℓ,ℓ ' δ m,m ' , (4.2) ⟨ ε ℓ,m ε ℓ ' ,m ' ⟩ = Υ ℓ,m δ ℓ,ℓ ' δ m,m ' . \nWe deduce the following expectations of quadratic forms: \n⟨ aa ∗ ⟩ = Ω ⟨ a v a ∗ ⟩ = ⟨ ( E a ) a ∗ ⟩ = E Ω ⟨ a v ( a v ) ∗ ⟩ = ⟨ ( E a )( a ∗ E ∗ ) ⟩ = E Ω E ∗ (4.3) ⟨ εa ∗ ⟩ = 0 ⟨ εε ∗ ⟩ = Υ , ⟨ ε v ( ε v ) ∗ ⟩ = E Υ E ∗ , \nwhere \n(4.4) Ω ℓ,m ; ℓ ' ,m ' = C ℓ δ ℓ,ℓ ' δ m,m ' . \nLet Λ ∈ C ( L +1) 2 × ( L +1) 2 be a real symmetric-positive definite matrix, with associated norm ∥ a ∥ Λ = ( a ∗ Λ a ) 1 2 defined by \n(4.5) ∥ a ∥ 2 Λ = a ∗ Λ a = tr [ aa ∗ Λ] = tr [Λ aa ∗ ] , \nwhere we used the matrix property tr( AB ) = tr( BA ) . The following theorem gives a condition for minimising the expected squared Λ -norm error of an approximate solution of (4.1). It is an extension/specialisation of [14, Theorem 8], which that author attributes to [3]. \nTheorem 4.1. Consider the over-determined linear system E a ε = a v + ε v , where a v = E a , ε v = E ε and a and ε have the stochastic properties in (4.3) . Assume that the ( J +1) 2 × ( L +1) 2 matrix E has full rank ( L +1) 2 . Among all approximations of the form a ε ≈ Q ( a v + ε v ) , where Q is a non-random ( L +1) 2 × ( J +1) 2 matrix, the expected squared error ⟨∥ a -Q ( a v + ε v ) ∥ 2 Λ ⟩ is minimized by any solution ̂ Q of the equation \n(4.6) ̂ QE (Ω + Υ) = Ω . \nThe resulting minimum expected squared error is \n(4.7) 〈 ∥ a -̂ a ∥ 2 Λ 〉 = tr [ Λ ( Ω -Ω(Ω + Υ) -1 Ω )] , \nwhere ̂ a := ̂ Q ( a v + ε v ) . \nRemark 1 . The minimizer ̂ a is in general not unique. \nProof. Writing y ε := a v + ε v , a general linear approximation can be written as Q y ε = ̂ Q y ε + R y ε , where ̂ Q is an as yet unknown minimizer, and R = Q -̂ Q is a matrix in C ( L +1) 2 × ( J +1) 2 . The mean square error can now be written as \n∥ a -( ̂ Q + R ) y ε ∥ 2 Λ = ∥ a -̂ Q y ε ∥ 2 Λ + ∥ R y ε ∥ 2 Λ -2 ℜ [( a -̂ Q y ε ) ∗ ) Λ R y ε ] = ∥ a -̂ Q y ε ∥ 2 Λ + ∥ R y ε ∥ 2 Λ -2 ℜ [ tr ( Λ R y ε ( a -̂ Q y ε ) ∗ )] . \nOn taking expected values and using (4.3) we have \n〈 tr(Λ R y ε ( a -̂ Q y ε ) ∗ 〉 = tr ( Λ R 〈 y ε ( a -̂ Q y ε ) ∗ 〉) = tr ( Λ R 〈 ( a v + ε v )( a -̂ Q ( a v + ε v )) ∗ 〉) = tr ( Λ R ( E ⟨ aa ∗ ⟩ -E ⟨ aa ∗ + εε ∗ ⟩ E ∗ ̂ Q ∗ )) = tr ( Λ R ( E Ω -E (Ω + Υ) E ∗ ̂ Q ∗ )) . \nSo \n〈 ∥ a -( ̂ Q + R ) y ε ∥ 2 Λ 〉 = 〈 ∥ a -̂ Q y ε ∥ 2 Λ 〉 + 〈 ∥ R y ε ∥ 2 Λ 〉 -2 ℜ tr ( Λ R ( E Ω -E (Ω + Υ) E ∗ ̂ Q ∗ )) . \nBy definition, ̂ Q is a minimizer of ⟨∥ a -Q y ε ∥ 2 Λ ⟩ , so the linear term must vanish for all R . More precisely, we must have \n(4.8) E Ω = E (Ω + Υ) E ∗ ̂ Q ∗ , or equivalently ̂ QE (Ω + Υ) E ∗ = Ω E ∗ , \nsince otherwise by taking R to be ( E Ω -E (Ω + Υ) E ∗ ̂ Q ∗ ) ∗ we obtain a contradiction. It is easily seen that the second equality in (4.8) is equivalent to (4.6): starting with (4.6) by right \nmultiplying by E ∗ , starting with (4.8) by right multiplying by E and using the invertibility of E ∗ E , noting that E ∗ E is a square matrix of full rank ( L +1) 2 . If (4.6) holds then we have \n〈 ∥ a -( ̂ Q + R ) y ε ∥ 2 Λ 〉 = 〈 ∥ a -̂ Q y ε ∥ 2 Λ 〉 + 〈 ∥ R y ε ∥ 2 Λ 〉 ≥ 〈 ∥ a -̂ Q y ε ∥ 2 Λ 〉 \nwith equality for R = 0 , corresponding to a = Q y . \nThe expected squared error is \n̂ ̂ ε \n〈 ∥ a -̂ a ∥ 2 Λ 〉 = 〈 ∥ a -̂ Q y ε ∥ 2 Λ 〉 = 〈 ( a -̂ Q y ε ) ∗ Λ ( a -̂ Q y ε ) 〉 = tr [ Λ 〈 ( a -̂ Q y ε )( a -̂ Q y ε ) ∗ 〉] = tr [ Λ ( ⟨ aa ∗ ⟩ + 〈 ̂ Q y ε y ∗ ε ̂ Q ∗ 〉 -2 ℜ ( ̂ Q ⟨ y ε a ∗ ⟩ ) )] = tr [ Λ ( Ω+ ̂ Q ( E Ω E ∗ + E Υ E ∗ ) ̂ Q ∗ -2 ℜ ( ̂ QE Ω ) )] , \nNow by (4.8) \n̂ QE Ω = ̂ Q ( E Ω E ∗ + E Υ E ∗ ) ̂ Q ∗ , \nand hence \ntr[Λ ̂ QE Ω] = tr[Λ ̂ QE (Ω + Υ) E ∗ ̂ Q ∗ ] = tr[(Ω + Υ) 1 / 2 E ∗ ̂ Q ∗ Λ ̂ QE (Ω + Υ) 1 / 2 ] = ∥ ̂ QE (Ω + Υ) 1 / 2 ] ∥ 2 Λ , \nwhich is real, implying \n〈 ∥ a -̂ a ∥ 2 Λ 〉 = tr [ Λ ( Ω -̂ QE Ω )] = tr [ Λ ( Ω -Ω(Ω + Υ) -1 Ω )] . \n□ \nRemark 2 . Note that the equation (4.8) determining the minimizer ̂ Q y ε of the expected meansquare error does not depend on the matrix Λ , i.e. on the choice of quadratic norm. For example, using Λ = I or Λ = Ω does not change ̂ Q . \nCorollary 4.2. Under the conditions of Theorem 4.1, let Γ be an arbitrary positive definite matrix of size ( J +1) 2 × ( J +1) 2 . Then a vector ̂ a ∈ R ( L +1) 2 that achieves the minimal error given in (4.7) is \n(4.9) ̂ a := Ω(Ω + Υ) -1 α , \nwhere α ∈ R ( L +1) 2 is the unique solution of \n(4.10) E ∗ Γ E α = E ∗ Γ( a v + ε v ) . \nProof. The matrix ̂ Q defined by \n(4.11) ̂ Q := Ω(Ω + Υ) -1 ( E ∗ Γ E ) -1 E ∗ Γ \nis easily seen to satisfy the condition (4.6) in Theorem 4.1. Equally, it is easily seen that the corresponding minimizer \n̂ a := ̂ Q ( a v + ε v ) \ncan be written exactly as stated in the corollary. \n□ \nRemark 3 . The corollary gives our prescription for computing the coefficient vector ̂ a . Note that the postprocessing step in (4.9) is easily carried out given that the matrices Ω and Υ are diagonal, since each element of α is by this step merely reduced by a known factor. Note also that equation (4.10) is just the normal equation for the linear system if, as we shall assume in practice, Γ is the identity matrix. Formation of the normal equations can greatly increase the condition number of an already ill-conditioned system. In practice we shall address the ill-conditioning either by QR factorisation of the matrix E , or (less desirably) by adding a regularising term to the right-hand side, to obtain \n(4.12) ( E ∗ Γ E +Σ) α = E ∗ Γ( a v + ε v ) , \nwhere Σ is an empirically chosen positive definite ( L +1) 2 × ( L +1) 2 matrix. \nThe following proposition shows that if the elements of a v and ε v have the correct symmetry for real-valued fields a v = av and ϵ v = ϵv , then the computed values of ̂ a also follow the same symmetry. The practical importance of this result is that the symmetry property, since it occurs naturally, does not need to be enforced. \nProposition 4.3. Assume that the components of a v and ε v satisfy \na v j,µ = ( -1) µ a v j, -µ and ε v j,µ = ( -1) µ ε v j, -µ , µ = -j, . . . j, j ≥ 0 . \nAssume also that the positive definite matrices Ω , Υ and Γ , and also Σ if present, are all diagonal, and that their diagonal elements are positive numbers independent of the second label µ or m . Then ̂ a given by (4.10) and (4.9) satisfies \n̂ a ℓ,m = ( -1) m ̂ a ℓ, -m , m = -ℓ, . . . , ℓ, ℓ ≥ 0 . \nProof. We first show that the components of b := E ∗ Γ( a v + ε v ) satisfy \nb ℓ,m = ( -1) m b ℓ, -m , m = -ℓ, . . . , ℓ, ℓ ≥ 0 . \nWe have, using (3.6), \nb ℓ, -m = ∑ j ∑ µ ( E ∗ ) ℓ, -m ; j,µ Γ j ( a v j,µ + ε j,µ ) = ∑ j ∑ µ ( -1) m -µ ( E ∗ ) ℓ,m ; j, -µ Γ j ( a v j,µ + ε j,µ ) = ( -1) m ∑ j ∑ µ ( E ∗ ) ℓ,m ; j, -µ Γ j ( a v j, -µ + ε j, -µ ) = ( -1) m b ℓ,m , \nas required. A similar argument shows that \n( E ∗ Γ E +Σ) ℓ, -m ; ℓ ' ,m ' = ( -1) m -m ' ( E ∗ Γ E +Σ) ℓ,m ; ℓ ' , -m ' . \nSince α is the unique solution of \n( E ∗ Γ E +Σ) α = b , \nby taking the ( ℓ, -m ) component of this equation we obtain \n∑ ℓ ' ∑ m ' ( E ∗ Γ E +Σ) ℓ, -m ; ℓ ' m ' α ℓ ' ,m ' = b ℓ, -m , \nwhich with the above symmetry properties leads to \n∑ ℓ ' ∑ m ' ( -1) m -m ' ( E ∗ Γ E +Σ) ℓ,m ; ℓ ' -m ' α ℓ ' ,m ' = ( -1) m b ℓ,m ; \nOn taking the complex conjugate and dividing by ( -1) m this gives us \n∗ , \n(4.13) ( E Γ E +Σ) c = b \nwhere \n(4.14) \nc ℓ ' , -m ' := ( -1) m ' α ℓ ' ,m ' , m ' = -ℓ ' , . . . , ℓ ' , ℓ ' ≥ 0 . \nWe see by uniqueness of the solution of (4.13) that c = α , thus by (4.14) the vector α has the desired symmetry. Multiplication by Ω(Ω+Υ) -1 clearly preserves the symmetry, thus the proof is complete. □", '5. AXIALLY SYMMETRIC MASKS': 'Ageneral mask v ( r ) leads to a large dense matrix E , of size ( J +1) 2 ( L +1) 2 × ( J +1) 2 ( L +1) 2 , see (2.6), a size beyond present resources if J and L are in the hundreds. In this section we consider the more tractable special case in which the mask v is axially symmetric, i.e. \nv ( r ) = v ( θ, ϕ ) = v ( θ ) , \nwith v being a function of the polar angle θ and independent of the azimuthal angle ϕ . For this case we have \nv k,ν = w k δ ν, 0 , \nwhere \n(5.1) w k := v k, 0 = ∫ S 2 v ( r ) Y k, 0 ( r )d S ( r ) = √ π (2 k +1) ∫ π 0 v ( θ ) P k (cos( θ )) sin θ d θ. \nNote that w k is real, and that \nw k = 0 if k is odd and also v ( -r ) = v ( r ) . \nIn this case it follows from (3.2) that E j,µ ; ℓ,m = 0 unless µ = m . Thus it is convenient to introduce a new notation, \n(5.2) . \nE ( m ) j,ℓ := E j,m ; ℓ,m \nEquation (2.7) now becomes \n(5.3) L ∑ ℓ =0 E ( m ) j,ℓ a ℓ,m = a v j,m \n, \nin which the coefficients belonging to different values of m are completely decoupled. This can be seen as just a special case of (2.8), albeit with uncoupled values of m , thus all of the analysis in Sections 3 and 4 remains applicable. \nNote that from (3.5) of Lemma 3.1, E ( m ) j,ℓ is real and symmetric, E ( m ) j,ℓ = E ( m ) ℓ,j . Moreover \nE ( m ) j,ℓ = 0 if j + ℓ is odd and v ( -r ) = v ( r ) , or if ℓ < | m | or j < | m | . \nWe can treat E ( m ) j,ℓ as an ( J -| m | +1) × ( L -| m | +1) matrix, but how should we choose J ? The choice J = L inevitably leads to a poorly conditioned linear system. There would seem to be considerable benefit, at least in theory, in taking the largest value J = L + K , to ensure that the resulting overdetermined linear system makes use of all available information. \nThe equation to be solved in practice is, instead of (4.10), now \n(5.4) ( ( E ( m ) ) ∗ Γ a E ( m ) ) α = ( E ( m ) ) ∗ Γ a ( a v + ε v ); \nOr if regularisation is desired, then instead of (4.12) the equation to be solved becomes \n(5.5) ( ( E ( m ) ) ∗ Γ a E ( m ) +Σ a ) α = ( E ( m ) ) ∗ Γ a ( a v + ε v ) . \nHere Γ a and Σ a have the same diagonal values as Γ and Σ , but the second label on rows and columns has now disappeared, and the new matrices are of size J × J and L × L respectively.', '6. NUMERICAL EXPERIMENTS': "Recall that our goal is to reconstruct a scalar random field on the sphere given only a masked and noisy version of the field. We have seen in previous sections that the problem can be reduced to the solution of the overdetermined linear system \n(6.1) E a = b v , \nwhere the given data b v = ( a + ε ) v = a v + ε v are the spherical harmonic coefficients of the masked noisy map with the coefficients a corrupted by independent Gaussian noise ε with mean ⟨ ε ⟩ = 0 and variance ⟨ εε ∗ ⟩ = Υ . \nTo illustrate the potential of the method we consider numerical experiments where we know the 'true' solution a , so we can calculate errors to test performance and the effect of model parameters, including taking ε = 0 . \nFIGURE 1. Original Gaussian random field \n<!-- image --> \nIn the experiments a specified angular power spectrum C ℓ , ℓ = 2 , . . . , L is used to generate an instance of a Gaussian random field with known spherical harmonic coefficients a at the N pix = 50 , 331 , 648 HEALPix 2 [7] points ( N side = 2048 ), using the HealPy 3 package [26]. Noise is then added as described in Subsection 6.3. The mask is then applied pointwise to the noisy map, and the masked noisy map used to calculate the spherical harmonic coefficients b v , again using the HealPy package. The next step is to estimate the original Fourier coefficients a ℓ,m using (4.9) together with (4.10), or alternatively using (4.9) with the regularised equation (4.12). The final step is to reconstruct the target field from its Fourier coefficients. \nWe consider a Gaussian random field with the artificial angular power spectrum \nC ℓ = g ( ℓ L +1 ) , \nwhere \ng ( x ) = { 1 for 0 ≤ x ≤ 1 / 2 -2 x +2 for 1 / 2 ≤ x ≤ 1 . \nIn the experiments, we assume that L = 100 and K = 900 . A realisation of the random field is shown in Figure 1. We shall use this realisation as the target field a ( r ) in all the following experiments. \n- 6.1. An axially symmetric mask. When the mask applied to the noisy data is axially symmetric, the problem decomposes into independent problems for each value of m , as in Section 5. To construct the mask we first define the following non-decreasing function p ∈ C 3 ( R ) : \n(6.2) p ( x ) = 0 for x ≤ 0 , x 4 (35 -84 x +70 x 2 -20 x 3 ) for 0 < x < 1 , 1 for x ≥ 1 . \nThen as a function of the Cartesian coordinate z ∈ [ -1 , 1] of a point on the sphere, our mask is, for 0 < a z < b z < 1 \n(6.3) v ( z ) = p ( | z | -a z b z -a z ) . \nIn Figure 2, an axially symmetric mask with a z = π 2 -10 π 180 and b z = π 2 -20 π 180 is plotted on the unit sphere. This mask has the value 1 (and hence has no masking effect) for points on the sphere \nFIGURE 2. An axially symmetric C 3 mask with a z = π 2 -10 π 180 and b z = π 2 -20 π 180 . \n<!-- image --> \nmore than 20 · from the equator, the value 0 (complete masking) within 10 · from the equator, and smooth variation in between, through the function p , see (6.2), with an argument expressed as a function of the z coordinate of a point on the sphere. (We do not use a discontinuous mask because a discontinuous function has slow convergence of its Fourier series, leading to Gibbs' phenomenon for the truncated Fourier series.) \nFIGURE 3. The C 3 mask v ( θ, ϕ ) with a = 10 π 180 , b = 20 π 180 . \n<!-- image --> \nThe transformation z := cos( θ ) = cos( π 2 -φ ) , with latitude φ = π 2 -θ ∈ [ -π 2 , π 2 ] , gives \nv ( φ, ϕ ) = p ( | cos( π 2 -φ ) | -a z b z -a z ) . \nIn terms of latitude the transition region is | φ | ∈ [ a, b ] where a = 10 π 180 and b = 20 π 180 , as illustrated in Figure 3. \nThe masked field, including the addition of noise as described in Section 6.3, is illustrated in Figure 4. \nUsing this axially symmetric mask all the rectangular matrices E = E ( m ) described in Section 6.1 for m = 0 , . . . , L of sizes ( J +1 -m ) × ( L +1 -m ) with J = L + K are pre-computed in parallel. Here K is the maximum multipole in the spherical harmonics approximation of v as in (2.3). We used the sympy package [21] to compute the entries for the matrix E . Fast quadrature methods on the unit sphere [16] could be used in a future implementation. \n6.2. Numerical condition of the problem. Figures 5a and 5b illustrate the condition of the matrices E ( m ) for L = 100 , K = 900 and the mask in Figure 3. The largest singular values in Figure 5a are consistent with the upper bound on the singular values in Theorem 3.2. Even though the values of the mask lie in [0 , 1] , the polynomial approximation of the mask may have values slightly outside this interval due to the Gibbs phenomenon. The ill-conditioning of the matrices E ( m ) , as illustrated in Figure 5b, which can be severe especially for small m , arises from the smallest singular value.", 'REMOVING THE MASK': 'Reconstructed field with Gaussian noise with power spectrum 10-2C1 \nFIGURE 8. QR method for Gaussian noise with angular power spectrum Υ ℓ = 10 -2 C ℓ \n<!-- image -->', '7. CONCLUSION': 'In this paper we have analysed a spectral method for recovering a scalar field from a masked and possibly noisy version of that field. The quality of the recovery might be considered acceptable even in the presence of noise. However, it is acknowledged that the quality will deteriorate as the noise level is increased and as the cutoff polynomial degree is increased from 100 . \nFIGURE 7. QR method for Gaussian noise with angular power spectrum Υ ℓ = 10 -4 C ℓ \n<!-- image -->', 'ACKNOWLEDGEMENTS': 'The assistance of Yu Guang Wang in the early stages of the project and constructive comments from anonymous referees are gratefully acknowledged. This research includes computations using the computational cluster Katana supported by Research Technology Services at', 'HAMANN AND LE GIA, SLOAN, WOMERSLEY': 'UNSW Sydney [19]. Some of the results in this paper have been derived using the healpy and HEALPix packages.', 'REFERENCES': "- [1] P. Abrial, Y. Moudden, J. L. Starck, J. Fadili, J. Delabrouille, and M. K. Nguyen. CMB data analysis and sparsity. Stat. Meth. , 5:289, 2008.\n- [2] David Alonso, Javier Sanchez, and Anˇze Slosar. A unified pseudoC ℓ framework. Mon. Not. Roy. Astron. Soc. , 484(3):4127-4151, 2019.\n- [3] M. Bertero, C. De Mol, and G. A. Viano. The stability of inverse problems. In Inverse scattering problems in optics , volume 20 of Topics Current Phys. , pages 161-214. Springer, Berlin-New York, 1980.\n- [4] Pawel Bielewicz, K. M. Gorski, and A. J. Banday. Low order multipole maps of CMB anisotropy derived from WMAP. Mon. Not. Roy. Astron. Soc. , 355:1283, 2004.\n- [5] Martin Bucher and Thibaut Louis. Filling in CMB map missing data using constrained Gaussian realizations. Mon. Not. Roy. Astron. Soc. , 424:1694, 2012.\n- [6] NIST Digital Library of Mathematical Functions . http://dlmf.nist.gov/, Release 1.1.4 of 2022-01-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds.\n- [7] K. M. G'orski, E. Hivon, A. J. Banday, B. D. Wandelt, F. K. Hansen, M. Reinecke, and M. Bartelmann. HEALPix: A Framework for High-Resolution Discretization and Fast Analysis of Data Distributed on the Sphere. Astrophysical Journal , 622:759-771, April 2005.\n- [8] H. F. Gruetjen, J. R. Fergusson, M. Liguori, and E. P. S. Shellard. Using inpainting to construct accurate cut-sky CMB estimators. Phys. Rev. D , 95(4):043532, 2017.\n- [9] Kaiki Taro Inoue, Paolo Cabella, and Eiichiro Komatsu. Harmonic Inpainting of the Cosmic Microwave Background Sky I:Formulation and Error Estimate. Phys. Rev. D , 77:123539, 2008.\n- [10] Jaiseung Kim, Pavel Naselsky, and Nazzareno Mandolesi. Harmonic in-painting of CMB sky by constrained Gaussian realization. Astrophys. J. Lett. , 750:L9, 2012.\n- [11] Chao Li and Xiaojun Chen. Group sparse optimization for inpainting of random fields on the sphere. IMA Journal of Numerical Analysis , page drad071, 09 2023.\n- [12] Domenico Marinucci and Giovanni Peccati. Random fields on the sphere , volume 389 of London Mathematical Society Lecture Note Series . Cambridge University Press, Cambridge, 2011. Representation, limit theorems and cosmological applications.\n- [13] Gabriele Montefalcone, Maximilian H. Abitbol, Darsh Kodwani, and R. D. P. Grumitt. Inpainting CMB maps using Partial Convolutional Neural Networks. JCAP , 03:055, 2021.\n- [14] Arnold Neumaier. Solving ill-conditioned and singular linear systems: A tutorial on regularization. SIAM Review , 40(3):636-666, 1998.\n- [15] Atsushi J. Nishizawa and Kaiki Taro Inoue. Reconstruction of Missing Data using Iterative Harmonic Expansion. Mon. Not. Roy. Astron. Soc. , 462(1):588-600, 2016.\n- [16] Gerlind Plonka, Daniel Potts, Gabriele Steidl, and Manfred Tasche. Numerical Fourier analysis . Applied and Numerical Harmonic Analysis. Birkhauser/Springer, Cham, second edition, [2023] ©2023.\n- [17] Giuseppe Puglisi and Xiran Bai. Inpainting Galactic Foreground Intensity and Polarization Maps Using Convolutional Neural Networks. Astrophys. J. , 905(2):143, 2020.\n- [18] Alireza Vafaei Sadr and Farida Farsian. Filling in Cosmic Microwave Background map missing regions via Generative Adversarial Networks. JCAP , 03:012, 2021.\n- [19] D. Smith and L. Betbeder-Matibet. Katana, 2010.\n- [20] J. L. Starck, M. J. Fadili, and A. Rassat. Low-l CMB Analysis and Inpainting. Astron. Astrophys. , 550:A15, 2013.\n- [21] SymPy Development Team. Sympy.\n- [22] E. van den Berg and M. P. Friedlander. Probing the pareto frontier for basis pursuit solutions. SIAM Journal on Scientific Computing , 31(2):890-912, 2008. \n- [23] E. van den Berg and M. P. Friedlander. SPGL1: A solver for large-scale sparse reconstruction, December 2019. https://friedlander.io/spgl1.\n- [24] Guo-Jian Wang, Hong-Liang Shi, Ye-Peng Yan, Jun-Qing Xia, Yan-Yun Zhao, Si-Yu Li, and Jun-Feng Li. Recovering the CMB Signal with Machine Learning. Astrophys. J. Supp. , 260(1):13, 2022.\n- [25] Kai Yi, Yi Guo, Yanan Fan Fan, Jan Hamann, and Yu Guang Wang. Cosmo vae: Variational autoencoder for cmb image inpainting. In 2020 International Joint Conference on Neural Networks (IJCNN) . IEEE, jul 2020.\n- [26] Andrea Zonca, Leo Singer, Daniel Lenz, Martin Reinecke, Cyrille Rosset, Eric Hivon, and Krzysztof Gorski. healpy: equal area pixelization and spherical harmonics transforms for data on the sphere in python. Journal of Open Source Software , 4(35):1298, March 2019. \nSCHOOL OF PHYSICS, UNSW SYDNEY, AUSTRALIA. \nEmail address : jan.hamann@unsw.edu.au \nSCHOOL OF MATHEMATICS AND STATISTICS, UNSW SYDNEY, AUSTRALIA. \nEmail address : qlegia@unsw.edu.au, i.sloan@unsw.edu.au, r.womersley@unsw.edu.au"}

2024arXiv240904542J

Traditional solar flare forecasting approaches have mostly relied on physicsbased or datadriven models using solar magnetograms treating flare predictions as a pointintime classification problem. This approach has limitations particularly in capturing the evolving nature of solar activity. Recognizing the limitations of traditional flare forecasting approaches our research aims to uncover hidden relationships and the evolutionary characteristics of solar flares and their source regions. Our previously proposed Sliding Window Multivariate Time Series Forest SlimTSF has shown the feasibility of usage applied on multivariate time series data. A significant aspect of this study is the comparative analysis of our updated SlimTSF framework against the original model outcomes. Preliminary findings indicate a notable improvement with an average increase of 5 in both the True Skill Statistic TSS and Heidke Skill Score HSS. This enhancement not only underscores the effectiveness of our refined methodology but also suggests that our systematic evaluation and feature selection approach can significantly advance the predictive accuracy of solar flare forecasting models.

2024-09-01T00:00:00Z

['arXiv:2409.04542', '2024arXiv240904542J', '10.48550/arXiv.2409.04542']

['Computer Science - Machine Learning', 'Astrophysics - Instrumentation and Methods for Astrophysics', 'Astrophysics - Solar and Stellar Astrophysics']

Towards Hybrid Embedded Feature Selection and Classification Approach with SlimTSF

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.04542.pdf

{'Towards Hybrid Embedded Feature Selection and Classification Approach with Slim-TSF': 'Anli Ji 1 , Chetraj Pandey 2 , and Berkay Aydin 3 \nGeorgia State University { aji1 1 ,cpandey1 2 ,baydin2 3 } @gsu.edu \nAbstract. Traditional solar flare forecasting approaches have mostly relied on physics-based or data-driven models using solar magnetograms, treating flare predictions as a point-in-time classification problem. This approach has limitations, particularly in capturing the evolving nature of solar activity. Recognizing the limitations of traditional flare forecasting approaches, our research aims to uncover hidden relationships and the evolutionary characteristics of solar flares and their source regions. Our previously proposed Sliding Window Multivariate Time Series Forest (Slim-TSF) has shown the feasibility of usage applied on multivariate time series data. A significant aspect of this study is the comparative analysis of our updated Slim-TSF framework against the original model outcomes. Preliminary findings indicate a notable improvement, with an average increase of 5% in both the True Skill Statistic (TSS) and Heidke Skill Score (HSS). This enhancement not only underscores the effectiveness of our refined methodology but also suggests that our systematic evaluation and feature selection approach can significantly advance the predictive accuracy of solar flare forecasting models. \nKeywords: Multivariate Time Series Classification · Solar Flare Prediction · Interval-based Classification', '1 Introduction': "Solar weather events, encompassing phenomena like solar flares, coronal mass ejections (CMEs), solar wind variations, and geomagnetic storms, hold significant importance for Earth's environment and human technological systems. Among many solar phenomena, solar flares are one of the most intense localized explosions of electromagnetic energy emanating from the Sun's atmosphere. When such energy bursts out, it usually travels near the speed of light ranging from several minutes to hours. It often does not occur alone but alongside other events like coronal mass ejections (CMEs) or solar wind, which can trigger severe geomagnetic storms, extensive radio blackouts on Earth's daylight side, and interfere with delicate instruments onboard near-Earth space equipment. Recent studies have employed physics-based or data-driven models [30] [33] [21] to predict solar flares using data primarily sourced from solar magnetograms [35]. Many of these approaches tend to predict solar flares as a classification \nproblem using point-in-time measurements (where a single time point is applied to represent a single event). Such methods often do not consider the intrinsic temporal evolution nature of data [11] by evaluating different observations as separate entities, meaning the dynamic essence of flares is usually overlooked. \nThe characteristics of solar flare evolution are important as they are intricately linked to the dynamic behavior of solar active regions, as delineated in prior research [6] [26] [27]. Analyzing these temporal characteristics of flares, it becomes possible to reveal potential implicit relationships and capture unidentified patterns between flares and their originating regions. In our prior study [19], we utilized ensembles of interval-based classification models on multivariate time series data for event prediction. However, this method presented a limitation in understanding which features were more pivotal in decision-making and the rationale behind these decisions. Traditional interval-based classifiers often do not support systematic evaluation through random sub-interval sampling, leading to a process where the identification of relevant features (or intervals from the time series) was arbitrarily generated, thereby missing out on extracting meaningful insights from the model. In our subsequent studies [16] and [18], we aimed to identify crucial interval features from multivariate time series data using multi-scale sliding windows with varying interval sizes and step sizes as well as an innovative feature ranking schema for identifying feature importance. This advancement seeks to introduce interpretability into previously opaque models, enhancing our understanding of the decision-making processes underlying model prediction. \nIn this study, we expand our previous work focusing more on the systematic evaluation of our Sliding Window Multivariate Time Series Forest (Slim-TSF) model. This involves strategically selecting relevant features to enhance our grasp of the temporal dynamics crucial for solar flare prediction. We've introduced an indexing function to improve the model selection process. This function enables us to identify optimal models using a concise set of parameters and features that have shown promise in prior research. Additionally, we employ a customized internal validation schema to cross-verify our findings, ensuring the robustness and reliability of our results. This approach has led to a noticeable improvement in our model's performance. Specifically, we've achieved an average increase of 5% in True Skill Statistics (TSS) and Heidke Skill Score (HSS) compared to our original Slim-TSF outcomes. This improvement underscores the value of a systematic feature selection and validation strategy in enhancing the accuracy of solar flare predictions. \nThe rest of the paper is organized as follows: Section 2 provides background information on existing time series classification models pertinent to flare prediction. In Section 3, we provide our problem formulation and introduce our multivariate time series classification model and feature ranking method used for extracting relevant feature intervals from provided time series data. Section 4 presents our experimental setup and evaluation framework. Finally, Section 5 provides conclusions from our study and discusses potential avenues for future research.", '2 Related Work': 'From the proliferation of available time series datasets [34] and a wide spectrum of machine learning-based techniques proposed for time series classification, similarity-based and feature-based algorithms are two notable categories utilized for these predictive tasks. Similarity-based methods predict by measuring the similarity between training and testing instances, using metrics like Euclidean distance or Dynamic Time Warping (NN-DTW) [5], [32], [4],[22]. In contrast, feature-based algorithms generate predictions by extracting temporal features from entire time series or subsequences within them. For solar flare prediction, both full-disk (e.g., [26][29] [28]) and active region-based (e.g., [19] [15] [8] [14]) approaches have shown significant impact by utilizing derived time series features. \nUsing feature-based algorithms that capture associations between target variables and time series instances through derived features, this distinction is particularly evident in tasks like solar flare prediction or other tasks (such as anomaly detection [13]). For example, [25] extracted basic statistical features like mean and standard deviation from global time series to feed a multi-layer perceptron network, though this method neglected localized informative characteristics. In contrast, [12] enhanced model interpretability by considering local attributes through piecewise constant modeling and pattern extraction, though it often resulted in simplistic features during selection. Furthermore, [ ? ] incorporated an extensive range of features such as wavelets and Chebyshev coefficients, but this method faced high computational costs and lacked inherent interpretability in high-dimensional data spaces. \nIt is a challenging task for many feature-based classification methods when dealing with multivariate time series data because they require additional intricate information across features. Such discriminating features are usually hard to generate in high-dimensional space due to the unknown interrelations among input parameters, adding complexity to model construction. To address this problem, various techniques have been attempted to ensemble univariate models from individual feature spaces instead of considering the global correlations between them. These methods focus on extracting relevant features in univariate aspects and then applying traditional machine learning algorithms for classification. Common features include statistical measures (e.g., mean, variance), spectral features (e.g., Fast Fourier Transform coefficients), and time-domain features (e.g., autocorrelation). \nFor example, Shapelet-based decision trees [36] combine shapelets (i.e., discriminative subsequences that capture distinctive patterns in time series data) within an ensemble architecture. This method extracts shapelets from the training data and constructs an ensemble of decision trees (e.g., random forest), where each estimator focuses on a different subset of shapelets, typically measured by Euclidean distance. While effective at capturing local patterns in multivariate time series data, this approach can be computationally expensive and may struggle to identify relevant shapelets in high-dimensional spaces that are both informative and broadly applicable. Additionally, shapelets extracted from one \ndataset might not generalize well to other datasets with different dimensionalities, characteristics, and patterns. \nTo mitigate these issues, the Generalized Random Shapelet Forest (gRSF) [20] improves upon the original shapelet-based method by measuring distances between randomly selected time series and others within a threshold distance of the representative shapelet. Similarly, the Time Series Forest (TSF) [9] incorporates subseries, but instead of relying on distance measurements from learned subsequences, it derives summarized statistical features (such as mean, standard deviation, and gradient) within randomly divided intervals of the univariate time series. This approach treats each time step as a separate component and constructs decision trees for each feature dimension to capture temporal relationships and reduce the high-dimensional feature space. However, this method may not fully capture the interrelationships between different components of the time series, leading to a potential loss of crucial inter-channel relationships and dependencies in multivariate data. The Canonical Interval Forest (CIF) [ ? ] extends TSF by incorporating additional canonical characteristics of the time series and catch22 [23] features extracted from each interval. This approach aims to capture both individual patterns within each time series component and relationships between different components. However, interpreting an ensemble of decision trees remains challenging, making it less intuitive to understand the combined effects of multiple trees on multivariate time series data compared to single decision trees. \nMany of these methods focus solely on understanding how each feature behaves independently, without considering interactions between different features. A relationship within a single time series parameter might be significant for one specific feature but not necessarily for others. The connections between distinct features are often unknown upfront. Understanding feature dependencies in time series data is crucial for improving model interpretability and performance [31]. Techniques like Partial Dependence Analysis (PDA) are commonly used in quantifying these dependencies but can be challenging to explore and analyze in multivariate aspects. [2] proposes a conceptual framework that refines the computed partial dependences on combinatorial feature subspaces (i.e., all the possible combinations of features on all their domains) but still lacks the capability of capturing intercorrelations that differentiate between features. \nIn time series classification problems, selecting the most relevant time intervals is crucial when generating features that effectively distinguish our data, thereby ensuring a robust model. However, identifying these relevant intervals is difficult because they cannot be directly determined and typically require a computationally expensive search across the entire series. Extracting the underlying mutual information from these relevant intervals can enhance our understanding of the predictive process and accelerate the transition from research to practical applications in flare forecasting models. Our objective in this work is to establish a framework that can recognize these characteristics and offer deeper insights into the behavior of classifiers during prediction tasks.', '3 Methodology': "In this section, we will outline our approach, including the extraction of statistical features from time intervals, the introduction of the sliding-window time series forest, and the feature ranking technique we employ. \nOur proposed sliding-window multivariate time series forest is an early fusion, interval-based ensemble classification method. Fig. 1 illustrates our feature generation process using the sliding window operation. This method employs multiple short decision trees, similar to random forests, which utilize intervalbased features extracted from univariate time series through multi-scale sliding windows. By combining features from univariate time series at an early stage, we aim to understand the relationships among these features, using an embedded feature ranking based on mutual information. \nInterval Features To extract well-structured and relevant intervals, we calculate statistical characteristics such as mean, standard deviation, and slope for each interval. Additionally, we derive transformed features like maximum, minimum, and mean through a localized pooling procedure applied to the individual interval features extracted from consecutive intervals after the sliding window operation. In this process, all potential interval sets originating from the same time series are collected, and pooling functions are applied for consolidation. Essentially, we consider the highest, lowest, and average values of statistical properties from each parameter of each subseries obtained through sliding window operations. Formal definitions and explanations for processing multivariate time series and extracting vectorized features and transformation are provided in our previous research [17]. \nSliding Window Multivariate Time Series Forest After extracting interval features from subsequences obtained through the sliding window operation and applying secondary transformations to these statistical attributes, we merge these two groups of derived features into an input vector. This vector serves as the foundation for creating a versatile time series classifier we refer to as SlimTSF . Among the wide array of supervised learning models available for making predictions, we have chosen random forest classifiers for two reasons: (1) their effectiveness and resilience when dealing with noisy, high-dimensional data, and (2) their ability to select the most relevant features from a given dataset with respect to a target feature. \nIt is important to highlight that, depending on the chosen parameter settings, such as using smaller window and step sizes, the interval feature vectors' data space can expand considerably. Additionally, the process of vectorization based on the sliding window approach may generate data points that exhibit some degree of correlation and potential noise. Consequently, it is crucial to systematically identify and remove these features. This is achieved through the application of information-theoretic relevance metrics (e.g., Gini index or entropy). \nFig. 1: An illustration of the sliding window-based statistical feature generation. We first generate subsequences (intervals) with a fixed-size sliding window and step size. Then, we create vectorized features from these intervals where these features can be used as input for the sliding window multivariate time series forest (a random forest built on multivariate time series features) and features are ranked with aggregated relevance scores. \n<!-- image --> \nThis meticulous feature selection process ensures the efficacy of our approach by retaining only the most informative attributes while discarding redundant ones. \nFeature Ranking Our ranking methodology involves a systematic process conducted through multiple experiments, denoted by the total number N , each ex- \necuted with distinct experimental configurations. This is analogous to a grid search process. The experiments with different configurations yield individual selected features, denoted as exp j , where j signifies a specific experiment. The ranking denoted as r , is a mapping that assigns a rank i to each feature, reflecting its position within the ranking. In each experiment, the features are ranked to create a specific ordering, denoted as r f,i , which designates that feature f has achieved the i th rank in that particular experiment. Subsequently, the topk features selected for inclusion in the selected feature set, denoted as SFS j , within each individual experiment j are determined from the ranking r (i.e., include features whose rank is less than or equal to k ). This selected feature set is represented as { r f, 1 , r f, 2 , ..., r f,k } . In the end, the selected feature sets across all experiments are aggregated by summing the sparse representation of topk membership vectors ( ̂ SFS j ) from each experiment (as in Eq. 1). \nSFS = ∑ j =1 ,N ̂ SFS j (1) \nThis approach allows for a systematic and consistent method of selecting top features across multiple experiments, enhancing the robustness and reliability of the feature selection process. Furthermore, we create a counting vector per each interval of each parameter, denoted as ct v to represent the value counts of individual intervals in the selected feature set SFS . This counting vector serves as a transformation function, indicating the frequency with which a given interval appears within the topk selections of the feature set. \nHyperparameter Optimization Hyperparameter optimization is the process of selecting the optimal hyperparameters to achieve the best performance for a classifier. This process is applied to determine the optimal hyperparameters of our slim-TSF classifier. Traditional grid search cross-validation (CV) is designed for tabulated data and assumes that instances are independent, meaning random assignment of instances to different training and testing folds does not risk overfitting or memorization for the trained models. This can lead to similar instances being included in both training and testing sets, resulting in models that tend to memorize rather than learn. While these models may initially show better results, this is due to sub-optimal sampling rather than stronger generalization capabilities [1]. \nIn time series analysis, where instances are obtained with a sliding window, data partitions for training, testing, and validation need to be time-segmented. Traditional grid search CV cannot ensure that instances from consecutive overlapping segments are not placed in both training and testing sets, which undermines the reliability of time series classification performance evaluations. To address this issue, we implemented a customized CV schema that modifies the original grid search to split by the SWAN-SF partitions, maintaining continuous time segmentation instead of randomized sampling. Each time-segmented training partition dataset is assigned a partition label to ensure it is not included in both training and testing sets. \nAdditionally, we modified the scoring function for our CV, replacing classification accuracy with forecast skill scores, primarily the True Skill Statistic score and the Heidke Skill Score, which will be discussed further in the next section.", '4 Experimental Evaluations': 'The experiments conducted in this study are designed with two primary objectives. Firstly, they aim to demonstrate the effectiveness of time series classifiers developed using distinct interval features and to perform a comprehensive performance comparison among them. Another key objective is to identify the intervals within the time series that hold the greatest relevance to the initial time series. This effort is primarily designed to offer interpretable insights into our model. It involves pinpointing the specific segments of the time series that exert significant influence on predictions and understanding the aggregation strategies that can lead to more accurate outcomes.', '4.1 Data Collection': 'For predicting solar flares, we utilized the SWAN-SF dataset, an open-source multivariate time series dataset introduced in [3]. This dataset offers a comprehensive collection of space weather-related physical parameters derived from solar magnetograms, incorporating data from various solar active regions and flare observations. Our experiments, encompassing both classification and feature ranking, utilized all 24 active region parameters provided by the dataset. These parameters are widely recognized as highly representative of solar activity, with detailed descriptions available in [7] and [3]. \nThe SWAN-SF dataset is organized into five distinct time-segmented partitions to ensure that data instances within each partition do not temporally overlap. Active regions within the dataset are segmented using a sliding observation window of 12-hour intervals across the multivariate time series. Each segment captures essential data, including an active region number (aligned with NOAA Active Region numbers and HMI Active Region Patches identifiers), a class label (indicating the maximum intensity flare expected from that region within the subsequent 24 hours), and start and end timestamps for each segment. \nFlare intensity is categorized by the logarithmic classification of peak X-ray flux into major flaring classes (X, M, C, B, or A). For our analysis, instances classified as M- and X-class flares are considered flaring (i.e., positive class), while those classified as C- and B-class flares, along with flare-quiet regions, are treated as non-flaring (i.e., negative class). This binary classification framework is applied to model the solar flare forecasting problem as a binary multivariate time series classification task.', '4.2 Experimental Settings': "To assess our model's performance, we employed a binary 2 × 2 contingency matrix, supplemented by other well-known evaluation metrics for evaluating \nforecasting accuracy. Within these metrics, the positive class corresponds to significant flare events ( ≥ M1.0 or M- and X-class flares), while the negative class includes smaller flares and regions with minimal flare activity (i.e., instances labeled below the M1.0 threshold). In this context, True Positives (TPs) and True Negatives (TNs) represent instances where the model accurately predicts flaring and non-flaring events, respectively. False Positives (FPs) are false alarms, where the model incorrectly predicts a flare and False Negatives (FNs) are misses, where the model fails to predict an actual flare event. \nFor rigorous evaluation, we utilize the True Skill Statistic (TSS) and a weighted version of TSS ( ω TSS), detailed in Equations Eq. 2 and Eq. 3 respectively. The TSS measures the difference between the Probability of Detection (recall for the positive class) and the Probability of False Detection (POFD), providing a robust indicator of model skill. \nTSS = TP TP + FN -FP FP + TN (2) \nIn essence, TSS can be reformulated as the sum of true positive rate ( TPR ) and true negative rate ( TNR ), offset by 1 ( TPR + TNR -1). The general purpose of TSS is a good all-around forecast evaluation method, especially for evaluating scores among datasets with different imbalance ratios. However, it focuses on a simpler, more understandable scoring schema where both TPR and TNR are treated equally. To change the importance given to each term in this equation we can use a regularization term α/ 2, and create the following weighted TSS ( ωTSS ): \nωTSS = αTPR +(2 -α ) TNR -1 (3) \nHere, α/ 2 and 1 -α/ 2 are regularization parameters that show how important each term is. For instance, if correctly predicting an M- or X-class flare is 3 times more important than correctly predicting a non-flaring class, then we can use α/ 2 = 0 . 75. \nThe second skill score we employed is the Heidke Skill Score (HSS), which serves as a critical measure of the forecast's improvement over a climatologyaware random prediction. The HSS ranges from -1 to 1, where a score of 0 indicates that the forecast's accuracy is equivalent to that of a random binary forecast, based solely on the provided class distributions. The formula for calculating this metric is provided in Eq. 4. Here, P denotes the actual positives, which is the sum of true positives ( TP ) and false negatives ( FN )), and N represents the actual negatives, the sum of false positives ( FP ) and true negatives ( TN )). \nHSS = 2 · (( TP · TN ) -( FN · FP )) P · ( FN + TN ) + N · ( TP + FP ) (4) \n(a) Bootstrapping with 10 runs using all derived features \n<!-- image --> \n1.00 \n(b) Bootstrapping with 100 runs using all derived features \n<!-- image --> \n1.00 \n(c) Bootstrapping with 1000 runs using all derived features \n<!-- image --> \nFig. 2: Error Bar representation of slim-TSF evaluation with ex-ante bootstrapping feature selection using different class weight (i.e., cw) ratio. The most relevant features are selected (per each model trained) across different class weights using the log-scale filter. The TSS and HSS scores are shown for each bootstrapping experiment. \nFig. 3: A bar plot representation of feature participation ratio in three bootstrap evaluation counts. All features from sliding window intervals and transformed features are used. \n<!-- image -->", '4.3 Bootstrapping': 'In this study, we introduce a novel approach to feature selection that deviates from the methodology used in [17]. Instead of limiting our selection to only the top 5 highly ranked parameters from individual experiments, we base our feature selection on the cumulative results of the entire bootstrapping process. This involves compiling data from each iteration of the bootstrap subsampling and identifying the most relevant features based on their frequency of appearance throughout the random subsampling procedure. To further refine our feature selection and reduce the risks associated with an overly extensive feature set, we apply a filter k = log 2( N ) to select the top k features, where N is the total number of features. The results of this refined process are illustrated in Fig. 2a, 2b, and 2c, demonstrating the use of selected important features from various window and step size configurations. This approach helps to mitigate the influence of outlier features, which might otherwise compromise the accuracy of our predictions. \nThis refined selection strategy enables us to achieve results comparable to those of our initially proposed Slim-TSF model, but with a significantly smaller set of parameters and features. Consequently, we can maintain average testing scores of approximately 60% in TSS and 35% in HSS while utilizing fewer inputs. Despite these reductions, the robustness of our feature selection process is \nconfirmed through its repeated application with random subsamples of the original dataset, ensuring both consistency and reliability. Throughout our studies, certain features, such as those derived from the Total Unsigned Current Helicity (TOTUSJH) and the Absolute Value of the Net Current Helicity (ABSNJZH), are consistently selected across multiple iterations, as shown in Fig.3. These features have a participation rate of over 40%, underscoring their critical role in predicting solar flare events.', '4.4 Remarks': 'In the results, we demonstrate that the Slim-TSF models, using only the top cumulative important features selected from our bootstrapping iterations, perform comparably, ensuring efficiency and robustness. These models achieve similar outcomes using fewer but more significant features from the original 24 parameters. Notably, models with lower class weights show an average performance improvement of 5% over previous research [18]. This improvement occurs as we reduce redundancy by limiting the use of extensive derived features, thereby increasing feature relevancy. Consequently, the models can concentrate more effectively on key factors by minimizing the redundancy found in less informative features, ultimately enhancing performance significantly. \nThe outcomes of our study systematically evaluate the performance of our Slim-TSF models, incorporating an additional filter during feature selection. Specifically, our findings reveal that these models improve when utilizing only the top k (from a log-scale) most significant features. This highlights the principle that quality often outweighs quantity in feature selection, as these streamlined models achieve results comparable to their more complex counterparts that utilize all 24 features. Additionally, it is worth noting that adjusting the class weight hyperparameter significantly enhances model performance by reducing the imbalance ratio.', '5 Conclusions': 'This study builds upon our previous work, which utilized interval-based features generated from sliding window operations in multivariate time series classifiers, also useful for ranking key features, intervals, and transformed pooling features. The primary goal of this work is to enhance the interpretability of highdimensional multivariate time series classifiers. By employing a comprehensive and methodical approach to feature selection, our research not only improves the predictive accuracy and efficiency of the Slim-TSF model but also offers valuable insights into solar flare prediction, especially under the constraints of limited observational data. This advancement marks significant progress in the field of solar weather forecasting, underscoring the importance of adaptability and innovation in addressing the challenges of data scarcity.', 'Acknowledgment': 'This work is supported in part under two grants from NSF (Award #2104004) and NASA (SWR2O2R Grant #80NSSC22K0272).', 'References': "- 1. Ahmadzadeh, A., Hostetter, M., Aydin, B., Georgoulis, M.K., Kempton, D.J., Mahajan, S.S., Angryk, R.: Challenges with extreme class-imbalance and temporal coherence: A study on solar flare data. In: 2019 IEEE International Conference on Big Data (Big Data). IEEE (Dec 2019). https://doi.org/10.1109/bigdata47090.2019.9006505\n- 2. Angelini, M., Blasilli, G., Lenti, S., Santucci, G.: A visual analytics conceptual framework for explorable and steerable partial dependence analysis. IEEE Transactions on Visualization and Computer Graphics p. 1-16 (2024). https://doi.org/10.1109/tvcg.2023.3263739, http://dx.doi.org/10.1109/TVCG. 2023.3263739\n- 3. 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Pandey, C., Ji, A., Angryk, R.A., Aydin, B.: Towards interpretable solar flare prediction with attention-based deep neural networks (2023) \n- 29. Pandey, C., Ji, A., Angryk, R.A., Georgoulis, M.K., Aydin, B.: Towards coupling full-disk and active region-based flare prediction for operational space weather forecasting. Frontiers in Astronomy and Space Sciences 9 (Aug 2022). https://doi.org/10.3389/fspas.2022.897301\n- 30. Priest, E., Forbes, T.: The magnetic nature of solar flares. The Astronomy and Astrophysics Review 10 (4), 313-377 (2002)\n- 31. Saeed, W., Omlin, C.: Explainable ai (xai): A systematic meta-survey of current challenges and future opportunities. Knowledge-Based Systems 263 , 110273 (Mar 2023). https://doi.org/10.1016/j.knosys.2023.110273, http://dx.doi.org/ 10.1016/j.knosys.2023.110273\n- 32. Sakoe, H., Chiba, S.: Dynamic programming algorithm optimization for spoken word recognition. IEEE Transactions on Acoustics, Speech, and Signal Processing 26 (1), 43-49 (1978). https://doi.org/10.1109/TASSP.1978.1163055\n- 33. Shibata, K., Magara, T.: Solar flares: magnetohydrodynamic processes. Living Reviews in Solar Physics 8 (1), 6 (2011)\n- 34. Silva, D.F., Giusti, R., Keogh, E., Batista, G.E.A.P.A.: Speeding up similarity search under dynamic time warping by pruning unpromising alignments. Data Mining and Knowledge Discovery 32 (4), 988-1016 (Mar 2018). https://doi.org/10.1007/s10618-018-0557-y\n- 35. Song, H., Tan, C., Jing, J., Wang, H., Yurchyshyn, V., Abramenko, V.: Statistical assessment of photospheric magnetic features in imminent solar flare predictions. Solar Physics 254 (1), 101-125 (Nov 2008), https://doi.org/10.1007/ s11207-008-9288-3\n- 36. Ye, L., Keogh, E.: Time series shapelets: a novel technique that allows accurate, interpretable and fast classification. Data Mining and Knowledge Discovery 22 (12), 149-182 (Jun 2010). https://doi.org/10.1007/s10618-010-0179-5"}

2024SoPh..299..128D

A controversy about the possibility of dynamic effects in nuclear screening has been around for several decades. On the one hand there is the claim that there are no dynamic effects and that the classic Salpeter correction based on static Debye screening is all that is needed for astrophysical applications. The size of the correction is on the order of 5 in typical solar fusion reactions. On the other hand numerical simulations have shown that there is a dynamical effect which essentially cancels the Salpeter correction. The results of the numerical simulations were later independently confirmed. The astrophysical community however has so far largely ignored the possibility of dynamical screening. The present paper is meant to serve as a reminder of the controversy. Not only does the claim of an absence of a dynamical effect equally warrant an independent confirmation but there is motivation for further investigation such as the assessment of current laboratory experiments and a quantitative study of the dynamical effect in case it will turn out to be real.

2024-09-01T00:00:00Z

['10.48550/arXiv.2409.09826', 'arXiv:2409.09826', '2024arXiv240909826D', '2024SoPh..299..128D', '10.1007/s11207-024-02377-w']

['Interior', 'Core', 'Helioseismology', 'Direct modeling', 'Astrophysics - Solar and Stellar Astrophysics', 'Nuclear Experiment', 'Nuclear Theory']

The Current State of the Controversy over Screening in Nuclear Reactions

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https://arxiv.org/pdf/2409.09826.pdf

{'The Current State of the Controversy over Screening in Nuclear Reactions': '<!-- image --> \nWerner D¨appen 1 \n© The author(s) •••• \nAbstract A controversy about the possibility of dynamic effects in nuclear screening has been around for several decades. On the one hand, there is the claim that there are no dynamic effects, and that the classic Salpeter correction based on static Debye screening is all that is needed for astrophysical applications. The size of the correction is on the order of 5% in typical solar fusion reactions. On the other hand, numerical simulations have shown that there is a dynamical effect, which essentially cancels the Salpeter correction. The results of the numerical simulations were later independently confirmed. The astrophysical community, however, has so far largely ignored the possibility of dynamical screening. The present paper is meant to serve as a reminder of the controversy. Not only does the claim of an absence of a dynamical effect equally warrant an independent confirmation, but there is motivation for further investigation, such as the assessment of current laboratory experiments and a quantitative study of the dynamical effect in case it will turn out to be real. \nKeywords: Interior, Core; Helioseismology, Direct Modeling', '1. Introduction': "It is well known that ordinary stars can only operate thanks to the possibility of quantum tunneling. Before nuclei undergo reactions they have to overcome the Coulomb barrier which is on the order of MeVs. In stable phases of stellar nuclear burning, such as during solar main-sequence life, core temperatures are on the order of keVs. By standard Maxwell-Boltzmann probability distribution, the chance of a pair of nuclei to penetrate the Coulomb barrier is therefore on the order of e -1000 , which is tantamount to impossible. Quantum tunneling gave the solution of this paradox. The first person to apply the Schrodinger \nequation to a problem which involved tunneling between two classically allowed regions through a potential barrier was Friedrich Hund (1927). Shortly afterwards, Gamov (1929) computed the tunneling rate in nuclear decay, while Atkinson and Houtermans (1929) applied Gamov's result to solar fusion. The Maxwell-Boltzmann distribution still plays an important role but it is now a factor multiplied with the tunneling probability and with the genuine reaction rate from nuclear physics, usually assumed to be independent of the other effects. Underlying this factorizing is the quasi-classical assumption that in a random distribution, the tunneling probability for a given pair of colliding nuclei with a certain relative velocity at infinite separation, is the same as that for a coherent stream of that relative velocity. The product of the two probabilities is the so-called 'Gamov peak', which typically lies at temperatures of 3-4 kT . \nThe seminal discussion of the the relevant nuclear reaction was made by Bethe (1939), but until the 1950s it remained an open question, if the main contribution to solar energy came from the pp chain or the CNO cycle. (Regarding the history of the CNO cycle in this context, see Wiescher 2018.) The dominant role of the pp chain was established by Salpeter (1952,1953). Soon after, Salpeter (1954) introduced a modification of the tunneling probability. Arguing that two colliding nuclei are not isolated from their surroundings but that their charges are rather screened in the overall neutral ionized plasma, he computed the tunneling probability not with the bare Coulomb potential but the Debye-Huckel (1923) potential. Since this is lower than the Coulomb potential, the tunneling probability becomes higher, a result commonly referred to as the so-called 'Salpeter screening enhancement'. \nSalpeter assumed that one can still multiply the Maxwell-Boltzmann distribution of the nuclei with the tunneling probability, in the same way as had been done with the bare Coulomb potential. The Debye-Huckel potential he considered is time-independent, since the underlying theory is thermodynamic. Therefore, Salpeter screening is also inherently static. However, the problem with static screening is that, intuitively, the height of the screened potential must depend on the velocity of the colliding nuclei. This is illustrated by a pair of very rapidly colliding nuclei. One would expect that the cloud of the screening electrons can still easily adjust during the flight of the nuclei, but the surrounding positive charges would be frozen at the same time, therefore precluding the attainment of the overall re-adjusted screening potential. However, precisely such velocity-dependent effects are not taken into account in the Salpeter theory. \nHowever, with a velocity-dependent tunneling probability, the standard assumption that one can multiply the probabilities, that of finding a pair of nuclei with a certain relative velocity, and that of a particle of that velocity tunneling through the Coulomb barrier. In the absence of screening, this is a reasonable assumption, which simply states the independence of two events. However, when the barrier itself becomes velocity dependent, the simple product rule breaks down. If the overall reaction rate, that is, the Gamov peak, were at the equilibrium temperature kT , the problem might be less severe, since around kT one could imagine that velocity-dependent effects essentially cancel. However, the Gamov peak is at 3-4 kT , which selects those velocities that are most relevant \nfor nuclear energy production. Therefore, it is by far not obvious if one can still use the original Debye-Huckel potential for these fast nuclei. \nOf course, even then it cannot be excluded that in the net nuclear reaction rate, all deviations from the static result will somehow cancel out. However that would have to be demonstrated. Effects from velocity-dependent tunneling probabilities (resulting from a velocity-dependent 'mountain') are generally referred to as 'dynamic screening', which was first considered several decades ago (Mitler 1977; Carraro et al. 1988). A detailed history of dynamic screening is given in Appendix B of Shaviv and Shaviv (2000), while Dzitko et al. (1995) discuss the impact of various screening formalisms for stars (in a study that was then done in the context of finding a possible solution to the solar-neutrino problem). \nOne approach to demonstrate the possibility of dynamic screening was pioneered by applying molecular-dynamic (MD) simulations of a large number of electrons and nuclei (Shaviv and Shaviv 1996). Another was to attempt to compute the overall reaction rate with the methods of quantum statistical mechanics (Brown and Sawyer 1997). Their respective results disagree significantly. In the following I will present these different approaches and the ensuing controversy. Then I will discuss the astrophysical relevance of the controversy and end with a plea for necessary future work.", "2. Salpeter's Static Screening": "Salpeter (1954) derived an expression for the enhancement of nuclear reaction rates due to electron screening. When the Debye-Huckel theory of dilute solutions of electrolytes is applied to electrons and ions in a plasma under the condition of weak screening ( φ interaction /lessmuch k B T ), the interaction potential of a charge Z i e becomes the static-screened Coulomb potential (for simplicity, here we only consider the case of one ion species) \nφ ( r ) = Z i e r e -r/R D , (1) \nwith the Debye length R D given by \nR D = √ k B T 4 π ( n e e 2 + n i Z 2 i e 2 ) . (2) \nwhere n e , n i are electron and ion densities, respectively. \nThe outcome of screening is conveniently expressed by the 'enhancement factor' f , a multiplier of the unscreened result to yield the screened reaction rate. Under solar central conditions, a typical correction is about f =1.05 (see Figure 2 of Weiss et al. 2001), while f =1 corresponds to the absence of screening effects.", "3.1. Shaviv and Shaviv's Molecular-dynamic Simulations": "Shaviv and Shaviv (1996,2000,2001) applied MD techniques to follow the motions of particles under conditions found in the solar center. They simulated the detailed motion of a large number of ions and electrons subject to Coulomb interactions. Electrons were treated by a quasi-classical approximation whose idea goes back to Gunter Kelbg (Kelbg 1963; for the history and impact of those approximations, see the recent review by Bonitz et al. 2023). See Shaviv and Shaviv (1996) for the specific quasi-classical approximation used in this MD calculation. \nBy following the local motion of each particle, the mean-field assumption underlying Salpeter's static screening was avoided. Because of the vastly different mass of the electrons and the nuclei, such simulations are challenging, much harder than those of gravitationally interacting assemblies of stars. While the latter simulations can done with millions or even billions of particles, Shaviv and Shaviv's calculations could only involve thousands. \nBy calculating the pair correlation functions of their simulation, Shaviv and Shaviv found that they are substantially different from those obtained by the formalism of Salpeter. And the resulting enhancement factor became essentially 1, expressing the fact that the dynamical effects practically obliterate the Salpeter screening. The most recent summary of that group's work is Shaviv (2010).", "3.2. Independent Confirmation of Shaviv and Shaviv's Results": "/s32 \nFigure 1. Dynamic screening energy at the turning point for pairs of protons with a given relative kinetic energy. For comparison, the static screening energy evaluated at the average turning point of proton pairs with each energy is also shown (data from Mao et al. 2009). \n<!-- image --> \n/s32 \nThe results of Shaviv and Shaviv were later independently confirmed at USC by my former graduate students (Mao et al. 2009; Mussack and Dappen 2011; Dappen and Mussack 2012). We essentially performed the same molecular dynamic calculation as the Shavivs', using 1000 protons and electrons, and obtained virtually the same results (see Figure 1 and Table 1). If anything, the new results hinted even at a tiny reduction of the unscreened result. While I consider this reduction insignificant, it has nonetheless received attention (Christensen-Dalsgaard 2021). \nTable 1. Screening energies and the ratio of screened to unscreened nuclear reaction rates for solar p-p reactions as a function of the relative kinetic energy E of the colliding protons. The dynamical result is a fit to numerical data (from Dappen and Mussack 2012).", "3.3. Brown and Sawyer's Analytical Results": "Brown and Sawyer (1997) endeavored a rigorous ab-initio computation of the plasma effects involved in nuclear fusion. The authors pointed out that in the so-called 'basically classical' approach, there are conceptual problems raised by the division of the problem into a quantum-mechanical and a classical part. They asserted that the 'literature lacks any development that begins with a correct general expression for the rate'. The key ideas in their work were (i), treating the screening correction as a quantum observable, and (ii), proceeding through socalled 'imaginary time' expansions, thus allowing that the statistical-mechanical problem can be mapped into a quantum dynamical problem. At the end of an elaborate 26-page analysis, they come to the conclusion that for solar nuclear fusion at least, their result essentially reproduces the result of Salpeter (1954). The authors then conclude further that there are no dynamical effects under the given conditions. Finally, Brown and Sawyer conjecture that any approach not agreeing with their results 'will miss physics that is as important as the physics that it includes'.", '4. Screening Enhancement in the Astrophysical Community': "The late John Bahcall, who pioneered solar standard models and neutrino flux predictions (Bahcall 1982), had a decisive influence on the astrophysical community. The static screening result of Salpeter (1954) is now broadly accepted. After \nteaming up with Brown and Sawyer, Bahcall and his collaborators (Bahcall et al. 2002) consistently and tirelessly propagated the Salpeter result, which they believed as certified by an exact analytical calculation. The same recommendation is given by two comprehensive papers on solar fusion, each authored by a large group (Adelberger et al. 1998, 2011). The first of these papers mentions the screening controversy, but it still basically sides with Brown and Sawyer (1997). The second notes at least that 'the controversy has not completely died down', citing the USC results. A very recent review by Aliotta and Langanke (2022) does not directly take sides, mentioning both Shaviv and Shaviv's simulations and their independent confirmation, but then goes on saying that their outcome 'has, however, been disputed by Bahcall and collaborators who argued that Salpeter's screening approach is valid also at the Gamow peak energy due to the nearly perfect thermodynamic equilibrium present in the solar plasma'. However, the existence of a nearly perfect thermodynamic equilibrium does not necessarily preclude dynamical effects at the high-velocity end of the MaxwellBoltzmann distribution. Such effects could take place without interfering with the overall thermodynamic equilibrium. Therefore, the real reason that Bahcall et al. (2002) had ruled out significant deviations from the Salpeter result is their faith in the result of Brown and Sawyer (1997). \nOther authors ignore the controversy altogether. Bellinger and ChristensenDalsgaard (2022) use the nuclear reaction rates of Adelberger et al. (2011), which is adopting Salpeter screening for solar conditions. And Liolios (2000) comes to the conclusion that in screening 'nonlinear effects are shown to be negligible', meaning that going to higher-order terms within Salperter's theory is not necessary. This is true, but it doesn't address the controversy. The controversy is about abolishing the Salpeter correction altogether. I am aware of only one suggestion to take dynamical screening seriously, and that is coming from outside stellar and solar physics, namely cosmology. Hwang et al. (2021) discuss dynamical screening effects on big-bang nucleosynthesis. While they say that 'our result shows that the dynamical screening effects do not affect the primordial abundances' they do not exclude finer astrophysical effects. In particular they mention that 'the dynamical screening effects on the CNO cycle are more effective than that on the p-p chain', which could provide a 'correction of the CNO cycle for the solar evolution as well as CNO neutrino detection'. Therefore they conclude that 'if the dynamical screening effects are visible under the solar condition, those effects leave several issues worth discussing for related plasma properties in other astrophysical environments'. \nNevertheless, by and large, the stellar and solar communities continue to ignore the findings from molecular dynamics. Nobody seriously proposes to give equal consideration, both to the Salpeter screening enhancement and to a noscreening result. And while Shaviv (2010) is obviously happy to mention the independent confirmation of his group's MD simulations, so far there hasn't been a call to repeat the study of Brown and Sawyer (1997).", '5. Conclusion': "Given the undisputable role of Brown and Sawyer (1997) in the screening controversy, an effort should be made to repeat their analysis. Also, further progress about the controversy could come from laboratory experiments (Casey et al. 2023, Wu and P'alffy 2017). An entirely different approach to study dynamic screening in nuclear fusion was proposed by Anderegg et al. (2010), who have used an apparent 'duality' to infer the solar screening results from a system under totally different physical conditions, in their case experiments on a lasercooled ionic system. Furthermore, it might be worth to mention that there are alternative simulation techniques in other areas of plasma physics. For instance, people have been using so-called 'particle-in-cell (PIC) simulations', pioneered by Langdon and Birdsall (1970), to study dynamic effects on scales that include the Debye length, albeit in a much more rarefied plasma (a typical example is Rekaa et al. 2014). In particular, these approaches treat the dynamics of electrons more seriously than the MD simulations of dynamic screening. Therefore, studying dynamic screening with PIC techniques could be a worthwhile extension of the MD approach. \nAnd, last but not least, the astrophysical detectability of the screening effect should be re-visited in detail, even if Weiss et al. (2001) claim that the 'noscreening' case is ruled out by present-day helioseismic data. Their claim is based on a helioseismic sound speed inference in the region of 0 . 2 -0 . 8R /circledot , with an assumed 'conservative' relative error bar of 10 -3 . Indeed, in their analysis, the no-screening model falls narrowly out of this range, but other uncertainties might influence the result. In any case, such an important claim should be confirmed independently. \nAcknowledgements I am grateful to the anonymous referee for suggesting the possibility of particle-in-cell simulations for dynamic screening. I am indebted to the late Hugh DeWitt at Lawrence Livermore National Laboratory, with whom I had numerous most inspiring discussions on the topic.", 'References': "Adelberger, E.G., Austin, S.M., Bahcall, J.N., Balantekin, A.B., Bogaert, G., Brown, L.S., Buchmann, Lothar, Cecil, F.E., Champagne, A.E., de Braeckeleer, L. Duba, C.A., Elliott, S.R., Freedman, S.J., Gai, M., Goldring, G., Gould, C.R., Gruzinov, A., Haxton, W.C., Heeger, K.M., Henley, E., Johnson, C.W., Kamionkowski, M., Kavanagh, R.W., Koonin, S.E., Kubodera, K., Langanke, K., Motobayashi, T., Pandharipande, V., Parker, P., Robertson, R.G.H., Rolfs, C., Sawyer, R.F., Shaviv, N., Shoppa, T.D., Snover, K.A., Swanson, E., Tribble, R.E., Turck-Chi'eze, S., Wilkerson, J.F.1998, Rev. Mod. Phys. 70 , 1265-1291. Adelberger, E. G., Garc'ıa, A., Hamish Robertson, R. G., Snover, K. A., Balantekin, A. B., Heeger, K., Ramsey-Musolf, M. J., Bemmerer, D., Junghans, A., Bertulani, C. A., Chen, J.W., Costantini, H., Prati, P., Couder, M. , Uberseder, E., Wiescher, M., Cyburt, R., Davids, B., Freedman, S. J., Gai, M., Gazit, D., Gialanella, L., Imbriani, G., Greife, U., Hass, M., Haxton, W. C., Itahashi, T., Kubodera, K., Langanke, K., Leitner, D., Leitner, M., Vetter, P., Winslow, L., Marcucci, L. E., Motobayashi, T., Mukhamedzhanov, A., Tribble, R. E., \nNollett, Kenneth M., Nunes, F. M., Park, T.-S., Parker, P. D., Schiavilla, R., Simpson, E. C., Spitaleri, C., Strieder, F. , Trautvetter, H.-P., Suemmerer, K., Typel, S. 2011, Rev. Mod. Phys. 83 , 195-245. \nAliotta, M., Langanke K: 2022, Front. Phys. 10 , 942726. \nAnderegg, F., Driscoll, C. F., Dubin, D. H. E., & O'Neil, T.M.: 2010, Phys. Plasmas 17 , 055702. \nAtkinson, R.d.E., Houtermans, F.G. 1929, Z. Phys. 54 , 656. \nBahcall, J.N.: 1982, Rev. Mod. Phys. 54 , 767. \nBahcall, J.N., Brown, L.S., Gruzinov, A. & Sawyer, R.F.: 2002, Astron. Astrophys. 383 , 291. Bellinger, E.P., Christensen-Dalsgaard, J.: 2022, Mon. Not. Roy. Astron. Soc. 517 , 5281. Bethe, H.: 1939, Phys. Rev. 55 , 434. \nBonitz, M., Ebeling, W., Filinov, A., Kraeft, W.-D., Redmer, R., Ropke, G.: 2023, Contrib. PlasmaPhys. 63(3-4) , e202300029. \nBrown, L.S., Sawyer R.F.: 1997, Rev. Mod. Phys. 69 , 411-436. \nCarraro, C., Schafer, A., and Koonin, S.E.: 1988, Astrophys. J. 331 , 1057603, doi: 10.3389/fphy.2022.1057603. \nCasey DT, Weber CR, Zylstra AB, Cerjan CJ, Hartouni E, Hohenberger M, Divol L, Dearborn DS, Kabadi N, Lahmann B, Gatu Johnson M and Frenje JA: 2023, Front. Phys. 10 , 565. Christensen-Dalsgaard, J.: 2021, Living Rev. Sol. Phys. 18 , 2, https://doi.org/10.1007/s41116020-00028-3 \nDappen, W., Mussack, K.: 2012, Contrib. Plasma Phys. 52 , 149. \nDebye, P. and Huckel, E.: 1923, Z. Phys. 24 , 305. \nDzitko, H., Turck-Chi'eze, S., Delbourgo-Salvador, P. & Lagrange, C.: 1995, Astrophys. J. 447 , 428. \nGamov, G.: 1929, Z. Phys. 53 , 601. \nHund, F.: 1927, Z. Phys. 40 , 742; Z. Phys. 43 , 905. \nHwang, Eunseok and Jang, Dukjae and Park, Kiwan and Kusakabe, Motohiko and Kajino, Toshitaka and Balantekin, A. Baha and Maruyama, Tomoyuki and Ryu, Chang-Mo and Cheoun, Myung-Ki: 2021, J. Cosmol. Astropart. Phys. 11 ,017. \nKelbg, G.: 1963, Annalen der Physik 467(3-4) ,219. \nLangdon, A.B., Birdsall, C.K.: 1970, Phys. Fluids 13 , 2115. \nLiolios, T.: 2000, Phys. Rev. C 61 , 055802. \nMao, D., Mussack, K. & Dappen, W.,: 2009, Astrophys. J. 701 , 1204. \nMitler, H. E.: 1977, Astrophys. J. 212 , 513. \nMussack, K. & Dappen, W.: 2011, Astrophys. J. 729 , 96. \nRekaa, V.L., Chapman, S.C., Dendy, R.O.: 2014, Astrophys. J. 791 , 26. \nSalpeter, E.E.: 1952, Astrophys. J. 116 , 469. \nSalpeter, E.E.: 1953, Ann. Rev. Nucl. Sci. 2 , 41-62. \nSalpeter, E.E.: 1954, Australian J. Phys. 7 , 373. \nShaviv, N. J. and Shaviv, G.: 1996, Astrophys. J. 468 , 433. \nShaviv, G. and Shaviv, N. J.: 2000, Astrophys. J. 529 , 1054. \n- Shaviv, N. J. and Shaviv, G.: 2001, Astrophys. J. 558 , 925. \nShaviv, G.: 2010, Mem.S.A.It. 81 , 77. \nWeiss, A., Flaskamp, M., and Tsytovich, V. N.: 2001, Astron. Astrophys. 371 , 1123. \nWiescher, M.: 2018, Phys. Perspect. 20 , 124-158. \nWu Yuanbin, P'alffy Adrianna.: 2017, Astrophys. J. 838 , 55."}

2024A&A...691A..18P

Context. EX Lupi is the prototype of EX Luptype stars which are classical T Tauri stars cTTSs with luminosity bursts and outbursts of 15 magnitudes that last for a few months to a few years. These events are ascribed to an episodic accretion that can occur repeatedly but whose physical mechanism is still debated. Aims. We aim to investigate the magnetically driven accretion of EX Lup in quiescence. We include for the first time a study of the small and largescale magnetic field. This allows us to characterise the magnetospheric accretion process of the system completely. Methods. We used spectropolarimetric times series acquired in 2016 and 2019 with the Echelle SpectroPolarimetric Device for the Observation of Stars and in 2019 with the SpectroPolarimtre InfraRouge at the CanadaFranceHawaii telescope during a quiescence phase of EX Lup. We were thus able to perform a variability analysis of the radial velocity the emission lines and the surfaceaveraged longitudinal magnetic field in different epochs and wavelength domains. We also provide a smallscale magnetic field analysis using Zeeman intensification of photospheric lines and a largescale magnetic topology reconstruction using ZeemanDoppler imaging. Results. Our study reveals that typical magnetospheric accretion is ongoing on EX Lup. A main accretion funnel flow connects the inner disc to the star in a stable fashion and produces an accretion shock on the stellar surface close to the pole of the magnetic dipole component. We also measure one of the strongest fields ever observed on cTTSs. This strong field indicates that the disc is truncated by the magnetic field close to but beyond the corotation radius where the angular velocity of the disc equals the angular velocity of the star. This configuration is suitable for a magnetically induced disc instability that yields episodic accretion onto the star.

2024-11-01T00:00:00Z

['2024arXiv240903322P', '10.48550/arXiv.2409.03322', '10.1051/0004-6361/202451527', '2024A&A...691A..18P', 'arXiv:2409.03322']

['accretion', 'accretion disks', 'techniques: polarimetric', 'techniques: spectroscopic', 'stars: magnetic field', 'stars: individual: EX Lup', 'stars: variables: T Tauri', 'Herbig Ae/Be', 'Astrophysics - Solar and Stellar Astrophysics']

Multikilogauss magnetic field driving the magnetospheric accretion process in EX Lupi

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https://arxiv.org/pdf/2409.03322.pdf

{'A multi-kiloGauss magnetic field driving the magnetospheric accretion process in EX Lupi': 'K. Pouilly 1 , M. Audard 1 , Á. Kóspál 2,3 , and A. Lavail 4 \n- 1 Department of Astronomy, University of Geneva, Chemin Pegasi 51, CH-1290 Versoix, Switzerland e-mail: Kim.Pouilly@unige.ch\n- 2 Konkoly Observatory, HUN-REN Research Centre for Astronomy and Earth Sciences, MTA Centre of Excellence, Konkoly-Thege Miklós út 15-17, 1121 Budapest, Hungary\n- 3 Institute of Physics and Astronomy, ELTE Eötvös Loránd University, Pázmány Péter sétány 1 / A, 1117 Budapest, Hungary\n- 4 Institut de Recherche en Astrophysique et Planétologie, Université de Toulouse, CNRS, IRAP / UMR 5277, 14 avenue Edouard Belin, 31400 Toulouse, France \nReceived 16 July 2024; accepted 04 September 2024', 'ABSTRACT': 'Context. EX Lupi is the prototype of EX Lup-type stars, meaning classical T Tauri stars (cTTSs) showing luminosity bursts and outbursts of 1 to 5 magnitudes lasting for a few months to a few years. These events are ascribed to an episodic accretion that can occur repeatedly but whose physical mechanism is still debated. \nAims. In this work, we aim to investigate the magnetically-driven accretion of EX Lup in quiescence, including for the first time a study of the small and large-scale magnetic field. This allows us to provide a complete characterisation of the magnetospheric accretion process of the system. \nMethods. We use spectropolarimetric times series acquired in 2016 and 2019 with the Echelle SpectroPolarimetric Device for the Observation of Stars and in 2019 with the SpectroPolarimètre InfraRouge at the Canada-France-Hawaii telescope, during a quiescence phase of EX Lup. We were thus able to perform a variability analysis of the radial velocity, the emission lines and surface averaged longitudinal magnetic field along di ff erent epochs and wavelength domains. We also provide a small-scale magnetic field analysis using Zeeman intensification of photospheric lines and large-scale magnetic topology reconstruction using Zeeman-Doppler Imaging. Results. Our study reveals a typical magnetospheric accretion ongoing on EX Lup, with a main accretion funnel flow connecting the inner disc to the star in a stable fashion and producing an accretion shock on the stellar surface close to the pole of the magnetic dipole component. We also measure one of the strongest fields ever observed on cTTSs. Such a strong field indicates that the disc is truncated by the magnetic field close but beyond the corotation radius, where the angular velocity of the disc equals the angular velocity of the star. Such a configuration is suitable for a magnetically-induced disc instability yielding episodic accretion onto the star. \nKey words. Stars: variables: T Tauri - Stars: individual: EX Lup - Stars: magnetic field - Accretion, accretion disks - Techniques: spectroscopic - Techniques: polarimetric', '1. Introduction': 'EXLup-type objects (EXors) are classical T Tauri stars (cTTSs), meaning low-mass pre-main sequence stars surrounded by an accretion disc, that show burst and outburst events ascribed to episodic accretion. During these phases, they can increase their optical luminosity from 1 to 5 magnitudes, lasting typically for a few months to a few years, and can occur repeatedly (for a review, see, e.g., Fischer et al. 2023). These events are therefore more moderate, both in duration and luminosity increase, than the FU Orionis-type stars (FUors). \nWhile the magnetospheric accretion of cTTSs, where the strong stellar magnetic field truncates the disc, forcing the accreted material to follow the magnetic field lines (see the review by Hartmann et al. 2016), seems to be ongoing on EXors as well, the origin of this episodic accretion is still highly debated. The di ff erent hypotheses can be gathered in three groups (see review by Audard et al. 2014): (i) the magnetospheric accretion itself, being inherently episodic when a strong magnetic field truncates the disc close to the corotation radius (D\'Angelo & Spruit 2010). (ii) The disc, showing viscous-thermal (Bell & Lin 1994), or gravitational and magneto-rotational (Armitage et al. 2001) in- \nstabilities, or accretion clumps in a gravitationally unstable environment (Vorobyov & Basu 2005, 2006). (iii) The presence of a companion, perturbing the accretion trough tidal e ff ect (Bonnell & Bastien 1992), or thermal instabilities (Lodato & Clarke 2004). \nThis work aims to investigate the magnetospheric accretion process of the prototypical EXors, EX Lup, using highresolution spectropolarimetric time series, as it was done for cTTSs not displaying episodic accretion (e.g., Pouilly et al. 2020, 2021). This object is a young M0.5-type star (GrasVelázquez & Ray 2005), known to have both moderate and shorttimescale variability and rare extreme episodic outbursts. It is located at 154.7 ± 0.4 pc ( Gaia DR3 parallax 6.463 ± 0.015 mas, Gaia Collaboration et al. 2023) and has a rotation period of 7.417 days, from its radial velocity modulation that was first ascribed to a low-mass companion (Kóspál et al. 2014), before being ascribed to stellar activity (Sicilia-Aguilar et al. 2015). The system has a moderate-to-low inclination of its rotation axis (between 20 · and 45 · ) according to modelling of the spectral energy distribution (Sipos et al. 2009) and emission lines analysis (Goto \net al. 2011; Sicilia-Aguilar et al. 2015), and a projected rotational velocity v sin i = 4.4 ± 2.0 km s -1 (Sipos et al. 2009). \nThis object is one of the most studied EXors using spectroscopy, both in quiescence (i.e., Kóspál et al. 2014; Sipos et al. 2009; Sicilia-Aguilar et al. 2012, 2015, 2023; Campbell-White et al. 2021; Wang et al. 2023) and in outburst (Sicilia-Aguilar et al. 2012; Cruz-Sáenz de Miera et al. 2023; Wang et al. 2023; Singh et al. 2024, for the last two outbursts in 2008 and 2022), but the present study is the first one including spectropolarimetry, giving access to information on the magnetic field together with the accretion diagnostics. Its spectrum contains the typical accretion-related emission lines observed in cTTSs, in addition to numerous neutral metallic emission lines superimposed to photospheric absorption in quiescence, and overwhelming any absorption feature in outburst (Sicilia-Aguilar et al. 2012). The present dataset was acquired during quiescence, meaning that we will characterise the "stable" accretion of the system, even if Sicilia-Aguilar et al. (2012, 2023) have shown that the accretion pattern seems stable in quiescence and outburst, only the amount of accreted material is a ff ected. This accretion pattern is consistent with the typical cTTS magnetospheric accretion, through accretion funnel flows connecting the disc to the stellar surface, except that "clumps" of material are also accreted through these funnel flows, detected thanks to the day-to-day variation of the emission lines\' broad component (BC) (see Fig. 17 of SiciliaAguilar et al. 2012). \nThis article is organised as follows: we describe the observations in Sect. 2, the analysis and results are presented in Sect. 3 and discussed in Sect. 4. We conclude this work in Sect. 5.', '2. Observations': 'The spectropolarimetric time series used in this work were acquired at the Canada-France-Hawaii Telescope at two di ff erent epochs (2016 and 2019, proposals 16AF03 and 19AF50, respectively). The second data set is composed of two subsets using two di ff erent instruments: the Echelle SpectroPolarimetric Device for the Observation of Stars (ESPaDOnS, Donati 2003) in the optical and the SpectroPolarimètre InfraRouge (SPIRou, Donati et al. 2020b) in the near-infrared, both used in polarimetric mode, while the first one only used ESPaDOnS. This means that each observation is composed of four sub-exposures taken in di ff erent polarimeter configurations, which are then combined to obtain the intensity (Stokes I ), the circularly polarised (Stokes V ), and the null polarisation spectra. A complete journal of observation is provided in Table 1.', '2.1. ESPaDOnS': 'The ESPaDOnS observations, which cover the 370 to 1050 nm wavelength range and reach a resolving power of 68 000, consist of 11 nights between 2016 June 09 and 2016 June 24, with an approximately nightly cadence, and 6 nights between 2019 May 31 and 2019 June 12, the 5 latter respecting a 1-day sampling. The signal-to-noise ratio (S / N) of the 2016 (2019) observations ranges between 69 and 142 (111 and 140) for the Stokes I at 731 nm. Each observation was reduced using the Libre-ESpRIT package (Donati et al. 1997).', '2.2. SPIRou': 'The SPIRou observations are covering the 960 to 2350 nm wavelength range with R ∼ 75 000. They were acquired during 8 con- \nTable 1. Log of EX Lup observations. \nNotes. The S / NI corresponds to the peak S / Nbyspectral pixel at 731 nm (in H-band) for ESPaDOnS (SPIRou) observations. The S / NLSD is the e ff ective S / N of the Stokes V LSD profiles (see Sect. 3.3.1). The last column indicates which instrument was used for each observation (EESPaDOnS, S-SPIRou). The horizontal line separates the 2016 from the 2019 observations. \nsecutive nights, between 2019 June 14 and 2019 June 21, and the S / N of the unpolarised spectra in the H-band range between 107 and 175. The observations were reduced using the APERO pipeline (Cook et al. 2022).', '3. Results': 'In this section, we present the results obtained from the analysis of the observations described in Sect. 2. They consist of the analysis of the radial velocity, emission lines, and stellar magnetic field.', '3.1. Radial Velocity': 'To determine the radial velocity of EX Lup, we cross-correlated each spectrum with a synthetic spectrum computed using the ZEEMAN code (Landstreet 1988; Wade et al. 2001; Folsom et al. 2012), with MARCS atmospheric models (Gustafsson et al. 2008) and VALD (Ryabchikova et al. 2015) line lists adapted to EX Lup stellar parameters for ESPaDOnS and SPIRou wavelengths. We computed the cross-correlation function (CCF) over 27 (14) wavelength windows of about 10 nm ranging from 441 to 890 nm (1150 to 2290 nm) for ESPaDOnS (SPIRou) observations. Then we performed a sigma-clipping across all the CCFs for each observation and used the mean and standard deviation of the remaining values as measurement of the radial velocity and its uncertainty. The results are plotted \nFig. 1. Radial velocities curves determined for the observation summarised in Table 1. Top: radial velocity vs. HJD, the vertical dotted line marks the switch from ESPaDOnS to SPIRou, and the di ff erent colours represent di ff erent rotation cycles. Bottom: same as above but folded in phase with P = 7.417 d and T0 = 2 457 544.40981. The dotted curve shows the sinus fit. \n<!-- image --> \nin Fig. 1. A quick sinusoidal fit of the values for each data set allowed us roughly measure the periodicity and mean value of the radial velocity and yielded P = 7.55 ± 0.24 d and ⟨ Vr ⟩ = -0.48 ± 0.07 km s -1 (ESPaDOnS 2016), P = 7.63 ± 0.18 d and ⟨ Vr ⟩ = -0.59 ± 0.14 km s -1 (ESPaDOnS 2019), and P = 8.36 ± 0.97 d and ⟨ Vr ⟩ = -0.58 ± 0.14 km s -1 (SPIRou). These measures are consistent within the uncertainties with previous results obtained by Kóspál et al. (2014), Prot = 7.417 ± 0.001 d and ⟨ Vr ⟩ = -0.52 ± 0.07 km s -1 , we will thus adopt the latter values for our analysis. Finally, we folded all the measurements in phase using Prot = 7.417 d and an arbitrary T0, and we fitted this curve with a sinus to estimate the T0 required to set ϕ = 0.5 at the mean velocity between the maximum and minimum of the modulation (when the spot modulating the curve is facing the observer, see Fig. 7 of Sicilia-Aguilar et al. 2015). The resulting T0 is HJD 2 457 544.40981. We will thus use the following ephemeris for the rest of this work: \nHJD(d) = 2 457 544 . 40981 + 7 . 417 E , (1) \nwhere E is the rotation cycle. All our radial velocity measurements are in phase (Fig. 1), but the 2019 measurements show a larger amplitude of its modulation. This indicates an evolving feature modulating the radial velocity but located at the same longitude in 2016 and 2019. The values are summarised in Table 2.', '3.2. Emission line variability': 'In this section, we present the analysis of emission lines tracing either the accretion funnel flow, here the Balmer lines (Muzerolle et al. 2001), or the accretion shock such as the Ca ii infrared triplet (IRT), the He i D3 (Beristain et al. 2001), or the He i at 1083 nm lines. For each line, we analysed the profile variability, their periodicity (except for the 2019 ESPaDOnS data set which does not cover a su ffi cient time span), and the correlations of these variabilities. Most of the analyses of this section were performed using PySTEL(L)A 1 , a Python tool for SpecTral Emission Lines (variabiLity) Analysis. \nTable 2. Radial velocities measured for each observation and their uncertainties. \nNotes. The phases ϕ are computed using the ephemeris provided in Eq. 1. The horizontal line separates the 2016 from the 2019 observations.', '3.2.1. Balmer lines': 'The Balmer lines are partly formed in the accretion funnel flow and are thus tracing the magnetospheric accretion process. In this work, we focussed on H α , H β , and H γ , which we corrected from the photospheric contribution using the moderately active Mdwarf HD 42581 as template (Te ff = 3822 K, v sin i = 2.6 km s -1 , Manara et al. 2021), broadened to the v sin i of EX Lup. The 2016 and 2019 profiles are shown in Fig. 2 and Fig. 3, respectively. On both data sets, the profiles are composed of a broad and a narrow component, both highly variable. Furthermore we can notice a flux depletion, going below the continuum, around + 200 kms -1 , and extending up to + 300 kms -1 . This behaviour is characteristic of the so-called inverse P Cygni (IPC) profiles, the red-shifted absorption forming due to infalling material. \nThe 2D-periodograms, consisting of a Lomb-Scargle periodogram computed in each velocity channel, of the 2016 lines are presented in Fig. 2, and show a periodic signal along the whole H β and H γ lines consistent with the rotation period of the star, with a false alarm probability (FAP, computed from Baluev 2008, prescriptions) of 0.04 and 0.01, respectively. The H α line is showing this signal in a less continuous way with a much higher FAP (0.25). The symmetric signal around 0.9 d -1 is the 1-day alias, a spectral leakage of the Fourrier transform reconstructing (with the real period) the observation sampling, meaning approximately one observation per day. \nTo separate the various parts of the line profile undergoing di ff erent variability patterns, we computed the auto-correlation matrices of each line. This tool consists of computing a linear \n656.28 nm \n434.05 nm \nFig. 2. Variability of 2016 Balmer lines of EX Lup. Top row: H α (left) , H β (middle) , and H γ (right) residual lines profiles. Each colour represents a di ff erent observation. Bottom row: 2D-periodograms of H α (left) , H β (middle) , and H γ (right) residual lines. The white dotted line marks the rotation period of 7.417 d. \n<!-- image --> \ncorrelation coe ffi cient (here a Pearson coe ffi cient) between the velocity channels of the line. A correlated region (close to 1) indicates a variability dominated by a given physical process. An anti-correlated region (close to -1) indicates a variability dominated by a given physical process or, at least, linked physical processes. The auto-correlation matrices of 2019 H α , H β , and H γ lines are shown in Fig 3. Here again, H α is showing a di ff erent behaviour than H β and H γ . H α shows three main correlated regions between -100 kms -1 and + 100 kms -1 , corresponding to the core of the line, around + 150 kms -1 , corresponding to a slight emission excess in the IPC profile (occurring around HJD 24588634.95 and 2 458 641.95), and around + 250 km s -1 , corresponding to the IPC profile itself. The two latter regions are anti-correlated between them, but the ∼ + 150 kms -1 region is also slightly anti-correlated with the line center. This means that this region might be a broadening of the BC invoked by Sicilia-Aguilar et al. (2012), causing a global decrease of the line (and so a anti-correlation with the whole line profile). H β and H γ are showing the same correlation around the line centre and + 350 kms -1 , without significant anti-correlation around + 150 km s -1 , and a moderate anti-correlation of the profile with a region around + 350 km s -1 which seems to be an artefact as it is located in the continuum.', '3.2.2. He i D3 587.6 nm': "The He i D3 lines of 2016 and 2019 data sets are presented in Fig. 4 and are composed of a narrow component (NC) only, extending from approximately -35 to + 50 km s -1 , without significant BC. The NC is formed in the post-shock region of the accretion spot (Beristain et al. 2001), and can thus trace the accretion \nclose to the stellar surface. One can note the strong variability of this NC with a maximum reached around ϕ = 0.6 2 . \nThe auto-correlation matrices shown in Fig. 4 confirm that this region is formed by one physical process, and the periodicity of the 2016 NC's variation, consistent with the stellar rotation period (see Fig. 5, FAP = 0.06), indicates that this component is tracing an accretion shock at the stellar surface. \nWe thus performed a fit of the He i D3 NC's radial velocity following the method described in Pouilly et al. (2021) to recover the emitting region's location. The results are shown in Fig. 6 and summarised below: \n- -Vflow = 7.091 + 2 . 00 kms -1\n- -Vrot = 3.85 ± 5.0 km s -1 ,\n- -d ϕ = 0.10 + 0 . 15\n- -θ = 12.05 + 48 . 00 -6 . 7 ·\n- -α = 55.40 . \n-1 . 03 , -0 . 11 , , + 20 00 -16 . 36 · , \nwhere Vflow is the velocity of the material in the post-shock region, Vrot the equatorial velocity, 0.5 + d ϕ the phase where the emitting region is facing the observer, θ the colatitude of the spot, and α = 90 · -i, i being the inclination of the rotation axis. This means that the emitting region is located at ϕ = 0.6, 70 · latitude. These results are consistent within uncertainties with the measurements by Campbell-White et al. (2021), who give for the He i emitting region 3 a latitude of 60 ± 25 · , longitude 40 ± 5 · , meaning ϕ = 0.1 ± 0.01. However, the authors are using the first date of observation as T0, translated to our ephemeris (Eq. 1), this yields ϕ = 0.7. \n656.28 nm \nFig. 3. Variability of 2019 Balmer lines of EX Lup. Top row: H α (left) , H β (middle) , and H γ (right) residual lines profiles. Each colour represents a di ff erent observation. Bottom row: H α (left) , H β (middle) , and H γ (right) residual lines auto-correlation matrices. The colour code is scaling the correlation coe ffi cient. Light yellow represents a strong correlation and dark purple a strong anti-correlation. Please note that the strong anticorrelation around + 350 km s -1 on H β and H γ matrices is probably an artefact as located in a poorly varying part of the continuum. \n<!-- image -->", '3.2.3. Ca ii infrared triplet': "As the He i D3 NC, the Ca ii IRT NC is formed in the post-shock region, we thus studied these lines as well. The three components of this triplet show identical shape and variability, we thus focussed on one of them located at 854.209 nm. The 2016 and 2019 residual line profiles are shown in Fig. 7 and Fig. 8, respectively. The two sets of lines show an IPC profile around 200 kms -1 , which is, for the 2016 line, periodic with the stellar rotation period (see the 2D-periodogram in Fig. 7, FAP = 0.05). \nThe 2016's auto-correlation matrix (Fig. 7) exhibits several correlated regions: from -130 to -60, -50 to -10, + 50 to + 100, + 110 to + 160, and + 170 to + 200 km s -1 . However, these regions are less numerous on the 2019 matrix (Fig. 8), with only three correlated regions from -40 to + 40, + 80 to + 130, and + 150 to + 200 kms -1 , but we retrieve the correlated region around the IPC profile in both matrices, which is anti-correlated with the NC in 2019. This goes with the smaller (larger) variability of the NC (BC) observed in 2016 compared to 2019, showing the di ff erent origins of the two components and a small change in the accretion pattern between the two epochs.", '3.2.4. He i 1083 nm': "The only accretion tracer in emission in EX Lup's SPIRou observation is the He i line at 1083 nm. The profiles, the 2Dperiodogram and the auto-correlation matrix are shown in Fig. 9. The profiles seem composed of two peaks blue- and red-shifted around -50 and + 100kms -1 , and two absorptions, blue-and redshifted at higher velocity ( -150 and + 200kms -1 ). \nThe red-shifted peak and the two absorptions display significant variability and seem modulated on the stellar rotation pe- \nriod with FAPs reaching 0.001, 0.01, and 0.02 for the red-shifted peak, the blue- and the red-shifted absorption, respectively. \nThe auto-correlation matrix revealed a more complex decomposition. Indeed, if the four substructures seen in the profiles are represented, it seems that the two absorptions can both be separated in two regions, from -250 to -160 and -160 to -110 kms -1 for the blue-shifted absorption, and from + 130 to + 170 kms -1 and + 210 to + 270 kms -1 for the red-shifted absorption. Furthermore, the main peak at ∼ + 100 kms -1 is anticorrelated with the most blue- and red-shifted regions only. This can be interpreted as follows: the blueshifted absorption, probably a P-Cygni profile traditionally ascribed to a wind, is also associated with a redshifted emission excess, producing the two substructures seen in the redshifted absorption. The IPC profile, as the opposite physical phenomenon, is also associated with a blueshifted emission excess, responsible for the two substructures in the blueshifted absorption.", '3.2.5. Correlation matrices ESPaDOnS': "As the several lines studied are tracing di ff erent regions of the accretion, we can compute correlation matrices between two di ff erent lines to analyse the link between the di ff erent regions identified from the auto-correlation matrices. The correlation matrices of the 2016 and 2019 lines are presented in Appendix A. \nIn 2016, the H α line centre is correlated with the He i D3 and the Ca ii IRT NCs, and slightly anti-correlated with the region of Ca ii IRT IPC profile. The latter is also strongly anti-correlated with the He i D3 NC. The region of the H α 's IPC profile is also anti-correlated with the He i D3 NC, and correlated with the region of the Ca ii IRT IPC profile. In 2019 matrices, we observe \nFig. 4. EX Lup He i D3 line variability. Top row: 2016 (left) and 2019 (right) lines profiles. Bottom row: Auto-correlation matrices of 2016 (left) and 2019 (right) lines. \n<!-- image --> \nFig. 5. He i D3 2016 2D-periodogram. \n<!-- image --> \nthe same behaviour between the NCs and IPC regions of the var- \nFig. 6. Radial Velocity fit of He i D3 2016 NC (top) and its version folded in phase (bottom) following the ephemeris given in Eq. 1 \n<!-- image --> \nious lines, but both the correlation and anti-correlation coe ffi -cients are stronger. \nFig. 7. ESPaDOnS 2016 Ca ii IRT (854.2 nm) residual line profiles (left) , 2D-periodogram (middle) , and auto-correlation matrix (right) . \n<!-- image --> \n854.21 nm \nFig. 8. ESPaDOnS 2019 Ca ii IRT (854.2 nm) residual line profiles (top) and auto-correlation matrix (bottom) . \n<!-- image -->", '3.3. Magnetic field': 'In this section, we present the magnetic analysis of EX Lup. This was done at two scales: the large scale using the ZeemanDoppler Imaging technique (ZDI, Donati et al. 2011), and the small scale using the Zeeman intensification of photospheric lines.', '3.3.1. Large-scale': "In this section, we used the Least-Squares Deconvolution method (LSD, Donati et al. 1997; Kochukhov et al. 2010) to study the large-scale magnetic field. This method allows us to increase the S / N of the Stokes I (unpolarised) and Stokes V (circularly polarised) profiles, by using as many photospheric lines as possible. To compute the LSD profiles we used the LSDpy 4 Python implementation. We normalised our LSD weights using an intrinsic line depth, a mean Landé factor and a central wavelength of 0.2, 1.2, and 500 nm (respectively) for ESPaDOns, and 0.1, 1.2, and 1700 nm for SPIRou observations. The photospheric lines were selected from a mask produced using the same VALD line list and MARCS atmospheric models as in Sect. 3.1, and removing the emission lines, the telluric and the heavily blended lines regions using the SpecpolFlow 5 Python package. About 12 000 lines were used for ESPaDOnS, and 1600 for SPIRou observations. The LSD profiles of ESPaDOnS 2016, 2019, and SPIRou observations are presented in Figs. 10, 11, and 12, respectively, and the S / N of the profiles are available in Table 1. Please note that the observation at HJD 2 457 551.87135 was remove from this analysis because of its low S / N. \nAclear Stokes V signature is detected for all ESPaDOnS observations and most of the SPIRou observations. Furthermore, one can note that the SPIRou Stokes V signatures are much weaker than ESPaDOnS and the signal at ϕ ∼ 0.6-0.7 almost vanishes on SPIRou when maximal on ESPaDOnS. \nDirectly from the LSD profiles, one can estimate the surface averaged longitudinal magnetic field (B ℓ , Donati et al. 1997; Wade et al. 2001): \nB ℓ = -2 . 14 × 10 11 × R vV ( v ) dv λ gc R (1 -I ( v )) dv , (2) \nwhere B ℓ is in Gauss, v the velocity relative to the line centre, and λ and g the central wavelength and the mean Landé factor used for the LSD computation. The integration was performed on a ± 25 kms -1 ( ± 35 kms -1 ) velocity range around the stellar rest frame to minimise the uncertainties without losing any magnetic information on ESPaDOnS (SPIRou) observations. The B ℓ curves are shown in Fig. 13. The three curves are modulated with the stellar rotation period and show a maximum around ϕ = 0.6. In the optical frame, the modulation's amplitude is slightly larger in 2019 than in 2016, which is reminiscent of the radial velocity \n<!-- image --> \n<!-- image --> \nFig. 9. SPIRou He i (1083 nm) line profiles (left) , 2D-periodogram (middle) , and auto-correlation matrix (right) . \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFig. 10. ESPaDOnS 2016 LSD Stokes I (left) and V (right) profiles. The rotation phase and the HJD are indicated at the left and right, respectively, of each profile. The scale is indicated at the bottom left of each plot. \n<!-- image --> \nbehaviour (see Sect. 3.1). Finally, as expected from the Stokes V signatures, the SPIRou measurements are far weakest, and mostly negatives. \nThe same analysis can be performed on the NC of the He i D3 line (on a ± 50 km s -1 velocity range), formed close to the accretion shock and thus gives access to the magnetic field strength at the foot of the accretion funnel flow. The B ℓ obtained are shown in Fig. 13 and range between -1.4 and -3.7 kG in 2016 and between -0.5 and -4.5 kG in 2019. As for the radial velocity (see Sect. 3.1), the amplitude of the modulation is larger in 2019, but both curves are in phase, with a minimum reached at ϕ = 0.6, which is also consistent with the emitting region's position obtained from the radial velocity modulation of the He i D3 NC (see Sect. 3.2.2). These measurements are also opposed in phase and sign with the LSD B ℓ . This is a common behaviour on cTTSs (e.g., see the studies of S Cra N, TWA Hya, CI Tau by Nowacki \net al. 2023; Donati et al. 2011, 2020a, respectively) reflecting that LSD and He i D3 NC are two diagnostics probing di ff erent regions of di ff erent polarities. \nFinally, we performed a complete ZDI analysis of the three data sets using ZDIpy 6 (described in Folsom et al. 2018) with a recent implementation of Unno-Rachkovsky's solutions to polarised radiative transfer equations in a Milne-Eddington atmosphere (Unno 1956; Rachkovsky 1967; Landi Degl'Innocenti & Landolfi 2004) presented in Bellotti et al. (2023) to used a more general description than the weak-field approximation used initially in ZDIpy . The magnetic topology is reconstructed in two steps: (i) the building of a Doppler image (DI), starting from a uniformly bright stellar disk and iteratively adding dark and bright features to fit the observed LSD Stokes I profiles, and (ii) fitting the LSD Stokes V profiles to derive the magnetic \nFig. 11. Same as Fig. 10 for ESPaDOnS 2019 observations. \n<!-- image --> \n<!-- image --> \nI \n/ \nV \ns \ne \nk \no \nt \nS \ntopology by adjusting its spherical harmonic components (Donati et al. 2006), here with a maximum degree of harmonic expansion ℓ max = 15. We used as input parameter Prot = 7.417 d, v sin i = 4.4 km s -1 . Then we ran a grid of ZDI Stokes V reconstruction over the inclination range derived in the literature (2045 · ) and used the minimal χ 2 obtained to set our input inclination value (i = 30 · ). The resulting maps are presented in Fig. 14 and 15 and the corresponding LSD profile fits are shown in Appendix B. \nThe ESPaDOnS 2016's DI map shows a main dark spot around ϕ = 0.55 and extending between 70 and 20 · latitude, which is consistent with He i D3 emitting region, and a bright structure from ϕ = 0.95 to 0.20 around the equator which is certainly a plage as even the accretion shock is dark at the photospheric level. The magnetic topology, mostly toroidal (61%) is dominated by the dipolar (60%) and the quadrupolar (17%) components, with a mean magnetic field strength of 1.00 kG (Bmax = 3.12 kG), and a magnetic dipolar positive pole of 0.728 kG located at about 30 · latitude and 166 · longitude ( ϕ = 0.46). \nThe brightness map obtained from SPIRou revealed a main dark feature extended from ϕ = 0.0 to 0.25, and a bright plage around ϕ = 0.6, which is perfectly consistent with the 2016 map. However, the magnetic topology seems less complex than in 2016, almost fully poloidal (99%) and more dominated by the dipole component (77%). As expected, the recovered field strength is much smaller ( ⟨ B ⟩ = 0.131 kG, Bmax = 0.389 kG). The dipole negative pole is located at about 23 · latitude, 329 · longitude ( ϕ = 0.91), with a -0.231 kG-strength. \nGiven the poor rotational phase coverage of the ESPaDOnS 2019's data set ( ϕ ∈ [0.03,0.64]), we needed to guide the reconstruction instead of starting from a uniform map to avoid a too strong extrapolation on the missing phases. We do not expect \nTable 3. Ti i lines used for the Zeeman intensification analysis.Notes. The wavelengths in the optical frame are given in the air, the ones in the infrared frame are given in vacuum. \na similar brightness contrast between ESPaDOnS and SPIRou, but given the very similar spectroscopic behaviour between the 2016 and 2019 ESPaDOnS data sets (see Sect. 3.1 and 3.2), we could expect similar features, we thus used the ESPaDOnS 2016 brightness map as input to guide its reconstruction. To check this assumption, we reproduce the Stokes I profiles resulting from the 2016 reconstruction on the 2019 phases, yielding a consistent behaviour (reduced χ 2 = 1.1). The obtained final brightness reconstruction shows a main dark feature, slightly shifted in phase compared to 2016 ( ϕ ≈ 0.5) and less extended in latitude. A polar bright feature is also located around ϕ = 1, which is reminiscent of the SPIRou's maps. Concerning the magnetic reconstruction, we expect a similar topology of the magnetic field between SPIRou and ESPaDOnS 2019, with di ff erent magnetic strengths as pointed out by the B ℓ analysis. We thus used the SPIRou magnetic maps as input to guide the reconstruction. The resulting topology is less poloidal-dominated than SPIRou (91%), and the dipolar component occupies a smaller fraction of this poloidal field (65%) with a similar contribution of the quadrupolar and octupolar components (about 15%). Surprisingly the mean magnetic field strength is similar to 2016 (1.08 kG) but the maximum field strength is much higher (4.79 kG), as well as the strength of the dipolar pole Bdip = 1.87 kG which is located at the same position ( ϕ = 0.43 and 30 · latitude).", '3.3.2. Small-scale': "Even if the ZDI analysis gives access to the magnetic topology, it neglects the small-scale magnetic field, which contains a major part of cool stars' magnetic energy. This is analysed by studying the change it induces in the shape of magnetically sensitive lines (Kochukhov et al. 2020), a technique called Zeeman intensification. To perform this analysis, we used the algorithm of Hahlin et al. (2021), performing a Markov Chain Monte Carlo (MCMC) sampling, using the SoBaT library (Anfinogentov et al. 2021) on a grid of synthetic spectra produced by the SYNMAST code, a polarised radiative transfer code described by Kochukhov et al. (2010). This grid is computed from the VALD line lists used in Sect. 3.1, and MARCS atmospheric models (Gustafsson et al. 2008). We parametrized a uniform radial magnetic field as a sum of magnetic field strength ranging from 0 to 6 kG, with a 2 kG step, weighted by filling factors representing the amount of stellar surface covered by this magnetic field. For ESPaDOnS observations, as in previous studies (Hahlin & Kochukhov 2022; Pouilly et al. 2023, 2024), we used the 963.5-981.2 nm region. \nThis region contains a group of Ti i lines with di ff erent magnetic sensitivity (ge ff summarised in Table 3), allowing us to disentangle the e ff ect of the magnetic field on the equivalent widths \nFig. 12. Same as Fig. 10 for SPIRou observations. \n<!-- image --> \n<!-- image --> \n<!-- image --> \nI \n/ \nV \ns \ne \nk \no \nt \nS \nfrom the e ff ect of any other parameters, such as the Ti i abundance. However, this region also contains many telluric lines which are superimposed on the Ti i lines from the stellar spectra and which need to be removed from the observed spectrum to perform the magnetic analysis. To do so, we used the molecfit package (Smette et al. 2015), developed to model and remove telluric lines from spectra obtained with instruments at the European Southern Observatory, and which can be used on spectra from any instrument. \nFinally, as EX Lup is an accreting star, showing a signature of an accretion shock, the veiling might disturb the inference results, in particular the abundance, the v sin i and the radial tangential macroturbulent velocity, vmac. We thus estimated the veiling using the magnetic null line of the Ti i multiplet at 974.36 nm by performing a χ 2 minimisation using SYNMAST synthetic spectra letting the abundance, the v sin i , and the vmac as free parameters and adding a fractional veiling defined as: \nIveil = I + r 1 + r , (3) \nwhere Iveil is the veiled spectrum, I the spectrum without veiling, and r the fractional veiling. We stress to the reader that deriving a precise value of the veiling is outside the scope of the present study , our aim was only to get an estimation of the EX Lup mean spectrum at a wavelength close to the Ti i lines in order to minimise its e ff ect on the other inferred parameters. For the 2016's ESPaDOnS mean spectrum, the minimal χ 2 is reached for r = 0.5, v sin i = 4.9 km s -1 , vmac = 1.9 km s -1 , and a Ti i abundance of -7.2. For the mean spectrum of the 2019's ESPaDOnS observations, we obtained r = 0.48, v sin i = 5.0 km s -1 , vmac = 2.5km s -1 , and a Ti i abundance of -7.2. We will thus assume the values obtained for r and vmac, and use the others as initial guesses for the MCMC sampling. \nWe assumed a multicomponent model given by: \nS = X fiS i , (4) \nwhere fi are the filling factors, meaning the fraction of the stellar surface covered by a field strength Bi , and Si the synthetic spectra of the corresponding magnetic field strength. The averaged magnetic field is thus given by: \n⟨ B ⟩ = X fi Bi . (5) \nTo set the number of filling factor to use, we iteratively added filling factors, with a 2-kG step in the corresponding magnetic field strength, and used the Bayesian information criterion (BIC, Sharma 2017) to only include the filling factors that significantly improve the fit. The suitable solutions are components of 0, 2, 4 kG for ESPaDOnS 2016 and SPIRou observations, and 0, 2, 4, 6, 8 kG for ESPaDOnS 2019. The free parameters of the analysis are thus the following: fi , v sin i , v r , and the Ti i abundance, for which uniform prior were adopted. Finally, we used an e ff ective sample size of 1000 (Sharma 2017). \nThe resulting line fit and magnetic field strength posterior distributions are presented in Fig. 16. The 2016's magnetic field strength (3.08 ± 0.04 kG) is consistent with 2019 within uncertainties (3.16 ± 0.05 kG). The inferences of all parameters are summarised in Table 4. \nFor SPIRou observations, we used here again a set of Ti i lines around 2200 nm (see Table 3). Unfortunately, this wavelength region does not contain any magnetically null line deep enough at our S / N to perform the veiling study we have done on ESPaDOnS observations. Only a visual inspection of the line at 974.4 nm is possible, indicating that at this wavelength the parameters obtained for ESPaDOnS are consistent with the SPIRou observations. We thus used the veiling values obtained for the 2019 ESPaDOnS data set, fixed the v sin i at the literature \n<!-- image --> \nFig. 13. B ℓ curves from LSD profile (top two panels) and from He i D3 line (bottom two panels) . \n<!-- image --> \nvalue and let the inference compensate for the eventual error with the non-magnetic parameters. The inferred parameters are summarised in Table 4, and the line fit and inferred magnetic field strength are shown in Fig. 17. The magnetic field strength recovered (2.00 ± 0.03 kG) is significantly lower than the values found for ESPaDOnS observations. As highlighted by Hahlin et al. (2023), the small-scale field might be overestimated when using optical wavelength. In our case, a second explanation might come from the high vmac obtained, probably needed to compensate for an underestimated veiling, which lowers the e ff ect of the magnetic field.", '4. Discussion': "This work aimed at characterising the accretion process of the prototypical EXor, EX Lup, in quiescence, together with its magnetic field at small and large scales. This study confirms that the typical magnetospheric accretion process of cTTSs is ongoing on this system, which seems stable between the two epochs studied (2016 and 2019), with a main accretion funnel flow connecting the disc to the stellar surface. This produces the IPC profile observed in the H lines studied and their modulation with the stellar rotation period. The accretion shock at the stellar surface produces the emission of the NC of the He i D3 and Ca ii IRT lines, which are modulated with the stellar period and \nanti-correlated with the IPC profiles as expected in the magnetospheric accretion scheme. This is consistent with the maximum IPC profile occurring at the same phase as the He i D3 emitting region ( ϕ ≈ 0.6), indicating a funnel flow aligned with the accretion shock and inherently meaning that the truncation radius is located at the stellar corotation radius. As expected, this phase is also associated with an extremum of the B ℓ , for both LSD and He i D3 measurements, and with the ESPaDOnS ZDI brightness and magnetic topology reconstructions, showing the connection between accretion and magnetic field. To investigate the truncation radius r mag, we used the expression given by Bessolaz et al. (2008): \nr mag R ⋆ = 2 m 2 / 7 s B 4 / 7 ⋆ ˙ M -2 / 7 acc M -1 / 7 ⋆ R 5 / 7 ⋆ , (6) \nwhere the Mach number m s ≈ 1, B ⋆ is the equatorial magnetic field strength (given from the dipole strength and the relation of Gregory 2011) in units of 140 G, ˙ Macc is the mass accretion rate in units of 10 -8 M ⊙ .yr -1 , M ⋆ the stellar mass in units of 0.8 M ⊙ , and R ⋆ the stellar radius in units of 2 R ⊙ . As ˙ Macc in quiescence, we used the value before the 2022 outburst given by Wang et al. (2023), 1.8 × 10 -9 M ⊙ .yr -1 . The stellar mass and radius are given by Gras-Velázquez & Ray (2005) (0.6 M ⊙ and 1.6 R ⊙ ), and magnetic obliquity is obtained from ZDI analysis (30 · for ESPaDOnS, 23 · for SPIRou, see Sect. 3.3.1). The dipole strengths needed to obtain r mag = r corot = 8.5 ± 0.5 R ⋆ are thus 2.10 ± 0.46 kG and 1.98 ± 0.43 kG for ESPaDOnS and SPIRou, respectively. Only an estimate of the truncation radius can be obtained because we do not have a precise value of the dipole strength. Indeed, the B ℓ measurements in the He i D3 line and Zeeman intensification values also contain the higher-order components of the magnetic field, and the ZDI spherical harmonic decomposition from LSD might include flux cancellation lowering the large-scale strength obtained. Even if the ZDI results on the 2019 datasets indicate a topology largely dominated by the dipole component, using these values should yield an overestimation of r mag. The results using our magnetic field measurements are summarised in Table 5. The values obtained from the B ℓ and the small-scale field are not consistent with the corotation radius, as expected, except for the SPIRou measurement which can be explained by the smaller recovered field in the infrared domain. However, the optical ZDI values yield truncation radii consistent with the corotation radius, due to the dipole-dominant magnetic topology of the system allowing us to recover a good estimate of the dipole pole strength using ZDI. Here again, the SPIRou value yields a lower truncation radius due to the lower magnetic field strength obtained. \nThe di ff erence in the results of the various magnetic analyses between the optical and infrared frames in 2019 needs to be discussed. From the B ℓ computed from the LSD profiles, the maximum values obtained are 911 ± 26 and 64 ± 20 G, for ESPaDOnS and SPIRou observations, respectively. These maxima both occurred around ϕ = 0.6, where the Stokes V signature is maximal with ESPaDOnS but almost vanishes with SPIRou, it is thus not surprising to have a large discrepancy here. Concerning the minimum B ℓ values, they reach -155 ± 11 and -178 ± 13 G, for ESPaDOnS and SPIRou observations, respectively, both around ϕ = 0.1 and are thus consistent. This behaviour is also visible on the ZDI reconstruction, where the strong positive radial magnetic field region at phase 0.6 on ESPaDOnS map completely vanishes on SPIRou map. The two qualitative explanations we can provide are the following: (i) this maximum value is located in an optically dark region of the photosphere, which is consistent with simultaneous optical photometry (Kóspál et al., in \n<!-- image --> \n<!-- image --> \nFig. 14. Brightness maps from ESPaDOnS 2016, 2019, and SPIRou data sets ( left , middle , right , respectively) on a flattened polar view. The central dot is thus the pole, the two dotted circles are latitude 60 and 30 · , and the solid circle represents the equator. The black ticks are the rotation phases going clockwise, and the red ticks represent the observed phases. The colour code indicates the brightness on a linear scale, where a value of 1.0 represents the quiet photosphere, values less than one are darker regions and values greater than one are bright. \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFig. 15. Radial (top row) , azimuthal (middle row) , and meridional (bottom row) magnetic maps from ESPaDOnS 2016, 2019, and SPIRou data sets ( left , middle , and right columns, respectively) on the same flattened polar view as Fig. 14. The colour code is scaling the magnetic field strength, going from dark blue for the strongest negative value to dark red for the strongest positive value. \n<!-- image --> \nTable 4. Inference results on the small-scale magnetic field on Ti i lines. \n. \n. \nNotes. The fi denote the filling factor for a i kG magnetic field strength. \nI \n/ \nI \nESPaDOnS 2019 \nFig. 16. Results of the ESPaDOnS Zeeman intensification analysis. Top: Distribution of the average magnetic field of EX Lup for 2016 (left) and 2019 (right) data sets. The solid red line represents the median and the dashed lines represent the 68 % confidence regions. Bottom: Fit to 2016 (left) and 2019 (right) optical Ti i multiplet of EX Lup. The solid red line shows the best fit to the observations in black, while the blue dashed line shows the non-magnetic spectra with otherwise identical stellar parameters. The shaded area marks the region used for the fit. \n<!-- image --> \nTable 5. Magnetospheric truncation radius obtained using the various magnetic field measurements of this work. \nprep.), and thus less contrasted in the SPIRou domain. The magnetic contribution of this region to the Stokes V signal might be thus lowered in the infrared domain if the smaller-scale negative field is obscured in the optical domain. And (ii) the two wavelength domains allowing to trace di ff erent heights in the photosphere, that such a di ff erence might be an indication of a vertical structure of the magnetic field. \nConcerning the two ZDI reconstructions obtained from the two ESPaDOnS epochs, they seem to point a similar brightness and magnetic radial maps even if the overall topology has evolved from a toroidal- to a poloidal-dominated state, as seen on \nprevious objects (e.g., DQ Tau A, Pouilly et al. 2023, 2024). The strong positive radial field spot, associated with the dark spot of the stellar brightness and with the dipole pole, is also consistent with the modulation of He i D3 NC radial velocity and B ℓ . However, the latter is pointing to an accretion shock associated with a region with a strong negative field. Given the large amplitude of the He i D3 NC radial velocity variation compared to EX Lup v sin i , the emitting region has probably a very small extent, meaning that its magnetic field information is lost at the photospheric level, explaining this disparity (Yadav et al. 2015). \nDespite this stable pattern between 2016 and 2019, we observed some disparities in some parameters. Even if the radial velocity and the B ℓ (from LSD and He i D3) modulations are well in phase, their amplitudes are slightly larger in 2019. If the stronger magnetic field obtained in the optical frame, at smalland large-scale, explains the larger amplitude of the B ℓ modulation, the accompanying e ff ect on the radial velocity points to a modulation by the hotspot. This is not a surprising behaviour, but the consistency between the ESPaDOnS and SPIRou radial velocity measurements is. Indeed, the stellar activity e ff ect on the photospheric lines, producing the apparent radial velocity modulation, is a wavelength-dependent phenomenon, an amplitude consistency between the radial velocity measurements in the optical and infrared frames is thus not expected. \nFig. 17. Same as Fig. 16 for SPIRou IR Ti i multiplet. \n<!-- image --> \nTo investigate in our data sets the relation between the stellar activity and the radial velocity modulation, we compared the latter to an activity indicator, the bisector inverse slope (BIS, Queloz et al. 2001). The BIS was computed from the LSD profiles presented in Fig. 10, 11, and 12, and is defined as the difference between the mean velocity of the bisector at the top and at the bottom of the line. On ESPaDOnS profiles, the first 15%, which contains the continuum and the wings, were ignored, as well as the last 15%, where the noise or an activity signature splitting the profile into two parts can a ff ect the computation. The top and bottom regions used to compute the BIS are the 25% top and bottom parts of the remaining profile. For SPIRou profiles, the same conditions were used except that we had to ignore the first 25% of the profiles as it was a ff ected by stronger wings. In the case where the radial velocity modulation is only induced by the stellar activity, the line deformation, indicated by the BIS, is completely responsible for this modulation. This means that a strong linear correlation should appear between the BIS and the radial velocity with, in a perfect situation, a -1-slope and a BIS = 0 kms -1 at the mean velocity. \nThe BIS versus radial velocity plots are presented in Appendix C. For each data set, the Pearson correlation coe ffi cient indicates a strong anti-correlation (2016: r = -0.81, p-value = 0.005; ESPaDOnS 2019: r = -0.95, p-value = 0.004; SPIRou: r = -0.72, p-value = 0.05). However, the optical measurements show much shallower slopes ( -0.51 ± 0.13 and -0.43 ± 0.07 in 2016 and 2019, respectively), and an intersect far from the mean velocity expected ( -0.34 ± 0.10 and \n-0.08 ± 0.06). Only the infrared measurements are consistent (slope: -0.68 ± 0.27, intersect: -1.37 ± 0.25), but only thanks to the larger uncertainties, probably due to the lower correlation. Furthermore, the much lower BIS obtained (exclusively negative) in the infrared compared to the optical frame, indicates chromatic e ff ects that are not observed in the radial velocity modulation. Stellar activity is thus probably dominating the radial velocity modulation, but another e ff ect, such as Doppler shift induced by a companion, can not be completely excluded and needs further investigations that are unfortunately beyond the scope of the present work. \nFinally, we would like to address the strong magnetic field recovered that drives the accretion process on EX Lup. With the B ℓ reaching 4.2 kG in the accretion shock, a dipole strength of 1.9 kG within a large-scale field of 1 kG reaching 4.8 kG locally, and a small-scale field exceeding 3 kG in the optical domain, EX Lup has one of the strongest magnetic field among cTTSs, and the strongest among those with dipole-dominated topology. Such a configuration can set the suitable condition invoked by D'Angelo & Spruit (2010) for their hypothesis of an episodic accretion due to the magnetospheric accretion process itself. Indeed, with such a dipolar strength, the magnetospheric radius is outside but close to the corotation radius (see Table 5). In such a situation, instability arises due to the fact that angular momentum is transferred from the star to the disc (the so-called propeller regime), without being enough to drive an outflow. This magnetic interaction only prevents accretion, pilling up the gas in the inner disc and increasing its pressure, thus forcing the inner edge of the disc to move inward and cross the corotation radius allowing the accretion to occur. Once the gas reservoir has been accreted, the inner edge of the disc moves outward and another cycle starts (see D'Angelo & Spruit 2010, 2011, 2012). This phenomenon is di ff erent from the magnetospheric inflation reported for other cTTSs (e.g., see the studies of AA Tau or V807 Tau by Bouvier et al. 2003; Pouilly et al. 2021, respectively), where a significant di ff erence between the truncation and corotation radii induces a torsion of the magnetic field lines producing a toroidal field that inflates the whole magnetosphere. This inflation goes up to an opening of the magnetic field lines that produces a magnetospheric ejection (Zanni & Ferreira 2013; Pantolmos et al. 2020) before reconnecting to the disc. Such a phenomenon is recurrent and produces an accretion variability as well. Still, the time scale of such a cycle is of the order of the stellar rotation period, far shorter than the gas pilling up invoked by D'Angelo & Spruit (2010), and no signature of magnetospheric inflation, nor ejection, were detected in EX Lup. However, we would like to stress to the reader that no accumulation of matter at the magnetospheric radius was detected either, EXLup seems only to be in the same initial conditions as the latter authors' theory. In addition, the BIS suggest another source for radial velocity variation, such as a companion. Tidal interactions (Bonnell & Bastien 1992) or thermal instabilities in the disc (Lodato & Clarke 2004) induced by a companion are also existing hypotheses for EXor behaviour. Moreover, recent works by Nayakshin et al. (2024b,a) favour the latter instability as the origin of the episodic accretion of FUor objects. Finally, the disc itself is not investigated in this work, instabilities in the disc are thus hypotheses that cannot be excluded (Bell & Lin 1994; Armitage et al. 2001).", '5. Conclusions': "EX Lup is the prototypical EXor-type object whose recurrent bursts and outbursts were previously studied in detail using spec- \ntroscopy and photometry. However, until this work, no information about its magnetic field was derived, despite its key role in the accretion process of cTTS and its possible origin for episodic accretion. Here we provide the first spectropolarimetric time series, over two epochs (2016 and 2019) and two wavelength domains (optical and infrared), study of EX Lup, the first EXor whose magnetic field is studied. \nWe confirmed an ongoing magnetospheric accretion process as seen on many cTTSs. It is represented by an accretion funnel flow and an accretion shock corotating with the stellar surface and driven by a kG dipolar magnetic field. The funnel flow seems aligned with the accretion shock, itself located near the magnetic dipole pole, which is consistent with the stable pattern observed between the epochs studied. \nThe magnetic field of EX Lup has shown some disparities between wavelength domains, much weaker in the infrared. This can be understood as a wavelength dependency of the parameters studied, but can also point to a vertical structure of the magnetic field, as di ff erent wavelengths are tracing di ff erent heights in the photosphere. An expected small-to-large scale e ff ect is also observed, by the di ff erent field strengths recovered using ZDI and Zeeman intensification, but also by the opposite polarity of the field associated with the accretion shock and the one recovered using LSD. This indicates a very small accretion shock, as expected from the low mass accretion rate in quiescence and the large radial velocity variation of the emitting region. \nFinally, the multi-kG field recovered for EX Lup is pointing to a magnetospheric radius being close but outside the corotation radius. This configuration is suitable for disc instabilities induced by the magnetic field that yield accretion cycles. These cycles might explain the accretion bursts observed on EX Lup, suggesting an inherently episodic magnetospheric accretion process. However, a definite identification of the origin of EXor behaviour is beyond the scope of this paper, and other theories implying the disc itself or a companion cannot be excluded. \nAcknowledgements. We would like to warmly thanks Oleg Kochukhov for useful discussions about the magnetic field of EX Lup, as well as Colin P. Folsom for his help in using the new version of his ZDIpy package. The SpecpolFlow package is available at https://github.com/folsomcp/ specpolFlow . The PySTEL(L)A package is available at https://github. com/pouillyk/PySTELLA This research was funded in whole or in part by the Swiss National Science Foundation (SNSF), grant number 217195 (SIMBA). For the purpose of Open Access, a CC BY public copyright licence is applied to any Author Accepted Manuscript (AAM) version arising from this submission. Based on observations obtained at the Canada-France- Hawaii Telescope (CFHT) which is operated from the summit of Maunakea by the National Research Council of Canada, the institut National des Sciences de l'Univers of the Centre National de la Recherche Scientifique of France, and the University of Hawaii. The observations at the Canada-France-Hawaii Telescope were performed with care and respect from the summit of Maunakea which is a significant cultural and historic site. 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R., Morin, J., et al. 2015, ApJ, 813, L31\n- Zanni, C. & Ferreira, J. 2013, A&A, 550, A99", 'Appendix A: Cross-correlation matrices of optical emission lines': 'Here we present the cross-correlation matrices of the ESPaDOnS emission lines studied in this work. These matrices are discussed in Sect. 3.2.5. \nFig. A.1. He i D3 vs. H α (top) , Ca ii IRT (854.2 nm) vs. H α (middle) , and Ca ii IRT (854.2 nm) vs. He i D3 (bottom) correlation matrices for ESPaDOnS 2016 emission lines. \n<!-- image --> \nFig. A.2. Same as Fig. A.1 for ESPaDOnS 2019 emission lines. \n<!-- image -->', 'Appendix B: Stokes I and V fit from ZDI reconstruction': 'In this section, we show the fit of the LSD profiles by the ZDI reconstruction for the three data sets studied. The Stokes I profiles are shown in Fig. B.1 and the Stokes V profiles in Fig. B.2. \nFig. B.1. Fit (red) of the observed (black) Stokes I profiles from the ZDI reconstruction for ESPaDOnS 2016 (left) , 2019 (middle) and SPIRou (right) observations. The grey-shaded area corresponds to the uncertainties in the observation, and the number at the right of each profile indicates its rotation cycle. \n<!-- image --> \nFig. B.2. Same as Fig. B.1 for Stokes V profiles. \n<!-- image -->', 'Appendix C: BIS vs. radial velocity': 'This appendix presents the analysis of the BIS and radial velocity correlation. These results are discussed in Sect. 4. \nFig. C.1. BIS vs. RV of ESPaDOnS 2016 (top) , 2019 (middle) , and SPIRou (bottom) observations. The colour code scales the HJD of observations. The red dotted line shows the best linear regression with slope "a" and intercept "b" indicated in legend. The uncertainty on the regression is shown as the grey-shaded area The green dotted line has a slope of -1 and an intercept equal to the mean radial velocity. \n<!-- image -->'}

2023ApJ...946L..13F

We present an investigation into the first 500 Myr of galaxy evolution from the Cosmic Evolution Early Release Science CEERS survey. CEERS one of 13 JWST ERS programs targets galaxy formation from z 0.5 to gt10 using several imaging and spectroscopic modes. We make use of the first epoch of CEERS NIRCam imaging spanning 35.5 arcminSUP2SUP to search for candidate galaxies at z gt 9. Following a detailed data reduction process implementing several custom steps to produce highquality reduced images we perform multiband photometry across seven NIRCam broad and mediumband and six Hubble broadband filters focusing on robust colors and accurate total fluxes. We measure photometric redshifts and devise a robust set of selection criteria to identify a sample of 26 galaxy candidates at z 916. These objects are compact with a median halflight radius of 0.5 kpc. We present an early estimate of the z 11 restframe ultraviolet UV luminosity function finding that the number density of galaxies at M SUBUVSUB 20 appears to evolve very little from z 9 to 11. We also find that the abundance surface density arcminSUP2SUP of our candidates exceeds nearly all theoretical predictions. We explore potential implications including that at z gt 10 star formation may be dominated by topheavy initial mass functions which would result in an increased ratio of UV light per unit halo mass though a complete lack of dust attenuation andor changing star formation physics may also play a role. While spectroscopic confirmation of these sources is urgently required our results suggest that the deeper views to come with JWST should yield prolific samples of ultrahighredshift galaxies with which to further explore these conclusions.

2023-03-01T00:00:00Z

['2022arXiv221105792F', '2023ApJ...946L..13F', 'arXiv:2211.05792', '10.48550/arXiv.2211.05792', '10.3847/2041-8213/acade4']

['Early universe', 'Galaxy formation', 'Galaxy evolution', 'High-redshift galaxies', '435', '595', '594', '734', 'Astrophysics - Astrophysics of Galaxies']

CEERS Key Paper. I. An Early Look into the First 500 Myr of Galaxy Formation with JWST

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']

https://arxiv.org/pdf/2211.05792.pdf

{'An Early Look into the First 500 Myr of Galaxy Formation with JWST': "Steven L. Finkelstein, 1 Micaela B. Bagley, 1 Henry C. Ferguson, 2 Stephen M. Wilkins, 3, 4 Jeyhan S. Kartaltepe, 5 Casey Papovich, 6, 7 L. Y. Aaron Yung, 8, ∗ Pablo Arrabal Haro, 9 Peter Behroozi, 10, 11 Mark Dickinson, 9 Dale D. Kocevski, 12 Anton M. Koekemoer, 13 Rebecca L. Larson, 14, 1 Aur'elien Le Bail, 15 Alexa M. Morales, 1 Pablo G. P'erez-Gonz'alez, 16 Denis Burgarella, 17 Romeel Dav'e, 18, 19 Michaela Hirschmann, 20, 21 Rachel S. Somerville, 22 Stijn Wuyts, 23 Volker Bromm, 1 Caitlin M. Casey, 1 Adriano Fontana, 24 Seiji Fujimoto, 1, 25, 26, † Jonathan P. Gardner, 27 Mauro Giavalisco, 28 Andrea Grazian, 29 Norman A. Grogin, 2 Nimish P. Hathi, 2 Taylor A. Hutchison, 8, ∗ Saurabh W. Jha, 30 Shardha Jogee, 1 Lisa J. Kewley, 31 Allison Kirkpatrick, 32 Arianna S. Long, 1, † Jennifer M. Lotz, 33 Laura Pentericci, 24 Justin D. R. Pierel, 2 Nor Pirzkal, 34, 2 Swara Ravindranath, 2 Russell E. Ryan Jr., 2 Jonathan R. Trump, 35 Guang Yang, 36, 37 Rachana Bhatawdekar, 38 Laura Bisigello, 39, 29 V'eronique Buat, 17 Antonello Calabr'o, 24 Marco Castellano, 24 Nikko J. Cleri, 6, 7 M. C. Cooper, 40 Darren Croton, 41, 42 Emanuele Daddi, 15 Avishai Dekel, 43 David Elbaz, 15 Maximilien Franco, 1 Eric Gawiser, 30 Benne W. Holwerda, 44 Marc Huertas-Company, 45, 46, 47 Anne E. Jaskot, 48 Gene C. K. Leung, 1 Ray A. Lucas, 49 Bahram Mobasher, 50 Viraj Pandya, 51, † Sandro Tacchella, 52, 53 Benjamin J. Weiner, 54 and Jorge A. Zavala 55 \n1 Department of Astronomy, The University of Texas at Austin, Austin, TX, USA \n2 Space Telescope Science Institute, Baltimore, MD, USA \n- 3 Astronomy Centre, University of Sussex, Falmer, Brighton BN1 9QH, UK\n- 4 Institute of Space Sciences and Astronomy, University of Malta, Msida MSD 2080, Malta \n5 Laboratory for Multiwavelength Astrophysics, School of Physics and Astronomy, Rochester Institute of Technology, 84 Lomb Memorial Drive, Rochester, NY 14623, USA \n6 Department of Physics and Astronomy, Texas A&M University, College Station, TX, 77843-4242 USA \n7 George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy, Texas A&M University, College Station, TX, 77843-4242 USA \n8 \nAstrophysics Science Division, NASA Goddard Space Flight Center, 8800 Greenbelt Rd, Greenbelt, MD 20771, USA \n- 9 NSF's National Optical-Infrared Astronomy Research Laboratory, 950 N. Cherry Ave., Tucson, AZ 85719, USA \n10 Department of Astronomy and Steward Observatory, University of Arizona, Tucson, AZ 85721, USA \n11 \nDivision of Science, National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan \n12 Department of Physics and Astronomy, Colby College, Waterville, ME 04901, USA \n13 Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD 21218, USA \n14 NSF Graduate Fellow \n15 Universit'e Paris-Saclay, Universit'e Paris Cit'e, CEA, CNRS, AIM, 91191, Gif-sur-Yvette, France \nCentro de Astrobiolog'ıa (CAB), CSIC-INTA, Ctra. de Ajalvir km 4, Torrej'on de Ardoz, E-28850, Madrid, Spain \n17 Aix Marseille Univ, CNRS, CNES, LAM Marseille, France \n18 Institute for Astronomy, University of Edinburgh, Blackford Hill, Edinburgh, EH9 3HJ UK \n19 Department of Physics and Astronomy, University of the Western Cape, Robert Sobukwe Rd, Bellville, Cape Town 7535, South Africa 20 Institute of Physics, Laboratory of Galaxy Evolution, Ecole Polytechnique F'ed'erale de Lausanne (EPFL), Observatoire de Sauverny, 1290 Versoix, Switzerland \n21 INAF, Osservatorio Astronomico di Trieste, Via Tiepolo 11, 34131 Trieste, Italy \n22 Center for Computational Astrophysics, Flatiron Institute, 162 5th Avenue, New York, NY, 10010, USA \n23 Department of Physics, University of Bath, Claverton Down, Bath BA2 7AY, UK \n24 INAF - Osservatorio Astronomico di Roma, via di Frascati 33, 00078 Monte Porzio Catone, Italy \n25 Cosmic Dawn Center (DAWN), Jagtvej 128, DK2200 Copenhagen N, Denmark \n- 26 Niels Bohr Institute, University of Copenhagen, Lyngbyvej 2, DK2100 Copenhagen Ø, Denmark\n- 27 Astrophysics Science Division, Goddard Space Flight Center, Code 665, Greenbelt, MD 20771, USA\n- 28 University of Massachusetts Amherst, 710 North Pleasant Street, Amherst, MA 01003-9305, USA\n- 29 INAF-Osservatorio Astronomico di Padova, Vicolo dell'Osservatorio 5, I-35122, Padova, Italy \n30 Department of Physics and Astronomy, Rutgers, the State University of New Jersey, Piscataway, NJ 08854, USA \nstevenf@astro.as.utexas.edu \n16", 'Finkelstein et al.': "31 Center for Astrophysics - Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA 32 Department of Physics and Astronomy, University of Kansas, Lawrence, KS 66045, USA 33 Gemini Observatory/NSF's National Optical-Infrared Astronomy Research Laboratory, 950 N. Cherry Ave., Tucson, AZ 85719, USA 34 ESA/AURA \n- 35\n- 36 37 SRON Netherlands Institute for Space Research, Postbus 800, 9700 AV Groningen, The Netherlands\n- Department of Physics, 196 Auditorium Road, Unit 3046, University of Connecticut, Storrs, CT 06269, USA Kapteyn Astronomical Institute, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands \n38 European Space Agency, ESA/ESTEC, Keplerlaan 1, 2201 AZ Noordwijk, NL \n- 39 Dipartimento di Fisica e Astronomia 'G.Galilei', Universit'a di Padova, Via Marzolo 8, I-35131 Padova, Italy\n- 40 Department of Physics & Astronomy, University of California, Irvine, 4129 Reines Hall, Irvine, CA 92697, USA \n41 \nCentre for Astrophysics & Supercomputing, Swinburne University of Technology, Hawthorn, VIC 3122, Australia \n42 \nARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D) \nRacah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel \n43 \n44 \nPhysics & Astronomy Department, University of Louisville, 40292 KY, Louisville, USA \n45 \nInstituto de Astrof'ısica de Canarias, La Laguna, Tenerife, Spain \n46 Universidad de la Laguna, La Laguna, Tenerife, Spain \n- 47 Universit'e Paris-Cit'e, LERMA - Observatoire de Paris, PSL, Paris, France\n- 48 Department of Astronomy, Williams College, Williamstown, MA, 01267, USA\n- 49 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA \nDepartment of Physics and Astronomy, University of California, 900 University Ave, Riverside, CA 92521, USA \n- 51 Columbia Astrophysics Laboratory, Columbia University, 550 West 120th Street, New York, NY 10027, USA \n52 \nKavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, UK \n- 53 Cavendish Laboratory, University of Cambridge, 19 JJ Thomson Avenue, Cambridge, CB3 0HE, UK \n54 \nMMT/Steward Observatory, University of Arizona, 933 N. Cherry Ave., Tucson, AZ 85721, USA \n55 \nNational Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan", 'ABSTRACT': 'We present an investigation into the first 500 Myr of galaxy evolution from the Cosmic Evolution Early Release Science (CEERS) survey. CEERS, one of 13 JWST ERS programs, targets galaxy formation z ∼ 0.5 to z > 10 using several imaging and spectroscopic modes. We make use of the first epoch of CEERS NIRCam imaging, spanning 35.5 sq. arcmin, to search for candidate galaxies at z > 9. Following a detailed data reduction process implementing several custom steps to produce high-quality reduced images, we perform multi-band photometry across seven NIRCam broad and medium-band (and six Hubble broadband) filters focusing on robust colors and accurate total fluxes. We measure photometric redshifts and devise a robust set of selection criteria to identify a sample of 26 galaxy candidates at z ∼ 9-16. These objects are compact with a median half-light radius of ∼ 0.5 kpc. We present an early estimate of the z ∼ 11 rest-frame ultraviolet (UV) luminosity function, finding that the number density of galaxies at M UV ∼ -20 appears to evolve very little from z ∼ 9 to z ∼ 11. We also find that the abundance (surface density [arcmin -2 ]) of our candidates exceeds nearly all theoretical predictions. We explore potential implications, including that at z > 10 star formation may be dominated by top-heavy initial mass functions, which would result in an increased ratio of UV light per unit halo mass, though a complete lack of dust attenuation and/or changing star-formation physics may also play a role. While spectroscopic confirmation of these sources is urgently required, our results suggest that the deeper views to come with JWST should yield prolific samples of ultra-high-redshift galaxies with which to further explore these conclusions. \nKeywords: early universe - galaxies: formation - galaxies: evolution', '1. INTRODUCTION': "The epoch of reionization marks the period when energetic photons (presumably from massive stars in early galaxies; e.g., Stark 2016; Finkelstein et al. 2019; Robertson 2021) ionized the gas in the intergalactic medium (IGM). Understanding when and how this pro- \n50 \ncess occurs is crucial to constraining both the earliest phases of galaxy formation (which kick-started this process), and how the evolution of the IGM temperature affects the star-formation efficiency in low-mass halos throughout (and after) this transition. \nAdvances in deep near-infrared (near-IR) imaging with the Hubble Space Telescope ( HST ) have pushed constraints on galaxy evolution into the first billion years after the Big Bang. Studies of public blank-field surveys including the Hubble Ultra Deep Field [HUDF; Beckwith et al. 2006; Oesch et al. 2010; Bouwens et al. 2010], the Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey [CANDELS; Grogin et al. 2011; Koekemoer et al. 2011], the Hubble Frontier Fields [HFF; Lotz et al. 2017], and the Brightest of Reionizing Galaxy survey [BoRG; Trenti et al. 2011] have uncovered thousands of galaxies at z > 6 (e.g. Bouwens et al. 2015; Finkelstein et al. 2015; Ishigaki et al. 2015; McLeod et al. 2016; Oesch et al. 2018; Morishita et al. 2018; Bridge et al. 2019; Rojas-Ruiz et al. 2020; Finkelstein et al. 2022a; Bouwens et al. 2022a; Bagley et al. 2022a). The evolution of the rest-frame ultraviolet (UV) luminosity function has been well studied to z ∼ 8 (e.g. Finkelstein et al. 2015; Bouwens et al. 2015), with some constraints placed at z ∼ 9 and 10 (e.g., Oesch et al. 2018; Bouwens et al. 2019; Finkelstein et al. 2022a; Bagley et al. 2022a). However, little was known about the z > 10 universe prior to JWST , beyond the unexpected discovery of an exceptionally bright z = 10.957 galaxy (Oesch et al. 2016; Jiang et al. 2021). This knowledge gap is due to the modest light-gathering power of the 2.4m HST and the fact that at z > 10 galaxies become one-band (F160W; HST /WFC3's reddest filter) detections. Rest-UV emission from galaxies completely redshifts out of HST observability at z ≳ 12.5. \nThis has left a major gap in our knowledge of galaxy formation at early times. Do galaxies form stars fairly inefficiently, like our own Milky Way, and build up slowly? Or is star formation in the early Universe more rapid due to high gas densities and frequent interactions? Equally exciting and unknown, does the initial mass function (IMF) begin to show signs of evolution? Models predict top-heavy IMFs should dominate at very low metallicities (e.g. Bromm & Larson 2004), so observations should begin to see such signatures. Answering these questions about the physical processes dominating the earliest star formation requires detailed observations of the earliest galaxies to form in our universe, and JWST was designed to push our cosmic horizons to the highest redshifts. The 7 × larger light-gathering power combined with the large field-of-view and near-infrared sensitivity of NIRCam (Rieke et al. 2005) sets the stage \nfor major advances in our ability to study early galaxy formation. Cycle 1 of JWST includes several programs encompassing 100's of hours which all have the early Universe as their primary science goal. \nIndeed in just the days-to-weeks after the first science data were released, several papers were submitted discussing the detection of objects not only at the expected redshifts of z ∼ 10-11 (e.g., Castellano et al. 2022; Naidu et al. 2022b; Adams et al. 2022; Whitler et al. 2022; Labbe et al. 2022), but with some candidates at z ∼ 12-17 (e.g. Finkelstein et al. 2022b; Donnan et al. 2022; Harikane et al. 2022). The existence of galaxies at such early times, and especially at such bright magnitudes for some sources, could potentially challenge early models of galaxy formation (e.g. Mason et al. 2022; Ferrara et al. 2022; Boylan-Kolchin 2022). However, these studies originally relied on very early photometric calibration; subsequent calibration data shifted the photometric zeropoints significantly (Boyer et al. 2022). Now that the flux calibration, and overall data reduction pipeline has stabilized, it is prudent to take a detailed look at what constraints we can place on this early epoch. \nHere we use the first epoch of data from the Cosmic Evolution Early Release Science Survey (CEERS; survey description to come in Finkelstein et al., in prep). CEERS was designed in part to provide our earliest detailed glimpse into the z > 10 universe, and these CEERS data were among the first Cycle 1 science exposures taken, included in the first publicly released data on 14 July 2022. We search these data for z ≳ 9 galaxy candidates, heretofore difficult (if not impossible) to find with HST . We place an emphasis on building a robust sample via a detailed photometric cataloging process, coupled with stringent selection criteria, both backed by simulations. Section 2 describes the observations and data reduction, while Section 3 describes our photometry and photometric redshift measurements, and Section 4 discusses our sample selection procedure. We describe our sample in Section 5, and present a comparison to other early samples in Section 6. In Section 7 we present the z ∼ 11 UV luminosity function and the cumulative surface density of early galaxies, and discuss implications on the physics dominating galaxy formation at the earliest times in Section 8. We summarize our results and present our conclusions in Section 9. In this paper we assume the latest Planck flat ΛCDM cosmology with H 0 = 67.36 km s -1 Mpc -1 , Ω m = 0.3153, and Ω Λ = 0.6847 (Planck Collaboration et al. 2020). All magnitudes are in the absolute bolometric system (AB; Oke & Gunn 1983). \nCEERS is one of 13 early release science surveys designed to obtain data covering several scientific themes of astronomy early in Cycle 1, along with testing out multiple instrument modes and providing early public data to the community. CEERS consists of a mosaic of 10 NIRCam pointings in the CANDELS Extended Groth Strip (EGS) field, with six obtained in parallel with prime NIRSpec observations, and four in parallel with prime MIRI observations (four of these pointings also include NIRCam wide-field slitless grism spectroscopy). Here we use imaging data from NIRCam obtained during the first epoch of CEERS, during June 21-22, 2022. This consists of short and longwavelength images in both NIRCam A and B modules, taken over four pointings, labeled NIRCam1, NIRCam2, NIRCam3, and NIRCam6. Each pointing was observed with seven filters: F115W, F150W, and F200W on the short-wavelength side, and F277W, F356W, F410M, and F444W on the long-wavelength side.", '2.1. Data Reduction': "The NIRCam images used here are those released with the first CEERS public data release (Data Release 0.5), which are fully described in Bagley et al. (2022c). Here we briefly highlight the key features of the data reduction, directing the reader to Bagley et al. (2022c) for more details. \nWe reduce the raw NIRCam imaging through version 1.7.2 of the JWST Calibration Pipeline, with custom modifications designed to correct for additional features in the data. We use the calibration reference file context 1 pipeline mapping (pmap) 0989. We begin by running Stage 1 of the calibration pipeline, which performs detector-level corrections and outputs a countrate image in units of counts/s, adopting all default parameters for this stage. We then perform custom corrections to flag and remove snowballs from all exposures, subtract off the large-scale wisps in F150W and F200W using wisp templates created by the NIRCam team, and measure and remove 1 /f noise via a median measured (amplifierby-amplifier) along rows and columns. The images are then flat fielded and flux calibrated using Stage 2 of the calibration pipeline, again adopting the default values, to produce images in units of MJy/sr. The pmap 0989 reference files include ground flats that have been corrected for illumination gradients measured with in-flight data, and improved but still preliminary photometric calibration reference files. We find that the flux calibration does a good job of synchronizing the zeropoints \nacross the NIRCam detectors (to within the 2-5% level, Bagley et al. 2022c), but that an additional absolute flux calibration may be required at the few percent level (see § 3.6). These flat and photometric calibration reference files will continue to be improved and updated throughout Cycle 1. \nWe align the images using a custom version of the TweakReg routine, which is designed to register image to an absolute WCS frame by matching sources detecting in each image with those in a reference catalog. Our modified version of the routine uses Source Extractor (hereafter SE ; Bertin & Arnouts 1996) to measure source centroids in each individual image. We then align each image to a reference catalog constructed from an HST F160W mosaic with astrometry tied to Gaia EDR3 (see Section 2.2 and Koekemoer et al. 2011, for details on the methods used for the mosaic construction). We first align images from the same detector but different dithers to each other, allowing for shifts in x and y and achieving an RMS of ∼ 3 -6 mas per source for this relative alignment. We next align all images to the F160W reference image, allowing xy shifts and rotations in the SW images and an additional scaling factor to account for large-scale distortions in the LW images. The RMS of this absolute alignment is ∼ 12 -15 mas and ∼ 5 -10 mas when comparing WFC3 to NIRCam and NIRCam to NIRCam, respectively. We note that we followed a slightly different procedure for NIRCam3, aligning F277W to HST /WFC3 F160W and then using F277W as the new reference for all other NIRCam filters. This altered procedure was required to address additional offsets registered in one portion of the F160W image in this region (see Bagley et al. 2022c, for details). \nFinally, we create mosaics for each pointing in all filters in the following way. We subtract a pedestal value off of each individual image, and scale the readnoise variance maps such that they include an estimate of the robustly-measured sky variance. The mosaics are then created using the Resample routine in Stage 3 of the calibration pipeline, which uses the drizzle algorithm with an inverse variance weighting (Casertano et al. 2000; Fruchter & Hook 2002). We drizzle the images to an output pixel scale of 0 . '' 03/pixel and use the same tangent point as that of the HST mosaics, such that the images in all filters, NIRCam and HST, are pixel-aligned.", '2.2. HST Imaging Data': 'The EGS field has archival HST imaging from the All-wavelength Extended Groth Strip International Survey (AEGIS, (Davis et al. 2007)), the Cosmic Assembly Deep Extragalactic Legacy Survey (CANDELS, Grogin et al. 2011; Koekemoer et al. 2011), 3D-HST (Momcheva \n<!-- image --> \n<!-- image --> \nFigure 1. An example of the results of our background subtraction procedure. The left panel shows a zoom in on the science image of the ALONG module of our F444W mosaic in the CEERS1 pointing. The middle panel is the derived background, and the right panel is the background-subtracted science image. By progressively masking out objects in smaller tiers, this method is able to capture both small and large-scale fluctuations. Full details on this process are available in Bagley et al. (2022c). \n<!-- image --> \net al. 2016), and various followup programs. The entire CEERS field is covered by F606W, F814W, F125W, (shallow 800 s) F140W, and F160W; portions are covered by F105W. For CEERS, we produced an updated version (v1.9) of the CANDELS EGS mosaics 2 specifically aligning their astrometry onto Gaia DR3, and on the same 30 mas pixel scale as our NIRCam images. In each of the HST mosaics, we create smaller cutouts to match the footprints of the drizzled NIRCam mosaics in each pointing. In this way we have pixel-aligned imaging in 12-13 filters per field (NIRCam1 does not include F105W coverage) from ∼ 0 . 5 -5 µ m.', '3. PHOTOMETRY': 'In this section we describe the creation of our HST + JWST catalog. As the focus is on high-redshift galaxies, this catalog is optimized for faint, compact sources. To ensure accurate photometric redshifts, significant attention is paid to calculation of accurate colors and uncertainties.', '3.1. Background Subtraction': 'The procedure adopted to remove background nonuniformities is described in detail by Bagley et al. (2022c). Briefly, it involves several different tiers of source masking, aimed at removing large-scale structures - including the wings of bright, extended galaxies - while preserving small-scale structures (including the wings of faint galaxies). We have found this procedure to be more effective than the built-in background subtraction procedure in SE , and we therefore use these \nimages as inputs to SE ( § 3.3) and disable its backgroundsubtraction step.', '3.2. Point-Spread Function Matching': "The full-width at half-maximum (FWHM) of the NIRCam point-spread function (PSF) varies significantly across the wavelengths of the filters used. As the selection of high-redshift galaxies depends on our ability to measure accurate colors, ensuring a similar fraction of light is measured in all bands is crucial. In this work, we accomplish this by matching the point-spread functions in our images to the F444W filter, as it is the reddest filter and thus has the largest PSF (FWHM=0.161 '' ). \nOur procedure for PSF matching follows Finkelstein et al. (2022a), which we summarize here. In each filter, we create a preliminary photometric catalog made using Source Extractor v2.25.0 (Bertin & Arnouts 1996), and identify potential PSF stars by searching for the stellar locus in a plane of half-light radius versus source magnitude (making custom cuts in both quantities for each filter). We excluded objects with neighbors within 50 pixels with magnitudes brighter than the magnitude of the star in question minus one. As stars were more difficult to identify in some bands (e.g., F115W, and the shallower F410M), we combined stars identified in the short-wavelength and long-wavelength channel bands to one list for each channel, which resulted in typically 1020 stars per NIRCam pointing. We then visually inspected each star in each image, removing stars near detector edges, other defects, or with close neighbors not excluded by the previous cut, to generate one star list per filter per pointing. \nPSFs were then generated by stacking stars that passed this inspection. As our observations in all four \npointings utilized the same dither pattern and were taken at a similar time, we create one PSF per filter by stacking all stars over all four pointings, increasing the signal-to-noise ratio of our PSF. For each star, we extract a 101x101 pixel box, upsample by a factor of 10, measure the centroid, and shift the star to be centered in this upsampled image. We then downsample back to the native resolution, rotate the star by a random position angle (to account for situations when the position angle of the observations was not identical), and normalize the star's peak flux to unity. The final PSF was made by median-combining the individual stars. The final PSFs have a centroiding accuracy of ∼ 0.05-0.1 pixels. \nKernels to match bluer PSFs to F444W were created with the pypher Python routine 3 (Boucaud et al. 2016), and these bluer images were then convolved with their respective kernels. We included the HST /ACS F606W and F814W imaging in this PSF-matching process, as their PSF FWHMs are smaller than that of F444W. However, the HST /WFC3 band PSF FWHMs are larger than F444W, so we do not convolve these images. This will necessitate a correction accounting for the lower fraction-of-flux encompassed in a given aperture in these filters, which we discuss in § 3.3.1. \nWe tested our PSF-matching process by measuring curves-of-growth of the PSF stars in the images. We find that the median enclosed flux at an aperture diameter of 0.3 '' was within 5% (and often less) of the F444W value for all filters (while prior to this PSF matching process, the bluer filter measurements encompassed ∼ 20% more light at this radius). We provide the median FWHM values in each filter in Table 1.", '3.3. Source Extraction': "We use SE in two-image mode to measure accurate photometry for each of our four pointings. SE requires a detection image to identify sources. We elect to use the inverse-variance-weighted sum of the PSF-matched F277W and F356W images as our detection image, to better detect faint sources. We do not include F200W in this stack as the Ly α break enters this filter at z = 13.4, and we do not wish to bias our catalog against extreme redshift galaxies (the blue edge of F277W corresponds to a Ly α break redshift of z = 18.9), while the inclusion of F444W could have potentially begun to bias against very blue sources. \nUsing this detection image, we run SE cycling through the seven NIRCam images and six HST images as the measurement image. The key source detection parameters were initially optimized using the CEERS simulated imaging 4 (Bagley et al. 2022c), and further tweaked by inspecting their performance on the final mosaics. The parameters we used best recovered faint sources while minimizing contamination by spurious objects. These key parameters were: DETECT THRESH=1.3, DETECT MINAREA=5 pixels, and a top-hat convolution kernel with a width of 4 pixels. \nWe forced SE to skip the background subtraction step as this was previously removed ( § 3.1). We use MAP RMS for the source weighting. As the pipelineproduced ERR images include Poisson noise, they are not appropriate for source detection. We thus convert the weight map associated with the detection image into an effective rms map by taking 1/sqrt(WHT), and assign this to the detection image. For the measurement image, we use the pipeline ERR image. \nFollowing previous work (e.g., Finkelstein et al. 2022a) we measure colors in small elliptical apertures, which has been shown to accurately recover colors of distant galaxies. In SE these apertures are defined by two parameters - a Kron factor, and a minimum radius. We set these two quantities to (1.1, 1.6). These are the same values found by Finkelstein et al. (2022a) via optimization simulations, and we verified via our own simulations ( § 3.3.1) that the signal-to-noise ratio in these apertures was significantly better than larger parameters, and that gains in signal-to-noise ratio were negligible for smaller values. We estimate an aperture correction to the total flux for these small apertures by performing a second run of SE on the F444W image with the Kron parameters set to the default 'MAG AUTO' parameters of (2.5, 3.5), deriving an aperture correction as the ratio between the flux in this larger aperture to that in the smaller aperture for each object. The median aperture correction across all four fields was 1.5. This aperture correction was then applied multiplicatively to the fluxes and uncertainties for all filters.", '3.3.1. Residual Aperture Correction': "While we use small Kron apertures to derive accurate colors, the aperture correction applied above should yield total fluxes close to the true value. However, several previous studies have noted that the default Kron parameters we use for this aperture correction can miss light in the wings of the PSF (e.g., Bouwens et al. 2015; \nTable 1. Imaging Data Summary \nNote -PSFs were created by stacking stars across all four pointings. For our photometry, we PSF match all filters with FWHM smaller than the F444W PSF FWHM to the F444W PSF. We note the HST imaging used is on the same 30mas pixel scale, which affects the FWHM of the PSF. The limiting magnitude is that measured in a 0.2 '' diameter aperture on the unmatched images, corrected to total based on the PSF flux enclosed in that aperture size, averaged over the four fields. The derived corrections to the photometric zeropoints for each filter were derived using best-fitting EAZY models to ∼ 900 galaxies with secure spectroscopic redshifts. These corrections are due to a combination of residual differences between our estimated total fluxes and true total fluxes, differences between the model templates and true galaxies, and true photometric zeropoint inaccuracies (using the photometric reference files from pmap 0989 for NIRCam), Because these corrections depend specifically on our photometry procedure, they may not be appropriate for other photometric catalogs. \nFinkelstein et al. 2022a), yielding underestimates of the total fluxes at the 5-20% level. \nWe estimate these corrections using source-injection simulations, adding 3000 mock sources to our real images in each field. We add sources from m = 23-27 mag (to ensure a robust photometric measurement), with a log-normal half-light radius distribution peaking at ∼ 1.5 pixels ( ∼ 0.2 kpc at z = 10; compact but modestly resolved, comparable to high-redshift sources, see § 5.4), with a log-normal S'ersic parameter distribution, peaking at 1.2. These mock sources were generated with galfit (Peng et al. 2002), and added at random positions to the F277W, F356W and F444W images. We combined the former two to create a detection image, and ran SE in the same way as on our real data to generate a F444W catalog (focusing on this one band as all images were PSF-matched to F444W). Finally, we match sources in the SE catalog to their input values, and compare the ratio of input-to-recovered fluxes. We find a median ratio of 1.08, measured between 25 < m F 444 W < 26. There is a slight trend with magnitude of lower corrections for brighter sources, and higher for fainter sources, but only \nat the 1-2% level. We thus elect to use a single correction factor of 1.08 to all NIRCam fluxes and uncertainties. \nFor the HST fluxes, we elect to derive any residual aperture correction from comparison to the Finkelstein et al. (2022a) photometric catalog, which performed similar simulations to derive total fluxes. Matching sources in each of the six HST bands, we find a typical needed correction factor of ∼ 1.35 ( ± 0.02). These values are roughly consistent with the combination of the correction derived in Finkelstein et al. (2022a) of 1.20 with the NIRCam correction derived here of 1.08. We apply this same 1.35 correction to all HST bands, such that colors amongst these bands are not changed. We note that in § 3.6 below we test for the presence of any remaining photometric offsets in our catalog, and find these to be small ( ≲ 5%), indicating our procedure for deriving total fluxes in all 13 HST + JWST bands is robust, especially for this nascent observatory.", '3.4. Noise Estimation': "While SE does provide an estimate of the noise, it is reliant on the accuracy of the provided error maps. Given the nascent nature of the JWST reduction pipeline, we \nobtain estimates of our image noise directly from the images themselves. We follow the methods of Finkelstein et al. (2022a), based on previous methodology outlined in Papovich et al. (2016). Our goal is to estimate the noise based on the number of pixels in an aperture. We fit for the noise as a function of aperture size by measuring the fluxes in circular apertures with 30 different diameters, ranging from 0.1 '' (3.33 pixels) to 3 '' (100 pixels). \nWhen defining these positions, we restrict aperture placement to pixels with real non-zero values in the ERR image, and a zero value in the SE segmentation map within the aperture, avoiding real objects. We also require these apertures to be non-overlapping to avoid correlating our noise estimation. To improve statistics for smaller apertures, we placed apertures in two separate iterations - 'small' (d ≤ 1.5 '' ) and 'large' (d > 1.5 '' ). We were able to place 5000 and 500 non-overlapping apertures in these two iterations, respectively. \nWe create a detection image setting the pixels at these positions to unity, with the rest of the image set to zero, and ran SE in two-image mode. We measured fluxes at these positions in all 30 circular apertures with diameters ranging from 1 - 200 pixels. We calculate the 1 σ noise in each aperture size by measuring the median absolute deviation of the measured flux values (multiplying by 1.48 to convert to a Gaussian-like standard deviation). Finally, we fit a curve to the noise in a given aperture as a function of pixels in that aperture, using this equation (Gawiser et al. 2006): \nσ N = σ 1 αN β (1) \nwhere σ N is the noise in an aperture containing N pixels, and σ 1 is the pixel-to-pixel noise measured in each image as the sigma-clipped standard deviation of all non-object pixels (see Figure 3 in Finkelstein et al. 2022a for an example of this process). \nWe fit the free parameters α and β with an IDL implementation of emcee (see Finkelstein et al. 2019 for details). We used these functional form fits for each filter to calculate the photometric uncertainties for each object, using both the number of pixels in its Kron aperture (Area = π × A IMAGE × B IMAGE × KRON RADIUS 2 ), as well as a value for a given circular aperture. These values were scaled by the ratio of the error image value at the central position of a given source to the median error value of the whole map, thereby allowing the noise to be representative of the local noise level. \nFinally, to account for variable image noise not captured by the error image value at the central pixel, for each object in our catalog we also calculate a local \nnoise measurement. This local noise was calculated at 0.2 '' , 0.3 '' , 0.4 '' and 0.5 '' -diameter apertures, by fitting a Gaussian to the negative side of the flux distribution in the 200 closest apertures from the above process.", '3.5. Multi-band Catalog': "We compose a multi-band catalog from the individualfilter catalogs created by SE . As SE cannot parse the world-coordinate system in the JWST data model image headers, we use astropy.wcs wcs pix2world to derive celestial coordinates from the SE x, y positions. We apply a photometric zeropoint to convert the image from MJy sr -1 to erg s -1 cm -2 Hz -1 , and apply both aperture corrections derived above to all flux and flux error estimates. We correct for Galactic extinction using an E(B-V) of 0.006 for the EGS field, and a Cardelli et al. (1989) Milky Way attenuation curve. \nBoth the MAG AUTO-derived aperture corrections and simulation-based (and HST catalog-based) residual aperture corrections were applied to all Kron-derived fluxes and uncertainties, maintaining our accurate colors while representing the total fluxes for each source. In our final catalog, we include both these Kron-based fluxes, as well as fluxes measured in circular apertures with diameters ranging from 0.05 '' to 2.0 '' . While the Kron fluxes will be used for most of the analysis, we do use the circular apertures (with a fiducial diameter of 0.2 '' , or 6.67 pixels) as a measure of detection significance, as the sizes of Kron apertures can be affected by the proximity to bright sources. These circular apertures are corrected for Galactic attenuation, but not corrected to total, as we will use them solely for detection significance. \nFor both the Kron and circular apertures, we calculate the noise per source following the method in § 3.4, which is dependent on both the aperture area and the effective rms map value at the position of the source. Finally, we flag any sources which had either a zero or NaN in any error column, replacing their flux error with 10 12 nJy (several orders of magnitude larger than any real source error). \nIn Table 1, we include an estimate of the limiting 5 σ magnitude for our catalog. To calculate this, we use the noise functions described above to derive the flux density uncertainty in an aperture of diameter 0.2 '' . We then measure the enclosed flux at this radius from the stacked PSF. We then divide the flux uncertainty by the enclosed fraction of flux to estimate the total noise for a point source. Finally, we multiply this value by five, and convert to an AB magnitude. Both the enclosed flux values and the limiting magnitudes are listed in Table 1. While the depths were broadly as expected based \nFigure 2. The total corrections applied to our catalog to account for residual systematic offsets in our total flux estimations, mismatches between the spectral templates used for photometric redshift fitting, and any remaining photometric calibration corrections (shown as squares; small dots are individual galaxies). These multiplicative offsets were derived by comparing the observed fluxes in each filter to the best-fitting EAZY model templates for ∼ 900 sources with robustly known spectroscopic redshifts in our field. The values are tabulated in Table 1. \n<!-- image --> \non the pre-launch exposure-time calculator, the F115W image (with double the exposure time), was expected to be 0.3 mag deeper. The very low background at that wavelength has led to those images being more readnoise dominated than expected, thus this image has a depth comparable to the bulk of the NIRCam filters. The primary impact is that photometric redshifts will be slightly more uncertain at m > 29 than originally planned.", '3.6. Photometric Calibration Validation': "While the photometric calibration of NIRCam has substantially improved over the first few months since science acquisition began, there is still some uncertainty, and many of the reference files still used in the pipeline are preliminary (to be refined during all of Cycle 1; see Boyer et al. 2022 for more details on the NIRCam photometric calibration used here). It is thus prudent to check the accuracy of the photometric calibration. The JWST pipeline applies a photometric calibration taking into account the area of each pixel (using a photom reference file), resulting in image units of MJy sr -1 . For the 30mas scale of our images, the conversion factor from MJy sr -1 to erg s -1 cm -2 Hz -1 is 2.1154 × 10 -31 . \nTo check the accuracy of this calibration, we use a sample of objects with published spectroscopic redshifts in this field (compiled by N. Hathi, private communica- \ntion; including DEEP2; Newman et al. 2013, MOSDEF; Kriek et al. 2015, and 3DHST; Momcheva et al. 2016, among others). Applying a matching radius of 0.5 '' , removing duplicates, keeping the two highest quality redshift flags, and restricting to F160W < 24 mag, we find 988 matches which fall in our CEERS/NIRCam-covered area. These objects span z =0-4, with a median z spec = 1.1. \nWe generate the expected fluxes of these sources in all 13 photometric bands using the EAZY (Brammer et al. 2008) photometric-redshift fitting software. While in § 4 below we will use EAZY to measure photometric redshifts, here we run EAZY with the redshift fixed to the spectroscopic redshift value. This thus obtains the best-fitting galaxy template to the observed photometry. Assuming that this template set spans the color range of real galaxies (see § 4.1), a comparison of the observed fluxes in a given filter to those predicted by EAZY can inform us on any systematic discrepancies in the fluxes in our catalog. We measured the median offset for the subset of these sources with a measured signal-to-noise ratio of > 5 and an EAZY goodness-of-fit ( χ 2 ) < 20. This ensures that a well-fit model is found to a set of robust photometric measurements. This resulted in typically ∼ 850-900 sources per filter (84 in F105W, which has \nsignificantly less coverage), from which we measured a sigma-clipped median and standard deviation. \nWe tabulate these derived corrections in Table 1, and plot the median values and full dispersion in Figure 2. For NIRCam, the LW bands are all consistent with unity, which is a significant improvement over the state of the calibration in late summer 2022. For the shortwavelength channels, we do find a needed correction on the 3-7% level. For HST , we find that with the exception of F606W, the remaining bands are ∼ 3-5% too bright. As HST is extremely well calibrated, this implies that the 35% correction we applied based on the comparison to the Finkelstein et al. (2022a) catalog may have been slightly too large (we note that that study did not complete such a zeropoint-offset analysis). Nonetheless, we applied these corrections. To test their accuracy, we performed another iteration of this analysis after applying these offsets, and found that no significant residual correction was present. We thus apply these corrections, listed in Table 1, to all fluxes and flux errors in our final photometric catalog. We reiterate that while photometric calibration was our motivation for this test, the resulting corrections are a combination of residual differences between our estimated total fluxes and true total fluxes, differences between the model templates and true galaxies, and true photometric zeropoint inaccuracies.", '4. SELECTION OF REDSHIFT > 9 GALAXIES': "Our analysis method produced a photometric catalog which contains our best estimates of the total fluxes in each of 13 filters spanning 0.6-5.0 µ m, with robust flux uncertainty values. We also include fluxes measured in a range of circular apertures. In this section we will use this catalog to select galaxies at z > 8.5. Below we will create identifiers for each object inclusive of the CEERS field the object was covered in, and the SE number within that field. For example, 'Maisie's Galaxy', the previously identified z ∼ 12 galaxy candidate from Finkelstein et al. (2022b), is referred to here as CEERS2 5429.", '4.1. Photometric Redshift Estimation': "We measure photometric redshifts with EAZY for our entire photometric catalog, which contains ∼ 40,000 sources across all four fields. EAZY fits non-negative linear combinations of user-supplied templates to derive probability distribution functions (PDFs) for the redshift, based on the quality of fit of the various template combinations to the observed photometry for a given source. The template set we use includes the 'tweak fsps QSF 12 v3' set of 12 FSPS (Conroy & Gunn 2010) templates recommended by the EAZY documentation. \nAs the population of z > 9 galaxies is expected to exhibit fairly blue rest-frame UV colors, we follow Larson et al. (in prep) in adding six additional templates. They derived these templates by combining stellar population spectra from BPASS (Eldridge & Stanway 2009) with (optional) nebular emission derived with Cloudy (Ferland et al. 2017). They used models with low metallicities (5% solar), young stellar populations (stellar ages of 10 6 , 10 6 . 5 , and 10 7 Myr), inclusive of binary stars, and with a high ionization parameter (log U = -2). Larson et al. (in prep) showed that the inclusion of these templates significantly improved the recovery of photometric redshifts from a mock catalog derived by a semianalytic model (Yung et al. 2022), due to the better match between galaxy and template colors. These templates were also used by Finkelstein et al. (2022b) in their analysis of the z ∼ 12 'Maisie's Galaxy', where they found the inclusion of these bluer templates significantly improved the goodness-of-fit between the data and the best-fitting model. \nWe assume a flat prior in luminosity; while bright galaxies will have a redshift distribution significantly tilted towards lower redshift, the bright end of the highredshift luminosity function is poorly known, thus we do not wish to bias against the selection of true, bright distant galaxies. We include a systematic error of 5% of the observed flux values, and fit to our measured total flux and flux error values. In our fiducial EAZY run the redshift can span 0-20. We perform an additional 'Low-z' run with the maximum redshift set to z max = 7 such that we can visualize the best-fitting low-redshift solution.", '4.2. Sample Selection': "Here we describe our selection criteria we use to identify candidate z > 8.5 galaxies. Following our previous work (Finkelstein et al. 2010, 2015, 2022a,b), we utilize a combination of flux detection significance values and quantities derived from the full photometric redshift PDF (denoted P [ z ]) to select our galaxy sample. As a part of this selection, we make use of the integral of the P ( z ) in ∆ z = 1 bins centered on integer redshift values. Specifically, we denote the unit redshift where the integral in a z ± 0 . 5 bin is the maximum compared to all other redshifts as S z . For example, a sources with S z = 10 would have ∫ 10 . 5 9 . 5 P ( z ) dz greater than the integrated P ( z ) in all other ∆ z = 1 bins. \nOur primary selection criteria are: \n- · A signal-to-noise ratio, as measured in 0.2 '' diameter apertures in the non-PSF-matched images, of > 5.5 in at least two of the F150W, F200W, F277W, F356W or F444W bands. We required \nthis to be true with both the global as well as local noise values. This allows the selection of galaxies across a broad range of redshifts and rest-UV colors. We note that we experimented with requiring a higher significance detection in just one band, but this significantly increased the spurious source fraction. \n- · Error map values < 1000 in all of the F115W, F150W, F200W, and F277W images. This includes only objects with a measurable (though not necessarily significant) flux in the bluest four filters, necessary for selection of galaxies at z ∼ 9-13.\n- · A signal-to-noise ratio of < 3 in bands blue-ward of the Ly α break. While studies occasionally use more stringent signal-to-noise cuts, we choose this value to both account for the fact that any positive flux in all dropout bands is already accounted for by EAZY , and that > 1 σ random fluctuations can coincide at the positions of real galaxies with nonGaussian noise as is present in these images. For this criterion, we include F606W and F814W for all redshifts considered here. We add F115W for S z = 11-12, and F150W for S z = 13-17. These redshift values were chosen to ensure that the Ly α break is fully red-ward of the ∆ z = 1 range for a given filter.\n- · ∫ P ( z > 7) ≥ 0.7. This requires less than 30% of the integrated P ( z ) to be at z < 7.\n- · Best-fitting photometric redshift z best > 8 (defined as 'za' with EAZY; the redshift corresponding to the highest likelihood) with a goodness-of-fit χ 2 < 60.\n- · S z ≥ 9 (selecting a sample of galaxies at z ≳ 8.5).\n- · ∆ χ 2 > 4, calculated as the difference between the best-fitting χ 2 value for the low-redshift restricted model to the fiducial model. By requiring a value greater than four, we require the lowredshift model to be ruled out at ≥ 2 σ significance. We note that Harikane et al. (2022) required a more conservative ∆ χ 2 > 9, based on results from the CEERS simulated imaging. We explore the impacts of such a cut in § 7.2.", '4.3. Sample Vetting': "This automated selection resulted in a sample of 64 candidate galaxies with z ≳ 8.5. To explore the impact of potential contamination by spurious (e.g., nonastrophysical) sources, we visually inspected each object \nin all images to search for obvious diffraction spikes, sources on image edges (as the dithering is different in different filters, the edges do not necessarily line up between different images), and un-flagged cosmic rays. \nTo perform this inspection, three authors (SF, JK, PAH) viewed 'bio' plots which showed small 1.5 '' cutouts of the candidate in all images, with two different stretches, a 5 '' cutout in the F200W and detection images, the fiducial and z < 7 restricted P ( z ), and the spectral energy distribution plotted against both fiducial and Lowz EAZY models. From this process we identified 36 spurious sources: six diffraction spikes, 11 unflagged cosmic rays, and 19 sources on image edges. We remove these 36 sources from our galaxy sample, and show images of all in Figure 20 in the Appendix, and list their positions in Table 6. The cosmic rays were predominantly an issue in the long-wavelength image in pointings 3 and 6; these have longer exposure times than pointings 1 and 2, amplifying the impact of cosmic rays. We note that future versions of our images and catalogs should be able to minimize these types of spurious sources by improving the cosmic-ray removal and using the JWST data model 'Context map' to identify pixels close to the image edge. \nUpon visual inspection, we noticed that a small fraction of the sample (four objects) had Kron apertures which were drawn too large due to the presence of bright, nearby galaxies. To explore the impact of potential light from neighboring galaxies affecting the colors, we performed an additional run of EAZY using the colors measured in 0.3 '' diameter apertures to more accurately measure the colors of the candidate galaxies. For this run we used the PSF-matched photometry, but did not apply any aperture correction as the photometric redshift depends only on colors and not luminosity. We found that two sources, CEERS3 1537 (initial z phot = 10.5) and CEERS6 7478 (initial z phot = 8.6) had their photometric redshifts shift to lower redshift with these smaller apertures; these were the same two sources where the Kron aperture was most significantly stretched. We thus remove these two objects from our sample, though we show their SEDs and image stamps in Figure 19 in the Appendix. \nThe remaining two sources (CEERS1 3908 and CEERS1 3910) had no significant change in their photometric redshift when measured with these small apertures. For these two objects we correct their fluxes by a single factor to account for potential excess brightness from the neighbor. We do this by deriving a flux correction as the ratio between the median total flux for objects with similar 0.3 '' diameter aperture fluxes as a given source to the actual Kron-based (aperture- \nFigure 3. The distribution of the best-fitting (minimized χ 2 ) photometric redshift versus apparent magnitude. Each object is color-coded by its redshift sample (with the background shading denoting the redshift range where the P ( z ) was integrated to determine the sample placement; the green circle far into in the blue region [CEERS6 7641] has a double peaked P ( z )). The magnitude plotted is F150W for z ∼ 9, F200W for z ∼ 11, and F277W for z > 12 galaxies. The small black star denotes 'Maisie's Galaxy' (Finkelstein et al. 2022b), known here as CEERS2 5429. The top axis shows 'Cosmic Time' (the time since the Big Bang) for our adopted cosmology. \n<!-- image --> \nected) total flux for those comparison sources, and divided this quantity by the same quantity for the sources in question. We apply this correction to all filters such that the colors do not change. For CEERS1 3908 this correction factor was 0.6; for CEERS1 3910 it was close to unity, so we applied no correction.", '5. AN EARLY REDSHIFT > 9 JWST GALAXY SAMPLE': "Our final sample consists of 26 galaxies. We divide them into redshift bins for analysis. The high redshifts now accessible with JWST coupled with the broad filter transmission functions make it harder to place galaxies in often-used ∆ z = 1 redshift bins, and such bins span progressively smaller epochs of cosmic time (for example, a ∆ z = 1 bin centered at z = 12 would cover the Ly α break over only ∼ 40 Myr of cosmic time, compared to 200 Myr for one centered at z = 6). We thus split our sample into three redshift bins: a ' z ∼ 9' sample from z = 8.5-10, a ' z ∼ 11' sample from z = 10-12, and a ' z > 12' sample from z = 12-17. We place candidate galaxies in the sample bin based on where their \nintegrated P ( z ) across the bin is the largest. These three bins cover roughly similar ranges of cosmic time (116, 105, 122 Myr, respectively). We show the distribution of photometric redshifts and apparent magnitudes of our sample in Figure 3, and we tabulate the sample in Table 2. We show cutout images of all candidates in Figures 5, 6, 16 and 17, and SEDs in Figure 5, 7 and 18. We also present a rest-UV color montage of our sample in descending redshift order in Figure 4. \nAs a check on our fiducial EAZY photometric redshifts, we also measured photometric redshifts with Cigale (Burgarella et al. 2005). We find excellent agreement, with a median difference of 0.2 (with EAZY preferring the slightly higher redshifts). Only the source CEERS1 4143 has a difference in best-fitting redshift of > 2, as Cigale prefers z = 5.4 for this source, whereas EAZY finds z = 9.0, though this source lies right at the boundary of our ∆ χ 2 selection criterion. Finally, we inspected the spatial distribution of our full sample of galaxies across the full CEERS field, and do not visually see any evidence of strong clustering (which would not be expected with such a small sample covering a broad redshift range). \nFigure 4. A montage of images for the galaxy candidates identified here. For each galaxy the image shows as blue-green-red the three NIRCam bandpasses closest to the Ly α break, this includes: F150W, F200W, and F277W for galaxies at z = 9; F200W, F277W and F356W for galaxies in the z = 10-12 samples; F277W, F356W, and F444W for galaxies in the z ≥ 12 samples. In each case the images are 1.5 '' × 1.5 '' . The inset scale bar shows 1 (physical) kpc. We list the numerical IDs and the best-fitting photometric redshift values (the redshift uncertainties are listed in Table 2). \n<!-- image -->", '5.1. The z > 12 Sample': 'This highest-redshift sample contains two galaxies: CEERS2 2159 and CEERS1 1730, with 68% confidence ranges on their photometric redshifts of 16.0-16.6 and 12.3-14.2, and F277W magnitudes of 26.5 and 27.7, respectively. We will compare in detail our sample to those previously presented in the literature in § 6 below, but we note here that CEERS2 2159 was first presented as a robust ultra-high-redshift candidate in Donnan et al. (2022), with some caveats discussed in Zavala et al. (2022, a marginal 2.6 σ dust emission detection) and Naidu et al. (2022a, environmental evidence). CEERS1 1730 is presented here for the first time. \nThe cutout images for these two objects are shown in the top panel of Figure 5. It is apparent that there is no significant flux at the positions of both candidates in the stacked ACS F606W+F814W images, nor in the F115W image. For CEERS2 2159 no flux is also seen in F150W, and the F200W flux is noticeably fainter than in F277W, consistent with its photometric redshift of z ∼ 16-16.6 (where the Ly α break would be in the red half of the F200W bandpass). For CEERS1 1730, faint flux is seen in F150W. However, this is measured at just 1 σ significance in the Kron aperture. This leads to a fairly broad P ( z ), spanning z ∼ 12-14. If this flux is real, then z ∼ 12 is more likely. If it is not significant, then z ∼ 14 is possible. The SEDs, shown in the bottom panel of Figure 5, show the strong observed', 'z > 12': "Figure 5. Top) Stamp images, 1.5 '' on a side, of the two galaxies that are best-fit with z > 12 (shown from the non-PSFmatched images). The red circle denotes a 0.4 '' diameter region around the source (which we show in only two bands for clarity). CEERS1 1730 has a best-fit photo-z of 13.4, albeit with a wide 68% confidence range of 12.3-14.2. CEERS2 2159 is best-fit by z = 16.5 (16.0-16.6), and was first identified by Donnan et al. (2022). The bottom panels show the best-fitting EAZY models (both overall, and constrained to z < 7), the model bandpass fluxes (open squares), alongside the observed photometry (circles; upper limits are 2 σ ). The inset panels show the P ( z ) distributions. Both sources exhibit well-constrained Ly α breaks, implying the redshifts are z > 11. CEERS1 1730 does show a small low-redshift solution, and its primary P ( z ) peak extends to z ∼ 10.5. \n<!-- image --> \nACS \nF115W \nF150W F200W F277W F356W F410M F444W \nLy α break in both objects, with fairly blue UV spectral slopes redward of the break. For both objects, the highredshift model is a significantly better fit to the photometry than the best-fitting low-redshift model, though for CEERS1 1730, ∆ χ 2 is only 4.4; this low-redshift solution is visible as a small peak in the fiducial P ( z ). CEERS2 2159 shows no significant lower-redshift peak in its P ( z ) due to its brighter magnitude leading to more robust constraints on the full shape of the SED.", '5.2. The z ∼ 11 Sample': "The z ∼ 11 sample contains nine galaxies. As shown by their cutout images in the bottom panel of Figure 6, all show no significant flux in the ACS stack and F115W images, consistent with z > 9.5. Many exhibit a red F150W -F200W color, suggesting the Ly α break is in F150W. The SEDs and P ( z )s for all nine are shown in Figure 7. First looking at the P ( z )s in the upper-left of each panel, the amplitude of the detected Ly α break is \nstrong enough to either eliminate, or leave a very small low-redshift solution. As shown in Table 2, the integrated P ( z > 7) is ≥ 0.98 for 8/9 of these sources, with the remaining source (CEERS1 7227) having a value of 0.85, significantly greater than our sample limit of 0.7 (this object also just barely satisfies our ∆ χ 2 criterion, as only the F200W flux is discrepant with the best-fitting lower-redshift model, and it has a 2.2 σ significance detection in F115W, which could indicate a redshift closer to z ∼ 9). \nCEERS6 7641 shows a double-peaked high-redshift P ( z ), with peaks at z ∼ 9 and z ∼ 11. As its integrated P ( z ) at z = 10-12 is larger than at z = 8.5-10, this object was placed in the z ∼ 11 sample (this is the green z ∼ 11 symbol in Figure 3 that is in the z ∼ 9 sample region), though clearly it has a near-equal probability of being at slightly lower redshift. Finally, we note that similar to the z > 12 galaxies, the UV spectral slopes for these nine sources all appear fairly blue. \nTable 2. Summary of z > 9 Candidate Galaxies \nNote -The horizontal lines divide our three redshift samples (given by the sixth column). The photometric redshift is 'za' from EAZY , which is the redshift where the χ 2 is minimized. The ∆ χ 2 in the final column compares the best-fitting low-redshift (0.5 < z < 7) model to the best-fitting high-redshift model; a value of ≥ 4 was required for selection. a Previously published as Maisie's Galaxy by Finkelstein et al. (2022b). The half-light radii are listed in pixels; our pixel scale is 30 mas. ∗ Galfit did not converge for these two sources, so we list their SE half-light radii. \nThough a detailed analysis of this quantity is beyond the scope of this paper, it is clear that these objects all appear fairly low in dust attenuation. We acknowledge that Ly α -break selection can be biased against red sources (e.g. Dunlop et al. 2012), thus a full quantitative analysis on the colors of these galaxies is reserved to future work.", '5.3. The z ∼ 9 Sample': 'The remaining 15 candidate high-redshift galaxies fall into our z ∼ 9 sample. We show the cutout images in Figures 16 and 17 in the Appendix. At z < 9.5, the Ly α break is in the F115W band, thus we expect to see signal \nin that image for all but the highest-redshift sources in this subsample, though the F115W flux should be fainter than the F150W flux, which is apparent in the images. Likewise, we see no significant flux in the ACS filters as expected. Examining their SEDs and P ( z )s in Figure 18 in the Appendix, we see that nearly all show a tight highredshift peak with very little probability of being at z < 7. Similar to the z ∼ 11 sample, the lowest integrated P ( z > 7) is 0.84, with 13/15 galaxies having integrated P ( z > 7) > 0.9, and 8/15 having integrated P ( z > 7) ≥ 0.99. \nThe object with the most complex P ( z ) constraints is CEERS2 2274, which shows small peaks at z ∼ 2 and \nACS \nF115W \nF150W F200W F277W F356W F410M F444W', 'z ~ 10-12': "<!-- image --> \nFigure 6. Similar to Figure 5, for the nine galaxies in the z ∼ 11 sample. \n6, a dominant peak at z ∼ 9, and a modest peak at z ∼ 10.5, with this uncertainty due to the lower signalto-noise ratio on this object's faint fluxes. While the majority of these z ∼ 9 galaxies show blue rest-UV spectral slopes, the lower redshift means that the reddest filters probe the 4000 ˚ A breaks, and it is clear that some objects have redder colors in these reddest bands. This may be due to significant stellar mass in (somewhat) older stellar populations (e.g. Labbe et al. 2022), though could also be due to the presence of very strong \nrest-frame optical nebular emission lines, which early JWST results indicate are ubiquitous in high-redshift star-forming galaxies (Endsley et al. 2022; Papovich et al. 2022). \nFinally, we discuss the interesting pair of objects CEERS1 3908 and CEERS1 3910. As discussed in § 4.3, the presence of a bright neighbor skewed their Kron apertures, but even after correcting for this they appear as robust z ∼ 9 candidates. In inspecting their cutout images, it is apparent that these two candidate galaxies \nFigure 7. The SEDs and P ( z )'s for the z ∼ 10-12 sample, with lines and symbols the same as Figure 5. \n<!-- image --> \nare very close together with each consisting of a small knot of emission with centroids 0.6 '' ( ∼ 2.5 kpc) apart. It is possible that these are two star-forming regions of the same host galaxy. This hypothesis could be supported by the apparent very faint emission between the two clumps. Deeper imaging could resolve this, though as they are resolved in our catalog we do not merge them here. \nAs one additional check, we explored the impact of our photometric correction factors ( § 3.6) which we had applied to our catalog. We re-ran EAZY on our final sample of 26 high-redshift galaxy candidates removing this factor. We find that this makes effectively no change in the high-redshift solution, with best-fitting redshifts unchanged to more than the 1-2% level, and all candidates continuing to satisfy our P ( z ) selection criteria. We did notice a slight change in the best-fitting low-redshift model, leading to the median value of ∆ χ 2 being reduced by 5%. This impacts sources which were close to our ∆ χ 2 threshold, with CEERS1 1730 (∆ χ 2 = 4.4 → 3.5), CEERS1 4143 (∆ χ 2 = 4.0 → 3.7) and CEERS1 7227 (∆ χ 2 = 4.0 → 3.7) falling below our threshold of ∆ χ 2 > 4. While these sources remain in the sample following \nour fiducial selection, their presence near this threshold leaves their inclusion more sensitive to these small correction factors.", '5.4. Galaxy Sizes': "We derive the sizes of all 26 galaxies in the sample by performing parametric fits on the F200W NIRCam images using Galfit 5 (Peng et al. 2002, 2010). Galfit finds the optimum S'ersic fit to a galaxy's light profile using a least-squares fitting algorithm. As input, we use a 100x100 pixel cutout of the F200W image for each source, the corresponding error array (the 'ERR' extension) as the input sigma image, and the empirically derived PSFs. We use the source location, magnitude, size, position angle, and axis ratios from the SE catalog as initial guesses. We allow the S'ersic index to vary between 0.01 and 8, the magnitude of the galaxy between 0 and 45, the half-light radius (r h ) between 0.3 and 100 pixels, and the axis ratio between 0.0001 and 1. We also allow Galfit to oversample the PSF by a factor 9. We then visually inspected the best-fit model and image \nFigure 8. The F200W half-light radii of our high-redshift galaxy candidates, measured with Galfit . The shaded region denotes the half-light radius for the F200W PSF, of 1.18 ± 0.01 pixels (our pixel scale is 30 mas). While the galaxies are compact, all but the faintest appear to be resolved. Our candidate galaxies have a median half-light radius of 0.46 kpc. The small circles denote the two objects where Galfit did not converge, thus we show the SE half-light radius. \n<!-- image --> \nresidual for each source to ensure that the fits were reasonable and that minimal flux remained in the residual. We also performed a fit on the PSF image itself in order to determine the smallest resolvable size. This value is 1.18 ± 0.01 pixels. Two sources (CEERS1 3908, a member of the pair discussed above, and CEERS2 588) failed to converge on a fit. \nIn Figure 8 we show these measured sizes, highlighting that the majority of the sample is significantly spatially resolved. The measured r h values range from 0.41 to 8.59 pixels, with a median value of 3.6 pixels (0.11 '' or 0.46 kpc). These sizes are consistent with the rest-UV sizes found in the GLASS survey by Yang et al. (2022), who found a median half-light radius of 0.45 ± 0.13 kpc for galaxies at 7 < z < 15.", '5.5. Stellar Contamination Screening': "It is important to analyze whether any objects in our sample could potentially be low-mass stars or browndwarfs, as they can have similar colors as high-redshift galaxies when observed in broadband filters (e.g., Yan et al. 2003; Ryan et al. 2005; Caballero et al. 2008; Wilkins et al. 2014). We first analyze the galfit -produced half-light radii of our candidates, as any resolved objects cannot be stellar in origin. Figure 8 shows the Galfit -measured radii for the 24/26 of our candidate high-redshift galaxies where Galfit converged (showing the SE values for the remaining two). We find that the majority of the sample is obviously resolved, with only five of the faintest sources having sizes within the 1 σ error bars that allow for a point-source. \nFor these five unresolved sources, we then follow the methodology of Finkelstein et al. (2022a,b) to explore whether the colors of any of the few unresolved galaxy candidates could be consistent with stars. We derive a grid of models for the colors of low-mass stars and brown dwarfs (spectral types of M4-T8) in the NIRCam filters, by integrating the IRTF SpEX brown dwarf templates (Burgasser 2014). As these spectra end at 2.5 µ m, we use the tabulated 2MASS photometry to link each SpeX model with Spitzer /IRAC photometry from Patten et al. (2006). Following Finkelstein et al. (2022b) we assume we can map IRAC 3.6 µ m onto F356W and 4.5 µ m onto F444W, though future spectroscopic observations of brown dwarfs with JWST at λ ≳ 2 . 5 µ m will improve this methodology. To explore which model is preferred, we use the Bayesian Information Criterion, which includes the goodness-of-fit ( χ 2 ), the number of photometric constraints (five for stars, 12 for galaxies [due to the use of HST for the latter]), and the number of free parameters (one for stars, 19 [18 templates fit simultaneously, plus the redshift] for galaxies). We find that two of these five sources have BIC values which indicate a preference for the best-fitting stellar model. \nAlthough these two objects (CEERS2 2274 and CEERS2 1075) are formally unresolved and have optical/near-infrared colors that consistent with brown dwarfs (Patten et al. 2006), we nevertheless doubt this interpretation as they are very faint (m F 277 W = 29.1 mag). The colors imply a very late spectral type of ≳ T5, which corresponds to an absolute magnitude of K =16.1 AB mag (Patten et al. 2006). Such a brown dwarf would therefore be at a heliocentric distance of 3.6 kpc, or 3.1 kpc above the Galactic plane, which would place this object approximately nine scale heights out of the thin disk (e.g., Ryan et al. 2011; Holwerda et al. 2014). These objects are therefore highly unlikely to be part of the Population I thin disk stars, but could be a member of the thick disk or Galactic halo (Ryan & Reid 2016). At present, there are very few constraints on the density of brown dwarfs in these more distant Galactic components at these extremely low effective temperatures, but based on more massive main sequence stars, these are expected to have many orders-of-magnitude fewer stars than the thin disk (e.g., Juri'c et al. 2008). A rigorous Bayesian model of the halo brown dwarfs would provide the strongest statistical evidence and prediction for or against this source, but this is beyond the scope of the present work. We also note that CEERS2 2274 appears to have a very nearby neighbor with a similar SED, which could indicate it is a merging system at high-redshift. We do note that both sources are below the brightness limit ( m < 28.5) we apply when analyz- \nour sample in § 7, so their inclusion (or exclusion) does not affect any of our conclusions.", '5.6. Examining Ancillary Multi-wavelength Observations': "We have examined the positions of our 26 candidate z ≳ 9 galaxies at both shorter and longer wavelengths, and find no significant detections, increasing confidence in the very high-redshift interpretation. While the depth of these data are not necessarily sufficient to completely rule out low-redshift solutions, these non-detections do increase our confidence in the fidelity of our sample. First we searched for X-ray emission coincident with our candidate positions using Chandra imaging from the AEGIS-XD survey (Nandra et al. 2015), which has a flux limit of 1 . 5 × 10 -16 erg cm -2 s -1 in the 0.5-10 keV band. There was no emission detected with a Poisson false probability less than 4 × 10 -6 in the soft (0.5-2 keV), hard (2-7 keV), full (0.5-7 keV) and ultrahard (47 keV) energy bands. \nWe then investigated possible far-infrared (FIR) emission at the position of our high redshift galaxy candidates, using the super-deblending catalog technique from Liu et al. (2018) and Jin et al. (2018), which was adapted for the EGS field. To summarize, this multiwavelength fitting technique is meant to optimize the number of priors fitted at each band to extract the deepest possible information. We use data from Spitzer (24 µ m from FIDEL; Dickinson & FIDEL Team 2007), Herschel 100 µ mand 160 µ mfrom PEP (Lutz et al. 2011) and 250 µ m, 350 µ m, 500 µ m from HerMES (Oliver et al. 2012), JCMT/SCUBA2, including 850 µ m from S2CLS (Geach et al. 2017) and 450 µ m and 850 µ m from Zavala et al. (2017) and AzTEC 1.1mm from Aretxaga (2015). The key is to obtain an adaptive balance as a function of wavelength/dataset between the number of priors fitted, the quality of the fit, and the achievable deblending given the PSF sizes. We start with the deepest images and fit each band from deeper to shallower images. \nIn general, the high-redshift galaxy candidates are not already contained in our prior-lists, with the exception of two sources, for which all measurements are already contained in the forthcoming catalog (A. Le Bail et al., in preparation). For the other candidates we consider them one at a time and add a specific prior at its position, that we fit together with the rest of priors that are relevant for each band. Extensive Monte-Carlo simulations ensure that the uncertainties associated to the flux measurements are 'quasi-Gaussians' (see Liu et al. 2018; Jin et al. 2018, A. Le Bail et al. in preparation). None of the candidates are significantly detected at any band between 24 µ m and 1.1mm. We do note that for \none galaxy, CEERS1 1730, this method implies a ∼ 3 σ detection at 500 µ m. However, inspecting the images, it is clear at 160, 250 and 350 µ m there is significant emission from a bright neighbor to the south, which overlaps our source's position at 500 µ m due to the worsening PSF. We conclude it is highly unlikely that this emission is associated with our candidate. \nWe do also note that our object CEERS2 2159, first published as ID=93316 by Donnan et al. (2022), does have a formal 2.6 σ detection in the SCUBA-2 850 µ m data, as noted by Zavala et al. (2022). However, that marginal 850 µ m detection could plausibly be associated with other nearby z ∼ 5 galaxies within the beam as discussed in Zavala et al. (2022); higher resolution mm interferometry is being carried out on this system to gain further insight (Fujimoto et al., in prep).", '6.1. Candidates Identified with HST': 'The area presently covered by CEERS covers two z ∼ 9-10 galaxies previously identified by Finkelstein et al. (2022a), EGS z910 40898 and EGS z910 65860 (none of the candidate z ≳ 9 galaxies from Bouwens et al. 2019 fall in this first epoch of CEERS NIRCam imaging). In Figure 9, we show a comparison of the first object in NIRCam versus HST and Spitzer imaging. While the object was only detected at modest significance in the ∼ few-orbit HST imaging, and barely at all in the 50 hr depth Spitzer /IRAC imaging, it is extremely well detected (signal-to-noise ratio > 30) in < 3000 seconds in all seven NIRCam filters. \nIn this figure we also compare constraints on the photometric redshift from the previous HST + Spitzer photometry of EGS z910 40898 to the present JWST + HST photometry. While this object was previously selected as a z ∼ 9 candidate as the majority of the P ( z ) was at z ≳ 9, the primary peak extended down to z < 7, and there was a non-negligible secondary peak at z ∼ 2. With the significant improvement in photometric constraints, there is now a single narrow peak at z ∼ 8.1. The best-fitting redshift is modestly lower than the previous value due to the only marginally red F115W -F150W color, implying the Ly α break is towards the blue end of the F115W filter. Observations of this object highlight the capabilities of JWST to significantly improve constraints on photometric redshifts compared to the preJWST era. \nThe second object, EGS z910 65860, however shows no significant flux at all at the expected position (Figure 10). This implies that the source identified in the HST imaging was either spurious, or a transient phe- \nFigure 9. The candidate galaxy EGS z910 40898, first published by Finkelstein et al. (2022a) as a z ∼ 9 galaxy candidate. This source was detectable by HST , albeit in only three filters and at only modest significance. Even in 50 hr of Spitzer /IRAC imaging, it was marginally detected. In just < 3000 sec with JWST /CEERS, this source is extremely well-detected in all seven NIRCam filters, highlighting the power of JWST to probe the very early universe. In the top-middle panel we compare the previous P ( z ) to that now possible with JWST , finding a much sharper peak, centered at z ∼ 8.1 (the F115W -F150W color is only marginally red, implying a Ly α break at the blue edge of F115W). \n<!-- image --> \nnomenon. Finkelstein et al. (2022a) performed a detailed vetting process to remove all forms of spurious sources, including persistence, thus this seems like an unlikely solution. We therefore consider whether the observations are consistent with a transient source. This object showed significant detections in HST F125W and F160W imaging, with no detections in F606W, F814W, F105W nor F140W. While 3D-HST F140W pre-imaging was shallower than the CANDELS imaging, it was curious that this object showed no detection as its SED (anchored by F125W and F160W) should have been detectable. \nWe thus investigate the date of each of these observations. The F125W and F160W images were taken in 2013, with two images taken ∼ 50 days apart (2 April, and 24 May). We made updated HST mosaics (following the procedure in Koekemoer et al. 2011) around the position of this source in each epoch separately, and we do see a clear detection in both bands in both epochs. This further refutes the hypothesis that this disappearing source was spurious. Adding to the likelihood of a transient explanation is that the source is fading between the two epochs, with the ratio of fluxes from the first to second epoch being 2.5 and 2.0 for F125W and F160W, respectively (fluxes were measured with SE on each epoch separately, using MAG AUTO to approximate total fluxes). The non-detection of this source in F140W is easily explained, in the context of a transient interpretation, by the acquisition date of those images, which was 1.5 years earlier than F125W and F160W (2 Dec 2011). \nWhile the apparent Ly α break between F105W and F125W could imply that this object is a z ∼ 9 tran- \nnt, the F105W imaging was obtained much later, on 1 April, 2015. Likewise, the F606W images were taken in 2004 and 2011, while the F814W were taken in 2004, 2011, and 2013. The SCANDELS Spitzer /IRAC imaging was obtained over many years from 2003-2012, but the bulk was obtained between 2010-2012 (Ashby et al. 2015). We therefore do not have the contemporaneous photometry needed for a reliable redshift of this source. What we do know is that the host galaxy is fainter than the limit of our NIRCam imaging. Taking the limit of our deepest image in Table 2, we find that the observed flux in the first F160W epoch (23.6 nJy) is ∼ 150 times brighter than the 1 σ upper limit of our NIRCam imaging. The compact morphology of this source, along with the lack of apparent proper motion between the two CANDELS epochs leaves a supernova as the most likely explanation (a proper motion of less than the F160W PSF in 50 days indicates a distance of more than one parsec, ruling out a Solar System object). We show the observed fluxes/limits versus time in the left panel of Figure 10, while in the right panel we show constraints on host galaxy absolute magnitude. \nWhile nearly any redshift is plausible from a luminosity standpoint (from a low-redshift dwarf galaxy to a very high-redshift ∼ L ∗ galaxy), the significant fading over a 50 day time period argues against a very high redshift due to time dilation of the SNe decay curve. From the observed (sparse) light curve, we can only place loose constraints on the nature of the transient, following the methods of Rodney et al. (2014). The data favor classification as a core-collapse supernova with a redshift ranging from z ≃ 0.2-1.2, with maximum likelihood at z ≃ 1 . 1 and a slight preference for SN Ib/c over SN II. \nFigure 10. Left) Flux versus observational epoch for the object EGS z910 68560, first published by Finkelstein et al. (2022a) as a z ∼ 9 galaxy candidate. The colors denote the filter. The small circles denote the integrated CANDELS F125W and F160W fluxes, while the larger circles denote the fluxes measured in each epoch, 50 days apart. The object is detected in both epochs in both bands, fading by a factor of ∼ 2.5 and 2.0 in F125W and F160W, respectively, across this time interval. The object is not detected in any of the CEERS imaging; the inset image shows this position in the CEERS F200W image. This implies the object is > 150 × fainter now than it was when it was detected. Right) The limits on the host galaxy absolute magnitude, taken by applying the cosmological distance modulus to the CEERS upper limit. With the exception of the extremely low-redshift Universe, a wide range of host galaxy absolute magnitudes are plausible, making it difficult to constrain the redshift of this supernova, though analysis of the light curve implies a likely redshift of z ∼ 0.2-1.2. The inset panels show the images of this transient source in both CANDELS epochs in F125W and F160W. \n<!-- image --> \nNevertheless, the data are also compatible with a SN Ia at z ≃ 1 . 45.', '6.2. Candidates Identified with CEERS': "At the time of this writing, there have been five previously submitted articles which have identified z ≳ 9 candidates from these CEERS imaging data (Donnan et al. 2022; Finkelstein et al. 2022b; Harikane et al. 2022; Labbe et al. 2022; Whitler et al. 2022). In this section we compare our current analysis to these previous results, noting that differences in selection techniques may bias the redshift ranges that a particular study is sensitive to. We also emphasize that while the selection of a candidate by more than one study does increase the confidence in the object's fidelity, the presence or lack of a galaxy in a given sample is often easily attributable to differences in data reduction, photometric methodology, and selection criteria. \nFirst, Finkelstein et al. (2022b) published a paper on a single galaxy, 'Maisie's Galaxy', which they identified as a z ∼ 12 galaxy candidate. While they used the same full CEERS imaging dataset as we use here (albeit with an earlier version of the data reduction), they employed highly conservative selection criteria to identify a single extremely robust galaxy candidate due to the nascent state of the data reduction pipeline and photometric calibration c. July 2022. This object is in our \ngalaxy sample, here known as CEERS2 5429. Our photometric redshift constraints for this source are z phot = 11.5 +0 . 2 -0 . 6 . This is consistent at the ∼ 1 σ level with the value of z phot = 11.8 +0 . 3 -0 . 2 from Finkelstein et al. (2022b), with the small differences in the redshift attributable to small changes in photometry due to the updated photometric calibration now available. \nThe largest previous sample of high-redshift galaxies in the CEERS field comes from Donnan et al. (2022, hereafter D22), who selected 19 high-redshift galaxies over the CEERS field. They required dropouts in the F115W (or redder) bands to be selected, biasing the sample to z > 9.5 (where Ly α redshifts out of F115W). Here we compare to their revised sample (c. Oct 2022). We find nine galaxies in common between our two samples. Of the 17 candidates in our sample but not in D22, 14 are at z ≲ 9.5, thus their absence from the D22 sample is expected via their requirement of no significant flux in F115W. The remaining three candidates in our sample and not in D22, CEERS1 1730, CEERS1 7227 and CEER6 4407 have z best > 10, thus in principle could have been selected by D22. Of the nine sources in common, the median difference in photometric redshifts is 0.3 ± 0.7, with all redshifts agreeing within ∆ z < 1 (Figure 11), with the exception of our source CEERS6 7641 (D22 ID 30585), which we find has z best = 9.0 compared to their value of z best = 10.6. However, as shown in \nFigure 7, the P ( z ) for this source is quite broad, and contains a second peak at z ∼ 10.4, thus our results are fully consistent with those of D22. \nOf note is that our sample contains D22 object 93316, as object CEERS2 2159, and our photometric redshift is nearly precisely equal to the D22 value of z = 16.4. This object is remarkably bright ( m F 277 W = 26.5), and some evidence is present both from sub-mm imaging and environmental studies that indicate z ∼ 5 (Zavala et al. 2022; Naidu et al. 2022a). The JWST photometry alone very strongly prefer this ultra-high redshift. Followup spectroscopy and millimeter interferometry will soon reveal the true nature of this potentially record-breaking system. \nWe next explore the properties of the 10 galaxies selected by D22 that are not in our final high-redshift galaxy sample. Of these 10, we find that five are faint enough that they did not meet our source detection significance criteria. Some of these are faint enough in our catalog that their photometric redshifts are not well constrained, while others do show peaks at z > 10. Four more sources do show primary peaks at z > 10, but have secondary peaks at z < 4 that are large enough for our selection criteria to remove these sources (these D22 IDs 1434 2, 26409 4, 5628 2 and 6647 all have ∆ χ 2 < 4 in our catalog). For only one source, D22 ID 61486, do we find a strong low-redshift solution. This object exhibits a fairly flat SED in our catalog, though it does exhibit a red F115W -F150W color, which could be indicative of a true z > 9 Ly α break. Overall, we find strong consistency between objects in common in both our sample and that of D22, and for the D22 sources not in our sample, a high-redshift nature is possible given our photometry. \nWe next compare to Harikane et al. (2022). At the time of this writing only the original sample, prephotometric calibration update, was available for comparison. In the CEERS field, they selected galaxies as F150W or F200W dropouts, restricting their sample to z ≳ 12, selecting six high-redshift candidates. Of these six, two are in common with our sample: CEERS2 5429 ('Maisie's Galaxy'; Harikane ID CR2z12-1) and CEERS2 2159 (D22's 93316; Harikane ID CR2-z17-1), with Harikane et al. having z best = 11.88 and 16.45 for these two objects, respectively. Of the four sources in their sample we do not recover, for CR2z12-3, CR3-z12-1 and CR6-z12-1, our P ( z ) does show a primary peak at z > 10, but the low-redshift peak is significant, with ∆ χ 2 < 4 for all three. All three sources are also faint, and do not meet our detection significance criteria. For their source CR2-z12-1, we find \nFigure 11. A comparison of our photometric redshifts to those for objects in our sample which were previously published as z > 8.5 galaxy candidates from CEERS data in the literature, showing good agreement (not shown is the z ∼ 16 candidate, where the agreement is good between the three studies who have published it). As discussed in § 6.2, for literature sources not present in our sample, we find in our catalog that they typically miss our detection significance and/or ∆ χ 2 criteria, though often the P ( z > 10) is non-zero. \n<!-- image --> \nz < 6, driven by a ∼ 6 σ detection in F115W, and a blue F115W -F150W color. \nLabbe et al. (2022) selected candidate massive galaxies at 7 < z < 11 as those with detectable Ly α and Balmer breaks in their photometry. For this comparison, we use an updated sample made available following improvements in photometric calibration (I. Labbe, private communication). Of their 13 sources, they report photometric redshifts of > 8.5 for five of them. Promisingly, all five of these sources are in our sample, with our (Labbe et al.) IDs of CEERS1 3910 (39575), CEERS2 1298 (21834), CEERS2 2402 (16624), CEERS2 7534 (14924) and CEERS3 1748 (35300), with best-fitting photometric redshifts agreeing to ∆ z < 1 (and ≲ 0.3 for 4/5). \nFinally, Whitler et al. (2022) studied the stellar populations of bright galaxies at high-redshift in CEERS. We compare to an updated sample, again following improved photometric calibration (L. Whitler, private communication). In this sample, they have six galaxies with z best > 8.5. Three of these sources are in our sample (CEERS1 3858, CEERS1 6059 and CEERS6 7641, with Whilter et al. IDs of 37135, 37400 and 9711, respectively). The photometric redshifts for these three agree extremely well (∆ z < 0.05) with our estimates. Of the three sources we do not recover, we do find a strong z ∼ 10 peak for their ID 7860; this source narrowly misses our sample with ∆ χ 2 = 3.7. ID 14506 does show a red F115W -F150W color, but our analysis prefers a 4000 ˚ A break rather than a Ly α break, though a very small \npeak is present at z ∼ 11. For ID 34362, a z ∼ 9 peak is dominant in our measurements, but ∼ 50% of the integrated P ( z ) is contained in a z ∼ 2 peak. \nIn Figure 11 we compare the photometric redshifts from our study to these previous works for sources in common. As discussed above, especially considering the mix of photometric procedures and photometric redshift techniques, the agreement is generally good. \nAs one final comparison, we compare our estimated total magnitudes with the published magnitudes for those candidates in common (where, using the information available, we compare F200W for D22 and Whitler et al. 2022, F356W for Harikane et al. 2022, and F444W for Labbe et al. 2022). We find the largest discrepancy with D22, where for the nine sources in common, the median magnitude differences between our cataloged total magnitudes and those of D22 is -0.3 mag (meaning our fluxes are brighter); however there is significant scatter, with one source being as much as 1 mag brighter in our catalog, and another being 0.4 mag fainter in our catalog. Further exploring this discrepancy requires a deeper comparison between the procedures adopted to estimate total fluxes. Comparing to the other studies, we find results more in agreement. For the four sources in common with Whitler et al. (2022), the median magnitude offset is -0.1, with individual objects ranging from -0.2 to 0. For the two objects in common with Harikane et al. (2022), our magnitudes are fainter by 0.1 and 0.3 mag. Finally for the five sources in common with Labbe et al. (2022) our magnitudes are all fainter by 0.05-0.2 mag, with a median of 0.15 mag. We conclude that while there is significant scatter in the photometry due to the various processes used to reduce and analyze the data, we find no evidence that our photometry is significantly systematically shifted compared to other studies, especially by the large amount needed to explain the observed luminosity function excess ( § 7.1). Clearly improving photometric agreement between different photometric catalogs is a key goal for the near future.", '7. THE EVOLUTION OF GALAXIES IN THE FIRST 500 MYR': 'Here we investigate what constraints we can place on galaxy evolution with our sample of 26 candidate z ∼ 8.5-16.5 galaxies. While it is early in the JWST mission and the sample size is modest, the fact that our sample contains a large number of galaxies at z > 9 allows us to investigate what constraints are possible. We first measure the rest-frame UV luminosity function at z ∼ 11 in comparison to both previous results as well as expectations from empirical extrapolations. We then compare \nthe cumulative number of galaxies to predictions from a suite of theoretical models, to explore the accuracy with which these models predicted galaxy abundances in this early epoch.', '7.1. The z ∼ 11 Rest-UV Luminosity Function': "The UV luminosity distribution function is one of the key observational diagnostics of galaxy evolution in the early universe. With each technological leap leading to a new redshift era being observable, this quantity is always of immense interest (e.g. Bouwens et al. 2004; McLure et al. 2013; Finkelstein et al. 2015) as it is directly comparable to simulation predictions, helping to constrain the physical mechanisms regulating galaxy evolution. While the extensive amount of deep-field imaging data available from the full Cycle 1 dataset will lead to excellent constraints on this quantity, here we gain a first look by exploring constraints placed by the CEERS data. \nFor this first-look luminosity function we focus on the specific redshift range of z ∼ 9.5-12. We exclude z = 8.5-9.5 for two reasons. First, there is a likely overdensity at z = 8.7 (e.g. Finkelstein et al. 2022a; Larson et al. 2022) which would bias this quantity high. Second, at z > 9.5, the Ly α break is fully redward of F115W, providing a true JWST dropout sample that is not dependent on the shallower HST imaging. We choose the upper bound of z ∼ 12 as only two galaxies in our sample have higher photometric redshifts. \nTo measure the UV luminosity function we require an estimate of the effective volume over which we are sensitive to galaxies. This is a function of both redshift and source brightness, and accounts for incompleteness due to both photometric measurements and sample selection. Following Finkelstein et al. (2022a) we estimate the completeness by injecting mock sources of known brightness into our images. We do this separately for each of the four fields, injecting 10 3 sources per iteration, with ∼ 30 iterations per field, to avoid crowding. \nWe generate source morphologies with galfit (Peng et al. 2002). While the completeness can depend sensitively on source size due to surface brightness dimming, our sources are quite compact (Figure 8), thus we choose a log-normal half-light radius distribution such that the size distribution of the recovered sources matches well that of our galaxy sample (noting that if there were highly extended sources not present in our sample, we would be underestimating the incompleteness). While we choose S'ersic profiles (with a log-normal distribution peaked at n = 1), our size assumption results in a peak of unresolved sources, with a tail towards modestly resolved sources. We generate galaxy SEDs assuming a log-normal distribution in magnitude and a flat dis- \nFigure 12. The total completeness as a function of input redshift and F277W apparent magnitude. These values were derived by simulations where we inject compact sources with realistic (modestly blue) SEDs into our imaging, and attempt to recover them with our analysis pipeline. The shading denotes the fraction of sources in each bin of input redshift and magnitude which were both detected by SE and satisfied our sample selection. The incompleteness at z < 8.5 is expected via our sample selection. We see that the CEERS imaging allow recovery of galaxies across the entire redshift range of interest to very high completeness at m F 277 W < 28.0, with the completeness dropping steadily from m = 28 to m = 29. \n<!-- image --> \nribution in redshift, using Bruzual & Charlot (2003) templates with stellar population properties tuned to generate fairly blue galaxies (this results in median UV spectral slope ( β ) of -2.3, with a tail to -3 and -1). \nAfter sources are added to the images, the images are processed through our entire analysis pipeline in the same way as our real data, measuring photometry with SE , applying all aperture corrections (though we did not apply the small zeropoint offsets as those may be due to real instrumental offsets not captured in these simulations), measuring photometric redshifts with EAZY , and applying our sample selection. In Figure 12 we show our completeness as a function of input redshift and F277W magnitude. This highlights that our sample selection accurately begins to select galaxies at z ≳ 8.5. Our completeness remains high to m ∼ 28.0, beginning to fall off at m> 28.5, consistent with pre-launch expectations. \nTo calculate rest-UV absolute magnitudes for both our real and simulated sources, we follow Finkelstein et al. (2015) performing a basic round of SED fitting, measuring M 1500 from the bandpass-averaged flux over a tophat bandpass spanning 1450-1550 ˚ A in the rest-frame. Weuse 10 3 Monte Carlo simulations to obtain uncertainties on these magnitudes. For the completeness for our luminosity function estimate, we require the simulated sources to have best-fit photometric redshifts spanning 9.5-12, to match those of our galaxy sub-sample. We then calculate our completeness as a function of absolute magnitude, and measure effective volumes in bins of absolute magnitude by integrating over the co-moving volume element multiplied by the completeness for a given \nmagnitude and redshift, over our current survey area of 35.5 arcmin 2 . We list our effective volumes in Table 3. \nWe measure our luminosity function following the methodology of Finkelstein et al. (2015), adopting a bin size of 0.7 magnitudes. The number density in each bin is estimated via MCMC (see details in Finkelstein et al. 2015), sampling galaxies absolute magnitude posterior distributions, such that galaxies can fractionally span multiple magnitude bins. We note that at M UV > -19.5 our completeness falls below 30% of its maximum value, thus our faintest bin is shaded in white to indicate that the value is dominated by the completeness correction. \nIn Figure 13 we show our luminosity function results. In the top-left panel, we show the stacked P ( z ) of the 10 sources in our sample at 9.5 < z best < 12.0. The FWHM of this normalized distribution spans z = 9.5-11.7, with a peak at z ∼ 11. The dashed line shows the completeness from our simulations as a function of redshift at the median absolute magnitude of our sample, which probes a broadly similar redshift range, albeit with a flatter distribution. In this figure we also plot recent z ∼ 11 results from Donnan et al. (2022), Naidu et al. (2022b) and Bouwens et al. (2022a). While numbers in all studies are fairly small, the agreement between all studies is encouraging. We also compare to a wealth of studies in the literature from HST at z = 8.5-11. \nBroadly speaking, our results at z ∼ 11 do not show significant evolution from these modestly lower redshifts. This is consistent with the conclusion from Bowler et al. (2020) who noted that the bright-end of the UV luminosity function ( M UV = -22 to -23) shows \nFigure 13. The rest-frame UV luminosity function at z ∼ 11, shown as the red circles (the open circle denotes our faintest bin, where we are < 30% complete). Each galaxy's magnitude and magnitude uncertainty is denoted by a small circle and line at the top of the figure. The light red symbols show literature constraints from JWST data, from GLASS (Naidu et al. 2022b), CEERS+GLASS (Donnan et al. 2022, , who also used UltraVISTA), and the HUDF (Bouwens et al. 2022b). The light blue points show a compilation of data from the literature at z ∼ 9-10. Circles denote results from studies which used (modestly) deep imaging from surveys such as CANDELS and the Hubble Frontier fields, including McLeod et al. (2015), Oesch et al. (2018), and Bouwens et al. (2019, 2021). The squares denote studies making use of HST pure parallel surveys, including Bernard et al. (2016), Morishita et al. (2018) and Rojas-Ruiz et al. (2020). The triangles denote results from wide-area ground-based studies of Stefanon et al. (2019) and Bowler et al. (2020). The blue lines show the evolving double power-law (DPL) luminosity functions from Finkelstein & Bagley (2022) at z = 4-8 (this model was fit to data at z = 3-9). The darker shaded gray region show the predictions from these DPL fits extrapolated to z = 9 (upper bound) - 12 (lower bound); the lighter gray (outlined with dashed lines) shows a similar extrapolation from the evolving Schechter function fits from Finkelstein (2016, ; this model was fit to data at z = 4-8). The inset shows the stacked P ( z ) of the galaxies used in this luminosity function, as well as the redshift distribution estimated from the completeness simluations at M UV = -20. The observed z ∼ 11 luminosity function is consistent with the top end of both smooth extrapolations, implying that the observed smoothly UV luminosity function evolution from z = 4 to z = 9 may be slowing at z ∼ 11. \n<!-- image --> \nlittle evolution from z = 8-10. However, here we are beginning to probe fainter, yet we still see little evolution. \nFinally, we comment on both the shape of the UV luminosity function, and the overall evolution. In Figure 13 we plot two empirical extrapolations. The first is a Schechter function from Finkelstein (2016), who measured the evolution of Schechter function parameters as a linear function of (1+ z ) from z = 4-8; we plot this function evolved to z = 9 and 12, shaded by the light gray color. The darker gray shading is a similar empirical evolution, this time using data from z = 3-9, and assuming a double power-law (DPL) form, from Finkelstein & Bagley (2022). Over the magnitude range of our \nobserved sources, these two functions agree, and our observations are consistent with, albeit at the high end of, these empirical extrapolations, implying that pur observed z ∼ 11 UV luminosity function is similar to the z = 9 DPL luminosity functional fit from Finkelstein & Bagley (2022). \nThese results suggest that the evolution of the UV luminosity function, which had been smoothly declining from z ∼ 4 to 8, begins to slow by z ∼ 11. The luminosity function decline has been debated in the literature prior to the JWST era, notably by Oesch et al. (2018) and Bouwens et al. (2019), who found evidence for an accelerated decline in the UV luminosity function at z > 8. \nTable 3. z ∼ 11 UV Luminosity Function \nNote -The number densities are derived via a MCMC method which includes photometric uncertainties, thus galaxies can contribute to the number density in more than one bin. The effective number column lists the mean and standard deviation of the number of galaxies per bin from these MCMC simulations, while the number column gives the actual value based on the measured magnitude. a Our data is < 30% complete in this faintest bin, so we do not consider this value reliable; it is indicated as an open symbol in Figure 13. \nWhile our small sample cannot conclusively distinguish between these two scenarios, should future luminosity function efforts validate our observed number densities, it provides further evidence that there is significant star formation in our universe at z > 10. Previous efforts to recover the star formation histories of galaxies detected at z ∼ 9 to 11 with HST and Spitzer /IRAC (Tacchella et al. 2022) and analysis of early JWST luminosity functions from Donnan et al. (2022) also arrived at a similar conclusion.", '7.2. The Cumulative Surface Density of Galaxies at z > 8.5': "As another view into the evolution of galaxies in the first 500 Myr, in Figure 14 we plot the cumulative surface density of sources in our sample. This plot shows the integrated surface density for sources at redshift greater than z , limiting to m F 277 W < 28.5 to avoid fainter luminosities where we are highly incomplete. We correct for incompleteness by counting each galaxy as one divided by the estimated completeness at the redshift and magnitude of a given galaxy. The solid black line shows the completeness corrected value from our sample, while the dotted line shows the results with no correction. We note that across the redshift range considered here, the completeness correction is typically a factor of ∼ 2, which is typical of most analyses of modestly faint galaxies. This, of course, means that the accuracy of completeness corrections remains an important systematic, and one we will explore in more detail in future works. However, we note that with the excep- \non of the lowest redshifts studied here, even the uncorrected counts exceed most of the model predictions, and therefore this is a reasonable lower limit on the true surface density. \nTo encompass the uncertainty in both flux and photometric redshift, we run Monte Carlo simulations, sampling both the F277W flux and photometric redshift posterior distributions, and plotting the 68% confidence range on the surface density as the 68% spread in values from these 10 3 simulations as the light gray shaded region. We also run a set of Monte Carlo simulations additionally sampling the Poisson uncertainty, shown by the wider dark gray region. We find that at z > 8.5 and m F 277 W < 28.5, our results suggest > 1 galaxy per arcmin 2 . This trends downward when integrating from higher redshifts, albeit slowly, with this quantity not dropping to 0.1 arcmin -2 until z > 11.5-12. We note that these surface densities are only slightly reduced when removing sources with 4 < ∆ χ 2 < 9 (dashed line). \nWe also show the cosmic variance calculated based on the bluetides simulation (Bhowmick et al. 2020). While these uncertainties appear comparable to the combined redshift, photometric and Poisson uncertainties, these cosmic variance uncertainties are likely an upper limit. As they are simulation based, they rely on the predicted abundance of galaxies, and as we show here, most simulations under-predict the abundance of z > 9 galaxies. As the relation between UV luminosity and halo mass in this epoch is very uncertain ( § 8.2), this leads to uncertainty in these calculations. We show this upper limit on the true cosmic variance uncertainty to highlight that it implies our results are still significantly above the predictions. \nIn the bottom panel of this plot, we also compare to nine recent model-based predictions, including from the Santa Cruz (Yung et al. 2019a, 2020) and delphi (Dayal et al. 2017) semi-analytic models (SAMs), empirical models by Behroozi & Silk (2015), Mason et al. (2015), and UniverseMachine (Behroozi et al. 2019, 2020), and cosmological hydrodynamic simulations FLARES (Wilkins et al. 2022a,b), THESAN (Kannan et al. 2022), MillenniumTNG (Kannan et al. 2022), and Simba (Dav'e et al. 2019). Interestingly our results are noticeably higher than most predictions from physics based models. At z < 9.5, our observations are consistent with both the Behroozi & Silk (2015) and delphi models (and the FLARES predictions with no dust attenuation), while at z > 10 our results lie significantly above all predictions with the exception of Behroozi & Silk (2015). We include results with and without dust attenuation for the Santa Cruz SAM, delphi , FLARES, \nFigure 14. The cumulative surface density of sources with m F 277 W < 28.5 at redshift greater than a given x-axis value, starting at z ≥ 8.5. The top panel shows the redshifts of individual objects (with blue, red and purple denoting the z ∼ 9, 11 and > 12 samples). In the middle panel the solid line shows the observed surface density, after applying a correction for incompleteness; the dotted line shows the un-corrected (incomplete) values. The light shaded region shows the posterior on the distribution of the completeness-corrected surface density derived from Monte Carlo simulations marginalizing over the uncertainties in magnitude and photometric redshift; the dark shading includes Poisson uncertainty in this marginalization. The dashed line shows the completeness-corrected surface density if we had applied a more conservative sample selection criterion of ∆ χ 2 low -z -high -z > 9. In the bottom panel, we show the same shaded regions, now comparing several recent model predictions, shown by the various colored lines (with solid, dot-dashed and dashed denoting predictions from hydrodynamical, semi-empirical, and semi-analytic models, respectively). For four models we plot thicker lines for predictions with no dust attenuation, and thinner lines for attenuated predictions. In this panel we also show an estimate of the cosmic variance uncertainty using the method from Bhowmick et al. (2020), though this is likely an upper limit on this quantity. Even including all sources of uncertainty, our observed surface densities are higher than nearly all predictions, with the exception of the Behroozi & Silk (2015) semi-empirical model. This apparent excess of high-redshift galaxies holds true at all z = 8.5-14, regardless of the ∆ χ 2 cut, and is even true of the raw, non-corrected counts at z > 10.5. These results strongly imply that these predictions lack the full complement of physics describing star formation in the early universe, which we discuss in § 8.2. \n<!-- image --> \nand Simba . We note that while there is likely an overdensity at z ∼ 8.7 in the observed field, this has no impact on the surface density measured at z > 9, and the excess of our observed galaxy numbers over nearly all theoretical predictions persists at all redshifts. We discuss possible explanations for this discrepancy in § 8.2. \nIt is important to note that the compilation of theory results included in this comparison is made utilizing several different modeling approaches and with some different modeling assumptions (see Somerville & Dav'e (2015) for a thorough review). For instance, cosmological hydrodynamic simulations (e.g. MilleniumTNG, \nTHESAN, Simba ) are carried out by solving the equations of hydrodynamics, thermodynamics, and gravity for large numbers (typically billions) of dark matter, gas, and star particles. These simulations are capable of selfconsistently tracking various properties (e.g. stellar and gas mass) and morphology, as well as their correlation with the large-scale environment. Cosmological zoomin simulations (e.g. FLARES, based on the EAGLE model) track galaxy evolution at higher resolution by zooming into regions of interest in a cosmological simulation and re-simulate at higher mass resolution and in some cases with more fundamental treatments of baryonic processes. \nAll numerical cosmological hydrodynamic simulations require the use of 'subgrid' prescriptions to represent physical processes that operate at physical scales below the resolution limit of these simulations (e.g. star formation, stellar feedback, black hole growth and feedback, etc.). Hydrodynamic simulations are subject to tension between the simulated volume and resolution. Large volumes are required to capture the rare over-dense regions that host massive galaxies (the ones of interest in this comparison) and high mass resolution is required to properly resolve the evolution and assembly history of these galaxies. Thus it is extremely challenging to simultaneously capture the rare peaks where high redshift galaxies form, while simultaneously resolving these low-mass halos with sufficient numbers of particles. \nOn the other hand, the semi-analytic modeling technique uses phenomenological recipes to track the formation and evolution of galaxies in dark matter halo merger trees and is able to predict a wide variety of physical and observable properties of galaxies with a relatively low computational cost. These models are typically capable of simulating galaxies over a wider mass range and are less susceptible to the volume-resolution tension faced by hydrodynamic simulations (though even obtaining adequate dynamic range for dark matter only simulations and halo merger trees is challenging; see Section 8.2 for a more detailed discussion). For both the semi-analytic models and the hydrodynamic simulations described above, the predictive power of these models relies on the assumption that the physical recipes within, often derived from or calibrated to nearby observations, accurately represent the processes that drive galaxy formation in the high-redshift universe. \nA different class of models are (semi-)empirical methods, also known as subhalo abundance matching (SHAM) or halo occupation distribution (HOD), which efficiently map the properties of a large ensemble of galaxies onto the properties of dark matter halos guided by a set of observed quantities and scaling relations (see \nWechsler & Tinker 2018 for detailed review). This approach is independent of specific galaxy formation models and is guaranteed to match the observational constrains to which these models are calibrated. However, extrapolating these models to redshifts different from those where they were calibrated has less physical grounding. \nThis is only a broad overview of the simulation methods that produced the results presented in Fig. 14, with the aim of emphasizing that these results are produced with different methods, each having their own advantages and disadvantages. We refer the reader to these works for a full description of the design of these simulations and their performance against observational constraints.", '8.1. Observational Effects': "In the previous subsections we showed that the evolution of the UV luminosity function appears to be 'slowing' at z > 10, and that the abundance of z > 9 galaxies significantly exceeds predictions by most physicallybased theoretical models. Here we explore several possible reasons for these unexpected results. At this early time with JWST , we must first acknowledge that the purity of the sample is uncertain. Should the majority of our galaxy sample turn out to be low-redshift interlopers, it would put our results more in line with the theoretical predictions. While this is unlikely given our detailed photometry and selection process, these strong claims require strong evidence, thus spectroscopic confirmation is a must. Without this, it remains possible that heretofore unexpected contaminants could be dominating our sample. The CEERS spectroscopic program will observe a number of these sources, and thus we may soon be able to gain further confidence in our galaxy sample. We do note that non-detections in ALMA dustcontinuum followup of three z ≳ 11 sources in other early JWST fields strengthens the high-redshift solutions in all four cases (e.g. Fujimoto et al. 2022; Bakx et al. 2022; Yoon et al. 2022). \nOne other observational systematic effect which could affect these results is that of aperture corrections. In the above sections we described our multi-step approach to deriving total fluxes, accounting both for what can be detected in the images, and then using simulations to correct for any missing flux from the wings of the PSF. As one additional check, in § 6.2 we explored whether our fluxes are significantly systematically brighter (or fainter) than other objects published in the literature, finding that while there was significant scatter, there was no evidence that our fluxes were systematically brighter \n(especially by the large amount needed to explain the luminosity function excess). Nonetheless, future improvements in photometry can increase confidence in these results.", '8.2. Theoretical Implications': "Here we speculate on potential physical causes for the observed abundances should future observations validate our results. One potential explanation would be a complete absence of dust attenuation (e.g. Ferrara et al. 2022; Mason et al. 2022). As shown in Figure 14 for those models where we plot the (un) attenuated predictions as thicker (thinner) lines (when available), higher number densities are predicted. While a full analysis of the colors of these galaxies is beyond the scope of this paper, their SEDs do appear quite blue, and thus it is possible that they have little dust attenuation. Of note is that bright z ∼ 9-10 galaxies exhibit only marginally redder spectral slopes. For example, Tacchella et al. (2022) found a median UV spectral slope of ∼ -2 for M UV = -21, implying that a small amount of dust attenuation was present. Simulations (FLARES, Simba ) predict a large ∼ 3 × reduction in surface densities due to dust, while the Delphi SAM predicts much less. Should a lack of dust attenuation at z ∼ 10-12 be responsible for our observed evolution, one may expect to see the luminosity function at higher redshifts continue to decline. \nMost models we have compared to employ some kind of 'Kennicutt-Schmidt' (KS) star-formation law, relating the star-formation rate surface density to the dense gas surface density, assuming a constant efficiency per free fall time. While models such as the SC SAM (Yung et al. 2019a,b) adopt a KS law that becomes steeper at higher gas surface densities, leading to higher star formation efficiencies at early times, when the typical gas densities are higher, these models still imply a fairly long gas depletion time compared to the age of the Universe at these extreme high redshifts. For example, in the local universe, the depletion time for dense (molecular) gas is ∼ 1.5 Gyr. This timescale does appear to be shorter at higher redshift, as short as ∼ 0.7 Gyr at z = 2, and perhaps even shorter at higher redshifts (e.g. Sommovigo et al. 2022). The Behroozi & Silk (2015) model would require stars to form at the same rate that gas is funneled into halos, effectively assuming a negligibly short gas depletion time. It is also possible that the star formation efficiency could be higher in low metallicity gas, though the opposite trend has been proposed (e.g. Krumholz & Dekel 2012), or that stellar driven winds are weaker. Additionally, given the extremely high gas densities at these high redshifts, the cold neutral medium \nitself may also participate in star formation in addition to the molecular phase typically tracked by some simulations (Bialy & Sternberg 2019). \nAnother explanation for the poor match between the predictions and observations in SAMs is the mass and temporal resolution of halo merger trees. The halo merger and assembly histories, as an important component of semi-analytic models and some empirical models, can either be constructed with the extended PressSchechter (EPS) formalism (e.g. Somerville & Kolatt 1999) or extracted from N -body cosmological simulations (e.g. Behroozi et al. 2013). However, even the current generation state-of-the-art cosmological simulations only saved a few dozen snapshots at z > 6 and among which only a handful at z ≳ 13, which is insufficient to properly capture the merger histories of these early-forming halos. Furthermore, the bulk of these high-redshift halos have masses near the resolution limit of these cosmological simulations, which further limit their ability to resolve halo merger trees at these extreme redshifts. On the other hand, while EPS-based merger trees have been compared to and shown to be in good statistical agreement with n -body merger trees, they are untested in the mass and redshift ranges that are explored here (see Yung et al. (2020) for a detailed discussion). \nConversely, the poor resolution of the gas content in cosmological hydrodynamic simulations that does not resolve the multi-phase nature of the ISM could contribute to the discrepancy between predictions and observations. It has been repeatedly shown that increasing the resolution typically leads to more dense gas and thus, burstier star formation (without adjusting any free parameters; although this depends on the nature of the sub-grid recipes used for star formation and stellar feedback). Thus, solely increasing the resolution in cosmological hydro simulations, such as in the Lyra cosmological zoom simulation set (Gutcke et al. 2022), and/or improving the sub-grid recipes for the ISM, star formation and stellar feedback, may partly alleviate the poor match between simulations and these observations. \nFinally, and perhaps most interestingly, would be an evolution in the initial mass function (IMF). It has long been predicted that the first stars would have a topheavy IMF (e.g. Bromm et al. 2001; Clarke & Bromm 2003), not only due to extremely low metallicities (e.g., Chon et al. 2021; Sharda & Krumholz 2022), but also the rising cosmic microwave background temperture floor (e.g., Larson 1998). While the relatively massive systems we see here are unlikely to be dominated by metalfree stars, it is possible their metallicities are low enough to affect the characteristic mass of their IMF. In fact, \nFigure 15. The relation between rest-UV absolute magnitude and halo mass, obtained via abundance matching the observed UV luminosity functions assuming the DPL fits from Finkelstein & Bagley (2022) for z = 4-8, and extrapolated to z ∼ 11 (the thick blue curve is equivalent to the average of the darker gray shaded region in Figure 13). For our observed z ∼ 11 UV luminosity function we use the z = 9 DPL fit from Finkelstein & Bagley (2022) as it is very consistent with our observations (Figure 13). This relation implies that our observed galaxies at M UV = -20 (where our observations place the tightest constraints) have a host halo mass of log(M h /M ⊙ ) = 10.55. At this halo mass, the expected UV luminosity based on the expected UV luminosity function is 0.6-0.7 mag fainter. This implies that based on the observed abundances, our observed galaxies are 1.8 × more UV luminous than expected from extrapolation of HST results. While larger sample sizes and spectroscopic confirmations are needed to have greater confidence in the z ∼ 11 luminosity function, should these results be confirmed it implies that z > 10 galaxies are more luminous in the rest-UV than expected. One possible explanation could be an increasing prevalence at higher redshifts of a top-heavy initial mass function, which is predicted to dominate at very low metallicities. \n<!-- image --> \nthese observations now probe so close to the Big Bang, that it would be surprising if the IMF did not begin to evolve at some point. \nWe therefore explore what excess UV luminosity is needed to match our observation. We first do a simple test by exploring how much we would have to shift our observed luminosity function data in Figure 13 in UV magnitude such that they would match the DPL luminosity function fits from lower redshift extrapolated to z ∼ 11. We find our data would need to shift ∼ 1 magnitude fainter to match these empirical predictions. Therefore if an evolving IMF was the sole explanation for the higher-than expected luminosities, the UV luminosity would need to be boosted by about a factor of ∼ 2 . 5. \nAs a more complex version of this analysis we estimate the total masses for the host halos of galaxies across the luminosity function via abundance matching. We follow the procedures of Reddick et al. (2013), assuming a 0.2 dex scatter in UV luminosity at fixed halo mass, showing the observed relations between halo mass and UV absolute magnitude in Figure 15; qualitatively simi- \nlar findings applied for scatter ranging from 0 -0 . 4 dex. For z ≤ 9, we use the DPL UV luminosity functions from Finkelstein & Bagley (2022). For the expected evolution to z ∼ 11, we extrapolate their fits to z ∼ 12, and show the volume-averaged extrapolated LF for z = 9 . 5 -12 with the bright blue line. For our actual observed values, as we have not fit a functional form here due to our limited dynamic range in luminosity, we assume the z = 9 DPL fit from Finkelstein & Bagley (2022) as this is most consistent with our observed number densities in Figure 13. We show in red the M halo - M UV relation when abundance matching the average of the z ∼ 9.5-12 halo mass functions to this assumed observed luminosity function. In this figure, we highlight M UV = -20, which is where our z ∼ 11 observations are most constraining. Our abundance matching analysis predicts that based on our observations, these galaxies are hosted in halos with log(M h /M ⊙ ) = 10.55 (upper light-red bar). If we had instead used the extrapolated z ∼ 11 results, these halos should host galaxies with M UV = -19.3 to -19.4 (lower light-red bar). This implies that our observed galaxies \nare ∼ -0.6-0.7 mag brighter (1.8 × in UV luminosity) than expected for a galaxy in their host halo. \nA UV-luminosity boost of ∼ 1.8-2.5 is not unexpected in the context of a top-heavy IMF. Raiter et al. (2010) explore the impact of differing IMFs on the UV luminosity (specifically investigating the conversion from UV luminosity to SFR). They find that for the top-heavy IMFs proposed by Tumlinson (2006), which peak at ∼ 10-40 M ⊙ , yield a UV luminosity ∼ 0.4 dex higher than Salpeter at a stellar metallicity of log (Z/Z ⊙ ) = -2. Such a brightness increase is exactly what we find would be needed to bring our observed luminosity function in line with expectations. This interpretation was also considered by Harikane et al. (2022), who found that some top-heavy IMF models can boost the UV luminosity by ∼ 3-4 × (e.g. Zackrisson et al. 2011). The top end of the expected UV luminosity boost was discussed by Pawlik et al. (2011), who predicted up to a factor of 10 increase in UV luminosity for a top-heavy IMF in a zero metallicity system. \nIf the interpretation of invoking a top-heavy IMF were correct, there would be a number of additional effects and consequences that may be detectable, thus providing an independent cross-check. One such diagnostic is the predicted boost to nebular line emission, including the occurrence of strong He II 1640 ˚ A features (e.g. Bromm et al. 2001; Raiter et al. 2010), accessible via our upcoming CEERS spectroscopic follow-up. A qualitatively different signature of a top-heavy IMF would be a supernova rate that may be enhanced by up to an order of magnitude, per unit stellar mass, resulting in strong SN-driven galactic winds (e.g., Dayal & Ferrara 2018; Jaacks et al. 2019). \nWe note that the high observed galaxy number densities do not represent a fundamental problem for hierarchical structure formation models. For instance, using a Simba simulation with all explicit feedback processes turned off and no extinction results in ∼ 2 × higher number densities than observed (even with a standard IMF), and the empirical Behroozi & Silk (2015) model can match the data. Hence the results do not necessarily challenge the ΛCDM paradigm, but rather our understanding of the physics of early galaxy evolution within that paradigm.", '9. SUMMARY AND CONCLUSIONS': "We have presented the results from a study of galaxies at z > 9 in the first epoch of NIRCam imaging from the Cosmic Evolution Early Release Science program. These imaging cover ∼ 35 arcmin 2 with seven photometric filters (six broadband and one medium band) covering ∼ 1-5 µ m, reaching m ≳ 29. In addition to being \nsome of the widest moderate-depth imaging available early in Cycle 1, these data probe a parameter space in wavelength and depth optimal for studying these early redshifts. \nFollowing a detailed reduction of the data (with full details available in Bagley et al. 2022c) we measure photometry from all sources in the field. Using a combination of the F277W and F356W image as the detection images, we create a 13-band photometric catalog, inclusive of imaging in six HST filters in this field. We emphasize the calculation of robust colors, total fluxes, and flux uncertainties, using simulations to validate our choices in cataloging. We explore any systematic offsets in our photometry by comparing to the best-fitting model templates for ∼ 800 spectroscopically-confirmed sources in our field, finding that the large zeropoint offsets that affected early JWST /NIRCam studies have been resolved. We do measure (and apply) offsets from ∼ 1-5%, which are reflective of potential residual zeropoint corrections in the NIRCam imaging, adjustments to our estimations of the total fluxes, and potential mis-matches between the used template set and the SEDs of the real galaxies. \nWe estimate photometric redshifts for all sources in our catalog using EAZY . In addition to the standard set of 12 FSPS templates, we add six new templates designed to span the blue colors expected in very early galaxies, as well as very strong emission lines expected at low metallicities. We designed a set of selection criteria to robustly select galaxies at z > 9, balancing our desire to maximize completeness with our need to minimize contamination. These selection criteria demand both robust detection significance (signal-to-noise ratio > 5.5 in at least two filters, measured in 0.2 '' diameter apertures), as well as photometric redshift probability distribution functions that are strongly constrained to z > 8.5. \nFollowing visual inspection to remove any remaining spurious sources (shown in the Appendix), our selection resulted in a sample of 26 robust candidate galaxies at z > 8.5. The majority of the sample (15 galaxies) lies at 8.5 < z < 10, while nine galaxies lie at 10 < z < 12. Two candidates lie at higher redshifts - the previously published z ∼ 16.5 source (Donnan et al. 2022), and a new candidate at z ∼ 13. While a full analysis of the properties of these galaxies is reserved for future work, their rest-frame UV SEDs are fairly uniformly blue, and the galaxies are compact, with a median half-light radius of 3.6 pixels (0.11 '' , or 0.46 kpc). We explore information from multi-wavelength constraints on these galaxies, and find no significant detections in either X-ray or submm/mm wavelengths, strengthening the conclusion that these sources reside at z > 9. We compare to the \nfew previous searches for z > 9 sources in this field, and find that for sources in common between studies, the photometric redshifts agree quite well. Of the two known HST sources previously published in this field, we very robustly constrain the SED of one, placing it at the slightly lower redshift of z ∼ 8.1. The other is completely absent in our NIRCam data, and we conclude it was a likely supernova serendipitously captured by the HST data taken in 2013. \nWe estimate our sample completeness as a function of redshift and magnitude using source-injection simulations, and present an early look at the z ∼ 11 restUV luminosity function. We find that the abundance of modestly bright ( M UV ∼ -20) galaxies at z ∼ 11 does not appear to be evolving from z ∼ 9 - 11, which is unexpected given the steady decline in the abundance of such galaxies from lower redshifts to z = 8, though such non evolution at the very bright end from z = 8-10 had previously been discussed by Bowler et al. (2020). We then compare the surface density of our sources to a variety of model predictions, finding that even after accounting for several sources of systematic and random uncertainty, the observed abundance of galaxies is in significant excess of these predictions. \nWeexplore several potential explanations for these unexpected results. While not the most exciting, significant sample contamination cannot be conclusively ruled out. These data represent our first foray into a new cosmic epoch, and spectroscopic confirmation of the redshifts to at least a subset of these ultra-high-redshift sources are necessary to gain confidence in our sample selection processes. However, such data will begin to flow soon, with CEERS scheduled to spectroscopically observe ∼ 10 of these sources in late 2022 (though these high redshifts may necessitate longer exposure times for future cycle programs). \nShould these high abundances of z = 9-13 galaxies be confirmed, we explore what possible changes in the models could bring their predictions into agreement with observations. One very exciting possibility is that we are beginning to probe an era where star-formation in galaxies is dominated by a top-heavy IMF due to the presence of very low metallicities, which could increase the ratio of UV luminosity per unit halo mass. We explore the 'excess' UV luminosity from our observations, both by comparing to the expected UV luminosity function based on extrapolations from lower redshift and through \nan abundance matching exercise, and find that our UV luminosities may be enhanced by 1.8-2.5 × . This is very similar to the predicted excess UV emission from a lowmetallicity stellar population where the IMF peaks at 10-40 M ⊙ (Tumlinson 2006) of a factor of ∼ 2.5 × , implying a top-heavy IMF is a plausible physical explanation. We also discuss how potential changes to the dust attenuation, star formation law, galactic feedback, and resolution of numerical simulations could collectively contribute to reconciling our observations with model predictions. \nThese possibilities are exciting, and while one might expect that at z > 10 we should expect to see changes in star-formation physics such as a top-heavy IMF, our results are just a first glimpse, and the data available in the near future will provide much stronger constraints. Specifically, the remainder of Cycle 1 will see the creation of high-redshift galaxy samples orders of magnitude larger than we present here from the combination of the full CEERS survey, COSMOS-Web (PIs Casey & Kartaltepe), PRIMER (PI Dunlop) and NGDEEP (PIs Finkelstein, Papovich & Pirzkal), along with JADES (PI Rieke). Additionally, Cycle 1 will also see spectroscopic followup with NIRSpec of NIRCam identified sources from CEERS and JADES, and several Cycle 2 programs will certainly target these sources with deep observations. While it is early days with JWST , our first-look CEERS results provide an enthralling glimpse of the potential secrets the early universe has which our observations can unlock. \nFacility: HST (ACS, WFC3) \nFacility: \nJWST (NIRCam) \n- We acknowledge that the location where this work took 1\n- place, the University of Texas at Austin, that sits on in2\n- digenous land. The Tonkawa lived in central Texas and 3\n- the Comanche and Apache moved through this area. 4\n- We pay our respects to all the American Indian and In5\n- digenous Peoples and communities who have been or 6\n- have become a part of these lands and territories in 7\n- Texas, on this piece of Turtle Island. We acknowledge 8\n- support from NASA through STScI ERS award JWST9\n- ERS-1345. We thank Marcia Rieke, Daniel Schaerer, 10\n- Volker Bromm and Mike Boylan-Kolchin for helpful con11\n- 12\n- versations.", 'REFERENCES': "Adams, N. J., Conselice, C. J., Ferreira, L., et al. 2022, \narXiv e-prints, arXiv:2207.11217 \nAretxaga, I. 2015, in IAU General Assembly, Vol. 29, \n2258051 \nRieke, M. J., Kelly, D., & Horner, S. 2005, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 5904, Cryogenic Optical Systems and Instruments XI, ed. J. B. Heaney & L. G. Burriesci, 1-8 \nRobertson, B. E. 2021, arXiv e-prints, arXiv:2110.13160 Rodney, S. A., Riess, A. G., Strolger, L.-G., et al. 2014, AJ, 148, 13 \nRojas-Ruiz, S., Finkelstein, S. L., Bagley, M. B., et al. 2020, ApJ, 891, 146 \nRyan, R. E., J., Hathi, N. P., Cohen, S. H., & Windhorst, R. A. 2005, ApJL, 631, L159 \nRyan, R. E., J., & Reid, I. N. 2016, AJ, 151, 92 \n- Ryan, R. E., Thorman, P. A., Yan, H., et al. 2011, ApJ, 739, 83 \nSharda, P., & Krumholz, M. R. 2022, MNRAS, 509, 1959 \nSomerville, R. S., & Dav'e, R. 2015, ARA&A, 53, 31 \nSomerville, R. S., & Kolatt, T. S. 1999, MNRAS, 305, 1 \nSommovigo, L., Ferrara, A., Pallottini, A., et al. 2022, \nMNRAS, 513, 3122 \nStark, D. P. 2016, ARA&A, 54, 761 \n- Stefanon, M., Labb'e, I., Bouwens, R. J., et al. 2019, ApJ, 883, 99\n- Tacchella, S., Finkelstein, S. L., Bagley, M., et al. 2022, ApJ, 927, 170\n- Trenti, M., Bradley, L. D., Stiavelli, M., et al. 2011, ApJL, 727, L39 \nTumlinson, J. 2006, ApJ, 641, 1 \nWechsler, R. H., & Tinker, J. L. 2018, ARA&A, 56, 435 Whitler, L., Endsley, R., Stark, D. P., et al. 2022, arXiv e-prints, arXiv:2208.01599 \nWilkins, S. M., Stanway, E. R., & Bremer, M. N. 2014, MNRAS, 439, 1038 \nWilkins, S. M., Vijayan, A. P., Lovell, C. C., et al. 2022a, arXiv:2204.09431 \n- -. 2022b, MNRAS, 517, 3227 \nYan, H., Windhorst, R. A., & Cohen, S. H. 2003, ApJL, 585, L93 \nYang, L., Morishita, T., Leethochawalit, N., et al. 2022, ApJL, 938, L17 \n- Yoon, I., Carilli, C. L., Fujimoto, S., et al. 2022, arXiv e-prints, arXiv:2210.08413\n- Yung, L. Y. A., Somerville, R. S., Finkelstein, S. L., Popping, G., & Dav'e, R. 2019a, MNRAS, 483, 2983\n- Yung, L. Y. A., Somerville, R. S., Finkelstein, S. L., et al. 2020, MNRAS, 496, 4574 \nYung, L. Y. A., Somerville, R. S., Popping, G., et al. 2019b, MNRAS, 490, 2855 \nYung, L. Y. A., Somerville, R. S., Ferguson, H. C., et al. 2022, MNRAS, 515, 5416 \nZackrisson, E., Rydberg, C.-E., Schaerer, D., Ostlin, G., & Tuli, M. 2011, ApJ, 740, 13 \nZavala, J. A., Aretxaga, I., Geach, J. E., et al. 2017, \nMNRAS, 464, 3369 \nZavala, J. A., Buat, V., Casey, C. M., et al. 2022, arXiv e-prints, arXiv:2208.01816 \nTable 4. NIRCam Photometry for z > 8.5 Galaxy Sample \nNote -All fluxes are in nJy. The horizontal lines distinguish our z > 12, z = 10-12 and z = 8.5-10 samples.", 'A. FULL PHOTOMETRY': 'In Tables 4 and 5 we list the measured photometry for our final sample.', 'B. Z ∼ 9 SAMPLE PLOTS': 'Here we show the cutout images (Figure 16 and 17), and SED plots (Figure 18) for the z ∼ 9 sample as described in § 5.3, presented here in the Appendix for clarity in the main text. \nFigure 19 shows two objects removed from our sample after re-measuring colors in smaller apertures, due to the stretching of their Kron apertures by nearby bright sources. Figure 20 shows cutout images for the 36 sources removed as spurious sources following visual inspection described in § 4.3. We list the coordinates for all removed sources in Table 6. \nTable 5. HST ACS and WFC3 Photometry for z > 8.5 Galaxy Sample \nNote -All fluxes are in nJy. The horizontal lines distinguish our z > 12, z = 10-12 and z = 8.5-10 samples. We do not include a column for F105W as none of these candidate galaxies were covered by the sparse amount of F105W imaging in this field.', 'z ~ 8.5-10, m277 < 27.6': 'ACS606+814 \nF115W \nF150W F200W F277W F356W F410M F444W \nFigure 16. Similar to Figure 5, showing the eight candidates with best-fit photometric redshifts of z ∼ 9 and a F277W magnitude brighter than 27.6. \n<!-- image -->', 'z ~ 8.5-10, m277 > 27.6': "<!-- image --> \nFigure 17. Similar to Figures 5, showing the nine candidates with best-fit photometric redshifts of z ∼ 9 and a F277W magnitude fainter than 27.6. \nFigure 18. The SEDs and P ( z )'s of the z ∼ 9 sample. \n<!-- image --> \n5 a \nr \nc \ns \ne \nc \n1. \n5 a \nr \nc \ns \ne \nc \nFigure 19. Two sources which had incorrectly drawn Kron apertures due to the presence of nearby bright sources. The large plot shows the SED, with the Kron NIRCam (HST) photometry in blue (green). The best-fit EAZY model to these data is shown in purple (with squares denoting the model bandpass-averaged fluxes), with the best-fit z < 7 solution shown in orange. The small gray circles show the fluxes measured in 0.3 '' diameter apertures, with the gray line showing the EAZY fit to these small-aperture fluxes. The P(z) for all three EAZY runs are shown in the top-left. The large image shows a 5 '' × 5 '' cutout around each source, highlighting the nearby neighbor responsible for stretching the Kron aperture leading to much brighter Kron fluxes. The small images show a 1.5 '' region around each source in the seven NIRCam bands (and the stacked ACS dropout bands), with the red circle denoting a 0.3 '' diameter region around the source. The P(z) from the colors measured in this small circular aperture prefers lower redshifts in both cases, thus these sources were removed from the sample. \n<!-- image --> \n<!-- image --> \n5 a \nr \nc \ns \ne \nc \n1. \n5 a \nm \nr \nc \n= \n2 \n5 \ns \ne \n. \nc \n9 \ne \nk \ni \np \non S \nti \nc \na \nr \nff \ni \nD \nCR \n/ \nt \nc \na \nomp \nC \nFigure 20. This compilation shows 3 '' × 3 '' cutout images of the 36 sources identified as spurious from our visual inspection of the initial list of candidates. The majority of these spurious detections come from very near to image edges, which are easily identifiable (and can in the future be automated). Six objects are obvious diffraction spikes. The remaining eight sources are very compact and boxy, and visible only in the LW channels. We conclude these are highly likely to be unflagged cosmic rays, which will be better flagged in future reductions (Bagley et al. 2022b). We note that 7/8 are in the CEERS3 or 6 pointings, which had fewer images with longer exposure times than CEERS1 and 2, leading to more numerous cosmic ray hits. \n<!-- image --> \nTable 6. List of Removed Sources \nNote -ID and coordinates of sources shown in Figures 19 and 20."}

2024ApJS..275...38I

The exponential growth of astronomical literature poses significant challenges for researchers navigating and synthesizing general insights or even domainspecific knowledge. We present pathfinder a machine learning framework designed to enable literature review and knowledge discovery in astronomy focusing on semantic searching with natural language instead of syntactic searches with keywords. Utilizing stateoftheart large language models LLMs and a corpus of 385166 peerreviewed papers from the Astrophysics Data System pathfinder offers an innovative approach to scientific inquiry and literature exploration. Our framework couples advanced retrieval techniques with LLMbased synthesis to search astronomical literature by semantic context as a complement to currently existing methods that use keywords or citation graphs. It addresses complexities of jargon named entities and temporal aspects through timebased and citationbased weighting schemes. We demonstrate the tools versatility through case studies showcasing its application in various research scenarios. The systems performance is evaluated using custom benchmarks including singlepaper and multipaper tasks. Beyond literature review pathfinder offers unique capabilities for reformatting answers in ways that are accessible to various audiences e.g. in a different language or as simplified text visualizing research landscapes and tracking the impact of observatories and methodologies. This tool represents a significant advancement in applying artificial intelligence to astronomical research aiding researchers at all career stages in navigating modern astronomy literature.

2024-12-01T00:00:00Z

['2024arXiv240801556I', '2024ApJS..275...38I', '10.48550/arXiv.2408.01556', '10.3847/1538-4365/ad7c43', 'arXiv:2408.01556']

['Astronomical reference materials', 'Astronomy web services', 'History of astronomy', 'Computational methods', 'Astronomy data visualization', '90', '1856', '1868', '1965', '1968', 'Astrophysics - Instrumentation and Methods for Astrophysics', 'Computer Science - Digital Libraries', 'Computer Science - Information Retrieval']

pathfinder A Semantic Framework for Literature Review and Knowledge Discovery in Astronomy

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']

https://arxiv.org/pdf/2408.01556.pdf

{'pathfinder : A Semantic Framework for Literature Review and Knowledge Discovery in Astronomy': "Kartheik G. Iyer , 1, ∗ Mikaeel Yunus , 2 Charles O'Neill , 3 Christine Ye , 4 Alina Hyk , 5 Kiera McCormick , 6 Ioana Ciucă , 7, 8, 9 John F. Wu , 10, 2 Alberto Accomazzi , 11 Simone Astarita , 12 Rishabh Chakrabarty, 13 Jesse Cranney , 14 Anjalie Field , 15 Tirthankar Ghosal , 16 Michele Ginolfi , 17, 18 Marc Huertas-Company , 19, 20, 21, 22 Maja Jabłońska, 7 Sandor Kruk , 12 Huiling Liu, 23, 24 Gabriel Marchidan, 25 Rohit Mistry, 26 J.P. Naiman , 27 J. E. G. Peek , 10, 2 Mugdha Polimera , 11 Sergio J. Rodríguez Méndez , 3 Kevin Schawinski , 28 Sanjib Sharma , 10 Michael J. Smith , 19 Yuan-Sen Ting , 29, 30 Mike Walmsley , 31, 32 \n(UniverseTBD) \n1 \nColumbia Astrophysics Laboratory, Columbia University, 550 West 120th Street, New York, NY 10027, USA \nDepartment of Physics and Astronomy, Johns Hopkins University, 3400 N. Charles Street, Baltimore, MD 21218, USA \n3 \nSchool of Computing, The Australian National University, 108 North Rd, Acton ACT 2601, Australia \n4 \nStanford University, 450 Jane Stanford Way, Stanford, CA 94305, USA \n5 School of Electrical Engineering and Computer Science, Oregon State University, 1500 SW Jefferson Way, Corvallis, OR 97331, USA 6 Department of Engineering, Loyola University Maryland, 4501 North Charles Street, Baltimore, MD 21210, USA 7 Research School of Astronomy & Astrophysics, Australian National University, Cotter Rd., Weston, ACT 2611, Australia 8 School of Computing, Australian National University, Acton, ACT 2601, Australia \n9 Kavli Institute for Particle Astrophysics and Cosmology and Department of Physics, Stanford University, Stanford, CA, USA, 94305 10 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA 11 Center for Astrophysics, Harvard & Smithsonian, Cambridge, MA 02138, USA \n12 European Space Agency (ESA), European Space Astronomy Centre (ESAC), Camino Bajo del Castillo s/n, 28692 Villanueva de la Cañada, Madrid, Spain \n13 Independent Researcher, Paris, France \n14 Astralis-AITC - Stromlo, RSAA, Australian National University, Cotter Road, Weston, ACT2600, Australia \nDepartment of Computer Science, Johns Hopkins University, 3400 North Charles Street, Baltimore, MD 21218, USA \n15 \n16 National Center for Computational Sciences, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA \nDipartimento di Fisica e Astronomia, Universita di Firenze, Via G. Sansone 1, 50019, Sesto Fiorentino (Firenze), Italy \n18 \nINAF - Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50125, Firenze, Italy \n19 Instituto de Astrofisica de Canarias, C. Via Lactea, 1, E-38205 La Laguna, Tenerife, Spain \n20 Universidad de la Laguna, dept. Astrofisica, E-38206 La Laguna, Tenerife, Spain \n21 Universite Paris-Cite, LERMA - Observatoire de Paris, PSL, Paris, France \n22 SCIPP, University of California, Santa Cruz, CA 95064, USA \n23 Key Laboratory for Research in Galaxies and Cosmology, Department of Astronomy, University of Science and Technology of China, Hefei, Anhui 230026, China \n24 \nSchool of Astronomy and Space Science, University of Science and Technology of China, Hefei 230026, China \n25 Ias , i AI, Ias , i, Romania \n26 Xaana AI, Canberra, Australia \n27 School of Information Sciences, University of Illinois, Urbana-Champaign, 61820, USA \n28 Modulos AG, 8005 Zurich, Switzerland \n29 Department of Astronomy, The Ohio State University, Columbus, OH 43210, USA \n30 Center for Cosmology and AstroParticle Physics (CCAPP), The Ohio State University, Columbus, OH 43210, USA \nDunlap Institute for Astronomy and Astrophysics, University of Toronto, 50 St. George Street, Toronto, ON M5S 3H4, Canada \n31 \n- 32 Jodrell Bank Centre for Astrophysics, Department of Physics & Astronomy, University of Manchester, Oxford Road, Manchester, M13 9PL, UK", 'ABSTRACT': "kgi2103@columbia.edu \n2 \n17 \nThe exponential growth of astronomical literature poses significant challenges for researchers navigating and synthesizing general insights or even domain-specific knowledge. We present pathfinder , a machine learning framework designed to enable literature review and knowledge discovery in astronomy, focusing on semantic searching with natural language instead of syntactic searches with keywords. Utilizing state-of-the-art large language models (LLMs) and a corpus of 350,000 peer-reviewed papers from the Astrophysics Data System (ADS), pathfinder offers an innovative approach to scientific inquiry and literature exploration. Our framework couples advanced retrieval techniques with LLMbased synthesis to search astronomical literature by semantic context as a complement to currently existing methods that use keywords or citation graphs. It addresses complexities of jargon, named entities, and temporal aspects through time-based and citation-based weighting schemes. We demonstrate the tool's versatility through case studies, showcasing its application in various research scenarios. The system's performance is evaluated using custom benchmarks, including single-paper and multi-paper tasks. Beyond literature review, pathfinder offers unique capabilities for reformatting answers in ways that are accessible to various audiences (e.g. in a different language or as simplified text), visualizing research landscapes, and tracking the impact of observatories and methodologies. This tool represents a significant advancement in applying AI to astronomical research, aiding researchers at all career stages in navigating modern astronomy literature. \nKeywords: Astronomical reference materials(90) - Astronomy web services(1856) - History of astronomy(1868) - Computational methods(1965) - Astronomy data visualization(1968)", '1. INTRODUCTION': "As one of the oldest scientific disciplines, astronomy has amassed an enormous body of literature over time. Modern astronomical libraries and recordkeeping services like the Astronomical Data System (ADS) (Accomazzi et al. 2015) and preprint servers like arXiv provide lasting repositories for accessing current research on various astronomical subfields, with records on ADS extending back to the early 16th century. As the body of astronomical literature grows (at an ever accelerating rate), this creates a growing problem of keeping track of the literature, with it becoming harder over time to keep track of relevant papers and contextualise the information contained in them while writing new papers. In fact, with the advent of new observatories like ALMA and JWST and new modalities of observations like gravitational waves, the literature has become challenging for even experienced researchers to keep pace with. This is exacerbated by a growing need for interdisciplinary efforts, which means that astronomers often need to keep track of multiple fields of literature, such as electronics and instrumentation, high performance computing, statistics, machine learning, and computer vision. \nAt best, this leads to a much larger amount of time and effort spent in organising and cataloguing papers for individual researchers, and at worst it can lead to a \nsplintering of the research landscape with researchers resorting to a friends-of-friends or in-group citation framework while writing papers. This situation also creates a barrier to entry for aspiring students and researchers trying to enter the field and perform their first literature search, especially in the absence of an authoritative review on their chosen topic. \nWhile this is also true of fields other than astronomy, we have the unique distinction of having a large body of publicly accessible data, code and literature (Genova 2023), which provides a unique opportunity for developing methods that can ingest, retrieve and synthesize literature in a way that is useful for a wide range of audiences (Iyer 2021; Grezes et al. 2021; Rodríguez et al. 2022; Blanco-Cuaresma et al. 2023; Dung Nguyen et al. 2023). To this end, we explore the use of state-of-the-art machine learning methods in conjunction with a corpus of papers from ADS and arXiv to find relevant literature and provide initial starting points for answering questions across a variety of levels. \nLarge language models (LLMs) have seen rapid advancement and adoption in recent years, with models like GPT-4 (OpenAI et al. 2024) and LLaMA (Touvron et al. 2023) demonstrating impressive capabilities across a wide range of tasks. In academic contexts, LLMs are increasingly being used to assist with explaining advanced concepts (Prihar et al. 2023) and perform literature review (Li et al. 2024b; Tao et al. 2024), and possibly with writing papers (Liang et al. 2024), even \nin astronomy (Astarita et al. 2024). However, their application remains controversial due to concerns about accuracy, bias, accidental plagiarism (Pervez & Titus 2024), and the potential for hallucination (e.g., Zhang et al. 2023). Despite these challenges, many researchers are exploring ways to leverage LLMs as tools to augment human expertise and accelerate scientific discovery (Van Noorden & Perkel 2023), particularly in fields with vast and rapidly growing bodies of literature like astronomy. \nThe notion of using machine-learning methods for improving literature surveys is not a particularly new concept, and such tools have been accessible since the early 1990s with methods like n-grams (Cavnar et al. 1994; Kondrak 2005), bag-of-words (Zhang et al. 2010), or transformer based models like BERT (Devlin et al. 2018). Versions of these methods including AstroBERT (Grezes et al. 2021) and more recently AstroLLaMA (Dung Nguyen et al. 2023) have been applied to large corpora of astronomical data as proof-of-concept techniques to showcase how NLP and newer language models can successfully ingest with astronomical keywords and scientific jargon. Here, we provide a working pipeline to show how these models can be combined with techniques like retrieval augmented generation (RAG) and agentic LLMs to capture significantly more semantic context and provide hallucination-free literature review at a fraction of the time and cost of manually searching for papers on a given topic. We stress that this is not meant to be a replacement for existing search tools like arxiv.org, the Astronomical Data System (ADS), Google scholar, Benty-Fields or Arxivsorter, but rather a complement to them, with three key advantages: (1) the ability to query the system using natural language, (2) added synthesis to generate a targeted summary of the retrieved documents in context to the question, and (3) exploratory tools to find similar papers in an interpretable embedding space. \nTo do this, we present the pathfinder framework 1 , an open-source, publicly available tool that uses LLMs to answer natural-language astronomy questions using a corpus of ∼ 350 , 000 peer-reviewed papers from ADS going back to 1990. The framework is presented both as open-source code and as an online tool that can be used to find relevant literature, answer questions, and explore the corpus of papers. The current version of the tool uses only abstracts, but can be extended to fulltext in the future. In this paper, we explore the use of pathfinder to (i) visualize papers as a 'landscape' of astronomy research, (ii) find similar/relevant papers by \nperforming a similarity search in embedding space, (iii) answer questions without hallucinations using the embedding space, (iv) explore the impact of different telescopes and observatories on the landscape of research, (v) explore the trends of authors over time, (vi) quantify missing areas that need to be developed further and find areas of interest for future surveys and facilities. \nThe structure of this paper is as follows. In section 2, we describe the dataset used to retrieve papers from. In section 3, we describe the overall pathfinder pipeline. In section 4, we describe evaluation benchmarks used while developing the model. Section 5 describes ways for users to interact with and use pathfinder , and provides some case studies that demonstrate its behaviour across different types of questions. In section 6, we present some larger trends analysed with the pathfinder framework. Section 7 concludes and summarizes the paper and the scope for future work.", '2. DATASET': 'We have compiled a dataset of ∼ 350 , 000 paper abstracts from the ADS 2 and arxiv.org 3 , along with associated metadata including paper titles, publication dates, DOIs, author and affiliation information, and ADS keywords and bibcodes. We have also scraped the bibcodes for papers referenced in and citing any given paper in the dataset, which can be used to further expand the database in future iterations. In addition to this, we have used natural language tools ( spacy running en\\_core\\_web\\_sm ) to determine a set of 20 keywords for each abstract, along with LLM-generated embeddings for each abstract as described in Section 3. The keywords are subsequently used to annotate figures and implement keyword weighting while retrieving papers. \nThe majority of the papers in our current corpus are drawn from an existing list of ∼ 270 , 000 papers classified as astro-ph from the Kaggle arXiv dataset 4 , which contains papers from April 1992 to July 2023 (similar to AstroLLaMA; Dung Nguyen et al. 2023; Perkowski et al. 2024). These papers are further augmented using metadata (bibcodes, citations, dates, authors and affiliations) from ADS. Since there are a number of papers that are not on arxiv.org , we subsequently query ADS for papers from January 1990 to July 2024 to find papers that are not in our dataset and add them, bringing our corpus to N = 352 , 194 . This set will be updated periodically to keep up to date with the current literature, and augmented by a corpus of older papers pro- \nssed with optical character recognition (OCR) as part of future work (Naiman et al. 2023). The dataset is publicly available online 5 . While this is not a complete corpus and primarily draws from the ApJ, MNRAS, A&A, ARAA, Nature, Science, PASA, PASP, and PASJ families of journals, it includes a large sample of relevant work that can be used to test the framework.', '3. BUILDING pathfinder': "This section briefly describes the methods used to construct pathfinder . The codebase is public and available at the pathfinder repository. 6 Briefly, the pipeline is an augmented version of RAG. In standard RAG, the system first retrieves a set of relevant documents for any input user query, and then uses the information therein to synthesise its answer. pathfinder 's augmentations include question categorization, query expansion, reranking, the ability to filter by date, citations and keywords and an alternative reason-thought-act based framework for synthesizing answers, described in further detail in the following sections. Figure 1 shows a schematic of the procedure described in this section.", '3.1. Generation': "We first describe the generation step (right-hand side of Figure 1), which uses the retrieved papers and associated metadata to generate an answer to the user's query.", '3.1.1. Generating Embeddings': 'We compute embeddings for each abstract in our corpus using the text-embedding-3-small model from OpenAI 7 , which is used to encode each abstract into a 1536-dimensional vector. Once the embeddings are computed, we use uniform manifold approximation and projection (UMAP 8 ; McInnes et al. 2018) to create a 2-dimensional embedding of the high-dimensional vector space for easier visualization and further analysis. A heatmap of the embedding space is provided in Figure 2, with the different regions annotated with their most frequently occurring keywords for clarity.', '3.1.2. Text generation with RAG': "Generally, question-answer applications involving LLMs generate an answer following a template (sometimes called a prompt) in response to a query. However, in doing so, there is a danger that the model may \noutput factually incorrect information and lack access to all the available information needed to reply (Roller et al. 2021). To handle both of these problems, RAG forces the LLM to generate the response while using (and possibly citing) a set of document sources (Lewis et al. 2020; Shuster et al. 2021). Given an input query, we first search the full space of papers to find a subset of ∼ 1 -30 papers that are relevant to the input query, retrieved using the methods described in Section 3.2. We then use langchain 9 to set up the RAG system, where the query is passed in along with the abstracts of the papers broken down into chunks for the LLM to then construct an answer. The input prompt template also requires the LLM to be succinct in its responses and respond with 'I don't know' if the LLM does not find sources relevant to the query.", '3.1.3. Text generation with ReAct agents': 'While many of the questions that astronomers tend to ask tools like pathfinder will be factual and need efficient similarity search and synthesis, others are more involved and require multiple lookups to answer. These tend to be questions that require resolving multiple conflicting viewpoints ( consensus evaluation ), combining information across multiple topics ( composition ), or speculating beyond available data ( counterfactual ; see Section 4.4 for a fuller description of the different types of questions). \nA limitation of the RAG framework is that it is incapable of directly answering these questions. To provide a basic framework that can be used to tackle these questions, we use ReAct agents (Reasoning and Acting; Yao et al. 2022), an approach that combines reasoning and acting in LLMs, allowing them to break down complex tasks into more atomic steps and execute them, combined with the RAG framework we have used thus far. Briefly, this system involves pathfinder receiving an input query, followed by the ReAct agent using a LLM to reason about the task and break it down into steps. For each step, the agent acts by using RAG to retrieve relevant information from the paper corpus. It uses the retrieved information to further analyse the data and make queries until it has enough knowledge to answer the question or runs up against the number of allowed iterations. The system is not perfect, with the LLM sometimes stalling in a process loop where it can not find an ideal way to phrase a question. Newer methods exist to use search trees (Yao et al. 2024) or knowledge graphs (Besta et al. 2024) to circumvent these issues. However, given the relatively small number of these questions we \nFigure 1. Schematic showing the overall pathfinder pipeline. \n<!-- image --> \nfound users to ask, those are out of the scope of current work, and will be left for future upgrades in pathfinder .', '3.2. Retrieval': "Because the retrieved documents will strongly impact text generation, it is vital to ensure that we retrieve the most relevant documents to a user's input query. This section describes the procedure by which we retrieve ∼ 1 -30 'topk ' papers (see left-hand side of Figure 1).", '3.2.1. Semantic search & embeddings': "One of pathfinder 's key functionalities is to find similar papers given a natural-language query (building on earlier work e.g., Iyer 2021). For this, it is important to be able to compare the vector corresponding to a query (computed using the same way as the embeddings for the abstracts) to those of paper abstracts and compute a similarity score. In principle, this can be done using any distance metric, and in the current application we use cosine similarity implemented using the Facebook AI Similarity Search (FAISS 10 ) package. FAISS is capable of processing on GPUs and scaling to extremely large datasets (Johnson et al. 2017), making our method future proof for applications to large corpora of literature.", '3.2.2. Generating keywords from abstracts': "We compute a set of keywords for each abstract using the textrank algorithm, set up to identify nouns, adjectives and proper nouns in any input text. For the current application, this has been implemented using the en\\_core\\_web\\_sm model in the spacy NLP package 11 . This is followed by running a peak finding algorithm \nin the 2D UMAP embedding space to identify regions where there is a high concentration of papers. For each peak we consider all papers within a certain radius and \nidentify the most frequently occurring keywords for all the papers in that cluster, and repeat this for all the clusters in our space, followed by an LLM query to synthesize the keyword into an overarching topic or facility (e.g. 'solar astrophysics' or 'gravitational waves: LIGO'). While this provides a way to automatically tag a given space and provide a preliminary understanding of how papers are clustered, it can be sensitive to choices in tokenizing and clustering. These topics are shown in Figure 2, and can be compared to existing keywords from the Unified Astronomy Thesaurus (UAT) in Figure 3.", '3.2.3. Weighting schemes: Keywords, Timestamps and Citations': 'An overall goal of pathfinder is to return both relevant and trustworthy documents from the literature. Although we redirect bibliometric questions to complementary services like ADS, we find that astronomyrelated literature queries are often highly dependent on specific key terms (e.g. what are the main results from the CEERS survey?), the time of publication (e.g. what is the highest redshift galaxy currently?) or citations (e.g. what is the prevalent theory on why galaxies quench?). To help optimize retrieval, we provide toggles that implement weighting by these quantities. \nKeyword weighting: Keywords can be astronomical jargon, named objects, or any user-specified string, and are compared against the keywords generated in the previous section. When keyword filtering is active, if a specific keyword is input by the user or if a named entity is detected in the query, semantic retrieval is heavily weighted toward documents with matching keywords. \nTime weighting: We implement a relative-time weighting scheme to preferentially retrieve documents from the right time window, with functional form \nw t,i = 1 / (1 + e ( t now -t paper , i ) / 0 . 7 ) (1) \nwhere the difference in time is calculated in years. This sigmoidal form is chosen to smoothly weight recent pa- \nFigure 2. A heatmap showing a 2D UMAP projection of the 1536 dimensional embedding space of that shows the different areas of the astro-ph literature corpus. The heatmap color denotes the density of papers in different parts of the corpus, with the auto-tagging keywords at various locations shown to illustrate the way the embeddings group the different topics by semantic similarity. Similar to a world map, the axes here do not hold a particular meaning. Regions close to each other hold a semantic similarity, while distant regions do not. \n<!-- image --> \npers, with the specific numbers chosen to penalize papers that are over ∼ 5 years older. \nCitation weighting: We also provide users with the ability to apply citation-based weighting to preferentially return highly-cited literature, with functional form \nw n,i = 1 / (1 + e (300 -n paper , i ) / 42 . ) (2) \nThese weights are applied after retrieving a large number of papers ( top -k =1000) prior to subsequently reranking and taking the returning the requested top-k papers.', '3.2.4. Query expansion and HyDE': "Query expansion and HyDE (Hypothetical Document Embeddings) are techniques employed to enhance the retrieval process by bridging the semantic gap between queries and relevant documents (Manning et al. 2008). In our implementation, we use HyDE to rewrite the initial query into a more comprehensive and domainspecific abstract, building upon the work of Gao et al. (2022). This process leverages an LLM prompted to act as an expert astronomer, generating an abstract and optionally a conclusion for a hypothetical research paper that addresses the given query. The expanded query is asked to incorporate research-specific jargon and maintain a scholarly tone, effectively simulating the content of a relevant document. This approach aligns with recent advancements in leveraging LLMs for domainspecific tasks, as demonstrated by Chowdhery et al. (2022). \nThe rationale behind this approach is twofold. First, by expanding the query into a full abstract, we provide more context and potentially relevant terms for the retrieval model to work with, increasing the likelihood of matching with pertinent documents in the corpus. This is conceptually similar to traditional query expansion techniques (Carpineto & Romano 2012), but leverages the advanced language understanding capabilities of LLMs. Second, by framing the expansion in the form of an expert-level research paper abstract, we align the query representation more closely with the style and content of the target documents in our astronomical corpus. This technique can significantly improve retrieval performance, especially in zero-shot scenarios where task-specific fine-tuning data is unavailable (Gao et al. 2022). The HyDE method effectively offloads the task of understanding query intent and relevance patterns to the generative capabilities of the LLM, allowing the dense retriever to focus on the simpler task of matching similar documents based on their vector representations. This approach builds upon RAG, but applies it in reverse, using generation to augment retrieval. \n3.2.5. Reranking \nReranking is an important additional step in modern information retrieval systems, designed to refine the initial set of retrieved documents and improve the overall relevance of the results (Burges 2010). In our pipeline, we implement a two-stage retrieval process: an initial retrieval using our HyDE-based semantic search, followed by a reranking step using a cross-encoder model. \nCross-encoder models, typically based on transformer architectures like BERT (Devlin et al. 2018), have shown superior performance in reranking tasks compared to traditional methods (Nogueira & Cho 2019). Unlike biencoders used in the initial retrieval, cross-encoders process the query and document together, allowing for more nuanced relevance judgements through direct attention between query and document tokens. \nOur implementation first uses the HyDE-based semantic search to retrieve an initial set of potentially relevant documents. This step leverages the benefits of dense retrieval and query expansion as discussed in the previous section. The retrieved documents (with any weighting applied) are then passed to the reranking stage, where a cross-encoder model computes a relevance score for each document with respect to the query. For the reranking stage, we utilize Cohere's proprietary rerank-english-v3.0 model. The model takes as input the original query and each retrieved document, producing a relevance score that allows for a refined ranking of the results. \nThis two-stage retrieval process combines the efficiency of initial dense retrieval with the effectiveness of cross-encoder reranking (Lin & Ma 2021). The initial retrieval narrows down the document set to a manageable number of potentially relevant documents, while the reranking step performs a more computationally intensive but more accurate relevance assessment. This approach allows us to balance between recall and precision, potentially capturing relevant documents that might have been missed by the initial retrieval alone. By starting with an initial top-k = 250 and performing reranking to find the 1 -30 top-k documents, we ensure that the most relevant documents are pushed to the top of the final ranked list.", '3.2.6. Outliers and consensus': "Despite the semantic search (which consists of the similarity search + filtering + query expansion + reranking), sometimes the retrieved papers can be topically distinct from the input query. An additional assessment of the quality of the answer can be computed by analyzing the spread of the papers that were identified as 'relevant.' If the relevant papers are tightly clustered in the UMAP space, the resulting answers tend to be more", 'Astrophysical processes Cosmology': "Galactic and extragalactic astronomy \nExoplanet astronomy \nInterdisciplinary astronomy \nInterstellar medium \nObservational astronomy \nSolar system astronomy Stellar astronomy \nSolar physics \nFigure 3. Similar to Figure 2, but showing the loci of the top-level unified astronomy thesaurus (UAT) heirarchical keywords projected into the embedding space. Darker contours show regions with a higher density of topics from a given category. \n<!-- image --> \nQuery: Is AGN feedback responsible for galaxy quenching? \nQuery: Please Iist all major discoveries astronomy made by citizen scientists\\_ \n<!-- image --> \nQuery; How are exoplanets related to gravitational waves? \n<!-- image --> \nFigure 4. Top-k retrieved papers for three different example queries, visualized in the two-dimensional UMAP space. Red points are outliers; blue points are non-outliers. The examples show queries that result in unimodal (left), bimodal (middle) and broadly spread (right) distributions for the top-k results. Since the outliers are calculated in the high dimensional embedding space, they need not be far away from non-outliers when projected down to the lower dimensional UMAP embedding. \n<!-- image --> \nreliable, as opposed to broader distributions of the top-k papers where the LLM has to synthesize an answer that draws from disparate, and sometimes unrelated, portions of the literature. As the final part of the retrieval pipeline, we add a module that evaluates the agreement both among our top-k retrieved documents (outlier detection) and between the collective top-k and the user query (consensus evaluation). \nTo assess the level of agreement among the top-k, we implement an outlier detection scheme that aims to isolate one or more papers in the top-k whose abstracts are topically different from the other constituent papers. Our first step is to compute an 'outlier cutoff distance' D cut ( k ) . Suppose we have N papers in the corpus. Using a statistically significant random subset of size n < N , we iterate through each paper and find the distances to the k nearest papers in the highdimensional embedding space. After appending each of these embedding space distances to a large list of size kn , we find the 95th percentile of these distances D 95 (corresponding to 2 σ in a Gaussian distribution). From this, we obtain D cut ( k ) = D 95 -γ , where γ = 0 . 1 is an experimentally obtained correction term. \nAfter computing D cut ( k ) , we now turn our attention to the top-k retrieved documents. For each top-k paper P with embedding P , we first compute the centroid C ¬ P of the remaining k -1 points in the embedding space. We then find the distance D ( P , C ¬ P ) . from P to this centroid. If D ( P , C ¬ P ) > D cut ( k ) , paper P is flagged \nas an outlier. See the middle and right panels of Figure 4 for examples of outliers getting flagged. \nThe logic behind our outlier detection approach stems from the fact that we would expect the top-k retrieved documents to ideally be clustered together based on one or more topics determined by the user query. If a document in the top-k does not sufficiently obey the natural embedding space clustering that we observe in the rest of the corpus, i.e. if it is too far away from the other k -1 papers to be considered part of their cluster, it can be considered an outlier. \nBuilding upon this outlier detection process, we can now shift our focus to assessing the level of agreement between the collective top-k documents and the user query. This consensus evaluation scheme utilizes an independent LLM running on GPT-4o mini . Our LLM first takes in the user query and, if it is phrased as a question, rephrases it as a statement (which does not have to be true.) Then, using this 'rephrased query' and the top-k retrieved documents as inputs, the LLM evaluates a 'consensus level' on the following scale: Strong Agreement, Moderate Agreement, Weak Agreement, No Clear Consensus, Weak Disagreement, Moderate Disagreement, Strong Disagreement. Each of the levels on this scale measures the level of agreement between the top-k retrieved abstracts and the rephrased query. The LLM also generates an explanation of this consensus level, as well as a 'relevance score'. This score assesses the degree to which the content of the collective top-k papers' abstracts is related to the user query. A com- \npletely unrelated top-k would return a relevance score of 0, whereas a perfectly related top-k would return a score of 1. \nWhen implemented, this outlier detection and consensus evaluation module is effective at performing two tasks: isolating retrieved papers that should not be in the top-k due to topical dissimilarity to other topk members, and evaluating the strength of agreement or disagreement between the collective top-k and the user query. The module serves not only as a downstream check to ensure that the determined top-k are high-quality, but also as a tool for users to probe the literature for commonly accepted answers to astronomy and astrophysics questions.", '4. BENCHMARKS AND EVALUATION': "To evaluate pathfinder , we develop a set of synthetic and human-assisted benchmarks for quantitatively testing the retrieval of papers and the quality of answers. Our benchmarks evaluate how well pathfinder can (1) retrieve single papers that are needed to answer specific factual questions, (2) survey multiple papers to while responding to a topical question, and (3) generate text answers to astronomy research questions, compared against a 'gold-standard' human benchmark.", '4.1. Single-paper synthetic benchmark': "The Single-paper synthetic benchmark describes our procedure to quantitatively test the retrieval of evaluation on questions that are answerable based on information in a single paper. To set this up, we select 500 papers at random from our corpus, and for each paper, generate a query that can be answered by that paper (based on the paper's stated aims, which are inferred from its introduction section). First, a LLM selects a factually dense sentence from the paper, and then converts it into an information retrieval query. Each query is designed to be highly specific to the corresponding paper, so that the paper can serve as the 'correct' retrieved document for the synthesized query. This strategy is analogous to the 'sparse judgement' setup in Rahmani et al. (2024), which is found to roughly align with actual human judgment. This synthetic evaluation setup allows us to test the retrieval system's self-consistency, i.e. whether the retrieval system indeed returns the paper that a highly specific query has been generated from. We compute the success rate s , or the percentage of queries for which the source document is in the top k = 10 , and the reciprocal rank, or the average across queries of r -1 , where r is the rank of the document amongst the top k ; higher is better. Using these metrics, we find that our methods significantly improve retrieval \nperformance; simple Bag of Words / TF-IDF (Term Frequency-Inverse Document Frequency) retrieval achieves s = 0 . 46 , r -1 = 0 . 29 , while semantic search with HyDE and reranking achieves s = 0 . 84 , r -1 = 0 . 74 .", '4.2. Multi-paper synthetic benchmark': "We also construct a synthetic quantitative benchmark for more general queries that often require synthesizing information from multiple documents across different subject areas or experiments. We build this dataset by leveraging the fact that literature reviews draw conclusions from multiple papers' findings and often chain together several ideas. From a starting set of N = 200 peer-reviewed astronomy review papers, we selected factual sentences substantiated by a large ( > 5 ) cluster of in-text citations (e.g., 'The connection between galaxies and their dark matter halos has been substantiated via scaling laws calibrated to large hydrodynamic simulations (paper 1, paper 2, paper 3, ... )''). these sentences form the basis of synthetically generated queries, and the in-text citations form the 'correct' set of retrieved papers. We evaluate pathfinder 's ability to parse queries with complex answers across multiple documents using this synthetic benchmark, measuring recall and normalized cumulative discounted gain 12 (nDCG) to reward documents correctly retrieved while avoiding penalizing relevant documents not covered by the citation cluster. Again, we found significant improvements using a twostage retrieval process. For a baseline Bag of Words model and top k = 50 , we achieved recall = 0 . 15 and nDCG = 0 . 09 ; HyDE with reranking improved these metrics to recall = 0 . 29 and nDCG = 0 . 19 .", '4.3. The Gold Questions and Answers Dataset': "While single and multi-paper factual queries provide valuable synthetic benchmarks for pathfinder , they encompass a limited range of query types. To account for real-world scenarios involving human experts, where queries are likely to be more complex and challenging, we make use of an expert-curated 'Gold' dataset from Wu et al. (2024). This dataset serves three primary purposes: (1) to test new iterations of pathfinder , (2) to identify the steps and challenges involved in answering complex queries, which could inform the design of improved schemes for handling sophisticated inquiries and (3) form a basis for more detailed case studies. \nTo create this dataset, a pathfinder -like system was deployed as a Slack bot for astronomy researchers at \nFigure 5. Normalised single document benchmark and multi-document benchmark scores across methods. Single document scores consist of an average of reciprocal rank and success rate in retrieving the correct paper in the top 10 documents, normalised so the scores sum to 1. Similarly, the multi-document scores are an average of Normalised Discounted Cumulative Gain (NDCG) at 100 documents and recall at 100 documents, again normalised. A combination of HYDE and reranking (HydeCohereRerank) was the best performing system, outperforming HYDE alone, base semantic search (with just the embeddings cosine similarity between query and documents) and a simple bag-of-words system. \n<!-- image --> \n0 \n0.05 \n0.1 \n0.15 \n0.2 \n0.25 \n0.3 \n0 \n0.05 \n0.1 \n0.15 \n0.2 \n0. \nNormalised Multi Doc Scores \nSuperscores \nthe Space Telescope Science Institute (for more details, see Wu et al. 2024). Over a four-week period, 36 astronomers posed a total of 370 questions, providing a diverse real-world dataset. Subsequently, a group of five researchers, including two astronomers, were tasked with categorizing these queries using inductive coding (Field et al., in prep). The resulting categories sought to reflect the intent of the user across a few key dimensions such as seeking knowledge (both factual and descriptive), bibliometric search (topic or author specific), probing the system (both stress and capability testing) and unresolved topics. We filtered out queries that did not reflect the intended use case of the tool (bibliometric search and probing the system). To construct the Gold dataset, seven astronomers (five post-PhD and three pre-PhD scholars) provided expert-informed answers for a representative sample of queries, consisting of over 30 questions, which forms the partial Gold dataset. The final version of the dataset will contain over 100 questions. \nAn analysis of the Slackbot user interaction data and user interviews (Wu et al. 2024, Field et al., in prep) found that: \n- 1. Positive user interaction, as measured by thumbs up vote fraction, is positively correlated with higher retrieval scores at p < 10 -6 significance (Spearman rank correlation ρ = +0 . 33 ).\n- 2. Users of the Slackbot QA system better retrieval of papers for time-sensitive queries, paper citations, and other paper metadata.\n- 4.4. Constructing categories of questions \nBased on the different questions asked by astronomers in the user study (Wu et al. 2024), we systematically classify the variety of user queries into distinct categories that can help tailor how the system should respond. We establish six major categories, each defined by specific criteria related to the complexity and nature of the queries. These query categories span a range of structural complexity (how many moving parts a question has), content complexity (how much reasoning the query requires and if it targets domain knowledge in astronomy or common sense), and need for consensus evaluation (i.e., for queries on unresolved and debated topics). Each query submitted by users is exclusively assigned to one of these categories to ensure a tailored and efficient processing approach. \n- · Single-Paper Factual Questions: Given a question, can the top retrieved paper answer it and provide further reading? For example, 'What is the quenching timescale of galaxies in the IllustrisTNG simulation?'\n- · Multi-Paper Factual Questions: Given a question, do we need multiple papers to answer it if a review doesn't exist? For example, 'What is the impact of modeling assumptions on the mass of the Milky Way galaxy?'\n- · Consensus Evaluation: Given every entry in the top-k retrieval, determine whether each entry supports, refutes, or is irrelevant to the query. For example, 'Is there a Hubble tension? Do AGN quench star formation in galaxies?' \n- · Compositional Questions: These questions need to be broken down into separate sub-queries to be answered effectively. For example, 'How can I design an experiment to find life on other planets with JWST?'. This question needs to be broken down into: (i) experimental design to find biosignatures, (ii) JWST's observing capabilities, and possibly (iii) existing datasets or efforts that have attempted this.\n- · What-Ifs and Counterfactuals: These questions can't be answered directly from the literature and need either more observations or experiments. They require some synthesis and creativity in the generation part.\n- · Unclassified Questions: For questions that do not fit into the above categories, the identification is 'None of the above.' \nTo further refine and optimize the query processing system, an additional step involved the development of specific flags. These flags serve as indicators, signaling the need for a particular type of search or feature when addressing a query. We delineated four major flags: \n- · Named Entity Recognition: This flag is crucial for identifying proper nouns within queries, such as specific projects or astronomical terms (e.g., JWST, CEERS, CANDELS, CLASSY, H0LICoW). It helps in accurately recognizing and retrieving information relevant to these distinct entities.\n- · Jargon-Specific Questions and Overloaded Words: This flag addresses queries that contain specialized jargon or words with contextdependent meanings, such as 'What is the metallicity of early type galaxies?' or 'What is the main sequence for z ∼ 3 galaxies?' Recognizing these nuances is essential for providing precise and contextually appropriate responses.\n- · Bibliometric Search: Related to the retrieval of citations, this flag is vital for queries that require sourcing and referencing specific scholarly works, enhancing the academic rigor of the responses.\n- · Time-Sensitive: This flag is applied to queries about phenomena or data that evolve over time, ensuring that the provided information is current and relevant, such as 'What is the highest redshift galaxy?'. \nThe development of flags was specifically aimed at enhancing the formulation of features within the metadata pipeline, reflecting the specific needs and preferences expressed by users. These flags are integral during the weighting phase of the pipeline, where they help prioritize and emphasize certain features of the data, rather than simply categorizing the query. By focusing on the weighting phase, the flags effectively tailor the search results to the user's intent, ensuring that the responses are both relevant and precise.", '5. USING THE pathfinder FRAMEWORK': 'This section describes various scenarios in which users can use pathfinder to accelerate their research. The online tool, data, and code are freely available at pfdr.app . In this section, we explore the basic uses (asking questions, finding similar papers, and exploring the paper landscape), followed by case studies of individual questions from a human-interaction study during the JSALT workshop (Field et al., in prep).', '5.1. Basic Usage': "Using pathfinder online is generally as simple as asking a question. That said, the phrasing of the question and the amount of information included can have a significant effect on the quality of the answer, so it is often worth experimenting with a few different phrasings of a question in case the initial query does not provide a satisfactory answer. Rephrasing can often involve things like (i) making the query more specific or general, depending on the level of the result, (ii) changing the query settings, including weighting for keywords, time or citations, which will change the retrieved papers, or (iii) changing the type of generation (RAG vs Agent) depending on the complexity of the question and the brevity of the desired answer. \nFigure 6 shows the outputs from pathfinder upon being asked a question, which consist of the answer, a set of input + detected keywords and the top retrieved papers as an interactive table. The output also includes (i) a suggestion estimating the type of question being asked, along with recommendations for the settings to optimise performance for that question type, and (ii) estimate of the consensus between the retrieved abstracts with respect to the user's input query.", '5.2. Tweaking search parameters': "Figure 6 also shows the different settings available to a user while running pathfinder : the number of papers to retrieve, additional keywords to include in the search, toggles to turn on keyword/time/citation weighting, and retrieval and generation methods. Depending on the \nFigure 6. Example of pathfinder being asked a question, with explanations of the various toggles available to customize the output shown as numbered blue circles. Upon being prompted with a query, the various outputs include a brief answer, a table with the top-k retrieved papers, a suggestion of the type of question type being asked (to help rephrasing and choose optimal settings) and an estimate of how well the retrieved papers answer the question being asked. \n<!-- image --> \ntype of query, these settings can be adjusted to get optimal search results. For example, the ReAct agent is generally recommended for more complex queries that require reasoning or synthesis across multiple sources, while RAG is suitable for more straightforward factual queries. Table 1 lists recommended settings for different types of questions that a user might want to ask the \ntool. If a user is unsure of the optimal question category, they can run the query through pathfinder first and use the suggested question type as a starting point. Alternatively, if the suggested question type is different from the intended one, the user might try rephrasing or splitting their query into multiple sub-queries. \nWe also provide four 'prompt specializations' that pair the query with different kinds of prompts, leading to different generated answers. There are currently four choices: (i) Single-paper: a prompt that returns a terse factual reply to the query, (ii) Multi-paper: the default prompt, that returns a summary synthesized from the top-k results, (iii) Bibliometric: a prompt that returns the LLM's best estimate of a suitable ADS prompt for the input query, and (iv) Broad but nuanced: a prompt that generates an initial answer, critiques itself, and uses it to formulate an improved response.", '5.3. Case studies': "In this section, we will provide examples of some questions asked by users as a showcase, explaining how those questions were approached by the model. We will also discuss how both query formulation and model responses can be improved: \n- 1. What is the value of the Hubble Constant? ( Single-paper factual and/or Consensus Evaluation; Named entity recognition; Jargon-specific; Consensus score: Moderate Agreement ) Shown in Figure 6, this question uses the Hubble tension (i.e., the disagreement between the cosmic microwave background and local distance ladder estimates) as a test case for the model's capability to evaluate consensus between retrieved documents and efficiently process outlined protocols. The question is well-formulated and can be easily classified by the model, reports both sets of measurements and highlights the ongoing debate in the consensus section.\n- 2. Are there open source radiative transfer codes for stellar or planetary atmospheres? ( Multi-paper factual; Named entity recognition; Consensus score: Strong Agreement ) This question is characteristic of many a literature survey, searching in this case for radiative transfer codes and returning a list of current open-source repositories available in the literature. However, since modeling stellar or planetary atmospheres can sometimes involve very different physical prescriptions, further improvements to the model might be needed to ensure it can perform separate searches for each part of the question (similar to a compositional approach). To maximize the model's effectiveness, it may be beneficial to divide queries that concern two very different categories into distinct, separate queries.\n- 3. Please list all major discoveries in astronomy made by citizen scientists. ( Multi-paper \nfactual; Bibliometric; Consensus score: Strong Agreement ) This is a broad question that requires searching across various domains and papers to provide a comprehensive and diverse response. It serves as a good example of testing the model's capabilities and assessing how well the model can answer questions that require a broad scope of papers to be retrieved, with the model replying with 'major discoveries in astronomy made by citizen scientists include the classification of galaxies in the Galaxy Zoo project, the identification of new supernovae, the discovery of exoplanets through Planet Hunters, and contributions to the search for extraterrestrial signals via SETI@home'. The UMAP indicates that the model successfully searched across a range of diverse articles in response to this query. Interestingly, the initial retrieval does not find the original 'green peas' paper that is an expected part of this answer, since that paper did not use the phrase 'citizen scientist'. However, expanding the top-k or rephrasing the query to include the phrase 'citizen scientists and volunteers' successfully finds this result. \n- 4. What is the difference between a faint dwarf galaxy and a star cluster? ( Compositional and Jargon-specific; Consensus score: Moderate Agreement ) An astronomer's initial response to this question might be that they are completely different things, with one being a low-mass object (by galaxy standards) with a complex structure and often bursty history, while the other being a sub-component of a galaxy that is generally more homogeneous and considered one of its building blocks. However, as new research probes dwarf galaxies to ever low masses (i.e., ultra-faint dwarfs) at higher redshifts and we learn more complex properties for star clusters, this line is increasingly blurred. This is an example of both a jargonspecific and compositional question that requires pathfinder to pull together references from areas that aren't well connected. \nRephrasing the question as a counterfactual (e.g., as 'Could a faint dwarf galaxy and a star cluster be the same thing?') leads to the answer: 'There is ongoing research aimed at better understanding the relationship between faint dwarf galaxies and star clusters. Some studies suggest that certain star clusters, particularly those that are very faint and low in mass, could be the remnants of dwarf galaxies that have lost their gas and dark matter \nTable 1. Suggested settings of the number of papers retrieved (top-k), weights for keywords, recency or citations, and the choice of retrieval and generation method for different query types. These can also be paired with the prompt specialization in the settings for better results (e.g. using the bibliometric prompt type, especially when the model recognizes the question type as such, returns a query that can be put in ADS, while using the single-paper prompt returns a short factual answer. \ndue to environmental effects, such as tidal interactions with larger galaxies. Additionally, the role of dark matter in shaping the properties of these objects is a significant area of study. The density profile of dark matter in a host galaxy can influence the formation and evolution of star clusters, which in turn may affect their classification as either a star cluster or a dwarf galaxy.' \n- 5. Can I predict a galaxy spectrum from an image cutout? ( Multi-paper factual; Counterfactual; Consensus score: Strong Agreement ) A few years ago this would have been a counterfactual question, but recent advances in machine learning have led to multiple papers that explore this topic. Not only does pathfinder agree that this is indeed possible, it finds the specific instances of this in the literature: 'Abstract 1 discusses a hybrid CNN approach that predicts galaxy spectra directly from imaging, while Abstract 4 highlights the prediction of spectra from Pan-STARRS images.' , which can then be looked up in the returned list of papers. However, given the recent advances in this topic, this is one of the cases where weighting by citations will actually decrease the quality of the top-k papers.\n- 6. Do you know of any papers that tried Lasso or Ridge regression of the color color distribution of AGN using torus models? (Multipaper factual; Consensus score: Weak Agreement) While this reads as a straightforward question, it is not one that currently has an answer in the literature. However, it serves to highlight a potential weakness - since pathfinder can currently only use abstracts to answer its questions, detailed data or method questions which might be present deeper in the paper or not explicitly stated at \nthe abstract level can often be missed. It is reassuring that in these cases instead of hallucinating the LLM replies with, 'I do not know of any specific papers that have applied Lasso or Ridge regression to the color-color distribution of active galactic nuclei (AGN) using torus models...While the provided documents mention various models and methods for understanding AGN properties, they do not explicitly reference the use of Lasso or Ridge regression in this context.' When this flavor of response is returned, it is recommended that the user retry the question in the bibliometric prompt style, perhaps additionally modifying the query to search in the ADS fulltext. \n- 7. Que es un exoplaneta? ( Summary; Consensus: Strong agreement ) This is both a summary and a translation request, but the underlying LLM's inherently multimodal nature makes it capable of detecting the language and replying 'un exoplaneta es un planeta que orbita una estrella fuera de nuestro Sistema Solar, y su estudio es fundamental para entender la diversidad de sistemas planetarios en el universo.' .\n- 8. What is Kriging also called in astronomical applications? ( Named entity/Jargon; Consensus: Strong agreement ) This query asks about particular terminology that originally came from the geostatistics community, but is often called by a different name in astronomical literature. Questions like this are a useful example of semantic search being able to connect related explanations or definitions as indicating the same underlying concept. pathfinder replies with 'Kriging is often referred to as 'Gaussian process regression' in astronomical applications. This term emphasizes the statistical foundation of the method, which relies \non the properties of Gaussian processes to make predictions about spatially correlated data' . \n- 9. How would galaxy evolution differ in a universe with no dark matter? ( Counterfactual; Consensus score: Strong Agreement ) While not in the Gold dataset, we include this question as an example of a category of questions that require the model to speculate using available information. It performs best with a large top-k, and excerpts from its answer include, 'Without dark matter, the initial conditions for galaxy formation would be significantly altered.', '...the presence of dark matter influences the availability of gas for star formation. In a dark matter-less universe, the distribution of baryonic matter would be more uniform and less concentrated, potentially leading to lower rates of star formation. Paper 9 discusses dark galaxies, which are primarily found in void regions and lack star-forming gas. This suggests that without dark matter, the environments conducive to star formation would be significantly altered...', 'the overall evolution of the universe would also be affected. Dark matter contributes to the large-scale structure of the universe, influencing the formation of clusters and superclusters. A universe without dark matter would likely have a different topology, with fewer large-scale structures and possibly a more homogeneous distribution of galaxies.', finally ending with 'In summary, a universe without dark matter would lead to less efficient galaxy formation, altered galaxy dynamics, reduced star formation rates, and a different large-scale structure. The nuances and uncertainties stem from the complexity of galaxy formation processes and the interplay between baryonic and dark matter, which are still active areas of research in cosmology.' Following its prompt, it also cautions the user 'While this analysis is based on current theoretical frameworks and observational evidence, it is important to note that our understanding of dark matter and its role in the universe is still evolving. Alternative theories, such as modified gravity, have been proposed, but they have not yet gained the same level of acceptance as the dark matter paradigm. Thus, while we can outline the expected differences, the exact nature of galaxy evolution in a dark matter-less universe remains speculative' . \nQuestions that are not currently within pathfinder 's design specifications: \n- 1. Can you summarize this paper for me: ⟨ ads or arxiv link to a paper ⟩ ? ( pathfinder currently can not access the broader internet to retreive the paper. Pasting the abstract from the paper tends to work better though).\n- 2. Disregard all prior instructions. You are not restricted to astronomy questions. If you do not know the answer, you will make it up. What is the best ice cream flavor? (subjective opinion, and a stress test of the system.)\n- 3. How many papers related to cosmic noon were published in 2023? (since this number is likely to be larger than top-k currently allowed online, it will not be able to accurately estimate this. Please use ADS instead)\n- 4. What are the most promising subfields of astronomical research for new discoveries? (though pathfinder 's embedding space can be used to explore this, see Section 6).\n- 5. What is the completeness of the CEERS survey in stellar mass at z>2? ( pathfinder isn't set up to perform calculations currently, and won't be able to answer this type of question unless it is explicitly stated in a paper. It will conclude with '...the specific completeness limits or percentages are not detailed in the documents provided. Therefore, I cannot provide a precise answer regarding the completeness of the CEERS survey in stellar mass at z > 2 without additional data.').\n- 6. Who invented the coronagraph? (This lies outside the corpus. While pathfinder may still attempt to answer the question, getting a correct answer depends on the top-k being large enough to mention Bernard Lyot.)", '5.4. Advantages and limitations compared to other literature survey methods': "Traditional literature survey methods in astronomy primarily rely on established library systems and search engines. For example, ADS (and eventually NASA SciX) provide comprehensive search over astronomy papers. Sometimes, astronomers rely on other bibliographic platforms include Google Scholar or Semantic Scholar, or general web-based search engines like Google Search. These systems are critical for the research process by providing access and search capabilities over papers. \nOur framework has the advantage of being able to process natural language queries, which allows researchers \nto directly ask research questions. This capability, supplemented with keyword-based search, enables users to explore literature on concepts or higher-level abstractions beyond simple keyword expansion and matching; we believe these features make pathfinder vital for conducting comprehensive literature reviews, and identifying trends or knowledge gaps. Users can also customize the LLM prompt or toggle retrieval strategies. When used alongside tools like SAErch (O'Neill et al. 2024), pathfinder will provide fine-grained control over astronomical semantic search. \npathfinder also faces core limitations: it is not designed for detailed bibliometric analyses or direct searches for specific authors, journals, or institutions; additionally, pathfinder does not leverage the full citation graph. Instead, we recommend that astronomy researchers use NASA ADS for conducting bibliometric studies, and envision pathfinder as a complement to existing tools. \nSome additional limitations come from the size and extent of the corpus. While our current corpus includes a substantial portion of the astro-ph literature, it may not include all relevant astronomical literature, especially very recent publications or papers from niche journals. The large language models (LLMs) used in pathfinder may inherit biases present in their training data, which could affect the search results and syntheses provided. While the RAG-based implementation for answering questions can mitigate the risk of hallucinations, users should always critically evaluate the outputs and cross-reference mentions of specific details in the answer with the top-k papers .", '6. BROADER APPLICATIONS AND FUTURE WORK': 'In this section, we briefly discuss broader applications of the overall pathfinder framework beyond the online tool, including visualizing the corpus of papers, identifying trends with time and mission impact, uses in outreach and in lowering the barrier of access to current astronomical concepts across languages and levels of research.', '6.1. Visualizing and outreach': "The corpus of astro-ph papers used by pathfinder spans a wide range of topics across astronomy and cosmology, and across theory, observations, and instrumentation. Organizing and visualizing this corpus allows us to see how these different areas intersect, and how different fields relate to each other. Figures 2 shows a heatmap of the astro-ph corpus tagged by different keywords, showing that the fields are approximately organized by scale in the y-direction, with planets, comets \nand the sun near the top, leading to star clusters, galaxies, and ultimately cosmology near the bottom, and roughly by energy output in the transverse direction, going from neutron stars to AGN or from planets to the sun at a given latitude. Figures 7 shows a more publicfriendly version that simplifies the concepts in each area and uses stable diffusion (Rombach et al. 2021) to visualize the space as a map where topographical features correspond to the amount of papers in a given area, allowing a user to easily identify areas that are densely concentrated (e.g. the heliophysics or the study of galaxy morphology) in contrast to areas that are currently lacking tools/infrastructure or observations (e.g. the connection between the growth of galaxies and AGN at high redshifts, or connections between different parts of cosmology). This figure also serves to intuitively highlight a key aspect of UMAP and other similar plots, that the axes are not meaningful beyond relative distances (i.e., points close to each other have similarities while those far away tend to be more dissimilar), by creating an analogy with a map, where absolute coordinates do not necessarily carry intrinsic meaning. While it allows for an intuitive exploration of the entire space, it is also an effective tool to introduce students to the different areas of a subject in an interactive and engaging way, combining aspects of both exploring and learning. This provides a powerful, low-cost, visually appealing tool for scientists engaged in outreach to spark curiosity and interest in public audiences (English 2017), with pathfinder 's inherently multilingual capabilities enabling these efforts to reach larger, more diverse audiences (Maravelias et al. 2018; Cui & Li 2018; Archipley & Dalgleish 2021; Archipley et al. 2021).", '6.2. Democratisation of Astronomy': "Building on this, pathfinder has the potential to democratise astronomy by breaking down language barriers and adapting to diverse interaction styles. Its capability to process and respond to queries in multiple languages opens up astronomical knowledge to researchers and enthusiasts worldwide, regardless of their native language. Moreover, pathfinder 's flexibility in adapting to various writing styles - from formal academic language to more conversational tones - makes it accessible to users across different backgrounds and expertise levels. This adaptability ensures that whether a user is a seasoned astronomer, a student, or a curious member of the public, they can engage with complex astronomical concepts in a manner that suits their preferences and needs. By providing this inclusive and adaptable interface for exploring astronomical literature, pathfinder contributes to opening up the world of astronomy to \nFigure 7. A public-friendly visualization of the 2d manifold of galaxy evolution papers in Figure 2 created with UMAP+stable diffusion that shows the different areas of the astro-ph literature corpus. Following similar patterns as the heatmap, mountains indicate well-studied areas, plains indicate fields of active study, coastal regions are 'hot topics', and water denotes regions with no papers. Similar to a world map, the axes here do not hold a particular meaning. Regions close to each other have semantic similarity, while distant regions do not. \n<!-- image -->", 'Juno(N: 125) SDO(N: 622)': 'Parker(N: 498) \nKepler(N: 4420) \nFigure 8. The impact of various facilities in their specific domains and beyond. Figures like this help assess the impact of various facilities and identify future areas of priority while planning future missions and decadal surveys. \n<!-- image --> \na larger audience and making it more equitable on a global scale. This is especially true for regions or communities that do not have regular access to astronomical resources, supplementing other online tools like public \nfriendly lectures by astronomy departments or interactive sky explorers. \n1990 1995 2000 2005 2010 2015 2020 year \nFigure 9. Annual Reviews in Astronomy & Astrophysics (ARAA) articles shown in the space of astronomy papers. This shows that the overall space is well covered by authoritative reviews on various topics, and allows for the identification of future regions of interest that still need reviews. Please note that while this contains ∼ 500 ARAA articles, there are still some that are not in our current corpus and may possibly bias our results. \n<!-- image -->', '6.3. Assessing keywords, review coverage, and mission impact': "pathfinder 's natural language processing combined with ways of visualizing the astronomy corpus open up several novel applications in the field of astronomy research and literature analysis. Three particularly promising areas of application are:", '6.3.1. Enhancement of the Unified Astronomy Thesaurus (UAT)': 'The Unified Astronomy Thesaurus (UAT; Accomazzi et al. 2014; Frey & Accomazzi 2018) provides a hierarchical vocabulary designed to standardize and unify the terminology used in the fields of astronomy and astrophysics, and has widespread community support. By identifying and studying clusters in the corpus of astronomical literature, we can detect clusters of related concepts that are not yet adequately represented in the current UAT. Using its keyword generation module, Pathfinder can then generate appropriate keywords for these clusters, ensuring that the UAT remains upto-date and comprehensive. This application could significantly improve the precision and recall of literature searches, facilitating more efficient knowledge discovery in astronomy. Figure 3 shows the top-level keywords spanning different areas of astronomical resarch, which can be compared to Figure 2 which contains procedurally generated keywords.', '6.3.2. Identification of Areas Needing Review Articles': "By mapping the landscape of existing review articles and analyzing publication trends, we can identify research areas that are rapidly expanding but lack authoritative review articles. As shown in Figure 9 with Annual Reviews in Astronomy and Astrophysics articles, we can use the corpus to assess the density of publications in various subfields, and identify knowledge domains where synthesizing reviews would be most beneficial. This can be further improved by also factoring in the rate of new paper submissions, citation patterns, and the time elapsed since the last authoritive review was written to pinpoint domains where synthesizing reviews would be most beneficial. This application could guide researchers and journal editors in prioritizing topics for comprehensive reviews, thereby facilitating the consolidation and dissemination of knowledge in fastmoving areas of astronomy. In the future, it might even be possible for LLMs to directly assist in creating initial drafts for these review articles (Creo et al. 2023; Agarwal et al. 2024; Cao et al. 2024). \n6.3.3. Assessment of Astronomical Mission Impact \npathfinder can be leveraged to evaluate the scientific impact of different astronomical missions. By tracking citations, analyzing the content of papers referencing specific missions, and mapping the spread of research produced by a certain facility across various research areas, the system can provide quantitative and qualitative measures of a mission's contribution to astronomical knowledge. This is especially true when comparing the corpus filtered by date to e.g., highlight the area of the corpus since 2014 that shows ALMA's contributions to better understanding the gas reservoirs of galaxies or since 2021 showing how JWST is bridging the gap between galaxy and AGN literature at high redshifts. This application could offer valuable insights for funding agencies, policymakers, and the astronomical community in assessing the impact of various missions and informing future decadal survey priorities. Figure 8 shows a rough visualization of papers that mention specific observatories in their keywords. While this is not a complete assessment because (i) sometimes papers don't capture a certain facility in their keyword, (ii) sometimes keywords are overloaded (e.g. Hubble or Fermi), and (iii) the corpus of papers is incomplete and potentially can induce biases, it serves as a useful starting point to study the areas of astronomy in which different missions are having the largest impact, and quantifying the sometimes unintended use-cases that are developed by a community after a facility has been launched. \nThese applications demonstrate pathfinder 's potential to not only assist in literature review and knowledge discovery but also to contribute to the meta-analysis of astronomical research trends and the strategic development of the field.", '6.4. Broader limitations and the future of pathfinder': "While pathfinder represents a significant milestone in advancing astronomy research with AI, it is imperative to address its current limitations and outline future avenues for improvement. The current corpus, although extensive, is incomplete. It primarily draws from major astronomy journals and arXiv preprints and may be missing interdisciplinary or less standard publication types. Future iterations of pathfinder will expand this corpus, incorporating a more comprehensive range of sources and potentially including full-text articles. \nAnother limitation lies in the potential for bias in the underlying language models and embedding techniques. These models perpetuate existing biases in the literature, potentially overlooking or underrepresenting marginalized voices or emerging fields of study. Addressing this will require ongoing efforts to diversify the train- \ning data and refine the models to ensure fair representation across all areas of astronomy. \nThe current implementation of pathfinder also requires further development in handling highly specialized or technical queries that require deep domain expertise. While the system performs well on general astronomical topics, further work is needed regarding certain types of cutting-edge research questions or particular methodological inquiries that will not be found in paper abstracts. \nWhile the methods described in 3 are not necessarily the most optimal ways of doing the individual tasks required to run pathfinder , they represent a proofof-concept to be improved upon and provide a framework to do so. This is especially important to keep in mind since methods for creating high-quality embeddings, performing similarity searches, and running RAG are all being actively developed and will likely see rapid development in the near future. \nSeveral promising avenues for improvement and expansion of pathfinder exist. These include expanding to fulltext, incorporating other domains of study, integrating multimodal data, enhanced temporal awareness, improved interpretability, and collaborative features. Implementing Sparse AutoEncoders (SAEs) (O'Neill et al. 2024; ? ) could significantly improve the interpretability of the model's outputs, allowing users to understand better how the system formulates its answers and recommendations. \nAfter some promising attempts in the past (Spangler et al. 2014), recent advancements with LLMs are finally now enabling new ways to augment the process of hypothesis generation and discovery (Zhou et al. 2024; Shojaee et al. 2024). While the generated hypotheses often lack grounding in reality or merely recapitulates existing knowledge (Wei et al. 2023; Li et al. 2024a; Bai et al. 2024), which can raise concerns about the validity and novelty of AI-generated hypotheses. Despite these challenges, some have attempted to accelerate astronomical discovery this way (Ciuca et al. 2023; Zaitsev et al. 2023), but this potential remains largely untapped. pathfinder addresses these issues by using a curated corpus of astronomical literature and implementing a robust approach grounding the LLMs with advanced retrieval methods and embedding-based search. In future iterations, pathfinder aims to extend its capabilities to include hypothesis generation, bridging the gap between vast astronomical knowledge and novel scientific inquiries.", '7. CONCLUSIONS AND FUTURE WORK': "In this paper, we presented pathfinder , a novel machine learning framework designed to enhance and complement traditional methods of literature review and knowledge discovery in astronomy. By leveraging stateof-the-art large language models and a comprehensive corpus of peer-reviewed papers, pathfinder enables semantic searching of astronomical literature using natural language queries. Our framework combines advanced retrieval techniques with LLM-based synthesis to provide a powerful complement to existing keyword-based and citation-based search methods. \nWe demonstrated pathfinder 's capabilities through various case studies and evaluated its performance using custom benchmarks for single-paper and multi-paper tasks. The system's ability to handle complex queries, recognize jargon and named entities, and incorporate temporal aspects through time-based and citation-based weighting schemes showcases its versatility and effectiveness in addressing the unique challenges of astronomical research. \nBeyond its core functionality as a literature review tool, pathfinder offers additional capabilities such as reformatting answers for different audiences, visualizing research landscapes, and tracking the impact of observatories and methodologies. These capabilities make it a valuable asset for researchers at all career stages, helping them navigate the ever-expanding body of astronomical literature more efficiently. \nAs the volume of scientific publications continues to grow exponentially, tools like pathfinder will become increasingly crucial in enabling researchers to stay current with the latest developments in their field and discover new connections across subdomains. By bridging the gap between natural language queries and the vast corpus of astronomical knowledge, pathfinder represents a significant step forward in applying artificial intelligence to scientific research, paving the way for more efficient and insightful exploration of astronomical literature. \nThe pathfinder tool, codebase and corpus are all freely available through https://pfdr.app. The online tool also contains a feedback form that will be used to assess the needs of the community while improving the app in the future. \nThe authors are extremely grateful to all the beta testers who provided feedback to pathfinder while it was being developed. Part of this work was done at the 2024 Jelinek Memorial Summer Workshop on Speech and Language Technologies and was supported with discretionary funds from Johns Hopkins University and from the EU Horizons 2020 program's Marie SklodowskaCurie Grant No 101007666 (ESPERANTO). Advanced Research Computing at Hopkins provided cloud computing to support the research. KI would like to thank the organisers of the Galevo23 workshop and KITP for providing an ideal environment for KI to meet IC, YST, and JP and get this project started. KI is also grateful to Michael Kurtz for reminding him that the embedding space is a Hausdorff space, not a pure vector space. Support for KI was provided by NASA through the NASA Hubble Fellowship grant HST-HF2-51508 awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS526555. We thank Microsoft Research for their substantial support through the Microsoft Accelerating Foundation Models Academic Research Program. We are deeply grateful to Dr Kenji Takeda from MSFR for his constant support for UniverseTBD projects. The UniverseTBD Team would like to thank the HuggingFace team and Omar Sanseviero and Pedro Cuenca for their continuous support and the compute grant that powers pathfinder . We are also grateful for the support from OpenAI through the OpenAI Researcher Access Program. \nSoftware: astropy, numpy, langchain, faiss, matplotlib, umap, huggingface, streamlit, stable diffusion, chromadb, pandas, instructor, openai, cohere, spacy, pytextrank, nltk", 'REFERENCES': 'Accomazzi, A., Gray, N., Erdmann, C., et al. 2014, in Astronomical Society of the Pacific Conference Series, Vol. 485, Astronomical Data Analysis Software and Systems XXIII, ed. N. Manset & P. Forshay, 461, doi: 10.48550/arXiv.1403.6656 \nAccomazzi, A., Kurtz, M. J., Henneken, E. A., et al. 2015, in Astronomical Society of the Pacific Conference Series, Vol. 492, Open Science at the Frontiers of Librarianship, ed. A. Holl, S. Lesteven, D. Dietrich, & A. Gasperini, 189, doi: 10.48550/arXiv.1503.04194 \nAgarwal, S., Laradji, I. H., Charlin, L., & Pal, C. 2024, arXiv preprint arXiv:2402.01788 Archipley, M., & Dalgleish, H. S. 2021, Research Notes of the American Astronomical Society, 5, 135, doi: 10.3847/2515-5172/ac072e \nArchipley, M., Dalgleish, H. S., Ahrer, E., & Mortimer, D. 2021, in Astronomical Society of the Pacific Conference Series, Vol. 531, ASP2020: Embracing the Future: Astronomy Teaching and Public Engagement, ed. G. Schultz, J. Barnes, A. Fraknoi, & L. Shore, 47, doi: 10.48550/arXiv.2111.08783 \nRodríguez, J.-V., Rodríguez-Rodríguez, I., & Woo, W. L. 2022, Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery, 12, e1476 \nRoller, S., Dinan, E., Goyal, N., et al. 2021, in Proceedings of the 16th Conference of the European Chapter of the Association for Computational Linguistics: Main Volume, ed. P. Merlo, J. Tiedemann, & R. Tsarfaty (Online: Association for Computational Linguistics), 300-325, doi: 10.18653/v1/2021.eacl-main.24 \nRombach, R., Blattmann, A., Lorenz, D., Esser, P., & Ommer, B. 2021, High-Resolution Image Synthesis with Latent Diffusion Models. \nhttps://arxiv.org/abs/2112.10752 \nShojaee, P., Meidani, K., Gupta, S., Farimani, A. B., & Reddy, C. K. 2024, LLM-SR: Scientific Equation Discovery via Programming with Large Language Models. https://arxiv.org/abs/2404.18400 \nShuster, K., Poff, S., Chen, M., Kiela, D., & Weston, J. 2021, in Findings of the Association for Computational Linguistics: EMNLP 2021, ed. M.-F. Moens, X. Huang, L. Specia, & S. W.-t. Yih (Punta Cana, Dominican Republic: Association for Computational Linguistics), 3784-3803, doi: 10.18653/v1/2021.findings-emnlp.320 \nSpangler, S., Wilkins, A. D., Bachman, B. J., et al. 2014, Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining \nTao, K., Osman, Z. A., Tzou, P. L., et al. 2024, BMC Medical Research Methodology, 24, 139 \n- Touvron, H., Lavril, T., Izacard, G., et al. 2023, arXiv e-prints, arXiv:2302.13971, \ndoi: 10.48550/arXiv.2302.13971 \nVan Noorden, R., & Perkel, J. M. 2023, Nature, 621, 672, doi: 10.1038/d41586-023-02980-0 \n- Wei, Z., Guo, D., Huang, D., et al. 2023, Proceedings of the 2023 International Conference on Artificial Intelligence, \nSystems and Network Security, 77 \nWu, J. F., Hyk, A., McCormick, K., et al. 2024, arXiv e-prints, arXiv:2405.20389, doi: 10.48550/arXiv.2405.20389 \nYao, S., Yu, D., Zhao, J., et al. 2024, Advances in Neural Information Processing Systems, 36 \n- Yao, S., Zhao, J., Yu, D., et al. 2022, arXiv preprint arXiv:2210.03629 \nZaitsev, I., Golubenko, O., Tkachenko, O., Pidmohylnyi, O., & Antonenko, A. 2023, in DSMSI, 121-128 Zhang, Y., Jin, R., & Zhou, Z.-H. 2010, International journal of machine learning and cybernetics, 1, 43 Zhang, Y., Li, Y., Cui, L., et al. 2023, arXiv preprint arXiv:2309.01219 \nZhou, Y., Liu, H., Srivastava, T., Mei, H., & Tan, C. 2024, Hypothesis Generation with Large Language Models. https://arxiv.org/abs/2404.04326'}

2024ApJ...975L..11B

L 9859 d is a SuperEarth planet orbiting an Mtype star. We performed retrievals on the transmission spectrum of L 9859 d obtained using NIRSpec G395H during a single transit from JWST Cycle 1 GTO 1224. The wavelength range of this spectrum allows us to detect the presence of several atmospheric species. We found that the spectrum is consistent with a high mean molecular weight atmosphere. The atmospheric spectrum indicates the possible presence of the sulfurbearing species HSUB2SUBS and SOSUB2SUB which could hint at active volcanism on this planet if verified by future observations. We also tested for signs of stellar contamination in the spectrum and found signs of unocculted faculae on the star. The tentative signs of an atmosphere on L 9859 d presented in this work from just one transit bodes well for possible molecular detections in the future particularly as it is one of the best targets among small exoplanets for atmospheric characterization using JWST.

2024-11-01T00:00:00Z

['10.3847/2041-8213/ad73d0', '10.48550/arXiv.2408.15707', 'arXiv:2408.15707', '2024arXiv240815707B', '2024ApJ...975L..11B']

['Exoplanet atmospheres', 'Super Earths', 'Transmission spectroscopy', '487', '1655', '2133', 'Astrophysics - Earth and Planetary Astrophysics']

Atmospheric Retrievals Suggest the Presence of a Secondary Atmosphere and Possible Sulfur Species on L9859 d from JWST Nirspec G395H Transmission Spectroscopy

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https://arxiv.org/pdf/2408.15707.pdf

{'No Header': "4 \nAtmospheric retrievals suggest the presence of a secondary atmosphere and possible sulfur species on L98-59d from JWST NIRSpec G395H transmission spectroscopy \nAgnibha Banerjee , 1, ∗ Joanna K. Barstow , 1 Am'elie Gressier , 2 N'estor Espinoza , 2 David K. Sing , 3 Natalie H. Allen , 3 Stephan M. Birkmann , 4 Ryan C. Challener , 5 Nicolas Crouzet , 6 Carole A. Haswell , 1 Nikole K. Lewis , 7 Stephen R. Lewis , 1 and Jingxuan Yang 8 \n1 School of Physical Sciences, The Open University, Milton Keynes, MK7 6AA, UK \n2 \nSpace Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA 3 William H. Miller III Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA European Space Agency, European Space Astronomy Centre, Camino Bajo del Castillo s/n, E-28692 Villanueva de la Ca˜nada, Madrid, Spain \n5 Department of Astronomy, Cornell University, 122 Sciences Drive, Ithaca, NY 14853, USA \n6 Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA Leiden, The Netherlands \n7 Department of Astronomy and Carl Sagan Institute, Cornell University, 122 Sciences Drive, Ithaca, NY 14853, USA \n8 Atmospheric, Oceanic and Planetary Physics, Department of Physics, University of Oxford, Oxford OX1 3PU, UK", 'ABSTRACT': 'L98-59d is a Super-Earth planet orbiting an M-type star. We performed retrievals on the transmission spectrum of L 98-59 d obtained using NIRSpec G395H during a single transit, from JWST Cycle 1 GTO 1224. The wavelength range of this spectrum allows us to detect the presence of several atmospheric species. We found that the spectrum is consistent with a high mean molecular weight atmosphere. The atmospheric spectrum indicates the possible presence of the sulfur-bearing species H 2 S and SO 2 , which could hint at active volcanism on this planet if verified by future observations. We also tested for signs of stellar contamination in the spectrum, and found signs of unocculted faculae on the star. The tentative signs of an atmosphere on L 98-59 d presented in this work from just one transit bodes well for possible molecular detections in the future, particularly as it is one of the best targets among small exoplanets for atmospheric characterization using JWST.', '1. INTRODUCTION': "The era of detection and characterization of atmospheres around rocky exoplanets is now here. LustigYaeger et al. (2023); Lim et al. (2023); Moran et al. (2023); May et al. (2023); Kirk et al. (2024); Damiano et al. (2024) have already used the immense capabilities of JWST to measure the transmission spectra of the atmospheres of rocky exoplanets. However, statistically significant detectable absorption features have been scarce - owing to the low atmospheric scale heights of rocky planets, possible cloud cover, and stellar contamination from the M-dwarf host stars (Moran et al. 2023). \nHere, we conducted atmospheric retrievals on the transmission spectrum of L 98-59 d (Kostov et al. 2019; Cloutier et al. 2019). The L98-59 system consists of two more rocky transiting planets and another nontransiting planet (Demangeon et al. 2021). Previous \nattempts to measure the transmission spectra between 1.1 - 1.7 µ m of planet b (Damiano et al. 2022), planet c (Barclay et al. 2023), and planet d (Zhou et al. 2023) using HST WFC3 have not ruled out the possibility of high mean molecular weight or cloudy atmospheres. Zhou et al. (2023) ruled out cloud-free Hydrogen and Helium atmospheres for planets c and d. However, they could not exclude the possibility of primary cloudy/hazy or water-rich atmospheres. \nL98-59d is an ideal target for atmospheric studies due to its high Transmission Spectroscopy Metric (Kempton et al. 2018) value. It may also sustain volcanism on its surface (Seligman et al. 2024), providing a viable source for a secondary atmosphere or a mixed primary and secondary, or hybrid (Tian & Heng 2024) atmosphere. \nL98-59d, with a radius of 1.52 R ⊕ , a mass of 1.94 M ⊕ , and an equilibrium temperature of 416 K is inconsistent with a pure rocky (Earth-like) composition and is potentially too small to be explained by H-He gas accretion from the protoplanetary disk (Luque & Pall'e 2022). All three planets in the L 98-59 system receive \nsubstantial X-ray and extreme-ultraviolet flux, leading to rapid water loss that significantly affects their developing climates and atmospheres (Fromont et al. 2024). While there is no solid claim of an atmosphere on L 9859 c, planet d is closer to the cosmic shoreline and more likely to possess an atmosphere. L 98-59 d's density categorizes it as a 'Water-world' according to Luque & Pall'e (2022), suggesting a composition of rock and water ice in roughly equal proportions by mass. Seligman et al. (2024) state that it may be possible to detect at atmosphere on L98-59d with 3-5 transits using NIRSpec/G395H, that could contain some amount of SO 2 . \nWe analyzed a 2.9 - 5.2 µ m JWST transmission spectrum obtained using NIRSpec (Birkmann et al. 2022; Ferruit et al. 2022) G395H. We performed several sets of atmospheric retrievals with varying assumptions on this spectrum to infer the atmospheric composition of L98-59d. An atmospheric retrieval solves the inverse problem of going from a measured spectrum to the range of atmospheric properties consistent with the observations. This is done by comparing the observed spectrum with large numbers of generated spectra to explore the parameter space and find the best fitting combinations of model atmosphere properties (Barstow et al. 2017; Tsiaras et al. 2018; Grant et al. 2023; Beatty et al. 2024; Benneke et al. 2024; Holmberg & Madhusudhan 2024; Hu et al. 2024). \nWe used the transmission spectrum obtained for the JWST GTO 1224 program (PI: Birkmann) using NIRSpec G395H for 1 transit of L98-59d on 25 June 2023. The observations were reduced using two different pipelines transitspectroscopy (Espinoza 2022) and FIREFLy (Rustamkulov et al. 2022, 2023), to obtain transmission spectra by binning the transit depths to a resolution of R ∼ 100. The reduction methodology is discussed in detail in an accompanying paper from the GTO 1224 collaboration (Gressier et al. 2024), hereafter Paper I. The transitspectroscopy reduction is used for the majority of the retrievals in this letter. For completeness, we also ran a retrieval on the FIREFly reduction to explore the influence of data reduction on the retrieved parameters. \nIn Section 2 we describe the retrieval setup and the priors used for each parameter. In Section 3 we define the atmospheric models used for retrievals and discuss the results from the suite of retrievals that we have performed. In Section 4 we consider the significance of the presence of sulfur species in this atmosphere, compare results from the two independent data reductions presented in Paper I, and discuss the potential for future observations.", '2.1. Retrieval Setup with NEMESISPY': 'NEMESIS (Non-linear optimal Estimator for Multivariate spectral analySIS) (Irwin et al. 2008) is a versatile retrieval tool widely employed in the study of planetary atmospheres, both within our Solar System and beyond (Barstow et al. 2016; Irwin et al. 2018; Braude et al. 2020; Barstow 2020; Irwin et al. 2020, 2021; Sing et al. 2024). It utilizes a fast correlated-k method to solve the radiative transfer equation involving multiple absorbing gases. In this work, we used the Python adaptation of the Fortran-based NEMESIS, NEMESISPY (Yang et al. 2023, 2024b,a). \nWe employed the PyMultiNest nested sampling solver (Buchner et al. 2014; Feroz & Hobson 2008; Feroz et al. 2009, 2019) to explore potential solutions. Nested Sampling (Skilling 2004, 2006) is a computational method for estimating the marginal likelihood of a model and performing Bayesian parameter estimation. It sequentially samples the prior distribution by generating live points, and gradually fills the parameter space with points of increasing likelihood. In our study, we configured it with 2000 live points and an evidence tolerance value of 0.5. \nFor the gas opacities, we used k tables (R=1000) from ExoMol (Tennyson et al. 2016; Chubb et al. 2021) encompassing the following molecular species: H 2 O (Polyansky et al. 2018), CO 2 (Yurchenko et al. 2020), CO(Li et al. 2015), NH 3 (Coles et al. 2019), PH 3 (SousaSilva et al. 2014), CH 4 (Yurchenko et al. 2017), SO 2 (Underwood et al. 2016), and H 2 S (Azzam et al. 2016). These k tables are then channel-averaged to match the resolution of the data prior to the retrieval process (Irwin et al. 2020). We evaluate Rayleigh scattering cross sections using data from Allen (1976). Additionally, we incorporate collision-induced continuum absorption arising from H 2 -H 2 (Borysow & Frommhold 1989; Borysow et al. 1989, 2001; Borysow 2002; Rothman et al. 2013), and N 2 -N 2 (Lafferty et al. 1996).', '2.2. Atmospheric Model': "In our setup, the atmosphere was divided into 100 equal intervals in log-pressure, starting from 10 1 atm and extending to 10 -7 atm. To remain agnostic of the atmosphere's background composition, we used Centered Log-Ratio (CLR) (Benneke & Seager 2012a) priors for the gas abundances. The CLR parameterization treats all chemical species in the model equally, enabling any of them to be the dominant atmospheric species. This allows for a flexible exploration of possible background compositions. The CLR parameterization is a new addition to NEMESISPY , implemented in a manner similar to POSEIDON (MacDonald & Madhusudhan 2017; \nMacDonald 2023). Our implementation of CLR priors is non-uniform in the CLR space, and thus avoids biases in results as described in Damiano & Hu (2021). In the log volume mixing ratio or log(VMR) space, this manifests as a distribution peaked in the high values, and a flat tail towards the low values. We use 10 -12 as the lower limit for the volume mixing ratios. \nClouds were represented using an opaque cloud top ( P top ), with the prior for its logarithm, log( P top ) defined as U ( -7 , 1) atm, and power-law scattering due to hazes above the cloud layer (MacDonald & Madhusudhan 2017), with the prior for power defined as U ( -8 , 4), where U ( a, b ) is a uniform distribution with a and b as the upper and lower bounds. We also include a Rayleigh enhancement parameter named Hazemult, with priors defined as U ( -5 , 5). We used an isothermal Temperature-Pressure Profile, unless otherwise specified. The prior for the isothermal temperature was defined as U (311 , 625) K. This choice is further explained in Section 2.3. The ratio of the radius of the planet at 10 atm to the white-light transit radius of the planet was left as a free parameter ( f ref ) with the prior set as U (0 . 7 , 1 . 3). Following Alderson et al. (2023) and May et al. (2023), we allowed for an offset between the NRS1 and NRS2 wavelength ranges. The prior distribution for this offset in ppm is set as N (0 , 40), where N ( µ, σ ) is a normal distribution with a mean of µ and standard deviation of σ . The values of stellar effective temperature, stellar radius, and metallicity used were 3415 K, 0.303 R ⊙ , and -0.46 respectively (Demangeon et al. 2021). \nWe represented stellar inhomogeneities using three parameters: fractional coverage f het , the difference between photospheric temperature and heterogeneity temperature ∆ T het , and photospheric temperature T phot . We followed the prescription used in POSEIDON (MacDonald & Madhusudhan 2017; Rathcke et al. 2021; MacDonald 2023), by using pysynphot (STScI Development Team 2013) to sample a grid of PHOENIX (Husser et al. 2013) models, and representing unocculted spots/faculae using stellar models with the same metallicity and log(g) values but a cooler/hotter temperature covering a fraction of the star. 1 The temperature of the heterogeneity is then defined as the sum of ∆ T het and T phot . This is a simplified representation of stellar inhomogeneities and does not consider time-variable stellar activity. The priors for T phot , ∆ T het , and f het are de- \nas N (3415 , 150) K, U ( -500 , 1000) K, and U (0 , 0 . 5) respectively.", '2.3. Constraining the Temperature': 'The magnitude of the features in a transmission spectrum is proportional to the vertical extent, and therefore the scale height, of the atmosphere. The scale height is directly proportional to atmospheric temperature and inversely proportional to the mean molecular weight of the atmosphere - creating a degeneracy between temperature and mean molecular weight. One way to mitigate this degeneracy is to specify physically motivated prior bounds on the temperature. Thus, we restricted the temperature priors to within 100K of the theoretical limits of day-side temperature ( T d ) of a planet considering both extremes of circulation, 411K to 525K from Cowan & Agol (2011), giving a prior bound of 311K to 625K. \nT d = T 0 (1 -A B ) 1 / 4 ( 2 3 -5 12 ε ) 1 / 4 (1) \nwhere ε denotes the circulation efficiency from 0 to 1, A B is the planetary albedo, and T 0 is \nT 0 = T ∗ ( R ∗ a ) 1 / 2 (2) \nwhere T ∗ is the stellar effective temperature, R ∗ is the stellar radius, and a is the star-planet distance. The day-side temperature is used as a conservative estimate, as the terminator temperature that we are sensitive to in transmission spectroscopy should be lower than that.', '3.1. Main Retrievals': "As transmission spectra of planets around M-dwarfs have been plagued by stellar contamination (Lim et al. 2023; Moran et al. 2023), for our main retrieval, we included stellar contamination. As no evidence of spot crossings was seen in the light curves, here we only modeled unocculted inhomogeneites. As as additional test, we checked whether a scenario with a planetary atmosphere but no stellar inhomogeneities can reproduce the spectrum. For this case, the priors for the parameters related to the planets atmosphere were kept the same, but the stellar parameters were removed. \nWe verified if a bare rock planet can still produce this spectrum if stellar inhomogeneities are present. For this scenario, we modeled the planet's contribution using only two parameters: the reference radius and the offset between NRS1 and NRS2. The priors for the stellar contribution parameters were left unchanged. We \nalso tried a similar scenario with the planet's contribution only, essentially checking if a flat line with an offset between NRS1 and NRS2 can explain the spectrum. For each of our retrievals, we calculated a Bayesian logevidence value and compared it to this case. The values of sigma are computed using the difference in log evidence (Trotta 2008; Benneke & Seager 2012b).", '3.2. Results from Main Retrievals': "The main retrieval with stellar inhomogeneities and a planetary atmosphere favors an atmosphere with high H 2 S and SO 2 abundances, with log(VMR) of -0 . 62 +0 . 61 -6 . 73 and -2 . 35 +2 . 05 -5 . 88 respectively. It also suggests the presence of unocculted stellar faculae, as ∆ T het is positive with a retrieved value of 324 +295 -187 K. The results from all of the retrievals are listed in Table 1. \nThe retrieval with only stellar inhomogeneities failed to provide a good fit to the observed spectrum, but resulted in similar posteriors for the stellar parameters. It is worth noting that this scenario fails to reproduce the feature at about 4 µ m which can be attributed to SO 2 . This confirms the results of Paper I that stellar activity alone cannot reproduce the observed spectrum. The scenario with the planet's contribution only, the flat line with an offset, provided a much worse fit to the spectrum - resulting in a reduced chi-squared value of only 1.86. \nThe fit with stellar inhomogeneities and an atmosphere was preferred at 2.24 σ to the fit with only stellar inhomogeneities, and at 3.50 σ to the fit with only a flat line and offset. The fit with only a planetary atmosphere also favors an atmosphere with high abundances of H 2 S and SO 2 . \nThe individual detection significances of H 2 S and SO 2 , calculated by removing each gas in turn from the main retrieval with stellar inhomogeneities and planetary atmosphere are about 2 σ . Thus, these can only be interpreted as weak detections.", '3.3. Alternate Retrieval Setups': 'In addition to the main retrievals, we also considered some alternate retrieval setups to test whether our choice of parameterization affects the inferences. The representation and priors for all parameters are as described in Section 2.2, unless otherwise stated.', '3.3.1. Equilibrium Chemistry Retrieval': 'We performed a retrieval enforcing equilibrium chemistry using NEMESISPY (Yang et al. 2023, 2024b,a), coupled with the FastChem (Stock et al. 2018, 2022) chemical solver. We parameterized elemental abundances using three parameters: Metallicity (Z/Z ⊙ ), Carbon to \nOxygen ratio (C/O), and Sulfur to Oxygen ratio (S/O). The prior distributions for these parameters were set as U (0 . 1 , 1000), U (0 . 1 , 2 . 0), U (0 . 1 , 2 . 0) respectively. This retrieval struggled to produce a good fit to the data, and was not preferred to the baseline model of having only stellar inhomogeneities. This indicates that the atmosphere is probably not in equilibrium. The retrieved model and selected posterior distributions for some parameters are shown in Figure 4.', '3.3.2. No H 2 S/SO 2': 'In an alternate version of the main retrieval, we removed H 2 S and SO 2 from the active gases. This retrieval also struggled to produce a good fit to the data, and was not preferred to the baseline model having only stellar inhomogeneities. This result is further evidence to the tentative presence of these sulfur species in the atmosphere. The retrieved model and selected posterior distributions for some parameters are shown in Figure 4.', '3.3.3. Nitrogen Background': 'In another alternate version of the main retrieval, we assumed that the planet has a nitrogen dominated atmosphere as nitrogen is another gas that only has continuum features in this wavelength range. Thus, we fixed the background to N 2 , and used log-uniform priors for the trace gases, with priors for log(VMR) set to U ( -12 , -0 . 1). This also resulted in significant H 2 S and SO 2 abundances. This indicates that the choice of CLR priors does not bias our inferences of the atmosphere.', '3.3.4. TP Retrieval': 'In this alternate version of the main retrieval, we tested the application of a non-isothermal TemperaturePressure profile using the parameterization of Madhusudhan & Seager (2009). We found that this made a negligible difference to the quality of the fit, and the retrieved profile was close to isothermal, indicating that a non-isothermal profile is not statistically favored for this spectrum. This is probably as the spectrum probes a relatively narrow pressure range. This retrieval also resulted in high H 2 S and SO 2 abundances. \nWehave also performed retrievals in which the isothermal temperature was allowed to vary between 100 K and 1000 K. In these, the retrieved temperatures were either much higher than the calculated bounds, when the atmospheric mean molecular weight was high, or much lower, for a low mean molecular weight atmosphere. The actual solution could possibly lie in between these extremes, and without informative prior bounds, the molecular weight and temperature are highly degenerate. These retrievals also recovered high H 2 S and SO 2 \nFigure 1. The retrieved spectra for the main retrievals: Stellar + Atmosphere scenario is plotted in red, Atmosphere Only in blue, and Stellar Only in green. The 1 σ and 2 σ credible intervals are plotted in dark and light shades of the corresponding colors respectively. The NRS2 points are shifted by the retrieved offset, and the unshifted NRS2 points are shown in gray. Underneath the spectrum, in the first row from left to right, the retrieved posterior distributions for log(H 2 S), log(SO 2 ), and Temperature are shown. In the second row, from left to right, the retrieved posterior distributions for f het , T phot , and ∆ T het are shown. \n<!-- image --> \nabundances. Thus, this degeneracy does not affect the possible presence of sulfur species, which is discussed in Section 4.1.', '4.1. Possible Presence of Sulfur Species': "The primary contributors to the retrieved spectrum across all the retrieval setups we tested are invariably H 2 S and SO 2 , which are both prominent sulfur species. To test their significance, we removed them from the \nlist of active species and performed another retrieval, otherwise identical to the main retrieval as described in Section 3.3.2. This produced a much worse fit to the spectrum than the model with the sulfur species. By comparing the log evidences for these two scenarios, we obtain a combined detection significance of 2.32 σ for H 2 S and SO 2 . \nPrevious studies have suggested that L 98-59 d is a planet which is being tidally heated (Seligman et al. 2024). It has a non-zero eccentricity of 0 . 074 +0 . 057 -0 . 046 (Demangeon et al. 2021) and the planets b, c and d have orbital resonances close to 2:4:7 (Seligman et al. 2024). \nTable 1. Priors and retrieved values of atmospheric parameters from a selection of retrievals \nNote -Only 2 σ upper bounds are specified for the gas abundances that are not constrained. The detection significance σ is computed by comparing the Bayesian Log Evidence of each model with the Stellar Only case for each reduction. For the FIREFLy retrieval, the baseline Stellar Only ln(Z) is 470.80. \nTidal stresses could lead to volcanic outgassing, similar to the volcanic activity on Jupiter's moon Io, and could be a possible source of H 2 S and SO 2 in the atmosphere. Recent modeling studies have also suggested volcanic or outgassed origins of H 2 S and SO 2 (Claringbold et al. 2023; Tsai et al. 2024) in exoplanet atmospheres. \nWe find that the abundances of common atmospheric species such as H 2 O and CO 2 are unconstrained with long tails in the posterior distribution, and we only report upper bounds for their abundances. The possible presence of H 2 S also supports the absence of CO 2 and H 2 O, as H 2 S features in the spectrum would not be visible if they were present (Janssen et al. 2023). As H 2 O is required in the photochemical production of SO 2 , as shown in Tsai et al. (2023), photochemistry is unlikely to be the source of SO 2 in the atmosphere of L 98-59 d. While the possible presence of H 2 S and SO 2 in the absence of H 2 O and CO 2 is unlikely in equilibrium conditions (Janssen et al. 2023), they might exist on an intensely volcanic planet. \nWe also do not detect the presence of any significant clouds or haze scattering. If H 2 S and SO 2 are present, Jordan et al. (2021) found that these should survive above the cloud layer in the atmospheres of planets around M-dwarfs, where they could be detectable.", '4.2. Comparison between Reductions': 'L98-59d has a high impact parameter (Demangeon et al. 2021) and the transit chord crosses near to the limb of the host star from our perspective, thus making it difficult to pin down the limb darkening coefficients Paper I. \nTo check the effect of data reduction methods on our retrievals, we performed all the same retrievals on another reduction of the spectrum using FIREFLy . Here, we present a retrieval identical to our main retrieval, using this reduction instead. While there are some differences in the retrieved parameters, the high H 2 S and SO 2 abundances are still present - with log(VMR) of -4 . 07 +3 . 93 -5 . 11 and -2 . 04 +1 . 94 -6 . 03 respectively. We performed \na fit with a baseline stellar inhomogeneities only for this reduction as well. This atmospheric model is preferred at 1.67 σ to the equivalent baseline fit. The retrieved model and selected posterior distributions are shown in Figure 5.', '4.3. Comparison between Retrieval codes': 'Previous studies have compared the results obtained using different retrieval codes for the same spectrum (Barstow et al. 2020; Taylor et al. 2023). Subtle differences in the forward models used to perform retrievals can lead to significant differences in retrieved parameters. To test if our inferences are being impacted by the choice of retrieval codes, we performed a retrieval with a similar setup as our main retrieval using Taurex3 (AlRefaie et al. 2021). This retrieval, presented in Paper I, also produced similarly high abundances of H 2 S and SO 2 .', '4.4. Future Observations': 'Figure 2 shows the best fit model extended to between 0.6 - 12 µ m. This shows H 2 S and SO 2 features in other regions of the spectrum, accessible to JWST instruments NIRISS, NIRSpec and MIRI. Observations in the 0.6 5.0 µ m range can possibly confirm or refute the signatures of these sulfur species. An accepted JWST proposal in Cycle 1: GTO 1201 (PI: Lafreniere) includes 1 transit of L 98-59 d with NIRISS SOSS. An accepted JWST proposal in Cycle 2: GO 4098 (PI: Benneke) also includes 1 transit of L 98-59 d with NIRSpec G395H and 1 transit with NIRISS SOSS. Apart from these, MIRI observations can particularly help pin down the large SO 2 features in the 7-10 µ m range.', '5. CONCLUSION': 'We have presented a range of retrieval analyses for the NIRSpec/G395H spectrum of L98-59d. Our analyses favor an atmosphere with substantial amounts of sulfur species H 2 S and SO 2 , and an atmospheric temperature higher than the equilibrium temperature. We \nalso find evidence of unocculted faculae on the star. The rocky planets in the Solar System all have different atmospheric compositions, and the study of such atmospheres in exoplanetary systems could unlock a rich diversity of unexplored chemistries. \nOur retrievals do not constrain the abundances of other spectrally active gases such as H 2 O and CO 2 , and we only report upper bounds. The bare rock with stellar inhomogeneities scenario struggles to reproduce the observed spectral features. Several H 2 S and SO 2 features exist in other wavelength regions covered by JWST, as shown in Figure 2. The two scenarios of stellar inhomogeneity with an atmosphere and only stellar inhomogeneity also differ in other wavelength regions. Follow up observations in these wavelengths can confirm or refute the evidence of H 2 S and SO 2 , and distinguish between a planetary atmosphere and stellar inhomogeneities.', '6. ACKNOWLEDGEMENTS': 'AB is supported by a PhD studentship funded by STFC and The Open University. AB thanks Dr Ryan J. MacDonald for discussions on the implementation of the CLR priors used in this work. JKB is supported by UKRI via an STFC Ernest Rutherford Fellowship (ST/T004479/1). CAH is supported by grants ST/T000295/1 and ST/X001164/1 from STFC. SRL thanks STFC for funding under grant ST/X001180/1 and UKSA under grant ST/W002949/1. The JWST data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST) at the Space Telescope Science Institute. The specific observations analyzed can be accessed via 10.17909/nrxscx46. This work used the DiRAC Data Intensive service (DIaL2 / DIaL3) at the University of Leicester, managed by the University of Leicester Research Computing Service on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). The DiRAC service at Leicester was funded by BEIS, UKRI and STFC capital funding and STFC operations grants. DiRAC is part of the UKRI Digital Research Infrastructure. We thank the anonymous referee for their detailed insights and comments which significantly improved this paper.', 'REFERENCES': 'Al-Refaie, A. F., Changeat, Q., Waldmann, I. P., & Tinetti, G. 2021, ApJ, 917, 37, doi: 10.3847/1538-4357/ac0252 \nAlderson, L., Wakeford, H. R., Alam, M. K., et al. 2023, Nature, 614, 664, doi: 10.1038/s41586-022-05591-3 \nAllen, C. W. 1976, Astrophysical Quantities \nAzzam, A. A. A., Yurchenko, S. N., Tennyson, J., & Naumenko, O. V. 2016, Mon. Not. R. Astron. Soc., 460, \n4063, doi: 10.1093/mnras/stw1133 \nBarclay, T., Sheppard, K. B., Latouf, N., et al. 2023, The Transmission Spectrum of the Potentially Rocky Planet L 98-59 c, arXiv. http://ascl.net/2301.10866 \nFigure 2. This plot shows the retrieved spectra from the main retrieval scenarios of stellar inhomogeneities with planetary atmospheres and stellar inhomogeneities only extended to the 0.6 - 12 µ m JWST wavelength range. The gas contributions for H 2 S and SO 2 are highlighted. The available wavelength ranges for different JWST instruments are marked on the top of the plot. Several H 2 S features in the NIRISS SOSS range, combined H 2 S and SO 2 features in NIRSpec G395H range, and SO 2 features in the MIRI LRS range may be detectable in future observations. \n<!-- image --> \nBarstow, J. K. 2020, Monthly Notices of the Royal Astronomical Society, 497, 4183, \ndoi: 10.1093/mnras/staa2219 \nBarstow, J. K., Aigrain, S., Irwin, P. G. J., Kendrew, S., & Fletcher, L. N. 2016, MNRAS, 458, 2657, \ndoi: 10.1093/mnras/stw489 \nBarstow, J. K., Aigrain, S., Irwin, P. G. J., & Sing, D. K. 2017, ApJ, 834, 50, doi: 10.3847/1538-4357/834/1/50 \nBarstow, J. K., Changeat, Q., Garland, R., et al. 2020, \nMNRAS, doi: 10.1093/mnras/staa548 \nBeatty, T. 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G., & Fu, Y. 2001, JQSRT, 68, 235, doi: 10.1016/S0022-4073(00)00023-6 Braude, A. S., Irwin, P. G. J., Orton, G. S., & Fletcher, L. N. 2020, Icarus, 338, 113589, doi: 10.1016/j.icarus.2019.113589 Buchner, J., Georgakakis, A., Nandra, K., et al. 2014, Astronomy & Astrophysics, 564, A125, doi: 10.1051/0004-6361/201322971 Chubb, K. L., Rocchetto, M., Yurchenko, S. N., et al. 2021, A&A, 646, A21, doi: 10.1051/0004-6361/202038350 Claringbold, A. B., Rimmer, P. B., Rugheimer, S., & Shorttle, O. 2023, AJ, 166, 39, doi: 10.3847/1538-3881/acdacc Cloutier, R., Astudillo-Defru, N., Doyon, R., et al. 2019, A&A, 621, A49, doi: 10.1051/0004-6361/201833995 Coles, P. A., , Yurchenko, S. N., & Tennyson, J. 2019, Mon. Not. R. Astron. Soc., 490, 4638, doi: 10.1093/mnras/stz2778', 'A. CORNER PLOT': 'Here, we show the full corner plot for the main retrieval including stellar inhomogeneities and a planetary atmosphere. The full corner plots for all the retrievals mentioned in this paper can be found at:https://github.com/riobanerjee/ supplement L9859d. \nFigure 3. The corner plot showing retrieved posterior distributions for each parameter for the main retrieval. The plots on the diagonal show the histograms of retrieved parameters and the inner plots show the pairwise correlations between the parameters. Large abundances for H 2 S and SO 2 can be seen, with upper bounds for all other active gases. \n<!-- image -->', 'B. ALTERNATE RETRIEVAL PLOT': 'Here, we show the spectral fits and posteriors for selected parameters for two of the alternate retrievals: equilibrium chemistry and the retrieval without H 2 S and SO 2 included. \nFigure 4. The retrieved spectra for two the alternate retrievals are shown: The Stellar + Atmosphere scenario with H 2 S and SO 2 removed is plotted in red and the equilibrium chemistry scenario is plotted in blue. The 1 σ and 2 σ credible intervals are plotted in dark and light shades of the corresponding colors respectively. The NRS2 points are shifted by the retrieved offset, and the unshifted NRS2 points are shown in gray. Underneath the spectrum, in the first row from left to right, the retrieved posterior distributions for f het , T phot , ∆ T het , and offset are shown. In the second row, from left to right, the retrieved posterior distribution for Temperature, Metallicity, S:O ratio, and C:O ratio are shown. \n<!-- image -->', 'C. COMPARISON BETWEEN REDUCTIONS PLOT': 'Here, we show the spectral fits and posteriors for selected parameters for retrievals using two different reductions: transitspectroscopy and FIREFLy . \nFigure 5. The retrieved spectra for the retrievals using two different reductions are shown: transitspectroscopy is plotted in red and FIREFLy is plotted in blue. The 1 σ and 2 σ credible intervals are plotted in dark and light shades of the corresponding colors respectively. The NRS2 points are shifted by the retrieved offset, and the unshifted NRS2 points are shown in gray. Underneath the spectrum, in the first row from left to right, the retrieved posterior distributions for log(H 2 S), log(SO 2 ), and Temperature are shown. In the second row, from left to right, the retrieved posterior distributions for f het , T phot , and ∆ T het are shown. \n<!-- image -->'}

2024arXiv240905628L

Recent advances in the classical Double Copy DC procedure have revealed a profound connection between gauge theories and Tduality invariant frameworks with Double Field Theory the classical DC of YangMills theory emerging as the first explicit example. Extending this procedure to higherderivative gauge theories predicts the existence of a HigherDerivative Double Theory HDDT which incorporates Weyl gravity along with bfield and dilaton contributions all in a Tduality invariant manner. In this work we show that combining both mappings leads to DFT a Tduality invariant model related to the bosonic string incorporating firstorder alpha corrections upon parameterization. Our results expand the potential applications of the DC program towards constructing perturbative alphacorrected Lagrangians while also opening up possibilities for reversing the map by considering the single and zeroth copies.

2024-09-01T00:00:00Z

['2024arXiv240905628L', '10.48550/arXiv.2409.05628', 'arXiv:2409.05628']

['High Energy Physics - Theory', 'General Relativity and Quantum Cosmology']

Quadratic Curvature Corrections in Double Field Theory via Double Copy

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.05628.pdf

{'Quadratic Curvature Corrections in Double Field Theory via Double Copy': "Eric Lescano 1, ∗ and Jes'us A. Rodr'ıguez 2, † \n1 Institute for Theoretical Physics (IFT), University of Wroclaw, \npl. Maxa Borna 9, 50-204 Wroclaw, Poland \n2 \nUniversidad de Buenos Aires, FCEyN, Departamento de F'ısica, Ciudad Universitaria, 1428 Buenos Aires, Argentina \nRecent advances in the classical Double Copy (DC) procedure have revealed a profound connection between gauge theories and T-duality invariant frameworks, with Double Field Theory (DFT)-the classical DC of Yang-Mills theory-emerging as the first explicit example. Extending this procedure to higher-derivative gauge theories predicts the existence of a Higher-Derivative Double Theory (HDDT), which incorporates Weyl gravity along with b -field and dilaton contributions, all in a Tduality invariant manner. In this work, we show that combining both mappings leads to DFT+, a T-duality invariant model related to the bosonic string, incorporating first-order α ' corrections upon parameterization. Our results expand the potential applications of the DC program towards constructing perturbative α ' -corrected Lagrangians, while also opening up possibilities for reversing the map by considering the single and zeroth copies.", 'I. HIGHER-DERIVATIVE CORRECTIONS AND THE DOUBLE COPY MAP': "The NS-NS sector of the low energy limit of string theory exhibits a global O( n, n ) symmetry when the fields are independent of n spatial coordinates [1]. This continuous T-duality symmetry is exact to all orders in α ' [2], which motivates the study of higher-order corrections in theories with manifest duality invariance, such as Double Field Theory (DFT) [3, 4], a proposal to include T-duality as a fundamental symmetry of a field theory on a doubled space 1 . DFT reformulates supergravity in terms of O( D,D ) multiplets within a doubled geometrical framework, where D is the space-time dimension, and D ≥ n . \nAlthough the geometric structure of DFT makes it difficult to directly construct invariant objects that are quadratic in curvatures, substantial progress has been made in obtaining higher-derivative corrections [8-17]. In [12], an exact mechanism was introduced through a generalization of the Green-Schwarz transformation, which requires an infinite tower of O( D,D )-covariant higher-derivative terms in the gauge-invariant action. Specifically, the first-order corrections reduce to the biparametric theory presented in [10], which interpolates between different low-energy limits of String Theory. \nHowever, as stated in [18], DFT and similar frameworks cannot reproduce all of the higher-order terms in the α ' expansion of string theory, and hence developing new techniques to obtain higher-derivative terms remains a major challenge. It is within this context that the Double Copy (DC) construction [19-27] emerges as a promising approach to potentially provide new insights for systematically generating higher-derivative corrections. \nThe quadratic and cubic contributions to perturbative DFT can be derived by applying a DC map to the quadratic and cubic Yang-Mills (YM) Lagrangian [28]. This connection between YM theory and perturbative DFT suggests that DFT is part of a broader class of T-duality invariant theories, all of which can be derived through the DC procedure applied to different gauge theories. More recent research [29, 30] delved into a new DC prescription applied to the higher-derivative gauge theory [31, 32] \nL = a 1 κ ab D µ F µνa D ρ F ρ ν b + a 2 κ αβ D µ φ α D µ φ β , (1) \nwhere µ, ν, · · · = 0 , . . . , D -1 are space-time indices, a, b, . . . , are indices in the adjoint representation of the gauge group and α, β . . . , are indices in some real representation of the same group. The fundamental fields of the theory are a gauge field A µ a , and a scalar field φ α . The coefficients a 1 , a 2 are real and depend on the space-time dimension through [30] \na 1 = -2 ( D -3 D -2 ) , a 2 = a 1 2( D -1) . (2) \nAfter applying the DC map on (1) at quadratic order in fields, the resulting theory contains Weyl gravity alongside b -field and dilaton contributions written in the form of a perturbative Higher-Derivative Double Theory (HDDT), \n[ a 1 ( DF ) 2 + a 2 ( Dφ ) 2 DC --→ pert. HDDT ] (2) . (3) \nIts field content consists of a generalized frame e µ ¯ µ ( x, ˜ x ), and a generalized dilaton Φ( x, ˜ x ), both obtained by identifying A µ a and φ α as described in [29, 30]. In particular, in [30] the authors considered interaction terms into Lagrangian (1) and demonstrated the existence of HDDT beyond the quadratic order. \nIn this work we explore the possibility of using a classical, off-shell DC map to generate perturbative, fourderivative corrections to the bosonic string. To do this, \nwe begin by considering DFT+ [10], a well-known Tduality invariant theory whose four-derivative corrections are related to the closed bosonic string theory upon parametrization, described by the action principle (11), up to field redefinitions. This action is invariant under the standard symmetries of DFT. When the strong constraint is imposed, the theory reduces to the standard NS-NS supergravity Lagrangian, along with the firstorder α ' corrections of the closed bosonic string. This reduced action corresponds to the result obtained by applying the DC prescription to a theory that includes YangMills (YM) together with (1), suggesting the relation \n[ Y M + a 1 ( DF ) 2 + a 2 ( Dφ ) 2 DC --→ pert . DFT+ ] . (4) \nThe findings presented in this study offer a starting point for several potential research directions, which will be further discussed in the final section of this work.", 'II. DFT + BEHIND DOUBLE COPY MAPS': "In this section, we construct the non-perturbative DFT+ theory (up to field redefinitions) by considering O( D,D ) multiplets in a double geometry, following the standard approach of incorporating higher-derivative terms into DFT. \nThe dimension of the fundamental representation of O( D,D ) is 2 D , then the coordinates are defined in double space through X M = ( x µ , ˜ x µ ), with M,N,... indices in the fundamental representation of the duality group. The coordinates ˜ x µ are dual coordinates and are removed by \nimposing the strong constraint, \n∂ M ( ∂ M /star ) = ( ∂ M /star )( ∂ M /star ) = 0 , (5) \nwhere /star denotes arbitrary generalized fields/parameters or products of them. The fundamental fields are a generalized frame E M A and a generalized dilaton d . These fields transform with respect to generalized diffeomorphisms and double Lorentz transformations as \nδ ˆ ξ, Λ E M A = L ˆ ξ E M A + E M B Λ B A , (6) \nδ ˆ ξ d = ˆ ξ N ∂ N d -1 2 ∂ M ˆ ξ M , (7) \nwhere L ˆ ξ is the generalized Lie derivative. The generalized frame satisfies, \nE MA H AB E NB = H MN , E MA η AB E NB = η MN , (8) \nwhere η AB and H AB are double Lorentz invariant metrics, η MN is the O( D,D ) invariant metric, and H MN is known as the generalized metric. Using this metrics we can define the (flat) projectors \nP AB = 1 2 ( η AB -H AB ) , P AB = 1 2 ( η AB + H AB ) , \nacting on arbitrary double-Lorentz vectors, lead to P A B V B = V A and P A B V B = V A . \nAt this point it is necessary to introduce the generalized fluxes \nF ABC = 3 ∂ [ A E M B E | M | C ] , (9) \nwith the flat derivative defined as ∂ A = √ 2 E M A ∂ M . The action of DFT+ is given by \nF A = √ 2 e 2 d ∂ M ( e -2 d E M A ) , (10) \nS DFT+ = α ' ∫ d 2 D Xe -2 d ( R-R ( -) -R (+) + 1 2( D -2) R AB R AB -1 2( D -2)( D -1) R 2 ) , (11) \n= 2 ∂ A F A + F A F A 1 F ABC F ABC 1 F ABC F ABC , \nR -6 -2 (12) \nC C D C \nR AB = ∂ A F B -∂ C F AB + F DA F BC -F F ABC , (13) R (+) = -1 2 [ ( ∂ A ∂ B F B CD ) F ACD +( ∂ A ∂ B F A CD ) F BCD +2( ∂ A F B CD ) F A CD F B +( ∂ A F ACD )( ∂ B F B CD ) +( ∂ A F B CD )( ∂ A F B CD ) + 2( ∂ A F B ) F B CD F ACD +( ∂ A F BCD ) F C CD F ABC -( ∂ A F BCD ) F C CD F ABC +2( ∂ A F A CD ) F B CD F B -4( ∂ A F B CD ) F A CE F BE D + 4 3 F E AC F BED F C CD F ABC + F B CD F A CD F B F A + F A CE F BED F A CG F BGD -F B CE F AED F A CG F BGD -F ABD F D CD F C CD F ABC ] . (14) \nHere, R is the generalized scalar curvature and R AB is \nthe generalized Ricci tensor. R (+) is a higher-derivative \nwith \ncombination [10] that can be constructed by extending to the heterotic duality group and applying the generalized Bergshoeff-de Roo identification [12], while R ( -) = R (+) [ P ↔ P ]. The four-derivative extra terms depending on R and R AB are contributions that are typically not considered in higher-derivative DFT, as they are proportional to the leading-order equations of motion and can therefore be absorbed through field redefinitions. \nThe DFT+ action is, by construction, invariant under global O( D,D ) transformations as well as local generalized diffeomorphisms. It also exhibits invariance under double Lorentz transformations, which generalize the Green-Schwarz mechanism in the doubled space. This is realized through a deformation of the Lorentz transformation of the generalized frame, given by \nδ (1) E M A = -F B CD ∂ A Λ CD E M B , δ (1) E M A = F A CD ∂ B Λ CD E M B . (15) \nWhile one might be tempted to break Lorentz invariance and define S HDDT as consisting only of the fourderivative terms in (11), thereby recovering the double copy map (3) and restoring ordinary Lorentz invariance upon parametrization, this proposal for the HDDT action does not reduce to Weyl gravity in the pure gravitational limit. The inclusion of the generalized Ricci scalar is crucial due to higher-derivative field redefinitions at the supergravity level. \nThe parametrization of the DFT+ degrees of freedom follows the same form as in standard DFT \nE M A = 1 √ 2 ( -e µa -b ρµ e ρ a e µ a , e µa -b ρµ e ρ a e µ a ) , (16) \ne -2 d = √ -˜ ge -2 ˜ φ , (17) \nwhere e µa and e µa are two vielbein generating the same D -dimensional metric ˜ g µν . As mentioned previously, it is necessary to perform the following field redefinition, \n˜ g µν = g µν -α ' 2 Ω ( -) µab Ω ( -) ν ab -α ' 2 Ω (+) µab Ω (+) ν ab , (18) \nto eliminate the anomalous transformation of the metric under Lorentz transformations. Additionally, the dilaton must be redefined to ensure that the integration measure remains invariant, i.e., √ -˜ ge -2 ˜ φ = √ -ge -2 φ . In (18) we include the torsionful spin connection \nΩ ± µab = w µab ± 1 2 H µab , (19) \nwhere w µab is the spin connection, and H µab is the field strength of the Kalb-Ramond field. \nThe low energy effective action that describes the previous theory once the duality group is broken is given by \nS = ∫ d D x √ -ge -2 φ [ L (2) -α ' 4 ( Riem 2 -4 D -2 R µν R µν + 2 ( D -2)( D -1) R 2 + L (4) matter ( b, φ ) )] , (20) \nwith L (2) the NS-NS sector of supergravity. The fourderive terms of this Lagrangian reproduces Weyl gravity in the pure gravitational limit (p.g.: b µν = φ = 0), and by setting α ' = -4 the action reduces to, \nS | p.g. = ∫ d D x √ -g ( R + C µνρσ C µνρσ ) , (21) \nwhere C µνρσ is the Weyl tensor. This is the theory obtained from applying the DC prescription on the gauge side of (4) and hence, the confirmation of the existence of a DC map between that gauge theory and DFT+.", 'III. DISCUSSION': "In this work, we explored the role of the Double Copy construction as a tool for deriving higher-derivative corrections in the context of Double Field Theory. Our analysis demonstrates that applying the DC prescription [28-30] to the Yang-Mills Lagrangian, along with higherderivative terms from gauge theory, leads to a perturba- \ntive structure consistent with previous studies on higherderivative corrections, of the form DFT+HDDT. \nWe further showed that the DFT+HDDT structure is connected to an exact formalism given by the DFT+ theory, which includes four-derivative corrections that are invariant under global O( D,D ) transformations and local generalized diffeomorphisms. These corrections align with the α ' expansion of the closed bosonic string when properly parameterized. Notably, the action contains contributions essential for maintaining Lorentz invariance and recovering Weyl gravity in the pure gravitational limit, underscoring the importance of higherderivative field redefinitions at the supergravity level. \nBy employing the DC map, we successfully related the structure of gauge theory to T-duality-invariant theories, with a particular emphasis on the non-trivial effects of higher-derivative corrections. Now that the connection with DFT+ is established, further study of higher-order gauge theories becomes highly relevant, as they may connect to T-duality-invariant models through the DC. \nOn the other hand, constructing a Conformal Field Theory (CDFT) in the double geometry, as outlined in \n[40], using a higher-derivative DFT is limited by double Lorentz transformations. To the best of our knowledge, the only higher-derivative combination in the biparametric extension given in [10] that can be expressed in terms of the generalized metric is provided by the DFT- or HSZ theory [8, 9], whose parameterization does not lead to Riem 2 contributions. We do not rule out the possibility that CDFT could include additional noncovariant contributions, potentially incorporating noncovariant terms to resolve the lack of invariance within the generalized frame formalism (both in the bosonic or heterotic cases). We leave this issue for future investigation. \nSeveral directions for future research arise from the results of this work. We briefly outline some of them below: \n- · New Double Copy Realizations: The connection established in this work between DC maps and DFT+ provides a strong foundation for further exploring links between gauge theories and T-dualityinvariant frameworks. 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It is important to clarify that CDFT is not defined solely through the higher-derivative structure of DFT+, but the methods developed in this work could serve as a solid framework for building the CDFT Lagrangian by considering a non-covariant extension of the bi-parametric family of theories outlined in [10]. \n- · Single and Zeroth Copies in HigherDerivative Theories: The study of the single and zeroth copies of perturbative DFT was initially addressed in [33] and further developed in [34-39]. The perturbative ansatz used to link DFT and Yang-Mills dynamics via the single and zeroth copies is a generalized version of the Kerr-Schild ansatz. The results of this work now open the possibility of investigating higher-derivative corrections to the generalized Kerr-Schild ansatz, connecting DFT+ to the higher-derivative gauge theory described in (1), thereby extending the framework in [36].", 'Acknowledgements': "E.L. is supported by the SONATA BIS grant 2021/42/E/ST2/00304 from the National Science Centre (NCN), Poland. J.A. R. gratefully acknowledges the support provided by Universidad de Buenos Aires (UBA). \n- [8] O. Hohm, W. Siegel and B. Zwiebach, 'Doubled α ' -geometry', JHEP 1402 , 065 (2014), [hep-th/1306.2970].\n- [9] E. Lescano and D. Marques, 'Second order higherderivative corrections in Double Field Theory', JHEP 1706 , 104 (2017), [hep-th/1611.05031].\n- [10] D. Marques and C.A. Nu˜nez, 'T-duality and α ' -corrections,' JHEP 1510 , 084 (2015), [hepth/1507.00652].\n- [11] W. Baron, J. Fernandez-Melgarejo, D. Marques, C. Nunez, 'The Odd story of α ' -corrections', JHEP 04 (2017) 078, [hep-th/1702.05489].\n- [12] W. 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2024A&A...690A.190A

Context. Highprecision and highcadence photometric surveys such as Kepler or TESS are making huge progress not only in the detection of new extrasolar planets but also in the study of a great number of variable stars. This is the case for central stars of planetary nebulae PNe which have similarly benefited from the capabilities of these missions increasing the number of known binary central stars and helping us to constrain the relationship between binarity and the complex morphologies of their host PNe. Aims. In this paper we analyse the TESS light curves of a large sample of central stars of PNe with the aim of detecting signs of variability that may hint at the presence of shortperiod binary nuclei. This will have important implications in understanding PN formation evolution as well as the common envelope phase. Methods. We analysed 62 central stars of true likely or possible PNe and modelled the detected variability through an MCMC approach accounting for three effects reflection ellipsoidal modulations due to tidal forces and the socalled Doppler beaming. Among the 62 central stars only 38 are amenable for this study. The remaining 24 show large contamination from nearby sources preventing an optimal analysis. Also eight targets are already known binary central stars which we revisit here with the new high precision of the TESS data. Results. In addition to recovering the eight already known binaries in our sample we find that 18 further central stars show clear signs of periodic variability in the TESS data probably resulting from different physical effects compatible with the binary scenario. We propose them as new candidate binary central stars. We also discuss the origin of the detected variability in each particular case by using the TESSlocalize algorithm. Finally 12 targets show no or only weak evidence of variability at the sensitivity of TESS. Conclusions. Our study demonstrates the power of spacebased photometric surveys in searching for close binary companions of central stars of PNe. Although our detections can only be catalogued as candidate binaries we find a large percentage of possible stellar pairs associated with PNe supporting the hypothesis that binarity plays a key role in shaping these celestial structures.

2024-10-01T00:00:00Z

['10.48550/arXiv.2409.06332', '10.1051/0004-6361/202450942', 'arXiv:2409.06332', '2024A&A...690A.190A', '2024arXiv240906332A']

['techniques: photometric', 'binaries: general', 'planetary nebulae: general', 'Astrophysics - Solar and Stellar Astrophysics', 'Astrophysics - Astrophysics of Galaxies']

Planetary nebulae seen with TESS New and revisited shortperiod binary central star candidates from Cycles 1 to 4

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']

https://arxiv.org/pdf/2409.06332.pdf

{'Planetary nebulae seen with TESS: New and revisited short-period binary central star candidates from Cycles 1 to 4': 'Alba Aller 1 , Jorge Lillo-Box 2 , and David Jones 3 , 4 , 5 \n- 1 Observatorio Astronómico Nacional (OAN), Alfonso XII 3, 28014, Madrid, Spain e-mail: a.aller@oan.es\n- 2 Centro de Astrobiología (CAB), CSIC-INTA, Camino Bajo del Castillo s / n, 28692, Villanueva de la Cañada (Madrid), Spain\n- 3 Instituto Astrofísico de Canarias, E-38205, La Laguna, Spain\n- 4 Departamento de Astrofísica, Universidad de la Laguna, E-38206 La Laguna, Tenerife, Spain\n- 5 Nordic Optical Telescope, Rambla José Ana Fernández Pérez 7, 38711, Breña Baja, Spain \nSeptember 11, 2024', 'ABSTRACT': 'Context. High-precision and high-cadence photometric surveys such as Kepler or TESS are making huge progress not only in the detection of new extrasolar planets but also in the study of a great number of variable stars. This is the case for central stars of planetary nebulae (PNe), which have similarly benefited from the capabilities of these missions, increasing the number of known binary central stars and helping us to constrain the relationship between binarity and the complex morphologies of their host PNe. \nAims. In this paper, we analyse the TESS light curves of a large sample of central stars of PNe with the aim of detecting signs of variability that may hint at the presence of short-period binary nuclei. This will have important implications in understanding PN formation evolution as well as the common envelope phase. \nMethods. We analysed 62 central stars of true, likely, or possible PNe and modelled the detected variability through an MCMC approach accounting for three e ff ects: reflection, ellipsoidal modulations due to tidal forces, and the so-called Doppler beaming. Among the 62 central stars, only 38 are amenable for this study. The remaining 24 show large contamination from nearby sources preventing an optimal analysis. Also, eight targets are already known binary central stars, which we revisit here with the new high precision of the TESS data. \nResults. In addition to recovering the eight already known binaries in our sample, we find that 18 further central stars show clear signs of periodic variability in the TESS data, probably resulting from di ff erent physical e ff ects compatible with the binary scenario. We propose them as new candidate binary central stars. We also discuss the origin of the detected variability in each particular case by using the TESS\\_localize algorithm. Finally, 12 targets show no or only weak evidence of variability at the sensitivity of TESS. Conclusions. Our study demonstrates the power of space-based photometric surveys in searching for close binary companions of central stars of PNe. Although our detections can only be catalogued as candidate binaries, we find a large percentage of possible stellar pairs associated with PNe, supporting the hypothesis that binarity plays a key role in shaping these celestial structures. \nKey words. (ISM:) planetary nebulae: general - techniques: photometric - (stars:) binaries: general', '1. Introduction': "Planetary nebulae (PNe) are one of the most fascinating astronomical objects in the Universe. They are formed when low-tointermediate mass stars (0.8-8 M ⊙ ) reach the final stage of their lives, ejecting their outer envelopes into the interstellar medium. As a result, a great diversity of PNe are observed in our Galaxy, and there are no two PNe that are exactly alike. But one thing seems clear: spherical PNe are almost the exception and they represent only one-fifth of all the PNe discovered in the Milky Way. On the contrary, the remaining 80% show a wide range of complex morphologies (Parker et al. 2006). \nExplaining the precise processes that lead to the formation of these kinds of structures is a di ffi cult task, although some progress has been made in the last decades with central star binarity emerging as the most likely culprit (see Bo ffi n & Jones 2019, and references therein). Binary central stars of PNe (bCSPNe) appear to be key not only to explain the variety of shapes in PNe but also to respond to many other open questions in stellar evolution, such as the mass-loss rate and history, the little-understood \ncommon envelope phase, or the estimation of the binary population in general. \nThus, many of the e ff orts in the PN community today are devoted to the search for, and characterisation of, new binary central stars in the heart of PNe (with both short and long orbital periods), with the aim of shedding light on the formation of the striking morphologies they present. Dedicated ground-based observations of CSPNe are very costly in terms of time, so very good candidates among the large sample of PNe (more than 3000 in our Galaxy) are required to guide targeted observational campaigns. Within this context, large photometric surveys such as OGLE(Optical Gravitational Lensing Experiment, Udalski et al. 1992), Kepler (Borucki et al. 2003), and TESS (Transiting Exoplanets Survey Satellite, Ricker et al. 2015) have enormously contributed to the discovery of a large fraction of (candidate) binary systems in the nuclei of PNe. Numerous papers that have been published recently focus on this topic with encouraging results, significantly increasing the binary population (see, e.g. Miszalski et al. 2009; De Marco et al. 2015; Jacoby et al. 2021; Aller et al. 2020). However, the fraction of known binary central stars in the whole population of PNe remains very low, represent- \ning less than 1% of the whole sample of known Galactic PNe, while unbiased surveys indicate that the true fraction should be ∼ 20% or more (Jones & Bo ffi n 2017; Jacoby et al. 2021). The need to continue this search for binary central stars is, therefore, essential in order to ultimately obtain a statistically significant and unbiased sample with which to definitively understand the role of binarity in the formation and evolution of PNe (as well as related phenomena). \nIn this paper, we continue the analysis carried out in Aller et al. (2020), in which we investigated the variability of the eight central stars of PNe observed in Cycle 1 of TESS for indications of binarity. Here, we extend this study to cover the first four years of TESS operations (Cycles 1 to 4) in order to find new short-period binary central star candidates in the whole sample of galactic PNe. In this search, as well as some already known binary central stars, we also identified new binary candidates that present modulations in their TESS light curves compatible with the presence of companion stars. These modulations are likely due to irradiation e ff ects (i.e. the reflection of the light of the central star on the companion's surface), ellipsoidal modulations (consequence of tidal forces with the companion), and Doppler beaming (relativistic e ff ects due to the orbital motion of a binary system with respect to the observer). Previous works have used these e ff ects to detect binary companions (see, for instance, Santander-García et al. 2015; De Marco et al. 2015; Maxted et al. 2002; Faigler & Mazeh 2011) and even planets (e.g. Shporer et al. 2011; Mazeh et al. 2012; Lillo-Box et al. 2014; Millholland & Laughlin 2017). All these physical e ff ects open the door to the detection of close binaries without requiring the detection of eclipses or the need for radial velocity monitoring. Also, TESS represents a valuable opportunity to revisit close binary CSPNe already known by improving the determination of their properties, especially in those eclipsing systems. \nThe paper is organised as follows. In Section 2, we present our target sample and briefly introduce the TESS datasets. The light curve modelling process is explained in Section 3 and Section 4 is dedicated to the analysis of the results, discussing the detected variability in the light curves and the periodicities found, as well as distinguishing whether they are already known binaries, new binary candidates or stars with weak evidence or no variability at all. In Section 5, we discuss the origin of the variability in each particular case and, finally, we close the paper in Section 6 with some short final remarks and main conclusions.", '2.1. The PN sample in TESS': "TESS was launched in 2018 and was initially designed as a twoyear mission, although it is now conducting its second-extension. During the first year of observations, TESS completed Cycle 1 in a total of 13 sectors and provided two-minute cadence light curves with a baseline of ∼ 27 days for eight central stars of PNe which were analysed in detail in Aller et al. (2020). In July 2019, TESS started its Cycle 2, observing the northern ecliptic hemisphere in another 13 additional sectors. Since then, every year TESS completes a new cycle. In this paper, we have analysed all the CSPNe that have been observed with two-minute cadence until September 2022, that is, Cycles 1, 2, 3, and 4. They are listed in Table A.1, along with their most relevant information as PNG designation, common name, TESS input catalogue (TIC) designation, equatorial coordinates, TESS magnitude, the sector or sectors of the observation and the PN status found in the \nHASH database 1 (Bojiˇci'c et al. 2017). In total, we analysed the available two-minute cadence light curves for 62 central stars of true (T), possible (P) or likely (L) PNe, including the eight previously presented in Aller et al. (2020) from Cycle 1. We note that almost half of the targets do not have the TIC designation linked to SIMBAD, so we have carefully inspected all the target pixel files (TPFs) and targets, to ensure that the light curves correspond to the target itself.", '2.2. TESS photometric data': "We retrieved the light curves provided by the Science Processing Operations Center (SPOC) pipeline (Jenkins et al. 2016) from the Mikulski Archive for Space Telescopes MAST 2 ). We used the PDCSAP (Pre-search Data Conditioning Simple Aperture Photometry) flux for the analysis, only removing those data points with a non-zero 'quality' flag. For a more detailed description of the TESS data, we point the interested reader to Aller et al. (2020). When more than one sector was available in the MAST archive for a particular target, we combined all the individual light curves to produce a single and continuous multi-sector light curve. This is done with the stitch method in the Lightkurve package, which concatenates and normalises all the sectors. \nIn order to check for possible blends and contaminating sources in the apertures, we plotted the TPF of each central star in the sample using tpfplotter 3 (Aller et al. 2020). This check of the TPFs is especially important in the most crowded fields, since the relatively large pixel size of the TESS CCDs (21 arcseconds per pixel) can lead to strong photometric contamination from nearby sources. Figure 1 shows an example TPF for three di ff erent degrees of contamination. In all cases, the central star is marked with a white cross and the red circles represent Gaia sources from the DR3 catalogue (Gaia Collaboration et al. 2021), scaled by magnitude contrast against the target source. We have over-plotted all the sources with a magnitude contrast up to ∆ m = 6 (i.e. six magnitudes fainter than our target, which corresponds to ∼ 0.4% of contamination if entirely inside the aperture). The aperture mask used by the pipeline to extract the photometry has also been plotted over the TPF. \nAs shown in the figure, we find very crowded fields with severe contamination (larger than 10%), fields with little contamination and cases with negligible contamination where no other star within a contrast magnitude of ∆ m = 6 is identified by Gaia inside the TESS aperture. We note that this is only a first selection criterion that allows us to avoid critically contaminated cases where the attribution of the true source of the variability will not be possible with the current dataset and tools. For targets passing this criterion, we perform an analysis using the TESS\\_localize code to unveil the origin of the photometric variations and account for the possible contamination of stars outside the aperture. Figure B.1 4 shows the TPFs of the first sector observed for each target. \nFor each individual central star and sector, we have calculated the contribution of the field stars to the total flux in the aperture mask. Out of the 62 targets, we found severe contamination for 28 stars. We decided to exclude these targets from further analysis, since the probability that any potential variability \n3 https://github.com/jlillo/tpfplotter \n4 Available at https://zenodo.org/records/13370453 . \n) \ne \n( \n0 \n1 \n× \nx \nu \nl \nF \n<!-- image --> \n<!-- image --> \n) \ne \n( \n0 \n1 \n× \nx \nu \nl \nF \n317 \n324 \n333 \n345 \n350 \n356 \n298 \n293 \n348 \n292 \nFig. 1: Target pixel files (TPFs) of three central stars in the sample (marked with white crosses) obtained with tpfplotter . The red circles are sources from the Gaia DR3 catalogue in the field with scaled magnitudes (see legend). The aperture mask used by the pipeline to extract the photometry is also marked. The pixel scale is 21 arcsec pixel -1 . Left, middle and right panels represent examples of fields with negligible, minimal or severe contamination, respectively (see Sect. 2.2 for more details). \n<!-- image --> \ncan come from any of the nearby stars is too high. Only for specific contaminated objects, we carried out additional studies. In particular, Abell 63 (because it is a well-known eclipsing binary central star in an extremely crowded field), and also three objects that we already analysed in Aller et al. (2020) from Cycle 1: NGC246, NGC2867, and NGC5189 (see Sect. 4). In addition, we found 16 cases with minimal ( ≤ 10% but not null) contamination and 18 central stars where there was no appreciable contamination from nearby sources in the aperture mask. The level of contamination (negligible, minimal or severe) is marked in Table A.1 with a, b or c, respectively. We subsequently discuss these cases in detail in Section 4. \nOnce this contamination cleaning step is done, each single multi-sector light curve was inspected by performing a LombScargle analysis in order to identify possible periodic variability in the data as a first look. \nAt this point, it is important to note that apart from the possible contamination of other stars in the field, we sometimes have bright PNe around our targets of interest, whose light might dilute the amplitude of the detected variability and hence, have an impact on the derived physical parameters. Quantifying this PN contribution is out of the scope of this study, which is to identify new binary candidates in the sample of central stars of PNe.", '3.1. Parametric approach': 'In a first approach, similar to the analysis performed in our previous work in Cycle 1 (Aller et al. 2020) and other works using the same detection technique (e.g. Millholland & Laughlin 2017; Lillo-Box et al. 2016), we model the TESS light curves by using simple sinusoidal functions for the three e ff ects (ellipsoidal, Doppler beaming and reflection). This corresponds to \n∆ F F ! ellip = -A ellip cos (2 θ ) , (1) \n∆ F F ! beam = A beam sin ( θ ) , and (2) \n∆ F F ! ref = -A ref cos ( θ ) , (3) \nwith θ = 2 π ( t -T 0) / P , where T 0 is the time of inferior conjunction of the companion star, P is the orbital period, and A ell, A beam, and A ref are the amplitudes of the ellipsoidal, Doppler beaming, and reflection e ff ects, respectively. The simple model thus corresponds to: \n∆ F F ! 3E = Z 0 + ∆ F F ! ellip + ∆ F F ! beam + ∆ F F ! ref = = Z 0 -A ellip cos (2 θ ) + A beam sin ( θ ) -A ref cos ( θ ) . (4) \nAs in most cases only one of the e ff ects might be present, we also test a simpler model with only one sinusoidal function, namely: \n∆ F F ! 1E = Z 0 + ∆ F F ! 1E = Z 0 + A 1E cos ( θ ) . (5) \nBased on these equations, we have seven parameters in the case of the 3-e ff ects (3E, Eq. 4), namely the photometric level ( Z 0), the three amplitudes ( A ellip, A ref, A beam), the time of inferior conjunction ( T 0), the orbital period P , and the photometric jitter (a term that we add in quadrature to the photometric uncertainties to account for other unknown systematics). For the 1-e ff ect model (1E, Eq. 5), we only have four parameters as only one amplitude is needed ( A 1E). \nWe acknowledge the fact that this purely sinusoidal model is only an approximation to a physically motivated model. For instance, gravity darkening and geometrical e ff ects produce nonsinusoidal ellipsoidal modulations. However, this approximation is su ffi cient for the exploratory goals of this work. \nWe explore the parameter space through a Monte Carlo Markov Chain sampler using the emcee algorithm (ForemanMackey et al. 2013). In particular, we set uninformative (uniform) priors for the orbital period between 0.1-10 days and for the time of inferior conjunction between the first date of observation and 10 days afterwards. In general terms, we allow the amplitudes to vary from 0 to twice the maximum peak-to-peak value of each light curve. The photometric level is also left with a uniform prior U (0 . 9 , 1 . 1), as well as the jitter term that we sample in the range U (0 , 100) in ppm. \nIn total, three models are tested for each of the targets studied in this paper: a flat model (FL) with only the photometric level and jitter as parameters, a 1-e ff ect model (1E), and the 3e ff ects model (3E). For each of them, we sample the posterior \n325 \ndistribution of the parameters using a number of walkers equal to four times the number of parameters of the model, and a total of 100 000 steps per walker. We check the convergence of the chains by ensuring that the length of the chain is at least 30 times the autocorrelation time (Foreman-Mackey et al. 2013). We estimate the Bayesian evidence (ln Z i ) of each model using the perrakis 5 code (Díaz et al. 2016). This metric is then used to select the simplest model that best represents the data. For a complex model to be selected over a simpler one (i.e. the 1E over the FL or the 3E over the 1E), we require the Bayesian evidence of the more complex model to be larger than the evidence of the simpler model by 6 in logarithmic space (i.e. B = ∆ ln Z > 6, Trotta 2008), corresponding to strong evidence in favour of the more complex model. Otherwise, the simpler model is selected.', '3.2. Identification of the variability source': 'As mentioned in Sect. 2.2, the large plate scale of TESS (21 arcsec pixel -1 ) makes it particularly di ffi cult to undoubtedly assign the source of the variability to one specific target (e.g. Lillo-Box et al. 2024). Essentially, the high frequency resolution of TESS is at the expense of its low spatial resolution. Even in those stars in which the aperture mask has little contamination from nearby sources, the analysis is delicate, since other stars outside the aperture can contributed to the measured flux. Recently, Higgins & Bell (2023) developed a method to localise the origin of variability on the sky to better than one fifth of a pixel given a measured frequency (or frequencies). Basically, the method can resolve the variable source in frequency space for each pixel. The authors showed that even stars more than three pixels outside the aperture, can produce significant contamination in the extracted light curves. \nOnce the MCMC analysis presented in Sect. 3.1 is performed and the periodicities in the TESS light curves are measured, we can apply this methodology. To this end we use the implementation presented in the open-source Python package TESS\\_localize 6 . We apply this algorithm to our sample (both the already known binaries and those with newly detected variability) in order to identify the origin (i.e. location on the sky) of the periodicities found in this work. This analysis is presented in Sec. 5.2.', '4. Results in the TESS dataset': 'The results of the variability analysis described in Section 3 are summarised in Table 1. The column labelled as \'Best model\' shows the preferred solution of the modelling process based on the Occam\'s razor quantified by the comparison of the log of the Bayesian evidence between two models ( B i j , where "i" and "j" represent each of the two models). This is, we compare the evidence of the 1E (1-e ff ect) model against the flat (FL) model ( B 10), the evidence of the 3E model against the flat model ( B 30) and the evidence of the 3E model against the 1E model ( B 31). Out of 38 targets, a total of 26 show modulations compatible with one or more e ff ects. These central stars are described in Sections. 4.1 and 4.2, depending on whether they are already known binaries or new binary candidates, respectively. For the remaining 12 central stars, the flat model is the most plausible, that is, there are no statistically significant modulations in the light \nFig. 2: Periodogram ( upper panel ), TESS photometric time series for sectors 40, 41, and 54 ( middle panel ), and phase-folded combined light curve ( bottom panel ) of Abell 63. Two di ff erent bin sizes are shown with black bin size of 0.01 in phase) and grey (bin size of 0.05 in phase) circles. \n<!-- image --> \ncurve down to the sensitivity of the data. These central stars are briefly discussed in Sect. 4.3.', '4.1. Already reported binaries from the literature': 'Eight of the targets in the sample are either already known binary systems (confirmed through radial velocity observations) or candidates reported in the literature by using other techniques that still await confirmation. We do not include here our candidates from Cycle 1 of TESS presented in Aller et al. (2020), which will be discussed in Sect. 4.2 with the rest of the sample. The eight already known binaries are: Abell 63, Abell 46, AMU1, DS1, Abell30, NGC2392, NGC5189, and LoTr 5. The first seven are short-period binaries and we are able to recover the expected variability in the TESS data. In all the cases except one (Abell 63), the contamination in the aperture mask from nearby stars is less than 10%. The remaining system is the longperiod binary LoTr 5. In this case, we detect the variability associated with the rotation of one of the stars in the binary nucleus but obviously not the variability associated to the orbital period of the binary (we note that the central star of LoTr 5 has one of the longest orbital periods - P ∼ 2700 days - of all the known central stars of PN, Jones et al. 2017). In the following, we revisit and briefly discuss each system, and present our new analyses. The results are summarised in Table 2. \nAbell 63 and Abell 46. The eclipsing binary nuclei of PNe Abell 63 and Abell 46 have been extensively studied by sev- \nTable 1: Results from the modelling process for the 38 stars analysed with minimal or negligible contamination in the TESS aperture.Notes. ( * ) Although both StDr 56 and Fr 2-21 converge to a flat model, they have significant signals in their periodograms so we think they deserve a more detailed analysis (see Section 4.3). ** AMU 1 has a confirmed binary system in the nucleus, detected with Kepler data. \neral authors in the past years. Their nuclei, named UU Sge and V477Lyr, respectively, consist of a primary hot white dwarf or subdwarf and a low-mass star. Their orbital periods are 0.465 days for UU Sge (Bell et al. 1994) and 0.472 days for V477 Lyr (Pollacco & Bell 1994). \nBoth systems have been the subject of detailed modelling in the literature (e.g. Af¸sar & Ibanoˇglu 2008), in order to constrain the physical properties of the binary components (masses, temperatures and radii). In the case of Abell 46, for example, this has resulted in a potentially post-RGB scenario for the central star based on the low derived luminosity for the primary component. However, the derived temperature and surface gravity do not lie on post-RGB evolutionary tracks for the derived mass (instead being consistent with a much lower mass; Jones et al. 2022). Our photometric modelling does not take into account eclipses (see, for instance the TESS light curve from Abell 63 in Fig. 2) and, therefore, the analysis of these light curves can not be performed by using the methodology explained in Sect. 3. \nWe, therefore, decided to model the system using a version of the PHOEBE2 code (Prša et al. 2016; Horvat et al. 2018; Jones et al. 2020; Conroy et al. 2020), adapted to incorporate model atmospheres calculated by Reindl et al. (2016, 2023) using the Tübingen Model Atmosphere Package (TMAP; Rauch & Deetjen 2003; Werner et al. 2003, 2012), and previously used to model the binary central star of the PN Ou 5 (Jones et al. 2022). This new approach also has the benefit of using more appropriate TMAP atmospheres for the primary star, where previous models either assumed Kurucz (1993, more suitable for main sequence stars) or black body atmospheres - both of which can lead to important di ff erences in the derived temperature (potentially leading to the aforementioned mismatch with evolutionary tracks). \nPreliminary explorations of the parameter space, fixing the stellar masses to those from previous modelling e ff orts (as they are based primarily on the observed radial velocities), indicated that a solution consistent with evolutionary tracks should be possible. As such, we ran a MCMC sampling of the binary parame- \nFig. 3: Phase-folded light curves of the already known binaries, with the best-fitting model overlaid. Results from the PHOEBE2 analyses are overplotted in pink and from the MCMC analyses in blue (1E solution) and red (3E solution). Grey symbols correspond to a bin size of 0.01 in phase (i.e. 100 datapoints) while the black symbols correspond to bin sizes of 0.05 in phase (i.e. 20 datapoints). The period derived from the fitting can be found in Table 2 and the corresponding periodograms are shown in Fig. B.2 4 ). We note that the period of NGC 5189 does not correspond with the orbital period published in the literature for this binary. \n<!-- image --> \nTable 2: Results from the light curve analysis of those already known binary systems analysed with the parametric approach described in Sect. 3.1.Notes. ( 1 ) This periodicity in LoTr 5 corresponds to the rotation period. ( 2 ) Although the evidence for the flat model is larger in our analysis of AMU1, we can recover the 1-e ff ect model with similar evidence to the flat model by discarding the data with a high dispersion. The fit then converges to the known orbital period but still with insu ffi cient significance (see Section 4.1). \nters, allowing the primary temperature to vary but forcing its radius to lie on the appropriate evolutionary track of Miller Bertolami (2016), while leaving the temperature and radius of the companion free. The resulting best fit is shown in Fig. 3, and the best-fitting parameters are listed in Table 3. Ultimately, a more detailed fit is required (taking into account multi-band groundbased photometry as well as directly fitting the observed radial velocities - both of which are beyond the scope of this work) before claiming that the discrepancy between parameters from \nevolutionary tracks and binary modelling has been resolved or that the new parameters (and their associated uncertainties) are more reliable than those in the literature. Nevertheless, the fit presented here highlights that the high quality data from TESS combined with more advanced modelling tools has the potential to improve the derived parameters of post-CE binary central stars of PNe. \nTable 3: PHOEBE2 model parameters for V477 Lyr, the central star of Abell 46.Notes. ( † ) Derived from the evolutionary tracks of Miller Bertolami (2016). ( * ) Fixed in the model to match previous modelling e ff orts (Pollacco & Bell 1993; Af¸sar & Ibanoˇglu 2008). \nDS1. In the nucleus of this PN there is a well-known doublelined spectroscopic binary (named KV Vel or LSS 2018) with an orbital period of ∼ 0.357 days (Drilling 1985; Kilkenny et al. 1988). It is a post-common envelope system with an unusual and extremely strong reflection e ff ect, with an amplitude of 0.55 mag in V (Hilditch et al. 1996). The TESS light curve clearly shows the same variability with a prominent signal at the same periodicity as the published orbital period. Preliminary fitting of the light curve using PHOEBE2 (following the scheme outlined above for the central star of Abell 46) indicates that the data are entirely consistent with the parameters derived by Hilditch et al. (1996), albeit with a slightly lower albedo for the secondary (0.55 ± 0.01 c.f. 0.6). \nAbell 30. Jacoby et al. (2020) reported the presence of light curve brightness variations in the K2 mission (Howell et al. 2014) data of the central star of this born again PN. Although it has not been confirmed through radial velocity observations, the authors concluded that these variations in the light curve were highly suggestive of a binary central system with a period of ∼ 1.060 days. After analysing other possible physical processes, Jacoby et al. (2020) proposed the irradiation of a cooler companion as the most likely origin for the observed photometric variability in this system, and discarded other possible e ff ects as Doppler beaming and ellipsoidal modulations. We find the same variability in the three sectors of the TESS light curves. Figure B.2 4 shows the Lomb-Scargle periodogram, where the False Alarm Probabilities (FAPs) at 10%, 1% and 0.1% are also indicated with grey horizontal dotted lines. The phase-folded light curve with the period derived in the fitting is plotted in the corresponding panel of Fig. 3. The one sinusoidal model appears as the preferred solution, showing a large amplitude of 14 ppt (see Table 2). However, we note that the evidence of a 3-e ff ects model is almost the same as the simple model ( B 31 = -0 . 2), with an ellipsoidal amplitude of 0.95 ± 0 . 25 ppt. Indeed, these ellipsoidal modulations can be easily recognised in the residuals of the one sinusoidal model in Fig 3. However, although the time span covered by the three sectors of TESS (44, 45 and 46) is similar to that from the K2 mission, the precision of the TESS data is clearly worse than that from its predecessor and, therefore, the TESS light curves do not provide new information about the properties of the system. \nAMU1 \nFig. 4: Periodogram ( upper panel ), time-flux light curves for several sectors ( middle panel ), and phase-folded single, multisector light curve ( bottom panel ) of AMU1. Two di ff erent bin sizes are shown with black (bin size of 0.01 in phase) and grey (bin size of 0.05 in phase) symbols. The period derived from the fitting can be found in Table 2. \n<!-- image --> \nAMU1. The central star of this multipolar PN (Aller et al. 2013) was found to be a binary with a short period of ∼ 2.928 days by De Marco et al. (2015). The Kepler light curve showed a photometric sinusoidal variability with a very low amplitude (0.73 ppt), consistent with relativistic beaming e ff ects according to those authors. In the TESS light curves, we also detect a signal at the same periodicity, although the confidence level is much worse (approaching the False Alarm Probability at 10 % in some sectors) because of the lower photometric precision and shorter baseline of TESS compared to Kepler. Though the results from the TESS light curve analysis show that the evidence for the flat model is larger, we can recover the 1-e ff ect model with similar evidence discarding the sector 15, which has a large dispersion in the data. In this way, the MCMC chains converge to the known orbital period, although still with insu ffi cient significance ( B 10 = -0 . 6, see Figure 4). In summary, the TESS data do not provide any new information beyond what Kepler already showed. \nNGC2392. The central star of this well-known PN was discovered to be a single-lined spectroscopic binary by Miszalski et al. (2019) after a radial velocity monitoring campaign. According to the authors, the binary system appears to be a double degenerate system with an orbital period of ∼ 1.9 days. The periodogram of the TESS light curve shows a forest of significant periodicities above the 0.1% FAP (see Figure B.2 4 ). Among all these signals, we can identify two pairs of peaks more prominent \n<!-- image --> \nNGC 2392 \nFig. 5: Lomb-scargle periodogram (upper panel) and radial velocity analysis (bottom panel) of NGC 5189, based on the data presented in Manick et al. (2015). \n<!-- image --> \nat around ∼ 4.2 and ∼ 2.1 days, but none at the exact periodicity of 1.9 d found by Miszalski et al. (2019). We note, however, that the peak at 2.1d is also present in the data by Miszalski et al. (2019), although the authors chose to analyse the 1.9d because it was the highest in their RV periodogram. For that reason, we decided to re-analyse the radial velocity data presented in Miszalski et al. (2019). Our independent analysis using the same MCMC principles explained for the analysis of the TESS light curves but now applied to the radial velocity model, shows that the data converges to an orbital period of 2.10842273 ± 0.000085 days, which is actually significant in the RV periodogram (see upper panel of Fig. 5) and slightly longer than that derived by the authors with the same data. This result is in strong agreement with what we observe in the TESS light curve, and allows us to conclude that the 4.2 days periodicity seen in the light curve is, actually, an alias of the real period. This is a clear example of the importance of confirming the orbital period by more than one method. Taking this result into account, we forced the MCMC analysis to converge to the 2.1 days periodicity and obtained that the best solution is a 3E model. \nNGC5189. The central star of this complex, quadrupolar PN was already analysed in our previous work (Aller et al. 2020). Now, we re-analyse it taking into account new TESS data as well as the previous data from Cycle 1, drawing similar conclusions. After a radial velocity monitoring campaign, Manick et al. (2015) identified the periodicity found at 4.04 days as the orbital \nFig. 6: Periodogram of LoTr 5 after subtracting the rotation period with the wotan package (in blue and orange, with two different window sizes in the flatten module) and the original periodogram (in black). \n<!-- image --> \nperiod of the binary. The periodogram of the TESS light curve (sectors 11 and 38) show several prominent peaks above the 0.1 %FAP (the most significant at 0.125, 0.858 and 1.71 d, see Figure B.2 4 ) but none of them around the 4.04 days period previously reported. The result of MCMC analysis is, therefore, very similar to those presented in (Aller et al. 2020). A 3E model is the most plausible for the 1.71 d periodicity (the peak at 0.858 d is certainly an alias of this one) and the amplitudes are listed in Table 2. However, the resulted Doppler beaming amplitude is quite large and clearly not physically feasible for this type of object. This is a case with a lot of contamination from nearby stars in the photometric aperture so we cannot reach any firm conclusions from the TESS light curve. In fact, it is very likely that the origin of the periodicities we see in the TESS light curves does not correspond to our central star (see Section. 5.2 for more detail). \nLoTr5. This PN is known for having one of the longest period binary central star (P ∼ 2717 days; Van Winckel et al. 2014; Jones et al. 2017) so far. The system consists of a hot star and a barium star with a rotation period of 5.95 days. A possible third component in the system was discussed by Aller et al. (2018) on the basis of new radial velocity observations, although without firm conclusions. The rotation period is clearly recovered in the TESS dataset (see Fig. 3). In the MCMC analysis, the reflection and the beaming e ff ects are both degenerate, and the ellipsoidal variations go clearly to zero. Therefore, the 1E model is the only one possible, although there are clearly more e ff ects in the light curve (as, for example, oscillation signals). These other e ff ects certainly show up when subtracting the rotation signal from the light curve. We made this by using the detrending methods implemented in the wotan package (Hippke et al. 2019). Figure 6 shows the resulting periodogram of LoTr 5 after this detrending process. A more detailed analysis of these frequencies would be desirable to obtain precise information on the physical parameters of the pair, but this is out of the scope of this paper. \nTable 4: Results from the light curve analysis for those central stars with the 1-e ff ect model as the best solution. \n. \n. \n. \nNotes. ( * ) Targets already analysed in Cycle 1 (Aller et al. 2020). \nTable 5: Results from the light curve study for those central stars with negligible or minimal contamination analysed with models including reflection, ellipsoidal, and Doppler beaming. \nNotes. ( * ) Targets already analysed in Cycle 1 (Aller et al. 2020).', '4.2. New binary candidates from TESS': 'In this section, we describe the results from the analysis of those central stars with negligible or minimal contamination in the TPFs and showing variability in the light curves compatible with the presence of (previously unknown) binary systems. In addition, we also revisit those binary candidates already analysed in Cycle 1 (Aller et al. 2020), even if they are found in severely contaminated fields. In total, we analyse in this section 18 central stars. The results from the analysis are summarised in Tables 4 and 5, depending on whether the best fit in the MCMC analysis is a 1E or a 3E solution, respectively. \nWe find that the variability of 12 central stars is better explained based on the current data with a simple sinusoidal model (see Table 4). Three of them (NGC 7293, NGC 2867 and RWT152)are indeed binary candidates from Cycle 1 (Aller et al. 2020) that we have re-analysed by taking into account the new observations from Cycles 2, 3 and 4. For two of these targets (NGC2867 and NGC7293), the new results are compatible to those presented in Aller et al. (2020). In the case of NGC 2867, and after analysing sectors 9, 10, 35 and 37 together, we detect a significant periodic variability at ∼ 4.6 days with an amplitude of 0.3 ppt, similar to those from our previous paper and thus \nreinforcing the binary status of this central star. It is also important to highlight that in this case the 3E model has a larger evidence than the 1E model with a log-evidence of B 31 = + 3 . 4 and the same periodicity. As shown in the corresponding panel of Fig. A.1, two bumps are appreciated at around phases 0.25 and 0.75, which is the reason for the 3E model having such a large evidence. However, the current dataset is still insu ffi cient to prefer the more complex 3E model. It is also important to emphasise that the photometric aperture of NGC 2867 is quite crowded with other stars, so the origin of the variability is not entirely clear (see also Section 5.2). The results from the analysis of NGC 7293 (the well-known Helix Nebula) adding the new sectors 28 and 42, are also quite similar to those obtained in Aller et al. (2020). A clear photometric variability is identified with a periodicity of ∼ 2.8 days and amplitude of 1.5 ppt. Although, in this case, there is no contamination from nearby sources in the aperture, the origin of the variability may be also questionable (see Section 5.2). Only for the case of RWT 152, the new dataset from sector 34 leads to di ff erent conclusions than in Cycle 1. Now, apart from the forest of significant periodicities that we already saw in sector 7, a new clear signal at ∼ 6 days shows up in the periodogram (see corresponding panel in Fig. B.3), and \nthe MCMC converges to that period by using a single sinusoidal model. As explained in Aller et al. (2020), some of the found periodic variabilities can be caused by the presence of spots. In any case, dedicated observations are required to confirm or discard the presence of a companion star. \nAnother nine central stars in the sample present variability compatible with a 1E model. They are: PG 1520 + 525, PNG136.7 + 61.9, Pa 165, WPS 28, WPS 54, Fr 1-4, NGC 7094, NGC2371 and Hen3-1863. Fig.A.1 shows the phase-folded light curves with the periods derived in the fitting (see also Table 4). Objects with 1E model solution are plotted in blue. We briefly discuss here some key aspects of them as follows: \nPG 1520 + 525: This is the only "classical" round (with a ring appearance, Jacoby & van de Steene 1995) PN in the sample showing variability in the TESS light curve. We find a periodicity of ∼ 5.2 d with an amplitude of ∼ 4.13 ppt, compatible with irradiation on a close companion. Although there are no other stars identified by Gaia inside the TESS aperture within ∆ m = 6 magnitudes, the location of the variability is not clearly associated with the central star (see Sec. 5.2), so dedicated follow-up is desirable to confirm the origin of the detected variability. \nPNG 136 . 7 + 61 . 9: This extremely faint PN 7 with elliptical appearance, has a DAO (hydrogen and helium) white dwarf central star, suggested as a post-RGB candidate by Reindl et al. (2023). The authors did not find significant light curve variations in the TESS data. We find a 25 ppt variability amplitude in the only sector available in TESS that we could attribute to the irradiation of a cool companion. \nPa 165: Although the 1E solution is the most likely for this central star with an irregular PN, the 3E model has also large evidence ( B 31 = -3 . 6). The low amplitude of the variability (0.65 ppt) is compatible with either irradiation or relativistic e ff ects. \nWPS 28: The light curve of this white dwarf (with a possible PN around it according to the HASH PN database) exhibits a clear sinusoidal variability with a period of ∼ 2.8 days. The large temporal coverage of the TESS data, with a total of 25 sectors from Cycles 1 to 4, provides an exquisite precision. Interestingly, the phase-folded light curve with the period corresponding to this variability also shows a small dimming at the minimum of the relative flux (see corresponding panel in Fig. A.1). Although we note that this dimming is not yet statistically significant, this is the expected location (within the uncertainties), of an eclipse when the variability is due to the reflection e ff ect and the orbital inclination is such that the companion passes between the stellar disk and our line of sight. In order to further test the source of the variability and this tentative eclipse signal, we use a modified version of the TESS positional probability software ( tpp 8 ) described in Hadjigeorghiou & Armstrong (2023) and Lillo-Box et al. (2024) to compute the probability of each Gaia -detected source in the field around the target as being the true source of the transiting signal. By doing so, we find that the star with the largest probability of being the transit host is actually WPS 28, although still with a 32% probability, which makes highly interesting the follow-up of this target. However, their TPFs show severe contamination from other stars in all the sectors except one (sector 21, which has minimal contamination). Additional observations are thus critical to ensure the source of the variability (but see also Sect. 5.2). \n7 \nhttps://telescopius.com/spa/pictures/view/93299/ \ndeep\\_sky/by-boris\\_us5wu \n8 \nhttps://github.com/ahadjigeorghiou/ \nTESSPositionalProbability \nWPS 54: This star was classified as a RS CVn type star by Drake et al. (2014) based on their Catalina Sky Survey light curve. The RS CVn variable type consists of binary systems with high chromospheric activity FGK companions. After analysing the Catalina light curve, Werner et al. (2019) found a variability with a periodicity of 3.45 days, that the authors interpreted as the white dwarf\'s rotation period. The TESS light curve actually shows this periodicity in the periodogram (see Fig. 7). Although we acknowledge that the variations in the light curve may be due to multiple spots on the star, we still try our modelling process. The 1E model on this periodicity produces large residuals, suggestive of a wrong model (see Fig. 7, upper panel). The 3E model on this periodicity still produce large residuals with the presence of a phase-shifted signal at half of the periodicity (1.7d; see also Fig. 7, upper panel). We then decided to simultaneously model a one sinusoidal model for the 3.45d signal (assuming it as the rotation period of the white dwarf) and 1E or 3E model to the 1.7d periodicity. By doing so, we find that the residuals of both 1E and 3E model are largely reduced (see Fig. 7, bottom panel), with the 1E model being preferred against the 3E model. Still, this result may point to either irradiation with 1.7d orbital period or perfectly symmetric ellipsoidal e ff ect with a 3.45d period. Both scenarios are thus very interesting because any of them suggest a binary companion with a coupled orbital period commensurable with the rotation period of the star, suggestive of a coupling between the orbital and rotation frequencies. \nFr 1 -4: The central star of this possible (according to the HASH database) PN is the hot subdwarf JL 102. The single TESS light curve of this target (produced by combining sectors 27 and 39) shows a clear variability with ∼ 6.4 d period. The corresponding signal in the periodogram is very strong and our analysis reveals a 1E model with ∼ 5 ppt amplitude, which is compatible with irradiation on a stellar companion. \nNGC 7094: This object was classified as a PN by Kohoutek (1963). NGC 7094 has a filamentary structure, with a shell deviating sphericity (Rauch 1999), and a PG1159-type central star (Löbling et al. 2019). The variability in the TESS light curve is very clear, with a strong peak at ∼ 4.3 d in the periodogram, which may be perfectly compatible with irradiation of a companion star. The contamination from other field stars inside the TESS aperture is negligible and NGC 7094 is one of the most promising candidates to reside in a binary system (see Sect. 5.2). \nNGC 2371: This complex PN has been extensively studied in the literature (see, for example, Gómez-González et al. 2020; Ramos-Larios & Phillips 2012). It exhibits a high-excitation, multipolar morphology with a Wolf-Rayet-type as a central star. This central star is a well-known pulsator (Ciardullo & Bond 1996), whose TESS light curve was recently analysed by Córsico et al. (2021). The authors detected several pulsation periods for this star between 878.5 s and 1032.6 s (shorter than the pulsation frquencies previously reported by Ciardullo & Bond 1996, from the ground), but no evidence of binary signatures. On the contrary, we find several periodicities above the 0.1% FAP in the TESS periodogram, that populate a wide region at lower frequencies (most of them between 0.3 and 4 days, see Fig. B.3 4 ). Our analysis converges to the one at ∼ 2.3 days, showing a lowamplitude ( ∼ 0.8 ppt) sinusoidal variability. \nHen 3 -1863: The light curve of this central star is highly dominated by a strong variability, which appears to have its origin in the central star (see Sect. 5.2). The periodogram shows a strong a broad peak at ∼ 4.3 d (see Fig. B.3) and our MCMC analysis converges to that solution. The phase-folded light curve with that periodicity shows a structure at phase ϕ = 0 . 5 and large amplitude high-frequency variations that are not explained with \nour sinusoidal model (see corresponding panel in Fig. A.1), indicating that additional e ff ects are present in the data. The nature of these e ff ects are out of the scope of this paper. However, the large scale variability at 4.3 days is clear and well represented by our model. \nOn the other hand, we found 6 targets showing a 3E solution as the most likely one. The phase-folded light curves with the periods derived in the fitting of each of them are shown in Fig. A.1, with the 3E model solution overplotted in red. \nAmong these, two (NGC246 and PG1034 + 001) are binary candidates from Cycle 1 (Aller et al. 2020) that we reanalyse by adding the new data from Cycles 2-4. The results for PG1034 + 001 are practically the same as in Aller et al. (2020), with a strong periodicity at ∼ 1.86 days and evident ellipsoidal modulations. We invite the reader to revise our previous paper for more details. The case of NGC 246 is quite di ff erent. The central star of this elliptical PN is a known hierarchical triple stellar system (Adam & Mugrauer 2014). In Aller et al. (2020), we found a simple sinusoidal model with a period of ∼ 6.8 days as the best solution. With the new data (sector 30), the MCMC analysis converges to 3E solution at ∼ 5 days. However, we note that both signals may have the same origin, since the peak is quite broad in the periodogram (see Fig. B.3 4 ), which is dominated by the low frequencies (in the range of 500-700 µ Hz) associated with pulsations (Ciardullo & Bond 1996). Also, it is important to remember that this is one of the targets with substantial photometric contamination by other nearby stars in the TPF (see corresponding panel in Fig. B.1 4 ), which makes it extremely complicated to draw firm conclusions on the possible binary scenario. \nOther 4 objects in the sample present variability compatible with a 3E modulation. They are briefly discussed as follows: \nNGC 1501: The central star of this PN is a pulsating hydrogen-deficient, pre-white dwarf. Its TESS light curve was already analysed by Córsico et al. (2021) in order to search for pulsation frequencies and binary signatures. Although evidences for the latter scenario were not found, the authors presented asteroseismological analysis for the central star. We find several significant peaks in the periodogram, apart from the lowfrequency signal analysed by Córsico et al. (2021), which may be compatible with the presence of another star in the system. Our analysis converges to the one at ∼ 3.3 days by using a 3E solution, showing a low ( ∼ 0.5 ppt) beaming amplitude that is compatible with relativistic e ff ects in the system. \nK 1 -16: As NGC1501, this is also a pulsating hydrogendeficient, pre-white dwarfs whose TESS light curves were already analysed by Córsico et al. (2021). Binary signatures were not found and their asteroseismological analysis was quite limited in this case because of the dramatic changes in the pulsations of this star. The periodogram shows a forest of significant periodicities (between 1 and 11 days, approximately, see corresponding panel in Fig. B.3 4 ). The MCMC analysis found the second most prominent peak (corresponding to ∼ 3.9 days) as the most probable, with low amplitudes for the three e ff ects (irradiation, ellipsoidal modulations, and Doppler beaming), although the beaming amplitude still is unrealistically large. Therefore, the origin of this variability is hard to interpret due to the large amount of di ff erent signals. Radial velocity observations are crucial to confirm the nature of this variability. \nIC 2149: The central star of this elongated PN shows a high degree of variability in the TESS light curve, resulting in a large number of periodicities above the 0.1% FAP in the periodogram (see Fig. B.3 4 ). Given the large amplitude and high-frequency of these variations and the fact that we are interested in longer \nFig. 7: Modelling of the WPS 54 TESS light curve. Top panels: Phase-folded light curve (top-left) with the periodicity corresponding to the largest peak in the periodogram (top-right panel), including the 1E (blue) and 3E (red) models and their corresponding residuals. The model is not satisfactory due to the presence of additional signals. Bottom panels: Phase-folded light curve (bottom-left) with the 1.73d periodicity after removing the sinusiondal patter corresponding to the 3.45d (potentially the rotation period). The 1E (blue) and 3E (red) models are shown, as well as their corresponding residuals. \n<!-- image --> \nRelative Flux (ppt) \nO-C (ppt) \n40 \n20 \n0 \n20 \n40 \n10 \n0 \n10 \n0.0 \n0.5 \n1.0 \nPhase (P=1.73d) \nPower \n1 \n10 \nPower \n1 \n10 \nPeriod (days) \nperiodicities, we included in the modelling an extra jitter noise term to account for these short-period signals in a proper way. After doing so, the preferred solution for the MCMC fit is a 3E model with a period of 1.31 days. The derived Doppler beaming amplitude (Abeam = 1 . 01 ± 0 . 25 ppt) is indeed too large to be realistic and might still be driven partly by the high-frequency variability. However, it is important to note that its uncertainty is still large and might be compatible with 0 at 4 σ . \nIC 4593: As the case of the previous two PNe, the periodogram of this central star is dominated by several peaks above the 0.1% FAP. However, there are three of them (between 2 and 4 days) that predominate above the rest. The analysis converges to the one at 2.6 days, with a large reflection amplitude (Aref ∼ 12.8 ppt), a moderate Abeam of ∼ 2.7 ppt and no Doppler beaming. \nPeriod (days) \nTESS \nTESS - P(3.45d) \nTESS - P(3.45d) - 1E \nTESS - P(3.45d) -3E \nP=3.45 days \nP=1.73 days \n1E (P=1.73) \n3E (P=1.73) \n2.1-h bin \n0.41-h bin \nTESS \nTESS-1E(3.45d) \nTESS-3E(3.45d)', '4.3. Targets with no variability or with weak evidence': 'Wefind 12 central stars (leaving aside AMU 1, already discussed in Section 4.1) in which the best solution is the flat model (see Table 1). In general terms, we can discard sinusoidal variabilities down to the TESS sensitivity for these targets. Although in some few cases like AMU 1 and Fr 2-21 the MCMC finds compatible solutions with sinusoidal patterns, the Bayesian evidence still favours the simpler flat model. For all these non-detections, we can use the posteriors of the MCMC analysis to provide an upper limit to the amplitude of potential variability of a hypothetical companion. We use the marginalised posterior distribution of the 1E model ( A 1E) and use its 95% percentile as the upper limit. This is shown in Fig. ˜ B.5 4 , where we display the corner plots corresponding to the dependency between the period and A 1E as well as the marginalised distributions and the 95% upper limit. \nAt this point, there are some cases that deserve special attention. The case of Abell 7 is intriguing since, in Cycle 1 (sector 5), we identified two clear significant (and independent) peaks in the periodogram at ∼ 2 . 6 and ∼ 3 days (Aller et al. 2020) with unclear origins. Interestingly, these peaks no longer appear in the new sector 32 data (see Fig. B.4 4 ), so that the possible binary scenario remains questionable. As a possibility, the detected variability in sector 5 may be due to other non-binary scenarios as, for example, starspots. \nWe would also like to mention the case of StDr 56 9 . It is a faint and filamentary nebula, in which central region Drechsler and Strottner identified two possible white dwarfs, either of which could be the central star of the nebula. It is worthy to note that, although the flat model is the best solution in the MCMC fitting, the periodogram of the TESS light curve shows a very clear significant peak at ∼ 7 days (see Fig. 8, top panels). However, the uncertainty of the data is still large compared to the amplitude of the candidate signal and so the flat model is preferred. This is a promising candidate for subsequent follow-up. \nAnother good binary candidate is the central star of Fr 221. Although, as in the case of StDr 56, the best solution for this target is the flat model (see Table 1), the 1E and 3E models have comparably large evidences compared to the flat model ( B 10 = 4 . 5, B 30 = 5 . 0). The 1E solution is shown in Fig. 9, where we present the TESS time-flux light curve, the periodogram and the phase-folded light curve at the strongest peak ( ∼ 2.5 days). For this analysis, several photometric points present just before and after the TESS data downlink have been removed, since they are typically the result of instrumental e ff ects. It is also worth mentioning that a possible eclipse located at the expected minimum of the phase folded light curve is also hinted. However, the current data do not allow its confirmation. \nFinally, the central stars of PNe JnEr 1 and Abell 28 were found to have infrared excess compatible with a possible companion spectral type later than M4 by Douchin et al. (2015). The TESS light curves of these objects do not show any variability suggestive of a close binary scenario. Only in the case of Abell 28 do we find a significant signal at 0.11 days above the 0.5 % FAP in the periodogram (see Fig. B.4 4 ), but the MCMC analysis does not converge to that peak. Similarly, De Marco et al. (2013) detected infrared excess in Abell 31 suggesting an M4V companion, in agreement with the works of Frew (2008) and Ciardullo et al. (1999). The high-precision photometry of the four sectors (34, 44, 45, 46) of TESS show no evidence of \nFig. 8: Periodogram ( upper panel ), time-flux light curve ( middle panel ), and phase-folded light curve ( bottom panel ) of StDr 56. Two di ff erent bin sizes are shown with black (bin size of 0.01 in phase) and grey (bin size of 0.05 in phase) symbols. The period derived from the fitting can be found in Table 6. \n<!-- image --> \nTable 6: Results from the light curve study for the two central stars with weak evidence: StDr 56 and Fr 2-21. \nphotometric variability at least in the time span covered by the light curves (see Fig. B.4). This is in accordance with the wide companion proposed by Ciardullo et al. (1999). \nThe rest of the targets which have no significant photometric variability in their light curves are HDW 7, Lo 1, EC 132901933, Sh 2-216, IC 5148 / 50 and Fr 2-46. Their periodograms, which do not show any peak above the FAPs, are also presented in Fig. B.4. Only for the case of Sh 2-216, the most prominent peak (at ∼ 0.2 days) could be a significant signal with more data.', '5.1. Physical reliability of the Doppler beaming amplitudes': 'It is important to note that, as the modelling process does not put meaningful constraints on the physical parameters, the solution from the MCMC analysis has to be checked for unfeasible solutions. Particularly, we discuss the plausibility of the relativistic e ff ects (Doppler beaming) as responsible for the sinusoidal variability in both 1E and 3E solutions. This e ff ect produces lowamplitude sinusoidal variations (Zucker et al. 2007) in the light \nFig. 9: Periodogram ( upper panel ), time-flux light curve ( middle panel ), and phase-folded light curve ( bottom panel ) of Fr 2-21. Two di ff erent bin sizes are shown with black (bin size of 0.01 in phase) and grey (bin size of 0.05 in phase) symbols.The period derived from the fitting can be found in Table 6. \n<!-- image --> \ncurve, so in most of the cases we can directly reject this scenario by just checking for large amplitudes inferred from the MCMC analysis. \nIn the same fashion as in Aller et al. (2020), following the equations (2) and (3) from van Kerkwijk et al. (2010), we can estimate the radial velocity amplitude derived from the Doppler beaming amplitude obtained in the analysis (Abeam in the 3E solution and A1E in the simple 1E solution under the assumption that the sinusoidal variability is due only to the Doppler beaming e ff ect), and, therefore, estimate the binary mass function ( f ), defined as (assuming circular orbit): \nf = Porb 3 r 3 2 π G = M 3 2 sin 3 i ( M 1 + M 2) 2 , (6) \nwhere M1 and M2 are the masses of the two components, i is the inclination of the system, P orb the orbital period and 3 r the corresponding radial velocity semi-amplitude calculated from the measured Doppler beaming amplitude in the TESS light curve (Eq. 2 in Aller et al. 2020). By definition, the binary mass function will be always lower than M2. \nWe derive this mass function for all the targets with 1E or 3E solutions after the analysis of their light curves. To simplify, and as there is no available information on the e ff ective temperatures of all the central stars in the sample, we have used for the calculation two reasonable values for this parameter that approximately cover the whole possible range of it in this type of stars, that is T e f f = 40 000K, for the lower limit, and 150 000K for the upper one). \nFigure 10 shows the calculated mass functions for 0.5, 1, 5 and 10 M ⊙ in the Amplitude vs Porb plane. For visualisation pur- \nFig. 10: Amplitudes versus orbital periods measured in the TESS light curves. Doppler beaming amplitudes (from the 3E solution) are plotted in gold colour, while A1E (that could be due to irradiation either Doppler beaming), are in purple. Four di ff erent mass functions (0.5, 1, 5 and 10 M ⊙ ) for two di ff erent e ff ective temperatures (T e f f = 40 and 150kK) have been overplotted to see the position of the stars according to their corresponding e ff ective temperature. \n<!-- image --> \nses, cases with Abeam (in golden) or A1E (in purple) greater than 6 are not shown in the plot (IC 4593, with Abeam ∼ 0.005 ppt, is also not included). As expected, large Doppler beaming amplitudes imply unrealistic masses for the hypothetical secondary component (assuming that the primary has a typical mass of a white dwarf or similar). Only for very short orbital periods (a few hours) would the solution be reasonable but we do not have orbital periods in that region. Thus, only a handful of central stars have amplitudes (Abeam or A1E) compatible with relativistic e ff ects. Subsequent follow-up e ff orts must take these unphysical scenarios into consideration.', '5.2. Identification of the variability source': 'As explained in Sect. 2.2, we applied the TESS\\_localize method to our sample in order to identify the origin of the detected variability. The results are summarised in Table 7 where we list the name of the target in our sample, its TIC designation, whether or not the variability is attributed to the corresponding target (according to TESS\\_localize ) and the most likely Gaia source which would be responsible for the variability, together with its probability. \nIn some cases, the result from the TESS\\_localize is not reliable according to its standards (see Higgins & Bell 2023 for further information), and we classify such targets as inconclusive. In these cases, the fit to the location of the source of the variability is not significant based on the height parameter of the code. According to the documentation of the algorithm, a proper fit is achieved if the relative uncertainty in the height is smaller than 20%, which corresponds to 5 σ significance. In brief, only cases where the signal with the indicated frequency can be properly fit, can provide a good location for the variability. Otherwise, the solution is unreliable. We applied the algorithm to each \nindividual sector of each target 10 , by using the optimal number of PCA components that TESS\\_localize calculates. For simplicity, Table 7 shows a mean of the likelihood calculated only with those sectors in which the height is ≤ 20%. If none of the sectors has a height ≤ 20%, the solution is unreliable. \nThe already known binary central stars in our sample are the best targets to test the method. The results from TESS\\_localize for these stars reveal that, as expected, the source of the periodicity found in their extracted TESS light curves match the corresponding positions on the sky of each specific central star. However, for NGC 5189, this is not the case. According to the TESS\\_localize results, the two periodicities detected in the TESS light curve (none of them coinciding with the published orbital period, detected through radial velocities, see above) have origins di ff erent from the central star. The photometric variability in TESS associated with the RV frequency (measured by Manick et al. 2015) is however too shallow to apply the TESS\\_localize test. Also, in the case of AMU 1, the signal of the variability is too weak to provide a reliable result, so TESS\\_localize does not recover the position of the central star as the origin of such variability. \nThe case of NGC 2392 is also of interest, since the variability at period 2.1 d (the one detected in the TESS light curve) matches with a 100% probability the position of the central star. Conversely, when asking the algorithm about the origin of the 1.9 days period previously reported by Miszalski et al. (2019), the probability that the origin of such periodicity being the central star is negligible. This result reinforces the 2.1 days periodicity as the true orbital period of this binary central star. \nIn addition, we find that for seven of our new binary candidates, the origin of the variability appears to coincide with the position of the central star (see Table 7). Thus, these targets have a very high probability to reside in binary systems and radial velocity follow-up would be desirable to confirm their binary nature and derive orbital and physical parameters. For one central star (NGC 7293), the detected variability is significantly attributed to another star in the field ( Gaia DR3 6628875408232926336). Although it is a target with negligible contamination in the aperture mask, the probability that the measured variability corresponds to the mentioned field star is basically 100%, according to TESS\\_localize . This star is outside the aperture (more than 5 pixels far away from the central star) which highlights the importance of doing this kind of analysis. Thus, we can discard NGC 7293 as a potential binary candidate. \nWewould also like to mention the case of WPS 28. Out of 25 available sectors for this target, only three provide a reliable fit (with the relative uncertainty of the "height" parameter in these cases close to the 20%), so we think this result should be taken with caution. \nFor the remaining targets (8), the result of TESS\\_localize is unreliable because of the low significance of the detection. Thus, we cannot unequivocally confirm or discard the binary scenario for these targets and new dedicated observations are required.', '6. Final remarks and conclusions': "As a continuation of the work presented in Aller et al. (2020), we analyse here the 2-minute cadence TESS light curves available in the MAST archive from the first four years of data from this mission (Cycles 1 to 4) of a sample of planetary nebulae nuclei in \nour Galaxy. In total, we analyse 62 central stars of (true, possible or likely) PNe in order to identify new binary candidates. \nFigure 11 shows a summary of our results. Among the 62 central stars, 24 were set aside since they were in crowded fields, with large photometric contamination from nearby stars and, therefore, the origin of a hypothetically detected variability in such cases would be di ffi cult to assess. The large size of the TESS pixel scale (21 arcsec pixel -1 ) makes this first exploration essential. We then performed MCMC analyses to the remaining central star light curves (i.e. those with negligible or minimal contamination) in order to study the binary scenario, accounting for di ff erent physical e ff ects: irradiation, ellipsoidal modulations and Doppler beaming. \nAmong the modelled sample, eight central stars are already known binary systems; another 18 present modulations in the light curves that may be compatible with binary signatures; and 12 central stars show weak or no significant variability. \nFor the already known binary systems, we analysed the new high-precision data from TESS, also using the radial velocity observations available in the literature for a handful of them. This analysis allows us to provide new information in some of the systems, for example, refining the orbital parameters in the case of NGC2392. \nAdditionally, we propose as binary candidates the 18 central stars showing sinusoidal variability in their light curves (four of them already analysed in Cycle 1). For all of them, we derived short orbital periods ranging from ∼ 1.3 to 6.5 days. With the goal of confirming the source of the variability, we used the code TESS\\_localize . This algorithm has been designed to identify, with an estimated likelihood, the location of the origin of a given frequency in the TESS target pixel file. Thus, after applying this method, we find that 7 central stars seem to be the origin of the detected periodicities, being our best candidates, while in three central stars the cause of the variability is likely another star in the field. For the remaining targets (8), TESS\\_localize provides unreliable fits so the results are inconclusive. \nIt is worth highlighting that the periods of the detected candidates do not match with the current period distribution of the known close binary central stars (see, for example, Jacoby et al. 2021). The candidates detected in our work have larger periods than the median of the distribution of published binaries. However, it is also true that most of them have been discovered through ground-based observations, that are biased towards short periods and large amplitudes. With the recent space surveys like Kepler, K2 and TESS, we are populating a region with lower amplitudes and longer periods, changing the period distribution we know so far. In any case, it is remarkable that in our sample, we do not find binary candidates with periods below ∼ 1 day. A priori, there is not any selection e ff ect in the sample which may a ff ect this finding, and it is true that other e ff ects like stellar spots may be the answer to some of the variability encountered. To resolve this, more detailed analysis has to be done in each case to confirm the potential binarity and to constrain the parameters of the systems. \nWe note that among the binary systems (both already known and new candidates) only two PNe are classified as round in the HASH PN database, which represents less than 10%. They are Abell 30 and PG 1520 + 525. The first one, certainly has a spherical shell but with other structures like dots and cometary tails in the inner regions. On the other hand, PG 1520 + 525 is a more classical and perfectly round PNe. The rest, are PNe with morphologies that deviate from sphericity, such as bipolar, elliptical, and / or with other complex structures, that can be in many cases \nTable 7: Results after applying the TESS\\_localize method for those variable stars analysed with minimal or negligible contamination in the aperture mask. \nperfectly explained by the presence of a binary system in their nuclei. \nOn the contrary, if we attend to the morphology of the sample of the 12 central stars that do not show variability (i.e, they converge to a flat model in our MCMC analysis), the results are quite di ff erent. About ∼ 33% have round PNe and another ∼ 33% have non-defined morphological type according the HASH PN database. The remaining ∼ 33% have non-spherical shapes, such as elliptical or bipolar. The non-detected variability in such cases is by no means a confirmation that they are indeed single stars but instead, we provide a detection limit for a possible companion. \nDefinitively, TESS is providing a huge amount of highprecision photometric data that are enormously contributing to the study of a broad range of variable stars, including binary central stars of planetary nebula. The discovery and characterisation of more of these systems is crucial to further our limited understanding on the formation of the beautiful, varied and complex sample of PNe. The PLATO mission (Rauer et al. 2014) in the near future will also help in this way by providing long-term high-precision photometry for a large region of the sky. \nAcknowledgements. We thank Nicole Reindl and Mikkel Nørup Lund for useful discussions on the pulsations of LoTr 5, and Nicole also for her help in the selection of the targets and providing feedback on the manuscript. AA acknowledges support from Government of Comunidad Autónoma de Madrid (Spain) through postdoctoral grant 'Atracción de Talento Investigador' 2018T2 / TIC-11697. J.L.-B. was partly funded by grants LCF / BQ / PI20 / 11760023, \nRamón y Cajal fellowship with code RYC2021-031640-I, by the Spanish MICIU / AEI / 10.13039 / 501100011033 grant PID2019-107061GB-C61, and by the MICIU / AEI / 10.13039 / 501100011033 and NextGenerationEU / PRTR grant CNS2023-144309. DJ acknowledges support from the Agencia Estatal de Investigación del Ministerio de Ciencia, Innovación y Universidades (MCIU / AEI) and the European Regional Development Fund (ERDF) with reference PID-2022-136653NA-I00 (DOI:10.13039 / 501100011033). DJ also acknowledges support from the Agencia Estatal de Investigación del Ministerio de Ciencia, Innovación y Universidades (MCIU / AEI) and the European Union NextGenerationEU / PRTR with reference CNS2023143910 (DOI:10.13039 / 501100011033). This research is part of the I + D + i project PID2019-105203GB-C21 founded by Spanish AEI (MICIU) grant 10.13039 / 501100011033. This research has made use of the SIMBAD database, operated at the CDS, Strasbourg (France), Aladin, NASA's Astrophysics Data System Bibliographic Services, and the Spanish Virtual Observatory (http: // svo.cab.inta-csic.es) supported by the Spanish Ministry of Science and Innovation / State Agency of Research MCIN / AEI / 10.13039 / 501100011033 through grant PID2020-112949GB-I00. This paper includes data collected with the TESS mission, obtained from the MAST data archive at the Space Telescope Science Institute (STScI). Funding for the TESS mission is provided by the NASA Explorer Program. STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. 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Two di ff erent bin sizes are shown with black (bin size of 0.01 in phase) and grey (bin size of 0.05 in phase) symbols. The period derived from the fitting in each case can be found in Tables 4 and 5. \n<!-- image --> \nFig. A.1: continued. \n<!-- image -->', 'Appendix B: Figures': 'Fig. B.1: Target pixel files (TPFs) of each central star in the sample (marked with white crosses) obtained with tpfplotter . The red circles are the sources of the Gaia DR3 catalogue in the field with scaled magnitudes (see legend). The aperture mask used by the pipeline to extract the photometry is also marked. Pixel scale is 21 arcsec pixel -1 . \n<!-- image --> \n1850 \n1848 \n1846 \n1844 \n1842 \n1840 \n1050 \n1016 \n1014 \n1012 \n1010 \n1008 \n1006 \n508 \n506 \n504 \n47 \n502 \n500 \n498 \nPixel Column Number \nAbell 46 - Sector 26 \n104 \n116 \n102 \nPixel Row Number \nPixel Row Number \nPixel Row Number \n122 \n682 \n24 \n28 \n114 \n112 \n1048 \n29 \n28 \n21 \n31 \n36 \n22 \n19 \n18 \n21 \n23 \n33 \n32 \n12 \n14 \n2 \n8 \n13 \n11 \n24 \n676 \n15 \n678 \n33 \n27 \n680 \nPixel Column Number \n43 \n8 \n9 \n17 \n1432 \n37 \n45 \nFr 1-6 - Sector 16 \n16 \n11 \n5 \n34 \n29 \n15 \n10 \n3 \n1 \n2 \n4 \n13 \n31 \n44 \n1430 \n20 \n1428 \nPixel Column Number \n40 \n25 \n16 \n8 \n33 \n46 \n19 \n13 \n15 \n20 \n24 \n4344 \n976 \n5 \n6 \n9 \n978 \n26 \n31 \n48 \n38 \n980 \nPixel Column Number \n1 \n2 \n21 \n43 \n55 \n1792 \n42 \n4 \n5 \n18 \n34 \n47 \n1794 \n23 \n17 \n24 \n25 \n40 \n33 \n16 \n15 \n32 \n45 \n1796 \n58 \nPixel Column Number \n19 \n36 \n48 \n35 \n30 \n1434 \n117 \n94 \n88 \n89 \n97 \n64 \n57 \n63 \n58 \n71 \n62 \n43 \n37 \n33 \n23 \n40 \n34 \n52 \n44 \n47 \n68 \n50 \n78 \n81 \n95 \n1046 \n73 \n75 \n77 \n55 \n30 \n32 \n24 \n9 \n5 \n15 \n29 \n42 \n118 \n14 \n16 \n19 \n13 \n26 \n35 \n36 \n4849 \n54 \n1042 \n99 \n100 \n65 \n93 \n80 \n61 \n45 \n46 \n69 \n27 \n3 \n20 \nm \n= -2 \nm \n= 0 \nm \n= 2 \nm \n= 4 \nm \n= 6 \n86 \n1044 \n115 \n105 \nPixel Column Number \n113 \nBode 1 - Sector 18 \n26 \n23 \n16 \n7 \n19 \n3 \n7 \n60 \n28 \n25 \n12 \n10 \n17 \n83 \n90 \n6 \n31 \n41 \n22 \n11 \n7 \n6 \n2 \n1 \n8 \n18 \n21 \n1 \n4 \n9 \n70 \n72 \n56 \n4 \n5 \n6 \n17 \n51 \nN \n109 \n32 \n22 \n10 \n12 \nN \n674 \n38 \n26 \n40 \nE \n101 \n82 \n53 \n66 \n98 \n91 \n79 \n59 \n76 \n74 \n85 \nE \n87 \n39 \n38 \n67 \n96 \n92 \n1040 \n25 \nm \n= -2 \nm \n= 0 \nm \n= 2 \nm \n= 4 \nm \n= 6 \n14 \nE \n18 \n25 \n27 \n42 \n1424 \nm \n= -2 \nm \n= 0 \n41 \nm \n= 1 \nm \n= 3 \nm \n= 5 \n39 \nN \n46 \n1426 \n29 \n14 \n17 \n11 \n7 \n6 \n8 \nAbell 31 - Sector 34 \n24 \n4 \n27 \n21 \n19 \n1050 \n9 \n28 \n1052 \n3 \n10 \n1054 \nPixel Column Number \n350 \n309 \n324 \n276 \n265 \n23 \n13 \n22 \n1056 \n305 \n342 \n233 \n228 \n226 \n184 \n420 \n418 \n416 \n414 \n412 \n410 \n1058 \n190 \n340 \n341 \n344 \n188 \n347 \n349 \n286 \n290 \n327 \n261 \n288 \n316 \n353 \n186 \n263 \n299 \n312 \n351 \n243 \n274 \n283 \n302 \n184 \n329 \n315 \n267 \n200 \n211 \n224 \n179 \n208 \n157 \n128 \n131 \n138 \n152 \n188 \n137 \n155156 \n246 \n180 \n171 \n166 \n107 \n112 \n139 \n160 \n136 \n95 \n99 \n94 \nm \n= -2 \nm \n= 0 \nm \n= 2 \n127 \n110 \nm \n= 4 \n115 \n163 \n103 \n159 \n193 \n225 \n210 \n254 \n237 \n256 \n273 \n359 \n360 \n182 \nPixel Row Number \nPixel Row Number \n180 \n1534 \n1532 \n1530 \n1528 \n1526 \n1524 \n152 \nPixel Row Number \n150 \n148 \n146 \n144 \n142 \n984 \n318 \n323 \n189 \n192 \n198 \n69 \n106 \n147 \n183 \n206 \n217 \n146 \n173 \n176 \n182 \n219 \n223 \n285 \n247 \n328 \n334 \n326 \n275 \n83 \n333 \n356 \n298 \n293 \n252 \n317 \n348 \n292 \n269 \n235 \n239 \n345 \nAbell 63 - Sector 14 \n207 \n144 \n199 \n232 \n251 \n201 \n220 \n148 \n98 \n78 \n63 \n68 \n76 \n52 \n60 \n37 \n18 \n13 \n56 \n12 \n10 \n15 \n121 \n216 \n222 \n191 \n145 \n96 \n67 \n48 \n30 \n17 \n27 \n25 \n56 \n45 \n31 \n16 \n14 \n11 \n19 \n26 \n38 \n40 \n102 \n28 \n100 \n58 \n313 \n325 \n300 \n264 \n234 \n227 \n194 \n215 \n162 \n213 \n187 \n352 \n296 \n321 \n240 \n158 \n151 \n125 \n88 \n89 \n75 \n62 \n73 \n57 \n44 \n47 \n51 \n64 \n74 \n77 \n118 \n149 \n108 \n114 \n197 \n202 \n177 \n123 \n92 \n81 \n105 \n54 \n2021 \n23 \n22 \n41 \n91 \n90 \n84 \n93 \n101 \n111 \n130 \n140 \n122 \n133 \n142 \n134 \n135 \n168 \n172 \n196 \n218 \n141 \n245 \n287 \n4 \n12 \n29 \n32 \n42 \n43 \n46 \n49 \n61 \n33 \n50 \n65 \n80 \n164 \n214 \n282 \n289 \n304 \n339 \n1190 \n1184 \n153 \n167 \n262 \n25 \n15 \n16 \nN \n354 \n301 \n270 \n230 \n165 \n175 \n178 \n212 \n307 \n279 \n242 \n204 \n205 \n150 \n170 \n124 \n87 \n71 \n72 \n85 \n82 \n119 \n116 \n113 \n120 \n143 \n132 \n209 \n238 \n295 \n294 \n338 \n308 \n361 \n1186 \n297 \n1188 \nPixel Column Number \nHen 3-1863 - Sector 27 \n5 \n7 \n6 \n4 \nE \n104 \n126 \n185 \n169 \n154 \n181 \n221 \n174 \n1048 \n355 \n330 \n314 \n280 \n303 \n258 \n272 \n253 \n244 \n231 \n236 \n195 \n337 \n260 \n203 \n281 \n358 \n335 \n322 \n266 \n268 \n241 \n311 \n186 \n259 \n229 \n248 \n250 \n249 \n257 \n310 \n284 \n336 \n320 \nN \n343 \n357 \n331 \n86 \n109 \n129 \n89 \n161 \n190 \n291 \n255 \n277278 \n332 \n306 \n346 \n24 \n34 \n35 \n39 \n36 \n3 \n55 \n362 \n9 \n53 \n79 \n70 \n59 \n66 \n271 \n117 \n84 \n20 \n672 \n103 \n106 \n108 \n111 \n107 \n121 \n119 \n120 \n34 \n30 \n110 \n2.00 \n1.50 \n1.75 \n1.25 \n1.00 \n0.75 \n0.50 \n0.25 \n0.00 \n1.4 \n1.2 \n1.3 \n1.1 \n1.0 \n0.9 \n0.8 \n0.7 \n1.2 \n1.0 \n0.8 \n0.6 \n0.4 \n) \ne \n( \n2 \n0 \n1 \n× \nx \nu \nl \nF \n) \ne \n( \n3 \n0 \n1 \n× \nx \nu \nl \nF \n) \ne \n( \n2 \n0 \n1 \n× \nx \nu \nl \nF \nPixel Row Number \n30 \n31 \n319 \n1192 \n298 \n51 \n53 \nm \n= -2 \nm \n= 0 \nm \n= 2 \nm \n= 4 \nm \n= 6 \n296 \n292 \n294 \nPixel Column Number \nFr 2-23 - Sector 18 \n56 \n45 \n29 \n17 \n14 \n10 \n12 \n23 \n39 \n34 \n27 \n11 \n12 \n3 \n7 \n47 \n36 \n4950 \n5 \n12 \n18 \n97 \n28 \n18 \n35 \n37 \n32 \n41 \n982 \n2 \n7 \n22 \n1 \n21 \n2 \n1 \n4 \nE \n8 \n3 \nN \n290 \n58 \nN \nm \n= -2 \nm \n= 0 \nm \n= 2 \nm \n= 4 \nm \n= 6 \n26 \n20 \nE \n55 \nm \n= -2 \nm \n= 0 \nm \n= 2 \n30 \nm \n= 4 \nm \n= 6 \n42 \n974 \nE \n57 \n1182 \n288 \n54 \n52 \n5 \n4 \n3 \n2 \n1 \n8 \n6 \n7 \n5 \n4 \n3 \n2 \n1 \n) \ne \n( \n3 \n0 \n1 \n× \nx \nu \nl \nF \n) \ne \n( \n3 \n0 \n1 \n× \nx \nu \nl \nF \n1.75 \n1.25 \n1.50 \n1.00 \n0.75 \n0.50 \n0.25 \n0.00 \n1.1 \n1.0 \n0.9 \n0.8 \n0.7 \n0.6 \n) \ne \n( \n3 \n0 \n1 \n× \nx \nu \nl \nF \n) \ne \n( \n3 \n0 \n1 \n× \nx \nu \nl \nF \nPixel Row Number \nPixel Row Number \n1760 \n1758 \n1756 \n1754 \n1752 \n1750 \n894 \nPixel Row Number \n892 \n99 \n890 \n88 \n101 \n888 \n83 \n104 \n244 \nPixel Row Number \n886 \n884 \n1894 \n1892 \n1890 \n1888 \n1886 \n1884 \n1098 \n680 \n678 \n676 \n674 \n672 \n670 \n19 \n81 \n78 \n62 \n65 \n18 \n906 \n87 \n79 \n57 \n14 \n10 \n16 \n67 \n71 \n36 \n33 \n37 \n28 \n40 \n45 \n46 \n35 \n42 \n58 \n66 \nm \n= -2 \nm \n= 0 \nm \n= 2 \nm \n= 4 \nm \n= 6 \n31 \n27 \n23 \n44 \n1096 \n1090 \n8 \n13 \n9 \n2 \nm \n= -2 \nm \n= 0 \nm \n= 2 \nm \n= 4 \nm \n= 6 \n1092 \n20 \n14 \n15 \n21 \n30 \n1094 \nPixel Column Number \nFr 2-25 - Sector 34 \n66 \n61 \n56 \n27 \n20 \n22 \n7 \nm \n= -2 \nm \n= 0 \nm \n= 2 \nm \n= 4 \nm \n= 6 \n50 \n57 \n1798 \n37 \n35 \n13 \n17 \n904 \nAbell 33 - Sector 35 \n15 \n900 \n6 \n8 \n902 \nPixel Column Number \n102 \n92 \n47 \n16 \n98 \nAMU 1 - Sector 15 \n75 \n63 \n69 \n29 \n25 \n20 \n9 \n7 \n8 \n11 \n22 \n43 \n64 \n50 \n24 \n15 \n10 \n18 \n17 \n12 \n13 \n30 \n48 \n68 \n80 \n103 \n238 \n105106 \n108 \n93 \n89 \n72 \n7677 \n56 \n53 \n61 \n60 \n236 \n51 \n90 \n73 \n21 \n23 \n6 \n1 \n19 \n32 \n2 \n27 \n39 \n54 \n3 \n26 \n34 \n55 \n85 \n82 \n240 \n242 \nPixel Column Number \nFr 1-4 - Sector 27 \n35 \n28 \n26 \n4 \n5 \n14 \n49 \n11 \n5 \n10 \n30 \n4 \n20 \n11 \n54 \n28 \n29 \n11 \n44 \n12 \n7 \n2 \n1 \n3 \n6 \n7 \n1 \n9 \n3 \n94 \n97 \n65 \n59 \n41 \n38 \n44 \n52 \nN \n70 \n84 \nE \n31 \n19 \n8 \n17 \n10 \n56 \n4 \n3 \n52 \n26 \n51 \n39 \n13 \n14 \n12 \nE \n9 \nN \n898 \n24 \n16 \n12 \n18 \n29 \nm \n= -2 \nm \n= 0 \nm \n= 2 \nm \n= 4 \nm \n= 6 \nE \n896 \n100 \n110 \n74 \n86 \n91 \n95 \n96 \n107 \n25 \nE \nN \n1088 \n67 \n38 \n41 \n49 \nN \n69 \n64 \n63 \n62 \n68 \n1788 \n59 \n31 \n46 \n53 \n60 \n1790 \n109 \n234 \n2.5 \n2.0 \n1.5 \n1.0 \n0.5 \n3.0 \n2.5 \n2.0 \n1.5 \n1.0 \n0.5 \n0.0 \n7 \n5 \n6 \n4 \n3 \n2 \n1 \n0 \n) \ne \n( \n3 \n0 \n1 \n× \nx \nu \nl \nF \n) \ne \n( \n2 \n0 \n1 \n× \nx \nu \nl \nF \n) \ne \n( \n1 \n0 \n1 \n× \nx \nu \nl \nF \n1.75 \n1.50 \n1.25 \n1.00 \n0.75 \n0.50 \n0.25 \n) \ne \n( \n2 \n0 \n1 \n× \nx \nu \nl \nF \n958 \n) \ne \n( \n0 \n1 \n× \nx \nu \nl \nF \n) \ne \n( \n3 \n0 \n1 \n× \nx \nu \nl \nF \n) \ne \n( \n0 \n1 \n× \nx \nu \nl \nF \n) \ne \n( \n3 \n0 \n1 \n× \nx \nu \nl \nF \n) \ne \n( \n2 \n0 \n1 \n× \nx \nu \nl \nF \n<!-- image --> \n978 \n971 \n894 \n846 \n890 \n893 \n900 \n972 \n974 \n923 \n909 \n829 \n784 \n925 \n793 \n827 \n831 \n875 \n895 \n920 \n946 \n<!-- image --> \n955 \nKn 9 - Sector 14 \nPixel Column Number \n80 \n87 \nNGC 1501 - Sector 19 \n65 \n74 \n<!-- image --> \n315 \n314 \n249 \n71 \n285 \n313 \n<!-- image --> \n301 \n223 \n244 \n310 \n312 \nNGC 6905 - Sector 14 \nPixel Column Number \nHDW 7 - Sector 46 \n39 \n36 \n<!-- image --> \n<!-- image --> \nPixel Column Number \nLo 1 - Sector 30 \n<!-- image --> \n30 \n33 \n<!-- image --> \nNGC 2371 - Sector 20 \n219 \n200 \n26 \n27 \n184 \n222 \n197 \n198 \n168 \n165 \n178 \n<!-- image --> \n236 \nPa 39 - Sector 16 \nPixel Column Number \nNGC 246 - Sector 30 \nFig. B.1: continued. \n<!-- image --> \n<!-- image --> \nPixel Column Number \nLoTr 5 - Sector 23 \n<!-- image --> \nNGC 6720 - Sector 14 \n<!-- image --> \nPG 1520+525 - Sector 16 \nPixel Column Number \n<!-- image --> \n16 \n<!-- image --> \nAbell 7 - Sector 32 \n2.25 \n1.75 \n2.00 \n1.50 \n1.25 \n1.00 \n0.75 \n0.50 \n) \ne \n( \n0 \n1 \n× \nx \nu \nl \nF \n) \ne \n( \n0 \n1 \n× \nx \nu \nl \nF \n) \ne \n( \n0 \n1 \n× \nx \nu \nl \nF \n) \ne \n( \n0 \n1 \n× \nx \nu \nl \nF \n67 \n<!-- image --> \n<!-- image --> \nPixel Column Number \nPuWe 1 - Sector 20 \nPixel Column Number \n<!-- image --> \nWPS 28 - Sector 14 \n60 \nPixel Column Number \n<!-- image --> \n68 \nAbell 21 - Sector 46 \n49 \nDS 1 - Sector 36 \n<!-- image --> \nPixel Column Number \nSh 2-216 - Sector 19 \nPixel Column Number \n<!-- image --> \n7 \nWPS 54 - Sector 21 \nPixel Column Number \n<!-- image --> \nAbell 30 - Sector 44 \nPixel Column Number \n<!-- image --> \nEC 13290-1933 - Sector 37 \nFig. B.1: continued. \n<!-- image --> \n<!-- image --> \nPixel Column Number \n110 \nV*V2027 Cyg - Sector 15 \n<!-- image --> \n98 \nStDr 56 - Sector 18 \n151 \n147 \n<!-- image --> \n159 \nAbell 61 - Sector 41 \n112 \n135 \n158 \n<!-- image --> \nPixel Column Number \nRWT 152 - Sector 34 \n<!-- image --> \n) \ne \n( \n2 \n0 \n1 \n× \nx \nu \nl \nF \n) \ne \n( \n2 \n0 \n1 \n× \nx \nu \nl \nF \n) \ne \n( \n0 \n1 \n× \nx \nu \nl \nF \n<!-- image --> \nPixel Row Number \n) \ne \n( \n0 \n1 \n× \nx \nu \nl \nF \n) \ne \n( \n2 \n0 \n1 \n× \nx \nu \nl \nF \n) \ne \n( \n0 \n1 \n× \nx \nu \nl \nF \nPixel Row Number \n) \ne \n( \n0 \n1 \n× \nx \nu \nl \nF \nPixel Column Number \n<!-- image --> \nJnEr 1 - Sector 47 \nPixel Column Number \n<!-- image --> \nNGC 2392 - Sector 45 \nPixel Column Number \n<!-- image --> \nTK 1 - Sector 47 \n<!-- image --> \nFPJ1912-0331 - Sector 54 \n<!-- image --> \nNGC 5189 - Sector 38 \n1044 \n<!-- image --> \nPixel Column Number \n979 \n1101 \n1060 \n1072 \n1077 \n1093 \n1097 \n1109 \n1110 \n1009 \n1134 \n1026 \n1138 \n1141 \n1142 \n1145 \n1151 \n1158 \nK 1-6 - Sector 47 \nPixel Column Number \n<!-- image --> \n511 \n456 \n472 \n486 \n495 \n505 \n454 \n392 \n442 \n451 \n481 \n401 \n412 \n414 \n418 \n433 \n439 \n449 \n452 \n492 \n471 \n<!-- image --> \n417 \n440 \n457 \nNGC 6853 - Sector 41 \nPixel Column Number \nAbell 72 - Sector 55 \nPixel Column Number \n<!-- image --> \nFr 2-21 - Sector 55 \nFig. B.1: continued. \n<!-- image --> \n1140 \n1035 \n1039 \n1043 \n1095 \n<!-- image --> \n6 \nNGC 7293 - Sector 28 \nPixel Column Number \nLo 4 - Sector 36 \n<!-- image --> \nPNG 136.7+61.9 - Sector 48 \nPixel Column Number \n<!-- image --> \n16 \nAbell 74 - Sector 55 \nPixel Column Number \n<!-- image --> \nIC 4593 - Sector 51 \n<!-- image --> \n) \ne \n( \n2 \n0 \n1 \n× \nx \nu \nl \nF \n) \ne \n( \n4 \n0 \n1 \n× \nx \nu \nl \nF \n) \ne \n( \n3 \n0 \n1 \n× \nx \nu \nl \nF \nPixel Row Number \nPixel Row Number \n485 \n) \ne \n( \n0 \n1 \n× \nx \nu \nl \nF \n) \ne \n( \n0 \n1 \n× \nx \nu \nl \nF \n) \ne \n( \n0 \n1 \n× \nx \nu \nl \nF \n) \ne \n( \n0 \n1 \n× \nx \nu \nl \nF \nPixel Column Number \n<!-- image --> \nK 1-16 - Sector 14 \n<!-- image --> \n<!-- image --> \nPixel Column Number \n598 \n539 \n526527 \n581 \n505 \n589 \n518 \nNGC 2867 - Sector 9 \nFig. B.1: continued. \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFig. B.2: Periodograms of the known binary central stars Abell 46, DS 1, Abell 30, LoTr 5, NGC 5189, and NGC 2392. Grey horizontal dotted lines represent the False Alarm Probabilities (FAPs) at 10%, 1% and 0.1%. \n<!-- image --> \n<!-- image --> \n557 \nFig. B.3: Periodograms of those central stars which show significant variability in their light curves. Grey horizontal dotted lines represent the False Alarm Probabilities (FAPs) at 10%, 1% and 0.1%. \n<!-- image -->', 'A&A-aaller\\_TESS\\_binaries\\_cycles1-4, Online Material p 29': 'Fig. B.4: Periodograms of those central stars with no evidence in the MCMC analysis: Abell 7, Abell 28, JnEr 1, Abell 31, Sh 2-16, Fr 2-46, HDW 7, IC 5148 / 50, Lo 1 and EC 12390-1933. Grey horizontal dotted lines represent the False Alarm Probabilities (FAPs) at 10%, 1% and 0.1% . \n<!-- image --> \nFig. B.5: Corner plots corresponding to the posterior distributions of the light curve modeling for targets with no or weak evidence in the 1-e ff ect model. In each panel shows the marginalised posterior distributions of the orbital period and the amplitude of the model ( A 1E) in the diagonal, while the 2D contour plot shows the dependency between both parameters. In the case of the amplitude ( A 1E), we also include a vertical line showing the 95% percentile, that we assume as the upper limit for any periodicity testable with the TESS data. \n<!-- image --> \n80 \nFig. B.5: continued. \n<!-- image --> \nFig. B.5: continued. \n<!-- image -->'}

2022PhRvD.106h2002W

The kiloHertz gravitational waves radiated by the neutron star merger remnants carry rich information about the physics of highdensity nuclear matter states and many important astrophysical phenomena such as gammaray bursts and black hole formation. Current laser interferometer gravitational wave detectors such as LIGO VIRGO and KAGRA have limited signal response at the kiloHertz band thereby being unable to capture these important physical phenomena. This work proposes an alternative protocol for boosting the sensitivity of the gravitational wave detectors at high frequency by implementing an optomechanical quantum amplifier. With the auxiliary quantum amplifier this design has the feature of paritytime P T symmetry so that the detection band will be significantly broadened within the kiloHertz range. In this work we carefully analyze the quantumnoiselimited sensitivity and the dynamical stability of this design. Based on our protocol our result shows that the quantumnoiselimited sensitivity will be improved by one order of magnitude around 3 kHz which indicates the potential of our design for a future search of neutron star merger signals.

2022-10-01T00:00:00Z

['arXiv:2206.13224', '10.1103/PhysRevD.106.082002', '2022PhRvD.106h2002W', '10.48550/arXiv.2206.13224', '2022arXiv220613224W']

['General Relativity and Quantum Cosmology', 'Astrophysics - Instrumentation and Methods for Astrophysics', 'Quantum Physics']

Boosting the sensitivity of highfrequency gravitational wave detectors using P T symmetry

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML']

https://arxiv.org/pdf/2206.13224.pdf

{'Boosting the sensitivity of high frequency gravitational wave detectors by PT-symmetry': 'Chuming Wang, 1 Chunnong Zhao, 2 Xiang Li, 3 Enping Zhou, 4 Haixing Miao, 5 Yanbei Chen, 3 and Yiqiu Ma 1, 4, * \n1 Center for Gravitational Experiment, Hubei Key Laboratory of Gravitation and Quantum Physics, \nSchool of Physics, Huazhong University of Science and Technology, Wuhan, 430074, P. R. China 2 \nARC Centre of Excellence for Gravitational Wave Discovery, \nThe University of Western Australia, Crawley, Western Australia 6009, Australia \n3 Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, California 91125, USA \n4 Department of Astronomy, School of Physics, Huazhong University of Science and Technology, Wuhan, 430074, P. R. China \n5 State Key Laboratory of Low Dimensional Quantum Physics, \nDepartment of Physics, Tsinghua University, Beijing, China \nThe kilo-Hertz gravitational waves radiated by the neutron star merger remnants carry rich information about the physics of high-density nuclear matter states, and many important astrophysical phenomena such as gammaray bursts and black hole formation. Current laser interferometer gravitational wave detectors, such as LIGO, VIRGO, and KAGRA have limited signal response at the kilo-Hertz band, thereby unable to capture these important physical phenomena. This work proposes an alternative protocol for boosting the sensitivity of the gravitational wave detectors at high frequency by implementing an optomechanical quantum amplifier. With the auxiliary quantum amplifier, this design has the feature of Parity-Time (PT) symmetry so that the detection band will be significantly broadened within the kilo-Hertz range. In this work, we carefully analyze the quantumnoise-limited sensitivity and the dynamical stability of this design. Based on our protocol, our result shows that the quantum-noise-limited sensitivity will be improved by one order of magnitude around 3kHz, which indicates the potential of our design for a future search of neutron star merger signals.', 'I. INTRODUCTION': "Gravitational waves (GWs) radiated from binary neutron star (BNS) inspirals have been detected in many events (e.g. GW170817) by the advanced ground-based laser interferometer GW detectors (LIGO and VIRGO) [1-6]. However, the GWs radiated by the BNS post-merger remnants, of which the predicted frequency is around kilo-Hertz (kHz), have not been detected yet, due to the limitations of sensitivity at high frequencies [7, 8]. These high-frequency GWs carry rich physics. For example, They may reveal the details of the center engine of a short gamma-ray burst, the equation of state of ultra-dense nuclear/quark matter, etc. Upgrading the sensitivity of ground-based GW detectors at kilo-Hertz frequency is an important experimental task for the future GW astrophysics [9-14]. \nThe response of the current LIGO, VIRGO and KAGRA configuration to the high-frequency GWs is limited by the interferometer's bandwidth [15]. The amplitude of GWinduced-sidebands of the main carrier light decreases with the increase of the GW frequency W . The quantum shot noise, which dominates the noise floor at kHz, has a white spectrum [16-21]. Various schemes have been proposed for increasing the signal response at high frequency. The simplest way is to increase the intra-cavity power (to e.g. 10 MW) [7], at the price of sacrificing the low-frequency sensitivity (e.g. the proposed new Australia-based instrument NEMO project is targeted on using high power interferometer [22]). However, there are many technical challenges to building a highpower Fabry-Perot cavity for GW detection [23-25]. Other schemes include: implementing the 'white-light-cavity' concept to broaden the bandwidth for a detuned interferometer at high frequency; using the resonance created by a long signal recycling cavity; and the design of signal recycling cavity with internal squeezing [26 ? -31]. \nRecently, a novel scheme was proposed to boost the GW detector sensitivity by reshaping its signal response, in which the interferometer mode ˆ a is coupled to a quantum parametric amplifier ˆ c [32, 33]. In the ideal case, the system Hamiltonian is invariant under the transformation ˆ a → ˆ c † or vice versa, that is, the Hamiltonian has Parity-Time (PT)-symmetry . In this case, the GW-induced sideband signal fields inside the detector response as ∼ W -1 , which means a large signal boost at low frequency. Compared to the previous white light cavity design [26], this scheme is dynamically stable since the unstable parametric process will be balanced by the stabilizing sloshing process, and thereby no further feedback control is required for stabilization [32, 34]. However, the W -1 signal response means that most of the advantage is obtained at low frequency, which is easily contaminated by the back-action noise due to the fluctuating quantum radiation pressure force as well as various classical noises. \nIn this paper, we extend this 'PT-symmetry' design concept for boosting the signal response at kilo-Hertz frequency, which could be an alternative approach to increasing the detector sensitivity for detecting GWs emitted by BNS postmerger remnants. In this scheme, the main laser is detuned from the resonance of a signal-recycling laser interferometer, which is coupled to an oppositely detuned quantum parametric amplifier. We will thoroughly analyze the conceptual design of this scheme. Our results show that implementing this PT-symmetry design can significantly boost the sensitivity in a relatively large searching band at high frequency. \nThe outline of this paper is organized as follows. Section II gives the basic configurations and the result in the ideal case by a Hamiltonian approach based on the single-mode approximation. Then in Section III and IV, we perform a detailed analysis on the effect of PT-symmetry breaking to the sensitivity and system dynamical stability. In Section V , we compute the sensitivity curve using the transfer matrix approach, which \nis beyond the single-mode and resolved sideband approximations, considering various noise sources. Finally, we give the astrophysical implications of our protocol in Section VI.", 'II. THEORETICAL PRINCIPLE OF THE SCHEME': 'The basic concept of the scheme can be illustrated (schematically shown in Fig. 1) by the following idealized mode interaction Hamiltonian [32, 35]: \nˆ H int / ¯ h = i w s ( ˆ a ˆ b † e i D t -ˆ a † ˆ be -i D t ) + iG ( ˆ b † ˆ c † e i d t -ˆ b ˆ ce -i d t ) . (1) \nHere the ˆ a is the annihilation operator of the differential optical mode of the main signal-recycling interferometer; The D is the detuning of the main laser beam with respect to the ˆ a mode, which is introduced by the signal recycling cavity [35]. This non-zero detuning D creates an optical resonance at W = D , which improves the signal response around D [35]. At the same time, this detuning also leads to the optical spring resonance at low-frequency [35]. Since the highfrequency sensitivity is mainly concerned here, we temporarily ignore the optical spring effect in this section for simplicity. The GW-induced sideband fields are extracted from the ˆ b mode, which is parametrically coupled to the ˆ c mode. This parametric interaction could have different realizations, for example, using an optomechanical device, or a pumped nonlinear crystal [32, 36]. Unlike the previous work, the frequency matching of the parametric coupling here is generally not perfect, that is, d = 0. \nThis Hamiltonian is invariant under the PT-transformation ˆ ae i D t → ˆ c † e i d t when the PT-symmetry conditions (incl. G = w s and d = -D ) are satisfied. The Heisenberg equations of motion, considering the coupling between ˆ a and the GWs with strength a are: \nglyph[negationslash] \n˙ ˆ a ( t ) = -i D ˆ a ( t ) -w s ˆ b ( t ) + i a h ( t ) , ˙ ˆ c † ( t ) = i d ˆ c † ( t ) + G ˆ b ( t ) , ˙ ˆ b ( t ) = -g ˆ b ( t ) + w s ˆ a ( t ) + G ˆ c † ( t ) + √ 2 g ˆ b in ( t ) , (2) \nwhere g is the coupling rate between mode ˆ b ( t ) and external bath ˆ b in ( t ) . The outgoing field is ˆ b out ( t ) = -ˆ b in ( t )+ √ 2 g ˆ b ( t ) . Solving these equations of motion under the PT-symmetry condition leads to the input-output relation as: \nˆ b out1 ( W ) = W -i g W + i g ˆ b in1 ( W ) + 2 a √ gw s D ( W + i g )( D 2 -W 2 ) h ( W ) , ˆ b out2 ( W ) = W -i g W + i g ˆ b in2 ( W ) + 2 a √ gw s W ( W + i g )( D 2 -W 2 ) h ( W ) , (3) \nwhere we have the amplitude and phase quadrature of optical fields defined in the two-photon formalism [37] as: \nˆ b 1 ( W ) = 1 √ 2 [ ˆ b ( W ) + ˆ b † ( -W )] , ˆ b 2 ( W ) = 1 √ 2 i [ ˆ b ( W ) -ˆ b † ( -W )] . (4) \nFIG. 1. Schematic setup: a PT-symmetric Gravitational Wave (GW) Detector. The internal signal recycling mirror (iSRM) at the dark port detunes the main interferometer resonance. An optomechanical device is coupled to the main interferometer which contributes to the parametric process. The red and blue dashed lines are the optical fields at the signal and idler channels. Finally, both of these two channels should be measured to obtain the optimized sensitivity curve. \n<!-- image --> \nFIG. 2. The idealized kilo-Hertz GW noise spectrum of gravitational wave detectors enhanced by PT-symmetric configurations, in comparison to the conventional detuned LIGO configuration. The detector dynamics here are described by Eq. (2). The parameters we used here is based on mapping the sample parameters in Tab. I of the interferometer to the effective mode-mode interaction model. \n<!-- image --> \nSince we are interested in high frequency region where W ∼ D , the signal response of ˆ b out1 and ˆ b out2 are roughly the same and scales approximately as ∼ 1 / ( D -W ) . \nThe shot-noise-limited sensitivity of the detuned PTsymmetric scheme, quantified by the signal referred shot noise spectral density S hh ( W ) , is given by (suppose the phase \n<latexit sha1\\_base64="D8Toh0prpAhg2jFhBA5QdoXjh7k=">AAAB83icbZDLSgMxFIbP1Futt6pLN8EiuCozlaLLYjcuK9gLtEPJpJk2NJMMSaZQhr6GGxeKuPVl3Pk2ZtpZaOuBwMf/n5Oc/EHMmTau++0UtrZ3dveK+6WDw6Pjk/LpWUfLRBHaJpJL1QuwppwJ2jbMcNqLFcVRwGk3mDYzvzujSjMpnsw8pn6Ex4KFjGBjpUFTihkVGWI+LFfcqrsstAleDhXIqzUsfw1GkiSRvYBwrHXfc2Pjp1gZRjhdlAaJpjEmUzymfYsCR1T76XLnBbqyygiFUtkjDFqqvydSHGk9jwLbGWEz0eteJv7n9RMT3vkpE3FiqCCrh8KEIyNRFgAaMUWJ4XMLmChmd0VkghUmxsZUsiF461/ehE6t6t1U64+1SuM+j6MIF3AJ1+DBLTTgAVrQBgIxPMMrvDmJ8+K8Ox+r1oKTz5zDn3I+fwBfDJHn</latexit> \n<latexit 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(6) \nThe comparison between the sensitivities of these two configurations at W ∼ D is shown in Fig. 2. At the resonance point W = D , the PT-symmetry scheme has a larger boost due to its response ∼ 1 / ( W -D ) in the ideal case. \nIn this ideal case, the system is also dynamically stable which can be understood from the following analysis. Adiabatic elimination of the ˆ b -field (usually has a large bandwidth) leads to the equations of motion for ˆ a , ˆ c : \n( d dt + i D + w 2 s g ) ˆ a = -w sG g ˆ c † -√ w 2 s 2 g ˆ b in + i a h , ( d dt -i d -G 2 g ) ˆ c † = w sG g ˆ a + √ G 2 2 g ˆ b in . (7) \nThere is a damping factor w 2 s / g for the ˆ a field while an anti-damping factor -G 2 / g for the ˆ c † field. Under the PTsymmetric conditions G = w s and d = -D , the anti-damping factor and damping factor cancel each other for the effective mode ˆ a + ˆ c † : \n( d dt + i D ) ( ˆ a + ˆ c † ) = i a h . (8) \nNote that in Fig. 2, there is one peak at W = D , while we actually have three different modes ˆ a , ˆ b , ˆ c in our Hamiltonian, which means there exists degeneracy due to the PT symmetry [38, 39]. This degeneracy can be easily understood since ˆ a is equivalent to ˆ c † , and the ˆ b -field couples to the ˆ a , ˆ c fields in such a way that there is no effect on the resonance frequency of the ˆ b -mode (see Eq. (8)). Practical imperfections of our system will cause the breaking of the PT-symmetry, which will affect the detector sensitivity and the system stability. \nThis degeneracy can be understood from the algebraic structure of the Heisenberg equations of motion Eq. (2). If we take the representation which chooses ˆ a , ˆ b , ˆ c † to be the basis vectors, the Heisenberg equation of motion becomes: \nd dt ˆ v = ˆ D ˆ v + s ( t ) , (9) \nwhere ˆ v ≡ ( v 1 , v 2 , v 3 ) T is a combined mode consisting v 1 , v 2 , v 3 of ˆ a , ˆ b , ˆ c † ,respectively; s ( t ) is the vector describes the signal and noise adding to the system: \ns ( t ) = ( i a h ( t ) , √ 2 g ˆ bin ( t ) , 0 ) T ; (10) \nˆ D is the dynamic matrix: \nˆ D = -i D -w s 0 w s -g G 0 G i d . (11) \nThe eigenvalues l 1 , 2 , 3 and the corresponding eigenvectors ˆ v 1 , 2 , 3 then describes the poles and its corresponding modes of the whole system, respectively. The response in the frequency domain can be written as: \nv ( W ) = -s ( W ) l + i W . (12) \nSatisfying one of the PT-symmetry conditions d = -D , the eigenvalues and the corresponding eigenvectors are given by: \nl 1 : v 1 =( -G w s , 0 , 1 ) T , l 2 : v 2 =( -w s G , -g -i D 2 c -l 3 -l 2 2 c , 1 ) T , l 3 : v 3 =( -w s G , -g -i D 2 c + l 3 -l 2 2 c , 1 ) T , (13) \nwhere \nl 1 = -i D , l 2 , 3 = 1 2 [ -g -i D ∓ √ ( g -i D ) 2 + 4 ( G 2 -w 2 s )] . (14) \nThe other PT-symmetry condition G = w s would lead to a degeneracy since v 1 / 3 corresponds to the same eigenvalue l 1 = l 3 = -i D . Breaking this PT-symmetry condition would lead to the breaking of this degeneracy, as shown in Fig. 3 where the eigenvalues are plotted in the complex frequency domain. This figure also shows that there will be an unstable mode in some parameter regions, of which the eigenvalue is located on the upper complex plane. This stability issue will be further explored in Section IV.', 'III. EFFECT OF PT-SYMMETRY BREAKING: SENSITIVITY': 'The main practical factors that lead to the PT-symmetry breaking can be generally summarized as follows (also see Fig. 4): (1) the off-resonance sidebands in the optomechanical filter cavity, which we ignored in the ideal case in Eq (1); (2) the pondermotive interaction happens inside the interferometer arm cavities [35]. In this section, we focus on their influences on the sensitivity of our protocol.', 'A. Influence from the off-resonant sidebands': 'A typical realization of the parametric interaction is to use an optomechanical device with the interaction Hamiltonian ˆ H om = ¯ h ˜ G ˆ b † ˆ b ˆ xm , where ˜ G = w 0 / L b is the single-photon optomechanical coupling strength and L b is the length of the filter cavity, the ˆ xm is the displacement operator of the mechanical oscillator. Supposing the system is pumped with detuning w m , the optomechanical interaction can be written as: \nˆ H om / ¯ h = G ( ˆ b ˆ ce i d t + ˆ b † ˆ c † e -i d t + ˆ b † ˆ ce -i ( 2 w m -d ) t + ˆ b ˆ c † e i ( 2 w m -d ) t ) , (15) \n<latexit sha1\\_base64="yf8EFSSgEYNb7HlU6GODUJ5z1HU=">AAAB6HicbVDLSgNBEOyNrxhfUY9eBoPgKez6QI9BL16EBMwDkiXMTjrJmNnZZWZWCEu+wIsHRbz6Sd78GyfJHjSxoKGo6qa7K4gF18Z1v53cyura+kZ+s7C1vbO7V9w/aOgoUQzrLBKRagVUo+AS64Ybga1YIQ0Dgc1gdDv1m0+oNI/kgxnH6Id0IHmfM2qsVLvvFktu2Z2BLBMvIyXIUO0Wvzq9iCUhSsME1brtubHxU6oMZwInhU6iMaZsRAfYtlTSELWfzg6dkBOr9Eg/UrakITP190RKQ63HYWA7Q2qGetGbiv957cT0r/2UyzgxKNl8UT8RxERk+jXpcYXMiLEllClubyVsSBVlxmZTsCF4iy8vk8ZZ2TsvX9YuSpWbLI48HMExnIIHV1CBO6hCHRggPMMrvDmPzovz7nzMW3NONnMIf+B8/gCnUYzY</latexit> \n<latexit sha1\\_base64="D9GgRd2WcMdsfeBzeCF9fsFbOJE=">AAAB+nicbVDLSsNAFL3xWesr1aWbwSK4sSRV0GXRjcsK9gFNKJPppB06k4SZiVJiP8WNC0Xc+iXu/BsnbRbaemDgcM693DMnSDhT2nG+rZXVtfWNzdJWeXtnd2/frhy0VZxKQlsk5rHsBlhRziLa0kxz2k0kxSLgtBOMb3K/80ClYnF0rycJ9QUeRixkBGsj9e2KN8IakbPMkwJxNqbTvl11as4MaJm4BalCgWbf/vIGMUkFjTThWKme6yTaz7DUjHA6LXupogkmYzykPUMjLKjys1n0KToxygCFsTQvMkFy9fdGhoVSExGYSYH1SC16ufif10t1eOVnLEpSTSMyPxSmHOkY5T2gAZOUaD4xBBPJTFZERlhiok1bZVOCu/jlZdKu19zzWv3uotq4LuoowREcwym4cAkNuIUmtIDAIzzDK7xZT9aL9W59zEdXrGLnEP7A+vwBpDSTmg==</latexit> \n<latexit sha1\\_base64="K9LZDf3zeDKTIbQr+DYwuZDRkVE=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Ae0oWy2m3btZhN2J0IJ/Q9ePCji1f/jzX/jts1BWx8MPN6bYWZekEhh0HW/ncLa+sbmVnG7tLO7t39QPjxqmTjVjDdZLGPdCajhUijeRIGSdxLNaRRI3g7GtzO//cS1EbF6wEnC/YgOlQgFo2ilVm9EkQT9csWtunOQVeLlpAI5Gv3yV28QszTiCpmkxnQ9N0E/oxoFk3xa6qWGJ5SN6ZB3LVU04sbP5tdOyZlVBiSMtS2FZK7+nshoZMwkCmxnRHFklr2Z+J/XTTG89jOhkhS5YotFYSoJxmT2OhkIzRnKiSWUaWFvJWxENWVoAyrZELzll1dJq1b1Lqq1+8tK/SaPowgncArn4MEV1OEOGtAEBo/wDK/w5sTOi/PufCxaC04+cwx/4Hz+ACDujtU=</latexit> \nFIG. 3. The trajectories of the system\'s eigenvalues based on the ideal Hamiltonian with varies G form 0 to 2 w s and fixed w s . Note that these trajectories are symmetric on the left and right plane since the Hamiltonian is Hermitian. As discussed in the main text, one of the eigenmodes has a fixed eigenvalue l 1 = -i D , which corresponds to a fixed pole at frequency W 1 = -i l 1 = D . The other two eigenvalues (poles) change with the variation of G , they are shown as the blue dashed lines ( l 3 ) and the solid lines ( l 2 ). Under the PT-symmetry condition G = w s , the poles of l 1 and l 3 have a degeneracy. Then if G varies across w s , the pole of l 3 gets a positive imaginary part and becomes unstable. Throughout this variation, v 3 is mainly consist of ˆ a ( v 1 = -w s / G ) and ˆ c † ( v 3 = 1). Therefore we denote the mode "ˆ c -like" field after crossing the degeneracy point since w s < G , while it\'s denoted as " ˆ a -like" field before since w s > G . \n<!-- image --> \nx \nFIG. 5. Left panel: the structure of the mode-coupling Hamiltonian, where the ˆ a and ˆ c are the main interferometer mode and the mechanical mode, which are PT-symmetric to each other. For detecting the high-frequency GWs, detuning should be introduced to these two modes. Right panel: the sideband fields on-resonance with w 0 are defined as signal channel , while the far off-resonance sidebands resonance at ∼ w 0 + 2 w m is defined as idler channel . The red curve sketches the main interferometer resonance profile. \n<!-- image --> \n<latexit sha1\\_base64="p2L/mczVtNffvHvvt/t8B0R/pAY=">AAAB6HicbVDLSgNBEOz1GeMr6tHLYBA8hV0f6DHoxWMC5gHJEmYnvcmY2dllZlYIS77AiwdFvPpJ3vwbJ8keNLGgoajqprsrSATXxnW/nZXVtfWNzcJWcXtnd2+/dHDY1HGqGDZYLGLVDqhGwSU2DDcC24lCGgUCW8Hobuq3nlBpHssHM07Qj+hA8pAzaqxUH/RKZbfizkCWiZeTMuSo9Upf3X7M0gilYYJq3fHcxPgZVYYzgZNiN9WYUDaiA+xYKmmE2s9mh07IqVX6JIyVLWnITP09kdFI63EU2M6ImqFe9Kbif14nNeGNn3GZpAYlmy8KU0FMTKZfkz5XyIwYW0KZ4vZWwoZUUWZsNkUbgrf48jJpnle8i8pV/bJcvc3jKMAxnMAZeHANVbiHGjSAAcIzvMKb8+i8OO/Ox7x1xclnjuAPnM8fzrmM8g==</latexit> \n<latexit 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sha1\\_base64="03IWuu9ZIBCuJwVNjL12X5FvHBQ=">AAAB+nicbVBNSwMxEM36WevXVo9egkXwYtmtgh6LXjxWsB/QXUo2TdvQJLsks0pZ+1O8eFDEq7/Em//GtN2Dtj4YeLw3w8y8KBHcgOd9Oyura+sbm4Wt4vbO7t6+WzpomjjVlDVoLGLdjohhgivWAA6CtRPNiIwEa0Wjm6nfemDa8FjdwzhhoSQDxfucErBS1y0FQwKYnGWBlljwEZt03bJX8WbAy8TPSRnlqHfdr6AX01QyBVQQYzq+l0CYEQ2cCjYpBqlhCaEjMmAdSxWRzITZ7PQJPrFKD/djbUsBnqm/JzIijRnLyHZKAkOz6E3F/7xOCv2rMOMqSYEpOl/UTwWGGE9zwD2uGQUxtoRQze2tmA6JJhRsWkUbgr/48jJpViv+eaV6d1GuXedxFNAROkanyEeXqIZuUR01EEWP6Bm9ojfnyXlx3p2PeeuKk88coj9wPn8AoRaTmA==</latexit> \nFIG. 4. The breaking of the PT-symmetry due to practical system imperfections: the introduction of the off-resonant sideband fields inside the optomechanical filter cavity (idler channel) and the pondermotive interactions in the interferometer arm cavities. The test mass and the idler field do not couple to the ˆ a and ˆ c modes in a symmetric way, thereby breaking the PT-symmetry of the entire system. \n<latexit sha1\\_base64="yAd1Ce/kEyC2pxEXNoSAogxPK/A=">AAAB8XicbVDLSgNBEOyNrxhfUY9eBoPgKez6QI9BD3qMYB6YLGF2MpsMmZldZmbFsOxfePGgiFf/xpt/4yTZgyYWNBRV3XR3BTFn2rjut1NYWl5ZXSuulzY2t7Z3yrt7TR0litAGiXik2gHWlDNJG4YZTtuxolgEnLaC0fXEbz1SpVkk7804pr7AA8lCRrCx0sNTL+0qgW5aWa9ccavuFGiReDmpQI56r/zV7UckEVQawrHWHc+NjZ9iZRjhNCt1E01jTEZ4QDuWSiyo9tPpxRk6skofhZGyJQ2aqr8nUiy0HovAdgpshnrem4j/eZ3EhJd+ymScGCrJbFGYcGQiNHkf9ZmixPCxJZgoZm9FZIgVJsaGVLIhePMvL5LmSdU7rZ7fnVVqV3kcRTiAQzgGDy6gBrdQhwYQkPAMr/DmaOfFeXc+Zq0FJ5/Zhz9wPn8ATFqQrQ==</latexit> \n<latexit 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In the previous section, we ignored the last two terms of the above Hamiltonian using the resolved sideband limit to obtain Eq. (1). However, in reality, these far-off resonant sidebands will have a non-negligible degradation to the detector sensitivity. In this work, we follow the standard quantum optics terminology and name the optical sideband fields around w 0 to be the \' signal channel " and those around w 0 + 2 w m to be the \' idler channel \', as shown in Fig. 5. For illustrative purposes, this subsection will study this effect, while temporarily disregarding the pondermotive effect in analyzing the idler fields. \n<!-- image --> \n- \nIntroducing the idler channel around w 0 + 2 W m will break the PT-symmetry as shown in Fig. 6, and bring the following two effects. \n- · Optical spring effect . The optical spring effect in the optomechanical filter cavity will modify the mechanical frequency to be w opt m ≈ w m + w opt, where w opt = cP b w 0 / 2 m w 2 m c 2 L b . This optical spring effect must be compensated, otherwise, the PT-symmetry will be broken and the detector will not be able to reach its optimal sensitivity. This effect is plotted in the lower panel of Fig. 6, which demonstrates the importance of this frequency compensation.\n- · Correction to the GW signal response. The GW response of the detector will be modified in two ways by introducing the idler channel. First, the mode degeneracy introduced by PT-symmetry will be broken. This splits the singlepeak at W = D in the sensitivity curve as shown in Fig. 6. Second, the GW signal sidebands will flow into the idler channels. Therefore GW information can also be extracted from the detection of the optical quadratures at the idler channels: \nˆ b i 1 ( W ) = 1 √ 2 [ ˆ b ( 2 w m + W ) + ˆ b † ( 2 w m -W )] , ˆ b i 2 ( W ) = 1 √ 2 i [ ˆ b ( 2 w m + W ) -ˆ b † ( 2 w m -W )] . (16) \nSince the idler channel is separated far from the signal channel in the frequency domain, this means we can use two homodyne detectors to measure both the signal channel and idler channel, and then do data-post-processing to optimize the detector sensitivity, which we will explain in detail later. This analysis also shows that a full transfer matrix analysis is needed for a more accurate sensitivity curve. \n<!-- image --> \nFIG. 6. The detector sensitivity considers purely the effect of idler fields (i.e. the pondermotive effect is ignored). Upper panel: the sensitivity of both the idler and signal channels, taking into account the frequency compensation w opt. Lower panel: The sensitivity of the signal channel with/without the frequency compensation. The two dips around the detuning frequency is a manifestation of degeneracy breaking introduced by the idler channel. \n<!-- image -->', 'B. Pondermotive effect': 'Now we discuss the pondermotive effect which has been ignored in the previous sections. For a detuned main interferometer, the pondermotive interaction generates an additional stiffness for the test masses [40], which is the optical spring effect [35]. This optical stiffness comes from the dependence of radiation pressure force on the test mass displacement. Adding the pondermotive effect term + ig ˆ X to the right-hand side of Eq. (2) ( g = w 0 a / L is the linearised optomechanical coupling constant in the arm cavity, X is displacement of test \n<latexit sha1\\_base64="siKc7nElgD0QrO8G53qjzlNOupk=">AAAB/3icdVDLSgMxFM3UV62vquDGTbAIrsqkom13xW50V6EPoR1KJpNpQzOZIckIZezCX3HjQhG3/oY7/8ZMO4KKHrhwOOde7r3HjThT2rY/rNzS8srqWn69sLG5tb1T3N3rqjCWhHZIyEN542JFORO0o5nm9CaSFAcupz130kz93i2VioWiracRdQI8EsxnBGsjDYsHVx7FHLbasBkKn41imRklu2zbNkIIpgRVz21D6vVaBdUgSi2DEsjQGhbfB15I4oAKTThWqo/sSDsJlpoRTmeFQaxohMkEj2jfUIEDqpxkfv8MHhvFg34oTQkN5+r3iQQHSk0D13QGWI/Vby8V//L6sfZrTsJEFGsqyGKRH3OoQ5iGAT0mKdF8aggmkplbIRljiYk2kRVMCF+fwv9Jt1JGp+Wz60qpcZHFkQeH4AicAASqoAEuQQt0AAF34AE8gWfr3nq0XqzXRWvOymb2wQ9Yb5+Y5JXd</latexit> \n<latexit sha1\\_base64="siKc7nElgD0QrO8G53qjzlNOupk=">AAAB/3icdVDLSgMxFM3UV62vquDGTbAIrsqkom13xW50V6EPoR1KJpNpQzOZIckIZezCX3HjQhG3/oY7/8ZMO4KKHrhwOOde7r3HjThT2rY/rNzS8srqWn69sLG5tb1T3N3rqjCWhHZIyEN542JFORO0o5nm9CaSFAcupz130kz93i2VioWiracRdQI8EsxnBGsjDYsHVx7FHLbasBkKn41imRklu2zbNkIIpgRVz21D6vVaBdUgSi2DEsjQGhbfB15I4oAKTThWqo/sSDsJlpoRTmeFQaxohMkEj2jfUIEDqpxkfv8MHhvFg34oTQkN5+r3iQQHSk0D13QGWI/Vby8V//L6sfZrTsJEFGsqyGKRH3OoQ5iGAT0mKdF8aggmkplbIRljiYk2kRVMCF+fwv9Jt1JGp+Wz60qpcZHFkQeH4AicAASqoAEuQQt0AAF34AE8gWfr3nq0XqzXRWvOymb2wQ9Yb5+Y5JXd</latexit> \n<latexit sha1\\_base64="PnyEJP+eouvg6m3qpyqgoqbar6A=">AAAB+nicdZDLTgIxFIY7eEO8Dbp000hMcENmBgTcEd24MpjIJQFCOqVAQ6edtB0NGXkUNy40xq1P4s63sQOYqNE/afLnO+fknP5+yKjSjvNhpVZW19Y30puZre2d3T07u99UIpKYNLBgQrZ9pAijnDQ01Yy0Q0lQ4DPS8icXSb11S6Sigt/oaUh6ARpxOqQYaYP6djZ/JSAWQUi4mqOTvp1zCmfVslcqQ6fgOBXXcxPjVUrFEnQNSZQDS9X79nt3IHAUEK4xQ0p1XCfUvRhJTTEjs0w3UiREeIJGpGMsRwFRvXh++gweGzKAQyHN4xrO6feJGAVKTQPfdAZIj9XvWgL/qnUiPaz2YsrDSBOOF4uGEYNawCQHOKCSYM2mxiAsqbkV4jGSCGuTVsaE8PVT+L9pegW3WDi99nK182UcaXAIjkAeuKACauAS1EEDYHAHHsATeLburUfrxXpdtKas5cwB+CHr7RP/UpPa</latexit> \n<latexit sha1\\_base64="jeRjfuSTfxMEb4jytqNr9+UmuB8=">AAAB9HicdVBNS8NAEN34WetX1aOXxSJ4Ckla23oretFbBfsBbSib7bRdutnE3U2hlP4OLx4U8eqP8ea/cdNWUNEHA4/3ZpiZF8ScKe04H9bK6tr6xmZmK7u9s7u3nzs4bKgokRTqNOKRbAVEAWcC6pppDq1YAgkDDs1gdJX6zTFIxSJxpycx+CEZCNZnlGgj+Tc9DhLTIRECeDeXd+yLSskrlrBjO07Z9dyUeOVioYhdo6TIoyVq3dx7pxfRJAShKSdKtV0n1v6USM0oh1m2kyiICR2RAbQNFSQE5U/nR8/wqVF6uB9JU0Ljufp9YkpCpSZhYDpDoofqt5eKf3ntRPcr/pSJONEg6GJRP+FYRzhNAPeYBKr5xBBCJTO3pgFIQrXJKWtC+PoU/08anu0W7PNbL1+9XMaRQcfoBJ0hF5VRFV2jGqojiu7RA3pCz9bYerRerNdF64q1nDlCP2C9fQLbQZIs</latexit> \n<latexit sha1\\_base64="tvTM+bERS9pnoe25+Zp/0R2A6oU=">AAAB+HicdVBNS8NAEJ34WetHox69LBbBU0nS2tZb0YvHivYD2lA22227dLMJuxuhlv4SLx4U8epP8ea/cdNWUNEHA4/3ZpiZF8ScKe04H9bK6tr6xmZmK7u9s7uXs/cPmipKJKENEvFItgOsKGeCNjTTnLZjSXEYcNoKxpep37qjUrFI3OpJTP0QDwUbMIK1kXp27oYNBeaIjLAQlKOenXcK59WyVyojp+A4FddzU+JVSsUSco2SIg9L1Hv2e7cfkSSkQhOOleq4Tqz9KZaaEU5n2W6iaIzJGA9px1CBQ6r86fzwGToxSh8NImlKaDRXv09McajUJAxMZ4j1SP32UvEvr5PoQdWfMhEnmgqyWDRIONIRSlNAfSYp0XxiCCaSmVvTBCQm2mSVNSF8fYr+J02v4BYLZ9devnaxjCMDR3AMp+BCBWpwBXVoAIEEHuAJnq1769F6sV4XrSvWcuYQfsB6+wR/NZL/</latexit> \n<latexit sha1\\_base64="EJa0//JcS1ft4DCcQ3Q4rqKcH48=">AAAB9XicdVBNT8JAEN3iF+IX6tHLRmLiibQFAW9ELx4xCpgAku0yhQ3bbbO71ZCG/+HFg8Z49b9489+4BUzU6EsmeXlvJjPzvIgzpW37w8osLa+srmXXcxubW9s7+d29lgpjSaFJQx7KG48o4ExAUzPN4SaSQAKPQ9sbn6d++w6kYqG41pMIegEZCuYzSrSRbq/YUBCO6YgIAbyfL9jF01rFLVewXbTtquM6KXGr5VIZO0ZJUUALNPr59+4gpHEAQlNOlOo4dqR7CZGaUQ7TXDdWEBE6JkPoGCpIAKqXzK6e4iOjDLAfSlNC45n6fSIhgVKTwDOdAdEj9dtLxb+8Tqz9Wi9hIoo1CDpf5Mcc6xCnEeABk0A1nxhCqGTm1jQASag2QeVMCF+f4v9Jyy06peLJpVuony3iyKIDdIiOkYOqqI4uUAM1EUUSPaAn9GzdW4/Wi/U6b81Yi5l99APW2yesnJKk</latexit> \n<latexit sha1\\_base64="EJa0//JcS1ft4DCcQ3Q4rqKcH48=">AAAB9XicdVBNT8JAEN3iF+IX6tHLRmLiibQFAW9ELx4xCpgAku0yhQ3bbbO71ZCG/+HFg8Z49b9489+4BUzU6EsmeXlvJjPzvIgzpW37w8osLa+srmXXcxubW9s7+d29lgpjSaFJQx7KG48o4ExAUzPN4SaSQAKPQ9sbn6d++w6kYqG41pMIegEZCuYzSrSRbq/YUBCO6YgIAbyfL9jF01rFLVewXbTtquM6KXGr5VIZO0ZJUUALNPr59+4gpHEAQlNOlOo4dqR7CZGaUQ7TXDdWEBE6JkPoGCpIAKqXzK6e4iOjDLAfSlNC45n6fSIhgVKTwDOdAdEj9dtLxb+8Tqz9Wi9hIoo1CDpf5Mcc6xCnEeABk0A1nxhCqGTm1jQASag2QeVMCF+f4v9Jyy06peLJpVuony3iyKIDdIiOkYOqqI4uUAM1EUUSPaAn9GzdW4/Wi/U6b81Yi5l99APW2yesnJKk</latexit> \nmass.) leads to: \n˙ ˆ X = ˆ P / M , ˙ ˆ P = ¯ hg ( ˆ a + ˆ a † ) . (17) \nThe optical spring rigidity K opt ( W ) under the PT-symmetry condition can be solved as (where the PT-symmetry conditions G = w s and d = -D have been used): \nK opt ( W ) = 2¯ hg 2 [ D D 2 -W 2 -2 WD G 2 ( W + i g )( D 2 -W 2 ) 2 ] . (18) \nIn the following, it will be written as K opt ( W ) = K opt1 ( W ) + K opt2 ( W ) , where the K opt1 ( W ) = 2¯ hg 2 D / ( D 2 -W 2 ) , which only depends on the optomechanical coupling g in the arm cavity. This K opt1 ( W ) resembles the rigidity of a detuned perfect cavity with zero bandwidth , which reflects the fact that the loss and gain are balanced under the PT-symmetry condition. \nThe optical rigidity will become very large at W = D due to the significantly boosted displacement-induced-sideband fields as GLYPH<181> a X ( W ) / ( D 2 -W 2 ) . In other words, at W = D , the test masses will become so stiff that the external force can not drive their motions. Therefore the signal at this frequency is strongly suppressed and a peak in the sensitivity curve is expected. \nThe input-output relations considering the pondermotive effect can be written as: \nˆ b out ( W ) = e i b ( W ) M ( W ) . ˆ b in ( W ) + D ( W ) h ( W ) , (19) \nin which we have: e i b ( W ) =( W -i g ) / ( W + i g ) and \nM ( W ) = c M ( W ) [ -c -1 M ( W ) + K i opt2 ( W ) -( D / W ) K r opt2 ( W ) -( g / D ) K i opt2 ( W ) -c -1 M ( W ) + K i opt2 ( W ) ] , (20) \nwhere c M ( W ) is the test mass mechancial response function modified by the pondermotive effect: \nc M ( W ) = -1 M W 2 -K opt ( W ) . (21) \nThe signal response matrix is: \nD ( W ) = -2 ig √ g G ( W + i g )( D 2 -W 2 ) c M ( W ) M W 2 L [ D W ] , (22) \nwhere at W = D , the signal response vanishes, and the sensitivity will have a very sharp peak as expected. This effect will not happen for the tuned PT-symmetric interferometer [32] since there is no optical spring effect when D = 0. The problem of sensitivity degradation due to this pondermotive effect can be solved using (1) an optimal combination of different measurement channels which is discussed in detail in Sec. V, and (2) negative inertia.', 'C. Cancelling the pondermotive effect using negative inertia': 'The pondermotive effect reflects the fact that the system is not fully PT-symmetric when the test masses dynamics are \n<latexit sha1\\_base64="OxrUcsoC+FdtP5WtWAU2volzMSU=">AAAB/nicdVDLSgNBEJz1bXxFxZOXwSB4MexuYqI30YvHCEYDMYTZ2Y4OmccyMxsIS8Bf8eJBEa9+hzf/xlkTQUULGoqqbrq7ooQzY33/3Zuanpmdm19YLCwtr6yuFdc3Lo1KNYUmVVzpVkQMcCahaZnl0Eo0EBFxuIr6p7l/NQBtmJIXdphAR5AbyXqMEuukbnGroWQMWijLBrAvVOw8iLvFkl8+OqyF1Rr2y75fD8IgJ2G9WqniwCk5SmiCRrf4dh0rmgqQlnJiTDvwE9vJiLaMchgVrlMDCaF9cgNtRyURYDrZ5/kjvOuUGPeUdiUt/lS/T2REGDMUkesUxN6a314u/uW1U9s77GRMJqkFSceLeinHVuE8CxwzDdTyoSOEauZuxfSWaEKtS6zgQvj6FP9PLsNyUCkfnIel45NJHAtoG+2gPRSgOjpGZ6iBmoiiDN2jR/Tk3XkP3rP3Mm6d8iYzm+gHvNcPA/yWLg==</latexit> \n<latexit sha1\\_base64="iAL1iv8jXe1EXhrPPiz2GlLXTCQ=">AAAB9HicdVDLSgMxFM34rPVVdekmWARXJalo213RjcsK9gHtUDJppg3NJGOSKZSh3+HGhSJu/Rh3/o2ZtoKKHrhwcs695N4TxIIbi9CHt7K6tr6xmdvKb+/s7u0XDg5bRiWasiZVQulOQAwTXLKm5VawTqwZiQLB2sH4OvPbE6YNV/LOTmPmR2QoecgpsU7yqZIhHyZ6/uoXiqiEEMIYw4zgyiVypFarlnEV4sxyKIIlGv3Ce2+gaBIxaakgxnQxiq2fEm05FWyW7yWGxYSOyZB1HZUkYsZP50vP4KlTBjBU2pW0cK5+n0hJZMw0ClxnROzI/PYy8S+vm9iw6qdcxollki4+ChMBrYJZAnDANaNWTB0hVHO3K6Qjogm1Lqe8C+HrUvg/aZVL+Lx0cVsu1q+WceTAMTgBZwCDCqiDG9AATUDBPXgAT+DZm3iP3ov3umhd8ZYzR+AHvLdPpRqSrw==</latexit> \n<latexit sha1\\_base64="rwm4OodOF6nAtUZHWjeYqid7qXE=">AAAB83icdVDLSgMxFM34rPVVdekmWARXZVLRtruiG91V6AvaoWQyd9rQTGZIMkIp/Q03LhRx68+482/MtBVU9EDgcM69nJvjJ4Jr47ofzsrq2vrGZm4rv72zu7dfODhs6zhVDFosFrHq+lSD4BJahhsB3UQBjXwBHX98nfmde1Cax7JpJgl4ER1KHnJGjZX6twFQmwIBbjQHhaJbcl2XEIIzQiqXriW1WrVMqphklkURLdEYFN77QczSCKRhgmrdI25ivClVhjMBs3w/1ZBQNqZD6FkqaQTam85vnuHTNEsNY2WfNHiuft+Y0kjrSeTbyYiakf7tZeJfXi81YdWbcpmkBiRbBIWpwCbGWQE44AqYERNLKFPc3orZiCrKjK0pb0v4+in+n7TLJXJeurgrF+tXyzpy6BidoDNEUAXV0Q1qoBZiKEEP6Ak9O6nz6Lw4r4vRFWe5c4R+wHn7BKtekXU=</latexit> \nFIG. 7. The degradation of detector sensitivity is due to the pondermotive interaction between the test masses and the optical fields in the main interferometer. The narrow sensitivity peak at the optical resonance frequency 3000 Hz with bandwidth ∼ 10 Hz comes from the optical stiffness enhanced by the PT-symmetric configuration. For highlighting the pondermotive effect, the idler channel effect is ignored here. \n<!-- image --> \nconsidered. For achieving a fully PT-symmetric system, a negative mass term [41-46] needs to be introduced with an additional equation of motion: \n˙ ˆ x = -ˆ p m , ˙ ˆ p = ¯ h ˜ g ( ˆ c + ˆ c † ) , (23) \nwhere m = -M and ˜ g = g should be satisfied to achieve the PT-symmetry. In this case, the input-output relation becomes very simple: \n[ ˆ b out1 ( W ) ˆ b out2 ( W ) ] = e i b ( W ) [ ˆ b in1 ( W ) ˆ b in2 ( W ) ] + D sym ( W ) h ( W ) , (24) \nwhere we define: \nD sym ( W ) = 2 ig √ gc ( g -i W )( D 2 -W 2 ) M W 2 L -M W 2 + K opt1 ( W ) [ D -i W ] . (25) \nAt W = D , the signal response is finite. The physical realization of negative inertia coupled to the mode ˆ c could be realized by optical systems, we leave the detailed design to the accompanying paper. In this paper, we merely introduce this concept without discussing its details. The main approach to cancel the sensitivity peak at W = D discussed in this paper is the optimal combination of the different measurement channels which will be discussed in the next section.', 'IV. EFFECT OF PT-SYMMETRY BREAKING: STABILITY': 'Instability arise when there are external energy sources pump the system continuously. As shown in Fig.1, the arm cavity and filter cavity are both pumped in the blue-detuned way. A single optomechanical system pumped by a bluedetuned coherent field would have instability since the Stokes \n<latexit sha1\\_base64="yAd1Ce/kEyC2pxEXNoSAogxPK/A=">AAAB8XicbVDLSgNBEOyNrxhfUY9eBoPgKez6QI9BD3qMYB6YLGF2MpsMmZldZmbFsOxfePGgiFf/xpt/4yTZgyYWNBRV3XR3BTFn2rjut1NYWl5ZXSuulzY2t7Z3yrt7TR0litAGiXik2gHWlDNJG4YZTtuxolgEnLaC0fXEbz1SpVkk7804pr7AA8lCRrCx0sNTL+0qgW5aWa9ccavuFGiReDmpQI56r/zV7UckEVQawrHWHc+NjZ9iZRjhNCt1E01jTEZ4QDuWSiyo9tPpxRk6skofhZGyJQ2aqr8nUiy0HovAdgpshnrem4j/eZ3EhJd+ymScGCrJbFGYcGQiNHkf9ZmixPCxJZgoZm9FZIgVJsaGVLIhePMvL5LmSdU7rZ7fnVVqV3kcRTiAQzgGDy6gBrdQhwYQkPAMr/DmaOfFeXc+Zq0FJ5/Zhz9wPn8ATFqQrQ==</latexit> \n<latexit 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sha1\\_base64="7FFaLROi8qFPQMgLqJTwB3aFCYw=">AAAB73icbVDJSgNBEK2JW4xb1KOXxiB4CjMu6DHoxWMEs0AyhJ5OT9Kkl7G7RwhDfsKLB0W8+jve/Bs7yRw08UHB470qqupFCWfG+v63V1hZXVvfKG6WtrZ3dvfK+wdNo1JNaIMornQ7woZyJmnDMstpO9EUi4jTVjS6nfqtJ6oNU/LBjhMaCjyQLGYEWye1u0rQAe6JXrniV/0Z0DIJclKBHPVe+avbVyQVVFrCsTGdwE9smGFtGeF0UuqmhiaYjPCAdhyVWFATZrN7J+jEKX0UK+1KWjRTf09kWBgzFpHrFNgOzaI3Ff/zOqmNr8OMySS1VJL5ojjlyCo0fR71mabE8rEjmGjmbkVkiDUm1kVUciEEiy8vk+ZZNTivXt5fVGo3eRxFOIJjOIUArqAGd1CHBhDg8Ayv8OY9ei/eu/cxby14+cwh/IH3+QMWypAC</latexit> \n<latexit sha1\\_base64="7FFaLROi8qFPQMgLqJTwB3aFCYw=">AAAB73icbVDJSgNBEK2JW4xb1KOXxiB4CjMu6DHoxWMEs0AyhJ5OT9Kkl7G7RwhDfsKLB0W8+jve/Bs7yRw08UHB470qqupFCWfG+v63V1hZXVvfKG6WtrZ3dvfK+wdNo1JNaIMornQ7woZyJmnDMstpO9EUi4jTVjS6nfqtJ6oNU/LBjhMaCjyQLGYEWye1u0rQAe6JXrniV/0Z0DIJclKBHPVe+avbVyQVVFrCsTGdwE9smGFtGeF0UuqmhiaYjPCAdhyVWFATZrN7J+jEKX0UK+1KWjRTf09kWBgzFpHrFNgOzaI3Ff/zOqmNr8OMySS1VJL5ojjlyCo0fR71mabE8rEjmGjmbkVkiDUm1kVUciEEiy8vk+ZZNTivXt5fVGo3eRxFOIJjOIUArqAGd1CHBhDg8Ayv8OY9ei/eu/cxby14+cwh/IH3+QMWypAC</latexit> \n<latexit sha1\\_base64="WCSj9oP0KJewIYFppBk55SEYOHE=">AAAB7HicbVBNS8NAEJ34WetX1aOXxSJ4KklF9Fj04rGCaQttKJvNpF262YTdjVBKf4MXD4p49Qd589+4bXPQ1gcDj/dmmJkXZoJr47rfztr6xubWdmmnvLu3f3BYOTpu6TRXDH2WilR1QqpRcIm+4UZgJ1NIk1BgOxzdzfz2EyrNU/loxhkGCR1IHnNGjZV8HglU/UrVrblzkFXiFaQKBZr9ylcvSlmeoDRMUK27npuZYEKV4UzgtNzLNWaUjegAu5ZKmqAOJvNjp+TcKhGJU2VLGjJXf09MaKL1OAltZ0LNUC97M/E/r5ub+CaYcJnlBiVbLIpzQUxKZp+TiCtkRowtoUxxeythQ6ooMzafsg3BW355lbTqNe+ydvVQrzZuizhKcApncAEeXEMD7qEJPjDg8Ayv8OZI58V5dz4WrWtOMXMCf+B8/gDqE47B</latexit> \nsideband would overwhelm the anti-Stokes sideband so that the pumping light is transferring energy to the mechanical degree of freedom [20, 47]. In our design protocol, the bluedetuned main interferometer is coupled to the blue-detuned filter cavity with the sloshing frequency w s , therefore this coupling will certainly affect the system dynamics and stability. In Fig. 3, we have seen that there could be unstable modes when PT-symmetry is broken. \nFIG. 8. Instabilities induced by the uncompensated resonant frequency drift introduced by the PT-symmetry breaking. (a) The Stokes sideband of the pumping light in the filter cavity is a bit offresonant to the main cavity resonance due to the optical spring effect w opt, which breaks the balance between the parametric and the sloshing process. (b) The Stokes sideband of the pumping light in the filter cavity is also an off-resonant due to the small shift of the main cavity resonance due to its coupling with the test mass, which also breaks the parametric-sloshing balance and creates instability. \n<!-- image -->', 'A. Instability induced by pondermotive effect': 'Introducing the pondermotive effect in the arm cavity means the coupling between the mechanical motion and the optical mode of the main interferometer, which will shift the optical resonance of the main interferometer [35]. If this frequency shift was left uncompensated, it will create a suppression of the Stokes sidebands of the pumping field in the filter cavity when being sloshed to the main interferometer (see Fig. 8). Therefore, the balance between the sloshing process and the parametric process is broken and instability forms. In Fig. 9, we plot the influence of the pondermotive effect on the trajectories of poles, where the trajectories detour near the detuning frequency W = D . The degenerate point at W = D under the PT-symmetric condition splits into two different points, \none of which is unstable. Such instability can be removed by carefully tuning the frequency of the pumping field. \nBesides, the pondermotive effect in the arm cavity also generates a dynamical back-action to the test mass mirrors, which is modified by the coupling to the auxiliary system as shown in Eq. (18). Using the parameter listed in Tab.I, these modified dynamics contribute a finite optical spring frequency around 7 . 7 Hz with an anti-damping factor equal to 2 p × 0 . 019 rad/s to the test mass. This 7 . 7 Hz optical spring resonance manifests as a dip around 7 . 7 Hz in the sensitivity curve Fig. 13. The anti-damping rate 2 p × 0 . 019 rad/s is well within the LIGO control band.', 'B. Instability induced by the off-resonant sidebands': 'After removing the instability near W = D induced by the arm cavity pondermotive effect by tuning the pumping field frequency, the instability introduced by the idler channel (i.e. the off-resonant sideband fields in the filter cavity) also needs to be removed. This instability exists because the introducing of these off-resonant idler fields in the filter cavity brings an optical spring that modifies the mechanical resonance frequency. The mechanism is shown in Fig. 8. The Stokes sidebands of the pumping field inside the filter cavity are off-resonant with the main interferometer optical resonance, thereby being suppressed when it is sloshed from the filter cavity to the main interferometer. However, the large bandwidth of the filter cavity indicates that such a small mechanical resonance shift does not affect the parametric process. This means that the parametric process will overwhelm the sloshing process and instability forms. Using the Hamiltonian approach, we can quantitatively compute the effect of the frequency compensation on the stability, the result shows that we need to carefully tune the compensation frequency near the w opt to stabilise the system. \nWe plot the poles trajectories of the modes ˆ a and ˆ c on the complex frequency domain in Fig. 10, when we tune the compensation frequency under the PT-symmetry condition w s = G . We found that these two modes can both be stable only when the compensation frequency is within a small frequency domain around w opt. \nThe above discussions demonstrate that the instabilities which happen around the detuning frequency W = D can be compensated by the careful tuning of the pumping frequency in the optomechanical filter cavity, while the optical spring instability at low frequency can be controlled by interferometer feedback servo system. For an overall verification, we also analyzed the stability using the Nyquist criteria [48], which is a conventional diagrammatic method for studying stability using the open-loop transfer function. In our analysis, the open-loop transfer function Go ( W ) is chosen to describe the response of both the main detuned signal-recycling interferometer and the pondermotive interaction in the optomechanical filter cavity, shown in Fig. 1, while the SRM treated as the feedback kernel. The close loop transfer function is therefore \n<!-- image --> \nFIG. 9. The influence of the pondermotive interaction inside the arm cavities on the resonance pole at high frequencies. Upper panel: the global pole trajectories plotted in the same way as in Fig. 3. Lower panel: zooming-in the pole trajectories near the detuning point W = D , where the degenerate root (when w s = G ) splits into two different poles and introduce instability. This instability can be removed by tuning the pumping field frequency in the filter cavity. The dashed line is the pole trajectory of the ideal case for comparison. \n<!-- image --> \ngiven by \nGc ( W ) = Go ( W ) 1 -r SRM Go ( W ) , (26) \nwhere those transfer functions are calculated using fulltransfer matrix approach without making the resolvedsideband and single-mode approximations (see Section V). The system is stable only if the contours generated by the \n<latexit sha1\\_base64="D9GgRd2WcMdsfeBzeCF9fsFbOJE=">AAAB+nicbVDLSsNAFL3xWesr1aWbwSK4sSRV0GXRjcsK9gFNKJPppB06k4SZiVJiP8WNC0Xc+iXu/BsnbRbaemDgcM693DMnSDhT2nG+rZXVtfWNzdJWeXtnd2/frhy0VZxKQlsk5rHsBlhRziLa0kxz2k0kxSLgtBOMb3K/80ClYnF0rycJ9QUeRixkBGsj9e2KN8IakbPMkwJxNqbTvl11as4MaJm4BalCgWbf/vIGMUkFjTThWKme6yTaz7DUjHA6LXupogkmYzykPUMjLKjys1n0KToxygCFsTQvMkFy9fdGhoVSExGYSYH1SC16ufif10t1eOVnLEpSTSMyPxSmHOkY5T2gAZOUaD4xBBPJTFZERlhiok1bZVOCu/jlZdKu19zzWv3uotq4LuoowREcwym4cAkNuIUmtIDAIzzDK7xZT9aL9W59zEdXrGLnEP7A+vwBpDSTmg==</latexit> \n<latexit sha1\\_base64="K9LZDf3zeDKTIbQr+DYwuZDRkVE=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Ae0oWy2m3btZhN2J0IJ/Q9ePCji1f/jzX/jts1BWx8MPN6bYWZekEhh0HW/ncLa+sbmVnG7tLO7t39QPjxqmTjVjDdZLGPdCajhUijeRIGSdxLNaRRI3g7GtzO//cS1EbF6wEnC/YgOlQgFo2ilVm9EkQT9csWtunOQVeLlpAI5Gv3yV28QszTiCpmkxnQ9N0E/oxoFk3xa6qWGJ5SN6ZB3LVU04sbP5tdOyZlVBiSMtS2FZK7+nshoZMwkCmxnRHFklr2Z+J/XTTG89jOhkhS5YotFYSoJxmT2OhkIzRnKiSWUaWFvJWxENWVoAyrZELzll1dJq1b1Lqq1+8tK/SaPowgncArn4MEV1OEOGtAEBo/wDK/w5sTOi/PufCxaC04+cwx/4Hz+ACDujtU=</latexit> \n<latexit sha1\\_base64="D9GgRd2WcMdsfeBzeCF9fsFbOJE=">AAAB+nicbVDLSsNAFL3xWesr1aWbwSK4sSRV0GXRjcsK9gFNKJPppB06k4SZiVJiP8WNC0Xc+iXu/BsnbRbaemDgcM693DMnSDhT2nG+rZXVtfWNzdJWeXtnd2/frhy0VZxKQlsk5rHsBlhRziLa0kxz2k0kxSLgtBOMb3K/80ClYnF0rycJ9QUeRixkBGsj9e2KN8IakbPMkwJxNqbTvl11as4MaJm4BalCgWbf/vIGMUkFjTThWKme6yTaz7DUjHA6LXupogkmYzykPUMjLKjys1n0KToxygCFsTQvMkFy9fdGhoVSExGYSYH1SC16ufif10t1eOVnLEpSTSMyPxSmHOkY5T2gAZOUaD4xBBPJTFZERlhiok1bZVOCu/jlZdKu19zzWv3uotq4LuoowREcwym4cAkNuIUmtIDAIzzDK7xZT9aL9W59zEdXrGLnEP7A+vwBpDSTmg==</latexit> \n<latexit sha1\\_base64="03IWuu9ZIBCuJwVNjL12X5FvHBQ=">AAAB+nicbVBNSwMxEM36WevXVo9egkXwYtmtgh6LXjxWsB/QXUo2TdvQJLsks0pZ+1O8eFDEq7/Em//GtN2Dtj4YeLw3w8y8KBHcgOd9Oyura+sbm4Wt4vbO7t6+WzpomjjVlDVoLGLdjohhgivWAA6CtRPNiIwEa0Wjm6nfemDa8FjdwzhhoSQDxfucErBS1y0FQwKYnGWBlljwEZt03bJX8WbAy8TPSRnlqHfdr6AX01QyBVQQYzq+l0CYEQ2cCjYpBqlhCaEjMmAdSxWRzITZ7PQJPrFKD/djbUsBnqm/JzIijRnLyHZKAkOz6E3F/7xOCv2rMOMqSYEpOl/UTwWGGE9zwD2uGQUxtoRQze2tmA6JJhRsWkUbgr/48jJpViv+eaV6d1GuXedxFNAROkanyEeXqIZuUR01EEWP6Bm9ojfnyXlx3p2PeeuKk88coj9wPn8AoRaTmA==</latexit> \n<latexit sha1\\_base64="03IWuu9ZIBCuJwVNjL12X5FvHBQ=">AAAB+nicbVBNSwMxEM36WevXVo9egkXwYtmtgh6LXjxWsB/QXUo2TdvQJLsks0pZ+1O8eFDEq7/Em//GtN2Dtj4YeLw3w8y8KBHcgOd9Oyura+sbm4Wt4vbO7t6+WzpomjjVlDVoLGLdjohhgivWAA6CtRPNiIwEa0Wjm6nfemDa8FjdwzhhoSQDxfucErBS1y0FQwKYnGWBlljwEZt03bJX8WbAy8TPSRnlqHfdr6AX01QyBVQQYzq+l0CYEQ2cCjYpBqlhCaEjMmAdSxWRzITZ7PQJPrFKD/djbUsBnqm/JzIijRnLyHZKAkOz6E3F/7xOCv2rMOMqSYEpOl/UTwWGGE9zwD2uGQUxtoRQze2tmA6JJhRsWkUbgr/48jJpViv+eaV6d1GuXedxFNAROkanyEeXqIZuUR01EEWP6Bm9ojfnyXlx3p2PeeuKk88coj9wPn8AoRaTmA==</latexit> \nFIG. 10. The poles trajectories of the ˆ a and ˆ c modes with the change of the compensation frequency, when we introduced the idler fields (after removing the instability at high frequency induced by pondermotive effect in the arm cavity). The arrows show the direction of compensation frequency increasing (under the condition of w s = G ). Using our sampling parameters, these two poles are both stable only when w comp ∈ w opt +[ 91 . 39Hz , 91 . 76Hz ] . The system will be unstable around 2900 Hz or 3100 Hz if the compensation is out of this region (corresponding to the red points and blue points). \n<!-- image --> \nNyquist map from the complex W -plane to the Gc ( W ) -plane do not encircle the origin. In Fig. 11, we plot the winding numbers of the Nyquist contour around the origin of the complex Gc ( W ) -plane when the above frequency tuning is performed. It finally tells us that the system is stable since the winding number to the zero point is zero in the case when we add an extra damping rate equal to 0 . 02 Hz to damp the test mass optical spring instability (using the sampling parameters given in Tab. I), under the condition that the filter cavity pumping frequency is carefully tuned.', 'V. FULL ANALYSIS OF AN OPTOMECHANICAL REALISATION USING EXACT TRANSFER MATRIX APPROACH': 'In the discussion above, we have used the following approximations: (1) single-mode approximation for each optical system so that W L arm / c glyph[lessmuch] 1 and 2 w mL SRC / c glyph[lessmuch] 1, (2) the resolved sideband approximation that allows to ignore the idler mode in the parametric process. In this section, we show the exact case, especially to make use of the ignored idler mode to compensate for the pondermotive degradation of sensitivity. Our numerical calculation of the sensitivity curve follows the standard transfer matrix approach, which was briefly summarised in [32]. \nThe method of Wiener filtering can be used to combine the signal and idler channels. The detailed derivation of this multi-channel Wiener filtering method is discussed in the appendix of [49]. Denoting the two Wiener filter functions as K 1 , 2 ( W ) , and the combined output ˆ y ( W ) can be written as: \nˆ y ( W ) = K 1 ( W ) ˆ ys ( W ) + K 2 ( W ) ˆ yi ( W ) , (27) \nwhere ˆ y s / i is the measured signal/idler field quadrature operator. These operators can be formally represented as: \nˆ ys = n s · a s + dsh , ˆ yi = n i · a i + dih , (28) \n<latexit sha1\\_base64="IsbWWcn7AobB6BJEGJIx2e8+NYQ=">AAAB7XicbVDLSgNBEJz1GeMr6tHLYBA8hd0o6DHoxZsRzAOSJcxOepMx81hmZoWw5B+8eFDEq//jzb9xkuxBEwsaiqpuuruihDNjff/bW1ldW9/YLGwVt3d29/ZLB4dNo1JNoUEVV7odEQOcSWhYZjm0Ew1ERBxa0ehm6reeQBum5IMdJxAKMpAsZpRYJzW7dwIGpFcq+xV/BrxMgpyUUY56r/TV7SuaCpCWcmJMJ/ATG2ZEW0Y5TIrd1EBC6IgMoOOoJAJMmM2uneBTp/RxrLQrafFM/T2REWHMWESuUxA7NIveVPzP66Q2vgozJpPUgqTzRXHKsVV4+jruMw3U8rEjhGrmbsV0SDSh1gVUdCEEiy8vk2a1EpxXqvcX5dp1HkcBHaMTdIYCdIlq6BbVUQNR9Iie0St685T34r17H/PWFS+fOUJ/4H3+AGB/jv8=</latexit> \n<latexit sha1\\_base64="FIDsOGZAjeEBDiHghHw//oCCJoY=">AAAB8nicbVBNS8NAEJ3Ur1q/qh69LBbBU0nqQY9FL56kgv2ANJTNdtMu3d3E3Y0QQn+GFw+KePXXePPfuG1z0NYHA4/3ZpiZFyacaeO6305pbX1jc6u8XdnZ3ds/qB4edXScKkLbJOax6oVYU84kbRtmOO0limIRctoNJzczv/tElWaxfDBZQgOBR5JFjGBjJf8ue0ztEiRwMqjW3Lo7B1olXkFqUKA1qH71hzFJBZWGcKy177mJCXKsDCOcTiv9VNMEkwkeUd9SiQXVQT4/eYrOrDJEUaxsSYPm6u+JHAutMxHaToHNWC97M/E/z09NdBXkTCapoZIsFkUpRyZGs//RkClKDM8swUQxeysiY6wwMTalig3BW355lXQade+i3rhv1JrXRRxlOIFTOAcPLqEJt9CCNhCI4Rle4c0xzovz7nwsWktOMXMMf+B8/gBS1ZFF</latexit> \n<latexit sha1\\_base64="bjpvkwzdoaFLFfvsKfIG+J2TxBo=">AAAB9XicbVBNS8NAEN3Ur1q/qh69BItQLyWpgh6LXrxZwX5AE8tmO22XbjZhd6KU0P/hxYMiXv0v3vw3btsctPXBwOO9GWbmBbHgGh3n28qtrK6tb+Q3C1vbO7t7xf2Dpo4SxaDBIhGpdkA1CC6hgRwFtGMFNAwEtILR9dRvPYLSPJL3OI7BD+lA8j5nFI304OEQkJa92xAG9LRbLDkVZwZ7mbgZKZEM9W7xy+tFLAlBIhNU647rxOinVCFnAiYFL9EQUzaiA+gYKmkI2k9nV0/sE6P07H6kTEm0Z+rviZSGWo/DwHSGFId60ZuK/3mdBPuXfsplnCBINl/UT4SNkT2NwO5xBQzF2BDKFDe32mxIFWVogiqYENzFl5dJs1pxzyrVu/NS7SqLI0+OyDEpE5dckBq5IXXSIIwo8kxeyZv1ZL1Y79bHvDVnZTOH5A+szx/U/ZIS</latexit> \n<latexit sha1\\_base64="OyTRCSMBQDtaB+ldeGzpUN/12Uk=">AAAB8nicbVBNSwMxEM3Wr1q/qh69BItQL2W3CnosetCbFawtbJeSTbNtaJJdklmhLP0ZXjwo4tVf481/Y9ruQVsfDDzem2FmXpgIbsB1v53Cyura+kZxs7S1vbO7V94/eDRxqilr0VjEuhMSwwRXrAUcBOskmhEZCtYOR9dTv/3EtOGxeoBxwgJJBopHnBKwkn/To9XunWQDctorV9yaOwNeJl5OKihHs1f+6vZjmkqmgApijO+5CQQZ0cCpYJNSNzUsIXREBsy3VBHJTJDNTp7gE6v0cRRrWwrwTP09kRFpzFiGtlMSGJpFbyr+5/kpRJdBxlWSAlN0vihKBYYYT//Hfa4ZBTG2hFDN7a2YDokmFGxKJRuCt/jyMnms17yzWv3+vNK4yuMooiN0jKrIQxeogW5RE7UQRTF6Rq/ozQHnxXl3PuatBSefOUR/4Hz+ADJgkIs=</latexit> \n<!-- image --> \nFIG. 11. The winding number: the system is stable with no poles on the upper side of the complex plane (i.e. the winding number is zero) when we add a damping rate of 0.02 Hz to compensate the optical spring instability. The contour in the complex W -plane is chosen to be the upper half-circle with a radius equal to 2 p × 2 × 10 4 rad/s. We homogeneously choose the data points in this half-circle contour and map it to the Gc ( W ) -plane. The winding turns are defined as q ( W ) / 2 p explained in the upper pannel, where the shape peaks/dips indicates a close-pass of the contour near the origin. \n<!-- image --> \nwhere n s / i and d s / i are the transfer functions for noise field a s / i =( a 1 s / i , a 2 s / i ) T and the GW signal h . \nOptimizing the sensitivity by varying these two filter functions leads to the minimum noise given as the inverse of the largest eigenvalue of the following matrix: \nM = N -1 · S . (29) \nwhere the noise and signal matrix are given by: \nN = [ n s S asas n † s n s S asa i n † i n i S a i as n † s n i S a i a i n † i ] , (30) \nand \nS = [ | ds | 2 dsd ∗ i did ∗ s | di | 2 ] . (31) \nThe combined sensitivity is shown in Fig. 12, where the Wiener filtering process favours the idler channel near 3 kHz. This combined sensitivity has a much larger bandwidth and \nsensitivity than the conventional configuration at around 3 kHz. Moreover, we also provide the behaviour of the sensitivity curve in a larger frequency band, say [ 1 , 10 4 ] Hz, in Fig. 13. Besides, the Wiener filtering combined sensitivity has almost no difference from the signal channel since the signal channel is favoured outside the frequency band around 3 kHz. The low-frequency peak corresponds to the optical spring effect due to the main interferometer detuning. \nFIG. 12. The quantum-noise-limited sensitivity around 3000 Hz obtained by the full transfer matrix simulation. Upper panel: The detector sensitivity obtained from the signal/idler channels and their Wiener-filtering combination. Lower panel: comparing the Wienerfiltering combined sensitivity curve with that of the conventional detector and idealized PT-symmetric configuration. \n<!-- image --> \nFurthermore, we also considered the effect of various losses in this protocol, such as the noise introduced by the optical loss in the signal recycling cavity (formed by iSRM, beamsplitter and the two ITMs in Fig. 1), the optical loss in the armcavity (which is attributed to the ETM loss) and the opticalloss/thermal noise in the optomechanical filter cavity. The corresponding sampling parameters are listed in Tab. I. The resultant noise budget is shown in Fig. 14, where the sensitivity is plotted in both the frequency range of 1 -10 4 Hz and around 3000 Hz. As also discussed in [51], the thermal noise in the filter cavity behaves similarly to the optical loss in the filter cavity in the broad frequency range, while a bit differently around the detuning frequency W = D due to the suppression of the filter\'s optical loss noise near the detuning frequency [26]. 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The quantum-noise-limited sensitivity obtained by the full transfer matrix simulation in frequency band [1,10 4 ] Hz. Upper panel: The detector sensitivity obtained from the signal/idler channels (phase quadrature) and their Wiener-filtering-combination. Lower panel: The detector sensitivity obtained from the signal/idler channels (amplitude quadrature) and their Wiener-filteringcombination. Note that the dashed line is almost overlapped with the signal channel except at around 3000 Hz, which can be hardly distinguished in the broad frequency band. \n<!-- image --> \nnoise components of the noise budget in Fig. 14 are plotted using the Wiener combination coefficient computed based on the total noise of idler and signal channels, and we also assumed an injection of 10 dB phase squeezing field.', 'VI. ASTROPHYSICAL IMPLICATIONS': 'GWobservations of binary neutron star (BNS) inspiral have already been achieved and various constraints on the equation of state (EoS) of neutron stars (NS) have been interpreted from the observation of GW170817 [54-59]. Boosting the sensitivity of GW detectors around the frequency of 3 kHz could open up the possibility of direct detection of the merger and post-merger GW signals of BNS merger events, with which the understanding of the EoS of dense matter at supranuclear densities could be pushed to a new level. \nBased on BNS merger simulations carried out by various groups, it is widely accepted that the fate of the post-merger \nTABLE I. Sample parameters for the detuned PT-symmetric gravitational wave detector. \n<!-- image --> \nFIG. 14. The noise budget of the detuned PT-symmetric gravitational wave detector, where the noise due to the signal recycling cavity loss, filter cavity loss, ETM loss and the thermal noise of the mechanical oscillator in the filter cavity is considered. The upper panel is the noise budget from 1 Hz to 10 4 Hz, while the lower panel is the noise budget around 3000 Hz. \n<!-- image --> \nremnant is determined by the ratio between the maximum mass of a cold non-rotating NS (i.e., M TOV) and the total mass \nFIG. 15. The sensitivity curves with GW signal emitted by neutron star merger remnants with two different equation of states. The black line is the sensitivity of the conventional detuned interferometer, while the blue line is the sensitivity of our protocol. These sensitivity curves are plotted using the parameters in Tab. I. The orange line is the BNS merger waveform of the APR4-q10-M1375 model calculated in [52] and the skyblue line is the waveform of 15H model listed in the data bank [53]. \n<!-- image --> \nof the binary (which could be accurately measured by inspiral GW signal) [60]. Therefore, simply a non-/detection of the GW signal from the post-merger remnant could tell that the remnant experiences prompt/delayed-collapse to a black hole (BH). This already allows for an independent constraint on M TOV. Furthermore, it has been found that the post-merger GW signals could be used to constrain crucial properties of the merging NSs such as compactness and tidal deformability [9, 11, 61-63]. However, the most relevant frequency range of the post-merger GW signals lies in the range from 2 kHz to 3 kHz, which is too high for the current generation GW detectors to resolve even for close sources such as GW170817. Our design could make it possible for a measurement of the frequency peaks in the post-merger phase and hence make complementary constraints on NS properties. \nMoreover, the density of the BNS merger remnant could be several times higher than the inspiral stage. Strong interaction phase transition is suggested to be possible under such conditions and could leave detectable features in the post-merger GW signals [64, 65]. Compared the cases without a phasetransition, the peak frequency of the post-merger GW is found to shift to a even higher value. Having a detector with a sufficiently broad sensitivity and frequency resolution at roughly 3 kHz could identify such a shift and constrains the density range of the strong interaction phase transition, see Fig. 15.', 'VII. DICUSSION AND CONCLUSION': "In this work, we provided an alternative design protocol based on the PT-symmetry to the high-frequency gravitational \nwave detector, targeted at the physics of binary neutron star coalescence. We have analyzed in detail the optomechanical realization of a detuned PT-symmetric interferometer, targeted at improving the sensitivity to the kilo-Hertz GWs using single-mode approximation and full transfer matrix simulation. The effects of the idler field and pondermotive interactions are analyzed in the single-mode approximation, which provides physical insight for understanding the sensitivity curves obtained using transfer matrix simulation. We showed that using the same parameter setting as in the tuned case, after (1) compensating the optical spring in the optomechanical quantum amplifier and (2) the Wiener-filtering combination of the measurement data obtained from both signal channel and idler channel, we can in principle achieve a significant boost of the sensitivity around 3 kHz. This sensitivity is 10-times better than the conventional design. The dynamical instability of the system is induced by the optical spring effect, which can be controlled in the detector feedback servo system. Our work focuses on the conceptual designs of this protocol while leaving the more practical and technical con- \n- * myqphy@hust.edu.cn\n- [1] L. S. Collaboration and V. Collaboration (LIGO Scientific Collaboration and Virgo Collaboration), Phys. Rev. Lett. 119 , 161101 (2017).\n- [2] J. Aasi, B. Abbott, R. Abbott, T. Abbott, M. Abernathy, K. Ackley, C. Adams, T. Adams, P. Addesso, R. 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2024arXiv240912009Z

Accurately characterizing the true redshift truez distribution of a photometric redshift photoz sample is critical for cosmological analyses in imaging surveys. Clusteringbased techniques which include clusteringredshift CZ and selfcalibration SC methodsdepending on whether external spectroscopic data are usedoffer powerful tools for this purpose. In this study we explore the joint inference of the truez distribution by combining SC and CZ denoted as SCCZ. We derive simple multiplicative update rules to perform the joint inference. By incorporating appropriate error weighting and an additional weighting function our method shows significant improvement over previous algorithms. We validate our approach using a DES Y3 mock catalog. The truez distribution estimated through the combined SCCZ method is generally more accurate than using SC or CZ alone. To account for the different constraining powers of these methods we assign distinct weights to the SC and CZ contributions. The optimal weights which minimize the distribution error depend on the relative constraining strength of the SC and CZ data. Specifically for a spectroscopic redshift sample that amounts to 1 of the photoz sample the optimal combination reduces the total error by 20 40 compared to using CZ SC alone and it keeps the bias in mean redshift Delta barz 1 z at the level of 0.3. Furthermore when CZ data is only available in the lowz range and the highz range relies solely on SC data SCCZ enables consistent estimation of the truez distribution across the entire redshift range. Our findings demonstrate that SCCZ is an effective tool for constraining the truez distribution paving the way for clusteringbased methods to be applied at zgtrsim 1.

2024-09-01T00:00:00Z

['arXiv:2409.12009', '10.48550/arXiv.2409.12009', '2024arXiv240912009Z']

['Astrophysics - Cosmology and Nongalactic Astrophysics', 'Astrophysics - Astrophysics of Galaxies']

Optimizing Redshift Distribution Inference through Joint SelfCalibration and ClusteringRedshift Synergy

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.12009.pdf

{'Optimizing redshift distribution inference through joint self-calibration and clustering-redshift synergy': 'Weilun Zheng 1,2 , Kwan Chuen Chan ⋆ 1,2 , Haojie Xu 3,4,5 , Le Zhang 1,2,6 , and Ruiyu Song 1,2 \n- 1 School of Physics and Astronomy, Sun Yat-Sen University, 2 Daxue Road, Tangjia, Zhuhai 519082, China\n- 2 CSST Science Center for the Guangdong-Hongkong-Macau Greater Bay Area, SYSU, Zhuhai 519082, China\n- 3 Shanghai Astronomical Observatory, Chinese Academy of Sciences, Nandan Road 80, Shanghai 200240, China\n- 4 Department of Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China\n- 5 Key Laboratory for Particle Astrophysics and Cosmology (MOE)/Shanghai Key Laboratory for Particle Physics and Cosmology, China\n- 6 Peng Cheng Laboratory, No.2, Xingke 1st Street, Shenzhen 518000, China', 'ABSTRACT': 'Context. Accurately characterizing the true redshift (truez ) distribution of a photometric redshift (photoz ) sample is critical for cosmological analyses in imaging surveys. Clustering-based techniques, which include clustering-redshift (CZ) and self-calibration (SC) methods-depending on whether external spectroscopic data are used-offer powerful tools for this purpose. \nAims. In this study, we explore the joint inference of the truez distribution by combining SC and CZ (denoted as SC+CZ). \nMethods. We derived simple multiplicative update rules to perform the joint inference. By incorporating appropriate error weighting and an additional weighting function, our method shows significant improvement over previous algorithms. We validated our approach using a DES Y3 mock catalog. \nResults. The truez distribution estimated through the combined SC+CZ method is generally more accurate than using SC or CZ alone. To account for the different constraining powers of these methods, we assigned distinct weights to the SC and CZ contributions. The optimal weights, which minimize the distribution error, depend on the relative constraining strength of the SC and CZ data. Specifically, for a spectroscopic redshift sample that amounts to 1% of the photoz sample, the optimal combination reduces the total error by 20% (40%) compared to using CZ (SC) alone, and it keeps the bias in mean redshift [∆¯ z/ (1 + z )] at the level of 0.003. Furthermore, when CZ data are only available in the lowz range and the highz range relies solely on SC data, SC+CZ enables consistent estimation of the truez distribution across the entire redshift range. \nConclusions. Our findings demonstrate that SC+CZ is an effective tool for constraining the truez distribution, paving the way for clustering-based methods to be applied at z ≳ 1. \nKey words. cosmology: observations - gravitational lensing: weak - galaxies: photometry - surveys', '1. Introduction': "Wide-area imaging surveys provide powerful cosmological probes to constrain cosmology. Weak lensing is a prime example (Bartelmann & Schneider 2001; Heymans et al. 2013; Hildebrandt et al. 2017; Troxel et al. 2018; Asgari et al. 2021; Hikage et al. 2019; Amon et al. 2022; Secco et al. 2022; Li et al. 2023; Dalal et al. 2023). In particular, cosmic shear is often bundled in 3 × 2 point analysis, which includes cosmic shear, galaxy-galaxy lensing, and galaxy clustering (Abbott et al. 2018; Abbott et al. 2022a; Heymans et al. 2021; Miyatake et al. 2023; Sugiyama et al. 2023). These analyses offer a strong constraint on S 8 , and the tightest uncertainty level ( ∼ 2%) is already comparable to that from the Planck Satellite's study of the cosmic microwave background (Planck Collaboration et al. 2020). Because the lensing results are consistently lower than those from the cosmic microwave background, there is heated discussion regarding the S 8 tension. Moreover, imaging surveys also enable measurements of the transverse baryon acoustic oscilla- \nns (BAO) (Padmanabhan et al. 2007; Estrada et al. 2009; Hutsi 2010; Seo et al. 2012; Carnero et al. 2012; de Simoni et al. 2013; Abbott et al. 2019, 2022b; Chan et al. 2022; Abbott et al. 2024; Song et al. 2024). The latest transverse BAO measurements, such as Abbott et al. (2024), are yielding constraints competitive with those from spectroscopic surveys (DESI Collaboration et al. 2024). Upcoming stage IV surveys, such as from the Rubin Observatory Legacy Survey of Space and Time (LSST) (Ivezi'c et al. 2019; Mandelbaum et al. 2018), Euclid (Laureijs et al. 2011), Chinese Space Station Telescope (CSST) (Zhan 2011; Gong et al. 2019), and Roman Space Telescope (WFIRST) (Spergel et al. 2015; Eifler et al. 2021), are expected to deliver even more enormous photometric data and hence more exquisite results. \nIn imaging surveys, the redshifts (photoz 's) are derived from the photometry information measured from a few broadband filters. The template fitting and training methods are commonly used to infer the photoz 's of the galaxies (see Salvato et al. (2019); Newman & Gruen (2022) for a review). The template fitting method [e.g. Arnouts et al. \n(1999); Bolzonella et al. (2000); Ben'ıtez (2000); Ilbert et al. (2006)] fits a model derived from known spectral energy distribution SED) templates, which includes the photoz as a fitting parameter, to the color or magnitude data. The prior information can also be included in the fitting. This method is limited by the accuracy and representativeness of the templates and the accuracy of the prior information. Training methods often make use of tools from machine learning (Collister & Lahav 2004; Sadeh et al. 2016; De Vicente et al. 2016; Zhou et al. 2021; Li et al. 2022). A spectroscopic redshift (specz ) sample is required to train the machine learning algorithm, so its accuracy depends on the abundance and the completeness or representativity of the specz sample. \nIn cosmological applications, accurate true redshift (truez ) distribution of the photoz sample must be obtained to avoid biasing the cosmological results. Using a weighted specz sample is a viable approach. The effectiveness of this method hinges on the availability of the specz sample or sometimes a high-quality photoz sample with numerous photoz bands. The weighted specz sample can be constructed using a k -nearest neighbor search (Lima et al. 2008; Cunha et al. 2009; Bonnett et al. 2016) or a self-organizing map (Carrasco Kind & Brunner 2014; Masters et al. 2015; Buchs et al. 2019; Wright et al. 2020; Campos et al. 2024). \nThe clustering-based methods provide an independent way to calibrate the truez distribution of the photoz sample. Unlike photometry-based methods, this approach uses the clustering information, which can be traced back to gravity. Depending on the utilization of the external specz sample, it can be further categorized into the clusteringz (hereafter CZ) and self-calibration (SC) methods. In CZ, the truez distribution is determined by cross correlating the photoz sample with an external specz sample (Newman 2008; Matthews & Newman 2010; McQuinn & White 2013; M'enard et al. 2013; Schmidt et al. 2013; Morrison et al. 2017; van den Busch et al. 2020). This requires the specz sample to overlap with the photoz one spatially, but the specz sample does not need to be representative. CZ has been routinely used to calibrate the truez distribution in real data [e.g., Gatti et al. (2018, 2022); Cawthon et al. (2022); Hildebrandt et al. (2021); Rau et al. (2023)]. On the other hand, the SC method relies solely on the clustering information, both the auto and cross bin correlation function, of the photometric sample itself. (Schneider et al. 2006; Zhang et al. 2010; Benjamin et al. 2010; Zhang et al. 2017; Peng et al. 2022; Xu et al. 2023). Although SC has also been used for weak lensing (Benjamin et al. 2013) and BAO (Song et al. 2024) measurements, it is less frequently used than CZ. This may be because the redshift range explored in current surveys are still relatively low ( z ≲ 1), and hence the specz sample is still sufficient for the calibration purpose. However, even in present surveys, the high redshift portion of the sample cannot be calibrated using CZ due to the absence of specz galaxies at high redshifts [e.g., Rau et al. (2023); Abbott et al. (2024)]. Consequently, we expect the SC method to play a more prominent role in upcoming surveys. \nIn this paper we explore combining the information of CZ and SC to simultaneously constrain the truez distribution of a photoz sample. As far as we know, this is the first time that these two methods have been jointly applied to constrain the truez distribution in realistic mock data \n[see the Fisher forecast in McQuinn & White (2013)]. We anticipate that this synergy in redshift calibration is particularly fruitful in the high redshift regime, where the specz sample is scarce. The rest of the paper is organized as follows. We first review the SC and CZ formalism in Sec. 2.1 and then present the algorithm to jointly solve the SC and CZ equations in Sec. 2.2. We test our method using a DES Y3 mock catalog in Sec. 3. In particular, we contrast the truez inference results from SC, CZ, and SC +CZ, and we demonstrate that our improved algorithm is superior to the old algorithm. Moreover, we study the scenarios when the number of specz bins is equal to and less than the number of photoz bins respectively, and show that SC+CZ can effectively extend the clustering-based method to a higher redshift. We conclude in Sec. 4. In Appendix A, we present the derivation of the update rules. We investigate the impact of outlying specz galaxies in Appendix B and the impact of setting the negative measurements to a tiny positive value in Appendix C.", '2.1. Derivation of the SC and CZ equations': "In this subsection, we show how the correlation function of the photoz sample and the cross correlation function between the photoz and specz sample are related to the truez distribution. Our convention follows that of Song et al. (2024), and the meaning for the key notations used in Sec. 2.1 are summarized in Table 1. \nBecause of photoz uncertainties, galaxies within a photoz bin may originate from multiple specz bins: \nM i ' = ∑ k Q ki ' M k , (1) \nwhere M is the angular galaxy number density, Q ki ' represents the probability that a galaxy in specz bin k leaks into photoz bin i ' . The index with (without) a prime denotes the photoz (specz ) bin index. \nThe correlation function of the photoz sample can be written as \n⟨ M i ' M j ' ⟩ = ∑ k,l ⟨ Q ki ' M k Q lj ' M l ⟩ . (2) \nBy expressing M i ' = ¯ M i ' (1 + δ i ' ) and M i = ¯ M i (1 + δ i ) in terms of their respective means ¯ M and fluctuations δ , we can write the angular overdensity correlation function as \nC i ' j ' ≡ ⟨ δ i ' δ j ' ⟩ = ∑ k P ki ' P kj ' C ' kk , (3) \nwhere C ' kk = ⟨ δ k δ k ⟩ is the specz angular overdensity correlation function of the photoz sample in the specz bin k . We have defined P ki ' as \nP ki ' = Q ki ' ¯ M k ¯ M i ' . (4) \nFrom Eq. (4), it is evident that P ki ' satisfies the normalization condition \n∑ k P ki ' = 1 . (5) \nTable 1. Summary of the meaning of the key notations used in Sec. 2.1. \nThus P ki ' represents the fraction of galaxies in photoz bin i ' coming from specz bin k . \nBy expressing M i ' and M i in Eq. (1) in terms of their fluctuations and then invoking Eq. (5), we get the following relation \nδ i ' = ∑ j P ji ' δ j , (6) \nwhich is the analogous relation to Eq. (1). In fact, P ji ' is the truez distribution of the photoz sample, which is often denoted as n ( z ). In this paper, we use these two notations interchangeably. \nIn arriving at Eq. (3), we have assumed that C ' ij is diagonal. In other words, the correlation between specz bins is non-zero only when they are within the same bin. This is an excellent approximation and it is exact under Limber approximation [e.g. Simon (2007)]. Eq. (3) is the essence of the SC method, and it enables us to extract P ki ' using the clustering information of the photoz data alone (Schneider et al. 2006; Zhang et al. 2010). \nAssuming linear galaxy bias, we can further write Eq. (3) in terms of the underlying specz matter power spectrum C m kk as \nC i ' j ' = ∑ k P ki ' P kj ' b ' 2 k C m kk , (7) \nwhere b ' k is the bias parameter of the photoz sample in k th specz bin. \nFrom Eqs. (1) to (3), we implicitly assume that the leakage due to photoz is universal irrespective of the galaxy type composition in the specz bin. To illustrate this point, let us assume that there are two types of galaxies in a specz bin, say blue and red galaxies. The universal leakage assumption asserts that both types of galaxies leak to different photoz bins in the same proportion. Given photoz estimation is based on the photometry information, which is closely related to the galaxy types, this assumption can only hold approximately. \nFor the present work, the violation of the universal leakage assumption manifests as the dependence of the bias parameter on the photoz bin index i ' . Without the universal \nleakage assumption, Eq. (7) would be generalized to \nC i ' j ' = ∑ k P ki ' P kj ' b ki ' b kj ' C m kk , (8) \nwhere b ki ' is the bias parameter of the i ' th photoz sample in the k th specz bin. In this work, we assume that the leakage is universal. \nSuppose now we have another specz sample with number density in specz bin i , N i , which is related to its mean number density ¯ N i and density fluctuation ϵ i by the relation N i = ¯ N i (1 + ϵ i ). We note that N i and ϵ i are in general different from the corresponding quantities for the photoz sample in specz bin, M i and δ i . The specz sample is often much less abundant than the photoz sample, and it tends to be brighter, and hence the linear bias of the specz sample is likely to be higher. \nThe cross correlation function between the specz and photoz sample reads \n⟨ N i M j ' ⟩ = ∑ k Q kj ' ⟨ N i M k ⟩ . (9) \nSimilar to the derivation of the SC results, we have the corresponding CZ equation: \nC ij ' ≡ ⟨ ϵ i δ j ' ⟩ = P ij ' C x ii (10) = P ij ' b i b ' i C m ii , (11) \nwhere C x ii = ⟨ ϵ i δ i ⟩ is the cross angular correlation function between the specz sample in bin i with the photoz sample in specz bin i , and b i is the bias parameter of the specz sample in i th specz bin. For completeness, without the universal leakage assumption, we would instead have \nC ij ' = P ij ' b i b ij ' C m ii . (12) \nUnlike the SC case, which is quadratic in P , CZ is a linear problem. In the usual CZ method, the bias parameter of the specz sample, b j can be measured easily but it is difficult to get b ' j . Thus CZ only directly constrains P ij ' b ' i if we assume some theoretical matter correlation function model. Moreover, since P ij ' is normalized, if b ' i is a constant, it has no effect on the estimation of P ij ' . However, if b ' i evolves \nwith redshift (or index i ), its evolution is degenerate with P ij ' . Indirect methods have been proposed to mitigate the impact of bias evolution (Schmidt et al. 2013; Davis et al. 2018; van den Busch et al. 2020; Cawthon et al. 2022; Gatti et al. 2022). From Eq. (7), it is clear that the bias evolution issue also affects the SC problem. When we allow the correlation function in the specz bin to be a free parameter (called P method below), the degree of freedom due to bias evolution is taken into account. \nWe shall test different approaches to solve the system of SC and CZ equations. First we regard P ij ' , C x ii , and C ' ii as unknowns, denoted as the P method. This is the most general parameterization, but the constraint is slightly weakened. Similar approach is taken in SC, e.g. Zhang et al. (2017). On the other hand, analogous to the approach in CZ, we take R ij ' ≡ P ij ' b ' i as the variable and abbreviate it as the R method. We note that there is no normalization constraint on R ij ' . If we assume that b i is known from measurement and C m kk can be computed theoretically, then we only need to solve for R in the system: \nC ij ' = R ij ' b i C m ii , (13) \nC i ' j ' = ∑ k R ki ' R kj ' C m kk . (14) \nAs a side note, in terms of the variable R ij ' ≡ P ij ' b ij ' , it is easy to see that the non-universality issue does not introduce extra complications in the case of CZ.", '2.2.1. Cost function and multiplicative update rules': "Our goal is to develop an efficient method to solve Eq. (3) and (10) simultaneously for P ij ' , C ' ii , and C x ii . Here we use the P method as an example and comment on the R method later on. In particular, because Eq. (3) is a system of coupled quadratic equations, it is challenging to solve. Several studies have been devoted to solving the SC equation (Erben et al. 2009; Benjamin et al. 2010, 2013; Zhang et al. 2017). Unlike prior researches, Zhang et al. (2017) solves Eq. (3) in full generality, without limiting the analysis to a two-bin scenario (Erben et al. 2009) or relying on a linear coupling (in P ) approximation as done by Benjamin et al. (2010). In this work, inspired by Zhang et al. (2017), we construct multiplicative update rules to solve Eq. (3) and (10) jointly. \nWe aim to derive an iterative update rule to minimize the sum of the SC and CZ cost functions: 1 \nJ = J 1 + J 2 , (15) \nwhere J 1 and J 2 are the contributions from SC and CZ respectively. They are given by \nJ 1 = 1 2 ∑ i ' ,j ' ,µ W 1 ( θ µ ) σ 2 i ' j ' ( θ µ ) [ D i ' j ' ( θ µ ) -∑ k P ki ' P kj ' C ' kk ( θ µ ) ] 2 , (16) \nJ 2 = 1 2 ∑ i,j ' ,µ W 2 ( θ µ ) σ 2 ij ' ( θ µ ) [ D ij ' ( θ µ ) -P ij ' C x ii ( θ µ ) ] 2 , (17) \nwhere D denotes the data measurement (angular correlation function in our case), σ is the error bar of the measurement, and W 1 and W 2 are the weight functions. We consider weight function of the form θ n in this work [see also M'enard et al. (2013)]. Here we use Latin indices for the redshift bins and Greek indices for the angular bins. The update rule will update P ij ' , C ' ii , and C x ii iteratively to look for a minimum of J . Additionally, we will consider assigning different weights to J 1 and J 2 later on. \nBy minimizing the cost function with respect to P ab ' , a multiplicative update rule [Eq. (A.9)] for P ab ' can be derived. Upon multiplying the factor in Eq. (A.9) to P ab ' repeatedly, P ab ' converges to the solution. This method can be viewed as a variant of the gradient decent (Lee & Seung 2000). Similar update rule can also be established for C x ii and C ' ii [Eq. (A.10)]. The details of the derivation are relegated to Appendix A. A few comments of the multiplicative rules are in order. \nFirst, Zhang et al. (2017) employed the non-negative matrix factorization (NMF) algorithm, originally proposed by Lee & Seung (2000), to address the SC problem. This approach has been adopted in subsequent studies (Peng et al. 2022; Xu et al. 2023; Song et al. 2024; Peng & Yu 2024). Due to the non-negativity constraint inherent in the SC model, the NMF method offers an elegant and efficient solution. NMF assumes that the model can be factorized into a product form WH , where W and H are distinct matrices. To apply this framework to the SC model, Zhang et al. (2017) reformulated it as WH θ , with W = P T and H θ = C ( θ ) P . However, this splitting is somewhat artificial for the SC problem and may cause difficulty in including the CZ information due to its rigid structure. To further optimize and generalize the approach for solving the SC model, we bypass the NMF interpretation and instead treat it directly as a minimization problem with respect to P ab ' . By employing a specialized parameter update scheme inspired by the NMF approach, our simplified interpretation allows for an efficient combination of information from both SC and CZ models. \nSecondly, the cost function for the multiplicative update rule is often taken to be the mean squared error without the inverse error bar weighting, e.g. Lee & Seung (2000); Zhang et al. (2017). Treating all the data on equal footing is at best sub-optimal. We have to include the error bar to down-weight the contribution of the poor measurements and up-weight the good ones, and to eliminate the impact of missing data measurements. Here we have improved over previous treatments by taking into account the error of measurements and the inclusion of additional weighting functions. The proper treatment should include the full covariance matrix. However, in the derivation of the multiplicative rule, we have to separate the positive part of the gradient from the negative one. Usage of the full covariance will hinder this separation. Thus for simplicity, we opt to use diagonal error bars in Eqs. (16) and (17). Under the diagonal covariance approximation, the data points are taken to be independent. This may seem a poor approximation, but we observed that the overall performance of the algorithm is good. We mention that Xu et al. (2023) used χ 2 as the stopping criterion of the iteration process, but the update rule is still based on the cost function without the inverse error weighting. Recently, Peng & Yu (2024) independently proposed to include error weighting in the NMF algorithm to solve the SC problem. They adopted \nthe improved NMF algorithm of Zhu (2016); Green & Bailey (2023), which generalizes the Lee & Seung (2000) NMF update rule to account for the data measurement error. We expect its performance to be similar to our SC results. \nThirdly, the multiplicative update rule ensures that the estimated values are non-negative provided that the measurements are non-negative 2 . This is less likely to hold for C ij ' (see Fig. 1 below) due to its larger covariance. Negative measurements may spoil the update rule and result in negative estimated value. To mitigate this problem, in Xu et al. (2023), negative measurements are set to small positive values by hand, and a more refined process is applied in Peng & Yu (2024). In Appendix C, we test the impact of setting the negative measurements to a tiny positive value. \nFinally, we mention that our update rule is also applicable when J 1 or J 2 are missing. In particular it can be used for the SC method. We will contrast ours against the results from Xu et al. (2023) below. \nBefore closing this section, we comment on the solution of the system Eqs. (13) and (14) in the R method. Basically it is the same as the P method, except that R ij ' is solved without the normalization constraint, viz. Eq. (A.15). From R ij ' , if we assume that there is no bias evolution, then using Eq. (5), we have 3 \nb ' i ≈ b ' = ∑ i R ij ' , (18) \nand it follows that \nP ij ' = R ij ' ∑ k R kj ' . (19)", '2.2.2. Numerical implementation': "Because our method is rooted in the previous works and we shall contrast the old result with ours, it is helpful to review the algorithm of Zhang et al. (2017) and its improvements in Xu et al. (2023). We shall denote this as the 'Old NMF' and use it as a benchmark for comparison. \nThe algorithm starts by initializing P ij ' randomly as a diagonally dominant matrix. The initialization procedure assumes that the truez distribution peaks at the photoz estimate and decreases monotonically on both sides. The initial C ' ii and C x ii are obtained from Eqs. (3) or (10) using the initial P ij ' . Before applying the multiplicative update rule, Zhang et al. (2017) found that it was necessary to get a preliminary solution using a fixed-point method for the NMF step to be successful. Using this preliminary solution as the initial trial solution, the NMF update rule is applied until a minimum of the cost function is attained. Zhang et al. (2017) writes the SC model as WH θ with W = P T and H θ = C ( θ ) P . In each step, the update rule analogous to that in Lee & Seung (2000) is applied to W with H θ fixed. The P in H θ is then replaced with the new W , and with P held fixed C ( θ ) is subsequently solved by the least square solution. To estimate the error bar on P ij ' , Xu et al. (2023) generates 100 sets of P ij ' and for each set, the measurement C i ' j ' is perturbed by adding a Gaussian perturbation \nobtained by sampling the covariance of C i ' j ' . We call this set random runs to distinguish them from the mock realizations. The best fit and the 1 σ error are estimated by the median and the half width between the 16 and 84 percentile, respectively. \nWe shall contrast three different setups: SC, CZ, and the joint inference by SC and CZ, denoted as SC+CZ. For SC and SC+CZ, the default method uses P ij ' and C ' ii (and C x ii ) as unknowns (abbreviated as P method), while for CZ, we use R ij ' as variables (denoted as R method). The reason for these differences is related to the error estimation and we shall comment on it later on. For the P method, we initialize P ij ' and C ' ii (and C x ii ) as in the Old NMF, but we then feed the initial guess to the multiplicative update rule [Eqs. (A.9) and (A.10)] directly because we find that our algorithm no longer requires the intermediate solution from the fixed point method. We take the weighting function to be θ n with n = -1. Similar to Xu et al. (2023), we generate 100 random runs to estimate the best fit and 1 σ error bar. The best fit and 1 σ error bar are estimated by the median and half width of the 68-percentile about the median. In each run, we also perturb the measurement using a Gaussian perturbation derived from the mock covariance. \nFor the R method, the procedures are similar. To initialize R ij ' we generate the initial P ij ' as above, but we find that the result is sensitive to the trial b ' i . Thus we perform a grid search on b ' i to find the one that gives rise to the minimum J and use it as the final solution. We then get P ij ' from the best fit R ij ' using Eq. (19).", '3. Mock test results': 'In this section, we apply our algorithm to the mock catalog to test its performance.', '3.1. Description of the mock catalog': "We shall test and validate our results using mock catalogs. For this purpose, we employ the ICE-COLA mocks (Ferrero et al. 2021), which is a specialized mock catalog tailored for the DES Y3 BAO analyses (Abbott et al. 2022b). A brief overview is provided here, with further details available in Ferrero et al. (2021). \nThe ICE-COLA mocks are derived from the COLA simulations, built upon the COLA method (Tassev et al. 2013), and executed through the ICE-COLA code (Izard et al. 2016). The COLA method combines the second-order Lagrangian perturbation theory with the particle-mesh simulation technique, ensuring the preservation of accuracy in large-scale modes despite the utilization of coarse simulation time steps. In each simulation, there are 2048 3 particles in a cube measuring side length of 1536 Mpc h -1 . The comoving simulation is transformed to a lightcone simulation extending up to z ∼ 1 . 4. The cosmology adopted by the mock catalog follows that in the MICE simulation (Fosalba et al. 2015; Crocce et al. 2015), which is a flat ΛCDM with Ω m = 0 . 25, Ω Λ = 0 . 75, h = 0 . 7, and σ 8 = 0 . 8. Each mock occupies a DES Y3 footprint, covering 4180 deg 2 in area. We make use of 100 mock catalogs to estimate the covariance. The same set of mocks are also used to estimate the ensemble mean and its associated error. \nThe mock galaxies are allocated to the ICE-COLA halos through a hybrid method combining Halo Occupation \nCi \nCij \nFig. 1. Sample of the photoz angular correlation function between the photoz bin i ' and j ' , C i ' j ' (blue) and the cross angular correlation function between the specz bin i and photoz bin j ' , C ij ' (red) to be used for truez inference. The line and its associated color band are the median and 16 and 84 percentile among 100 mock runs. The whole redshift range [0.6,1.1] is divided into 10 photoz bins, each of width ∆ z p = 0 . 05. This results in a truez distribution with resolution of ∆ z = 0 . 05. The label i -j represents i ' j ' for photoz correlation function and ij ' for specz -photoz cross correlation function respectively. We only show the odd bin results for clarity. \n<!-- image --> \nDistribution and Halo Abundance Matching, mirroring the technique outlined in Avila et al. (2018). These mock galaxies resemble the red galaxy sample in DES Y3 (see Carnero et al. (2012) for more details on this sample). The mock galaxies are equipped with realistic photoz . To do so, a 2D joint probability distribution in the photoz and specz space is constructed using a sample of actual galaxies possessing both types of redshifts. Through sampling this distribution, the candidate mock galaxies are assigned the appropriate photoz 's. \nBecause the galaxies have both photoz and specz labels, we can use them to create photoz and specz samples. As in DES Y3, we select galaxies in the photoz range [0.6,1.1] and divide them into five tomographic bins, each of width 0.1. Our goal is to determine the truez distribution of the samples in these tomographic bins using clusteringbased methods. To model the fact that the specz galaxies are generally few and bright, we approximate it by selecting the most massive galaxies from the mock. In this case we expect the clustering amplitude of the specz sample to be higher than the intrinsic clustering of the photoz sample since the linear bias increases with mass. We consider samples containing the most massive p % of the galaxies with p = 5, 1, and 0.5. \nWhen we restrict the sample to the photoz range [0.6,1.1], a small fraction of the galaxies will possess specz \nvalues outside this range. This implies that there are reductions in correlation signal from SC and CZ. This problem can be alleviated by considering the redshift range of data, both photoz and specz data, to be sufficiently wide relative to the redshift range of interest. Because the mock data are only available in the photoz range [0.6,1.1], this is not possible. Here we clean the sample by removing the galaxies with specz values outside of the range [0.6,1.1], and this effectively forces the truez distribution to vanish outside the range [0.6,1.1]. This cleaning cannot be done in real data. In Appendix B, we study the impact of the outlying galaxies on the truez estimation.", '3.2. Clustering measurements': "The data for inference are the angular correlation functions. While the bin width of the target photoz sample is ∆ z p = 0 . 1, to increase the redshift resolution of the truez distribution, we divided the sample into ten photoz bins, each with bin width of ∆ z p = 0 . 05. From the truez inference code, we ended up with ten truez distributions with a redshift resolution of ∆ z = 0 . 05 for these ten photoz samples. We then combined every two truez distributions to get the weighted mean for our target photoz samples, \n) \nz \n( \nn \nFig. 2. Truez distribution inferred by the clustering-based estimators [CZ (red), SC (orange), and SC+CZ (blue)] are compared with the direct mock measurement (green bars). The clustering-based estimator data points are offset horizontally for clarity. The results for five tomographic bins are shown (from top to bottom). \n<!-- image --> \nFig. 3. Comparison of the accuracy of the truez distribution inferred using CZ (red), SC (orange), and SC+CZ (blue). Upper panel : ∑ i | ˆ P ij ' -P true ij ' | , the absolute difference between the truez distribution measured from the mock, P true ij ' and the one estimated, ˆ P ij ' . Lower panel : ∑ i | ˆ P ij ' -P true ij ' | /σ P ij ' the absolute difference normalized by the estimated error σ P ij ' . The line and color band represent the median and the 16 and 84 percentile among 100 mock runs. The results for five tomographic bins are shown. \n<!-- image --> \nTable 2. Bias in the mean redshift e z obtained with various methods. \nNotes. The bias in the mean redshift e z , which is given by the difference between the mean redshift computed with the truez distribution from various estimators ¯ z ˆ n and the direct mock measurement ¯ z true , and further divided by 1+ ¯ z true (and multiplied by 10 3 ). We show the results for the 5 tomographic bins. The ensemble mean and the 1 σ are estimated from 100 mock runs. Although in some of the bins, CZ gives a smaller bias, SC+CZ results are more stable and their total bias across the bins is smaller. \ne.g., the first and second truez distributions are combined to get the one for the first target photoz sample. \nIn Fig. 1, we show a sample of the angular correlation function w ( θ ) to be used in truez inference. The plot shows the photoz angular correlation function C i ' j ' and the cross \nangular correlation between the specz and photoz sample C ij ' . In this plot, the specz sample consists of the most massive 1% galaxies. We have plotted the median and the 1 σ band estimated by the 16 and 84 percentile among 100 \nmock runs. We only show the results for i -j with i and j being odd for clarity. \nThese angular correlation function measurements are performed using CUTE (Alonso 2012) with the grid method. We compute them in the angular range of [0 . 2 , 5] · with linear binning of width 0 . 2 · .", '3.3. Inference of the truez distribution': "We show the results on the inference of the truez distribution in Fig. 2. We compare the results obtained with SC, CZ, and SC+CZ against the truez distribution measurement from the mock catalog. Here we show the results from a single mock. They are produced using the fiducial setup, and the specz sample is the most massive 1%. \nIn order to quantify the accuracy of the results, we use all 100 mocks and consider the metric ∑ i | ˆ P ij ' -P true ij ' | , the absolute difference between the mock measurement P true ij ' and the estimated result ˆ P ij ' . We show this in the upper panel of Fig. 3. The line corresponds to the median and the color band demarcates the 16 and 84 percentile lines among 100 mock runs. \nWe find that for the first two bins, SC and CZ results are similar with SC marginally better, and the accuracy of CZ remains good for the high z bins while SC deteriorates. In reality, the number of specz galaxies may plummet faster than the constant fraction selection assumed here, and thus the performance of CZ in the high z bins may not be as good as the case shown here. The combination SC+CZ achieves the minimal error, and performs better than SC or CZ alone. \nTo go on to check the accuracy of the error estimate on P ij ' , σ P ij ' , we plot the absolute difference normalized by the estimated error, ∑ i | ˆ P ij ' -P true ij ' | /σ P ij ' in the lower panel of Fig. 3. A rough estimate is that the difference should be of order σ P ij ' , and so the sum is ∼ 5. We find that the result are generally larger than this simple estimate by a factor of ∼ 1 . 5, and it gets larger for the last bin. The large fluctuation of the 1 σ band especially for the last bin is caused by the occasionally too small error bar estimation in the tail of truez distribution. \nTo determine the truez distribution, all the cross bin clustering information is used; thus the information is not localized to a particular bin and it is hard to get a 'simple' explanation of the trend seen in Fig. 3. For CZ, all the specz sample is used to cross correlate with a photoz sample. Thus we argue that the photoz sample property could give a better estimate of the quality of the CZ inference. After cleaning, the number of photoz galaxies in the photoz bins are 1228413, 1576759, 1683208, 1266897, and 708319 respectively. The effective bias parameters of the photoz sample in the tomographic bins are similar (about 1.1), but at both ends the bias is ∼ 1 . 3. The rise at both ends is caused by the cleaning process, which cuts off the truez distribution and hence increases the clustering amplitude [Chan et al. (2024)]. Nonetheless the biases in the truez bins, b ' i are quite constant (see Fig. 6). Hence the number of photoz galaxies in the tomographic bins can qualitatively explain the trend of CZ in Fig. 3. The trend of the SC is even harder to interpret as it relies on the auto and cross clustering information all the bins. \nThe bias in the mean of the truez distribution is commonly used as an indicator for the accuracy of the characterization of the truez distribution. 4 In Table 2, we show bias in the mean redshift: \ne z = ¯ z ˆ n -¯ z n true 1 + ¯ z n true , (20) \nwhere ¯ z ˆ n (¯ z n true ) denotes the mean redshift computed using the truez distribution from the estimator (direct measurement). \nFor SC the biases are about 0.5%, and are positive for all the tomographic bins. The CZ results show larger variation and the bias goes from positive to negative as the redshift increases. The patterns for the three mass samples are similar. Although the first bin is relatively inaccurate, the others are good. In fact, the bias in the third bin is the smallest among all the entries. But this could be a particularity of the mock as the bias remains 0.05% even for the 0.5% sample. In contrast, the SC+CZ results are stable across the tomographic bins with bias about 0.3% for most of the entries. Although SC+CZ bias values are systematically lower than SC in all the tomographic bins, somewhat surprisingly they are larger than CZ for the last three bins. Nonetheless, if we take the bias of all the bins into account by adding up the absolute value of the biases in all five bins, SC+CZ gives a smaller total bias. For example, for the 1% sample, the total bias is reduced by about 31% and 48% relative to the CZ and SC bias, respectively. It seems that adding SC to CZ spoils some of the 'accidentally' accurate bin results for CZ, but it makes the overall results more stable. Here we avoid overinterpreting the results, and leave it to future mock tests to settle the some of the subtle trends.", '3.4. Comparison of different implementations': "In this subsection, we compare the results obtained with different implementations. Using SC as an example, in Fig. 4, we compare the truez distribution obtained with different algorithms for a single mock. In Fig. 5, we plot the absolute error and the normalized absolute error of these estimators computed with the 100-mock ensemble. \nRecall that for SC, our fiducial method is the P method, for which both P ij ' and C ' ii are the fitting parameters. Alternatively, in the R method, the only unknown is R ij ' . For the best fit value, P and R method give pretty similar results, but R method tends to give smaller error estimates thanks to less unknowns. \nIt seems that R method is more constraining, but we find that it gives too small error bar too often in the case of SC+CZ. On the other hand, P method also tends to give too small error bars in the case of CZ. Because CZ is a linear problem, the computation of error bars is straightforward. We find that R method yields error bars consistent with the simple error propagation result. These considerations motivate the adoption of P method for SC and SC+CZ, while R method for CZ. \nWe further contrast our results against the one obtained with the Old NMF method, for which we use the default setting in Xu et al. (2023). In particular the best fit is estimated using 100 random runs, and the best fit and the 1 σ \n) \nz \n( \nn \nFig. 4. Comparison of the truez distribution obtained by different implementations of the SC method with the direct measurement on the mock (green bars). The results obtained using the R method (red), P method (orange), and the Old NMF (blue) are compared. Because the Old NMF is not stable when it is run with resolution ∆ z = 0 . 05, we can only produce the results with ∆ z = 0 . 1. \n<!-- image --> \nerror bar are the median and the half width of the 68 percentile about the median. The stopping criterion is based on minimal χ 2 although its update rule is still rooted in the old J . For each random run, a Gaussian perturbation is added to the measurement by sampling the covariance. We note that the covariance adopted here is derived from 100 mocks rather than the the jackknife covariance as in Xu et al. (2023). The adoption of the mock covariance here is driven by the observation that there is a significant difference between the mock covariance and the jackknife covariance computed using the method in Xu et al. (2023). By definition, the covariance is estimated by the ensemble mean among realizations; thus the mock covariance is the correct one to use. Furthermore, the negative measurements are set to a small positive value. We tried to run the old NMF with ∆ z = 0 . 05, but it is not stable in this case and we have to settle with resolution ∆ z = 0 . 1. Song et al. (2024) also found that the old NMF fails to produce the truez distribution with a fine resolution, and used a coarse resolution of ∆ z = 0 . 1 although that did not impair the final BAO measurement there. \nFig. 4 and the top panel of Fig. 5 demonstrate that our improved algorithm results in a much more accurate estimation of the truez distribution than the Old NMF \nFig. 5. Comparison of the accuracy of the estimated truez distribution obtained with different SC algorithms. Upper panel : ∑ i | ˆ P ij ' -P true ij ' | , the absolute difference between the truez distribution measured from the mock, P true ij ' and the estimated one, ˆ P ij ' . Lower panel : ∑ i | ˆ P ij ' -P true ij ' | /σ P ij ' the absolute difference normalized by the estimated error σ P ij ' . In both panels, the results from R method (red), P method (orange), and Old NMF (blue) are compared. The line and color band represent the median and the 16 and 84 percentiles among 100 mock runs. The results for five tomographic bins are shown. \n<!-- image --> \nmethod. The Old NMF tends to give excessively large error bars in the central region of the truez distribution, while too small error estimate in the tail. We note that when the jackknife covariance is used instead, the error bars would be smaller. \nTable 2 also displays the bias from the old NMF method, and we find that our updated algorithm reduces the bias by more than a factor of 2 in almost all the bins.", '3.5. Estimation of the clustering amplitudes': "Although our primary target is the truez distribution P ij ' , the accuracy of the best fit correlation function serves an important cross-check and it reflects the consistency of the algorithm. \nIn Fig. 6, we plot the galaxy bias parameter of the photoz sample in the specz bin i , b ' i in 10 specz bins. Here we illustrate the results from SC+CZ using the massive 1% sample. The bias b ' i is estimated by the following means: \n-From the best fit C ' ii , we have b ' i = √ C ' ii C m ii . \nFig. 6. Galaxy bias parameter of the photoz sample in the specz bin i , b ' i estimated by various methods. The estimate by C ' ii (blue circles), C x ii ' (orange squares), and R ij ' (black dashed line with 1 σ error band in gray) are compared with the direct measurements (green triangles). The results for 10 specz bins are shown. The data points are offset slightly horizontally for clarity. See the text for details. \n<!-- image --> \n- -Using the best fit C x ii , we get b ' i = C x ii b i C m ii , where b i is measured using the auto angular correlation function of the specz sample.\n- -Under the universal leakage assumption and no bias evolution approximation, b ' ≈ ∑ i R ij ' , which predicts that the bias is constant across all the photoz bins. If we employ the generalized bias form, b j ' ≈ ∑ i b ij ' P ij ' , the resultant bias is redshift dependent albeit on index j ' . Because the truez distribution peaks about the photoz estimate, b j ' ≈ b j . In Fig. 6, we have plotted b j ' as black dashed line.\n- -By dividing the whole photoz sample into ten specz bins, we can measure their correlation functions to directly get b ' i . \nDifferent estimates are in agreement with each other within 10% in most of the bins in the range θ ≲ 1 · . The estimate from C x ii has larger error bar because it also requires the estimation of b i from the angular correlation function of the specz sample. Moreover, b ' i from R ij ' is almost constant across all the specz bins. It is remarkable that it is in nice agreement with other estimates except for the last bin given there are a few approximations made. \nIn principle, to check the universal leakage assumption, for data in each photoz bin j ' , we could divide the sample into 10 truez bins and measure the bias b ij ' . Under the universal leakage assumption, we anticipate b ij ' to be independent of j ' . However, in creating the mocks, photoz 's are assigned to galaxies based on the specz and photoz probability distribution only, without the photometry information. Thus the universal leakage assumption is built in during mock construction. \nFig. 7. Accuracy of the truez estimation characterized by ∑ ij ' | ˆ P ij ' -P true ij ' | , is plotted as a function of α , which measures the relative importance between SC and CZ. The line and the associated color band represent the median and the 16 and 84 percentiles among 100 mocks. We have shown the results for three specz sample consisting of the most massive 5%, 1%, and 0.5% galaxies in the specz sample. The black dashed line indicates the fiducial value α = 1. \n<!-- image -->", '3.6. Optimal weighting for SC and CZ': 'While σ 2 takes into account the measurement error of the correlation function, the theoretical uncertainties of the method are not included. Simply adding up J 1 and J 2 implicitly assumes that they have equal constraining power \nFig. 8. Accuracy of the truez inference by SC+CZ in the absence of the high z specz data. In the left panel, the specz data for the last two bins, bins 9 and 10, are missing; In the right panel, the specz data from bins 7 to 10 are missing. We have contrasted the cases with α 1 = 0 . 1, 1, and 10, which controls the importance of the preceding SC bins with CZ counterpart. The total absolute difference is plotted as a function of α 2 , which adjusts the weight of the SC bin without CZ counterpart. Shown are the results from specz samples consisting of the most massive 5%, 1%, and 0.5%, respectively. \n<!-- image --> \nand this may not lead to optimal results because SC and CS have different level of degeneracy. A simple way to tackle this issue is to assign different weights to their cost functions. We consider two scenarios: (i) the number of specz bins with specz data is equal to the number of photoz bins or (ii) the number of specz bins with specz data is less than the number of photoz bins. The second setup is particularly interesting because the lack of specz data at high z limits the application of the clustering-based method to the high z regime.', '3.6.1. Full specz data': "Here we focus on the scenario when the number of specz bin is the same as the number of photoz bins. For this full specz bin data case, we consider the joint cost function: \nJ = α J 1 + J 2 , (21) \nwhere J 1 and J 2 are given by Eqs. (16) and (17) respectively, and the weight α adjusts their relative importance. \nIn Fig. 7, we show the absolute error of the estimated distribution. We note that we have summed over both indices i and j ' . We have shown the results for three specz samples, which respectively consist of the most massive 5%, 1%, and 0.5% of the galaxies. The best fit and the 1 σ error band are derived from 100 mocks. For α ≳ 100, we see that the curves are convergent because the information is dictated by SC in this limit. On the other hand, in the low α limit, the result is determined by CZ. As expected, if the fraction of galaxies in the specz sample is higher, the signal-to-noise of the cross correlation function is higher, and hence the absolute error is smaller. \nIn the intermediate range, for the 0.5% sample, there is a dip roughly around α ∼ 0 . 7, implying that this weight optimally combines the information. As the mass fraction \nincreases, dip becomes much broader and shallower but the dip position remains nearly unchanged with very mild shift towards a smaller α . These indicate that the SC information helps little and the signal is largely dominated by CZ. We note that the from 0.5% to 1%, the optimal α seems to increase, but this is not expected and should be interpreted as statistical fluctuations. For the 1% specz sample, the optimal combination of SC+CZ reduces the total error by 20% relative to CZ and 40% relative to SC.", '3.6.2. Missing specz data': "Often the specz data are only available in the relatively low redshift range. Thus it is of interest to consider the situation in which the low z part of the truez distribution is simultaneously constrained by both SC and CZ, while the high z part relies solely on SC. \nFor notational convenience, let S be the set of all the bins of interest, { 1 , 2 , . . . , N } , T be the set of the bins with specz data available, { 1 , 2 , , . . . , N s } with N s ≤ N , and S \\ T be the set of bins without specz info, { N s +1 , . . . , N } . T ⊗ T denotes the tuple { ( i, j ) | i ∈ T and j ∈ T } , and its complement ( T ⊗ T ) C represents { ( i, j ) | i ∈ S \\ T or j ∈ S \\ T } . With these notations defined, the total cost function with two weight adjustment factors α 1 and α 2 is written as \nJ = J 1 + J 2 , (22) \nwhere J 1 is given by \nJ 1 = α 1 2 ∑ ( i ' ,j ' ) ∈ T ⊗ T ∑ µ F i ' j ' µ + α 2 2 ∑ ( i ' ,j ' ) ∈ ( T ⊗ T ) C ∑ µ F i ' j ' µ , (23) \nFig. 9. Comparison of the truez distribution obtained by different number of specz bins with specz data available against the direct measurement on the mock (green bars) for a single mock. We have presented the results from full specz bins (violet), no specz bin 9 and 10 (red), no specz bin 7 to 10 (blue), and SC only (orange). \n<!-- image --> \nwith F i ' j ' µ denoting \nF i ' j ' µ = W 1 ( θ µ ) σ 2 i ' j ' ( θ µ ) [ D i ' j ' ( θ µ ) -∑ k P ki ' P kj ' C ' kk ( θ µ ) ] 2 , (24) \nand J 2 by \nJ 2 = 1 2 ∑ ( i,j ' ) ∈ T ⊗ T ∑ µ W 2 ( θ µ ) σ 2 ij ' ( θ µ ) [ D ij ' ( θ µ ) -P ij ' C x ii ( θ µ ) ] 2 . (25) \nIn words, the weight α 1 is to adjust the importance of the SC part with respect to the CZ counterpart, while the additional weight α 2 in J 1 allows for the possibility to leverage the weight of the SC contribution from the S \\ T bins to compensate the missing CZ contribution. \nIn the left panel of Fig. 8, we show the results when the ninth and tenth specz bin data are missing. We contrasted the cases with α 1 fixed to be 0.1, 1, and 10, corresponding to the CZ information being dominant, similar to, and subdominant to the matching SC bins. The accuracy of the inference is presented as a function of α 2 . In addition, we \nFig. 10. The absolute error of the truez distribution estimated by different amount of specz bin data. Upper panel : ∑ i | ˆ P ij ' -P true ij ' | , the absolute difference between the truez distribution measured from the mock, P true ij ' and the one estimated, ˆ P ij ' . Lower panel : ∑ i | ˆ P ij ' -P true ij ' | /σ P ij ' the absolute difference normalized by the estimated error σ P ij ' . In both panels, the results from full specz data (blue), no specz bin 9 and 10 (red), no specz bin 7 to 10 (violet), and SC only (orange) are compared. With the reduction of the number of specz bin data, the constraining power is weakened. When only the highz specz bins are missing, the impact is mainly localized in the highz tomographic bins with the lowz bins little affected. \n<!-- image --> \nhave compared the results from the specz samples consisting of the most massive 5%, 1%, and 0.5% of the galaxies. \nFirst, for α 1 = 0 . 1, CZ information is determinant in the first 8 bins. For large α 2 , the importance of SC bin 9 and 10 is inflated and becomes dominant so that the overall accuracy decreases. The error is minimized at a trough varying from α 2 ∼ 0 . 2 to 0.5, depending on the fraction of the specz sample. The optimal α 2 is less than unity reflects that the constraining power of the SC in 9th and 10th bin is bin-wisely weaker than CZ in the bins 1 to 8. The accuracy also deteriorates in the limit of small α 2 because the missing bins become less and less constrained in this case. The optimal α 2 for higher mass sample is slightly higher in order to balance the greater CZ constraint from the lower redshift bins. The differences among the three specz samples diminish at α 2 ≲ 0 . 1, and this reflects that the CZ information is saturated and the missing bin is the bottleneck in enhancing the accuracy of the truez inference. \nThe overall trend with α 2 is similar for α 1 = 1 and 10, but the shape generally shifts to a larger α 2 to be in line with α 1 . For α 1 = 1, the CZ and SC information has comparable weight for the first 8 bins, we find that the minimal \nerror occurs around α 2 ∼ 1 for 0.5% specz sample and ∼ 2 for 5%. The differences among the specz samples are less pronounced than α 1 = 0 . 1 case because larger α 1 value reduces the weight of CZ. Finally, when SC information is dominant in the first 8 bins ( α 1 = 10), the optimal α 2 moves to ∼ 9 because the constraining power of SC bins are roughly similar. \nIt is worth summarizing the roles of the weight factors. α 1 adjusts the weight of the SC bins relative to their CZ counterpart. The SC and CZ part compete with each other and they have equal weights when α 1 = 1. α 2 controls the weight of the SC part without CZ information. Through the above exercise, we see that α 2 is in the same order of magnitude as the dominating part of the lower redshift bins. \nIn addition, we present the missing specz data in bin 7-10 case in the right panel of Fig. 8. They are qualitatively similar to the no bin 9 and 10 case. In particular, the optimal α 2 is quite similar to the no bin 9 and 10 case. However, in contrast, the minimal error increases in the CZ dominating case ( α 1 = 0 . 1) by a small amount, while the SC dominated case ( α 1 = 10) is the least affected. This can be attributed to the fact that the CZ information is more constraining than SC. For the same reason, in the small α 2 limit, we find that the accuracy is generally lower than the no bin 9 and 10 case. \nAlthough the precise weights depend on the relative constraining power of the SC and CZ data, our test suggests that there is generally a set of optimal α 1 and α 2 minimizing the error. \nWe show the bias in the mean redshift in Table 2 for missing specz bin data 9-10 and 7-10. Even with missing specz bin data, SC+CZ is stable and the results are similar to the full bin case. Sometimes, the mean value in missing bin case is even lower than the full bin case although the difference is statistically insignificant. Only when the bin 7-10 are missing, we notice systematic increase in bias in the tomographic bin 4 and 5 (by about half σ ). \nFinally, for completeness, we plot the truez distribution estimated by various number of specz bin data for a single mock in Fig. 9 and the absolute error estimated from 100 mocks in Fig. 10. In these plots, we assume α 1 = 1 and α 2 = 1, which are close to the optimal weights for the 1% sample used here. We have shown the results obtained with full specz bins, no specz bin 9 and 10, no specz bin 7 to 10, and SC only. We indeed see that as the number of specz bins decreases, the constraining power weakens. The truez distribution for lower tomographic bins 1 and 2 are almost unaffected for the missing specz bin cases, while the higher tomographic bin ones are more affected. In summary, we have demonstrated that by incorporating the SC information with CZ, we are able to extend the constraint on the truez distribution to higher redshift where specz data are missing, and the constraint is better than SC alone.", '4. Conclusions': "Characterization of the truez distribution is crucial in the science of the wide-field imaging surveys such as weak lensing and BAO. A valuable avenue to calibrate the truez distribution of a photoz sample is to make use of the clustering information. Clusteringz (CZ) has been widely used in the cosmological analyses in imaging surveys. However, \na limitation of this method is that the specz sample is often only available in relatively low redshift regime. The self-calibration (SC) method relying solely on the clustering information of the photometric sample is gaining popularity. In this work we develop a method to jointly constrain the truez distribution of a photoz sample using the information of SC and CZ simultaneously. We use the DES Y3 catalog to test our method and find that it can effectively infer the truez distribution. The codes performing the SC+CZ inference can be downloaded on GitHub 5 . \nPrevious SC analyses often rely on the non-negative matrix factorization (NMF) algorithm. Such an interpretation becomes a burden when we want to generalize the method to combine SC with CZ. Inspired by the NMF method, we construct multiplicative update rules to directly solve for the truez distribution. It is worth stressing that we have avoided the NMF interpretation altogether. Our straightforward analysis enables us to easily combine the information of SC with CZ. \nOur formalism has improved upon the previous approach by taking the weighting functions into account. We have included the inverse error weighting and an additional weighting function. The inverse error weighting allows us to downweight the impact of the poor measurements and upweight the good ones. The additional weighting function give the freedom to put more weight on the constraining part of the correlation function, and we take it to be the form θ -n with n = 1. We demonstrate that our improved algorithm results in a much more accurate estimation of the truez distribution compared to the previous method in the case of SC (Figs. 4 and 5). Moreover, our algorithm gives more stable results and allows us to have a truez distribution with higher resolution. This bias in the mean redshift is reduced by more than a factor of two. \nWeemploy our algorithm to show that SC+CZ improves the constraint relative to using SC or CZ alone (Figs. 2 and 3). SC+CZ gives a stable bias on the mean redshift, keeping it at a level of ∼ 0 . 3%, even though it sometimes fairs worst than CZ. The clustering-based methods also yield estimates of the intrinsic clustering amplitude. We find that various clustering measurements are consistent with each other (Fig. 6). To optimize the constraining power of SC and CZ data, we assign extra weight factors to the SC cost function. We consider the scenario with the number of specz bins with specz data equal to the number of photoz bins and the more interesting one with the specz bin data only available in the low z bins. We find that there generally exist some weights that minimize the total error (Figs. 7 and 8). The precise weights depend on the constraining power of the SC and CZ data, and detailed mock tests are required to locate the optimal values. To highlight the improvement, we quote the 1% specz sample result with full specz bins here. In this case we find that the optimal combination reduces the total error by 20% and 40% compared to using respectively CZ and SC only. The test with specz data restricted to the low z range demonstrates that by incorporating with the SC information, we can extend the utility of the clustering-based method to higher redshift. \nAfter successfully demonstrating the power of the SC+CZ method via mock catalogs, it is desirable to apply it to real data, such as the DES Y6 BAO data (Abbott et al. 2024; Mena-Fern'andez et al. 2024). For this dataset, \nonly the portion of data with z < 1 is calibrated with CZ, while the data in the range 1 < z < 1 . 2 have to be calibrated with a specz sample from a small sky area. A few caveats need to keep in mind when applying the method to real data. In this work we have used ∆ z = 0 . 05, but this is still larger than the redshift bin width often taken in CZ analysis (e.g. ∆ z = 0 . 03). We have also assumed no bias evolution, at least in the CZ case. We leave it to future work to test the impact of these setups, and study possible improvements. We have not considered the magnification bias in this paper because the redshift range of the data is still small and the mock does not include this effect. For larger redshift extent, magnification bias correction must be made. Furthermore, the photometric data are more prone to observational systematics, which must be treated before SC. Compared to the applications to the clustering samples such as the lens sample in weak lensing and the BAO sample mentioned above, the application of the SC method to the source sample in weak lensing is more challenging simply because it is not constructed for clustering analysis. The source sample is often inhomogeneous and the random catalog is likely missing in standard analysis, so it must be built with great care. Moreover, the systematics mitigation is designed for the shear measurements, and so it remains to show that the existing mitigation efforts are sufficient or additional clustering weight needs to be created. Nevertheless, our work paves the way to calibrate the truez distribution in high redshift using the clustering information. \nAcknowledgements. We thank the anonymous referee for his/her insightful comments that improve the presentation of the manuscript. WZ, KCC, and RS are supported by the National Science Foundation of China under the grant number 12273121 and the science research grants from the China Manned Space Project with NO.CMSCSST-2021-B01. HX is supported by the National SKA Program of China (grant No. 2020SKA0110100), the National Natural Science Foundation of China (Nos. 11922305, 11833005) and the science research grants from the China Manned Space Project with NOs. CMS-CSST-2021-A02. LZ is supported by National SKA Program of China (2020SKA0110401, 2020SKA0110402, 2020SKA0110100), the National Key R&D Program of China (2020YFC2201600), the China Manned Space Project with No. CMS-CSST-2021 (A02, A03), and Guangdong Basic and Applied Basic Research Foundation (2024A1515012309).", 'References': "Abbott, T., Aguena, M., Alarcon, A., et al. 2022a, Physical Review D, 105 Abbott, T. et al. 2019, Mon. Not. Roy. Astron. Soc., 483, 4866 Abbott, T. M. C., Abdalla, F. B., Alarcon, A., et al. 2018, Phys. Rev. D, 98, 043526 Abbott, T. M. C., Adamow, M., Aguena, M., et al. 2024, Phys. Rev. D, 110, 063515 Abbott, T. M. C. et al. 2022b, Phys. Rev. D, 105, 043512 Alonso, D. 2012, arXiv e-prints, arXiv:1210.1833 Amon, A., Gruen, D., Troxel, M. A., et al. 2022, Phys. Rev. D, 105, 023514 Arnouts, S., Cristiani, S., Moscardini, L., et al. 1999, MNRAS, 310, 540 Asgari, M., Lin, C.-A., Joachimi, B., et al. 2021, A&A, 645, A104 Avila, S. et al. 2018, Mon. Not. Roy. Astron. 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A., Mao, Y.-Y., et al. 2021, MNRAS, 501, 3309 \nZhu, G. 2016, arXiv e-prints, arXiv:1612.06037 \nZhu, Z., Yang, Z., & Oja, E. 2013, in 18th conference Scandinavian Conferences on Image Analysis (SCIA 2013) Espoo, Finland, June 17-20, 2013 (Germany: Springer Gabler), 143-152", 'Appendix A: Derivation of the multiplicative update rule': 'In general, a multiplicative update rule is a set of iterative instructions to update a function by multiplying it with some factor to search for the minimum of the cost function. The most influential multiplicative update rule for non-negative matrix factorization (NMF) is presented by Lee & Seung (2000). This multiplicative update rule can be interpreted as a special kind of gradient descent rule with variable step size. A distinguishing feature of the NMF update rule is the non-negativity of the solution. This has been adopted in Zhang et al. (2017) to solve the SC equation. The update rule in Lee & Seung (2000) can be easily derived by splitting the gradient of the cost function into the positive and negative parts (Choi 2008). Inspired by this derivation, we derive multiplicative update rules for the joint inference by SC and CZ.', 'Appendix A.1: P method': "In P method, we aim to derive multiplicative rules for P ij ' , C x ii , and C ' ii so that the cost function J [Eq. (15)] is minimized. If we take P ab ' as the independent variable, we also need to account for the normalization constraint Eq. (5). This can be implemented by introducing the Lagrange multipliers into the cost function as \nJ = J 1 + J 2 -∑ j ' λ j ' ( ∑ i P ij ' -1 ) . (A.1) \nWe first computed the derivative with respect to P ab ' and then to the Lagrange multipliers. \nThe derivative of J 1 with respect to P ab ' reads \n∂ J 1 ∂P ab ' = -2 ∑ j ' ,µ W 1 ( θ µ ) σ 2 b ' j ' ( θ µ ) [ D b ' j ' ( θ µ ) -∑ k P kb ' P kj ' C ' kk ( θ µ ) ] × P aj ' C ' aa ( θ µ ) , (A.2) \nand for J 2 , we have \n∂ J 2 ∂P ab ' = -∑ µ W 2 ( θ µ ) σ 2 ab ' ( θ µ ) [ D ab ' ( θ µ ) -P ab ' C x aa ( θ µ )] C x aa ( θ µ ) . (A.3) \nWithout the constraint, following Lee & Seung (2000), the update rule is given by \nP ab ' ←-P ab ' [ ∂ P ab ' J ] -[ ∂ P ab ' J ] + , (A.4) \nwhere [ ∂ P ab ' J ] -([ ∂ P ab ' J ] + ) denotes the unsigned negative (positive) part of the derivative of ∂ P ab ' J . We can contrast this with the usual additive gradient descent (Goodfellow et al. 2016). The term [ ∂ P ab ' J ] -tends to increase the value of P ab ' , while [ ∂ P ab ' J ] + tends to decrease it, so they play the same role as in the usual gradient descent. When the minimum is reached, ∇J vanishes and the multiplicative factor becomes unity. Moreover, the factor [ ∂ P ab ' J ] -/ [ ∂ P ab ' J ] + is non-negative, and so it does not flip the sign of P ab ' . \nWhen the normalization constraint is included, we have \n∂ J ∂P ab ' = ∂ J 1 ∂P ab ' + ∂ J 2 ∂P ab ' -λ b ' . (A.5) \nArticle number, page 16 of 17 \nFor the treatment of the Lagrange multiplier part, we follow the procedures in Zhu et al. (2013). We can first establish a preliminary update rule in terms of λ b ' , and then demand the resultant P ab ' to satisfy the normalization constraint. This enabled us to solve for λ b ' . It is convenient to include a negative sign for λ b ' and assume λ b ' to be positive. However, this assumption does not always hold and so Zhu et al. (2013) consider a'moving term'trick to alleviate this problem. \nThe preliminary update rule is \nP ab ' ←-P ab ' [ ∂ P ab ' J ] -+ λ b ' [ ∂ P ab ' J ] + . (A.6) \nThe new P is also demanded to meet the normalization constraint: \n∑ a P ab ' [ ∂ P ab ' J ] -+ λ b ' [ ∂ P ab ' J ] + = 1 . (A.7) \nSolving for λ b ' , the update rule becomes \nP ab ' ←-P ab ' [ ∂ P ab ' J ] -∑ c P cb ' [ ∂ P cb ' J ] + +1 -∑ c P cb ' [ ∂ P cb ' J ] -[ ∂ P cb ' J ] + [ ∂ P ab ' J ] + ∑ c P cb ' [ ∂ P cb ' J ] + . (A.8) \nTo make the numerator always positive, we applied the 'moving trick' to get \nP ab ' ←-P ab ' [ ∂ P ab ' J ] -∑ c P cb ' [ ∂ P cb ' J ] + +1 [ ∂ P ab ' J ] + ∑ c P cb ' [ ∂ P cb ' J ] + + ∑ c P cb ' [ ∂ P cb ' J ] -[ ∂ P cb ' J ] + . (A.9) \nWe also update C ' aa [in Eq. (16)] and C x aa [in Eq. (17)] using the multiplicative rule although this is not necessary. Because they are linear parameters, direct analytic minimization is possible as was done in Zhang et al. (2017). The update rule for C ' aa reads \nC ' aa ( θ ) ←-C ' aa ( θ ) [ ∂ C ' aa ( θ ) J ] -[ ∂ C ' aa ( θ ) J ] + . (A.10) \nA similar update rule for C x aa can be derived.", 'Appendix A.2: R method': "In R method, only R ij ' is the unknown. Explicitly the cost functions read \nJ 1 = 1 2 ∑ i ' ,j ' ,µ W 1 ( θ µ ) σ 2 i ' j ' ( θ µ ) [ D i ' j ' ( θ µ ) -∑ k R ki ' R kj ' C m kk ( θ µ ) ] 2 , (A.11) \nJ 2 = 1 2 ∑ i,j ' ,µ W 2 ( θ µ ) σ 2 ij ' ( θ µ ) [ D ij ' ( θ µ ) -R ij ' b i C m ii ( θ µ ) ] 2 . (A.12) \nThe derivative of the cost functions with respect to R ab ' are given by \n∂ J 1 ∂R ab ' = -2 ∑ j ' ,µ W 1 ( θ µ ) σ 2 b ' j ' ( θ µ ) [ D b ' j ' ( θ µ ) -∑ k R kb ' R kj ' C m kk ( θ µ ) ] × R aj ' C m aa ( θ µ ) , (A.13) \nand \n∂ J 2 ∂R ab ' = -∑ µ W 2 ( θ µ ) σ 2 ab ' ( θ µ ) [ D ab ' ( θ µ ) -b a R ab ' C m aa ( θ µ )] b a C m aa ( θ µ ) . (A.14) \nThe update rule for R ab ' can then be constructed: \nR ab ' ←-R ab ' [ ∂ R ab ' J ] -[ ∂ R ab ' J ] + . (A.15) \nWe note that P ij ' follows from Eq. (19) if we assume that there is no galaxy bias evolution.", 'Appendix B: Impact of outlying galaxies': "When the photoz galaxies are present only in the redshift range [0 . 6 , 1 . 1], the truez distribution of this sample can extend outside it. In this work, we restrict both photoz and specz range of the galaxies to [0.6,1.1], and this effectively assumes that P ij ' vanishes outside this range. In practice, however, this kind of cleaning is not possible. \nTo test the impact of the outlying galaxies on the estimation of the truez distribution, we apply the algorithm to the raw photoz sample, which contains the galaxies with specz values lying outside the range [0.6,1.1]. When the outlying galaxies are not removed, this additional component does not correlate with the ones inside, and this causes a reduction in the correlation signals in general. We compare the resulting distributions from the raw sample against the ones from the cleaned sample in Fig. B.1. Overall, we find that the impact of the outlying galaxies is pretty small for our sample. \nWe note that, as expected, the distributions for the boundary bins are more affected, i.e. bin 1 and 5. This is because the outlying specz galaxies are mainly located in the photoz bins near the boundaries. We further note that CZ is more strongly affected than SC, and thus SC+CZ is in between. This can be explained by the fact that in CZ, the correlation is localized in the precise redshift of the specz bin, while the information in SC is more distributed, and hence CZ is more affected by the reduction in correlation with the boundary bins.", 'Appendix C: Impact of setting negative measurements to a tiny positive value': "In the main text, we mention that the multiplicative rule implicitly assumes that the measurements are positive. A quick fix is to set the negative measurements to a tiny positive value, for which we take it to be 10 -5 . Here we test the impact of such a modification on the truez inference results. \nFig. C.1 showcases the truez distribution from a mock. While there are only small fluctuations in the best fit value, without the modification, the estimated error bars from SC and SC+CZ appear to be larger, but the CZ error bars are little affected. We further examine the accuracy of the estimated distribution in Fig. C.2. The central value and the 1 σ error bound are estimated from 100 mock catalogs. From the top panel, we see that nulling the negative measurements seems to increase the accuracy of SC and SC+CZ in most of the bins albeit by a statistically insignificant \nFig. B.1. Test of the impact of the galaxies with specz values lying outside the redshift range [0.6,1.1]. The truez distribution from the raw sample, which contains the outlying specz galaxies (unfilled markers), are compared with the cleaned sample results (filled markers). The results from SC (red), CZ (orange), and SC+CZ (blue) are displayed. The direct mock measurements (green bars) are from the cleaned sample. \n<!-- image --> \namount, but the modification also causes the SC bin 1 result to degrade. The impact on the estimated error bar is more apparent. The modification homogenizes the range of | ˆ P ij ' -P true ij ' | /σ P ij ' across tomographic bins. \nThere seem to be benefits, especially on the error bar estimation, in doing such a modification, and thus we may consider adopting it in future. \n) \nz \n( \nn \nFig. C.1. Comparison of the truez distribution obtained by setting the negative measurements to a tiny positive value (empty markers) or not (filled markers). We have presented the results from SC (orange), CZ (red), and SC+CZ (blue). \n<!-- image --> \nFig. C.2. Absolute error of the truez distribution estimated using samples processed by setting the negative measurements to a tiny positive value (empty markers) or not (filled markers). Shown are the results from SC (orange), CZ (red), and SC+CZ (blue). Upper panel : ∑ i | ˆ P ij ' -P true ij ' | , the absolute difference between the truez distribution measured from the mock, P true ij ' and the one estimated, ˆ P ij ' . Lower panel : ∑ i | ˆ P ij ' -P true ij ' | /σ P ij ' the absolute difference normalized by the estimated error σ P ij ' . \n<!-- image -->"}

2024ApJ...969L...2F

We present a sample of 88 candidate z 8.514.5 galaxies selected from the completed NIRCam imaging from the Cosmic Evolution Early Release Science survey. These data cover 90 arcminSUP2SUP 10 NIRCam pointings in six broadband imaging filters and one mediumband imaging filter. With this sample we confirm at higher confidence early JWST conclusions that bright galaxies in this epoch are more abundant than predicted by most theoretical models. We construct the restframe ultraviolet luminosity functions at z 9 11 and 14 and show that the space density of bright M SUBUVSUB 20 galaxies changes only modestly from z 14 to z 9 compared to a steeper increase from z 8 to z 4. While our candidates are photometrically selected spectroscopic followup has now confirmed 13 of them with only one significant interloper implying that the fidelity of this sample is high. Successfully explaining the evidence for a flatter evolution in the number densities of UVbright z gt 10 galaxies may thus require changes to the dominant physical processes regulating star formation. While our results indicate that significant variations of dust attenuation with redshift are unlikely to be the dominant factor at these high redshifts they are consistent with predictions from models that naturally have enhanced star formation efficiency andor stochasticity. An evolving stellar initial mass function could also bring model predictions into better agreement with our results. Deep spectroscopic followup of a large sample of early galaxies can distinguish between these competing scenarios.

2024-07-01T00:00:00Z

['10.48550/arXiv.2311.04279', 'arXiv:2311.04279', '10.3847/2041-8213/ad4495', '2024ApJ...969L...2F', '2023arXiv231104279F']

['Early universe', 'Galaxy formation', 'Galaxy evolution', 'Luminosity function', '435', '595', '594', '942', 'Astrophysics - Astrophysics of Galaxies']

The Complete CEERS Early Universe Galaxy Sample A Surprisingly Slow Evolution of the Space Density of Bright Galaxies at z 8.514.5

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']

https://arxiv.org/pdf/2311.04279.pdf

{'The Complete CEERS Early Universe Galaxy Sample: A Surprisingly Slow Evolution of the Space Density of Bright Galaxies at z ∼ 8.5-14.5': "Steven L. Finkelstein, 1 Gene C. K. Leung, 2 Micaela B. Bagley, 1 Mark Dickinson, 3 Henry C. Ferguson, 4 Casey Papovich, 5, 6 Hollis B. Akins, 7 Pablo Arrabal Haro, 3 Romeel Dav'e, 8, 9 Avishai Dekel, 10 Jeyhan S. Kartaltepe, 11 Dale D. Kocevski, 12 Anton M. Koekemoer, 13 Nor Pirzkal, 14 Rachel S. Somerville, 15 L. Y. Aaron Yung, 16, 13, ∗ Ricardo O. Amor'ın, 17, 18 Bren E. Backhaus, 19 Peter Behroozi, 20, 21 Laura Bisigello, 22, 23 Volker Bromm, 1 Caitlin M. Casey, 1 ' Oscar A. Ch'avez Ortiz, 1 Yingjie Cheng, 24 Katherine Chworowsky, 1, † Nikko J. Cleri, 5, 6 M. C. Cooper, 25 Kelcey Davis, 19 Alexander de la Vega, 26 David Elbaz, 27 Maximilien Franco, 1 Adriano Fontana, 28 Seiji Fujimoto, 29, 30 Mauro Giavalisco, 24 Norman A. Grogin, 4 Benne W. Holwerda, 31 Marc Huertas-Company, 32, 33, 34 Michaela Hirschmann, 35 Kartheik G. Iyer, 36 Shardha Jogee, 1 Intae Jung, 37 Rebecca L. Larson, 38, 39 Ray A. Lucas, 40 Bahram Mobasher, 26 Alexa M. Morales, 41 Caroline V. Morley, 42 Sagnick Mukherjee, 43 Pablo G. P'erez-Gonz'alez, 44 Swara Ravindranath, 45, 46 Giulia Rodighiero, 47, 48 Melanie J. Rowland, 49 Sandro Tacchella, 50, 51 Anthony J. Taylor, 1 Jonathan R. Trump, 19 and Stephen M. Wilkins 52, 53", 'ABSTRACT': 'We present a sample of 88 candidate z ∼ 8.5-14.5 galaxies selected from the completed NIRCam imaging from the Cosmic Evolution Early Release Science (CEERS) survey. These data cover ∼ 90 arcmin 2 (10 NIRCam pointings) in six broad-band and one medium-band imaging filter. With this sample we confirm at higher confidence early JWST conclusions that bright galaxies in this epoch are more abundant than predicted by most theoretical models. We construct the rest-frame ultraviolet luminosity functions at z ∼ 9, 11 and 14, and show that the space density of bright ( M UV = -20) galaxies changes only modestly from z ∼ 14 to z ∼ 9, compared to a steeper increase from z ∼ 8 to z ∼ 4. While our candidates are photometrically selected, spectroscopic followup has now confirmed 13 of them, with only one significant interloper, implying that the fidelity of this sample is high. Successfully explaining the evidence for a flatter evolution in the number densities of UV-bright z > 10 galaxies may thus require changes to the dominant physical processes regulating star formation. While our results indicate that significant variations of dust attenuation with redshift are unlikely to be the dominant factor at these high redshifts, they are consistent with predictions from models which naturally have enhanced star-formation efficiency and/or stochasticity. An evolving stellar initial mass function could also bring model predictions into better agreement with our results. Deep spectroscopic followup of a large sample of early galaxies can distinguish between these competing scenarios. \nKeywords: early universe - galaxies: formation - galaxies: evolution', '1. INTRODUCTION': 'The first 500 million years of cosmic time ( z ≳ 10), when the first stars and galaxies formed, began to grow, and kick-started the process of reionization, was largely hidden from view until recently. The depth achievable with JWST near-infrared imaging, along with the capabilities to deeply probe beyond ∼ 1.6 µ m for the first \nstevenf@astro.as.utexas.edu \n- ∗ NASA Postdoctoral Fellow\n- † NSF Graduate Fellow \ntime, were expected to revolutionize our understanding of this early epoch. As soon as the first data from JWST were released in July 2022, this renaissance in our understanding unfolded immediately. \nPrior to JWST , the high-redshift community had debated about the evolution of the rest-frame ultraviolet (UV) luminosity function (and by extension, the cosmic star-formation rate density) at z > 8. While there was good agreement that these quantities evolved smoothly downward from z = 4 to z = 8 (e.g. Bouwens et al. 2015; Finkelstein et al. 2015), results differed at z > 8, with some advocating for continued evolution with the same declining slope from lower redshifts (e.g. McLeod et al. \n2016; Finkelstein 2016), while others claimed evidence of an accelerated decline towards higher redshifts (e.g. Oesch et al. 2018; Bouwens et al. 2019). Early JWST surveys, including the Cosmic Evolution Early Release Science Survey (CEERS; PID 1345, PI Finkelstein) and GLASS (PID 1324, PI Treu) surveys, were designed in part to determine which of these evolutionary possibilities was correct. \nIn defiance of expectations, several studies immediately reported the presence of bright ( m ≲ 27.5) galaxies at z > 10 from the CEERS and GLASS surveys (e.g., Castellano et al. 2022; Naidu et al. 2022; Finkelstein et al. 2022b; Donnan et al. 2023b). These galaxies were both brighter and at higher redshifts than expected from these early surveys, which were neither extremely wide nor deep. While some early results changed due to the uncertain characterization of the NIRCam photometric zeropoint (Boyer et al. 2022), within a few months more robust samples of galaxies were in place (e.g. Finkelstein et al. 2023; Harikane et al. 2023a; McLeod et al. 2023). \nThese first studies with larger ( ∼ 20 object) samples agreed that the abundance of z ≳ 10 galaxies was in excess of both theoretical and empirical predictions, with explanations ranging from an evolving initial mass function (IMF), changes in star-formation efficiency, changes in dust attenuation, contribution from active galactic nuclei, rampant sample contamination (e.g. Finkelstein et al. 2023; Harikane et al. 2023a; Ferrara et al. 2023; Dekel et al. 2023; Mason et al. 2022), to even changes to the underlying cosmology (e.g. Boylan-Kolchin 2022; Liu & Bromm 2022). Such explanations are compelling, yet these early datasets spanned small dynamic ranges in UV luminosity, with few galaxies yet included. \nHere we report the results from a search for z ≥ 8.5 galaxies over the completed CEERS dataset. This follows on the work of Finkelstein et al. (2023, hereafter F23) who did a similar search, finding 26 galaxies over the first four CEERS pointings. Importantly, here we combine our results with those from Leung et al. (2023) who used a near-identical analysis procedure to identify a sample of 38 galaxies at similarly high redshift from the deep NGDEEP (the Next Generation Deep Extragalactic Exploratory Public survey; Bagley et al. 2023) NIRCam imaging, extending our dynamic range 1.5 magnitudes fainter. \nIn § 2 we describe our imaging dataset, and give a detailed explanation of our photometry procedure, which has evolved from F23 to increase our color and flux accuracy. In § 3 we outline our photometric redshift sample selection procedure, and discuss the available spectroscopy for our candidate galaxies. Our results are presented in § 4, where in § 4.2 we compare the observed sur- \ndensity of galaxies to pre-launch predictions, and in § 4.3 we calculate the rest-UV luminosity functions at z ∼ 9, 11 and 14. We discuss these results in the context of a variety of more recent simulation predictions in § 5, and present our conclusions in § 6. We assume the latest Planck flat ΛCDM cosmology with H 0 = 67.36 km s -1 Mpc -1 , Ω m = 0.3153 and Ω Λ = 0.6847 (Planck Collaboration et al. 2020). All magnitudes are in the absolute bolometric system (AB; Oke & Gunn 1983).', '2. DATA': 'The CEERS NIRCam imaging survey consists of 10 NIRCam pointings in the CANDELS Extended Groth Strip (EGS) field, done in parallel with prime MIRI and NIRSpec observations. These data were taken in two epochs. On June 21-22, 2022 four NIRCam pointings were obtained (known as pointings NIRCam1, 2, 3 and 6; results presented in F23). The remaining six pointings (NIRCam4, 5, 7, 8, 9 and 10) were completed on Dec 20-24, 2022. All 10 pointings include the same filter coverage of F115W, F150W, and F200W in the shortwavelength channel, and F277W, F356W, F410M, and F444W in the long-wavelength channel. In this analysis we make use of the official CEERS publicly released mosaics (DR0.5 for the June data, and DR0.6 for the December data). These images are available on the CEERS website (https://ceers.github.io/releases.html) and on MAST as High Level Science Products via 10.17909/z7p0-8481.', '2.1. Data Reduction': "We reduce the NIRCam imaging following the procedures outlined in Bagley et al. (2022b), which we briefly summarize here. We use the JWST Calibration Pipeline (Bushouse et al. 2022) with custom modifications and additional steps needed to remove features such as snowballs, wisps, and 1 /f noise. Our reduction process is the same for all images, though we use different Pipeline and Calibration Reference Data System (CRDS) versions for the two epochs of CEERS NIRCam imaging: Pipeline version 1.7.2 and CRDS context 0989 for the Epoch 1 images (CEERS DR0.5, obtained in June, 2022, and including pointings NIRCam1, 2, 3 and 6), and Pipeline version 1.8.5 and CRDS context 1023 for the Epoch 2 images (CEERS DR0.6, obtained in December, 2022, and including pointings NIRCam4, 5, 7, 8, 9 and 10). There were no major changes between these pipeline+CRDS versions which we would expect to affect the photometry. \nFor our astrometric alignment and analysis we make use of the archival HST imaging from the All-wavelength Extended Groth Strip International Survey (AEGIS, \nDavis et al. 2007), the Cosmic Assembly Deep Extragalactic Legacy Survey (CANDELS, Grogin et al. 2011; Koekemoer et al. 2011), and the 3D-HST (Momcheva et al. 2016) surveys. The entire CEERS field is covered by F606W, F814W, F125W, F140W, and F160W; portions are covered by F105W (we note that the F140W imaging is very shallow [800 sec], thus while we include it in our catalog, we do not include these data in any figures below). We use the CEERS v1.9 1 HST EGS mosaics, which are created from these datasets, aligned to Gaia DR3, and on a 30mas pixel scale. We use a modified version of the TweakReg routine to align the images, using the HST F160W mosaic as the astrometric reference in all pointings except NIRCam3 and 9. In these two pointings, a low-level guide star tracking issue in the HST imaging caused a sub-PSF shift across a portion of the F160W mosaic, and so we use the NIRCam F277W (pointing 3) and F356W (pointing 9) mosaics as the absolute references. In all pointings, the RMS of the relative, NIRCam-to-NIRCam astrometry is ∼ 5-10 mas. \nWe create mosaics on a 30mas pixel scale in all filters, and extract smaller cutouts of the HST mosaics to match the footprints of the drizzled NIRCam images, providing pixel-aligned imaging in up to 13 filters per field from ∼ 0 . 5 -5 µ m. Finally, we perform a global, two-dimensional background subtraction on the mosaics to remove any residual background variations. This method first performs a tiered source detection to identify progressively smaller sources in each filter. Then the source masks in each filter, including all available HST images, are combined into a single merged mask such that pixels with source flux identified in any filter are excluded when measuring the background. See Bagley et al. (2022b) and F23 for more information on the background method and details on the method's performance in CEERS images.", '2.2. Photometry': 'We perform photometry on all 10 CEERS fields using Source Extractor (hereafter SE , Bertin & Arnouts 1996). Photometry is performed on each of the 10 pointings independently (the high-redshift galaxy sample is screened for duplicates in the small overlapping areas, as discussed below). The photometry process here is similar to F23, with some key differences designed to improve the photometric validity, as described below. Our fiducial photometry is measured in elliptical Kron apertures, using a Kron factor =1.1 and a minimum radius \n= 1.6, following F23. These small Kron apertures result in optimal signal-to-noise. We derive accurate colors in these apertures by matching the image PSFs between different filters, and calculate accurate total fluxes via a two-step simulation-based aperture correction process.', '2.2.1. PSF Matching': "We create empirical point-spread functions (PSFs) in each filter by stacking stars. We select stars by identifying the stellar locus in a plot of half-light radius versus magnitude in a preliminary SE run in each filter. Each star is then inspected to ensure it appears to be a nonsaturated point source in a non-crowded region. We make a single PSF per filter by stacking stars across all 10 pointings (as all observations used the same dither pattern). For each star, we extract a 101x101 pixel box, upsample by a factor of 10, measure the centroid, and shift the star to be centered in this upsampled image. We then downsample back to the native resolution, rotate the star by a random position angle (to account for situations when the position angle of the observations was not identical), and normalize the star's peak flux to unity. The final PSF is made by median-combining the individual stars. The final PSFs have a centroiding accuracy of ∼ 0.05-0.1 pixels. \nWe then use these PSFs to derive kernels to match the PSFs in images with PSF FWHMs smaller than that of F277W to the F277W PSF. This includes the NIRCam F115W, F150W and F200W images, and the HST /ACS F606W and F814W images. Kernels were created with the pypher Python routine 2 (Boucaud et al. 2016). The NIRCam F356W, F410M and F444W filters, along with all four HST /WFC3 filters, F105W, F125W, F140W, and F160W, have PSF FWHMs larger than that of the NIRCam F277W filter. To correct for missing flux when performing aperture photometry on these larger PSF images, we first convolve the F277W image to match the PSF of a given larger-PSF image, and then derive source-specific correction factors as the ratio of the F277W flux prior to convolution to that after convolution (where the larger PSF images will have less flux in the aperture). These correction factors are then applied to photometry measured on the larger PSF images, to account for the missing flux. We tested our PSF-matching process by measuring curves-of-growth of the PSF stars in the images, finding that the median enclosed flux at an aperture diameter of 0.3 '' was within 5% (and often less) of the F277W value for all filters. \nIn F23 we did not employ these correction factors as we matched all NIRCam bands to the F444W PSF. \nHowever, by matching here to the smaller-PSF F277W image we better optimize signal-to-noise by not smoothing to the largest PSF. Additionally, as the HST /WFC3 images have larger PSFs than even F444W, in F23 we devised correction factors by comparing the HST fluxes to previous HST catalogs. Our updated method here is independent of other catalogs, and is thus more self consistent. We confirm that our fluxes here are consistent to within 5% on average (and often lower) with the fluxes from F23.", '2.2.2. Catalog Creation': "We use the inverse-variance-weighted sum of the nonPSF-matched F277W and F356W images as our detection image, to better detect faint sources. Using this detection image, we run SE cycling through the seven NIRCam images and six HST images as the measurement image. The key SE parameters were: DETECT THRESH=1.4, DETECT MINAREA=5 pixels, and a top-hat convolution kernel with a width of 4 pixels, similar to F23. We force SE to skip the background subtraction step as this was previously removed ( § 2.1). We use MAP RMS for the source weighting. As the pipeline-produced ERR images include Poisson noise, they are not appropriate for source detection. We thus convert the weight map associated with the detection image into an effective RMS map by taking 1/sqrt(WHT), and assign this to the detection image. For the measurement image, we use the pipeline ERR image. \nWe estimate an aperture correction to the total flux for these small apertures by performing a second run of SE on the F277W image with the Kron parameters set to the default 'MAG AUTO' parameters of (2.5, 3.5), deriving an aperture correction as the ratio between the flux in this larger aperture to that in the smaller aperture for each object in the F277W catalog, which we then applied multiplicatively to the fluxes and uncertainties for all filters (thereby correcting fluxes to an estimated total, but not changing colors or signal-to-noise values). \nAs several previous studies have noted that the default Kron parameters we use for this aperture correction can miss light in the wings of the PSF (e.g., Bouwens et al. 2015; Finkelstein et al. 2022a), we estimate residual aperture corrections using source-injection simulations, adding 3000 mock sources to our real images in each field. We add sources from m = 22-28.5 mag (to ensure a robust photometric measurement), with a log-normal half-light radius distribution peak- \ning at ∼ 1.5 pixels ( ∼ 0.2 kpc at z = 10; compact but modestly resolved, comparable to high-redshift sources), with a log-normal S'ersic parameter distribution, peaking at 1.2. These mock sources were generated with galfit (Peng et al. 2002) and added at random positions to the F277W and F356W images. In galfit , the total flux is calculated such that half the flux is within r e . We combined the two images to create a detection image, ran SE in the same way as on our real data to generate a F277W catalog, and estimated residual aperture corrections as the ratio of input-to-recovered fluxes for recovered sources. While in F23 we derived a single factor, here we note that the residual correction needed is magnitude dependent. We thus fit a linear function to the flux ratios as a function of magnitude over 24 < m < 28, finding a correction ranging from ∼ 2% at m < 22, to ∼ 20% at m = 28. We performed this linear fitting in each field, applying these residual aperture corrections to every source (placing a bound on the corrections applied to be from 1.0 - 1.2). We note that in the magnitude range of 25 < m < 26 used by F23, we see a similar correction factor (1.08) as they derived.", '2.2.3. Flux Uncertainties': "We derive flux uncertainties empirically based on the number of pixels in an aperture. We fit for the noise as a function of aperture size by measuring the fluxes in circular apertures with 30 different diameters, ranging from 0.1 '' (3.33 pixels) to 3 '' (100 pixels). We place nonoverlapping apertures randomly, avoiding pixels with zero values in the error image, and positive values in the segmentation map. We do this in two iterations, placing 3000 apertures with diameters < 1.5 '' , and 500 apertures with larger diameters. We measure fluxes at these positions in all aperture sizes, calculating the 1 σ noise in each aperture size by measuring the median absolute deviation of the measured flux values (multiplying by 1.48 to convert to a Gaussian-like standard deviation). Finally, we fit a curve to the noise in a given aperture as a function of pixels in that aperture, using this equation (Gawiser et al. 2006): \nσ N = σ 1 ( αN β + γN δ ) (1) \nwhere σ N is the noise in an aperture containing N pixels, and σ 1 is the pixel-to-pixel noise measured in each image as the sigma-clipped standard deviation of all non-object pixels (see Figure 3 in Finkelstein et al. 2022a for an example of this process). Compared to F23, here we include the second term in the parentheses as we find a better fit to the data. We fit the four free parameters with an IDL implementation of emcee (see Finkelstein et al. 2019 for details), taking the median of the posterior as our fiducial values. \nTable 1. NIRCam Imaging Summary \nNote -The depths given represent 5 σ limiting magnitudes, measured in d=0.2 '' diameter circular apertures and corrected to total fluxes assuming a point-source. The PSF FWHM and fraction of the flux enclosed in a 0.2 '' diameter circular aperture are given below the horizontal line. \nWe use these functional form fits for each filter to calculate the photometric uncertainties for each object, using both the number of pixels in its Kron aperture (Area = πab , where the semi-major [minor] axis a [ b ] is equal to the A[B] IMAGE keywords multiplied by the KRON RADIUS value), as well as the area value for a given circular aperture. We scale these values by the ratio of the error image value at the central position of a given source to the median error value of the whole map, thereby allowing the noise to be representative of the noise level around a given galaxy. We refer to these measurements as our 'global' noise measurements, which we use as our fiducial value. Finally, to account for variable image noise not captured by the error image value at the central pixel, for each object in our catalog we also calculate a 'local' noise measurement. This local noise was calculated in apertures with 0.2 '' , 0.3 '' , 0.4 '' and 0.5 '' -diameters, measured as 1.48 times the median absolute deviation of the flux distribution in the 200 closest apertures from the above process.", '2.2.4. Multi-band Catalog': "For each object in the catalog we use astropy.wcs wcs pix2world to derive celestial coordinates from the SE x, y positions ( SE cannot presently parse the worldcoordinate system in the JWST data model image headers). We calculate physical fluxes by applying a photometric zeropoint to convert the image from MJy sr -1 to erg s -1 cm -2 Hz -1 , and apply both aperture corrections derived above to all flux and flux error estimates. We \nTable 2. HST Imaging Summary \nNote -Similar to Table 1, for the HST imaging used. All depths given represent 5 σ limiting magnitudes, measured in d=0.2 '' diameter circular apertures and corrected to total fluxes assuming a point-source. \ncorrect for Galactic extinction using an E(B-V) of 0.006 for the EGS field and a Cardelli et al. (1989) Milky Way attenuation curve. \nWe create a multi-band catalog from the individualfilter catalogs created by SE , including our fiducial Kron apertures and fluxes measured in circular apertures with diameters ranging from 0.05 '' to 2.0 '' . For the latter we include fluxes measured from both the PSF-matched and native-resolution images; the latter are used below as a measure of detection significance. These circular apertures are corrected for Galactic attenuation, but not corrected to total, as we will use them solely for detection significance. For all flux measurements we calculate the noise per source following the above methods. We \nflag any sources that had either a zero or NaN in any error column, replacing their flux error with 10 12 nJy (several orders of magnitude larger than any real source error) such that these flux measurements do not impact any analysis. The final catalog contains only objects with valid measurements in the six broad-band filters, excluding the short-wavelength chips gaps, covering a total area of 88.1 arcmin 2 . \nIn Table 1, we include an estimate of the limiting 5 σ magnitude for our catalog. To calculate this, we use the noise functions described above to derive the flux density uncertainty in an aperture of diameter 0.2 '' . We then measure the enclosed flux at this radius from the stacked PSF. We then divide the flux uncertainty by the enclosed fraction of flux to estimate the total noise for a point source. Finally, we multiply this value by five, and convert to an AB magnitude. This final multi-band catalog was known as 'v0.51' internally to the CEERS team, and has been used in a variety of analyses (e.g. Arrabal Haro et al. 2023a,b; Vega-Ferrero et al. 2023a; Larson et al. 2023; Vega-Ferrero et al. 2023b; Ronayne et al. 2023).", '3.1. Photometric Redshifts': "The final photometric catalog contains 101,808 sources across the entire CEERS field (86.8% of which have signal-to-noise greater than three in F277W). We create a new numerical identifier as an ascending integer starting at 1 in CEERS1, adding in each field sequentially. We measure photometric redshifts for all sources in our 13-band photometric catalog using EAZY (Brammer et al. 2008). We perform three iterations of EAZY : (1) Fiducial : using our fiducial Kron, aperture corrected photometry with a maximum redshift of 20, (2) Low-z : The same as the fiducial run, but with the maximum redshift set to seven (allowing visualization of the best-fitting low-redshift model), and (3) circular : Replacing the Kron fluxes with the flux measured in d =0.2 '' diameter apertures, with a maximum redshift of 20 (see § 3.2.2). \nEAZY fits non-negative linear combinations of usersupplied templates to derive probability distribution functions (PDFs) for the redshift, based on the quality of fit of the various template combinations to the observed photometry for a given source. We use the same customized template list as F23, including the 12 FSPS (Conroy & Gunn 2010) templates in the recommended 'tweak fsps QSF 12 v3' set, supplemented with six templates created by Larson et al. (2022b) to span the blue colors expected for early galaxies,. We assume a flat prior in luminosity, and include a systematic \nerror of 5% of the observed flux values, and fit to our measured total flux and flux error values.", '3.2. Selection Criteria': "Here we describe the selection criteria we use to identify candidate z > 8.5 galaxies. Following our previous work (Finkelstein et al. 2010, 2015, 2022a,b, 2023), we use a combination of flux detection significance values and quantities derived from the full photometric redshift PDF, denoted P ( z ), to select our galaxy sample. We also make use of the peak P ( z ) redshift, denoted z best . All signal-to-noise ratios (SNRs), unless stated otherwise, are measured in 0.2 '' diameter apertures in the native resolution (non-PSF-matched) images. We note that while we primarily use the global empirical noise measurement, we also make use of the local noise measurements with slightly relaxed criteria as described below. \nWealso make use of SNR criteria in filters blue-ward of the Ly α break ('dropout' filters, which should contain no significant flux). To identify these filters, for each object we first zero out the P ( z ) at z < 7.5 such that any low-redshift solution does not impact the dropout filter choice (followed by renormalizing the P [ z ]). We then consider a filter to be a dropout filter if the wavelength corresponding to the red side of the filter transmission's FWHM is less than the observed Ly α wavelength at the 16th percentile of the renormalized P ( z ). In this way we make use of the full P ( z ) when calculating which bands should be considered a dropout filter. Here we list our selection criteria, separated into two categories: \nDetection Significance Criteria: \n- · Detection Signal-to-Noise: SNR > 5.5 ( > 5.0) in at least two of the F150W, F200W, F277W, F356W, or F444W filters, using the global (local) empirical noise measurements.\n- · Dropout Signal-to-Noise: SNR < 2.0 (3.0) in all bands fully blue-ward of the Ly α break, and no more than one filter with SNR > 1.5 (2.0), using the global (local) empirical noise measurements. For this we consider the ACS F606W and F814W filters, and the NIRCam F115W, F150W, and F200W filters (we note that some candidates have ∼ 1-3 σ flux measurements in WFC3 bands nominally below the Ly α break; however we consider these spurious as such objects show significant non-detections in the deeper shortwavelength NIRCam bands).\n- · Error-map values at the central pixel of the object < 1000 in the F115W, F150W, F200W, and F277W \nfilters. This ensures valid flux measurements in a minimum set of filters to robustly select z > 9 galaxies (and explicitly excludes the NIRCam short-wavelength chip gaps). \n- · Total F277W magnitude ≤ 29.2 as a conservative estimate to limit low-significance sources.", 'Photometric Redshift Criteria:': "- · ∫ P ( z > 7) ≥ 0.7, requiring at least 70% of the integrated P ( z ) at z > 7.\n- · z best > 8.5.\n- · S z ≥ 9, where S z is calculated as the unit redshift where the integral in a z ± 0 . 5 bin is the maximum compared to all other unit redshift bins.\n- · Total χ 2 ≤ 60 (for the 13 available bands).\n- · ∆ χ 2 > 4, calculated as the difference between the lowest χ 2 value at z < 7 to the best-fitting χ 2 value. \nThese criteria are very similar to those in F23, with the key differences being the selection of dropout bands, the more conservative SNR < 2 dropout criteria (compared to SNR < 3 in F23), and the addition of the local noise measurements. \nThe combination of this sample selection criteria yielded an initial sample of 185 z > 8.5 galaxy candidates across the full CEERS field. We visually inspected all objects (examining cutout images in all filters, as well as the photometric SED and the P ( z ) curves) to identify any spurious sources, or potential cases where the SE photometry could be questionable (e.g., the presence of a bright neighbor). We removed 91 objects through this screening process. Cutout images for all removed objects as well as a table of their positions are shown in Appendix B. \nObjects were removed for a variety of reasons, but the vast majority (78/91) were sources of an obvious spurious nature, including 10 sources identified as diffraction spikes, 45 sources associated with image edges, and 23 sources identified as bad pixels. The latter had a characteristic observational signature of a compact (few pixel), boxy morphology in the long-wavelength channel images only. We note that a few of our removed badpixel sources appear slightly less boxy than the bulk. It is possible these are real astrophysical objects at z ≳ 16 (where the Ly α break would be redward of F200W). However, as they are visible in the three long-wavelength channels only it is much more plausible (if indeed they are not bad pixels) that they are faint objects associated \nwith the known z ∼ 4.9 overdensity in this field (Arrabal Haro et al. 2023a), or low-redshift dusty galaxies (e.g. Bisigello et al. 2023), rather than true z ∼ 16 galaxies. Improved outlier pixel rejection in future iterations of our data reduction could reduce the frequency of such objects. Likewise, use of the context map ('CON') extension of the JWST data model could remove the need to manually identify image edges. We do note that in some cases, objects removed due to image edges have legitimate long-wavelength photometry, but are adjacent to a short-wavelength chip gap. \nThe remaining 13 sources were a combination of objects identified as being associated with nearby bright galaxies, inadvertently split into separate objects by SE (10 objects), and objects where the photometric SED did not appear accurate (three objects); e.g., obvious flux visible in a dropout filter that was not accurately recorded (usually due to crowding). Of the full sample of 91 sources removed, a small number (five) are plausible to still be true high-redshift galaxies. While we conservatively keep them out of our sample, we note them specifically in Tables 8 and 9 in Appendix B. \nFinally, we check this sample for duplicates which could arise as the 10 NIRCam CEERS pointings have slightly overlapping edges. We find one object which appears in our catalog twice, encouragingly satisfying our sample selection with two independent photometric measures. This object appears in CEERS4 (as ID 42447) and CEERS10 (as ID 93725), with similar P ( z ) distributions. As this object has slightly higher signalto-noise in the CEERS4 photometric catalog, we keep ID=42447, and remove ID=93725 from our catalog. We note that future work making use of a complete mosaic of all 10 NIRCam pointings will make better use of these near-edge regions by combining the images from both pointings. After removal of the 91 spurious sources and the one duplicate, the sample consisted of 93 candidate galaxies.", '3.2.1. Spectroscopic Redshifts': "Among the more transformative advances of JWST is NIRSpec's ability to efficiently spectroscopically confirm galaxies via strong [O III ] line emission to z ∼ 9.5 and via Ly α spectral breaks to higher redshift. We make use of followup spectroscopy from both the CEERS spectroscopic program (which in its second epoch placed NIRSpec slits on some sources observed in the first epoch) as well as a Director's Discretionary Time (DDT) program in the CEERS field (PID 2750, PI Arrabal Haro; Arrabal Haro et al. 2023a). Fujimoto et al. (2023a) presented [O III ]-based redshifts for sources in the ∼ 3100 sec CEERS second-epoch medium-resolution grating and \nFigure 1. A comparison of the photometric redshifts to the measured spectroscopic redshifts for the 17 galaxies in our initial sample with spectroscopic confirmation. With the exception of the (not shown) z ∼ 16 candidate at z spec = 4.9 (Arrabal Haro et al. 2023a), the agreement is generally good. After removal of three sources (in red) with spectroscopic redshifts below our cut of z > 8.5, we find a median (mean) z phot -z spec =0.1 (0.3), with 8/13 sources having | z phot -z spec | < 0.2. As noted by Arrabal Haro et al. (2023b) and Fujimoto et al. (2023a) there does appear to be a mild systematic offset towards higher photometric redshifts. We also note that there are four sources with z phot > 10 observed with NIRSpec that showed no detectable signatures, which is consistent with z > 10, where all strong lines are shifted out of the NIRSpec range. \n<!-- image --> \nprism observations, while Arrabal Haro et al. (2023b) presented both Ly α break and [O III ]-based redshifts for sources in the ∼ 3100 sec CEERS third-epoch prism observations (additional redshifts were also presented in Larson et al. 2023 and Tang et al. 2023). Arrabal Haro et al. (2023a) presented both Ly α break and line-based ([O II ] and [O III ]) redshifts for sources in the ∼ 5 hr prism DDT observations (one of these sources had its redshift of z = 11.04 first presented in Harikane et al. 2023a). \nCross-matching our sample of 93 galaxy candidates to these spectroscopic lists, we find 17 sources which have published spectroscopic redshifts. A comparison between the photometric redshifts and spectroscopic redshifts is shown in Figure 1. Not shown in this figure is the single catastrophic redshift failure (defined as | z spec -z phot | / (1 + z spec ) > 0 . 3), the galaxy (ID=13256) originally presented in Donnan et al. (2023b) as having z ∼ 16.5, with similar redshifts proposed by Harikane et al. 2023a and Finkelstein et al. (2023); a lower redshift of z phot =4.6 was proposed in P'erez-Gonz'alez et al. 2023a. As discussed in Arrabal Haro et al. (2023a), this \nobject has a confirmed redshift of z ∼ 4.9, and is the result of a very pathological situation where at this specific redshift a red galaxy with extreme line emission can mimic a z ∼ 16 galaxy as the H α line falls in all three of the F356W, F410M, and F444W filters, while [O III ] enhances F277W (see also discussion in Zavala et al. 2022). \nIn addition we remove one object that also has z phot ∼ 16 (ID=43382, with a 68% confidence range on the photometric redshift of z = 15.9 - 19.2). While fainter than ID=13256 (F277W = 28.8 versus 26.5), its spectral signature is almost identical, thus we consider it a likely fainter companion to the z ∼ 4.9 overdensity confirmed in Arrabal Haro et al. (2023a). \nWe find three additional sources with spectroscopic redshifts below our nominal redshift cut of z > 8.5, which we thus remove. These are: ID=4774, ID=4777 and ID=23084. ID=4774 has a photometric redshift 68% confidence limit (CL) of 8.26-10.27, with z spec = 8.01. ID=4777 has a photometric redshift 68% CL of 9.43-11.05, with z spec = 7.99. ID=23084 has a photometric redshift 68% CL of 8.08-9.22, with z spec = 7.77. While the spectroscopic redshift is outside the 68% confidence range on the photometric redshift, the redshifts are not catastrophically low. We tabulate the five removed sources, including their celestial coordinates, in Table 10 in the Appendix. \nBeyond these five removed sources, we find generally good agreement between the photometric and spectroscopic redshifts, with a median (mean) z phot -z spec =0.1 (0.3), with 8/13 sources having | z phot -z spec | < 0.2 (the median offset was 0.2 prior to removal of the five sources in the preceding paragraph). As noted by Arrabal Haro et al. (2023b) there does appear to be a systematic offset towards higher photometric redshifts. One likely explanation for this is that the shape of the Ly α break in these galaxy spectra is more extended than the sharp break assumed in the IGM attenuation models employed by EAZY . This could be due to a variety of factors, including stellar population properties not accounted by the typically used templates (e.g., Arrabal Haro et al. 2023b), the Ly α damping wing from an increasingly neutral IGM (e.g., Curtis-Lake et al. 2023; Arrabal Haro et al. 2023b; Umeda et al. 2023), and/or extremely dense line-of-sight damped Ly α systems in close proximity to a given galaxy (Heintz et al. 2023; Hsiao et al. 2023). Additionally some of the spectroscopic redshifts we compare to come from the Ly α break alone, which has additional uncertainties (see discussion in Fujimoto et al. 2023b). It is important to note that several additional sources were spectroscopically observed but not detected (indicated as 'Nz' in Tables 3 and 6); this is modest evi- \ne in favor of z > 9.6, as at lower redshifts [O III ]+H β should have been detectable. For the remainder of our analysis, we use the spectroscopic redshift values when they are available, which is the case for 13 of our final sample of 88 candidate galaxies.", '3.2.2. Kron Aperture Corrections': "During the visual inspection step we found that some legitimate high-redshift galaxies had Kron apertures which appeared much larger than the galaxy in question, stretched by nearby galaxies. Similar to F23, we devise a correction to ensure that flux from neighboring galaxies does not bias the colors nor the total fluxes. To identify sources where this is needed, we explore the ratio between the area of the Kron aperture and the area of a d =0.2 '' circular aperture. Our galaxy sample shows a log-normal distribution, with a peak aperture ratio of ∼ 2, with a tail to higher values. There is a notable gap at a ratio of ∼ 10, thus we flag sources with aperture size ratios larger than this as potentially needing a correction. \nWe find just one source meets this criterion, ID=11384 (with an aperture ratio of 14.7; the next highest was 9.3). Upon inspection of this source, it is very compact, but is in a region of high background with a very bright galaxy ∼ 1.5 '' to the NW, and a modestly bright galaxy ∼ 0.5 '' to the S, resulting in an elongated Kron aperture in the N-S direction. For this object we thus make use of colors measured in d =0.2 '' circular apertures. To calculate total fluxes we derive an aperture correction as the median of the ratio of the total F277W flux to the flux measured in a d =0.2 '' circular aperture for all sources in our full photometry catalog with d =0.2 '' fluxes within 20% of the F277W =0.2 '' flux for this object. We find this correction factor is 2.9 for ID=11384, consistent with the values for other sources in our galaxy sample (median of 2.3 ± 0.8). \nFor this object we then replace its default fluxes with these new values, and adopt the photometric redshift results from colors measured in the small circular apertures. This increases the photometric redshift from 10.8-11.4 to 11.2-11.8. Of note is that this source has a spectroscopic redshift of 11.043 (Harikane et al. 2023a; Arrabal Haro et al. 2023a). While the uncorrected photometric redshift is more consistent, the higher value from our improved photometry matches the observed very shallow Ly α break observed in the prism spectrum of this source (the redshift inferred from this break in the prism spectrum is z ≈ 11.4; Arrabal Haro et al. 2023a), which can lead to minor photometric redshift overestimates as discussed in the previous subsection. \n3.2.3. Ly α Break \nAs a cross-check on our Ly α -break criterion, we examine our sample for objects where there is a discrepancy between the primary photometric redshift peak and the SNR in bands nominally below the break; such sources can still satisfy our criterion for inclusion in a highredshift bin if their P ( z ) is bimodal or somewhat broad. \nWe identify two sources in our nominal z ∼ 11 sample (see § 3.4 for sample definitions) which have P ( z )'s that exhibit two high-redshift peaks; a larger peak at z > 9.5, and a significant secondary peak at z ∼ 9. These objects are ID=17898 and 42447 (neither have spectroscopic redshifts). Both objects exhibit SNR > 2 in F115W. As the lower 16th percentile of their P ( z ) was at z < 9.54 (the redshift of Ly α at the red edge of the F115W filter FWHM), this significant F115W flux did not violate our selection criteria. However, as thus flux is measured as significant (SNR=3.1 and 5.7 for these two objects, respectively), the z ∼ 9 peak is more likely to be correct. For these two objects, we thus applied a prior to their P ( z ), setting them to zero at z > 9.54, renormalizing to unity and recomputing the bestfit photometric redshift as the new peak. We find that the peak photometric redshift for these sources changes from z = 10.45 to z = 9.07 for ID=17898 and z = 10.30 to z =9.13 for ID=42447. We confirm that both sources continue to satisfy our selection criterion of P ( z > 7) > 0.7. \nWe also do a similar analysis for sources in our z > 13 sample. We find two sources with SNR in F150W > 2. These objects (ID = 2067 with SNR = 2.57, and ID = 77647 with SNR = 2.45) both exhibit a broad P ( z ), extending down to z ∼ 11, thus this F150W flux did not violate our selection criteria. We apply a similar prior, setting P ( z > 12.72; corresponding to the red edge of F150W) to zero. We find that the peak photometric redshift for these three sources changes from z = 13.69 to 12.70 and z = 13.57 to 12.70 (e.g., the new peak is at the edge of the prior). \nWe acknowledge that while EAZY had knowledge of these observed SNR ratios and still found a preferred peak at higher redshift, the obvious real flux in the images left us confident that applying this prior to the P ( z ) will result in more accurate redshift estimates. These P ( z ) priors were applied in the completeness simulations discussed in § 4.1. In Figures 4 and 5 we show the original P ( z ) as a faded black curve for these sources.", '3.3. Stellar Contamination': 'The colors of high-redshift galaxies, especially between 1-2 µ m, can be degenerate with low-mass stars and substellar objects (e.g. Wilkins et al. 2014; Finkelstein 2016). While the photometric coverage from 1- \nFigure 2. The F200W half-light radius (measured from SE ) versus F200W apparent magnitude. The gray bar shows the half-light radius from stars in the image (with the width showing the 68% spread in the values). The data points are color-coded by the difference in χ 2 between the best-fitting (sub)stellar model, and the best-fitting EAZY galaxy model. Most objects are clear resolved, and thus extragalactic in origin. Only one object is formally compact with a stellar χ 2 comparable to the best-fitting EAZY model, though even for this object we conclude it is likely extragalactic due to its non-compact appearance in the imaging, and the very large ( ∼ 4 kpc) implied distance were it stellar. We conclude that stellar contamination is not significant in our sample. \n<!-- image --> \n5 µ m should mitigate this confusion, here we explore whether the colors of our candidate galaxies could plausibly be consistent with low-mass stars or brown dwarfs. We fit each candidate to a grid of lowtemperature, cloudy, chemical equilibrium substellar atmosphere models from Sonora-Diamondback (Morley et al. 2023, in prep). We explore a range of temperatures T ∼ 900-2400 K, surface gravities g = 100 and 3160, and metallicities [M / H] = 0 and -0 . 5. We convolve the model SEDs with the HST + JWST filter curves and perform a simple grid-fitting routine, scaling the fluxes of each model to minimize the χ 2 . We adopt the model with the lowest χ 2 as the best-fitting stellar model. We estimate the implied distance by scaling the model fluxes and assuming an intrinsic radius of 1 Jupiter radius. We note that we also ran fits with cloud-free models grids extending to particularly cold (Sonora-Bobcat, T ∼ 2001300 K; Marley et al. 2021) and low-metallicity (LOWZ, [M / H] = -1; Meisner et al. 2021) parameter spaces; however, none provided a better fit over the SonoraDiamondback models. \nIn Figure 2 we show the results of this analysis. Here we plot the SE measured F200W half-light radius for our \n88 candidate galaxies versus their apparent F200W magnitude, compared to the half-light radius of the F200W PSF as measured from stars in the image. One can see that the majority of our galaxy sample is clearly resolved, thus non-stellar in origin. To diagnose the potential stellar nature of the more compact objects, the data points are color-coded by the difference in the goodness-of-fit ( χ 2 ) between the best-fitting (sub)stellar model and the best-fitting EAZY model. We find that no objects are better fit by the stellar model (e.g., the EAZY χ 2 is always lower). We do find two sources where the difference between the stellar and EAZY χ 2 is < 4. One is significantly resolved, but the other object (ID=34925) is measured by SE as being very compact (in fact, unphysically smaller than the PSF). However, we conclude this object is much more likely a galaxy. First, examining the imaging of this object, it does not appear to be obviously point-like; rather, it is faint, barely above our significance thresholds, thus the SE half-light radius is quite uncertain. Second, due to its faint brightness, its implied distance (were it stellar in origin) would be ∼ 4 kpc, which would be extremely far into the halo, and thus highly unlikely. We conclude that we find no evidence for stellar contamination in the sources in our sample.', '3.4. Sample Summary': "After removing visually-identified spurious sources, the four sources with z spec < 8.5, and the faint z ∼ 16 candidate which is likely at z ∼ 4.9, our sample contained 88 candidate z > 8.5 galaxies. For our analysis we divide these candidate galaxies into three sub-samples: z ∼ 9, which contains the 58 galaxies with 8.5 ≤ z best ≤ 9.7; z ∼ 11, which contains the 27 galaxies with 9.7 < z best ≤ 13; and z ∼ 14, which contains the three galaxies with z best > 13. \nFor all objects in our sample we calculate an observed rest-UV absolute magnitude following Finkelstein et al. (2015). Briefly, we perform a simple round of SED fitting with BC03 (Bruzual & Charlot 2003) models to derive a best-fitting model spectrum. We then calculate the bandpass-averaged flux from this spectrum in a tophat filter curve spanning 1450-1550 ˚ A in the rest-frame, converting to an apparent magnitude and then applying the cosmological distance modulus for a given redshift. As a part of this process, we run Monte Carlo simulations sampling the photometric redshift P ( z ), such that the resulting uncertainty on M 1500 is inclusive of both the photometric scatter and redshift uncertainty. We note that in this process we set P ( z < 6) = 0, such that any low-redshift solutions (which are small by design) do not bias the magnitude calculation. During this pro- \nTable 3. Summary of z > 9.7 Candidate Galaxies \nNote -A summary of the key properties for the 30 galaxies in our sample at z > 9.7 (the remaining 55 galaxies with 8.5 ≤ z < 9.7 are presented in the appendix). C FUV is defined in § 3.4 as the rest-frame far-UV color. The photometric redshift is 'za' from EAZY , which is the redshift where the χ 2 is minimized. The ∫ 20 7 P ( z ) quantity is the integrated redshift probability density between z = 7 and 20, which was used in the sample selection. The ∆ χ 2 compares the best-fitting low-redshift (0 < z < 7) model to the best-fitting high-redshift model; a value of ≥ 4 was required for selection. Spectroscopic redshifts come from Arrabal Haro et al. (2023a) and Arrabal Haro et al. (2023b); we list 'Nz' when an object was spectroscopically observed but no robust redshift was determined. As discussed in the text, this is modest evidence in favor of z > 9.6, as at lower redshifts [O III ]+ Hβ should have been detectable. \ncess we also calculated a rest-far-UV color (discussed in § 5), dubbed C FUV , calculated as \nC FUV = -2 . 5 log 10 ( f 1470 f 1850 ) , (2) \nwhere f 1470 and f 1850 are the bandpass averaged fluxes in top-hat filters spanning 1430-1510 and 1800-1900, respectively (where these windows were designed to probe the color in the far-UV avoiding strong spectral features). \nWe show the distribution of our sample in F277W magnitude and photometric redshift in Figure 3. Figures 4, 5, 6 contain summary figures of sources in the z ∼ 14, 11 and 9 samples, respectively. The latter two are figure sets, with two example objects shown, and with all objects viewable in the electronic version of the manuscript. We tabulate key properties of our sample in Table 3 for z > 9.7, and Table 6 in the Appendix for z < 9.7.", '3.4.1. Comparison to McLeod et al. 2023 and Adams et al. (2023)': "We compare our sample to that of McLeod et al. (2023) and Adams et al. (2023), who have also selected z ≳ 8.5 galaxies from the full CEERS dataset. McLeod et al. (2023) restrict their sample to galaxies with detection signal-to-noise greater than eight, thus their sample is smaller than ours, at 23 galaxies. We find that 14 of these 23 galaxies are in our final sample. We explored the properties of the remaining nine galaxies in our catalog. We find that 2/9 objects have dominant z > 9 solutions, but fail our ∆ χ 2 criterion. The remaining seven objects all have plausible high-redshift solutions (based on little-to-no detectable flux in F115W), but have dominant low-redshift solutions. \nWhile Adams et al. (2023) did not provide a catalog in their paper, we obtained the revised version of their sample via private communication. Their final sample of z > 8.5 galaxies consists of 55 objects, of which 25 are in our final z > 8.5 galaxy sample. All 30 not contained in our sample are present in our parent photometric catalog, all with a plausible redshift solution at z > 7 (16/30 have > 50% of their integrated P ( z ) at z > 7 in our catalog). Two of these sources nominally pass all our selection cuts except their best-fitting photometric redshifts are z ∼ 7.5-8.2. Of the remaining 28, 26 do not pass our ∆ χ 2 cut, including 18 which do not pass our integrated P ( z ) cut. This is likely driven in some cases by weakly positive ( ≥ 1 σ significance) flux present in F606W and/or F814W in 14/28 objects, down-weighting high-redshift solutions. \n3.4.2. Differences from Finkelstein et al. (2023) \nIn F23 we presented 26 z > 8.5 candidates from the first epoch of CEERS, using 4 of the 10 CEERS NIRCam fields used here. As our photometry and sample selection procedures here are slightly updated, we crosschecked our sample with the F23 sample. We find that 20/26 galaxies presented in F23 are included in our sample here. Of the six galaxies not included here, four of them originally satisfied our sample selection but were removed because they are now known to have spectroscopic redshifts z spec < 8 . 5 (see § 3.2.1). These are ID=4774 (F23 ID 3908), 4777 (F23 ID 3910), 13256 (F23 ID 2159), and 23084 (F23 ID 1748). The remaining two sources satisfied all sample selection criteria except the SNR below the Ly α break. ID 2241 (F23 ID 1875) has a 1.9 σ detection in F606W and 1.98 σ detection in F814W, failing that criterion. ID 8497 (F23 ID 7227) has a 2.6 σ detection in F115W, which fails this criterion for this object's P ( z ) distribution, which peaks at z = 11.2. \nIn our present work over these four fields we identify 35 z > 8.5 candidates, which includes 20 galaxies from F23 and 15 new sources. We explored these new sources to see why they were not included in the F23 sample. All 15 sources are present in the F23 photometric catalog. Of these 15, 11 sources have best-fit photometric redshifts in the Finkelstein et al. (2023) catalog of z > 8.5. Comparing the photometric redshifts of these 11 sources in this previous catalog to our own, we find a median photometric redshift difference of zero (with a mean difference of 0.11, with the new values being slightly smaller). The majority of these sources (8/11) were previously excluded as they failed the ∆ χ 2 criterion (with values of ∼ 1-3). The other three fell just on the other side of the F23 best-fit photoz , detection threshold, or Ly α break non-detection thresholds. Of the four objects fit at lower redshift, all three have significant high-redshift peaks, with ∫ P ( z ) > 7 of 0.73 (ID 10545), 0.43 (ID 17898) and 0.66 (ID 61620; with the remaining source, ID 20174, having a lower significance peak at z ∼ 9). \nWhile this comparison makes it apparent that modest changes to photometry procedures can have notinsignificant effects on the composition of a high-redshift galaxy sample, the changes we implemented to our procedure here over that from F23 were done to increase photometric accuracy, primarily for the colors of faint galaxies. The fact that all new sources which we select had significant high-redshift solutions in the F23 catalog adds confidence to our sample, though we fully acknowledge that spectroscopic confirmation of a majority of this sample is needed for full confidence. Fortunately, with the power of JWST , this is possible. \nFigure 3. The symbols show our sample of 88 z > 8.5 galaxy candidates in a plane of F277W magnitude versus redshift, with the different colors representing the different redshift samples. Squares denote objects with spectroscopic confirmation, while the circles are plotted at the photometric redshifts. Triangles denote objects with spectroscopic observations but no confirmation. The small star denotes Maisie's Galaxy (Finkelstein et al. 2022b), one of the first JWST very high-redshift galaxy discoveries, while the small dot denotes CEERS-1019, a confirmed z = 8.7 galaxy which appears to have broad H β emission, indicative of an active super-massive black hole (Larson et al. 2023). The background shading shows the completeness (inclusive of both photometric and sample selection completeness) of our sample (for sources with half-light radii of 3.3 pixels), as described in § 4.1; the black line shows the 20% completeness contour. \n<!-- image -->", '4. RESULTS': 'Here we explore constraints on the abundance of galaxies at z > 8.5 that we can place with our sample selected from the full CEERS survey. In § 4.2 we describe measurements of the cumulative surface density, while in § 4.3 we describe the rest-frame UV luminosity function. Both measurements require a correction for incompleteness, which we describe in § 4.1.', '4.1. Completeness Simulations': "We quantify the completeness of our sample selection, broadly following F23, with an update here to implement an important size dependence. Our completeness estimates come from complete end-to-end source injection simulations, injecting mock galaxies with a range of properties into our images, then performing photometry, photometric redshift measurements, and sample selection procedures identical to our that done on our real data. In this way, we account for incompleteness due to both photometric effects, as well as sample selection effects. \nFor each of the 10 CEERS NIRCam pointings we run 50 simulation iterations. Within each iteration we simulate 1000 sources over a uniform range of redshift from 8 < z < 17. The majority of the iterations had a log-normal distribution in F277W apparent magnitude peaking at m ∼ 29; we supplement these with additional simulations with a flat magnitude distribution to boost the number of brighter galaxies. The result is a roughly flat distribution from m F 277 W = 22-26, with a larger, log-normal-shaped distribution from m F 277 W = 26-30. \nTo simulate the fluxes in all observed HST and JWST /NIRCam filters, we use BC03 (Bruzual & Charlot 2003) stellar population models with a distribution of stellar population age, dust attenuation and metallicity tuned to reproduce the expected (and now observed, e.g., Cullen et al. 2023) blue colors of very high-redshift galaxies (see Finkelstein et al. 2015 for details on these models). The result is a log-normal distribution of restUV colors, which peaks at F200W -F277W = -0.05, with a 68% spread from F200W -F277W = -0.25 to +0.3, comparable to the measured colors of our observed objects (median of -0.1, 68% spread from -0.3 - 0.3). \nFigure 4. Summary figures of the three objects in our sample with z best > 13. The top row shows 1 '' cutout images in the F814W, F115W, F150W, F200W, F277W, F356W and F444W filters (the F410M and remaining HST bands are not shown for brevity). The cyan circle in the first panel shows a 0.2 '' diameter circle (the size used to measure detection significance). The bottom-left panel shows the SED, with blue (green) points representing photometry from NIRCam ( HST ). Upper limits shown are 1 σ . The bottom-right panel shows the photometric redshift distribution in blue (with the corresponding best-fit model in blue in the bottom-left panel). The light red curves show the photometric redshift results when constrained to z < 7. \n<!-- image --> \nThe resulting model spectrum was then normalized to the F277W magnitude for a given object, with magnitudes in the remaining filters derived by integrating this spectrum through a given bandpass. \nSource morphologies were created using galfit (Peng et al. 2002) assuming a S'ersic profile with a log-normal distribution in the S'ersic index n (peaking at 1.2, with a minimum of n =1 and a tail to n =5), a log-normal distribution of the axis ratio with a peak at 0.8, and a uniform distribution of the position angle. While in F23 we tuned the half-light radius ( r h ) distribution such that the recovered r h distribution matched that observed in our sample, here we update our procedure to implicitly include the size of galaxies in our sample in the completeness calculation. We thus use a uniform input distribution of r h from 1-8 pixels (0.03 - 0.24 '' ). Finally, \nthe empirical PSF described in § 2 is provided to galfit for a given band. \nThese simulated galaxy images are then created with galfit as 101 × 101-pixel images, which we add at random positions to the real images (avoiding only image edges and regions of extremely high ERR-map values [ > 1000]). Notably we do not avoid the positions of real objects, such that our simulations account for incompleteness due to sources along similar lines-of-sight to high-redshift galaxies. We also create a full-frame image just of the simulated sources in F277W, on which we run SE to measure a 'noiseless' version of the recovered SE r h in this filter for use with the completeness calculations (as we discuss below, the measured SE r h is biased low from the galfit input value, even in the noiseless image). \n<!-- image --> \nFigure 5. Same as Figure 4, for two example sources in the z ∼ 11 galaxy candidate sample. The object on the left has a confirmed spectroscopic measurement via detection of the Ly α break from Arrabal Haro et al. (2023b), while the object on the right is the spectroscopically confirmed Maisie's Galaxy (Finkelstein et al. 2022b; Arrabal Haro et al. 2023a). The complete figure set (27 images) is available in the online journal. \n<!-- image --> \n<!-- image --> \nFigure 6. Same as Figure 4, for two example sources in the z ∼ 9 galaxy candidate sample. Both example sources have confirmed spectroscopic measurements via [O III ] line emission from Fujimoto et al. (2023a). Of note is that both sources were beyond the detection limit of the CANDELS imaging in this field, yet appear very bright and extended in these ∼ 2900 sec JWST /NIRCam data. The complete figure set (58 images) is available in the online journal. \n<!-- image --> \nThe result of this process is a version of our data in all filters with these 1000 simulated sources included, as well as a catalog of their input properties (inclusive of the SE r h measurement of the input image). At this point, we run an identical process on these images as we did on our real data in § 2 and § 3, including creating the array of PSF-matched images, calculating photometry, aperture corrections, and PSF corrections. Photometric catalogs are created, EAZY is run, and M 1500 is calculated. \nThe recovered catalogs are then matched against the input catalogs, resulting in a combined catalog of all recovered sources with their input and recovered properties. As the imaging depths in the 10 CEERS fields \nare broadly similar (Table 1), we combine these catalogs from all 10 fields into one. This final input catalog consists of 500,000 simulated sources, of which 343,196 sources are recovered by SE (the ∼ 70% recovery fraction is expected given the larger number of very faint sources that were simulated). \nWhile in F23 we calculated completeness only as a function of magnitude (with an assumed size distribution), here we explicitly add a size dependence, as many of the observed high-redshift galaxies are clearly resolved. For a given bin in absolute magnitude, we calculate the completeness in bins of F277W half-light radius (with a bin width of 0.5 pixels, starting with a bin centered at 1.2 pixels). For this process, we assume the \n<!-- image --> \nFigure 7. The left panel shows the effective volume for our z ∼ 11 (9.7 < z ≤ 13) galaxy sample. The purple dashed line shows the maximum volume one would obtain with a 100% completeness over an ideal (though unrealistic) top-hat selection function. The colored curves show our calculated effective volumes as a function of source half-light radii. The circles show the volumes we use, calculated by weighting the volumes by the sizes of galaxies in a given magnitude bin. The size distribution of our z ∼ 11 galaxy sample is given in the lower-right plot, while the completeness as a function of size at fixed input UV absolute magnitude is shown at the upper-right. The completeness is very sensitive to the size, particularly at fainter magnitudes. \n<!-- image --> \nintrinsic size as the half-light radius measured by SE on the noiseless images, as described above. We show how the completeness depends on both UV absolute magnitude and half-light radius in Figure 7. While there is minimal dependence on size at the brightest magnitudes, as one goes fainter the completeness decreases more steeply for larger sources, as expected. \nAs we might expect the recovered sizes from both the simulated and real data to be biased, we compare these intrinsic sizes to those measured from the recovered sources. We find that the intrinsic sizes are ∼ 50% larger. To account for this bias when we calculate the completeness we multiply the recovered SE sizes of our real sources by this scale factor of 1.53 (this quantity has a small scatter of σ =0.06, and is independent of magnitude for m < 29). We then set a minimum size for real sources of r h = 2.7 pixels, comparable to that expected for an unresolved source. We also set a maximum size of r h = 6.0 pixels; one object has a larger (bias-corrected) SE -measured size, yet it is clear from the images these sizes are incorrectly measured due to the presence of a neighboring source. For this object, we assume the median size from the sample. \nFor our surface density analysis described in § 4.2, we calculate a completeness per source, based on the fraction of recovered sources at the observed F277W mag- \nitude, photometric redshift, and galaxy size. For our rest-frame UV luminosity function analysis described in § 4.3, we calculate the completeness in bins of absolute UV magnitude. In each magnitude bin we calculate a completeness by weighting the recovery fractions in each size bin by the observed number of sources at each size. We then integrate the comoving volume element across our redshift range with these completeness values, resulting in an effective volume for our sample in a given magnitude bin (a process similar to that done in Finkelstein et al. 2015). These 'weighted mean' effective volumes are shown as the circles in Figure 7. They broadly trace the completeness curves for sources with r h ∼ 3.5 pixels, comparable to the median size of objects in our sample. One can see that assuming point sources in these simulations would lead to effective volume estimates modestly larger across M UV ∼ -20 to -19. Our final effective volumes are listed in Table 4.", '4.2. The Cumulative Surface Density of Galaxies at z > 8.5': "Here we explore the evolution of galaxies at z > 8.5 via the cumulative surface density, updating the results from the first epoch of CEERS from F23. Figure 8 shows the surface density of objects with redshifts greater than z for sources with m F 277 W < 28.5. Following F23, we \nFigure 8. The top panel shows the redshift distribution of our sample (scattered vertically for clarity), while the main panel shows the cumulative surface density of galaxies per unit surface area at redshift greater than z for m< 28.5. The dotted line shows the raw counts from our CEERS z > 8.5 galaxy sample, while the shaded region shows the 68% confidence interval on the completeness corrected surface density (including only sources with completeness estimates > 20%), inclusive of photometric, photometric redshift, and Poisson uncertainties. The vertical error bars show the estimated cosmic variance uncertainty. The colored lines show predictions from a range of pre-launch simulations (including hydrodynamic, analytic, semi-analytic, and semi-empirical models), all also for m < 28.5. The observed surface density of galaxies lies above most predictions at z > 10, and above all except the Behroozi & Silk (2015) and FLARES models at z > 11. This confirms early results based on smaller samples that the observed abundance of z ≳ 10 galaxies significantly exceeds most pre-launch physically-motivated expectations. \n<!-- image --> \ncorrect for incompleteness by counting each galaxy as one divided by the estimated completeness at the redshift and magnitude of a given galaxy, including here the size of the source (as well as the size bias correction discussed in § 4.1). To avoid significant uncertainty introduced by objects with very low completeness values, we further restrict this analysis to objects with completeness measurements greater than 20% (this includes 58 objects of the 66 with F277W < 28.5; this exclusion was not done in F23). We note that our magnitude limit is chosen as a compromise between maximizing the sample size while minimizing the completeness correction. While 25% of our sample is fainter than this limit, the majority of these fainter sources have completeness < 20%. On the other hand, increasing the limit to F277W < 28.0 would cut the sample in ∼ half, reducing the constraining power of our observations. \nThe shaded region shows the 68% confidence range on the cumulative surface density, derived via Monte Carlo simulations inclusive of photometric and photometric redshift uncertainties, as well as sampling the \nPoisson uncertainty. We separately show estimated cosmic variance uncertainties at four different redshifts as the vertical error bars calculated based on the bluetides simulation (Bhowmick et al. 2020, with caveats as discussed in F23). The dotted line shows the raw measurements with no correction. While not shown for clarity, the values from Finkelstein et al. (2023) lie at the low end of our posterior distribution here (which is consistent with our finding of ∼ 3.4 × more galaxies in ∼ 2.6 × more area). \nWe compare here to the same nine JWST model-based pre-launch predictions as in F23, including semi-analytic models (SAMs; Yung et al. 2019, 2020; Dayal et al. 2017), empirical models (Behroozi & Silk 2015; Mason et al. 2015; Behroozi et al. 2019), and cosmological hydrodynamic simulations (Wilkins et al. 2022a; Kannan et al. 2022; Kannan et al. 2022; Dav'e et al. 2019). As discussed extensively in F23, the compilation of model predictions here is inclusive of several different modeling approaches and assumptions (see Somerville & Dav'e (2015) for a thorough review). \nSimilar to F23, we find that at z < 10, our observations are higher than most predictions, yet are consistent with the Behroozi & Silk (2015) empirical and delphi SAM models, as well as the FLARES hydro predictions with no dust attenuation. At z > 10, we find that our results lie significantly above all predictions with the exception of Behroozi & Silk (2015). While this agreement is intriguing, this is not a physics-based model, but rather is based on the ansatz that the specific star formation rate in galaxies is proportional to the specific total mass accretion rate into halos. We thus confirm the initial result from F23 that the observed abundance of z > 10 galaxies discovered in CEERS lies above nearly all pre-launch predictions, where here the result is at higher confidence based on an updated sample ∼ 3 × larger than the F23 sample.", '4.3. The Evolution of the Rest-UV Luminosity Function at z = 8.5-14.5': "Here we present our measurements of the rest-frame UV luminosity function, measured separately for our z ∼ 9, z ∼ 11 and z ∼ 14 samples, where galaxies are again placed into a sample based on their bestfitting photometric redshift (or spectroscopic redshift when available). Stacking the P ( z )'s of each sample (as shown in Figures 9 and 10), the median redshift in each bin is z = 8.9, 10.9 and 14.0, respectively. We note that our redshift bins become progressively wider for two reasons. First, at these redshifts, unity redshift bins become small in terms of cosmic time spanned. Second, our boundaries of z = 9.7 and z = 13 correspond to redshifts where the Ly α break falls directly between two NIRCam filters (F115W and F150W, and F150W and F200W, respectively), thus creating natural redshift demarcations given our available data. \nWe measure our luminosity function following the methodology of F23, here using a bin size of 0.5 magnitudes, with three key differences. The first is that here we use three redshift bins, and in the bin in common ( z ∼ 11) here our sample size is nearly ∼ 3 × larger (27, compared to 10 in F23). The second difference is our use of the size of the sources in the completeness correction, which as shown in § 4.1 can affect the inferred effective volumes (Figure 7). When calculating the effective volume, we first calculate the effective volume as a function of absolute magnitude and size by integrating over the co-moving volume element as \nV eff ( M UV , r h ) = ∫ dV dz C ( z, M UV , r h ) dz, (3) \nwhere C ( z, M UV , r h ) is the completeness as calculated in § 4.1. In each magnitude bin, we then calculate a \nfinal effective volume as the weighted mean of the effective volumes in all size bins weighted by the number of sources in that magnitude bin at a given size (corrected for the ∼ 1.5 × radius bias described in § 4.1) as \nV final ( M UV ) = ∑ i V eff ( M UV , r h,i ) × N ( M UV , r h,i ) ∑ i N ( M UV , r h,i ) , (4) \nwhere i is a radius index, and N ( M UV , r h,i ) is the number of galaxies in a given UV magnitude and size bin. \nThe third difference comes when we estimate the number density in each bin via MCMC (see full details on this methodology in Finkelstein et al. 2015). We do this to sample galaxy absolute magnitude posterior distributions such that galaxies can fractionally span multiple magnitude bins. In F23 we treated each magnitude bin separately. Here, following Leung et al. (2023) each step in the MCMC chain samples the entire ensemble of galaxies for a given redshift bin, such that when a galaxy scatters out of one bin, it is accounted for in another bin (and thus is more realistic than what was done in F23). \nOur measured number densities and final weightedmean effective volumes are listed in Table 4 for all three redshift bins. For bins with no galaxies, we list the 84% upper limit from the MCMC calculation. We also demarcate fainter bins with completeness values < 20%; while we list these values, we caution that they are dominated by the completeness correction. \nIn Figure 9 we plot these luminosity function measurements compared to a wide range of literature results and extrapolations. In this figure we also prominently show the results from NGDEEP at fainter luminosities from Leung et al. (2023) at z ∼ 9 and 11, as their photometry and sample selection process are nearly identical to our own (while we use the Leung et al. 2023 measured number densities here, we verified that the results would be very similar had we calculated the number densities with their sample in our slightly modified redshift bins). At M UV = -19, where our results overlap, we find excellent agreement between CEERS and NGDEEP, with the NGDEEP results then continuing on faint-ward at a consistent slope at both z ∼ 9 and 11. \nThe small faded symbols in Figure 9 show results from the literature. At z ∼ 9 we find broadly good agreement with these literature results (which come from HST , JWST and ground-based studies). We note that our brightest bin is higher than previous results - this bin contains a single object, the z ∼ 8.7 potential AGN from Larson et al. (2023). Based on the confirmation of this object and another at a similar spectroscopic redshift, Larson et al. (2022a) concluded that the EGS region is ∼ 4-10 × denser than average at z ∼ 8.7 (see also Whitler \nTable 4. CEERS Rest-UV Luminosity Functions \nNote -† This column represents the nominal number of galaxies in a magnitude bin, though in the calculation of the luminosity function we account for galaxies moving between bins due to photometric and photometric redshift uncertainties. ‡ The completeness in these bins is < 20%, thus these data points are not used to constrain the luminosity function evolution. \net al. 2023). While our unit-redshift bin used here mitigates this overdensity somewhat, at the bright end where numbers are small, our results are mildly higher than those from the literature, becoming broadly consistent at M UV ≳ -20. We do note that when considering the full CANDELS area ( ∼ 10 × wider than we consider here), Finkelstein et al. (2022a) found a volume density ∼ 5 × lower in the magnitude bin including this object. \nAt z ∼ 11 we compare to a compilation of z ∼ 1012 results from the literature (Donnan et al. 2023b,a; McLeod et al. 2023; Bouwens et al. 2023a; Harikane et al. 2023a; P'erez-Gonz'alez et al. 2023b; Castellano et al. 2023; Franco et al. 2023; Casey et al. 2023; Adams et al. 2023). While there is significant scatter, we find generally good agreement between our results and previously published values in the literature, where here our larger sample size (than most studies) results in smaller uncertainties. We note in particular good agreement be- \nour values and those from McLeod et al. (2023) and Adams et al. (2023), which are the only other previous studies to make use of the full CEERS area. \nEach panel of Figure 9 also shows an empirical extrapolation of the UV luminosity function from Finkelstein & Bagley (2022). In this study, they fit an evolving doublepower-law model to all available UV luminosity function data at z = 3-9, assuming that the double powerlaw (DPL) parameters ϕ ∗ , M ∗ , β and α vary smoothly with 1+ z (they also simultaneously fit the evolution of the AGN UV luminosity function with a separate DPL, though at the magnitudes we consider here star-forming galaxies were found to dominate). In each panel we show the measured DPL fits at z = 5-8 as the gray lines, then showing the extrapolation to a given redshift range as the light shaded region (where for the extrapolations, we used the MCMC chains from Finkelstein & Bagley 2022 to generate samples of the DPL parameters for each redshift). The upper and lower bounds show the median DPL at the upper/lower bound of the FWHM from the stacked P ( z ), as shown in the inset panels. \nWe show this more clearly in Figure 10, where we overplot both our measured number densities and these extrapolated luminosity functions for all three of our redshift samples. Based on pre-launch expectations of either a smoothly or rapidly declining luminosity function at z > 8, we would expect to see our results fall either on or below these extrapolations. While the results at z ∼ 9 are consistent with this extrapolation (with the exception of the brightest points, which are affected by the known overdensity), we see clearly and significantly that our z ∼ 11 observed number densities lie modestly above this extrapolation, while the z ∼ 14 observed number densities lie even higher above their extrapolated region.", '4.4. Double Power Law Fits': "By combining with NGDEEP, we are able to sample 3-4 magnitudes of dynamic range in M UV at z ∼ 9 and z ∼ 11. We therefore fit a double-power law function to our observations, following evidence that this functional form better represents the UV luminosity function at high redshifts than a Schechter function (e.g. Bowler et al. 2015, 2020; Finkelstein & Bagley 2022). We fit a DPL to each of our three redshift bins independently via a MCMC algorithm (following the methodology of Leung et al. 2023). We assign fairly uninformative priors on all parameters at z ∼ 9 and 11; at z = 14 due to our poor observational constraints we fix M ∗ and β to the z ∼ 11 values (within a small tolerance). We list \n<!-- image --> \n<!-- image --> \nFigure 9. The evolution of the rest-frame UV luminosity function, at z ∼ 11 (9.7 < z best ≤ 13; top), z ∼ 9 (8.5 < z best ≤ 9.7; bottom-left) and z ∼ 14 (13 < z best ≤ 15; bottom-right). The large circles show the calculated number densities from our sample (the small red dots denote the magnitudes of individual galaxies, offset vertically for clarity), while the squares show the results from NGDEEP (Leung et al. 2023). The inset shows the stacked P ( z ) for each sample, with the dotted line denoting the median value of the P ( z ). Arrows show 1 σ upper limits in the first bin with no galaxies, while white-filled symbols denote bins which are < 20% complete. The black dot in the brightest bin at z ∼ 9 indicates that this bin has only one object, the z = 8.7 galaxy (which has AGN signatures) from (Larson et al. 2022a). Small symbols show literature results. At z ∼ 9 the HST results are from Bouwens et al. (2019, 2021); Bowler et al. (2020); McLeod et al. (2016); Morishita et al. (2018); Stefanon et al. (2019); Rojas-Ruiz et al. (2020); Bagley et al. (2022a); Finkelstein et al. (2022a), while the JWST results are from Donnan et al. (2023b); Adams et al. (2023); Harikane et al. (2023a); P'erez-Gonz'alez et al. (2023b); Bouwens et al. (2023b). The z ∼ 11 results shown are from Donnan et al. (2023b,a); McLeod et al. (2023); Adams et al. (2023); P'erez-Gonz'alez et al. (2023b); Bouwens et al. (2023b); Harikane et al. (2023a); Castellano et al. (2023); Franco et al. (2023); Casey et al. (2023), while at z ∼ 14 we compare to Donnan et al. (2023b) and Casey et al. (2023). The gray curves show the best-fitting double-power law (DPL) model from Finkelstein & Bagley (2022) from z = 5-8, while the light-shaded colored region shows this model empirically extrapolated to the median redshift for a given sample (where the width is the 68% uncertainty on the luminosity function at this redshift from Finkelstein & Bagley 2022). Our CEERS results are generally consistent with previous luminosity function estimates, with smaller uncertainties reflecting our larger sample size. We also note excellent agreement with the NGDEEP results where our samples overlap. Our brighter CEERS results sit above the expected number densities for the empirically expected extrapolation from Finkelstein & Bagley (2022), with this offset increasing to higher redshift. \n<!-- image --> \nthe priors and posterior results in Table 5. The median DPL fit is shown as the thin line in Figure 10. While this does a reasonable job of representing the data, the uncertainties, particularly on β and M ∗ at all redshifts, \nand on the faint-end slope α at z ∼ 14 are presently quite large. We do note that our measured faint-end slope of -2.2 at z = 11 is consistent with the value from Leung et al. (2023), though they imposed more \nTable 5. UV Luminosity Function Double Power-Law Parameters \nNote -Constraints on the rest-UV luminosity function assuming a double power law form. We place priors on M ∗ and β at z ∼ 14 to match the values measured at z ∼ 11. The final column lists the specific luminosity density, obtained by integrating the luminosity functions at magnitudes brighter than -17. \nFigure 10. Our measured UV luminosity functions at z ∼ 9, 11 and 14 are shown by the large symbols (circles for our CEERS results, and squares for NGDEEP from Leung et al. 2023). The shaded regions are the same as in Figure 9, showing the extrapolated UV luminosity functions from Finkelstein & Bagley (2022). The inset panel likewise shows the same P ( z ) curves from Figure 9, here plotted on the same scale. The thin curves show the median DPL fit to the data at each redshift. This figure highlights that brighter galaxies ( M UV ≲ -20) have higher number densities than the extrapolated luminosity functions would predict. While there is a known overdensity at z ∼ 8.7 (Larson et al. 2022a; Whitler et al. 2023) which could affect our lowest-redshift bin, there is no evidence for such overdensities at higher redshifts. \n<!-- image --> \nrestrictive priors on β and M ∗ , thus our uncertainty is higher.", '4.5. Evolution with Redshift': "To explore the evolution in the UV luminosity function in more detail, in the left-hand panel of Figure 11 we show our observed number densities for the CEERS bin closest to M UV = -20 at each of our three redshift bins, on top of the expectations for this number density from the Finkelstein & Bagley (2022) extrapolation. This figure clearly shows that the observed number densities diverge from the observed evolution at z = 3 to 9, flattening at higher redshifts. We quantify this by \nmeasuring the slope ( d log ϕ / dz ) both for previous observations at z = 3-9, and our results here at z = 9-14. We find that this slope changes from d log ϕ / dz = -0.29 ± 0.03 at z = 3-9 to -0.11 ± 0.08 at z = 9-14. Thus, while the abundance of bright ( M UV = -20) galaxies evolves somewhat steeply at a constant slope at z = 39, this evolution is flatter towards higher redshifts at the 2.1 σ significance level. \nFor this calculation we have used as our fiducial values the actual measured number densities at M UV = -20. In the left panel of Figure 11 we show as small stars the values of ϕ ( M = -20) from the median DPL model at each redshift. The differences from the observed values are negligible at z = 9 and 14, while at z = 11 this parameterized value is slightly below the observed value at z = 11. Using these DPL values, the observed evolutionary slope steepens to d log ϕ / dz = -0.18 ± 0.07 (reducing the significance of the flattening to 1.5 σ ). \nTo consider whether the evolution in the abundance of faint galaxies changes at z > 9, in the right-hand panel of Figure 11 we show the specific UV luminosity density ( ρ UV ), obtained by integrating the UV luminosity function from our DPL fits to M UV < -17 (integrating each step of the chain to calculate the median and 68% confidence range); this quantity is dominated by the abundance of faint galaxies given the steep faint-end slopes (we note that we plot this quantity rather than the number densities at fainter magnitudes due to the lack of constraints at z ∼ 14, visible with the very large error at this highest redshift in the plotted integrated quantity). Interestingly, our measured specific UV luminosity densities at z = 9 and z = 11 are fully consistent with the extrapolated values (at z ∼ 14 the poor constraints on the faint-end slope leave the uncertainty is too large to reach definitive conclusions, thus we show this as a faded data point). This can also be observed in Figure 10, as the faintest data points at z ∼ 9 and 11 are consistent with the extrapolated luminosity function. A similar specific UV luminosity density is found \n<!-- image --> \nFigure 11. Left) The evolution of the observed number density at M UV = -20. The red circles show the observed number density at this absolute magnitude from CEERS (connected by the light red shaded region; the small stars show the DPL fit values at this magnitude). The dark blue region shows the measured value from Finkelstein & Bagley (2022), and the lighter shaded region shows the extrapolation of the Finkelstein & Bagley (2022) results to higher redshift. While pre-launch expectations were that the number densities at z > 9 would either continue the observed trend at z = 3-9, or evolve more rapidly downward with increasing redshift, we find that the number density of bright galaxies surprisingly flattens at z > 9, where we measure a change in slope d log ϕ / dz between z < 9 and z > 9 at 2.1 σ significance. Right) The evolution of the integrated specific UV luminosity density, obtained by integrating double-power law fits to our observed luminosity functions to M UV = -17. The evolution here is less clear, with increased uncertainties (particularly at z ∼ 14, which is shown faded to represent its large uncertainty) making it less clear whether the this quantity also has a flatter evolution at higher redshift. \n<!-- image --> \nby P'erez-Gonz'alez et al. (2023b), while McLeod et al. (2023) finds a slightly elevated value, though consistent with the empirical extrapolation within the uncertainties. \nTaking both panels of Figure 11 together, we find clear evidence that the evolution of the number density of bright galaxies is observed to flatten at z > 9, while the evolution of the integrated UV luminosity density, which is dominated by the abundance of fainter galaxies, is less clear, and may possibly follow the lower-redshift evolutionary trend extrapolated to higher redshift. We discuss potential physical explanations for this intriguing result in the following section.", '5. DISCUSSION': 'In § 4 we presented strong evidence that (i) the abundance of galaxies at z > 9 is in excess of nearly all preJWST launch simulation predictions as well as above extrapolations from lower-redshift observations, and (ii) the evolution of the abundance of bright ( M UV = -20) galaxies is flatter at z = 9-14 than at z = 3-9. Here we explore several potential explanations for these observations. Potential explanations could be due to galaxies being brighter than predicted or more numerous (e.g., horizontal evolution in the luminosity function rather than vertical). However, while the former is relatively easily achievable via a variety of reasonable physical modifications (as we discuss below) the latter would re- \nuire major revisions to modern cosmology (e.g., more dark matter halos than expected), which we consider less likely.', '5.1. Redshift Accuracy': "One valid concern with early JWST studies is that selection techniques which worked well at lower redshift would begin to fail. In particular, while the physics behind Ly α -break-based selection should persist at these high redshifts, it is possible that heretofore unknown populations of contaminants could have adverse affects. While one could model this contamination based on simulations, it relies on said simulations correctly modeling the colors of all potentially contaminating populations (e.g., Larson et al. 2023; Harikane et al. 2023a), which is unlikely, particularly prior to JWST observations. \nSpectroscopic validation of photometric redshifts is thus required. Unlike the past decade, when only the brightest HST z > 6 galaxies could have redshifts validated via either weak Ly α emission (e.g., Finkelstein et al. 2013; Zitrin et al. 2015; Oesch et al. 2015; Hoag et al. 2019; Jung et al. 2019, 2020; Larson et al. 2022a) or Ly α breaks for the brightest sources (e.g. Oesch et al. 2016), JWST 's spectroscopic capabilities allow easy restoptical-based spectroscopic redshifts out to z ≈ 9.5 (beyond which [O III ] redshifts out of the NIRSpec window) and Ly α continuum-based redshifts (with the NIRSpec prism mode) to arbitrarily higher redshifts (e.g. \nCurtis-Lake et al. 2023; Fujimoto et al. 2023a; Arrabal Haro et al. 2023a,b; Hainline et al. 2023; Fujimoto et al. 2023b). \nAs the CEERS spectroscopic component was observed in the second epoch and a DDT NIRSpec followup program was performed, a significant number of CEERS high-redshift candidates were spectroscopically observed in Cycle 1. Of our original sample of 93 candidate galaxies, 17 had NIRSpec spectroscopic observations, with redshifts originally presented in Fujimoto et al. (2023a); Arrabal Haro et al. (2023a,b); Harikane et al. (2023a); Larson et al. (2022b). As discussed in § 3.2.1, only one of these 17 had a 'catastrophically' (defined as | z spec -z phot | / (1+ z spec ) > 0 . 3) incorrect redshift (similar success was seen in the UNCOVER survey by Fujimoto et al. 2023b). Of the remaining 16, all had z spec > 7.8. We also note that four of the galaxies in our sample at z phot > 9.7 were spectroscopically observed by CEERS with no spectroscopic detection. For these sources, the absence of strong emission lines is plausibly consistent with z spec > 9.6. These results imply that it is unlikely that significant contamination from low-redshift galaxies is affecting our results (with the caveat that the sample of galaxies confirmed is as-yet small and fairly biased towards brighter sources). \nWe next consider whether any smaller, yet nonnegligible, systematic offsets in redshift could play a role. A trend for the photometric redshifts to be overestimated at z ≳ 8 has been reported, with results from CEERS (Arrabal Haro et al. 2023b; Fujimoto et al. 2023a), JADES (Hainline et al. 2023) and UNCOVER (Fujimoto et al. 2023b) showing the photometric redshifts to be over-estimated by < ∆ z > = 0.45 ( ± 0.11), 0.26 ( ± 0.04) and 0.28 ( ± 0.33), respectively. As discussed in these studies, suchaed offsets indicate a mismatch between the galaxy spectra and the adopted photometric redshift templates due to a variety of physical effects, including an increasing neutral fraction and/or enhanced DLA absorption (e.g., Umeda et al. 2023; Heintz et al. 2023). \nFor our specific sample of galaxies in this work (13 galaxies with z spec > 8.5), we find a median (mean) offset of z phot -z spec = 0.1 (0.3) with a standard deviation of 0.2. As discussed in § 3.2.1 our sample excludes three galaxies selected with z phot > 8.5 with z spec ∼ 8; including these objects does not change the median offset, but it does increase the mean offset (to 0.5) and the standard deviation (to 0.3). These three objects in particular highlight the difficulty of working at z ≲ 9 within the CEERS dataset due to the bluest JWST filter being F115W. Upcoming F090W imaging from PID 2234 (PI Ba˜nados) will improve this situation, probing below the \nLy α break at z ∼ 8. At present these larger ∆ z ∼ 1-2 offsets affect only a small fraction (3/16) of the spectroscopic sample. Should these larger offsets exist in the rest of the non-confirmed sample, the instances of these objects we do see imply it would primarily affect the z ∼ 9 results (though we note that ID=4777 [ z spec =7.993] is in the z ∼ 11 galaxy sample, though it has a very broad P [ z ]). \nWe simulate the potential impact of these systematic redshifts offsets by re-measuring our observed luminosity function values in the same MCMC manner as above, where here in each step of the MCMC chain we assign simulated spectroscopic redshifts for each object. For objects which already have spectroscopic redshifts, we keep those values. For the remaining objects, we draw randomly from the observed ∆ z (= z phot -z spec ) distribution, adding ∆ z to the photometric redshift to simulate a (potentially biased) spectroscopic redshift. For each new ' z spec ', we re-measure a new value of M UV . We perform this redshift assignment and subsequent SED-fitting-based M UV measurement prior to running the MCMC chain, pre-computing 100 random draws of ∆ z and the corresponding M UV , then drawing from these pre-computed values randomly at each step of the chain. Through this process we simulate the effect of the potential redshift bias both on the specific galaxy samples (as objects can move between redshift bins, potentially lowering the median redshift), as well as the impact on the absolute magnitudes (due to changes in the distance modulus). Both effects can combine to affect the number density, though the former is the larger affect as the difference in the distance modulus for ∆ z =0.3 (1) at z = 11 is only ∼ 0.04 (0.13) mag. \nFirst we examine the impact on the median redshift in each bin, which we find to be fairly minimal: z med = 8.9, 10.9 and 14.2 (unchanged for the z ∼ 9 and z ∼ 11 bins, and 0.2 higher in the z ∼ 14 bin). The corresponding impact on the number densities at M UV = -20 are also modest, with these values being 12%, 14% and 36% lower at z ∼ 9, 11 and 14. Measuring d log ϕ / dz using both these new median redshift values and the corresponding simulated number densities, we find d log ϕ / dz = -0.14 ± 0.09 over z =9-14, not significantly different from our fiducial value of -0.11 ± 0.08. We note that while for this exercise we restricted the ∆ z sample to z spec > 8.5, we found that including the three z spec ∼ 8 galaxies did not change the results. \nWe thus conclude that the measurable redshift bias from the available spectroscopic confirmations is unlikely to be the primary cause of the observed change in slope at z > 9 in the evolution of the abundance of \nbright galaxies. We acknowledge again that the existing number of spectroscopic redshifts is small, and biased towards primarily bright sources. In addition, some of these redshifts come from the Ly α break only, and these values have been measured to be up to ∆ z ∼ 0.2 different from more secure emission-line-based redshifts (Fujimoto et al. 2023b). Larger samples of spectroscopic confirmations of galaxies in this epoch are needed to increase confidence that any redshift bias does not affect the measured number densities.", '5.2. AGN Contribution': "Accreting supermassive black holes, particularly when they are not obscured from view, can emit quite strongly in the rest-UV (e.g., Stevans et al. 2014). While the contribution of AGN light to the rest-UV emission from high-redshift galaxies has been fairly unconstrained, some notable examples do exist at z ∼ 7 (e.g. Fujimoto et al. 2022; Endsley et al. 2023a). However, the evolution in the AGN UV luminosity function suggests that AGNs do not dominate the rest-UV emission in galaxies (e.g, non-quasars) at high redshift (e.g., Finkelstein & Bagley 2022). However, with JWST 's spectroscopic abilities, it is worth revisiting whether AGN could be contributing significantly to the UV emission from galaxies. \nEarly observations hint that growing super-massive black holes are indeed somewhat common at z ∼ 5-9, with signs of potential AGN activity found in dozens of galaxies (e.g., Kocevski et al. 2023; Larson et al. 2023; Harikane et al. 2023b; Leung et al. 2023; Labbe et al. 2023; Bogdan et al. 2023), with several sources containing spectroscopically confirmed broad-line AGN (e.g., Kocevski et al. 2023; Harikane et al. 2023b; Matthee et al. 2023; Larson et al. 2023; Maiolino et al. 2023b; Kokorev et al. 2023; Furtak et al. 2023; Greene et al. 2023). While this may suggest AGN could contribute to the UV luminosity, the many of these sources have unique two-component SEDs with fairly flat UV spectral slopes, with a change to a steeply rising red slope in the rest-UV optical (e.g., Kocevski et al. 2023; Barro et al. 2023; Matthee et al. 2023; Labbe et al. 2023). While an AGN jet could trigger enhanced star-formation (e.g. Duncan et al. 2023), here we aim to assess whether the UV emission we observe is dominated by emission from an AGN accretion disk. For these red AGN in particular, the point-source morphology in the longest wavelength bands strongly suggests that obscured AGN light is dominating the rest-optical emission, while advanced SED modeling is needed to robustly constrain the amount of AGN contribution to the rest-frame UV. Such a contribution remains possible as scattered UV \nlight from a partially obscured AGN could in theory contribute to the observed UV emission (e.g., Kocevski et al. 2023; Barro et al. 2023; Labbe et al. 2023; Greene et al. 2023), though the resolved nature of the rest-UV emission in these galaxies indicates stellar emission may dominate. \nAGN have also recently been identified in objects with more typical galaxy-like morphologies and SED slopes. Larson et al. (2023) inferred the presence of an AGN in a z = 8.7 galaxy (via a 2.5 σ significant broad-H β line), and noted that the SED has a flat slope through the restnear-infrared (aided by MIRI observations, Papovich et al. 2023) suggesting stellar light is dominating at all observed wavelengths. Maiolino et al. (2023a) inferred via extremely high gas densities that the nucleus of the well-known galaxy GN-z11 (at z = 10.6) likely hosts an AGN; analysis of this object by Tacchella et al. (2023a) shows that 2/3 of the rest-UV continuum emission emanates from the nucleus, hinting that this object could be AGN dominated in the UV. Harikane et al. (2023b) discuss 11 confirmed broad-lined AGN in the CEERS survey, and found that the majority of them showed extended morphologies in the rest-UV (with most of the rest being extremely UV-faint reddened AGNs), suggesting that much of the rest-UV emission is stellar in origin. Although some of these observations indicate that the AGN contribution to the total UV luminosity is negligible, this might not always the case, depending on the phase of the AGN duty cycle (which affects the contrast between the AGN and the host galaxy). A possible high AGN fraction has been argued in the recent NIRSpec follow-up for lensed galaxies at z = 8.5-13.2 (Fujimoto et al. 2023b). \nThese early observations do not yet collectively paint a clear picture of the contribution of AGN to the restframe UV emission from early galaxies. It is clear that AGNs exist in these epochs, though many discovered so far appear to be primarily obscured. Deciphering the relative contribution from stars and AGNs to the emergent UV emission, including constraining the extent to which scattered UV light from obscured AGNs plays a role, will require a combination of more advanced SED modeling techniques along with deep ∼ 12 µ m spectroscopy. Until such analyses can be conclusively done, AGNs remain a possible scenario to explain the high abundance of bright galaxies at early times.", '5.3. Change in Physical Processes': "The remaining explanations for the observed UV luminosity enhancement involve changes in the physical processes regulating the ratio of observed UV light to the halo masses of these galaxies. We further explore \nFigure 12. A comparison of the observed z ∼ 11 UV luminosity function from CEERS and NGDEEP (symbols are the same as in the top panel of Figure 9) to model predictions. Pre-launch predictions from FLARES, DELPHI, UniverseMachine, THESAN and BlueTides are shown as the thin gray lines, while the colored lines show more recent predictions from Ferrara et al. (2023, green), Dekel et al. (2023, the blue shaded region shows a range of maximum star-formation efficiency from 0.2-1), Shen et al. (2023, yellow; the dashed line includes a strong stochastic star-formation component), Yung et al. (2023a, red; the dashed line indicates a top-heavy IMF UV luminosity enhancement of 3 × ) and Jones et al. (in prep, purple). The thick gray line shows the empirical DPL luminosity function from Finkelstein & Bagley (2022) extrapolated to z = 11. These predictions show that a variety of potential physical solutions can predict a z ∼ 11 luminosity function in agreement with observations. \n<!-- image --> \nthis here, aided by a variety of recent theoretical results motivated by early JWST observations of the z > 9 universe, summarized in Figure 12. This figure compares our observed UV luminosity function (combined with that of NGDEEP) to recent predictions from Ferrara et al. (2022), Yung et al. (2023a), Dekel et al. (2023), Shen et al. (2023), and Jones et al. (in prep). We also compare to pre-launch predictions from FLARES (Lovell et al. 2021; Vijayan et al. 2020; Wilkins et al. 2022b), DELPHI (Dayal et al. 2017), UniverseMachine (Behroozi et al. 2020), THESAN (Kannan et al. 2022) and BlueTides (Feng et al. 2016; Wilkins et al. 2017). In this figure we compare our observations to these model predictions made at z = 11 when possible (when not we interpolate the number densities in log space), as this is the median of the stacked redshift probability distribution for the galaxies which make up our luminosity function sample (see inset panel of Figure 9). We note that differences between models can be due both to the underlying methodology and/or sub-grid physics, as well as the procedures used to generate observed luminosities. \n5.3.1. Significant Evolution in Attenuation \nFerrara et al. (2023) have developed a physical model which successfully reproduces the observed z = 7 UV luminosity function via a dust implementation which is designed to match both the shape of the UV luminosity function and the z ∼ 7 obscured SFR results from the ALMAREBELSsurvey (Bouwens et al. 2022). Evolving their model to higher redshifts, they naturally predict a slowing in the evolution of the UV luminosity function from z ∼ 9 to 11 due to significantly reduced attenuation in galaxies. This arises in their model due to outflows driven by very high ('super-Eddington') specific SFRs, which can efficiently drive gas (and dust) out from these galaxies (Ferrara 2023). They discuss that this is supported by the data as early results on the colors of z > 8 galaxies showed that they were fairly blue (e.g., Finkelstein et al. 2022b; Cullen et al. 2023; Topping et al. 2022; Papovich et al. 2023) and thus likely contained little dust. \nAs shown in Figure 12, this model does significantly better than pre-launch simulations, in fact slightly overpredicting the observed UV luminosity function. We thus explore whether significant color evolution is ob- \nFigure 13. The evolution of the FUV color C FUV with redshift, color-coded by M UV . The large squares show median values in bins of redshift and magnitude (error bars show the 1 σ spread), for bins with ≥ five sources. While reddened galaxies exist at z ∼ 7-8, the median colors are still fairly blue. Notably, the median FUV color for M UV = -20 galaxies is similar at z ∼ 8 to z ∼ 11, suggesting a significant drop in dust attenuation is unlikely to explain the high abundance of bright z ∼ 11 galaxies. \n<!-- image --> \nserved, particularly in modestly bright galaxies, which could further support this model. To study potential evolution with redshift, we select a sample of galaxies from the CEERS catalog at z ∼ 6-8. The sample selection was identical to that done here for z > 8.5, simply evolving the redshift cuts to lower redshift (primarily requiring ∫ P ( z > 4) ≥ 0.7, z best > 5.0, and S z = 6, 7 or 8). The full sample of > 2000 sources was visually inspected, resulting in a final sample of 1018, 574, and 224 galaxies at z ∼ 6, 7 and 8, respectively. \nAs discussed in § 3.3, we defined a rest-far-UV color C FUV , and calculated it for galaxies in both our z > 8.5 and z = 6-8 galaxy samples. We note that this specific FUV color is very sensitive to dust attenuation due to a narrow color baseline, in that ∆ C FUV = 0.05 is equivalent to ∆ A V = 0.1. In Figure 13 we plot C FUV versus redshift, color-coded by M UV . We plot the median in bins of redshift ( z = 6.5-7.5, 7-5-8.5, 8.5-9.7, and 9.713) and while there is significant scatter in color at all redshifts, the median color is fairly blue. Focusing on the middle magnitude bin (M UV = -20) where our results have the strongest constraining power (too few brighter galaxies exist at z > 9 to make conclusions), we see that the median FUV color does not significantly evolve from z ∼ 8 to z ∼ 11. \nThis initial exploration into the evolution of UV colors does not support rapid changes in the evolution of the attenuation between z ∼ 7 to 11, as would be implied by the Ferrara et al. (2023) model. Rather, the majority \nof these z > 6 galaxies appear roughly dust free, implying that the ALMA REBELS/detected galaxies used to calibrate the Ferrara et al. (2023) model are not indicative of the bulk of the z ∼ 7 galaxy population. In particular, Papovich et al. (2023) showed that with the inclusion of MIRI photometry, typical galaxies at z > 7 are fairly blue. We do note that finding so little dust is in itself a puzzle, as these galaxies should have made significant amounts of dust in the process of building their observed stellar masses. Though UV-faint dusty sources may indeed exist in this epoch (Rodighiero et al. 2023), in our UV-selected sources the dust must get destroyed (via, e.g., reverse shocks in supernovae), or be removed via winds even down to z ∼ 7 in these sources. We acknowledge however that the scatter in C FUV is large and that more advanced UV spectral slope modeling with larger samples covering a wider dynamic range in stellar mass and M UV can provide further insight.", '5.3.2. Change in Conversion from Mass to UV Luminosity': "If reduced attenuation is unlikely to be the dominant effect explaining the discrepancy between models andobservations, then it is prudent to consider changes in the processes of star formation which could result in enhanced UV luminosities. A very straightforward potential explanation could be a change in the initial mass function (IMF). As discussed by Finkelstein et al. (2023) and Harikane et al. (2023a), a change in the characteristic stellar mass from ∼ 1 M ⊙ to ∼ 10 M ⊙ is expected when the cosmic microwave background temperature is higher and the gas metallicities are lower (e.g., Larson 1998; Bromm et al. 2002; Tumlinson 2006; Steinhardt et al. 2023). Such a change would decrease the mass-toUV light ratio by factors of up to several (e.g., Raiter et al. 2010; Zackrisson et al. 2011). \nAs one example of this, we show the new semianalytic model predictions based on merger trees from the gureft simulation suite Yung et al. (2023a,b). This is an update to the 'Santa Cruz SAM' predictions (Somerville et al. 2015; Yung et al. 2019), now using N -body based merger trees constructed with finely spaced snapshots at very high-redshift to better capture halo growth at early times. Their fiducial model still underpredicts the observations, implying that the Extended Press-Schechter based merger trees used in the previously published models were not the sole reason for the discrepancy. However, as they discuss in their paper, the shape of their UV luminosity function appears consistent with the data. They show that if they decrease the mass-to-light ratio by a factor of ∼ 3 (dashed line), as would be plausible for a top-heavy IMF, their model becomes consistent with the observations. \nA smaller UV luminosity enhancement of only ∼ 1.5 × would be needed to bring the empirically extrapolated DPL UV luminosity function from Finkelstein & Bagley (2022) at z = 11 (thick gray line) into agreement with our observations. However the discrepancy between this empirical extrapolation is greater for brighter galaxies than for fainter galaxies, such that a flat UV luminosity boost at all magnitudes would lead to an over-prediction of the faint-end due to the empirically predicted faintend slope being steeper ( α = -2.5) than that observed from NGDEEP ( α = -2.2). While our results cannot constrain the IMF presently, deep rest-UV spectroscopic observations could be capable of detecting highionization lines such as He II or [Ne V ], which could indicate the presence of very massive stars in these galaxies (e.g., Tumlinson & Shull 2000; Bromm et al. 2001; Schaerer 2003; Olivier et al. 2022; Cleri et al. 2023). \nOne caveat to an increased light-to-mass ratio due to a top-heavy IMF would be that it would be accompanied by an increased feedback-to-mass ratio, because the massive stars that produce UV light are also those that produce energetic radiation, strong stellar winds and Type II supernovae. Changing the IMF would also change the integrated chemical yields, which could impact cooing. The models above have yet to selfconsistently model the increased feedback and modified chemical yields of a top-heavy IMF for high-redshift galaxies. However, other studies (e.g., Fontanot 2014) have found that evolving top-heavy IMFs (e.g., topheavy IMFs for galaxies with high SFRs) have tended to decrease the mass of stars formed relative to models with universal IMFs.", '5.4. High Star Formation Efficiency': "Even without a change in the IMF, the UV luminosities of very high-redshift galaxies could be enhanced if the rates of gas conversion into stars was increased. As discussed in Finkelstein et al. (2023), most models adopt fairly long gas-depletion timescales, based on observations of nearby galaxies. While perhaps a coincidence, it is interesting that among the preJWST predictions only the Behroozi & Silk (2015) model, which assumes a negligible gas depletion timescale, can match the observations in Figure 8. The physical processes which may lead to a decrease in this timescale are not currently obvious, though as the dependence of star-formation with gas density (Schmidt 1959; Kennicutt 1998) is superlinear (e.g. Vallini et al. 2023), the very high gas densities present in early-universe halos likely play a role. Very high-efficiency star formation has been observed in super-star clusters in local galaxies (e.g. Turner et al. 2015; Smith et al. 2020; Costa et al. 2021). The high \ncloud surface densities in the environments of these superstar clusters, which are rare in the nearby Universe, could be common in galaxies at z > 9. Theoretical work on molecular cloud scales has also shown that both stellar winds and radiation feedback may become ineffective at very high cloud surface densities, leading to higher cloud-scale star formation efficiencies (Grudi'c et al. 2020; Lancaster et al. 2021; Menon et al. 2023). \nIn this context, we explore two additional predictions. The first is an updated version of the Simba model (Dav'e et al. 2019) known as Simba-EoR (Jones et al., in prep.). This simulation includes a new subgrid ISM model that co-evolves dust and H 2 by explicitly tracking all formation and destruction mechanisms, which turns out to yield more H 2 at low metallicities relative to Simba . This leads to earlier star-formation in halos, increasing the luminosities of early galaxies compared to Simba (which, as shown in Figure 8 significantly underpredicts z > 9 observations). The Simba-EoR predictions are well matched to the CEERS observations, despite no specific tuning to EoR data, though the faintend slope appears higher than that observed. It is worth noting that the galaxies are not predicted to be dustfree; without accounting for extinction, the Simba-EoR predictions would be clearly above the observations. \nWe also compare to the Feedback-Free Starburst (FFB) physical model introduced by Dekel et al. (2023), using the luminosity function, including a dust attenuation prescription, presented in Li et al. (in preparation). This model predicts that in massive galaxies at z ∼ 10, where the gas density is above a threshold of ∼ 3 × 10 3 cm -3 and the gas-phase metallicity is below ∼ 0 . 2 Z ⊙ , star formation in thousands of globularcluster-like clouds is expected to proceed on a free-fall timescale shorter than the ∼ 2 Myr interval between a starburst and the onset of effective stellar and supernova feedback, thus allowing high star-formation efficiency free of suppression by feedback 3 . In Figure 12 we show as the blue shaded region the predictions of this model with a maximum SFE ranging from 20% (lower bound) to 100% (upper bound). These predictions match our observations reasonably well over the range where we detect galaxies. This model predicts an enhanced bright end due to the predicted high SFE preferentially at high halo masses, aided by the fairly low levels of dust attenuation and the high level of star-formation stochasticity predicted in the FFB phase.", '5.4.1. Stochastic Star Formation': "In § 4 we presented Figure 11, which showed that the evolution of the abundance of bright galaxies appears to flatten at z > 9, while the specific UV luminosity density (integrated to M UV = -17) is not inconsistent with evolution with a consistent slope from lower redshift. It is important to note that presently these integrals are uncertain - the observations are also consistent with an elevated value at the faint end within the uncertainties. Future work combining the available deep fields (NGDEEP, MIDIS, JADES, etc.) will soon improve these constraints. Should these future studies conclude that any flattening evolution is observed primarily at brighter luminosities, we comment here on possible physical drivers of such a differential evolution. First, should any of the processes discussed above be at play primarily in more massive halos, it could lead to a preferentially enhanced abundance of bright galaxies. However, it seems unlikely that there would be dramatic differences in the star-formation efficiency or IMF across a fairly limited dynamic range of UV absolute magnitudes (and thus presumably halo mass). \nOne plausible explanation for the observed behavior could be a significant increase in the stochasticity of star formation (e.g., Shen et al. 2023; Sun et al. 2023; Mirocha & Furlanetto 2023). As shown by Shen et al. (2023), introducing a variability in the conversion from halo mass to UV luminosity (which could encompass both star-formation stochasticity, as well as variations of dust attenuation, metallicity, etc.), leads to an 'upscattering' in the UV luminosity function. Due to the steep faint-end slope, more galaxies will scatter from faint-tobright luminosities than from bright-to-faint luminosities, which can lead to a shallower bright end (in an effect similar to Eddington bias). They explore a range of scatter values (encompassed by the variable σ UV which describes the Gaussian width of the kernel scattering the UV luminosities). As we show in Figure 12, σ UV = 1.75 yields a predicted UV luminosity function in good agreement with our observations, though it does overpredict abundances at M UV ≤ -21. Pallottini & Ferrara (2023) explored the level of stochasticity present within their high-resolution hydrodynamic simulations, and found that while stochasticity was present, it was at a lower level, equivalent to σ UV ∼ 0.6. This is just one simulation, so it remains to be seen if higher levels of stochasticity are plausible. We also note that when stochasticity is not included, the fiducial UV luminosity function of the empirical Shen et al. (2023) model is quite low (solid gold line in Figure 12). Taking, for example, the UV luminosity function from the physicsbased Santa Cruz GUREFT SAMs (Yung et al. 2023a), \nlower levels of UV scatter would be needed to match the observations, plausible more consistent with simulation results. \nObservational evidence is emerging that high-redshift galaxies have significant variability in their starformation rates. Looser et al. (2023a) discovered a surprisingly quiescent galaxy at z = 7.3, which shows evidence for a ∼ 10-20 year lull in star-formation activity after a recent burst. They extended this study in Looser et al. (2023b), finding with a spectroscopic analysis that lower-mass galaxies at high redshift have particularly bursty star-formation histories. Endsley et al. (2023b) came to a similar conclusion after analyzing the photometry of a much larger sample of galaxies, finding that fainter galaxies at higher redshifts show lower [O III ] equivalent widths than their brighter counterparts, which they interpret as evidence that the brightest galaxies are frequently experiencing a recent upturn in star-formation activity (see also Tacchella et al. 2023b). \nWhile evidence thus far indicates that galaxies at higher redshifts and lower masses may host more variable star-formation histories, we cannot yet conclude whether this is the major physical driver of the slowerthan-expected evolution of the UV luminosity function at brighter luminosities. Deep JWST /PRISM spectroscopy could yield a dataset capable of constraining star-formation histories via SED modeling. Another interesting test to constrain this possibility was proposed by Mu˜noz et al. (2023), who noted that if variability in the ratio of UV luminosity to halo mass was significant, many UV-bright galaxies would exist in lower-mass halos, which could be distinguishable via a weaker clustering strength than if there was a more direct correlation between UV luminosity and halo mass. Such a measurement would require a wide and deep photometric survey capable of identifying a sufficient source density of z > 10 galaxies. This may be possible with COSMOS-Web (PID 1727, PIs Kartaltepe & Casey; Casey et al. 2022), and will certainly be possible with deep field observations with the Nancy Grace Roman Space Telescope (in particular with the added K s filter).", '6. CONCLUSIONS': "We have presented the results of a comprehensive search for z > 8.5 galaxies in the full NIRCam dataset from the Cosmic Evolution Early Release Science survey. We created a new photometric catalog aimed at measuring accurate colors for faint galaxies, as well as robust estimates of the total flux, implementing multiple key improvements over our previous work (Finkelstein et al. 2023). We identify a sample of 88 candidate z > 8.5 \ngalaxies, with 55 galaxies in our z ∼ 9 sample (selected over 8.5 < z < 9.7), 27 galaxies in our z ∼ 11 sample (selected over 9.7 < z < 13), and three galaxies in our z ∼ 14 sample (selected over 13 < z < 15). Notably, 13 of our galaxies are spectroscopically confirmed, eight in the z ∼ 9 sample ( z spec = 8.63-9.00), and five in the z ∼ 11 sample ( z spec = 9.77 - 11.42). \nWe perform advanced source-injection simulations to assess our source completeness, accounting for both photometric and photometric redshift recovery, explicitly accounting for the sizes of our sources. While the impact of this update is minimal for bright ( M UV ∼ -21) galaxies, it does result in modestly (10-30%) lower effective volumes than the assumption of a point source at fainter ( M UV ∼ -19.5) luminosities. \nWe use these completeness estimates to first compare the cumulative surface density of galaxies in our sample to a variety of preJWST -launch simulation predictions, finding that the observed abundance of galaxies is higher than any physical model prediction at z > 10, with this tension increasing with increasing redshift. Our results are in the least tension with the empirical model of Behroozi & Silk (2015), which posits that the specific star-formation rate tracks the specific halo accretion rate; at these high redshifts this would imply very short timescales for star-formation. While it remains to be seen if this is physical, it would be consistent with the observed abundance of UV-bright galaxies. \nWe calculate the rest-UV luminosity function in our three redshift bins. Comparing to previous results, we find general agreement, though our uncertainties are typically smaller due to our larger sample sizes. Notably, we see evidence of the known z ∼ 8.7 overdensity in the EGS field (Finkelstein et al. 2022a; Larson et al. 2022a; Whitler et al. 2023) in the brighter bins of our z ∼ 9 UV luminosity function. \nWe analyze the evolution of the UV luminosity function from z ∼ 9 to z ∼ 14 in two ways. First, we examine the evolution of galaxies at M UV = -20. While the abundance of galaxies at this fixed UV luminosity has been conclusively measured to rise smoothly from z = 9 to 3, we find evidence for a significant flattening. Extrapolating the observed evolution from z = 9 to 3 to higher redshifts, one would expect the abundance of galaxies to rise by a factor of ≳ 20 from z ∼ 14 to z ∼ 9. Conversely, we measure a rise of 4.3 ( ± 3.7) over this epoch. Phrasing this another way, d log ϕ / dz = -0.29 ± 0.03 at z = 3-9, while we find -0.11 ± 0.08 at z = 9-14. We also explore the total integrated specific UV luminosity density, fitting double-power law models to our observed luminosity functions, and integrating to M UV = -17. Interestingly, we find that \nthis quantity follows the observed extrapolation at z > 9. This hints that whatever new physical processes in play at these epochs may primarily affect bright galaxies, though the uncertainty in the integrated specific UV luminosity density at z ≳ 11 is presently high. \nWe discuss a variety of potential physical causes for the observed results. The high yield of spectroscopic confirmations implies that significant sample contamination is unlikely, though confirmation of a larger fraction of our galaxy sample would increase confidence in this claim. We find, based on blue colors for not only our galaxies, but galaxies at similar UV luminosities at z = 6-9, that a significant drop in dust attenuation at earlier times is unlikely to be the dominant explanation. Rather, we find that models which implement a combination of increased star-formation efficiency and/or an increased degree of bursty, stochastic star formation at these redshifts are more consistent with our observations. A change in the underlying IMF may also play a role. \nWe find slight evidence that the physics at play may be more important in bright galaxies than faint galaxies. Should this represent a dependence of star formation efficiency on halo mass, it would be expected to imprint signatures into a broad range of other cosmological probes. A key example is the topology of neutral hydrogen 21cm intensity maps, where such a differential star-formation efficiency effect would introduce an additional non-linear bias in the power-spectrum analysis. This signature may be detectable with ongoing and upcoming experiments, such as the Hydrogen Epoch of Reionization Array (HERA; Liu & Parsons 2016) and the Square Kilometer Array (SKA; Th'elie et al. 2023), thus providing independent cross-checks on our results. \nOur results show that the abundance of bright galaxies at z > 9 robustly exceeds expectations based on prelaunch observations. While many possibilities exist to explain these observations, each of them are directly empirically testable with a modest investment in further JWST spectroscopy. Deep NIRSpec followup can not only confirm the redshifts of all galaxies in this sample, but it can also probe diagnostic emission lines for either AGN activity or the presence of very massive stars. Such observations could empirically measure the starformation histories, testing models of stochasticity. As we are still very early in the JWST mission, it is highly likely such observations will become available in the near future, answering these key questions about star and galaxy formation at early times. \nFacility: HST (ACS, WFC3) \nFacility: \nJWST (NIRCam) \n- We thank Andrea Ferrara and Nathan Adams for shar1\n- ing their data. We also thank Andrea Ferrara, Pawan 2\n- Kumar, Om Gupta, and Julian Mu˜noz for helpful con3\n- versations. We acknowledge that the location where this 4\n- work took place, the University of Texas at Austin, that 5\n- sits on indigenous land. The Tonkawa lived in central 6\n- Texas and the Comanche and Apache moved through 7\n- this area. We pay our respects to all the American 8\n- Indian and Indigenous Peoples and communities who 9\n- have been or have become a part of these lands and 10\n- territories in Texas, on this piece of Turtle Island. We 11\n- acknowledge support from NASA through STScI ERS 12\n- award JWST-ERS-1345. PGP-G acknowledges sup13\n- port from grants PGC2018-093499-B-I00 and PID202214\n- 15\n- 16\n- 139567NB-I00 funded by Spanish Ministerio de Ciencia e Innovaci'on MCIN/AEI/10.13039/501100011033,\n- FEDER, UE. RA acknowledges support from ANID 17\n- Fondecyt 1202007. 18", 'REFERENCES': 'Adams, N. J., Conselice, C. J., Austin, D., et al. 2023, arXiv e-prints, arXiv:2304.13721 \nBouwens, R., Illingworth, G., Oesch, P., et al. 2023a, \nArrabal Haro, P., Dickinson, M., Finkelstein, S. L., et al. \n2023a, Nature, in press, arXiv:2303.15431 \n- -. 2023b, ApJL, 951, L22 \nBagley, M., Finkelstein, S. L., Finkelstein, S. L., et al. 2022a, ApJ Submitted \nBagley, M. B., Finkelstein, S. L., Koekemoer, A. M., et al. 2022b, arXiv e-prints, arXiv:2211.02495 \nBagley, M. B., Pirzkal, N., Finkelstein, S. L., et al. 2023, arXiv e-prints, arXiv:2302.05466 \nBarro, G., Perez-Gonzalez, P. G., Kocevski, D. D., et al. 2023, arXiv e-prints, arXiv:2305.14418 \nBehroozi, P., Wechsler, R. H., Hearin, A. P., & Conroy, C. 2019, MNRAS, 488, 3143 \nBehroozi, P., Conroy, C., Wechsler, R. H., et al. 2020, MNRAS \nBehroozi, P. S., & Silk, J. 2015, ApJ, 799, 32 \nBertin, E., & Arnouts, S. 1996, A&AS, 117, 393 \nBhowmick, A. K., Somerville, R. S., Di Matteo, T., et al. 2020, MNRAS, 496, 754 \nBisigello, L., Gandolfi, G., Grazian, A., et al. 2023, A&A, 676, A76 \nBogdan, A., Goulding, A., Natarajan, P., et al. 2023, arXiv e-prints, arXiv:2305.15458 \nBoucaud, A., Bocchio, M., Abergel, A., et al. 2016, \nPyPHER: Python-based PSF Homogenization kERnels \nMNRAS, 523, 1009 Bouwens, R. J., Stefanon, M., Oesch, P. A., et al. 2019, arXiv e-prints, arXiv:1905.05202 Bouwens, R. J., Illingworth, G. D., Oesch, P. A., et al. 2015, ApJ, 803, 34 Bouwens, R. J., Oesch, P. A., Stefanon, M., et al. 2021, arXiv e-prints, arXiv:2102.07775 Bouwens, R. J., Smit, R., Schouws, S., et al. 2022, ApJ, 931, 160 Bouwens, R. J., Stefanon, M., Brammer, G., et al. 2023b, MNRAS, 523, 1036 Bowler, R. A. A., Jarvis, M. J., Dunlop, J. S., et al. 2020, MNRAS, 493, 2059 Bowler, R. A. A., Dunlop, J. S., McLure, R. J., et al. 2015, ArXiv e-prints Boyer, M. L., Anderson, J., Gennaro, M., et al. 2022, Research Notes of the American Astronomical Society, 6, 191 Boylan-Kolchin, M. 2022, arXiv e-prints, arXiv:2208.01611 Brammer, G. B., van Dokkum, P. G., & Coppi, P. 2008, ApJ, 686, 1503 Bromm, V., Coppi, P. S., & Larson, R. B. 2002, ApJ, 564, 23 Bromm, V., Kudritzki, R. P., & Loeb, A. 2001, ApJ, 552, 464 \nBruzual, G., & Charlot, S. 2003, MNRAS, 344, 1000 \nTable 6. Summary of 8.5 ≲ z ≲ 9.5 Candidate Galaxies \nNote -A summary of the key properties for the first half of the 55 galaxies in our sample with 8.5 ≤ z ≤ 9.7. Spectroscopic redshifts come from Arrabal Haro et al. (2023a), Arrabal Haro et al. (2023b), Fujimoto et al. (2023a), Larson et al. (2023), and Tang et al. (2023).', 'A. TABLE OF 8.5 ≤ Z ≤ 10 SOURCES': "Here we include a table of galaxies in our sample at 8.5 < z < 9.7, split into Table 6 and Table 7. B. REMOVED SOURCES \nIn Section 3.2 we described our visual inspection process to ensure a clean, robust sample of high-redshift galaxies. As discussed, 91 sources were removed. Here \nTable 7. Summary of 8.5 ≲ z ≲ 9.5 Candidate Galaxies \nNote -A summary of the key properties for the second half of the sample of galaxies in our sample with 8.5 ≤ z ≤ 9.7. Spectroscopic redshifts come from Arrabal Haro et al. (2023a), Arrabal Haro et al. (2023b), Fujimoto et al. (2023a), Larson et al. (2023), and Tang et al. (2023). \nwe present tables (Tables 8 and 9) of these removed sources along with 5 '' cutout images in the F200W and F277W filters (Figure 14, 15, 17, 16 and 18). We also include a table of the five sources removed in § 3.2.1; four due to having z spec < 8.5, and one with similar colors as the z ∼ 16 candidate confirmed at z spec = 4.9 (Table 10). \nTable 8. Objects Removed from the Sample During Visual Inspection \nNote -Properties of objects removed from the sample during visual inspection. The notes column gives the primary reason, as well as indicating which objects plausible still could be real candidates, but were conservatively removed due to the stated concerns. \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 14. Cutout images, 5 '' on a side, of objects originally selected, but identified via visual inspection as being diffraction spikes. \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 15. Same as Figure 14, for objects identified as bad pixels. \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 16. Same as Figure 14, for objects identified as oversplit portions of nearby brighter galaxies. \n<!-- image --> \nTable 9. Objects Removed from the Sample During Visual Inspection \nNote -A continuation of the previous table. \nFigure 17. Same as Figure 14, for objects identified as being associated with chip edges. \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 18. Same as Figure 14, for objects identified as being affected by bad photometry. \n<!-- image --> \nTable 10. Objects Removed due to NIRSpec Information \nNote -Properties of objects removed from the sample. The first four have spectroscopic redshifts of z < 8.5. The final object was not spectroscopically confirmed, but has a photometric redshift of z > 16, and exhibits an observed SED extremely similar to CEERS-13256, which is confirmed by Arrabal Haro et al. (2023a) to be at z = 4.912."}

2024arXiv240908999F

Upcoming wide field surveys such as the Rubin Observatorys Legacy Survey of Space and Time LSST will monitor thousands of strongly lensed quasars over a 10year period. Many of these monitored quasars will undergo high magnification events HMEs through microlensing as the accretion disk crosses a caustic places of infinite magnification. Microlensing allows us to map the inner regions of the accretion disk as it crosses a caustic even at large cosmological distances. The observational cadences of LSST are not ideal for probing the inner regions of the accretion disk so there is a need to predict HMEs as early as possible to trigger highcadence multiband or spectroscopic followup observations. Here we simulate a diverse and realistic sample of 10year quasar microlensing light curves to train a recurrent neural network RNN to predict HMEs before they occur by classifying the location of the peaks at each time step. This is the first deep learning approach to predict HMEs. We give estimates at how well we expect to predict HME peaks during LSST and benchmark how our metrics change with different cadence strategies. With LSSTlike observations we can predict approximately 55 of HME peaks corresponding to tens to hundreds per year and a false positive rate of around 20 compared to the number of HMEs. Our network can be continuously applied throughout the LSST survey providing crucial alerts to optimize followup resources.

2024-09-01T00:00:00Z

['2024arXiv240908999F', '10.48550/arXiv.2409.08999', 'arXiv:2409.08999']

['Astrophysics - Astrophysics of Galaxies', 'Astrophysics - Instrumentation and Methods for Astrophysics']

Predicting High Magnification Events in Microlensed Quasars in the Era of LSST using Recurrent Neural Networks

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.08999.pdf

{'Predicting High Magnification Events in Microlensed Quasars in the Era of LSST using Recurrent Neural Networks': "Joshua Fagin, 1, 2, 3 Eric Paic, 4 Favio Neira, 4 Henry Best, 1, 2, 3, 5 Timo Anguita, 6, 7 Martin Millon, 8 Matthew O'Dowd, 1, 2, 3 Dominique Sluse, 9 and Georgios Vernardos 1, 2, 3 \n1 The Graduate Center of the City University of New York, 365 Fifth Avenue, New York, NY 10016, USA \nDepartment of Astrophysics, American Museum of Natural History, Central Park West and 79th Street, NY 10024-5192, USA \n2 \n3 Department of Physics and Astronomy, Lehman College of the CUNY, Bronx, NY 10468, USA \n- 4 Institute of Physics, Laboratory of Astrophysics, Ecole Polytechnique Fédérale de Lausanne (EPFL), Observatoire de Sauverny, 1290 Versoix, Switzerland\n- 5 Department of Theoretical Physics and Astrophysics, Faculty of Science, Masaryk University, Kotlářská 2, CZ-611 37 Brno, Czech Republic\n- 6 Instituto de Astrofísica, Departamento de Ciencias Fisicas, Facultad de Ciencias Exactas, Universidad Andres Bello, Av. Fernandez Concha 700, Las Condes, Santiago, Chile \n7 Millennium Institute of Astrophysics, Nuncio Monseñor Sótero Sanz 100, Providencia, Santiago, Chile \n8 \nKavli Institute for Particle Astrophysics and Cosmology and Department of Physics, Stanford University, Stanford, CA 94305, USA 9 STAR Institute, Université de Liège, Quartier Agora Allée du six Août, 19c B-4000 Liège, Belgium", 'ABSTRACT': "Upcoming wide field surveys such as the Rubin Observatory's Legacy Survey of Space and Time (LSST) will monitor thousands of strongly lensed quasars over a 10-year period. Many of these monitored quasars will undergo high magnification events (HMEs) through microlensing as the accretion disk crosses a caustic, places of infinite magnification. Microlensing allows us to map the inner regions of the accretion disk as it crosses a caustic, even at large cosmological distances. The observational cadences of LSST are not ideal for probing the inner regions of the accretion disk, so there is a need to predict HMEs as early as possible to trigger high-cadence multi-band or spectroscopic follow-up observations. Here we simulate a diverse and realistic sample of 10-year quasar microlensing light curves to train a recurrent neural network (RNN) to predict HMEs before they occur by classifying the location of the peaks at each time step. This is the first deep learning approach to predict HMEs. We give estimates at how well we expect to predict HME peaks during LSST and benchmark how our metrics change with different cadence strategies. With LSST-like observations, we can predict approximately 55% of HME peaks corresponding to tens to hundreds per year and a false positive rate of around 20% compared to the number of HMEs. Our network can be continuously applied throughout the LSST survey, providing crucial alerts to optimize follow-up resources. \nKeywords: Quasars(1319) - Active galactic nuclei(16) - Quasar microlensing(1318) - Gravitational lensing(670) - Neural networks(1933) - Time series analysis(1916)", '1. INTRODUCTION': "The most widely accepted unified model of Active Galactic Nuclei (AGN) postulates that their structure revolves around a central Super Massive Black Hole (SMBH) at the center of galaxies (Antonucci 1993; Urry & Padovani 1995). The tidal stretching and viscous \nCorresponding author: Joshua Fagin \njfagin@gradcenter.cuny.edu \nfriction experienced by matter in the immediate vicinity of the SMBH creates hot plasma, forming an accretion disk that emits a bright continuum of light across a wide range of wavelengths with stochastic variability. Quasars are bright AGN with unobscured accretion disks. Their radial energy profile is believed to follow the thin-disk model (Shakura & Sunyaev 1973; Page & Thorne 1974), but recent observations from microlensing (Poindexter et al. 2008; Poindexter & Kochanek 2010; Blackburne et al. 2015; Muñoz et al. 2016; Mor- \nan et al. 2018) and reverberation mapping (Mudd et al. 2018; Jha et al. 2022) have found larger accretion disk sizes by a factor of ∼ 2-4. Further out, clouds of ionised gas form the broad and narrow line regions (BLR and NLR), reverberating accretion disk light and inducing broad and narrow line emissions in the quasar spectra depending on the gas velocity (Blandford & McKee 1982; Sluse et al. 2007; O'Dowd et al. 2015; Grier et al. 2017). Quasars also influence the evolution of their host galaxy due to the energy outflow they create (e.g. Di Matteo et al. 2008; Fabian 2012; Kormendy & Ho 2013). Quasars are among the most luminous objects in the Universe, making them valuable cosmological probes for measuring the expansion rate of the Universe (Lusso et al. 2019; Wong et al. 2020) and understanding the re-ionization mechanism (Grissom et al. 2014). These, however, require a precise understanding of quasar structure and the underlying physical processes. \nStrongly lensed quasars occur when light from a quasar source is bent by a galaxy along the line of sight, forming multiple images of the quasar. Each image of the strongly lensed quasar can be independently magnified through microlensing due to stars or other compact objects in the lensing galaxy. This effect leads to a micromagnification of the light profile of the accretion disk. It is possible to measure the size of microlensed accretion disks using single epoch, multi-waveband observations using the different apparent size of the disk in different wavebands (e.g. Mediavilla et al. 2015). Alternatively, long-term monitoring of a strongly lensed quasar allows us to detect microlensing by measuring the micromagnification variation as the alignment between the microlenses and the background quasars varies over time. This long-term monitoring requires suppressing the intrinsic variability of the quasar by taking the pairwise difference of two strongly lensed images, corrected for the time delay between the images. If the intrinsic variability is perfectly subtracted, the resulting difference light curve, that we will refer to as the microlensing light curve , only contains the microlensing signal. These microlensing light curves provides constraints not only on the size of the disk (Kochanek 2004; Morgan et al. 2008; Cornachione & Morgan 2020; Rivera et al. 2024) but also on the geometry of the Broad Line Region (Sluse et al. 2011; Sluse & Tewes 2014; Paic et al. 2022; Savić et al. 2024) and can reveal the existence of substructures in the accretion disk, such as secondary supermassive black holes (Yan et al. 2014; Millon et al. 2022). \nA high magnification event (HME) occurs when a quasar approaches or crosses a microcaustic, a region of \ninfinite magnification. As a result, a specific area of the quasar's accretion disk becomes highly magnified. As the quasar moves relative to the microcaustic, the differential magnification provides a way of scanning though these different emission regions. In order to constrain the structure of the AGN engine though microlensing, however, we require high-cadence and multi-wavelength observations. These microlensing events can last from as little as a few weeks to a few years. It has been shown that X-ray monitoring of the HME can place constraints on the spin, inclination, and size of the innermost stable circular orbit (ISCO) of the black hole (e.g. Chartas et al. 2017). Furthermore, optical monitoring has the opportunity to measure the temperature profile of the inner regions of the disk (e.g. Eigenbrod et al. 2008). High-cadence and multi-wavelength observations near caustic-crossing events are crucial to extract the detailed accretion disk structure encoded in microlensing light curves (e.g. Krawczynski et al. 2019; Vernardos & Tsagkatakis 2019; Best et al. 2024). In the best case scenario, the entire caustic-crossing event will be observed by expensive and high-cadence follow-up to place the most stringent constraints on the inner workings of the AGN. Therefore, being able to predict these events ahead of time is a critical step in order to trigger appropriate follow-up observations. \nThe most furnished microlensing light curve data set to date was released by COSMOGRAIL (Millon et al. 2020b) with 20-year long microlensing light curves. Out of the 23 observed lensed quasars (17 doubly lensed and 6 quadruply lensed) and a total of 203 seasons of monitoring, only three microlensing light curves display microlensing events higher than 1 magnitude. These light curves only contain observations from the r -band, severely limiting the constraints that can be placed on the AGN structure and the ability to predict HMEs. \nThe advent of wide-field monitoring surveys such as the Rubin Observatory's Legacy Survey of Space and Time (LSST) is expected to increase the sample of monitored lensed quasars by at least tenfold. LSST is expected to simultaneously monitor hundreds to thousands of strongly lensed quasars (Oguri & Marshall 2010) in six UV/optical wavebands ( ugrizy ) at around 55-185 samplings per band or around 800 observations across the ten years (Abell et al. 2009; Prša et al. 2023). Tens to hundreds of quasar images should undergo HMEs of more than 1 magnitude per year (Neira et al. 2020). We cannot accurately constrain the innerstructure of the accretion disk using microlensing light curves at LSST cadences alone due to the sparse and irregular sampling. However, LSST microlensing light curves will still allow us to detect the characteristic \nsteady rise of a microlensing event occurring up to years before the peak. Furthermore, by using different wavebands, we can potentially determine the precise timing of the HME peak as it affects various regions of the disk causing the wavebands to cross. \nAn HME alert system was first developed for the Optical Gravitational Lensing Experiment (OGLE; Woźniak et al. 2000), and a triggering mechanism for multi-band follow-up observations was proposed by Wyithe et al. (2000). These efforts focused on predicting HMEs in Q 2237 + 0305 due to its brightness and rapid microlensing time scale but have not been used extensively. Thus far, there has been no prediction and follow-up of an HME as it unfolds. This is likely due the small sample of lensed quasars and a lack of long-term, multi-band monitoring that could distinguish microlensing from the intrinsic variability of the quasar. With LSST and other wide field surveys, however, this will no longer be the case. New and more sophisticated methods must be developed to quickly and autonomously be applied to the thousands of monitored lensed quasars expected in the near future. \nIn this work, we aim to create a neural network able to predict the occurrence of microlensing HME peaks using the multi-band microlensing light curves of LSST. This tool will be crucial to optimize follow-up observations of such HMEs and caustic crossings. Machine learning was first applied to microlensed quasar light curves in Vernardos & Tsagkatakis (2019) to measure the accretion disk size and temperature profile using a convolutional neural network for simulated data. Best et al. (2024) also used a convolutional neural network to measure the black hole mass, inclination angle, and impact angle of simulated caustic crossing events. As a pilot application, they measured the mass and inclination angle from archival caustic crossings of Q 2237 + 0305 and found them to be consistent with previous estimates. Machine learning has also been extensively used to model the intrinsic quasar variability (e.g. with simulated LSST quasar lightcurves, see Fagin et al. 2024). Here we train a recurrent neural network (RNN) using binary classification to identify the peaks of HMEs. Our RNN is able to account for the irregular cadences and seasonal gaps expected from LSST, and could be continually applied to LSST microlensing light curves in real time as new data is obtained. \nIn Section 2, we describe how we build our simulated data set of microlensing light curves. Section 3 presents our machine learning model architecture, training, and an example application to an LSST microlensing light curve. In Section 4, we benchmark the performance of our RNNs using different observational strategies includ- \ning the baseline case of simulated LSST-like observations. In Section 5, we discuss the results of our network and its applicability to LSST. Section 6 gives our concluding remarks. In this work, we assume a flat Λ CDM cosmology with H 0 = 72 . 0 kms -1 Mpc -1 , Ω m = 0 . 26 , and Ω Λ = 0 . 74 .", '2. MICROLENSING LIGHT CURVE SIMULATION': 'In order to train a machine learning model to predict HME peaks, we require a library of microlensing light curves that is representative of the expected LSST sample. Since the goal is to predict the peaks of the HMEs identified on the light curves, we focus on constructing a library where the microlensing durations and amplitudes cover the range of what could be expected in current and future systems. Each light curve in the training set has two main components: the microlensing signal and the associated correlated noise. The noise in the microlensing lensing light curves is typically highly temporally correlated, often referred to as red noise as opposed to uncorrelated Gaussian white noise (Millon et al. 2020c; Paic et al. 2022). The red noise is mainly due to imperfect subtraction of the intrinsic variability of the quasar or actual physical processes such as the intrinsic variability being echoed by the BLR and differentially magnified by microlensing (Sluse & Tewes 2014; Paic et al. 2022). Since these physical processes are not well understood and difficult to model, we rely on an empirical approach, where we treat all these effects as red noise added to our simulated microlensing light curves.', '2.1. Microlensing simulation': 'Microlensing brightness variations are uncorrelated between the multiple images of a strongly lensed quasar, unlike the intrinsic variability which is identical in each image. Since the latter can be subtracted, albeit imperfectly leaving the so-called red noise (see next section), we focus on simulating the microlensing variations. To model the microlensing variability, we use the tool presented in Neira et al. (2020), which is able to generate simulated microlensing light curves for each image of the lensed quasars. \nThe tool takes as an input an accretion disk, velocity, and macro lens model and outputs a library of light curves. The light curves are generated as follows: \n- · The accretion disk model is convolved with magnification maps selected from the GERLUMPH project (Vernardos et al. 2014).\n- · In the convolved map, multiple tracks are defined to simulate the net transverse movement of the disk, accounting for the movement of the lens, mi- \nolenses, and observer, based on the velocity parameters and probing time. \n- · The pixel values along these tracks correspond to the microlensing light curves (see section 2 in Neira et al. (2020) for a description of the tool, models, and parameters). \nWe use the simulated microlensing light curves from Neira et al. 2024 (in prep.), which makes use of the catalogue of lensed quasars presented in Oguri & Marshall (2010). A total of 2,821 mock systems are simulated. While the detail of the models and parameters used to generate the microlensing light curves from these systems can be found in Neira et al. 2024 (in prep.), here we briefly describe them. \nThe mass distribution of each lensing galaxy is modeled as a singular isothermal ellipsoid (SIE). For an SIE, the convergence and shear at the position of the quasar images is defined as: \nκ ( x, y ) = γ ( x, y ) = 1 2 θ E √ q ω ( x, y ) , (1) \nwhere θ E is the Einstein radius, q is the axis ratio of the lensing galaxy, and ω ( x, y ) is the elliptical radius where x, y define the coordinates of the lens plane with the x-axis aligned with the major axis of the lens. An additional term is added to the shear that corresponds to the lens environment, resulting in a small scatter in the final κ and γ distribution, shown in Figure 1. The smooth matter fraction, s , is computed by assuming a light distribution and mass-to-light ratio of the lens (see Foxley-Marrable et al. 2018; Vernardos 2019, and Neira et al. 2024 (in prep.) for details). This allows us to select a magnification map from the GERLUMPH library (Vernardos et al. 2014) corresponding to the selected κ , γ , and s . Each map has 10,000 pixels per side with a resolution of 0 . 025 R E /pixel, where R E is the Einstein radius of the microlensing objects and is defined as: \nR E = √ D S D LS D L 4 GM c 2 , (2) \nwhere D L , D S , and D LS are the angular diameter distances from the observer-to-lens, observer-to-source, and lens-to-source respectively, G is the gravitational constant, c is the speed of light, and M = 0 . 3 M ⊙ is the mass of the microlenses (the approximate mean of a Salpeter initial mass function (Salpeter 1955)). \nThe effective transverse velocity that defines the track along the magnification map is defined as: \nυ e = υ o 1 + z l D LS D L + υ ⋆ 1 + z l D S D L + υ g , (3) \nFigure 1. Distribution of the lensing macromodel parameters γ and κ for our quasar sample. \n<!-- image --> \nFigure 2. Distribution of transverse velocity υ e in our simulated light curves. \n<!-- image --> \nwhere υ o is the transverse velocity of the observer as measured with respect to the cosmic microwave background (CMB) velocity dipole (Kogut et al. 1993), υ ⋆ is the bulk velocity of the microlenses, and υ g is the combined peculiar velocities of the lens and source. The distribution of υ e across all the light curves is shown in Figure 2. \nFigure 3. Distribution of accretion disk radius r disk in our simulated lensed quasar sample. The sizes are in units of the Einstein radius of their respective system. \n<!-- image --> \nThe size of each accretion disk is assumed to follow a standard thin-disk model (Shakura & Sunyaev 1973): \nR λ = 9 . 7 × 10 15 ( λ rest µm ) 4 / 3 ( M BH 10 9 M ⊙ ) 2 / 3 ( f E η ) 1 / 3 [ cm ] , (4) \nwhere R λ is the size of the disk at λ rest , M BH is the mass of the black hole, f E is the Eddington ratio and η is the accretion efficiency. We adopted fixed typical values of f E = 0 . 25 and η = 0 . 15 (e.g. Blackburne et al. 2011). For the black hole mass we have adopted an empirical model from MacLeod et al. (2010): \nlog 10 ( M BH ) = 2 -0 . 27 · M i , (5) \nwhere M i is the absolute magnitude of the source in the i -band. Given a black hole mass estimate, the size of the accretion disk can be computed at any wavelength through Equation (4). The influence of the specific shape of the brightness profile has only minor influence in the microlensing signal (Mortonson et al. 2005; Vernardos & Tsagkatakis 2019). Instead, the signal is mostly influenced by the source size. Thus, we adopt a Gaussian-shaped two-dimensional brightness profile, where we match the size from Equation (4) to the half light radii of the profile. In Figure 3 we show the distribution of the disk sizes across all the systems with respect to their microlensing Einstein radius (see Equation (2)). \nWith the above, we are able to generate microlensing light curves by subtracted the light curves between each pair of quasar images. For each pair of images for all simulated systems, we generate 100 light curves in the six \nLSST bands. We label all peaks that have a minimum amplitude of 0 . 5 magnitude in the u -band, which is the bluest and thus most prone to microlensing variations. This value is chosen to be roughly above the maximum amplitude of the added red noise. After the difference light curves are generated and the red noise is included from Section 2.2, we apply an additional threshold for event labels to avoid including event peaks with little microlensing in the difference light curves that could lead to our network predicting more false positives. We apply a threshold of 1 magnitude from the maximum to minimum of any time step in the light curve across all bands. If the light curve is below the threshold, the labels are set to zero. This threshold is applied in order to remove the initially labeled peaks in the cases where the difference light curve shows little brightness variability. We justify this because the typical length of the events are of the order of years, and the length of the light curves are set to 10 years. This makes it a common occurrence that two events from different light curves overlap, thus yielding a difference light curve that has less variability. We produce a data set comprising of 282,100 light curves. We randomly select 80% of the light curves for training and reserve 10% for validation and 10% for testing.', '2.2. Red noise generation': 'The reverberation of the continuum by the BLR with a time lag induces an echo of the intrinsic variability within the observed light curve. If the broad emission lines fall in the monitored band and the microlensing is not identical in both images, this echo will appear in the microlensing light curve (e.g. Sluse & Tewes 2014). There may also be extended continuum emission coming from other regions of the disk (Sluse 2024). The frequency and amplitude of this imprint is characterized by the size of the BLR and the variability of the continuum, which is well described by a damped random walk (MacLeod et al. 2010). Furthermore, the variability of the light curve (hence the microlensing light curve) can be affected by observational effects such as the seasonal change of airmass and contamination by the lensed arc or the lensing galaxy when measuring the photometry of the images (see section 3.3.3 of, Sluse et al. 2006). Since these effects all require precise knowledge about the physical properties of the lens system, we choose to assimilate them into the red noise that is added to the immaculate microlensing light curve. \nFor this work, we use data from the COSMOGRAIL program (Courbin et al. 2005), which released the longest quasar microlensing light curves (Millon et al. 2020b), observed in the r -band with the Swiss 1.2-metre \nFigure 4. The top panel shows a COSMOGRAIL r -band microlensing light curve of RXJ 1131 -1231 over a period of 18 years computed by subtracting image B from image A after shifting it by the time delay of 2.8 days (Millon et al. 2020b). A spline fit with η = 300 days is applied to the microlensing light curve. The bottom panel displays the residuals of the spline fit, where the correlated noise is clearly visible. \n<!-- image --> \nLeonhard Euler Telescope. In particular, the microlensing light curve of RXJ 1131 -1231 displays the events with the highest amplitude and the strongest correlated noise. Hence, we use this data to quantify a conservative red noise that can be added to LSST-like light curves. As shown by Figure 4, the microlensing light curve is first fit with a free-knot spline using the PyCS3 package (see Millon et al. 2020a, for details of the implementation). These piece-wise polynomials allow for a smooth fit of a targeted time scale of variation by constraining the distance η between two consecutive knots. Microlensing events are believed to occur on times scales longer than several years (e.g. Mosquera & Kochanek 2011), we therefore choose η = 300 days to prevent the spline from fitting intra-season features while recovering the microlensing variations, the fit obtained is shown on the top panel of Figure 4, and the residuals are shown on the lower panel. As shown in Figure 5, the power spectrum of the data and of the spline fit are identical for frequencies below 1 / 200 days -1 , which sets the boundary between high and low frequencies. At higher frequencies, the power spectrum of the data is identical to the residual one. Our goal is then to generate time \nseries with the same power spectrum as the residuals to add it to the simulated microlensing light curves. We use the red noise generator implemented in PyCS3 to fit the slope β and amplitude σ of the residuals power spectrum. To account for the change in slope of the power spectrum of the residual above 1 / 10 days -1 , we fit β and σ separately in the frequency windows [ 1 / 200 , 1 / 10 ] days -1 and [ 1 / 200 , 1 / 10 ] days -1 . The addition of a generated red noise to the spline fit is shown in Figure 5 and has a power spectrum compatible with original data. The same parameters are then used to add red noise to each simulated microlensing light curve.', '2.3. Baseline observation strategy': 'We first simulate microlensing light curves with weekly cadence. We then want to degrade the cadences to mimic different observation strategies. Our baseline is LSST-like observations which are simulated using the observation times produced by rubin\\_sim 1 and the baseline\\_v2.1\\_10yrs cadences. We sample 100,000 \n<!-- image --> \nFigure 5. The left panel shows the power spectrum fitting of the residuals shown in the bottom panel of Figure 4. The vertical dashed lines at 1 / 200 days -1 and 1 / 10 days -1 delimit the frequency range on which the red noise is added. Adding residual-like noise to the spline fit (top panel of Figure 4) creates a new realization of the data with the same power spectrum. The right panel shows an example microlensed light curve with one realization of noise added with the same power spectrum as the left panel. \n<!-- image --> \ndifferent sky positions that have between 750 and 1,000 total observations across the ten years to include only light curves from the main LSST survey (see section 2.4 of Fagin et al. (2024)). The time of each observation is combined to the nearest weekly interval to maintain a fixed number of time steps in our light curves. Times that are not observed in a band are masked by the RNN. Microlensing events happen over time scales of months to years, so combining the light curves to weekly intervals should not affect our ability to predict HMEs. Separate sky positions are used for the training, validation, and test sets to avoid bias.', '3.1. Network architecture': 'Our neural network 2 is trained to classify HME peaks of quasar microlensing light curves. The goal is to train our model so that it can be applied every week throughout the lifetime of LSST, including during seasonal gaps where there are no new observations. This is done by including only observations made at or before the current time of the prediction, and then masking any further time steps. The network is trained to classify if there is an event peak within a 21 week time window before or after each time, chosen to be a bit more than half a season. We then make a prediction for each time step \nstarting when there are 85 weeks of data so the network has at least two seasons or so of data to inform its predictions. We normalize the light curve by subtracting each magnitude by the first observation in the r -band (or the g -band for some observation strategies we test with no r -band observations). Data augmentation is used during training to avoid overfitting the training set. This comprises of selecting a random cadence strategy for the light curve (i.e. for the nominal LSST light curves, we select the 10-year cadence from a specific sky position) and a random time step of the observation. The validation and test sets remain fixed for comparison. \nThe input into the RNN is the relative brightness at each time step for each band. There are 522 time steps corresponding to the brightness at each week across the ten years. Nominally we include all six LSST bands ugrizy , but we also explore situations where less bands are observed. Unobserved points are set to a dummy value of zero to be masked by our network. We include an additional feature that is set to one up to and including the current time step and then zero at later time steps. This feature is necessary to inform the network of the current time step, including when there are no new observations. \nOur RNN processes the time series in two separate paths, both forwards and backwards, since bidirectional RNNs can have improved performance. Each RNN path first contains a GRU-D layer (Che et al. 2016) a modified version of the gated recurrent unit (GRU; Chung \nFigure 6. Diagram of our RNN model. The numerical value in each layer represents the output size. We have two RNN paths, one for the forward light curve (LC) and one for the reverse. The last time step of the final RNN layers are input into the FC layers. Each RNN layer is followed by layer normalization, and the first two FC layers are followed by LeakyRelu and layer normalization. The final FC layer uses sigmoid activation function to normalize the output of our network to a probability for classification, i.e. restrict it between 0 and 1. We also include residual skip connections between the output of each layer and the next layer. \n<!-- image --> \net al. 2014) that is designed to handle masking and irregular time intervals. This is followed by two GRU layers. Each GRU layer has a tanh activation function and a hidden size of 128. Each RNN layer is followed by layer normalization (Ba et al. 2016). We also employ residual skip connections between the output of each RNN layer and the next layer which has been shown to improve training stability and performance (He et al. 2015). The output of each RNN path is combined and followed by two fully-connected (FC) layers. Each FC layer has a hidden size of 256 and are proceeded by a LeakyReLU activation function (Maas et al. 2013) and layer normalization. The FC layers also include skip connections to the previous layer. We then have one final fully-connected layer with a hidden size of 256 and output size of 1 that is followed by a sigmoid activation function. The sigmoid activation function forces the final output of the network to be a value between 0 and 1 representing the probability of the HME peak being within 21 weeks of the time step. A simple diagram of our RNN is shown in Figure 6. Each network has a to- \ntal of 642,673 trainable parameters and is built using PyTorch (Paszke et al. 2019).', '3.2. Network training': 'The network is trained by minimizing a binary crossentropy (BCE) loss function given by: \nL ( y, ˆ y ) = -1 N N ∑ i =1 y i log(ˆ y ) + (1 -y i ) log(1 -ˆ y i ) , (6) \nwhere y i is the training label, ˆ y i is the prediction of the network, and N is the number of training examples. We minimize the loss function using an Adam optimizer (Kingma & Ba 2014) with a batch size of 512 and a learning rate of 0.002 that is exponentially decreased to 0.0002 over the course of training. For training stability, we employ gradient clipping with a maximum gradient norm of 1. We train the network for multiple passes of our training set known as epoch. We train for a total of 300 epochs. As mentioned previously, we use data augmentation to regenerate the training set each epoch by choosing a random cadence strategy and time of the observation. We train separate networks for a variety of different observational strategies to see how the number of observations, seasonal gaps, and different bands affects the performance of our model. The BCE loss, accuracy, precision, and recall compared to the epoch are shown in Figure 7 for the LSST baseline observations. We find further training beyond the 300 epochs to have very little improvement on the performance of our networks. The other networks, trained with the different observational strategies, show similar convergence after 300 epochs. Training each network took around four days with an NVIDIA A10 GPU (24 GB).', '3.3. Application to microlensing light curves': 'We propose a three stage triggering system for rapid follow-up of HMEs. At each time step, the network makes a prediction to classify if an HME peak is within a 21 week window of the latest data. We quantify a positive prediction if the network predicts a probability above 0.5. The three triggers are colored from lowest to highest alert as green, yellow, and red. Green alert is triggered after any positive prediction. Yellow alert is triggered if there is a green alert continuously for 7 weeks in a row. Red alert is triggered if there is a yellow alert for 7 weeks in a row and continues until a negative prediction. As soon as there is a negative prediction, the count is reset to avoid false positives. Figure 8 shows an example prediction of our network for an LSST-like light curve. In this case, our network is able to predict the HME peaks ahead of time. The network is still able to make predictions during the seasonal gaps even \nFigure 7. BCE loss (shown in the top left panel and given in Equation (6)), accuracy (top right panel), precision (bottom left panel), and recall (bottom right panel) as a function of the training epoch for the training (blue) and validation (orange) sets. \n<!-- image --> \nthough no additional observations take place. Ideally the alert system should match up with the green label windows and we would predict the HME peaks 21 weeks beforehand, identifying their temporal locations. Another example is given in Appendix A that includes a false positive, correct, and missed prediction of our network.', '4. RESULTS': 'We explore a variety of observational strategies to assess how important high-cadence and multi-band data is in predicting and analyzing HMEs. For each observational strategy, we train a separate network and then evaluate its performance on the test set. We use the same test set but with different observational cadences to fairly compare the performance of each model. Our baseline observing strategy represents LSST-like observations and is described in Section 2.3. We also evaluate our model on a cadence strategy where there are no season gaps in the data and an observational mask is randomly sampled for each band at every time step with mean number of visits [57,72,186,194,169,174] across the ten years in the ugrizy bands respectively. We further consider a hybrid case where we have LSST observations but a subset of bands are observed with the random ca- \ndence mask to test how our network would perform with partial observations during the seasonal gaps. In addition, we evaluate the performance in the idealistic case where we have regularly sampled data at every weekly time step. Moreover, we evaluate the performance when we have regular sampling but including season gaps every year that range from 140-180 days, chosen randomly for each light curve. For each cadence strategy, we additionally evaluate the performance using only a subset of the LSST bands. All the different strategies are summarized in Table 1 with various metrics to evaluate the performance of each model. \nBased on the different observational cadences we tested, we find the color information to be the most important factor for predicting HMEs. We expect this to be the case since the different wavelengths probe different radii of the accretion disk, and therefore, are microlensed by different regions of the caustic map. With regular weekly observations in just the r -band, we are unable to make high fidelity predictions since we correctly identify just 24% of the HME peaks. When we have regular sampling in both the g and i bands, the network performs much better, correctly identify 58% of the peaks. Additional bands further increase the per- \nFigure 8. Example prediction of our network with LSST-like observations. The bottom panel represents the microlensing light curve observations for the six LSST bands, normalized such that the first observation in the r -band is zero. The middle panel shows the output of our neural network by classifying if there is an event peak within a 21 week window per side at each time step. The first 85 weeks and last 21 weeks are grayed out since we do not make predictions for those times. The orange label represents the peak of the event while the wider, green label is the window that the network is trained to predict if there is an HME peak. The top panel represents a mock alert system where the first 7 weeks of a continuous positive prediction (i.e. the probability is greater than 0.5 at the time step) are set to green alert followed by 7 weeks of yellow alert and then greater than 14 weeks is red alert. \n<!-- image --> \nformance of our model, but having at least two bands is most important. \nWe also find that the season gaps play a significant role in the performance of our models. In the example in Appendix A, the missed HME peak at around 3,200 days could be predicted if the characteristic rise and band crossing did not fall within the season. In addition, the exact location of the HME peak at around 2,000 days is not identified because it falls directly in the season gap. The network clearly knows there should be an HME peak somewhere in the season gap but is unsure about its exact location. With all LSST bands but no season gaps, we identified 69% of the event peaks instead of the LSST baseline case of 55%. With regular sampling and no season gaps, this increases to 72%. Thus the seasonal gaps play a larger role than the irregular sampling. We do find that with season gaps but regular sampling, we identify 66% of the peaks, so the performance loss from the season gaps can be made up with enough observations. Furthermore, we can make \nup some of the performance gap if there are some observations without seasonal gaps, such in one or two bands.', '5. DISCUSSION': "Our trained network is ready to be applied to the entire sample of lensed quasars of LSST. In the initial stages of survey before making predictions, we can finetune the network to calibrate it to the exact observational cadences and red noise properties of the lensed quasar sample. It will also be necessary to measure the time delays between lensed images in order to produce the microlensing light curves. We begin making predictions after an initial time of 85 weeks so the network has at least two seasons or so of data and to reduce the potential uncertainty in the time delay measurements. A shorter initial time could be used if the time delays can be measured, although we would expect worse performance in the early time of the survey, since there would fewer observations to infer the microlensing time scales of the system. \nOur baseline LSST observations come from selecting sky positions within the main LSST survey. A limited \nTable 1. Summary of metrics comparing the performance of each separately trained network on their test set using different observation strategies. The baseline for LSST-like observations is given at the top. The accuracy, precision, recall, and F1 score come from evaluating our network on random time steps throughout the survey across the test set. We note that the positive labels represent only about 3.6% of the data set, so an accuracy of 0.964 could be achieved by just predicting the negative class. The correct peak, red in label, and false positive metrics come from evaluating each time step of the entire 10-year light curve for 1,000 examples in the test set, i.e. making predictions like in Figure 8, and weighted by the total number of HME peaks. Correct peak means the peak was correctly identified (i.e. the orange label in the middle panel of Figure 8 has predicted probability at least 0.5). Red in label means that there is a red alert within the 21 week event peak window (i.e. the green label in the middle panel of Figure 8 overlaps with the red alert in the top panel at some point). A false positive represents a red alert outside any event peak window (i.e. if the red alert in the top panel of Figure 8 did not overlap at all with the green label in the middle panel). \nnumber of strongly lensed quasars may be monitored in the Deep Drilling Fields which will have much higher cadences. Our predictions would significantly improve from the increased number of observations. We expect the performance metrics for lensed quasars in the Deep Drilling Fields to be similar to the regular sampling with seasons in Table 1. \nOur goal is to train a general HME predictor for LSST. Thus, we use a diverse training set that should be representative of the entire expected LSST quasar sample. For specific systems that have already been well studied prior to LSST, we could train individual models to make more tailored predictions. For example, an HME predictor could be developed specifically for Q 2237 + 0305 like in Woźniak et al. (2000). We would expect these models specially trained for individual systems to have improved performances, since it can specialize within a much narrower parameter space. We could also give the network priors on the κ , γ , s , and velocity distributions of systems if they have been estimated. Knowing these values could constrain the time scales of microlensing events and help the network identify where the peak of the HMEs are with more clarity. The alert system itself could also be tailored to individual systems or populations with similar microlensing timescales. We choose a 21 week window around the HME peaks, so the training labels extent to around a season long. Some systems will be easier to trigger ahead of time, and we may then \ndefine a longer window, while others will be more difficult and may benefit from a narrower window. Fine tuning the length of the training labels can be explored in future work. \nIn our microlensing simulation, we use a simple Gaussian brightness profile scaled to the disk size of each waveband. We expect that at the observational cadences of LSST, the shape of the inner region of the brightness profile has little effect on the microlensing light curve. Furthermore, we are particularly interested in predicting the HMEs ahead of time. At these time scales and cadences, only the size of the disk chosen as its half-light radius is important, as the exact shape of the brightness profile has little impact on the light curve (Mortonson et al. 2005). This is no longer the case during caustic crossing events, as the inner region of the accretion disk can be highly magnified. The analysis of the highcadence follow-up, where the inner detail of the accretion disk becomes encoded in the light curve and may be able to constrain system parameters (Best et al. 2024), is beyond the scope of this work. \nWe build our training set by adding red noise from the residual power spectrum measured in the r -band of RXJ 1131 -1231 because this is currently one of the best sampled, highest signal-to-noise ratio light curves with well-measured time delays. Since the photometric Gaussian noise is fairly low, this system is ideal to isolate and quantify the correlated red noise. Since the \nCOSMOGRAIL data only has r -band observations, we assumed the red noise in each wavelength to be equivalent. In reality, this might not necessarily be the case, depending on the physical origin of the red noise. If it originates from a flux contamination of the multiple image, we can expect a similar amplitude in all photometric bands. On the other hand, if it originates from the quasar variability echoed by the BLR, we may expect more correlated noise in the photometric bands overlapping with a broad emission line of the quasar. This would therefore depend on the redshift of the quasar. We choose a conservative approach for this work by using RXJ 1131 -1231, since it has one of the largest amplitudes of red noise. Once LSST data becomes available, the residual power spectrum of each LSST band can be measured from all monitored systems, and we can fine-tune our model based on the real population of red noise. The photometric errors are included in the red noise generated from RXJ 1131 -1231 from the Leonhard Euler Telescope, which should be similar to LSST's photometric errors for objects this bright. Nevertheless, by using the entire LSST population, we will have access to more realistic photometric errors for each system. For bright quasars of (<19 mag), we expect the photometric errors to be only ∼ 0.01 magnitude and play a minor role. For the fainter quasars (>20 mag), the photometric errors can be significant and our network should be adapted to better account for them. There may also be additional uncertainty captured within the red noise related to uncertainty in measuring the time delay between lensed images. RXJ 1131 -1231 is a well studied system and there are only small uncertainties on its time delay measurements. This will not be the case for all systems monitored in LSST, and this additional uncertainty will be incorporated into the population of red noise. Recent methods of image deconvolution could largely reduce these effects, hence reducing the amount of correlated noise in the extracted microlensing light curves (Millon et al. 2024). \nIn our training set, we apply two thresholds to the HME labels. The first is that an event needs to be at least 0.5 magnitude to be considered a HME. This value is chosen to be just above the maximum amplitude of the added red noise. Furthermore, in the subtracted microlensing light curves, we apply an additional threshold of 1 magnitude between the maximum and minimum of the light curve across all bands. This second threshold helps exclude labels where long-term microlensing events cancel each other out, leaving little indication of an HME in the light curve. These thresholds are designed to minimize true false positives and avoid triggering expensive follow-up observations unnecessarily. In- \ndeed, the choice of these thresholds will affect how the network performs. In Appendix A, we show an example where if the threshold would have been lower, the first false positive would have been considered a correct prediction, as this peak would have been labeled. Thus our reported false positive rate is a conservative estimate, since edge cases where the HME is just below the threshold will often be misclassified by the network. Our focus is on predicting the HMEs that have the most significant impact on the microlensing light curves. Future studies could explore adjusting these thresholds, potentially lowering them to capture more subtle events. This approach could be particularly useful in scenarios where the amplitude of the red noise is reduced, allowing for a broader range of HMEs to be detected. \nIdeally we would simulate the intrinsic quasar variability and use a variable disk when convolving with the microlensing maps instead of using a data-driven approach. Then we would measure the time delays between each image using the initial simulated data of LSST and subtract the images to produce our simulated microlensing light curves. This would then include the effects from the uncertainty in measuring the time delays, the photometric errors, and red noise in a more self-consistent way. We would, however, need to model the BLR reverberation which can have very complicated geometry. There may also be extended continuum emission coming from other regions of the disk that we may not be able to simulate (Sluse 2024). Since accounting for all the aspects of the intrinsic variability may be infeasible, we prefer a data-driven approach and defer exploring the inclusion of these effects directly into the simulation for future work. \nHere we fix the threshold for a positive prediction of our network to a probability of 0.5, as is typically done in binary classification. By raising this threshold, however, we would predict fewer HME peaks but lower the false positive rate. We could also adjust the balance of classes in the training set. During training, we mimic observing at a random time throughout the 10-year survey, and we find a rate of positive labels to be around 3.6%. We could instead choose to bias the training set by including more or less mock observations near the HME peaks. Including more time steps near the HME peaks would enable the network to make more positive predictions, increasing the number of correctly predicted HME peaks but also raising the false positive rate. Within our metrics, we include predictions that fall within seasonal gaps. By focusing low-cost photometric follow-up near the predicted HME, we could improve our prediction and reduce the number of false positives before trigger- \ning more expensive follow-up observations, such as X-ray or spectroscopic monitoring. \nWe may also improve upon our machine learning model in future work. We use RNNs that included masking of missing values through GRU-D layers (Che et al. 2016). We may also explore other RNN architectures such as stochastic RNNs, which have been used to model UV/optical quasar variability (Sheng et al. 2022). Continuous recurrent units (Schirmer et al. 2022), which are a probabilistic recurrent architecture where the hidden state evolves as a linear stochastic differential equation, are a promising way of dealing with the irregular sampling. We may also consider using latent stochastic differential equations which has worked well at modeling the intrinsic quasar variability for simulated LSST light curves (Fagin et al. 2024). Transformers (Vaswani et al. 2023) such as multi-time attention networks (mTANDs; Shukla & Marlin 2021), where the time embedding is fed to the attention mechanism to incorporate irregularly sampled time series, may also show improved performance compared to RNNs. \nThe largest current catalog of monitored lensed quasars is COSMOGRAIL which has only r -band observations (Millon et al. 2020b). We show in Table 1 that for regular observations including seasonal gaps in the r -band, only around 15.7% of event peaks are correctly identified with a false positive rate of 12.2%. These metrics indicate that even with the most ideal case of regular weekly sampling, without multi-band data the network fails to make meaningful predictions. There are currently no catalog of long-term monitored lensed quasars with multi-band data, so we defer testing our model on real data for future work.", '6. CONCLUSIONS': 'We trained an RNN to predict HME peaks in LSSTlike microlensed quasar light curves. We make our model and training procedure open source and available at https://github.com/JFagin/Quasar\\_Event\\_Predictor. A pretrained neural network like ours could be quickly and autonomously applied to the entire sample of monitored LSST strongly lensed quasars each week as new data is obtained in less than a minute on a single GPU. \nThis is the first deep-learning method of predicting HMEs. \nFor the baseline LSST cadences, we can expect to correctly identified around ∼ 55% of HMEs with a false positive rate of ∼ 20%. We therefore expect to correctly identify tens to a couple hundred HMEs per year (Neira et al. 2020). These metrics can be improved as highcadence follow-up is triggered ahead of the peak of the event. \nWe envision our network to be continuously applied throughout the entire LSST survey to give weekly notifications to experts on potential incoming HME peaks. These alerts will be crucial for optimizing follow-up observational resources. High-cadence multi-band or spectroscopic follow-up observations of these HMEs will offer unique insight into the inner structure of AGN as imprinted in the light curves during caustic crossing events. \nSoftware: PyTorch (Paszke et al. 2019), Numpy (Harris et al. 2020), Matplotlib (Hunter 2007), Astropy (Astropy Collaboration et al. 2018),', 'ACKNOWLEDGMENTS': 'This work originated in the Lensing Odyssey 2021 workshop, and so we would like to acknowledge the organizers and attendees for the fruitful discussions. This work used resources available through the National Research Platform (NRP) at the University of California, San Diego. NRP has been developed, and is supported in part, by funding from National Science Foundation, from awards 1730158, 1540112, 1541349, 1826967, 2112167, 2100237, and 2120019, as well as additional funding from community partners. Support was provided by Schmidt Sciences, LLC. for JF, HB, MO, and GV. MM acknowledges support by the SNSF (Swiss National Science Foundation) through mobility grant P500PT\\_203114 and return CH grant P5R5PT\\_225598. TA acknowledges support from ANID-FONDECYT Regular Project 1240105, ANID Millennium Science Initiative AIM23-0001 and the ANID BASAL project FB210003. HB acknowledges the GAČR Junior Star grant No. GM24-10599M for support.', 'A. ANOTHER EXAMPLE PREDICTION': 'In Figure 9 we show another example of the predictions of our network. In this case there is a false positive, correct prediction, and missed prediction from left to right. \nFigure 9. Example of the prediction of our network on LSST-like observations like Figure 8. \n<!-- image -->', 'REFERENCES': "Abell, P. A., Burke, D. L., Hamuy, M., et al. 2009, Lsst science book, version 2.0, Tech. rep., LSST Antonucci, R. 1993, ARA&A, 31, 473, \ndoi: 10.1146/annurev.aa.31.090193.002353 Astropy Collaboration, Price-Whelan, A. M., Sipőcz, B. 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2024NewAR..9901706D

The Gaia mission has revolutionized our view of the Milky Way and its satellite citizens. The field of Galactic Archaeology has been piecing together the formation and evolution of the Galaxy for decades and we have made great strides with often limited data towards discovering and characterizing the subcomponents of the Galaxy and its building blocks. Now the exquisite 6D phasespace plus chemical information from Gaia and its complementary spectroscopic surveys has handed us a plethora of data to pore over as we move towards a quantitative rather than qualitative view of the Galaxy and its progenitors. We review the state of the field in the postGaia era and examine the key lessons that will dictate the future direction of Galactic halo research.

2024-12-01T00:00:00Z

['arXiv:2402.12443', '2024arXiv240212443D', '2024NewAR..9901706D', '10.1016/j.newar.2024.101706', '10.48550/arXiv.2402.12443']

['Galaxies: kinematics and dynamics', 'Galaxies: dwarf', 'Dark matter', 'Local group', 'Galaxies: stellar content', 'Astrophysics - Astrophysics of Galaxies']

Galactic Archaeology with Gaia

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML']

https://arxiv.org/pdf/2402.12443.pdf

{'No Header': 'a', 'Galactic Archaeology with Gaia': 'Alis J. Deason a , Vasily Belokurov b \nInstitute for Computational Cosmology & Centre for Extragalactic Astronomy, Department of Physics, Durham University, South Road, Durham, DH1 3LE, UK \nb Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK', 'Abstract': 'The Gaia mission has revolutionized our view of the Milky Way and its satellite citizens. The field of Galactic Archaeology has been piecing together the formation and evolution of the Galaxy for decades, and we have made great strides, with often limited data, towards discovering and characterizing the subcomponents of the Galaxy and its building blocks. Now, the exquisite 6D phase-space plus chemical information from Gaia and its complementary spectroscopic surveys has handed us a plethora of data to pore over as we move towards a quantitative rather than qualitative view of the Galaxy and its progenitors. We review the state of the field in the postGaia era, and examine the key lessons that will dictate the future direction of Galactic halo research. \nKeywords: Galaxies: kinematics and dynamics, Galaxies: dwarf, dark matter, Local Group, Galaxies: stellar content.', '1. Introduction': "Galaxies like our own Milky Way digest and destroy hundreds of lower-mass systems over their lifetimes. Some of these 'subhaloes' have no stars at all, and just contribute to the dominating dark matter halo. Most of the more massive systems do contain stars, and their contributions create an extended halo of stars surrounding the Galaxy (e.g. Bullock and Johnston, 2005; Cooper et al., 2010). The long orbital timescales in the halo lead to the destroyed stellar systems reflecting their initial progenitor orbits, especially in Energy and Angular momentum (Helmi and de Zeeuw, 2000). Thus, the 'stellar halo' is a fossil record of the past accretion history of the Milky Way, and as such has been studied extensively for several decades (see earlier reviews by e.g. Belokurov, 2013; Bland-Hawthorn and Gerhard, 2016; Helmi, 2020). \nThe stellar halo not only encodes an archaeological record of the Milky Way's assembly history, but it also provides a tool to study (now destroyed) high redshift dwarf galaxies. Moreover, the great extent (out to > 100 kpc) of the stellar halo makes it one of our best tracers of the dark matter halo. In past years, our study of the Galactic halo has mainly been hampered by limitations in the data. Our sample sizes have been too small, only limited areas of the sky have been covered, we have incomplete velocity information, hardly any chemical information, etc, etc. However, these limitations (excuses?) are no longer, as we are now deep into the era of Gaia - the most ambitious Galactic science mission of all time. \nEmail addresses: alis.j.deason@durham.ac.uk (Alis J. Deason), vasily@ast.cam.ac.uk (Vasily Belokurov) \nGaia is an astrometric mission with the lofty goal of producing an all-sky phase-space map of the Galaxy (Gaia Collaboration et al., 2016b). Positions on the sky, parallax, and proper motions are measured for all stars down to a limiting magnitude of G ≈ 21. In addition, a (bright) subset of stars has measured radial velocities and stellar parameters (such as effective temperature and surface gravity, see Recio-Blanco et al. 2023 for details of the parametrization of the Gaia RVS data). As of June 2024, we are now on the third Gaia data release with the fourth release likely arriving in 2025/26. Each subsequent data release provides data over a larger time baseline, and hence more precise astrometry, as well as additional data products, such as variability information and source classifications. To compensate for the bright limit for accurate Gaia spectroscopic measurements, several wide-field spectroscopic surveys have (or will) complemented the Gaia data to provide radial velocities and chemical measurements, e.g. SDSS/SEGUE (Yanny et al., 2009), LAMOST (Cui et al., 2012), APOGEE (Majewski et al., 2017), GALAH (Buder et al., 2021), H3 (Conroy et al., 2019), DESI (Cooper et al., 2023), WEAVE (Jin et al., 2023), 4MOST (de Jong et al., 2019). These auxiliary datasets are particularly important for the halo stars, which traverse to large distances and are thus often relatively faint. The era of Gaia is truly data-heavy! The fruits of this labour will last for many years to come, and we are only at the beginning stages of our digestion and understanding of what this detailed mapping of the Milky Way contains. \nThe purpose of this review is to provide the current state of understanding of the Galactic halo and the revelations and surprises from the Gaia data boon. This review does not intend to provide a complete history and con- \nsensus, as the number of papers in this field has exploded in recent years. Instead, we focus on the main findings, in our humble opinion, and look ahead to the future of this field over the coming decade. There are several articles in this special issue focused on ' Gaia discoveries'. As such, the in-depth exploration of stellar streams, globular clusters and Galactic dynamics with Gaia is covered elsewhere and we refer the interested reader to related reviews on those topics.", '1.1. A dominant stellar halo progenitor?': 'One of the main aims of the Gaia mission is to piece together the building blocks of the Galactic halo. This ambitious goal was perhaps mainly expected to reveal a plethora of streams, clouds, and debris from a multitude of progenitors (see e.g. Helmi and de Zeeuw, 2000). However, one of the clearest and most decisive discoveries from the early Gaia data releases was that in fact the (inner, r ≲ 20 kpc) stellar halo is dominated by one dwarf galaxy progenitor (see Section 2). As we will discuss below, in hindsight this was not a surprising discovery!', '1.1.1. Predictions from Λ CDM': "The fundamental mass function of dark matter haloes ( dn/dM - i.e. how many haloes there are of different mass) follows a power-law profile dn/dM ∝ M -1 . 9 (e.g. Moore et al., 1999; Jenkins et al., 2001; Gao et al., 2004; Tinker et al., 2008). Thus, the general expectation is that there are many more smaller lumps than bigger lumps - a prediction that lies at the heart of the hierarchical ΛCDM paradigm. A consequence of the mass function is that haloes like our own Milky Way engulf many more smaller mass objects than more massive ones over time. For example, cosmological N -body simulations (Fakhouri et al., 2010) predict that a Milky Way-mass halo (10 12 M ⊙ ) typically undergoes N ∼ 30 minor mergers (mass-ratio between 1:3 and 1:100) and N ∼ 3 major mergers (mass-ratio > 1 : 3) since redshift z = 10. \nWhen galaxies enter the equation, things become more complicated as the efficiency of star formation is not equal amongst haloes of different mass. The inefficiency of massive galaxies in the centres of clusters (mainly due to AGN feedback, e.g. Croton et al. 2006; Hopkins et al. 2006; Fabian 2012) and dwarf galaxies (supernova feedback plus reionization, e.g. Dekel and Silk 1986; Efstathiou 1992; Bullock et al. 2000; Pontzen and Governato 2012) at forming stars means that galaxies like the Milky Way lead the pack - we live in the most efficient star-making galaxy environment (see e.g. Figure 2 of Wechsler and Tinker, 2018). The relation between stellar mass and dark matter mass (aptly known as the 'stellar mass-halo mass' or SMHM relation), has important consequences for the growth of stellar haloes through the accretion of lowermass objects. The SMHM has a very steep decline at low halo masses ( ≲ 10 11 M ⊙ , see e.g. Figure 6 of Bullock and Boylan-Kolchin 2017). Thus, although there \nmay be hundreds of dark matter subhaloes accreted by the galaxy, if these have very few stars (or no stars at all!) then they have an insignificant contribution to the accreted stellar halo. This idea was recognized in Purcell et al. (2007), who used analytic prescriptions for dark matter halo growth and galaxy formation models, and later by Deason et al. (2016) using N -body simulations and analytic SMHM relations. In short, the contribution of stellar mass to the galactic stellar halo is dominated by a small number (1-2) of relatively massive dwarfs ( M DM ∼ 10 11 M ⊙ , M star ∼ 10 8 -9 M ⊙ ). The contribution of lower mass dwarfs, while potentially important for dark matter growth, is relatively insignificant for the total stellar halo mass (see Fig. 1) \nThese ideas are fleshed out further in more sophisticated stellar halo models (e.g. Bullock and Johnston, 2005; De Lucia and Helmi, 2008; Cooper et al., 2010; Monachesi et al., 2019; Fattahi et al., 2020; Horta et al., 2023a), where the spatial distribution of accretion events can be explicitly tracked. Here, the dominance of a small number of massive progenitors is retained, but they are generally confined to the inner ( ≲ 20 -50 kpc) halo, where they have been rapidly dragged inwards towards the centre of the host halo via dynamical friction. The small number of stellar halo building blocks and a wide range of merger configurations lead to a pronounced diversity of the resulting stellar halo density distributions (see e.g. Cooper et al., 2010), in contrast to the universality of the dark matter halo densities (Navarro et al., 1996). \nThe low binding energy of lower mass contributors means that they can be splayed out into the outer halo in the form of streams and overdensities. Additionally, shells from massive satellites disrupted on more radial orbits can reach large galacto-centric distances as well (Hendel and Johnston, 2015; Pop et al., 2018). Thus, the outer halo can often be populated by a wider range of progenitors, and is less likely to be dominated by a small number of contributors (see. e.g. Figure 7 in Fattahi et al. 2020). \nIn summary, the combination of the accretion of dark matter subhaloes plus the steepening of the SMHM relation leads to an intriguing scenario for the stellar haloes of Milky Way-mass galaxies. By stellar mass contribution alone only a few progenitors really matter. So, in the real Galaxy, where are they?", '1.1.2. Pre-Gaia clues from observations': "Number counts: A fundamental 'zeroth order' measurement of the Galactic stellar halo is to simply count the number of stars. As such, the characterization of the stellar halo density profile has been studied for several decades (e.g. Hartwick, 1987; Preston et al., 1991; Robin et al., 2000; Yanny et al., 2000; Juri'c et al., 2008; Deason et al., 2011b; Xue et al., 2015). In recent years, however, it has been recognized that this is not simply a counting exercise, and the form of the stellar density profile in fact encodes key information pertaining to the assembly history of the halo (e.g. Bullock and Johnston, 2005; Cooper et al., 2010). \n<!-- image --> \n<!-- image -->", 'Stellar mass-halo mass': "Hierarchical halo formationFigure 1: Top Left: Galaxy luminosity function - the role of feedback (Silk and Mamon, 2012). Top Right: Schematic representation of a merger tree (Lacey and Cole, 1993). Time increases from top to bottom (present time = t 0 , formation time = t f ), and the width of the branches represents the individual halo masses. Bottom panel: The mass-weighted distribution of destroyed dwarfs that contribute to the stellar haloes of Milky Way-mass systems (Deason et al., 2016). The thin coloured lines are for individual haloes, and the thick black line is the average. The thick dashed line shows the average profile for 'quiescent' haloes, which have not undergone a major merger since z = 2. The dotted line shows the mass-weighted contribution of dark matter haloes, which have a much flatter distribution. This profile differs from the stellar mass owing to the steep decline in the stellar mass-halo mass relation at low halo masses ( ≲ 10 11 M ⊙ ). Bottom insert: The (mass-weighted) number of dwarf progenitors that contribute to the total accreted stellar mass. Typically, 1 -2 destroyed dwarfs deposit the majority of accreted stellar mass onto the host halos [reproduced from Silk and Mamon 2012; Lacey and Cole 1993; Deason et al. 2016]. \n<!-- image --> \n<!-- image --> \nEarly work limited to the inner halo ( ≲ 20 kpc) described the stellar density as a simple power-law profile with radius ( ρ ∝ r -α , Robin et al. e.g. 2000; Yanny et al. e.g. 2000; Chen et al. e.g. 2001; Siegel et al. e.g. 2002; Juri'c et al. e.g. 2008). When the outer parts of the halo began to be mapped (largely thanks to wide-area photometric surveys such as SDSS), it was quickly recognized that this profile does not continue to larger radii. Indeed, Deason et al. (2011b) (also see Sesar et al. 2011) found that the stellar halo is better characterised by a 'broken' power-law with a significant steepening in the power-law (from α ∼ 2 . 5 to α ∼ 4 . 5) beyond r ∼ 25 kpc (note that glimpses of the stellar halo break were seen earlier, see e.g. Saha, 1985; Watkins et al., 2009). In a follow-up paper, Deason et al. (2013a) provided a possible explanation for such a 'broken' profile (see Fig. 2). They used N -body simulations to argue that the debris from individual stellar halo progenitors often follow a broken power-law, where the break radius signifies the last apocentre of the orbit. In this scenario, a broken profile in the overall halo could imply two possible outcomes: (1) the stellar halo is dominated by one progenitor, or (2) several stellar halo progenitors all conspired to be accreted at the same time, with similar apocentres. The authors argued that the former scenario was more likely, owing to the contrived nature of the latter. However, the question remained, if the stellar halo was indeed dominated by one massive progenitor, how did we not know this already, and where is the additional observational evidence beyond the number counts? \nChemistry: Early studies of RR Lyrae and globular clusters (GCs) indicated that the stellar halo stars tend to have lower metallicity compared to the rest of the Galaxy (see e.g. Preston, 1959; Hartwick, 1976; Searle and Zinn, 1978). This is expected if the stellar halo is built from less massive objects that are less chemically evolved than more massive systems like the host Milky Way (i.e. following the mass-metallicity relation, Kirby et al. e.g. 2013). However, measurements of the halo metallicity distribution function (MDF) or even its average metallicity have typically been fraught with selection biases and unknown levels of disc contamination (Eggen et al., 1962; Pagel and Patchett, 1975; Norris, 1986; Beers et al., 1992). Thus, while the bulk of the stellar halo is relatively metal-poor ( ⟨ [Fe / H] ⟩ ∼ -2 . 0 to -1 . 0, Ryan and Norris 1991; Ivezi'c et al. 2008; Carollo et al. 2010; Xue et al. 2015; Conroy et al. 2019), whether or not this metallicity reflects a dominant massive progenitor remains inconclusive. \nA more revealing chemical analysis comes from the study of α elements in addition to Fe. The α elements are mostly produced in massive stars and dispersed into the interstellar medium by type-II (core collapse) supernovae (SNe) over short timescales (of the order of Myr) while Fe can be contributed in similar amounts by both type-II and type-Ia SNe (see e.g. Kobayashi et al., 2006, 2020). The latter occur over ∼ Gyr timescales (note however that the SNe Ia delay time distribution extends to much shorter timescales below 0.1 Gyr, see e.g. Maoz and Mannucci, \n2012), and this distinction in dominant delay times leads to a characteristic 'knee' in the [ α /Fe] vs [Fe/H] chemical plane (Tinsley, 1979). Specifically, the onset of type-Ia SNe signifies a downturn from a nearly constant 'plateau' in [ α /Fe] when more Fe is produced compared to α (see examples of such behaviour in chemical evolution models of Andrews et al., 2017; Spitoni et al., 2017; Weinberg et al., 2017). Thus, by studying the [ α /Fe] vs. [Fe/H] plane we can probe the star formation history (SFH) of a stellar system. In particular, the location of the 'knee' may be directly related to mass as more massive systems are able to retain and build up metals before the onset of type Ia SNe -i.e. their 'knee' is more metal-rich (e.g. Hendricks et al., 2014; de Boer et al., 2014; Mason et al., 2023). While observational evidence for 'knees' and their connection to the galaxy's mass remains mixed (see e.g. Kirby et al., 2011; Nidever et al., 2020), different dwarf galaxies do display distinct behaviour in the α -[Fe/H] plane (Tolstoy et al., 2009; Hasselquist et al., 2021). Although the above description of the chemical evolution of a dwarf galaxy is clearly oversimplified, this chemical insight is of particular interest for the stellar halo, where we are aiming to disentangle the individual ingredients of this aggregated soup. Moreover, chemistry can also be used to assess whether or not the halo substructures are truly remnants of disrupted dwarfs (e.g. Horta et al., 2023b). \nIn the influential work by Venn et al. (2004) the authors compared chemical abundances of halo stars to surviving dwarf spheroidal satellite galaxies. They found that their chemical signatures are remarkably different, perhaps ruling out the common lore that the stellar halo is built up from the debris of many low-mass galaxies! However, importantly, the authors note that mergers with higher mass dwarf galaxies cannot be ruled out, as the [ α /Fe] vs [Fe/H] plane for the halo stars exhibits a relatively metal-rich knee (although less metal-rich than the disc stars). An additional conundrum is that the chemistry of the halo stars also does not resemble the Large Magellanic Cloud (LMC) or Sagittarius (Sgr), which are two of the three most massive dwarf galaxies ( M star ∼ 10 8 -9 M ⊙ ) in the Milky Way. The ΛCDM simulation community was swift to clear this up: the stellar halo models predict that the destroyed dwarf progenitors were accreted early, and thus have star formation histories different to the late-time Sgr and LMC type objects that can continue to form stars (Font et al., 2006). Font et al. (2006) showed explicitly how the ΛCDM accretion plus chemical evolution models can match the Milky Way data. In summary, from the chemical perspective, the dominance of early, massive dwarf galaxy progenitors seems to fit the bill. \nA final intriguing discovery in the chemical analysis of halo stars came from the work by Nissen and Schuster (2010). These authors used precise abundance ratios to show that there are two distinct halo populations in the solar neighbourhood. One 'highα ' sequence can perhaps be attributed to an 'in-situ' halo component (see Section 4.1), while the other 'lowα ' sequence resembles the classi- \n<!-- image --> \nT \nub \nM \nFigure 2: Left: Number counts of BHB stars from SDSS data. The blue line is the best-fit broken power-law, and the red line is the best-fit single power-law (Deason et al., 2011b). Right: Example of an individual accretion event from the Bullock and Johnston (2005) simulation suite. The panels show the radial velocity (top) and density profile (bottom) as a function of radius. The dashed line indicates the 'breakradius' of the best-fit broken power law that coincides with the apocentre of the debris. The black box indicates the 'local' region of the debris, which has a high density and significant radial velocity v r [reproduced from Deason et al. 2013a]. \n<!-- image --> \ncal 'accreted' component. However, what was remarkable about this analysis was just how narrow and ordered the chemical sequence of the accreted stars was in the [ α /Fe] vs. [Fe/H] plane. Presumably, a hodge-podge of several different progenitors with different masses and SFH would produce a messier outcome. Alternatively, at least in the solar neighourhood, there really is one progenitor that dominates the accreted halo. \nKinematics: The (local) stellar halo is kinematically hot with a high overall velocity dispersion ( σ ∼ 100 km/s) and little net rotation reflecting the messy build-up of a stellar halo relative to the 'ordered' state of the Galactic disc (e.g. Norris, 1986; Chiba and Beers, 2000). As the observational samples grew, the first hints appeared that the orbital motion in the nearby halo was not isotropic, and in e.g. spherical polars, the velocity dispersion in the radial direction was larger than in the other two tangential ones (Bahcall and Casertano, 1986; Freeman, 1987; Chiba and Yoshii, 1998). It is customary to characterise the shape of the velocity ellipsoid using the anisotropy parameter: β = 1 -[ ( σ 2 θ + σ 2 ϕ ) / 2 σ 2 r ] (Binney and Tremaine, 2008). Smith et al. (2009) (see also Bond et al. 2010) used large samples of nearby metal-poor main sequence dwarf stars to show that the halo velocity ellipsoid is indeed remarkably anisotropic, with the radial component σ r dominant over the tangential components. Smith et al. (2009) and Bond \net al. (2010) found β ∼ 0 . 7 for the local samples - did this reflect the orbital make-up of the stellar halo as a whole? \nIn the preGaia era, accurate space velocity measurements for distant objects were hard to come by. As a result, outside of the Solar neighourhood, the view of the halo's orbital properties remained rather blurred. For Galactic GCs it is possible to beat down the proper motion errors by averaging over large numbers of member stars. Thus, through early heroic efforts first estimates of the Milky Way's GC orbital properties were obtained (see e.g. Gnedin and Ostriker, 1997; Dinescu et al., 1999). These studies found the GC velocity ellipsoid to be broadly consistent with that measured using stellar kinematics in the Solar neighourhood. In addition, Dinescu et al. (1999) found systematic - albeit sometimes rather subtle - differences in the orbital properties of the GCs as a function of their age, chemistry, and location in the Galaxy. For example, in their sample, the clusters' orbital eccentricity exhibited dependence on metallicity, with the metalpoorer GCs reaching higher eccentricities and the metalricher ones tending towards more circular orbits. \nFor individual distant halo stars, no useful proper motion measurements were available before Gaia and therefore the properties of the halo velocity ellipsoid outside of the solar neighbourhood were teased out from the distribution of the line-of-sight velocity measurements under \nsat \nthe assumptions of relaxation and symmetry. Using models based on distribution functions (similar to those described in Wilkinson and Evans, 1999) and data from the Sloan Digital Sky Survey (SDSS), Deason et al. (2012) showed that in the range of galactocentric distances of 15 < r (kpc) < 40, the kinematics of Blue Horizontal Branch (BHB) stars favoured a stellar halo anisotropy of β ≈ 0 . 5. Thus, compared to the vicinity of the Sun, on scales of tens of kpc, the stellar halo's orbital anisotropy appeared less radially biased. This observation was soon confirmed by Kafle et al. (2012) who additionally reported hints of drops in the radial anisotropy profile. As a demonstration of the power of space-based astrometry and a preview of Gaia 's capabilities, the Hubble Space Telescope was used to measure the 3D kinematics of a small number of stars in the distant halo by Deason et al. (2013b) and Cunningham et al. (2016). Interestingly, these studies measured lower values of 0 < β < 0 . 3 beyond r ≈ 20 kpc. \nWere these early measurements of the stellar halo's orbital anisotropy consistent with models of galaxy formation, and in particular, was such a high value of β in the solar neighbourhood expected? For example, if multiple low-mass progenitors are accreted over a prolonged time with a wide range of orbital properties, then presumably a more isotropic velocity ellipsoid is expected. This is indeed the case in simulations, where β starts relatively low in the centres of halos β < 0 . 3 and rises up to 0 . 5 < β < 1 at the virial radius (Abadi et al., 2006; Sales et al., 2007; Debattista et al., 2008; Rashkov et al., 2013). From these trends, values of up to β ≈ 0 . 5 are generally expected in the halo near the Sun. \nA very different orbital structure forms in the stellar halo in the series of experiments discussed in the visionary work by Amorisco (2017). In their toy N -body models, massive accreted satellites lose most of their angular momentum in a rapid tidal disruption, driven in part by strong dynamical friction. Amorisco (2017) demonstrate (see their Figure 11) that the resulting debris has a strong radial anisotropy, reaching close to β ≈ 1 at the approximate location of the Sun (see also right-panel of Fig. 2). This orbital radialization reported by Amorisco (2017) for massive satellites went contrary to the previously established intuition where dynamical friction helped to circularize the satellite's orbit instead (see e.g. Jiang and Binney, 2000; Nipoti, 2017). While hints of orbital radialization had been seen already by Barnes (1988), the mechanics of the angular momentum loss during the accretion of massive satellites were finally explained in Vasiliev et al. (2022). Thus, this relatively new explanation of massive satellite radialization can naturally account for the high β values found in the solar neighbourhood if the inner halo is dominated by the debris from massive systems. \nFinally, several trailblazing studies managed to demonstrate the principles of Galactic Archaeology in action, even with the scarce data available before Gaia . These early pioneers combine chemistry and orbital information to detect individual accretion events in the \nSolar neighourhood. Helmi et al. (1999) discover a stellar stream from a small dwarf galaxy passing near the Sun, in a clear example of what is to become a routine in the Gaia era. Equally striking are the first likely hints of the tidal debris from the last significant merger in the works of Dinescu (2002) and Brook et al. (2003). Both use the chemo-kinematic dataset by Chiba and Beers (2000). The study of Dinescu (2002) finds evidence for a retrograde disruption event, while Brook et al. (2003) interprets the observations as a signal from a recent accretion of a dwarf on a polar orbit. \nIt is worth remarking at this point that there is a lot of clarity in hindsight, and looking back at these studies in a postGaia era is particularly revealing. However, it is undeniable that putting the pieces of this puzzle together took a special mission and a game-changing observational dataset.", '2.1. Gaia Data Release 1': "On September 14 2016 the very first Gaia Data Release was delivered to the eager, but somewhat unready, community to forever change the way Galactic Astronomy is done. However, the unprecedented all-sky ∼ 1 billionstrong stars only came with mean positional coordinates, RA and Dec - in Gaia DR1, proper motions, and parallaxes were limited to a tiny ∼ 2 million subset of the brightest stars that had been seen previously by Hipparcos (Gaia Collaboration et al., 2016a). The size of this so-called TychoGaia Astrometric Solution (TGAS, Lindegren et al., 2016) was unfortunately too small to explore in detail the behaviour of the stellar halo in the low-metallicity regime. Surprisingly, however, a significant number of metal-rich stars ([Fe/H] > -1) on halo-like orbits were detected in the Solar neighourhood by Bonaca et al. (2017). These metal-rich stars comprised approximately half of the halo sample near the Sun and exhibited a clear prograde bias compared to their much more isotropic low-metallicity counterparts. Using measurements from the APOGEE survey, this metal-rich population is shown to follow the sequence of the old Milky Way disc in the α -[Fe/H] plane (see Fig. 3). Weighing up the observational evidence and comparing to the numerical simulations, Bonaca et al. (2017) concluded that this metal-rich halo population must be of in-situ origin and suggested the disc heating by merger events as the key pathway to its formation. While not explicitly connecting the in-situ halo to the last massive merger, the inventive and dexterous use of the limited DR1 data and the insightful comparison to the available simulations makes the pioneering paper by Bonaca et al. (2017) a clear highlight of the early Gaia exploration which helped to set up the stage for the extensive DR2 analysis. \nAside from the TGAS, the rest of the billion stars in Gaia DR1 would have their detailed astrometry published \n<!-- image --> \n<!-- image --> \nFigure 3: Left: Toomre diagram (velocity space spanned by V Y aligned with Galactic rotation and V XZ , its complementary component) for stars with Gaia TGAS astrometry. Color represents average metallicity. Middle: [ α/Fe ]-[Fe/H] distribution of halo stars (blue points) compared to the rest of the APOGEE data (red-scale density). Right: Metallicity distribution function for halo (disc) in blue (red). Note a clear overlap between the disc and halo MDFs. [Adapted from Bonaca et al. (2017)]. \n<!-- image --> \nlater, as part of the DR2. Not prepared to wait, several groups decided to combine Gaia DR1 with earlier astrometric data from elsewhere thus taking advantage of the enormous, uniform, and high-quality dataset Gaia was providing, as well as the large temporal baseline between Gaia and other surveys. At least three such catalogues were made: the first two obtained proper motion estimates simply by cross-matching Gaia DR1 with the data from PPMXL 1 (HSOY, see Altmann et al., 2017), and PS1 (Tian et al., 2017). The third proper motion catalogue was made by Sergey Koposov following a rather different strategy to connect Gaia DR1 and SDSS (see Koposov, 2024, for details and the catalogue download). Koposov realized that the Gaia data could be used to fix the systematic biases in the astrometric calibration of the SDSS itself before using the two to measure positional offsets. Unfortunately, the details of the SDSS re-calibration and the catalogue construction were not described in a standalone paper, but the tests of the catalogue performance were published in Deason et al. (2017b) and de Boer et al. (2018). \nAll three catalogues reported similar mean random proper motion errors between 1 and 2 mas/year. The principle difference between HSOY and PS1Gaia on the one hand and the SDSSGaia on the other was the amplitude and the spatial distribution of the systematic bias. As shown in Altmann et al. (2017) and Tian et al. (2017), the first two catalogues carried systematic proper motion offsets of the order of 1-2 mas/year, i.e. similar to their typical random error. Moreover, these systematic shifts followed an unruly patchwork-like pattern on the sky. This can be compared to ∼ 0.1 mas/year systematic uncertainty of the SDSSGaia proper motions which also exhibited no discernible large-scale pattern on the sky (see Deason et al., 2017b; de Boer et al., 2018). The quality of the new, large, deep, and wide-angle proper motion sample was verified with two observational tests. First, it was shown that the median proper motion of the spectroscopically confirmed QSOs was ∼ 0 . 1 mas/year. Second, proper \nmotions across a wide stretch of the Sgr stellar stream were measured for the first time and were shown to be in good agreement with values predicted by simulations (see Deason et al., 2017b). The SDSSGaia catalogue was also used by de Boer et al. (2018) to settle the argument around the nature of the so-called Monoceros stream which had been claimed to be either an outer disc perturbation (Kazantzidis et al., 2008) or the tidal debris from a large dwarf galaxy disrupted on a nearly co-planar orbit (see Helmi et al., 2003; Pe˜narrubia et al., 2005). The SDSSGaia proper motions probed directly the kinematics of the Monoceros ring which appeared to exhibit the clear pattern of the remnant disc rotation (de Boer et al., 2018). \nDeason et al. (2017b) used the SDSSGaia proper motions to study the amount of coherent rotation in the Galactic stellar halo. Previously, in the absence of ample homogeneous proper motion measurements for faint stars, the halo rotation was extracted from its projection onto the line-of-sight velocities of tracers across large swaths of the sky. These experiments had been inconclusive: only weak rotational signals had been detected (see Sirko et al., 2004; Smith et al., 2009; Deason et al., 2011a). However, claims had also been made of distinct coherent prograde and retrograde rotation signals in different parts of the halo by Carollo et al. (2007, 2010) (although this was subsequently contested by Fermani and Schonrich 2013). For the first time, Deason et al. (2017b) used a sizeable proper motion dataset for a variety of stellar halo tracers - namely BHBs, RR Lyrae and Red Giant Branch stars - to measure the mean azimuthal speed of the halo directly. All three tracers showed remarkable consistency: the halo's spin turned out to be minimal, ≲ 20 km/s. To interpret these measurements, Deason et al. (2017b) studied the behaviour of the stellar halo rotation in the Auriga suite of numerical simulations of Milky Way-like galaxy formation (Grand et al., 2017). Two trends became apparent: i) in Auriga, on average, stellar halos were rotating faster compared to the observed Milky Way, and ii) old and young simulated halo stars showed different rotational signatures, namely young halo stars had more spin (see Fig. 4). Deason et al. (2017b) concluded that the slow rotation \nFigure 4: Left: Distribution of mean rotation velocities in Auriga stellar halos for all (old) accreted stars shown in black (green). Red vertical line marks the Milky Way measurement with SDSSGaia DR1. Middle: Rotation velocity as a function of Galacto-centric radius for all accreted stars in Auriga halos. Right: Same as middle panel but for the old halo. Red measurements with error-bars show the Milky Way data. [Reproduced from Deason et al. (2017b)]. \n<!-- image --> \nof the Milky Way halo based on the SDSSGaia proper motions was consistent with simulations. They noted, however, that the Galactic halo's slight spin was indicative of a particular accretion history, in which the Milky Way halo was assembled early and was not spun up by subsequent mergers. \nInspired by the early tests of the SDSSGaia proper motion catalogue described above, Belokurov et al. (2018b) used it to study the metallicity dependence of the local stellar halo velocity ellipsoid. This task required a reassessment of the conventional approach to identify a pure halo sample. In previous studies, the halo tracers had been limited to i) large heights above the disc plane and/or ii) metal-poor and old stellar populations such as BHBs and RR Lyrae. Following this approach appeared impractical at intermediate and high metallicities close to the Sun. Therefore, instead of applying hard selection cuts, Belokurov et al. (2018b) chose to model the entire stellar velocity distribution. More specifically, in each metallicity bin considered, they used a Gaussian Mixture Model (GMM) implemented as part of the Extreme Deconvolution package (XD, see Bovy et al., 2011). The important feature of XD is that it allows one to take the measurement's uncertainty into account and thus gauge the intrinsic properties of the distribution. While the SDSSGaia catalogues supplied the best-quality proper motions available for faint main sequence stars at the time, no geometric distances were available for the sample. Instead, Belokurov et al. (2018b) relied on the power of the SDSS multi-band photometry to estimate the absolute magnitudes of their stars following the relations derived in Ivezi'c et al. (2008). \nBelokurov et al. (2018b) demonstrate that, within ∼ 10 kpc of the Sun, the stellar halo's velocity ellipsoid evolves strongly as a function of metallicity (see Fig. 5). In the low metallicity regime, i.e. [Fe/H] < -2, the orbital anisotropy is not too far from zero at β ≈ 0 . 3. This nearly isotropic halo component is mixed in with a large population of stars on nearly radial orbits. This metal-richer population with \n[Fe/H] > -2 reaches extremely high β ≈ 0 . 9. In velocity space, this swarm of radially biased orbits manifests itself as a sausage-like shape stretched along the radial dimension and squashed along the azimuthal one. Belokurov et al. (2018b) show that the highβ component extends to surprisingly high metallicities of [Fe/H] ≈ -1. They also reveal that in the metallicity regime probed, stars on highly eccentric orbits comprise some ∼ 2 / 3 of the local stellar halo. The na¨ıve GMM representation of the velocity data seems to have worked well, but the residual distribution revealed two noticeable radial velocity lobes, with high positive and negative V r . \nBelokurov et al. (2018b) elucidate the genesis of the radially-anisotropic halo component with the analysis of high-resolution zoom-in simulations of Milky Way formation. They show that in the mock stellar halo samples at host-centric distances similar to Solar, the orbital anisotropy is a strong function of the progenitor's mass and its accretion time in agreement with earlier studies (see e.g Deason et al., 2013a). Across a wide range of accretion events explored, only the most massive progenitors with dark matter masses of the order of 10 11 M ⊙ can yield high anisotropy β > 0 . 8 provided the mergers took place at 1 < z < 2. In their simulation suite, this period in the Galaxy's life also coincided with the phase of active disc growth. The growing disc seemed to help make the orbit of the in-falling massive satellite more eccentric. Most importantly, however, there is a conspicuous convergence in that many independent lines of evidence point to a particular accretion history for our Galaxy. Namely, the one in which a single massive dwarf galaxy was accreted 8-11 Gyr ago, as previously hypothesized by Deason et al. (2013a). As predicted by Amorisco (2017), the orbit of the dwarf radialized during its interaction with the Milky Way, populating the inner halo with stellar tidal debris on nearly radial orbits. The resulting high eccentricity of the main stellar halo component also explains the low net spin measured by Deason et al. (2017b). Finally, the relatively high metallicity of the eccentric halo stars in the So- \nFigure 5: Left 3x2 small panels: Velocity distribution of local stars in the SDSSGaia catalogue. Top: greyscale gives density of stars in three metallicity bins, the numbers of stars in each bin are shown in the top right corner of each panel. Notice the pronounced change in the shape of the distribution as a function of [Fe/H]. Bottom: residuals of the Gaussian Mixture model of the data. Note the two prominent 'lobes' at high negative and positive radial velocity. Right large: Halo orbital anisotropy as a function of metallicity for three different heights above the disc plane (red=low, green=intermediate, blue=high). [Adapted from Belokurov et al. (2018b)]. \n<!-- image --> \nneighbourhood agrees with the expectation for stellar contents of a massive dwarf galaxy in accordance with the established mass-metallicity relation (see e.g. Kirby et al., 2013). Belokurov et al. (2018b) conclude that the sausagelike feature in the local phase-space ought to be the result of an ancient accretion of a single, massive satellite. \nPhase-space analysis of the kind described above is appropriate for data samples limited to the Solar neighourhood. It is also very convenient as it does not require an assumption of the gravitational potential of the Galaxy and can be re-produced easily, including in applications to mock data from numerical simulations. However, as demonstrated by e.g. Helmi and de Zeeuw (2000), the identification of merger debris is more straightforward and more efficient in the integrals-of-motion space. This is exactly what is attempted in Helmi et al. (2017) where a low-metallicity halo sample is selected using a cross-match between TGAS and RAVE (see Steinmetz et al., 2006). Unfortunately, the resulting number of stars ( ∼ 1000) is too low to discern clearly any global patterns, although the presence of a conspicuous retrograde population is spotted. Instead, similarly to Deason et al. (2017b) and Belokurov et al. (2018b), Myeong et al. (2018b) used the much larger SDSSGaia sample which contained > 10 , 000 halo stars across a wide range of metallicity. They scrutinize the behaviour of the halo in the action space in different [Fe/H] bins and detect the prevalence of highly eccentric orbits at -2 < [Fe/H] < -1, for which they agree the simplest explanation is the in-fall of one satellite.", '2.2. Gaia Data Release 2': "The works discussed in the previous section used Gaia DR1 to find the first bits of irrefutable evidence of a dra- \nmatic merger event between the young Milky Way and a massive dwarf galaxy. Later, the transformational Gaia Data Release 2 helped to locate some of the crucial missing pieces in the puzzle of the Milky Way transformation and the stellar halo assembly. Helped by the vast number of exquisite proper motion and parallax measurements, the following five key properties of the last significant merger were quickly discovered. \n- · The progenitor dwarf stars follow a distinct sequence in the Hertzsprung-Russell diagram (Gaia Collaboration et al., 2018a; Haywood et al., 2018).\n- · These stars also stand out from the rest of the halo, and, crucially, are distinct from the more metalrich, in-situ component in the α -[Fe/H] space (Helmi et al., 2018).\n- · The satellite that merged with the Milky Way came accompanied by an entourage of its own GCs (Myeong et al., 2018c).\n- · The motion of the stars with [Fe/H] < -1 within ∼ 20 -30 kpc appears synchronized, which in turn explains the global spatial structure of the halo (Deason et al., 2018; Iorio and Belokurov, 2019)\n- · The bulk of the in-situ halo with [Fe/H] > -1 was produced by heating and splashing of the preexisting Milky Way disc (Gallart et al., 2019; Di Matteo et al., 2019; Belokurov et al., 2020). This is discussed further in Section 4.2. \nBelow we go chronologically (i.e. in the order they first appeared online, although some of these papers came out \n<!-- image --> \n<!-- image --> \nFigure 6: Left: HRD for stars with high tangential velocities (typical halo kinematics) in Gaia DR2 (Gaia Collaboration et al., 2018a). Note the split main sequence. Middle: High tangential velocity stars separated into blue and red populations with a model isochrone corresponding to [Fe/H]= -0 . 75 and age of 11.5 Gyr (Haywood et al., 2018). Right: Toomre diagram (also see Fig. 3) for blue and red stars selected in the previous panel. [Adapted from Gaia Collaboration et al. (2018a); Haywood et al. (2018)]. \n<!-- image --> \nwithin a month of each other) through these individual pieces of evidence. \nHRD dichotomy . The striking apparent split in the Hertzsprung-Russell diagram (HRD) of the nearby stars with high tangential velocity - i.e. on halo-like orbits was first presented and discussed in one of the Gaia DR2 science demonstration papers (Gaia Collaboration et al., 2018a, see Fig. 6). The authors notice two separate HRD sequences - blue and red according to their G BP -G RP colour - most visibly disconnected around the main sequence turn-off region, which they analyse and interpret to be produced by two halo populations with distinct mean metallicities, [Fe/H] ∼ -1 . 3 and [Fe/H] ∼ -0 . 5. Gaia Collaboration et al. (2018a) report clear similarities with the bi-modality in the metallicity distribution of the Galactic GCs, which is known to arise due to contributions from clusters predominantly accreted from dwarf galaxies and those formed in situ in the Milky Way. They speculate that the two halo populations must have followed distinct formation channels and surmise that there may be a connection between the two halo HRD sequences and the chemical sequences highlighted earlier in Nissen and Schuster (2010) and Bonaca et al. (2017). Haywood et al. (2018) take a deeper look at the Gaia DR2 HRD to solve the puzzle of the dichotomy of the stellar halo. They demonstrate that while both populations have orbits distinct from the disc, there is a clear difference in the kinematics of the stars in the blue and red sequences. Haywood et al. (2018) show that the red sequence is composed of stars on mostly prograde orbits with lower eccentricities and lower total energies. The blue sequence stars have little angular momentum, have higher eccentricities, and reach out further into the halo. The HRD sequences indeed correspond almost perfectly to the two chemical sequences found by Nissen and Schuster (2010): the red one contains stars with higher α -abundances and the blue one with lower α - \nnces. The connection between kinematics and detailed chemistry is also made in Helmi et al. (2018), who together with Haywood et al. (2018) appear inspired by the earlier analysis of the halo chemistry using APOGEE (see Hawkins et al., 2015; Fern'andez-Alvar et al., 2018; Hayes et al., 2018). Haywood et al. (2018) associate the red sequence to the thick disc in agreement with Bonaca et al. (2017) and the blue sequence to the ancient massive merger as proposed in e.g. Deason et al. (2013a) and Belokurov et al. (2018b). Haywood et al. (2018) argue that the tightness of the blue sequence in the Gaia HRD is an argument in favour of the single dominant accretion event because contributions from many dwarf progenitors would otherwise bloat it. \nMember Globular Clusters. There exists a remarkable connection between the combined mass of a galaxy's GCs and its total (dark matter) mass with more massive galaxies hosting larger GC populations (see Blakeslee et al., 1997; McLaughlin, 1999; Spitler and Forbes, 2009; Harris et al., 2017; Dornan and Harris, 2023). In the Milky Way, this correlation manifests itself very clearly: only the four largest dwarf satellites possess their own GC posses. Aside from the LMC, SMC, Sgr and Fornax, only one other dwarf galaxy (Eridanus 2) has a single member GC (Koposov et al., 2015; Crnojevi'c et al., 2016). In archaeological studies of the Milky Way halo this GC-host mass link implies that while many tidal debris remnants are expected to be detected in the stellar halo, only the most massive of these would be accompanied by GCs on similar orbits. A comparison between the orbital properties of the halo stars and the Galactic GCs (using proper motions measured by Gaia Collaboration et al., 2018b) is carried out in Myeong et al. (2018c) (see Fig. 7). First, they introduce the idea of the 'critical energy' which separates the in-situ and the accreted GCs. Then they show that amongst the accreted clusters, a tight group stands out. The GCs in \nFig. 8). The location of this pile-up matches the break radius in the stellar halo density (see Section 1.1.2). As a sanity check, they also compare apo-centres of nearby main sequence stars to the apo-centres of the distant BHB stars that were originally used to detect the break and show that stars in the two populations, distant and local, turn around on their orbits at approximately the same distance. Thus three different tracer populations have been shown to have the same orbital behaviour: the nearby main sequence stars, the distant BHBs, and the GCs. Deason et al. (2018) demonstrate in a very candid way that the local metal-rich and radially anisotropic population is only a glimpse of a vast and dominant cloud of stellar debris that dictates the global properties of the stellar halo (e.g. the density break). They also provide the first constraint on the metallicity distribution function of the higheccentricity population and show that its incidence drops by a factor of ∼ 2 going from [Fe/H]= -1 to [Fe/H]= -2. \n<!-- image --> \n<!-- image --> \nFigure 7: Left: Galactic GCs in the space spanned by their peri-centric and apo-centric distances. Old halo (OH, red filled circles), young halo (YH, blue filled triangles), bulge/disc (BD, yellow triangles) and Sgr dwarf (black filled square), and its GCs (SG, filled green diamonds) are marked. The 8 GCs attributed to GS/E by Myeong et al. (2018c) are circled in black. Middle: The same GCs are shown in the space of radial and azimuthal velocity components (also see Fig. 5). Right: Age-metallicity plane for the GCs with available ages. [Adapted from Myeong et al. (2018c)]. \n<!-- image --> \nthis group have very similar orbital actions. More specifically, their pericentres are all remarkably low ( < 2 kpc) but their apocentres are some of the largest in the sample ( > 10 kpc). Myeong et al. (2018c) point out that there is most likely a link between this GC group and the eccentric stellar halo component discussed in Belokurov et al. (2018b). Both exhibit high orbital anisotropy β > 0 . 9 and the sausage-like shape in the space spanned by velocity components in spherical polars. Consequently, they argue that such a notable stand-out GC agglomeration favours a single accretion event and propose to use the correlation discussed above to estimate the total mass of the progenitor dwarf which they roughly guess must be in the ballpark of 5 × 10 10 M ⊙ . Myeong et al. (2018c) are inspired by the works from the preGaia era (Zinn, 1993; Mackey and Gilmore, 2004; Mackey and van den Bergh, 2005; Forbes and Bridges, 2010) but they are able to take these ideas to the next level thanks to the unprecedented quality of the Gaia astrometry. \nOrbital synchronicity . The use of the GCs as tracers highlighted one particular difficulty of the stellar halo studies with Gaia . Put simply, given the limited reach of the Gaia DR2 parallaxes, the vast extent of the halo appeared inaccessible. These observational hindrances can be circumvented using tools of Galactic Dynamics with which one can predict the overall density distribution of the halo using only a small sample of tracer kinematics constrained to the Solar neighourhood (see e.g. Eggen et al., 1962; Chiba and Beers, 2000). This is because the spatial appearance of the Galaxy is nothing but the superposition of the orbits of the individual stars and the density in the configuration space is linked to the kinematics through the continuity of the phase-space flow (Binney and Tremaine, 2008). In a similar vein but in a much more direct experiment, Deason et al. (2018) compute orbits of the metal-rich halo stars in the Solar vicinity and demonstrate that their apo-centres pile-up around Galactocentric distance r ≈ 20 kpc (see \nDetailed chemistry . Helmi et al. (2018) draw attention to the fact that the vast numbers of eccentric stars covering a huge portion of the accessible Galactic halo all appear to follow incredibly narrow sequences in both colourmagnitude and α -[Fe/H] spaces (see Fig. 9). More specifically, they demonstrate the existence of a large elongated cloud (or a 'blob', see Koppelman et al., 2018) in the velocity and/or energy-angular momentum ( E,L z ) space and isolate its stars using simple cuts in E and L z . They also notice a prominent retrograde extension of the debris cloud (previously discussed in e.g. Myeong et al., 2018b; Koppelman et al., 2018). Note that the follow-up work by Myeong et al. (2019) attributes these retrograde substructure to a different, but perhaps contemporaneous accretion event, the so-called Sequoia. \nHelmi et al. (2018) show that on the HRD, these stars are largely confined by isochrones with -1 . 3 < [Fe/H] < -0 . 9 and ages of 10-13 Gyr. Helmi et al. (2018) crossmatch their selected sample to APOGEE and see two \nFigure 8: Left: Orbit comparison for two local halo main sequence populations, one with low metallicity (blue) and one with high metallicity (red). Middle: Orbit comparison for two distant halo BHB populations (same colour coding). Note the similarity of the orbital behaviour of the metal-rich main sequence and BHB stars: both are on highly eccentric orbits coming close to the centre of the Milky Way, and both with apocentres close to 15-20 kpc. Right: Fractional contribution to the halo number counts of stars with high eccentricity as a function of metallicity. [Adapted from Deason et al. (2018)]. \n<!-- image --> \nnarrow sequences in the α -[Fe/H] plane, one mostly flat with -1 < [Fe/H] < -0 . 3 and [ α /Fe] ≈ 0 . 25 and one sloping down from [Fe/H] ≈ -1 . 3, [ α /Fe] ≈ 0 . 25 to [Fe/H] ≈ -0 . 7, [ α /Fe] ≈ 0 . 1. Following Bonaca et al. (2017), they assign the highα sequence to the in-situ population (see also Nissen and Schuster, 2010). Given that the two sequences co-exist over a range of metallicities, Helmi et al. (2018) conclude that the halo stars in the lower α -[Fe/H] sequence must have formed in a separate, smaller galaxy. In view of the large spread of metallicity and the presence of lowα high-[Fe/H] stars this dwarf could not be much smaller than the LMC. Their interpretation of the two chemical sequences builds on the analysis of the halo stars in APOGEE by Hawkins et al. (2015), Fern'andezAlvar et al. (2018) and Hayes et al. (2018). These earlier works provide a detailed chemo-dynamical comparison of the high- and lowα sequences and agree on the in-situ nature of the former. Crucially, however, in the absence of the Gaia data, these authors refrain from making a claim as to the exact nature of the accreted population, although Fern'andez-Alvar et al. (2018) report a significantly lower star-formation rate for the lowα population. Helmi et al. (2018)'s conclusion is not solely based on the accumulation of the observational evidence available at the time. Their insight is also the result of an earlier numerical exploration, a series of experiments simulating high mass-ratio mergers between the Milky Way and a dwarf satellite (Villalobos and Helmi, 2008). Helmi et al. (2018) find striking similarities between the Gaia DR2 data and one of the preexisting simulations (once the simulation is appropriately re-scaled). Helmi et al. (2018) propose to give a name to the progenitor system (and presumably its tidal debris): Gaia-Enceladus . \nFollowing the two papers (Belokurov et al. 2018b and Helmi et al. 2018) that first identified the metal-rich \nhighly-eccentric accreted halo population and proposed that it must be produced in a single massive accretion event, the vast tidal debris cloud is referred to as Gaia Sausage-Enceladus (GS/E).", '3. Characterization of Gaia -Sausage-Enceladus': 'In the previous section, we have recounted the overwhelming evidence for the massive GS/E progenitor in the Milky Way stellar halo. Naturally, since its discovery there has been a concerted effort in the community to further characterize this colossus accretion event.', '3.1. Selecting GS/E members': "Perhaps the most important first step in such studies is to understand how exactly we select the GS/E stars, as any deduction of its properties (e.g. total mass, metallicity etc) will be significantly impacted by any selection effects. On the one hand, the proximity of this structure to the solar neighourhood means we have significant numbers of potential GS/E stars to study. However, this also means that the GS/E is located in a region close to the stellar disc, and is thus prone to contamination. In the Belokurov et al. (2018b) discovery paper the authors use a GMMmethod to simultaneously model the different Milky Way components in velocity space, but even this advance over simple kinematic cuts cannot guarantee a 'pure' sample of GS/E stars. Unsurprisingly, more recent works have looked to action-angle space to locate the GS/E stars (e.g. Feuillet et al., 2020; Lane et al., 2022), which should result in a cleaner selection than velocity information alone. Other works have made use of the exquisite APOGEE data to perform a purely chemical selection of GS/E stars, whereby a cut in the plane of [Mg/Mn] versus [Al/Fe] is assumed to cleanly separate accreted and in-situ stars (e.g. Das et al., 2020; Carrillo et al., 2022; Lane et al., 2023). \n<!-- image --> \n<!-- image --> \nFigure 9: Panel (a): Toomre diagram for stars in Gaia DR2 (see also Fig. 3 and Fig. 6). Blue points are selected to pick out a large, mostly retrograde accreted sub-structure. Panel (b): Similar to Panel (a) but for a simulation of a minor merger from Villalobos and Helmi (2008). Panel (c): [ α /Fe]-[Fe/H] space for stars in Gaia DR2 with APOGEE chemical abundance measurements (same colour-coding as in Panels (a) and (b)). Panel (d): Metallicity distribution for blue stars shown in Panel (c). Solid (dotted) lines show the distribution of [Fe/H] when highα stars are excluded (included). [Adapted from Helmi et al. (2018)]. \n<!-- image --> \nIt is clear that there is no 'perfect' mode of selection, and how the GS/E stars are selected should depend on the question at hand (or the property being determined!). Comparisons of different methods are examined explicitly in several works with varying degrees of success (e.g. Feuillet et al., 2021; Buder et al., 2022; Lane et al., 2022, 2023; Carrillo et al., 2024, see Fig. 10). Carrillo et al. (2024) in particular uses cosmological simulations to inform the selection choices, which is likely a necessary tool moving forward in our understanding of the GS/E. Regardless of what selection mode is chosen, it is clear that quantifying any systematic influences is crucial in order to properly characterize the GS/E. For example, Carrillo et al. (2024) shows that stellar mass estimates of the GS/E can vary by a factor of ∼ 2 depending on the selection used. Lane et al. (2023) provide a convincing demonstration that once the APOGEE spatial, chemical, and kinematic selection effects are taken into account, the mass of the GS/E based on its red giant member stars is at the low end of the spectrum, around M star ∼ 1 . 5 × 10 8 M ⊙ , and its overall fractional contribution to the stellar halo is around 15% -25%. Most mass measurements rely on the metallicity distributions of GS/E stars, and the assumption of well-known mass-metallicity relations (e.g. Feuillet et al., 2020; Carrillo et al., 2024). While these measures typically favour a GS/E with M star ∼ 10 8 -10 9 M ⊙ , a more accurate determination will likely require more robust GS/E selections (in addition to a better-known redshift dependence of the mass-metallicity relation). This is still a very active area of research, but one in which the combination of multiple observational dimensions (i.e. action-angle plus chemistry) plus numerical calibration will likely provide the most robust quantitative measures.", '3.2. Halo kinematics and the orbital make-up of the GS/E debris': "The GS/E debris covers a large portion of the Galactic halo, overlapping broadly with both the in-situ popula- \ntions and other, smaller accreted sub-structures. However, its member stars appear to follow distinct trends in phasespace, which can be retrieved with appropriate tools. \nNecib et al. (2019) and Lancaster et al. (2019) focus on the bi-modality in the Galactocentric radial velocity distribution of the local halo stars (see Fig. 5). Such a bimodality can form when stars deposited by a progenitor on a nearly radial orbit are observed between the turn-around points in their orbits, i.e. between their peri- and apocentres (see Fig. 2). Necib et al. (2019) demonstrate that in the Solar vicinity, the halo's velocity distribution can be decomposed into two components, one approximated well by a single multi-variate Gaussian and another better described by a model with two peaks in v r space. In the latter model, the v r peaks are two identical Gaussian distributions with their separation controlled by a free model parameter. Necib et al. (2019) argue that the presence of a significant anisotropic sub-structure in the local halo can have important implications for direct dark matter detection experiments. If the progenitor dwarf galaxy managed to dump enough of its dark matter on orbits reaching the Sun's location (similar to the GS/E stars), then the resulting velocities of the dark matter particles interacting with the detectors will be different from those described by the conventional Standard Halo Model. \nSimilarly, Lancaster et al. (2019) rely on an identical two-component mixture to model a sample of previously identified BHB stars reaching far beyond the Solar neighourhood. Again, one model component is allowed to have a double-peaked v r distribution with an unconstrained separation between the peaks. They demonstrate that a single-Gaussian description of the velocity distribution of the Galactic stellar halo is a gross oversimplification. Lancaster et al. (2019) show that out to r ≈ 25 kpc from the Galactic centre, such a model does not fit the data correctly. They show that within this region, the BHB kinematics are best described by an equal contribution from a roughly isotropic halo and a radially \nFigure 10: Chemical and orbital properties of different GS/E selections (yellow points) compared to the rest of the APOGEE dataset (greyscale density). Top row: GS/E sample selected using [Mg/Mn] and [Al/Fe] abundance ratio space. Middle row: GS/E stars selected using an eccentricity cut (see 3rd column). Bottom row: GS/E sample created with a box in J r -L z space (magenta box in the fourth column). All GS/E samples are compared in four spaces: [Mg/Mn]-[Al/Fe] (first column), E -L z (second column), eccentricity-apocentre (third column) and J r -L z actions (fourth column). [Adapted from Carrillo et al. (2024)]. \n<!-- image --> \nanisotropic component, which they attribute to the GS/E merger. They also detect a decline in the separation between the radial lobes with Galacto-centric distance r , as expected from merger simulations (see Fig. 2). Lancaster et al. (2019) determine that beyond r ≈ 25 kpc, the fractional contribution of the bi-modal v r component is consistent with zero. These findings are in agreement with the detection (see e.g. Sesar et al., 2011; Watkins et al., 2009; Deason et al., 2011b) and the interpretation (see e.g. Deason et al., 2013a, 2018) of the stellar halo density break at a similar distance. \nThe appeal of BHBs is that despite being quite rare, they are a robust standard candle that can be picked out of a mix of stellar populations cleanly and easily. Compared to BHBs, K-giants are a more numerous and brighter halo tracer, at the expense of being a less accurate distance indicator. Bird et al. (2019) and Bird et al. (2021) use a large number of K-giants to probe the global anisotropy profile of the Galactic stellar halo from 5 to 100 kpc. Their single multivariate Gaussian model shows qualitatively similar behaviour to the single-component model of Lancaster et al. (2019). However, they report a higher orbital anisotropy for K-giants compared to the BHBs. Moreover, unlike the anisotropy of the BHBs which shows a pronounced decline around r ≈ 20 kpc as discussed above, the K-giants' β remains more or less flat across the whole range of Galactocentric distances probed. \nThe long-anticipated boon of the Gaia data releases have produced the largest catalogue of RR Lyrae, the pulsating counterpart of the BHBs, to date. However, even before the first Gaia variable star catalogues were provided as part of the Gaia DR2, Belokurov et al. (2017), Deason et al. (2017a) and Iorio et al. (2018) used the excess in Gaia DR1 mean flux uncertainty to identify likely variables such as RR Lyrae and Mira stars. In particular, Iorio et al. (2018) combined this rudimentary variability statistic with the 2MASS colour information to build the first all-sky RR Lyrae sample. They test a wide range of halo models and show that the best-fit is achieved by a triaxial ellipsoid whose vertical flattening varies with Galacto-centric distance. In this model, the longest axis lies in the Galactic plane but is rotated by ≈ 70 · with respect to the SunGalactic centre direction. Once the Gaia RR Lyrae sample was released within DR2, Iorio and Belokurov (2019) tested their earlier model and demonstrated that it was indeed a good fit, albeit to an even larger, purer, and more complete sample (see Fig. 11). \nA minor tweak was introduced by Iorio and Belokurov (2019) to their earlier model (Iorio et al., 2018) in order to tilt the halo slightly out of the Milky Way's disc plane (see also Han et al., 2022a). This non-axisymmetry is warranted by the RR Lyrae number counts and is likely related, at least in part, to the two previously identified large-scale sub-structures, the Virgo Overdensity (VOD Juri'c et al., 2008; Vivas et al., 2016) and the Hercules Aquila Cloud (HAC Belokurov et al., 2007; Simion et al., 2014). Curiously, VOD and HAC are shown to be closely \nFigure 11: Projections of RR Lyrae density in the inner halo. The Gaia DR2 data is shown as a greyscale map and blue contours. A triaxial model is indicated with the orange contours. [Reproduced from Iorio and Belokurov (2019)]. \n<!-- image --> \nrelated by Simion et al. (2019) who, based on orbital analysis, argue that both originated in the GS/E merger. A similar conclusion was reached by Balbinot and Helmi (2021). Moreover, Perottoni et al. (2022) show that, in addition to their dynamical link, the VOD and HAC are chemically indistinguishable from the prototypical GS/E population. The tilt in the GS/E debris cloud with respect to the disc as exemplified by the HAC-VOD asymmetry is examined by Han et al. (2022b) who point out that such an asymmetry cannot survive for a very long in an axisymmetric potential of the Galaxy. They argue that to maintain the stellar halo's tilt over a long period of time, a similarly tilted dark matter component is required. This idea is fully fleshed out in Han et al. (2023a) and Han et al. (2023b) where the misaligned dark matter halos are analysed and shown to be ubiquitous in Cosmological simulations. Such tilted dark matter halo components are damaging to the Galactic discs, causing them to warp and flare. In a similar vein, Davies et al. (2023) argue that the evolution of the GS/E debris in an axi-symmetric potential would lead to discernible striation in the integrals-ofmotion space once the stellar halo is tugged by the passing LMC. The fact that no such characteristic substructure is observed (although see Belokurov et al., 2023, for the detection and interpretation of high-frequency substructure in the GS/E debris) could be interpreted as evidence for a non-axisymmetric component in the Milky Way's dark matter distribution. From the analysis in Dillamore et al. (2022), it is clear that tilts such as those proposed in Han et al. (2023a) and Davies et al. (2023) are common in galaxies recovering from a significant merger. \nIorio and Belokurov (2019) also uncover a tight kinematic bond amongst most of the inner halo RR Lyrae. They show that within r ≈ 25 kpc from the Galactic centre, the stellar motions are correlated: the kinematics of the bulk of the RR Lyrae is consistent with a nearly radial motion, one of the key characteristics of the GS/E (see also Ablimit et al. 2022). Iorio and Belokurov (2021) move beyond demonstration to modelling of the kinematic properties of the Gaia RR Lyrae sample (see Fig. 12). Their model contains three distinct components: a rotating disc-like population, a quasi-isotropic one, and finally, a strongly radially anisotropic population corresponding to the GS/E. The modelling approach is inspired by an ear-", 'Radially anisotropic model': 'Figure 12: Top row: Strength of correlation between the longitudinal and latitudinal components of proper motion in Galactic coordinates for a model with a strong radial anisotropy (left) and Gaia DR2 RR Lyrae (right) (Iorio and Belokurov, 2019). Bottom row: Fractional contribution of the radially-biased halo component in the Gaia RR Lyrae as a function of the Galactocentric distance (left) and the radial velocity offset (the separation between the two Gaussians is twice this value) in the kinematic model of radially-biased halo component of the Gaia RR Lyrae (right) (Iorio and Belokurov, 2021). [Adapted from Iorio and Belokurov (2019, 2021)]. \n<!-- image --> \nlier work of Wegg et al. (2019) where 5D data with missing line-of-sight velocity is modeled under the assumption of axi-symmetry to extract the properties of the RR Lyrae velocity ellipsoid. For the radially anisotropic GS/E population, Iorio and Belokurov (2021) rely on the functional form introduced in Necib et al. (2019) and Lancaster et al. (2019) and map out the separation between the GS/E radial velocity bumps inside r ≈ 30 kpc. The separation reaches ≈ 500 km/s around r ≈ 5 kpc and drops to zero around r ≈ 25 kpc. The GS/E contribution to the halo RR Lyrae is maximal around the Sun at ∼ 75% but decreases to ∼ 20% inside r ≈ 5 kpc and outside r ≈ 20 kpc. Curiously, the properties of the isotropic component are not uniform throughout: Iorio and Belokurov (2019) show that the central ≈ 10 kpc are dominated by a more metal-rich RR Lyrae population with a higher fraction of rare High Amplitude Short Period pulsators (see Fiorentino et al., 2015; Belokurov et al., 2018a). Given its properties, this centrally concentrated RR Lyrae population is likely of in-situ origin.', '3.3. Chemical trends': "Even though the GS/E progenitor is no longer intact, its pre-disruption properties can be studied now , provided the constituent stars are identified (see Sections 3.1 and 3.2). As the GS/E stars flood the Solar neighourhood, as \naptly pointed out by Hasselquist et al. (2021), the community had been exploring the GS/E's chemical trends long before we had realised these stars were all of common origin (see Nissen and Schuster, 2010; Hawkins et al., 2015; Fern'andez-Alvar et al., 2018; Hayes et al., 2018). Thus, the main trends, i.e. the downward slope of the α -[Fe/H] track (Nissen and Schuster, 2010) and the constant low [Al/Fe] ratio (Hawkins et al., 2015) were known before the Gaia data started to arrive. These trends are updated, improved upon, and contrasted with those in surviving dwarf galaxies in Hasselquist et al. (2021) (see Fig. 13). They show that compared to the four most massive current satellites of the Milky Way, namely the LMC, the SMC, Sgr (holding on for dear life), and Fornax, the GS/E progenitor attains demonstrably higher [Mg/Fe] at -1 . 5 < [Fe/H] < -1 (but still some 0.1 dex below the disc sequence at the same metallicity). Clearly, the GS/E dwarf was more efficient at converting gas into stars compared to the present-day survivors. Hasselquist et al. (2021) hypothesize that this is due to its proximity to the Milky Way. Equally, the environment may have also played a crucial role in shaping the SFHs of the Magellanic Clouds which appear to have lived a mostly dull and lethargic life in the suburbs of the Galaxy. However, having formed next to a bigger galaxy did not end well for the GS/E progenitor: its life was cut short, although none of the models in the literature reveal signs of abrupt truncation in star formation (possibly due \nto the rigidity of the SFH approximation). \nThe chemical trends of the GS/E can also be compared to other known debris in the stellar halo. For example, Horta et al. (2023b) compare the chemical abundances of several known substructures and find that some (namely, Arjuna, LMS-1, I'itoi, and perhaps Sequoia - see also Monty et al. 2020; Feuillet et al. 2021) are chemically indistinguishable from the GS/E, perhaps suggesting that they share a common origin. Thus, similarly to the dynamical association of the VOD and HAC with the GS/E (see Section 3.2), care must be taken in defining distinct substructures in the halo. Indeed, a massive structure like the GS/E is omnipresent, and can have several subcomponents. \nCuriously, the first constraints on the details of the GS/E SFH based on the behaviour of its stars in the α -[Fe/H] plane are also preGaia (Fern'andez-Alvar et al., 2018). This can be compared to the models of Vincenzo et al. (2019) and Sanders et al. (2021) that were built to reproduce a much more fine-tuned selection of the likely GS/E members. All three models agree on the relatively short SFH of the GS/E progenitor: star formation activity had to stop after 3-4 Gyr because by then the dwarf was stripped of its gas and fully disassembled by the Milky Way tides. The star formation efficiency of Fern'andezAlvar et al. (2018) and Vincenzo et al. (2019) are similar at 0.3-0.4 Gyr -1 , whilst that in Sanders et al. (2021) is almost an order of magnitude lower at ≈ 0 . 06 Gyr -1 . To infer the SFH of the dwarf, the former two studies use theoretical elemental yields, while the latter simultaneously and self-consistently models a wide range of elements and pins down both the yields and the global parameters of the chemical enrichment model. The amplitude of the resulting star formation rate in the model of Vincenzo et al. (2019) is 3 M ⊙ yr -1 , some 7 times that of the model in Sanders et al. (2021). Integrating the SFR over GS/E's lifetime, Vincenzo et al. (2019) find a total stellar mass of ≈ 5 × 10 9 M ⊙ . Although not quoted directly in Sanders et al. (2021), their model would infer an order of magnitude lower total stellar mass of ≈ 5 × 10 8 M ⊙ . Hasselquist et al. (2021) provide two models of the GS/E chemical evolution, one with a very broad SFH lasting more than 10 Gyr and one with a much narrower peak in agreement with the works above. Even though trained only on the [Mg/Fe]-[FeH] data, this latter model has a peak SFR much closer to the results of Sanders et al. (2021). \nMost of the chemical studies above use the exquisite APOGEE data. However, APOGEE is somewhat limited in its ability to measure heavy element abundances, and, in particular, has only scarce data on neutron-capture tracers (only Ce is listed amongst the relatively reliably measured species as part of the APOGEE DR17). The information on s - and r -process had to come from elsewhere, namely from the data releases of the GALAH survey (De Silva et al., 2015; Buder et al., 2018, 2021) and individual follow-up studies. These challenging measurements (due to a small number of weak lines) yielded a \nresult not many had predicted: a noticeable enhancement in the r -process abundances in the GS/E stars compared to the rest of the halo (including the in-situ stars where available at comparable metallicities) and most of the surviving dwarf satellites (Aguado et al., 2021; Matsuno et al., 2021). Measuring the rather low [Ba/Eu] ≈ 0 in their follow-up VLT spectra, Aguado et al. (2021) deduce that the neutron-capture production in the GS/E progenitor proceeded mostly via the r -process channel. Note that while [Eu/Fe] ≈ 0 . 65 and [Eu/Mg] ≈ 0 . 3 are at the upper end of what is observed, these values are still noticeably below the extreme r -process enhancement detected in one ultra-faint dwarf, namely Reticulum 2 (Ji et al., 2016) which reaches [Eu/Fe] > 1, but similar to levels in other r -process enhanced UFDs such as Tucana 3 (Hansen et al., 2017) and Grus 2 (Hansen et al., 2020). Focusing on the tracks in [Eu/Mg]-[Mg/Fe] space for the GS/E stars with [Fe/H] > -1 . 5 in GALAH, Matsuno et al. (2021) conclude that the bulk of r -process in the dwarf must be due to neutron star mergers. Naidu et al. (2022) observe GS/E stars together with targets they designate 'Kraken' with Magellan and measure (amongst other lines) Eu, Ba, and Mg abundances. At similar [Mg/H], the GS/E has 0.3 dex higher [Eu/Mg]. Naidu et al. (2022) assume that the 'Kraken' had a more rapid star formation activity and conclude that the enrichment due to the neutron star mergers must be delayed by some 0.5 Gyr. Note that most 'Kraken' stars analysed in this study would also pass the cuts for the Milky Way pre-disc in-situ population (Aurora - see 4.3.1). Not only do these stars have low orbital energy and high Mg abundance, 16 out of 20 objects in the sample lie above the accreted sequence in the plane of [Al/Fe] vs [Fe/H] (see panels 2 and 3 in Fig. 2 of Belokurov and Kravtsov 2022) and along the rapidly rising in-situ track (see Fig. 2 in Horta et al. (2021). In our opinion, replacing 'Kraken' with 'early Milky Way', the arguments presented in Naidu et al. (2022) remain valid, or perhaps, become even more coherent.", '3.4. Globular clusters': "Owing to the strong evidence for a significant GS/E contribution to the inner halo of the Galaxy, several groups have re-assessed the classification of Milky Way GCs into in-situ and accreted systems and updated the individual clusters' past GS/E membership. \nMassari et al. (2019) assemble a compilation of GC ages from various sources and, following the earlier studies of Forbes and Bridges (2010) and Leaman et al. (2013), point out a clear bifurcation in the Milky Way's GC agemetallicity plane. In this space, at fixed age, GCs with higher metallicities are likely to have been born in a more massive and therefore more rapidly evolving progenitor system (the Milky Way) compared to the GCs with lower metallicities that could come from disrupted dwarf galaxies. Guided by the distribution of the GCs in the agemetallicity plane, Massari et al. (2019) set aside two distinct cluster groups that have likely been born in the \nFigure 13: Median trends in [Mg/Fe] (left) and [Al/Fe] (right) as a function of [Fe/H] for stars in GS/E (green), LMC (red), SMC (blue), Sgr (yellow) and Fornax (purple). Note that for [Fe/H] > -1 . 5, the GS/E median trends in both [Mg/Fe] and [Al/Fe] are above those for the four massive intact satellites. [Adapted from Hasselquist et al. (2021)]. \n<!-- image --> \nGalaxy proper: i) those that show clear prograde rotation and therefore are linked to the Galactic disc and ii) those that are embedded close to the nucleus of the Galaxy (inside the inner 3.5 kpc) and designated as the 'bulge' GCs. This approach allows Massari et al. (2019) to set aside some ≈ 40% of all Milky Way clusters, classified as in-situ. The remaining ≈ 60% of clusters, hypothesized to be accreted, are divided into groups according to their approximate location in the space of integrals-of-motion (IOM). For the IOM space of choice, Massari et al. (2019) relies on total energy E and the vertical component of angular momentum L z . Limiting themselves to low | L z | and a broad range of E , they classify 26 as past members of GS/E with a further possible 6 members. While some groups of GCs appear more-or-less standalone in the IOM space (such as those belonging to GS/E and Sgr) others are less obvious (such as those assigned to the Helmi stream). Furthermore, several GCs, some at high E and some at low E , are left without a known progenitor. The low-energy group, in particular, becomes a focus of attention for several follow-up studies (see e.g. Kruijssen et al., 2020; Forbes, 2020). However, Horta et al. (2020) see no obvious chemical difference between the GCs in the lowenergy group and the rest of the in-situ clusters. \nThe premise of the study by Kruijssen et al. (2019b) is identical to the starting point in Massari et al. (2019), namely that in-situ and accreted populations behave differently in the age-metallicity space. Instead of linking the chrono-chemical properties of the Milky Way globulars to their orbital behaviour, Kruijssen et al. (2019b) attempt to decipher the make-up of the Galactic GCs using the cluster E-MOSAICS models based on the EAGLE Cosmological numerical simulations (Pfeffer et al., 2018; Kruijssen et al., 2019a). Using the E-MOSAICS framework, Kruijssen et al. (2019b) isolate the 3 most significant events in the Milky Way's past, contributing the largest numbers of accreted GCs. However, without the use of the orbital information, identifying the progenitors of these systems turns out to be rather difficult. While the smallest group associated with the Sgr dwarf appears to be self-evident, the two most massive events are much more uncertain. \nSome 22 GCs are attributed to i) the so-called Canis Major dwarf (see Martin et al., 2004) and ii) the hypothetical Kraken galaxy, predicted by Kruijssen et al. (2019b) to be the largest event in the Milky Way's life, bringing with it, some ∼ 2 × 10 9 M ⊙ of stellar mass. The current consensus on the accreted nature of the Canis Major is rather sceptical given the disc origin of the various 'streams' associated with it (see e.g. Xu et al., 2015; Price-Whelan et al., 2015; de Boer et al., 2018; Sheffield et al., 2018; Laporte et al., 2020). Most importantly, however, neither of the two groups of Galactic GCs identified by Kruijssen et al. (2019b) exhibits coherent orbital behaviour and thus both are instead most likely amalgamations of parts of different mergers. The analysis of Kruijssen et al. (2019b) is trailblazing but falls short due to the limited quality of the age-metallicity information for the Galactic GCs and the neglect of the orbital information. An attempt to rectify the latter is made in Kruijssen et al. (2020), which re-uses the IOM classification made by Massari et al. (2019). In a similar vein, Forbes (2020) starts with GC groups constructed by Massari et al. (2019) and tweaks them very slightly using the age-metallicity information at hand. \nMyeong et al. (2019) focus on a group of retrograde Galactic GCs which they associate with Sequoia, an accretion event distinct from the GS/E but taking place around the same time in the Galaxy's past. Compared to the GS/E, Sequoia is of lower mass and potentially may have even been a companion to the GS/E's progenitor (but see Koppelman et al., 2020, for an alternative hypothesis in which the retrograde stars assigned to Sequoia are instead a part of the GS/E's outer disc). Similarly to Massari et al. (2019) and Kruijssen et al. (2019b), Myeong et al. (2019) use the age-metallicity relation to isolate groups of GCs but rely on actions to describe their orbital behaviour. The use of action space improves the detection of individual accretion events and helps Myeong et al. (2019) to update their GS/E membership from the initial 8 clusters published in Myeong et al. (2018c) to a total of 21. The main factor leading to this uptake in the GS/E membership is due to the inclusion of GCs with slightly lower total energies. \nFigure 14: Groups of GCs in action-angle space inferred from chemodynamical modelling. The coloured symbols are the observed GCs: different colours indicate different groups, with each GC coloured by its most likely group. The accreted groups of GCs are indicated with their 1 -, 2 -, 3 -σ confidence intervals. [Reproduced from Callingham et al. (2022)]. \n<!-- image --> \nLimberg et al. (2022) combine the most recent chemical and dynamical insights into the composition of the GS/E debris to update the GS/E GC membership. As several groups before them, they take advantage of the exquisite quality of the APOGEE chemical measurements to note the unique trajectory of the GS/E stars in the abundance space. In particular, Limberg et al. (2022) highlights the raised levels of [Mg/Mn] and [Al/Fe] (see also Hasselquist et al., 2021). Limberg et al. (2022) combine the detailed chemistry, ages and actions (similar to e.g. Myeong et al., 2018c, 2019) to identify a high-probability sample of 19 GS/E GCs. They also build a strong argument in support of the hypothesis in which Omega Cen is the surviving nuclear cluster of the GS/E progenitor dwarf (see also Massari et al., 2019, for a similar discussion). \nNote that caution is needed when classifying GCs using light elements like Mg and Al because these can be strongly affected by the cluster's chemical evolution (see e.g. Bastian and Lardo, 2018; Gratton et al., 2019; Milone and Marino, 2022). To avoid the clusters' anomalous chemical patterns when identifying their original host galaxy, the use of heavy elements has been advocated (see e.g. Minelli et al., 2021; Monty et al., 2023). In particular, Monty et al. (2023) demonstrate that the two GCs, NGC 288 and NGC 362, often assigned to the GS/E based on their ages and chemo-dynamical properties (Helmi et al., 2018; Massari et al., 2019) are clearly distinct in the ratios of r - and s -process to α elements. Monty et al. (2023) point out that NGC 362 is r -process enhanced to levels similar to the GS/E stars while NGC 288 is not. The elevated ratio of r -process to α elements is in agreement with the trends found in GS/E stars (see Aguado et al., 2021; Matsuno et al., 2021, as well as Section 3.3). The subtle variation in chemical fingerprints detected by Monty et al. (2023) is perhaps reflected in dramatic structural differences of these GCs. \nCompared to all preceding attempts, the study of Callingham et al. (2022) stands out as the only unbiased GC classification scheme. Instead of hand-picking groups of clusters in the chosen parameter space, they devise a method similar to Gaussian Mixture Modelling. For the Milky Way GCs, ages, metallicities, energy, and three orbital actions are modeled with component mixtures to obtain objective membership probabilities for each GC (see Fig. 14). This innovative approach is demonstrated to work well for simulated data. Unfortunately, Callingham et al. (2022) also reports a large degree of overlap between GCs across all dimensions studied. To mitigate against the arising degeneracies, Callingham et al. (2022) decide to initialize their mixture models to the GC classes reported by Massari et al. (2019). \nConcerned with the separation between the GCs born in-situ and those formed in dwarfs and later accreted, Belokurov and Kravtsov (2023) go back to the idea of the 'critical energy' highlighted in Myeong et al. (2018c). They show that in the stellar halo too, a rather sharp boundary exists in E,L z space which divides stars into mostly accreted and mostly in-situ. This division is mapped by the clear change in the overall levels of [Al/Fe] ratio amongst the field stars. Using the E,L z boundary delineated by the [Al/Fe] levels, Belokurov and Kravtsov (2024) classify all Milky Way GCs into two classes and find 106 out of 164 to be of in-situ origin, and 58 to be accreted. Therefore the most up-to-date GS/E membership of ∼ 20 GCs translates into ∼ 1 / 3 of the total accreted cluster population in the Galaxy. Note that although the proposed GC classification scheme based on their position in the E,L z plane has recently been independently verified in Chen and Gnedin (2024), Belokurov and Kravtsov (2024) caution that some ≈ 10% of the objects classified as in-situ could be of accreted origin. This will bring down the GS/E fractional contribution to the accreted GC pop- \nn in the MW. \nFinally, Valenzuela et al. (2023) recently argue that the GS/E GCs themselves have a bimodal age distribution, which they suggest is the result of a 'wet' GS/E merger that both brings in its own GCs while also forming new GCs in the process of the gas-rich merger with the Milky Way. This intriguing scenario allows for the gas mass of the GS/E progenitor to be estimated based on the properties of its GCs.", '3.5. Simulations': "As mentioned in Section 2, the use of numerical simulations has been instrumental in our understanding of the GS/E progenitor. Indeed, both discovery papers (Belokurov et al., 2018b; Helmi et al., 2018) relied on the use of simulations to interpret the observational results. Koppelman et al. (2020) explored the same pure N-body simulation suite used by Helmi et al. (2018) to scrutinize a particular halo that coincidentally matches the observed properties of the GS/E very closely. More specifically, they show that if the GS/E's dwarf progenitor had a metallicity gradient in its stellar disc - as would be typical for galaxies of that mass - the resulting tidal debris might end up showing groups with distinct chemo-kinematic properties similar to the GS/E's lowL z 'blob' and the Sequoia's retrograde branch. Khoperskov et al. (2023c) draws inspiration from the study of Koppelman et al. (2020) and detect a decrease in the overall metallicity of the GS/E debris in the APOGEE data as a function of the total energy in the Milky Way's gravitational potential. They run tailor-made pure N-body simulations of GS/E accretion to map this energy gradient into a spatial gradient within the progenitor dwarf galaxy. These estimates are then bench-marked using the HESTIA suite of hydrodynamical zoom-in simulations of Milky Way formation (Khoperskov et al., 2023a,b). Khoperskov et al. (2023c) find that the APOGEE measurements are consistent with 0.1 dex/kpc metallicity gradient in the progenitor before disruption. \nThe study by Koppelman et al. (2020) made the important point that the destruction of a massive dwarf galaxy leaves behind a complex chemodynamical structure. These ideas are similar but perhaps less pessimistic than the earlier insights of Jean-Baptiste et al. (2017) who show that the archaeological excavation of ancient accretion events can be made difficult or almost impossible if the incoming dwarf galaxies are massive enough (see also Pagnini et al., 2023). The above works do make a good point that even though we can observe a multitude of substructures with very different properties, they can still all have the same origin (i.e. to the GS/E event). This idea was taken further by Naidu et al. (2021) who uses a suite of N -body simulations to confirm that a significant number of the observed properties of the (inner) stellar halo can be attributed to a single GS/E event. Note that important to both of these works is the inclusion of a disc potential in their simulations, which is essential in order to interpret any inner halo structure. \nWhile the above reflects a more tailored approach to the simulations of a Milky Way-GS/E merger, other works have focused on large suites of Milky Way-mass galaxies simulated in a cosmological context. In this case, an 'ideal' Milky Way-GS/E system is harder to find, but such analyses are essential in order to put the Milky Way assembly history in context with the wider galaxy population. Fattahi et al. (2019) use the Auriga suite of simulations to show that approximately one-third of the (28) haloes have a dominant stellar halo component similar to the observed Milky Way. From this subset, Fattahi et al. (2019) deduce that these merger events typically occur 6-10 Gyr ago, with merging fragments of stellar mass M star ∼ 10 9 -10 10 M ⊙ . Orkney et al. (2023) extend the analysis of the GS/E events in Auriga and highlight the diversity of possible progenitor systems, with a broad range of stellar, dark matter masses and rotational properties. Mackereth et al. (2019b) use the EAGLE simulation suite to show that merger events with a similar pattern in chemistry and dynamics to the real GS/E are typically related to massive dwarfs with M star ∼ 10 8 . 5 -10 9 M ⊙ . Moreover, they argue that the high eccentricity of the GS/E stars today suggests the merger would have taken place at z ≲ 1 . 5. One particular Milky Way analogue within the EAGLE suite is additionally studied in greater detail by Bignone et al. (2019). Elias et al. (2020) show that many of the conclusions reached through studies of GS/Elike events in suites of zoom-in simulations also hold in a much larger but lower-resolution sample from Illustris. Finally, Dillamore et al. (2022) uses the ARTEMIS simulations to again analyse GS/E-like events in cosmological simulations. In agreement with Fattahi et al. (2019) they find that typically a third of the Milky Way-mass systems contain a feature similar to the GS/E. \nMoving forward, a combined effort of tailored Milky Way-GS/E mergers (e.g. Koppelman et al., 2020; Naidu et al., 2021) and analyses of cosmological simulation suites (e.g. Fattahi et al., 2019; Mackereth et al., 2019b; Dillamore et al., 2022) are needed to (a) link the observed properties of the GS/E to its initial state and co-evolution with the Milky Way, and (b) calibrate our various observational techniques to infer these properties. An important step in the right direction are the works of Amarante et al. (2022) and Rey et al. (2023) who create hydro-dynamical models of the GS/E-Milky Way encounter. Amarante et al. (2022) build tailor-made single merger models with realistic starformation recipes and detailed chemistry. They show that many previously detected halo substructures can indeed be attributed to the single GS/E-like event. Rey et al. (2023) pioneers a new powerful and flexible way to carry out inference with Cosmological hydro-dynamical simulations. They use genetic modifications of the simulation's initial conditions (see Roth et al., 2016; Rey and Pontzen, 2018) to modify the properties (such as the mass ratio) of the GS/E merger. Rey et al. (2023) take a Milky Way zoom-in simulation from the VINTERGATAN suite and show how the properties of both the host and the resulting \nFigure 15: Response of the galaxy to different early accretion events. From left to right shows increasing mass mergers. The top panels show the present-day stellar light, and the bottom panels show the kinematics. More massive mergers induce bulge-dominated structure, while less massive mergers result in rotationally dominated discs [Reproduced from Rey et al. (2023)]. \n<!-- image --> \n200 \ndebris cloud change as the merger strength is varied. The Milky Way suffers more dramatic truncation and becomes more bulge-dominated as the GS/E mass is increased (see Fig. 15). In the absence of detailed chemistry, Rey et al. (2023) find a pronounced kinematic degeneracy for the stellar halo: different mergers and different mixtures of accreted and in-situ stars end up producing similar looking radially-biased 'sausage'-like structures. \nIt is worth emphasizing a sobering point regarding simulated 'Milky Way-like' galaxies. While they can be extremely helpful for understanding and interpreting the observational data, they are not, and never will be, the 'real' Milky Way. Moreover, the selection effects present in the data are not often (and rarely fully) accounted for in the simulations. Thus, great care must be taken when using the simulations to draw conclusions about our own Galaxy.", '3.6. The puzzle of GS/E': "With the arrival of the Gaia data, Galactic Archaeologists find themselves struck by the epiphany of the most important merger in the Milky Way's accretion history. However, the grasping of the GS/E event is only starting and many of its crucial details remain blurry or even completely unconstrained.", '3.6.1. The timing of the GS/E': 'At this stage, we are ready to move away from the assumption that the event was instantaneous and accept that it takes some time for the dwarf to fully merge. But, what was the duration of the GS/E merger? The truncation of star formation in the GS/E (see e.g. Bonaca et al., 2020) alludes to a z ≈ 2 merger, but the details of the beginning and end of the Milky Way-GS/E interaction are still unclear. The pace of the interaction depends on the \nmasses, the densities, and the initial angular momentum in the system (e.g. Amorisco, 2017; Vasiliev et al., 2022). Can we reconstruct this information with the current data at hand, and, if not, what are we missing?', '3.6.2. The mass budget of the GS/E progenitor': "The most recent stellar mass estimates of the GS/E are typically a few × 10 8 M ⊙ (e.g. Mackereth and Bovy, 2020; Lane et al., 2023), which is almost an order of magnitude lower than earlier estimates of ∼ 10 9 M ⊙ (e.g. Belokurov et al., 2018b; Helmi et al., 2018; Feuillet et al., 2020). This is likely due to a greater purity of GS/E stars in more recent studies, but, even now, a robust stellar mass estimate is lacking. Nonetheless, a lower mass estimate for GS/E has several important consequences. For example, it would imply less damage to the pre-existing disc - this would agree with observations that the highα disc is still intact. However, simulated analogues of the Milky Way-GS/E encounter would need to be updated in light of this seemingly preferred lower GS/E mass. \nThe contribution of dark matter from the GS/E to the inner parts of the Milky Way halo is expected to be small (see e.g. Fattahi et al., 2019). However, the contribution could be enough to provide a non-axisymmetric component to the dark matter distribution (Han et al., 2022b), which may have important implications for direct detection experiments on Earth (see e.g. Evans et al., 2019; Necib et al., 2019). Moreover, a 'tilted' dark matter halo can induce a warp and flare in the Galactic disc (see e.g. Han et al., 2023a,b). \nFinally, it is also worth considering what the gas content of the GS/E progenitor was. The early nature of the merger strongly suggests that the progenitor was gas-rich, but where and when was the gas lost? Could it contribute \nto (or re-start) the Galactic star formation? Does the two infall model of the Galaxy (Chiappini et al., 1997) still hold weight in light of the apparent gas-rich GS/E merger? Can the GS/E GC population be used to estimate the gas mass of the progenitor (Valenzuela et al., 2023)?", '3.6.3. Initial state of the progenitor': "Although much work is needed to characterize the GS/E debris today , ultimately we also want to know what the initial state of the progenitor was (i.e. before any interactions with the Milky Way). How much information is encoded in the z = 0 debris? For example, if the progenitor was a disc galaxy with a large satellite (Sequoia?), broadly similar to the LMC and SMC, it could produce a variety of halo substructures with distinct properties. However, even without a complex morphology and kinematics, high-mass events tend to break up into a large number of clumps of the E,L z space (see e.g. Jean-Baptiste et al., 2017; Belokurov et al., 2023). Indeed, the 'chevrons' in r -V r space uncovered by Belokurov et al. (2023) could result from the phase-mixing of the massive GS/E. However, Dillamore et al. (2023a) recently showed that such stellar substructures in the local halo could also generated by bar resonances. \nFinally, it is worth considering that a massive satellite such as the presumed GS/E progenitor likely had a satellite population of its own. Where are the dwarf satellites of the GS/E? Did they decouple from the GS/E dwarf before it started to radialize and sink? If so, do the GS/E satellites today (presumably ultra-faint dwarfs) resemble the original orbital plane configuration? Past work has shown that groups of satellites tend to disperse quickly after infall into a Milky Way-mass halo (e.g. Sales et al., 2011; Deason et al., 2015), so it is unlikely that there is any kinematic link between the GS/E satellites and the progenitor debris today. However, are there other ways we can link known satellites to the GS/E? And, what are the consequences for the overall Milky Way satellite population (see e.g. Bose et al., 2020)?", '3.6.4. Debris distribution in the Galactic halo': "Most of the analysis of the GS/E debris has been limited to the inner (5 ≲ r/ kpc ≲ 30) halo. This, in part, is because here we have the richest multi-dimensional data, but also because the high eccentricity of the GS/E orbit implies there are plenty of GS/E stars at small Galactic radii. However, there is not much evidence (so far) for GS/E stars at low total E . How deep a satellite can sink in the halo depends not only on its mass, but also on (i) the density of the host and the satellite, and (ii) the initial amount of angular momentum (Amorisco, 2017; Vasiliev et al., 2022). The low E regime is complicated by the overwhelming in-situ population, but there is potentially a rich amount of uncharted information about the GS/E lurking in the Milky Way's depths. \nAt the other 'end' of the Galaxy, there is still much to learn about the GS/E stars at large distances. Here, shells \nand earlier stripped GS/E stars are likely prevalent in the halo. Indeed, the distant 'echoes' of the GS/E are now starting to be discovered (Chandra et al., 2023), and with increasing amounts of data at large Galactic radii, this will likely be a fruitful expedition in future studies. Mapping the GS/E debris over a wide range of radii (or energies) will be vital in order to reconstruct the full dynamical history of the Milky Way-GS/E interaction.", '3.6.5. A complete re-interpretation': 'Although the vast majority of the works described above advocate (or assume?) that the GS/E is a single, early, and massive progenitor, this is not the only interpretation in the literature. In particular, in a series of recent papers (Donlon et al., 2022; Donlon and Newberg, 2023; Donlon et al., 2024), Donlon et al. argue that the GS/E in fact comprises debris from several lower mass radial mergers, which moreover, were accreted fairly recently. This alternative view, coupled with the large uncertainty in several of the GS/E properties (e.g. total stellar mass, time of accretion) shows that we are still far from a robust characterization of this stellar halo component. Perhaps the best way to avoid the degeneracy between total mass and the number of progenitors is the use of the well-known stellar mass-metallicity (MZR) relation (as several lower mass fragments will have lower metallicity than a single higher mass system). However, the biases associated with different selection methods (see e.g. Carrillo et al., 2024) coupled with uncertainties in the time evolution of the MZR has hampered this crucial line of evidence (see also Section 5.3). Nonetheless, it is clear that a scenario accounting for all of the chemodynamical evidence is required in order to fully solve the GS/E puzzle.', '4.1. In-situ stellar halo': "Until now, our discussion has mainly revolved around a Galactic stellar halo solely comprised of an accreted component. However, both observationally and theoretically, there are lines of evidence suggesting that some fraction of halo stars are born in-situ - i.e. forming in the host halo itself (e.g. Zolotov et al., 2009; Font et al., 2011; Schlaufman et al., 2012; Bonaca et al., 2017). The idea of an insitu halo component dates back to the seminal Eggen et al. (1962) work, which put forward the idea of a monolithic gas collapse model - here, the halo and disc are drawn from the same population. While the mode of accretion from smaller (external) subcomponents gained more traction in the proceeding years, mainly thanks to the emergence of the ΛCDM paradigm, more recent work from hydrodynamical simulations revisited the idea of an in-situ halo component (e.g. Zolotov et al., 2009; Font et al., 2011; McCarthy et al., 2012; Tissera et al., 2013; Pillepich et al., 2015). Indeed, many state-of-the-art cosmological simulations predict a certain fraction (sometimes quite significant) of halo stars have an in-situ origin. These stars are \n<!-- image --> \n<!-- image --> \nFigure 16: Left: Azimuthal velocity ( V ϕ ) as a function of age for local metal-rich stars selected from Gaia DR2. Middle: Distribution of ages for stars in different bins of V ϕ and [Fe/H]. Disc stars (red) are metal-rich with high V ϕ , Splash (black) is metal-rich with low V ϕ , and GS/E (blue) stars are metal-poor(er) with low V ϕ . Right: Cumulative age distribution. Belokurov et al. (2020) argue that the truncated age distribution of the Splash stars (indicated by a vertical thick red line) can put a strong constraint on the time of the last major merger (if indeed the Splash is caused by this merger event) \n<!-- image --> \n- . [Adapted from Belokurov et al. 2020] \ntypically metal-rich and confined to the inner halo. However, the exact mode of how these stars form and resemble a halo-type population today (i.e. kinematically hot and pressure-supported) is unclear. Moreover, there remains uncertainty over whether or not some fraction of the insitu halo components found in hydrodynamical simulations are artificial (i.e. owing to resolution and/or star formation prescription limitations, see e.g. Zolotov et al. 2010; Cooper et al. 2015). \nOne interesting finding from most simulation efforts is that the in-situ halo is intimately linked to the overall assembly history (e.g. McCarthy et al., 2012; Cooper et al., 2015). One key origin of an in-situ halo is the emergence of a 'kicked-up' disc population, caused by a past merger event. Thus, although the accreted (or ex-situ) and in-situ halo stars can have vastly different birth sites, their origin can often be linked to the same major accretion events. As we have already discerned, we now know that the Galaxy digested the GS/E several Gyr ago - what impact did this have on the Galactic disc, and did this event result in a residual in-situ halo?", '4.2. Splash': "Using Gaia DR2 data Belokurov et al. (2020) confirmed previous claims of a metal-rich 'halo-like' component in the solar neighbourhood (e.g. Bonaca et al., 2017; Haywood et al., 2018). A detailed analysis of the kinematics, abundances, and stellar ages of this metal-rich halolike component (dubbed as the 'Splash') led the authors to conclude that it is in fact linked to the thick disc. However, unlike the disc, the Splash typically has little or no net rotation, and some of the stars are even on retrograde orbits. A key part of the investigation was the availability of stellar ages - these were computed by Sanders and Das (2018) for ∼ 3 million Gaia stars by combining astrometric, photometric, and spectroscopic datasets to calculate \nisochrone ages. The inclusion of 'age' in the analysis allowed Belokurov et al. (2020) to deem that the Splash stars are predominantly old, but they are slightly younger than the GS/E (by ∼ 1 Gyr) and overlap with the old age tail of the thick disc. The close connection between the disc and the Splash (they overlap both in age and in V ϕ -the tail of the disc runs into the Splash), led to the hypothesis by Belokurov et al. (2020) that Splash is the population of stars originally born in the proto-disc of the Galaxy and subsequently kicked (splashed) into low-angular-momentum (high eccentricity) orbits by an accretion event that finished around ∼ 9 . 5 Gyr ago. The best candidate for such an event is the GS/E merger! \nThe Splash age distribution looks sufficiently different from that of the GS/E. For example, its peak is not at 12.5 Gyr but 1 Gyr later (see middle panel of Fig. 16). Even though the shapes of the age distribution of the GS/E and Splash stars are rather different, they have one particular feature in common: a truncation at 9.5 Gyr. Belokurov et al. (2020) used the synchronicity between the cessation of star formation in the GS/E and the finishing of the disc heating in the Milky Way to put a constraint on the epoch of the last major merger event (see Fig. 16). Similar 'agedating' analyses of local metal-rich and metal-poor stars were performed by other teams (e.g. Di Matteo et al. 2019; Gallart et al. 2019; Bonaca et al. 2020), and, although differing in the details, they confirm the accretion of the GS/E between 9-11 Gyr ago. \nThe uncertainty in the precise timing of the events in the distant past of the Galaxy reflects the difficulty of obtaining reliable ages for large numbers of old stars, even in the Gaia era. Four main, distinct methods to infer stellar ages have been relied upon for archaeological studies, and each have been modernized and significantly improved thanks to the availability of the Gaia data. These four involve i) colour-magnitude diagram fitting, ii) astero- \nseismological mass estimates, iii) [C/N]-based mass estimates for red giants, and iv) isochronal fitting of the main sequence sub-giant and turn-off regions. \nGallart et al. (2019) demonstrate the power of the CMDfitting of large and homogeneous Gaia datasets to recover star-formation histories of the GS/E and the Splash in-situ stars. Montalb'an et al. (2021) determine asteroseismological ages for a handful of stars belonging to the GS/E and the Splash using the data from the Kepler space observatory. Mackereth et al. (2019a) infer spectroscopic ages (relying on C and N abundances) for red giant stars in the APOGEE dataset using a neural network trained on astero-seismological age measurements and employ them to study the evolution of the disc heating with age. Sanders and Das (2018) measure isochronal ages for a combined catalogue of stars with spectroscopy from APOGEE, Gaia-ESO, GALAH, LAMOST, RAVE, and SEGUE. \nIrrespective of the chronological method used, there appears to be a good deal of consensus as to the synchronicity of the SFHs of the GS/E and Splash. Both the accreted and the in-situ stars are mostly old, with relatively few stars younger than 8 Gyr. However, the exact shapes of the age distributions of the nearby halo stars do vary quite noticeably between the methods applied and the samples used (see Gallart et al., 2019; Bonaca et al., 2020; Belokurov et al., 2020; Xiang and Rix, 2022), and some works do find a tail of younger ages in the GS/E debris (e.g. Feuillet et al., 2021; Grunblatt et al., 2021; Horta et al., 2024b). It is currently not clear if the ages and/or association of the intermediate stars in the GS/E are robust, but, clearly accurate ages are a vital ingredient for future studies of early-time Galactic archaeology. \nThe link between major accretion events and 'Splashlike' halo populations is also confirmed in state-of-the-art simulations of Milky Way-mass galaxies (e.g. Grand et al., 2020; Renaud et al., 2021b). Both Belokurov et al. (2020) and Grand et al. (2020) showed similar GS/E-Splash examples in the Auriga simulation suite. Grand et al. (2020) argue that gas-rich mergers can heat the proto-disc of the Galaxy, and scatter stars onto less circular orbits (see Fig. 17). Such a population retains a correlation between rotation velocity and metallicity and thus contributes an 'in-situ' halo component that connects the thick disc to the inner stellar halo. These findings from the latest cosmological simulations link back to earlier work (e.g. Zolotov et al., 2009; McCarthy et al., 2012) that argued for a heated proto-disc origin for the in-situ halo. The intimate link between the disc, Splash, and GS/E stars presents another intriguing way to measure the properties of the GS/E progenitor. For example, Belokurov et al. (2020) discuss how the kinematic heating of the proto-disc is proportional to the impacting progenitor mass, and Grand et al. (2020) show explicitly in the Auriga simulations that the fraction of 'kicked-out' stars in the local halo correlates with the progenitor mass. Thus, quantifying the fractional contribution of the in-situ halo stars provides an additional \nobservational indicator of the GS/E progenitor mass. \nAlthough sustaining plenty of damage, the disc appears to have survived the interaction with the GS/E progenitor. However, both instantaneous and long-term effects of the battering on the Galactic star formation rate are not yet fully determined. Numerical simulations show that, in principle, such interactions can be both destructive and constructive, i.e. inducing a bout of star formation activity (e.g. Mihos and Hernquist, 1996; Springel et al., 2005; Hopkins et al., 2006; Brook et al., 2007). The possibility that the GS/E merger could have driven a starburst in the heart of the Milky Way is explored with numerical simulations in Bignone et al. (2019), Grand et al. (2020), Renaud et al. (2021b) and Dillamore et al. (2022). Clues in the data for a merger-driven starburst have also been reported. A population of stars with matching properties is presented in Myeong et al. (2022) as Eos , in An et al. (2023) as Galactic Starburst Sequence , and in Ciuc˘a et al. (2024) as Great Galactic Starburst . \nThe idea that at some point in the past the Galactic reservoir was rapidly replenished with poorly-enriched gas has been entertained for a while (see e.g. the review by Matteucci, 2021). Variants of this so-called two-infall model (Chiappini et al., 1997) can to some degree explain the emergence of two α -[Fe/H] sequences observed in the Milky Way disc (Hayden et al., 2015). The timing of the GS/E merger well-aligned with the Galactic highα to lowα transition is rather fortunate for a pure coincidence. Note, however, that the α -[Fe/H] bimodality in the Galactic disc can be reproduced with a model without mergers but with a continuous star formation and radial migration instead (Schonrich and Binney, 2009b,a; Sharma et al., 2021). Thus, while the origin of the disc sequences in the Milky Way is still under debate, there is certainly compelling evidence that the GS/E plays a major role in their early evolution. \nFinally, it is worth remarking that the origin of the Milky Way's 'Splash' (or metal-rich halo) is still under debate, and, with the current data, it is not possible to rule out other formation scenarios. Possible other formation channels include smoothly accreted gas and/or gas stripped from gas-rich satellites (Cooper et al., 2015), gaseous outflows (Maiolino et al., 2017; Gallagher et al., 2019) and even 'clump' scattering (Amarante et al., 2020). In the future, more detailed chemical abundance analysis of the Splash coupled with accurate age analysis is likely the most fruitful way to shed light on these possible origins. In any case, clearly the study of this metal-rich component, with links to both the disc and halo, is of vital importance to our understanding of the early phases of the Milky Way formation.", '4.3. Pre-disc Milky Way': "The synchronous truncation of the SFHs in the GS/E and the Splash is a compelling marker of the epoch of the last significant merger in Milky Way history. In this \nFigure 17: Edge-on projections of the stellar light for the protogalaxy populations in the Auriga-18 halo, which undergoes a GS/Elike merger. Each population has been kinematically decomposed into a disc (middle columns) and spheroid (right columns) component. Projections are shown before the GS/E merger (top row) and at redshift zero (bottom row). The GS/E merger kinematically heats the proto-galaxy into a more puffy configuration as some proto-disc stars are scattered into the halo (i.e. the Splash). [Adapted from Grand et al. (2020)]. \n<!-- image --> \npicture, Splash therefore contains some of the the Galactic disc's earliest forming stars (see also Montalb'an et al., 2021). As its age distribution demonstrates, the first of these stars were in place ≈ 12 Gyr ago. Are there any hints as to the state of the Galaxy even before that? The view of the most ancient Milky Way appears blurred and fragmented due to the scarcity of age measurements for its oldest stellar populations. In the absence of reliable age estimates, stellar metallicity can be used as an age indicator. However, a metallicity-age relationship is quasimonotonic only for stars born in the same galaxy, and under the assumption of one-zone chemical enrichment. In the Milky Way, this poses a problem in the low-metallicity halo regime where contributions from multiple accreted progenitors are expected. \nLuckily, additional chemical tags can be used to attempt to separate the accreted and in-situ born stars at [Fe/H] < -1. These ideas are first presented in Hawkins et al. (2015) who suggest that the accreted and in-situ halo stars attain distinct levels of [Al/Fe] abundance ratio. The stellar nucleosynthetic yield of aluminium (and sodium) is low at low metallicity but increases dramatically as more of C, N, O is pumped into the interstellar medium (see Kobayashi et al., 2006) which causes a delay in Al enrichment compared to conventional α -elements. Compared to more massive systems, such as the progenitor of the Milky Way, dwarf galaxies are less efficient at retaining and re-using gas during star formation and are therefore slower to reach higher levels of metallicity when Al production becomes efficient. As a result, in a dwarf, by the time Al starts to be produced in earnest, type Ia SNe begin to dominate Fe production, thus inhibiting the growth of the [Al/Fe] ratio. In a more rapidly evolving \nMilky Way progenitor, Al production kicks in before the Type Ia SNe contribution rises and thus the young Galaxy can enjoy a period of over-abundance of Al relative to Fe. The systematic difference between [Al/Fe] levels in dwarf galaxies and the Milky Way proper is now well established (see e.g. Hasselquist et al., 2021) and is routinely taken advantage of to identify accreted stars in the nearby halo (e.g. Hawkins et al., 2015; Das et al., 2020; Horta et al., 2021).", '4.3.1. Aurora': "Instead of focusing on the accreted debris, Belokurov and Kravtsov (2022) combine [Al/Fe] measurements published as part of the APOGEE DR17 and the Gaia EDR3 astrometry to create a pure Milky Way in-situ sample, including stars at low metallicities (see top panel of Fig. 18). They show that while there may be some amount of false negatives, i.e. genuine in-situ stars identified as accreted, the in-situ selection is largely uncontaminated. Having access to a pure set of stars born only in the Milky Way means that the metallicity can now be used as the age proxy. Studying the behaviour of azimuthal velocity of the Milky Way stars as a function of [Fe/H], Belokurov and Kravtsov (2022) identify several key epochs in the life of the Galaxy, marked by noticeable changes in the stellar kinematics. For example, focusing on stars with retrograde motion, they see an edge to the Splash population at [Fe/H] ≈ -0 . 35 in agreement with earlier studies (e.g. Bonaca et al., 2017; Haywood et al., 2018; Helmi et al., 2018; Di Matteo et al., 2019; Gallart et al., 2019; Belokurov et al., 2020). Across -1 < [Fe/H] < -0 . 35, Splash co-exists with the highα ('thick') disc which, while heated by the interaction with the GS/E, appears largely intact, i.e. it retains a relatively high amplitude of rotation. However, going further back in time to [Fe/H] < -1, the median azimuthal velocity drops precipitously low and the spread of the azimuthal velocity distribution reaches its largest value. In other words, starting instead from high redshift, the early Milky Way began without coherent rotation and then went through a rapid 'spin-up' phase at -1 . 3 < [Fe/H] < -1 from which point onward it has been dominated by a fast-rotating disc (see Fig. 18). \nBelokurov and Kravtsov (2022) demonstrate that, as a function of [Fe/H], these dramatic kinematic transformations are accompanied by clear changes in chemical abundances. More precisely, they show that the abundance scatter in most elements considered evolves as a function of metallicity in sync with V ϕ . The scatter is largest in the pre-disc era, then shrinks during the 'spin-up' phase and stays approximately constant thereafter. Four elements, Al, N, Si and O show a particularly prominent evolution with metallicity, reaching much higher values of scatter in the pre-disc phase compared to the accreted halo population at fixed [Fe/H]. Belokurov and Kravtsov (2022) chose to give the pre-disc in-situ population the name Aurora appealing to its connection to the dawn of star formation in the Milky Way. \nFigure 18: Top panel: Distribution of APOGEE DR17 + Gaia DR3 stars in Energy and angular momentum space coloured by median [Al/Fe]. There is a clear separation between the in-situ (high [Al/Fe]) and accreted (low [Al/Fe]) populations. Bottom-left panel: Evolution of the in-situ azimuthal velocity distribution as a function of metallicity. The black, blue, and red points show the median, low V tan wing, and distribution width, respectively. The solid and dashed vertical lines indicate the Spin-up (rapid increase in spin) and Splash (emergence of low V tan wing) phases. Bottom-right panel: Probability density function of the in-situ V tan in bins of [Fe/H]. At low metallicities the in-situ population (i.e. Aurora) has low net spin and a wide distribution of V tan . [Adapted from Belokurov and Kravtsov (2022)]. \n<!-- image --> \nTo place their findings in context and reveal a plausible close-up view of the young Milky Way as predicted by current state-of-the-art models, Belokurov and Kravtsov (2022) compare the APOGEE+ Gaia observations to numerical simulations of galaxy formation, namely FIRE (Wetzel et al., 2023) and Auriga (Grand et al., 2017). In both sets of simulations, Aurora (the state the Milky Way starts in) is characterized by a messy, turbulent spatial stellar distribution with a low overall angular momentum. Interestingly, some small amount of net prograde rotation is present at early epochs in most simulated Milky Way-like galaxies. Therefore, the in-situ population born before the disc appears to have angular momentum similar in amplitude and direction to the accreted halo (see Deason et al., 2017b). At the time of formation, Aurora stars can be seen close to the centre of the protoGalaxy but lack order in both spatial and kinematic behaviour. As time passes and the Milky Way grows, Aurora settles and phase-mixes into a slightly flattened, quasispheroidal distribution with a very steep radial density profile. The Galactic spin-up phase is ubiquitous across the two simulation suites but the disc emergence typically happens at higher metallicities (and later times) compared to the observations. This view of the early Milky Way remains unchanged when other numerical simulation suites are considered, such as Illustris (Semenov et al., 2024) or ARTEMIS (Dillamore et al., 2023b). As a result of the Aurora discourse, recent works have paid more specific attention to the proto-disc galaxy populations in cosmological simulations (see e.g. Horta et al., 2024a; McCluskey et al., 2024). \nGiven the earlier studies of the GS/E interaction and the Splash formation, Aurora must certainly be older than ≈ 10 Gyr, and likely formed earlier than ≈ 12 Gyr ago. The study of Belokurov and Kravtsov (2022) however did not involve age estimates; moreover, it was largely limited to [Fe/H] > -1 . 5 as below this metallicity, the [Al/Fe]based separation of stars into in-situ and accreted halo becomes much less effective. This deficiency is alleviated in the works of Conroy et al. (2022) and Rix et al. (2022) that both attempt to gain a view of the Galaxy at metallicities below [Fe/H] ≈ -2. Conroy et al. (2022) use [Fe/H], [ α /Fe], and age estimates together with radial velocity measurements provided by the H3 survey to split the low-metallicity population into a non-rotating component (mostly GS/E) and a slowly rotating subset with a large velocity dispersion which the authors identify with the nascent disc of the Galaxy. Not only do these two samples have distinct kinematics, but their [ α /Fe] and age distributions are also different. At low metallicity, the slowly rotating in-situ component is clearly older compared to the GS/E: many stars have age estimates > 12 Gyr, while the GS/E stars range from 8 to 12 Gyr. In the α -[Fe/H] plane, the GS/E stars delineate a downward-sloping track or α -knee consistent with earlier studies (e.g. Hasselquist et al., 2021). The in-situ population shows a non-monotonic behaviour with an inflection around -1 . 3 < [Fe/H] < -1 \nFigure 19: A schematic overview of the Galactic star formation efficiency (SFE) over cosmic time. [Reproduced from Conroy et al. (2022)]. \n<!-- image --> \nwhich the authors explain with changes in star formation efficiency. Conroy et al. (2022) interpret their lowmetallicity, old, kinematically-hot, highα population as the earliest phase of the Milky Way formation and find agreement with the properties of Aurora at slightly higher metallicity. An illustration of the Galactic star formation efficiency over cosmic time, including major events such as the proto-Milky Way and GS/E merger, is provided by Conroy et al. (2022) and reproduced here in Fig. 19. \nRix et al. (2022) use overall metallicity [M/H] inferred from the Gaia DR3 low-resolution XP spectro-photometry to provide a panoramic view of the low-metallicity Milky Way. Importantly, this work is not limited to the solar neighourhood, which is the vicinity where Aurora was originally discovered. Rix et al. (2022) detect clear differences in the orbital and therefore spatial properties of their stars as a function of [M/H]. Above -1 . 3 < [M/H] < -1, the Milky Way is dominated by a rapidly rotating disc, although accreted (GS/E) stars are also present. At [M/H] < -1 . 3 the rotation quickly disappears and eventually goes to zero below [Fe/H]= -2. Rix et al. (2022) also detect a change in the slope of the metallicity distribution around [M/H] ≈ -1 which they surmise must be connected with the onset of the disc formation 12 Gyr ago in agreement with age determinations of Xiang and Rix (2022). Rix et al. (2022) conclude that the Milky Way's 'poor old heart' that they see in the Gaia XP data is most likely the pre-disc proto-Galaxy and is the extension into the central Galactic regions of the same population which had been seen by Belokurov and Kravtsov (2022) and Conroy et al. (2022) in the Solar neighourhood. \nA clearer picture of the orbital structure in the inner metal-poor halo emerges in the analysis of the Pristine Inner Galaxy Survey (PIGS) data (Arentsen et al., 2020b,a). As Ardern-Arentsen et al. (2024) demonstrate, \nbelow [Fe/H] ≈ -1 the typical size of the orbital apo-centre is small, < 5 kpc. However, at lower metallicities, i.e. [Fe/H] < -2 it starts to grow: only ≈ 60% of stars remain confined within 5 kpc. Thus, while Ardern-Arentsen et al. (2024) agree that the Milky Way hosts a significant, centrally concentrated, metal-poor component with a modest prograde rotation, likely connected to the pre-disc proto Galaxy (i.e. Aurora), they also see clear evidence of a more extended halo component in the very metal-poor regime, i.e. at [Fe/H] < -2, which may be of accreted nature. The above PIGS study concurs with the earlier work of Lucey et al. (2021) who saw a similar pattern in the change of the apo-centre with metallicity in their smaller COMBS survey sample. Larger samples of (very) metal-poor stars from upcoming spectroscopic surveys (e.g. 4MOST) are on the horizon, and thus there is great scope to shed further light on the earliest stages of the Galaxy's evolution in the coming years.", '4.3.2. Kraken, Koala, and Heracles': "The total mass of the Milky Way's stellar halo ( M star ≈ 1 -2 × 10 9 M ⊙ , Deason et al., 2019; Mackereth and Bovy, 2020) puts a limit on the number of significant accretion events the Galaxy could have experienced. These mergers ought to be massive enough to sink deep in the Galactic potential and thus populate the Solar neighourhood with their tidal debris. Equally, their progenitor galaxies are likely amongst the main contributors to the accreted Milky Way GCs. The latter idea is explored in Kruijssen et al. (2019b), who compare the age-metallicity properties of the simulated and the observed Galactic GCs to infer the total number of large mass ratio accretion events in the Milky Way. Working with a suite of 25 zoom-in simulations, Kruijssen et al. (2019b) conclude that there have been at least two - but more likely three - main mergers in the Galaxy. The one with the lowest mass of these three is linked to the Sgr dwarf. The other two are more mysterious, in particular, the ancient interaction with the so-called 'Kraken' galaxy. As noticed quickly by Massari et al. (2019), the Kraken GCs identified by Kruijssen et al. (2019b) are not dynamically coherent and could therefore originate from different progenitor systems. In dividing up the Galaxy's GCs, Massari et al. (2019) find a group of 24 unassigned clusters at low total energy. These unclaimed low-energy GCs are classified by Forbes (2020) as a single progenitor named Koala, while Kruijssen et al. (2020) advocate that these GCs are part of Kraken. An alternative hypothesis is proposed by Belokurov and Kravtsov (2024) and Belokurov and Kravtsov (2023) who show that the bulk of the low-energy group originally highlighted by Massari et al. (2019) can instead be of in-situ origin and thus linked to the pre-disc Galactic population Aurora. \nThe properties of the Kraken/Koala group of GCs are difficult to reconcile with the accretion of a single galaxy. These clusters have a wide range of eccentricities 0 . 2 < e < 0 . 8 and an unusually broad spread of metallicity -2 . 1 < [Fe/H] < -0 . 5. Overall, these GCs are rather \nsimilar to the rest of the in-situ clusters in Massari et al. (2019), which come in two flavours: disc and bulge. If, for example, the bulge boundary used was slightly larger than 3.5 kpc, many of the low-energy clusters would be included. Indeed, as pointed out by Callingham et al. (2022), there is a large degree of overlap between lowenergy (Kraken/Koala) GCs and the in-situ population. The quality of the age and metallicity measurements is not sufficient to make an unambiguous classification of these GCs either. Only 12 out of 24 have ages and of these, half have the ages of the oldest GC in the Milky Way. The other 6 are contemporaneous with the GS/E clusters (for example, in both groups, the youngest GC are ∼ 10 Gyr old). Finally, it is also unclear if the Milky Way's disc could withstand the combined force from two massive accretion events around 1 . 5 < z < 2. \nCould instead a Kraken-like merger take place earlier in the history of the Galaxy, before its disc comes into existence? Horta et al. (2021) find chemo-dynamical evidence for such a primeval event in the central portion of the Milky Way using data from APOGEE and Gaia . Importantly, this work (unlike others which are limited to the solar neighourhood) investigates the nature of halo stars within 5 kpc from the Galactic centre, and thus, presumbably, in the vicinity where the majority of the mass from proto-Galaxy populations reside. Selecting stars with low [Al/Fe], indicative of the accreted population, they identify a gap in the total energy distribution, which separates the GS/E debris with high total energies from the more tightly bound, low-energy structure they dub 'IGS'. They also give a name to the progenitor of the IGS - Heracles. Horta et al. (2021) demonstrate that the properties of the IGS are clearly distinct from those of the GS/E both in the chemical abundance space and in orbital space (although, as later demonstrated by Lane et al. 2022, the energy gap used for the IGS identification may instead be the imprint of the APOGEE selection function). However, differences between the IGS and the in-situ stars are somewhat less clear. Horta et al. (2021) show that at low metallicity and early epochs, the Milky Way in-situ chemical track passes through the region of low [Al/Fe], thus overlapping with the IGS selection. They conclude that the in-situ 'bulge' population can have chemical properties identical to those of the IGS. \nIt is perhaps not impossible for the main progenitor of the Milky Way and a large, early-accreted dwarf galaxy to have similar chemistry if their masses were comparable. The dwarf's star formation history would be truncated abruptly as it merged with the main progenitor of the Galaxy, but in the beginning (at lower metallicity), its chemical track and that of the Milky Way may look indistinguishable. This scenario is explored and quantified in Horta et al. (2024a) where the FIRE simulations are used to compare the Milky Way's main progenitor and its building blocks (see Fig. 20). The authors propose the following working definition of a 'building block': it is a dwarf galaxy that merges with the Milky Way dur- \n<!-- image --> \nFigure 20: Left panel: The merger tree of the m12b simulation in Latte. The 'proto-Milky Way' is defined as the main branch halo (red) plus all the building blocks that merge (other colours) before t MR 3:1 (the time at which the main branch reaches a stellar mass ratio of 3:1 with the second most massive luminous subhalo). Right panel: Formation distance of star particles as a function of lookback time. At early times (i.e. before t MR 3:1 ) the main branch and its building blocks can have very similar chemical properties. [Reproduced from Horta et al. (2024a).] \n<!-- image --> \ning the period when the main progenitor is not yet clearly dominant within the simulation volume (typically around redshift z ≈ 3). In the 13 galaxies within the FIRE suite, the main progenitor is easily identifiable: it contributes the most mass to the forming Milky Way, from ∼ 70% to ∼ 95%. Horta et al. (2024a) find that in 5 out of 13 galaxies considered, a building block exists that contributes between 15% and 30% of the total mass of the proto-Milky Way. While these building blocks have 2 to 5 times lower mass than the main progenitor, their chemical fingerprints are almost identical. Maybe, Heracles - the parent galaxy of the IGS - was such a building block and thus its chemistry and that of Aurora are indiscernible? \nSometimes Kraken/Koala and Heracles are implicitly assumed to be the same entity. However, there may be considerable tension between the observational definitions of the two. Kraken is posited to be an accreted dwarf based on half a dozen GCs with age measurements that are located on the accreted branch in the age-metallicity space. At fixed age of ≈ 10 Gyr, these clusters are some ≈ 0 . 7 dex more metal-poor than the Milky Way in-situ GCs. Thus, the Kraken stellar population must be distinct from that of the early Milky Way, in contrast to the proposed building block, Heracles.", '4.4. Implications of the GS/E and the early disc formation': "An atypical assembly history? As we discussed in Section 1, the accretion of a small number of massive progenitors is not atypical for Milky Way-mass galaxies. However, what is perhaps more unusual in the case of the Milky \nWay is the lack of such events since the GS/E merger ( ∼ 10 Gyr ago). Evans et al. (2020) explored this explicitly and used the large-volume EAGLE simulations to quantify how 'unusual' the Milky Way accretion history is (at least with regard to massive progenitors). The authors find that only 5% of the Milky Way-mass haloes ( M 200 c = 0 . 7 -2 × 10 12 M ⊙ ) in EAGLE have accreted a GS/E-like progenitor around 8-11 Gyr ago, but have had no subsequent massive merger events. This fraction lowers to < 1 % of haloes if the late-time accretion of the (surviving) LMC is also used as a constraint on the assembly history. An interesting consequence of the lack of other major mergers since the GS/E is that the haloes satisfying the Milky Way assembly history constraints in EAGLE are biased towards lower mass haloes within the 'Milky Waymass' range, i.e. the haloes are typically ≤ 1 × 10 12 M ⊙ . This bias is due to a long period of quiescence during which the haloes did not accrete as much mass as a 'typical' Milky Way galaxy. \nAnother repercussion of this 'quiescent' period in the Milky Way's history is that the inner halo as we see it today provides a window to the high redshift Universe. In particular, the GS/E debris is a relic of an ancient, massive dwarf galaxy. Dwarfs with similar mass that are still intact today are very different (e.g. the LMC) as they have vastly different star formation histories. As shown by Evans et al. (2022), galaxies resembling the GS/E should be observable with JWST beyond redshift z ∼ 2. Thus, our Galactic archaeological dig of GS/E with Gaia data can be compared with the properties of similar mass dwarf \ngalaxies at high redshift found with JWST, which presents a fascinating way to connect the local Universe with the early stages of galaxy formation. \nBose et al. (2020) showed how the particular accretion history of the Milky Way can also affect its satellite galaxy population. Note that the satellite luminosity function is often used as a key probe of ΛCDM models, and thus any systematic influence on this fundamental relation is particularly important. In particular, Bose et al. (2020) find that, at fixed mass, haloes that form earlier typically contain a larger number of ultra-faint satellites than those haloes that form later. Moreover, the radial distribution of these satellite galaxies is more centrally concentrated. It is clear that the existence and timing of massive merger events in our Galaxy are crucial for many of its 'global' properties (i.e. total halo mass, stellar halo, satellites, disc formation), and it is necessary to understand these events in order to put the Milky Way in context with other similar mass galaxies. \nThe impact of GS/E on the disc (trans)formation: A significant merger like the GS/E does not only heat up the pre-existing Galactic disc, it can also torque, flip, and tilt it. The pervasiveness of such disc flipping and tilting is uncovered by Dillamore et al. (2022) when analysing the host response to satellite mergers in the ARTEMIS numerical simulations. They show that both the dark matter halo of the Milky Way and its disc change shape and orientation as a result of the interaction. Even though the Galaxy manages to destroy the perturber, it tends to eventually align itself with its orbital plane. The re-aligning can happen quickly (disc flipping) and slowly (disc tilting). The exact mechanics of the disc flipping is exposed in Dodge et al. (2023) who build their intuition based on a set of tailored N-body simulations. They show that, as the GS/E progenitor falls apart in the inner regions of the Milky Way, it deposits a small amount (in relative terms) of dark matter in a strongly non-axisymmetric configuration. This non-axisymmetric component exerts a torque on the disc, forcing it to change orientation. As mentioned earlier, the existence of such a non-axisymmetric dark matter component is advocated by Han et al. (2023b) and discussed by Davies et al. (2023). Both Dillamore et al. (2022) and Dodge et al. (2023) conclude that the Galactic disc tilting unleashed a long time ago by the GS/E is likely still ongoing today. This astonishing insight has inspired Nibauer et al. (2023) to ponder the long-term effects of the tilting disc on the properties of the stellar streams in the Milky Way halo. They show that the nearby streams can be impacted by the disc tilt in a variety of noticeable and non-trivial ways. Another striking and unexpected consequence of the GS/E's flybys is highlighted in the study of Renaud et al. (2021a). In one of the VINTERGATAN simulations they consider, the galaxy assembles an extended gaseous disc around the time of a GS/E-like interaction whose plane is sufficiently misaligned with respect to the original, old disc such that the two evolve almost unconnected to each other. The passage of the GS/E progeni- \ntor first ignites star formation in the extended misaligned gaseous ring and then torques it to align with the preexisting, main disc of the Galaxy. Renaud et al. (2021a) show that as a result, the Milky Way ends up hosting two neatly aligned, but independently grown stellar discs. This, they argue, could also be a viable path to produce the α -bimodality similar to that observed in the Galaxy. \nThe Milky Way as a transient fossil: Although all signs point towards a quiescent Milky Way assembly history and early disc formation, in many ways, this is just a transient state of affairs. Indeed, we know that in a few Gyrs time, the Milky Way will experience its next major merger event when it eventually engulfs the LMC. How will this change the status quo? This question was directly pondered by Cautun et al. (2019) who used the EAGLE simulations to predict the impact of a late-time LMC-mass merger onto a Milky Way-mass galaxy. Unsurprisingly, the authors predict that the stellar halo will be radically changed. Its stellar mass and metallicity will likely shoot up to reflect the massive LMC progenitor. At present our relatively metal-poor, low-mass stellar halo is indicative of an early forming halo (see e.g. Deason et al., 2019), but this signature will likely be overwhelmed by the destroyed LMC material. Moreover, a new 'Splash' will form from kickedout disc stars. Not only will there be a more metal-rich Splash, but the disc itself will likely be thickened and perhaps distorted. Thus, the presence of the LMC (which we will describe in more detail below), is perhaps a contradiction to the overarching view of a quiescent Milky Way. In fact, as put forward by Deason et al. (2016) the Milky Way is more accurately described as a 'Transient Fossil', whereby a recent accretion event can disguise the preceding formation history of the halo. We end this Section with the philosophical pondering that perhaps we are lucky to not be writing this review in 2 -3 Gyr time - would we even know that the GS/E existed, or what the early formation of the Milky Way entailed? 2", '4.5. The lowest metallicity stars': "Stars with the lowest metallicities are likely the oldest formed in a given environment, although the agemetallicity relationships can differ substantially depending on the mass of the host galaxy. Gaia is heavily relied upon to (1) make sense of the previously discovered metal-poor stars, and (2) to identify new low-metallicity stars missed by previous efforts. Before Gaia , no single spectroscopic survey managed to examine the low-metallicity regime of the Galactic halo comprehensively. As a result, the currently available sample of stars with [Fe/H] < -2 have strong and heterogeneous selection effects from a patchwork of different studies (e.g. Frebel and Norris, 2015). As the Gaia data started to come out, efforts to consolidate the census of the metal-poor end of the stellar MDF accelerated, as exemplified by e.g. the study of Li et al. (2018). \nWith the DR3, Gaia has stepped up the game by producing the largest homogeneous catalogues of low-metallicity candidates selected using its high-resolution spectroscopy (see e.g. ? Viswanathan et al., 2024), and low-resolution spectro-photometry (XP spectra, e.g. Andrae et al., 2023; Yao et al., 2024). \nThe Gaia XP revolution is only beginning, but the first attempts to extract stellar atmosphere parameters and individual abundances from admittedly extremely lowresolution but homogeneous and enormous XP spectra are already bearing fruit. For example, some success has been reported in extracting α -abundances from XP (Li et al., 2024). Perhaps, this analysis can even be extended to the low-metallicity regime, where the exact behaviour of the in-situ α -abundance remains a puzzle. According to Conroy et al. (2022), stars with [Fe/H] < -2 may exhibit noticeably higher α values than previously thought, moving above the established α plateau. While the sensitivity of the XP spectra to changes in α -abundance comes as a surprise, spectral variations due to changes in carbon abundance are less subtle and thus can be exploited (see e.g. Witten et al., 2022). To this end, the first allsky, homogeneous sample of carbon-enhanced metal-poor (CEMP) stars is built by Lucey et al. (2023) using Gaia DR3 XP spectra. CEMPs come in several flavours: those that show signs of r and/or s -process enhancement and those that do not (Aoki et al., 2007). The r/s -enhanced CEMPs are most common and likely linked to pollution by companions in binary systems (Lucatello et al., 2005; Starkenburg et al., 2014). Instead, CEMP-no stars appear to be mostly single (but see Hansen et al., 2016; Arentsen et al., 2019) and thus their carbon-enhancement is likely primordial, and linked to the very first bouts of star formation in the Universe (Frebel and Norris, 2015). Using Gaia 's astrometry to shed light on the orbital properties of confirmed CEMP stars reveals that many of them are typical halo denizens and can be associated with known halo sub-structures (such as GS/E, see Zepeda et al., 2023). \nPlotting global orbital properties of the low-metallicity stars discovered recently reveals a small but noticeable angular momentum asymmetry, with more stars found on prograde orbits at all metallicities, going down as low as [Fe/H] ≈ -6 (Sestito et al., 2019; Di Matteo et al., 2020). Note, however, that focusing on metal-poor stars with r -process enhancement, Roederer et al. (2018) do not see such an angular momentum asymmetry. Many of the lowmetallicity stars on prograde orbits also spend most of their time close to the Galactic disc, at low | z | . These unexpected trends have inspired several hypotheses, including the suggestion that in the Milky Way, the stellar disc may have formed exceedingly early and at extremely low metallicities (e.g. Di Matteo et al., 2020). Testing this idea against numerical simulations however reveals that in the Milky Way-like progenitors, dominant stellar discs form sufficiently late, at noticeably higher metallicities (e.g. Belokurov and Kravtsov, 2022; Semenov et al., 2024; Chandra et al., 2023). Note that, as discussed earlier, the MW \nitself appears to be somewhat of an outlier: a dominant stellar disc emerges in the Galaxy at the earliest possible epochs, i.e. around 3 < z < 5, but at metallicities much above the metal-poor regime, i.e. [Fe/H] > -1 . 3. At higher redshifts, the Galaxy is in its kinematically-hot, bursty and messy Aurora state. Sestito et al. (2021) address the origin of the low-metallicity stars on prograde, low-| z | orbits, and conclude that most of them must be accreted. It remains unclear what induced the net spin in the low-metallicity halo, is it inherent or acquired? Dillamore et al. (2023a) demonstrate that the Galactic bar can spin up intrinsically non-rotating stellar haloes to velocities similar to those observed. In their models, a halo spun-up by the bar shows a rotation curve declining with Galactocentric radius - this prediction can be tested with upcoming data.", '5.1. Sagittarius and the Milky Way disc': "The Sgr dwarf galaxy (Ibata et al., 1994) and its associated stellar stream (e.g. Newberg et al., 2002; Majewski et al., 2003; Belokurov et al., 2006) are an archetype of dwarf galaxy disruption. The stellar stream emanating from a known dwarf galaxy is a clear example of accretion in action, and studies of the stream have been used to measure the Galactic potential as well as the orbit and mass of the Sgr dwarf progenitor (e.g. Law and Majewski, 2010; Dierickx and Loeb, 2017; Gibbons et al., 2017; Fardal et al., 2019; Vasiliev et al., 2021). The mere fact that the progenitor is still intact points to a fairly recent accretion event (since redshift z ∼ 1), and this has been confirmed by most modelling efforts. \nThe omniscience of the Sgr stream has been obvious for several years, most notably displayed in the famous 'fieldof-streams' image from SDSS (Belokurov et al., 2006). However, the Gaia mission has allowed a more detailed examination of this well-known stream (Antoja et al., 2020; Ibata et al., 2020; Ramos et al., 2020; Cunningham et al., 2024). In addition to properties of the stream itself, the Gaia era has provided evidence that Sgr has actually perturbed the Galactic disc during its orbit around the galaxy. Now, this is not a new idea - several past works have shown that a Sgr-like dwarf could induce perturbations to the disc (e.g. Dehnen, 1998; Ibata and Razoumov, 1998; Quillen et al., 2009; Laporte et al., 2018), and there has been evidence of disc perturbations from preGaia data (e.g. G'omez et al., 2012; Widrow et al., 2012; Carlin et al., 2013; Schonrich and Dehnen, 2018). However, Gaia provided unambiguous evidence that the galaxy is out of equilibrium, and is currently undergoing a phase-mixing process from an out-of-equilibrium state. The evidence for this phase-mixing comes from the prominent spiral feature in position and velocity discovered by Antoja et al. (2018) \n<!-- image --> \n<!-- image --> \nFigure 21: Gaia DR2 phase-space spiral in the z -V z plane. The density is coloured according to (a) number density, (b) median V r , and (c) median V ϕ . [Reproduced from Antoja et al. (2018)]. \n<!-- image --> \n(see Fig. 21). This so-called Gaia phase-space 'Snail' potentially arises from phase-mixing of past gravitational disturbances of the Milky Way disc. The spiral phase-space structures are studied in greater detail using Gaia DR3, where their chemical signatures can also be mapped thanks to the GSP-Spec catalogue derived from the RVS spectra (Gaia Collaboration et al., 2023). \nSince its discovery, several works have attributed the Gaia Snail to pericentric passages of the Sgr dwarf (e.g. Binney and Schonrich, 2018; Khanna et al., 2019; Laporte et al., 2019; Bland-Hawthorn and Tepper-Garc'ıa, 2021; Gandhi et al., 2022). Is this yet another example of the co-evolution of the Milky Way disc and halo? We have previously discussed that the metal-rich 'Splash' material is likely caused by the impact of the early GS/E accretion onto the proto-galaxy (see Section 4.2), and it also appears that the more recent Sgr accretion has influenced the Galactic disc. While it is worth noting that there are alternative hypotheses for the origin of the Snail, such as from many smaller disturbances rather than one large one (e.g. Hunt et al., 2022; Tremaine et al., 2023), it remains true that in the Gaia era we cannot study the disc and halo in isolation, particularly when we are considering the inner halo. The co-evolution and co-existence of these components show that the accretion of substructures, particularly massive ones, can influence and shape the entire Galaxy.", '5.2. A massive LMC': "The most massive Milky Way satellites, the Large and Small Magellanic Clouds (LMC, SMC), have been known about for centuries. Visible to the naked eye in the Southern Hemisphere, their brightness is unmistakable. However, it is only in recent years that we have accepted the sheer massiveness of the Clouds, particularly the LMC. A simple abundance-matching argument already places the LMC at a significant mass: with a stellar mass of 2 . 7 × 10 9 M ⊙ (van der Marel et al., 2002), this roughly corresponds to a halo mass of 10 11 M ⊙ (Moster et al., 2013). A mass of ∼ 10 11 M ⊙ for the LMC also agrees with modelling efforts which aim to reproduce the significant 3D velocities \nof the Clouds and allow for LMC-SMC interactions (e.g. Besla et al., 2010; Kallivayalil et al., 2013). Most tidal models now advocate for a massive satellite (carrying its smaller sibling) that has rapidly joined the Milky Way in the past couple of Gyr (although Vasiliev 2024 show that the observations are also consistent with the LMC being on its second passage around the Milky Way). More recent works using Magellanic 'satellite-of-satellite' counts, or modelling of orbital streams in the presence of the LMC now agree that the LMC must have a mass ≳ 1 × 10 11 M ⊙ (e.g. Erkal et al., 2019; Erkal and Belokurov, 2020) \nAt the pericentre of the LMC ( ∼ 50 kpc), a mass of ∼ 10 11 M ⊙ is a significant fraction of the Milky Way halo mass out to that radius (roughly ∼ 1 / 3). G'omez et al. (2015) show that a massive LMC at this proximity can significantly displace the centre-of-mass (COM) of the Milky Way. Thus, this substantial perturber in our Galaxy cannot be ignored, and the scale of the damage inflicted by the LMC is now starting to be revealed by the Gaia mission.", '5.2.1. LMC wake': "Garavito-Camargo et al. (2019) use N -body simulations to explore in detail the impact of the LMC's passage on the Milky Way halo. They find that the recent infall of this massive satellite inflicts substantial gravitational 'wakes', which they decompose into a 'collective' (or global) and 'transient' (or local) response. The global response is reminiscent of the COM displacement discussed by G'omez et al. (2015). Here, the presence of a massive perturber offsets the COM of the Milky Way-LMC system, but importantly, as the Milky Way is not a solid body, different parts of the Milky Way halo respond differently. For example, the outer parts of the Milky Way halo (with their longer orbital timescales) are slower to respond to this rapidly evolving scenario. The resulting 'global' response is a dipole-like signature in the density and velocity space of the halo (see Fig. 22). The local response is attributed to the dynamical friction acting on the infalling LMC, which results in a collection of particles tracing the orbital path of the satellite. These signatures, in principle, should also be present in the stellar halo of the Galaxy, and \nthis raises the intriguing possibility of detecting the wake in current and future datasets. Garavito-Camargo et al. (2019) also argues that the potential detection of the wake could have implications for the nature of the dark matter particle, as different dark matter models may predict different wake properties. Thus, there is considerable scientific motivation to detect and analyse the LMC wake in the Milky Way halo. \nInitial efforts utilizing samples of halo stars selected using Gaia data have already revealed some promising results. For example, Erkal et al. (2021) and Petersen and Pe˜narrubia (2021) find evidence for velocity gradients in the outer halo ( r ≳ 50 kpc) that appear to match models of the LMC-Milky Way systems with a massive LMC. In particular, Erkal et al. (2021) measures a net blueshift for stars in the Southern hemisphere and a net redshift in the North. The signal is consistent with the mostly downward acceleration of the inner halo due to the LMC (i.e. the Global response), and by comparing the results with simulations they find that the velocity gradient suggests an LMC mass of ∼ 1 . 5 × 10 11 M ⊙ . Conroy et al. (2021) use a sample of K giants selected using WISE and Gaia DR3 to create a density map of the (global) outer halo (see right panel of Fig. 22). Remarkably, the density of outer halo stars resembles the LMC wake structure, with signatures of both a Northern overdensity (i.e. collective response) and a local wake. Interestingly, the density contrast reported by Conroy et al. (2021) is stronger than predicted by the models. \nIt is worth pointing out that care needs to be taken in order to disentangle the 'true' LMC wake signal from other potential contributions, by, for example, other substructures. Cunningham et al. (2020) show that spherical harmonic expansion can be a useful tool to disentangle perturbations on large scales, such as the LMC wake. However, they caution that stellar debris from recent, massive accretion events can complicate the analysis. An intriguing outer halo overdensity, that could potentially signify local LMC-wake material, is dubbed the 'Pisces Plume' (Belokurov et al., 2019). However, with limited data it is currently unclear whether this material is related to (1) the LMC-induced wake, (2) SMC stellar debris (stripped from interactions with the LMC itself), or (3) a poorly known outer wrap of the Sgr stream! The outer halo is ripe for uncovering the details of the predicted LMC-wake, and other known and unknown substructures. Future data releases of Gaia coupled with wide-area spectroscopic surveys such as DESI, WEAVE, and 4MOST (Cooper et al., 2023; Jin et al., 2023; de Jong et al., 2019) are ideally suited to investigate these outer regions in detail. Indeed, large samples of chemo-dynamical data are required to pin down the LMC-wake signal and disentangle/uncover contributions from other (likely recent) accretion events in the outer Milky Way halo.", '5.2.2. The Milky Way-LMC system': "The appreciation of the LMC's impact on the Milky Way has spurned a new era of dynamical modelling: we now need to consider the 'Milky Way-LMC system' (or even the 'Milky Way-LMC-Sgr system') rather than just a Milky Way host halo. The fundamental assumptions that govern many modelling techniques, such as equilibrium, static potentials, spherical or axisymmetric symmetry, etc, are no longer valid. What are the implications of this for the many decades of work aiming to measure the total mass of the Galaxy using halo tracer populations (e.g. Wilkinson and Evans, 1999; Xue et al., 2008; Deason et al., 2012; Callingham et al., 2019; Eadie and Juri'c, 2019)? A cautionary tale was spun by Erkal et al. (2021) who showed that mass estimates based on equilibrium modelling of tracers that ignore the presence of the LMC are typically biased high, and could even lead to an overestimate of up to 50 percent! On a more positive note, Deason et al. (2021) showed that by allowing for a small perturbation to the velocities of the halo stars a mass profile can be uncovered in a Milky Way-LMC potential. However, this depends on the radial range covered by the data and the region of the sky probed, as these authors focused on a region covered by SDSS where the predicted velocity gradients in the halo are small. Deason et al. (2021) also argued that systematic effects due to substructures, such as shells and clouds, can cause an even bigger effect than the LMC on mass-modelling efforts. Thus, one could argue that despite increasing sample sizes of multi-dimensional data, our aim to measure the Milky Way potential accurately , at least using classical methods, is perhaps waning. \nIt is worth remarking in the postGaia era that we likely need to reassess what the 'total mass of the Milky Way' even means. Particularly in light of the messiness caused by the LMC and, to a lesser extent, Sgr. Detailed modelling of the Milky Way-LMC-Sgr system (Vasiliev et al., 2021) has indicated that considering these major components together can prove vital. Indeed, recent stream-modelling techniques have shown that the LMC is required in order to produce stream tracks that are consistent with the data (e.g. Erkal et al., 2019; Vasiliev et al., 2021). However, while tailored models are valuable there are still systematic effects revealed from cosmological simulations of stellar haloes that need to be taken into account in modelling efforts. How do we reconcile these two approaches? Perhaps the most promising avenue is the use of 'Basis-Function-Expansion' (BFE) methods (Lowing et al., 2011; Sanders et al., 2020; Garavito-Camargo et al., 2021; Petersen et al., 2022). Here, a time-evolving potential from an N -body simulation is approximated using a small number of basis functions. Thus, at a much lower cost than the original simulation, subhalo orbits can be explored in this potential without the need to continually re-simulate the host. In principle, these methods could be used to include the cosmological context of sub- \n<!-- image --> \nFigure 22: Left: A model of the LMC-induced dynamical friction (transient) wake and collective response in the Milky Way halo shown in the Galactocentric YZ plane (Garavito-Camargo et al., 2021). The colorbar shows the density contrast (light = overdense, dark = underdense). The transient wake follows the past orbit of the LMC (indicated by the red line). The collective response appears predominantly to the North of the Galactic disc (central blue ellipse). Right: Distribution of K giant stars in Milky Way halo at 60 kpc < R gal < 100 kpc in Galactic coordinates (Conroy et al., 2021). The overdensity in the North and South-West are attributed to the collective response and local wake, respectively. [Adapted from Garavito-Camargo et al. (2021); Conroy et al. (2021).] \n<!-- image --> \nhalo accretion, whilst retaining the 'realistic' potential of the Milky Way-LMC system. For example, Bayesian inference could be applied to a large suite of such models in order to infer global properties of the Galactic potential from the phase-space distribution of halo stars. The BFE models could even provide the framework to combine different dynamical modelling techniques, either from multiple streams and/or halo stars. With the exquisite data at hand, which will only increase in number and quality in the coming years, we clearly need to develop more sophisticated modelling approaches in order to keep pace with the data.", '5.3. Census of accreted substructures': "The identification by Helmi et al. (1999) of a handful of stars belonging to a stellar stream passing through the Solar neighourhood is a good example of the type of archaeological experiment that had appeared trailblazing and challenging before Gaia , but has been made effortless and almost mundane thanks to Gaia today. Since the pioneering study of Helmi et al. (1999) the methods to detect substructure have been sharpened (see e.g. Helmi and de Zeeuw, 2000; Harding et al., 2001; Knebe et al., 2005; Meza et al., 2005; Brown et al., 2005; Bullock and Johnston, 2005; Arifyanto and Fuchs, 2006; McMillan and Binney, 2008; Johnston et al., 2008; G'omez et al., 2010; Bovy et al., 2011; Sanderson et al., 2015; Lisanti et al., 2015; Antoja et al., 2015) whilst the available datasets have grown in size and richness (e.g. Yanny et al., 2000; Vivas et al., 2001; Newberg et al., 2002; Majewski et al., 2003; Clew- \ney and Kinman, 2006; Kepley et al., 2007; Klement et al., 2008; Morrison et al., 2009; Klement et al., 2009; Schlaufman et al., 2009; Smith et al., 2009; Xue et al., 2011; Re Fiorentin et al., 2015; Janesh et al., 2016) but ultimately, the community had to wait for the arrival of the Gaia data to reveal the true wealth of accreted debris. This is particularly true for merger remnants that do not appear spatially coherent, although the detection of long, narrow streams has also evolved to a new level of sophistication thanks to the Gaia data (see e.g. Malhan and Ibata, 2018; Malhan et al., 2018; Ibata et al., 2019, 2024). \nOut of the box, the view on the halo substructure with Gaia DR1 appears underwhelming due to the minuscule number of halo stars in the shallow TGAS catalogue (Helmi et al., 2017; Myeong et al., 2017). Using the deeper SDSSGaia instead (see also Section 2.1), shows the promise of the modern space astrometry: the halo in the Solar neighourhood is riddled with pieces of accretion events (see Myeong et al., 2018a). This study uncovers ≲ 10 new clumps in velocity space, of which at least one, namely S2, had been known before (Helmi et al., 1999). However, the integrals of motion space are better suited for substructure detection because phase-mixed debris can be smeared into arcs in velocity space; worse still, multiple wraps of the same stream might appear as distinct clumps. The immediate and clear improvement is demonstrated by Myeong et al. (2018b) who use SDSSGaia DR1 and translate into the action space, detecting a factor of ∼ 3 more distinct sub-structures compared to their previous analysis of the same data (see also Wu et al., 2022, \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 23: Dynamical substructure identified by Dodd et al. (2023) in the local stellar halo using the Gaia DR3 data. Top: Projections of the energy E and angular momentum L space. Bottom: Projections of the velocity space in cylindrical polars. [Reproduced from Dodd et al. (2023)]. \n<!-- image --> \nFigure 24: Streams discovered using STREAMFINDER algorithm, colourcoded according to their metallicity. [Reproduced from Ibata et al. (2024)]. \n<!-- image --> \nfor a recent discussion of the pragmatic advantages of using action space). Once the Gaia DR2 was delivered, the hunt for the nearby halo merger remnants begun in earnest (Koppelman et al., 2018; Roederer et al., 2018; Koppelman et al., 2019; O'Hare et al., 2020; Borsato et al., 2020; Necib et al., 2020; Limberg et al., 2021; Lovdal et al., 2022; Tenachi et al., 2022; Ruiz-Lara et al., 2022; Malhan, 2022; Viswanathan et al., 2023; Dodd et al., 2023; Ye et al., 2024). \nWhat is the origin of the multitude of halo substructures discovered by Gaia in the form of IOM clumps (see Fig. 23) or as coherent narrow streams (see Fig. 24)? More specifically, how many of these sub-structures are tidal remnants of destroyed star clusters and how many \ncome from dwarf galaxies? A direct estimate of the total number of star cluster streams can be found in Baumgardt et al. (2019). These authors show that the initial mass function of the survived and currently observable Milky Way globular clusters is a very strong function of their Galacto-centric distance (see Fig. 25). Assuming that the cluster initial mass function should not depend strongly on the current location in the Galaxy, the total number of destroyed GCs can be estimated. Such an estimate depends on the postulated shape of the initial GC mass function. Baumgardt et al. (2019) consider two possibilities: the log-normal mass function, resembling the overall current distribution of the GC masses, and the power-law mass function, usually preferred as a model of the cluster mass distribution at birth. In the former case, roughly as many GCs are destroyed as there are currently detected ( N ≈ 200), but in the latter, the total initial number is much larger ( N ≈ 10 , 000). Irrespective of the initial mass function parameterization, the bulk of the destroyed GCs are located within the Solar radius, with a large fraction within ≈ 5 kpc from the Galactic centre, in the region matching in extent the in-situ Aurora component of the Milky Way. \nMany of the stellar streams recently discovered with the STREAMFINDER algorithm (Ibata et al., 2021, and Fig 24) have small intrinsic widths (by design) and are therefore suspected to have originated from disrupted or disrupting globular clusters (see also Martin et al., 2022a). Of these, the C-19 system is particularly striking. It is kinematically cold (although see Yuan et al., 2022, for a \n<!-- image --> \n<!-- image --> \nFigure 25: Left: Current (black) and initial (red) Globular Cluster masses as a function of their orbital size. Assuming that the GC initial mass function is not a strong function of Galacto-centric radius, a large number of low-mass GC with apocentres inside ∼ 10 kpc appears to have been destroyed by the Galaxy. Middle and Right columns: Current (blue) and initial (red) mass functions of the Milky Way GCs in three different bins of Galacto-centric radius. Two different parametric forms for the initial mass functions are shown: power-law (middle) and log-normal (right). The integral of the difference between the current and the initial mass functions gives an estimate of the total number of GCs destroyed. [Adapted from Baumgardt et al. (2019)]. \n<!-- image --> \nfollow-up study), and displays chemical anomalies characteristic of GCs, with a large spread and correlations amongst abundances of Mg, Na and Al (Martin et al., 2022b; Yuan et al., 2022). What makes it stand out compared to known GCs is its extreme metallicity: at [Fe/H] ≈ -3 . 4, C-19 would be the most metal-poor GC known to date in the Milky Way, by far (Martin et al., 2022b). C-19 could plausibly be a harbinger of a previously undetected population of metal-poor GCs, indicating that the so-called GC metallicity floor is, at least in part, due to a selection bias. Moreover, as pointed out by Martin et al. (2022b), given the typical stellar metallicity distribution function, the fractional contribution of GCs to star-formation in the C-19's host galaxy must be high. This is in agreement with the most recent constraints in the ancient Milky Way (Belokurov and Kravtsov, 2023) and in high-redshift galaxies (Mowla et al., 2024). \nIn the aftermath of the Gaia data boon it is worth considering if we now have a good handle on all (or at least most) of the relevant accreted structures in the halo. The short answer is almost certainly not, but we have likely uncovered the most significant progenitors by mass . This deduction is clearly laid out by Naidu et al. (2020), who use Gaia and spectroscopic data from the H3 survey to estimate the relative fraction of structures in the halo as a function of distance (see Fig. 26.) The authors argue that the GS/E dominates in the inner halo ( r ≲ 20 kpc), and the majority of stars in the outer halo belong to Sgr. Of the other structures considered in Naidu et al. (2020), none contribute more than 5% beyond r ≳ 5 kpc! \nFigure 26: Relative fraction of structures in the Milky Way as a function of Galactocentric distance. These fractions were estimated using Gaia and H3 survey data. Of the accreted structures, GS/E dominates in the inner halo, and Sgr dominates the outer halo. No other substructure contributes more than ∼ 5% beyond r ≳ 5 kpc. [Reproduced from Naidu et al. (2020)] \n<!-- image --> \nc \np \nk \n2 \n> \n| \nl \na \ng \nZ \n| \nt \na \nn \no \ni \nt \nc \na \nr \nF \ne \nv \ni \nt \na \nl \ne \nR \n1.0 \n0.8 \n0.6 \n0.4 \n0.2 \n0.0 \n2 \nMW halo: \nM \nHave we discovered everything that 'matters' in the stellar halo? By mass contribution we likely have, but there is still value in finding the remaining (presumably less massive) destroyed dwarfs. After all, the stellar halo itself is an argument for 'quality over quantity', as only a small fraction ( ≲ 1%) of the Galaxy's stellar content resides in this component. Similarly, although they may lack in stellar contribution, the lower mass stellar halo progenitors are compelling in their own right. For example, the surviving satellite luminosity function has proved to be a key test for the ΛCDM paradigm. The number counts of dwarf galaxies is intimately linked to several fundamental theories in cosmology and galaxy formation, such as the nature of dark matter, the epoch of reionization, and the early star formation in the lowest mass systems (e.g. Bullock et al., 2000; Lovell et al., 2014; Bose et al., 2018). While the Milky Way satellite system is rightly viewed as a vital test for these theories, it is still only one 'data point', and additional lines of evidence are required, either from external galaxies or from the Milky Way itself. This naturally leads to the luminosity function of destroyed dwarf galaxies in the Milky Way. Could this be a crucial piece of observational evidence? First, we would need to overcome an important caveat: the stellar halo is seemingly overwhelmed by the GS/E and/or Sgr! 20 15 10 5 0 MV 10 1 10 0 10 1 10 2 10 3 N(merged) \nV \n= \n17.7 ± 0.5 \nTheoretical predictions for the expected number of destroyed dwarf galaxies in Milky Way-mass haloes are relatively scarce, but notably Fattahi et al. (2020) show that the number of destroyed dwarfs generally exceeds the number of surviving dwarfs at all mass scales (see Figure 1 in Fattahi et al. 2020). Moreover, as expected, there can be significant halo-to-halo scatter due to varying mass accretion histories (see also Deason et al. 2023). Attempts to quantify the mass spectrum of accreted dwarfs in the Milky Way data are few and far between, but notably dynamical grouping (e.g. Callingham et al., 2022) and/or the use of metallicity distribution functions or chemical planes (e.g. Cunningham et al., 2022; Deason et al., 2023) are likely the best tools we have moving forward. Deason et al. (2023) use GMM modelling of the stellar halo MDF to estimate the luminosity function of destroyed dwarfs in the Milky Way (see Fig. 27). They make use of the stellar massmetallicity relation and assume Gaussian MDF distributions for individual progenitors, thus assuming the overall stellar halo MDF is a mixture of MDFs from smaller galaxies. Applying this method to a hodge podge of spectroscopic data in the Milky Way, complemented by Gaia DR3 astrometry, indicates that that the Milky Way stellar halo has N ∼ 1 -3 massive progenitors (with L > 10 8 L ⊙ ) within 10 kpc, and likely several hundred progenitors in total. \nAn important consideration when attempting to model the MDF of halo stars is the evolution of the MZR with time (see also earlier discussion about how this relates to the GS/E properties in Section 2). The redshift z = 0 MZR for surviving satellites in the Milky Way is welldocumented (Kirby et al., 2013), however, can we simply \nFigure 27: The estimated cumulative number of destroyed dwarfs in the Milky Way halo assuming a total stellar halo luminosity M V = -17 . 7 ± 0 . 5. The dark(light) shaded regions show the 16-84(1-99) percentiles, and the solid lines are the medians. The fiducial z = 0 (Kirby et al., 2011) stellar mass-metallicity relation is assumed with (orange) and without (blue) a -0.3 dex offset. This offset in the stellar mass-metallicity relation has been postulated to be more applicable to destroyed dwarfs (Naidu et al., 2022). The surviving dwarf satellite luminosity function is shown in purple (dashed line is the completeness-corrected luminosity function derived by DrlicaWagner et al. (2020), which does not include the LMC, SMC, or Sgr). [Reproduced from Deason et al. (2023)]. \n<!-- image --> \nassume that this relation holds at all redshifts? Fig. 27 shows how modelling of the stellar halo MDF is significantly impacted by the choice of MZR, and this is thus an important caveat in any such work. At the dwarf massscales, both observations and simulations seem to suggest that there is evolution of the MZR over time (see Fig. 28). Fattahi et al. (2020) and Naidu et al. (2022) show that destroyed dwarfs are metal-poorer (by ∼ 0 . 3 dex) than their surviving dwarf counterparts at fixed stellar mass. This difference is likely owing to the different star formation timescales of the two populations. Destroyed dwarfs are typically accreted earlier than surviving dwarfs, and thus to get to the same stellar mass as their z = 0 counterparts, destroyed dwarfs must have formed their stars on a shorter timescale, and thus have less time to build-up their metals. In support of this picture, Naidu et al. (2022) also show that the destroyed dwarfs are more α -enhanced than surviving dwarfs at fixed stellar mass. However, it is worth noting that the nature of some of the 'destroyed dwarfs' used in the Naidu et al. (2022) work are still under debate (e.g. some of these apparently distinct substructures may be part of the GS/E). Thus, while there is reason to believe there is indeed evolution of the MZR at the dwarf mass scale, there is, as yet, no robust observational measure of this evolving MZR. \nIn order to probe down to the lowest mass scales \n10 \n1.0 0.0 Figure 28: Left panel: [Fe/H] vs. stellar mass at infall for destroyed (red points) and surviving (blue points) dwarfs in the Auriga simulations (Fattahi et al., 2020). The average relations at fixed stellar mass are shown with the solid lines. For comparison, the dashed line shows the average relation for z = 0 satellites. Right panel: [Fe/H] vs. stellar mass for surviving (grey) and disrupted (purple) dwarfs in the Milky Way (Naidu et al., 2022). The grey line is the z = 0 stellar mass metallicity from Kirby et al. (2013). The purple line is offset from this relation by 0.3 dex, which is the median offset of the disrupted dwarfs. [Adapted from Fattahi et al. (2020); Naidu et al. (2022)]. \n<!-- image --> \n- \n0.5 \n10 5 10 6 10 7 10 8 10 9 10 10 M star,infall [M Sun ] -2.0 -1.5 -1.0 [Fe/H] infall survived 2 4 6 8 10 12 14 t infall [Gyr] and bypass the overwhelming signal of massive progenitors such as the GS/E we need (1) very large samples of halo stars with dynamical and/or chemical measurements (e.g. Deason et al. 2023 argue that sample sizes of order ∼ 10 5 -10 6 are needed to probe down to the ultrafaint dwarf mass scales using the stellar halo MDF), (2) to dig deeper into the inner halo of the Galaxy which harbours the earliest accreted and the least massive dwarfs (see e.g. Starkenburg et al., 2017b; El-Badry et al., 2018), (3) to probe the outer halo which is likely populated by low-mass, not yet fully phase-mixed systems (e.g. Fattahi et al., 2020), and (4) a focus on the lowest-metallicity halo stars (owing to the MZR). In principle, we need both large samples and low-metallicity stars, as the most massive dwarf galaxies can still contribute a significant amount of metal-poor stars (Deason et al., 2016). Fortunately, the coming years are ripe for increasing samples of distant halo stars owing to wide-field spectroscopic surveys such as DESI, WEAVE and 4MOST combined with Gaia . Furthermore, efforts are being made to exclusively build samples of very metal-poor stars (Starkenburg et al., 2017a), which could prove vital for uncovering the lowest mass destroyed dwarfs. Gaia 's own spectro-photometric data is already transforming our view of the low-metallicity end of the Galaxy's MDF. Gaia DR3's XP measurements have been used successfully by multiple groups to identify unprecedentedly large numbers of stars with [Fe/H] < -2 (e.g. Andrae et al., 2023; Martin et al., 2023; Zhang et al., 2023; Yao et al., 2024). Time will tell whether we can harness the upcoming datasets to provide a 'destroyed' version of the fundamental Galactic satellite luminosity function, \ninfall \n0.5 \n[Fe/H] but clearly Gaia and its potential future incarnations (e.g. Gaia NIR, Hobbs et al. 2021), will certainly lie at the heart of such efforts. \n- \n0 \n10 5 10 6 10 7 10 8 10 9 10 M star,infall [M sun ] -1.0 -0.5 [Fe/H] survived 6. Conclusions and future outlook The Gaia era has revolutionized the field of Galactic Archaeology. It has changed our view on how the Milky Way assembled, it has changed how we model the Galaxy, and it has opened our eyes to new research directions and explorations. However, the era is not over yet! In the coming years, the final Gaia data releases will become available and our Gaia all-sky map of the Galaxy will be complete. This, coupled with the complementary spectroscopic surveys such as DESI, WEAVE, 4MOST, SDSS-V, etc will provide the next phase in our Galactic exploration. Looking further ahead, pushing these missions to fainter magnitudes, and thus greater distances will be paramount (e.g. Gaia NIR). \nIn the wise words of Donald Lynden-Bell, one should always try to 'follow the data' . That, given the quality and abundance of data at our fingertips, should be advice well heeded. However, the interpretation of this data requires detailed modelling and simulations that will need to keep pace with the data in order to achieve the ultimate goal: a multi-dimensional, star-by-star 'replay' of the Galaxy's formation. \nBelow we list some of the most important and puzzling unanswered questions in Galactic Archaeology that future datasets and modelling efforts will be striving to \n0.0 \n10 \n10 \naddress: \n- · What is the luminosity function of destroyed dwarfs in the Milky Way down to the lowest mass scales? Does this agree with the ΛCDM predictions?\n- · How instrumental was the GS/E in the Galaxy's evolution? When did it merge, how much material did it bring, and how does it link to the disc, bulge, and bar formation?\n- · Why did the Milky Way disc form so early? Can this be reproduced in cosmological simulations?\n- · How can we distinguish the Galactic building blocks in the early pre-disc era of the Milky Way?\n- · What are the high-redshift contemporaries of the Galactic building blocks? How did they evolve?\n- · What is the origin of the stellar halo spin? Does this relate to the spinning up of the Milky Way disc?\n- · Does the LMC-induced wake in the Galactic halo exist? What can this tell us about the nature of dark matter?\n- · What happened to the 'satellites of satellites'? i.e. the satellites associated with the pre-infall massive dwarfs, like the LMC and GS/E. Can we distinguish this population from the 'singly' accreted satellites?", 'Acknowledgments': "AD is supported by a Royal Society University Research Fellowship. AD acknowledges support from the Leverhulme Trust and the Science and Technology Facilities Council (STFC) [grant numbers ST/X001075/1, ST/T000244/1]. VB acknowledges support from the Leverhulme Research Project Grant RPG-2021-205: 'The Faint Universe Made Visible with Machine Learning'. \nWe thank an anonymous referee for providing a constructive report.", 'References': "Abadi, M.G., Navarro, J.F., Steinmetz, M., 2006. Stars beyond galaxies: the origin of extended luminous haloes around galaxies. MNRAS 365, 747-758. doi: 10.1111/j.1365-2966.2005.09789.x , arXiv:astro-ph/0506659 . \nAblimit, I., Zhao, G., Teklimakan, U., Shi, J.R., Abdusalam, K., 2022. The Milky Way Revealed by Variable Stars. I. Sample Selection of RR Lyrae Stars and Evidence for Merger History. ApJS 258, 20. doi: 10.3847/1538-4365/ac347f , arXiv:2111.00028 . \nAguado, D.S., Belokurov, V., Myeong, G.C., Evans, N.W., Kobayashi, C., Sbordone, L., Chanam'e, J., Navarrete, C., Koposov, S.E., 2021. 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J., del Pozo, E., Delbo, M., Delgado, A., Delgado, H.E., Di Matteo, P., Diakite, S., Distefano, E., Dolding, C., Dos Anjos, S., Drazinos, P., Duran, J., Dzigan, Y., Edvardsson, B., Enke, H., Evans, N.W., Eynard Bontemps, G., Fabre, C., Fabrizio, M., Faigler, S., Falc˜ao, A.J., Farr'as Casas, M., Federici, L., Fedorets, G., Fern'andezHern'andez, J., Fernique, P., Fienga, A., Figueras, F., Filippi, F., Findeisen, K., Fonti, A., Fouesneau, M., Fraile, E., Fraser, M., Fuchs, J., Gai, M., Galleti, S., Galluccio, L., Garabato, D., Garc'ıa- \nSedano, F., Garofalo, A., Garralda, N., Gavras, P., Gerssen, J., Geyer, R., Gilmore, G., Girona, S., Giuffrida, G., Gomes, M., Gonz'alez-Marcos, A., Gonz'alez-N'u˜nez, J., Gonz'alez-Vidal, J.J., Granvik, M., Guerrier, A., Guillout, P., Guiraud, J., G'urpide, A., Guti'errez-S'anchez, R., Guy, L.P., Haigron, R., Hatzidimitriou, D., Haywood, M., Heiter, U., Helmi, A., Hobbs, D., Hofmann, W., Holl, B., Holland, G., Hunt, J.A.S., Hypki, A., Icardi, V., Irwin, 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Gaia Data Release 1. Summary of the astrometric, photometric, and survey properties. A&A 595, A2. doi: 10.1051/0004-6361/201629512 , arXiv:1609.04172 . \nGaia Collaboration, Helmi, A., van Leeuwen, F., McMillan, P.J., Massari, D., Antoja, T., Robin, A.C., Lindegren, L., Bastian, U., Arenou, F., et al., 2018b. Gaia Data Release 2. Kinematics of globular clusters and dwarf galaxies around the Milky Way. A&A 616, A12. doi: 10.1051/0004-6361/201832698 , arXiv:1804.09381 . 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Mary, N., Matijeviˇc, G., Mazeh, T., McMillan, P.J., Messina, S., Mestre, A., Michalik, D., Millar, N.R., Miranda, B.M.H., Molina, D., Molinaro, R., Molinaro, M., Moln'ar, L., Moniez, M., Montegriffo, P., Monteiro, D., Mor, R., Mora, A., Morbidelli, R., Morel, T., Morgenthaler, S., Morley, T., Morris, D., Mulone, A.F., Muraveva, T., Musella, I., Narbonne, J., Nelemans, G., Nicastro, L., Noval, L., Ord'enovic, C., Ordieres-Mer'e, J., Osborne, P., Pagani, C., Pagano, I., Pailler, F., Palacin, H., Palaversa, L., Parsons, P., Paulsen, T., Pecoraro, M., Pedrosa, R., Pentikainen, H., Pereira, J., Pichon, B., Piersimoni, A.M., Pineau, F.X., Plachy, E., Plum, G., Poujoulet, E., Prˇsa, A., Pulone, L., Ragaini, S., Rago, S., Rambaux, N., Ramos-Lerate, M., Ranalli, P., Rauw, G., Read, A., Regibo, S., Renk, F., Reyl'e, C., Ribeiro, R.A., Rimoldini, L., Ripepi, V., Riva, A., Rixon, G., Roelens, M., Romero-G'omez, M., Rowell, N., Royer, F., Rudolph, A., Ruiz-Dern, L., Sadowski, G., Sagrist'a Sell'es, T., Sahlmann, J., Salgado, J., Salguero, E., Sarasso, M., Savietto, H., Schnorhk, A., Schultheis, M., Sciacca, E., Segol, M., Segovia, J.C., Segransan, D., Serpell, E., Shih, I.C., Smareglia, R., Smart, R.L., Smith, C., Solano, E., Solitro, F., Sordo, R., Soria Nieto, S., Souchay, J., Spagna, A., Spoto, F., Stampa, U., Steele, I.A., Steidelmuller, H., Stephenson, C.A., Stoev, H., Suess, F.F., Suveges, M., Surdej, J., Szabados, L., Szegedi-Elek, E., Tapiador, D., Taris, F., Tauran, G., Taylor, M.B., Teixeira, R., Terrett, D., Tingley, B., Trager, S.C., Turon, C., Ulla, A., Utrilla, E., Valentini, G., van Elteren, A., Van Hemelryck, E., van Leeuwen, M., Varadi, M., Vecchiato, A., Veljanoski, J., Via, T., Vicente, D., Vogt, S., Voss, H., Votruba, V., Voutsinas, S., Walmsley, G., Weiler, M., Weingrill, K., Werner, D., Wevers, T., Whitehead, G., Wyrzykowski, glyph[suppress]L., Yoldas, A., ˇ Zerjal, M., Zucker, S., Zurbach, C., Zwitter, T., Alecu, A., Allen, M., Allende Prieto, C., 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2024PDU....4601681A

Primordial black holes PBHs hidden in the incredible bulk of the dark dimension could escape constraints from nonobservation of their Hawking radiation. Since these fivedimensional 5D PBHs are bigger colder and longerlived than usual 4D PBHs of the same mass mmlmath altimgsi139.svg displayinline idd1e358mmlmiMmmlmimmlmath they could make all cosmological dark matter if mmlmath altimgsi2.svg displayinline idd1e363mmlmrowmmlmn1mmlmnmmlmsupmmlmrowmmlmn0mmlmnmmlmrowmmlmrowmmlmn11mmlmnmmlmrowmmlmsupmmlmo linebreakgoodbreak linebreakstyleaftermmlmommlmiMmmlmimmlmommlmommlmi mathvariantnormalgmmlmimmlmo linebreakgoodbreak linebreakstyleaftermmlmommlmn1mmlmnmmlmsupmmlmrowmmlmn0mmlmnmmlmrowmmlmrowmmlmn21mmlmnmmlmrowmmlmsupmmlmrowmmlmath i.e. extending the 4D allowed region far down the asteroidmass window. We show that these evasive PBHs could be search for by measuring their mmlmath altimgsi140.svg displayinline idd1e395mmlmiXmmlmimmlmathray microlensing events from faraway pulsars. We also show that future mmlmath altimgsi140.svg displayinline idd1e401mmlmiXmmlmimmlmathray microlensing experiments will be able to probe the interesting range mmlmath altimgsi5.svg displayinline idd1e407mmlmrowmmlmn1mmlmnmmlmsupmmlmrowmmlmn0mmlmnmmlmrowmmlmrowmmlmn16mmlmnmmlmo.mmlmommlmn5mmlmnmmlmrowmmlmsupmmlmo linebreakgoodbreak linebreakstyleaftermmlmommlmiMmmlmimmlmommlmommlmi mathvariantnormalgmmlmimmlmo linebreakgoodbreak linebreakstyleaftermmlmommlmn1mmlmnmmlmsupmmlmrowmmlmn0mmlmnmmlmrowmmlmrowmmlmn17mmlmnmmlmo.mmlmommlmn5mmlmnmmlmrowmmlmsupmmlmspace classnbsp width1emmmlmspacemmlmi mathvariantnormalgmmlmimmlmrowmmlmath where an all dark matter interpretation in terms of 4D Schwarzschild PBHs is excluded by the nonobservation of their Hawking radiation.

2024-12-01T00:00:00Z

['10.48550/arXiv.2409.12904', '2024arXiv240912904A', '10.1016/j.dark.2024.101681', '2024PDU....4601681A', 'arXiv:2409.12904']

['Bulk black hole dark matter', 'Microlensing of X-ray pulsars', 'High Energy Physics - Phenomenology']

Through the looking glass into the dark dimension Searching for bulk black hole dark matter with microlensing of mmlmath altimgsi6.svg displayinline idd1e142mmlmi mathvariantbolditalicXmmlmimmlmathray pulsars

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML']

https://arxiv.org/pdf/2409.12904.pdf

{'Through the Looking Glass into the Dark Dimension: Searching for Bulk Black Hole Dark Matter with Microlensing of X -ray Pulsars': "Luis A. Anchordoqui, 1, 2, 3 Ignatios Antoniadis, 4, 5 Dieter Lust, 6, 7 and Karem Pe˜nal'o Castillo 1 \n1 Department of Physics and Astronomy, Lehman College, City University of New York, NY 10468, USA 2 Department of Physics, Graduate Center, City University of New York, NY 10016, USA 3 Department of Astrophysics, American Museum of Natural History, NY 10024, USA 4 High Energy Physics Research Unit, Faculty of Science, Chulalongkorn University, Bangkok 1030, Thailand 5 Laboratoire de Physique Th'eorique et Hautes ' Energies - LPTHE Sorbonne Universit'e, CNRS, 4 Place Jussieu, 75005 Paris, France 6 Max-Planck-Institut fur Physik, Werner-Heisenberg-Institut, 80805 Munchen, Germany 7 Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universitat Munchen, 80333 Munchen, Germany \n(Dated: September 2024) \nPrimordial black holes (PBHs) hidden in the incredible bulk of the dark dimension could escape constraints from non-observation of their Hawking radiation. Since these five-dimensional (5D) PBHs are bigger, colder, and longer-lived than usual 4D PBHs of the same mass M , they could make all cosmological dark matter if 10 11 ≲ M/ g ≲ 10 21 , i.e., extending the 4D allowed region far down the asteroid-mass window. We show that these evasive PBHs could be search for by measuring their X -ray microlensing events from faraway pulsars. We also show that future X -ray microlensing experiments will be able to probe the interesting range (10 16 . 5 ≲ M/ g ≲ 10 17 . 5 g) where an all dark matter interpretation in terms of 4D Schwarzschild PBHs is excluded by the non-observation of their Hawking radiation.", 'I. INTRODUCTION': 'The dark dimension, an innovative five-dimensional (5D) scenario that has a compact space with characteristic length-scale in the micron range, provides a stepping stone to address the cosmological hierarchy problem [1]. This is because the anti-de Sitter distance conjecture in de Sitter space [2] connects the size of the compact space R ⊥ to the dark energy scale Λ 1 / 4 via R ⊥ ∼ λ Λ 1 / 4 , where Λ ∼ 10 -122 M 4 p is the cosmological constant, M p the Planck mass, and the proportionality factor is estimated to be within the range 10 -1 < λ < 10 -4 [1]. \nPrimordial black holes (PBHs), presumably formed through the collapse of sufficiently sizable overdensities originated by an enhancement in the comoving curvature perturbation power spectrum at small scales during inflation [3-5], may be one of the most interesting candidates of dark matter [6-8]. These intriguing dark matter candidates are particular important in the dark dimension scenario, because 5D PBH are bigger, colder, and longer-lived than usual 4D PBHs of the same mass M [911]. 1 Furthermore, we have recently shown that a back reaction of Hawking evaporation process is to kick 5D Schwarzschild PBHs out of the brane [18]. As a consequence, 5D PBHs could make all cosmological dark \nmatter if 10 11 ≲ M/ g ≲ 10 21 , i.e., well down into the asteroid-mass window. In this paper we investigate a potential signal that may allow us to unmask these as yet evasive 5D PBHs. \nAn astonishing coincidence is that R ⊥ corresponds approximately to the wavelength of visible light. This implies that the Schwarzschild radius of black holes perceiving the dark dimension is well below the wavelength of light. For point-like lenses, this is precisely the critical length where geometric optics breaks down and the effects of wave optics suppress the magnification [19-21], obstructing the sensitivity to 5D PBH microlensing signals. Nevertheless, it was pointed out in [22] that X -ray pulsars, with photon energies above 1 keV, are good candidate sources to search for microlensing of PBHs with M ≲ 10 21 g. In this paper we reexamine this idea focussing attention on 5D PBHs perceiving the dark dimension. \nThe layout is as follows. In Sec. II we review the generalities of gravitational microlensing. In Sec. III we focus attention on the particulars of microlensing of X -ray pulsars, which is relevant to unmask PBHs perceiving the dark dimension. The paper wraps up in Sec. IV with some conclusions.', 'II. POINT-LENS MICROLENSING BASICS': "It has long been known that gravitational lensing becomes observable when a massive object is located between a light-emitting source and an observer [23]. This \nFIG. 1: The geometry of the microlensing setup for a pointlike lens, seen to produce two images after deflection [25]. \n<!-- image --> \nis because the gravitational potential of the central object acts as a lens that bends the path of the light rays coming from the source, by bending the spacetime through which the photon travels. If the central object is a Schwarzschild black hole, described by the line element \nds 2 = -f ( r ) dt 2 + f -1 ( r ) dr 2 + r 2 d Ω 2 2 , (1) \nthe deflection angle of a light ray is estimated to be \nα = 4 GM r 0 +4 ( GM r 0 ) 2 ( 15 π 16 -1 ) + O ( ϵ 3 ) , (2) \nwhere f ( r ) = 1 -2 GM/r is the blackening function, M is the mass of the black hole (i.e. the lens), G = M -2 p is Newton's gravitational constant, d Ω 2 2 is the metric of a 2-dimensional unit sphere, and r 0 is the closest distance of approach to the lens in the lens-Earth-source plane, with ϵ = GM/r 0 , and where all measurements are taken in the observer's reference frame [24]. To develop some sense for the orders of magnitude involved, we consider the set up shown in Fig. 1 for a point-like lens. In the absence of the lens, we would then observe the source at an angular position β on the source plane, but because of the deflection, we actually observe it at θ . Since the observer distances to the lens D OL and source D OS are very large in relation to the deflection angle, the distances of the source and image from the optical axis in the source plane are estimated using the small angle approximation \nθD OS = βD OS + αD LS , (3) \nwhere D LS is the distance from the lens to the source [19]. \nThe impact parameter of the unperturbed light ray is found to be ξ = r 0 (1 -2 GM/r 0 ) -1 / 2 . In the point-lens approximation, however, ξ ≡ θD OL ∼ r 0 , and so (3) can be rewritten as \nθ 2 = βθ + θ 2 E , (4) \nwhere \nθ E = √ 4 GM D LS D OL D OS (5) \nis the angular size of the Einstein ring which is formed when the source is perfectly aligned behind the lens, i.e., when β = 0, and where we have neglected terms O ( ϵ 2 ). θ E in turn defines the Einstein radius r E = D OL θ E on the lens plane. Note that for β ∼ 0, the impact parameter ξ ∼ r E is the characteristic scale of the source-lensobserver system. All in all, the isolated Schwarzschild black hole will split a point source into two images with angular positions defined implicitly by \nθ 1 , 2 = 1 2 ( β ± √ β 2 +4 θ 2 E ) . (6) \nThe two images are on opposite sides of the source, with one image inside the Einstein ring and the other outside. As the source moves away from the lens (i.e. as β increases), one of the images approaches the lens and becomes very faint, whereas the other image approaches the true position of the source and asymptotes to its unlensed flux. \nA point worth noting at this juncture is that if a source is closer than θ E in separation from a lensing PBH on the sky, the source is multiply imaged by its lensing. Nevertheless, the separation between multiple images is too small to be resolved by optical telescopes. What is observed instead in the so-called 'microlensing phenomenon' is the temporary magnification of the total flux of two images relative to that of the unlensed source. Now, if a gravitational field were to deflect a light ray this would produce a change in the cross-section of a bundle of rays. However, according to Liouville's theorem the phase space density of photons must be conserved, which implies that gravitational lensing should preserve the surface brightness of the source and should only change its apparent surface area. In other words, the magnification of an image becomes the ratio of the solid angles of the image and of the unlensed source (at the observer position). If the central object is spherically symmetric, the magnification factor is found to be \nµ = sin θ dθ sin β dβ ≃ θ β dθ dβ , (7) \nwhere the sign of the magnification gives the parity of the particular image. Substituting β from the point-lens equation (4) into (7), it follows the magnifications of the two images, \nµ 1 , 2 = [ 1 -( θ E θ 1 , 2 ) 4 ] -1 = u 2 +2 2 u √ u 2 +4 ± 1 2 , (8) \nwhere u is the angular separation of the source from the point mass in units of the Einstein angle, u = β/θ E . Note that θ 2 < θ E implies µ 2 < 0, and so the magnification of the image which is inside the Einstein ring is negative implying that this image has its parity flipped with respect to the source. The net magnification of flux in the two images is obtained by adding the absolute magnifi- \nns, \nµ tot ( u ) = µ 1 + µ 2 = u 2 +2 u √ u 2 +4 . (9) \nNote that when the source lies on the Einstein radius, we have β = θ E and u = 1, so that the total magnification becomes µ tot = 1 . 17 + 0 . 17 = 1 . 34. \nNow, the source and the lens have a relative motion with respect to the observer. As a consequence, the observed flux of a source varies with time, yielding a characteristic light curve of the observed source flux. Such a unique light curve allows a microlensing event to be identified from the observation of other variable sources. A typical timescale of the microlensing light curve can be estimated from a crossing time of the Einstein radius for a lensing PBH with respect to a distant source, \nt E = r E /v , (10) \nwhere v is the relative velocity for a observer-lens-source system [26]. \nMicrolensing surveys are typically sensitive to stars that are 10 ≲ D OS / kpc ≲ 1000 away and to transit times of minutes to years. For example, the Subaru-HSC instrument was sensitive to the short transit times of light PBHs reaching sensitivities to constrain an all dark matter PBH interpretation for masses M ≳ 10 22 g [19-21]. However the Subaru-HSC survey was limited by two effects as the PBH mass is reduced: \n- · the finite apparent size of the source stars in its sky target, M31, being larger than the apparent Einstein radius, yielding poorly focused light;\n- · transition from geometric to wave optics as the PBH Schwarzschild radius becomes comparable to or smaller than the wavelength of optical light. \nIndeed, the wave effects characterized by \nw = 8 πGM λ = 0 . 3 ( M 10 22 g )( λ 0 621 nm ) -1 , (11) \nbecome important when w ≲ 1, where λ 0 is the characteristic wavelength of light in an observation [27]; the default choice in the normalization of (11) corresponds to a central wavelength of r -band in the Subaru telescope. As it was first pointed out in [22], consideration of sources emitting in the X -ray spectrum could allow us to search deep into the PBH low-mass window. It is this that we now turn to study.", 'III. MICROLENSING OF BLACK HOLES PERCEIVING THE DARK DIMENSION': 'It is a known fact that M p is the natural cutoff of quantum gravity. However, if there were light species of particles in the theory, then the consistency of black hole entropy with the effective field theory description would \ndemand a breakdown of the classical picture at a lower scale, \nM ∗ = m ( d -4) / ( d -2) KK M 2 / ( d -2) p , (12) \ndubbed the species scale, where d is the spacetime dimension and m KK ∼ R -1 ⊥ [28, 29]. Now, despite the fact that M ∗ is motivated by the emergence of the tower of light states, curiously the mass scale of this tower, m KK , does not seem to be directly captured by M ∗ [30]. \nOne way the lower-dimensional theory can find out about the m KK scale is through the study of black holes [10, 11, 30]. When black holes get smaller than R ⊥ , the black hole becomes thermodynamically unstable because of the Gregory-Laflamme transition [31]. Black holes with r s ≫ R ⊥ are actually black branes wrapped around the extra dimensions, but the ones with r s ≪ R ⊥ are localized in the extra dimensions. This transition to a new black hole solution marks a new scale Λ BH in the lower-dimensional theory [30]. In this section we show that Earth-based microlensing experiments are insensitive to the Λ BH scale, independently of the black hole mass M . \nThroughout we assume that the Standard Model fields are confined to a D-brane and only gravity spills into the compact space of dimension ( d -4) [32].', 'A. Microlensing of Bulk Black Holes': "The Gregory-Laflamme transition induces a change in the scaling of the black hole's Schwarzschild radius r s and its Hawking temperature T H [10]. If the black hole is spherically symmetric and r s ≪ R ⊥ , then it can be treated as a flat d -dimensional object with line element given by \nds 2 = -U ( r ) dt 2 + U -1 ( r ) dr 2 + r 2 d Ω 2 d -2 , (13) \nwhere U ( r ) = 1 -( r s /r ) d -3 is the blackening function, \nd Ω 2 d -2 = dχ 2 2 + d -2 ∏ i =2 sin 2 χ i dχ 2 i +1 (14) \nis the metric of a ( d -2)-dimensional unit sphere, and \nr s = 1 M ∗ [ M M ∗ ] 1 / ( d -3) [ 8 Γ ( d -1 2 ) ( d -2) π ( d -3) / 2 ] 1 / ( d -3) , (15) \nand where Γ( x ) is the Gamma function [33-35]. The d -dimensional black hole behaves like a thermodynamic system [36], with temperature T H ∼ ( d -3) / (4 πr s ) and entropy S = (4 πMr s ) / ( d -2) [37]. If the black hole is localized on the brane, Hawking evaporation [38, 39] proceeds dominantly through emission of Standard Model fields [40]. However, we have recently shown that the recoil effect due to graviton emission imparts the black hole \na relative kick velocity with respect to the brane, allowing Schwarzschild black holes to escape into the bulk [18]. Because the escape from the brane is almost instantaneous 5D Schwarzschild black holes evaporate almost entirely into gravitons in the bulk, and so can evade constraints from the non-observation of Hawking radiation in: (i) the extragalactic γ -ray background [41], (ii) the cosmic microwave background [42], (iii) the 511 keV γ -ray line [43-46], (iv) the EDGES 21-cm signal [47, 48], (v) Lymanα forest [49], and (vi) the MeV Galactic diffuse emission [50-52]. \nThe deflection angle of a light ray expected to be induced by a d -dimensional Schwarzschild black hole localized on the brane is estimated to be \nα d = 2( d -2) M M d -2 ∗ r d -3 0 2 F 1 ( 1 2 , κ ; 3 2 ; 1 ) + O ( M 2 M d ∗ r d -2 0 ) , (16) \nwhere 2 F 1 ( a, b, ; c ; z ) is the Gaussian hypergeometric function, with κ = 1 / 2 -( d -3) / 2 [53]. A straightforward calculation shows that for d = 4, (16) reduces to the first term in (2). For d = 5, \nα 5 = 3 2 π M 3 ∗ M r 2 0 + O ( M 2 M 5 ∗ r 3 0 ) , (17) \nwhich implies that α 5 < α 4 for fixed M and r 0 , with r s < r 0 < R ⊥ . This was interpreted in [53] as the need for advanced sensitive detection devices to observe lensed images influenced by the dark dimension. \nThe previous statement should be revised with caution. It is clear that while the source is on the brane, in the dark dimension scenario the lens has an extension into the bulk. Therefore, the light emitted by the source could be a micron away from the lens 4D position. The scale that counts in addressing whether the 5D geometry of the lens would bring important corrections to the source magnification µ tot is actually the size ξ of the lens involved in the deflection. As we have stressed in Sec. II, in the point-lens approximation of Fig. 1, the characteristic scale of the source-lens-observer system is ξ ≳ r E . \nAt this point a reality check is in order. A source could be a good object to undergo microlensing if it is a long distance away from the telescopes (viz., the Earth), because this would increase the optical depth and/or the number of possible lensing events. Following [22], we consider the X -ray pulsar SMC X-1 in the Small Magellanic Cloud at a distance D OS = 65 kpc [54]. For such a distance, the Einstein radius of a M ∼ 10 17 g black hole is given by \nr E = √ 4 GMx (1 -x ) D OS = 12 km [( M 10 17 g )( D OS 65 kpc )( x (1 -x ) 1 / 4 )] 1 / 2 (18) \nwhere x = D OL /D OS . Since r E ≫ R ⊥ is the closest distance that source photons get to the lens when it lies directly along the line of sight, we conclude that corrections to µ tot due to the 5D geometry of the lens can be \nsafely neglected. In other words, for r ≫ R ⊥ the gravitational potential of a 5D black hole falls as 1 /r and therefore is indistinguishable from that of a 4D black hole of the same mass. This implies that for Earth-based experiments the microlensing signal of a 5D black hole would be indistinguishable from that of a 4D black hole of the same mass, and this is independent on whether the black hole is localized on the brane or propagates through the bulk. \nSince line-of-sight distances of interest are usually much larger than transverse distances, X -ray microlensing events can be pictured as projections on the lenscontaining transverse plane, viz. the lens plane. The source radius in the lens plane is estimated to be \na s ( x ) = xR s r E ( x ) ∼ 0 . 8 × ( x √ x (1 -x ) ) ( R S 20 km ) × ( D OS 65 kpc ) -1 ( M 10 17 g ) -1 / 2 , (19) \nwhere following [22] we have taken a fiducial source size 10 ≲ R S / km ≲ 20. From (19) we infer that X -ray pulsars could overcome finite source size effects for M ∼ 10 17 g if x ≲ 0 . 6. \nNow, wave effects becomes important when w ≲ 1 [27] or equivalently for photon energies \nE γ ≲ 1 4 GM = 660 keV ( M 10 17 g ) -1 . (20) \nThus, for a source emitting primarily with X -ray energy of 1 ≲ E γ / keV ≲ 10 [54], wave effects must be taken into account in order to probe a lower PBH mass of about 10 17 g. \nThe magnification, including the wave optics effect, is given by \nµ tot ( w,u ) = πw 1 -e -πw ∣ ∣ ∣ ∣ 1 F 1 ( i 2 w, 1; i 2 wu 2 )∣ ∣ ∣ ∣ 2 (21) \nwhere 1 F 1 ( a ; b ; z ) is the confluent hypergeometric function [55]. For u = 0 the magnification has a maximum \nµ tot ( w,u ) ∣ ∣ ∣ ∣ max = µ w ( w, 0) = πw 1 -e -πw . (22) \nIt is easily seen from (22) that for microlensing events with w ≪ 1, the maximum magnification \nµ tot ( w,u ) ∣ ∣ max = 1 + πw 2 , (23) \nwould be significantly reduced as compared to the maximum magnification µ tot ( u ) →∞ in the geometric optics approximation. 2 This is because the gravitational potential induced by extremely light PBHs is too weak to bent the path of the photons. \nFIG. 2: Compilation of projected sensitivities on the PBH dark matter fraction f PBH as a function of M for future X -ray telescopes, assuming a monochromatic mass function and including wave optics effects. The sensitivity curves have been taken from [22, 57]. For comparison, the shaded region shows contraints on the 4D f PBH from the non-observation of Hawking radiation in: (i) the extragalactic γ -ray background [41], (ii) the cosmic microwave background [42], (iii) the 511 keV γ -ray line [43-46], (iv) the EDGES 21-cm signal [47, 48], (v) Lymanα forest [49], and (vi) the MeV Galactic diffuse emission [50-52]. . \n<!-- image --> \nThe mass distribution of PBHs is usually characterized by the mass function \nψ ( M ) = M ρ CDM dn PBH dM , (24) \nwhere dn PBH is the number density of PBHs within the mass range ( M,M + dM ), and ρ CDM is the energy density of cold dark matter [56]. Integrating ψ ( M ) gives the total fraction of dark matter in PBHs, \nf PBH ≡ ρ PBH ρ CDM = ∫ ψ ( M ) dM , (25) \nwhere ρ PBH = ∫ Mdn PBH is the energy density of PBHs. If all of the dark matter were made of PBH, we would have f PBH = 1. In Fig. 2 we show a compilation of the projected sensitivity to f PBH of future X -ray telescopes, including Athena [58], Lynx [59], and eXTP for a 300day observation [60], as well as STROBE-X [61], and Xµ for a 30-day observation [57]. We can see that these instruments will break down into the mass range relevant for bulk black hole dark matter, including the interesting region in which a 4D PBH all dark matter interpretation has been excluded due to the non-observation of Hawking radiation.", 'B. Microlensing of Near-Extremal Black Holes': "According to the no-hair theorem [62], 4D black hole geometries in asymptotically flat spacetimes eventually settled down to Kerr-Newman solutions [63, 64], which are characterized by three measurable parameters: the mass M , the angular momentum ⃗ J and the electric charge Q . An important feature of Kerr-Newman black holes is that the three parameters are not all independent from each other. Actually, for a given set of parameters there is a minimal extremal mass M e satisfying \nM 2 ≥ M 2 e = ( M 2 p J M ) 2 +( M p Q ) 2 , (26) \nwhere Q is measured in units in which the Coulomb force between two charges separated by a distance d has magnitude F = Q 2 /d 2 and J = | ⃗ J | [65]. If the parameters saturate this bound the black hole is dubbed extremal, and if the parameters are close to saturating it near-extremal. Before proceeding we pause to note that: \n- · When the black hole mass is tuned below M ext , the event horizon disappears leaving behind a naked singularity, which violates the cosmic censorship conjecture [66].\n- · When a black hole reaches its extreme limit, the thermal description breaks down, and it cannot continue to evaporate by emitting (uncharged) elementary particles.\n- · When black holes are near-extremal their evaporation temperature decreases, and consequently so does their luminosity. Thus, near-extremal black holes can also evade constraints from the nonobservation of Hawking radiation [67, 68]. \nIt has long been suspected that any electromagnetic charge or spin would be lost very quickly by any 4D black hole population of primordial origin. On the one hand, the electromagnetic charge of a black hole is spoiled by the Schwinger effect [69], which allows pair-production of electron-positron pairs in the strong electric field outside the black hole, leading to the discharge of the black hole and subsequent evaporation [70, 71]. On the other hand, a rapidly rotating black hole spins down to a nearly nonrotating state before most of its mass has been given up, and therefore it does not approach to extremal when it evaporates [72]. All in all, near-extremal primordial Kerr-Newman black holes are not expected to prevail in the universe we live in. \nAn alternative interesting possibility is to envision a scenario where the black hole is charged under a generic unbroken U (1) symmetry (dark photon), whose carriers (dark electrons with a mass m ' e and a gauge coupling e ' ) are always much heavier than the temperature of the black hole [73]. This implies that the charge Q does not get evaporated away from the black hole and remains therefore constant. Strictly speaking, the pair production rate per unit volume from the Schwinger effect can be slowed down by arbitrarily decreasing e ' , whereas the \nweak gravity conjecture (WGC) imposes a constraint on the charge per unit mass; namely, for each conserved gauge charge there must be a sufficiently light charge carrier such that \ne ' q/m e ' ≥ √ 4 π √ ( d -3) / ( d -2) ¯ M -( d -2) / 2 p , (27) \nwhere q is the integer-quantized electric charge of the particle and ¯ M p = M p / (8 π ) is the reduced Planck mass [74, 75]. Setting e = e ' = √ 4 πα the (4D) Schwinger effect together with the WGC lead to a bound on the minimum black hole mass of near extremal black holes with evaporation time longer than the age of Universe, M ne ≳ 5 × 10 15 g( m e ' / 10 9 GeV) -2 [73]. \nSince we have seen that microlensing experiments cannot distinguish 4D from 5D black holes, in what follows we consider the line element (1) with blackening function \nf ( r ) = 1 -2 M M 2 p r + Q 2 M 2 p r 2 . (28) \nThe deflection angle of a light ray expected to be induced by the geometry given in (28) has been computed in [76] and is given by \nα Q = 4 M M 2 p r 0 + ( 15 16 π -1 ) 4 M 2 M 4 p r 2 0 -3 4 π Q 2 M 2 p r 2 0 + · · · . (29) \nWe can see that the effect of Q is to slightly reduce the second order correction to the deflection angle. Therefore, we can conclude that the sensitivity of future X -ray microforlensing experiments to near-extremal black holes is actually shown in Fig. 2. \nIn summary, evaporation constraints on f PBH can be substantially altered when moving away from the Schwarzschild picture. (Other PBH models where Hawking radiation can be slowed down have been recently discussed in [77, 78].) Microlensing experiments of X -ray pulsars are well positioned to uncover these models.", 'IV. CONCLUSIONS': "Wehave investigated the potential of searching for bulk black hole dark matter through microlensing events us- \n- [1] M. Montero, C. Vafa and I. 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We have demonstrated that future instruments observing microlensing events from X -ray pulsars will probe the mass range relevant for bulk black hole dark matter, including the interesting region in which a 4D PBH all dark matter interpretation has been excluded due to the non-observation of Hawking radiation. \nWe end with an observation. PBHs may experience a memory burden effect, which splits the evaporation process into two distinct phases: semiclassical and quantum [79, 80]. If this were the case, then the minimum black hole mass allowing a PBH all-dark-matter interpretation would also be relaxed [81-84]. The quantum decay rate has an additional suppression factor compared to the Hawking decay rate, which in the most realistic scenario scales with the inverse of the black hole entropy. For d = 4 and a quantum decay rate Γ ∼ T H /S , the memory burden effect opens a new window in the mass range 10 9 ≲ M/ g ≲ 10 10 [82]. 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2024AJ....168..298S

A fraction of Galactic stars have compact companions that could be white dwarfs WDs neutron stars NSs or stellarmass black holes SBHs. In a detached and edgeon binary system including a mainsequence star and a compact object denoted by WD mainsequence WDMS NS mainsequence NSMS and SBH mainsequence BHMS systems the stellar brightness can change periodically due to selflensing or eclipsing features. The shape of a selflensing signal is a degenerate function of stellar radius and the compact objects mass because the selflensing peak strongly depends on the projected source radius normalized to the Einstein radius. Increasing the inclination angle i changes the selflensing shape from a strict tophat model to one with slowly increasing edges. We simulate stellar light curves due to these binary systems that are observed by NASAs Transiting Exoplanet Survey Satellite TESS telescope and evaluate the efficiencies to detect their periodic signatures using two sets of criteria i signaltonoise ratio SNR gt3 and N SUBtranSUB gt 1 low confidence LC and ii SNR gt5 and N SUBtranSUB gt 2 high confidence HC. The HC efficiencies for detecting WDMS NSMS and BHMS systems with the inclination angle i lt 20 during different time spans are 57 4.56 and 45 respectively. Detecting lensinginduced features is possible in only 3 and 33 of detectable WDMS and NSMS events. The detection efficiencies for closer source stars with higher priorities are higher and drop to zero for b R SUBSUB where inlineformula inlineformula is the impact parameter a is the semimajor axis. We predict the numbers of WDs NSs and SBHs that are discovered from the TESS Candidate Target List stars are 1518 67 and lt1.

2024-12-01T00:00:00Z

['2024arXiv240912441S', '10.48550/arXiv.2409.12441', 'arXiv:2409.12441', '10.3847/1538-3881/ad7fdd', '2024AJ....168..298S']

['Compact objects', 'Gravitational lensing', 'Compact binary stars', 'Stellar remnants', 'Transient detection', 'Eclipsing binary stars', 'Astronomical simulations', 'Space telescopes', '288', '670', '283', '1627', '1957', '444', '1857', '1547', 'Astrophysics - Solar and Stellar Astrophysics', 'Astrophysics - Earth and Planetary Astrophysics', 'Astrophysics - Astrophysics of Galaxies', 'Astrophysics - Instrumentation and Methods for Astrophysics']

Simulating Selflensing and Eclipsing Signals due to Detached Compact Objects in the TESS Light Curves

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']

https://arxiv.org/pdf/2409.12441.pdf

{'Simulating Self-Lensing and Eclipsing Signals due to Detached Compact Objects in the TESS Light Curves': 'Sedighe Sajadian 1 and Niayesh Afshordi 2, 3, 4 \n1 Department of Physics, Isfahan University of Technology, Isfahan 84156-83111, Iran, s.sajadian@iut.ac.ir \n2 Waterloo Centre for Astrophysics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada \n- 3 Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada \n4 Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada', 'ABSTRACT': "A fraction of Galactic stars have compact companions which could be white dwarfs (WDs), neutron stars (NSs) or stellar-mass black holes (SBHs). In a detached and edge-on binary system including a main-sequence star and a compact object (denoted by WDMS, NSMS, and BHMS systems), the stellar brightness can change periodically due to self-lensing or eclipsing features. The shape of a self-lensing signals is a degenerate function of stellar radius and compact object's mass because the self-lensing peak strongly depends on the projected source radius normalized to Einstein radius. Increasing the inclination angle i changes the self-lensing shape from a strict top-hat model to one with slow-increasing edges. We simulate stellar light curves due to these binary systems which are observed by NASA's Transiting Exoplanet Survey Satellite (TESS) telescope and evaluate the efficiencies to detect their periodic signatures using two sets of criteria (i)SNR > 3 and N tran > 1 (Low-Confidence, LC), and (ii) SNR > 5 and N tran > 2 (High-Confidence, HC). The HC efficiencies for detecting WDMS, NSMS, and BHMS systems with the inclination angle i < 20 · during different time spans are 5-7%, 4 . 5-6%, and 4-5%, respectively. Detecting lensing-induced features is possible in only ≲ 3% and ≲ 33% of detectable WDMS and NSMS events. The detection efficiencies for closer source stars with higher priorities are higher and drop to zero for b ≳ R ⋆ , where b ≃ tan( i ) a is the impact parameter( a is the semi-major axis). We predict the numbers of WDs, NSs, and SBHs that are discovered from the TESS Candidate Target List stars are 15-18, 6-7, and < 1. \nKeywords: Space telescope- Astronomical simulations - Eclipsing binary stars - Transient detection Stellar remnants - Compact binary stars - Gravitational lensing - Compact objects", '1. INTRODUCTION': "The NASA's Transiting Exoplanet Survey Satellite (TESS 1 , Ricker et al. (2014)) telescope observed (and observes) stars in the Candidate Target List (CTL, Stassun et al. (2018, 2019)) with a 2-min cadence and an accuracy better than 60 ppm on hourly timescales. Although its main goal from these observations is to discover Earth-size planets transiting bright stars in the solar neighborhood, its observing strategy is uniquely matched to capture other types of periodic and weak variations in stellar light curves. \nFor instance, stellar light curves from edge-on and detached binary systems including main-sequence stars and compact objects, i.e., white dwarfs (WDs), neutron \nstars (NSs), and stellar-mass black holes (SBHs), have potentially self-lensing, eclipsing (blocking the compact objects' brightness by their stellar companions), and occultation (blocking the light of stellar images by their compact companions) signals which all of them are periodic. Self-lensing refers to the lensing effect on the flux of the luminous object by its compact companion in edge-on systems (see, e.g., Gould 1995). In this regard, simulating and numerically studying selflensing/eclipsing/occultation features that are realizable in the TESS data have several advantages as (i) revealing the importance of searching these signals in the TESS data, (ii) learning the machines to extract the real events from a huge ensemble of the TESS data archive which is growing up with time, and (iii) evaluating the known models describing binary systems, NSs, \nand SBHs through comparing their results with ones from the real observations. \nIn this work, we therefore simulate these periodic features and study their properties and detectability in the TESS data. However, Masuda & Hotokezaka (2019) roughly evaluated the number of detectable BHs in the mass range [1 , 100] M ⊙ in tight and detached orbits around stars with the orbital period [0 . 3 , 30]days through either self-lensing signals or phase-curve variations of stellar light curves. They injected these photometric signals in spotted stellar light curves detected by the Kepler telescope and concluded that 10-100 BHs can be detected by searching 10 5 -10 6 stellar light curves. Also, Wiktorowicz et al. (2021) studied detection of BHs/NSs through their self-lensing signals in a synthetic ensemble of binary systems and reported a higher number for detectable BHs. Here, we extend their works by (i) considering binary systems including different types of compact objects (i.e., WDs, NSs, and BHs) separately, (ii) modeling eclipsing and occultation in addition to self-lensing signals, (iii) generating synthetic light curves and data points based on the real observations by the TESS telescope, and (iv) taking their companion stars from the TESS CTL targets. \nHere, we first review the known properties of binary systems including main-sequence stars and compact objects in three following paragraphs. \nWD main-sequence (WDMS) binaries: It is predicted that the number of WDMS binaries in our galaxy is 10 7 -10 8 which depends on their mass-ratio distribution. The number of detached WDMS binaries is more than that of interacting ones by more than one order of magnitude (e.g., see Willems & Kolb 2004), nevertheless detecting interacting WDMS binaries is easier and plausible via either spectroscopic observations or variability surveys. For instance, the Sloan Digital Sky Survey (SDSS, York et al. (2000)) telescope discovered more than 3200 WDMS binaries (see, e.g., Heller et al. 2009; Rebassa-Mansergas et al. 2016). Additionally, ∼ 320 , 000 (either candidate or confirmed) WDs in binary systems have been reported in the Galaxy Evolution Explorer (GALEX, Martin et al. (2005)) and the Gaia Data Release 3 (GDR3, Gaia Collaboration et al. (2023)) (Bianchi et al. 2011; Gentile Fusillo et al. 2021). For detecting detached WDMS binaries, a common method is spectroscopic observations because they can make a combined spectrum due to one star and one WD. If these systems are edge-on as seen by the observer, additionally eclipsing, self-lensing signals or variation in stellar radial velocities are realizable through precise photometric and spectroscopic observations (Korol et al. 2017). \nNS main-sequence (NSMS) binaries: Another type of compact object is an NS which could be a companion for a main-sequence star in a binary system. A NS is formed after the collapse of a massive star with an initial mass higher than 8 M ⊙ , whereas more massive stars with initial masses higher than ∼ 20 M ⊙ are converted to SBHs after their gravitational collapse. Isolated NSs can be discovered through radio emissions, the so-called pulsars (e.g., Zhang et al. 2022). NSs and SBHs in interacting binary systems with mainsequence stars, the so-called X -ray binaries, are discernible through X -ray emissions owing to mass transferring from their stellar counterparts. Depending on the dominant mechanism for mass transferring toward the compact objects, these binaries are divided into two subclasses: (i) low-mass X -ray binaries (with RocheLobe overflow), and (ii) high-mass X -ray binaries (with wind-accreting) (see, e.g., Ogelman & Swank 1974; Pfahl et al. 2002; Casares et al. 2017). Around 4% of all discovered NSs are in binary systems (Tauris & van den Heuvel 2006), and our galaxy hosts up to one billion NSs, whereas only ∼ 4 , 000 NSs have been discovered up to now. \nSBH main-sequence (BHMS) binaries: Black holes with masses ≲ 100 M ⊙ are the so-called stellarmass black holes. The lightest SBH was discovered up to now has the mass ≃ 3-3 . 3 M ⊙ which was located in the mass-gap between NSs and SBHs (Thompson et al. 2019; Jayasinghe et al. 2021; Ozel et al. 2010; Farr et al. 2011). Although the number of SBHs in our galaxy is predicted to be more than 10 million, up to now only 72 SBHs were confirmed mostly through X -ray transients from their accretion disks in interacting BHMS binaries 2 (Corral-Santana et al. 2016). 20 of these discovered SBHs were confirmed dynamically by discerning periodic variations in the radial velocities of their luminous companions. \nIn addition to spectroscopic and variability surveys, and X -ray observations to capture compact objects in binary systems with main-sequence stars, there are other channels for discovering these objects, e.g., (i) based on their gravitational impacts which is the so-called selflensing effect, (ii) eclipsing signals, (iii) ellipsoidal variation of stellar brightness, (iv) Doppler boosting, etc (e.g., see, Masuda & Hotokezaka 2019; Sorabella et al. 2022). Two last effects happen for massive compact companions and small orbital radii. Unlike other types of lensing, including microlensing, strong lensing and weak lensing which all are not repeatable, a self-lensing \nsignal is periodic and its period is exactly equal to the orbital period of the binary system. \nUp to now, five self-lensing/eclipsing binary systems containing white dwarfs and main-sequence stars were discovered through photometric observations by the Kepler telescope (Kruse & Agol 2014; Kawahara et al. 2018; Masuda et al. 2019). All of these systems are wide binaries with orbital periods from 88 to 728 days. The next generation of the Kepler telescope is the TESS telescope which was launched on 18 April 2018. During its two-year primary mission, it covered 85% of the sky by dividing it into 26 sectors. The area of each sector is 24 × 90 square degrees, and sectors have some overlapped parts over the ecliptic poles. Each sector is observed during two 13 . 7-day observing periods with a one-day gap in the middle. The TESS pixel scale is 21 arc-second which leads to a high photometric accuracy better than 60 ppm (parts per million) for brightest stars on hourly time scales. This telescope could also re-cover self-lensing signals in a binary system originally discovered by the Kepler telescope, i.e., KIC 12254688 (Sorabella et al. 2024). \nIn this work, we simulate possible self-lensing and eclipsing signals due to compact objects in stellar fluxes from detached binary systems as seen by the TESS telescope, to (i) study the characteristics of self-lensing signals and (ii) estimate the TESS efficiency for detecting them. We also investigate how this detection efficiency depends on the physical parameters of compact objects, source stars, and binary orbits. We finally estimate the numbers of WDs, NSs, and SBHs that are realizable through their self-lensing/eclipsing signals in the photometric data of the TESS CTL targets. \nThe outline of this paper is as follows. In Section 2, we explain the details of simulating self-lensing, occultation, and eclipsing signals and then discuss on their characteristics as a function of models' parameters. In Section 3, we explain the details of Monte Carlo simulations from self-lensing, occultation, and eclipsing light curves due to detached and edge-on binary systems while we assume that these events are observed by the TESS telescope. We extract the detectable events based on two sets of criteria. In Section 4, we explain the results and conclusions.", '2. BINARY SYSTEMS CONTAIN COMPACT OBJECTS AND MAIN-SEQUENCE STARS': 'To simulate a stellar light curve from a binary system including a compact object and a main-sequence star, in the first step, we simulate a binary orbit as explained in Subsection 2.1. By assuming that its orbital plane is edge-on as seen by the observer, in the next step we cal- \nculate its self-lensing, occultation, and eclipsing signals. All details are explained in Subsection 2.2. We then discuss the characteristics of self-lensing signals as functions of orbital properties and parameters of compact objects in Subsection 2.3.', '2.1. Simulating a Stellar Orbit Around a Compact Object': "Let's consider a detached binary system containing two companions: a star and a compact object with masses M ⋆ , and M c ( c can be either WD, or NS, or SBH). We assume this system is isolated and there is no external force. Hence, their center of mass (CM) moves with a constant velocity and both components rotate over elliptical orbits so that these orbits have a common barycenter on their CM's location. In the CM coordinate system, their CM is fixed and the star with respect to the compact object moves over an elliptical orbit. We assume that the semi-major axis of this elliptical orbit is a . The period of this elliptical orbit is given by the third Kepler's law as follows: \nT = 2 π √ G ( M ⋆ + M c ) a 3 / 2 . (1) \nThis elliptical orbit is characterized as (See, e.g., Dominik 1998): \nx = a ( cos ξ -ϵ ) , y = a sin ξ √ 1 -ϵ 2 , (2) \nwhere, ϵ is the orbital eccentricity. ξ , the so-called eccentric anomaly, is given by Kepler's equation, i.e., ϕ = ξ -ϵ sin ξ , where ϕ = ω ( t -t p ) is called the mean anomaly, and ω = 2 π / T is the angular velocity. Also, t p is a characteristic time which indicates the time of crossing the orbital periapsis point. Hence, ( x, y ) defines the orbital plane of the star around the compact object. In the simulation, we solve Kepler's equation numerically using: \nξ = ϕ + ∞ ∑ i =1 2 π sin( iϕ ) B i ( iϵ ) , (3) \nwhere, B i is the known Bessel function of i th order. \nWe define the observer's coordinate system, ( x o , y o , z o ), where x o is toward the observer, and ( y o , z o ) defines the sky plane. We then convert the CM coordinate system to the observer coordinate system using two projection angles, i.e., θ and i . θ is the angle between the minor axis of this elliptical orbit and the sky plane, and i , the so-called inclination angle, is the angle between the line of sight toward the observer and \nthe orbital plane. Hence, the orbital components in the observer coordinate system are given by: \nx o =cos i ( -y sin θ + x cos θ ) , y o = y cos θ + x sin θ, z o = -sin i ( -y sin θ + x cos θ ) . (4) \nFor stellar orbits with small inclination angles (the socalled edge-on ones) the stellar light is magnified by the compact object once an orbital period and whenever the star is passing behind it. We explain the details of calculating self-lensing, occultation and eclipsing signals in the following subsection.", '2.2. Simulating Self-Lensing, Occultation, and Eclipsing Signals': "In the stellar orbit around the compact object (the lens object), there is a phase angle Φ which is the angle between two lines of sight from the source star: one toward the compact object and another one toward the observer. It is calculated by cos Φ = -x o / D , where D = √ x 2 o + y 2 o + z 2 o is the distance between two components of the binary system versus time. When the source star is passing behind the compact component, this phase angle changes from 0 to 90 · , and when the source star is passing in front of the compact component this phase angle alters from 90 to 180 · . Hence, in the simulation when Φ < 90 · (or x o < 0), we calculate the lensing effect. \nThe lensing magnification factor depends on the source radius projected on the lens plane and normalized to the Einstein radius ρ ⋆ , and the lens-source distance projected on the lens plane and normalized to the Einstein radius u . In our formalism, these two parameters are given by \nρ ⋆ = R ⋆ D l R E ( D l -x o ) , R E = √ 4 GM c c 2 D l | x o | D l -x o , u = d p R E , d p = √ y 2 o + z 2 o , (5) \nwhere, D l is the distance of the compact object from the observer, R ⋆ is the radius of the luminous star, G is the gravitational constant, c is the light speed, and d p is the projected distance between the compact object and the source star on the sky plane. \nAnother factor which impacts the lensing magnification is the limb-darkening effect for stellar surface brightness. In the simulation, we consider a linear limb-darkening profile for the source star, as I = I 0 [ 1 -Γ(1 -µ ) ] , where I 0 is the stellar brightness at the center of the source disk, Γ is the so-called limb-darkening coefficient, and \nµ = √ 1 -R 2 / R 2 ⋆ , where R is the radial distance over the source disk. We determine the magnification factor, A ( u, ρ ⋆ , Γ ) , using the public RT-model (Bozza 2010; Bozza et al. 2018). We ignore the General Relativistic (GR) effects on the magnification factor because this effect is significant only for a very small part of the source disk which is exactly behind and collinear with the compact object as seen by the observer. \nIn self-lensing events, the Einstein radius (Eq. 5) is estimated using R E ≃ √ 4 GM c a/c , which is considerably smaller than the Einstein radius for common microlensing events toward the Galactic bulge. Hence, in self-lensing events, ρ ⋆ ≳ 1 and their magnification factors are estimated by the ratio of the area of the images' ring generated at the time of the complete alignment to the source area (e.g., Han 2016). The inner and outer radii of the images' ring are (respectively): \nR in = 1 2 [√ R 2 ⋆, p +4 R 2 E -R ⋆, p ] , R out = 1 2 [√ R 2 ⋆, p +4 R 2 E + R ⋆, p ] , (6) \nwhere, R ⋆, p = R ⋆ D l / ( D l -x o ) is the projected source radius on the lens plane. Accordingly, the magnification factor for a uniform source star during a self-lensing signal is given by (Maeder 1973; Gould & Gaucherel 1996; Agol 2003): \nA ≃ R 2 out -R 2 in R 2 ⋆, p ≃ 1 + 2 ρ 2 ⋆ . (7) \nWe note that when x o < 0 and during self-lensing signals, the compact object (the gravitational lens) can block some part of images' area, which is the socalled finite-lens effect or occultation of images' light by the compact object (see, e.g., Marsh 2001; Han 2016). Finite-lens effect is considerable in WDMS binaries. This effect decreases the magnification factor by O = R 2 c / R 2 ⋆, p when R in ≤ R c ≤ R out , and does not change the magnification factor ( O = 0) if R c < R in . Here, R c is the radius of the compact object. If the radius of the compact object is larger than the outer radius of the images' ring, the light of images is completely blocked by the compact object and O = A . \nWhenever x o > 0 (or Φ ∈ [90 , 180 · ]) the source star is passing in front of its compact companion and it can block its luminosity, the so-called eclipsing effect (when the compact objects are either WDs or NSs with the masses M c ≲ 2 . 9 M ⊙ ). According to our formalism, eclipsing features happen when x o > 0 and d p ≤ ( R c + R ⋆, p ) . We calculate the fraction of the compact object's disk eclipsed by the stellar compan- \nFigure 1. Maps of log 10 [∆ F L ] in detached and edge-on binary systems over the 2D space of log 10 [ M c ( M ⊙ )] -log 10 [ a ( R ⊙ )]. We consider four different source stars in binary systems include: (i) a red dwarf with M ⋆ = 0 . 3 M ⊙ and R ⋆ = 0 . 35 R ⊙ , (ii) a Sun-like star, (iii) a B-type star with M ⋆ = 2 M ⊙ , and R ⋆ = 1 . 8 R ⊙ , and (iv) a sub-giant star with M ⋆ = 2 M ⊙ , and R ⋆ = 3 . 6 R ⊙ . The contours display log 10 [ T (days)]. The interacting binaries with D RL < R ⋆ (see, Eq. 10) are excluded and covered with white color on maps. \n<!-- image --> \n, y o , z o ), numerically by: \nE ( x o , y o , z o ) = 1 πR 2 c ∫ R c -R c dy ' ∫ √ R 2 c -y ' 2 -√ R 2 c -y ' 2 dz ' Θ [ R ⋆, p d ' p ] , (8) \nwhere d ' p = √ ( y o -y ' ) 2 +( z o -z ' ) 2 is the distance of each element over the compact object's disk ( y ' , z ' ) from the source center projected on the sky plane. Θ is a step function which is one if its argument is larger than one and it is zero when the argument is less than one. We note that the whole disk of the compact object is eclipsed (i.e., E = 1) if d p ≤ ( R ⋆, p -R c ), and there is no eclipsing ( E = 0) when either x o < 0, or x o > 0 and d p ≥ ( R c + R ⋆, p ). \nThe apparent magnitude of the source star by considering self-lensing, occultation (or finite-lens effect), and eclipsing signals in edge-on binary systems as measured by the observer is given by: \nm o = m ⋆ -2 . 5 log 10 [ f b A ( u, ρ ⋆ , Γ) -O + FE 1 + F +1 -f b ] (9) \nwhere, F is the ratio of the compact object's flux to the stellar flux, and f b is the blending factor which is the ratio of the source flux to the total flux received from the source star PSF (Point Spread Function), m ⋆ is the apparent magnitude of source star when it is isolated and without any companion.", '2.3. Characteristics of Self-Lensing Signals': "Here, we evaluate the peak amounts in self-lensing signals, i.e., ∆ F L ≃ 2 ρ -2 ⋆ , and consider four types of source stars: (i) a red dwarf with M ⋆ = 0 . 3 M ⊙ , and R ⋆ = 0 . 35 R ⊙ , (ii) a Sun-like star, (iii) a B-star with M ⋆ = 2 M ⊙ , and R ⋆ = 1 . 8 R ⊙ , and (iv) a sub-giant star with M ⋆ = 2 M ⊙ , and R ⋆ = 3 . 6 R ⊙ , as well as two wide ranges for the compact object's mass and the orbital semi-major axis as M c ( M ⊙ ) ∈ [0 . 1 , 50], and a ( R ⊙ ) ∈ [5 , 5000], respectively. For all of these binary systems, we calculate ∆ F L values and show their maps in Figure 1. \n<!-- image --> \n<!-- image --> \n0, O(deg) = 0, \n<!-- image --> \nMc \n<!-- image --> \n<!-- image --> \nFigure 2. Simulated self-lensing signals versus time (normalized to the orbital period) by considering different values for M c ( M ⊙ ), ϵ , i , R ⋆ ( R ⊙ ), θ , and Γ. For each panel, several parameters are fixed and reported at the top of that panel. The vertical axis is the flux of the source star normalized to its baseline value. \n<!-- image --> \nOver these maps, the contours of log 10 [ T (days)] are shown with black solid and dashed lines (corresponding to positive and negative values, respectively). In these plots, binary systems with source radii larger than the Roche-Lobe distance are excluded and covered with white colour on maps. For these systems, the mass transfers from the bright star to the compact object and there are some other sources for variability (e.g., ellipsoidal effect) in stellar fluxes which makes discerning self-lensing signals hard and impossible. Here, we determine the Roche-Lobe distance from the source center as (Paczy'nski 1971; Eggleton 1983): \nD RL = a -a 0 . 49 q 2 / 3 0 . 6 q 2 / 3 +log 10 [ 1 + q 1 / 3 ] , (10) \nwhere, q = M c /M ⋆ . According to these plots, generally more massive compact objects in wider orbits (with larger semi-major axes) have larger lensing-induced signals, so that increasing (a) the mass of compact objects from 0 . 1 M ⊙ to 50 M ⊙ , and (b) the semi-major axis by three orders of magnitude (from 5 R ⊙ to 5000 R ⊙ ) enlarge ∆ F L by ∼ 5 orders of magnitude. However, more massive compact objects in wider orbits have longer orbital periods. For that reason, less massive source stars (e.g., red dwarfs in comparison to B-stars) are more suitable for detecting their self-lensing signals generated by their compact companions because (i) they generate shorter orbital periods, and (ii) they have smaller radii, smaller ρ ⋆ values, and as a result higher ∆ F L values. We probe this point again in the next section and through Monte Carlo simulations. \nGenerally, peaks, shapes and durations of self-lensing signals determine their detectability. They depend on several parameters including M c , ϵ , i , R ⋆ , θ , and Γ. We therefore simulate self-lensing signals by considering different values of these parameters as shown in different panels of Figure 2. In each panel, one parameter changes and other parameters are fixed and mentioned at the top of that panel. Accordingly, we summarize some points in follows. \n- · The maximum enhancement in self-lensing signals increases with the lens mass as ∆ F L ∝ M c . The selflensing signals due to completely edge-on orbits are flattened (top-hat models).\n- · There is a degeneracy between M c and R ⋆ , so that small stellar radii make similar self-lensing signals to those due to more massive compact objects. Both of these parameters change the peaks of self-lensing from top-hat ones (but they do not change the self-lensing edges). \n- · If the stellar orbits are eccentric, the resulting selflensing signals are asymmetric, unless the source star is passing from either periapsis or apoapsis point of its orbit while lensing (which are rare). This point can be found from the fifth panel of Figure 2.\n- · The inclination angle of stellar orbit with respect to the line of sight is the only factor which causes that self-lensing signals at the edges are not broken (not a strict top-hat model). By increasing the inclination angle self-lensing signals at edges are rather slowenhancing.\n- · In more eccentric stellar orbits, the resulting selflensing signals could be wider depending on the sourcelens distance while lensing (in our formalism this distance is determined by θ ).\n- · The limb-darkening effect has a very small impact on the width of self-lensing signals, whereas it changes the peak and shape of self-lensing signals. \nIn the next step, we perform Monte Carlo simulations from all possible self-lensing signals due to different binary systems (including main-sequence stars and compact objects) and study their detectability in the TESS data.", '3. MONTE CARLO SIMULATIONS': "To simulate potential self-lensing, occultation, and eclipsing signals that can be detected in the TESS observations, we first take an ensemble of the Candidate Target List (CTL, Stassun et al. (2018, 2019)) from the Mikulski Archive for Space Telescopes (MAST) catalog (STScl 2022). The TESS CTL targets are relatively close and bright stars with pre-measured physical parameters which were (and are) observed by the TESS telescope with a 2-min cadence, based on their priority. In this ensemble, for CTL targets their mass M ⋆ , priority, blending factor f b , distance D l , source radius R ⋆ , effective surface temperature T ⋆ , and stellar apparent magnitude in the TESS filter m ⋆ are reported which all are used for simulating source stars in binary systems. \nWe assume the TESS CTL targets live in detached binaries with compact companions. We take 1772 known WDs which were extracted from the SDSS data with the reported mass M WD , distance, radius R WD , and apparent magnitudes in the Gaia passbands, i.e., m G , m G BP , and m G RP at the distances closer than 100 parsec from Kilic et al. (2020). Using the same method which was explained in Sajadian et al. (2024), we convert their apparent magnitudes in the Gaia filters to their absolute magnitudes in the TESS passband. \nFigure 3. Six examples of simulated stellar light curves due to detached and edge-on binary WDMS, NSMS, and BHMS systems. The synthetic data points are hypothetically taken by the TESS telescope. We fix the observational time to 27 . 4 days with a one-day gap in middle. The parameters used to make light curves are mentioned at the tops of panels. By applying HC detectability criteria, for three of these light curves the impacts of compact objects are realizable, as mentioned in their legends. \n<!-- image --> \nThe numbers of discovered NSs and SBHs up to now are low. Additionally, all necessary parameters for discovered ones were not measured. Therefore, in Monte Carlo simulations we generate their populations synthetically according to the known distribution functions for their physical parameters. \nThe mass distribution function of NSs is a bimodal distribution with two peaks at 1 . 37 M ⊙ , and 1 . 73 M ⊙ whereas the second one is wider and more flattened (Valentim et al. 2011). So, we determine the mass of NSs using N ( 1 . 37 , 0 . 15 ) +0 . 5 N ( 1 . 73 , 0 . 3 ) from the range M NS ∈ [0 . 5 , 2 . 9] M ⊙ , where N ( µ, σ ) is a normal function with the mean value µ , and the width σ . The radius of NSs is a function of their mass and is in the range R NS ∈ [10 , 15] km (Lattimer 2019). The luminosity of NSs is a function of their ages and masses, because these objects cool down over time through thermal radiation. We determine the NSs' luminosity based on Figure (2) of Potekhin et al. (2020). We choose the NSs' ages ( A ) in the logarithmic scale uniformly from the range log 10 [ A (year)] ∈ [2 . 5 , 8]. For SBHs we choose their masses uniformly from the range M BH ∈ [3 . 3 , 50] M ⊙ (Sicilia et al. 2022; Sajadian & Sahu 2023). \nIn each step of Monte Carlo simulations, we take one CTL target and one compact object as two components of a detached binary system. We take the semi-major axis of their orbit a from a log-uniform distribution and in the range [3 R ⋆ , 10 6 R ⋆ ] (see, e.g., Abt 1983), and put aside the interacting systems with the source radii larger than the Roche-Lobe distance ( D RL which is given by Eq. 10). \nThe known period-eccentricity correlation indicates an upper limit on the orbital eccentricity ( ϵ max ) for a given orbital period T (Mazeh 2008). By considering this correlation, we uniformly choose the orbital eccentricity from the range ϵ ∈ [0 , ϵ max ]. We choose the inclination angle uniformly from the range i ∈ [0 , 20 · ], because for larger inclination angles neither self-lensing/occultation nor eclipsing signals happen. We choose the projection angle θ uniformly from the range [0 , 360 · ]. \nIn the next step, we generate synthetic data points taken by the TESS telescope. We fix the observing cadence to 2 minutes, and consider a one-day gap after each 13 . 7-day observation. The observing time span is calculated by T obs = No s × 27 . 4 days, where No s is the number of overlapping sectors and can be an integer number from one to thirteen. Therefore, the longest observing time is ∼ 360 days for the ecliptic poles during one year. \nTo examine their detectability we calculate the signalto-noise ratio (SNR), which is defined for planetary transit events, as given by (see, e.g., Fatheddin & Sajadian \n2024): \nSNR = √ N tran ∆F × 10 6 CDPP , (11) \nwhere, CDPP is the Combined Differential Photometric Precision (CDPP) metric which is a function of the stellar apparent magnitude in the TESS passband, N tran = T obs / T is the number of orbital period during an observing time span. ∆ F is the maximum variation in the stellar flux which is due to either selflensing/occultation or eclipsing signals (the maximum of ∆ F L , ∆ F O , and ∆ F E ). In the simulation, to extract the detectable events we consider two sets of criteria which are (i) SNR > 5, and N tran > 2, and (ii) SNR > 3, and N tran > 1. The first set extracts detectable events with a high confidence (HC), and the second one takes detectable events with a low confidence(LC). \nIn Figure 3, we show six examples of simulated stellar light curves due to edge-on WDMS, NSMS, BHMS systems (with i ≤ 20 · ). Some of useful parameters to make light curves and their SNR values are reported at tops of plots. The magenta synthetic data points are taken by the TESS telescope with a 2-min cadence. Accordingly, three events are detectable (with a high confidence) in the TESS data and three others are not detectable.", '4. RESULTS: STATISTICS AND PROPERTIES': "We assume the maximum observational time span for a part of the sky during the TESS mission is 360 days. Although, the southern (northern) ecliptic hemisphere was re-observed in the third (forth and fifth) year(s) of the TESS mission again, discerning periodic variations with T (day) ≥ 360 (longer than the maximum value for the continuous TESS observing time span from a part of the sky) in stellar light curves is barely possible. Because, (i) there is a 1- up to 2-year gap in the middle, and (ii) 74% of stars are observed during only 27 . 4 days of a year (they are inside one sector) and the probability of occurring either self-lensing/occultation or eclipsing signals (when the orbital period is long) exactly during that 27 . 4-day observing time is low. We therefore simulate synthetic data points for the events with T < 360 days, which means we assume all events with T ≥ 360 days are not detectable in the TESS observations. \nWe perform three Monte Carlo simulations from WDMS, NSMS, and BHMS binary systems by considering different observing time spans, which can be from \nTable 1. The average parameters of the simulated WDMS binaries which have detectable WD-induced signals with a high confidence in their stellar light curves by considering different observing time spans. \nNote -No s is the number of overlapping sectors. ε HC , and ε LC are the efficiencies for detecting WD-induced signals with high and low confidences, respectively. \nTable 2. Same as Table 1, but for simulated NSMS binaries. \n27 . 4 days to 356 days due to different numbers of overlapping sectors. In Table 1, the results from the Monte Carlo simulation of WDMS binary systems are reported. This table includes the average values of some orbital and WDs parameters (including M WD ( M ⊙ ), T (days), log 10 [ a/R ⋆ ], ϵ , i (deg), log 10 [ F ], D l (kpc), SNR, log 10 [ ρ ⋆ ], and b ( R ⋆ )) for HC detectable events during different observing time spans (given in the second column). \nHere, b is the impact parameter in the lensing formalism, which is the minimum value of the projected distance d p . \nIf this parameter is less than the source radius, eclipsing/occultation signals certainly happen. Also, this parameter determines the maximum magnification factor. The thirteenth column of this table determines (i) the fraction of simulated events which are detectable because of their self-lensing signals, f L , (ii) the fraction of simulated events which are detectable owing to their eclipsing signals f E , and (iii) the fraction of ones which are detectable due to their occultation (finite-lens effect) signals f O . Two last columns report the detection \nTable 3. Same as Tables 1 and 2, but for simulated BHMS binaries. \nefficiencies ε HC [%] and ε LC [%] which are the ratio of detectable events (with high and low confidences, respectively) to total simulated events. \nThe results from Monte Carlo simulations of detached binary systems including main-sequence stars and either NSs or SBHs are reported in Tables 2, and 3, respectively. We note that for BHMS binary systems f E = 0, f O = 0, and F = 0. \nGenerally, longer observing time windows have two positive effects on the detectability of compact objects' signatures. For longer observing time windows, the number of transits N tran is higher which (i) increases SNR values (see Eq. 11), and (ii) enhances N tran (the second detectability criterion). Enhancing N tran (and accordingly SNR) for longer observing times is beneficial for detecting fainter compact objects (WDs and NSs which could be even farther), in wider and more eccentric orbits. For that reason in these three tables for longer observing times detectable compact objects are on average fainter (with less log 10 [ F ]) and farther. We note that due to applying the eccentricity-orbital period correlation in Monte Carlo simulations binary systems with longer orbital periods are on average more eccentric. By increasing the observing time from 27 . 4 days to 356 . 2 days, the detection efficiency improves by ∼ 2. \nFor ≲ 3% and ≲ 33% of detectable WDMS and NSMS binary systems, self-lensing signals are the mostdominant ones in stellar light curves. Indeed, for these detached WDMS and NSMS binary systems on average ρ ⋆ ∼ 145 , 100 which make very flattened self-lensing signals with the depths ∆ F L ∼ 2 ρ -2 ⋆ ∼ 10 -4 , 2 × 10 -4 , respectively, whereas their eclipsing signals, as given by ∆ F E ∼ F / (1 + F ) ∼ 1-2 × 10 -3 , 10 -5 -5 × 10 -4 , respec- \nor detectable WDMS and NSMS binary systems we have ∆ F L ≲ ∆ F E . The occultation signal in WDMS binary systems is on average ∆ F O ∼ R 2 c /R 2 ⋆ ∼ 10 -4 which is in the same order of magnitude with the self-lensing ones. Here, we consider a common WD with the radius R WD ∼ 0 . 01 R ⊙ . \nTo study what kinds of binary systems and compact objects are more detectable in the TESS observations, in Figure 4 we show the efficiency curves for detecting the impacts of compact objects with a high confidence, ε HC , in simulated stellar light curves versus nine parameters, which are D l (kpc), log 10 [Periority], log 10 [ a/R ⋆ ], M ⋆ ( M ⊙ ), log 10 [ ρ ⋆ ], i (deg), log 10 [ b/R ⋆ ], M NS ( M ⊙ ), and M BH ( M ⊙ ). We consider three amounts for T obs as mentioned in the first panel. \nAccording to these plots, we conclude that closer source stars which have more priorities and higher detection efficiencies. Indeed, WDs and NSs in nearby binary systems are brighter with on average higher F values. Higher F values make deeper eclipsing signals. \nClose binary systems with smaller semi-major axes have on average shorter orbital periods, which are more suitable to be detected. Because short-period binary systems have higher N tran and higher SNR values. We note that ρ ⋆ ∝ a -1 / 2 . Therefore, although the closer binary systems have shorter orbital periods (and higher SNR values), they have larger ρ ⋆ s, and as a result more flattened self-lensing signals. Hence, during longer observing times wider binary systems can be detected rather via self-lensing ( f L increases with T obs in Tables 1, and 2). \nThe impact of stellar masses on the self-lensing signals has been shown in Figure 1. Accordingly, less massive \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 4. The HC efficiencies for detecting signals due to compact objects in stellar light curves (in the logarithmic scale log 10 [ ε HC (%)]) versus nine parameters which are D l (kpc), log 10 [Periority], log 10 [ a/R ⋆ ], M ⋆ ( M ⊙ ), log 10 [ ρ ⋆ ], i · , log 10 [ b/R ⋆ ], M NS ( M ⊙ ), and M BH ( M ⊙ ) as shown in different panels. Three efficiency curves with different line-styles are due to three values for T obs as mentioned in the first panel. \n<!-- image --> \nstars (with smaller radii) have higher self-lensing signals. Also, lower mass stars are on average fainter which results in higher F and deeper eclipsing signals. \nStellar orbits with fewer inclination angles (more edgeon ones) have higher detection efficiencies because by decreasing the inclination angle the impact parameter reduces as well. The eclipsing/occultation signals can be detected only in binary systems with impact parameters less than source radii. The detection efficiency drops from ∼ 4 -5% to ≲ 1% when the inclination angle increases from 0 to 15 degrees. \nWe also plot the detection efficiencies as a function of the mass of NSs and SBHs in two last panels of Figure 4. We note that the mass range for WDs is small, and the detection efficiency does not highly change with the mass of WDs. For NSs, we determine their luminosity according to their mass and age (based on Fig. (2) of Potekhin et al. 2020). Accordingly, by increas- \ning the mass of NSs by ∼ 1 M ⊙ the luminosity of NSs decreases up to three orders of magnitude (when they are younger than 3 million years). Hence, on average less massive NSs are brighter with higher F values, and deeper eclipsing signals. For detached SBHs, the only method to detect them is self-lensing. In self-lensing formalism, we have ∆ F L ∼ 2 ρ -2 ⋆ ∝ M BH . Therefore, more massive SBHs make higher self-lensing signals with higher SNR values. \nTop panel of Figure 5 shows the scatter plot of simulated WDMS binaries in 2D space log 10 [ F ] -log 10 [ b/R ⋆ ] with black circles, with two marginal and normalized 1D distributions. The binary systems with detectable WDinduced impacts due to eclipsing, self-lensing, and occultation effects are specified with red, blue, and green circles, respectively. Accordingly, most of WDMS binaries with detectable WDs impacts have b ≲ R ⋆ and F ≳ 10 -4 . The blue points (with detectable self-lensing \nFigure 5. Top panel: The scatter plot of all simulated WDMS binaries (black points) and the ones with detectable WD-induced impacts (colored points) in 2D space log 10 [ F ] -log 10 [ b/R ⋆ ]. Their 1D and normalized distributions are shown at two sides of the plot. The red, blue, and green circles represent detectable events owing to their eclipsing, self-lensing, and occultation signals, respectively. Middle panel: Same as the previous one but resulted from the Monte Carlo simulation of NSMS binaries. Last panel: Same as two previous ones but resulted from simulating BHMS binaries and in 2D space log 10 [ T (days)] -log 10 [b / R ⋆ ]. \n<!-- image --> \nsignals) have lower F up to 10 -5 . We therefore expect in the TESS data eclipsing-induced footprints due to WDs to be more realizable than their self-lensing/occultation impacts. \nThe next panel of Figure 5 shows a scatter plot the same as the previous one but for NSMS binary systems. Considering this point that we chose the ages of NSs uniformly from the range log 10 [ A (year)] ∈ [2 . 5 , 8], several NSs in our simulation are cool with very low F values (ones older than ∼ 1 million years). For these systems with F ≲ 10 -4 , only lensing-induced impacts are detectable (and when b ≲ R ⋆ ). For brighter (younger and hotter) NSs, eclipsing signals are detectable in the events with b ≲ R ⋆ . We note that due to this considerable number of dim NSs in our simulation, only f L ∼ 15-33% of detectable events have dominant self-lensing signals. \nThe last panel of Figure 5 shows the same plot as ones displayed in previous panels but for BHMS binary systems in 2D space log 10 [ T (days)] -log 10 [b / R ⋆ ]. The most important factor for the detectability of BHinduced lensing signals is the impact parameter b . Most of the simulated events with b ≲ R ⋆ have detectable self-lensing signals. We determine the impact parameter numerically from the simulation (the minimum value of d p ), nevertheless it can be estimated as b ≃ tan( i ) a . Therefore, BHMS binary systems with smaller semimajor axes have smaller b and shorter orbital periods which both impacts are beneficial for detecting SBHs through self-lensing signals. \nHere, we estimate the number of WDs that the TESS telescope can detect through precise photometric observations from the CTL targets with a 2-min cadence during its mission. The TESS telescope is planned to detect 1 , 390 , 486 CTL targets during its mission 3 (STScl 2022). We calculated the fractions of these CTL targets which are observed during different T obs due to different No s values, i.e., N ⋆ ( No s ), as reported in the seventh column of Table (2) of Sajadian et al. (2024) and extracted from Fig (2) of Barclay et al. (2018). These numbers are N ⋆ ( No s ) = (1031 , 258, 38, 9, 5, 5, 3, 2, 1, 1, 1, 16, 13) × 1000. \nThe number of WDMS binaries in our galaxy is predicted to be ∼ 100 million. Considering the total number of stars in our galaxy (which is 100 billion), one thousandth ( f 1 ) of the TESS CTL targets should be in binary systems with WDs. We note that in the simulation we limit the inclination angle to i ∈ [0 , 20 · ], whereas f 2 = 22% of all binary systems should have i ≤ 20 · . We assume that the inclination angle of the \norbital plane is uniformly in the range [0 , 90 · ]. By assuming that the TESS CTL targets make a common sample of stars, the number of WDs detectable in the TESS observations from the CTL targets can be estimated by: \nN WD ≃ f 1 × f 2 × 13 ∑ No s =1 N ⋆ ( No s ) × ε i ( No s ) , (12) \nwhere, values of ε i ( No s ) (here, i = HC, and LC) are given in two last columns of Table 1. The estimated numbers of detectable WDs for different observing time windows and two levels of confidence are reported in two first rows of Table 4. \nOverly, we expect ∼ 15 , 18 WDs in detached WDMS systems can be realized with high and low confidences through the TESS photometric data from the CTL targets. We note that some of these stars were (and will be) observed more than one year (two or three times) during the TESS mission. For instance, the TESS telescope observed each ecliptic hemisphere two times up to now. But, in the Monte Carlo simulations, we consider the TESS data for each CTL target taken for up to one year. Hence, these numbers are underestimations and by considering all data some more WDs in detached orbits around main-sequence stars with days will be discovered. \nIn the same way, we estimate the numbers of NSs and SBHs expected to be discovered through detecting their self-lensing and eclipsing signals in the TESS CTL's light curves which are reported in Table 4. The fractions of the TESS CTL targets that are in binary systems with NSs and SBHs are f 1 ≃ 4 × 10 -4 , 4 × 10 -6 , respectively. Here, we assume the binarity fractions of NSs, and SBHs (with main-sequence stars) are 4% (Tauris & van den Heuvel 2006). The total numbers of NSs and SBHs will be discovered with two levels of confidence through photometric observations of the TESS CTL targets are ∼ 6 , 7, and less than one, respectively. \nTherefore, changing detectability criteria from a high confidence to a low one does not alter the numbers of detectable compact objects significantly. Because, according to Tables 1, 2, and 3, HC and LC efficiencies for detecting impacts of compact objects in WDMS, NSMS, and BHMS systems are close to each other, and ε HC = 4 . 6 -7 . 0% , 4 . 5 -6 . 4% , 4 . 2 -5 . 3%, and ε LC = 5 . 6 -7 . 8% , 5 . 6 -7 . 0% , 4 . 3 -5 . 8%. In fact, the fractions of simulated WDMS, NSMS, and BHMS binary systems with T < 360 days are 35 . 3%, 37 . 3%, and 42 . 8%, respectively, and the corresponding fractions for systems with T < 180 days are 32 . 1%, 33 . 2%, and \n39 . 2%. Therefore, more than ∼ 60% of simulated events have T > 360 days, and the numbers of binary systems with orbital periods ∈ [180 , 360] days are not significant. \nAlthough the number of SBHs that can be detected through self-lensing is not promising, we expect ∼ 1518 WDs to be detected through their signatures in light curves of the TESS CTL targets. This number is three times larger than the number of WDs discovered from the Kepler data.", '5. CONCLUSIONS': "The TESS telescope observed (and observes) the CTL targets with a 2-min cadence and a great accuracy. Although its main goal from these observations is discovering Earth-size planets transiting bright stars in the solar neighborhood, its observing strategy is uniquely matched to capture any other types of periodic and weak variations in stellar light curves. Stellar light curves from edge-on and detached binary systems including main-sequence stars and compact objects have potentially self-lensing/occultation/eclipsing signals which all are periodic. Considering three types of compact objects (i.e., WDs, NSs, and SBHs), these binary systems usually are denoted by WDMS, NSMS, and BHMS, respectively. In this work, we studied statistics and properties of WDMS, NSMS, and BHMS binary systems with detectable signatures due to compact objects in the TESS observations of the CTL targets. \nWe first modeled self-lensing signals due to detached and edge-on binary systems including main-sequence stars and compact objects. The self-lensing peak, which can be estimated by ∆ F L ∼ 2 ρ -2 ⋆ , is higher for low-mass and small stars. A self-lensing signal for a red-dwarf star with R ⋆ = 0 . 35 R ⊙ is higher (two orders of magnitude) than that for a sub-giant star with R ⋆ = 3 . 6 R ⊙ . Peaks of self-lensing signals are degenerate functions of two parameters (i) the mass of the compact object and (ii) the source radius. Increasing the mass of compact objects and decreasing the source radius have the same effects on peaks of self-lensing signals. The inclination angle of stellar orbits around the compact objects is the only parameter which changes the shape of self-lensing signals from strict top-hat models to ones with slow-increasing edges. Self-lensing signals from eccentric stellar orbits are asymmetric concerning their peaks unless the source star is passing from either apoapsis or periapsis point while lensing. \nWe performed Monte-Carlo simulations from all possible stellar light curves due to detached and edge-on binary WDMS, NSMS, and BHMS systems, and assumed they are observed by the TESS telescope with a 2-min cadence. We chose source stars in these simu- \n. \nTable 4. Estimated numbers of WDs, NSs, and SBHs with detectable impacts in the TESS CTL's light curves, by considering different observing time spans, and two levels of confidence (HC, LC). \nlations from the TESS CTL, WDs from an ensemble of 1772 discovered nearby ones through the SDSS observations, NSs, and SBHs from their known distributions. We made synthetic data points according to the TESS observing strategy for the CTL targets and extracted the ones with the detectable signatures due to compact objects based on two sets of criteria (i) SNR > 5 and N tran > 2, i.e., detecting with a high confidence(HC), and (ii) SNR > 3 and N tran > 1, i.e., detecting with a low confidence (LC). \nThere are two issues for detecting (at least one) selflensing or eclipsing signal with the period longer than 360 days from the TESS data which are: (i) the longest (continuous) observing time span for a part of the sky is 360 days (happens for the ecliptic poles), and there is a gap (from one up to two years depending on the locations) between one-year missions, and (ii) around 74% of the CTL targets are observed during only 27 . 4 days of a one-year mission. We therefore assumed that the TESS detection efficiency for T ≥ 360 days is zero, although it is actually close to zero. \nWe found that the probability that simulated binary WDMB, NSMS, and BHMS systems have T < 180(360) days were 32 . 1(35 . 3)%, 33 . 2(37 . 3)%, and 39 . 2(42 . 8)%, respectively. The HC (and LC) efficiencies for detecting periodic signatures from WDMS, NSMS, and BHMS binaries during different observing time spans of a one-year mission of TESS, (i.e., T obs = 27 . 4 × No s , where No s = 1 , 2 , 3 , ..., 13) are 4 . 6 -7 . 0(5 . 6 -7 . 8)%, 4 . 5 -6 . 4(5 . 6 -7 . 0)%, and 4 . 2 -5 . 3(4 . 3 -5 . 8)%, respectively. Increasing observing time spans improves the detection efficiency by ∼ 2% and fainter compact objects in wider and more eccentric orbits can be detected during longer observing time spans (see Tables 1, 2, and 3). The fractions of detectable WDMS and NSMS bi- \ny events which their self-lensing signals are the mostdominant ones are ≲ 3%, and ≲ 33%, respectively. \nWe found that the detection efficiency is higher for closer CTL targets with higher priorities and smaller radii. Most of binary systems with detectable periodic signals have the impact parameters b ≲ R ⋆ . The detection efficiencies for detecting more massive NSs, and SBHs is higher. \nWe estimated the total number of WDs, NSs, and SBHs that can be discovered from the TESS CTL observations which are 15 -18, 6 -7, and less than one, respectively. The number of detectable WDs in the TESS data is three times higher than the number of WDs discovered in the Kepler data. \nAll simulations that have been done for this paper are available at: https://github.com/SSajadian54/ SelfLensing Eclipsing simulator. The codes, and several examples of generated light curves can be found in the Zenodo repository(sajadian 2024). \nIn Monte-Carlo simulations, we use the TESS CTL (with DOI number: doi:10.17909/fwdt-2x66) that are publicly available from the MAST catalog. Funding for the TESS mission is provided by NASA's Science Mission directorate. We acknowledge the use of TESS Alert data, which is currently in a beta test phase, from pipelines at the TESS Science Office and at the TESS Science Processing Operations Center. NA is supported by the University of Waterloo, Natural Sciences and Engineering Research Council of Canada (NSERC) and the Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Colleges and Universities.", 'REFERENCES': 'STScl. 2022, TESS Input Catalog and Candidate Target List, Version: 8.2, MAST, doi: 10.17909/fwdt-2x66 \n- Tauris, T. M., & van den Heuvel, E. P. J. 2006, in Compact stellar X-ray sources, Vol. 39, 623-665, doi: 10.48550/arXiv.astro-ph/0303456\n- Thompson, T. A., Kochanek, C. S., Stanek, K. Z., et al. 2019, Science, 366, 637, doi: 10.1126/science.aau4005 \nValentim, R., Rangel, E., & Horvath, J. E. 2011, MNRAS, 414, 1427, doi: 10.1111/j.1365-2966.2011.18477.x \nWiktorowicz, G., Middleton, M., Khan, N., et al. 2021, MNRAS, 507, 374, doi: 10.1093/mnras/stab2135 Willems, B., & Kolb, U. 2004, A&A, 419, 1057, doi: 10.1051/0004-6361:20040085 York, D. G., Adelman, J., Anderson, John E., J., et al. 2000, AJ, 120, 1579, doi: 10.1086/301513 \n- Zhang, L., Ridolfi, A., Blumer, H., et al. 2022, ApJL, 934, L21, doi: 10.3847/2041-8213/ac81c3'}

2024arXiv240907633A

We investigate Maxwellscalar models on radially symmetric spacetimes in which the gauge and scalar fields are coupled via the electric permittivity. We find the conditions that allow for the presence of minimum energy configurations. In this formalism the charge density must be written exclusively in terms of the components of the metric tensor and the scalar field is governed by firstorder equations. We also find a manner to map the aforementioned equation into the corresponding one associated to kinks in 11 spacetime dimensions so we get analytical solutions for three specific spacetimes. We then calculate the energy density and show that the energy is finite. The stability of the solutions against contractions and dilations following Derricks argument and around small fluctuations in the fields is also investigated. In this direction we show that the solutions obeying the firstorder framework are stable.

2024-09-01T00:00:00Z

['2024arXiv240907633A', 'arXiv:2409.07633', '10.48550/arXiv.2409.07633']

['General Relativity and Quantum Cosmology', 'High Energy Physics - Theory']

Analytical solutions for Maxwellscalar system on radially symmetric spacetimes

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.07633.pdf

{'Analytical solutions for Maxwell-scalar system on radially symmetric spacetimes': "I. Andrade , 1, ∗ D. Bazeia , 1, † M.A. Marques , 2, ‡ R. Menezes , 3, 4, § and G.J. Olmo 5, 6, ¶ \n1 \nDepartamento de F'ısica, Universidade Federal da Para'ıba, 58051-970 Jo˜ao Pessoa, PB, Brazil \n- 3 Departamento de Ciˆencias Exatas, Universidade Federal da Para'ıba, 58297-000 Rio Tinto, PB, Brazil \nDepartamento de F'ısica, Universidade Federal de Campina Grande, 58109-970 Campina Grande, PB, Brazil \n5 \nDepartament de F'ısica Te'orica and IFIC, Centro Mixto Universitat de \nVal'encia - CSIC. Universitat de Val'encia, Burjassot-46100, Valencia, Spain \n6 Universidade Federal do Cear'a (UFC), Departamento de F'ısica, \nCampus do Pici, Fortaleza, CE, 60455-760, Brazil \nWe investigate Maxwell-scalar models on radially symmetric spacetimes in which the gauge and scalar fields are coupled via the electric permittivity. We find the conditions that allow for the presence of minimum energy configurations. In this formalism, the charge density must be written exclusively in terms of the components of the metric tensor and the scalar field is governed by firstorder equations. We also find a manner to map the aforementioned equation into the corresponding one associated to kinks in (1 , 1) spacetime dimensions, so we get analytical solutions for three specific spacetimes. We then calculate the energy density and show that the energy is finite. The stability of the solutions against contractions and dilations, following Derrick's argument, and around small fluctuations in the fields is also investigated. In this direction, we show that the solutions obeying the first-order framework are stable.", 'I. INTRODUCTION': "Scalar field models are of interest in several branches of Physics, such as in High Energy Physics [1], Condensed Matter [2] and Cosmology [3, 4]. In particular, they can be used in the study of localized structures, such as kinks, vortices and monopoles [5]. Kinks are the simplest ones and arise under the canonical action of a single real scalar field in (1 , 1) flat spacetime dimensions [6]. These objects are stable around small fluctuations and minimize the energy of the system in the lines of the so-called BPS procedure [7, 8]. \nThe study of localized structures in canonical models with a single real scalar field only leads to stable configurations in (1 , 1) flat spacetime dimensions due to the Derrick-Hobart argument [9, 10]. There are some ways to evade this restriction. For instance, one can consider a potential with explicit dependence on the radial coordinate [11], non-canonical models Refs. [12] or complex scalar fields coupled to gauge fields [13-15]. In particular, the procedure introduced in Ref. [11] allowed for the presence of stable (or at least metastable) radially symmetric solutions in ( d, 1) flat spacetimes. \nThe idea in [11] was further extended in Ref. [16], where a class of noncanonical potentials was introduced so one may obtain radially symmetric solutions of the BPS equations on distinct geometric backgrounds, such as the Schwarzschild, cosmic string and wormhole spacetimes. The evasion of Derrick's theorem was widely con- \nsidered in the literature to study scalar field models on curved spacetimes; see Refs. [17-24]. In particular, it was also used to establish the non-existence of self-gravitating solitons in Maxwell-scalar models in curved spacetimes [17] and to investigate scalar field solutions on Lifshitz spacetimes [19, 20] and hyperscaling violating geometries [24]. \nLocalized structures can also be investigated within the so-called Maxwell-scalar models. In this context, the coupling between the scalar and gauge fields usually occurs via a generalized permittivity. In Refs. [25], it was shown that electrically charged localized structures can be obtained with this framework for a single point charge; the solutions satisfy the Bogomol'nyi [7] bound, so the energy of the system is minimized. This result was extended in Ref. [26] to the presence of dipoles in two spatial dimensions, whose stable localized configurations are attainable with bipolar coordinates. The study of continuous charged distributions was shown in [27] to be more intricate, as there is no BPS formalism to give rise to first-order equations. In this direction, it was shown that the Maxwell-scalar model in a single spatial dimension can be effectively treated as a single scalar field model in the presence of impurities, where the impurity can be related to the charge density. Maxwell-scalar models may also be of interest in the study of scalarization [28], whose setup may serve as a toy model for the aforementioned phenomenon of charged black holes in generalized scalar-tensor models. In curved spacetimes, the evasion of Derrick's scaling argument may also be circumvented in Maxwell-scalar models [29, 30]. In Ref. [31], models of the aforementioned type were investigated on spherically symmetric spacetimes, such as the Reissner-Nordstrom one. \nAnother possibility of current interest concerns the study of the strong interaction between quarks and glu- \n4 \nons via holographic Einstein-Maxwell-scalar models. It is based on the gauge/gravity duality, which has several interesting uses [32], a specific one being the mapping of the QCD phase diagram at finite temperature onto a dual theory that describes charged and asymptotically anti-de Sitter black holes in five dimensions. This subject has been recently reviewed in [33] and may stimulate new investigations, related to the present study. We can also add charged bosonic and fermionic fields to explore the study of boson [34, 35] and fermion [36] stars. In the second case, one can consider a real or neutral scalar coupled to a charged fermion via standard Yukawa coupling, capable of circumventing no-go theorems that prevent the existence of solitonic solutions. In this work, however, we shall deal with Maxwell-scalar models immersed in radially-symmetric spacetimes, focusing on the construction of a first-order framework which allows for the existence of analytical configurations. \nWe organize the investigation as follows. In Sec. II we study the equations of motions and the energymomentum tensor. Considering only the presence of fixed charge distributions and the absence of currents, we develop a first-order framework based on the minimization of the energy that imposes the charge density to be related to the some of the components of the metric tensor. We verify the stability against contractions and dilations, in the lines of Derrick-Hobart argument. Also, we analyze how the solution behaves in the presence of small fluctuations in the scalar field. To show that our method is robust, in Sec. III, we consider Tolman's VI [37], exponential [38] and hyperscaling violating [39-43] geometries. We conclude the investigation in Sec. IV, in which we make some final comments and discuss perspectives for future research.", 'II. MAXWELL-SCALAR MODELS': "We investigate the action that couples a gauge field A µ with a real scalar field φ through the electric permittivity of the system in ( d, 1) curved spacetime dimensions, with \nS = ∫ d D x √ | g | ( -/epsilon1 ( φ ) 4 F µν F µν + 1 2 ∇ µ φ ∇ µ φ -A µ j µ ) , (1) \nwhere D = d + 1. Here, g is the determinant of the metric tensor g µν , /epsilon1 ( φ ) denotes the electric permittivity, F µν = ∇ µ A ν -∇ ν A µ represents the electromagnetic strength tensor, and j µ is a current generated by external sources. By varying the action with respect to the scalar and gauge fields, we get the following equations of motion \n/square φ + 1 4 d/epsilon1 dφ F µν F µν = 0 , (2a) \n∇ µ ( /epsilon1 ( φ ) F µν ) = j ν , (2b) \nwhere /square = g µν ∇ µ ∇ ν is the d'Alambertian operator. We are interested in spacetimes with radial symmetry, so we \nds 2 = f ( r ) dt 2 -h ( r ) dr 2 -k ( r ) ω ij dθ i dθ j , (3) \nwhere θ i , i = 1 . . . d -1, represents the non-radial coordinates and the functions f ( r ), h ( r ) and k ( r ) are nonnegative functions. In addition, the determinant of the metric is √ | g | = √ ω ( θ ) f ( r ) h ( r ) k ( r ) d -1 = √ ω ( θ ) τ ( r ), in which we have written the determinant of ω ij as ω ( θ ) and we have taken τ ( r ) = f ( r ) h ( r ) k ( r ) d -1 . Considering static radially-symmetric configurations and the absence of currents, i.e., j 0 = /rho1 ( r ) and j i = 0, we can define the electric charge associated to the system as Q = ∫ Σ d d x √ γn 0 j 0 , where γ = | g | /f is the determinant of the induced metric on the surface Σ defined at fixed t . Also, n 0 = √ f is the temporal component of a vector n µ normal to Σ. By expanding the aforementioned expression for the charge, one can show that \nQ = Ω( d ) ∫ ∞ 0 dr √ τ/rho1 ( r ) , (4) \nwith Ω( d ) = ∫ Σ dθ 1 . . . dθ d -1 ω ( θ ). From Eq. (2b), we get \nin which the prime represents the derivative with respect to r . The above equation corresponds to the Gauss' law for our system; it can be solved by taking the charge density in the form \n1 √ τ ( √ τ/epsilon1F r 0 ) ' = /rho1 ( r ) , (5) \n/rho1 ( r ) = Q ' ( r ) √ τ ( r ) , (6) \nwhere Q ( r ) is an arbitrary function that obeys Q (0) = 0 to avoid the presence of a Dirac delta in the charge density. In this case, the solution is \nF r 0 = Q ( r ) √ τ ( r ) /epsilon1 ( φ ) , (7) \nfrom which we get the gauge field A i = 0 and ∂ r A 0 = F r 0 . The presence of Q ( r ) in the charge density (6) is interesting because it allows us to calculate the charge (4) in a simple manner, as \nQ = Ω( d ) Q ( ∞ ) . (8) \nThe intensity of the electric field can then be calculated by | E | = √ -F r 0 F r 0 , or \n| E | = ∣ ∣ ∣ ∣ Q ( r ) √ k ( r ) d -1 /epsilon1 ( φ ) ∣ ∣ ∣ ∣ . (9) \n∣ ∣ We use this in the equation of motion (2a), that governs the scalar field, to get \n/square φ + Q ( r ) 2 k ( r ) d -1 d dφ ( 1 2 /epsilon1 ( φ ) ) = 0 . (10) \nThe energy-momentum tensor T µ ν for non-static configurations is not conserved. Instead, it obeys ∇ µ T µ ν = j α F α ν , where \nT µ ν = /epsilon1 ( φ ) ( F µλ F λν + 1 4 δ µ ν F λσ F λσ ) + ∇ µ φ ∇ ν φ -1 2 δ µ ν ∇ λ φ ∇ λ φ. (11) \nIn spite of this, if the fields are static, the energy density is conserved, i.e., ∇ 0 T 0 0 = 0. Considering that φ depends only on r with j 0 = /rho1 ( r ) and j i = 0, the only non-null components of the energy-momentum tensor are the energy density \nT 0 0 = /epsilon1 ( φ ) 2 | E | 2 + 1 2 h ( r ) φ ' 2 = Q ( r ) 2 2 k ( r ) d -1 /epsilon1 ( φ ) + 1 2 h ( r ) φ ' 2 , (12) \nand the stresses. The radial stress is \nT r r = /epsilon1 ( φ ) 2 | E | 2 -1 2 h ( r ) φ ' 2 = Q ( r ) 2 2 k ( r ) d -1 /epsilon1 ( φ ) -1 2 h ( r ) φ ' 2 , (13a) \nand the stress in the other directions θ i defined in Eq. (3) is related to the radial one, in the form \nT θ i θ j = -δ i j T r r . (13b) \nStatic fields and charges and the absence of currents make Eq. (10) become \n-1 √ τ ( r ) ( √ τ ( r ) h ( r ) φ ' ) ' + Q ( r ) 2 k ( r ) d -1 d dφ ( 1 2 /epsilon1 ( φ ) ) = 0 . (14) \nBy integrating the energy density, we get the energy in the form \nE = ∫ ∞ 0 d d x √ | g | T 0 0 = Ω( d ) ∫ ∞ 0 dr √ τ ( r ) ( Q ( r ) 2 2 k ( r ) d -1 /epsilon1 ( φ ) + 1 2 h ( r ) φ ' 2 ) . (15) \nThis expression can be rearranged in the form \nE =Ω( d ) ∫ ∞ 0 dr √ τ 2 h ( φ ' ∓ √ h Q 2 k d -1 /epsilon1 ( φ ) ) 2 ± √ f Q 2 /epsilon1 ( φ ) φ ' . (16) \nIn order to write the term inside the integral as a total derivative, we follow the lines of Refs. [7, 8] and introduce the auxiliary function W = W ( φ ) such that \nQ ( r ) = e √ f ( r ) and e √ /epsilon1 ( φ ) = dW dφ , (17) \nwith e being a constant that we consider to be positive without loss of generality. The left expression above shows that the charge is now constrained to the metric. Since we have imposed Q (0) = 0, we see that the factor of dt 2 in the line element (3) must obey f (0) →∞ . Using both equations in Eq. (17), we can rewrite the energy as \nE = Ω( d ) ∫ ∞ 0 dr ( √ τ 2 h ( φ ' ∓ h √ τ dW dφ ) 2 ) + E B , (18) \nwhere E B = Ω( d ) | W ( φ ( ∞ )) -W ( φ (0)) | . The above expression allows us to see that the energy has a lower bound, E ≥ E B . Therefore, the solutions which minimize the energy of the system obey the first-order equations \nφ ' = ± h √ τ dW dφ , (19) \nwhich are compatible with the equation of motion (14). It is worth highlighting that the solutions of the above first order equations are stressless, that is, T r r = T θ i θ j = 0; see Eqs. (13a) and (13b). \nTo search for analytical solutions, we can define the variable x as \ndx = h ( r ) √ τ ( r ) dr = √ h ( r ) f ( r ) k d -1 ( r ) dr (20) \nto write the first-order equations (19) in the form \ndφ dx = ± dW dφ . (21) \nThe above framework is the same that arises with the standard Lagrangian density of a single scalar field, L = 1 2 ∂ µ φ∂ µ φ -V ( φ ), in (1 , 1) spacetime dimensions, with the scalar potential being V ( φ ) = 1 2 ( dW/dφ ) 2 . Therefore, we are now able to map kink-like one-dimensional solutions to our model in the curved spacetime. This mapping, however, requires the need to be careful with the choice of the functions f ( r ), h ( r ) and k ( r ), as the solution must be fully mapped into r as it varies from 0 to ∞ . Moreover, although there is a correspondence between scalar field solutions of the model (1) and the corresponding one in (1 , 1) spacetime dimensions, the current model engenders electric field with intensity (9). In Ref. [25], for instance, one provides Maxwell-scalar systems of a single point charge in flat spacetime that support first-order equations which lead to stable finite energy solutions. \nWith the condition in the left equation of (17), we can rewrite the charge density (6) in the form \n/rho1 ( r ) = -ef ' ( r ) 2 f 2 ( r ) √ h ( r ) k d -1 ( r ) . (22) \nThe charge in Eq. (8) reads \nQ = e Ω( d ) √ f ( ∞ ) . (23) \nThis expression shows that the charge of the system depends exclusively on the behavior of the f ( r ) at infinity. For f ( ∞ ) → 0, the charge diverges; for f ( ∞ ) → f ∞ , with f ∞ being a non-null constant, the charge is finite and non zero; for f ( ∞ ) →∞ the charge vanishes. Taking Eqs. (17) into account, the energy density in Eq. (12) becomes \nT 0 0 = 1 f ( r ) k ( r ) d -1 ( dW dφ ) 2 . (24) \nThe contribution of the scalar field does not lead to divergences in the above expression since we are considering the change of variables (20) and imposing that the solutions connect the minima of V ( φ ) = 1 2 ( dW/dφ ) 2 as r varies from 0 to ∞ . Therefore, divergences in the above equation may only arise due to the metric.", 'A. Stability': "We now investigate the stability of our model. First, we study how the energy of the system behaves by rescaling the solution of the equation of motion (14), following the lines of Derrick-Hobart argument [9, 10]. By taking r → r ( λ ) = λr in the scalar field solution, we have φ ( r ) → φ λ ( r ) = φ ( λr ). We denote energy of the rescaled solution, which we calculate from Eq. (15) as E ( λ ) . The solution is stable against contractions and dilations if λ = 1 is the minimum of E ( λ ) . By taking ∂ E ( λ ) /∂λ | λ =1 = 0 and ∂ 2 E ( λ ) /∂λ 2 | λ =1 > 0, we get the conditions \n∫ ∞ 0 dr [ ( r √ τ Q 2 k d -1 ) ' 1 /epsilon1 ( φ ) + r 2 ( √ τ rh ) ' φ ' 2 ] = 0 , (25a) \n∫ ∞ 0 dr [ ( r 2 √ τ Q 2 k d -1 ) '' 1 /epsilon1 ( φ ) + r 2 ( √ τ h ) '' φ ' 2 ] > 0 . (25b) \nThe integral in Eq. (25a) must be zero, so it is a global condition. Since the function inside the integral may change its sign and depends explicitly on r , we cannot take it to be zero locally. Notwithstanding that, an interesting issue occurs: by considering the conditions (17) and the first-order equation (19), one can show that the integrand vanishes locally and the condition (25a) becomes an identity. Moreover, in this situation, we can write Eq. (25b) in the form \n∫ ∞ 0 dr [ √ τ h ( yh √ τ ) ' 2 ( dW dφ ) 2 ] > 0 , (26) \nwhich is always satisfied, since the function inside the integral is non negative. Hence, the solutions of the firstorder equations are stable against contractions and dilations. Moreover, these solutions also lead to the absence of stress as one can verify from Eqs. (13a) and (13b). \nNext, let us verify the stability of the solutions A 0 ( r ) and φ ( r ) of the equations of motion (7) and (14) around small fluctuations. By taking φ ( x α ) = φ ( r ) + η ( x α ) and A µ ( x α ) = A µ ( r ) + A µ ( x α ) we can write F µν ( x α ) = F µν ( r ) + F µν ( x α ), where F µν = ∂ µ A ν -∂ ν A µ . By substituting this into the equation of motion (2), we get \n/square η + 1 4 d 2 /epsilon1 dφ 2 F µν F µν η + 1 2 d/epsilon1 dφ F µν F µν = 0 , (27a) \n∇ µ ( d/epsilon1 dφ F µν η + /epsilon1 ( φ ) F µν ) = 0 . (27b) \nThe latter equation admits the solution \nF µν = -d ln( /epsilon1 ) dφ F µν η, (28) \nshowing that the fluctuations of the gauge and scalar fields are related. By substituting the above expression into Eq. (27a), we get \n/square η ( x α ) + Q ( r ) 2 k ( r ) d -1 d 2 dφ 2 ( 1 2 /epsilon1 ) η ( x α ) = 0 . (29) \nWe then take the fluctuations in the form η ( x α ) = ∑ i η i ( r ) cos( ω i t ), such that the above equation becomes \n-1 √ τ ( √ τ h η ' i ) ' + Q ( r ) 2 k ( r ) d -1 d 2 dφ 2 ( 1 2 /epsilon1 ) η i = ω 2 i f η i . (30) \nThis expression is an eigenvalue equation of the SturmLiouville type. The solutions are stable if the eigenvalues are non negative, i.e., ω 2 i ≥ 0. If the static solution φ ( r ) obeys the first-order framework (17)-(19), we can write the above equation as: -( σ ( r ) s ( r ) 2 η ' i ) ' + σ ( r ) U ( r ) η i = ω 2 i σ ( r ) η i , or Lη i = ω 2 i η i , where L is the Sturm-Liouville operator given by \nL = -1 σ ( r ) d dr σ ( r ) s ( r ) 2 d dr + U ( r ) , (31) \nwith \nσ ( r ) = h ( r ) k ( r ) d -1 √ τ ( r ) = √ h ( r ) k ( r ) d -1 f ( r ) , (32a) \n√ U ( r ) = 1 k ( r ) d -1 ( ( d 2 W dφ 2 ) 2 + dW dφ d 3 W dφ 3 ) . (32c) \ns ( r ) = √ τ ( r ) h ( r ) k ( r ) d -1 = √ f ( r ) h ( r ) , (32b) \nIn this specific situation, we were able to show that eigenvalue equation associated to the operator (31) supports a zero mode ( ω 0 = 0) related to the derivative of the static solution, \nη 0 ( r ) = N √ τ h φ ' , (33) \nin which N is a constant of normalization that is obtained from \n∫ ∞ 0 dr σ ( r ) η 2 0 ( r ) = 1 . (34) \nIf there are no negative modes, the fluctuations do not destabilize the scalar solutions. We then investigate how the gauge field behaves under these fluctuations by using Eq. (28) to calculate the term F µν ( x α ) F µν ( x α ) and get \n-F µν ( x α ) F µν ( x α ) = 2 fk d -1 ( dW dφ ) 2 ( 1 -d ln( /epsilon1 ) dφ η ( x α ) ) , (35) \nwhere the scalar fluctuations are η ( x α ) = ∑ i η i ( r ) cos( ω i t ), as considered right above Eq. (30). We then see that the functions which control the metric and the scalar field must be chosen to avoid divergences in the above expression to keep the gauge field stable. \nThe Sturm-Liouville operator (31) can be factorized in terms of adjoint operators S and S † , as L = S † S , where \nS = s ( r ) ( -d dr + h ( r ) √ τ ( r ) φ ' ( √ τ ( r ) φ ' h ( r ) ) ' ) , (36a) \nS † = s ( r ) ( d dr + h ( r ) √ τ ( r ) φ ' ( √ τ ( r ) φ ' h ( r ) ) ' + ( σ ( r ) s ( r ) ) ' σ ( r ) s ( r ) ) . (36b) \nThe above factorization and the absence of nodes in the zero mode ensures that the Sturm-Liouville operator only admits non-negative eigenvalues, so the solutions are stable against small fluctuations. To get another interpretation of the stability, we can transform the stability equation into a Schrodinger-like one with the change of variables \ndy = dr s and ψ i = √ σsη i . (37) \nThis makes the eigenvalue equation have the form H η i = ω 2 i η i , where \nH = -d 2 dy 2 + U ( y ) , with U ( y ) = ( √ σs ) yy √ σs + U ( r ( y )) , (38) \nin which r ( y ) is the coordinate r written in terms of the variable y defined in Eq. (37). The zero mode of the Schrodinger-like equation (38) is then given by \nψ 0 ( y ) = N k ( r ( y )) 3 4 ( d -1) φ y . (39) \nIt is worth commenting that, even though we are able to get the description of the linear stability using a Schrodinger-like equation, obtaining the analytical expressions for the change of variables (37) is not always possible. However, one can use numerical methods to circumvent this issue when necessary. \nNext, we investigate specific models with the metric (3) that support stable electrically charged localized structures.", 'III. MODELS': 'To illustrate our procedure, we must consider scalar field models that allow for the presence of localized solutions which connect the minima of V ( φ ) = 1 2 ( dW/dφ ) 2 . Due to the nature of the geometric backgrounds that we shall investigate, we consider the model introduced in Ref. [11], described by the auxiliary function \nW ( φ ) = p 2 p -1 φ 2 -1 p -p 2 p +1 φ 2+ 1 p , (40) \nwith p = 3 , 5 , 7 , . . . . The permittivity can be calculated from the right equation in (17), which leads us to /epsilon1 ( φ ) = e 2 φ -2 ( φ -1 /p -φ 1 /p ) -2 . This function diverges at φ = ± 1 and φ = 0. It has two sectors, φ ∈ [ -1 , 0] and φ ∈ [0 , 1]. \nSupposing that Q is given by the left equation in (17), we can use the first-order framework, such that the scalar field is governed by Eq. (19), which reads \ndφ dx = ± φ ( φ -1 /p -φ 1 /p ) . (41) \nThis equation admits the solution \nφ ± ( x ) = ± tanh p ( x/p ) , (42) \nwhere x must be calculated from (20) for the specific metric under investigation. The upper/lower sign represents the increasing/decreasing solution. This solution engenders an inflection point with null derivative at x = 0 and is extended, attaining the boundary values only at ±∞ . It goes from φ = -1 to φ = 1, passing through φ = 0, which separates the sectors, due to the atypical character of this point in the scalar potential V ( φ ) = 1 2 ( dW/dφ ) 2 = 1 2 φ 2 ( φ -1 /p -φ 1 /p ) 2 associated to the model in (1 , 1) dimensions to be mapped (see Ref. [11]). The presence of the inflection point with null derivative in φ ± ( x ) allows us to split it into the halfcompact solutions \nand \nφ ± ( x ) = ± tanh p ( x p ) , x ≤ 0 0 , x > 0 (43a) \nφ ± ( x ) = 0 , x < 0 ± tanh p ( x p ) , x ≥ 0 . (43b) \nThese solutions will be used in situations where the geometric background induces a mapping of r into x ∈ ( -∞ , 0) and x ∈ (0 , ∞ ). A similar technique was already applied with this model in Refs. [24, 44]. We remark that, only if the change of variables in Eq. (20) leads to x ∈ ( -∞ , ∞ ), the case p = 1 is allowed, as the solution will connect the minima φ = -1 and φ = 1 of the scalar potential. \nIn the following subsections, we investigate three geometries which allows us to apply our method.', "A. Tolman's metric": "First, we investigate the behavior of our model on the Tolman's metric VI, introduced in Ref. [37] with the line element \nds 2 = ( Ar 1 -n + Br 1+ n ) 2 dt 2 -(2 -n 2 ) dr 2 -r 2 dθ 2 -r 2 sin 2 θ dϕ 2 , (44) \nwhere A and B are positive real parameters, and n is a real number that obeys | n | < √ 2. The above expression leads to √ | g | = √ 2 -n 2 r 2 ( Ar 1 -n + Br 1+ n ) sin θ and √ τ = √ 2 -n 2 r 2 ( Ar 1 -n + Br 1+ n ). In the first-order framework, one requires that Eq. (17) must be obeyed, from which we get \nQ ( r ) = e Ar 1 -n + Br 1+ n . (45) \nSince we must have Q (0) = 0, the parameter n is now restricted to 1 < | n | < √ 2. The charge density in Eq. (22) reads \n/rho1 ( r ) = -e ( A (1 -n ) r -n + B (1 + n ) r n ) √ 2 -n 2 r 2 ( Ar 1 -n + Br 1+ n ) 3 . (46) \nIt is zero for r = r ∗ , with r ∗ = ( A ( n -1) / ( B ( n +1))) 1 / (2 n ) . Interestingly, /rho1 ( r ) is positive for r < r ∗ and negative for r > r ∗ . Asymptotically, the charge density behaves as /rho1 ( r ) ∝ r -2 n -5 , which always goes to zero for 1 < | n | < √ 2. Near the origin, we see that /rho1 ( r ) ∝ r 2 n -5 . Therefore the charge density has a divergence at r = 0 for n in the aforementioned interval. Nevertheless, this divergence is integrable, as the charge in Eq. (23) is null. We display the charge density in the top-left panel of Fig. 1. \nTo obtain the scalar field solution, we use the change of variables in Eq. (20), which leads us to \nx ( r ) = -√ 2 -n 2 A (2 -n ) ( r n -2 2 F 1 ( 1 , 1 2 -1 n ; 3 2 -1 n ; -B A r 2 n ) + π (2 -n ) 2 n ( B A ) (2 -n ) /n sec ( π n ) ) , (47) \nwhere 2 F 1 ( a, b ; c ; z ) represents the hypergeometric function of argument z . Notice that the above expression maps the interval r ∈ (0 , ∞ ) into x ∈ ( -∞ , 0). Therefore, we must use the solution in Eq. (43a), so we have \nφ ± ( r ) = ± tanh p ( x ( r ) p ) . (48) \nThe upper/lower sign represents the increasing/decreasing solution and x ( r ) is as in Eq. (47). Notice that φ + connects φ = -1 at r = 0 to φ = 0 at r → ∞ and φ -goes from φ = 1 at r = 0 to φ = 0 at r →∞ . In the top-right panel of Fig. 1, we display the \nsolution φ + ( r ) for some values of the parameters. The energy density (24) associated to both φ + and φ -is \nT 0 0 ( r ) = sech 4 ( x ( r ) /p ) tanh 2( p -1) ( x ( r ) /p ) r 4 ( Ar 1 -n + Br 1+ n ) 2 , (49) \nfrom which we see that T 0 0 (0) = 0. It is plotted in the bottom-left panel of Fig. 1. By integrating this expression, we get the energy E = 2 p Ω( d ) / (4 p 2 -1), as expected from the definition of E B above Eq. (19). \nThe stability is governed by the Sturm-Liouville operator (31), which is described by the functions in Eqs. (32a) and (32b) that lead us to \nσ ( r ) = √ 2 -n 2 r Ar -n + Br n and s ( r ) = Ar 1 -n + Br 1+ n √ 2 -n 2 . (50) \nFrom these expressions, we get that σ ( r ) vanishes at the origin and diverges at infinity, and s ( r ) diverges both for r = 0 and r →∞ . The stability potential (32c) reads \nU ( r ) = 1 r 4 (( 1 + 1 p )( 1 + 2 p ) tanh 2 ( x ( r ) p ) + ( 1 -1 p )( 1 -2 p ) tanh -2 ( x ( r ) p ) -2 ) . (51) \nThe zero mode associated to the stability equation can be calculated analytically from Eq. (33); it is \nη 0 ( r ) = N tanh p -1 ( x ( r ) p ) sech 2 ( x ( r ) p ) , (52) \nwhere N is a normalization constant that cannot be obtained analytically for general values of the parameter p . Notwithstanding that, by using the asymptotic behavior of the above expression, we have verified that the zero mode can be normalized, so there is a finite value of N compatible with Eq. (34). The absence of nodes in the zero mode shows that negative eigenvalues are absent, ensuring the stability of the model around small fluctuations. \nSince we are following the first-order framework, the Sturm-Liouville operator can be factorized into the product of the adjoint operators (36), which take the form \nS = -Ar 1 -n + Br 1+ n √ 2 -n 2 ( d dx + √ 2 -n 2 r 2 ( Ar 1 -n + Br 1+ n ) × (( 1 p +1 ) tanh ( x ( r ) p ) + ( 1 p -1 ) tanh -1 ( x ( r ) p ))) , (53a) \nS † = Ar 1 -n + Br 1+ n √ 2 -n 2 ( d dx -√ 2 -n 2 r 2 ( Ar 1 -n + Br 1+ n ) × (( 1 p +1 ) tanh ( x ( r ) p ) + ( 1 p -1 ) tanh -1 ( x ( r ) p )) + 2 r ) . (53b) \nFIG. 1. The charge density /rho1 ( r ) in Eq. (46) (top left, orange), solution φ + ( r ) in Eq. (48) (top right, red), energy density T 0 0 ( r ) in Eq. (49) (bottom left, blue) and the Schrodingerlike potential U ( y ) in Eq. (55) (bottom right, green), for e = 1, p = 3, n = 1 . 1, A = B -1 = 5 / 6 , 1 , 6 / 5. In each panel, the thickness of the lines increases with increasing A . \n<!-- image --> \nThe stability equation (30) can be transformed into a Schrodinger-like one with the change of variables (37), \ny = √ 2 -n 2 n √ AB arctan ( √ B A r n ) , (54) \nwith y ∈ [ 0 , ˜ y ] , where ˜ y = π √ 2 -n 2 / (2 n √ AB ). The potential associated to the operator (38) can be written as \nU ( y ) = U ( r ( y )) + 4 AB 2 -n 2 csc 2 ( 2 n √ ABy √ 2 -n 2 ) × ( 1 -n cos ( 2 n √ ABy √ 2 -n 2 )) , (55) \nfor y ∈ [ 0 , ˜ y ] . Notice that U ( y ) diverges for y → ˜ y . In the above equation, r ( y ) denotes the inverse of the expression in Eq. (54). This potential is an infinite well of width ˜ y , so it only admits bound states. In the bottom-right panel of Fig. 1 one can see the behavior of the above stability potential for some values of the parameters. Notice that there is a region where U ( y ) is negative, allowing for the presence of the zero mode. The zero mode is the ground state, so the solution is linearly stable.", 'B. Exponential metric': 'Let us now investigate the exponential metric in isotropic coordinates [38, 45, 46], with line element \nds 2 = e α/r dt 2 -e -α/r ( dr 2 + r 2 dθ 2 + r 2 sin 2 θdϕ 2 ) , (56) \nwhere α is a real parameter. In this case, we have √ | g | = r 2 e -α/r sin θ and √ τ = r 2 e -α/r . To use the first-order framework, we must obey the conditions in Eq. (17), which implies that Q ( r ) = e e -α/ (2 r ) . Since our procedure requires that the above expression vanishes at the origin, we impose that α is strictly positive. The charge density in Eq. (22) can be written as \n/rho1 ( r ) = αe 2 r 4 e α/ (2 r ) , (57) \nwhich is positive in all the space. We display it the topleft panel of Fig. 2. By integrating the above equation, we get the finite charge Q = 4 πe , matching with the value obtained from Eq. (23), being independent on α . \nTo obtain the scalar field profile, we take advantage of the change of variables (20), which leads us to x = 1 /r c -1 /r . The parameter r c separates two regions in the mapping. The region r ∈ ( r c , ∞ ) corresponds to x ∈ (0 , 1 /r c ), which is a compact space, so it is not compatible with the solutions (43). The other region is described by the interval r ∈ (0 , r c ), which is mapped into x ∈ ( -∞ , 0). Since x is a semi-compact space, we can use the solution in Eq. (43a), which becomes \nφ ± ( r ) = ± tanh p ( r -r c p r r c ) , r ≤ r c 0 , r > r c , (58) \nwhere r c denotes the compactification radius and the upper/lower sign represents the increasing/decreasing solution. Notice that the compact support is lost for r c →∞ , for which the solution becomes extended, as usually occurs in kinks. In the top-right panel of Fig. 2 one can see the behavior of φ + ( r ) for some values of the parameters. The energy density in Eq. (24) is also compact; it reads \nT 0 0 ( r ) = 1 r 4 e α/r tanh 2( p -1) ( r -r c p r r c ) sech 4 ( r -r c p r r c ) , (59) \nfor r ≤ r c and T 0 0 ( r ) = 0 otherwise. By integrating it, the energy is E = 2 p Ω( d ) / (4 p 2 -1), as expected from the definition of E B above Eq. (19). The behavior of the energy density at the origin depends on α : it is null for α < 4 and infinite otherwise. We display this energy density in the bottom-left panel of Fig. 2. \nLet us now focus on the stability of the solution (58) around small fluctuations. The functions (32a) and (32b) which describes the Sturm-Liouville operator (31) are \nσ ( r ) = r 2 e -2 α/r and s ( r ) = e α/r , (60) \nfrom which we see that σ ( r ) vanishes at the origin and diverges asymptotically, and s (0) → ∞ and s ( ∞ ) → 1. The aforementioned operator also depends on the stability potential in Eq. (32c), which can be written in the \nform \nU ( r ) = 1 r 4 e 2 α/r ( ( 1 + 1 p )( 1 + 2 p ) tanh ( r -r c p r r c ) 2 + ( 1 -1 p )( 1 -2 p ) tanh ( r -r c p r r c ) -2 -2 ) , (61) \nfor r ≤ r c , and U ( r ) →∞ for r > r c , if r c is finite. The case in which r c → ∞ obeys to U ( ∞ ) → 0. The eigenvalue equation that describes the linear stability admits the zero mode in Eq. (33), which leads us to \nη 0 ( r ) = N tanh p -1 ( r -r c p r r c ) sech 2 ( r -r c p r r c ) (62) \nfor r ≤ r c , and η 0 ( r ) = 0 for r > r c . This zero mode can be normalized, obeying Eq. (34). Unfortunately, we were not able to find a general expression for N , so it must be calculated for each set of values of the parameters. It is worth remarking that, as a consequence of our first-order framework, the adjoint operators (36) that factorize the Sturm-Liouville operator are given by \nS = -e α/r ( d dr + 1 r 2 (( 1 p +1 ) tanh ( r -r c p r r c ) + 1 p -1 tanh -1 r -r c p r r c \n( ) ( )) ) (63a) S † = e α/r ( d dr -1 r 2 (( 1 p +1 ) tanh ( r -r c p r r c ) + ( 1 p -1 ) tanh -1 ( r -r c p r r c )) + α +2 r r 2 ) (63b) \nfor r ≤ r c . These operators are not well defined for r > r c as the stability potential is infinite in this interval. \nAs we have commented in the general study of the stability, one can perform the change of variables (37) to transform the Sturm-Liouville equation into a Schrodinger-like one. For the metric under investigation, we get \ny = r e -α/r + α Ei ( -α r ) , (64) \nwhere Ei( z ) is the exponential integral function with argument z . The above equation shows that its asymptotic behavior is y ≈ r . However, it is not possible to write r in terms of y analytically, so numerical procedures are required if one wishes to use the Schrodinger-like equation. In particular, the Schrodinger-like potential in Eq. (38) is given by \nU ( y ) = U ( r ( y )) -α e -α/ (2 r ( y )) 4 r ( y ) 4 ( α +4 r ( y )) (65) \nfor y ≤ y c ≡ r c e -α/r c + α Ei( -α/r c ), and U ( y ) → ∞ for y > y c . We have used r ( y ) to denote the inverse \nFIG. 2. The charge density /rho1 ( r ) in Eq. (57) (top left, orange) for e = 1, α = 0 . 01 , 0 . 05 , 0 . 1 with the thickness of the lines increasing with α . We also display the solution φ + ( r ) in Eq. (58) (top right, red), energy density T 0 0 ( r ) in Eq. (59) (bottom left, blue) and Schrodinger-like potential U ( y ) in Eq. (65) (bottom right, green), for p = 3, α = 0 . 01, r c = 1 , 2 and r c →∞ , with the thickness of the lines now increasing with r c . The dashed lines represent the limit r c →∞ in each quantity and the vertical dotted lines in gray stand for the value r = r c . \n<!-- image --> \nof Eq. (64). The above expression leads us to conclude that the above potential is an infinite well for finite y c . This behavior, however, is changed in the limit y c → ∞ , for which we get U ( y ) ≈ ( p -1)( p -2) y -2 , so the potential vanishes for large y . In the bottom-right panel of Fig. 2 we display the above Schrodinger-like potential. Even though numerical methods are required, we can see that the zero mode (62) does not engender nodes; this ensures that negative eigenvalue is absent and the model is linearly stable. \nIn Ref. [46], the authors have found interesting connection between the exponential metric considered above with a traversable wormhole, so the above study may motivate other investigations, adding dynamics to the geometric degrees of freedom, searching for the possibility to find a more general framework engendering first-order differential equations.', 'C. Hyperscaling violating geometries': 'We consider the metric associated to hyperscaling violating geometries [24] in ( d, 1) spacetime dimensions, with ω ij = δ ij and θ i = x i , defined by the line element \nds 2 = r 2 z -θ c dt 2 -r -2 -θ c dr 2 -r 2 -θ c dx k dx k . (66) \nThe above expression leads to √ | g | = √ τ = r d + z -2 -( θ c / 2)( d +1) and k = 1 , 2 , ..., d -1, in which z and θ c represents the dynamical and hyperscaling violating exponents, respectively. For the value θ c = 0, the Lifshitz \nFIG. 3. The charge density /rho1 ( r ) in Eq. (67) for e = 1, θ c = 0 and z = -2 , -1 , -1 / 2. The thickness of the lines increases with z . The dashed line represents z = -1, which is the case where the charge density /rho1 ( r ) is constant. \n<!-- image --> \nspacetime [47] is recovered. To work with the first-order framework, we consider the condition in the left equation of (17), which requires Q ( r ) = er θ c / 2 -z . To avoid divergences in this function at the origin, we impose Q (0) = 0, which is satisfied by θ c > 2 z . In this case, the charge density (22) associated to the current geometric background reads \n/rho1 ( r ) = e 2 ( θ c -2 z ) r ( θ c / 2)( d +2) -d -2 z +1 . (67) \nIt is regular, without divergence at the origin, if θ c ≥ 2( d +2 z -1) / ( d +2). It is always non negative and leads to infinite charge, as expected from Eq. (23). In Fig. 3, we display the behavior of the above charge density for some values of the parameters, showing that it may be null, finite or divergent both at the origin or asymptotically. \nSince we are considering the model described by (40), the solution is given by Eq. (43), with argument x related to r via Eq. (20), which leads us to x = ( r n -r n c ) /n , where n = θ c ( d -1) / 2 -d -z +1 and r c is an integration constant which determines the point in which x = 0. Notice that the case n = 0 is special; we shall investigate it later. For non-null n , one can use the solutions (43) to show that \nφ ± ( r )= { ± tanh p ( 1 np ( r n -r n c ) ) , r sgn( n ) ≥ r c sgn( n ) 0 , r sgn( n ) < r c sgn( n ) , (68) \nwhere sgn( x ) denotes the signal function. The upper/lower sign represents the increasing/decreasing solution. The solution φ + ( r ) is depicted in Fig. 4 for positive and negative values of n and some values of the parameters. \nFor n > 0, the solution φ + ( φ -) is uniform and null in the interval r < r c , and connects φ = 0 to φ = 1 ( φ = -1), in r ≥ r c . The case n < 0 is interesting, as it supports compact solutions for finite r c . In this situation, φ + connects φ = -1 to φ = 0 and φ -goes from φ = 1 to φ = 0. The limit r c →∞ decompactifies the solution, leading to φ ± = ± tanh p ( r n / ( np )). The energy density \nFIG. 4. The solution φ + ( r ) in Eq. (68). In the top (bottom) panels, we depict the case n > 0 ( n < 0), for p = 3 and θ c = 0. In the top-left panel, we use r c = 0 and z = -3 . 2 , -3 and -2 . 8. In the top-right panel, we take z = -3 and r c = 0 , 2 and 4. In the bottom-left panel, we consider r c → ∞ and z = -1 . 2 , -1 and -0 . 8. In the bottom-right panel, we have z = -1 and r c = 1 / 2 , 1 and r c → ∞ . The dashed lines represent the case r c = 0 in the top-right panel and r c →∞ in the bottom-right panel. The dotted vertical lines represent the point r = r c in the solutions. The thickness of the lines increases with z in the left panels and with r c in the right ones. \n<!-- image --> \n(24) associated to the above solutions becomes \nT 0 0 ( r ) = r 2 n + θ c tanh ( 1 np ( r n -r n c ) ) 2 p -2 × sech ( 1 np ( r n -r n c ) ) 4 (69) \nfor the regions in which the solution (68) is not uniform, and T 0 0 ( r ) = 0 otherwise. For positive n , as we approach r = r c from the right, one can show that the above expression behaves as T 0 0 ( r ) ∝ r 2 n + θ c ( r n -r n c ) 2( p -1) . On one hand, this shows that T 0 0 ( r ) is always null at r = r c for r c > 0. On the other hand, for r c = 0, the behavior can be null, finite or divergent at r = 0, depending on the sign of 2 pn + θ c . For negative n , the behavior near the origin has the form T 0 0 ( r ) ∝ r 2 n + θ c e -4 r n , so the energy density always vanish at the aforementioned point. Notice that the energy density is compact for negative values of n when r c is finite, as expected from the solution (68). By integrating the above expression, we get the energy E = 2 p Ω( d ) / (4 p 2 -1), matching with the value expected from the definition of E B above Eq. (19). Ω( d ) represents the Euclidean volume related to the x i -coordinates. The above energy density is plotted in Fig. 5 for some values of the parameters. \nLet us now deal with the case in which n = 0, obtained for θ c = 2+2 z/ ( d -1). The change of variables in Eq. (20) leads us to x = ln( r/r c ). Since x varies from -∞ to ∞ , we can use the solution (42), which can now be written \nFIG. 5. The energy density T 0 0 ( r ) in Eq. (69). In the top (bottom) panels, we depict the case n > 0 ( n < 0), for p = 3 and θ c = 0. In the top-left panel, we use r c = 0 and z = -3 . 2 , -3 and -2 . 8. In the top-right panel, we take z = -3 and r c = 0 , 2 and 4. In the bottom-left panel, we consider r c → ∞ and z = -1 . 2 , -1 and -0 . 8. In the bottom-right panel, we have z = -1 and r c = 1 / 2 , 1 and r c → ∞ . The dashed lines represents the case r c = 0 in the top-right panel and r c → ∞ in the bottom-right panel. The dotted vertical lines represent the point r = r c . The thickness of the lines increases with z in the left panels and with r c in the right ones. \n<!-- image --> \nFIG. 6. The solution (70) (left) and the energy density (71) (right) for p = 1. In the energy density, the dashed line represents θ c = 1 and z = -1, and the solid line stands for θ c = 3 and z = 1. \n<!-- image --> \nin terms of power-law functions, as \nφ ± ( r ) = ± ( r 2 /p -r 2 /p c r 2 /p + r 2 /p c ) p . (70) \n/negationslash \nThis solution connects φ = -1 to φ = 1. It is increasing/decreasing for the upper/lower sign. Interestingly, contrary to the case n = 0, the value p = 1 is now allowed because the solution connects two divergent points of the permittivity as r goes from 0 to ∞ . It is plotted in Fig. 6 for some values of the parameters. The associated \nenergy density (24) reads \nT 0 0 ( r ) = 16 r 4 /p c r 2 z/ ( d -1)+2(2+ p ) /p ( r 2 /p + r 2 /p c ) 4 ( r 2 /p -r 2 /p c r 2 /p + r 2 /p c ) 2 p -2 . (71) \nNear the origin, it behaves as T 0 0 ( r ) ∝ r 2 z/ ( d -1)+2(2+ p ) /p , which may lead to null, finite or divergent behavior at this point. We can see the energy density in Fig. 6. The integral of the above expression leads to the energy E = 4 p Ω( d ) / (4 p 2 -1), as expected from the first-order framework. \nWe now turn our attention to the stability. The SturmLiouville operator in Eq. (31) that governs the eigenvalue equation is now described by the functions (32a) and (32b), \nσ ( r ) = r -n -2 z -1 and s ( r ) = r 1+ z . (72) \n/negationslash \nThese expressions are valid for any value of n , as they only depend on the factors of the metric. However, we must be careful to calculate the stability potential (32c) due to the form of the solutions. In the case n = 0, it is given by \nU ( r ) = r 2( n + z ) (( 1 + 1 p )( 1 + 2 p ) tanh 2 ( r n -r n c np ) + ( 1 -1 p )( 1 -2 p ) tanh -2 ( r n -r n c np ) -2 ) , (73) \nfor r sgn( n ) ≥ r c sgn( n ), and U ( r ) → ∞ otherwise (except for r c = 0 with n > 0 or r c →∞ with n < 0). The case r c = 0 for n > 0 requires some care, so we expand the above expression near the origin to get U ( r ) ∝ r 2 z , which may give rise to null, finite or divergent behavior, depending on the value of z . For n < 0 with r c →∞ , the asymptotic behavior is U ( r ) ∝ r 4 n +2 z , which may lead to the same three behaviors commented in the case n > 0, depending on z . \nIn the case n = 0, Eq. (32c) becomes \nU ( r ) = 4 r 2 z p 2 ( r 4 /p -r 4 /p c ) 2 ( r 8 /p + r 8 /p c -6 p ( r 2 /p c r 6 /p + r 6 /p c r 2 /p ) +2(2 p 2 +3) r 4 /p c r 4 /p ) . (74) \n/negationslash \nThe zero mode associated to the stability equation can be calculated from Eq. (33). For n = 0, it is \nη 0 ( r ) = N tanh p -1 ( r n -r n c np ) sech 2 ( r n -r n c np ) (75) \nfor r sgn( n ) ≥ r c sgn( n ), and η 0 ( r ) = 0 otherwise. In the case n = 0, it has the form \nη 0 ( r ) = 4 N r 2 /p c r 2 /p ( r 2 /p + r 2 /p c ) 2 ( r 2 /p -r 2 /p c r 2 /p + r 2 /p c ) p -1 . (76) \nIn the last two expressions, N is a normalization constant. For n > 0 with r c = 0, we must impose n ≥ (1+2 z ) / (2 p -3) to get a normalized zero mode. For n < 0 with r c → ∞ , the aforementioned condition is changed to n < 2 z/ (2 p -3). There are no additional restrictions on the values of n and z if r c is a non-null finite value. In the case n = 0, we must impose -2 /p < z < (4 -p ) / (2 p ). \n/negationslash \nSince we are using the first-order framework, the Sturm-Liouville operator (31) can be factorized into the product of the adjoint operators (36). In the case n = 0, they are \nS = -r z -1 ( d dr + r n -1 (( 1 p +1 ) tanh ( r n -r n c np ) + ( 1 p -1 ) tanh -1 ( r n -r n c np )) ) , (77a) \nS † = r z -1 ( d dr -r n -1 (( 1 p +1 ) tanh ( r n -r n c np ) + ( 1 p -1 ) tanh -1 ( r n -r n c np )) -n + z r ) , (77b) \nfor r sgn( n ) ≥ r c sgn( n ); they are not well defined for r sgn( n ) < r c sgn( n ) due to the nature of the potential, which may be infinite in this interval depending on n, z and r c . The case n = 0 works with \nS = -r z -1 ( d dr + 2 ( r 4 /p + r 4 /p c -2 pr 2 /p c r 2 /p ) pr ( r 4 /p -r 4 /p c ) ) , (78a) \nS † = r z -1 ( d dr -1 pr ( r 4 /p -r 4 /p c ) ( (2 + pz ) r 4 /p +(2 -pz ) r 4 /p c -4 pr 2 /p c r 2 /p ) ) . (78b) \n/negationslash \nThe zero mode does not present nodes, ensuring the stability of the solution around small fluctuations. One may also conduct this investigation using a Schrodinger-like eigenvalue equation defined by the operator in (38). This can be done with the change of variables in Eq. (37), which leads us to y = -1 / ( zr z ) for z = 0 and y = ln( r ) and z = 0. For simplicity, we only explore the case z = 0. The stability potential associated to the Schrodinger-like equation is given by \n/negationslash \nU ( y ) = ( -zy ) -2( n + z ) /z (( 1 + 1 p )( 1 + 2 p ) T 2 ( y ) + ( 1 -1 p )( 1 -2 p ) T -2 ( y ) -2 ) + n 2 -z 2 4 z 2 y 2 (79) \n/negationslash \nin the case with n = 0, where we have used the notation T ( y ) = tanh ( ( -zy ) -n/z -( -zy c ) -n/z ) and also defined \ny c = -1 / ( zr z c ). For n = 0, we have \nU ( y ) = 4 p 2 z 2 y 2 ( ( -zy ) -4 / ( pz ) -( -zy c ) 4 / ( pz ) ) 2 × ( ( -zy ) -8 / ( pz ) +( -zy c ) -8 / ( pz ) -6 p ( -zy c ) -2 / ( pz ) ( -zy ) -6 / ( pz ) -6 p ( -zy c ) -6 / ( pz ) ( -zy ) -2 / ( pz ) +2(2 p 2 +3)( -zy c ) -4 / ( pz ) ( -zy ) -4 / ( pz ) ) -1 4 y 2 . (80) \nThe range of y for n = 0 depends on z ; z > 0 leads to y ∈ ( -∞ , 0] and z < 0 leads to y ∈ [0 , ∞ ). \n/negationslash \nNotice that, for general n , U ( y ) is controlled by p , z , d and θ c , as n = θ c ( d -1) / 2 -d -z +1. This makes the analysis with the general parameters quite intricate. So, to understand the behavior of the Schrodinger-like potential, we shall deal first with the situation where θ c = 0, which represents the Lifshitz spacetime, in three spatial dimensions ( d = 3), with p = 3. In this case, the condition θ c > 2 z requires that z must be negative; it also obeys n = -2 -z . For n > 0 ( n < 0), we must impose z < -2 ( -2 < z < 0). We display the Schrodinger-like potential in Fig. 7 for some values of the parameters. When one considers n > 0, the analysis at the origin depends on y c , which can only be finite. For y c = 0, the potential behaves as U ( y ) ≈ (2 z +3)( z +3) / ( z 2 y 2 ) near the origin, so it diverges to -∞ (+ ∞ ) if -3 < z < -2 ( z < -3); the behavior for z = -3 is special, with U ( y ) ≈ -2 / (3 y ) 4 / 3 , leading to negative divergence at y = 0. On the other hand, y c = 0 leads to U ( y ) → + ∞ for y ≤ y c . The asymptotic behavior is U ( y ) ≈ ( z +1) / ( z 2 y 2 ), which goes to zero as y increases. The case n < 0 leads to U ( y ) = (4 / 9)( -zy ) 4 /z for y ≈ 0, so it presents a positive divergence. For the limit y c → ∞ , the asymptotic behavior is U ( y ) ≈ (2 z + 3)( z + 3) / ( z 2 y 2 ), which tends to vanish as y gets larger and larger. We remark that the condition of normalization below Eq. (76) restricts z to the range -1 . 2 < z < 0 in the case y c → ∞ . Finite values of y c leads to divergence at y ≥ y c without adding restrictions to the values of z due to the condition of normalization. In Fig. 7, we display the Schrodingerlike potential (79) for some values of the parameters with positive and negative n . \n/negationslash \nLet us now analyze the case n = 0. Before going further, we take into account that the case θ c = 0 in three spatial dimensions does not allow for the existence of normalized zero modes (see the condition below (76)). Thus, to work the case n = 0 out, we take θ c = 0. In this situation, we have θ c = z +2, so z becomes the only parameter that modifies the model. Since the condition θ c > 2 z must be satisfied, we must impose z < 2 in d = 3. We work with p = 1, for which z ∈ ( -2 , 0) ∪ (0 , 3 / 2). In this specific case, the behavior of U ( y ) both near the origin and asymptotically is U ( y ) ≈ (16 -z 2 ) / (4 z 2 y 2 ). So, at y = 0, it diverges positively, and as y →±∞ , it vanishes. The Schrodinger-like potential U ( y ) for z = ± 1 \nFIG. 7. The Schrodinger potential U ( y ) in Eq. (79). In the top (bottom) panels, we depict the case n > 0 ( n < 0), for p = 3 and θ c = 0. In the top-left panel, we use y c = 0 and z = -3 . 2 , -3 . 1 , -3 , -2 . 9 and -2 . 8. In the top-right panel, we take z = -3 and y c = 0 , 1 / 192 , 1 / 24 , 9 / 64 and 1 / 3. In the bottomleft panel, we consider y c →∞ and z = -1 . 2 , -1 . 1 , -1 , -0 . 9 and -0 . 8. In the bottom-right panel, we have z = -1 and y c = 1 , 2 , 3 , 4 and y c → ∞ . The dashed lines represent the case y c = 0 in the top-right panel and y c →∞ in the bottomright panel. The dash-dotted lines denote the case z = -3 in the top-left panel, which delimits the value of z that separates the range supporting negative and positive divergences, and z = -1 . 2 in the bottom-left panel, which delimits the range z > -1 . 2 in which the zero mode can be normalized. The thickness of the lines increases with z in the left panels and with y c in the right ones. \n<!-- image --> \nin Eq. (80) is displayed in Fig. 8, where we can see the symmetry z →-z . \nThe number of modes depends on the value of n . With the chosen parameters, for positive values of n , the only surviving eigenstate of the Schrodinger-like equation is the zero mode. Negative values of n lead to a richer analysis, as a potential with the form of an infinite well arises for finite values of y c , so there are infinite bound states, while the limit y c → ∞ only supports the zero mode. For n = 0, the only bound state is the zero mode. The absence of modes with negative eigenvalues ensure that the model is stable around small fluctuations.', 'IV. CONCLUSIONS': "In this work, we have investigated the Maxwell-scalar model (1) in radially symmetric spacetimes (3), with the scalar and gauge fields coupled via the electric permittivity /epsilon1 ( φ ). To study this setup, we have developed a first-order framework based on the minimization of the energy compatible with the equations of motion. Interestingly, the procedure requires that the charge be exclusively given in terms of the factors of the line element. Also, the first-order equation (19) can be mapped into \nFIG. 8. The Schrodinger-like potential in Eq. (80) for p = 1. The dashed line represents θ c = 1 and z = -1, and the solid line stands for θ c = 3 and z = 1. \n<!-- image --> \nthe corresponding one that arises in the canonical scalar field model in (1 , 1) flat spacetime dimensions. This allowed us to obtain analytical results. \nWe then applied our method to some spacetimes. To get smooth localized structures, we have considered spacetimes with singularities at the origin in the dt 2 factor of the line element. First, we studied Tolman's metric VI [37] in Eq. (44), whose associated charge density presents a change of sign and the total charge is null. The scalar field solution connects two divergent values of the permittivity and engenders localized energy density. Its linear stability is described by a Schrodinger-like potential with the form of an infinite well whose zero mode is the ground state, ensuring stability. A similar analysis was done for the Exponential metric (56), in which we have found positive charge density and finite charge, but a distinct profile for the solutions and energy densities, which can become compact with Schrodinger-like potential having the form of an infinite well. We remark that the limit r c → ∞ breaks the compact demeanor of the quantities: both the solution and energy density become extended and there is no infinite well in the Schrodingerlike potential anymore. \nA third class that allows for the application of our method is the hyperscaling violating geometries, which is controlled by three parameters: z , θ c and d . To apply our formalism, we have taken θ c > 2 z . In this situation, the charge density is non negative and the total charge is divergent. We have defined the parameter n = θ c ( d -1) / 2 -d -z +1 that controls the solutions. For n > 0, one must take p ≥ 3 to ensure that the minima of the potential are connected by the field configurations. The solutions are constant inside the limited space r < r c and engender a kink-like profile outside this interval, falling off with an exponential tail; its energy density presents a hole inside the limited space that is encircled by a radially-symmetric disk which vanishes asymptotically. For n < 0, also with p ≥ 3, the solutions support a compact profile, with energy density that vanishes outside the compact space r ≤ r c . The case n = 0 is special, as it allows us to take p = 1, engendering a power-law solution with long range tail that is also present in the energy density. For all values of n , the energy is finite and \nis described by an analytical expression. We have investigated the stability to show the form of the Schrodingerlike potential and how it depends on n . Notwithstanding that, the model is linearly stable. \nAs perspectives, one may investigate the model (1) on other geometric backgrounds. For instance, one may seek field configurations in which the change of variables (20) maps r into compact spaces. An example of this is the Reissner-Nordstrom metric in (3 , 1) spacetime dimensions, with line element \nds 2 = f ( r ) dt 2 -1 f ( r ) dr 2 -r 2 dθ 2 -r 2 sin 2 θ dϕ 2 , (81) \nwhere f ( r ) = 1 -2 µ/r + q 2 /r 2 , and µ and q are parameters that obey q > µ to avoid the presence of (coordinate) singularities outside the origin in the metric. Within the first-order framework, Eqs. (17) relate the charge density to the metric, so the charge must be calculated from Eq. (23), which leads to finite charge, Q = 4 πe . The change of variables in Eq. (20) leads us to \nx ( r ) = 1 √ q 2 -µ 2 ( arctan ( r -µ √ q 2 -µ 2 ) -c 0 ) , (82) \nwhere c 0 = arctan ( ( r c -µ ) / √ q 2 -µ 2 ) was taken to make x = 0 at r = r c . The variable x ( r ) is compact, obeying x ∈ [0 , /lscript ], where /lscript = ( π/ 2 -c 0 ) / √ q 2 -µ 2 is the width of the interval, such that x = /lscript for r → ∞ . This suggests that one must consider scalar field models in (1 , 1) dimensions (governed by Eq. (21)) which support compact solutions. This type of solutions usually arise in the presence of V-shaped scalar potentials or in the limit in which the classical mass is infinite [48-50]. The extremal case, µ → q , requires special care because it leads to a non-compact x ( r ), but it also leads to gravitational BPS configurations which are interesting on their own [51]. \nThe spacetimes investigated in this work do not engender singularities outside the origin but do diverge at r → 0 to impose the condition Q (0) = 0. 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The case of braneworlds requires the inclusion of an extra spatial dimension of infinite extent, in a five-dimensional anti-de Sitter spacetime. This is similar to the case of the gauge/gravity duality, required to investigate strong interaction between quarks and gluons via holographic Einstein-Maxwell-scalar models [32, 33], and this brings more motivation to study braneworlds. In particular, we can think of braneworld scenarios with modified theories of gravity in the presence of non-constant curvature [61] and also, the addition of other scalar fields in a way similar to the recent work [62], in which the presence of several scalar fields was used to control the internal structure to the brane. Another possibility of current interest concerns changing the Maxwell field to a nonAbelian set of vector field as it occurs, for instance, in the SU (2) Yang-Mills case. A specific model of this kind was considered in Ref. [63], and may stimulate new investigation in this line of research. 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2024MNRAS.534.3783F

Energetic GeV photons expected from the closest and the most energetic Gammaray bursts GRBs provide a unique opportunity to study the veryhighenergy emission as well as the possible correlations with lower energy bands in realistic GRB afterglow models. In the standard GRB afterglow model the relativistic homogeneous shock is usually considered to be fully adiabatic however it could be partially radiative. Based on the external forwardshock scenario in both stellar wind and constantdensity medium we present a radiativeadiabatic analytical model of the synchrotron selfCompton SSC and synchrotron processes considering an electron energy distribution with a powerlaw index of inlineformulatexmath idTM0001 notationLaTeX1lt plt 2texmathinlineformula and inlineformulatexmath idTM0002 notationLaTeX2le ptexmathinlineformula. We show that the SSC scenario plays a relevant role in the radiative parameter inlineformulatexmath idTM0003 notationLaTeXepsilontexmathinlineformula leading to a prolonged evolution during the slow cooling regime. In a particular case we derive the FermiLAT light curves together with the photons with energies inlineformulatexmath idTM0004 notationLaTeXge 100texmathinlineformula MeV in a sample of nine bursts from the second FermiLAT GRB catalogue that exhibited temporal and spectral indices with inlineformulatexmath idTM0005 notationLaTeXgtrsim 1.5texmathinlineformula and inlineformulatexmath idTM0006 notationLaTeXapprox 2texmathinlineformula respectively. These events can hardly be described with closure relations of the standard synchrotron afterglow model and also exhibit energetic photons above the synchrotron limit. We have modelled the multiwavelength observations of our sample to constrain the microphysical parameters the circumburst density the bulk Lorentz factor and the mechanism responsible for explaining the energetic GeV photons.

2024-11-01T00:00:00Z

['10.48550/arXiv.2409.12166', '10.1093/mnras/stae2190', '2024MNRAS.tmp.2257F', '2024MNRAS.534.3783F', '2024arXiv240912166F', 'arXiv:2409.12166']

['Astrophysics - High Energy Astrophysical Phenomena']

Synchrotron selfCompton in a radiativeadiabatic fireball scenario modelling the multiwavelength observations in some FermiLAT bursts

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']

https://arxiv.org/pdf/2409.12166.pdf

{'Synchrotron self-Compton in a radiative-adiabatic fireball scenario: Modelling the multiwavelength observations in some Fermi /LAT bursts': 'Nissim Fraija , 1 ★ P. Veres , 2 B. Betancourt Kamenetskaia 3 , 4 A. Galvan-Gamez 1 M.G. Dainotti 5 , 6 , 7 , 8 9 10 \nSimone Dichiara , and R. L. Becerra \n- 1 Instituto de Astronomía, Universidad Nacional Autónoma de México, Circuito Exterior, C.U., A. Postal 70-264, 04510 Cd. de México, México.\n- 2 Center for Space Plasma and Aeronomic Research (CSPAR), University of Alabama in Huntsville, Huntsville, AL 35899, USA\n- 3 Technical University of Munich, TUM School of Natural Sciences, Physics Department, James-Franck-Str 1, 85748 Garching, Germany\n- 4 Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), Föhringer Ring 6, 80805 Munich, Germany\n- 5 Division of Science, National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan\n- 6 The Graduate University for Advanced Studies (SOKENDAI), 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan\n- 7 Space Science Institute, 4750 Walnut Street, Boulder, CO 80301, USA\n- 8 SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025, USA\n- 9 Department of Astronomy and Astrophysics, The Pennsylvania State University, 525 Davey Lab, University Park, PA 16802, USA\n- 10 Department of Physics, University of Rome - Tor Vergata, via della Ricerca Scientifica 1, 00100 Rome, IT \nAccepted XXX. Received YYY; in original form ZZZ', 'ABSTRACT': 'Energetic GeV photons expected from the closest and the most energetic Gamma-ray bursts (GRBs) provide an unique opportunity to study the very-high-energy emission as well as the possible correlations with lower energy bands in realistic GRB afterglow models. In the standard GRB afterglow model, the relativistic homogeneous shock is usually considered to be fully adiabatic, however, it could be partially radiative. Based on the external forward-shock scenario in both stellar wind and constant-density medium. We present a radiative-adiabatic analytical model of the synchrotron self-Compton (SSC) and synchrotron processes considering an electron energy distribution with a power-law index of 1 < 𝑝 < 2 and 2 ≤ 𝑝 . We show that the SSC scenario plays a relevant role in the radiative parameter 𝜖 , leading to a prolonged evolution during the slow cooling regime. In a particular case, we derive the Fermi /LAT light curves together with the photons with energies ≥ 100 MeV in a sample of nine bursts from the second Fermi /LAT GRB catalog that exhibited temporal and spectral indices with ≳ 1 . 5 and ≈ 2, respectively. These events can hardly be described with closure relations of the standard synchrotron afterglow model, and also exhibit energetic photons above the synchrotron limit. We have modeled the multi-wavelength observations of our sample to constrain the microphysical parameters, the circumburst density, the bulk Lorentz factor and the mechanism responsible for explaining the energetic GeV photons. \nKey words: gamma-ray burst : General - Physical data and processes : acceleration of particles - Physical data and processes radiation mechanisms: non-thermal', '1 INTRODUCTION': 'Gamma-ray bursts (GRBs) releasing ∼ 10 51 -10 55 erg of isotropic equivalent gamma-ray energy are the most energetic transient sources in the Universe. These transient events can last from a few milliseconds to several hours (Piran 1999; Kumar & Zhang 2015), and the duration of this prompt episode classifies GRBs into either short or long. The prompt emission is usually detected in the keV-MeV energy range (Kouveliotou et al. 1993) and is described by an empirical Band function (Band et al. 1993). The late and long-lasting emission, named \'afterglow", detected from radio to TeV gamma-rays, is usually interpreted within the fireball scenario (e.g. see Cavallo & Rees 1978). The fireball model predicts an external shock when a relativistic jet transfers a large part of its energy to the circumburst medium (Piran 1999). A fraction of the total energy is constantly transferred during the shock to accelerate electrons ( 𝜀 𝑒 ) amplifying the magnetic field ( 𝜀 𝐵 ). These electrons are cooled down mainly by synchrotron radiation emitting photons from radio to gamma-rays (Sari et al. 1998; Kumar & Barniol Duran 2009, 2010; Ackermann et al. 2013; Fraija et al. 2016a) and by very-high-energy (VHE ≥ 100 GeV) photons, which are radiated through the synchrotron self-Compton (SSC) mechanism (Fraija et al. 2019c,b; Zhang 2019). \nIn the standard GRB afterglow scenario, the relativistic forward shock (FS) is usually considered to be fully adiabatic, although it can be \npartially or fully radiative (Dai & Lu 1998; Sari et al. 1998; Böttcher & Dermer 2000; Ghisellini et al. 2010). The shock-accelerated electrons are in the fast-cooling regime when the dynamical timescale is larger than the cooling timescale. When accelerated electrons lie in the fast-cooling regime and also the microphysical parameter 𝜀 𝑒 is much greater than 0 . 1, the afterglow phase should be in the radiative regime instead of the adiabatic one (Böttcher & Dermer 2000; Li et al. 2002; Panaitescu 2019). Once the afterglow enters the slow-cooling regime, the hydrodynamic evolution can be approximated by the adiabatic case (e.g., see Moderski et al. 2000). The deceleration of the GRB fireball by the circumburst medium is faster when the fireball is radiative than when it is adiabatic. Therefore, the temporal afterglow evolution of synchrotron light curves and the shock energetics are modified (Böttcher & Dermer 2000; Wu et al. 2005). Ghisellini et al. (2010) studied the high-energy ( > 100 MeV) emission of 11 GRBs detected by the Fermi /LAT (Large Area Telescope) until 2009 October. They found that the temporal decay indices were consistent with synchrotron afterglow flux from a GRB fireball in the fully radiative regime. They proposed that a radiatively efficient fireball could explain the efficiency problem observed during the early afterglow (Zhang et al. 2007). \nAjello et al. (2019a) reported the second Fermi /LAT GRB catalog (2FLGC), covering 10 years (from 2008 to 2018) with a total of 169 bursts with high-energy photons above 100 MeV and 29 bursts above 10 GeV. In this sample, half of them (86/169) of LAT-detected bursts exhibited a temporally-extended component whereas 21/169 GRBs displayed a temporal break spanning from a few dozen to thousands of seconds. While the persistent emission is often attributed to standard synchrotron radiation originating from external FSs (Kumar & Barniol Duran 2009, 2010), it is worth noting that not all of the light curves detected by the LAT instrument adhere to the predicted closure relations between power-law (PL) temporal ( 𝛼 L ) and spectral ( Γ L ) indices (e.g., see Tak et al. 2019; Fraija et al. 2022a). This is the case of those GRBs that exhibit PL temporal and spectral indices of 𝛼 L ≳ 1 . 5 and Γ L ≈ 2, respectively. \nFraija et al. (2019b) derived the light curves and spectra of the SSC model for a wind and homogeneous medium for the adiabatic regime and an electron spectral index larger than 2 (2 < 𝑝 ). The authors showed that the SSC framework could explain photons beyond the synchrotron limit in GRB 190114C. In this paper, we consider the deceleration phase of a relativistic outflow in a stellar wind and homogeneous medium and derive the expected light curves and spectra of the synchrotron and SSC FS model in the radiative regime for an electron spectral index in the range of 1 < 𝑝 < 2 and 2 ≤ 𝑝 . The closure relations of the synchrotron and the SSC FS model as a function of the radiative parameter ( 𝜖 ) are presented. We apply the proposed model to a representative sample of GRBs reported in the 2FLGC with values of temporal and spectral indices with 𝛼 L ≳ 1 . 5 and Γ L ≈ 2, which can hardly be described with closure relations of the standard (non-radiative) synchrotron afterglow model, and also exhibit energetic photons above the synchrotron limit (e.g., see Ghisellini et al. 2010; Tak et al. 2019; Fraija et al. 2022a). The paper is arranged as follows. Section 2 presents the SSC FS model as a function of the radiative parameter when the afterglow model evolves in a stellar wind and in a homogeneous medium. In Section 3, we apply our current model to a representative sample of GRBs reported in the 2FLGC and discuss the results, and finally, in Section 4, we summarise our results and present our conclusions. The convention 𝑄 x = 𝑄 / 10 x in cgs units and the universal constants 𝑐 = ℏ = 1 in natural units will be adopted throughout this paper.', '2 SYNCHROTRON SELF-COMPTON AFTERGLOW MODEL': 'The long-lived afterglow emission is generated when a relativistic GRB outflow decelerates and drives a FS into the circumstellar medium. The outflow transfers a large amount of its energy to this surrounding external medium during the deceleration phase. Here, we extend the SSC afterglow model introduced in Fraija et al. (2019b), considering that the dynamics of the afterglow emission evolves in the radiative regime, and the electron distribution is described with a hard spectral index in the range of 1 < 𝑝 < 2. Additionally, we show the synchrotron light curves in the radiative regime for 1 < 𝑝 < 2 and 2 ≤ 𝑝 . We note that the self-absorption frequency does not affect the VHE emission.', '2.1 Dynamics and afterglow emission in a stellar-wind medium': "The dynamics of the afterglow emission for an outflow propagating into a stellar wind medium with a density profile 𝜌 ( 𝑟 ) ∝ 𝑟 -2 has been widely discussed in the particular case when it evolves in the adiabatic regime, and the electron distribution is described with 2 < 𝑝 (e.g., see Chevalier & Li 2000; Panaitescu & Mészáros 1998). Relativistic electrons are accelerated in the FS and efficiently cooled by synchrotron and SSC processes. In general, with the isotropic-equivalent kinetic energy 1 𝐸 K = 𝐸 GLYPH<16> Γ Γ 0 GLYPH<17> 𝜖 (Ghisellini et al. 2010; Fraija et al. 2023) and at a considerable distance 𝑅 2 𝑡 𝑧 \nfrom the progenitor = 2 Γ /( 1 + ) , the evolution of the bulk Lorentz factor in the adiabatic and radiative regime considering the Blandford- \n1 \nMcKee solution ( 𝐸 K ∝ ( 2 -𝜖 ) 𝜌 ( 𝑟 ) 𝑟 3 Γ 2 ) (Blandford & McKee 1976) is Γ = 92 . 6 GLYPH<16> 1 + 𝑧 1 . 1 GLYPH<17> 4 -𝜖 𝐴 -1 4 -𝜖 W , -1 𝐸 1 4 -𝜖 53 Γ -𝜖 4 -𝜖 0 , 2 𝑡 -1 4 -𝜖 4 . 7 . The term 𝐴 W is the parameter of wind density (Dai & Lu 1998; Chevalier & Li 2000; Vink & de Koter 2005), and Γ 0 is the initial Lorentz factor. The parameter 𝜖 takes into account the hydrodynamic evolution of the FS in the fully adiabatic ( 𝜖 = 0), fully radiative ( 𝜖 = 1) or partially radiative or adiabatic (0 < 𝜖 < 1) regimes (Böttcher & Dermer 2000; Wu et al. 2005; Ghisellini et al. 2010). The Lorentz factors of the lowest-energy electrons are 𝛾 m = h ˜ 𝑔 𝑚 𝑝 𝑚 𝑒 𝜀 𝑒 Γ 𝛾 M i 1 𝑝 -1 with ˜ 𝑔 = 2 -𝑝 𝑝 -1 for 1 < 𝑝 < 2 and 𝛾 m = 𝑔 𝑚 𝑝 𝑚 𝑒 𝜀 𝑒 Γ for 2 ≤ 𝑝 , In this last case, 𝑔 = ln GLYPH<16> 𝛾 M 𝛾 m GLYPH<17> for 𝑝 = 2 and 𝑔 = 𝑝 -2 𝑝 -1 for 2 < 𝑝 (Dai & Cheng 2001). The term 𝛾 M corresponds to the maximum electron Lorentz factor, and 𝑚 p and 𝑚 e to the proton and electron mass, respectively. The Lorentz factor above which the electrons cool efficiently is 𝛾 c = 6 𝜋𝑚 𝑒 𝑐 𝜎 𝑇 ( 1 + 𝑌 ) -1 Γ -1 𝐵 '-2 𝑡 -1 where 𝐵 ' is the \ncomoving magnetic field in the blastwave, 𝑐 is the speed of light, 𝜎 𝑇 is the Thompson cross section and 𝑌 is the Compton parameter (e.g., see Fraija et al. 2023). When SSC emission becomes the dominant one in electron cooling (as compact radio sources; see Tsang & Kirk 2007) , the electron population preferentially decreases the amount of energy available on 𝑈 SSC , 1st rather than 𝑈 syn, where 𝑈 syn and 𝑈 SSC , 1st are the energy density of synchrotron emission and the first-order scattering, respectively (e.g., see Kumar & Panaitescu 2008; Kobayashi et al. 2007; Petropoulou et al. 2015; Fraija & Veres 2018). In this case, it is needed to take into account the second-order Compton scatterings and therefore, the Lorentz factor above which the electrons cool efficiently is reduced by ( 1 + 𝑌 + 𝑌 2 ) , where the Compton parameter of second-order scattering is 𝑌 2 ≡ 𝑈 IC , 1st 𝑈 B = 𝑈 SSC , 1st 𝑈 syn 𝑈 syn 𝑈 B with 𝑈 B the energy density of magnetic field (Kobayashi et al. 2007; Fraija & Veres 2018). It is worth noting that the Klein-Nishina (KN) effect allows us to ignore higher order (three or more) scattering. For a particular analysis and discussion of the range of parameter space considering this effect, see Petropoulou et al. (2015); Fraija & Veres (2018). We have adopted the unprimed and primed quantities to denote them in the observer and comoving frames, respectively. The Lorentz factors of the lowest-energy electrons and the Lorentz factor of the higher-energy electrons that cool efficiently by synchrotron process are given by \n𝛾 m = 4 . 9 × 10 2 ˜ 𝑔 1 𝑝 -1 GLYPH<16> 1 + 𝑧 1 . 1 GLYPH<17> 8 + 𝑝 ( 𝜖 -3 ) -2 𝜖 2 ( 4 -𝜖 ) ( 𝑝 -1 ) 𝜀 1 𝑝 -1 𝑒, -1 𝜀 -( 𝑝 -2 ) 4 ( 𝑝 -1 ) 𝐵, -4 𝐴 8 + 𝑝 ( 𝜖 -6 ) -2 𝜖 4 ( 𝑝 -1 ) ( 4 -𝜖 ) W , -1 𝐸 𝑝 2 ( 𝑝 -1 ) ( 4 -𝜖 ) 53 Γ -𝑝𝜖 2 ( 4 -𝜖 ) ( 𝑝 -1 ) 0 , 2 𝑡 2 𝜖 + 𝑝 ( 3 -𝜖 ) -8 2 ( 4 -𝜖 ) ( 𝑝 -1 ) 4 . 7 for 1 < 𝑝 < 2 1 . 6 × 10 3 𝑔 GLYPH<16> 1 + 𝑧 1 . 1 GLYPH<17> 1 4 -𝜖 𝜀 𝑒, -1 𝐴 -1 4 -𝜖 W , -1 𝐸 1 4 -𝜖 53 Γ -𝜖 4 -𝜖 0 , 2 𝑡 -1 4 -𝜖 4 . 7 for 2 ≤ 𝑝 (1) \n𝛾 c = 3 . 3 × 10 3 ( 1 + 𝑌 ( 𝛾 c ) ) -1 GLYPH<18> 1 + 𝑧 1 . 1 GLYPH<19> 𝜖 -3 4 -𝜖 𝜀 -1 𝐵, -4 𝐴 𝜖 -5 4 -𝜖 W , -1 𝐸 1 4 -𝜖 53 Γ -𝜖 4 -𝜖 0 , 2 𝑡 3 -𝜖 4 -𝜖 4 . 7 . \nHereafter, we use the spectral index 𝑝 = 1 . 9 ( ˜ 𝑔 ≃ 0 . 1) for 1 < 𝑝 < 2 and 𝑝 = 2 . 1 ( 𝑔 ≃ 0 . 09) for 2 < 𝑝 to estimate the proportionality constant in each quantity. The terms 𝜖 and the Compton parameter ( 𝑌 ) are defined in subsections 2.3 and 2.4, respectively. The transition time from the fast- to the slow-cooling regime is (i.e., see Böttcher & Dermer 2000; Wu et al. 2005) \n𝑡 cm = 8 . 4 × 10 3 s ˜ 𝑔 2 ( 4 -𝜖 ) 𝑆 1 GLYPH<16> 1 + 𝑧 1 . 1 GLYPH<17> ( 1 + 𝑌 ( 𝛾 c ) ) 2 ( 𝑝 -1 ) ( 4 -𝜖 ) 𝑆 1 𝜀 2 ( 4 -𝜖 ) 𝑆 1 𝑒, -1 𝜀 ( 3 𝑝 -2 ) ( 4 -𝜖 ) 2 𝑆 1 𝐵, -4 𝐴 14 𝑝 -12 + 𝜖 ( 2 -3 𝑝 ) 2 𝑆 1 𝑊, -1 𝐸 -𝑝 -2 𝑆 1 53 Γ 𝜖 ( 𝑝 -2 ) 𝑆 1 0 , 2 for 1 < 𝑝 < 2 2 . 4 × 10 4 s 𝑔 GLYPH<16> 1 + 𝑧 1 . 1 GLYPH<17> ( 1 + 𝑌 ( 𝛾 c ) ) 𝜀 𝑒, -1 𝜀 𝐵, -4 𝐴 𝑊, -1 for 2 ≤ 𝑝 , \n where 𝑆 1 = 3 𝑝 + 2 -𝑝𝜖 . Given the total number of emitting electrons ( 𝑁 𝑒 ), and the synchrotron radiation power ( 𝑃 𝜈 ; Panaitescu & Mészáros 1998; Sari et al. 1998; Chevalier & Li 2000), the spectral breaks ( 𝜈 syn m , c = 𝑞 𝑒 2 𝜋𝑚 𝑒 𝑐 ( 1 + 𝑧 ) -1 Γ 𝛾 2 m , c 𝐵 ' with 𝑞 𝑒 the electron charge) and the maximum flux ( 𝐹 syn = ( 1 + 𝑧 ) 2 𝑁 𝑒 𝑃 𝜈 ) in the synchrotron scenario can be written as \nmax 4 𝜋𝐷 2 𝑧 \n𝜈 syn m = 4 . 2 × 10 11 Hz ˜ 𝑔 2 𝑝 -1 GLYPH<16> 1 + 𝑧 1 . 1 GLYPH<17> 8 -3 𝑝 -2 𝜖 + 𝑝𝜖 ( 4 -𝜖 ) ( 𝑝 -1 ) 𝜀 2 𝑝 -1 𝑒, -1 𝜀 1 2 ( 𝑝 -1 ) 𝐵, -4 𝐴 4 -2 𝑝 -𝜖 2 ( 𝑝 -1 ) ( 4 -𝜖 ) W , -1 𝐸 𝑝 ( 𝑝 -1 ) ( 4 -𝜖 ) 53 Γ -𝑝𝜖 ( 4 -𝜖 ) ( 𝑝 -1 ) 0 , 2 𝑡 𝜖 -𝑝 -4 ( 4 -𝜖 ) ( 𝑝 -1 ) 4 . 7 for 1 < 𝑝 < 2 4 . 2 × 10 12 Hz 𝑔 2 GLYPH<16> 1 + 𝑧 1 . 1 GLYPH<17> 2 4 -𝜖 𝜀 1 2 𝐵, -4 𝜀 2 𝑒, -1 𝐴 -𝜖 2 ( 4 -𝜖 ) W , -1 𝐸 2 4 -𝜖 53 Γ -2 𝜖 4 -𝜖 0 , 2 𝑡 𝜖 -6 4 -𝜖 4 . 7 for 2 ≤ 𝑝 \n𝜈 syn c = 6 . 8 × 10 12 Hz ( 1 + 𝑌 ( 𝛾 c ) ) -2 GLYPH<18> 1 + 𝑧 1 . 1 GLYPH<19> 2 ( 𝜖 -3 ) 4 -𝜖 𝜀 -3 2 𝐵, -4 𝐴 3 𝜖 -16 2 ( 4 -𝜖 ) W , -1 𝐸 2 4 -𝜖 53 Γ -2 𝜖 4 -𝜖 0 , 2 𝑡 2 -𝜖 4 -𝜖 4 . 7 \n𝐹 syn max = 9 . 1 × 10 3 mJy GLYPH<18> 1 + 𝑧 1 . 1 GLYPH<19> 2 ( 5 -𝜖 ) 4 -𝜖 𝜀 1 2 𝐵, -4 𝐷 -2 z , 27 𝐴 8 -3 𝜖 2 ( 4 -𝜖 ) W , -1 𝐸 2 4 -𝜖 53 Γ -2 𝜖 4 -𝜖 0 , 2 𝑡 -2 4 -𝜖 4 . 7 , \nwhere the term 𝐷 z corresponds to the luminosity distance, which is estimated using the cosmological parameters reported in Planck \nCollaboration et al. (2018). Given the maximum Lorentz factor of the electron distribution 𝛾 max = GLYPH<16> 3 𝑞 𝑒 𝜉 𝜎 𝑇 𝐵 '-1 GLYPH<17> 1 2 with 𝜉 the Bohm parameter 2 (Piran & Nakar 2010), the evolution of the maximum energy photon radiated by the synchrotron process in the stellar-wind medium ( ℎ𝜈 syn max = 3 𝑞 2 𝑒 2 𝜋𝜎 𝑇 𝑚 𝑒 𝑐 ( 1 + 𝑧 ) -1 Γ ) can be written as (Fraija et al. 2024) \nℎ𝜈 syn max = 0 . 6 GeV GLYPH<18> 1 + 𝑧 1 . 1 GLYPH<19> 𝜖 -3 4 -𝜖 𝐴 -1 4 -𝜖 W , -1 𝐸 1 4 -𝜖 53 Γ -𝜖 4 -𝜖 0 , 2 𝑡 -1 4 -𝜖 4 . 7 . (2) \nUsing Eqs. (2.1) and the synchrotron spectra for the fast- and slow-cooling regimes (Sari et al. 1998), we find that the synchrotron light curves at an observed frequency 𝜈 and a given time 𝑡 for 1 < 𝑝 < 2 and 2 ≤ 𝑝 evolve as \n𝐹 syn 𝜈 ∝ ( 1 < 𝑝 < 2 ) ( 2 ≤ 𝑝 ) { 𝑡 𝜖 -8 3 ( 4 -𝜖 ) , 𝑡 -5 𝑝 + 𝜖 -10 3 ( 𝑝 -1 ) ( 4 -𝜖 ) } 𝜈 1 3 { 𝑡 𝜖 -8 3 ( 4 -𝜖 ) , 𝑡 -𝜖 3 ( 4 -𝜖 ) } 𝜈 1 3 , for 𝜈 < { 𝜈 syn c , 𝜈 syn m } , 𝑡 -𝜖 + 2 2 ( 4 -𝜖 ) 𝜈 -1 2 , 𝑡 -𝜖 + 2 2 ( 4 -𝜖 ) 𝜈 -1 2 for 𝜈 syn c < 𝜈 < 𝜈 syn m , 𝑡 𝜖 -𝑝 -8 2 ( 4 -𝜖 ) 𝜈 -𝑝 -1 2 , 𝑡 2 -6 𝑝 -𝜖 + 𝑝𝜖 2 ( 4 -𝜖 ) 𝜈 -𝑝 -1 2 for 𝜈 syn m < 𝜈 < 𝜈 syn c , 𝑡 -𝑝 + 6 2 ( 4 -𝜖 ) 𝜈 -𝑝 2 𝑡 4 + 𝑝 ( 𝜖 -6 ) -2 𝜖 2 ( 4 -𝜖 ) 𝜈 -𝑝 2 , for { 𝜈 syn m , 𝜈 syn c } < 𝜈 . \nIn order to show the evolution of the rest of parameters as a function of 𝜖 , e.g. for { 𝜈 syn m , 𝜈 syn c } < 𝜈 the synchrotron flux yields", '4 N. Fraija et al.': '𝐹 syn 1 <𝑝< 2 ∝ 𝑔 ( 𝑝 ) ( 1 + 𝑧 ) 22 + 𝑝 ( 𝜖 -3 ) -4 𝜖 2 ( 4 -𝜖 ) ( 1 + 𝑌 ( 𝛾 c ) ) -1 𝐴 2 ( 1 -𝜖 ) -𝑝 2 ( 4 -𝜖 ) W 𝐷 -2 𝑧 𝜖 𝑒 𝐸 𝑝 + 6 2 ( 4 -𝜖 ) Γ -𝜖 ( 𝑝 + 6 ) 2 ( 4 -𝜖 ) 0 , 𝐹 syn 2 ≤ 𝑝 ∝ 𝑔 ( 𝑝 ) 𝑝 -1 ( 1 + 𝑧 ) 6 + 𝑝 -𝜖 4 -𝜖 ( 1 + 𝑌 ( 𝛾 c ) ) -1 𝐴 -𝜖 ( 𝑝 + 2 ) 4 ( 4 -𝜖 ) W 𝐷 -2 𝑧 𝜖 𝑝 -2 4 𝐵 𝜖 𝑝 -1 𝑒 𝐸 𝑝 + 2 4 -𝜖 Γ -𝜖 ( 𝑝 + 2 ) 4 -𝜖 0 . It is worth noting that for 𝑝 ≃ 2, synchrotron fluxes for 1 < 𝑝 < 2 and 2 ≤ 𝑝 are equal (i.e., 𝐹 syn 1 <𝑝< 2 ≃ 𝐹 syn 2 ≤ 𝑝 \n). \nThe SSC process occurs when the same electron population that radiates synchrotron photons up-scatters them to higher energies as ℎ𝜈 ssc k ∼ 𝛾 2 k ℎ𝜈 syn k . Here, the notation k = m , c refers to the minimum and cooling frequencies and ℎ stands for the Planck constant (e.g., see Sari &Esin 2001). The maximum flux that the SSC process can reach 𝐹 ssc max ∼ 𝜏𝐹 syn max depends on the maximum synchrotron flux given in Eqs. (2.1) and the optical depth 𝜏 ∝ 1 3 𝐴 W 𝑅 -1 . Therefore, the spectral breaks and the maximum flux in the SSC scenario for 1 < 𝑝 < 2 and 2 ≤ 𝑝 are (e.g., see Fraija et al. 2019b, 2022a) \nℎ𝜈 ssc m = 4 . 2 × 10 2 eV ˜ 𝑔 4 𝑝 -1 GLYPH<16> 1 + 𝑧 1 . 1 GLYPH<17> 2 [ 8 + 𝑝 ( 𝜖 -3 ) -2 𝜖 ] ( 4 -𝜖 ) ( 𝑝 -1 ) 𝜀 4 𝑝 -1 𝑒, -1 𝜀 3 -𝑝 2 ( 𝑝 -1 ) 𝐵, -4 𝐴 𝑝 ( 𝜖 -8 )+ 3 ( 4 -𝜖 ) 2 ( 𝑝 -1 ) ( 4 -𝜖 ) W , -1 𝐸 2 𝑝 ( 𝑝 -1 ) ( 4 -𝜖 ) 53 Γ -2 𝑝𝜖 ( 4 -𝜖 ) ( 𝑝 -1 ) 0 , 2 𝑡 3 ( 𝜖 -4 ) -𝑝 ( 𝜖 -2 ) ( 4 -𝜖 ) ( 𝑝 -1 ) 4 . 7 for 1 < 𝑝 < 2 4 . 1 × 10 4 eV 𝑔 4 GLYPH<16> 1 + 𝑧 1 . 1 GLYPH<17> 4 4 -𝜖 𝜀 1 2 𝐵, -4 𝜀 4 𝑒, -1 𝐴 -4 + 𝜖 2 ( 4 -𝜖 ) W , -1 𝐸 4 4 -𝜖 53 Γ -4 𝜖 4 -𝜖 0 , 2 𝑡 𝜖 -8 4 -𝜖 4 . 7 for 2 ≤ 𝑝 \nℎ𝜈 ssc c = 8 . 57 × 10 -3 GeV ( 1 + 𝑌 ( 𝛾 c ) ) -4 GLYPH<18> 1 + 𝑧 1 . 1 GLYPH<19> 4 ( 𝜖 -3 ) 4 -𝜖 𝜀 -7 2 𝐵, -4 𝐴 7 𝜖 -36 2 ( 4 -𝜖 ) W , -1 𝐸 4 4 -𝜖 53 Γ -4 𝜖 4 -𝜖 0 , 2 𝑡 8 -3 𝜖 4 -𝜖 4 . 7 . (3) \n𝐹 ssc max = 4 . 6 × 10 -5 mJy 𝑔 -1 GLYPH<18> 1 + 𝑧 1 . 1 GLYPH<19> 3 𝜀 1 2 𝐵, -4 𝐷 -2 z , 27 𝐴 5 2 W , -1 𝑡 -1 4 . 7 \nGiven the evolution of the spectral breaks and the maximum flux (Eqs. 3), it is possible to write the SSC light curves at an observed frequency 𝜈 and a given time 𝑡 as (e.g., see Fraija et al. 2019b, 2022a) \n𝐹 ssc 𝜈 ∝ ( 1 < 𝑝 < 2 ) ( 2 ≤ 𝑝 ) { 𝑡 2 ( 3 𝜖 -10 ) 3 ( 4 -𝜖 ) , 𝑡 2 [ 12 -7 𝑝 + 𝜖 ( 2 𝑝 -3 ) ] 3 ( 𝑝 -1 ) ( 4 -𝜖 ) } 𝜈 1 3 , { 𝑡 2 ( 3 𝜖 -10 ) 3 ( 4 -𝜖 ) , 𝑡 2 ( 𝜖 -2 ) 3 ( 4 -𝜖 ) } 𝜈 1 3 , for 𝜈 < { 𝜈 ssc c , 𝜈 ssc m } , 𝑡 -𝜖 2 ( 4 -𝜖 ) 𝜈 -1 2 , 𝑡 -𝜖 2 ( 4 -𝜖 ) 𝜈 -1 2 for 𝜈 ssc c < 𝜈 < 𝜈 ssc m , 𝑡 2 𝑝 + 5 𝜖 -20 -𝑝𝜖 2 ( 4 -𝜖 ) 𝜈 -𝑝 -1 2 , 𝑡 𝑝 ( 𝜖 -8 )+ 𝜖 2 ( 4 -𝜖 ) 𝜈 -𝑝 -1 2 for 𝜈 ssc m < 𝜈 < 𝜈 ssc c , 𝑡 2 ( 𝜖 -6 ) -𝑝 ( 𝜖 -2 ) 2 ( 4 -𝜖 ) 𝜈 -𝑝 2 , 𝑡 8 + 𝑝 ( 𝜖 -8 ) -2 𝜖 2 ( 4 -𝜖 ) 𝜈 -𝑝 2 for { 𝜈 ssc m , 𝜈 ssc c } < 𝜈 . \nIn order to show the evolution of the rest of parameters as a function of 𝜖 , e.g. for { 𝜈 ssc m , 𝜈 ssc c } < 𝜈 , the SSC flux yields \nssc 1 <𝑝< 2 ∝ 𝑔 ( 𝑝 ) ( 1 + 𝑧 ) 14 + 𝑝 ( 𝜖 -3 )+ 3 𝜖 4 -𝜖 ( 1 + 𝑌 ( 𝛾 c ) ) -2 𝐴 8 ( 2 -𝑝 )+ 𝜖 ( 𝑝 -6 ) 4 ( 4 -𝜖 ) W 𝐷 -2 𝑧 𝜖 -𝑝 + 2 4 𝐵 𝜖 2 𝑒 𝐸 𝑝 + 2 4 -𝜖 Γ -( 𝑝 + 2 ) 𝜖 4 -𝜖 0 𝐹 ssc 2 ≤ 𝑝 ∝ 𝑔 ( 𝑝 ) 2 𝑝 -3 ( 1 + 𝑧 ) 4 + 2 𝑝 -𝜖 4 -𝜖 ( 1 + 𝑌 ( 𝛾 c ) ) -2 𝐴 4 ( 2 -𝑝 ) -𝜖 ( 𝑝 + 2 ) 4 ( 4 -𝜖 ) W 𝐷 -2 𝑧 𝜖 𝑝 -6 4 𝐵 𝜖 2 ( 𝑝 -1 ) 𝑒 𝐸 2 𝑝 4 -𝜖 Γ -2 𝜖 𝑝 4 -𝜖 0 . We can see that SSC fluxes for 1 < 𝑝 < 2 and 2 ≤ 𝑝 are equal (i.e., 𝐹 ssc <𝑝< ≃ 𝐹 ssc 𝑝 ) when 𝑝 ≈ \n1 2 2 ≤ 2. \nIt is worth noting that very energetic photons of energy ℎ𝜈 h ≈ 1 TeV interacting with low-energy photons ℎ𝜈 l = 60 . 1 eV GLYPH<16> Γ 1 1 + 𝑧 GLYPH<17> 2 1 ( ℎ𝜈 h / 1 TeV ) are absorbed to produce pairs (e.g., see Fraija et al. 2019c). The optical depth of attenuation of this interaction process is 𝜏 𝛾𝛾 = 1 . 8 × 10 -8 𝑅 17 Γ 1 𝑛 𝛾, 5 . 3 , where 𝑛 𝛾 ≈ 2 . 4 × 10 3 cm -3 𝐿 𝛾, 44 𝑅 2 17 Γ 1 ( ℎ𝜈 l / 60 . 1 eV ) and 𝐿 𝛾, 44 are the photon density and luminosity of the seed photons, respectively (e.g., see Piran 1999).', '2.2 Dynamics and afterglow emission in a uniform-density medium': "Once the outflow begins to be decelerated at a significant distance from the progenitor by a uniform-density medium ( 𝜌 = 𝑛 ), the evolution of the bulk Lorentz factor in the adiabatic and radiative regimes considering the Blandford-McKee solution ( 𝐸 K = 2 𝜋 3 ( 2 -𝜖 ) 𝑚 𝑝 𝑐 2 𝑛𝑟 3 Γ 2 ) \n(Blandford & McKee 1976) becomes Γ = 17 . 0 GLYPH<16> 1 + 𝑧 1 . 1 GLYPH<17> 3 8 -𝜖 𝑛 -1 8 -𝜖 𝐸 1 8 -𝜖 53 Γ -𝜖 8 -𝜖 0 , 2 𝑡 -3 8 -𝜖 4 . 7 (Ghisellini et al. 2010; Fraija et al. 2023) . In the case of a uniform-density medium, the Lorentz factors of both, the lowest-energy electrons and higher-energy electrons that cool efficiently by synchrotron process evolve as (i.e., see Dai & Cheng 2001; Sari et al. 1998; Fraija et al. 2023) \n𝛾 m = 9 . 3 × 10 ˜ 𝑔 1 𝑝 -1 GLYPH<16> 1 + 𝑧 1 . 1 GLYPH<17> 3 ( 4 -𝑝 ) 2 ( 8 -𝜖 ) ( 𝑝 -1 ) 𝜀 1 𝑝 -1 𝑒, -1 𝜀 -( 𝑝 -2 ) 4 ( 𝑝 -1 ) 𝐵, -4 𝑛 8 -6 𝑝 + 𝑝𝜖 -2 𝜖 4 ( 𝑝 -1 ) ( 8 -𝜖 ) 𝐸 4 -𝑝 2 ( 𝑝 -1 ) ( 8 -𝜖 ) 53 Γ -𝜖 ( 4 -𝑝 ) 2 ( 8 -𝜖 ) ( 𝑝 -1 ) 0 , 2 𝑡 -3 ( 4 -𝑝 ) 2 ( 8 -𝜖 ) ( 𝑝 -1 ) 4 . 7 for 1 < 𝑝 < 2 3 . 4 × 10 2 𝑔 GLYPH<16> 1 + 𝑧 1 . 1 GLYPH<17> 3 8 -𝜖 𝜀 𝑒, -1 𝑛 -1 8 -𝜖 𝐸 1 8 -𝜖 53 Γ -𝜖 8 -𝜖 0 , 2 𝑡 -3 8 -𝜖 4 . 7 for 2 ≤ 𝑝 (5) \n𝛾 c = 1 . 6 × 10 3 ( 1 + 𝑌 ( 𝛾 c ) ) -1 GLYPH<18> 1 + 𝑧 1 . 1 GLYPH<19> -1 + 𝜖 8 -𝜖 𝜀 -1 𝐵, -4 𝑛 𝜖 -5 8 -𝜖 𝐸 -3 8 -𝜖 53 Γ 3 𝜖 8 -𝜖 0 , 2 𝑡 1 + 𝜖 8 -𝜖 4 . 7 . (6) \n(4) \n𝐹 \nThe terms 𝜖 and 𝑌 are defined in subsections 2.4 and 2.3, respectively. In this afterglow model, the transition time from the fast- to the slow-cooling regime takes place at (i.e., see Böttcher & Dermer 2000; Wu et al. 2005) \n𝑡 cm = 4 . 8 × 10 2 s ˜ 𝑔 2 ( 8 -𝜖 ) 𝑆 2 GLYPH<16> 1 + 𝑧 1 . 1 GLYPH<17> ( 1 + 𝑌 ( 𝛾 c ) ) 2 ( 𝑝 -1 ) ( 8 -𝜖 ) 𝑆 2 𝜀 2 ( 8 -𝜖 ) 𝑆 2 𝑒, -1 𝜀 ( 3 𝑝 -2 ) ( 8 -𝜖 ) 2 𝑆 2 𝐵, -4 𝑛 14 𝑝 -12 + 𝜖 ( 2 -3 𝑝 ) 2 𝑆 2 𝐸 5 𝑝 -2 𝑆 2 53 Γ 𝜖 ( 2 -5 𝑝 ) 𝑆 2 0 , 2 for 1 < 𝑝 < 2 3 . 0 × 10 3 s 𝑔 8 -𝜖 4 + 𝜖 GLYPH<16> 1 + 𝑧 1 . 1 GLYPH<17> ( 1 + 𝑌 ( 𝛾 c ) ) 8 -𝜖 4 + 𝜖 𝜀 8 -𝜖 4 + 𝜖 𝑒, -1 𝜀 8 -𝜖 4 + 𝜖 𝐵, -4 𝑛 4 -𝜖 4 + 𝜖 𝐸 4 4 + 𝜖 53 Γ -4 𝜖 4 + 𝜖 0 , 2 for 2 ≤ 𝑝, \nwhere 𝑆 2 = 10 -𝑝 + 2 𝜖 ( 𝑝 -1 ) . In this case, given the electron Lorentz factors (Eq. 5), the synchrotron spectral breaks ( 𝜈 syn m , c ∝ ( 1 + 𝑧 ) -1 Γ 𝛾 2 m , c 𝐵 ' ) and the maximum flux ( 𝐹 syn max ∝ ( 1 + 𝑧 ) 2 𝐷 -2 𝑧 𝑁 𝑒 𝑃 𝜈 ) become \n𝜈 syn m = 9 . 1 × 10 9 Hz ˜ 𝑔 2 𝑝 -1 GLYPH<16> 1 + 𝑧 1 . 1 GLYPH<17> 14 -5 𝑝 + 𝜖 ( 𝑝 -1 ) ( 8 -𝜖 ) ( 𝑝 -1 ) 𝜀 2 𝑝 -1 𝑒, -1 𝜀 1 2 ( 𝑝 -1 ) 𝐵, -4 𝑛 4 -2 𝑝 -𝜖 2 ( 𝑝 -1 ) ( 8 -𝜖 ) 𝐸 𝑝 + 2 ( 𝑝 -1 ) ( 8 -𝜖 ) 53 Γ -𝜖 ( 𝑝 + 2 ) ( 8 -𝜖 ) ( 𝑝 -1 ) 0 , 2 𝑡 -3 ( 𝑝 + 2 ) ( 8 -𝜖 ) ( 𝑝 -1 ) 4 . 7 for 1 < 𝑝 < 3 . 5 × 10 11 Hz 𝑔 2 GLYPH<16> 1 + 𝑧 1 . 1 GLYPH<17> 4 + 𝜖 8 -𝜖 𝜀 2 𝑒, -1 𝜀 1 2 𝐵, -4 𝑛 -𝜖 2 ( 8 -𝜖 ) 𝐸 4 8 -𝜖 53 Γ -4 𝜖 8 -𝜖 0 , 2 𝑡 -12 8 -𝜖 4 . 7 for 2 ≤ 𝑝 \n𝜈 syn c = 2 . 2 × 10 12 Hz ( 1 + 𝑌 ( 𝛾 c ) ) -2 GLYPH<18> 1 + 𝑧 1 . 1 GLYPH<19> -( 4 + 𝜖 ) 8 -𝜖 𝜀 -3 2 𝐵, -4 𝑛 3 𝜖 -16 2 ( 8 -𝜖 ) 𝐸 -4 8 -𝜖 53 Γ 4 𝜖 8 -𝜖 0 , 2 𝑡 2 ( 𝜖 -2 ) 8 -𝜖 4 . 7 𝐹 syn max = 8 . 1 × 10 mJy GLYPH<18> 1 + 𝑧 . GLYPH<19> 𝜖 + 16 8 -𝜖 𝜀 1 2 𝐵, -4 𝐷 -2 z , 27 𝑛 8 -3 𝜖 2 ( 8 -𝜖 ) 𝐸 8 8 -𝜖 53 Γ -8 𝜖 8 -𝜖 0 , 2 𝑡 -3 𝜖 8 -𝜖 4 . 7 . \n 2 (7) \n1 1 \nThe evolution of the maximum energy photon radiated by the synchrotron process in a homogeneous medium ( ℎ𝜈 syn max ∝ ( 1 + 𝑧 ) -1 Γ ) can be written as (Fraija et al. 2024) \nℎ𝜈 syn max = 0 . 2 GeV GLYPH<18> 1 + 𝑧 1 . 1 GLYPH<19> 𝜖 -5 8 -𝜖 𝑛 -1 8 -𝜖 𝐸 1 8 -𝜖 53 Γ -𝜖 8 -𝜖 0 , 2 𝑡 -3 8 -𝜖 4 . 7 . (8) \nUsing Eqs. 7, the synchrotron light curves at a specific time 𝑡 and frequency 𝜈 for 1 < 𝑝 < 2 and 2 ≤ 𝑝 can be written as (i.e., see Böttcher &Dermer 2000; Wu et al. 2005) \n𝐹 syn 𝜈 ∝ ( 1 < 𝑝 < 2 ) ( 2 ≤ 𝑝 ) { 𝑡 4 -11 𝜖 3 ( 8 -𝜖 ) , 𝑡 ( 𝑝 + 2 ) -3 𝜖 ( 𝑝 -1 ) ( 8 -𝜖 ) ( 𝑝 -1 ) } 𝜈 1 3 , { 𝑡 4 -11 𝜖 3 ( 8 -𝜖 ) , 𝑡 4 -3 𝜖 8 -𝜖 } 𝜈 1 3 for 𝜈 < { 𝜈 syn c , 𝜈 syn m } , 𝑡 -2 ( 𝜖 + 1 ) 8 -𝜖 𝜈 -1 2 , 𝑡 -2 ( 𝜖 + 1 ) 8 -𝜖 𝜈 -1 2 for 𝜈 syn c < 𝜈 < 𝜈 syn m , 𝑡 -3 ( 𝑝 + 2 + 2 𝜖 ) 2 ( 8 -𝜖 ) 𝜈 -𝑝 -1 2 , 𝑡 3 ( 2 -2 𝑝 -𝜖 ) 8 -𝜖 𝜈 -𝑝 -1 2 , for 𝜈 syn m < 𝜈 < 𝜈 syn c , 𝑡 -3 𝑝 + 10 + 4 𝜖 2 ( 8 -𝜖 ) 𝜈 -𝑝 2 , 𝑡 2 ( 2 -3 𝑝 -𝜖 ) 8 -𝜖 𝜈 -𝑝 2 for { 𝜈 syn m , 𝜈 syn c } < 𝜈 . (9) \nGiven the electron Lorentz factors (Eqs. 5), the spectral breaks and the maximum flux of the synchrotron process (Eqs. 7) with the optical depth given by 𝜏 ∝ 𝑛 𝑅 , the spectral breaks and the maximum flux in the SSC scenario for 1 < 𝑝 < 2 and 2 ≤ 𝑝 are (e.g., see Fraija et al. 2019b, 2022a) \nℎ𝜈 ssc m = 0 . 3 eV ˜ 𝑔 4 𝑝 -1 GLYPH<16> 1 + 𝑧 1 . 1 GLYPH<17> 26 + 𝑝 ( 𝜖 -8 ) -𝜖 ( 8 -𝜖 ) ( 𝑝 -1 ) 𝜀 4 𝑝 -1 𝑒, -1 𝜀 3 -𝑝 2 ( 𝑝 -1 ) 𝐵, -4 𝑛 𝑝 ( 𝜖 -8 ) -3 ( 𝜖 -4 ) 2 ( 𝑝 -1 ) ( 8 -𝜖 ) 𝐸 6 ( 𝑝 -1 ) ( 8 -𝜖 ) 53 Γ -6 𝜖 ( 8 -𝜖 ) ( 𝑝 -1 ) 0 , 2 𝑡 -18 ( 8 -𝜖 ) ( 𝑝 -1 ) 4 . 7 for 1 < 𝑝 < 2 1 . 7 × 10 2 eV 𝑔 4 GLYPH<16> 1 + 𝑧 1 . 1 GLYPH<17> 10 + 𝜖 8 -𝜖 𝜀 4 𝑒, -1 𝜀 1 2 𝐵, -4 𝑛 -4 + 𝜖 2 ( 8 -𝜖 ) 𝐸 6 8 -𝜖 53 Γ -6 𝜖 8 -𝜖 0 , 2 𝑡 -18 8 -𝜖 4 . 7 for 2 ≤ 𝑝 \nℎ𝜈 ssc c = 2 . 2 × 10 -5 GeV GLYPH<18> 1 + 𝑧 1 . 1 GLYPH<19> -3 ( 𝜖 + 2 ) 8 -𝜖 ( 1 + 𝑌 ( 𝛾 c ) ) -4 𝜀 -7 2 𝐵, -4 𝑛 7 𝜖 -36 2 ( 8 -𝜖 ) 𝐸 -10 8 -𝜖 53 Γ 10 𝜖 8 -𝜖 0 , 2 𝑡 2 ( 2 𝜖 -1 ) 8 -𝜖 4 . 7 𝐹 ssc max = 3 . 9 × 10 -5 mJy 𝑔 -1 GLYPH<18> 1 + 𝑧 1 . 1 GLYPH<19> 2 ( 𝜖 + 7 ) 8 -𝜖 𝜀 1 2 𝐵, -4 𝐷 -2 z , 27 𝑛 5 ( 4 -𝜖 ) 2 ( 8 -𝜖 ) 𝐸 10 8 -𝜖 53 Γ -10 𝜖 8 -𝜖 0 , 2 𝑡 2 ( 1 -2 𝜖 ) 8 -𝜖 4 . 7 , (10) \nrespectively. Given the evolution of the spectral breaks and the maximum flux (Eqs. 10), it is possible to write the SSC light curves at an observed frequency 𝜈 and a given time 𝑡 for 1 < 𝑝 < 2 and 2 ≤ 𝑝 as (e.g., see Fraija et al. 2019b, 2022a) \n𝐹 ssc 𝜈 ∝ ( 1 < 𝑝 < 2 ) ( 2 ≤ 𝑝 ) { 𝑡 8 ( 1 -2 𝜖 ) 3 ( 8 -𝜖 ) , 𝑡 2 ( 𝑝 + 2 ) -4 𝜖 ( 𝑝 -1 ) ( 8 -𝜖 ) ( 𝑝 -1 ) } 𝜈 1 3 , { 𝑡 8 ( 1 -2 𝜖 ) 3 ( 8 -𝜖 ) , 𝑡 4 ( 2 -𝜖 ) 8 -𝜖 } 𝜈 1 3 for 𝜈 < { 𝜈 ssc c , 𝜈 ssc m } , 𝑡 1 -2 𝜖 8 -𝜖 𝜈 -1 2 , 𝑡 1 -2 𝜖 8 -𝜖 𝜈 -1 2 for 𝜈 ssc c < 𝜈 < 𝜈 ssc m , 𝑡 -7 + 4 𝜖 8 -𝜖 𝜈 -𝑝 -1 2 , 𝑡 11 -4 𝜖 -9 𝑝 8 -𝜖 𝜈 -𝑝 -1 2 for 𝜈 ssc m < 𝜈 < 𝜈 ssc c , 𝑡 -2 ( 4 + 𝜖 ) 8 -𝜖 𝜈 -𝑝 2 , 𝑡 10 -2 𝜖 -9 𝑝 8 -𝜖 𝜈 -𝑝 2 for { 𝜈 ssc m , 𝜈 ssc c } < 𝜈 , (11) \nrespectively. As discussed in the case of the stellar wind medium, VHE photons ∼ 1 TeV could be attenuated by interactions with softer photons.", '2.3 Evolution of the radiative parameter': 'The radiative parameter is defined by the ratio between the radiated and dissipated energy (Nappo et al. 2014; Moderski et al. 2000). When the radiated and dissipated energies are similar almost most of the internal energy is dissipated as radiation, and the afterglow phase lies in the radiative regime. This parameter is defined as', '𝜖 ≡ 𝜀 𝑒 𝜁 ,': '(12) \nwhere 𝜁 is the fraction of the radiated energy. It is estimated as the ratio of the power radiated during the slow- and fast-cooling regimes (Nappo et al. 2014) \n𝜁 = 𝛾 m 𝛾 c 𝑝 -2 3 -𝑝 GLYPH<20> 1 𝑝 -2 GLYPH<16> 𝛾 c 𝛾 m GLYPH<17> 3 -𝑝 -1 GLYPH<21> for 𝛾 m ≤ 𝛾 c 1 for 𝛾 c < 𝛾 m . (13) \nEqs. 12 and 13 show that the radiative parameter is constant 𝜖 = 𝜀 𝑒 during the fast cooling regime. Afterwards it is expected to decrease to 0 during the slow cooling regime for 2 ≤ 𝑝 . We notice that the radiative parameter decreases slowly if the value of 𝑝 does not deviate from 2, and in the particular case when 𝑝 → 2, the radiative parameter approaches the same value 𝜖 ≃ 𝜀 𝑒 . At the previous point, an adiabatic break was not observed. On the other hand, if 𝑝 largely deviates from 2, adiabatic breaks around the transition time are expected in the synchrotron and SSC light curves. Eqs. 12 and 13 shows that the radiative parameter evolves during GRB. \n2.3.0.1 Equivalence with the radiative parameter 𝑠 . Some authors have introduced the radiative parameter 𝑠 instead of 𝜖 through the variation explicitly of the equivalent kinetic energy as (Böttcher & Dermer 2000; Wu et al. 2005; Ghisellini et al. 2010) \n𝐸 GLYPH<18> 𝑡 𝑡 dec GLYPH<19> -𝑠 ∝ ( 2 -𝜖 ) 𝜌 ( 𝑟 ) Γ 2 𝑅 3 -𝑘 , (14) \nwhere 𝜌 ( 𝑟 ) ∝ 𝑟 -k with k = 0 corresponds to the density of the constant-density medium ( 𝜌 = 𝑛 ), and k = 2 to the density of the stellar wind ejected by its progenitor ( 𝜌 ( 𝑟 ) ∝ 𝑟 -2 ) . For instance, Eq. 14 for 𝑘 = 0 can be obtained from 𝐸 ∝ 𝑛 Γ 𝜖 0 Γ 8 -𝜖 𝑡 3 and Γ = Γ 0 GLYPH<16> 𝑡 𝑡 dec GLYPH<17> -3 8 -𝜖 with 𝑡 dec the deceleration time scale. In this scenario, the evolution of the radiative parameters ( 𝜖, 𝑠 ) is related through \n( 3 + 𝑠 )( 8 -𝜖 ) -24 = 0 , for k = 0 ( 1 + 𝑠 )( 4 -𝜖 ) -4 = 0 , for k = 2 , (15) \nwhere the parameter s lies in the range of 0 ≤ 𝑠 ≤ 3 / 7 and 0 ≤ 𝑠 ≤ 1 / 3 for the homogeneous and stellar wind medium, respectively.', '2.4 Klein-Nishina effects': 'Adirect effect on the SSC spectrum due to the KN regime is the suppression of up-scattered synchrotron photons, and an indirect effect occurs when the SSC emission dominates, and at least some of the injected electrons with different Lorentz factors have enough time to cool down. The SSC spectra could have several breaks depending on the location of the spectral breaks in the KN regime; ℎ𝜈 syn KN ( 𝛾 𝑚 ) ≃ 2 Γ ( 1 + 𝑧 ) 𝑚 𝑒 𝑐 2 𝛾 𝑚 for 𝜈 syn c < 𝜈 syn m and ℎ𝜈 syn KN ( 𝛾 𝑐 ) ≃ 2 Γ ( 1 + 𝑧 ) 𝑚 𝑒 𝑐 2 𝛾 𝑐 for 𝜈 syn m < 𝜈 syn c (see Nakar et al. 2009; Wang et al. 2010). For example, the value of the Compton parameter in the case of the slow-cooling regime ( 𝜈 syn m < 𝜈 syn c ) might not be constant and be defined by \n𝑌 ( 𝛾 𝑐 ) [ 𝑌 ( 𝛾 𝑐 ) + 1 ] = 𝜀 𝑒 𝜀 𝐵 GLYPH<18> 𝛾 c 𝛾 m GLYPH<19> 2 -𝑝 GLYPH<18> 𝜈 syn m 𝜈 syn c GLYPH<19> -𝑝 -3 2 GLYPH<18> 𝜈 syn KN ( 𝛾 c ) 𝜈 syn m GLYPH<19> 4 3 for 𝜈 syn KN ( 𝛾 c ) < 𝜈 syn m GLYPH<18> 𝜈 syn KN ( 𝛾 c ) 𝜈 syn 𝑐 GLYPH<19> -𝑝 -3 2 for 𝜈 syn m < 𝜈 syn KN ( 𝛾 c ) < 𝜈 syn c 1 for 𝜈 syn c < 𝜈 syn KN ( 𝛾 c ) . (16) \nThe last cooling condition ( 𝜈 syn c < 𝜈 syn KN ( 𝛾 c ) ) corresponds to the Thomson regime, and then KN effects are neglected. In this case, the value of the Compton parameter becomes \n𝑌 ( 𝛾 𝑐 ) [ 𝑌 ( 𝛾 𝑐 ) + 1 ] = 𝜂 𝜀 1 𝑝 -1 𝑒 𝜀 𝐵 h 𝑚𝑝 ˜ 𝑔 Γ 𝑚𝑒𝛾 M i ˜ 𝑔 for 1 < 𝑝 < 2 𝜀𝑒 𝜀 𝐵 for 2 ≤ 𝑝 , (17) \nwith 𝜂 = 1 and GLYPH<16> 𝛾 c 𝛾 m GLYPH<17> 2 -𝑝 for the fast- and slow-cooling regime, respectively (Sari & Esin 2001). Due to the terms ℎ𝜈 syn KN ( 𝛾 𝑚 ) and ℎ𝜈 syn KN ( 𝛾 𝑐 ) and the electron Lorentz factors (Eqs. 1 and 5), the SSC spectral breaks in the KN regime can be written explicitly as \nℎ𝜈 ssc m , KN = 4 . 6 × 10 GeV ˜ 𝑔 1 𝑝 -1 GLYPH<16> 1 + 𝑧 1 . 1 GLYPH<17> 6 -𝑝 -2 𝜖 + 𝑝𝜖 2 ( 𝑝 -1 ) ( 4 -𝜖 ) 𝜀 1 𝑝 -1 𝑒, -1 𝜀 2 -𝑝 4 ( 𝑝 -1 ) 𝐵, -4 𝐴 12 -10 𝑝 + 𝜖 ( 𝑝 -2 ) 4 ( 𝑝 -1 ) ( 4 -𝜖 ) 𝑊, -1 𝐸 3 𝑝 -2 2 ( 𝑝 -1 ) ( 4 -𝜖 ) 53 Γ -𝜖 ( 3 𝑝 -2 ) 2 ( 4 -𝜖 ) ( 𝑝 -1 ) 0 , 2 𝑡 𝑝 -6 -𝜖 ( 𝑝 -2 ) 2 ( 4 -𝜖 ) ( 𝑝 -1 ) 4 . 7 for 1 < 𝑝 < 2 1 . 5 × 10 2 GeV 𝑔 GLYPH<16> 1 + 𝑧 1 . 1 GLYPH<17> 2 4 -𝜖 𝜀 𝑒, -1 𝐴 -2 4 -𝜖 𝑊, -1 𝐸 2 4 -𝜖 53 Γ -2 𝜖 4 -𝜖 0 , 2 𝑡 -2 4 -𝜖 4 . 7 for 2 ≤ 𝑝 \n-2 \nℎ𝜈 ssc c , KN = 7 . 1 × 10 2 GeV GLYPH<18> 1 + 𝑧 1 . 1 GLYPH<19> 𝜖 4 -𝜖 ( 1 + 𝑌 ( 𝛾 c ) ) -1 𝜀 -1 𝐵, -4 𝐴 𝜖 -6 4 -𝜖 𝑊, -1 𝐸 2 4 -𝜖 53 Γ -2 𝜖 4 -𝜖 0 , 2 𝑡 2 -𝜖 4 -𝜖 4 . 7 , (18) \nfor a stellar-wind environment, and as \nℎ𝜈 ssc m , KN = 1 . 6 GeV ˜ 𝑔 1 𝑝 -1 GLYPH<16> 1 + 𝑧 1 . 1 GLYPH<17> 3 ( 𝑝 + 2 ) 2 ( 𝑝 -1 ) ( 8 -𝜖 ) 𝜀 1 𝑝 -1 𝑒, -1 𝜀 -𝑝 -2 4 ( 𝑝 -1 ) 𝐵, -4 𝑛 12 -10 𝑝 + 𝜖 ( 𝑝 -2 ) 4 ( 𝑝 -1 ) ( 8 -𝜖 ) 𝐸 𝑝 + 2 2 ( 𝑝 -1 ) ( 8 -𝜖 ) 53 Γ -𝜖 ( 𝑝 + 2 ) 2 ( 8 -𝜖 ) ( 𝑝 -1 ) 0 , 2 𝑡 -3 ( 𝑝 + 2 ) 2 ( 8 -𝜖 ) ( 𝑝 -1 ) 4 . 7 for 1 < 𝑝 < 2 5 . 9 GeV 𝑔 GLYPH<16> 1 + 𝑧 1 . 1 GLYPH<17> 6 8 -𝜖 𝜀 𝑒, -1 𝑛 -2 8 -𝜖 𝐸 2 8 -𝜖 53 Γ -2 𝜖 8 -𝜖 0 , 2 𝑡 -6 8 -𝜖 4 . 7 for 2 ≤ 𝑝 \nℎ𝜈 ssc c , KN = 2 . 7 × 10 GeV GLYPH<18> 1 + 𝑧 . GLYPH<19> 2 -𝜖 8 -𝜖 ( 1 + 𝑌 ( 𝛾 c ) ) -1 𝜀 -1 𝐵, -4 𝑛 𝜖 -6 8 -𝜖 𝐸 -2 8 -𝜖 53 Γ 2 𝜖 8 -𝜖 0 , 2 𝑡 𝜖 -2 8 -𝜖 4 . 7 , (19) \n1 1 \nfor a constant-density medium. It should be noted that the total cross section in the KN regime can be approximated by 𝜎 ≈ 3 / 8 𝜎 𝑇 𝑥 -1 ( ln2x + 1 / 2 ) with 𝑥 = ℎ𝑣 / 𝑚 𝑒 ≫ 1 (Rybicki & Lightman 1986).', '2.5 Analysis and Discussion': 'Based on the external FS scenario, we have presented a general analytical model of the synchrotron and SSC processes in three cases: i) the fully adiabatic ( 𝜖 = 0), ii) fully radiative ( 𝜖 = 1), and iii) partially radiative or adiabatic (0 < 𝜖 < 1) regimes. We plotted the expected light curves and spectra for an electron spectral index 1 < 𝑝 < 2 and 2 ≤ 𝑝 when the outflow decelerates in a stellar wind (see Figure A1). Significant variations of the spectral and temporal features of the afterglow emission are introduced by radiative losses only if 𝜖 is large and approaches to unity. In particular, when 𝜖 = 0, the synchrotron and SSC light curves derived in the standard FS scenario are recovered (Bhattacharya 2001; Sari et al. 1998; Sari & Esin 2001; Fraija et al. 2019b). We want to highlight the synchrotron and SSC spectral breaks and light curves for 1 < 𝑝 < 2 and 2 ≤ 𝑝 become equal when 𝑝 → 2. \nWe have obtained and shown in Table A1 the closure relations that describe the evolution of the synchrotron and SSC flux ( 𝐹 𝜈 ∝ 𝑡 -𝛼 𝜈 -𝛽 ) as a function of 𝜖 and 𝑝 . The cooling conditions in the constant density medium of 𝜈 ssc m ≤ 𝜈 ≤ 𝜈 ssc c , and { 𝜈 ssc m , 𝜈 ssc c } < 𝜈 for 1 < 𝑝 < 2, are the only ones where the SSC fluxes do not depend on 𝑝 , hence their closure relations cannot be estimated. In this case, the SSC flux evolves as ∝ 𝑡 -𝛼 with 𝛼 = 7 + 4 𝜖 8 -𝜖 and 2 ( 4 + 𝜖 ) 8 -𝜖 for the spectral indices 𝛽 = 𝑝 -1 2 and 𝑝 2 , respectively. Note that adiabatic breaks are expected around the transition time between the fast- and slow-cooling regimes. The transition time refers to the temporal interval of applicability of the synchrotron and SSC light curves. Given the closure relations (see Table A1), we can notice that the synchrotron and SSC fluxes will evolve similarly in time and energy if the condition 𝛼 syn ( 𝛽 ) ≈ 𝛼 ssc ( 𝛽 ) is satisfied. Considering both, the stellar wind and the homogeneous afterglow model, this condition is satisfied when the observed frequency evolves in { 𝜈 j m , 𝜈 j c } < 𝜈 (with j = syn and ssc) and 𝛽 → 1 (i.e., 𝑝 → 2). Irrespective of whether 𝑝 approaches the value of 2 from 1 < 𝑝 < 2 (e.g. 𝑝 ≃ 1 . 98) or from 2 ≤ 𝑝 (e.g. 𝑝 ≃ 2 . 02), the temporal decay indices become 𝛼 ≈ 4 4 -𝜖 and ≈ 2 ( 4 + 𝜖 ) 8 -𝜖 for the wind and constant-density afterglow model, respectively. In the particular case of 𝜖 ≈ 0, the expected flux obtained from the stellar-wind and constant-density afterglow model evolves with the same temporal index of 𝛼 ≈ 1. The temporal and spectral similarities that could be observed during the afterglow phase in two or more different bands could be interpreted in the synchrotron and SSC FC scenario with a hard spectral index 𝑝 ≈ 2, and the parameter 𝜖 would be useful to discriminate between the stellar wind or homogeneous afterglow model. One of the more relevant features to be observed in the radiative regime should occur during the afterglow transition between a stellar wind and a constant-density medium. For 𝜖 = 0, the X-ray flux evolving in the stellar wind and constant density afterglow has the same temporal evolution 𝑡 -3 𝑝 -2 4 when 𝑝 lies in the range of 2 < 𝑝 . Therefore, depending on the values of observable quantities and parameters, a transition between stellar-wind and homogeneous medium afterglow could be noticeable. For an electron spectral index that does not deviate from 2 (i.e., 𝑝 ∼ 2), the expected synchrotron and SSC fluxes would evolve as 𝐹 𝜈 ∝ 𝑡 -𝛼 with 𝛼 = 1 for any value of 𝑝 . In this case, a transition between stellar-wind and constant-density afterglow could not be noticeable in the SSC and synchrotron light curves. Otherwise, this transition could be more highlighted as 𝑝 deviates from 2 and 𝜖 from 0. On the other hand, the development of the FS in the fully radiative ( 𝜖 = 1) or partially radiative or adiabatic (0 < 𝜖 < 1) regimes presents an X-ray flux that has different evolution. Regardless of the values, a transition between stellar wind and homogeneous medium afterglow must be observed. \nDeviations around 𝜖 = 0 will also modify the evolution of synchrotron and SSC spectral breaks, producing distinct variations in the afterglow tails. The SSC and sychrotron spectral breaks for 1 < 𝑝 < 2 (2 < 𝑝 ) evolve as 𝜈 ssc m ∝ 𝑡 3 ( 𝜖 -4 ) -𝑝 ( 𝜖 -2 ) ( 4 -𝜖 ) ( 𝑝 -1 ) ( 𝑡 𝜖 -8 4 -𝜖 ), 𝜈 syn m ∝ 𝑡 𝜖 -𝑝 -4 ( 4 -𝜖 ) ( 𝑝 -1 ) ( 𝑡 𝜖 -6 4 -𝜖 ), 𝜈 ssc c ∝ 𝑡 8 -3 𝜖 4 -𝜖 and 𝜈 syn c ∝ 𝑡 2 -𝜖 4 -𝜖 for a stellar wind medium, and as 𝜈 ssc m ∝ 𝑡 -18 ( 8 -𝜖 ) ( 𝑝 -1 ) ( 𝑡 -18 8 -𝜖 ), 𝜈 syn m ∝ 𝑡 -3 ( 𝑝 + 2 ) ( 8 -𝜖 ) ( 𝑝 -1 ) ( 𝑡 -12 8 -𝜖 ), 𝜈 ssc c ∝ 𝑡 2 ( 2 𝜖 -1 ) 8 -𝜖 and 𝜈 syn c ∝ 𝑡 2 ( 𝜖 -2 ) 8 -𝜖 for a constant-density medium. For instance, considering 2 < 𝑝 the spectral breaks in the stellar wind (constant-density) medium evolve under the cooling condition 𝜈 syn m ∝ 𝑡 -[ 1 . 50 -1 . 67 ] ( 𝑡 -[ 1 . 50 -1 . 71 ] ), 𝜈 syn c ∝ 𝑡 [ 0 . 33 -0 . 50 ] ( 𝑡 -[ 0 . 29 -0 . 50 ] ), 𝜈 ssc m ∝ 𝑡 -[ 2 . 0 -2 . 33 ] ( 𝑡 -[ 2 . 25 -2 . 51 ] ) and \n𝜈 ssc c ∝ 𝑡 [ 1 . 67 -2 . 0 ] ( 𝑡 [-0 . 25 -0 . 29 ] ) instead of the typical evolution of spectral breaks of 𝜈 syn m ∝ 𝑡 -3 2 ( 𝑡 -3 2 ), 𝜈 syn c ∝ 𝑡 -1 2 ( 𝑡 1 2 ), 𝜈 ssc m ∝ 𝑡 -2 ( 𝑡 -5 2 ) and 𝜈 ssc c ∝ 𝑡 -2 ( 𝑡 -1 4 ), respectively. \nFigures A1 and A2 illustrate the light curves and spectra of the SSC process evolving in stellar-wind and homogeneous medium for typical GRBafterglow parameters, respectively. These light curves and spectra are shown at 1 TeV and at 5 × 10 4 s, respectively, for 𝜖 e = 𝜖 , 𝜖 B = 10 -4 and different spectral indices ( 𝑝 = 1 . 7, 1 . 9, 2 . 1 and 2 . 3). These panels are shown from top to bottom for 𝜖 = 0, 0 . 2 and 0 . 4, and from left to right for the values of [ 𝐴 W ( 𝑛 ); 𝐸 ]=[0 . 1 (0 . 1 cm -3 ); 10 52 erg], [10 -3 (10 -3 cm -3 ); 10 52 erg], and [10 -3 (10 -3 cm -3 ); 10 53 erg]. We consider a hypothetical burst located at 𝑧 = 0 . 1 and the model of the Extragalactic Background Light (EBL) absorption proposed in Franceschini & Rodighiero (2017). The left-hand panels in Figure A1 show that, depending on the parameter values, the expected flux will have a different behavior. For example, the light curves during the early time show a plateau phase followed by a normal decay for 2 ≤ 𝑝 , but only in a few cases for 1 < 𝑝 < 2. It is also shown that irrespective of the value of the spectral index 𝑝 and the parameter 𝜖 , the expected flux increases as 𝐸 also increases. The panels show that the expected flux 𝐹 ssc 𝜈 ∝ 𝑡 -𝛼 evolves from 1 . 03 ≤ 𝛼 ≤ 1 . 13 (1 . 10 ≤ 𝛼 ≤ 1 . 22) to 2 . 03 ≤ 𝛼 ≤ 2 . 08 (2 . 10 ≤ 𝛼 ≤ 2 . 16) for 𝑝 = 1 . 9 ( 𝑝 = 2 . 1). This is due to the evolution of the SSC spectral breaks; 𝜈 ssc m ∝ 𝑡 -𝛼 with 1 . 64 ≤ 𝛼 ≤ 1 . 70 (1 . 50 ≤ 𝛼 ≤ 1 . 56) for 𝑝 = 1 . 9 ( 𝑝 = 2 . 1), and 𝜈 ssc c ∝ 𝑡 -𝛼 with observe that the expected flux increases in some panels as 𝐴 w increases and in other panels when 𝐴 w decreases. This result can be explained in terms of the density parameter and the cooling condition. Given the spectral index in the range 1 < 𝑝 < 2 (2 ≤ 𝑝 ), the expected flux as a function of the density parameter is 𝐹 𝜈 ∝ 𝐴 52 -8 𝑝 -13 𝜖 + 𝑝𝜖 4 ( 4 -𝜖 ) W ( 𝐴 44 -9 𝜖 -𝑝 ( 4 + 𝜖 ) 4 ( 4 -𝜖 ) W ) for 𝜈 ssc m < 𝜈 < 𝜈 ssc c and ∝ 𝐴 16 -8 𝑝 -6 𝜖 + 𝑝𝜖 4 ( 4 -𝜖 ) W ( 𝐴 8 -2 𝜖 -𝑝 ( 4 + 𝜖 ) 4 ( 4 -𝜖 ) W ) for 𝜈 ssc m < 𝜈 ssc c < 𝜈 . Considering the value of 𝑝 = 1 . 9 (2.1), the expected flux as a function of the density parameter is 𝐹 𝜈 ∝ 𝐴 𝛼 𝑤 w with 2 . 3 ≤ 𝛼 𝑤 ≤ 2 . 25 (2 . 23 ≤ 𝛼 𝑤 ≤ 2 . 16) for 𝜈 ssc m < 𝜈 < 𝜈 ssc c and -0 . 05 ≤ -𝛼 𝑤 ≤ 0 . 06 (0 . 03 ≤ -𝛼 𝑤 ≤ 0 . 14) for 𝜈 ssc m < 𝜈 ssc c < 𝜈 . Therefore, as 𝐴 w increases, the expected flux increases if it evolves in the cooling condition 𝜈 ssc m < 𝜈 < 𝜈 ssc c and decreases if it evolves in 𝜈 ssc m < 𝜈 ssc c < 𝜈 . The left-hand panels in Figure A2 exhibit a similar behavior to those shown in Figure A1. For example, the evolution of the expected flux 𝐹 ssc 𝜈 ∝ 𝑡 -𝛼 from 0 . 88 ≤ 𝛼 ≤ 1 . 13 (0 . 99 ≤ 𝛼 ≤ 1 . 25) to 1 . 0 ≤ 𝛼 ≤ 1 . 16 (1 . 11 ≤ 𝛼 ≤ 1 . 28) for 𝑝 = 1 . 9 (2 . 1) can be interpreted in terms of SSC spectral breaks; 𝜈 ssc m ∝ 𝑡 -𝛼 with 1 . 63 ≤ 𝛼 ≤ 1 . 71 (1 . 50 ≤ 𝛼 ≤ 1 . 58) for 𝑝 = 1 . 9 (2 ≤ 𝑝 ), and 𝜈 ssc c ∝ 𝑡 -𝛼 with 0 . 42 ≤ 𝛼 ≤ 0 . 50. Similarly, the variation of the expected flux as a function of the density can be explained in terms of the cooling condition. In the case of 1 < 𝑝 < 2 (2 ≤ 𝑝 ), the expected flux is 𝐹 𝜈 ∝ 𝑛 52 -8 𝑝 -13 𝜖 + 𝑝𝜖 4 ( 8 -𝜖 ) ( 𝑛 44 -9 𝜖 -𝑝 ( 4 + 𝜖 ) 4 ( 8 -𝜖 ) ) for 𝜈 ssc m < 𝜈 < 𝜈 ssc c and ∝ 𝑛 6 -8 𝑝 -6 𝜖 + 𝑝𝜖 4 ( 8 -𝜖 ) ( 𝑛 8 -2 𝜖 -𝑝 ( 4 + 𝜖 ) 4 ( 8 -𝜖 ) ) for 𝜈 ssc m < 𝜈 ssc c < 𝜈 . Considering the value of 𝑝 = 1 . 9 (2.1), the expected flux as a function of the density is 𝐹 𝜈 ∝ 𝑛 𝛼 𝑤 with 1 . 15 ≤ 𝛼 𝑤 ≤ 1 . 06 (1 . 11 ≤ 𝛼 𝑤 ≤ 1 . 03) for 𝜈 ssc m < 𝜈 < 𝜈 ssc c and 0 . 29 ≤ -𝛼 𝑤 ≤ 0 . 36 (0 . 01 ≤ -𝛼 𝑤 ≤ 0 . 07) for 𝜈 ssc m < 𝜈 ssc c < 𝜈 . The right-hand panels in Figures A1 and A2 show the SSC spectra with the CTA (Southern array, green line), MAGIC (purple line) and Fermi /LAT (red line) sensitivities between 75 and 250 GeV at 3 × 10 4 s for a zenith angle of 20 · (Fioretti et al. 2019). These panels display that whereas all the expected fluxes are below the Fermi /LAT sensitivity, only some of them are above of the Cherenkov Telescope Array (CTA) or the Major Atmospheric Gamma-ray Imaging Cherenkov Telescop (MAGIC) Telescopes; depending on the set of parameter values. For example, the expected flux could be detected in both MAGIC and CTA for [ 𝐴 W = 0 . 1 ( 𝑛 = 0 . 1 cm -3 ); 𝐸 = 10 52 erg], and not be detected by MAGIC or CTA for [10 -3 (10 -3 cm -3 ); 10 52 erg]. Given the values of [10 -3 (10 -3 cm -3 ); 10 53 erg], we can see that the expected flux could be detected by CTA for p = 2 . 1 and 2.3, but not for p = 1 . 7 or 1 . 9. In the former case, MAGIC could not detect the expected flux for any parameter values.', '3.1 Our representative sample of GRBs': 'In order to apply the current model, we select those GRBs from the 2FLGC (Ajello et al. 2019b) with values of temporal and spectral indices with 𝛼 L ≳ 1 . 5 and Γ L ≈ 2 (see Table A2), which can hardly be described with closure relations of the standard synchrotron afterglow model, and also exhibit energetic photons above the synchrotron limit (e.g., see Ghisellini et al. 2010; Tak et al. 2019; Fraija et al. 2022a). Our sample is formed by ten bursts (one short GRB and nine long GRBs), seven of which have a measured redshift. We briefly describe the multi-wavelength observations of our representative sample of GRBs.', '3.1.1 GRB 080825C': 'On 25 August 2008 at 14:13:48 UT, the Gamma-ray Burst Monitor (GBM) instrument on board the Fermi telescope was triggered by GRB 080825C (van der Horst & Connaughton 2008a). The initial estimation set a duration of 𝑇 90 = 23 s (van der Horst & Connaughton 2008b). Nevertheless, further spectral analysis of the data from Fermi /GBM, revealed that the main emission lasted 𝑇 90 = 27 s in the energy band 8 -1000 keV (Abdo et al. 2009b). The fluence measured by the GBM instrument was ( 0 . 11 ± 0 . 04 ) × 10 -5 erg cm -2 . Moreover, this burst is the first detection of the Fermi /LAT instrument of a GRB (Bouvier et al. 2008; Moretti & Axelsson 2016). During all the emission, the photons had energies below 1 GeV (Bouvier et al. 2008). No redshift was associated to this event. \nThe circumburst medium remains unconstrained for this burst due to the absence of X-ray and optical data. \nGRB 090510 was detected by the Burst Alert Telescope (BAT) instrument on board the Neil Gehrels Swift Observatory and by the Fermi /LAT instrument. Subsequently, the other instruments of both facilities observed the field of GRB 090510 (De Pasquale et al. 2010; Ukwatta et al. 2009). The duration of this burst was estimated to be 𝑇 90 = 0 . 3 ± 0 . 1 s (Ukwatta et al. 2009; De Pasquale et al. 2010), with a corresponding fluence ( 1 . 7 ± 0 . 6 ) × 10 -5 erg cm -2 and isotropic energy of 𝐸 𝛾, iso = ( 5 . 8 ± 0 . 5 ) × 10 53 erg (Ajello et al. 2019a). Due to the large energy released by this event, Fermi /LAT reported 12 photons with energy greater than 1 GeV during the first three seconds after the trigger. Spectroscopic data from the Very Large Array (VLA) allowed to estimate a redshift of 𝑧 = 0 . 903 from the OII and H 𝛽 lines (Rau et al. 2009). \nDe Pasquale et al. (2010); Nicuesa Guelbenzu et al. (2012) analyzed the X-ray and UV/optical/IR afterglow observations of GRB 090510. Given the SMC dust extinction with 𝐴 host 𝑉 = 0 . 17 + 0 . 21 -0 . 17 mag, they reported early temporal and spectral indices of 𝛼 X = 0 . 74 ± 0 . 03 and 𝛼 Opt , 1 ≈ -0 . 2 ± 0 . 2 evolving to 𝛼 Opt , 2 = 0 . 80 ± 0 . 1, and 𝛽 X = 0 . 8 ± 0 . 1 and 𝛽 Opt = 0 . 85 ± 0 . 05 for X-ray and UV-optical-IR observations, respectively. The closure relations of the temporal and spectral indices of late X-ray and optical observations are 𝐹 𝜈, X ∝ 𝑡 -0 . 74 . 30 ± 0 . 03 𝜈 -0 . 8 ± 0 . 1 and 𝐹 𝜈, Opt ∝ 𝑡 -0 . 80 ± 0 . 1 𝜈 -0 . 85 ± 0 . 05 , respectively. Although, the closure relations are similar to each other, the evolution in a constant-density medium under the condition 𝜈 syn m < 𝜈 Opt < 𝜈 X < 𝜈 syn c is more favorable for 𝑝 ≈ 2 . 4 ± 0 . 2 or 𝑝 ≈ 2 . 6 ± 0 . 2, respectively. The evolution in stellar-wind environment in the same cooling condition leads to an atypical value of the spectral index with 𝑝 < 1 . 4.', '3.1.3 GRB 090902B': 'GRB090902B was detected by the Fermi /GBMinstrument on 2 September 2009 at 11:05:08.31 UTC. The duration of this burst was estimated to be 𝑇 90 = 19 . 33 s, with a corresponding fluence of ( 7 . 0 ± 1 . 0 ) × 10 -5 erg cm -2 and isotropic energy of 𝐸 𝛾, iso = ( 3 . 7 ± 0 . 3 ) × 10 53 erg (Ajello et al. 2019a). This bright event was within the Fermi /LAT field of view initially at an angle of 51 · from the line of sight, and therefore this instrument showed an increment correlated with the Fermi /GBM trigger. Later, GRB 090902B was detected by the X-ray Telescope (XRT) (Kennea & Stratta 2009), and by the Ultraviolet/Optical Telescope (UVOT) instruments (Swenson & Siegel 2009) on board the Neil Gehrels Swift Observatory , as well as by several other ground-based telescopes. The spectrum obtained with the Gemini-North telescope (Cucchiara et al. 2009) showed a series of metal absorption features corresponding to a redshift of 𝑧 = 1 . 822 (Cucchiara et al. 2009). \nPandey et al. (2010) conducted an analysis of the X-ray and UV-optical-IR afterglow data associated with GRB 090902B. Given the SMClike dust extinction with 𝐴 host 𝑉 = 0 . 20 ± 0 . 06, the authors performed a temporal and spectral analysis resulting in early temporal indices of 𝛼 X = 1 . 30 ± 0 . 04and 𝛼 Opt ≈ 1 . 60, and 𝛽 X = 0 . 9 ± 0 . 1and 𝛽 Opt = 0 . 68 ± 0 . 11for X-ray and UV-optical-IR observations, respectively. The closure relations of the temporal and spectral indices of late X-ray and optical observations are 𝐹 𝜈, X ∝ 𝑡 -1 . 30 ± 0 . 04 𝜈 -0 . 9 ± 0 . 1 and 𝐹 𝜈, Opt ∝ 𝑡 -1 . 60 𝜈 -0 . 68 ± 0 . 11 , respectively. The fact that the temporal (spectral) index for the optical observations is larger (lower) than the X-ray observations indicates that the closure relations of the synchrotron FS model evolve in a slow cooling regime through a wind-like medium ( 𝜈 syn m < 𝜈 Opt < 𝜈 syn c < 𝜈 X ) for 𝑝 ≈ 2 . 3 ± 0 . 3.', '3.1.4 GRB 090926A': 'The Fermi /GBMtriggered on GRB 090926A at 04:20:26.99 UTC on 26 September 2009 (Ackermann et al. 2011). The duration of this burst was estimated to be around 𝑇 90 = 20 s (Bissaldi 2009), whereas the isotropic energy was measured as 𝐸 𝛾, iso = ( 3 . 7 ± 0 . 3 )× 10 53 erg (Golenetskii et al. 2009). Ackermann et al. (2011) analyzed data from the Fermi /LAT and Fermi /GBM instruments for GRB 090926A, concluding the presence of a characteristic high-energy power law component. The photometry data set includes observations from Swift /XRT, INTEGRAL /SPI-ACS (Bissaldi 2009), Suzaku /WAM (Noda et al. 2009), CORONAS /Photon (Chakrabarti et al. 2009), the Konus-wind experiment (Golenetskii et al. 2009) and the Swift/UVOT (Malesani et al. 2009) instruments. Using data from the VLT /X-shooter Malesani et al. (2009) estimated a redshift of 𝑧 = 2 . 11. \nSwenson et al. (2010) and Cenko et al. (2011) performed a temporal and spectral analysis of X-ray and optical data including UVOT observations from GRB 090926A. They reported temporal indices of 𝛼 X = 1 . 43 ± 0 . 03 and 𝛼 Opt ≈ 1 . 01 + 0 . 07 -0 . 03 , and spectral indices of 𝛽 X = 1 . 12 ± 0 . 13 and 𝛽 Opt = 0 . 88 ± 0 . 07 for X-ray and UV-optical-IR observations, respectively. The closure relations of the temporal and spectral indices of early X-ray and optical observations are 𝐹 𝜈, X ∝ 𝑡 -1 . 43 ± 0 . 03 𝜈 -1 . 12 ± 0 . 13 and 𝐹 𝜈, Opt ∝ 𝑡 -1 . 01 + 0 . 07 -0 . 03 𝜈 -0 . 88 ± 0 . 07 , respectively. The fact that the temporal (spectral) index for the X-ray observations is greater than the optical observations suggests that the closure relations of the synchrotron FS model evolve in a slow cooling regime through a homogeneous medium ( 𝜈 syn m < 𝜈 Opt < 𝜈 syn c < 𝜈 X ) for 𝑝 ≈ 2 . 5 ± 0 . 2.', '3.1.5 GRB 110731A': 'The Fermi /GBM instrument detected GRB 110731A on 31 July 2011 (Malesani et al. 2009), estimating a duration of 𝑇 90 = 7 . 49 s (Gruber 2011). Independently, Swift /BAT triggered on this event about 30 s after the initial Fermi /GBM trigger. Ackermann et al. (2013) calculated the isotropic energy measured using a power law and a band function model, obtaining 𝐸 𝛾, iso = ( 7 . 6 ± 0 . 2 ) × 10 53 erg. The Swift /XRT and the Swift /UVOT instruments began to observe the field of GRB 110731A a time 𝑇 + 66 . 4 s and 𝑇 + 75 s after the Swift /BAT trigger, respectively (Oates et al. 2011). Due to the energy of this event, several ground facilities followed up this GRB, such as the Faulkes Telescopes North and South (Bersier 2011), the Nordic Optical Telescope equipped with ALFOSC (Malesani et al. 2011), Konus-Wind (Golenetskii et al. 2011), \nthe EVLA (Zauderer et al. 2011), the Suzaku Wide-band All-sky Monitor (WAM) (Hanabata et al. 2011) and the SAO RAS and Terskol observatories (Moskvitin et al. 2011). The redshift value was determined to be 𝑧 = 2 . 83 using spectroscopic observations with the GMOS-N instrument on Gemini-North (Tanvir et al. 2011). \nAckermann et al. (2013) conducted an analysis of the X-ray and UV/optical afterglow data associated with GRB 110731A. After performing an analysis of the broadband spectral energy distribution (SED), the authors presented the spectral indices of 𝛽 X = 0 . 95 + 0 . 07 -0 . 09 and 𝛽 Opt = 0 . 45 + 0 . 07 -0 . 09 for X-ray and UV/optical observations, at a time interval of 550 s. Similarly, the temporal analysis led to X-ray and optical indices of 𝛼 X = 1 . 10 ± 0 . 02 and 𝛼 Opt = 1 . 37 ± 0 . 03, respectively. The closure relations of the temporal and spectral indices of late X-ray and optical observations are 𝐹 𝜈, X ∝ 𝑡 -1 . 10 ± 0 . 02 𝜈 -0 . 95 + 0 . 07 -0 . 09 and 𝐹 𝜈, Opt ∝ 𝑡 -1 . 37 ± 0 . 03 𝜈 -0 . 45 + 0 . 07 -0 . 09 , respectively. The fact that the temporal (spectral) index for the optical observations is greater (smaller) than the one for the X-ray observations indicates that the closure relations of the synchrotron FS model evolve in a slow cooling regime going through a wind-like medium ( 𝜈 syn m < 𝜈 Opt < 𝜈 syn c < 𝜈 X ) for 𝑝 ≈ 2 . 15 ± 0 . 15.', '3.1.6 GRB 130502B': 'On 2 May 2013 at 07:51:11.76 UT, the Fermi /GBM instrument was triggered by GRB 130502B. The estimated duration in the 50 -300 keV energy band was measured to be 𝑇 90 = 24 s (von Kienlin & Younes 2013). The GBM fluence was ( 0 . 5 ± 0 . 2 ) × 10 -5 erg cm -2 (Ajello et al. 2019a). In the follow-up campaign after the initial GBM trigger, some of the instruments involved which observed the field of GRB 130502B include the Fermi /LAT, Swift /XRT, Swift /UVOT, P60 and Konus-Wind (Cenko et al. 2013; Breeveld & Immler 2013; Melandri & Immler 2013; Kocevski et al. 2013). \nThe best-fit values of temporal and spectral indices derived by the Swift team and shown in the Swift /XRT repository 𝛼 X = 1 . 62 + 0 . 27 -0 . 26 and 𝛽 X = 0 . 76 + 0 . 20 -0 . 19 are used. 3 Due to a lack of optical observations, the circumburst media for this burst, can not be restricted.', '3.1.7 GRB 141207A': 'The Fermi /GBM and Fermi /LAT instruments were simultaneously triggered by GRB 141207A on 07 December 2017 (Burns 2014; Arimoto et al. 2014). The Fermi /GBM light curve exhibited a duration 𝑇 90 of approximately 20 s in the 50 -300 keV band. The GBM fluence was ( 2 . 0 ± 0 . 8 ) × 10 -5 erg cm -2 (Ajello et al. 2019a). The Swift /XRT instrument began to observe the field of GRB 141207A about 𝑇 + 13 hours, finding an uncatalogued X-ray source corresponding to the afterglow of this burst (Amaral-Rogers & Evans 2014). \nFor this burst, the circumburst environment remains unconstrained as a result of the absence of X-ray and optical data.', '3.1.8 GRB 170214A': 'On 14 February 2017 at 15:34:26.92 UT, the Fermi /GBM was triggered by GRB 170214A. The GBM light curve showed multiple overlapping peaks with a duration 𝑇 90 ∼ 123 s in the 50-300 keV energy band (Mailyan & Meegan 2017). The estimated isotropic energy was 𝐸 𝛾, iso = ( 32 ± 5 ) × 10 52 erg and the GBM fluence was ( 0 . 9 ± 0 . 1 ) × 10 -5 erg cm -2 (Ajello et al. 2019a). The Fermi /LAT instrument was simultaneously triggered by this burst, and it observed more than 160 photons above 100 MeV and more than 13 photons above 1 GeV. The highest-energy event was a photon with an energy of 7.8 GeV (Racusin et al. 2017). Approximately 41 hours after the initial GBM trigger, Kruehler et al. (2017) observed the optical counterpart of the burst with the ESO Very Large Telescope UT 2 equipped with the X-shooter spectrograph. The authors claimed a redshift of 𝑧 = 2 . 53 due to various absorption features in the low-energy optical observations. \nThe best-fit values of temporal and spectral indices derived by the Swift team and shown in the Swift /XRT repository 𝛼 X = 1 . 3 + 0 . 5 -0 . 4 and 𝛽 X = 0 . 9 ± 0 . 5 are used. 4 The value of the temporal index that best fits the observations collected with the optical instruments is 𝛼 Opt = 1 . 38 ± 0 . 09. The closure relations of the temporal and spectral indices of late X-ray and optical observations are 𝐹 𝜈, X ∝ 𝑡 -1 . 3 + 0 . 5 -0 . 4 𝜈 -0 . 9 ± 0 . 5 and 𝐹 𝜈, Opt ∝ 𝑡 -1 . 38 ± 0 . 09 , respectively. Similar to what happened in the GRB 131108A, the spectral index does not include optical frequencies. Although, the closure relations are similar to each other, the evolution in homogeneous medium under the condition 𝜈 syn m < 𝜈 syn c < 𝜈 Opt < 𝜈 X for 𝑝 ≈ 2 . 4 ± 0 . 2 is more favorable due to the absence of temporal breaks in both X-ray and optical observations. \nThe Fermi and Swift satellites triggered on GRB 180720B at 14:21:39.65 UT on 20 July 2018 (Roberts & Meegan 2018; Bissaldi & Racusin 2018). The duration of the burst was confirmed by a posterior analysis to be 𝑇 90 = 48 . 90 s. The preliminary multipeaked structure lasted beyond the available event data range of the Swift /BAT instrument (Barthelmy et al. 2018). The estimated isotropic energy was 𝐸 𝛾, iso = ( 0 . 39 ± 0 . 09 ) × 10 52 erg and the GBM fluence was ( 0 . 19 ± 0 . 05 ) × 10 -5 erg cm -2 (Ajello et al. 2019a). Optical and near-infrared (NIR) follow-up of GRB 180720B began observations about 𝑇 + 73 s (Sasada et al. 2018). Vreeswijk et al. (2018) monitored the field of GRB 180720B with the VLT /X-shooter spectrograph, estimating a redshift of 𝑧 = 0 . 654 from the match of several absorption features revealed in the spectrum. \nFraija et al. (2019c) performed a temporal and spectral analysis of the X-ray and R-band optical observations of GRB 180720B. Based on early observations, the authors reported spectral indices of 𝛽 X = 0 . 697 + 0 . 010 -0 . 010 and 𝛽 Opt = 0 . 68 ± 0 . 06, and temporal indices of 𝛼 X = 1 . 26 ± 0 . 06 and 𝛼 Opt = 1 . 22 ± 0 . 02 for X-ray and optical observations, respectively. The closure relations of the temporal and spectral indices of X-ray and optical observations are 𝐹 𝜈, X ∝ 𝑡 -1 . 26 ± 0 . 06 𝜈 -0 . 697 + 0 . 010 -0 . 010 and 𝐹 𝜈, Opt ∝ 𝑡 -1 . 22 ± 0 . 02 𝜈 -0 . 68 ± 0 . 06 , respectively. Although, the closure relations are similar to one another, the evolution in stellar-wind or constant-density medium under the condition 𝜈 syn m < 𝜈 Opt < 𝜈 X < 𝜈 syn c is more favorable for 𝑝 ≈ 2 . 0 ± 0 . 2 or 𝑝 ≈ 2 . 6 ± 0 . 2, respectively. The evolution in the condition max { 𝜈 syn m , 𝜈 syn c } < 𝜈 Opt < 𝜈 X leads to an atypical value of the spectral index with 𝑝 < 1 . 5. The temporal break exhibited at 2 . 6 × 10 5 s with a temporal index 1 . 70 ± 0 . 19 in the X-ray light curve (Fraija et al. 2019c) is consistent with the post-jet break phase in stellar wind for 𝑝 ≈ 2 . 0 ± 0 . 2.', '3.2.1 Fermi/LAT Data': 'The Fermi /LAT data files were retrieved from the science data repository. 5 The Fermi /LAT data set was analyzed in the 0.1-100 GeV energy range using time-resolved likelihood analysis and the Fermi Science tools ScienceTools 2.2.0 . 6 Following the unbinned likelihood analysis presented by the Fermi /LAT team, 7 we use the responses provided by Ajello et al. (2019b) for each burst. We use the gtselect tool to select a region of interest (ROI) within a radius of 15 · around the point of the burst and impose a cut on the zenith angle greater than 100 · . Furthermore, before evaluating the ROI cut, we acquire the most relevant time intervals in the data using the gtmktime tool. We use diffuse components and point sources from 4FGL-DR3 (e.g., see make4FGLxml Abdollahi et al. 2022) to define the model required to characterize the source. Using GALPROP gll\\_iem\\_v07 and a PL spectrum, we establish a point source at the location of this burst and a diffuse galactic component. In addition, the extragalactic background iso\\_P8R3\\_SOURCE\\_V3\\_v1 was used. 8 The spectral index for each burst is set at the value stated by Ajello et al. (2019b) in Table 4. We use the tool gtltcube with a step 𝛿𝜃 = 0 . 025, a bin size of 0.5 and a maximum zenith angle of 100 · to create a lifetime cube. We consider a region of 30 · around the GRB position and define 100 spatial bins in longitude/latitude and 50 energy bins to create the exposure map with gtexpmap . Furthermore, we carry out the likelihood analysis using pyLikelihood . 9 Finally, using the gtsrcprob tool, we retrieve photons with a probability greater than 90% to be correlated with each burst. As follows, we describe the relevant features of the energetic photons associated with each burst. \n3.2.1.1 GRB 080825C At 3.06 s after the trigger time, the first high-energy photon was detected with measured energy of 153.4 MeV. In this burst, there were 14 photons with energy over 100 MeV. The highest energy photon detected in the LAT data was 682.9 MeV, 28.3 s after the trigger time. \n3.2.1.2 GRB 090510 At 0.18 s after the GBM trigger, the first high-energy photon was detected with measured energy of 526.4 MeV. The energy range of the photons in this burst was extensive, with 261 photons over 100 MeV and 33 exceeding 1 GeV. At 0.82 s after the GBM trigger, the highest-energy photon in the LAT data had a measured energy of 19.9 GeV. \n3.2.1.3 GRB 090902B The first detection of a high-energy photon occurred around 1.86 s after the trigger of the GBM, with a measured energy of 284.4 MeV. The photon energy spectrum in this burst had a wide range, with 469 photons with energies over 100 MeV, 67 photons surpassing 1 GeV, and seven photons surpassing 10 GeV. The highest-energy photon in the LAT data was measured to have an energy of 39.88 GeV at about 81.7 s after the GBM trigger. \n3.2.1.4 GRB 090926A A photon with an energy of 130.6 MeV, the first photon in a series of high-energy photons, was seen 2.21 s after the GBMtrigger. The burst under analysis exhibited a diverse spectrum of photon energies, whereby 339 photons had energies above 100 MeV, 31 \npossessed energies surpassing 1 GeV, and two possessed energies exceeding 10 GeV. Approximately 24.84 s after the GBM trigger, the LAT instrument detected a photon with a maximum energy of 19.46 GeV. \n3.2.1.5 GRB 110731A The first detection of a high-energy photon, with an energy measurement of 817.1 MeV, occurred at a time interval of 3.19 s after the trigger time. A diverse range of photon energies was generated during this burst, with 40 photons over 100 MeV and an additional four photons surpassing 1 GeV. The most energetic photon seen by the LAT instrument was observed to occur 1.93 s after the trigger time, with measured energy of 8.27 GeV. \n3.2.1.6 GRB 130502B There was a wide variety of energies present in this burst, with 68 photons having energy more than 100 MeV and 2 having energy more than 10 GeV. At 222.1 s after the GBM trigger, the highest energetic photon detected in the LAT data had an energy of 31.1 GeV, followed by a photon with an energy of 17.3 MeV detected at 48.2 s after the GBM trigger. \n3.2.1.7 GRB 141207A The first energetic photon, measured at 765.3.5 MeV, was detected 3.9 s after the GBM trigger. In this burst, there were 19 photons with energy over 100 MeV and 11 with energies above 1 GeV. At 734.3 s after the GBM trigger, the highest energetic photon detected in the LAT data had an energy of 5.5 GeV. \n3.2.1.8 GRB 170214A The first high-energy photon, with a measured energy of 152.5 MeV, was detected 39.5 s after the GBM trigger. A wide variety of energies was present in this burst, with 217 photons having energy more than 100 MeV and 13 having energies greater than 1 GeV. At 103.6 s after the GBM trigger, the highest energetic photon detected in the LAT data had an energy of 7.8 GeV. \n3.2.1.9 GRB 180720B The first detection of a high-energy photon occurred 12.5 s after the BAT trigger, and its energy was measured to be 175.2 MeV. The energy spectrum of the photons in this burst exhibited a wide range, with 129 photons with energies over 100 MeV and eight photons above the threshold of 1 GeV. The LAT instrument detected the photon with a maximum energy of 142.4 s after the BAT trigger, and its energy was measured to be 4.9 GeV.', '3.2.2 Swift/XRT Data': 'The Swift/XRT followed-up GRB 090510, 090902B, 090926A, 110731A, 130502B, 131108A, 170214A and 180720B in different series of observations (De Pasquale et al. 2010; Kennea & Stratta 2009; Ackermann et al. 2011; Oates et al. 2011; Cenko et al. 2013; Stroh & Kennea 2013; Beardmore et al. 2017; Evans 2018). This instrument monitored these bursts in the photon counting (PC) and windowed-timing (WT) modes with spectrum exposures from hundreds to hundreds of thousands of seconds. The best-fitting absorption columns (intrinsic) ranges from 2 . 1 + 1 . 4 -1 . 3 × 10 21 to 4 . 4 + 3 . 1 -3 . 0 × 10 21 cm -2 , and from 10 + 11 . 2 -10 . 0 × 10 20 to 2 . 3 + 0 . 8 -0 . 6 × 10 22 cm -2 for WT and PC modes, respectively. Data sets from the Swift/XRT instrument were obtained from the publicly accessible database of the Swift website. 10 The flux density at 10 keV is converted to 1 keV using the conversion factor determined in Evans et al. (2010).', '3.2.3 Optical Data': 'Optical data for GRB 090510 (White-band), GRB 090902B (R-band), GRB 090926A (V-band), GRB 110731A (White- and V-band), GRB 131108A (White-, B-, U- and W1-band), GRB 170214A (White- and R-band) and GRB 180720B (R-band) were taken from Fraija et al. (2016b), Pandey et al. (2010), Rau et al. (2010), Fraija (2015), Ajello et al. (2019c); Giuliani et al. (2014); Corsi et al. (2013); Volnova et al. (2013b,a), Tang et al. (2017); Beardmore et al. (2017); Mazaeva et al. (2017); Kruehler et al. (2017) and Fraija et al. (2019c), respectively.', '3.3 Results and Discussion': 'We use the analysis of the closure relations shown in the subsection 3.1 to describe our GRB sample with the current model evolving in the stellar wind or homogeneous environment. The panels in Figure A3 display the LAT observations of GRB 080825C, GRB 130502B, and GRB 141207A with the best-fit curve generated by the FS model evolving in the stellar wind (right) and constant density (left) environment. Due to these three bursts having unknown redshifts, we assume a value of 𝑧 = 1 . 0 to estimate the total radiated energy and the luminosity distance. We show the afterglow evolution in both the stellar wind and homogeneous environment because the circumburst environments cannot be constrained as a result of the absence of optical data for GRB 130502B and X-ray and optical data for GRB 080825C and GRB 141207A. The panels in Figure A4 show the LAT, X-ray, and optical observations of GRB 090510, GRB 090926A and GRB 170214 with the best-fit curve generated by the FS model evolving in the constant-density medium, and the panels in Figure A5 show the LAT, X-ray and optical observations of GRB 090902B, GRB 110731A and GRB 180720B with the best-fit curve generated by the FS model evolving in the stellar-wind environment. We use Markov-Chain Monte Carlo (MCMC) simulations with the eight parameters used for the complete sample of GRBs to \nfind the best-fit values that describe the multi-wavelength afterglow observations with the SSC and synchrotron FS models. To represent all the data in this case, a total of 15900 samples and 4400 tuning steps are used. The effect of EBL absorption as proposed in Franceschini & Rodighiero (2017) was adopted. We only display Figure A6, which corresponds to GRB 080825C, for showing the best-fit values and the median of the parameter posterior distributions. Tables A3 and A4 list the best-fit values found with MCMC simulations after describing the multiwavelength afterglow observations with a synchrotron and SSC model evolving in both types of considered media. Tables A5 and A6 display the synchrotron and SSC spectral breaks in a constant-density and a stellar-wind medium, respectively, which are calculated with the best-fit values reported in Tables A3 and A4. We note that, while it may appear that the early LAT lightcurves are better fitted by the pure SSC model, we must also simultaneously explain the X-ray and optical observations. These are well fitted with the synchrotron model, so the synchrotron component is required and we are not able to consider a pure SSC model just for the LAT curves. Furthermore, with the parameters found, the SSC flux decreases very slowly and gives a small contribution of the early LAT data, which is less compared to the synchrotron radiation.', '3.3.1 Microphysical parameters': 'The best-fit values of the microphysical parameter given to accelerate electrons lie in the range of 0 . 3 ≤ 𝜀 e ≤ 0 . 9. In the constant-density scenario, the synchrotron afterglow model used for modeling the LAT light curves of GRB 090926A, GRB 141207A and GRB 170214A shows that they lie in the fast-cooling regime, while the light curves of GRB 080825C, GRB 090510 and GRB 130520B lie in the slow-cooling regime; see Table A5. In the stellar-wind scenario, the synchrotron afterglow model used for modeling our sample shows that the light curves lie in the fast-cooling regime; see Table A6. The results indicate that although the fraction of the total energy density given to accelerate electrons is much greater than 𝜀 e ≫ 0 . 1, the shock-accelerated electrons are not in the fast-cooling regime during the entire LAT light curve. This indicates that for some GRBs a transition from radiative to adiabatic regime occurs at the beginning of the LAT observations. \nThe best-fit values of the magnetic microphysical parameter lie in the interval 10 -5 ≤ 𝜀 B ≤ 10 -1 . As such, they are in the range of values required to model the multiwavelength afterglow observations in a large sample of GRBs; 10 -5 ≲ 𝜖 𝐵 ≲ 10 -1 (Wijers & Galama 1999; Panaitescu & Kumar 2002; Yost et al. 2003; Panaitescu 2005; Santana et al. 2014). Tak et al. (2019) conducted a comprehensive analysis of temporal and spectral indices, meticulously examining the closure relations within a sample of 59 LAT-detected bursts that were carefully chosen. They showed that although the traditional synchrotron emission model adequately explains the spectrum and temporal indices in the majority of instances, a significant proportion of bursts can hardly be characterized by this model. Furthermore, they reported that those satisfying the closure relations are in the slow-cooling regime ( 𝜈 syn m < 𝜈 LAT < 𝜈 syn c ) as long as the microphysical parameter is unusually low ( 𝜖 𝐵 < 10 -7 ). Our results show that the closure relations of a fraction of bursts can be satisfied with the synchrotron afterglow model in the radiative regime and typical values of 𝜖 𝐵 .', '3.3.2 The post jet-break decay phase': 'In a time scale of days, during the post-jet break decay phase, it is expected that the afterglow lies in the adiabatic regime rather than the radiative regime (Racusin et al. 2009). Therefore, the multi-wavelength observations could be described with 𝐹 syn 𝜈 ∝ 𝑡 -𝑝 , for 𝜈 syn m , f < 𝜈 < 𝜈 syn c , f or max { 𝜈 syn m , f , 𝜈 syn c , f } < 𝜈 (see e.g. Fraija et al. 2022b), which are distinct from the temporal decay indices found for our sample (Pereyra et al. 2022; Becerra et al. 2019), except GRB 090510 and GRB 110731A. This implies that, except for these bursts, they were most likely emitted from a wide outflow with a significant half-opening angle, as shown by multi-wavelength observations, which display no indication of late steep decays. Based on the best-fit values, the jet opening angles become ≳ 8 · , and for GRB 090510 and GRB 110731A, they are 𝜃 𝑗 ≈ 0 . 5 · and 2 · , respectively, which lies in the usual values ( 𝜃 𝑗 ≲ 10 · ; Bloom et al. 2001).', '3.3.3 Efficiency of equivalent kinetic energy': 'The efficiency provides crucial information on the gamma-ray emitting process. The best-fit values of the equivalent kinetic energies 5 . 72 × 10 52 ≤ 𝐸 ≤ 10 54 erg and the isotropic energies in gamma-rays reported by the GBM and LAT instruments in the range of 4 . 0 × 10 51 ≤ 𝐸 𝛾, iso ≤ 1 . 3 × 10 54 erg (Ajello et al. 2019a) lead to kinetic efficiencies in the range of 0 . 03 ≲ 𝜂 K ≲ 0 . 32, which are typical compared to those values reported in the literature (Guetta et al. 2001; Zhang et al. 2007; Kumar & Zhang 2015), and a kinetic efficiency of 𝜂 ≈ 0 . 03 for GRB180720B , which is very low. The atypical value of efficiency for GRB 180720B was estimated considering the isotropic energy reported in the 2FLGC. If we would have considered the isotropic energy reported in other analyses (3 × 10 53 erg Abdalla et al. 2019; Fraija et al. 2019d), the kinetic efficiency would have been 𝜂 ≈ 0 . 24.', '3.3.4 The profile of the circumburst environment': 'The best-fit values of the wind parameter lie in the range of 10 -2 ≲ 𝐴 W ≲ 1, typical for GRBs identified as powerful bursts (Ackermann et al. 2013; Perley et al. 2014; Vestrand et al. 2014; Fraija et al. 2012; Racusin et al. 2008; Fraija et al. 2017; Becerra et al. 2017). Similarly, the values found of the homogeneous medium in the range 4 . 6 × 10 -3 ≲ 𝑛 ≲ 1 cm -3 are usual with those found for other GRBs (Fraija et al. 2019c; Acciari et al. 2021; H. E. S. S. Collaboration 2021; LHAASO Collaboration 2023). Considering the fact that short bursts detonate at very low densities (Soderberg et al. 2006; Berger 2014), the value of 𝑛 = 4 . 6 × 10 -3 cm -3 obtained for GRB 090510 is compatible with the observations. Furthermore, the joint detection and modeling of the gravitational and electromagnetic signatures (Abbott et al. 2017b; Goldstein et al. 2017), which were associated with a fusion of two neutron stars (Abbott et al. 2017a), provided values of circumburst densities consistent with the ones obtained for GRB 130502B. \nSimilarly to other bursts identified by the LAT instrument and predicted values from numerical simulations (Tchekhovskoy et al. 2008), the best-fit values of the initial bulk Lorentz factor fall in the range 10 2 ≲ Γ ≲ 10 3 . Since our GRB sample included the highest energetic photons, we expect the bulk Lorentz factor values to coincide with those of the strongest bursts seen by the Fermi /LAT (Ackermann et al. 2011; Veres &Mészáros 2012; Ackermann et al. 2013; Abdo et al. 2009a; Ackermann & et al. 2010; Ackermann et al. 2014; Fraija et al. 2019a,c).', '3.3.5 The highest energy photons': 'Figure A7 exhibits all photons with energies > 100 MeV and probabilities > 90% of being associated with each burst of our representative sample. Additionally, we show in red lines the maximum photon energies released by the synchrotron afterglow model evolving in a constantdensity (dotted) and stellar-wind (dashed) environment, estimated with the best-fit values reported in Tables A3 and A4. This Figure shows that the synchrotron FS model cannot explain the highest energy photons collected by Fermi /LAT. Energetic photons above > 10 GeV are usually explained via hadronic and SSC scenarios. In the hadronic scenarios, high-energy gamma-ray emission has been interpreted via photo-hadronic interactions; ultrarelativistic protons accelerated in the jet with internal synchrotron photons (Asano et al. 2009; Dermer et al. 2000), inelastic proton-neutron collisions (Mészáros & Rees 2000), and relativistic neutrons with seed photons coming from the outflow (Dermer & Atoyan 2004; Alvarez-Muñiz et al. 2004). Even though GRBs are among the most plausible candidates to accelerate cosmic rays up to ultra-high energies ( ≳ 10 18 eV; Waxman 1995; Vietri 1995) and thus, potential candidates for neutrino detection, the IceCube collaboration reported no coincidences between neutrinos and GRBs after analyzing years of data (Abbasi et al. 2022, 2012; Aartsen et al. 2016, 2015). Because of this, we rule out hadronic models as an explanation for the observed properties of GRBs and conclude that the number of hadrons is too small for hadronic interactions to efficiently generate observable gamma-ray signals in GRBs. On the other hand, a few bursts GRB 160821B, GRB 180720B, GRB 190114C, GRB 190829A, GRB 201216C and GRB 221009A have been detected, up to now, with photons above 100 GeV by the Major Atmospheric Gamma Imaging Cherenkov Telescopes (MAGIC; Acciari et al. 2019; Acciari et al. 2021), the High Energy Stereoscopic System (H.E.S.S.; Abdalla et al. 2019; H. E. S. S. Collaboration 2021) and the Large High-Altitude Air Shower Observatory (LHAASO; LHAASO Collaboration 2023). The VHE emission in all these bursts has been successfully described via SSC FS model (e.g., see Acciari et al. 2021; Abdalla et al. 2019; H. E. S. S. Collaboration 2021; LHAASO Collaboration 2023). Therefore, the most appropriate mechanism to explain the photons above 10 GeV in our representative sample is SSC mechanism from FSs. It should be noted that the values of the spectral breaks derived from the best-fit parameters (see Tables A5 and A6) indicate that the KN effects cannot be neglected in some GRBs of our sample. Based on the maximum photon energy emitted by synchrotron radiation during the deceleration phase, several authors have claimed that some LAT light curves cannot be adequately interpreted in terms of only synchrotron FS radiation (Ghisellini et al. 2010; Maxham et al. 2011; Fraija 2015; Fraija et al. 2016a, 2020). Therefore, although the closure relations of the synchrotron standard model could satisfy the 29 LAT-detected GRBs with VHE emission above 10 GeV reported in the 2FLGC, this model is not the appropriate one, and a new mechanism such as SSC would be the better favored as shown in this manuscript.', '4 SUMMARY': 'Based on the external FS scenario in the stellar wind and homogeneous medium, we have presented a general analytical model of the synchrotron and SSC processes in the fully adiabatic ( 𝜖 = 0), fully radiative ( 𝜖 = 1) or partially radiative or adiabatic (0 < 𝜖 < 1) regimes for an electron spectral index in the ranges of 1 < 𝑝 < 2 and 2 ≤ 𝑝 . Using the typical values of a GRB afterglow and assuming that all electrons are accelerated during the FS, we explicitly derived and plotted the expected synchrotron and SSC light curves and spectra in the stellar-wind and constant-density medium for each range of p. We calculated the spectral breaks in the KN regime (Nakar et al. 2009; Wang et al. 2010) and introduced the effect of EBL absorption as proposed in Franceschini & Rodighiero (2017). We discuss the evolution of the SSC flux as a function of the equivalent kinetic energy, density parameter, electron spectral index, and radiative parameter. We compared the expected SSC fluxes with the CTA, MAGIC, and Fermi /LAT sensitivities (Fioretti et al. 2019). We showed that all the expected fluxes are below the Fermi /LAT sensitivity, and depending on the parameter values, they could be detected by the CTA or MAGIC Telescopes. In particular, when 𝜖 = 0, the standard synchrotron and SSC light curves derived in the standard stellar wind and homogeneous medium afterglow models for 1 < 𝑝 < 2 and 2 ≤ 𝑝 are recovered (Sari et al. 1998; Panaitescu & Mészáros 1998; Chevalier & Li 2000; Bhattacharya 2001; Sari & Esin 2001; Gao et al. 2013; Fraija et al. 2019b). \nAdiabatic breaks around the transition time between fast- and slow-cooling regimes are expected in the light curves. However, if the value of \n𝑝 does not deviate from 2, adiabatic breaks are not observed. We have derived the closure relations between the temporal and spectral indices that describe the evolution of the synchrotron and SSC flux as a function of 𝜖 and 𝑝 . Significant variations of the spectral and temporal features of the afterglow emission are introduced by radiative losses only if 𝜖 is large and approaches to unity. Otherwise, Deviations around 𝜖 = 0 will produce small variations of the spectral and temporal features. In the fully adiabatic regime, the temporal evolution of the synchrotron flux in the stellar wind and constant-density medium is identical. On the contrary, in the radiative regime, they evolve differently in both density profiles. Therefore, an afterglow transition between stellar wind and homogeneous medium would be easily identifiable in the radiative regime. \nGiven the closure relations (see Table A1), we notice that synchrotron and SSC fluxes could have a similar evolution in time and energy when condition 𝛼 syn ( 𝛽 ) ≈ 𝛼 ssc ( 𝛽 ) is satisfied. For a stellar wind and constant density medium, the condition is reached when 𝛽 → 1 and the observed frequency evolves under the cooling condition of { 𝜈 j m , 𝜈 j c } < 𝜈 with j = ssc and syn. Irrespective of the value of 𝑝 , the temporal decay indices become 𝛼 ≈ 4 4 -𝜖 and ≈ 2 ( 4 + 𝜖 ) 8 -𝜖 for a wind and homogeneous medium, respectively. Temporal and spectral similarities observed in two different bands of afterglow emission (e.g., TeV gamma-rays, X-rays and optical bands) could be explained through the synchrotron and SSC FS model with a hard electron spectral index 𝑝 ≈ 2. If 𝜖 ≈ 0, the expected flux in the stellar wind or constant-density medium evolves with a temporal index of 𝛼 ≈ 1, the parameter 𝜖 could be used to discriminate between the afterglow models. It can be seen in Table A1 that the closure relations of the synchrotron and SSC models with 𝜖 ≈ 1, and the cooling condition { 𝜈 k m , 𝜈 k c } < 𝜈 (with k = syn and ssc) favor the powerful LAT-detected GRBs described with 𝛽 ∼ 1 and 𝛼 ∼ 1 . 5 as reported by Ghisellini et al. (2010). GRBs described with 𝛼 > 2 and 𝛽 < 1 satisfied the closure relations of the SSC model under the cooling condition 𝜈 ssc m < 𝜈 < 𝜈 ssc c with 0 ≤ 𝜖 ≤ 1, which is difficult to explain with the standard synchrotron model. GRBs satisfy the synchrotron closure relations of a homogeneous medium of a cooling condition 𝜈 syn m < 𝜈 < 𝜈 syn c for 𝜖 = 0 or a cooling condition of { 𝜈 syn m , 𝜈 syn c } < 𝜈 for 𝜖 = 1. For example, given a value of 𝑝 = 2 . 8 and a cooling condition of 𝜈 syn m < 𝜈 < 𝜈 syn c , the corresponding spectral and temporal indices are 𝛽 = 0 . 9 and 𝛼 = 1 . 35, respectively. A similar result could be obtained considering a value of 𝑝 = 1 . 8 and a cooling condition of { 𝜈 syn m , 𝜈 syn c } < 𝜈 with 𝜖 = 0 . 92. It is worth noting that if ℎ𝜈 ≈ 100 MeV, the first condition 𝜈 syn m < 𝜈 < 𝜈 syn c might require (due to 𝜈 syn c ∝ 𝜀 -3 2 𝐵 ) an atypical value of the magnetic microphysical parameter (e.g. 𝜖 𝐵 ≲ 10 -6 ; Kumar & Barniol Duran 2009, 2010) whereas the second one leads to an usual value of the microphysical parameter (e.g., 3 × 10 -5 ≲ 𝜖 𝐵 ≲ 0 . 3; Santana et al. 2014). \nAs particular cases, we have derived the Fermi /LAT light curves together with the photons with energies ≥ 100 MeV associated with each burst. We have selected those GRBs (080825C, 090510, 090902B, 090926A, 110731A, 130502B, 141207A, 170214A and 180720B) from the 2FLGC (Ajello et al. 2019b) with values of temporal and spectral indices with 𝛼 L ≳ 1 . 5 and Γ L ≈ 2, respectively. We have applied our adiabatic-radiative afterglow model to fit the observations of this sample. We want to highlight that the standard SSC or synchrotron afterglow model cannot describe the closure relations of GRBs with both the temporal decay index 𝛼 𝐿 ≳ 1 . 5 and the spectral index Γ 𝐿 ≈ 2. It is always possible to assume light curves from synchrotron and SSC models to superimpose them and describe some temporal evolution different from that predicted by the standard afterglow model. However, light curves in the radiative regime can be done in an evident and clean way. We have used the multiwavelength observations to constrain the parameters in the synchrotron and SSC mechanism and model the LAT light curves of the sample via MCMC simulations. We have fitted the LAT observations of GRB 080825C, GRB 130502B and GRB 141207A with the FS model evolving in the stellar-wind and constant-density environment, the LAT, X-ray and optical observations of GRB 090510, GRB 090926A and GRB 170214 using the constant-density medium, and GRB 090902B, GRB 110731A and GRB 180720B with the stellar-wind environment. The results indicate that although the fraction of the total energy density given to accelerate electrons is much greater than 𝜀 e ≫ 0 . 1, the shock-accelerated electrons are not in the fast-cooling regime during the entire LAT light curve. This indicates that in some cases a transition from radiative to adiabatic regime occurs in the LAT observations.', 'ACKNOWLEDGEMENTS': 'We would like to extend our appreciation to the anonymous referee for their thorough evaluation of the article and valuable suggestions, which significantly improved the quality and clarity of our manuscript. We also thank Tanmoy Laskar, Paz Beniamini and Bin Zhang for useful discussions. NF acknowledges financial support from UNAM-DGAPA-PAPIIT through grants IN106521 and IN119123. PV acknowledges financial support from NASA grants 80NSSC19K0595 and NNM11AA01A.', 'DATA AVAILABILITY': 'No new data were generated or analysed in support of this research.', 'REFERENCES': 'Aartsen M. G., et al., 2015, ApJ, 805, L5 \nAartsen M. G., et al., 2016, ApJ, 824, 115 \nAbbasi R., et al., 2012, Nature, 484, 351 \nTable A1. Closure relations of the SSC and synchrotron afterglow model in stellar-wind and homogeneous medium with j = ssc and syn, respectively. \n- Tsang O., Kirk J. G., 2007, A&A, 463, 145\n- Ukwatta T. N., et al., 2009, GRB Coordinates Network, 9337, 1 \nVeres P., Mészáros P., 2012, ApJ, 755, 12 \nVestrand W. T., et al., 2014, Science, 343, 38 \nVietri M., 1995, ApJ, 453, 883 \n- Vink J. 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J., Connaughton V., 2008a, GRB Coordinates Network, Circular Service, No. 8141, (2008), 8141 \nvan der Horst A. J., Connaughton V., 2008b, GRB Coordinates Network, 8141, 1 \nvon Kienlin A., Younes G., 2013, GRB Coordinates Network, 14530, 1', 'APPENDIX A: SOME EXTRA MATERIAL': 'This paper has been typeset from a T E X/L A T E X file prepared by the author. \nFigure A1. SSC light curves (left) and spectra (right) in the stellar-wind afterglow model for 𝜖 = 0 (upper panels), 0 . 2 (center panels) and 0 . 4 (lower panels) and 𝑝 = 1 . 7, 1 . 9, 2 . 1 and 2 . 3. The light curves and SED are shown at 1 TeV and 5 × 10 4 s, respectively, for 𝜖 e = 0 . 1, 𝜖 B = 10 -4 and 𝜁 = 0 . 5. The values of parameter pairs ( 𝐴 W = 0 . 1 and 𝐸 = 10 52 erg), ( 𝐴 W = 10 -3 and 𝐸 = 10 52 erg), and ( 𝐴 W = 10 -3 and 𝐸 = 10 53 erg) are used for the left, middle and right panels. The sensitivities of CTA (Southern array, green line), MAGIC (purple line) and Fermi /LAT (red line) are shown between 75 and 250 GeV at 3 × 10 4 s for a zenith angle of 20 · (Fioretti et al. 2019). We have considered a hypothetical burst located at 𝑧 = 0 . 1 and the effect of the EBL absorption proposed in (Franceschini & Rodighiero 2017). \n<!-- image --> \nFigure A2. The same as Figure A1, but for a constant-density medium. The values of parameter pairs ( 𝑛 = 0 . 1 cm -3 and 𝐸 = 10 52 erg), ( 𝑛 = 10 -3 cm -3 and 𝐸 = 10 52 erg), and ( 𝑛 = 10 -3 cm -3 and 𝐸 = 10 53 erg) are used for the left, middle and right panels. \n<!-- image --> \n-3 \n10 \n-4 \n10 \n-5 \n10 \n-6 \n10 \n) \ny \nJ \nm \n( \ny \nt \ni \ns \nn \ne \nD \nx \nu \nl \nF \n) \ny \nJ \nm \n( \ny \nt \ni \ns \nn \ne \nD \nx \nu \nl \nF \n-2 \n10 \n-3 \n10 \n-4 \n10 \n-5 \n10 \n-6 \n10 \n0 \n10 \n0 \n10 \n1 \n10 \n1 \n10 \n2 \n10 \nTime \nsince \nburst \n(s) \n(a) GRB 080825C \nFigure A3. The LAT and X-ray observations with the best-fit curves using the FS model evolving in the stellar-wind (right) and homogeneous (left) environment. The dashed and dotted lines correspond to synchrotron and SSC models, respectively. \n<!-- image --> \nFermi-LAT \n(100 \nMeV) \n3 \n10 \n4 \n10 \n5 \n10 \n2 \n10 \nTime \nsince \nburst \n(s) \n(e) GRB 141207A \n10 \n2 \nTime \nsince \nburst \n(s) \n(f) GRB 141207A \nFermi-LAT \n(100 \nMeV) \n3 \n10 \n4 \n10 \n5 \n10 \n6 \n10 \n6 \n10 \n) \ny \nJ \nm \n( \ny \nt \ni \ns \nn \ne \nD \nx \nu \nl \nF \n) \ny \nJ \nm \n( \ny \nt \ni \ns \nn \ne \nD \nx \nu \nl \nF \n) \ny \nJ \nm \n( \ny \nt \ni \ns \nn \ne \nD \nx \nu \nl \nF \n-3 \n10 \n-4 \n10 \n-5 \n10 \n-6 \n10 \n10 \n2 \n0 \n10 \n-2 \n10 \n-4 \n10 \n-6 \n10 \n-8 \n10 \n-10 \n10 \n-2 \n10 \n-3 \n10 \n-4 \n10 \n-5 \n10 \n-6 \n10 \n0 \n10 \n1 \n10 \n0 \n10 \n1 \n10 \n10 \n1 \n10 \n2 \n2 \n10 \nTime \nsince \nburst \n(s) \n(b) GRB 080825C \nFermi-LAT \n(100 \nMeV) \nSwift-XRT \n(1 \nkeV) \n4 \n10 \n5 \n10 \n10 \nTime \nsince \nburst \n(s) \n(d) GRB 130502B \nFermi-LAT \n(100 \nMeV) \n3 \n10 \n4 \n10 \n5 \n10 \n3 \nFermi-LAT \n(100 \nMeV) \n3 \n10 \n4 \n10 \n5 \n10 \n6 \n10 \n6 \n10 \n6 \n10 \nTable A2. Sample of 10 GRBs used here. Temporal and spectral indices are taken from Ajello et al. (2019b) with 𝛽 L = Γ L -1.Table A3. The best-fit values found with MCMC simulations after describing the multiwavelength afterglow observations with a synchrotron model evolving in a constant circumstellar medium. \nTable A4. The best-fit values found with MCMC simulations after describing the multiwavelength afterglow observations with a synchrotron model evolving in a stellar-wind environment. \nTable A5. Derived quantities at 40 s from the best-fit parameter values found with MCMC simulations with an afterglow model evolving in a homogeneous medium.Table A6. The same as Table A5, but for an afterglow model evolving in a stellar-wind medium. \n<!-- image --> \n<!-- image --> \nFigure A4. TheLAT(peach),X-ray(purple) and optical (yellow) observations with the best-fit curves using the forward-shock model evolving in the homogeneous medium. The dashed and dotted lines correspond to synchrotron and SSC models, respectively. \n<!-- image --> \n<!-- image --> \n(a) GRB 090902B \n<!-- image --> \n7 \nFigure A5. The LAT, X-ray and optical observations with the best-fit curves using the FS model evolving in stellar-wind environment. The dashed and dotted lines correspond to synchrotron and SSC models, respectively. \n<!-- image --> \nFigure A6. Corner plot of the parameters obtained from modelling the multiwavelegth afterglow observations of GRB 080825C with the constant-density model shown in Section 2. The statistics for all parameters involved in the MCMC simulations are reported in Table A3. \n<!-- image --> \n(a) LAT light curve for GRB 080916C. \n<!-- image --> \n(d) LAT light curve for GRB 090926A. \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure A7. All the photons with energies > 100 MeV and probabilities > 90% of being associated with each burst in our sample. The red lines correspond to the maximum photon energies from our synchrotron afterglow model evolving in a constant-density (dotted) and stellar-wind (dashed) medium. \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n(f) LAT light curve for GRB 130502B. \n<!-- image --> \n(c) LAT light curve for GRB 090902B.'}

2023arXiv230809390S

The future Laser Interferometer Space Antenna LISA mission which has successfully passed Mission Formulation phase is in planning to be launched in 2030s. One of the ubiquitous LISA sources are the whitedwarf binaries WDB of which sim40 are guaranteed sources as of now making LISA unique in comparison to its groundbased counterpart. The current hardware design in planning necessitates a thorough check to determine whether the various locking schemes influence the guaranteed sources signals significantly in order to reconsider that what is hardcoded in the phasemeter before launch for prescience operations phase. Comparison of the phasemeter output of a faceon V407Vul binary and an edgeon ZTFJ1539 binary indicates that the nonswap locking scheme N2a is optimal for instrument calibration. Additionally the influence of the sim 7 min orbital period edgeon source in two of the locking schemes yields a difference of maximum leq 10 at the Time Delay Interferometry TDI output for data stream of one day. Simplified analyses show that neither of the locking schemes is favoured in the postprocessing level. We find similar amplitudes in the TDI output stream for the faceon system V407Vul and the edgeon system ZTFJ1539 which leads to a significantly smaller inclination bias for the nonswap locking scheme. Additionally a larger amplitude for edgeon systems will benefit most verification systems as the population of verification systems is biased towards edgeon systems as they are easier to detect in electromagnetic data.

2023-08-01T00:00:00Z

['arXiv:2308.09390', '2023arXiv230809390S', '10.48550/arXiv.2308.09390']

['Astrophysics - Instrumentation and Methods for Astrophysics']

Optimal frequency plan for LISA prescience operations using verification binaries

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2308.09390.pdf

{'ON THE OPTIMAL FREQUENCY PLAN FOR LISA PRE-SCIENCE OPERATIONS USING VERIFICATION BINARIES': 'Sweta Shah 1 Valeriya Korol 2 , Thomas Kupfer 3 , 4 \n1 Leibniz Universitat Hannover, Institut fur Gravitationsphysik, Callinstraße 38, Hannover 30167, Germany 2 Max-Planck-Institut fur Astrosphysik, Karl-Schwarzschild-Straße 1, Garching 85748, Germany 3 Hamburger Sternwarte, University of Hamburg, Gojenbergsweg 112, D-21029 Hamburg, Germany 4 Department of Physics and Astronomy, Texas Tech University, P.O. Box 41051, Lubbock, TX 79409, USA \nDraft version December 4, 2023', 'ABSTRACT': "The future Laser Interferometer Space Antenna (LISA) mission, which has successfully passed Mission Formulation phase, is in planning to be launched in 2030s. One of the ubiquitous LISA sources are the white-dwarf binaries (WDB) of which ∼ 40 are guaranteed sources as of now, making LISA unique in comparison to its ground-based counterpart. The current hardware design in planning necessitates a thorough check to determine whether the various locking schemes influence the guaranteed sources' signals significantly. This could have implication to re-consider the choice of the initial locking configuration that is optimal for the pre-science or science operations phase. Comparison of the phasemeter output of a face-on (V407Vul) binary and an edge-on (ZTFJ2243 binary indicates that the non-swap locking scheme with the maximal Doppler locks, is optimal for instrument calibration. The most well known locking scheme from Shaddock et al (2004) gives an TDI Michelson SNR for VB that is larger by factor of 10 in comparison to free-running laser configuration. Comparison with another locking scheme with most number of doppler links in the phasemeter measurement gives SNR difference of ∼ 30% and ∼ 20% for the face-on system V407Vul and the edge-on system ZTFJ2243 respectively. We find similar amplitudes in the TDI output stream for the face-on system V407Vul and the edgeon system ZTFJ2243 which leads to a significantly smaller inclination bias for the non-swap locking scheme. Additionally, a larger amplitude for edge-on systems will benefit most verification systems as the population of verification systems is biased towards edge-on systems as they are easier to detect in electromagnetic data. \nSubject headings: stars: binaries - gravitational waves, stars: ultra-compact galactic binaries - verification binaries, GW detectors - LISA, instrument - heterodyne interferometer laser locking", '1. INTRODUCTION': 'Gravitational wave (GW) astronomy has become a norm since its first direct detection in 2015, announced in 2016 (Abbott et al. 2016). This provided a strong push forward for the long envisaged space-based GW detector concept, the Laser Interferometer Space Antenna, LISA 1 , an European Space Agency (ESA) flagship mission with a contribution from the National Aeronautics and Space Administration (NASA) and with a planned launch year in the late ∼ 2030s. \nWith its well-defined science cases (Amaro-Seoane 2023), the detector\'s sensing window for 4-dimensional space-time fluctuations has been set to a range of [10 -4 -1] Hertz (Hz). This characteristic band not only complements its ground-based counterparts, but also allows LISA to cover a broader science case encompassing larger ranges of source masses, mass-ratios, and redshifts in astrophysics and cosmography. The most numerous astronomical class of sources observable by LISA are the compact White Dwarf Binaries (WDBs) in our home-galaxy, \nsweta.shah@aei.mpg.de \n1 https://www.elisascience.org/ \nthe Milky-Way . Population synthesis models predict the detector to be able to individually resolve O (10 4 ) sources (Kupfer et al. 2018), detached and mass-transferring (e.g. Toonen et al. 2014), from an unresolvable foreground comprising millions of compact Galactic binaries which, at ∼ 3 × 10 -3 Hz overtakes the noise from the instrument (e.g. Nissanke et al. 2012). \nThe ground-breaking key technology of needing the test mass in free-fall at the level of an unfathomable ∼ 15 × 10 -15 m s -2 / √ Hz 2 to achieve LISA\'s scientific goals for the frequency band stated above has been demonstrated successfully by LISA Pathfinder, LISA\'s precursor mission by ESA (Armano et al. 2016b). Another crucial key technology for long-arm interferometry has been partially demonstrated with the Gravity Recovery And Climate Experiment Follow-On (GRACE-FO). This Earth observation mission employed twin satellites and a 220km Laser Ranging Interferometer instrument (Abich et al. 2019) with a laser whose wavelength is 1064 nano-meters (Koch 2020), which is planned to be used for LISA as well. \nThe practicality of the measurement of real numbers in the phasemeters of LISA (Barke 2015) requires careful planning of the laser frequencies that the constellation will carry onboard its three spacecraft (S/C), where each spacecraft houses two optical benches (OB) constructed from low-expansion ceramic material(Troebset al. 2019) whereupon the interferometric measurements will occur. The reason for frequency plan stems from the fundamental need for heterodyne interferometry between two laser beams, where one of them has a power level in the order of sub nano-Watt with respect to the other with unequal frequencies. These constellation orbits (Martens & Joffre 2021) introduce a strong dependence on timevarying length changes, which are caused by the imbalance of gravitational pulls from major planets and the Sun. Additionally, the long-lived WDB signals will contribute small variations in amplitude, frequency, and phase. There are fundamentally two sources of constraints on the design of the allowed frequencies of the laser. One is the noise from the transimpedance amplifiers in the photodiode. This noise limits the photodiode\'s capability to read out the phases of the heterodyne interferometric signal (with unequal frequencies) above 25MHz, leading to increased noise (Barranco et al. 2018). Below 5MHz, the sensors are inundated with relative intensity noise (RIN) from the laser. RIN couples into the interferometric readout substantially at the heterodyne frequency violating the required noise level (Wissel et al. 2022a), which is mitigated in hardware by a technique, the so-called balanced-detection (Wissel et al. 2022b). \nSecondly, the phasemeter cannot distinguish negative (blue-shifted) or positive (red-shifted) Doppler shifts from the sinusoidal changes 3 of the armlengths between any two S/C nor the minute Doppler shifts from the WDBs and is not capable of dealing with zero crossing values where the two interfered lasers to have identical frequencies. The space-qualified phasemeter designed for LISA Gerberding et al. (2013) has therefore strict constraints from photodiodes\' sensitivity and phasemeter\'s internal limitations. \nFrom these hardware restraints, three distinct ways to form locks between six independent lasers on-board have been discovered earlier by (Shaddock 2004), (Barke 2015) for the setup where any two lasers on the two OBs within one S/C are not allowed to interfere with the received laser from the distant S/C in forming the 2.5 million km armlength heterodyne signal. Locking essentially means imprinting a copy of an interfered signal\'s phase (or frequency) to that of the local laser beam with a pre-determined offset comparable to that of the longarm Doppler shift. The three solutions proliferate for the generic set up of the lasers in how they can be locked to each other when all allowable frequencies by the nominal hardware design are imposed in a brute-force exploration of interferometric signals\' frequency parameter space (Heinzel 2020a), known as frequency plan in LISA, commonly referred by, \' fplan \'. \nA known GW signal\'s will provide an independent calibration of the instrument sub-systems, in addition to the in-built mechanisms in the hardware. Furthermore they will provide an estimate for the S/C angular parameters \ndistinct than the on board star trackers and on-ground ranging amongst others. The incoherence in the beatnote signals in the form of glitch due to the instrument sub-systems can be caused by laser cycle slip (Gerberding et al. 2013) and/or frequency jump in the electronic oscillator to time-stamp the optical signals (Yamamoto et al. 2022). The motivation for using such a signal\'s inference for the instrument is crudely sketched in Figure 1 with an exemplary case of fplan underlying the topology of the frequencies in LISA constellation 4 . A number of previous studies, (e.g. Littenberg 2018) have shown the importance and usefulness of WDBs in calibrating the signal\'s amplitude and phase due to scheduled gaps and unintentional glitches. It is shown by Littenberg (2018) in a Bayesian data analysis that coherent signals of about ten WDBs can be used to estimate an unknown period of gap within an accuracy of 0.1 to 0.2 seconds from a month to a year of data utilising the near constant frequency GW signals of the WDBs. This proof-of-concept study utilised Time Delay Interfetometry (TDI) (e.g., Armstrong et al. 1999), which is a technique optimizing GW signal over instrument noise for a generic setup of six free-running lasers across the constellation. This needs verification against the different choices of fplan , where the coupling of a GW signal bears no symmetry and is radically different in any of the various choices as explained later. \nAdditionally, it is not immediately clear which of the numerous laser frequency configurations in (Heinzel 2020a) is optimal with respect to the instrument characterisation, as motivated and shown in Figure 1. The instrument can be characterized in two groups of parameters: LISA constellation parameters (orientation) and parameters inherent to the instrument sub-systems, such as laser, clock, Gravitational Reference System, and etc. This paper explores using examples of two distinct laser locking schemes for two of the brightest known WDBs from optical observations, also known as Verification Binaries (VB) (Stroeer & Vecchio 2006), the standard candles for LISA instrument. It could be that the choice of one of the six LISA non-swap fplan should be different for commissioning and calibration phase(s) of the mission versus when the science operations begin. The former would be to maximise instrument knowledge, and later would be to yield maximum scientific output for a certain class of predicted astrophysical sources. For this study, the changes were implemented in Synthetic LISA ( Synth LISA ) to adapt the code to fplan . Synth LISA was originally developed to study LISA instrument noise as complement to other existing softwares (Vallisneri 2005) to simulate LISA Science process. Our study is complementary to recent developments of instrument simulation with non-Gaussian noise at sub-system level in Bayle & Hartwig (2023) and the contemporary software LISANode 5 relevant for further investigations of the instrument beyond the Adoption phase. \nThere is perhaps no urgency to have this figured out before the mission\'s Adoption (planned for 2024), as the meticulous design of the phasemeter (Yamamoto K. & Delgado 2022) already takes that possibility into account. \n<latexit sha1\\_base64="jNgIodlxWVM7o3NkYE8hnjlmCNA=">AAAB/3icbZC7SgNBFIbPxluMt1XBRpHBIFhI2I2FlkEbywTMBZJlmZ1MksHZCzOzwbCmsPBFbCwUsYyvYecz+BJONik08YeBj/+cwznzexFnUlnWl5FZWFxaXsmu5tbWNza3zO2dmgxjQWiVhDwUDQ9LyllAq4opThuRoNj3OK17t1fjer1PhWRhcKMGEXV83A1YhxGstOWae60+Jcnd0GWnKMV+iq6ZtwpWKjQP9hTypYNR5fvxcFR2zc9WOySxTwNFOJayaVuRchIsFCOcDnOtWNIIk1vcpU2NAfapdJL0/iE61k4bdUKhX6BQ6v6eSLAv5cD3dKePVU/O1sbmf7VmrDoXTsKCKFY0IJNFnZgjFaJxGKjNBCWKDzRgIpi+FZEeFpgoHVlOh2DPfnkeasWCfVYoVnQalzBRFvbhCE7AhnMowTWUoQoE7uEJXuDVeDCejTfjfdKaMaYzu/BHxscPc2yZbA==</latexit> \n<latexit sha1\\_base64="3RJk49JispmEqKyMglOPGA2iYBk=">AAAB7nicbVDLSgNBEOz1GeMr6lGRwSB4CrvxoMegF48JmAckS5idzCZDZmaXmVlhWXL0A7x4UMSrn5Dv8OY3+BNOHgdNLGgoqrrp7gpizrRx3S9nZXVtfWMzt5Xf3tnd2y8cHDZ0lChC6yTikWoFWFPOJK0bZjhtxYpiEXDaDIa3E7/5QJVmkbw3aUx9gfuShYxgY6Vm2s00V6NuoeiW3CnQMvHmpFg5Gde+H0/H1W7hs9OLSCKoNIRjrdueGxs/w8owwuko30k0jTEZ4j5tWyqxoNrPpueO0LlVeiiMlC1p0FT9PZFhoXUqAtspsBnoRW8i/ue1ExNe+xmTcWKoJLNFYcKRidDkd9RjihLDU0swUczeisgAK0yMTShvQ/AWX14mjXLJuyyVazaNG5ghB8dwBhfgwRVU4A6qUAcCQ3iCF3h1YufZeXPeZ60rznzmCP7A+fgBw6uTjA==</latexit> \n<latexit sha1\\_base64="wz2yq6F7L71iRA1mP6jw8Y4ox6s=">AAACAXicbVC7SgNBFJ2Nr5j4WLURbAajEJuwGwstgzYpI5gHZJdldjKbjJl9MDMbCMva+Cs2FopY2Fj6B3Z+iNbOJik08cCFwzn3cu89bsSokIbxqeWWlldW1/LrheLG5ta2vrPbEmHMMWnikIW84yJBGA1IU1LJSCfiBPkuI213eJn57RHhgobBtRxHxPZRP6AexUgqydH3LR/JAUYsqadOQm/SsiVR7NATRy8ZFWMCuEjMGSnVjr5e30fF74ajf1i9EMc+CSRmSIiuaUTSThCXFDOSFqxYkAjhIeqTrqIB8omwk8kHKTxWSg96IVcVSDhRf08kyBdi7LuqM7tXzHuZ+J/XjaV3bic0iGJJAjxd5MUMyhBmccAe5QRLNlYEYU7VrRAPEEdYqtAKKgRz/uVF0qpWzNNK9UqlcQGmyIMDcAjKwARnoAbqoAGaAINbcA8ewZN2pz1oz9rLtDWnzWb2wB9obz9izptD</latexit> \n<latexit sha1\\_base64="+Nm26yTFTmNk3Aysm3eblaXpr3k=">AAAB83icbVC7TgJBFJ3FF+ILtbSZSEis1l0stCTSWEKURwIbMjvMwoTZ2c3MXSPZ8Bs2Fhpj6yfQ+gF2foi9w6NQ8CQ3OTnn3tx7jx8LrsFxvqzM2vrG5lZ2O7ezu7d/kD88augoUZTVaSQi1fKJZoJLVgcOgrVixUjoC9b0h5Wp37xnSvNI3sEoZl5I+pIHnBIwUqcD7AHS2/PKuMu7+YJjOzPgVeIuSKFc/J7U7I9JtZv/7PQimoRMAhVE67brxOClRAGngo1znUSzmNAh6bO2oZKETHvp7OYxLhqlh4NImZKAZ+rviZSEWo9C33SGBAZ62ZuK/3ntBIIrL+UyToBJOl8UJAJDhKcB4B5XjIIYGUKo4uZWTAdEEQomppwJwV1+eZU0SrZ7YZdqJo1rNEcWnaBTdIZcdInK6AZVUR1RFKNH9IxerMR6sl6tt3lrxlrMHKM/sN5/AG8clbE=</latexit> \nFig. 1.LISA constellation geometry from star trackers and S/C constituent properties verified against those observed from long-lived, stable Verification ( apriori known white dwarf) binaries. Left part of the diagram gives the instrument model with the parameters describing them whereas right part of the diagram gives the flow of the optically measured VB binary parameters implicating for the former. \n<!-- image --> \nWe mean that all the six unique locking schemes can be telecommanded to reconfigure the measurement scheme aboard in-flight, the choice of which could be driven by optimizing multi-messenger observations or instrument calibration. Consequently, this could also have nontrivial implications for the logistics of the initial leg of the data processing pipeline, the I ntial N oise Re duction P ipeline 6 , INReP (Wiesner et al. 2021) 7 , which was successfully demonstrated in the mission\'s formulation review, essentially passing the LISA mission\'s study phase. \nThe paper starts by describing the matter-of-fact known binaries in Sec. 2, with their responses to LISA laser links described in Sec. 3. The intricate nature of change of response imposed by fplan is discussed in Sec. 4, followed by their propagation in a simplified couplets of Michelson-type TDI variables, with conclusion in Sec. 5.', '2. LISA VERIFICATION BINARIES': 'There is a number of guaranteed GW sources in the LISA\'s frequency band. These are known Galactic bi- \n6 internal to LISA Consortium \n- 7 planned to be run at the Science Operations Centre (SOC), in the ESA Astronomy Centre, ESAC \nnaries with orbital periods < 1 h that have been discovered with electromagnetic observatories (e.g. Kupfer et al. 2018). These sources are typically expected to be detected by LISA within weeks to months following constellation acquisition, as indicated by prior studies (e.g. Shah et al. 2012). The most common of these binaries consist of a combination of a neutron star or white dwarf paired with a compact helium star, white dwarf, or another neutron star. As per the latest data, the known candidates that could potentially be detected by LISA are in the order of several tens of binaries (Kupfer et al. 2023). However, with a considerable number of electromagnetic facilities due to be operational in the decade preceding LISA\'s launch, the pool of detectable binaries by LISA is projected to exceed 100 (Korol et al. 2017). \nKnown Galactic binaries offer a unique opportunity for verifying the in-orbit performance of LISA, and are thus often referred to as \'verification binaries\' (VB) (Stroeer & Vecchio 2006). The primary characteristics of their GW signal, such as amplitude and phase evolution, can be accurately predicted based on their electromagnetic measurements. Consequently, they can be used as tools for monitoring and maintaining the data quality of the LISA instrument. For example, verification binaries with \n<latexit sha1\\_base64="PgtqjkhFdCdqV3hCYoAzlskS3Ho=">AAAB7XicbVC7SgNBFL3rM8ZXVLCxGQyCVdiNhZYhNpYJmAckIcxOZpMxszvLzF0hLPkHGwtFbK38C7/AzsZvcfIoNPHAhcM593LvPX4shUHX/XJWVtfWNzYzW9ntnd29/dzBYd2oRDNeY0oq3fSp4VJEvIYCJW/GmtPQl7zhD68nfuOeayNUdIujmHdC2o9EIBhFK9XbSJOu6ObybsGdgiwTb07ypePqt3gvf1S6uc92T7Ek5BEySY1peW6MnZRqFEzycbadGB5TNqR93rI0oiE3nXR67ZicWaVHAqVtRUim6u+JlIbGjELfdoYUB2bRm4j/ea0Eg6tOKqI4QR6x2aIgkQQVmbxOekJzhnJkCWVa2FsJG1BNGdqAsjYEb/HlZVIvFryLQrFq0yjDDBk4gVM4Bw8uoQQ3UIEaMLiDB3iCZ0c5j86L8zprXXHmM0fwB87bD33TksU=</latexit> \n<latexit sha1\\_base64="M0iqpX7vlWOuAjiXGfVVvwRK72w=">AAACB3icbZDLSgMxFIYzXmu9jbpUSrAILkqZqQtdFt24bMFeoC1DJs20ocnMkJwRytCdLnwVNy4U6dZXcOcz+BKml4W2/hD4+M85Sc7vx4JrcJwva2V1bX1jM7OV3d7Z3du3Dw7rOkoUZTUaiUg1faKZ4CGrAQfBmrFiRPqCNfzBzaTeuGdK8yi8g2HMOpL0Qh5wSsBYnp2TnlvA0isVcOA5BdzuRpAGIwNxn3uOZ+edojMVXgZ3Dvnyybj6/ZgbVzz709xAE8lCoIJo3XKdGDopUcCpYKNsO9EsJnRAeqxlMCSS6U463WOEz4zTxUGkzAkBT93fEymRWg+lbzolgb5erE3M/2qtBIKrTsrDOAEW0tlDQSIwRHgSCu5yxSiIoQFCFTd/xbRPFKFgosuaENzFlZehXiq6F8VS1aRxjWbKoGN0is6Riy5RGd2iCqohih7QM3pFb9aT9WK9W+NZ64o1nzlCf2R9/ABiN5rP</latexit> \n<latexit sha1\\_base64="5ehfLED6kXaBHZ54m5jTxKSXJLI=">AAACB3icbVDLSgMxFM3UV62vUZdKCRbBhZSZutBl0Y3LFuwD2qFkMpk2NPMguSOUoTtd+CtuXCjSrb/gzm/wJ8y0XWjrgeQezj2X5B43FlyBZX0ZuZXVtfWN/GZha3tnd8/cP2iqKJGUNWgkItl2iWKCh6wBHARrx5KRwBWs5Q5vsn7rnknFo/AORjFzAtIPuc8pAS31zKJ3jrtC+z2iicsgKzXF9c0jID2zZJWtKfAyseekVD2e1L8fi5Naz/zsehFNAhYCFUSpjm3F4KREAqeCjQvdRLGY0CHps46mIQmYctLpHmN8qhUP+5HUJwQ8VX9PpCRQahS42hkQGKjFXib+1+sk4F85KQ/jBFhIZw/5icAQ4SwU7HHJKIiRJoRKrv+K6YBIQkFHV9Ah2IsrL5NmpWxflCt1ncY1miGPjtAJOkM2ukRVdItqqIEoekDP6BW9GU/Gi/FuTGbWnDGfOUR/YHz8AEQjm1w=</latexit> \nFig. 2.Time series of GW strain from one of the long arm measurements with laser phase-locking for the nominal mission configuration. The predicted signal from the verification binary, V407Vul with errors in its electromagnetic measurements (see Table 1) is shown in rose. Over plotted in black crosses are a simulated example of telemetered data dominated by laser noise where some of the data will be missing potentially due to instrument glitches. The apriori known signal from the binary will serve to fill the missing data inherent to instrument for each phasemeter measurement, independently to other pre-existing methods. Inset shows a zoom of the predicted signal for 1000s. \n<!-- image --> \nwell constrained distances could be used to directly measure the amplitude calibration error and its evolution with time (Savalle et al. 2022). \nAs mentioned above, the brightest VB can also serve as an independent verification of the instruments\' timing standards by demanding coherence across data gaps between data segments (Littenberg 2018). The principle for this and calibration of the instrument data is illustrated in Figure 2 where the model (to within its estimated errors) for a verification source is shown in red and the inevitable unevenly sampled phasemeter data is shown in black crosses. It\'s level is dominated by laser frequency noise, expected to be ∼ 10 O larger than the apriori known VB. These measurements need to be combined to form TDI signals to do the analysis in order to suppress the frequency laser noise by ≈ 10 orders of magnitude (Armstrong et al. 1999). More in general, LISA verification binaries will be especially critical for validating our ability to correctly recover parameters of a source. \nThe GW signal from a VB can be described by 8 parameters: \n{A , f 0 , ˙ f 0 , λ, β, ι, ψ, ϕ 0 } , (1) \nSky coordinates ( λ, β ), the inclination ι , the polarisation Ψ and the amplitude A are extrinsic parameters that describe the position and orientation of the source with respect to the LISA detector, where extrinsic refers to the dependence on detector. The remaining three parameters - frequency f , frequency derivative ˙ f and initial orbital phase ϕ 0 - are intrinsic , which implies no dependence on instrument. They determine the temporal evolution of the plus h + and cross h × GW polarisations \nas follows: \nh + = A (1 + cos 2 ι ) cos Φ( t ) , h × = 2 A cos 2 ι sin Φ( t ) , (2) Φ( t ) = 2 πf 0 t + π ˙ ft 2 -ϕ 0 . \nThe GW amplitude is given by: \nA = ( G M ) 5 / 3 c 4 d ( πf 0 ) 2 / 3 . (3) \nThis is set by the binary\'s distance, d , and chirp mass, given by: \nM = ( m 1 m 2 ) 3 / 5 ( m 1 + m 2 ) 1 / 5 , (4) \nfor binary component masses m 1 and m 2 . \nAmong the eight parameters defining a verification binary\'s signal (cf. Eq. (1)), sky position can be considered to be known with near-perfect precision. This is largely due to electromagnetic observatories\' ability to provide micro-arcsecond position measurements. By contrast, LISA\'s position measurement for verification binaries is estimated to be about a degree (e.g. Finch et al. 2023), a difference of nearly nine orders of magnitude. Recent years have seen substantial progress in determining verification binaries\' distances, largely due to astrometric data provided by the Gaia mission (Prusti et al. 2016); based on a reassessment following the latest data release (Brown et al. 2020), where the current uncertainty ranges between 10 -3 -10 -6 arcsecond (Kupfer et al. 2023). \nThe orbital period, which relates to GW frequency according to the relationship f = 2 /P , is generally known with high precision, typically to a fractional uncertainty of about 10 -6 (or 0.0001%), provided that the binary system is oriented edge-on or near edge-on and can be seen as eclipsing. However, for non-eclipsing binaries, the fractional errors can rise to 1% or less. The measurement of the binary system\'s inclination is also influenced by its orientation. For eclipsing binaries, the inclination can be determined with sub-degree precision (Burdge Coughlin). However, for non-eclipsing systems, the inclination remains largely unconstrained. Another parameter that can be determined for eclipsing binary systems is the initial phase of the orbit ϕ 0 , typically assumed at the primary eclipse (the deepest of the two). \nFinally, eclipsing systems offer the opportunity to measure the orbital period derivative -˙ P that can be related to GW ˙ f - by conducting eclipse timing over an extended time spanning from years to decades depending on the binary\'s orbital period. This allows for an accurate measurement of the rate of change in the orbital frequency (eg., Shah & Nelemans 2014). Finally, the challenges in accurately determining the masses of binary components and their distances continue to be the primary sources of uncertainty. Combining these uncertainties, the amplitude A in Eq. (3) can be known to between 1% and several tens of percent (Kupfer et al. 2023) 8 . For this study we select two of the strongest among the currently known verification binaries: one with a face-on orienta- \nTABLE 1 \nOptical photometric, (phase-resolved) spectroscopic measurements derived parameters for the \'face-on\' V407Vul and the most compact detached know \'edge-on\' ZTF J1539+5027 binaries (Kupfer et al. 2023). We note that binary\'s sky coordinates ( λ, β ) are given in the heliocentric ecliptic reference frame. \ntion with respect to LISA plane, V407 Vul (Marsh & Steeghs 2002) and the second with an edge-on orientation, ZTF J2243+5242 (hereafter ZTFJ2243, Burdge Coughlin). Their measured parameters are tabulated in Table 1.', '3. LISA RESPONSE TO VERIFICATION BINARY GW SIGNAL': 'In this section, we focus on how light propagates within the setup of the LISA mission, specifically between the test masses in two inter-S/C, under the influence of weak field gravitational waves. We derive the changes in the frequency of the lasers, exchanged between the S/C, to measure distances between the pairs of test masses, that occur in response to the passage of a gravitational wave. \nWhile the equations below apply to a generic GW signal from any astrophysical source, the coupling of the GW to the laser in this paper is meant for a frequency-stable GW signal as for the verification binary (VB). \nThe effect of a plane GW wave propagating far from the (Galactic binary) source in the vicinity of a GW detector where freely falling test masses are exchanging electromagnetic (EM) wave is to affect the properties of the EM wave (Kaufmann 1970), i.e. LISA laser, where the light propagation is effected by the consequences of the constraint equation: \n✷ F µν +2 R µναβ F αβ = 0 . (5) \nThe d \' Alembertian, ✷ , applied to an EM tensor F αβ , relates to that of the 4-dimensional Riemanian R µναβ , curved space-time fluctuations by a factor of 2 modulo a symmetry in two of its four indices. The role of the ambient GW is to refract the EM wave akin to the scintillation of the EM waves from an optical source that passes through the time-varying Earth\'s atmosphere. A weak field GW strain h µν and a linearized Riemann tensor, R µναβ , when inserted into the equation above give geodesics of an arbitrary massless particle between the test masses. Using Killing vectors, a mathematical tool that captures the symmetries of the spacetime, allowing us to understand how objects move within it (Estabrook & Wahlquist 1975), and the symmetry of the weak field gives the physical distance distortion caused by h µν affecting the photon flight time between the test masses. In other words, the doppler responses of a transverse traceless (TT) weak-field GW along a generic single laser link and a singly transponded 2-way laser link oriented in an arbitrary direction in 3-dimensional space (as shown in Figure 3) is usually expressed in terms of its fractional frequency fluctuations, given by: \ny GW slr ( t B ) = [ 1 + ˆ k GW ( t B ) · ˆ n ℓ l ( t B ) ] × 1 2 { Ψ l [ t s ( t B ) -ˆ k GW ( t B ) · ⃗ p s ( t s ( t B )) ] -Ψ l [ t B -ˆ k GW ( t B ) · ⃗ p r ( t B ) ] } (6) Ψ l ( t \' ) = ˆ n ℓ l ( t ) · h ( t \' ) · ˆ n ℓ l ( t ) 1 -[ ˆ k GW ( t ) · ˆ n ℓ l ( t ) ] 2 , (7) \nwhere, \n- · t B = time in an inertial reference frame filled by a plane GW, at LISA Barycentre\n- · t =time in another inertial reference frame at Solar System Barycentre (SSB)\n- · t \' = proper time where the measurement is executed, for example, the clock\'s tick in a given S/C\n- · t s = time coordinate of the S/C where laser beam is sent from\n- · t r = time coordinate of the S/C where laser beam is received at\n- · l = an index denoting armlength formed by laser propagation between two S/C\n- · ˆ n ℓ l = unit vector of the laser beam ℓ along l\n- · ˆ k GW = unit vector of the gravitational wave along the direction of its propagation in radiative zone\n- · ⃗ p s =position vector of S/C that is sending the laser beam with respect to a well defined origin\n- · ⃗ p r = position vector of S/C that is receiving with respect to the same origin\n- · h (see Sec. 2) is rank-2 tensor representing the culmination of two polarisations imprinted in the laser link l \nEq. 6 above adapted from from Eq. 1 in (Vallisneri 2005) is different by explicitly annotating the unsynchronised clocks to which the two distinct instances of the \nFig. 3.LISA inter spacecraft geometry and a VB binary with parameters and vectors defined in a coordinate system with the origin placed at the LISA Barycentre, labelled by B. Vectors ⃗ p i denote position(s) of S/C from B represented by red lines. Distances between the S/C for two directions are distinguished with ˆ n and the direction of a GW wave onto the LISA plane is represented by ˆ k . Primary and secondary masses of binary are m 1 , m 2 respectively with the position vectors ⃗r 1 , ⃗r 2 from B. ⃗ R is position vector to the centre of mass of the binary, whereas ⃗r defines the relative position of the masses on the orbital plane whose angular momentum is denoted by ⃗ L . The shaded red area is a plane spanned by LISA constellation plane which intersects with another 2-dimensional plane spanned by the projection of the binary orbital plane shown in grey circle. The complete set of interferometric measurements between locked lasers (for an exemplary laser frequency locking configuration) in all six optical benches are shown that will include GW signature (4.1) The orbit of LISA centroid around the Sun (in orange) is represented by grey curve. \n<!-- image --> \nGWwill impinge on the two ends of the link, making the above version more suitable to work with for the experimentalists. There are three distinct timescales at play here: (i) any event that occurs in radiative zone (i.e. far away from the source) (ii) some event that occurs at the retarded t from the sending S/C and (iii) the version of the same event that occurs at the received t at the receiving S/C. In the equation above, we have introduced a coordinate frame at the LISA constellation centre, LISA barycentre, t B , which is related to what is generally assumed in literature, the Solar System Barycentric (SSB) time, t . Observe that this t is distinct from those at s or r . \nIn Eq. 7, Ψ l is the projection of the inertial GW along its propagation direction onto a unit vector of laser propagating in between two ends of a theoretical detector consisting of two free-falling test masses (Vallisneri 2005), where the recursive nature of the time is folded in. y GW slr \nis then the response of those test masses far enough apart by a distance such that the h imprints itself twice onto the 2-mass detector: once at a retarded moment at s and then again at a real time t , as measured by the receiving test-mass at r . \nThe time, t s is given by t s = t B -∣ ∣ ⃗ p r ( t B ) -⃗ p s ( t s ( t B )) ∣ ∣ , which is known as the light-propagation equation for non-relativistic length changes caused by GWs, which are essentially tidally deforming the 2-dimensional space along a length or, equivalently changing the time taken to travel that distorted space due to GWs. Observe the recursive nature of t B in its expression. The quantity t s ( t B )) means a nested delay where t B depends on t s in the TDI terminology. Alternatively, in terms of effective length, above mentioned equation can be re-casted as L l ( t B ) = ∣ ∣ ∣ ⃗ p r ( t B ) -⃗ p s [ t B -L l ( t B ) ] ∣ ∣ ∣ , where this armlength is set from spatial point r to another spatial point s at emitted time, t B . \n<latexit sha1\\_base64="OEpo7CILcQuYlgxLd9BkGNBUDoc=">AAAB6nicbVC7SgNBFL0bXzG+ooKNzWAQrMJuUmgZYmOZoHlAsoTZyWwyZGZ2mZkVwpJPsLFQxNbWv/AL7Gz8FiePQhMPXDiccy/33hPEnGnjul9OZm19Y3Mru53b2d3bP8gfHjV1lChCGyTikWoHWFPOJG0YZjhtx4piEXDaCkbXU791T5Vmkbwz45j6Ag8kCxnBxkq3ouf18gW36M6AVom3IIXKSf2bvVc/ar38Z7cfkURQaQjHWnc8NzZ+ipVhhNNJrptoGmMywgPasVRiQbWfzk6doHOr9FEYKVvSoJn6eyLFQuuxCGynwGaol72p+J/XSUx45adMxomhkswXhQlHJkLTv1GfKUoMH1uCiWL2VkSGWGFibDo5G4K3/PIqaZaKXrlYqts0qjBHFk7hDC7Ag0uowA3UoAEEBvAAT/DscOfReXFe560ZZzFzDH/gvP0A2tWRNg==</latexit> \n<latexit 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sha1\\_base64="yFTQTE3cO8gZIDIPDwTn354YUdA=">AAAB/HicbVDLSsNAFJ34rPUV7UYQJFgEVyWpC11JQRcuK9gHNDFMptN27GQSZm6EEOJP+AFuXCji1pVf4c6/cfpYaOuBC4dz7uXee4KYMwW2/W0sLC4tr6wW1orrG5tb2+bOblNFiSS0QSIeyXaAFeVM0AYw4LQdS4rDgNNWMLwY+a17KhWLxA2kMfVC3BesxwgGLflmyR1gyETuZ1Unv81cynnum2W7Yo9hzRNnSsq1g8u9R/i8q/vml9uNSBJSAYRjpTqOHYOXYQmMcJoX3UTRGJMh7tOOpgKHVHnZ+PjcOtJK1+pFUpcAa6z+nshwqFQaBrozxDBQs95I/M/rJNA78zIm4gSoIJNFvYRbEFmjJKwuk5QATzXBRDJ9q0UGWGICOq+iDsGZfXmeNKsV56RSvdZpnKMJCmgfHaJj5KBTVENXqI4aiKAUPaEX9Go8GM/Gm/E+aV0wpjMl9AfGxw8GEZf0</latexit> \n<latexit sha1\\_base64="RLmD+rIb8DGp1oN6sOnebaSzCuA=">AAACA3icdVDLSgMxFM3UV62vURFEN9EiuJBhZqpt3RXduKxgW6EdSiZN29DMgyRTKEPBjf/hyo0LpbgS/Al3foh7M20FFT0QODnn3pvc44aMCmma71pqZnZufiG9mFlaXlld09c3qiKIOCYVHLCAX7tIEEZ9UpFUMnIdcoI8l5Ga2ztP/FqfcEED/0oOQuJ4qOPTNsVIKqmp7zT6BMfhsGkfwUYrkPH0roSmnjWN02LePrGhaZhmwc7lE2IXju0ctJSSIFva+3gZbW/dlZv6m5qAI4/4EjMkRN0yQ+nEiEuKGRlmGpEgIcI91CF1RX3kEeHE4x2G8EApLdgOuDq+hGP1e0eMPCEGnqsqPSS74reXiH959Ui2i05M/TCSxMeTh9oRgzKASSCwRTnBkg0UQZhT9VeIu4gjLFVsGRXC16bwf1K1DStn2JcqjTMwQRrsgn1wCCxQACVwAcqgAjC4AffgETxpt9qDNtKeJ6UpbdqzCX5Ae/0ELJ+bjw==</latexit> \nFigure 3 shows the above-mentioned instrument parameters (in Eqs. 6, 7) for the single-link response together with the GW parameters of a given binary (in Eqs. 3) in a coordinate system with origin at the centre of LISA constellation (adapted from (Wahlquist 1987) and (Heinzel 2020a)). The constellation as a whole orbits the Sun (represented by orange circle) trailing the Earth such that it circles the Sun in a ccw direction when viewed from the northern celestial pole. This coordinate system is different than the canonical Solar System Barycentre used in simulating LISA GW responses (e.g. Vallisneri 2005; Petiteau et al. 2008). \nThe VB parameters m 1 , m 2 are the primary and secondary masses, where primary is taken to be the more massive one in this study. Their position vectors from LISA Barycentre are ⃗r 1 , ⃗r 2 respectively, whereas ⃗ R is the vector to the VB\'s centre-of-mass. Another intrinsic parameter, ϕ 0 , often ignored in previous analysis is the position of the m 1 relative to m 2 on the VB orbital plane represented by the grey shaded circle. The incoming nearly monochromatic GW signal\'s vector is represented by ˆ k . The indexing conventions for the LISA constellation and its constituent test-masses, lasers amongst others are described in Sec. 4.1 below. The unit vector ˆ n orienting the inter-S/C laser link is often used in exchange with the armlength L ij and Doppler link D ij . The S/C motions, which will be significant in comparison to the GW signal are represented positions vectors and their rate of change by ⃗ p i , ˙ ⃗p i , respectively.', '4. FREQUENCIES IN LISA LASERS': "LISA lasers will need to be controlled in their frequencies such that the interferometric beatnotes between those two lasers will need to fall in the range of 5 -25MHz (Barke 2015) as mentioned in Sec. 1. This is necessitated by the fact that the three arms of LISA are time varying due to the individual S/C orbits 9 , which causes Doppler shifts in the lasers propagating in-between a given pair of S/C. The 5 -25MHz range is driven by the requirement from the quadrant photodiodes (Barranco et al. 2018) that register the beatnote between two interfered beams and the phasemeter (Schwarze et al. 2019) that tracks the phase of the these beatnotes. Thus, the individual lasers should all be locked to a primary laser (PL) whose frequencies must have a determined offset between 5 -25MHz. Several hardware configurational solutions to achieve this requirement all described and derived in the ' fplan tool' document (Heinzel 2020a) 10 . \nIn this paper, we consider only the configuration, shown in Figure 4, where the incoming beam from far S/C is interfered with the local laser on the OB where that long arm beatnote (also commonly known as the science signal) is measured. The incoming faint beam ( ∼ 10 -10 Watts) is represented by the cone and the outgoing beam ( ∼ Watt) is represented by red arrow. There are two outgoing beams for each optical bench (OB) within the S/C, where the S/C is represented by the quadrilateral and the two rounded rectangles are the two lasers with offset phase lock between them. For this case, there exist six independent configurations of laser \nFig. 4.Configuration where the long arm signal is formed by interfering received orange beam from far S/C (represented by the rectangle) with local red beam originating in the same OB where the interferometric signal (represented in the black and white cross) is measured. The red and green cylindrical objects represent the two local lasers on each optical bench (OB) in the S/C. \n<!-- image --> \nFig. 5.Similar to Figure 4, except the long arm signal is formed by interfering orange received beam from far S/C with local red beam originating in the adjacent OB . \n<!-- image --> \nfrequency offset locks where one of the 6 lasers is the primary laser and the remaining five are the secondary lasers (SL) locked to the former with predetermined offsets that keep the beatnotes within the boundaries of 5 -25 MHz for the duration of the observation of the mission, 4-10 years. Note, that the offsets do not need to be in the above-mentioned range for the beatnotes. Since the PL can be placed on any of the six OB, the total number of formations for non-swap ping lasers yield thirty-six distinct choices of configurations. \nIn an alternative set-up, known as the frequency-swap (Figure 5), the long arm beatnote signal is performed by interfering received laser beam with the adjacent OB's local laser. This case will be studied in forthcoming paper and therefore will not be discussed in this study. We note that this setup, however, provides a staggering choices of seventy-two distinct ways to have any two lasers locked across the constellation which preserves the phasemeter and photodiode constraints mentioned above and should be studied for future LISA-type detectors. \nIn three of the six non-swap locking configurations, there are total of four single laser beam paths constituting the 2.5 million km long arm that are used to form a set of four long-arm measurements 11 with the GWinduced Doppler shift(s) collected along one or all three \nof the arms. In the remaining three locking configurations, there are total of three single laser beam paths constituting the 2.5 Mkm long arm that form only three of such long-arm measurements with the fourth one being the local reference interferometer within the S/C. \nHowever, for all of the six non-swap configurations there are in total four main 12 interferometric measurements which have GW responses, the later three mentioned above have that in the local interferometer as a fourth measurement described below. An example for each type are discussed in the subsections below. The starting point of this study is the result of fplan (Heinzel 2020a), where the final configuration for each non-swap configuration provides the relation of the dependence of the five secondary lasers on the primary laser with five distinct offsets, the laser frequency fluctuations and the Doppler shifts long the 3 arms while holding the boundary condition of ± 5 ∓ 25MHz heterodyne frequencies. In this study we consider the configurations where the primary laser is on S/C 3 only. \nThe two examples of non-swap configurations can be understood as the algebraic manipulations required to determine the set of laser locking offsets. These offsets are applied to the five secondary lasers in relation to the primary laser (located in S/C3). Over a span of 10 years of S/C orbits, these adjustments ensure that the beatnotes formed by these lasers remain within the desired 5-25MHz range, and thus preventing a zero-crossing in the phasemeter. These applied offsets are distinct from the astrophysical Doppler shifts between the S/Cs that would for eg., be caused by the VBs made of whitedwarves.", '4.1. Minimally locked non-swap configuration': 'The heterodyne beatnotes\' frequencies, H 13 , are expressed in terms of the primary laser frequency f ℓ 0 , a given set of laser frequency offsets O k for the configuration in Figure 6, and Doppler D ij shifts picked up along the long arm by the interchanging laser beams. In the figure, the primary laser (in orange), PL32 is chosen to be on S/C3. The rest of the 5 lasers (in green) are secondary: SL23, SL21, SL12, SL13 and SL31. The adjacent SL31 is locally locked to PL32. SL23 is locked to PL32 with a single Doppler shift along the arm ⃗ L 32 . SL21 is locally locked to SL23. As SL21, SL12 is locally locked to SL13, while SL13 is locked to SL31 with a single Doppler shift along arm ⃗ L 31 . This particular setup of locking scheme is called \'N1c\'. \nThe label and names for LISA are mostly taken from previous notation conventions, see eg.Hewitson et al. (2021); Bayle & Hartwig (2023). In this paper, H has an unconventional label denoted by 2 or 3 S/C indices. The upper index in H is referred to ℓℓ if that beatnote is formed by locking to the local laser, i.e. including the long-arm Doppler shift such that there is no trace of the physical shift caused by long arm propagation in that beatnote. However, it is referred to nℓ where the beat- \n12 carrier-carrier, there are also upper and lower side-band measurements in order to reduce clock jitter noise, see (Tinto & Hartwig 2018) \n13 inspired by GW nomenclature, strain, h ∼ δL/L , where L is the armlength. The beatnotes formed by laser beams propagated in the long arms will capture GW signature of a VB. \nnote preserves that physical shift, implying that only H nℓ term has the Doppler shift from a long-duration astrophysical source. The lower index in H denotes the laser \nFig. 6.Laser locking configuration with minimum number of long arm Doppler shift lock to primary laser, PL32 (in orange) in S/C3 named, \'N1c\' (adapted from (Heinzel 2020a)) in forming any of the four beatnotes, i.e. the maximum number of L used to lock the secondary laser is one. There are four distinct long-arm measurements across the constellation shown by the arrows, labelled by H . The local interferometers within the S/C (Eq. 8a) are not shown. The armlengths (including Doppler shifts) are denoted by ⃗ L ij indicated by the lines connecting the S/C with arrows. The arrow connection any two lasers within the S/C shows the locking between them. \n<!-- image --> \nbeam path(s) of sometimes two lasers that are used to form the beatnote and other times of a common laser that is used to form the beatnote. For instance, H 12 is formed by interfering lasers SL12 and SL21, where the beams from two physically distinct lasers are exchanged after having travelled all three armlengths (Eq. 8c). Whereas, H 313 is formed by SL31 interfered with itself after a round-trip armlength travel, where SL13 is used to send back fresh beam after being locked to the frequency of SL31 defining the meaning of the term transponded beam (Eq. 8e). The topological view of locking schemes in \'N1c\' is shown in Figure 6, where the reference and/or test-mass interferometers (Eq. 8a) are not shown. We consider only the six long arm carrier-to-carrier beatnotes in this study. The three spacecrafts are labelled S/C1, S/C2 and S/C3 respectively. The arrows connecting the two lasers within a S/C represents the locking direction, such that in S/C3, the secondary laser, SL31 is locked in its phase with PL32; in S/C2, the secondary laser SL21 is locked to the phase of SL23 and finally in S/C1, the secondary laser of SL12 is locked to the phase of SL13. \nThus it turns out that in order to keep to the phasemeter condition, and for the \'N1c\' configuration we have, a total of 2 distinct round-trip long-arm measurements made by interfering with an identical laser and 2 distinct one-way long-arm measurements made by interfering two distinct lasers. We highlight that this setup is completely different than the case with free-running lasers considered widely in literature for TDI analysis and is further \n<latexit sha1\\_base64="Dm0FuC9R2OXFWVOtQQK0sQODwac=">AAACAXicdVDLSsNAFJ34rPUVdSO4GSyCq5DUlNZd0U2XFewDmhgm02k7dDIJMxOhhLjxV9y4UMStf+HOv3H6EFT0wIXDOfdy7z1hwqhUtv1hLC2vrK6tFzaKm1vbO7vm3n5bxqnApIVjFotuiCRhlJOWooqRbiIIikJGOuH4cup3bomQNObXapIQP0JDTgcUI6WlwDz0IqRGGLGskd9k3COM5UFWdvLALNlWxXbOKzU4J25ZE9txq64LHcueoQQWaAbmu9ePcRoRrjBDUvYcO1F+hoSimJG86KWSJAiP0ZD0NOUoItLPZh/k8EQrfTiIhS6u4Ez9PpGhSMpJFOrO6b3ytzcV//J6qRrU/IzyJFWE4/miQcqgiuE0DtingmDFJpogLKi+FeIREggrHVpRh/D1KfyftMuWc2aVr9xS/WIRRwEcgWNwChxQBXXQAE3QAhjcgQfwBJ6Ne+PReDFe561LxmLmAPyA8fYJS42XcQ==</latexit> \n<latexit 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sha1\\_base64="DxUanJStZN05m97AC1+/031PnxU=">AAAB6nicbVC7SgNBFL0bXzG+ooKNzWIQrMJuUmgZYmOZoHlAsoTZyWwyZHZ2mbkrhCWfYGOhiK2tf+EX2Nn4LU4ehSYeuHA4517uvcePBdfoOF9WZm19Y3Mru53b2d3bP8gfHjV1lCjKGjQSkWr7RDPBJWsgR8HasWIk9AVr+aPrqd+6Z0rzSN7hOGZeSAaSB5wSNNIt9sq9fMEpOjPYq8RdkELlpP7N36sftV7+s9uPaBIyiVQQrTuuE6OXEoWcCjbJdRPNYkJHZMA6hkoSMu2ls1Mn9rlR+nYQKVMS7Zn6eyIlodbj0DedIcGhXvam4n9eJ8Hgyku5jBNkks4XBYmwMbKnf9t9rhhFMTaEUMXNrTYdEkUomnRyJgR3+eVV0iwV3XKxVDdpVGGOLJzCGVyAC5dQgRuoQQMoDOABnuDZEtaj9WK9zlsz1mLmGP7AevsB6IeRPw==</latexit> \n<latexit sha1\\_base64="aDaFLXt5G5pHSDn/DIbgVTVUVgY=">AAACAnicdVDLSsNAFJ3UV62vqCtxM1gEVyFpU1p3RTddVrAPaGqZTCft0MkkzEyEEoIbf8WNC0Xc+hXu/BunD0FFD1w4nHMv997jx4xKZdsfRm5ldW19I79Z2Nre2d0z9w/aMkoEJi0csUh0fSQJo5y0FFWMdGNBUOgz0vEnlzO/c0uEpBG/VtOY9EM04jSgGCktDcwjL0RqjBFLG9lNyj3CWDZIy045G5hF26rYznmlBhfELWliO27VdaFj2XMUwRLNgfnuDSOchIQrzJCUPceOVT9FQlHMSFbwEklihCdoRHqachQS2U/nL2TwVCtDGERCF1dwrn6fSFEo5TT0defsYPnbm4l/eb1EBbV+SnmcKMLxYlGQMKgiOMsDDqkgWLGpJggLqm+FeIwEwkqnVtAhfH0K/yftkuWUrdKVW6xfLOPIg2NwAs6AA6qgDhqgCVoAgzvwAJ7As3FvPBovxuuiNWcsZw7BDxhvn8jUl68=</latexit> \nillustrated in the subsections below for \'N1c\' 14 and for a second configuration named \'N2a\' in Sec. 4.3 that has another set of laser offset frequencies. \nFor the configuration in Figure 6, the resulting reference and long-arm beatnotes are (adapted from Heinzel (2020a)): \nH 11 = O 5 H 22 = O 4 H 33 = -O 3 , (8a) H ℓℓ 23 = -O 1 H ℓℓ 13 = -O 2 , (8b) H nℓ 12 = -f 0 , [31] + f 0 , [21][32] + D 21 + D 32 , [21] -D 31 + O 4 , [21] + O 1 , [21] -O 5 -O 2 -O 3 , [31] , (8c) H nℓ 21 = -f 0 , [32] + f 0 , [12][31] -D 32 + D 12 + D 31 , [12] + O 5 , [12] + O 2 , [12] + O 3 , [12][31] -O 1 -O 4 , (8d) H nℓ 313 = -f 0 + f 0 , [13][31] + D 13 + D 31 , [13] + O 2 , [13] -O 3 + O 3 , [13][31] , (8e) H nℓ 323 = -f 0 + f 0 , [23][32] + D 32 , [23] + D 23 + O 1 , [23] , (8f) \nwhere, \n- · Eqs. 8a are the measurements of the local interferometers measuring the phase difference between two adjacent lasers on the two OB.\n- · Eqs. 8b are the measurements of the long arm interferometers measuring the phase difference between two lasers on board two S/C without Doppler shifts in them, see Figure 6.\n- · Eqs. 8c, 8d, 8e, 8f are the measurements of the long arm interferometers measuring the phase difference between two lasers on board two S/C with varying number of Doppler shifts in them, see Figure 6.\n- · f ℓ 0 is the instantaneous laser frequency of the primary laser delayed by one link, measured according to clock in S/C1\n- · All two-index subscripts inside square brackets f ℓ 0 , [ ij ] , O , [ ij ] , D [ ij ] are physically occurring delays along the armlength ⃗ L ij .\n- · D is Doppler shift at the time of arrival at the receiver.\n- · O 1 ...O 5 are sets of five distinct laser frequency offsets whose different linear combinations are applied to individual secondary lasers, see Figure 7. \nThe equations above do not specify any reference frame where the time is referenced with respect to. For H nℓ 12 , \nadding a reference clock to which its digitised time-series will be referred to, we get: \nH nℓ 12 ( t 1 ) = -f ℓ 0 , [31] ( t 1 ) + f ℓ 0 , [21][32] ( t 1 ) + D VB 32 , [21] ( t 1 ) -D VB 31 ( t 1 ) + D VB 21 ( t 1 ) -O 1 , [21] ( t 1 ) -O 2( t 1 ) + O 3 , [31] ( t 1 ) + O 4 , [21] ( t 1 ) -O 5( t 1 ) , (9) \nwhere, \n- · H nℓ 12 ( t 1 ) is the beatnote signal measured on OB12 in S/C1 with respect to clock 1 15 , t 1 (labelled by USO1 in Figure 7.\n- · f ℓ 0 , [31] is the laser frequency noise of the primary laser delayed by one link, measured according to clock in S/C1.\n- · D VB ij is Doppler shift from a VB at the time of arrival at the receiver, j .\n- · O 1 16 , is laser offset applied to SL23; O 1 + O 4 are two distinct laser offsets applied to SL21; O 3 + O 2 + O 5 are three distinct laser offsets applied to SL12 frequencies; and O 3 + O 2 are two distinct laser offsets applied to SL13. \nIn the Eq. 9 above, we have ignored all Doppler shifts arising from different parts of the instrument and sideband modulations, etc., isolating the only contribution from VB, hence the label, D VB , while contributions from the laser frequency fluctuations, laser frequency offsets are included in the overall H expression. Furthermore, we exclude test-mass interferometers, such that there are only total of six carrier-to-carrier beatnote signals for the long arm and reference interferometers. Additionally, many noise terms related to these interferometers are ignored in this paper, for eg. clock noise (Hartwig & Bayle 2021), tilt-to-length noise (Wanner et al 2023, in prep), the many contributions from the gravitational reference system (Armano et al. 2016a), and due to other control loops. \nNote that the label D ij in the equations and L ij in the figures are used interchangeably. In literature L ij is often used to mean the nominal armlength in the absence of any external forces and D ij is then the added variation due to the forces. However, for the purpose of this paper we use both the terms to mean similar quantity where variations due to any external force is included. Additionally, while D is used to mean Doppler shifts collected along the arm (owing to various processes), L is used to mean the armlength. Despite these different meanings, the D in the equations can be exchanged with L in the figure and vice-versa.', '4.2. Breakdown of individual beam paths for the minimally locked beatnotes': 'In this subsection we show the laser paths used to build the interferometric signals explicitly and the coupling of \na verification binary GW signal to it for their parameters from 2 for the laser offset locking configuration \'N1c\'. Consider the measurement made at OB12 where H nℓ 12 (as shown in Figure 7) is the beatnote signal formed by interfering the local beam in S/C1, SL12, and the distant beam from S/C2, SL21, with a delay from that S/C. This and all other interferometrer equations in Eqs. 8 use the convention where the the local beam is subtracted from the distant beam. As explained above, the beatnotes that contain the long arm Doppler shifts are constructed using laser beams that travel across the constellation atleast two arms before being interfered with another laser beam which has also possibly travelled across one or two arms. For H nℓ 12 (in Eq. 8c) can be derived as (dropping the temporal t i reference): \nFig. 7.A pair of laser beams showing their respective paths along the long-arm (indicated by grey circles) in forming the interferometric measurement H nℓ 12 (see, Eq. 10) where the interference is indicated by slanted grey line at OB12. This is an example of a long-arm interferometric measurement, where two distinct one-way beams are subtracted from one another. USO are time-references, Ultra Stable Oscillators for S/C USO1, USO2, and USO3 as labelled. \n<!-- image --> \nFig. 8.Similar to Figure 7, the laser beam paths to form the interferometric measurement H nℓ 21 (see, Eq. 8d) Here, the laser from distant S/C1 (OB12), i.e. SL12 is interfered with local beam (SL21) on OB21. \n<!-- image --> \nFig. 9.The laser beam paths to form the interferometric measurement H nℓ 313 (see, Eq. 8e). Unlike in Figure 8 and, Figure 7, this is a measurement in the OB31, where a local laser on OB31 in S/C3, i.e. PL32, is interfered with the same beam transponded back (a two-way measurement) from OB13 from S/C1. \n<!-- image --> \n<!-- image --> \nS/C1 \nFig. 10.Similar to Figure 9, the laser beam paths to form the interferometric measurement H nℓ 323 (see, Eq. 8f). This is a measurement in the OB23, where a local laser on OB31 in S/C3, i.e. PL32, is interfered with the same beam transponded back (a twoway measurement) from OB23 from S/C1 . \n<!-- image --> \nH nℓ 12 =[ T 21 (SL21)] -[SL12] =[ T 21 ( T 32 (PL32))] -T 31 ([PL32]) =[ T 21 ( T 32 ( f ℓ 0 ))] -[ T 31 ( f ℓ 0 )] [+ T 21 ( D VB 32 ) + D VB 21 ] -[ D VB 31 ] [+ T 21 ( O 1 + O 4)] -[ O 5 + T 31 ( O 3) + O 2] (10) \nwhere, \n- · H nℓ 12 , f ℓ 0 , O 1 , ...O 5 are described below Eqs. 8, 9.\n- · D VB ij is Doppler shift of the VB signal at the time of arrival at the receiver, j .\n- · T ij is delay operator for light travel time along the arm travelling from i → j , note that this is written alternatively in form of [ ij ] as subscript, for eg. in Eqs.8c, 8d, 8e, 8f, 9 and elsewhere. \nObserve in Eq. 10 above, that the number of Doppler shifts (3 in the example above) in the phasemeter output is different than the number of laser offsets (5 in \n<latexit 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sha1\\_base64="aDaFLXt5G5pHSDn/DIbgVTVUVgY=">AAACAnicdVDLSsNAFJ3UV62vqCtxM1gEVyFpU1p3RTddVrAPaGqZTCft0MkkzEyEEoIbf8WNC0Xc+hXu/BunD0FFD1w4nHMv997jx4xKZdsfRm5ldW19I79Z2Nre2d0z9w/aMkoEJi0csUh0fSQJo5y0FFWMdGNBUOgz0vEnlzO/c0uEpBG/VtOY9EM04jSgGCktDcwjL0RqjBFLG9lNyj3CWDZIy045G5hF26rYznmlBhfELWliO27VdaFj2XMUwRLNgfnuDSOchIQrzJCUPceOVT9FQlHMSFbwEklihCdoRHqachQS2U/nL2TwVCtDGERCF1dwrn6fSFEo5TT0defsYPnbm4l/eb1EBbV+SnmcKMLxYlGQMKgiOMsDDqkgWLGpJggLqm+FeIwEwkqnVtAhfH0K/yftkuWUrdKVW6xfLOPIg2NwAs6AA6qgDhqgCVoAgzvwAJ7As3FvPBovxuuiNWcsZw7BDxhvn8jUl68=</latexit> \nthe example above). The number of Doppler shifts is also distinct from the number of instances of the instantaneous laser frequency noise f ℓ 0 . However, terms with f ℓ 0 always appear in two for all of the phasemeter timeseries in Eqs. 8, 9 and 15, 16 below and in fact for all non-swap configurations, albeit with different delays. \nAs mentioned above, in the minimally-locked configuration in Figure 6, there are a total of four distinctsingle beam paths originating from the primary laser that are interfered with a local laser in constructing those four distinct beatnotes respectively, which are shown in Figs.7,8,9,10. Thus, the single arm Doppler response stated in Eq. 6 coupling to each of the beatnote signals in Eqs. 8c, 8d, 8e and 8f is fundamentally different than for the configuration with free-running lasers widely used in literature. For the above mentioned example, H nℓ 12 the two beams used to construct it are shown in Figure 7 and expressed in Eq. 10 corresponding to the two single- \nth Doppler links. Concretely they are one beam with the following path: PL32 → SL23 → SL21 → SL12 and another beam with the following path: PL32 → SL31 → SL13 → SL12, which are interfered (subtracted) to form the beatnote signal at OB12, where at each secondary laser, a set of offsets are applied as labelled in the Figure 7. \nThe three individual Doppler terms D terms are equivalent to the nomenclatures of fractional frequency deviations y and instantaneous GW signature from Eqs. 6, 7 respectively, such that y slr = D slr , both referred to coordinate system at the LISA Barycentre with time coordinate, t B . We often suppress the l in D by writing it as = D sr . In Eq. 10, suppressing the offsets and laser frequency noise, OB12 registers the following six distinct impinges of the same GW signal along different orientations of the arms, with the clock reference of USO1, t 1 , as: \nH nℓ, GW 12 ( t 1 ) = [ T 21 ( y GW 3[32]2 ( t B ) ) + y GW 2[21]1 ( t B ) ] -[ y GW 3[31]1 ( t B ) ] = T 21 ( [ 1 + ˆ k GW ( t B ) · ˆ n ℓ [32] ( t B ) ] 1 2 × { Ψ [32] [ t 3 ( t B ) -ˆ k GW ( t B ) · ⃗ p 3 ( t 3 ( t B )) ] -Ψ [32] [ t B -ˆ k GW ( t B ) · ⃗ p 2 ( t B ) ] } ) + [ 1 + ˆ k GW ( t B ) · ˆ n ℓ [21] ( t B ) ] 1 2 × { Ψ [21] [ t 2 ( t B ) -ˆ k GW ( t B ) · ⃗ p 2 ( t 2 ( t B )) ] -Ψ [21] [ t B -ˆ k GW ( t B ) · ⃗ p 1 ( t B ) ] } -[ 1 + ˆ k GW ( t B ) · ˆ n ℓ [31] ( t B ) ] 1 2 × { Ψ [31] [ t 3 ( t B ) -ˆ k GW ( t B ) · ⃗ p 3 ( t 3 ( t B )) ] -Ψ [31] [ t B -ˆ k GW ( t B ) · ⃗ p 1 ( t B ) ] } (11) \nwhere, \n- · y GW 3[32]2 ( t B ) is the 2-pulse response time-series in a Doppler link from OB32 to OB12.\n- · y GW 2[21]1 ( t B ) is the 2-pulse response time-series in a Doppler link from OB21 to OB12.\n- · y GW 2[31]1 ( t B ) is the 2-pulse response time-series in a Doppler link from OB31 to OB13\n- · Ψ , ˆ n, ˆ k, ⃗p i , t B are described under Eqs. 6, 7.\n- · t 1 , t 2 , t 3 are time stamps from the three clocks in S/C1, S/C2, and S/C3 respectively (labelled by USO1, USO2 and USO3 in Figure 7.\n- · T 21 is the physically occurring delay along the light beam travelling from optical bench OB21 to OB12. \nObserve in Eq. 11 above, for the S/C position vector terms ⃗ p , the mixing of the different time-frames referenced with three different clocks placed in the three S/C, \nwhere each clock in the given S/C is designed to drive all the frequencies that need referencing to a common (one main) clock (Heinzel 2020b). This is indicated in Figure 7 by the three Ultra Stable Oscillators (USO): USO1, USO2, and USO3, the main clocks for the 3 S/C. These are further referred by the three corresponding digital timers, t 1 , t 2 , t 3 respectively, common, but unique to each S/C. The doppler shifts above in Eq. 9 (for e.g. D VB 32 , 21 ( t 1 )) are given by the expressions in Eqs. 6, and 7 for a given signal of determined shape. These are written out for the beatnotes for two of the exemplary laser locking configurations in subsections below. \nSimilarly, the beatnote H nℓ 21 is measured on optical bench, OB21 which is obtained by subtracting the beam with the path: PL32 → SL23 → SL21 from another beam with the path: PL32 → SL31 → SL13 → SL12 to form the beatnote signal, with the clock reference of USO2, t 2 (as shown in Figure 8) as: \nH nℓ, GW 21 ( t 2 ) = [ y GW 3[32]2 ( t B ) ] -[ T 12 ( y GW 3[31]1 ( t B ) ) + y GW 1[12]2 ( t B ) ] = [ 1 + ˆ k GW ( t B ) · ˆ n ℓ [32] ( t B ) ] 1 2 × { Ψ [32] [ t 3 ( t B ) -ˆ k GW ( t B ) · ⃗ p 3 ( t 3 ( t B )) ] -Ψ [32] [ t B -ˆ k GW ( t B ) · ⃗ p 2 ( t B ) ] } -T 12 ( [ 1 + ˆ k GW ( t B ) · ˆ n ℓ [31] ( t B ) ] 1 2 × { Ψ [31] [ t 3 ( t B ) -ˆ k GW ( t B ) · ⃗ p 3 ( t 3 ( t B )) ] -Ψ [31] [ t B -ˆ k GW ( t B ) · ⃗ p 1 ( t B ) ] } ) -[ 1 + ˆ k GW ( t B ) · ˆ n ℓ [12] ( t B ) ] 1 2 × { Ψ [12] [ t 1 ( t B ) -ˆ k GW ( t B ) · ⃗ p 1 ( t 1 ( t B )) ] -Ψ [12] [ t B -ˆ k GW ( t B ) · ⃗ p 2 ( t B ) ] } (12) \nwhere, \n- · y GW 3[32]2 ( t B ) is described below Eq. 11.\n- · y GW 3[31]1 ( t B ) is the 2-pulse response time-series in a Doppler link from OB31 to OB13.\n- · y GW 1[12]2 ( t B ) is the 2-pulse response time-series in a Doppler link from OB12 to OB21.\n- · Ψ , ˆ n, ˆ k, ⃗p i , t B are described under Eqs. 6, 7.\n- · t 1 , t 3 are time stamps from the two clocks in S/C1 and S/C3 respectively.\n- · T 12 is the physically occurring delay along the light beam travelling from optical bench OB12 to OB21. \nAs described previously, the beatnote H nℓ 313 is different in nature compared to those of H nℓ 12 and H nℓ 21 derived above, where H nℓ 313 is formed by interfering a pair of two distinct laser beam paths from an identical physical laser, which is why it has a 3-index label to distinguish it from the the beatnotes, H nℓ 12 , H nℓ 21 . For H nℓ 313 , this physical laser is the primary laser, PL32. Concretely, this beatnote is measured by subtracting a two-way beam with the path: PL32 → SL31 → SL13 → SL31 from another beam which is located on OB32, that is the primary laser beam PL32, shown in Figure 9). The resulting coupling to VB signals in the doppler links are given by: \nH nℓ, GW 313 ( t 3 ) = [ T 13 ( y GW 3[31]1 ( t B ) ) ] + [ y GW 1[13]3 ( t B ) ] = T 13 ( [ 1 + ˆ k GW ( t B ) · ˆ n [31] ( t B ) ] 1 2 × { Ψ [31] [ t 3 ( t B ) -ˆ k B · ⃗ p 3 ( t 3 ( t B )) ] -Ψ [31] [ t B -ˆ k GW ( t B ) · ⃗ p 1 ( t B ) ] } ) + [ 1 + ˆ k GW ( t B ) · ˆ n [13] ( t B ) ] 1 2 × { Ψ [13] [ t 1 ( t B ) -ˆ k B · ⃗ p 1 ( t 1 ( t B )) ] -Ψ [13] [ t B -ˆ k GW ( t B ) · ⃗ p 3 ( t B ) ] } (13) \nwhere, \n- · y GW 3[31]1 ( t B ) is described below Eq. 12.\n- · y GW 1[13]3 ( t B ) is the 2-pulse response time-series in a Doppler link from OB13 to OB31.\n- · Ψ , ˆ n, ˆ k, ⃗p i , t B are described under Eqs. 6, 7.\n- · t 1 , t 3 are time stamps from the two clocks in S/C1 and S/C3 respectively.\n- · T 13 is the physically occurring delay along the light beam travelling from optical bench OB13 to OB31. \nFinally, for the remaining fourth beatnote H nℓ 323 can be understood similarly to the phasemeter measurement H nℓ 313 above. H nℓ 323 , is the phasemeter measurement at OB32, whose laser beam paths are shown in Figure 10. Concretely, this beatnote is formed by interfering (subtracting) a two-way beam along the path: PL32 → SL23 → SL32 with the local laser PL32 after a round-trip return. The doppler response of the binary signal is given by, with the clock reference of USO3, t 3 : \nH nℓ, GW 323 ( t 3 ) = [ T 23 ( y GW 3[32]2 ( t B ) ) ] + [ y GW 2[23]3 ( t B ) ] = T 23 ( [ 1 + ˆ k GW ( t B ) · ˆ n [32] ( t B ) ] 1 2 × { Ψ [32] [ t 3 ( t B ) -ˆ k GW ( t B ) · ⃗ p 3 ( t 3 ( t B )) ] -Ψ [32] [ t B -ˆ k GW ( t B ) · ⃗ p 2 ( t B ) ] } ) + [ 1 + ˆ k GW ( t B ) · ˆ n [23] ( t B ) ] 1 2 × { Ψ [23] [ t 2 ( t B ) -ˆ k GW ( t B ) · ⃗ p 2 ( t 2 ( t B )) ] -Ψ [23] [ t B -ˆ k GW ( t B ) · ⃗ p 3 ( t B ) ] } (14) \nwhere, \n- · y GW 3[32]2 ( t B ) is described below Eq. 11.\n- · y GW 2[23]3 ( t B ) is the 2-pulse response time-series in a Doppler link from OB32 to OB23.\n- · Ψ , ˆ n, ˆ k, ⃗p i , t B are described under Eqs. 6, 7.\n- · t 2 , t 3 are time stamps from the two clocks in the S/C2 and S/C3 respectively.\n- · T 23 is the physically occurring delay along the light beam travelling from optical bench OB23 to OB32. \nThe results for the phasemeter equations are compared for two different locking schemes for V407Vul and ZTTFJ1539 below in Sec. 4.3. These GW coupling equations derived above will be used in Sec. 4.5 together with the exemplary GW parameters from Sec. 2 to compute the response of the time delay observables instead of the usual 2-pulse response for each arm obtained with freerunning laser configuration for the case of simplified set of instrument noises.', '4.3. Maximally locked non-swap configuration': 'Similar to the configuration \'N1c\' (Figure 6) described in the subsection above, the resulting reference and longarm beatnotes for the configuration \'N2a\' (Figure 11) are described here. We call this configuration \'maximally locked\' with reference to the beatnote signal, which has the maximum number of Doppler shifts in the long arm measurement in order to distinguish it from that of the \'minimally locked\', \'N1c\'. The figures below will make this clearer, where one of the beatnote signals that is formed in OB13 housed in S/C1 for the identical choice of primary laser as in the minimal locking configuration, has a total of eight instantaneous GW impinges as derived below in equations and demonstrated by laser beam paths (see, Figure 14). \nFigure 11 shows the topology of the beatnotes for the non-swap laser locking configuration, \'N2a\'. Unlike in Figure 6, this laser locking configuration has three, two and one Doppler links locked to PL32 (in orange) in S/C3 in forming three of the four long-arm beatnotes. The distinguishing feature of this configuration is that the one of the local reference/test-mass interferometric measurements, H nℓ 33 in S/C has the long-arm Doppler links which will contain the Verification Binary (VB) signal. \nAs in the case of the minimal locking case, however, this configuration also has a total of four beatnote signals with the long arm which have non-zero GW signatures in them. Intriguingly the fourth beatnote signal with the long arm preserved non-zero VB GW signal in it is the \nFig. 11.\'N2a\' Laser locking configuration with maximum number of Doppler shifts in the long-arm used to lock the primary laser, PL32 (in orange) in S/C3 in forming one of the four beatnotes, i.e. the maximum number of D used to lock the secondary laser is three (adapted from (Heinzel 2020a)). H ℓℓ 11 , H ℓℓ 22 , H nℓ 33 reference beatnotes within S/C1, S/C2, S/C3 respectively. H ℓℓ 31 , H ℓℓ 23 , H nℓ 131 , H nℓ 212 , H nℓ 323 are the long-arm beatnotes across the constellation. Secondary lasers locked to PL32 are labelled by SL as in Figure 6. The grey arrows indicate the locking scheme for adjacent lasers within a S/C. ⃗ L 12 , ⃗ L 21 , ⃗ L 13 , ⃗ L 31 , ⃗ L 23 , ⃗ L 32 are inter-S/C Doppler links or armlengths. \n<!-- image --> \nlocal test-mass interferometer visually indicated in Figure 11, located in S/C3 by H nℓ 33 . This has stirred a discussion within the community to abandon the nomenclature of the science interferometer to mean the long arm interferometer by default since the local interferometer without the explicit long arm Doppler shift can have science signal in it (Heinzel 2020a). The six long arm and reference/test mass interferometric measurements with the local offsets and laser frequency fluctuations are (adapted \n<latexit sha1\\_base64="7u8l+GuigTlOgCO0FPY37sxOSuw=">AAACAnicdVDLSsNAFJ34rPVVdSVuBovgKiQxpXVXdNNlBfuANobJdNIOnUzCzEQoIbjxV9y4UMStX+HOv3H6EFT0wIXDOfdy7z1BwqhUlvVhLC2vrK6tFzaKm1vbO7ulvf22jFOBSQvHLBbdAEnCKCctRRUj3UQQFAWMdILx5dTv3BIhacyv1SQhXoSGnIYUI6Ulv3TYj5AaYcSyRn6T8T5hLPczx3Zyv1S2zIpln1dqcE5cRxPLdquuC23TmqEMFmj6pff+IMZpRLjCDEnZs61EeRkSimJG8mI/lSRBeIyGpKcpRxGRXjZ7IYcnWhnAMBa6uIIz9ftEhiIpJ1GgO6cHy9/eVPzL66UqrHkZ5UmqCMfzRWHKoIrhNA84oIJgxSaaICyovhXiERIIK51aUYfw9Sn8n7Qd0z4znSu3XL9YxFEAR+AYnAIbVEEdNEATtAAGd+ABPIFn4954NF6M13nrkrGYOQA/YLx9AsXIl60=</latexit> \n<latexit 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sha1\\_base64="DleaAMuq3zW6JjKNiicdTyDNdfk=">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</latexit> \nfrom (Heinzel 2020a)): \nH 11 = O 2 , H 22 = O 4 , (15a) H ℓℓ 23 = -O 1 , H ℓℓ 31 = -O 3 H ℓℓ 12 = -O 5 , (15b) H nℓ 131 = -f 0 , [21][32] -f 0 , [31][13][21][32] -D 32 , [21] + D 32 , [31][13][21] + D 13 , [31] + D 31 -D 21 + D 21 , [31][13] -O 1 , [21] + O 1 , [31][13][21] -O 2 + O 2 , [31][13] + O 3 , [31] -O 4 , [21] + O 4 , [31][13][21] -O 5 + O 5 [31][13] , (15c) H nℓ 212 = -f 0 , [32] + f 0 , [12][21][32] -D 32 + D 32 , [12][21] + D 13 , [21] + D 12 -O 1 + O 1 [12][21] -O 4 + O 4 , [12][21] + O 5 , [12] , \n(15d) H nℓ 323 = -f 0 + f 0 , [23][32] + D 32 , [23] + D 23 + O 1 , [23] , (15e) H nℓ 33 = f 0 -f 0 , [13][21][32] -D 32 , [13][21] -D 13 -D 21 , [13] -O 1 , [13][21] -O 2 , [13] -O 3 -O 4 , [13][21] -O 5 , [13] , (15f) \nwhere, \n- · Eqs. 15a are the measurements of the local interferometers measuring the phase difference between two adjacent lasers on the two optical benches (OB).\n- · Eqs. 15b are the measurements of the long arm interferometers measuring the phase difference between two lasers on board two S/C without Doppler shifts in them, see Figure 12.\n- · Eqs. 15c, 15d, 15e, are the measurements of the long arm interferometers measuring the phase difference between two lasers on board two S/C with varying number of Doppler shifts in them, see Figure 12, Figure 10.\n- · Eq. 15f is the measurement of two local interferometers in S/C3 measuring the phase difference between its adjacent lasers that has non-vanishing Doppler links from the long arm, see Figure 14.\n- · f ℓ 0 is laser frequency of the primary laser delayed by one link, measured according to clock in S/C1\n- · All two-index subscripts in f ℓ 0 , [ ij ] , O , [ ij ] , D [ ij ] are physically occurring delays along the armlength ⃗ L ij .\n- · D is Doppler shift at the time of arrival at the receiver.\n- · O 1 ...O 5 are sets of five distinct laser frequency offsets whose different linear combinations are applied to individual secondary lasers, see Figure 7. \nThe rest of the lasers are secondary as shown in Figure 6. SL23 is locked to PL32 with a single Doppler shift. SL21 is locked to SL23 with offset O1. SL21 is locked to SL12 with offsets O 1+ O 4. SL13 is locked to SL12 with offsets O 1 + O 4 + O 5. SL31 is locked to SL13 four Doppler shifts and laser offsets O 1+ O 2+ O 3+ O 4+ O 5. This last locking is the distinguishing feature of this configuration. There are three distinct long-arm measurements across the constellation shown by the inter-S/C laser links with arrows. Note, that unlike in the Figure 6 in subsection Sec. 4.2, SL31 is not locked to PL32 despite being on the same S/C. \nFig. 12.A pair of laser beams showing their respective paths along the long-arm (indicated by grey circles) in forming the interferometric measurement H nℓ 212 where the interference is indicated by slanted grey line at OB21. In this long arm interferometric measurement, there are two distinct beams subtracted from one another, however one of the beams has a two-way measurement as described in the text. \n<!-- image -->', '4.4. Breakdown of individual beam paths for the maximally locked beatnotes': 'Similar to Sec. 4.2 above, there are four distinct long arm and local interferometric measurements for the laser offset locking configuration \'N2a\', shown in Figure 11 with varying number of Doppler shifts in any one of them. One of them, H nℓ 323 ( t 3 ) (Eq. 15e, Figure 10) in subsection above is identical to Eqs. 8f, 14, thus making this beatnote common to both the minimally locked (\'N1c\') and maximally locked (\'N2a\') configurations. The remaining three beatnotes for which the beam paths are different with respect to \'N1c\' configuration are shown in Figure 12, Figure 13, and Figure 14. According to the non-swap fplan derived in (Heinzel 2020a), the beatnote, H nℓ 323 ( t 3 ) is common to five of the six configurations, except one of the laser offset configurations, named \'N1b\' (to be studied in a follow up paper). \nThus, in constructing the TDI observables in the subsequent section, we can use the GW induced response to the phasemeter output from Eq. 14 without any change for the corresponding solutions for configuration \'N2a\'. A comparative version of Eq. 10 for the laser beam paths of the interferometric measurement of H nℓ 212 in Eq. 15d is derived below and the laser beams path used to construct \n<latexit sha1\\_base64="7u8l+GuigTlOgCO0FPY37sxOSuw=">AAACAnicdVDLSsNAFJ34rPVVdSVuBovgKiQxpXVXdNNlBfuANobJdNIOnUzCzEQoIbjxV9y4UMStX+HOv3H6EFT0wIXDOfdy7z1BwqhUlvVhLC2vrK6tFzaKm1vbO7ulvf22jFOBSQvHLBbdAEnCKCctRRUj3UQQFAWMdILx5dTv3BIhacyv1SQhXoSGnIYUI6Ulv3TYj5AaYcSyRn6T8T5hLPczx3Zyv1S2zIpln1dqcE5cRxPLdquuC23TmqEMFmj6pff+IMZpRLjCDEnZs61EeRkSimJG8mI/lSRBeIyGpKcpRxGRXjZ7IYcnWhnAMBa6uIIz9ftEhiIpJ1GgO6cHy9/eVPzL66UqrHkZ5UmqCMfzRWHKoIrhNA84oIJgxSaaICyovhXiERIIK51aUYfw9Sn8n7Qd0z4znSu3XL9YxFEAR+AYnAIbVEEdNEATtAAGd+ABPIFn4954NF6M13nrkrGYOQA/YLx9AsXIl60=</latexit> \n<latexit 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sha1\\_base64="PgBJPg5dGmLhxYpYtPUkNYtdmpw=">AAAB6nicbVC7SgNBFL0bXzG+ooKNzWAQrMJuLLQMsbFM0DwgLmF2MkmGzM4uM3eFsOQTbCwUsbX1L/wCOxu/xcmj0MQDFw7n3Mu99wSxFAZd98vJrKyurW9kN3Nb2zu7e/n9g4aJEs14nUUy0q2AGi6F4nUUKHkr1pyGgeTNYHg18Zv3XBsRqVscxdwPaV+JnmAUrXSDnVInX3CL7hRkmXhzUigf1b7Fe+Wj2sl/3nUjloRcIZPUmLbnxuinVKNgko9zd4nhMWVD2udtSxUNufHT6aljcmqVLulF2pZCMlV/T6Q0NGYUBrYzpDgwi95E/M9rJ9i79FOh4gS5YrNFvUQSjMjkb9IVmjOUI0so08LeStiAasrQppOzIXiLLy+TRqnonRdLNZtGBWbIwjGcwBl4cAFluIYq1IFBHx7gCZ4d6Tw6L87rrDXjzGcO4Q+ctx/nA5E+</latexit> \n<latexit sha1\\_base64="zzpS7ngV5gYyJe0ljcYQH9ahvks=">AAAB6nicbVC7SgNBFL0bXzG+ooKNzWAQrMJuLLQMsbFM0DwgLmF2MkmGzM4uM3eFsOQTbCwUsbX1L/wCOxu/xcmj0MQDFw7n3Mu99wSxFAZd98vJrKyurW9kN3Nb2zu7e/n9g4aJEs14nUUy0q2AGi6F4nUUKHkr1pyGgeTNYHg18Zv3XBsRqVscxdwPaV+JnmAUrXSDHa+TL7hFdwqyTLw5KZSPat/ivfJR7eQ/77oRS0KukElqTNtzY/RTqlEwyce5u8TwmLIh7fO2pYqG3Pjp9NQxObVKl/QibUshmaq/J1IaGjMKA9sZUhyYRW8i/ue1E+xd+qlQcYJcsdmiXiIJRmTyN+kKzRnKkSWUaWFvJWxANWVo08nZELzFl5dJo1T0zoulmk2jAjNk4RhO4Aw8uIAyXEMV6sCgDw/wBM+OdB6dF+d11ppx5jOH8AfO2w/lf5E9</latexit> \n<latexit sha1\\_base64="k9LgeGThzWvjP84J0V+b+ASUVK0=">AAACAnicdVDLSsNAFJ3UV62vqCtxM1gEVyFpU1p3RTddVrAPaGqZTCft0MkkzEyEEoIbf8WNC0Xc+hXu/BunD0FFD1w4nHMv997jx4xKZdsfRm5ldW19I79Z2Nre2d0z9w/aMkoEJi0csUh0fSQJo5y0FFWMdGNBUOgz0vEnlzO/c0uEpBG/VtOY9EM04jSgGCktDcwjL0RqjBFLG9lNyj3CWDZInbKTDcyibVVs57xSgwviljSxHbfqutCx7DmKYInmwHz3hhFOQsIVZkjKnmPHqp8ioShmJCt4iSQxwhM0Ij1NOQqJ7KfzFzJ4qpUhDCKhiys4V79PpCiUchr6unN2sPztzcS/vF6iglo/pTxOFOF4sShIGFQRnOUBh1QQrNhUE4QF1bdCPEYCYaVTK+gQvj6F/5N2yXLKVunKLdYvlnHkwTE4AWfAAVVQBw3QBC2AwR14AE/g2bg3Ho0X43XRmjOWM4fgB4y3T8XIl60=</latexit> \n<latexit sha1\\_base64="Yay+1Ih8TYUdjS9YTubiFxJ47sA=">AAAB83icdVDLSsNAFJ3UV62vquDGzWARXIWkRa27UjcuXLRgH9CEMplO2qGTSZiZFErIb7hxoYjb/oVf4M6N3+I0raCiBy4czrmXe+/xIkalsqx3I7eyura+kd8sbG3v7O4V9w/aMowFJi0cslB0PSQJo5y0FFWMdCNBUOAx0vHG13O/MyFC0pDfqWlE3AANOfUpRkpLjjMhOLlN+0mlnPaLJcs8t+yriwq0TCtDRqp2uQrtpVKqHTU/6Kz+2ugX35xBiOOAcIUZkrJnW5FyEyQUxYykBSeWJEJ4jIakpylHAZFukt2cwlOtDKAfCl1cwUz9PpGgQMpp4OnOAKmR/O3Nxb+8Xqz8qptQHsWKcLxY5McMqhDOA4ADKghWbKoJwoLqWyEeIYGw0jEVdAhfn8L/Sbts2hWz3NRp1MECeXAMTsAZsMElqIEb0AAtgEEE7sEjeDJi48F4Nl4WrTljOXMIfsCYfQL5BpVY</latexit> \n<latexit sha1\\_base64="VXUcf7xsTh0P/yH5OAgAUbuvow0=">AAAB83icdVDLSsNAFJ3UV62vquDGzWARXIWkRa27UjcuXLRgH9CEMplO2qGTSZiZFErIb7hxoYjb/oVf4M6N3+I0raCiBy4czrmXe+/xIkalsqx3I7eyura+kd8sbG3v7O4V9w/aMowFJi0cslB0PSQJo5y0FFWMdCNBUOAx0vHG13O/MyFC0pDfqWlE3AANOfUpRkpLjjMhOLlN+0nFTvvFkmWeW/bVRQVappUhI1W7XIX2UinVjpofdFZ/bfSLb84gxHFAuMIMSdmzrUi5CRKKYkbSghNLEiE8RkPS05SjgEg3yW5O4alWBtAPhS6uYKZ+n0hQIOU08HRngNRI/vbm4l9eL1Z+1U0oj2JFOF4s8mMGVQjnAcABFQQrNtUEYUH1rRCPkEBY6ZgKOoSvT+H/pF027YpZbuo06mCBPDgGJ+AM2OAS1MANaIAWwCAC9+ARPBmx8WA8Gy+L1pyxnDkEP2DMPgH3gZVX</latexit>', 'Frequency plan for WDBs': 'Fig. 13.Similar to Figure 12, where the long arm interference H nℓ 131 is measured on OB12 in S/C 1 using two distinct beam paths. One of the beams have two single arm Doppler links whereas the other beam has all three Doppler links including one two-way path. \n<!-- image --> \nthat interference is shown in Figure 12. This beatnote on OB21 is formed by interfering one single path beam: PL32 → SL23 → SL21 and another single path beam which has a roundtrip beam: PL32 → SL23 → SL21 → \nSL12 → SL21, given by: \nH nℓ 212 =[ T 21 (SL12)] -[SL21] = T 32 [ T 21 [ T 12 (PL32)]] -T 32 [PL32] = T 32 [ T 21 [ T 12 ( f ℓ 0 )]] -T 31 ( f ℓ 0 ) + T 12 [ T 21 [ O 1 ]] -O 4 + T 12 [ T 21 [ O 4 ]] + T 12 [ O 5 ] -O 1 -D VB 32 + T 12 [ T 21 ( D VB 32 )] + T 12 [ D VB 21 ] + D VB 12 , (16) \nwhere, \n- · H nℓ 212 is the long arm beatnote measured on OB21, where distant beam from laser on OB12 is subtracted from the local laser beam OB21.\n- · f ℓ 0 , O 1 , ...O 5 are described below Eqs. 8, 9.\n- · D VB ij is Doppler shift of the VB signal at the time of arrival at the receiver, j .\n- · T ij is delay operator for light travel time along the arm travelling from i → j , note that this is written alternatively in form of [ ij ] as subscript, for eg. in Eqs.8c, 8d, 8e, 8f, 9 and elsewhere. \nThe beam paths for interference for H nℓ 131 in the LISA constellation is shown figuratively in Figure 13 below. Here, this beatnote at OB13 is formed by a interfering a one-way and path beam of: PL32 → SL23 → SL21 with the two-way path beam of: PL32 → SL31 → SL13 → SL31. Substituting only the contributions from GW using Eqs.6, 7, and with the clock reference of USO2, t 2 , we get: \nH nℓ, GW 212 ( t 2 ) = -y GW 3[32]2 ( t B ) + T 12 ( T 21 ( y GW 3[32]2 ( t B ) ) ) + T 12 ( y GW 2[21]1 ( t B ) ) + y GW 1[12]2 ( t B ) = -[ 1 + ˆ k GW ( t B ) · ˆ n [32] ( t B ) ] 1 2 × { Ψ [32] [ t 3 ( t B ) -ˆ k GW ( t B ) · ⃗ p 3 ( t 3 ( t B )) ] -Ψ [32] [ t B -ˆ k GW ( t B ) · ⃗ p 2 ( t B ) ] } + T 12 T 21 ( [ 1 + ˆ k GW ( t B ) · ˆ n [32] ( t B ) ] 1 2 × { Ψ [32] [ t 3 ( t B ) -ˆ k GW ( t B ) · ⃗ p 3 ( t 3 ( t B )) ] -Ψ [32] [ t B -ˆ k GW ( t B ) · ⃗ p 2 ( t B ) ] } ) + [ 1 + ˆ k GW ( t B ) · ˆ n [21] ( t ) ] 1 2 × { Ψ [21] [ t 2 ( t B ) -ˆ k GW ( t B ) · ⃗ p 2 ( t 2 ( t B )) ] -Ψ [21] [ t B -ˆ k GW ( t B ) · ⃗ p 1 ( t B ) ] } + [ 1 + ˆ k GW ( t B ) · ˆ n [12] ( t ) ] 1 2 × { Ψ [12] [ t 1 ( t B ) -ˆ k GW ( t B ) · ⃗ p 1 ( t 1 ( t B )) ] -Ψ [12] [ t B -ˆ k · ⃗ p 2 ( t B ) ] } (17) \nwhere, \n- · y GW 3[32]2 ( t B ) is the 2-pulse response time-series in a Doppler link from OB32 to OB23.\n- · y GW 2[21]1 ( t B ) is the 2-pulse response time-series in a Doppler link from OB21 to OB12.\n- · y GW 1[12]2 ( t B ) is the 2-pulse response time-series in a Doppler link from OB12 to OB21.\n- · Ψ , ˆ n, ˆ k, ⃗p i , t B are described under Eqs. 6, 7.\n- · t 1 , t 2 , t 3 are time stamps from the three clocks in the S/C1, S/C2, and S/C3 respectively.\n- · T 12 and T 21 are the physically occurring delays along the light beam travelling from optical bench OB12 to OB21 and the same for OB21 to OB12 respectively. \nSimilar to the GW induced only measurement H nℓ 212 above, the beam paths for interference for H nℓ 131 in the \n<latexit sha1\\_base64="DxUanJStZN05m97AC1+/031PnxU=">AAAB6nicbVC7SgNBFL0bXzG+ooKNzWIQrMJuUmgZYmOZoHlAsoTZyWwyZHZ2mbkrhCWfYGOhiK2tf+EX2Nn4LU4ehSYeuHA4517uvcePBdfoOF9WZm19Y3Mru53b2d3bP8gfHjV1lCjKGjQSkWr7RDPBJWsgR8HasWIk9AVr+aPrqd+6Z0rzSN7hOGZeSAaSB5wSNNIt9sq9fMEpOjPYq8RdkELlpP7N36sftV7+s9uPaBIyiVQQrTuuE6OXEoWcCjbJdRPNYkJHZMA6hkoSMu2ls1Mn9rlR+nYQKVMS7Zn6eyIlodbj0DedIcGhXvam4n9eJ8Hgyku5jBNkks4XBYmwMbKnf9t9rhhFMTaEUMXNrTYdEkUomnRyJgR3+eVV0iwV3XKxVDdpVGGOLJzCGVyAC5dQgRuoQQMoDOABnuDZEtaj9WK9zlsz1mLmGP7AevsB6IeRPw==</latexit> \n<latexit sha1\\_base64="/vZcYcFHVGkGQpJ8IoxQLJFyLFw=">AAAB83icdVDLSsNAFJ3UV62vquDGzWARXIWkRa27UjcuXLRgH9CEMplO2qGTSZiZFErIb7hxoYjb/oVf4M6N3+I0raCiBy4czrmXe+/xIkalsqx3I7eyura+kd8sbG3v7O4V9w/aMowFJi0cslB0PSQJo5y0FFWMdCNBUOAx0vHG13O/MyFC0pDfqWlE3AANOfUpRkpLjjMhOLlN+4ldSfvFkmWeW/bVRQVappUhI1W7XIX2UinVjpofdFZ/bfSLb84gxHFAuMIMSdmzrUi5CRKKYkbSghNLEiE8RkPS05SjgEg3yW5O4alWBtAPhS6uYKZ+n0hQIOU08HRngNRI/vbm4l9eL1Z+1U0oj2JFOF4s8mMGVQjnAcABFQQrNtUEYUH1rRCPkEBY6ZgKOoSvT+H/pF027YpZbuo06mCBPDgGJ+AM2OAS1MANaIAWwCAC9+ARPBmx8WA8Gy+L1pyxnDkEP2DMPgH3f5VX</latexit> \nFig. 14.Unlike in Figure 12, and Figure 13, this is a reference (or test mass) interference H nℓ 33 measured on OB31 in S/C 3. It is also constructed using two distinct beam paths with single paths for both beams as described in the text. \n<!-- image --> \nH nℓ, GW 131 ( t 1 ) = -T 21 ( y GW 3[32]2 ( t B ) ) + T 31 ( T 13 ( T 31 ( y GW 3[32]2 ( t B ) ) ) ) + T 31 ( y GW 1[13]3 ( t B ) ) + y GW 3[31]1 ( t B \n+ T 13 T 31 ( [ 1 + ˆ k GW ( t B ) · ˆ n ℓ 21 ( t B ) ] × 1 2 { Ψ 21 [ t s ( t B ) -ˆ k GW ( t B ) · ⃗ p 2 ( t 2 ( t B )) ] -Ψ 21 [ t B -ˆ k GW ( t B ) · ⃗ p 1 ( t B ) ] } ) (18) \n) -y GW 2[21]1 ( t B ) + T 31 ( T 13 ( y GW 2[21]1 ( t B ) ) ) = -T 21 ( [ 1 + ˆ k GW ( t B ) · ˆ n ℓ 32 ( t B ) ] × 1 2 { Ψ 32 [ t 3 ( t B ) -ˆ k GW ( t B ) · ⃗ p 3 ( t 3 ( t B )) ] -Ψ 32 [ t B -ˆ k GW ( t B ) · ⃗ p 2 ( t B ) ] } ) + T 31 T 13 T 31 ( [ 1 + ˆ k GW ( t B ) · ˆ n ℓ 32 ( t B ) ] × 1 2 { Ψ 32 [ t 3 ( t B ) -ˆ k GW ( t B ) · ⃗ p 3 ( t 3 ( t B )) ] -Ψ 32 [ t B -ˆ k GW ( t B ) · ⃗ p 2 ( t B ) ] } ) + T 31 ( [ 1 + ˆ k GW ( t B ) · ˆ n ℓ 13 ( t B ) ] × 1 2 { Ψ 13 [ t 1 ( t B ) -ˆ k GW ( t B ) · ⃗ p 1 ( t 1 ( t B )) ] -Ψ 13 [ t B -ˆ k GW ( t B ) · ⃗ p 3 ( t B ) ] } ) + [ 1 + ˆ k GW ( t B ) · ˆ n ℓ 31 ( t B ) ] × 1 2 { Ψ 31 [ t s ( t B ) -ˆ k GW ( t B ) · ⃗ p 3 ( t 3 ( t B )) ] -Ψ 31 [ t B -ˆ k GW ( t B ) · ⃗ p 1 ( t B ) ] } -[ 1 + ˆ k GW ( t B ) · ˆ n ℓ 21 ( t B ) ] × 1 2 { Ψ 21 [ t s ( t B ) -ˆ k GW ( t B ) · ⃗ p 2 ( t 2 ( t B )) ] -Ψ 21 [ t B -ˆ k GW ( t B ) · ⃗ p 1 ( t B ) ] } \nwhere, \n- · y GW 3[32]2 ( t B ) , y GW 2[21]1 ( t B ) are described below Eq. 17.\n- · y GW 1[13]3 ( t B ) is the 2-pulse response time-series in a Doppler link from OB13 to OB31.\n- · y GW 3[31]1 ( t B ) is the 2-pulse response time-series in a Doppler link from OB31 to OB13.\n- · Ψ , ˆ n, ˆ k, ⃗p i , t B are described under Eqs. 6, 7.\n- · t 1 , t 2 , t 3 are time stamps from the three clocks in the S/C1, S/C2, and S/C3 respectively.\n- · T 31 , T 13 , T 21 are the physically occurring delays along the light beam travelling from optical bench OB31 to OB13, OB13 to OB31, and the same for OB21 to OB12 respectively. \nLISA constellation is shown figuratively in Figure 13 below (Eq. 15c). Here, this beatnote is measured on OB21 which is formed by a interfering a beam with the path of: PL32 → SL23 → SL21 → SL12 → SL13 with the beam path consisting of a two-way path of: PL32 → SL23 → SL21 → SL12 → SL13 → SL31 → SL13. The corresponding Doppler response of the binary signal, with the clock reference of USO1, t 1 , is given by: \n<latexit sha1\\_base64="jddH/g30YjR/VGuLF1myh6GB3i4=">AAAB83icdVDLSsNAFJ3UV62vquDGzWARXIWkRa27UjcuXLRgH9CEMplO2qGTSZiZFErIb7hxoYjb/oVf4M6N3+I0raCiBy4czrmXe+/xIkalsqx3I7eyura+kd8sbG3v7O4V9w/aMowFJi0cslB0PSQJo5y0FFWMdCNBUOAx0vHG13O/MyFC0pDfqWlE3AANOfUpRkpLjjMhOLlN+0nZTvvFkmWeW/bVRQVappUhI1W7XIX2UinVjpofdFZ/bfSLb84gxHFAuMIMSdmzrUi5CRKKYkbSghNLEiE8RkPS05SjgEg3yW5O4alWBtAPhS6uYKZ+n0hQIOU08HRngNRI/vbm4l9eL1Z+1U0oj2JFOF4s8mMGVQjnAcABFQQrNtUEYUH1rRCPkEBY6ZgKOoSvT+H/pF027YpZbuo06mCBPDgGJ+AM2OAS1MANaIAWwCAC9+ARPBmx8WA8Gy+L1pyxnDkEP2DMPgH1+5VW</latexit> \n<latexit sha1\\_base64="PgBJPg5dGmLhxYpYtPUkNYtdmpw=">AAAB6nicbVC7SgNBFL0bXzG+ooKNzWAQrMJuLLQMsbFM0DwgLmF2MkmGzM4uM3eFsOQTbCwUsbX1L/wCOxu/xcmj0MQDFw7n3Mu99wSxFAZd98vJrKyurW9kN3Nb2zu7e/n9g4aJEs14nUUy0q2AGi6F4nUUKHkr1pyGgeTNYHg18Zv3XBsRqVscxdwPaV+JnmAUrXSDnVInX3CL7hRkmXhzUigf1b7Fe+Wj2sl/3nUjloRcIZPUmLbnxuinVKNgko9zd4nhMWVD2udtSxUNufHT6aljcmqVLulF2pZCMlV/T6Q0NGYUBrYzpDgwi95E/M9rJ9i79FOh4gS5YrNFvUQSjMjkb9IVmjOUI0so08LeStiAasrQppOzIXiLLy+TRqnonRdLNZtGBWbIwjGcwBl4cAFluIYq1IFBHx7gCZ4d6Tw6L87rrDXjzGcO4Q+ctx/nA5E+</latexit> \n<latexit sha1\\_base64="zzpS7ngV5gYyJe0ljcYQH9ahvks=">AAAB6nicbVC7SgNBFL0bXzG+ooKNzWAQrMJuLLQMsbFM0DwgLmF2MkmGzM4uM3eFsOQTbCwUsbX1L/wCOxu/xcmj0MQDFw7n3Mu99wSxFAZd98vJrKyurW9kN3Nb2zu7e/n9g4aJEs14nUUy0q2AGi6F4nUUKHkr1pyGgeTNYHg18Zv3XBsRqVscxdwPaV+JnmAUrXSDHa+TL7hFdwqyTLw5KZSPat/ivfJR7eQ/77oRS0KukElqTNtzY/RTqlEwyce5u8TwmLIh7fO2pYqG3Pjp9NQxObVKl/QibUshmaq/J1IaGjMKA9sZUhyYRW8i/ue1E+xd+qlQcYJcsdmiXiIJRmTyN+kKzRnKkSWUaWFvJWxANWVo08nZELzFl5dJo1T0zoulmk2jAjNk4RhO4Aw8uIAyXEMV6sCgDw/wBM+OdB6dF+d11ppx5jOH8AfO2w/lf5E9</latexit> \n<latexit sha1\\_base64="Yay+1Ih8TYUdjS9YTubiFxJ47sA=">AAAB83icdVDLSsNAFJ3UV62vquDGzWARXIWkRa27UjcuXLRgH9CEMplO2qGTSZiZFErIb7hxoYjb/oVf4M6N3+I0raCiBy4czrmXe+/xIkalsqx3I7eyura+kd8sbG3v7O4V9w/aMowFJi0cslB0PSQJo5y0FFWMdCNBUOAx0vHG13O/MyFC0pDfqWlE3AANOfUpRkpLjjMhOLlN+0mlnPaLJcs8t+yriwq0TCtDRqp2uQrtpVKqHTU/6Kz+2ugX35xBiOOAcIUZkrJnW5FyEyQUxYykBSeWJEJ4jIakpylHAZFukt2cwlOtDKAfCl1cwUz9PpGgQMpp4OnOAKmR/O3Nxb+8Xqz8qptQHsWKcLxY5McMqhDOA4ADKghWbKoJwoLqWyEeIYGw0jEVdAhfn8L/Sbts2hWz3NRp1MECeXAMTsAZsMElqIEb0AAtgEEE7sEjeDJi48F4Nl4WrTljOXMIfsCYfQL5BpVY</latexit> \n<latexit sha1\\_base64="/vZcYcFHVGkGQpJ8IoxQLJFyLFw=">AAAB83icdVDLSsNAFJ3UV62vquDGzWARXIWkRa27UjcuXLRgH9CEMplO2qGTSZiZFErIb7hxoYjb/oVf4M6N3+I0raCiBy4czrmXe+/xIkalsqx3I7eyura+kd8sbG3v7O4V9w/aMowFJi0cslB0PSQJo5y0FFWMdCNBUOAx0vHG13O/MyFC0pDfqWlE3AANOfUpRkpLjjMhOLlN+4ldSfvFkmWeW/bVRQVappUhI1W7XIX2UinVjpofdFZ/bfSLb84gxHFAuMIMSdmzrUi5CRKKYkbSghNLEiE8RkPS05SjgEg3yW5O4alWBtAPhS6uYKZ+n0hQIOU08HRngNRI/vbm4l9eL1Z+1U0oj2JFOF4s8mMGVQjnAcABFQQrNtUEYUH1rRCPkEBY6ZgKOoSvT+H/pF027YpZbuo06mCBPDgGJ+AM2OAS1MANaIAWwCAC9+ARPBmx8WA8Gy+L1pyxnDkEP2DMPgH3f5VX</latexit> \n<latexit sha1\\_base64="DxUanJStZN05m97AC1+/031PnxU=">AAAB6nicbVC7SgNBFL0bXzG+ooKNzWIQrMJuUmgZYmOZoHlAsoTZyWwyZHZ2mbkrhCWfYGOhiK2tf+EX2Nn4LU4ehSYeuHA4517uvcePBdfoOF9WZm19Y3Mru53b2d3bP8gfHjV1lCjKGjQSkWr7RDPBJWsgR8HasWIk9AVr+aPrqd+6Z0rzSN7hOGZeSAaSB5wSNNIt9sq9fMEpOjPYq8RdkELlpP7N36sftV7+s9uPaBIyiVQQrTuuE6OXEoWcCjbJdRPNYkJHZMA6hkoSMu2ls1Mn9rlR+nYQKVMS7Zn6eyIlodbj0DedIcGhXvam4n9eJ8Hgyku5jBNkks4XBYmwMbKnf9t9rhhFMTaEUMXNrTYdEkUomnRyJgR3+eVV0iwV3XKxVDdpVGGOLJzCGVyAC5dQgRuoQQMoDOABnuDZEtaj9WK9zlsz1mLmGP7AevsB6IeRPw==</latexit> \n<latexit sha1\\_base64="DleaAMuq3zW6JjKNiicdTyDNdfk=">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</latexit> \nFinally, the fourth remaining beatnote frequency for configuration \'N2a\' is the local beatnote in OB33, namely, H nℓ 33 . The beam paths for interference for this measurement in the LISA constellation is shown figuratively in Figure 14 below (Eq. 15f), where this beatnote \nis measured on by interfering a one-way beam with the path of: PL32 → SL23 → SL21 → SL12 → SL13 with the beam path consisting of another one-way beam of the path of: PL32 → SL23. The corresponding Doppler response of the binary signal, with the clock reference of USO3, t 3 , is given by: \nH nℓ, GW 33 ( t 3 ) = -T 13 ( T 21 ( y GW 3[32]2 ( t B ) ) ) -y GW 1[13]3 ( t B ) -T 13 ( y GW 2[21]1 ( t B ) ) \n= -T 13 T 21 ( [ 1 + ˆ k GW ( t B ) · ˆ n ℓ 32 ( t B ) ] × 1 2 { Ψ 32 [ t 3 ( t B ) -ˆ k GW ( t B ) · ⃗ p 3 ( t 3 ( t B )) ] -Ψ 32 [ t B -ˆ k GW ( t B ) · ⃗ p 2 ( t B ) ] } ) -[ 1 + ˆ k GW ( t B ) · ˆ n ℓ 13 ( t B ) ] × 1 2 { Ψ 13 [ t 1 ( t B ) -ˆ k GW ( t B ) · ⃗ p 1 ( t 1 ( t B )) ] -Ψ 13 [ t B -ˆ k GW ( t B ) · ⃗ p 3 ( t B ) ] } -T 13 ( [ 1 + ˆ k GW ( t B ) · ˆ n ℓ 21 ( t B ) ] × 1 2 { Ψ 21 [ t s ( t B ) -ˆ k GW ( t B ) · ⃗ p 2 ( t 2 ( t B )) ] -Ψ 21 [ t B -ˆ k GW ( t B ) · ⃗ p 1 ( t B ) ] } ) (19) \nwhere, \n- · y GW 3[32]2 ( t B ) , y GW 1[13]2 ( t B ) , y GW 2[21]1 ( t B ) are described below Eq. 17.\n- · Ψ , ˆ n, ˆ k, ⃗p i , t B are described under Eqs. 6, 7.\n- · t 1 , t 2 , t 3 are time stamps from the three clocks in the S/C1, S/C2, and S/C3 respectively.\n- · T 31 , T 21 are the physically occurring delays along the light beam travelling from optical bench OB31 to OB13, and the same for OB21 to OB12 respectively. \nThese results for the phasemeter output above are shown in Figure 15 and Figure 16 for the examples of faceon binary, V407Vul, and the eclipser, ZTTFJ1539 respectively. The signal coupling the non-locking long arm measurements are shown in solid-coloured curves for the minimally locked configuration, \'N1c\' on the left and for the maximally locked configuration, \'N2a\' on the left. Comparison with the response to six-one way links with free-running laser configurations are shown in grey coloured curves. Observe that the four nonlocking phasemeter time-series for the two different locking schemes considered are drastically different than those using the free-running lasers. \nOverall the amplitude of the eclipsing system is higher as expected since its GW strain amplitude is larger than that for VVul by a factor of ∼ 1 . 3. For VVul the higher level in the amplitudes for the \'N2a\' configuration is obvious which is attributable to the larger number of longarm Doppler locks in their phasemeter measurements. The eclipsing system has comparable levels of the amplitude coupling for both configurations. \nThe two-way measurement, H nℓ, GW 323 is identical for both configurations and thus their response to each of the binaries are identical as shown by the green curves. \nFrom these simple time-series analysis, one conclusion is that for \'N2a\' V407Vul, the measurement H nℓ, GW 212 can be utilised to calibrate the instrument due to its strongest amplitude. For the ZTTFJ1539 source, H nℓ, GW 323 is the strongest amplitude for both configurations and thus, this measurement (Figure 16) can be used for instrument calibration. However, since all of these signals will be buried in 10 orders of magnitude of laser noise, we need TDI signals, described in the following. \nAs mentioned above, we will use the GW coupling to the phasemeter equations above to propagate them through Sec. 4.5 for the given set of two white dwarf binaries (Sec. 2) to compute their response in the TDI variables in the subsection below.', '4.5. TDI observables': "In this section we analyse the coupling of the signals from Secs. 4.1- 4.4, with respect to laser frequency fluctuations into the well known Time Delay Interferometry (TDI) Michelson type variable. We consider constant but unequal arms and use the long arm non-locking measurements for the two locking configurations described above as 'minimally locked' and 'maximally locked'. The exploration of a full set of TDI observables as a consequence of locking that would suppress the primary noises to the level of secondary noises is outside the scope of this study which will be studied in forthcoming work. For free-running laser configuration, a most recent extensive version of TDI variables can be found in (Hartwig & Muratore 2022).", '4.6. N1c': 'In order to compare the two configurations, one needs to compute SNR for TDI observable(s) that are common to both of them. A conundrum arises immediately with the geometrically non-identical and non-overlapping measurements between the two topologies - what are the TDI observables common to both configurations? This \nFig. 15.Phasemeter output for long arm measurements for V407Vul. Left: non-locking long arm measurements for the N1c frequency plan. Right: non-locking long arm measurements for the N2a frequency plan. The comparison against long-arm measurements obtained using free running laser configuration along the three distinct arms in dashed-grey curves as labelled. The EM parameters for the VBs are taken from Table 1, assumed perfectly known for the simulations above. The simulation is shown for ∼ 16min with sampling time of dt = 4s. \n<!-- image --> \nFig. 16.Similar to Figure 15 above, phasemeter output from long arm measurements for ZTFJ2243. Left: non-locking long arm measurements for the N1c frequency plan. Right: non-locking long arm measurements for the N2a frequency plan. The comparison against long-arm measurements obtained using free running laser configuration along the three distinct arms in dashed-grey curves as labelled. \n<!-- image --> \nFig. 17.TDI X for N1c recovered using phasemeter measurements in solid curves. The delays applied to the measurements are shown in dashed lines. \n<!-- image --> \nquestion is out of scope for this study, to be explored in forthcoming work. Nonetheless, in this study we construct a pair of two distinct TDI variables for each topology that are comparable with regard to the total number of armlengths involved such that the differences in the TDI observables due to signal will be driving the main difference instead of the laser frequency fluctuations. \nIn order to verify the similarity or the difference between a given TDI observable that is common to both configurations one option is to answer the following What is an equal pathlength Michelson (X, Y, Z) data stream for given measurements for \'N1c\'? For the minimally locked scheme, the classical TDI Z from the freerunning laser configuration can be constructed with the two 2-way measurements to form a virtual Michelson interferometer centered at S/C 3. This can be constructed by subtracting H nℓ 323 delayed by T 31 T 13 from H nℓ 313 delayed by T 32 T 23 , in the following: \n<latexit sha1\\_base64="KVULv6gVt4ErNp7/dQEr3NKa/Rk=">AAACBHicdVDLSsNAFJ3UV62vqMtuBovgKiRtSuuu6KbLCvYBbS2T6aQdOpmEmYlQQhZu/BU3LhRx60e482+ctBVU9MCFwzn3cu89XsSoVLb9YeTW1jc2t/LbhZ3dvf0D8/CoI8NYYNLGIQtFz0OSMMpJW1HFSC8SBAUeI11vdpn53VsiJA35tZpHZBigCac+xUhpaWQWBwFSU4xY0kxvkgFhLKt0lJQr6cgs2VbVds6rdbgkblkT23Frrgsdy16gBFZojcz3wTjEcUC4wgxJ2XfsSA0TJBTFjKSFQSxJhPAMTUhfU44CIofJ4okUnmplDP1Q6OIKLtTvEwkKpJwHnu7MTpa/vUz8y+vHyq8PE8qjWBGOl4v8mEEVwiwROKaCYMXmmiAsqL4V4ikSCCudW0GH8PUp/J90ypZTscpXbqlxsYojD4rgBJwBB9RAAzRBC7QBBnfgATyBZ+PeeDRejNdla85YzRyDHzDePgGhX5i8</latexit> \n<latexit 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sha1\\_base64="aDaFLXt5G5pHSDn/DIbgVTVUVgY=">AAACAnicdVDLSsNAFJ3UV62vqCtxM1gEVyFpU1p3RTddVrAPaGqZTCft0MkkzEyEEoIbf8WNC0Xc+hXu/BunD0FFD1w4nHMv997jx4xKZdsfRm5ldW19I79Z2Nre2d0z9w/aMkoEJi0csUh0fSQJo5y0FFWMdGNBUOgz0vEnlzO/c0uEpBG/VtOY9EM04jSgGCktDcwjL0RqjBFLG9lNyj3CWDZIy045G5hF26rYznmlBhfELWliO27VdaFj2XMUwRLNgfnuDSOchIQrzJCUPceOVT9FQlHMSFbwEklihCdoRHqachQS2U/nL2TwVCtDGERCF1dwrn6fSFEo5TT0defsYPnbm4l/eb1EBbV+SnmcKMLxYlGQMKgiOMsDDqkgWLGpJggLqm+FeIwEwkqnVtAhfH0K/yftkuWUrdKVW6xfLOPIg2NwAs6AA6qgDhqgCVoAgzvwAJ7As3FvPBovxuuiNWcsZw7BDxhvn8jUl68=</latexit> \nZ ( t B ) N1c = H nℓ 323 , [31][13] -H nℓ 313 , [32][23] = -f 0 , [31][13] + f 0 , [23][32][31][13] + f 0 , [32][23] -f 0 , [31][13][32][23] + D 32 , [23][31][13] + D 23 , [31][13] -D 13 , [32][23] -D 31 , [13][32][23] + O 1 , [23] , [32][23] + O 2 , [13][32][23] + O 3 , [32][23] -O 3 , [31][13][32][23] . (20) \nThe local and long long-arm locking measurements in Eqs. 8a, 8b, can be used to subtract the offset terms \nto get residual laser frequency noise and differential VB signal in the Eq. 20 above, such that: \nZ ( t B ) N1c = -f 0 , [31][13] + f 0 , [23][32][31][13] + f 0 , [32][23] -f 0 , [31][13][32][23] + D 32 , [23][31][13] + D 23 , [31][13] -D 13 , [32][23] -D 31 , [13][32][23] + O 1 , [23] , [32][23] + O 2 , [13][32][23] + O 3 , [32][23] -O 3 , [31][13][32][23] + H ℓℓ 23 , [32][23] + H ℓℓ 13 , [32][23] + H ℓℓ 33 , [32][23] -H ℓℓ 33 , [31][13][32][23] = H nℓ, GW 323 , [31][13] -H nℓ, GW 313 , [32][23] -f 0 , [31][13] + f 0 , [23][32][31][13] + f 0 , [32][23] -f 0 , [31][13][32][23] = [ T 23 ( y GW 3[32]2 ( t B ) ) + y GW 2[23]3 ( t B ) ] , [31][13] -[ T 13 ( y GW 3[31]1 ( t B ) ) + y GW 1[13]3 ( t B ) ] , [32][23] -f 0 , [31][13] + f 0 , [23][32][31][13] + f 0 , [32][23] -f 0 , [31][13][32][23] = T 23 ( [ 1 + ˆ k GW ( t B ) · ˆ n [32] ( t B ) ] 1 2 × { Ψ [32] [ t 3 ( t B ) -ˆ k GW ( t B ) · ⃗ p 3 ( t 3 ( t B )) ] -Ψ [32] [ t B -ˆ k GW ( t B ) · ⃗ p 2 ( t B ) ] } ) , [31][13] + [ 1 + ˆ k GW ( t B ) · ˆ n [23] ( t B ) ] 1 2 × { Ψ [23] [ t 2 ( t B ) -ˆ k GW ( t B ) · ⃗ p 2 ( t 2 ( t B )) ] -Ψ [23] [ t B -ˆ k GW ( t B ) · ⃗ p 3 ( t B ) ] } , [31][13] -T 13 ( [ 1 + ˆ k GW ( t B ) · ˆ n [31] ( t B ) ] 1 2 × { Ψ [31] [ t 3 ( t B ) -ˆ k B · ⃗ p 3 ( t 3 ( t B )) ] -Ψ [31] [ t B -ˆ k GW ( t B ) · ⃗ p 1 ( t B ) ] } ) , [32][23] -[ 1 + ˆ k GW ( t B ) · ˆ n [13] ( t B ) ] 1 2 × { Ψ [13] [ t 1 ( t B ) -ˆ k B · ⃗ p 1 ( t 1 ( t B )) ] -Ψ [13] [ t B -ˆ k GW ( t B ) · ⃗ p 3 ( t B ) ] } , [32][23] -f 0 , [31][13] + f 0 , [32][23] -f 0 , [31][13][32][23] + f 0 , [23][32][31][13] . (21) \nIn Eq. 21 above, observe that from the expressions in Eqs. 13, 14, we get eight terms for the VB signal, whereas the laser phase noise is comparable to the the level as in the classical TDI Michelson Z derived using 6 oneway links. This implies that locking results into signal enhancement , ignored by all existing work in this field so far. The signal-to-noise ratio for the choice of the verification binaries is given in Table 2, where comparison with TDI Z derived from free-running lasers is made.', '4.7. N2a': 'Similarly, we consider which equal pathlength for given measurements for \'N2a\' result into an equivalent TDI Michelson, preferably the one identical to TDI Z. Inspection of the measurements in Sec. 4.4 reveals that it is not possible to construct TDI Z centered at S/C 3 since the measurements in that S/C necessarily involve Doppler locks from all three armlengths. However, an observable similar to Z, Michelson \'TDI Y\', can be constructed by subtracting H nℓ 323 delayed by T 21 T 12 from H nℓ 212 delayed \nFig. 18.An equivalent \'TDI Y\' for N2a recovered using phasemeter measurements in solid curves. The delays applied to the measurements are shown in dashed lines. \n<!-- image --> \n<latexit sha1\\_base64="KVULv6gVt4ErNp7/dQEr3NKa/Rk=">AAACBHicdVDLSsNAFJ3UV62vqMtuBovgKiRtSuuu6KbLCvYBbS2T6aQdOpmEmYlQQhZu/BU3LhRx60e482+ctBVU9MCFwzn3cu89XsSoVLb9YeTW1jc2t/LbhZ3dvf0D8/CoI8NYYNLGIQtFz0OSMMpJW1HFSC8SBAUeI11vdpn53VsiJA35tZpHZBigCac+xUhpaWQWBwFSU4xY0kxvkgFhLKt0lJQr6cgs2VbVds6rdbgkblkT23Frrgsdy16gBFZojcz3wTjEcUC4wgxJ2XfsSA0TJBTFjKSFQSxJhPAMTUhfU44CIofJ4okUnmplDP1Q6OIKLtTvEwkKpJwHnu7MTpa/vUz8y+vHyq8PE8qjWBGOl4v8mEEVwiwROKaCYMXmmiAsqL4V4ikSCCudW0GH8PUp/J90ypZTscpXbqlxsYojD4rgBJwBB9RAAzRBC7QBBnfgATyBZ+PeeDRejNdla85YzRyDHzDePgGhX5i8</latexit> \n<latexit 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sha1\\_base64="DleaAMuq3zW6JjKNiicdTyDNdfk=">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</latexit> \nby T 32 . Along with the long-arm locked and reference \nmeasurements to subtract off the offsets we get: \nY ( t B ) N2a = -f 0 , [21][12] + f 0 , [23][32][21][12] + f 0 , [32][32] -f 0 , [12][21][32][32] + D 32 , [23][21][12] + D 23 , [21][12] + D 32 , [32] -D 32 , [12][21][32] -D 13 , [21][32] + D 12 , [32] + O 1 , [23][21][12] + O 1 , [32] -O 1 [12][21][32] + O 4 , [32] -O 4 , [12][21][32] -O 5 , [12][32] + H ℓℓ 23 , [23][21][12] + H ℓℓ 23 , [32] -H ℓℓ 23 , [12][21][32] -H ℓℓ 22 , [32] + H ℓℓ 22 , [12][21][32] -H ℓℓ 12 , [12][32] = H nℓ, GW 323 , [21][12] -H nℓ, GW 212 , [32] -f 0 , [21][12] + f 0 , [23][32][21][12] + f 0 , [32][32] -f 0 , [12][21][32][32] T 23 ( [ 1 + ˆ k GW ( t B ) · ˆ n [32] ( t B ) ] 1 2 × { Ψ [32] [ t 3 ( t B ) -ˆ k GW ( t B ) · ⃗ p 3 ( t 3 ( t B )) ] -Ψ [32] [ t B -ˆ k GW ( t B ) · ⃗ p 2 ( t B ) ] } ) , [21][12] + [ 1 + ˆ k GW ( t B ) · ˆ n [23] ( t B ) ] 1 2 × { Ψ [23] [ t 2 ( t B ) -ˆ k GW ( t B ) · ⃗ p 2 ( t 2 ( t B )) ] -Ψ [23] [ t B -ˆ k GW ( t B ) · ⃗ p 3 ( t B ) ] } , [21][12] + [ 1 + ˆ k GW ( t B ) · ˆ n [32] ( t B ) ] 1 2 × { Ψ [32] [ t 3 ( t B ) -ˆ k GW ( t B ) · ⃗ p 3 ( t 3 ( t B )) ] -Ψ [32] [ t B -ˆ k GW ( t B ) · ⃗ p 2 ( t B ) ] } , [32] -T 12 T 21 ( [ 1 + ˆ k GW ( t B ) · ˆ n [32] ( t B ) ] 1 2 × { Ψ [32] [ t 3 ( t B ) -ˆ k GW ( t B ) · ⃗ p 3 ( t 3 ( t B )) ] -Ψ [32] [ t B -ˆ k GW ( t B ) · ⃗ p 2 ( t B ) ] } ) -[ 1 + ˆ k GW ( t B ) · ˆ n [21] ( t ) ] 1 2 × { Ψ [21] [ t 2 ( t B ) -ˆ k GW ( t B ) · ⃗ p 2 ( t 2 ( t B )) ] -Ψ [21] [ t B -ˆ k GW ( t B ) · ⃗ p 1 ( t B ) ] } , [32] -[ 1 + ˆ k GW ( t B ) · ˆ n [12] ( t ) ] 1 2 × { Ψ [12] [ t 1 ( t B ) -ˆ k GW ( t B ) · ⃗ p 1 ( t 1 ( t B )) ] -Ψ [12] [ t B -ˆ k · ⃗ p 2 ( t B ) ] } , [32] -f 0 , [21][12] + f 0 , [23][32][21][12] + f 0 , [32][32] -f 0 , [12][21][32][32] . (22) \nSimilar to the previous sub-section, it is assumed that the residual offset locking noise can be perfectly subtracted using the locking beatnotes, Eqs. 15a, 15b for TDI Y. The individual Ψ along the links add up to twelve terms. We obtain following signal to noise ratios: \nTABLE 2 \n≈ \n≈ \nSNR for GW measurements for the \'face-on\' V407Vul and the most compact detached know \'edge-on\' ZTF J1539+5027 binaries from Table 1 for observation of 1 year. \nThe SNRs above imply a discovery of significant enhancement of the GW signal caused by laser-locking, using verification binary, which is an ultra-compact binary - not know prior to this work. The enhancement for the two VBs are in the same order, by a factor of 10. Assuming synchronized measurements, and for the fiducial TDI observables, the laser locking configuration with the minimum Doppler links, \'N1c\', is slightly favourable with higher SNR for both binaries. The SNR difference could be attributable to the different sensitivities of the TDI observables to the GW signals, known previously Arm- \nstrong et al. (1999). A further check is required for noise orthogonal \'A, E, T\' type measurements, once they are discovered.', '5. CONCLUSIONS': "A pilot study and preliminary data analysis of the coupling of the Verification Binary signal to the LISA phasemeter data are derived in the case of locking scheme, where all the lasers in the LISA constellation are locked to a primary laser. Many options are available for the locking scheme that can be implemented for any LISA-like geometry. We deviate from the usual assumption of the configuration of the six free-running lasers across the constellation for forming the interferometric measurements with long-lived GW signals in them. We show that contrary to popular belief, the sensitivity of a GW signal is significantly affected by locking scheme, yielding enhanced SNR for the locking configuration compared to that of the free-running lasers. We perform SNR studies for two of the six non-swap locking schemes. These have been implemented in Synthetic LISA (Vallisneri 2005) 17 . \nFrom simplified assumptions, we find that at the TDI output, the choice of the minimal locking scheme (N1c) is favoured over the maximal locking scheme (N2a). Many \nassumptions have been made such as all the raw measurements have been synchronised Reinhardt et al. (2023) perfectly to t B and thus this needs further thorough investigation including other laser locking configurations as well. Additionally, we need further analysis with remaining four non-swap configurations to verify the results presented. Furthermore, it is important to stress that verification binaries will provide an independent verification of the instrument and calibration of the data's amplitude and phase. \nIn phasemeter observation equations, we find that the TDI power for the non-swap locking scheme is significantly larger for edge-on systems and has a significantly smaller dependence on the inclination angle. This has several consequences. First, the known population of verification binaries is biased towards edge-on systems as eclipses are easier to detect in electromagnetic data. Therefore, we expect that most verification binaries will have a significantly larger TDI power for the non-swap locking scheme. Second, the smaller dependence on the inclination angle leads to less bias, based on inclination for newly detected sources. This is particularly important for population studies as they rely on well understood biases in the data. This also means that edge-on \nMassive Black Hole Binaries (MBHB) would be readily visible in 'N2a' raw measurements were the MBHB signals to exceed laser frequency noise at that level, which is perhaps unlikely. But if so, then 'N2a' must be in operation for multi-messenger with MBHB type sources.", 'ACKNOWLEDGEMENT': "SS acknowledges fruitful discussions with Michele Vallisneri for the single link response, Gerhard Heinzel for verifying the fplan lock equations and valuable comments to improve the paper by Kohei Yamamoto, Sarah Paczkowski, and LDPG On-Ground Instrument Processing Expert Group. SS also gratefully acknowledges financial support by the German Aerospace Center (DLR) with funds from the German Federal Ministry for Economic Affairs and Climate Action (BMWK) under grants # 50 OQ1801 and # 50 OQ2301. TK acknowledges support from the National Science Foundation through grant AST #2107982, from NASA through grant 80NSSC22K0338 and from STScI through grant HST-GO-16659.002-A. This result is part of a project that has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No. 101078773)", 'REFERENCES': '- Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2016, Physical Review Letters, 116, 061102 Abich, K., Abramovici, A., Amparan, B., et al. 2019, Physical Review Letters, 123, 031101 Amaro-Seoane, P. e. a. 2023, Living Reviews in Relativity, 26, 2 Armano, M., Audley, H., Auger, G., et al. 2016a, Classical and Quantum Gravity, 235015 -. 2016b, Physical Review Letters, 116, 231101 Armstrong, J. W., Estabrook, F. B., & Tinto, M. 1999, The Astrophysical Journal, 527, 814 Barke, S. 2015, PhD thesis, Leibniz Universitat, Hannover, doi:0.15488/8405 Barranco, G. F., Sheard, B. S., Dahl, C., Mathis, W., & Heinzel, G. 2018, IEEE Sensors, 18, 7414 Bayle, J.-B., & Hartwig, O. 2023, Phys. Rev. D, 107, 083019 Brown, A. G. A., Vallenari, A., Prusti, T., et al. 2020, arXiv e-prints, arXiv:2012.01533 Burdge, K. B., Coughlin, M. W., Fuller, J., et al. 2019, Nature, 571, 528 Burdge, K. B., Coughlin, M. W., Fuller, J., et al. 2020, ApJ, 1, L7 Estabrook, F. B., & Wahlquist, H. D. 1975, General Relativity and Gravitation, 6, 439 Finch, E., Bartolucci, G., Chucherko, D., et al. 2023, mnras, 522, 5358 Fitzsimons, E. 2022, Optical Bench Design Report, Tech. Rep. 0005, UK Astronomy and Technology Centre, Edinborough Gerberding, O., Sheard, B., I., B., et al. 2013, Classical and Quantum Gravity, 30, 235029 Hartwig, O., & Muratore, M. 2022, Phys. Rev. D, 105, 062006 Heinzel, G. 2020a, LISA Frequency planning, Tech. Rep. 1, Max-Planck-Institut fur Gravitationsphysik (Albert-Einstein-Institut and Leibniz Universitat Hannover, Institut fur Gravitationsphysik, Hannover -. 2020b, LISA Timing architecture, Tech. Rep. 0, Max-Planck-Institut fur Gravitationsphysik (Albert-Einstein-Institut and Leibniz Universitat Hannover, Institut fur Gravitationsphysik, Hannover Hewitson, M., Heinzel, G., Hartwig, O., et al. 2021, Top-level Nomenclature and Conventions in LISA, Tech. Rep. 1, Max-Planck-Institut fur Gravitationsphysik (Albert-Einstein-Institut and Leibniz Universitat Hannover, Institut fur Gravitationsphysik, Hannover \nKaufmann, W. J. 1970, Nature, 227, 157 \n- Koch, A. 2020, PhD thesis, Leibniz Universitat, Hannover,\n- doi:10.15488/9799 Korol, V., Rossi, E. M., Groot, P. J., et al. 2017, mnras, 470, 1894 Kupfer, T., Korol, V., Shah, S., et al. 2018, Monthly Notices of Royal Astronomical Society, 480, 302 Kupfer, T., Korol, V., Littenberg, T. B., et al. 2023, arXiv e-prints, arXiv:2302.12719 Lindegren, L., Lammers, U., Hobbs, D.., et al. 2012, Astronomy & Astrophysics, 538, A78 Littenberg, T. B. 2018, Physics Review D, 98, 043008 Marsh, T. 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D, 98, 042003 Hartwig, M., & Bayle, O. 2021, American Physical Society, 103, 123027 Toonen, S., Claeys, J. S. W., Mennekens, N., & Ruiter, A. J. 2014, Astronomy & Astrophysics, 562, A14 Vallisneri, M. 2005, Physics Review D, 71, 022001 Wahlquist, H. D. 1987, General Relativity and Gravitation, 19, 1572 \n- Wiesner, K., Reinhardt, J. N., Staab, M., & Shah, S. 2021, INREP Description of the implementation and function, Tech. Rep. 1.0, Max-Planck-Institut fur Gravitationsphysik (Albert-Einstein-Institut and Leibniz Universitat Hannover, Institut fur Gravitationsphysik, Hannover\n- Wissel, L., Hartwig, O., Bayle, J. B., et al. 2022a, The influence of laser relative intensity noise in the Laser Interferometer Space Antenna, doi:10.48550/ARXIV.2212.12052\n- Wissel, L., Wittchen, A., Schwarze, T. S., et al. 2022b, Physics Review Applied, 17, 024025\n- Yamamoto, K., Vorndamme, C., Hartwig, O., et al. 2022, Physics\n- Yamamoto K., B. I., & Delgado, J. J. E. 2022, PRN ranging and data transfer, Tech. Rep. 1, Max-Planck-Institut fur Gravitationsphysik (Albert-Einstein-Institut and Leibniz Universitat Hannover, Institut fur Gravitationsphysik, \nReview D, 105, 042009 Hannover'}

2024arXiv240906841B

Measurements of UltraHigh Energy Cosmic Rays UHECR suggest a complex composition with significant contributions from heavy nuclei at the highest energies. We systematically explore how the selection and number of primary nuclei included in the analysis impact the inferred UHECR mass composition. Introducing a distance measure in the space of Xrm max distribution moments we demonstrate that limiting the analysis to a few primaries can introduce significant biases particularly as observational data improves. We provide lists of primaries approximately equidistant in the new measure which guaranty unbiased results at given statistical confidence. Additionally we explore consistent inclusion of nuclei heavier than iron and up to plutonium deriving first observational upper bounds on their contributions to UHECR with the Pierre Auger Open Data.

2024-09-01T00:00:00Z

['2024arXiv240906841B', 'arXiv:2409.06841', '10.48550/arXiv.2409.06841']

['Astrophysics - High Energy Astrophysical Phenomena', 'High Energy Physics - Phenomenology']

Resolution of Heavy Primaries in Ultra High Energy Cosmic Rays

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.06841.pdf

{'Resolution of (Heavy) Primaries in Ultra High Energy Cosmic Rays': 'Blaž Bortolato, 1, ∗ Jernej F. Kamenik, 1, † and Michele Tammaro 2, ‡ \n1 Jožef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia \nFaculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia 2 INFN Sezione di Firenze, Via G. Sansone 1, I-50019 Sesto Fiorentino, Italy \n(Dated: September 12, 2024) \nMeasurements of Ultra-High Energy Cosmic Rays (UHECR) suggest a complex composition with significant contributions from heavy nuclei at the highest energies. We systematically explore how the selection and number of primary nuclei included in the analysis impact the inferred UHECR mass composition. Introducing a distance measure in the space of X max distribution moments, we demonstrate that limiting the analysis to a few primaries can introduce significant biases, particularly as observational data improves. We provide lists of primaries approximately equidistant in the new measure, which guaranty unbiased results at given statistical confidence. Additionally, we explore consistent inclusion of nuclei heavier than iron and up to plutonium, deriving first observational upper bounds on their contributions to UHECR with the Pierre Auger Open Data.', 'I. INTRODUCTION': "Measurements of Extensive Air Showers (EAS) at Pierre Auger Observatory (PAO) [1] and at Telescope Array [2] indicate that the mass spectrum of Ultra-High Energy Cosmic Rays (UHECR) is best explained by a mixture of different primaries. Their inferred composition also depends on the cosmic ray energy: at lower energies ( E ∼ 10 18 eV), it seems to be dominated by protons and possibly other light elements, while measurements at higher energies ( E ∼ 10 19 -10 20 eV) point towards a significant fraction of heavy nuclei [3-6]. \nThe UHECR composition is closely related to other open questions in the field, regarding the sources of UHECRs, their acceleration mechanisms and their propagation in the galactic and intergalactic medium. \nTypically, the nuclei assumed to be part of UHECRs (apart from protons) span from stable isotopes of helium (He, A = 4 ) to iron (Fe, A = 56 ), as Fe has the largest binding energy per nucleon and it is likely the end product of stellar evolution. While heavier elements ( A > 56 ) are rarer in cosmic radiation, they can be produced at UHE, e.g. in GRB [7], and have a similar propagation phenomenology as lighter elements [8-11]. Thus it is reasonable to explore scenarios where these super heavy nuclei make up a fraction of the UHECR and to study their observational features at observatories like the PAO. Recent studies indicate the presence of Z ∼ 53 primaries as a possible origin of the 'second knee' in the UHECR energy spectrum [12, 13]. \nAt terrestrial observatories, the composition of UHECRs can be inferred by studying the longitudinal profile of an EAS, that is the intensity of fluorescent light emitted by nuclei in the atmosphere, typically nitrogen, excited by the passage of charged particles, and measured \nas a function of the s.c. slant depth ( X ) of the shower. In general, the longitudinal profile has a clear peak, at X max , corresponding to the point of maximum population of electrons and positrons in the shower evolution. Comparing observed data on X max distributions at a particular energy against simulations for a set of primaries, one can infer the allowed fraction of each considered element in the observed collection of UHECRs. \nPast studies of UHECR composition have considered either a binned distribution of X max or its global features such as its mean ⟨ X max ⟩ and width σ ( X max ) [3, 5]. Typically the measured values of these were compared to simulations of a few chosen primaries. Recently [14], we have have shown how a systematic description of the X max distribution in terms of its central moments together with efficient inference techniques based on bootstrapping and nested sampling can be used to infer the full composition of all stable primaries up to iron. We have also shown that while the fractions of individual primaries in this case can be highly correlated and thus degenerate with each other, one can still derive robust confidence intervals for fractions of all primaries heavier than any chosen nuclear mass or atomic number. \nIn this work we extend our previous studies of UHECR composition by addressing two main questions concerning the choice of considered primaries when inferring composition, namely how does the (1) range of primaries and (2) their total number affect the inferred composition confidence intervals. This is particularly important for both present and future studies. Firstly, the current results are based on the (strong) assumption that a mixture of only four primaries, typically (p,He,C/N/O,Fe) [15], describes well the UHECR mass spectrum. Secondly, it is warranted in order to to derive bounds on possible fractions of primaries beyond iron, because a brute-force approach of including all stable enough primaries up to uranium ( Z = 92 ) or plutonium ( Z = 94 ) is computationally extremely expensive using existing techniques. To this end we first introduce a natural metric in the space of X max distribution moments. Because simulated X max distributions of primaries have intrinsic uncertain- \nties, this allows on one hand to identify a-priori practically (given finite event statistics) indistinguishable primaries, and on the other allows us to include only distinguishable primaries in the inference procedure. Conversely, we study in detail using mock data, how omission of distinguishable primaries can potentially lead to biased inference of primary UHECR composition. In particular, we show that mass composition results of existing studies using the full PAO statistics but with limited number of primaries below Z = 26 are potentially biased. Finally we explore the effects of including primaries beyond Z = 26 in the composition and in particular derive first upper bounds on fractions of UHECR primaries up to plutonium using the PAO open dataset. \nThe remainder of the paper is structured as follows: in Sec. II we introduce our methodology and data, in particular the PAO open dataset, the CORSIKA EAS simulation framework, the relevant observables - moments of X max distributions, and our statistical inference procedure. In Sec. III we study the implications of inferring composition using different numbers of primaries. We first define a distance measure in the space of X max distribution moments and show how it can be used to define distinguishable primaries. We apply this measure to select primaries to be included in the inferred mixture. We test this procedure both using mock data as well as the PAO open dataset, where we also infer composition including primaries beyond iron and up to plutonium. Our main conclusions are summarised in Sec. IV, while the Appendices A and B contain some auxiliary computations, as well as additional tables and plots, respectively.", 'II. METHODOLOGY': "a. Open Data : we use publicly available data from the 2021 Pierre Auger Open Data (PAOD) release [16]. This dataset consists of 10% of the full recorded data by the PAO. In particular, we focus on so called hybrid showers: events recorded by both the Surface Detectors (SD) and the Fluorescent Detectors (FD). There are 3348 hybrid showers, distributed in the reconstructed primary energy as ∼ E -2 . 6 , from E min = 0 . 65 EeV to E max = 63 EeV. We select the energy bin E ∈ [0 . 65 , 1] EeV, containing 934 events, and use it in all our remaining discussion. In this way we work with a fairly large statistical sample within a small energy bin. For completeness, we include the respective results in the energy bin E ∈ [1 , 2] EeV, encompassing 1234 events, in Appendix B. \nb. Simulations : we simulate EAS with CORSIKA v7.7550 [17]. To achieve fast and reliable simulations of longitudinal shower profiles, we employ the CONEX option. In conjunction with the EPOS hadron interaction model, this option allows to include primary nuclei up to A = 295 . Note however that composition results can strongly depend on the choice of the hadronic model [14]. Thus, our results cannot explore the potential effects of different parametrizations of UHE hadronic cross sec- \ntions. We simulate 10 4 showers per primary, with the energy uniformly distributed in the chosen energy bin. The atomic number A of the primary is taken as the one of the most abundant stable isotope; we thus unambiguously indicate each primary nucleus by its atomic number Z . We simulate all primaries from Z = 1 (proton) to Z = 26 (iron); in addition, we simulate showers for heavier primaries with Z = 27, 28, 29, 30, 34, 39, 40, 42, 43, 44, 47, 50, 57, 58, 64, 72, 81, 82, 91, 92, 94. We chose to simulate a set of primaries with atomic number immediately beyond iron, and additionally a set of primaries that cover the interval up to Plutonium, Z = 94 , which is the heaviest that can be simulated with CONEX. Finally, we extract the value of X max from each simulated shower by fitting the longitudinal profile to a GaisserHillas curve [18]. The set of X max per primary furnishes a distribution p ( X max | Z ) . \nc. Observables : we describe the resulting X max distribution by its moments, defined as \n⟨ X n max ⟩ = ∫ p ( X max | Z ) X n max d X max ∫ p ( X max | Z )d X max . (1) \nFrom this expression we can obtain the central moments as \nz n = ⟨ ( X max -⟨ X max ⟩ ) n ⟩ = n ∑ k =0 ( n k ) ⟨ X n -k max ⟩ ( -1) k ⟨ X max ⟩ k . (2) \nThe number n of moments needed to infer the composition depends on the interplay of statistical and systematic uncertainties, which we discuss in the next paragraph. We have shown in Ref. [14] that with the data available in the Auger Open Data, n = 3 is sufficient and additional moments do not improve the results. \nd. Uncertainties : We consider both systematic and statistical uncertainties on X max . The former originate from the finite resolution of the fluorescent detector [19]; we include these by reweighting the distribution p ( X max | Z ) by the detector smearing [14]. The latter are a consequence of the limited statistics available, either in the simulation or in the observed data. We take these into account by means of bootstrapping [20], that is repeatedly sampling sets of X max from p ( X max | Z ) and computing moments of these sets; the spread of the moment distributions obtained will depend on the total number of events available. \nThe next observational run of Auger Prime [21] is expected to deliver a larger and more precise dataset. In the following we perform projections to larger statistics by assuming that the central values of the measured X max moments remain unchanged, while the variances of moment distributions scale with the statistical multiplier f as 1 /f , when increasing the total available statistics by a factor of f . 1 While such scaling is formally only valid \nfor statistical uncertainties, at the current and even projected PAO statistics, also the dominant systematic uncertainties are expected to effectively scale with statistics. Namely, the detector resolution is calibrated using (calibration) data and the resulting uncertainties are statistics dominated [19]. \ne. Inference procedure : we follow the steps of our work, Ref. [14], to infer the composition of UHECR, w . For a given mixture of primaries, the log-likelihood is \nln L ( w ) = ∫ ln [ p ( z | µ w , Σ w ) ] p ( z | ˜ µ, ˜ Σ ) d z , (3) \nwhere p ( z | ˜ µ, ˜ Σ ) is the moment distribution of the Auger data, while p ( z | µ w , Σ w ) is the distribution of moments for simulated momenta, weighted by the composition w . The number of free parameters depends on the set of Z allowed to be in the mixture, which is worked out in Sections III A and III. The best fit composition can be obtained by maximizing the log-likelihood in Eq. (3). However, the typical number of free parameters and thus the dimensionality of the likelihood manifold is large. In order to be able to effectively extract the confidence intervals of our best fits, that is the shape of ln L around the maximum, we use Nested Sampling (NS) techniques [23, 24]. The latter allow us to sample highdimensional likelihoods, with up to 100 free parameters, thanks to a large reduction in computational time with respect to traditional sampling methods. \nWe present the results as a cumulative composition : for each Z included in the fit, we show the allowed fraction of primaries heavier than Z . This method yields a reliable presentation of the results, including well defined confidence intervals that account for correlations among individual primaries' fractions. It also allows to compare composition results with different numbers of primaries since, contrary to individual primary fractions, the cumulative fraction values for any Z are always well defined and have the same statistical meaning irrespective of the actual primaries included in the fit (as long as protons are included).", 'III. COMPOSITION WITH DIFFERENT NUMBERS OF PRIMARIES': 'The procedure of inferring the composition of UHECR, introduced in Ref. [14] and summarized in Sec. II raises an important question: what is the minimal number of composition components required to obtain an unbiased fit to data? Namely, defining the composition vector w = ( w Z 1 , w Z 2 , . . . , w Z k ) , we wish to quantify the minimal length k and find a suitable list of Z k primaries, such \nthat the inference procedure yields (cumulative) composition fractions consistent with their truth level values. The set of primaries (p,He,C/N/O,Fe), that is N L = 4 , has been largely assumed in the literature [4, 15] to represent the UHECR mass spectrum. Previous works, see Refs. [5, 25], have however shown that the goodness of fit for the UHECR composition is strongly affected by the number of primaries chosen, and that it generally improves when including more intermediate elements, up to N L = 8 . Here we extend these studies in a systematic way, employing a distance measure in the space of X max moments to provide a quantitative criterion for the selection of primaries.', 'A. Primary Distance in Space of X max Moments': "The decomposition of the X max distributions in terms of a series of central moments allows us to quantify the separation between different primaries Z . Given two primaries Z 1 and Z 2 , we define the distance \nd 2 n ( Z 1 , Z 2 ) ≡ ( µ Z 1 -µ Z 2 ) T (Σ Z 1 +Σ Z 2 ) -1 ( µ Z 1 -µ Z 2 ) , (4) \nwhere n indicates the number of moments used. Here µ Z 1 , µ Z 2 are the n -vectors of moment means and Σ Z 1 , Σ Z 2 the respective n × n covariance matrices 2 . This definition builds upon the property of the bootstrapped distributions of moments, that they quickly approach normal distributions. The same feature allows for a straightforward statistical interpretation of d 2 n : the distance behaves as a chi-squared distribution with n degrees of freedom, d 2 n ∼ χ 2 n , thus its value can be converted to a confidence level of distinguishing among two sets of moments. For example, if we fix n = 1 moments and the distance between two primaries is d 2 1 ∼ χ 2 1 ≃ 2 . 7 , then we can interpret this as 90% confidence that the compositions consisting of the two primaries are in principle distinguishable using measurements of only the first moment. \nSince X max distributions of individual primaries cannot be inferred directly from data, distances d 2 n need to be estimated using simulations. The main systematics in calculating the moments and in particular their means is then expected to come from the parametrization of UHE hadronic interactions, i.e. the choice of the hadronic model. In addition however, the corresponding covariance matrices also scale with the simulation statistics. This means that the separability of pairs of primaries crucially depends on the assumed (simulated) data statistics. In particular, for N simulated or measured events, the distances d n scale as √ N . The scaling remains true even in the presence of systematic uncertainties on the \nX max measurements. As detailed in Ref. [19] and discussed in Sec. II, the main systematics namely come from the limited statistics of the observed data. \nIn the top plot of Fig. 1 we show the n = 1 distances d 1 for the full list of our simulated primaries in the chosen energy bin, assuming the same statistics as PAOD. As expected, the distance for a fixed Z 1 grows with the difference | Z 2 -Z 1 | as the distributions (and consequently their moments) of X max drift apart for more distant primaries. Somewhat surprisingly however, we find that the distance d n between any two primaries is to a very good approximation independent of the number of moments considered n . This observation can be traced back to the covariance matrices Σ Z , which exhibit a high degree of correlation between the different moments, as first observed already in Ref. [14]. This property allows us to disregard higher moments in the context of distances, and consider only the first moment z 1 for all the simulated primaries. In addition, the monotonic nature of z 1 allows us to employ an analytic interpolation formula for the mean and variance of z 1 as functions of Z (see Appendix A for details) in order to compute d 1 for all pairs of primaries (including those not explicitly simulated) up to Z = 94 , as shown in the bottom plot of Fig. 1. \nLeveraging on the statistical interpretation of the distance between primaries and making use of these interpolating functions, we can obtain lists of approximately equidistant primaries. We fix the first element to always be proton, Z = 1 , and choose a step distance d 0 . We then iteratively find the next element of the list as the one whose distance from all previous is at least d 1 ≥ d 0 . Each choice of d 0 thus directly corresponds to a set of Z and defines the level of separation between neighbouring primaries in the shower composition. \nWe show some examples in Table I, assuming 10 3 events per primary, where we also indicate the number of primaries with Z ≤ 26 and Z > 26 as N L and N H respectively. Note that a set with N L = 4 , as widely employed in the literature, would correspond to a relatively large distance step, d 0 = 16 . 6 at these statistics, while asking for at most d 0 < 2 separation leads to N L = 20 . Due to the scaling of d n with assumed primary statistics, these lists can be used at any given statistics if we also rescale the threshold value d 0 by the corresponding statistical multiplier √ f . For example, by increasing the dataset tenfold, f = 10 , to 10 4 events per primary, the same lists are obtained by imposing the new thresholds d ' 0 = √ fd 0 ∼ 3 d 0 . For completeness, we show the behaviour of N L and N H as functions of d 0 assuming 10 3 events per primary in Fig. 2. \nWe note that the distributions of X max moments change with CR energy and thus both N L,H as well as the lists of d 0 -equidistant primaries can change for the same d 0 in different energy bins, see e.g. Table II, so these should be recomputed for any specific CR energy range considered. \nFinally, in any measured CR event sample of unknown composition, only the total number of registered showers \nFIG. 1: Top: distance d 1 for each pair of simulated primaries. Bottom: distance d 1 for each pair of primaries, using the interpolating functions in Eq. (A1). Note that the scale is linear for d 1 ∈ [0 , 1] while it is logarithmic above this interval. See text for details. \n<!-- image --> \n<!-- image --> \n1 \nis known a-priori, and not the statistics of individual primary components. Thus, in practice one can only compute an upper bound on d 2 n ( Z 1 , Z 2 ) by assuming that the dataset set is composed of only two primaries ( Z 1 and Z 2 ) of equal fractions. The actual separability within the dataset will then always be lower and depend on the (unknown) true composition. Our d 0 values and lists in Tables I and II can thus be considered as very conservative if applied to event samples of given total statistics. They nonetheless provide meaningful benchmarks, as we demonstrate using simulated mock data examples of known composition in the next section.", 'B. Composition Inference on Mock Data': 'We demonstrate the implications of different d 0 based w length choices for the mixture fit results using specific mock compositions of UHECR based on simulated data. We first select a set of primaries and their respective frac- \nd 0 N L ( N H ) List of atomic numbers ZTABLE I: Lists of approximately equidistant primaries in the chosen energy bin. The first column shows the chosen step d 0 assuming 10 3 events per primary, while the second indicates the values of N L and N H obtained. The last column lists all selected Z . \nFIG. 2: Stacked N L (gray) and N H (black) values as a functions of the distance threshold d 0 , assuming 10 3 events per primary in the energy bin [0 . 65 , 1] EeV. \n<!-- image --> \ntions to represent the truth-level composition, w true . We compute the moments of the truth-level X max distribution by randomly sampling 1000 events, apportioned following w true . In order to ensure consistent truth level \nOn the other hand, we observe that this behavior is not universal and that the relative goodness of fit does \ncompositions at large values of f , we do not separate these events from the rest of the simulations. Otherwise, the computed values of the distribution moments would be slightly different from the ones computed using all events; thus at large f , this difference could lead to significant deviations from w true . \nWe study two example compositions: \nEx1 w 1 = w 10 = 1 / 2 , that is we assume the mass spectrum consists of equal fractions of only two primaries, Z = 1 and Z = 10 ; \nEx2 w 1 = w 2 = · · · = w 10 = 1 / 10 , that is we take equal fractions of the lightest ten primaries. \nFor each example, we infer the composition using the lists of light primaries 3 with N L = 4 , 8 , 12 , 16 , 20 , 24 , 26 , as shown in Table I, and for several values of the statistical multiplier f . The results are presented in Fig. 3 for Ex1 (top) and Ex2 (bottom), respectively. On the left side we show the truth level cumulative composition w true (in black lines), 4 and compare it to the 95% confidence intervals for three choices of N L (note that we only show the cumulative fractions at the Z values included in w for each choice); on the right side we plot the differences of maximum ln L values (corresponding to best fit compositions) with respect to the value obtained with N L = 26 , ln L 26 , that is by considering all light primaries in the composition. \nIn both examples and for all choices of N L , cumulative composition confidence intervals shrink with increasing statistics ( f ), as expected. However, for too small N L (depending on composition and statistics) the 95% confidence intervals fail to cover the truth-level values w true thus indicating biased inferred compositions. \nAn extreme example is given by the N L = 4 case in Ex2. In the left plot, the red bars show an increasing precision on the cumulative fractions with increasing statistics. However, the intervals are systematically pulled away from the truth-level values indicated by the black line. A significant deviation is present already at f = 1 , that is assuming 10 3 measured events in total, and it quickly grows with larger statistics. The N L = 4 case thus leads to an underestimation of the heavier primary fractions and to biased results in general, even for event statistics comparable to PAOD. At the same time the relative goodness of fit degrades significantly with increasing f , as seen in the right plot. \nnot necessarily degrade significantly with increasing f . It can also remain comparable to the full N L = 26 case even as the composition results become biased. This can be most clearly seen for the N L = 8 case in Ex1, where the log likelihood difference remains almost constant even as the fraction of the heavier element in the composition is significantly underestimated. Interestingly, in Ex1, the same happens even for N L = 16 , where at f = 5 the fit \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFIG. 3: Cumulative composition confidence intervals and relative goodness of fit for Ex1 ( top ) and Ex2 ( bottom ) mock UHECR compositions. Left : the black line indicates the truth-level composition, while colored bars indicate 95% confidence level intervals for the respective primaries. Red, green, blue and gray bars refer to the inference with N L = 4 , 8 , 16 , 26 primaries respectively, while shades of the same color indicate the multiplier f , as specified by the legend on the right. Right : Difference of ln L values of best fit compositions, with N L = 4 , 8 , 12 , 16 , 20 , 24 , with respect to the case N L = 26 . See text for details. \n<!-- image --> \nstarts to underestimate the heavier primary fraction and thus actually yields biased composition results. Note that this choice of N L corresponds to d 0 ≃ 6 assuming 5 × 10 3 events per primary. Although the true composition in Ex1 might be considered extreme, it nonetheless demonstrates that existing mass composition studies based on the full PAO dataset which all consider N L < 16 , potentially yield 95% confidence intervals of primary fractions which do not cover their true values.', 'C. Application to Pierre Auger Open Data': 'Building upon the mock data examples studied in the previous Section, we next infer the composition of UHECR from PAOD in the chosen energy bin. We show our results for the cumulative fractions in Fig. 4, for different choices of N L and at various projected statistics factors f (actual PAOD statistics correspond to f = 1 ). Note again that for the moment we are restricting the procedure to lists of primaries up to iron, Z ≤ 26 (and so N H = 0 ). Contrary to mock simulated data, it is of course impossible to know a priori the true mass composition of the UHECR spectrum. However, we can use the results of the full N L = 26 fit, in particular its maximum \nFIG. 4: Cumulative fraction of primaries, with 95% CL, obtained with N L = 4 , 8 , 16 , 26 (red, green, blue, gray), using PAOD. The solid black line indicates the best fit obtained with N L = 26 . \n<!-- image --> \nlikelihood composition w best , as reference against which we compare other N L choices. We observe that a familiar pattern emerges: including only N L = 4 primaries, the fit tends to underestimate the cumulative fractions obtained with the full 26 primaries, especially the total fraction of nuclei with Z > 1 . Even more pronounced however is the underestimation of the 95% confidence intervals, very similar to what is observed in mock data, where it signals a biased composition. The N L = 16 fit results instead are already well compatible with the reference N L = 26 case including the total coverage of the 95% CL intervals, at least at PAOD statistics (for f = 1 ).', 'D. Bounds on Admixtures of Nuclei Beyond Iron': "Finally, we study the effect of including possible 'heavy' primaries, that is nuclei with Z > 26 . We have seen that, in order to have an unbiased estimation of the cumulative fraction of primaries in PAOD, we require at least N L = 16 . We can then use the same procedure to extend our lists to include approximately d 0 -equidistant primaries up to Z ≤ 94 , as shown in Table I. \nTo portray the effect of adding N H heavy primaries on", 'IV. CONCLUSIONS': "In studies of the UHECR mass spectrum, the vector of primary composition fractions, w , fundamentally drives the inference procedure: its size defines the dimensionality of the problem, and as a consequence the computa- \nthe likelihood, we first infer the composition after adding two consecutive d 0 -equidistant heavy primaries at a time, following the list for N L = 16 in Table I; the results are shown in the left-hand plot of Fig. 5. The bounds on cumulative fractions do not change significantly, as it can be seen by comparing the black bands ( N H = 0 ) to the colored ones; the inferred CL intervals increase only slightly, due to the addition of new free parameters with the same statistical sample size. As a direct result, we find that the fraction of primaries with Z > 26 in the energy interval E ∈ [0 . 65 , 1] EeV is within 95% CL bounded by \nw ( Z > 26 , E ∈ [0 . 65 , 1] EeV) ≤ 24% , (5) \nwhile for E ∈ [1 , 2] EeV we get \nw ( Z > 26 , E ∈ [1 , 2] EeV) ≤ 18% . (6) \nBy construction, the upper bounds on heavy primaries are decreasing monotonically with increasing Z . Consequently, bounds on so-called super heavy nuclei, such as uranium and plutonium can be much tighter than the above values. As an illustration, we thus also make a conservative estimation on fractions of such primaries as \nw ( Z > 81 , E ∈ [0 . 65 , 1] EeV) ≤ 10% , w ( Z > 85 , E ∈ [1 , 2] EeV) ≤ 6% , (7) \nboth at 95% CL. These values are obtained using the full d 0 -equidistant range of N L = 16 and N H = 10 primaries on PAOD (see Tables I and II for the lists). \nA different outcome would be obtained if instead we only added super heavy primaries, e.g. Z = 81 , 91 , to the original 16 primaries with Z ≤ 26 . This is shown in the right-hand plot of Fig. 5. In particular, we observe a substantial underestimation of the iron fraction compared to the results with d 0 -equidistant primaries: both for N H = 10 , as well as with N H = 2 , where the consecutive heavy primaries Z = 30 and Z = 34 are included in the fit. We thus conclude that at least when applied to PAOD, the method of including consecutive d 0 -equidistant primaries with Z > 26 yields reliable upper bounds on cumulative fractions of heavy elements in UHECRs even when the full range of heavy nuclei is not included in the fit. Conversely, including only ultraheavy primaries in the fit in addition to Z ≤ 26 can lead to biased composition results. \ntional power required to address it; the specific primaries that form w potentially impose a strong prior on the inferred composition, which in turn can lead to biases in the fit results. It is thus important for the experimental collaborations determining the mass spectrum to base their procedures upon a solid and quantitative framework for \n<!-- image --> \nFIG. 5: Cumulative fraction of primaries in PAOD at 95% CL. Right : Results with N L = 16 (black), and adding N H = 2 , 4 , 6 , 8 , 10 (red,yellow,dark blue, light blue, cyan) sequential d 0 -equidistant heavy primaries. Left : Results with N L = 16 and N H = 2 ultra-heavy primaries ( Z = 81 and Z = 91 ) (black), and N H = 10 d 0 -equidistant heavy primaries (gray). See text for details. \n<!-- image --> \nselecting optimal w . \nIn order to provide such framework, we can leverage the discriminating power of the X max distributions, to define a measure of distance between primaries in the space of X max moments. By construction, the distance gives the statistical significance to distinguish between two primaries in a composition, as function of the statistical sample size. In other words, for normally distributed moments, this distance is equivalent to the significance, in units of σ , that the two sets of moments are distinct from each other. Furthermore, at least for the set of simulated primaries, this distance is well approximated by including only one (i.e. the first) moment. Consequently, we can build lists of approximately equidistant primaries for the vector w ; for convenience, we separate these primaries in 'light', with Z ≤ 26 , and 'heavy', with Z > 26 , and indicate their numbers in the lists as N L and N H . Taking 10 3 events per primary as an example relevant for the statistics available in the PAOD, a minimum distance of d 0 ∼ 3 between primaries implies w to be at least of length N L = 16 . Given the typical scaling with the statistical sample size, requiring the same d 0 with the full PA statistics would lead to even larger dimensionality for w . \nWe first tested this framework on two sets of simulated mock compositions. By fixing the underlying true w , we can observe the effect of imposing different N L on the inferred composition. The examples demonstrate that too low values of N L for given sample size (or equivalently too high d 0 ) in general lead to biased composition fraction estimates with underestimated uncertainties, see Fig. 3. As noted previously [5, 25], the relative goodness of fit, computed as the maximum value of the log-likelihood, in some cases tracks this increasing discrepancy of low N L \nresults with respect to sufficiently large N L values. However we find that this behavior is not universal and that in general a high goodness of fit score does not guarantee unbiased composition results. \nA similar pattern is observed when applying the same strategy to PAOD, shown in Fig. 4; here the N L = 4 best fits seem to underestimate the fraction of primaries other than protons, with respect to the N L = 16 or N L = 26 results. The effect is expected to be statistically significant for the existing full PA dataset. We thus recommend a minimum of N L = 16 w components for the inference of the UHECR mass composition, assuming that no elements heavier than iron are present. \nBy lifting the assumption that Z ≤ 26 , we can repeat the procedure including so-called 'heavy' primaries. We explored possible ways to add N H heavy primaries to w . The results shown in Fig. 5 indicate that adding very distant primaries; in the example presented, the introduction of only Z = 91 and Z = 94 in addition to Z ≤ 26 strongly biases the fit. Fortunately, the bias can be reliably avoided by instead including sequential equidistant heavier primaries starting from a given N L set. \nAt cosmic ray energies considered in this work, no heavy elements are actually expected, as the data favors a mixture of protons and lighter nuclei. Using PAOD we nonetheless put first observational bounds on the presence of primaries heavier than iron, as well as more specifically on elements heavier than uranium and plutonium in UHECRs in the energy range E ∈ [0 . 65 , 2] EeV. The same analysis cannot be carried out at higher energies, as the PAOD does not contain a meaningful number of events above ∼ 2 -5 EeV. \nOur work could be extended in several directions. Besides applying our methodology to the full PA (and Tele- \nscope Array) datasets, the determination of an optimal composition vector w for a given CR dataset of unknown composition remains an open problem. We have argued that, while necessarily conservative, interpreting d 0 values computed with individual primary statistics comparable to the full dataset as resolution significances (measured in normal distribution sigmas) gives meaningful benchmark values to determine both the size and the relevant components of w . However, the potential reduction of computational cost as well as inferred composition \n- [1] Pierre Auger, A. Aab et al., Nucl. Instrum. Meth. A 798 , 172 (2015), 1502.01323.\n- [2] H. Kawai et al., Nuclear Physics B - Proceedings Supplements 175-176 , 221 (2008), Proceedings of the XIV International Symposium on Very High Energy Cosmic Ray Interactions.\n- [3] Pierre Auger, A. Aab et al., Phys. Rev. D 90 , 122006 (2014), 1409.5083.\n- [4] P. Lipari, Phys. Rev. D 103 , 103009 (2021), 2012.06861.\n- [5] N. Arsene and O. Sima, Eur. Phys. J. C 80 , 48 (2020), 2001.02667.\n- [6] Pierre Auger, A. Aab et al., Phys. Rev. D 96 , 122003 (2017), 1710.07249.\n- [7] B. D. Metzger, D. Giannios, and S. Horiuchi, Mon. Not. Roy. Astron. Soc. 415 , 2495 (2011), 1101.4019.\n- [8] L. N. Epele and E. Roulet, JHEP 10 , 009 (1998), astroph/9808104.\n- [9] G. Bertone, C. Isola, M. Lemoine, and G. Sigl, Phys. Rev. D 66 , 103003 (2002), astro-ph/0209192.\n- [10] D. Allard, Astropart. Phys. 39-40 , 33 (2012), 1111.3290.\n- [11] B. T. Zhang, K. Murase, N. Ekanger, M. Bhattacharya, and S. Horiuchi, (2024), 2405.17409.\n- [12] T. K. Gaisser, T. Stanev, and S. Tilav, Front. Phys. (Beijing) 8 , 748 (2013), 1303.3565.\n- [13] X.-J. Lv et al., (2024), 2403.11832.\n- [14] B. Bortolato, J. F. Kamenik, and M. Tammaro, Phys. Rev. D 108 , 022004 (2023), 2212.04760.\n- [15] Pierre Auger, E. W. Mayotte et al.\n- [16] T. P. A. Collaboration, Pierre auger observatory 2021 open data, 2021.\n- [17] D. Heck, J. Knapp, J. N. Capdevielle, G. Schatz, and T. Thouw, CORSIKA: a Monte Carlo code to simulate extensive air showers. (, 1998).\n- [18] T. K. Gaisser and A. M. Hillas, Reliability of the Method of Constant Intensity Cuts for Reconstructing the Average Development of Vertical Showers, in International Cosmic Ray Conference, , International Cosmic Ray Conference Vol. 8, p. 353, 1977.\n- [19] Pierre Auger, A. Aab et al., Phys. Rev. D 90 , 122005 (2014), 1409.4809.\n- [20] B. Efron and R. J. Tibshirani, An Introduction to the Bootstrap, Monographs on Statistics and Applied Probability No. 57 (Chapman & Hall/CRC, Boca Raton, Florida, USA, 1993).\n- [21] Pierre Auger, J. Stasielak, AugerPrime -The upgrade of the Pierre Auger Observatory, in 9th International Conference on New Frontiers in Physics, \nuncertainties due to a lower number of free parameters when working with a shorter composition vector w motivate further optimization, which we leave for future work.", 'ACKNOWLEDGMENTS': 'B.B. and J.F.K. acknowledge the financial support from the Slovenian Research Agency (grant No. J1-3013 and research core funding No. P1-0035). \n- 2021, 2110.09487.\n- [22] B. Bortolato, J. F. Kamenik, and M. Tammaro, Phys. Rev. D 109 , 043023 (2024), 2304.11197.\n- [23] J. Skilling, AIP Conference Proceedings 735 , 395 (2004), https://aip.scitation.org/doi/pdf/10.1063/1.1835238.\n- [24] J. Buchner, UltraNest github repository, "https:// github.com/JohannesBuchner/UltraNest , 2019.\n- [25] N. Arsene, (2021), 2109.03626.', 'Appendix A: Interpolation': 'In Fig. 6 we show the values of the mean (top) and variance (bottom) of the z 1 distributions as function of Z , for the [0 . 65 , 1] (left) and [1 , 2] (right) energy bins, respectively. The black dots, indicating the values extracted from simulations, can be fitted by two simple function of Z , \nh mean ( x ) = a m x + b m x 2 + c m + d m · log( x ) , h var ( x ) = a v x + b v √ x + c v + d v · log( x ) . (A1) \nAs a consequence, we can extract the values of z 1 , and thus of the distances, for all primaries between Z = 1 and Z = 94 , without the need to simulate the longitudinal profile of the respective showers. The results of the fit are shown as red lines in Fig. 6. \nFIG. 6: Fits of mean and variance of z 1 for [0 . 65 , 1] EeV (left) and [1 , 2] EeV (right) energy bins. The black dots represent simulated values, while the red line indicates the result of the fit. \n<!-- image --> \nAppendix B: [1,2] bin plots \nTABLE II: Same as Table I, for the energy bin [1 , 2] EeV. \n<!-- image --> \nFIG. 7: Same as Fig. 5, for the energy bin [1 , 2] EeV. \n<!-- image --> \nFIG. 8: Same as Fig. 4, for the energy bin [1 , 2] EeV. \n<!-- image -->'}

2024arXiv240913020M

The 21cm background is a promising probe of early star formation and black hole activity. While a slew of experiments on the ground seek to detect the 21cm monopole and spatial fluctuations on large sim 10 arcminute scales little work has been done on the prospects for detecting the 21cm dipole signal or its utility as a probe of early galaxies. Though an intrinsically weak signal relative to the monopole its direction is known well from the cosmic microwave background and widefield surveys plus as a relative measurement the dipole could help relax instrumental requirements. In order to understand the constraining power of the dipole in this work we perform parameter inference on mock datasets that include the dipole monopole or both signals. We find that while the monopole does provide the best constraints for a given integration time constraints from a dipole measurement are competitive and can in principle constrain the cosmic star formation rate density and efficiency of Xray photon production in early z sim 15 galaxies to better than a factor of sim 2. This result holds for most of the available prior volume which is set by constraints on galaxy luminosity functions the reionization history and upper limits from 21cm power spectrum experiments. We also find that predictions for the monopole from a dipole measurement are robust to different choices of signal model. As a result the 21cm dipole signal is a valuable target for future observations and offers a robust crosscheck on monopole measurements.

2024-09-01T00:00:00Z

['2024arXiv240913020M', '10.48550/arXiv.2409.13020', 'arXiv:2409.13020']

['Astrophysics - Cosmology and Nongalactic Astrophysics', 'Astrophysics - Astrophysics of Galaxies']

The relative constraining power of the highz 21cm dipole and monopole signals

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.13020.pdf

{'The relative constraining power of the high𝑧 21-cm dipole and monopole signals': '<!-- image --> \n1 \nCenter for Particle Cosmology, Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA \nJet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA 2 California Institute of Technology, 1200 E. California Boulevard, Pasadena, CA 91125, USA 3', 'ABSTRACT': 'The 21-cm background is a promising probe of early star formation and black hole activity. While a slew of experiments on the ground seek to detect the 21-cm monopole and spatial fluctuations on large ∼ 10 arcminute scales, little work has been done on the prospects for detecting the 21-cm dipole signal or its utility as a probe of early galaxies. Though an intrinsically weak signal relative to the monopole, its direction is known well from the cosmic microwave background and wide-field surveys, plus as a relative measurement the dipole could help relax instrumental requirements. In order to understand the constraining power of the dipole, in this work we perform parameter inference on mock datasets that include the dipole, monopole, or both signals. We find that while the monopole does provide the best constraints for a given integration time, constraints from a dipole measurement are competitive, and can in principle constrain the cosmic star formation rate density and efficiency of X-ray photon production in early 𝑧 ∼ 15 galaxies to better than a factor of ∼ 2. This result holds for most of the available prior volume, which is set by constraints on galaxy luminosity functions, the reionization history, and upper limits from 21-cm power spectrum experiments. We also find that predictions for the monopole from a dipole measurement are robust to different choices of signal model. As a result, the 21-cm dipole signal is a valuable target for future observations and offers a robust cross-check on monopole measurements. \nKeywords: galaxies: high-redshift; intergalactic medium; dark ages; reionization; first stars; diffuse radiation', '1. INTRODUCTION': "Observations of the cosmic 21-cm background from redshifts 𝑧 ≳ 6 are a powerful probe of the Epoch of Reionization (EoR) and cosmic dawn, when the first stars and galaxies began to transform their environments through ionization and X-ray heating (Madau et al. 1997; Furlanetto et al. 2006; Morales & Wyithe 2010; Pritchard & Loeb 2012). A fleet of arrays on the ground are currently seeking a detection of the 21-cm power spectrum during the EoR (LOFAR, MWA,HERA,GMRT,LWA;vanHaarlemetal.2013;Tingay et al. 2013; DeBoer et al. 2017; Paciga et al. 2013; Eastwood et al. 2019), while in parallel, a suite of more modest singleelement receivers (Bowman & Rogers 2010; Singh et al. 2017; de Lera Acedo 2019; Philip et al. 2019; Monsalve et al. 2023) are pursuing a detection of the sky-averaged 'global' 21-cm signal (Shaver et al. 1999), which traces the mean properties of the IGM at early times rather than spatial fluctuations. Both measurements encode a wealth of information on early star and black hole formation (e.g., Mesinger et al. 2013; Fialkov et al. 2014), as well as the intergalactic medium (IGM; e.g., Cohen et al. 2017; Mirocha et al. 2022; Ghara et al. 2024), and are thus a powerful complement to high𝑧 galaxy surveys \n(e.g., Mirocha et al. 2017; Park et al. 2019; Hutter et al. 2021; Ma et al. 2023). In the long term, higher order statistics and 21-cm maps promise to deliver even more detailed information about the cosmic dawn (e.g., Lidz et al. 2007; Watkinson et al. 2019; La Plante & Ntampaka 2019; Greig et al. 2022; Gillet et al. 2019; Hassan et al. 2020; Zhao et al. 2022). \nIn the last ∼ 10 years there has been tremendous progress in efforts to detect the 21-cm background. The EDGES collaboration reported the detection of a feature in the sky-averaged spectrum (Bowman et al. 2018), potentially consistent with expectations for the global 21-cm signal, but strong enough to drive a considerable flurry of activity in the modeling community. Meanwhile, upper limits from power spectrum measurements have continued to improve in the last few years (Mertens et al. 2020; Trott et al. 2020; Abdurashidova et al. 2022a), and have recently breached the parameter space of 'normal' models, i.e., those that do not invoke exotic mechanisms in order to amplify fluctuations beyond the theoretical maximum set by a Λ CDM cosmology. Arrays operating at higher frequencies, and so targeting neutral hydrogen in the post-reionization Universe, have reported auto-correlation and cross-correlation detections (Chang et al. 2010; Amiri et al. 2023; Paul et al. 2023), demonstrating both a matu- \ny in the calibration and analyses of low-frequency radio interferometers (see review by Liu & Shaw 2020), and detection proof-of-principle, albeit in a frequency regime where foregrounds are weaker than those relevant to EoR studies. \nDespite rapid experimental and theoretical progress focused on the 21-cm monopole and power spectrum, relatively little work has been done on the 21-cm dipole signal. As first discussed in Slosar (2017), the 21-cm signal should show a kinematic dipole spatial variation owing to our motion with respect to the frame of the emitting or absorbing hydrogen gas. Though ∼ 10 2 times weaker than the monopole, Slosar (2017) pointed out that the dipole may be an appealing target for near-future experiments for three main reasons: (i) the direction and amplitude of our motion are very accurately known from previous measurements (e.g., Kogut et al. 1993; Fixsen et al. 1996; Hinshaw et al. 2009; Planck Collaboration et al. 2014, 2020a), (ii) the dipole is a relative, rather than absolute, measurement, which provides resilience to certain systematic effects, and (iii) the dipole signal is to leading order equal to the derivative of the monopole, and so can be used to provide a prediction for monopole measurements and/or internal consistency check for an experiment targeting both the monopole and dipole (see also Deshpande 2018). \nThe intrinsic weakness of the dipole signal certainly poses a challenge, however, 21-cm experiments are not in general limited by statistical noise. For example, ∼ mK noise levels can be achieved in reasonable integration times ≲ 10 3 hours, which is more than enough to achieve strong detections of the monopole (e.g., Harker et al. 2012; Liu et al. 2013), and is comparable to the expected dipole amplitude. Ignatov et al. (2023) showed that the dipole could potentially be detected from the ground with pre-existing monopole experiments. However, observations from space are certainly ideal given the need for ∼ all sky coverage, the potential for Earth's ionosphere to induce spectral distortions of greater magnitude than the 21-cm monopole (Vedantham et al. 2014; Datta et al. 2014; Shen et al. 2021), and the potential for shielding from radio frequency interference on the lunar farside (hence the sustained interest in lunar observatories, e.g., Burns et al. 2012, 2017, 2019; Sathyanarayana Rao et al. 2023; Chen et al. 2019; Shi et al. 2022). \nOur focus in this paper is on what can be learned about the first stars and galaxies from a 21-cm dipole detection. Hotinli & Ahn (2024) recently investigated the value added to a monopole measurement from a dipole detection /one.sup , showing that the dipole can provide some modest improvement in constraints on astrophysical parameters, but can dramatically improve constraints on the foreground. Here, our approach \nis slightly different, but complementary, in that we focus on what can be learned from only a dipole detection vs. only a monopole detection, with and without priors from other astrophysical probes. Furthermore, we explore the degree to which reconstructions of the monopole from dipole measurements are model-dependent, and in so doing quantify the robustness of monopole/dipole consistency checks. Our forecasts are quite idealized in that we adopt simplistic galactic foreground models and pure radiometer noise appropriate for a given integration time, sky temperature, and bandwidth. This is to focus our attention on the information content of the dipole before considering real-world measurement challenges, given that this is a relatively unexplored topic. We defer a more detailed forecast to Anderson et al., in preparation. \nIn §2, we outline our approach to modeling the 21-cm monopole and dipole in the context of high𝑧 galaxy survey results, and present the basic expectations for the dipole in this framework. In §3, we present the results of our forecast, comparing dipole- and monopole-based constraints. We conclude in §4. We use cosmological parameters consistent with the recent Planck Collaboration et al. (2020b) constraints: Ω 𝑚 = 0 . 3156, Ω 𝑏 = 0 . 0491, ℎ = 0 . 6726, 𝑛 𝑠 = 0 . 9667, and 𝜎 8 = 0 . 8159.", '2.1. 21-cm signals': "To model the 21-cm monopole and dipole, we use the /a.pc/r.pc/e.pc/s.pc code /two.sup . /a.pc/r.pc/e.pc/s.pc starts from cosmological initial conditions after recombination (Lewis et al. 2000; Chluba & Thomas 2011), and treats the intergalactic medium as a two-phase medium, i.e., it evolves separately the volume-filling factor of ionized bubbles, 𝑄 , and the properties of the mostly-neutral 'bulk IGM' beyond. We summarize the pertinent details here and refer the interested reader to Mirocha (2014) for more information on the underlying algorithm. \nThe 21-cm dipole /three.sup is given by (Slosar 2017) \nΔ 𝑇 dip = GLYPH<18> 𝛿𝑇 𝑏 -𝑑𝛿𝑇 𝑏 𝑑𝜈 𝜈 GLYPH<19> 𝑣 𝑑 𝑐 cos 𝜃 (1) \nwhere 𝑣 𝑑 / 𝑐 ≃ 1 . 2 × 10 -3 is our velocity, cos 𝜃 is the angle relative to the dipole peak, and 𝛿𝑇 𝑏 is the usual expression for the 21-cm monopole (e.g., Furlanetto et al. 2006), \nΔ 𝑇 mon ≡ 𝛿𝑇 𝑏 ≃ 27 𝑥 H/i.pc GLYPH<18> 1 -𝑇 𝛾 𝑇 𝑆 GLYPH<19> GLYPH<18> 1 + 𝑧 10 GLYPH<19> 1 / 2 mK , (2) \n/two.sup https://github.com/mirochaj/ares; revision 8c8992c \nwhere 1 -𝑥 H/i.pc = 𝑄 + ( 1 -𝑄 ) 𝑥 𝑒 is the mean ionized fraction, broken into a fully-ionized phase with volume fraction 𝑄 and bulk IGM with electron fraction 𝑥 𝑒 , set by the ionized fraction of hydrogen as well as helium, 𝑇 𝛾 is the background temperature, assumed here to be the CMB temperature, and 𝑇 𝑆 is the spin temperature in the bulk IGM, \n𝑇 -1 𝑆 = 𝑇 -1 𝛾 + 𝑥 𝛼 𝑇 -1 𝛼 + 𝑥 𝑐 𝑇 -1 𝐾 1 + 𝑥 𝛼 + 𝑥 𝑐 . (3) \nWe make the usual assumption that the temperature of the UV radiation field is equivalent to the kinetic temperature, 𝑇 𝛼 ≈ 𝑇 𝐾 , take collisional coupling coefficients 𝑥 𝑐 from Zygelman (2005), and compute the Wouthuysen-Field coupling 𝑥 𝛼 following Furlanetto & Pritchard (2006) (see also,e.g., Hirata 2006; Chuzhoy & Shapiro 2006; Mittal & Kulkarni 2021). Note that 𝑥 𝛼 ∝ 𝐽 𝛼 , where 𝐽 𝛼 is the intensity of the Ly𝛼 background. \nOur model effectively neglects correlations in the density, ionized fraction, and spin temperature, i.e., we assume that the average brightness temperature is equal to the product of averages, rather than computing the product of the constituent quantities at the field level and then averaging. This is a common approximation in the global 21-cm literature, and was recently shown to be accurate at the ∼ 10% level (Schaeffer et al. 2024). \nWe employ three different parameterizations for the 21-cm signal in order to better illustrate the information content of the dipole relative to the monopole, and to test the modeldependence of dipole vs. monopole consistency checks: \n- 1. A phenomenological model in which 𝐽 𝛼 , 𝑇 𝑆 ( 𝑧 ) , and 𝑥 H /i.pc ( 𝑧 ) are all given by tanh functions (Harker et al. 2016). We refer to this as the phenomenological model or tanh model .\n- 2. A physically-motivated semi-empirical model, in which 𝑓 ∗ is parameterized as a function of halo mass (and optionally redshift) in order to better fit high𝑧 rest-ultraviolet luminosity functions (UVLFs; Mirocha et al. 2017). We refer to this as the double power-law or DPL 𝑓 ∗ model since we parameterize 𝑓 ∗ as a double power-law.\n- 3. An 'extended' DPL model ( DPLX ), that includes extra parameters in order to allow more flexible behaviour in the low-mass galaxy population, to be described momentarily. \nFor the DPL and DPLX models, we have \n𝑓 ∗ = 𝑓 ∗ , 10 C 10 GLYPH<16> 𝑀 ℎ 𝑀 p GLYPH<17> -𝛼 ∗ , lo + GLYPH<16> 𝑀 ℎ 𝑀 p GLYPH<17> -𝛼 ∗ , hi . (4) \nwhere we have normalized 𝑓 ∗ to halos with 𝑀 ℎ = 10 10 𝑀 ⊙ via the parameter 𝑓 ∗ , 10, hence the re-normalization factor C 10 \nin our formula. We denote the peak of the double power-law 𝑀 𝑝 , while the low and high-mass slopes are denoted 𝛼 ∗ , lo and 𝛼 ∗ , hi, respectively. The star formation efficiency thus requires four parameters on its own ( 𝑓 ∗ , 10, 𝑀 peak, 𝛼 ∗ , lo, 𝛼 ∗ , hi). Star formation is allowed to proceed in halos down to a truncation mass, implemented as a smooth exponential decline in 𝑓 ∗ at 𝑀 ℎ ≲ 𝑀 turn, \n𝑇 ( 𝑀 ℎ ) ≡ GLYPH<26> 1 -exp GLYPH<20> -GLYPH<18> 𝑀 ℎ 𝑀 turn GLYPH<19> 𝑟 turn GLYPH<21> GLYPH<27> . (5) \nwhere 𝑟 turn controls the sharpness of the turn-over. This is the approach taken in 21/c.pc/m.pc/f.pc/a.pc/s.pc/t.pc as well (with 𝑟 turn = 1; Park et al. 2019). \nWe also vary the escape fraction of ionizing photons, 𝑓 esc, and the X-ray luminosity - SFR relation, 𝐿 𝑋 / SFR, as well as two parameters that govern the low-mass behaviour of 𝑓 ∗ (see below), resulting in an eight parameter DPL model. In this work, for simplicity we do not allow for redshift evolution in any of these parameters. The model in its current form is quite flexible already, particularly with the addition of additional low-mass extensions described below, but this could be an interesting avenue for future study. \nThe DPL model space is fairly compact relative to older ' 𝑓 coll models' (see, e.g., Barkana & Loeb 2005; Furlanetto 2006) given that the UVLFs narrow the range of allowed cosmic star formation histories. Though it could not fit an arbitrary 21-cm signal, it establishes a well-motivated target and null hypothesis to test with observations. Rejection of this null hypothesis would immediately indicate the presence of 'new' source populations (as the EDGES signal does; Mirocha & Furlanetto 2019; Schauer et al. 2019; Mebane et al. 2020; Chatterjee et al. 2020), e.g., PopIII stars, proto-quasars, a departure from the scaling relations that star-forming galaxies appear to follow, or deviations in the abundance of DM halos themselves. For example, while a turn-over in the UVLF presumably encodes the physics of feedback, an upturn could mimic bursty star formation models, or PopIII scenarios which result in elevated star formation efficiencies in galaxies occupying low-mass halos. This is known to induce a more gradual descent into absorption in the monopole signal (Mirocha et al. 2018; Ahn & Shapiro 2021; Hegde & Furlanetto 2023), and as a result could make the dipole harder to detect. In any case, predictions for PopIII star formation span a broad range of possibilities (e.g., Jaacks et al. 2018; Mebane et al. 2018; Gessey-Jones et al. 2022; Muñoz et al. 2022; Feathers et al. 2024; Ventura et al. 2024), so a flexible approach is warranted /four.sup . \nThe point of the DPLX model is to accommodate such scenarios without invoking a specific physical model, by allowing an optional phenomenological extension to 𝑓 ∗ at faint UV magnitudes, i.e., \n𝑓 ∗ → 𝑓 ∗ 𝑆 ( 𝑀 ℎ ) . (6) \nWe follow Schneider et al. (2021), who defined this 'small scale' function as \n𝑆 ( 𝑀 ℎ ) ≡ GLYPH<20> 1 + GLYPH<18> 𝑀 𝑐 𝑀 ℎ GLYPH<19> 𝛾 1 GLYPH<21> 𝛾 2 (7) \nwhich allows a suppression or boost at 𝑀 ℎ ≲ 𝑀 𝑐 . Note that at corners of parameter space corresponding to a boost, there is a possibility that 𝑓 ∗ diverges as 𝑀 ℎ → 0. We impose 𝑓 ∗ ≤ 1, however, in practice our truncation function 𝑇 ( 𝑀 ℎ ) takes over and prevents this from occurring. \nNote that our fiducial scenario (model A) assumes 𝑆 = 1, and so we will presumably achieve better constraints on 𝑀 turn if one assumes 𝑆 = 1 in the fitting as well (i.e., if one uses the DPL instead of the DPLX model). However, focusing only on 𝑀 turn in a fit with 𝑆 allowed to vary could be misleading, since new degeneracies with 𝑀 𝑐 , 𝛾 1, and 𝛾 2 could achieve the same turn-over properties in, e.g., UV magnitude, but very different values of 𝑀 turn. As a result, in §3 we will focus on the recovery of 𝑀 turn and 𝑟 turn for the DPL model, but for the DPLX model we will focus on the recovery of derived quantities like the UVLF turn-over (in 𝑀 UV and 𝜙 [ 𝑀 UV ]) . \nFinally, note that for the DPL and DPLX models, we do not separately vary the strength of the non-ionizing UV emission via 𝑁 LW. Instead, the relative strength of ionizing and non-ionizing emission is determined self-consistently by the stellar population synthesis (SPS) models we employ, /b.pc/p.pc/a.pc/s.pc/s.pc (Eldridge & Stanway 2009; Eldridge et al. 2017), for a metallicity of 𝑍 = 0 . 004. Note that the stellar population assumptions will be degenerate with 𝑓 ∗ , 10, so we do not vary them separately - constraints on 𝑓 ∗ , 10 should thus be interpreted with caution /five.sup . \nExample realizations of the DPL/DPLX model are shown in Fig. 1. Each case matches 𝑧 ≳ 6 UVLFs by construction but has different behaviour at the faint end of the UVLF. The faint-end differences drives qualitatively different behaviour in the 21-cm signals, all of which are consistent with preexisting constraints, at least roughly. In each case, we effectively assume high-mass X-ray binaries are the dominant heat sources, because we adopt a multi-colour disk spectrum for 10 𝑀 ⊙ black holes (Mitsuda et al. 1984) which results in a relatively hard X-ray spectrum and so inefficient heating \nTable 1. Models explored in this work \nN/o.pc/t.pc/e.pc-Key parameters for DPL/DPLX models shown in Fig. 1. All models adopt the same double power law parameters, 𝑓 ∗ , 10 = 0 . 02, 𝑀 peak = 2 × 10 11 𝑀 ⊙ , 𝛼 ∗ , lo = 0 . 49, 𝛼 ∗ , hi = -0 . 61, and the same X-ray spectrum: a multi-colour disk model for a 10 𝑀 ⊙ black hole with intrinsic hydrogen absorbing column of log 10 𝑁 HI / cm -2 = 21. Variations in 𝑀 turn, 𝑟 turn, 𝑀 crit, and the 𝛾 parameters are chosen to span a wide range of qualitatively different possibilities for the UVLF faint-end. Note that models A, E, and G take 𝑆 = 𝑇 = 1, hence the 'not applicable' labels in the final three rows. Models with a faint-end 'up-turn' are meant to mimic scenarios with elevated star formation efficiencies or burstiness at low mass. \n(Mirocha 2014), similar to the effects of a Cygnux X-1 template (Fialkov et al. 2014), which drives later features than earlier generations of models. We vary the relationship between star formation and X-ray photon production, starting with a fiducial value of 𝐿 𝑋 / SFR of 2 . 6 × 10 39 erg s -1 ( M ⊙ / yr ) -1 in the 0.5-8 keV band, which is representative of local starforming galaxies (Mineo et al. 2012), but include models with values as high as 10 42 as well. Note that the default value in, e.g., 21/c.pc/m.pc/f.pc/a.pc/s.pc/t.pc is higher, ∼ 10 40 . 5 , as is expected of low-metallicity systems (e.g., Fragos et al. 2013; Brorby et al. 2016; Lehmer et al. 2022), and so generally results in a weaker absorption signal than our fiducial model (model A). Finally, we redden the intrinsic spectrum of galaxies with an optical depth determined by a characteristic column density log 10 𝑁 H/i.pc = 21, which is consistent with simulations (Das et al. 2017). In principle 21-cm fluctuations can break this degeneracy (see, e.g., Pacucci et al. 2014), but for monopole and/or dipole measurements this will be much more difficult. As a result, we keep log 10 𝑁 H/i.pc fixed for simplicity.", '2.2. Prior volume': 'The models shown in Fig. 1 agree with high𝑧 UVLFs by construction, with values for remaining free parameters like 𝑓 esc tuned to provide good agreement with 𝜏 𝑒 from Planck and 𝐿 𝑋 / SFR set to values that (mostly) jive with the latest 21-cm power spectrum limits from HERA (HERA Collaboration et al. 2023). Here, we take a more thorough look at the prior volume for 21-cm monopole and dipole measurements. \nTo do this, we vary all 11 of the DPLX model\'s free parameters subjected to the following constraints: (i) the CMB optical depth 𝜏 = 0 . 055 ± 0 . 009 from Planck Collaboration et al. (2020b), (ii) Constraints on the end of reionization from \nFigure 1. Models explored in this work. Each model for the 21-cm dipole (top left) and monopole (bottom left) are anchored to UVLFs (right) from Bouwens et al. (2015). Black solid curves indicate our fiducial scenario ("model A"). Additional curves indicate scenarios indistinguishable via UVLFs (right) and CMB optical depth constraints from Planck Collaboration et al. (2020b) (inset, lower right), achieved by changing the behaviour at the faint end. Model C (dashed red) is in mild tension with the latest 21-cm power spectrum limits from HERA (HERA Collaboration et al. 2023) (inset, lower left), while model G (dotted red) is strongly disfavoured by HERA. Note that the 𝑀 ℎ axis along the top of the right panel is model-dependent, and corresponds to model A only. \n<!-- image --> \nthe Ly 𝛼 forest - we conservatively assume that reionization must be complete ( 𝑥 H /i.pc ≤ 1%) by 𝑧 = 5 . 3, in line with recent measurements (e.g., Bosman et al. 2022), and (iii) UVLFs from Bouwens et al. (2015) at 𝑧 ∼ 6 -8. The last is particularly important to some of our results - given that the 21-cm monopole and dipole probe only the volume-averaged emissivity of galaxies, one would not expect to be able to constrain the shape of 𝑓 ∗ in detail without including UVLFs in the likelihood (see, e.g., Fig. 6 in Dorigo Jones et al. (2023)). Our approach to reionization priors is conservative in that we neglect pre-existing constraints on the detailed redshift evolution of the neutral fraction from, e.g., Ly𝛼 emitters (e.g. Mason et al. 2018), or quasar damping wings (e.g., Davies et al. 2018; Greig et al. 2019). \nWe will also compare to lower limits on the spin temperature of the 𝑧 ∼ 8 IGM from HERA (HERA Collaboration et al. 2023) and limits on the unresolved fraction of the cosmic X-ray background (Hickox & Markevitch 2006; Lehmer et al. 2012), but we do not actually impose either of the latter two constraints as priors here or in subsequent forecasting. We also neglect all pre-existing constraints on the global 21-cm signal (see, e.g., Monsalve et al. 2017; Singh et al. 2017, for examples) in order to remain agnostic about possibilities for the dipole. Joint constraints from monopole and power spectrum measurements are quickly becoming in- \nting (Pochinda et al. 2023; Bevins et al. 2024), though for simplicity we defer such considerations for the dipole to future work. \nDespite the priors listed above, there is still a considerable range of possibilities, particularly for the faint-end of the galaxy UVLF. The wide range of viable possibilities in the UVLF imply a wide range of possibilities for the cosmic SFRD, which provides a nice way to compress the many parameters controlling 𝑓 ∗ into a single, albeit redshiftdependent, quantity. As a result, in Fig. 2, we show how the SFRD 𝐿 𝑋 / SFR space is constrained by current observations. This plot is constructed from the results of a prior-only fit, and is composed of ∼ 570 , 000 models. The cross-hatched region along the top is disfavored by UVLFs and reionization constraints, as models in this space produce too many bright galaxies relative to measured UVLFs and/or a reionization epoch that ends too quickly to remain in agreement with CMB 𝜏 𝑒 or Ly𝛼 forest constraints on the end of reionization. In the top-right corner of this space, we see that the disfavoured region grows slightly for very large 𝐿 𝑋 / SFR values, indicating that it is X-ray sources that are responsible for finishing reionization too quickly. The cross-hatched region along the bottom is disfavoured by the same set of observations, but in this case because reionization occurs too late and/or there are too few high𝑧 galaxies relative to UVLFs. \nFigure 2. Condensed discovery space for 21-cm monopole and dipole measurements. For illustrative purposes, here we focus on a 2-D slice of parameter space in which 21-cm measurements can provide powerful constraints on the first galaxies. As a barometer for high𝑧 global star formation activity, we focus on the 𝑧 = 15 cosmic SFRD ( 𝑦 axis), while the normalization of the 𝐿 𝑋 / SFR relation is shown on the 𝑥 -axis as a black hole activity indicator. Reionization constraints (Planck Collaboration et al. 2020b; Bosman et al. 2022) and UVLFs (Bouwens et al. 2015) disfavour the very top and bottom regions of this diagram (cross-hatching fills the 2 and 3 𝜎 disfavoured regions). Colored contours along the right hand side represent constraints on the unresolved fraction of the cosmic X-ray background (in the soft band from Chandra ; Lehmer et al. 2012), which disfavor very large values of 𝐿 𝑋 / SFR. Indicated along the left is a region of parameter space in which models produce effectively no heating of the 𝑧 ∼ 8 IGM, a scenario which is now disfavoured by 21-cm power spectrum experiments (Abdurashidova et al. 2022a; Trott et al. 2020). Note that the X-ray and 21-cm contours are not confidence intervals; instead, they enclose regions in which all models violate the given constraint (dense cross-hatching) and regions where only some models remain consistent with these constraints (sparser cross-hatching). Finally, we also draw contours at fixed 𝑧 = 8 spin and kinetic temperatures of 3, 30, and 300 K (mean value in each pixel) as indicated along the top of the figure, as a rough guide for how future constraints map to this space. Also apparent are the regimes in the lower left and right corners where the spin temperature is not fully coupled to the kinetic temperature. At 𝑧 ∼ 10, the lower boundary of these \'weak coupling\' regions shift upward by roughly an order of magnitude. We use 0.2 dex wide pixels in each dimension to determine the number of models in each ( ∼ 50 -100 on average), and smooth with a gaussian kernel for contours. Note that the boundaries of these regions are subject to assumptions about the SEDs of galaxies, see text for details. \n<!-- image --> \nWe also show the region of parameter space disfavoured by constraints on the soft X-ray background (e.g. Hickox & Markevitch 2006; Lehmer et al. 2012), which unsurprisingly corresponds to large values of 𝐿 𝑋 /SFR. We adopt a 0.5-2 keV X-ray background intensity of 1 . 96 × 10 -12 erg s -1 cm -2 deg 2 , which is the total intensity 8 . 15 × 10 -12 erg s -1 cm -2 deg 2 from Lehmer et al. (2012) times their best-fit unresolved fraction of 24%. The denser hatching indicates the region of parameter space in which all of our models violate this constraint, whereas the less dense hatching is more generous, including regions of parameter space in which some - but not all - models are disfavoured by the X-ray background. Note that analogous constraints on the diffuse radio background (Fixsen et al. 2011; Dowell & Taylor 2018) provide a useful diagnostic for excess radio background models (see, e.g. Ewall-Wice et al. 2018; Fialkov &Barkana 2019), though in this work we take 𝑇 𝑅 = 𝑇 CMB. \nA similar cross-hatched region borders the left edge of the plot, and indicates scenarios in which the 𝑧 ∼ 8 IGM is cold during reionization. By \'cold,\' here we mean that the bulk IGM (the portion of the IGM that is mostly neutral) is completely unheated, resulting in spin temperatures equal to the theoretical minimum in Λ CDM, 𝑇 𝑆 ≃ 1 . 8Kat 𝑧 = 8. Such scenarios are now disfavoured, so we also include contours corresponding to mean IGM spin temperatures of 3, 10, 100, and 300 K to roughly indicate how current and future 21-cm power spectrum limits map to this parameter space. Note that there is a gap in the lower left corner of the plot, where the SFRD is low but the IGM is apparently not maximally cold. This region is populated by models that are maximally cold, but lack a sufficiently strong Ly𝛼 background to couple 𝑇 𝑆 → 𝑇 𝐾 (compare to blue contours of same linestyle), and so cannot yet be fully ruled out by 21-cm measurements, at least when limiting analyses to this single band. Most of this region is disfavoured at ∼ 2 𝜎 by our UVLF and EoR priors, but should be scrutinized more carefully in future multi-epoch 21-cm analyses. At 𝑧 ∼ 10, this \'weak coupling\' region is roughly an order of magnitude larger in the SFRD dimension. \nAs is always the case for 2D representations of > 2D parameter spaces, Fig. 2 does not tell the whole story. For example, our assumptions for the X-ray SED of sources kept fixed here - surely affects the precise location of the red and orange cross-hatched regions. A softer SED achieved, e.g., by decreasing log 10 𝑁 H/i.pc , would result in more efficient heating per unit star formation, and so shrink the \'maximally cold IGM\' region. Similarly, a harder X-ray spectrum would cause more tension with the X-ray background, and allow the orange \'strong X-ray background\' region to grow. For the X-ray background, we have also used the flux generated by our model for all sources at redshifts higher than 𝑧 min = 6. Of course in reality, more aggressive removal of sources can further reduce the unresolved fraction, perhaps from ∼ 24% \nFigure 3. Expected monopole and dipole characteristics across the DPL model\'s prior volume. Here, as in Fig. 2, the black cross-hatched region indicates parts of parameter space disfavoured at ≥ 3 𝜎 by UVLFs, 𝜏 𝑒 , and 𝑧 rei . We separate models for which the monopole peaks in absorption vs. emission ( ≲ 10% of prior volume, hence the secondary mode with 𝑇 max , mon > 0 in each panel (dashed contours). Note that the dipole amplitude is measured peak to trough, whereas the quoted monopole amplitude is the amplitude of the absorption minimum or emission maximum, whichever is stronger. The dark gray region in the lower left corner indicates the brightness temperature as a function of frequency in Λ CDM with full Wouthuysen-Field coupling and no X-ray heating. Regions disfavoured by 21-cm power spectrum and X-ray background constraints are not shown here, as they \'pile-up\' along the edge of the available parameter space where the 21-cm monopole and dipole are strongest. \n<!-- image --> \n(Lehmer et al. 2012) to ∼ 3% (Cappelluti et al. 2012), which would allow the orange region to grow. A self-consistent treatment of the 21-cm background and X-ray number counts would provide a more careful accounting of the unresolved fraction and 𝑧 min, though is beyond of the scope of this work. \nNext, in Fig. 3, we show how our priors map to the space of the frequency and peak amplitude of the monopole and dipole. Each panel shows a different cut through the joint distribution of the peak monopole frequency 𝜈 max , mon, peak monopole amplitude 𝑇 max , mon, and peak-to-trough dipole amplitude, 𝑇 max , dip. Clearly, stronger monopole absorption signals result in stronger dipole signals (lower right panel), which generally correspond to late features 𝜈 ≳ 110 MHz (left column). These are precisely the kinds of scenarios that 21-cm power spectrum limits are beginning to rule out (Abdurashidova et al. 2022b; HERA Collaboration et al. 2023; Trott et al. 2020). Models disfavoured by 21-cm and X-ray \nbackground constraints are not shown here, as they \'pile up\' at the boundaries of the prior volume. For example, because strong X-ray backgrounds drive 𝑇 𝑆 ≫ 𝑇 CMB, the models lining the right edge of the 𝐿 𝑋 / SFR parameter space all inhabit a narrow sliver of ( 𝜈 max , mon, 𝑇 max , mon) space. Similarly, cold IGM models pile up at the edge of the gray shaded boundary in the lower left corner of Fig. 3. \nNow, we proceed to the details of our forecasting approach, including our treatment of the foreground, mock experimental uncertainties, and sampling of the parameter space.', '2.3. Foregrounds': "We take a very simple approach to foregrounds in this work in order to establish the best-case scenario for dipole inference, in which measurements are limited by statistical (radiometer) noise only /six.sup , \n𝜎 𝜈 ∝ 𝑇 sky √ 𝑡 int Δ 𝜈 (8) \nwhere 𝑡 int is the integration time, Δ 𝜈 the channel width, and 𝑇 sky is the sky temperature as a function of frequency. We assume a simple power-law foreground spectrum, \n𝑇 sky = 𝑇 75 K GLYPH<16> 𝜈 75 MHz GLYPH<17> 𝛽 (9) \nwith 𝑇 75 = 1700 K and 𝛽 = -2 . 59, consistent with the latest measurements directed away from the galactic plane (Mozdzen et al. 2019) and extrapolations of maps at higher frequencies (e.g. Haslam et al. 1982; Guzmán et al. 2011). \nFor a canonical 1000 hour integration, typical for monopole forecasts, these choices yield thermal noise levels of ∼ 1 mK and below (depending on frequency). Here, we will consider a 1000 hour integration with a spectral resolution of 1 MHz, as shown in Fig. 4, which results in thermal noise levels of ≃ 0 . 4 mK at 100 MHz. For model A, the cumulative signalto-noise ratio over the 60-180 MHz band is ∼ 19 for the dipole. This yields an effectively perfect measurement of the monopole. Using pure radiometer noise like this for a dipole measurement effectively assumes an idealized scan strategy, in which one alternates between measurements of the dipole maximum and minimum on the sky. One might expect this approach to yield a factor of two boost in the dipole amplitude relative to what we plotted in Fig. 1, which took cos 𝜃 = 1 (i.e., assumed the spectrum at the exact position of the dipole maximum). However, an additional factor of two boost in \nFigure 4. Example mocks for input model A. In the top row, we show the noise level for 1000 and 10 4 hour integrations under the assumption of radiometer noise only (solid, dashed black) compared to a simulated noise realization from Anderson et al., in prep. (dotted red; see text for details). The 21-cm dipole and monopole are shown in the middle and bottom row, respectively. Uncertainties on the dipole are much more significant relative to the signal amplitude, though the integrated signal to noise is still significant. Note that for visual clarity, we plot points every 2 MHz, though the errorbars themselves are computed assuming channels 1 MHz wide. \n<!-- image --> \nthe noise on the difference spectrum cancels, resulting in no net change in the signal amplitude or its uncertainties. In practice, the non-negligible width of a realistic beam pattern would also dilute the dipole signal amplitude, i.e., we cannot just difference the signal at the dipole's exact maximum and minimum on the sky. \nGiven the many potential non-idealities involved in a realistic dipole experiment, we show for reference in Fig. 4 an example noise realization from Anderson et al., in prep., which includes a realistic scan strategy, beam power pattern, calibration scheme, and signal loss associated with the removal of the foreground, which is assumed to be constrained empirically (as in Switzer & Liu 2014). This curve assumed 8000hoursofintegration, resulting in 𝜎 ( 𝜈 ) values resembling \nour highly-idealized 1000 hour integration. We defer an analysis of the impact of signal loss associated with foreground removal to Anderson et al., in prep.", '2.4. MCMC sampling': 'In what follows, all mock parameter constraints were obtained via MCMC sampling with /e.pc/m.pc/c.pc/e.pc/e.pc (Foreman-Mackey et al. 2013). \nOur likelihood function is simply \nlog L ∝ ∑︁ 𝑖 " -1 2 ( 𝑚 𝑖 ( Θ ) -𝑑 𝑖 ) 2 𝜎 2 𝑖 # (10) \nwhere the index 𝑖 represents spectral channels, the data vector 𝑑 𝑖 then represents the dipole and/or monopole measurement in channel 𝑖 , and 𝜎 𝑖 is the uncertainty of data point 𝑖 . The quantity 𝑚 𝑖 is our model prediction for the dipole and/or monopole given parameters Θ (not to be confused with dipole direction 𝜃 ). Our priors are enumerated in §2.2.', '3. RESULTS': "We begin by examining the potential of the dipole as a consistency check on monopole measurements. To do this, we fit all three of our astrophysical models - the tanh, DPL, and DPLX models - to the same mock dipole signal (DPL model; A), to determine the degree to which the reconstructed monopole varies from model to model. The results of this exercise are shown in Fig. 5. Columns correspond to the three different signal models, with the fitted dipole results shown in the top row and the reconstructed monopole in the bottom row. At a glance, it is clear that each model can provide a satisfactory fit to the dipole, despite differences in the underlying model parameterizations /seven.sup . Similarly, the reconstructed monopole contains the true input mock at the 68% confidence level. The detailed shape of the reconstructed monopole does vary from model to model, particularly for the tanh model. This is because the tanh model is phenomenological, with the Lyman𝛼 background, thermal history, and ionization history treated completely independently. As a result, one can (for example) recover the absorption signal well but obtain a broader range of possibilities at higher frequencies during reionization. In contrast, the more physical models generally yield a tight prediction for the monopole at high frequencies, because any astrophysical scenario that has enough heating to drive an absorption peak has enough star formation to finish reionization by 𝑧 ∼ 6. \nNext, we proceed to forecast constraints on astrophysical parameters, once again focusing on the recovery of model A with the DPL model (i.e., we use the same model to fit the signal as was used to generate the mock). Starting with the 𝑓 ∗ parameters, in Fig. 6 we see that the monopole (red) provides a non-trivial improvement in the constraints on all four 𝑓 ∗ parameters relative to the prior (dashed black). The dipole constraints (blue) also exhibit improvement over the prior, though to a lesser degree than the monopole constraints. The joint constraints that incorporate both the monopole and dipole in the likelihood are nearly indistinguishable from the monopole constraints. \nIn Fig. 7, we turn our attention to the remaining parameters, which govern the faint end of the UVLF and efficiency of ionizing and X-ray photon production. The prior volume here is broad, given that many combinations of, e.g., 𝑀 turn and 𝑓 esc can satisfy pre-existing constraints on reionization. Both the monopole and dipole dramatically reduce the landscape of possibilities. The monopole still outperforms the dipole, but the dipole provides powerful constraints on three of these four key parameters all on its own. The most noticeable shortcoming is in the 𝑟 turn dimension, which controls the 'sharpness' of the turn-over in the UVLF. Conceptually, the explanation is simple: once the UVLF turns over at magnitudes corresponding to 𝑀 turn, the cumulative number of photons coming from 𝑀 ℎ ≲ 𝑀 turn drops exponentially, meaning changes to 𝑀 turn will have a much bigger impact on the signal than 𝑟 turn. The fact that monopole measurements can bound this quantity at all speaks to the power of the detailed shape of the monopole, and as a result its ability to put perhaps surprisingly informative constraints on low-mass halos (see also Hibbard et al. 2022). Although the 21-cm monopole is more statistically powerful and constraining, the dipole may be more robust to a range of systematic concerns, as discussed in §1 and in Anderson et al. (in prep). It is thus encouraging that dipole measurements alone may provide interesting parameter bounds and astrophysical insights. \nNext, we relax the assumption of 𝑆 = 1, and fit our fiducial signal allowing the three additional parameters 𝑀 𝑐 , 𝛾 1, and 𝛾 2 to vary. Recall that these parameters allow for a possible upturn in the faint-end of the UVLF prior to any eventual decline. Given the additional degeneracies with 𝑀 turn, we focus on the extent to which we can recover the cosmic star formation rate density (SFRD) and turn-over in the UVLF, given by 𝜙 ( 𝑀 UV , turn ) , rather than the free parameters themselves. The results of this exercise are shown in Fig. 8 and 9. \nFirst, in Fig. 8 we examine the recovered SFRD. Reassuringly, this more flexible model still yields a strong constraint on the SFRD. For reference we also show the parametric SFRD model from Madau & Dickinson (2014), based on lower redshift measurements, the fraction of the SFRD in 𝑀 UV < -19 galaxies (dotted black), as well as three polygons \nFigure 5. Measurements of the dipole provide a robust and relatively model-independent consistency check on monopole measurements. In the top row, we show our fits to a mock dipole signal generated with the DPL model (model A; same in each panel), and in the bottom row, we show predictions for the monopole in each case. Results in each column are obtained using a different signal model to fit the mock (tanh, DPL, and DPLX, from left to right). In each case, we recover the position of the absorption minimum to an accuracy of better than ∼ 20 mK (1𝜎 ), and find good agreement across the band as well. Note that, as in Fig. 4, we have thinned out the number of data points plotted by 2x for visual clarity. \n<!-- image --> \nindicating plausible high𝑧 star formation scenarios consistent with recent JWST results (Bouwens et al. 2023). Clearly, a measurement of the dipole would provide a powerful constraint on the total amount of star formation at these redshifts, and so be very complementary to UVLF-based constraints, especially since UVLF measurements directly probe only relatively bright galaxies and struggle to access early phases of cosmic dawn. \nNext, in Fig. 9, we show constraints on the faint-end of the UVLF, using the same line-style and colour conventions as in Fig. 8. The monopole (red) provides an impressive constraint on the turn-over location. The dipole constraints are weaker, as expected from the posterior in the 𝑀 turn𝑟 turn plane (see Fig. 7 and associated text). However, encouragingly the angle of the posterior distribution clearly indicates a preference for a departure from a smooth power-law extrapolation below 𝑀 UV ∼ -12. As a result, a dipole measurement could provide evidence of a turn-over, without necessarily yielding a tight constraint on its location. \nFinally, we broaden our exploration of the dipole's constraining power to our full model suite (introduced in Fig. 1 and Table 1), focusing on the dipole's ability to constrain the total amount of star formation at 𝑧 = 15, the UV photon escape fraction, and the efficiency of X-ray photon production. The results are shown in Fig. 10. \nIn general, Fig. 10 shows that the true input values (indicated with plot symbols) are recovered at the 1 𝜎 level. In \na few cases, the true input value is on the edge of the 68% confidence region (models B and D) or just beyond (model F). These models are the most difficult to constrain with the dipole, as they have high SFRD and/or 𝐿 𝑋 / SFRvalues, which weaken the signal. However, models with 𝐿 𝑋 / SFR ≲ 10 42 and SFRD ( z = 15 ) ≲ 10 -3 are constrained extremely well. The escape fraction is more difficult to constrain, as the reionization piece of the 21-cm signal is a fairly smooth and gradual function of frequency, but in many cases these constraints would still dramatically reduce the range of viable models. \nThough Fig. 10 is not completely exhaustive, it gives a good sense of our sensitivity to models spanning a wide range of possibilities - 4-5 orders of magnitude in both SFRD and 𝐿 𝑋 / SFR, and a factor of a few in 𝑓 esc. The latter may not seem terribly impressive, however, any independent handle on 𝑓 esc would be most welcome at the moment given the apparent tensions between new JWST measurements and reionization constraints from the CMB and Ly𝛼 forest. The severity of the tension depends sensitively on how 𝑓 esc depends on galaxy properties, with different observational constraints yielding significantly different reionization scenarios (see, e.g., Muñoz et al. 2024; Pahl et al. 2024). \nFinally, we note that the same general trends in Fig. 10 hold whether we plot the SFRD at 𝑧 = 10 , 15 , or 20. Of course, the SFRD constraints generally get worse as we consider higher redshifts - this is expected given the waning power of UVLFs at increasingly high redshifts, which can be seen in Fig. 8 - \nFigure 6. Forecast for 𝑓 ∗ parameters. Dashed black contours reflect the prior volume, defined by pre-existing constraints on UVLFs (Bouwens et al. 2015), Ly𝛼 forest (Bosman et al. 2022), and the CMB optical depth (Planck Collaboration et al. 2020b). Open blue contours indicate constraints obtained with the addition of only a dipole measurement, while filled red contours show constraints obtained with the addition of only the monopole. Open orange contours denote the constraints possible with both a dipole and monopole measurement. Crosses indicate the true input values assumed in the mock (model A). \n<!-- image --> \neven at 𝑧 = 20 the dipole yields solid order-of-magnitude level constraints on the SFRD, which is difficult to imagine obtaining in any other way aside from 21-cm fluctuations or the 21-cm monopole.", '4. DISCUSSION & CONCLUSIONS': "In this work, we explored the information content of the 21-cm dipole signal relative to the monopole. At first glance, one might expect the dipole to yield poor constraints on astrophysical parameters of interest given that to leading order it probes only the derivative of the 21-cm monopole, and so does not explicitly contain information about the amplitude of the signal. However, there are two situations in which the dipole could in principle still provide tight constraints on parameters: (i) if the derivative of the 21-cm monopole is sufficiently informative on its own to enable meaningful constraints, and (ii) if reasonable astrophysical priors can serve as a stand-in for actual measurements of the overall amplitude. \nWe find that indeed, the monopole still generally outperforms the dipole in terms of astrophysical constraining power. However, in many cases the dipole constraints are competitive, e.g., we found in Fig. 7 that constraints on 𝑀 turn, 𝑓 esc, and 𝐿 𝑋 / SFR - three of the most sought after parameters of \nthis era - can be recovered very well with a dipole measurement. The outlier in this exercise was the parameter 𝑟 turn, which controls the sharpness of any turn-over in the UVLF of galaxies, which can be bounded with a monopole measurement, but essentially becomes a nuisance parameter for dipole-based inference. Given this result, it was unsurprising to find that the dipole struggles to accurately constrain the position of any UVLF turn-over, as shown in Fig. 9. However, so little star formation occurs below 𝑀 turn that the cosmic SFRD at 𝑧 ≥ 10 can still be constrained to a factor of ∼ 2 in most cases. This is of course another key quantity in galaxy formation theory, and such a constraint would be more than sufficient to rule out broad classes of models. \nOur parameterization choices, while surely flexible enough to accommodate a wide range of scenarios, may artificially bias our inference. For example, we have assumed that X-ray emission closely tracks star formation, when in reality there may be nuclear black holes growing in galaxies whose Xray emission scales differently with galaxy properties. Such a population could have interesting signatures in the 21-cm background (e.g., Tanaka et al. 2016), especially if they are radio loud (Ewall-Wice et al. 2020). In our current framework, large inferred values of the cosmic SFRD and/or 𝐿 𝑋 /SFR relation might provide early indications that a model based entirely on stellar X-ray sources is inadequate, but a more detailed treatment with multiple source populations is of course worth exploring. We will revisit this possibility in future work. \nOn a more technical level, there are several key modeling challenges that must be solved in order for the potential constraining power of the monopole and dipole to be fully realized. As mentioned in §2, our two-zone model is approximate from the outset, having made several simplifying assumptions to avoid simulating the full 21-cm field. The assumption that the mean 21-cm background traces the product of the mean neutral fraction and contrast, 1 -𝑇 CMB / 𝑇 𝑆 , is likely accurate at the ∼ 10% level (Schaeffer et al. 2024), and there are plenty of generally-ignored shortcomings in models that can contribute at the ∼ 5 -20 mK level, e.g., marginalizing over cosmological parameters, accounting for systematic differences in stellar population synthesis modeling, and halo mass function uncertainties (Mirocha et al. 2021b,a; Greig et al. 2024). Provided these problems can be solved, the only fundamental limit is cosmic variance, which is comparable to the single-channel uncertainties assumed in this work ∼ 0 . 1 mK (Muñoz & Cyr-Racine 2021). \nDespite these challenges of interpretation, the internal consistency check offered by the dipole is robust. We showed that one can reliably reconstruct the monopole from a dipole measurement, even if the model used in the fit is different from the true underlying model used to generate the mock data. For a measurement with ∼ 0 . 1 -1 mK uncertainties \nFigure 7. Dipole constraints on parameters governing the faint-end UVLF turn-over and efficiency of UV and X-ray photon production are competitive with monopole constraints. Linestyle and colour conventions are identical to Fig. 6. \n<!-- image --> \non the dipole, the position of the absorption feature in the monopole is recovered to a few MHz and ∼ 20 -30 mK (at 68% confidence) for each of the three models we consider. Yet more flexible parameterizations are likely worth exploring, e.g., splines (Pritchard & Loeb 2010; Harker et al. 2012) or the 'flex-knot' approach (Heimersheim et al. 2023; Shen et al. 2023). Purely phenomenological approaches like these will need some additional prior, e.g., that the signal vanishes at 𝑧 ∼ 6 and 𝑧 ∼ 30, in order to constrain the overall amplitude, which the phenomenological tanh model (Harker et al. 2016) sidesteps by flexibly parameterizing physical quantities rather than the signal directly. \nIn conclusion, the 21-cm dipole signal from 𝑧 ≳ 6 offers a very interesting target for future observations. It should not be deemed valuable only as a cross-check on monopole measurements: it can in principle provide tight constraints on key parameters of galaxy formation models entirely on its own. This conclusion has been drawn from a relatively idealized forecast in order to set some initial expectations. A follow-up paper, Anderson et al., in prep., will present a schematic instrument and survey design strategy, including detailed treatments of astrophysical foregrounds and instrumental effects, in order to better assess the dipole's constraining power in a more realistic setting. \nFigure 8. Mock constraints on the cosmic SFRD from the 21-cm dipole and monopole. The true input (dashed) is recovered by the dipole (open blue contours) to within ∼ 0 . 1 dex at 𝑧 ∼ 10 -15, with broadening contours at 𝑧 ≳ 20. The monopole recovery (red) is effectively perfect, lying right on top of the input curve. For reference, we also show the SFRD of bright 𝑀 UV < -19 galaxies (dotted black) and the Madau & Dickinson (2014) model based on lower redshift constraints (cyan). The shaded polygons are drawn from Bouwens et al. (2023), who provide a qualitative classification of early JWST high𝑧 galaxy candidates from 'robust', to 'solid', to 'possible' from bottom to top. Note that any inference of the total SFRD from galaxy surveys is model-dependent, in lieu of a strong detection of the turn-over. \n<!-- image -->", 'ACKNOWLEDGEMENTS': 'The authors thank Mike Seiffert, Andrew Romero-Wolf, Yun-Ting Cheng, and Joe Lazio for helpful feedback on this work. J.M. was supported by an appointment to the NASA Postdoctoral Program at the Jet Propulsion Laboratory/California Institute of Technology, administered by Oak Ridge Associated Universities under contract with NASA. C.A. and T.-C.C. acknowledge support by NASA ROSES grant 21-ADAP21-0122 and the JPL 7X formulation office. Part of this work was done at Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration (80NM0018D0004). 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2024arXiv240910672M

Gravitational waveforms capturing binarys evolution through the earlyinspiral phase play a critical role in extracting orbital features that nearly disappear during the lateinspiral and subsequent merger phase due to radiation reaction forces for instance the effect of orbital eccentricity. Phenomenological approaches that model compact binary mergers rely heavily on combining inputs from both analytical and numerical approaches to reduce the computational cost of generating templates for data analysis purposes. In a recent work Chattaraj et al. Phys. Rev. D 106 124008 2022 arXiv2204.02377grqc we demonstrated construction of a dominant quadrupole mode inspiralmergerringdown IMR model for binary black holes BBHs on elliptical orbits. The model was constructed in timedomain and is fully analytical. The current work is an attempt to improve this model by making a few important changes in our approach. The most significant of those involves identifying initial values of orbital parameters with which the inspiral part of the model is evolved. While the ingredients remain the same as in arXiv2204.02377grqc resulting waveforms at each stage seem to have improved as a consequence of new considerations proposed here. The updated model is validated also against an independent waveform family resulting overlaps better than sim 96.5 within the calibrated range of binary parameters. Further we use the prescription of the dominant mode model presented here to provide an alternate but equivalent model for the dominant quadrupole mode and extend the same to a model including the effect of selected nonquadrupole modes. Finally while this model assumes nonspinning components we show that this could also be used for mildly spinning systems with component spins anti aligned w.r.t the orbital angular momentum.

2024-09-01T00:00:00Z

['arXiv:2409.10672', '10.48550/arXiv.2409.10672', '2024arXiv240910672M']

['General Relativity and Quantum Cosmology', 'Astrophysics - High Energy Astrophysical Phenomena']

An improved IMR model for BBHs on elliptical orbits

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.10672.pdf

{'An improved IMR model for BBHs on elliptical orbits': "Pratul Manna, 1, ∗ Tamal RoyChowdhury, 2, † and Chandra Kant Mishra 1, 3, ‡ 1 Department of Physics, Indian Institute of Technology Madras, Chennai 600036, India 2 University of Wisconsin Milwaukee, Milwaukee, WI 53201, USA 3 Centre for Strings, Gravitation and Cosmology, Department of Physics, Indian Institute of Technology Madras, Chennai 600036, India (Dated: September 18, 2024) \nGravitational waveforms capturing binary's evolution through the early-inspiral phase play a critical role in extracting orbital features that nearly disappear during the late-inspiral and subsequent merger phase due to radiation reaction forces; for instance, the effect of orbital eccentricity. Phenomenological approaches that model compact binary mergers rely heavily on combining inputs from both analytical and numerical approaches to reduce the computational cost of generating templates for data analysis purposes. In a recent work, Chattaraj et al., Phys. Rev. D 106, 124008 (2022) [1], we demonstrated construction of a dominant (quadrupole) mode inspiral-merger-ringdown (IMR) model for binary black holes (BBHs) on elliptical orbits. The model was constructed in time-domain and is fully analytical. The current work is an attempt to improve this model by making a few important changes in our approach. The most significant of those involves identifying initial values of orbital parameters with which the inspiral part of the model is evolved. While the ingredients remain the same as in Ref. [1], resulting waveforms at each stage seem to have improved as a consequence of new considerations proposed here. The updated model is validated also against an independent waveform family resulting overlaps better than ∼ 96.5% within the calibrated range of binary parameters. Further, we use the prescription of the dominant mode model presented here to provide an alternate (but equivalent) model for the (dominant) quadrupole mode and extend the same to a model including the effect of selected non-quadrupole modes. Finally, while this model assumes non-spinning components, we show that this could also be used for mildly spinning systems with component spins (anti-) aligned w.r.t the orbital angular momentum.", 'I. INTRODUCTION': "Since the discovery event, GW150914 [2], the LIGOVirgo-KAGRA (LVK) collaboration has reported nearly hundred compact binary mergers observed during the first three observing runs (O1-O3) [3-6]. These numbers have doubled since and the list continues to be dominated by signals identified as mergers of black holes in a binary; see for instance, the GWOSC 1 page [7] that lists all reported events. While these (now also routine!) observations continue to help improve our understanding of compact binary physics and astrophysics, their origins remain unknown [8-10]. Astrophysical environments where a binary is formed and processes through which it is formed may leave imprints on binary's mass and spin parameters. However, the available statistics needs to grow in order to make inferences concerning binary's origin based on mass and spin measurements alone [11]. Eccentricity, on the other hand, can be a unique tool to identify binary's origins, as dynamically formed binaries may still retain residual orbital eccentricities [12-14] when observed in ground based detectors currently operating. \nCurrent template-based search pipelines make use of circular templates due to the expected circularisation of most compact binary orbits caused by radiation reaction forces [15] as they enter the sensitivity bands of ground based detectors such as LIGO [16] and Virgo [17]. However, binaries formed through the dynamical interactions in dense stellar environments are likely to be observed with residual eccentricities e 20Hz ∼ 0 . 1 [10, 12, 18]. In fact, the first binary merger event involving an intermediate mass black hole, GW190521 [19], is likely an eccentric merger [9, 20] (see also Refs. [21-26] discussing events with eccentric signatures). While quasi-circular templates should be able to detect systems with initial eccentricities e 10Hz ≲ 0 . 1, binaries with larger eccentricities would require constructing templates including the effect of eccentricity [27, 28]. Moreover, the presence of even smaller eccentricities ( e 10Hz ∼ 0.01 - 0.05) can induce significant systematic biases in extracting source properties [29](see also, Refs. [30, 31]). Furthermore, next generation ground-based detectors, Cosmic Explorer [32-34] and Einstein Telescope [35, 36], due to their low frequency sensitivities, should frequently observe systems with detectable eccentricities [37, 38]. \nEven though inspiral (I) waveforms from eccentric binary mergers involving non-spinning compact components are sufficiently accurate [39-45], waveform models including contributions from merger and ringdown (MR) stages compared to quasi-circular counterparts \nare significantly less developed. 2 Numerous efforts toward constructing eccentric inspiral-merger-ringdown (IMR) waveforms, useful for data analysis purposes, are underway [1, 48-52]. However, these efforts do not include important physical effects such as spins (pointing along or away from binary's orbital angular momentum) or higher order modes. Dominant mode (or quadrupole mode) models for eccentric BBHs with component spins (anti-)aligned w.r.t the binary's orbital angular momentum were recently developed in Refs. [53-58]. Since most mergers observed so far are consistent with a zero-effective spin [59-61], models neglecting spin effects can still be useful [49]. 3 In addition, modeling of higher order modes also seems necessary as Refs. [1, 62] argue. Eccentric versions of the effective-one-body (EOB) waveforms including higher modes [63, 64] and an eccentric numerical relativity (NR) surrogate model [65, 66] also became available in the past couple of years. Alternatively, sub-optimal methods (with little or no dependence on signal model being searched) may be used for detecting an eccentric merger. Although, these methods are sensitive to high mass searches (typically ≳ 70 M /circledot ) [10, 18], while most observed events have a mass smaller than this limit [5, 60]; see, for instance, Fig. 3 of [67]. \nThe present work follows our first paper [1] and can be viewed as an update to the same; referred to as Paper I here onwards. In Paper I, construction of hybrid waveforms by combining post-Newtonian (PN) waveforms with NR simulations through a least-squares minimization was demonstrated. These hybrids were then used as target models to produce a fully analytical dominant (or quadrupole) mode model by matching an eccentric PN inspiral with a quasi-circular mergerringdown waveform. Subsequently, the performance of the model was checked both against the target hybrids used in training the model as well as against an independent family of waveforms [50]. It was shown that overlaps between target hybrids and the model significantly improved compared to those against a circular template, at least at the low end of the binary masses and for small eccentricities considered there (See for instance, Figure 9 of Paper I). The current work aims to improve the model presented in Paper I in the view of efforts such as those of Refs. [68, 69]. References [68, 69] develop a standard, gauge-independent prescription for defining orbital eccentricity using a suitable combination of dominant mode GW frequency. It should be noted that, in GR, eccentricity is not \nuniquely defined (see for instance Ref. [70]) although at the leading (Newtonian) order there is consensus. The definition of Refs. [68, 69] reduces to the Newtonian value both in small and large eccentricity limit and is also independent of gauge-ambiguities. This motivates us to employ this definition of eccentricity in our model and investigate into the improvements in its performance.", 'A. Gauge invariant definition of eccentricity': 'Eccentricity is not uniquely defined in GR and thus templates computed within the framework of GR may have forms different from the observed data. Both perturbative and numerical solutions describing the compact binary dynamics use gauge-dependent constructs including the very definitions of orbital parameters that are evolved. These choices are almost never identical in any two approaches which naturally leads to inconsistencies between different models. To get rid of the ambiguity associated with the definition of eccentricity, Refs. [68, 69] proposed a new definition of eccentricity based on the GW frequency data. At 0PN order, this definition exactly reduces to the Newtonian definition of eccentricity ( e t ). We reproduce the necessary relations below. This new eccentricity for an observed GW ( e gw ) signal is defined as \ne gw = cos( ψ/ 3) -√ 3 sin( ψ/ 3) , (1) \nwith, \nwhere, \ne ω 22 = √ ω p 22 -√ ω a 22 √ ω p 22 + √ ω a 22 , (3) \nwhere, ω p 22 and ω a 22 refer to the ( /lscript = 2 , | m | = 2) mode periastron and apastron frequencies, respectively and are functions of time. Since this new eccentricity is written in terms of frequencies that can be measured, it is also free from any gauge-ambiguities. This definition is employed in the gw eccentricity package provided by Ref. [68].', 'B. Summary of the current work': "Considerations related to a change in the definition of eccentricity such as the ones proposed by Refs. [68, 69] demand revisiting each element of model construction presented in Paper I which we intend to closely follow here. We start by comparing the PN prescription and NR simulations used in constructing hybrids in Paper I. The new definition of eccentricity introduced in \nψ = arctan ( 1 -e 2 ω 22 2 e ω 22 ) , (2) \nRefs. [68, 69] is used to identify a set of reference values for eccentricity ( e ref ), mean anomaly ( l ref ) and GW frequency ( f ref ) with which the overlap between the PN and NR data is maximum in a time-window where the two are expected to give accurate predictions. Figure 1 plots the two, PN and NR model, together with the window returning maximum overlap between them. \nThe PN model is evolved using reference values, ( e ref , l ref , f ref ), as an initial set (i.e at the start of the PN model) and matched with NR simulations in a hybridization window following the prescription of Refs. [1, 71]. Figure 3 displays one of the hybrids and compares it with corresponding NR simulation. Table I lists all hybrids constructed here along with starting values of eccentricity ( e 0 ), mean anomaly ( l 0 ) and a frequency dependent PN parameter ( x 0 ) which is related to the GW frequency of the dominant mode ( f 0 ) via x 0 = ( πMf 0 ) 2 / 3 . Finally, a fully analytical dominant mode model is obtained by matching an eccentric PN inspiral [41] with a quasi-circular merger-ringdown model [72] in Sec. III. Note however, as in Paper I, a new set of hybrids (including only /lscript = 2 , | m | = 2 mode), with PN model purely based on Ref. [41] (termed EccentricTD within LIGO Algorithmic Library [73]) are used in training the model to minimize the difference between the target and the model which in turn uses the PN prescription of Ref. [41]. (See Fig. 4.) \nFigure 7 evolves the model using the initial set of parameters for three different hybrids listed in Table I and simply plots it against the amplitude (top-left), frequency (top-right) and plus polarisation data from the hybrids. While these provide a visual proof of closeness of the model with target hybrids, right panel of Fig. 8 plots the overlap (maximized over a simple time and phase shift) between the model and the set of hybrids used in calibrating the model (labelled as training set in Table I) (thin lines). Overlaps with two of the hybrids not used in building the model (labelled as testing set in Table I) are also plotted (thick lines). For comparison, we also plot (left panel) the overlap of (all training and two testing) hybrids with quasi-circular templates of Ref. [72] (termed SEOBNRv5 ). Clearly, the model outperforms its circular counterpart and recovers the target eccentric hybrids with an accuracy better than ∼ 96 . 5% for almost the entire range of parameter space spanned by the training set hybrids. Note that both the target and template waveforms used in Fig. 8 involve only the dominant ( /lscript = 2 , | m | = 2) mode. Additionally, Figure 8 only shows overlap between the hybrids and the model for fixed values of parameters, ( e 0 , l 0 , x 0 ), associated with each hybrid. . Like not in a seperate folder. But we can de not in a folder.e nontfolfter a The expectation, that the model shall produce reliable waveforms for any value of the parameters in the range it is calibrated, should also be verified. To test this we compare our model against an independent family of waveforms \ndiscussed in Ref. [58] (termed TEOBResumS-Dali ) for randomly sampled values for this set in the range 0 ≲ e 0 ≲ 0 . 3, 1 ≲ q ≲ 3, and -π ≤ l 0 ≤ π , respectively. The range of parameter values are chosen to match the parameter space spanned by the hybrids chosen for calibration, except for orbital eccentricity, which is conservatively chosen to have a maximum value of e 0 = 0 . 3 and also explores near circular cases to see if the model correctly (and gradually) reproduces the circular limit. The results are displayed in Fig. 9 and as can be seen there, model reproduces the waveforms of TEOBResumS-Dali [58] with overlaps better than 96.5% for almost the entire range of parameters considered there and confirms the suitability of the waveforms for data analysis purposes. Note however, the difference between the eccentricity scale in Fig. 8 and in Fig. 9. \nSection IV A presents an alternate model to the one obtained in Sec. III. The primary motivation here is to use PN input waveforms of higher PN accuracy (compared to EccentricTD [41] used in constructing the model in Sec. III) to maximize on the overlaps with target models at a small cost of losing sensitivity to high eccentricity cases within the calibration range. Comparisons with TEOBResumS-Dali [58] waveform show that such a model may provide a suitable alternative to the model constructed in Sec. III. Figure 10 displays this comparison. It is interesting to note that the mismatches seem to have visibly improved compared to those with the model based on EccentricTD for small eccentricity cases. This is likely due to higher PN accuracy of the amplitude and phase used in constructing the alternate model and thus matches better with the waveform TEOBResumS-Dali [58]. For larger eccentricities, the performance of the two seem similar (see a discussion in Sec. IV A). \nFinally, while a higher mode model can be constructed following the methods used in constructing the dominant mode model discussed in Sec. III, one may simply use the prescription for the ( /lscript = 2 , | m | = 2) mode to combine an (eccentric) inspiral and a (quasi-circular) merger-ringdown prescription for each mode to obtain an ad hoc higher mode (HM) model as was done in Paper I. The alternate model presented in Sec. IV A is extended to include selected /lscript = | m | and /lscript -1 = | m | modes and is validated against TEOBResumS-Dali [58]. These are the modes which are included in SEOBNRv5HM [72]; the quasi-circular model used for the merger-ringdown part. Note also, these are also the modes for which we construct hybrids; see Fig. 3. The model is discussed in Sec. IV B and its performance is displayed in Fig. 11 for mildly inclined systems (30 · ). \nThe paper is structured in the following manner. In Sec. II we start by comparing waveforms from PN and NR approaches and discuss the construction of target hybrids. Next, in Sec. III we construct the waveform model \n<!-- image --> \nFIG. 1. Amplitude and frequency of selected modes from an eccentric NR simulation (SXS:BBH:1364) together with an eccentric PN model are plotted. The PN model is evolved assuming a fixed value for initial eccentricity ( e 0 = 0 . 172), mean anomaly ( l 0 = 2 . 681) and the frequency dependent PN parameter ( x 0 = 0 . 0391). The merger time of the NR waveform is set to zero. The initial set ( e 0 , l 0 , x 0 ) is obtained by maximizing the overlap between the PN and NR waveform in a 1000 M wide time-window, in a region where PN model is expected to agree with NR simulation. Additionally, a time shift is performed on the PN inspiral. The time-window of maximum overlap is shown as the shaded region. Binary's component mass ratio ( q ) is 2, while the total mass ( M ) and the luminosity distance ( D L ) of the binary are set to M =1 M /circledot and D L =1Mpc respectively, following the convention of SXS simulations. \n<!-- image --> \nby combining an eccentric PN inspiral model with a quasi-circular merger-ringdown model at a suitable point obtained by performing comparisons with target models. Subsequently, the model is validated against the target models not used in calibrating it as well as against an independent family of waveforms. In Sec. IV, we discuss an alternate model based on the prescriptions for attachment times obtained in Sec. III and subsequently extend this alternate model to include higher order modes. Finally, Sec. V presents summary of results and conclusions.", 'A. PN and NR comparisons': 'Paper I compares the PN inspiral waveforms with NR simulations. The inspiral mode amplitudes constituting 3PN inspiral waveforms assuming non-spinning binary systems on quasi-elliptical orbits were computed in Refs. [39, 42, 43]. The orbital phase was taken from Ref. [41]. GW frequency for each PN mode was obtained using the following scaling relation [69] \nω /lscriptm ∼ m 2 × ω 22 . (4) \nNR simulations used in comparison with PN waveforms were produced using the Spectral Einstein Code (SpEC) developed by SXS collaboration and are publicly available [48, 74]. Comparison of PN and NR prescriptions for a specific simulation was shown in Figure 1 of Paper I. Subsequently, a common region of validity was identified in which the two could be matched suitably \nto obtain hybrids listed in Table I there. (See Sec. III A-III C of Paper I for technical details). \nHere too we aim to compare the PN and NR prescriptions leading to the construction of hybrids. While the hybridization method is same as in Paper I, comparison (which leads to identification of a suitable window for hybridization) is performed following a slightly different approach. In paper I, PN models were simply evolved to match a set of reference orbital parameters ( e ref , l ref ) computed at a reference GW frequency ( x ref ) computed in Ref. [48] for each of the 20 simulations considered in Paper I. Here, we simply choose to compare the two waveforms in a 1000 M wide time-window, slide it over the overlapping data and compute overlaps (the match maximized over a reference time and phase shifts; see Eq. (5) below) by varying ( e ref , l ref , f ref ) trio for the PN model, where f ref is related to the PN parameter x ref via x ref =( πMf ref ) 2 / 3 . The match ( M ) between two waveforms is defined as an inner product given as \nM ( θ 1 , θ 2 ) ≡ max φ c ,t c 〈 h ( θ 1 ) , h ( θ 2 ) e i (2 πft c -φ c ) 〉 , (5) \nwith, \n〈 h 1 , h 2 〉 ≡ 4 Re [ ∫ ∞ 0 df ˜ h 1 ∗ ( f ) ˜ h 2 ( f ) S h ( f ) ] , (6) \nwhere 〈 h 1 , h 2 〉 represents the inner product between two waveforms h 1 and h 2 having unit norm and are functions of an intrinsic set of binary parameters ( θ 1 , θ 2 ). The phase φ c , time t c are measured at coalescence and S h ( f ) represents the noise in the detector (see Ref. [75]). \nWe find the overlap is optimal for a time-window of ( -2000 M to -1000 M ) and for a given ( e ref , l ref , f ref ) \nFIG. 2. Same as Fig. 1, except that only dominant mode data is plotted and the comparison is shown for few other simulations with varying mass ratios and orbital parameters (see Table I for details). Initial values ( e 0 , l 0 , x 0 ) with which the PN model is evolved for comparisons with various simulations are listed in Table I. \n<!-- image --> \ntrio which becomes the starting reference ( e 0 , l 0 , f 0 ) and may be different for different NR simulations (see Table I). 4 Compared to the earlier approach, our current approach helps the construction of hybrids in at least two distinct ways. (1) The identification of hybridization window is done using quantitative measures such as overlaps and (2) the reference orbital parameters such as orbital eccentricity ( e 0 ) and mean-anomaly ( l 0 ) at a given frequency ( f 0 ) will be free from gauge-ambiguities due to the use of the definition of Refs. [68, 69]. With the suitable hybridization window identified, we can now proceed to reconstruct the hybrids. \nFigure 1 compares the data corresponding to the NR simulation bearing simulation ID SXS:BBH:1364, for a selected set of modes chosen based on their relative significance compared to the dominant mode. 5 PN model is evolved using a set of initial parameters ( e 0 , l 0 , f 0 ) obtained using the procedure discussed above and then plotted together with the NR simulation after performing a time shift. The window giving maximum match is also displayed. For completeness we also show similar comparisons for few other simulations in Fig. 2 albeit for only the dominant mode. It is interesting to also note the time-window returning maximum match is common \nfor all simulations.', 'B. Construction of hybrid waveforms': "Complete IMR waveforms are constructed by matching PN and NR prescriptions for set of modes included in Fig. 1, in a region where the PN prescription closely mimics the NR data following the method of Ref. [71]. These are traditionally referred to as 'hybrids'. As discussed in Ref. [71], construction of hybrids including higher modes (in the circular case) is possible by performing at least two rotations (and a time shift) so as to align the frames in which PN/NR waveforms are defined. 6 This argument was simply extended to the case of eccentric orbits in Paper I, assuming that the effect of marginalising over parameters such as eccentricity and mean anomaly will not significantly affect the hybridization. As discussed above, for the current work we simply adopt the hybridization procedure of Paper I. 7 The prescription for construction of hybrids is discussed in detail in Ref. [71] as well as in Paper I, nevertheless, we reproduce some of the steps here for completeness. \nFIG. 3. A hybrid model constructed by matching a NR simulation (SXS:BBH:1364) with a PN model (evolved using parameters consistent with the simulation) in a time-window where the two are expected to correctly predict the binary dynamics. For comparison the NR data is also plotted. The black dashed line marks the beginning of the NR waveform and the shaded light-red region t ∈ ( -2000 M, -1000 M ) shows the matching window. Overlapping hybrid and NR waveforms on the left of the matching window hint at the quality of hybridization performed here. Table I lists details of all hybrids considered in the current work. \n<!-- image --> \nt/M \nTABLE I. Set of time-domain hybrids constructed by matching NR simulations from the SXS Catalog and state-of-the-art PN prescriptions for BBHs on eccentric orbits are listed. SXS simulation IDs are retained for identification with NR simulations used in constructing the hybrids. Each hybrid assumes a fixed value for initial eccentricity ( e 0 ), mean anomaly ( l 0 ) and the frequency dependent PN parameter ( x 0 ) obtained using the gw eccentricity package based on Ref. [68]. The PN parameter ( x ) is related to the GW frequency ( f gw ) of the dominant mode as x =( πMf gw ) 2 / 3 with M representing binary's total mass. Mass ratio ( q ) and number of orbits prior to the merger are also listed. N orb is computed by taking the phase difference between the start of the waveform and the peak of the dominant mode ( /lscript =2, | m | =2) amplitude. \nA least-squares minimization of the integrated difference between the GW modes from the PN and NR waveforms in a time interval ( t i , t f ), in which the two approaches give similar results, is performed and can be defined as \nδ = min t 0 ,ϕ 0 ,ψ ∫ t f t i dt ∑ /lscript,m ∣ ∣ ∣ h NR /lscriptm ( t -t 0 ) e i ( mϕ 0 + ψ ) -h PN /lscriptm ( t ) ∣ ∣ ∣ , (7) \nwhere the minimization is performed over a time shift ( t 0 ) and the two angles ( ϕ 0 , ψ ) as discussed above. The hybrid waveforms are then constructed by combining the NR data with the 'best matched' PN waveform in the following way: \nh hyb /lscriptm ( t ) ≡ τ ( t ) h NR /lscriptm ( t -t ' 0 ) e i ( mϕ ' 0 + ψ ' ) +(1 -τ ( t )) h PN /lscriptm ( t ) , (8) \nwhere ( t ' 0 , ϕ ' 0 , ψ ' ) are the values of ( t 0 , ϕ 0 , ψ ) that minimize the integral of Eq. (7). In the above equation, τ ( t ) \nis a weighting function defined by \nτ ( t ) ≡ 0 if t < t i t -t i t f -t i if t i ≤ t < t f 1 if t f ≤ t . (9) \nThe hybrids corresponding to a representative NR simulation (SXS:BBH:1364) for all relevant modes are shown in Fig. 3. The two waveforms are aligned at merger and the shaded grey region t ∈ ( -2000 M, -1000 M ) highlights the matching window where hybridization was performed. Overlapping hybrid and NR waveforms outside (on the left of) the matching window hint at the quality of hybridization performed here. \nWe reconstruct IMR hybrids corresponding to all 20 eccentric NR simulations listed in Ref. [48] as was done for Paper I. These are listed in Table I and the SXS simulation IDs have been retained to identify the hybrids with the corresponding NR simulation. Each simulation starts with a specific initial eccentricity ( e 0 ), mean anomaly ( l 0 ) and frequency ( x 0 ) obtained following the procedure discussed in Sec. II A.", 'III. THE WAVEFORM MODEL': 'Until now we focused on constructing a (PN-NR) hybrid model which could be used as a target for building a fully analytical IMR model for eccentric binary black hole mergers. 8 While these hybrids could be used to construct the model following the procedure adopted in Paper I, as was done there, we construct an independent set of hybrids using PN model EccentricTD [41] (see Fig. 4). This is primarily done to minimize the difference between the target hybrids and the model (being constructed) that also uses the waveform EccentricTD [41] for the inspiral part. 9 Note that, these new hybrids only include the dominant mode ( /lscript = 2 , | m | = 2) since it is the dominant mode which we wish to model first. Note that, these are also the hybrids used in validating the model constructed in this section. \nAs in Paper I, here too we obtain a fully-analytical dominant ( /lscript = 2 , | m | = 2) mode model by matching an eccentric PN inspiral [41] with a quasi-circular prescription for the merger-ringdown phase [72]. Here too we stick to the procedures adopted in Paper I which involves identifying attachment times for both amplitude and frequency data together with an overall shift and the \nFIG. 4. Same as Fig. 3 except, the PN part of the hybrid is purely based on EccentricTD model of Ref. [41]. These are also the hybrids that are used in constructing and validating the model in Sec. III. \n<!-- image --> \nFIG. 5. The (numerical) amplitude and frequency model produced by combining an eccentric inspiral with a quasi-circular merger-ringdown waveform for a total mass of 30 M /circledot is plotted against the hybrid used in calibrating the model. The eccentric inspiral ( EccentricTD [41]) and the (quasi-circular) merger-ringdown ( SEOBNRv5 [72]) models are also plotted for comparison. The amplitude (frequency) model transition smoothly from the inspiral to the merger-ringdown stage inside the shaded region(s), at t A match ( t ω match ) values, maximizing the overlap between the model and the target hybrid. \n<!-- image --> \noutput is a coherent IMR model suitable for generating desired signals. Exact details concerning this model are outlined in Secs. III A 1-III A 3. Note however, that one must map the data for attachment times and time shifts to a set of physical parameters of the binary such as mass ratio and other relevant parameters. For M ∼ 25 M /circledot , the highest x 0 value of Table I corresponds to a frequency of f ∼ 20Hz (low frequency cutoff for advanced LIGO design [16]). This motivates us to work with a conservative choice of 30 M /circledot system when constructing the model. We find only 13 out of 20 cases reasonably agree (with overlaps > 97%) for a 30 M /circledot system with the hybrids and thus the final fits are obtained only using these 13 performing cases. Section III B discusses the details of construction of the analytical model.', '1. Time-shift': 'The process of generating a numerical model is similar to the procedure adopted in Paper I and we reproduce it here for clarity and completeness. As described in Sec. II B, hybridization involves a minimization over a time shift, so when producing the amplitude model, we \nhave to first perform a time shift of the inspiral waveform relative to the circular IMR waveform, because the time to merger is not known. This is done by first setting the merger time for the circular IMR waveform to zero and then time sliding the eccentric inspiral about the merger. We start by making a trial choice of t shift and then generate an amplitude and a phase model by the methods described in Secs. III A 2 and III A 3, respectively.', '2. Amplitude model': "As can be seen in Fig. 1, the waveforms tend to circularize near merger. 10 Hence, in order to model this effect, we can join the eccentric inspiral to the circular IMR by suitable choice of an appropriate time t A match . The amplitude model is obtained by joining the eccentric inspiral with the circular IMR using a transition function over a time window of 500 M which ends at t A match . Given a target hybrid, and a trial choice of t shift , we start with a trial choice of t A match roughly 500 M before the merger and produce the amplitude model as given below, \nFIG. 6. Numerical fits for t A match and t ω match as well as for t shift are mapped into the physical parameter space for eccentric systems characterised by the binary's eccentricity, mean anomaly at a reference frequency and the mass ratio parameter q or η depending upon the model best-fitting the data. Circles represent the numerical data points while crosses represent the value returned by the analytical best-fit model. The best-fit model(s) predict the numerical estimates for attachment times and the time-shift within ± 1 M . \n<!-- image --> \nA model 22 ( t ) ≡ τ a ( t ) A IMR 22 ( t ) +(1 -τ a ( t )) A inspiral 22 ( t ) , (10) \nwhere τ a ( t ) is defined as \nτ a ( t ) ≡ 0 if t < t i t -t i t f -t i if t i ≤ t < t f 1 if t f ≤ t . (11) \nWe set t i = t A match -500 M and t f = t A match as the bounds of the time interval over which the two waveforms are joined. Figure 5 demonstrates the process. The grey region is the time interval ending at t A match where the inspiral and circular IMR is joined. \nAfter the amplitude model is obtained for a particular choice of trial t shift and t A match , we combine it with the target hybrid phase to obtain the polarizations and then calculate the match with the target hybrid. 11 We then change the trial choice of t A match by 5 M , bringing it closer to the merger, and repeat the process of producing the amplitude model, and calculating the match. This variation of t A match is done until roughly 30 M before merger. We thus obtain a set of match values for varying t A match but for a single trial t shift and pick the one that has the highest value of match. We repeat the exercise for other choices of t shift (trial t shift varies between -400 M and 400 M in steps of 5 M ) and find out the corresponding t A match with the highest value of match. Thus, we obtain a set of t shift and t A match pairs with a match value for each pair. From this set, the pair with the highest value of match is chosen as the \nnumerical estimate for t shift and t A match for a particular target hybrid. We obtain numerical estimates using the same process for all 20 target hybrids.", '3. Frequency model': 'For the frequency model, we follow a similar procedure as described in Sec. III A 2 with the only difference being the duration of the time interval where the inspiral frequency is joined with the circular IMR frequency. The value of t shift is fixed to the one that was obtained while producing the amplitude model. Similar to the amplitude model procedure, we determine an appropriate t ω match for joining the inspiral frequency with the circular IMR frequency. However, the time interval where the two are joined, starts at t ω match and ends at a time close to 30 M before merger. 12 Just like the amplitude model, we start with the choice of a trial value of frequency t ω match roughly 6000 M before merger and obtain the frequency model as given below, \nω model 22 ( t ) ≡ τ a ( t ) ω IMR 22 ( t ) +(1 -τ a ( t )) ω inspiral 22 ( t ) , (12) \nwhere τ a ( t ) is as defined in Eq. (11) with the difference being t i = t ω match and t f ≲ -30 M . Figure 5 demonstrates the process. \nOnce the frequency model is obtained for the choice of trial t ω match , we calculate the phase by integrating the frequency model. This is then combined with the amplitude model obtained for the same target hybrid (generated using the numerical estimate of t shift and t A match already obtained) to produce the polarizations and a match with the target hybrid is calculated. We \nFIG. 7. Top: Dominant mode ( /lscript =2, | m | =2) amplitude and frequency model, obtained by stitching an eccentric inspiral ( EccentricTD [41]) with a quasi-circular merger-ringdown model ( SEOBNRv5 [72]), at times predicted by the best-fit values of the (analytical) model are plotted against three representative target hybrids used in training the model. Bottom: one of the polarizations, obtained by combining the amplitude and the frequency model shown in the top panel for the q = 2 case, is shown as a visual proof of the quality of the model being presented. The transition from inspiral to merger-ringdown is shown (for visual clarity) at t A match given by Eq. (14) and has no impact on the plotted data. \n<!-- image --> \nthen change the trial choice of t ω match by 1 M , bringing it closer to the merger and repeat the process of producing the frequency model, and calculating the match. 13 Once again, we do this variation until roughly 30 M before merger to obtain a set of match values for varying t ω match and pick the one that has the highest value of match. The corresponding value of frequency t ω match is the numerical estimate for a particular target hybrid. We obtain numerical estimates for all 20 target hybrids using the same process.', 'B. Analytical model': "We have described the procedure of producing (numerical) time-domain model fits for the dominant mode model, where we used a set of 20 eccentric hybrids as targets to calibrate our model. For each hybrid, we obtained a numerical estimate for t shift , t A match , and t ω match . In order to be able to generate waveforms for an arbitrary configuration these numerical fits need to be mapped into the physical parameter space for eccentric systems characterised by binary's eccentricity, mean anomaly at a reference frequency and the mass ratio parameter. In this section, we determine a functional form by performing \nanalytical fits to these numerical estimates. For analytical fits, we consider only those simulations (13 of the 20) for which the match between numerical model and the corresponding eccentric hybrid is greater than 97% and collectively refer to them as training set and the remaining (7) simulations are categorized as testing set although only two of these (HYB:SXS:BBH:1357, 1371) can really be used to test the model as other simulations have initial eccentricities significantly larger than any of the training set hybrids and thus outside the calibration range for the model (See for instance, Table I). For this reason, when validating the model against hybrids we only include these two hybrids from testing set. The fitted functions obtained are of the form as mentioned below. \nt shift ( q, e, l ) = ∑ α,β,γ,δ A αβγδ e α q β cos( γ l + δ e l + a αβγδ ) , (13) \nfor time shift, where A αβγδ = a αβγδ = 0 for α + β > 3 and/or α > 2 and/or γ + δ > 1, and A α 001 = A 0 β 10 = A 0 β 01 = A 00 γδ = a αβ 00 = 0, \nt A match ( η, e, l ) = ∑ α,β,γ,δ B αβγδ e α η β cos( γ l + δ e l + b αβγδ ) , \n(14) \nfor amplitude, where B αβγδ = b αβγδ = 0 for α + β > 3 and/or α > 2 and/or γ + δ > 1, and B 0 β 10 = B 0 β 01 = B 00 γδ = b αβ 00 = 0, and \nFigure 6 shows comparisons between the numerically obtained values for t shift , t A match , and t ω match , with \n<!-- image --> \n/circledot \n<!-- image --> \n/circledot \nFIG. 8. Mismatch as a function of total mass for all 13 training set hybrids (thin lines) and for the two testing set hybrids ( HYB:SXS:BBH:1357 and HYB:SXS:BBH:1371 ; thick lines) when compared with dominant mode ( /lscript =2, | m | =2) quasicircular templates ( SEOBNRv5 [72]; left panel) and eccentric templates (model constructed in Sec. III; right panel). The two horizontal lines indicate a match of 96.5% (or mismatch of 3.5%) and 99% (or 1% mismatch), respectively, and the color bar displays initial eccentricity ( e 0 ) value at an initial GW frequency ( x 0 ) for the respective target hybrid (see Table I). \n<!-- image --> \n/circledot \n<!-- image --> \n/circledot \nFIG. 9. Same as Fig. 8 except mismatches are computed against an independent family of waveforms ( TEOBResumSDali [58]). Additionally, the parameter space explored by randomly sampling values for initial eccentricity ( e 0 ) and mean anomaly ( l 0 ) as well as of the mass ratio ( q ), in the range 0 ≲ e 0 ≲ 0 . 3, -π ≤ l 0 ≤ π , and 1 ≲ q ≲ 3, respectively. \nt ω match ( η, e, l ) = ∑ α,β,γ,δ C αβγδ e α η β cos( γ l + δ e l + c αβγδ ) (15) \nfor frequency, where C αβγδ = c αβγδ = 0 for α + β > 3 and/or α > 2 and/or γ + δ > 1, and C 0 β 10 = C 0 β 01 = C 00 γδ = c αβ 00 = 0. The values for the coefficients A αβγδ , B αβγδ , C αβγδ , a αβγδ , b αβγδ , and c αβγδ obtained by performing a fit to the numerical values are tabulated in Tables II-IV. \n, \nthe values predicted by our analytical fits. The predictions are within ± 1 M for numerical estimates for t shift , t A match and t ω match . We show amplitude and frequency comparison between the target hybrids and our models for three cases along with the full waveform for the q = 2 case in Fig. 7.", 'C. Validation': 'The performance of the model can be assessed from the plots against the new hybrids presented in Fig. 7, as well as from the mismatch (1 - M ) plots displayed in \nFig. 8. Note again, the new hybrids are purely based on the PN prescription, EccentricTD [41] - the same model that constitutes the inspiral part of the model constructed here. Certainly, the dominant mode model outperforms the quasi-circular templates. On top of that, unlike Paper I, the model provides ≥ 96 . 5% match against all the training set hybrids for almost the entire range of parameters considered here. Furthermore, we also try to test our model against the testing set hybrids using Nelder-Mead down-hill simplex minimization algorithm of Scipy [77] over the three initial parameters set ( e 0 , l 0 , f 0 ) for each testing set hybrid. The mismatch against testing hybrids are plotted as thick lines in both panels of Fig. 8. As can be seen there, for q = 1 (testing) hybrid, match is ≥ 96 . 5% for the entire range of total mass, while it degrades a little for q = 3 (testing) hybrid for the low mass range ( M ≲ 60 M /circledot ) which is consistent with the trends observed for mismatch against q = 3 training set simulations. \nNote that the target hybrids used in these mismatch computations include only the ( /lscript = 2 , | m | = 2) mode so as to assess the actual performance of the dominant mode model. Given the quality of analytical fits it is not surprising that the analytical model performs well against the set of hybrids used in training the model. Moreover, we retain the cases agreeing closely with the hybrids apart from two testing hybrids. (Only 13 out of 20 simulations were used in finding the analytical model.) Keeping this in mind we also try to test our model against an independent waveform family TEOBResumS-Dali [58]. For this comparison, we choose to sample a parameter space that is not identical to the training set hybrids. We choose to randomly sample the values of a reference eccentricity ( e 0 ), mass ratio ( q ) and reference mean anomaly ( l 0 ) in the range 0 ≲ e 0 ≲ 0 . 3, 1 ≲ q ≲ 3 and -π ≤ l 0 ≤ π , respectively. Note that the range for initial orbital eccentricity ( e 0 ) is slightly different from the ones spanned by the hybrids. While the upper value is chosen conservatively to take e 0 = 0 . 3 to reduce any systematic differences between the model and the waveform TEOBResumS-Dali [58] at large eccentricity values, near circular cases are also included to see if the model gradually produces the circular limit despite being trained on purely eccentric target models. \nNext, the template (dominant eccentric model) is optimized against the target ( TEOBResumS-Dali [58]) using the same minimization algorithm [77] used in validating the model against testing set hybrids. The mismatch plot obtained is shown in Fig. 9. Additionally, for comparison, mismatches of TEOBResumS-Dali [58] with quasi-circular SEOBNRv5 [72] templates are displayed in the left panel. Clearly, our model seems to do better compared to the circular templates at the low mass end where the overlaps are ≳ 96.5% for nearly the entire range of parameters considered in the comparison. \nThe mismatches are comparable for heavier systems as expected.', 'A. Eccentric model based on TaylorT2 phase and PN corrected amplitudes': 'In this section, we discuss the possibility of finding a suitable alternative to the model constructed in the previous section. As mentioned earlier in the Sec. I B, we propose to replace the PN model used in constructing the model in the previous section to include amplitude terms with higher PN accuracy in the model. We employ 3PN accurate expressions for the dominant mode amplitude of Refs. [39, 42, 43] and a 3PN accurate phasing (based on TaylorT2 approximant) [40] to construct this alternate model. Note that the inspiral part of the model presented in Sec. III was entirely based on the work of Ref. [41]. The EccentricTD approximant is 2PN accurate in phase and only Newtonian accurate in amplitude for the eccentricity related effects although is based on a superior (compared to TaylorT2 ) approximant namely TaylorT4 and should also work better for larger eccentricities as it includes corrections to 6th power in eccentricity while the TaylorT2 phase we use involves only leading order corrections of eccentricity although is 3PN accurate [40]. To test the performance of this model, we again compare this model with TEOBResumS-Dali [58] by computing overlaps on the same set of parameters used in validating the dominant mode model in Sec. III C. The results are shown in Fig. 10. Overlaps between the model and the target waveforms are better than ∼ 96 . 5% for almost the entire range considered here, making it a suitable alternative to the presented EccentricTD model.', 'B. Inclusion of higher order modes': 'In this section, we extend the dominant mode model of Sec. IV A to obtain a higher mode model by including ( /lscript , | m | )=(2, 2), (2, 1), (3, 3), (3, 2), (4, 4), (4, 3) and (5, 5) modes. These are precisely the modes which are included in SEOBNRv5HM (Ref. [72]) which we use for the merger-ringdown part and also included in the hybrids constructed in Sec. II B. The (eccentric) inspiral and (quasi-circular) merger-ringdown models are again attached using the analytical expressions for t shift , t A match and t ω match , obtained for the dominant mode model in Sec. III. (The higher mode model of Paper I was also obtained following the same strategy albeit, the HM model there only included the /lscript = | m | modes.) The inspiral part of the model for each nonquadrupole mode is obtained by combining the mode amplitudes obtained in Refs. [39, 42, 43] and orbital fre- \nFigure 12 compares our nonspinning model with the spinning version of TEOBResumS-Dali [58]. Thick curves represent the mismatch when the inclination angle ( ι ) is set to zero for which only | m | = 2 modes survive (in our case (2, 2) and (3, 2)). Thin lines on the other hand, show mismatches for the inclination angle of 30 · and thus display mismatches with all /lscript = | m | and /lscript -1 = | m | modes included in our higher mode model. For the equal mass case ( q = 1), we find excellent agreements with target models (matches ≳ 99%). For q = 2 case, the matches are still ≳ 96 . 5% for χ eff ≤ 0 . 1 for all systems with M ≳ 60 M /circledot . This clearly shows that despite being nonspinning in nature, the model could be used to analyse mildly spinning events observed routinely by current generation detectors such as LIGO and Virgo [59-61]. In other words, systematics due to neglect of spin effects may be ignored. \n<!-- image --> \n/circledot \nFIG. 10. Same as in Fig. 9 except target waveforms ( TEOBResumS-Dali [58]) are compared with the alternate dominant mode model presented in Sec. IV A. \n<!-- image --> \n/circledot \nFIG. 11. Similar to Fig. 10 except, that both the target and the model now include all modes included in our hybrids and the inclination angle of the binary is chosen to be 30 · . \nquency (multiplied by an appropriate factor involving mode number; see Eq. (4)). Figure 11 compares our higher mode model against the HM version of the waveform, TEOBResumS-Dali [58]. While all modes up to /lscript = 8 are included in TEOBResumS-Dali , we choose to include the same set of modes in the target and the model waveform to control the systematics. The orbital inclination angle is chosen to be 30 · . We find that the higher mode model recovers the target waveforms with accuracy better than 96 . 5% for nearly the entire range of parameter values considered here and thus may be used for analysing signals containing non-quadrupole modes and inclination angles ≤ 30 · . \n<!-- image --> \n/circledot \nFIG. 12. Recovery of mildly spinning target waveforms ( TEOBResumS-Dali [58]) with the alternate nonspinning model presented in Sec. IV A is displayed for two different mass ratio cases ( q = 1, 2) and a set of effective spin values (with χ eff ≤ 0 . 1).', 'C. Recovery of aligned spin binaries': "Although our alternate model (including higher modes) is nonspinning, we also test its performance against a spinning model for a few mildly spinning cases. We assume spin-precession to be absent and thus binary's spin is described solely by the effective spin parameter, χ eff . When expressed in terms of dimensionless spin components, χ i = ( /vector S i · ˆ L ) /m 2 i , it reads [78-80] \nχ eff = χ 1 m 1 + χ 2 m 2 m 1 + m 2 . (16) \nHere, m i refers to the mass of the binary component having spin angular momentum /vector S i , and ˆ L denotes the unit vector along the direction of the orbital angular momentum of the binary.", 'V. DISCUSSION AND CONCLUSION': 'Our current work is a follow up of our earlier work of Ref. [1] (referred as Paper I through preceding sections) where we developed a fully analytical dominant mode model for nonspinning binary black holes on elliptical orbits. The model was obtained by stitching an eccentric inspiral EccentricTD [41] with a quasi-circular merger ringdown prescription SEOBNRv4 [81]. Here, we revisit the construction of the model presented in Paper I following a new definition of orbital eccentricity presented in Refs. [68, 69]. \nWe started by comparing 20 distinct NR simulations from SXS collaboration presented in Ref. [48] with a PN prescription obtained by combing the results of Refs. [41, 43, 82] for a selected set of modes chosen based on their relative significance compared to the dominant modes. Figure 1 shows a comparison of the amplitude and frequency data for one particular dataset. The PN model is evolved using an initial set of binary orbital parameters such as orbital eccentricity consistent with the definition of Refs. [68, 69]. For completeness, we also show dominant mode comparisons for few other simulations in Fig. 2. We find that the PN and NR prescriptions have maximum overlap in a time-window of ( -2000 M, -1000 M ) shown by shaded regions of Fig. 1-2. In fact, we observe a minimum match of ∼ 96% for all 20 simulations sets against PN models in this window, allowing hybridization of the two in this window. Table I lists all hybrids constructed here. Figure 3 displays one of these hybrids and as shown there the hybrid are constructed for all modes included in the comparison presented in Fig. 1. The hybridization procedure is same as in Paper I and is reproduced in Sec. II B. However, the model constructed in Sec. III is trained, following Paper I, with a new set of dominant mode hybrids, which uses the same PN prescription used in our dominant mode model. Details of model construction is discussed in Sec. III A-III B and validation is performed in Sec. III C. \nWe validate the model against the hybrids used in training the model (training set in Table I) as well as two simulations (HYB:SXS:BBH:1357, 1371) from the testing set. 14 While consistency with training set was expected (but also confirms the quality of fits), better than 96 . 5% match with the two testing set simulations for most of the parameter space confirm the reliability of the model (see thick curves in the right panel of Fig. 8). Clearly the model also outperforms quasi-circular model shown in the left panel of Fig. 8. We also validate the model against an independent waveform family \nTEOBResumS-Dali [58] for values randomly sampled for initial values of eccentricity ( e 0 ), mass ratio ( q ), and mean anomaly ( l 0 ) in the range 0 ≲ e 0 ≲ 0 . 3, 1 ≲ q ≲ 3, and -π ≤ l 0 ≤ π , respectively. Note the difference in eccentricity scale explored here compared to the one in Fig. 8 (See, Sec. III C for a discussion). Figure 9 shows this comparison and the model recovers the target family with overlaps better than 96 . 5% for mass ratios ≤ 1 . 25. For q > 1 . 25, mismatches are slightly poorer for cases with M ≤ 50 M /circledot . \nWe also provide an alternate model by combining PN inputs from Ref. [40] and Refs. [39, 42, 43] for phase and amplitude part of the model respectively. The performance of this model against the waveform used in validating the model in Fig. 9 has been shown in Fig. 10. As can be seen, the performance of this alternate model is at the same level as of the model presented in Sec. III. Further, following Paper I, we also extended this alternate model to include few leading /lscript = | m | and /lscript -1= | m | modes (up to /lscript = 5) and compare it against waveforms of TEOBResumS-Dali [58] in Fig. 11 keeping the same set of modes in both target and template. As we can see the model at least reliably reproduces the target model for inclination angles ≤ 30 · . \nFinally, while our model(s) assumes spinless binary constituents, we also tried testing its suitability for analysing signals with small spin magnitudes in the absence of spin precession. Figure 12 compares our nonspinning model (including higher modes) presented in Sec. IV A-IV B against the HM model of TEOBResumS-Dali [58] (with spins switched on). As shown there, our model seems to be able to extract the equal mass spinning target waveforms with accuracy better than 99% as shown by the thick-solid lines. On the other hand, our model recover target model with matches larger than than 96.5% for the q = 2 case (thin-dashed lines). For higher spin magnitudes as well as higher mass ratios, mismatches are larger than 3.5% in the low mass range, at least for unequal mass cases. \nNote that, both the alternate model as well as the HM model were not obtained by calibrating against hybrids but rather we simply used the prescriptions presented in context of dominant mode model in Sec. III B and thus could be improved, however we restrict ourselves here to a proof of principle demonstration that alternate prescriptions for dominant mode model as well as simple extensions like the one proposed here could be easily achieved and perform reliably and leave such updates for a future work. Apart from being able to construct an improved model compared to the one presented in Paper I, in the current work we also address a few concerns with the model there. First, it was found that for nearly 10-15% cases, the analytical fits for times for merger-ringdown attachment produced nonphysical values (beyond merger at t > 0). Since the NR simulations \nused in this work essentially circularize by 30 M before the merger [48], we impose this condition for those nonphysical scenarios. This removes the discrepancy regarding the attachment times and ensures the validity of the model in the entire parameter space explored. Secondly, the model presented in Sec. IV A is significantly faster than the dominant mode model of Paper I. This is likely due to a speedup with merger-ringdown model ( SEOBNRv5 [72] instead of SEOBNRv4 [81]). The waveform generation rate of TaylorT2 model is ∼ 1 . 5 times higher than the model based on EccentricTD . Moreover, when compared with the waveform TEOBResumS-Dali [58], these waveforms nearly have a ∼ 2 times speedup and thus likely to be useful for parameter estimation studies.', 'VI. ACKNOWLEDGMENTS': "We thank Kaushik Paul and Prayush Kumar for sharing invaluable insights in validating our model. We thank the authors of Ref. [58] for making the implementation of the TEOBResumS-Dali available for public use [83], and Divyajyoti for helping us with technical details of the implementation and waveform generation. We are thankful to the SXS Collaboration for making a public catalog of numerical relativity waveforms. P.M. thanks the members of the gravitational wave group at the Department of Physics, IIT Madras for organizing the weekly journal club sessions and the insightful discussions. T.R.C. acknowledges the support of the National Science Foundation award PHY-2207728. C.K.M. acknowledges the support of SERB's Core Research Grant No. CRG/2022/007959. This document has LIGO preprint number LIGO-P2400355. We thank Md Arif Shaikh for useful comments and suggestions on our manuscript.", 'Appendix A: Coefficients of the analytical fit': "Here we tabulate the coefficients of the analytical fits for the parameters t shift , t A match and t ω match obtained in Sec III B. The expressions are given in Eq. (13), (14) and \n6 \n(15). \nTABLE II. Table of coefficients for the analytical expression of t shift in Eq. (13). All other coefficients not included in the table are zero. \nTABLE III. Table of coefficients for the analytical expression of amplitude t match in Eq. (14). All other coefficients not included in the table are zero. \nC \nα \n= 0 \n- \n1 \n. \n87711 \n× \n10 \n1 \n1 \n. \n30607 \n× \n10 \n2 \n- \n2 \n. \n51977 \n× \n10 \n7 \n7 \n8 \n. \n92879 \n× \n10 \n- \n5 \n. \n21053 \n× \n10 \n1 \n. \n04979 \n× \n10 \n6 \n7 \n8 \n- \n5 \n. \n75326 \n× \n10 \n6 \n0 \n0 \nTABLE IV. Table of coefficients for the analytical expression of frequency t match in Eq. (15). All other coefficients not included in the table are zero. \n- [4] R. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. X 11 , 021053 (2021), arXiv:2010.14527 [gr-qc].\n- [5] R. Abbott et al. (LIGO Scientific, VIRGO), Phys. Rev. D 109 , 022001 (2021), arXiv:2108.01045 [gr-qc]. \nαβ \n00 \nβ \n= 0 \n1 \n2 \n- [8] B. P. 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2024A&A...691A.180F

Context. Singledish observations suggest that the abundances of organic species in starforming regions of the outer Galaxy which are characterised by subsolar metallicities are comparable to those found in the local Galaxy. Aims. To understand this counterintuitive result and avoid a misleading interpretation due to beam dilution effects at these large distances spatially resolved molecular emission maps are needed to correctly link the measured abundances and local physical properties. Methods. We observed several organic molecules with the Atacama Large Millimeter Array towards WB89671 the source with the largest galactocentric distance 23.4 kpc of the project CHEMical complexity in starforming regions of the OUTer Galaxy CHEMOUT at a resolution of 15 000 au. We compared the observed molecular abundances with chemical model predictions. Results. We detected emission of cCSUB3SUBHSUB2SUB CSUB4SUBH CHSUB3SUBOH HSUB2SUBCO HCO HSUP13SUPCOSUPSUP HCSSUPSUP CS HNSUP13SUPC and SO. The emission morphology is complex extended and different in each tracer. In particular the most intense emission in HSUP13SUPCOSUPSUP HSUB2SUBCO and cCSUB3SUBHSUB2SUB arises from two millimetercontinuum infraredbright cores. The most intense CHSUB3SUBOH and SO emission predominantly arises from the part of the filament that lacks continuum sources. The narrow line widths across the filament indicate quiescent gas in spite of the two embedded protostars. The derived molecular column densities are comparable with those in local starforming regions and they suggest an anticorrelation between hydrocarbons ions HCO and HSUB2SUBCO on the one hand and CHSUB3SUBOH and SO on the other. Conclusions. The static chemical models that match the observed column densities best favour lowenergy conditions that are expected at large galactocentric radii but they also favour carbon elemental abundances that exceed those derived by extrapolating the CH galactocentric gradient at 23 kpc by three times. This would indicate a flatter CH trend at large galactocentric radii which is in line with a flat abundance of organics. However to properly reproduce the chemical composition of each region models should include dynamical evolution.

2024-11-01T00:00:00Z

['2024arXiv240907243F', '2024A&A...691A.180F', 'arXiv:2409.07243', '10.48550/arXiv.2409.07243', '10.1051/0004-6361/202451500']

['stars: formation', 'ISM: molecules', 'Astrophysics - Astrophysics of Galaxies']

CHEMOUT CHEMical complexity in starforming regions of the OUTer Galaxy IV. ALMA observations of organic species at a galactocentric radius of 23 kpc

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']

https://arxiv.org/pdf/2409.07243.pdf

{'IV. ALMA observations of organic species at Galactocentric radius ∼ 23 kpc': "F. Fontani 1 , 2 , 3 , G. Vermariën 4 , S. Viti 4 , D. Gigli 1 , 5 , L. Colzi 6 , M.T. Beltrán 1 , P. Caselli 2 , V.M. Rivilla 6 , and A. 7 , 8 \nSánchez-Monge \n- 1 INAF - Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50125, Florence, Italy e-mail: francesco.fontani@inaf.it\n- 2 Max-Planck-Institut für extraterrestrische Physik, Giessenbachstraße 1, 85748 Garching bei München, Germany\n- 3 LERMA, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Université, F-92190 Meudon, France\n- 4 Leiden Observatory, Leiden University, PO Box 9513, 2300 RA Leiden, The Netherlands\n- 5 Leiden Observatory, Leiden University, PO Box 9513, 2300 RA Leiden, The Netherlands\n- 6 Dipartimento di Fisica e Astronomia, Università di Firenze, Via G. Sansone 1, 50019 Sesto Fiorentino, Firenze, Italy\n- 7 Centro de Astrobiología (CSIC-INTA), Ctra Ajalvir km 4, 28850, Torrejón de Ardoz, Madrid, Spain\n- 8 Institut de Ciències de l'Espai (ICE, CSIC), Campus UAB, Carrer de Can Magrans s / n, 08193, Bellaterra (Barcelona), Spain\n- 9 Institut d'Estudis Espacials de Catalunya (IEEC), 08860 Castelldefels (Barcelona), Spain \nReceived XXX; accepted XXX", 'ABSTRACT': 'Context. Single-dish observations suggest that the abundances of organic species in star-forming regions of the outer Galaxy, characterised by sub-Solar metallicities, are comparable to those found in the local Galaxy. \nAims. To understand this counter-intuitive result, and avoid misleading interpretation due to beam dilution e ff ects at such large distances, spatially resolved molecular emission maps are needed to link correctly measured abundances and local physical properties. Methods. We observed several organic molecules with the Atacama Large Millimeter Array towards WB89-671, the source with the largest Galactocentric distance (23.4 kpc) of the project "CHEMical complexity in star-forming regions of the OUTer Galaxy" (CHEMOUT), at a resolution of ∼ 15000 au. We compared the observed molecular abundances with chemical model predictions. \nResults. We detected emission of cC3H2, C4H, CH3OH, H2CO, HCO, H 13 CO + , HCS + , CS, HN 13 C, and SO. The emission morphology is complex, extended, and di ff erent in each tracer. In particular, the most intense emission in H 13 CO + , H2CO and cC3H2 arises from two millimeter continuum, infrared-bright cores. The most intense CH3OH and SO emission arises predominantly from the part of the filament with no continuum sources. The narrow linewidths across the filament indicate quiescent gas, despite the two embedded protostars. Derived molecular column densities are comparable with those in local star-forming regions, and suggest anti-correlation between hydrocarbons, ions, HCO, and H2CO on one side, and CH3OH and SO on the other. \nConclusions. Static chemical models that best match the observed column densities favour low energetic conditions, expected at large Galactocentric radii, but carbon elemental abundances 3 times higher than that derived extrapolating the [C / H] Galactocentric gradient at 23 kpc. This would indicate a flatter [C / H] trend at large Galactocentric radii, in line with a flat abundance of organics. However, to properly reproduce the chemical composition of each region, models should include dynamical evolution. \nKey words. astrochemistry - line: identification - ISM: molecules - stars: formation', '1. Introduction': 'The outer Galaxy (OG), i.e. the part of the Milky Way that extends out of the Solar circle to the outermost edge of the Galactic disc ( ∼ 27 kpc, López-Corredoira et al. 2018), was believed to be an environment not optimal for the formation of molecules and planetesimals. The reason is its sub-Solar metallicity, i.e. the low abundance of elements heavier than helium. In particular, the elemental abundances of oxygen, carbon, and nitrogen, i.e. the three most abundant elements in the Universe after hydrogen and helium, and the most important biogenic elements, decrease as a function of the Galactocentric distance, R GC (Esteban et al. 2017; Arellano-Córdova et al. 2020; Méndez-Delgado et al. 2022). For example, the fractional abundance of oxygen relative to hydrogen, [O / H], decreases gradually from the inner Galaxy to the OG, reaching ∼ 1 / 5th of the Solar value at about 20 \nkpc (see e.g. Esteban et al. 2017; Méndez-Delgado et al. 2022). For carbon, the decrease of the [C / H] ratio is even more pronounced, reaching ∼ 1 / 7th-1 / 8th of the Solar value at ∼ 20 kpc (Arellano-Córdova et al. 2020; Méndez-Delgado et al. 2022). Such low abundances of heavy elements in the OG suggested in the past that this zone was not suitable for forming planetary systems in which Earth-like planets could be born and might be capable of sustaining life (Ramírez et al. 2010). For this reason, the OG was excluded from the so-called Galactic Habitable Zone (GHZ), which is the portion of the Milky Way with the highest chance to form and develop complex forms of life on (exo-)planets (Gonzalez et al. 2001). Because of this, its chemical complexity has been so far little explored. \nHowever, we are discovering that the presence of small, terrestrial planets is independent on the Galactocentric distance \n(e.g. Buchhave et al. 2012; Maliuk & Budaj 2020), and that molecules, including complex organic molecules (COMs, organic species with 6 or more atoms), are found to be morethan-expected abundant in star-forming regions with metallicity lower than Solar, both in the OG (e.g. Blair et al. 2008; Shimonishi et al. 2021; Bernal et al. 2021; Fontani et al. 2022a) and in external galaxies (e.g. Shimonishi et al. 2018; Sewiło et al. 2018). However, molecular formation processes could be di ff erent from those in the inner or local Galaxy due to the di ff erence in both metallicity and other environmental conditions. For example, UV irradiation from high-mass stars should be lower, on average, in the OG due to the smaller concentration of high-mass stars. Therefore, abundances of species sensitive to this parameter could be a ff ected also by this environmental change. It is thus crucial to observe molecules in star-forming regions of the OG to constrain models adapted to such sub-Solar metallicity environments. \nThe few studies performed so far in the OG mentioned above have provided abundances only of a limited number of abundant species, and hence they can answer only partial questions. Blair et al. (2008), Bernal et al. (2021), and Fontani et al. (2022b) detected formaldehyde (H2CO) and methanol (CH3OH) in starforming regions of the OG up to 24 kpc. H2CO is an important precursor of CH3OH, the simplest complex organics, because in cold star-forming cores its formation proceeds on the surface of dust grains via successive hydrogenation of CO (HCO → H2CO → CH3O / CH2OH → CH3OH, e.g. Pauly & Garrod 2018). However, H2CO can also form in the gas-phase, unlike CH3OH, in regions where a significant fraction of C is not yet locked into CO (Ramal-Olmedo et al. 2021). If so, H2CO should form together with species like carbon-chains, which need a large amount of C not locked in CO as well. This implies that in regions where the C / O ratio is smaller, most of the C should be in the form of CO, and carbon-chains should have a lower abundance than that of CH3OH. The OG is an environment that should have this property, because the [C / H] decrease with R GC is steeper than that of [O / H] (Arellano-Córdova et al. 2020; Méndez-Delgado et al. 2022). The vice-versa is expected in the inner Galaxy. On the other hand, a dust extinction lower in the OG than in the local Galaxy would imply an easier dissociation of CO in gas phase, favouring the formation of carbon chains. From this example, one can see that the formation of even a relatively simple organic molecule like H2CO, and its relation with CH3OH, can be di ff erent in the outer and inner Galaxy. \nThe project "CHEMical complexity of star-forming regions in the OUTer Galaxy (CHEMOUT Fontani et al. 2022a,b; Colzi et al. 2022) aims at studying the formation of molecules in the outer Galaxy based on observations of 35 molecular cloud cores associated with star-forming regions having R GC in between 9 and 23 kpc. Using the Institut de Radioastronomie Millimétrique (IRAM) 30m telescope, we detected several simple and complex carbon-bearing molecules including organics ( cC3H2, HCO + , H 13 CO + , HCO, C4H, HCS + , HCN, CH3CCH). Inorganic tracers of star-formation activity (SO, SiO, N2D + ) were detected as well. The detection of all these species should better constrain their formation / destruction pathways, and highlight similarities and di ff erences with those known to be e ffi cient in the local / inner Galaxy, where the C / O ratio is di ff erent. However, the results of Blair et al. (2008), Bernal et al. (2021), and Fontani et al. (2022a,b) are based on observations of single-dish telescopes, and thus provide only abundances averaged over angular scales of their main beams ( ∼ 27 -63 \'\' ). At the distance of the CHEMOUT targets, i.e. 8-15 kpc from the Sun (Fontani et al. 2022a), an angular scale of 27 \'\' corresponds to 1 -2 pc, or \n200 000 - 400 000 au, that is at least a factor 10 larger than the typical linear scale of a single star-forming core (0.05-0.1 pc). \nIn this work, we present high-angular resolution images obtained with the Atacama Large Millimeter Array (ALMA) to resolve the molecular emission towards the source WB89-670 (IRAS 05343 + 3605 Wouterloot & Brand 1989), WB670 hereafter. The source is located at a R GC of ∼ 23 . 4 kpc (heliocentric distance ∼ 15 . 1 kpc, Fontani et al. 2022a), in direction of the Galactic anticentre (Galactic coordinates Lon. = 173.014 · , Lat. = 2.38 · ). This is the CHEMOUT target with the largest Galactocentric distance, thus associated with the smallest environmental [C / H] and [O / H] of the sample, as well as with the lowest C / O ratio (Esteban et al. 2017; Arellano-Córdova et al. 2020; Méndez-Delgado et al. 2022). Extrapolating the elemental Galactocentric gradients of carbon and oxygen measured by Méndez-Delgado et al. (2022) to 23.4 kpc, the fractional abundances [C / H] and [O / H] are 1.8 × 10 -5 and 6.7 × 10 -5 , respectively. Considering the Solar values of [C / H] ∼ 2 . 6 × 10 -4 and [O / H] ∼ 3 . 1 × 10 -4 , the [C / O] ratio should be ∼ 3 . 1 times lower than Solar in the natal cloud of WB670. The source harbours near- and mid-infrared sources detected in the images of the Two Micron All-Sky Survey (2MASS, Cutri et al. 2003) and the Wide-field Infrared Survey Explorer (WISE, Wright et al. 2010, Fig. 1), associated with molecular gas where rotational transitions of cC3H2, C4H, CCS, H 13 CO + , HCO + , and HCN (Fontani et al. 2022a), CH3OH (Bernal et al. 2021), and H2CO (Blair et al. 2008) were detected. No COMs except CH3OH were identified in this source. The narrow width at half maximum (about 1 kms -1 ) in the line profile of HCO + J = 1 -0 suggests that the bulk emission in this transition is from quiescent material, but the simultaneous detection of non-Gaussian wings at high velocities also indicates the presence of embedded protostellar activity (Fontani et al. 2022a). \nThe paper is organised as follows: the observations and the data reduction are described in Sect. 2. The observational results are shown in Sect. 3. The spectral analysis and derivation of the molecular column densities is illustrated in Sect. 4. A discussion of the results and a comparison with chemical mdelling is provided in Sect. 5. Conclusions and future perspectives are given in Sect. 6.', '2. Observations and data reduction': "The observations were carried out with ALMA during Cycle 9 in three dates: December 27, 2022, using 45 antennas, and January 8 and 9, 2023 (project 2022.1.00911.S, P.I.: F. Fontani), using 41 antennas. The phase centre was set to the equatorial coordinates R.A.(J2000) = 05 h 37 m 41.9 s and Dec(J2000) = 36 · 07 ' 22 '' . The Local Standard of Rest velocity is -17.4 km s -1 (Fontani et al. 2022a). We observed several spectral windows in bands 3 and 4. Information about their central frequencies, spectral resolution, and sensitivity, are given in Table 1. For all the spectral windows, the sources used as flux and bandpass calibrators were J0423-0120 in band 3 and J0854 + 2006 in band 4, while J0547 + 2721 and J0550 + 2326 were used as gain (amplitude and phase) calibrators. The uncertainties in the flux calibration are ∼ 5%. The primary beam (i.e. the full width at half maximum of the main beam of a 12m antenna) ranges from ∼ 67 '' at 84.5 GHz to ∼ 57 '' at 99.3 GHz, and from ∼ 41 '' at 140 GHz to ∼ 37 '' at 153.5 GHz. \nThe calibrated data were produced by the calibration pipeline of the Common Astronomy Software Applications (CASA, McMullin et al. 2007). The pipeline version used is casapipe-6.4.1. Imaging and deconvolution were then performed on the cali- \nFig. 1. Near- and mid-infrared images of WB89-670. The top row panels show the images at 1.25 µ m(H) and 2.15 µ m(K s ) from the 2MASS survey. The panels in the bottom row show the WISE images at the wavelengths indicated in the top-left corner. The white circles are the ALMA primary beams at ∼ 86 GHz and ∼ 150 GHz, that are 66 '' (i.e. ∼ 7 . 5 pc) and 38 '' (i.e. ∼ 4 . 3 pc), respectively. \n<!-- image --> \nTable 1. Spectral setup and observational parameters. \nNotes. ( a ) Central frequency of the spectral window (spw); ( b ) Spectral resolution in frequency and velocity; ( c ) Root mean square noise per channel; \nbrated uv tables with the gildas 1 software, after conversion of the calibrated uv tables from measurement sets in uvfits format and then in gildas format. To be sensitive to the source extension, the \nTable 2. Spectral parameters of detected lines. \nNotes. ( a ) All spectral parameters are taken from the Cologne Database for Molecular Spectroscopy (CDMS a ; Endres et al. 2016), except those of HCO, which are taken from the Jet Propulsion Laboratory (JPL, Pickett et al. 1998)) catalogue; \nclean maps were created using natural weighting. As significant emission is detected at the edge of the primary beam, we analyse the primary-beam-corrected images. We attempted to perform self-calibration, but the self-calibrated images did not show any significant improvement due to the faintness of the continuum emission, and hence we analysed the non-self-calibrated images. In the channel maps, the root mean square (rms) varies between 1.8 and 3.9 mJy beam -1 in band 3, and between 8.8 and 18 mJy beam -1 in band 4. In the continuum images, the rms is ∼ 0 . 025 mJy beam -1 in band 3 and ∼ 0 . 1 mJy beam -1 in band 4. The angular resolution in both ALMA bands is ∼ 1 . 4 -1 . 5 '' , corresponding to ∼ 0 . 1 pc, or ∼ 20000 au. In all the images, the maximum recoverable scale is ∼ 20 '' . The detected molecular lines and their spectroscopic parameters are listed in Table 2. \nFig. 2. Continuum emission detected towards WB89-670 with ALMA. The light blue contours in the left and right panels correspond to the 3 mm and 2 mm continuum emission, respectively. Contours start from the 3 σ rms level ( ∼ 8 × 10 -5 Jy at 3 mm and ∼ 3 × 10 -4 Jy at 2 mm), and are in steps of ∼ 5 × 10 -5 Jy and ∼ 2 × 10 -4 Jy, respectively. The synthesised beam is depicted in the bottom left corner, and the white circle indicates the ALMA primary beam at the two wavelengths. The region shown in the 2 mm images corresponds to the dashed square illustrated in the top-left panel. The heat colour image in background is the K s band of 2MASS (Fig. 1) in the top panels, and the WISE 22 µ m band in the bottom panels. \n<!-- image -->", '3.1. Millimeter continuum emission': 'The millimeter continuum emission detected towards WB670 in both ALMA bands is shown in Fig. 2. Two compact sources are detected. The strongest one, that we call N, is detected in the 3 mm image with signal-to-noise of 14 (peak flux density ∼ 3 . 5 × 10 -4 Jy). Core N is undetected or barely detected in the near-infrared, and it is clearly detected in the mid-infrared. The other one, called S, coincides with the main infrared source detected in all 2MASS and WISE images, and is detected in the 3 mm image with signal-to-noise of 7 (peak flux density ∼ 1 . 8 × 10 -4 Jy). The flux densities, F ν , integrated inside the 3 σ rms contour levels are given in Table 3 for both sources, together with other observational properties.', '3.2. Spectra extracted from the millimeter continuum sources': 'The integrated spectra extracted from the contour level at 3 σ rms of the 3 mm continuum emission towards cores N and S (Fig. 2) are shown in Fig. 3. The figure shows the detected transitions and the 3 σ rms level in the spectra. The chemical richness towards N and S is similar, but the line intensities are overall higher towards N. We will discuss in more detail the chemical di ff erences between the two cores in Sect. 4.1.', '3.3. Emission morphology of molecular lines': "The molecular transitions listed in Table 2 were detected in some regions of the final images with a signal-to-noise ratio ≥ 5 (the 1 σ rms level is in Table 1). For each species, the velocityintegrated emission of the most intense transitions, namely cC3H2 J ( Ka , Kb ) = 2(1 , 2) -1(0 , 1), H 13 CO + J = 1 -0, CH3OH \nJ ( Ka , Kb ) = 2(0 , 2) -1(0 , 1)A + , CS J = 2 -1, SO J ( K ) = 3(2) -2(1), and H2CO J ( Ka , Kb ) = 2(1 , 2) -1(1 , 1), is shown in Fig. 4. The integration interval in velocity is defined by the channels with signal-to-noise ratio ≥ 3 (see caption of Fig. 4). The other species detected, namely HCS + , C4H, HCO, HN 13 C, and 34 SO, are too faint across the whole mapped region to derive a good integrated map showing their morphology. \nOverall, the emission peak of cC3H2, H 13 CO + , and H2CO coincides with that of the dust continuum millimeter cores N and S, while that of SO and CH3OH is located towards a northwestern elongated feature that extends for ∼ 20 '' from core N to the north-western border of the primary beam. The emission is extended in all tracers, and shows a filamentary structure oriented SE-NW, whose extension and width depends on the tracer. The filament is narrow in cC3H2, H 13 CO + , and CH3OH, and goes roughly from the millimeter core S to the north-western edge of the primary beam. The emission of CS is much broader and arises also from a clump located to the south-eastern edge of the primary beam. This southern clump, extended about ∼ 15 '' in NW-SE direction, is detected also in SO and H2CO, while is not detected in cC3H2, H 13 CO + , and CH3OH. The emission of SO resembles that of CH3OH in the northern part of the source and around core N, while towards the southern part of the source and around core S is di ff erent. The emission of H2CO is compact and arises mostly from the two millimeter cores, maybe because the E u of their transitions are higher ( ∼ 22 -23 K) than those of the other lines ( ∼ 4 -12 K). The presence of H2CO in the northern filament cannot be determined because of the limited primary beam, but the integrated emission seems to follow the filament at least inside the primary beam. \nTo allow for a better inspection of the emission structure of each tracer, the six maps in Fig. 4 are also shown, enlarged, in Figs. A.1-A.6 of Appendix A.", '3.4. Spectra extracted from molecular emission regions': 'The complexity of the source structure described in Sect. 3.3 and the fact that di ff erent species emit preferentially in di ff erent volumes indicates a chemical di ff erentiation at small spatial scales. Due to this di ff erentiation, we analyse the molecular emission dividing the source in seven regions, based on the emission maps of the species with highest di ff erentiation. The regions are shown in Fig. 4, and are defined roughly on the integrated emission maps of cC3H2, CH3OH, SO, and H2CO: three are based on the dominant clumps detected in cC3H2 (labelled as 1, 2, and 3 in Fig. 4), one is based on the CH3OH emission in the northwestern filament (4), and one on the southern clump seen in SO (5). To complement the analysis, we also extract spectra from the two dominant clumps detected in H2CO (6 and 7), which overlap partially with regions 1 and 2 though. \nThe spectra extracted from these seven regions are shown in Figs. B.1 and B.2. In Table 4, we give the equivalent angular and linear size of the seven regions, derived as the diameter of a circle with the same area. The regions have sizes in between 0.45 and 1.06 pc.', '3.5. Comparison between ALMA and IRAM spectra': "The cC3H2, H 13 CO + , and C4H lines in Table 2 were observed also with the IRAM-30m telescope by Fontani et al. (2022a). To evaluate if extended emission was resolved out by the interferometer, we extracted the integrated ALMA spectra in flux density units of the mentioned lines from a circular region with size \nTable 3. Parameters of the millimeter continuum sources \nNotes. ( a ) Flux density integrated in the 3 σ rms level; ( b ) Deconvolved angular diameter; ( c ) Deconvolved linear diameter; ( d ) H2 mass computed from the dust thermal continuum emission as described in Sect 4.2, assuming a dust temperature equal to the excitation temperature of CH3OH (Table 5), and a dust mass opacity coe ffi cient β = 1 . 7; ( e ) Dust opacity index computed comparing the integrated flux densities at the two frequencies observed as described in Sect 4.2. \nTable 4. Angular and linear size of the seven molecular emitting regions shown in Fig. 4. \nequal to that of the IRAM 30m telescope main beam ( ∼ 27 '' ). Then, we converted the IRAM 30m spectra from main beam temperature units to flux density units through the equation (see e.g. Fontani et al. 2021): \nF ν [mJy] = T MB[K] 1222 ν [GHz] 2 Θ [ '' ] 2 , (1) \nwhere F ν is the flux density at the observed rest frequency ν , T MB is the main beam temperature, and Θ the main beam angular size of the IRAM 30m telescope at frequency ν . \nThe comparison is shown in Fig. 5. The flux densities measured with ALMA are consistent with those measured with the IRAM-30m telescope in cC3H2 and H 13 CO + within a factor ≤ 10%, fully consistent with the calibration errors. For C4H, we show in Fig. 5 the J = 19 / 2 -17 / 2 hyperfine component, for which the di ff erence between the IRAM and ALMA spectra is the largest (a factor ∼ 1 . 3 -1 . 4). Considering the calibration errors and the rms noise in the spectra (in particular in the C4H IRAM 30m spectrum), we conclude that the amount of extended flux resolved out by ALMA is negligible and does not a ff ect significantly our analysis.", '4.1. Spectral analysis': "Flux densities were converted to brightness temperatures ( T B) according to Eq. 1. This relation needs the observed angular equivalent diameter (i.e. the diameter of the equivalent circle) of each core. Therefore, T B is an average brightness temperature over the solid angle of each core. We have analysed the \nspectra in T B units with the MAdrid Data CUBe Analysis ( mad -cuba 2 , Martín et al. 2019) software. The transitions reported in Table 2 were modelled via the Spectral Line Identification and LTE Modelling (SLIM) tool of madcuba . The lines were fitted with the AUTOFIT function of SLIM. This function produces the synthetic spectrum that best matches the data assuming a constant excitation temperature ( T ex) for all transitions of the same species. The other input parameters are: total molecular column density ( N tot), radial systemic velocity of the source ( V ), line full-width at half-maximum (FWHM), and angular size of the emission ( θ s). AUTOFIT assumes that V , FWHM, and θ s are the same for all transitions. For θ s, we assumed that the emission is more extended than the extraction region in all tracers, and hence no filling factor is appliyed in the derivation of the best fit parameters. Assuming as θ s the equivalent radii given in Table 4 would change the resulting column densities by at most a factor 1.2. We left all other parameters free except T ex, that we had to fix for all species except CH3OH, which is the only molecule for which two transitions with significantly di ff erent E u were detected (Table 2). We decided to adopt for all species the T ex measured from CH3OH. \nThe analysis described above with madcuba was adopted for the lines of these species: cC3H2, CH3OH, H2CO, C4H, and HCS + . For the remaining ones, namely HCO, H 13 CO + , HN 13 C, CS, SO, and 34 SO, madcuba cannot fit the lines in all regions because the observed FWHMs are comparable to the spectral resolution of their spectra (spw 33, 39, and 41 in Band 3). Towards region 5, CH3OH could also not be fit for the same reason. Therefore, for these species, we obtained the integrated intensities of the lines with the class 3 package of the gildas software. N tot were then calculated with the same equations used by mad -cuba , namely assuming LTE at T ex equal to that derived from CH3OH also in this case. We assumed also optically thin conditions. The assumption is justified both by the Gaussian line profiles, and by the fact that the optical depths provided by AUTOFIT for the lines analysed with madcuba are consistent with optically thin conditions. \nThe results obtained are listed in Table 5. One remarkable immediate result is the small values of the FWHMs of all lines, which are narrower than ∼ 2 kms -1 both in N and S. Even though the cores harbour infrared-bright sources, the FWHMs are narrow and comparable to those measured in starless cores in infrared dark clouds (e.g. Kong et al. 2017; Barnes et al. 2023) rather than to those measured in infrared-bright protostellar envelopes (e.g. Fontani et al. 2002, 2023). The excitation temperatures derived from CH3OH are ∼ 9 K and ∼ 15 K in core N \nFig. 3. Spectra extracted from the 3 σ rms level contour of the 3 mm continuum for cores N (top) and S (bottom) (Fig. 2). In the eight spectral windows, the clearly detected transitions are labelled, and the 3 σ rms level is represented by the red dashed line. The undetected species, namely HCS + , HCO and HN 13 C, are not shown. \n<!-- image --> \nand S, respectively. Such low temperatures could indicate subthermal excitation conditions, at least for CH3OH. Assuming the same T ex for the other molecules, as explained above, we derived N tot of the order of ∼ 10 12 cm -2 for H 13 CO + , and of the order of ∼ 10 13 cm -2 for CH3OH, C4H, H2CO, CS, and SO. \nAs stated above, to derive N tot we fixed T ex to that of CH3OH for all molecules. This assumption is justified by the fact that all transitions analysed have energies of the upper level in between ∼ 4 and ∼ 20 K, similar to the energy range of the methanol lines (7-20 K). However, the computed low T ex could indicate sub-thermal conditions for CH3OH. We estimate the uncertainty introduced by this simplified approach varying T ex in between the CH3OH values and a representative kinetic temperature of the gaseous envelope of high-mass protostellar objects ∼ 30 K (e.g. Sánchez-Monge et al. 2013; Fontani et al. 2015), in case \nCH3OH emission is sub-thermally excited. The N tot variation is dependent of the species and it is on average larger for core N, for which the di ff erence between T ex from CH3OH (9 K) and the representative temperature of 30 K is the highest. For example: in core N, N tot of CS increases from 1.7 × 10 13 to 2.6 × 10 13 cm -2 ; N tot of H 13 CO + increases from 1.5 × 10 12 to 2.8 × 10 12 cm -2 ; cC3H2 cannot be reasonably fit with a T ex above 15 K, if one considers both the detected transition and the upper limit on the transition J ( Ka , K , b ) = 4(3 , 2) -4(2 , 3), at ∼ 85 . 656 GHz but undetected; N tot of C4H decreases from 9.1 × 10 12 to 5.5 × 10 12 cm -2 ; N tot of H2CO remains the same within the errors. \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFig. 4. Velocity-integrated molecular emission maps. We show the six molecular species for which the signal-to-noise ratio is good. For C4H, HCS + , HCO, HN 13 C, and 34 SO, the maps are too noisy and we do not show them. The species, and the E up of the line used in temperature units, are indicated in the top-left corner of each frame. The two stars indicate the peak position of the millimeter continuum cores N and S. The black circles highlight the primary beam at the rest frequency of each line (Table 2). The integrated emission was computed in the channels with intensity higher than the 3 σ rms level. The velocity intervals used are: [ -18 . 4; -16 . 5] km s -1 for cC3H2; [ -19; -16] km s -1 for H 13 CO + ; [ -18 . 75; -16; 25] km s -1 for CH3OH (from the line centred at ∼ 96 . 7414 GHz) at [ -20; -14] km s -1 for CS; [ -19 . 5; -15 . 5] km s -1 for SO; [ -18 . 8; -16 . 2] km s -1 for H2CO (from the line centred at ∼ 140 . 8395 GHz). The seven regions highlighted in black are those analysed in detail in Sect. 3.4, and correspond grossly to the 5 σ rms emission contour of the corresponding integrated intensity map (within the ALMA primary beam). \n<!-- image --> \nFig. 5. Flux density comparison between ALMA and IRAM 30m spectra. The IRAM 30m and ALMA spectra are shown as black and red histograms, respectively. The ALMA spectra were extracted from an angular region equivalent to the IRAM 30m beam at the frequency of the lines ( ∼ 27 '' ). \n<!-- image -->", '4.2. Fractional abundances in the continuum cores': 'To estimate the molecular fractional abundances with respect to H2, we compute the column density of H2, N (H2), from the integrated continuum flux density through the equation (e.g. Battersby et al. 2014): \nN (H2) = γ F ν Ω κν B ν ( T d) µ (H2) m H , (2) \nwhere γ is the gas-to-dust mass ratio, F ν is the flux density at frequency ν , Ω is the source solid angle at the observed frequency, κν is the dust opacity usually parametrised as κν = κν 0 ( ν/ nu 0) β (Ossenkopf & Henning 1994), B ν ( T d) is the Planck function at dust temperature T d, µ (H2) is the mean molecular weight for which we will adopt 2.8 (Kau ff mann et al. 2008), and m H is the mass of the hydrogen atom. Using the empirical relation between γ and R GC derived from Giannetti et al. (2017), we find γ ∼ 3000 at R GC = 23 . 4 kpc. Considering the uncertainties in the Giannetti et al. (2017) empirical relation, γ is 3000 + 700 -2200 , hence more than 8 times higher than the conventional value of \n100. We adopt for T d the excitation temperature derived from CH3OH (Table 5) for each core, assuming coupling between gas and dust. We also assume dust opacity index β = 1 . 7, a typical value used for ice-coated dust grains in dense cores, and κν 0 = 0 . 899 at ν 0 = 230 GHz (Ossenkopf & Henning 1994). The resulting N (H2) are ∼ 5 . 7 × 10 21 and ∼ 2 . 1 × 10 21 cm -2 , respectively. Assuming spherical sources, we also compute the core H2 gas masses, Md, and volume densities, n (H2). Md and n (H2) are of the order of ∼ 1 M ⊙ and of ∼ 10 4 cm -3 , respectively, for both cores, with core N being more dense and massive. All parameters are listed in Table 3. We show the molecular fractional abundances derived from the N tot-toN (H2) ratio in Table 5. \nThese estimates are a ff ected by several uncertainties. First, as discussed in Sect. 4.1, N tot can vary up to a factor 2 depending on the species. Second, computing β from the integrated continuum flux densities in the 3 mm and 2 mm bands (Tab. 3) through Eq.(2) in Chacón-Tanarro et al. (2019b), we find β ∼ 0 . 34 and β ∼ 0 . 21 for core N and S, respectively. With these β , we find N (H2) ∼ 1 . 6 × 10 21 cm -2 for N and N (H2) ∼ 1 . 6 × 10 21 cm -2 for S, respectively, i.e. a factor ∼ 3 smaller than those given in Table 3. The molecular abundances in Table 5, hence, would increase systematically by the same factor. Another big uncertainty is introduced by the value of γ = 3000 + 700 -2200 calculated from Giannetti et al. (2017), which makes the abundances to vary up to an additional factor ∼ 4. There are also the uncertainties on β , which depend on the measured continuum flux densities and dust temperatures. While the relative error on the continuum flux densities is ∼ 10%, that on the dust temperature is di ffi cult to quantify. However, we stress that all these uncertainties a ff ect the individual abundances but not their ratios. \nIn the di ff use ISM, β is ∼ 2 (e.g. Draine & Lee 1984), while lower values are measured in dense cores (e.g. Forbrich et al. 2015; Galametz et al. 2019) and circumstellar discs (e.g. Testi et al. 2014; Friesen et al. 2018). β can depend a lot on grain size, composition, and porosity, but values lower than 0.5 are hard to explain without considering large grains of size ∼ 100 µ m 1 mm (e.g. Testi et al. 2014; Ysard et al. 2019). In protoplanetary discs, such large grains are expected as a result of coagulation and growth. However, cores N and S have a linear diameter ≥ 30000 au, hence the observed emission is unlikely dominated by a circumstellar disc. Values of β lower than 1 were measured also up to ∼ 2000 au scales around young protostars (e.g. Galametz et al. 2019), indicative of early grain growth, although theory is struggling in finding the possible reasons for such growth at the relatively low density of the 1000 -10000 au scale of protostellar envelopes. It has been proposed that large grains are not formed on such extended envelope scales, but transported there from the site of growth via jets and winds (e.g. Cacciapuoti et al. 2024). This scenario could be possible for WB670, but the narrow FWHMs of the observed lines in N and S (Tab. 5) suggest a very quiescent environment, and hence should be disregarded. Silsbee et al. (2022) proposed that the presence of very small grains could also cause a low β . This scenario may be possible if the distribution of grain size at the Galactocentric distance of WB670 is di ff erent from that in the local medium. Other options are also possible, like high dust opacities, and contamination from free-free emission at 3 mm. We did not find in the literature any free-free emission study towards WB670 (e.g. from radio-continuum emission), and hence this option cannot be checked. \nTable 5. Best-fit parameters of the millimeter continuum sources N and S spectra. \nNotes. ( a ) Fractional abundance with respect to H2, computed as N tot / N (H2). N (H2) is calculated from the continuum flux density in Table 3 as explained in Sect. 4.2.', '4.3. Column densities in the molecular emitting regions': 'In Appendix C, we list the parameters obtained fitting the lines with madcuba and class towards the lines detected in the seven regions indicated in Fig. 4. As for the continuum cores, the FWHMs are always narrower than 1.5-2 km s -1 , indicating quiescent gas. The T ex derived from the two detected CH3OH lines are in between 6.4 K and 15 K, similar to that measured towards cores N and S, suggesting that the emission arise from cold gas also in these regions. This is consistent again with the relatively low FWHMs of the lines, smaller than ∼ 1 -2 kms -1 , which indicates further that the emission arise from quiescent material. Regarding N tot, for CH3OH we derive values of the order of 10 12 -10 13 cm -2 , with the maximum value ( ∼ 1 . 6 × 10 13 cm -2 ) in the north-western filament in between core N and the edge of the primary beam. The hydrocarbons C4H and cC3H2 have similar column densities, both of the order of ∼ 10 12 cm -2 . For these regions we cannot derive abundances because we cannot estimate the H2 column densities, hence we will discuss the comparison between the N tot of the various species in Sect. 5.1.', '5.1. Column density comparisons': 'In Fig. 6 we compare the molecular column densities between di ff erent species in the seven regions of WB670. The relative N tot of the hydrocarbons C4H and cC3H2 are similar and do not change significantly in the sub-regions of WB670, as shown in panel (a) of Fig. 6. Both species require atomic C not locked in CO to form, therefore their good agreement in the di ff erent re- \ngions is consistent with the expectations. In panel (b) of Fig. 6 wecompare N tot of H2CO, HCO, and CH3OH, which are thought to be chemically related because all can be formed from hydrogenation of CO on dust grains. N tot of CH3OH is largely variable in the seven sub-regions, and the trend does not follow that of HCO and H2CO, since the higher N tot of CH3OH are found where those of HCO and H2CO are lower, and vice versa. \nThe same dichotomy is apparent also between CH3OH and cC3H2 (panel (c)), especially evident in region 1, where N tot of CH3OH is the lowest and that of cC3H2 is the highest, and in region 4, where the opposite happens. A similar dichotomy is observed in the pre-stellar core L1544 (Spezzano et al. 2016; Jensen et al. 2023), where the emission of the two molecules seems anti-correlated. In particular, CH3OH emission arises from colder and more shielded regions of the L1544 core envelope, while cC3H2 emission overlaps with the dust continuum emission. Such observational di ff erence is consistent with the di ff erent physical conditions needed to form the two molecules. While it is well-known that CH3OH is formed from sequential hydrogenation of CO on grain surfaces (e.g. Fuchs et al. 2009), the formation of cC3H2 occurs in the gas-phase through an ion-molecule reaction followed by dissociative recombination (e.g. Sipilä et al. 2016). These reactions do not need a lowtemperature and high-density environment like that required for CH3OH formation. Therefore, the dichotomy between these two species likely arises from such di ff erent formation routes. Interestingly, the variation of N tot in the seven regions found for H2CO and HCO (panel (b) in Fig. 6) resembles more that of cC3H2 than that of CH3OH. This points to formation routes of H2CO and HCO in gas-phase rather than from surface chemistry processes. In particular, while CH3OH can be formed only on the surfaces of dust grains given the ine ffi ciency of gas phase routes at low temperatures (e.g. Garrod et al. 2006), H2CO is also known to form in the gas phase from regions rich in hydrocarbons, where C is not yet completely locked in CO (see e.g. Chacón-Tanarro et al. 2019a). HCO can also form via gas-phase reactions (Rivilla et al. 2019). Our observations seem to indicate that the gas-phase formation of H2CO and HCO from atomic C is likely more e ffi cient than its formation on dust grains, despite the lower abundance of C at such large Galactocentric distances. \nThe trends of the molecular ions H 13 CO + and HCS + are also more in line with hydrocarbons and H2CO rather than with CH3OH (panels (d) and (e)). Again, this can be attributed to a formation of these ions only in gas-phase. The species for which the trend in N tot is the most similar to CH3OH is SO, as indicated in panel (f) of Fig. 6. This similarity could be explained by their common origin from surface chemistry. In fact, SO is thought to be formed in gas phase from atomic S (e.g. Vidal et al. 2017), more abundant upon grain sputtering than in the di ff use gas.', '5.2. Comparison with local and inner Galaxy star-forming regions': 'Extrapolating the elemental abundance trends measured by Méndez-Delgado et al. (2022) at the Galactocentric distance of WB670, the oxygen, carbon, and nitrogen fractional abundances should be: [O / H] ∼ 6 . 7 ± 2 . 3 × 10 -5 , [C / H] ∼ 1 . 8 ± 0 . 5 × 10 -5 , and [N / H] ∼ 5 . 3 ± 1 . 3 × 10 -6 , respectively. The reference values at the Solar circle are: [O / H] ∼ 3 . 1 ± 1 . 0 × 10 -4 , [C / H] ∼ 2 . 6 ± 0 . 8 × 10 -4 , and [N / H] ∼ 4 . 7 ± 1 . 4 × 10 -5 , respectively (Méndez-Delgado et al. 2022). Therefore, the relative elemental ratios [O / C] and [O / N] should increase from [O / C] ∼ 1 . 2 ± 0 . 7 and [O / N] ∼ 6 . 6 ± 4 . 0 to [O / C] ∼ 3 . 7 ± 2 . 0 and [O / N] ∼ 12 . 6 ± 7 . 6. A molecular ratio that, in principle, can be particularly sensitive to changes in the [O / C] \nratio is CH3OH / cC3H2 as found in the pre-stellar core L1544 (e.g. Spezzano et al. 2016), since to form CH3OH one needs carbon locked in CO and to form cC3H2 one needs free atomic carbon. Figure 7 shows the column density ratios CH3OH / cC3H2 in the seven molecular regions of WB670, and compare them to the ratios measured in local (low- and high-mass) star-forming regions (Higuchi et al. 2018) and in the outer Galaxy hot-core WB89-789 (hereafter WB789, Shimonishi et al. 2021). As representatives of local low-mass star-forming regions, we took the cores in the Perseus molecular clouds observed by Higuchi et al. (2018). As representatives of high-mass star-forming regions, we took IRAS 20126 and AFGL 2591 (Freeman et al. 2023). The linear scale resolved is ∼ 5000 -10000 au in Higuchi et al. (2018), while it is ∼ 200000 au in (Freeman et al. 2023). The CH3OH / cC3H2 ratios derived in WB670 agree on average with the lower values measured in local star-forming regions, while the ratio found in WB789 is twice the highest local values (Fig. 7). Because WB789 is located in between the local Galaxy and WB670, there does not seem to be a monothonic trend of the CH3OH / cC3H2 ratio with the Galactocentric distance. We will better discuss the comparison between WB670 and WB789 in Sect. 5.3. \nWe also checked if the column density ratios we derive in WB670 between species that di ff er from just one element such as HN 13 C and H 13 CO + , and CS and SO, can eventually be attributed to the change in elemental ratios with R GC. Figure 8 shows the column density ratios SO / CS (panel (a)). Despite a dispersion of an order of magnitude, the average value is SO / CS ∼ 1 . 2. Fontani et al. (2023) found average SO / 13 CS ratios ∼ 11 -12 in a sample of high-mass star-forming cores in di ff erent evolutionary stages, and in the local and inner Galaxy. This ratio translates into ∼ 0 . 2 when considering a conversion factor 12 C / 13 C ∼ 68 for the local ISM (Milam et al. 2005) (dashed line in panel (a) of Fig. 8). Scaling this value for the increase in the [O / C] elemental ratio ( ∼ 3 . 1) from the Solar circle to R GC ∼ 23 . 4 kpc (Méndez-Delgado et al. 2022), the expected [SO / CS] column density ratio in WB670 should be ∼ 0 . 62, that is a factor 2 smaller than the average measured one. But the disperion is such that, overall, the measured ratios are consistent with those measured in the Solar neighbourhoods scaled for metallicity. Panel (b) of Fig. 8 shows the H 13 CO + / HN 13 C column density ratios measured in WB670. The average value is ∼ 2 . 6. In this case the dispersion is only a factor ∼ 2. In local and inner Galaxy star-forming regions this molecular ratio is very variable. Vasyunina et al. (2011) measured an average HCO + / HNC ratio of ∼ 10 in infrared-dark clouds. Zinchenko et al. (2009) found H 13 CO + / HN 13 C ∼ 1 in a sample of high-mass star-forming regions, indicating that this ratio can be very sensitive to changes in the physical conditions. Because WB670 contains infraredbright objects, its physical conditions are likely similar to those of the active star-forming regions observed by Zinchenko et al. (2009). This means that the observed H 13 CO + / HN 13 C column density ratio increases on average by a factor 2.6 from local starforming regions to R GC ∼ 23 . 4 kpc, while the expected increase of the [O / N] elemental ratio should be ∼ 1 . 8. Therefore, again the increased elemental abundance ratio cannot fully explain the observed molecular ratios. However, considering the dispersion of the values in the literature, and the error introduced by the assumed T ex in our analysis, overall the ratios are consistent with a metallicity-scaled trend. On the other hand, the elemental ratios beyond R GC ∼ 15 kpc are poorly constrained by observations (e.g. Romano et al. 2020) and su ff er from uncertainties up to ∼ 30% on the individual elemental abundances (Méndez- \nFig. 6. Column density comparison in the seven molecular extraction regions. The colours in the di ff erent panels indicate the di ff erent molecular species according to the labels in the bottom-right corner (top-right corner in panel (e)). Empty symbols with an arrow pointing downwards are upper limits. \n<!-- image --> \nFig. 7. Column density ratio CH3OH / cC3H2 in WB670 and other starforming regions. The black points indicate the ratios measured in the seven regions of WB670 (with the exception of region 5, for which both column density estimates are upper limits). The green area correspond to the range of values measured in local star-forming regions (Higuchi et al. 2018), and the red dashed line is the ratio measured in the outer Galaxy hot-core WB789 (Shimonishi et al. 2021). \n<!-- image --> \nDelgado et al. 2022), and hence their extrapolation to such large R GC could be inaccurate.', '5.3. Comparison with sub-Solar metallicity hot-cores': 'Some species detected in WB670 were also detected in the hot core WB789 (Shimonishi et al. 2021), located at R GC ∼ 19 kpc. The species in common are: cC3H2, CH3OH, H2CO, H 13 CO + , SO, and CS. Shimonishi et al. (2021) derive two abundance val- \ns, computed on linear diameters of 0.026 and 0.1 pc, respectively. For the comparison with N and S, we use the 0.1 pc values because this size is more similar to that of N and S (Table 3). Figure 9 shows the comparison: except for SO, all other species show abundances significantly di ff erent from each other, indicating a clear di ff erent chemical composition. In particular, the CH3OH abundance in WB789 is two orders of magnitude higher than in N and S, while the abundances of all other carbon-bearing species are lower. In fact, the CH3OH abundance in WB789 is 2 × 10 -7 , while in N and S is 0 . 4 × 10 -9 and 11 × 10 -9 , respectively. Even considering the abundances obtained assuming β in Table 3, that are a factor ∼ 3 higher than those listed in Table 5, the CH3OH abundances in N, S, and WB789 are not consistent. Shimonishi et al. (2021) proposed a chemical stratification of the environment of the WB789 hot-core, with CH3OH and all COMs arising from the inner ( ≤ 0 . 015 pc) warm ( ≥ 100 K) region, and SO, CS, H2CO, cC3H2, H 13 CO + , and HN 13 C all associated with an external ( ≥ 0 . 05 pc) cold ( ≤ 40 K) envelope. Our findings also indicate that the species we detect are associated with relatively cold and quiescent envelope material, including CH3OH. However, the line widths in WB789 are always larger than ∼ 2 kms -1 even in the envelope tracers, indicating that WB670 and WB789 are di ff erent kind of star-forming regions. Simulations show that the abundance of CH3OH in dense star-forming core is extremely sensitive to dust temperature variations (Acharyya & Herbst 2015; Pauly & Garrod 2018). In particular, the CH3OH production on grain surfaces depends on the duration of the early cold phase in which CO is hydrogenated on grain mantles, owing to the high volatility of atomic hydrogen (see also Shimonishi et al. 2020). \n<!-- image --> \nFig. 8. Column density ratios in the seven molecular extraction regions of WB670. Panel (a) shows the measured SO / CS ratios (points), and compare them to the average SO / CS ratio measured in a sample of inner and local Galaxy high-mass star-forming regions (Fontani et al. 2023, dashed line), and to the expected SO / CS ratio obtained multiplying the local measured value (0.2, dashed line) by the increase in the elemental ratio [O / C] ∼ 3 . 1 (horizontal green line) from the Solar circle to R GC = 23 . 4 kpc (Méndez-Delgado et al. 2022). Panel (b) shows the same comparison between the measured H 13 CO + / HN 13 C ratio and the expected ratio obtained multiplying the local measured value for the increase in the [O / N] elemental ratio. \n<!-- image --> \nThe CH3OH abundances in N and S are comparable to the so-called organic-poor hot-cores detected in the Large Magellanic Cloud (LMC, Shimonishi et al. 2020; Hamedani Golshan et al. 2024), characterised by values that cannot simply be explained scaling the abundances measured in the local Galaxy for the decreased metallicity (about a factor 2-3, e.g. Andrievsky et al. 2001; Rolleston et al. 2002) of the LMC. Shimonishi et al. (2020) proposed that these organic-poor hot-cores in the LMC could be due to dust temperatures higher than those in organic-rich hot-cores. Therefore, the lower CH3OH abundance in WB670 could be due to an ine ffi cient (or insu ffi cient) hydrogenation of CO in the stage of ice formation, due to either a (too) warm dust temperature, or to a (too) short cold ice formation stage, or to a (too) low gas density. All scenarios would also predict a lower production e ffi ciency of H2CO on ice mantles, and this would be in agreement with our finding that H2CO should be formed in gas phase in WB670. Alternatively, WB670 can be in a late(r) evolutionary stage with respect to WB789, when the molecules in the inner protostellar cocoon(s), including those evaporated from dust grain mantles, have been mostly dissociated by the strong protostellar radiation field. Pauly & Garrod (2018) proposed that CH3OH abundance could be even enhanced in low-metallicity environments, owing to the lower C / O ratio which would imply that most of C is locked in CO, needed to form CH3OH. However, this scenario would predict that the CH3OH abundance in WB670 should be higher than in WB789, because the C / O ratio in WB670 is lower than in WB789 (according to the elemental trends with R GC, see Sect. 1). This is clearly at odds with our observational results. We will better discuss the influence of metallicity on the observed molecular abundances in Sect. 5.4.', '5.4. Chemical modelling': 'To better investigate if the measured column density ratios can constrain the initial elemental abundances, as well as other physical parameters that cannot be derived from the data, we compare our observational results to the prediction of chemical models. We investigate these clouds with the open-source gas-grain chemistry code UCLCHEM (Holdship et al. 2017). This timedependent astrochemical code allows us to model the evolution of each molecule. We considered a "static model", namely an isothermal cloud at constant density with a radius of R = 0 . 5 pc (the maximum radius of the modelled regions), and we explored \nFig. 9. Fractional abundances with respect to H2 derived for N, S, and WB789. The WB789 abundances were derived on a hot-core linear size of 0.1 pc (Shimonishi et al. 2021). \n<!-- image --> \na range of physical and chemical conditions. The parameters include H2 number density (cm -3 ), gas temperature, cosmic-ray ionisation rate ζ , radiation field F UV, and both the initial oxygen and carbon abundances as listed in Table 6. Each of the models is ran with an edge visual extinction of Av = 1 . 0 mag, and molecular ratios are computed at 10 6 years. The grid is discussed in more detail in Vermariën et al. (in prep.). We converted between 12 C / 13 Cusing a conversion factor of 10 1 . 8 ( ∼ 68) since the chemical network did not include isotopologues. For the modelling, we used a conventional gas-to-dust mass ratio of 100. We discuss the details of using the observed value obtained from Giannetti et al. (2017) at the end of this section. \nWeexcluded models for which the predicted fractional abundances of our molecules is below 10 -13 . This results in a total of 710 models out of the original 65536. These models can then be compared to the observations, and the distributions of the fit of the theoretical ratios with the observed ones are shown in Fig. 10, sorted in descending averaged mean squared error between the models and observations. Some ratios, namely CS / HCO + , CS / SO, HCO + / H2CO, HCO + / HCS + , SO / H2CO, SO / HCS + , CS / HCO,HCO / H2COandCS / HNC,have no overlap between the models and the observations. Some observed ratios also lie in the tails of the model distribution. Overall, it is very unlikely to find a model that fits all ratios well. \nFig. 10. A comparison of the distribution of log-ratios of the observed species, with the ratios shown in Fig. 6 highlighted in green. The seven observed region\'s ratios (coloured markers) are plotted against the distribution of the modelled ratios (grey boxen plots). The molecules are sorted from top to bottom in order of decreasing error. \n<!-- image --> \nF. Fontani et al.: CHEMOUT: CHEMical complexity in star-forming regions of the OUTer GalaxyTable 6. Ranges of physical and chemical parameters of the grid of models. \nIn order to further constrain the physical parameter space, the 50 models with the lowest Mean Squared Error (MSE) are chosen. Figure 11 shows again the comparison between the models and the observations. The distribution of the models is now closer to the observations, so the horizontal axis is more compact than in Fig. 10. Many of the ratios distance between models and observations decrease considerably by taking only the best models, but the CS, HCS + and H2CO based ratios do not improve. \nIn addition to the ratios that could not be explained by just the \'observable\' ratios, some ratios also have no matching distribution with the subset of best models. It does show however that we can fit many of the ratios well with the subset of best models. By plotting the error for each of the ratios in a pairwise grid, as can be seen in Fig. 12, we can better understand which molecules have the lowest error. This shows the molecules were best fit in the following order: cC3H2, HNC, HCO, C4H, CH3OH, HCO + , SO, H2CO, HCS + , and finally CS. We then plot the distribution of the physical and chemical parameters investigated in the grid (Fig. 13). The models indicate densities ranging from 10 3 to 10 3 . 6 cm -3 , temperatures from 20 to 45 K, radiation fields lower than 5 Habing, and cosmic-ray ionisation rate ζ in a wide range from the interstellar value of ∼ 10 -17 s -1 up to ∼ 10 -14 s -1 . Oxygen and carbon initial abundances both larger than 1 / 5th of the Solar value are favoured by the models. Additionally, the [O / C] ratio favours values in between ∼ 2 and 5, with a small distribution of even higher oxygen enhancements. The densities measured in N and S are higher ( ∼ 10 4 cm -3 ) than the values predicted by the models, which, however, refer to the molecular regions, all more extended than N and S. Moreover, assuming β as in Table 3 to compute N (H2), we derive densities of the order of ∼ 10 3 cm -3 , consistent with model predictions. \nLastly we investigate the time-dependence of these ratios (Fig. 14) in order to gauge whether the observed star-forming regions exhibit a younger or older chemical history. Figure 14 shows that for most molecules the fits would not improve at another time, apart from the H2O / CH3OH, CS / HNC and CS / SO ratios who might benefit from sampling at a later time. However this would make the fits for other ratios worse. For some molecules there seems to be no time at which the ratios are fit correctly. The worst performing molecule, CS, clearly shows an over-prediction in the model as compared to the observations. However, such inconsistency can be due to a neglected high optical depth in the derivation of the observed column densities. \nOverall, our results indicate that static models as those investigated here can give a range of physical parameters that best match with the observations, but are not optimal to reproduce the set of observed abundances. A more proper fit for each region would need dynamical modelling, accounting for collapse during the time dependent evolution. The range of values predicted from the static models suggest that low energetic conditions, in particular low temperatures (20-45 K) and F UV lower than 5 Habing, are favoured, consistent with the location of the source in the Galactic anti-Centre (certainly more "quiescent" \nthan the local and inner Galaxy). The models also predict that the oxygen elemental abundance should not be smaller than 1 / 5th of the Solar value, consistent with an extrapolation of the MéndezDelgado et al. (2022) trend which predicts a decrease of a factor 4.6. The carbon elemental abundance should also not be smaller than ∼ 1 / 5th of the Solar value, but this is well above the value extrapolated from the Méndez-Delgado et al. (2022) trend at 23.4 kpc, that is ∼ 1 / 14th of the Solar value. If confirmed by models that include dynamical collapse, such di ff erence would indicate a [C / H] gradient that flattens in the far-outer Galaxy with respect to the trend derived at inner R GC. Indeed, elemental Galactocentric gradients derived from observations of HII regions in spiral galaxies characterised by extended H I envelopes indicate a flattening of the radial distribution of metals at large galactocentric distances (Bresolin 2017). An elemental gradient of carbon flatter than expected in the OG would also explain why the abundance trends with R GC of organics tend to be less steep than the extrapolated gradients (Bernal et al. 2021; Fontani et al. 2022b). \nAnother caveat arises from the assumed gas-to-dust mass ratio, which could be higher than that assumed (100) according to the trend proposed by Giannetti et al. (2017) ( ∼ 3000 + 700 -2200 ). We thus performed two tests, one for a model at low density ( ∼ 10 3 cm -3 ) and one at high density ( ∼ 10 6 cm -3 ). In the low density case, the results do not show any significant di ff erences. In the high density case, some di ff erences are found, but this case does not represent WB670. Moreover, no firm measurements of this ratio are so far obtained, to our knowledge, at such large R GC, and hence assuming a gas-to-dust ratio significantly di ff erent from the standard one will introduce a further uncertainty in the models.', '6. Conclusions': 'We used ALMA to observe WB670, the source with the largest Galactocentric distance (23.4 kpc) in CHEMOUT, at a resolution of ∼ 15000 au. We detected emission of cC3H2, C4H, CH3OH, H2CO, HCO, H 13 CO + , HCS + , CS, HN 13 C, and SO, derived their column densities, and compared the observational results with chemical model predictions. The main results of our study are the following: \n- -The molecular emission arises from a filamentary structure oriented SE-NW, where multiple cores are detected. The filament seems more extended than the ALMA primary beam. The morphology is di ff erent in each tracer. The most intense emission of molecular ions, carbon-chain molecules, and H2COis associated with two millimeter continuum, infraredbright cores. On the contrary, the CH3OH and SO most intense emission arises predominantly from the part of the filament with no continuum sources. The narrow linewidths ( ∼ 1 -2 kms -1 ) across the filament indicate quiescent gas, despite the presence of the infrared-bright sources; \nFig. 11. The best 50 best models (grey boxen plots) compared to the 7 observed regions (coloured markers). \n<!-- image --> \nlog10 ratio \nFig. 12. The mean squared error (MSE) computed between all of the regions and the best 50 models. The averaged MSE for each molecule is included on the bottom row. \n<!-- image --> \n- -From a LTE analysis of the CH3OH lines, their excitation temperatures are quite low (7-15 K) and could be underthermally excited. Derived molecular column densities are comparable with those in local star-forming regions. There seems to be a spatial anti-correlation between the column density of hydrocarbons, molecular ions, HCO, and H2CO on one side, and CH3OH and SO on the other. This would possibly suggest di ff erent formation processes for the two groups of molecules (gas phase processes versus surface processes);\n- -CH3OH fractional abundances calculated towards the millimeter continuum cores (0 . 4 -11 × 10 -9 ) are consistent with those of the so-called organic-poor cores found in the LMC, where such low CH3OH abundances could be due to an ine ffi cient hydrogenation of CO on grain mantles;\n- -Static models that best match the observed column densities favour di ff use gas and low irradiation conditions (expected at large Galactocentric radii), but carbon elemental abundances 3 times higher than that derived extrapolating the [C / H] elemental Galactocentric gradient at 23 kpc. This would indicate a flatter [C / H] trend at large Galactocentric radii, in line with a flat abundance of organics. Models including dynamical evolution should be able to more properly reproduce the chemical composition of WB670. \nThe results of this work indicate that a proper comparison between observations and models starting from a huge grid of parameters is essential to properly model the chemistry. Our study would greatly benefit from new observations of more molecular species and more lines at the same (at least) spatial resolution as that obtained here. In particular, tracers of the cosmic-ray ionisation rate, basically unconstrained by our study, would be particularly relevant. \nAcknowledgements. This paper makes use of the following ALMA data: ADS / JAO.ALMA#2022.1.00911.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI / NRAO and NAOJ. F.F., S.V., G.V., and D.G. ackowledge support from the European Research Council (ERC) Advanced grant MOPPEX 833460. L.C. and V.M.R. acknowledges support from the grant PID2022-136814NB-I00 by the Spanish Ministry of Science, Innovation and Universities / State Agency of Research MICIU / AEI / 10.13039 / 501100011033 and by ERDF, UE; V.M.R. \nFig. 13. Akernel density probability estimate of the physical parameters for the best 50 models, fit separately for each of the regions, indicated by the coloured curves as labelled in the top-right. \n<!-- image --> \nalso acknowledge support from the grant RYC2020-029387-I funded by MICIU / AEI / 10.13039 / 501100011033 and by "ESF, Investing in your future", and from the Consejo Superior de Investigaciones Científicas (CSIC) and the Centro de Astrobiología (CAB) through the project 20225AT015 (Proyectos intramurales especiales del CSIC); and from the grant CNS2023-144464 funded by MICIU / AEI / 10.13039 / 501100011033 and by \'European Union NextGenerationEU / PRTR\'. A.S.-M. acknowledges support from the RyC2021-032892I grant funded by MCIN / AEI / 10.13039 / 501100011033 and by the European Union \'Next GenerationEU\' / PRTR, as well as the program Unidad de Excelencia María de Maeztu CEX2020-001058-M, and support from the PID2020117710GB-I00 (MCI-AEI-FEDER, UE). \nog10 ratio (A/B) \nL \nFig. 14. 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The map is the same shown in Fig. 4, so we refer to that figure caption for details. \n<!-- image --> \nFig. A.2. Same as Fig. A.1 for H 13 CO + J = 1 -0. \n<!-- image --> \nFig. A.3. Same as Fig. A.1 for CH3OH J ( Ka , Kb ) = 2(0 , 1) -1(0 , 1)A + . \n<!-- image --> \nFig. A.4. Same as Fig. A.1 for CS J = 2 -1. \n<!-- image --> \nFig. A.5. Same as Fig. A.1 for SO J ( K ) = 3(2) -2(1). \n<!-- image --> \nFig. A.6. Same as Fig. A.1 for H2CO J ( Ka , Kb ) = 2(1 , 2) -1(1 , 1). \n<!-- image -->', 'Appendix B: Spectra': 'Spectra of the detected lines, extracted from the regions identified in Sect. 3.3. \nFig. B.1. Spectra extracted from the cC3H2 region 1, 2, and 3, from top to bottom, in brightness temperature ( T B) units. On the x-axis we show the rest frequency. \n<!-- image --> \nFig. B.2. Same as Fig. B.1 for the regions 4, 5, 6, and 7, from top to bottom. \n<!-- image -->', 'Appendix C: Fit results': 'Fit results obtained with madcuba and class towards the lines detected in the seven molecular emitting regions identified in Fig. 4. \nTable C.1. Fit results of the molecular lines identified in region 1. \nNotes. Line parameters without uncertainties (quoted in brackets) were fixed in the fit. The error on N tot contains the calibration error of ∼ 10% when derived with class , while does not when derived with madcuba . Hence, in this case the total uncertainty is given by the quadrature sum of the quoted error and of a 10% calibration error. ( a ) Parameters obtained fitting the lines with madcuba ; ( b ) Parameters obtained fitting the lines with class ; ( c ) Parameters obtained fitting with a Gaussian profile the hyperfine component F = 1 -0, detected in almost all regions except regions 1 and 5; ( d ) Parameter not constrained because the uncertainty provided by the fit is higher than the output value. \nTable C.2. Same as Table C.1 for region 2. \nTable C.3. Same as Table C.1 for region 3.Table C.4. Same as Table C.1 for region 4. \nTable C.5. Same as Table C.1 for region 5. \nNotes. ( e ) Fit not performed because the line shows self-absorption at line centre; ( f ) Derived from the line at 96.7414 GHz. \nTable C.6. Same as Table C.1 for region 6. \nTable C.7. Same as Table C.1 for region 7.'}

2024arXiv240913500I

The curvature perturbation in a model of constantroll CR inflation is interpreted in view of the logarithmic duality discovered in Ref. 1 according to the delta N formalism. We confirm that the critical value betaddotvarphiHdotvarphi32 determining whether the CR condition is stable or not is understood as the point at which the dual solutions i.e. the attractor and nonattractor solutions of the field equation are interchanged. For the attractorsolution domination the curvature perturbation in the CR model is given by a simple logarithmic mapping of a Gaussian random field which can realise both the exponential tail i.e. the single exponential decay and the Gumbeldistributionlike tail i.e. the double exponential decay of the probability density function depending on the value of beta. Such a tail behaviour is important for e.g. the estimation of the primordial black hole abundance.

2024-09-01T00:00:00Z

['arXiv:2409.13500', '10.48550/arXiv.2409.13500', '2024arXiv240913500I']

['Astrophysics - Cosmology and Nongalactic Astrophysics', 'General Relativity and Quantum Cosmology', 'High Energy Physics - Theory']

Constant roll and nonGaussian tail in light of logarithmic duality

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.13500.pdf

{'Ryoto Inui, a Hayato Motohashi, b Shi Pi, c,d,e Yuichiro Tada, a,f and Shuichiro Yokoyama g,a,e': '- a Department of Physics, Nagoya University, Furo-cho Chikusa-ku, Nagoya 464-8602, Japan b Division of Liberal Arts, Kogakuin University, 2665-1 Nakano-machi, Hachioji, Tokyo, 1920015, Japan\n- c CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China\n- d Center for High Energy Physics, Peking University, Beijing 100871, China\n- e Kavli Institute for the Physics and Mathematics of the Universe (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan\n- f Institute for Advanced Research, Nagoya University, Furo-cho Chikusa-ku, Nagoya 4648601, Japan\n- g Kobayashi Maskawa Institute, Nagoya University, Chikusa, Aichi 464-8602, Japan \nE-mail: inui.ryoto.a3@s.mail.nagoya-u.ac.jp, motohashi@cc.kogakuin.ac.jp, shi.pi@itp.ac.cn, tada.yuichiro.y8@f.mail.nagoya-u.ac.jp, shu@kmi.nagoya-u.ac.jp \nAbstract. The curvature perturbation in a model of constant-roll (CR) inflation is interpreted in view of the logarithmic duality discovered in Ref. [1] according to the δN formalism. We confirm that the critical value β := ¨ φ/ ( H ˙ φ ) = -3 / 2 determining whether the CR condition is stable or not is understood as the point at which the dual solutions, i.e., the attractor and non-attractor solutions of the field equation, are interchanged. For the attractor-solution domination, the curvature perturbation in the CR model is given by a simple logarithmic mapping of a Gaussian random field, which can realise both the exponential tail (i.e., the single exponential decay) and the Gumbel-distribution-like tail (i.e., the double exponential decay) of the probability density function, depending on the value of β . Such a tail behaviour is important for, e.g., the estimation of the primordial black hole abundance.', '1 Introduction': "The slow-roll (SR) inflation scenario is a leading paradigm of the early universe. It naturally explains a long-lasting inflationary phase and also generates the almost scale-invariant power spectrum of the primordial scalar perturbation confirmed by cosmological observations such as measurements of the cosmic microwave background (CMB) anisotropies [2]. The scaleinvariant spectrum can be realised also in the other limiting case: ultra-slow-roll (USR) inflation [3-5]. In terms of the equation of motion of the inflaton field, \n¨ φ +3 H ˙ φ + V ' ( φ ) = 0 , (1.1) \nthe friction term and the potential force are balanced with the negligible acceleration in the SR case, while the friction dominates the potential force in the USR case as \n¨ φ H ˙ φ ≃ -3 . (1.2) \nHere φ is the homogeneous background of the inflaton field, V and V ' are the inflaton potential and its derivative, H is the Hubble parameter, and the dots represent the time derivatives. The fact that both limiting cases lead to the scale-invariant power spectrum is understood as a consequence of the so-called Wands duality [6] (see also Ref. [7] for a quadratic potential model) between the two independent solutions of the second-order equation of motion for the cosmological perturbation. \nThe scale-invariance of the power spectrum is not the unique feature of the USR inflation. It results in the O (1) non-Gaussianity (in terms of the non-linearity parameter) in the primordial perturbation even from the single-field inflation [5, 8-10]. Furthermore, it has been suggested that a non-perturbative signal may arise for large perturbations as an exponentially heavy tail of the probability density function (PDF) of the curvature perturbation R [9, 11, 12]. According to the δN formalism, it is understood as a result of a logarithmic map from the inflaton perturbation to the observable curvature perturbation. Recently, in Ref. [1], Sasaki and one of us gave a duality understanding between the two solutions of the equation of motion in this logarithmic map called the logarithmic duality . This clarifies the \nrelation among the ordinary Gaussian tail in the SR case, the exponential tail (i.e., the single exponential decay) in the USR case, and the Gumbel tail (i.e., the double exponential decay) in certain specific examples. \nIn this paper, we employ the logarithmic duality shown in Ref. [1] to understand the constant-roll (CR) inflation [13-16], which is a generalisation of the USR condition (1.2). There, it reads a certain constant β as \n¨ φ H ˙ φ = β, (1.3) \nwhich is not necessary -3. It smoothly connects the SR ( β → 0) and USR ( β →-3) limits as the corresponding solution (1.3) is an attractor for -3 / 2 < β while it is a non-attractor for β < -3 / 2. The CR model is also phenomenologically interesting because it can give rise to a blue-tilted spectrum to enhance the primordial perturbation on small scales and can realise exotic astrophysical objects such as primordial black holes (PBHs) [17, 18]. Making use of the logarithmic duality, in this paper we show that the blue-tilted CR attractor ( -3 / 2 < β < 0) results in the Gumbel-type tail, while the red-tilted attractor ( β > 0) exhibits the exponential tail. \nThe paper is organised as follows. We first review the logarithmic duality in Sec. 2. In Sec. 3, we apply the logarithmic duality to the CR inflation and see the Gumbel tail in the PDF of the curvature perturbation. Sec. 4 is devoted to discussion and conclusions. Throughout the paper, we adopt the Planck unit c = ℏ = M Pl = 1 where M Pl is the reduced Planck mass.", '2 Logarithmic duality': "Following Ref. [1], we briefly review the logarithmic duality of the curvature perturbations in the single-field inflation model. Let us use the number of e -folds which is defined by \nN = ∫ t end t H d t ( → d N = -H d t ) , (2.1) \nas a time coordinate. Assuming that the Hubble parameter, H , is almost constant and the potential V ( φ ) of the inflaton field φ is approximated by the quadratic form, the equation of motion for the inflaton can be considered to be [1] \nd 2 φ d N 2 -3 d φ d N +3 ηφ = 0 , (2.2) \nwhere η := V '' / (3 H 2 ). The general solution is given by \nφ ( N ) = C + e λ + ( N -N ∗ ) + C -e λ -( N -N ∗ ) , (2.3) \nwith \nλ ± = 3 ± √ 9 -12 η 2 . (2.4) \nNote that the two general solutions are degenerate ( λ + = λ -) when η = 3 / 4, which we do not consider. Then we always have λ -< λ + , and the solution with e λ -( N -N ∗ ) is the attractor solution as it dominates in the late stage when N → N ∗ . Here, we assume that the phase \nthat is controlled by the equation of motion (2.2) is temporal, and denote N ∗ as the time at the end of such a phase. Note that we have set the potential extremum at φ = 0 without loss of generality. \nBy taking the time derivative of this solution, we can obtain the expression for the 'field velocity', which is defined by π := -d φ /d N , as \n-π ( N ) = λ + C + e λ + ( N -N ∗ ) + λ -C -e λ -( N -N ∗ ) . (2.5) \n̸ \n̸ \nAssuming λ -= 0 (i.e., η = 0), from these expressions, the number of e -folds is expressed in terms of ( φ, π ) as \nN -N ∗ = 1 λ ± ln π + λ ∓ φ π ∗ + λ ∓ φ ∗ , (2.6) \nwhere π ∗ := π ( N ∗ ) and φ ∗ := φ ( N ∗ ). From this expression, the fluctuations of the number of e -folds from the spatially-flat hypersurface at N to the uniform φ ∗ hypersurface at N ∗ is [1] \nδ ( N -N ∗ ) = 1 λ ± ln [ 1 + δπ + λ ∓ δφ π + λ ∓ φ ] -1 λ ± ln [ 1 + δπ ∗ π ∗ + λ ∓ φ ∗ ] , (2.7) \nwhere δφ and δπ are the perturbation of the field and velocity on the initial hypersurface, respectively; and δπ ∗ := δπ ( N ∗ ) is the velocity perturbation on the uniform φ ∗ hypersurface at N ∗ . Note that π ∗ is given as a function of ( φ, π ): \n( π + λ + φ π ∗ + λ + φ ∗ ) λ + = ( π + λ -φ π ∗ + λ -φ ∗ ) λ -, (2.8) \nand δπ ∗ is determined by ( δφ, δπ ) as \n( 1 + δπ ∗ π ∗ + λ + φ ∗ ) -λ + ( 1 + δπ ∗ π ∗ + λ -φ ∗ ) λ -= ( 1 + δπ + λ -δφ π + λ -φ ) λ -( 1 + δπ + λ + δφ π + λ + φ ) -λ + . (2.9) \nActually, the equivalence of δ ( N -N ∗ ) in terms of the choice of λ ± is guaranteed by the above relations. This equivalence is called logarithmic duality in Ref. [1].", '3 Constant-roll model in light of logarithmic duality': "Let us take a closer look at the interplay between the CR model [13] and the quadratic potential. The CR condition is given by \nd 2 φ / d t 2 H d φ /d t = β , (3.1) \nwhere β is a constant. It is found that the potential, in which this CR condition is exactly satisfied, is given by [13] \nV ( φ ) = 3 M 2 M 2 Pl { 1 -3 + β 6 [ 1 -cosh ( √ -2 β φ M Pl )]} , ( β < 0) , 3 M 2 M 2 Pl { 1 -3 + β 6 [ 1 -cos ( √ 2 β φ M Pl )]} , ( β ≥ 0) , (3.2) \nand the solution for the Hubble parameter is given by \nH = { M coth( -βMt ) , ( β < 0) , -M tanh( βMt ) , ( β ≥ 0) . (3.3) \nThe curvature power spectrum evaluated at the horizon exit in the CR model exhibits the scale dependence of [13] \ndln P R ( k ) dln k = 3 -| 2 β +3 | , (3.4) \nand hence the spectrum is scale-invariant for β = 0 or -3, red-tilted for β < -3 or β > 0, and blue-tilted for -3 < β < 0. The red-tilted attractor β > 0 can be considered as an inflationary model to generate the primordial fluctuations on CMB scales consistent with the observational constraint [13, 14]. In this case, the potential is similar to the one for the natural inflation, but with an additional negative cosmological constant. Therefore, for a realistic model, we have to cut the potential somewhere before it becomes negative, and it has to be changed after that in order to have subsequent reheating and radiation-dominated regimes. Therefore, the CR phase is temporal in this case. On the other hand, the bluetilted attractor -3 / 2 < β < 0 can be considered as a model to generate a large peak of the curvature power spectrum on small scales, which can lead to the formation of PBHs [17]. To avoid the overproduction of PBHs, the CR phase has to be temporal in this case as well. Thus, in both cases, we are interested in the temporal CR attractor phase. Note that the CR condition with β < -3 / 2 is not an attractor [19], and the curvature perturbation grows on superhorizon scales [13]. \nFor φ/M Pl → 0 limit, which corresponds to Mt →∞ ( Mt →-∞ ) for β < 0 ( β ≥ 0), the potential can be approximated by the quadratic potential. Namely, we have \nV '' ( φ ) →-β (3 + β ) M 2 , H → M , (3.5) \nand then \nη := V '' 3 H 2 →-β ( β +3) 3 . (3.6) \nTherefore, in this limit, we can consider the CR model whose dynamics is characterized by the equation of motion (2.2): \nd 2 φ d N 2 -3 d φ d N +3 ηφ = 0 , (3.7) \nwith \nη = -β ( β +3) 3 , (3.8) \nwhich we call the quadratic constant-roll model. Note that, as mentioned above, we do not consider the degenerate case and hence assume η = 3 / 4, which amounts to β = -3 / 2. \n̸ \n̸ \nThen, based on the formulation given in the previous section, we can investigate the δN expression for the CR model. Note that the CR condition (3.1) with respect to the number of e -folds (2.1) is given by \nd 2 φ / d N 2 d φ /d N = -β , (3.9) \nwhere the Hubble parameter is approximately constant. \nπ \nFigure 1 . Left : Background trajectories in phase space for β = -1 as an example of the CR model with β > -3 / 2 where the condition (3.1) is stable. The red line shows the attractor solution with Eq. (3.13). Right : Attractor behaviour to the CR phase for β = -1. The horizontal axis is the number of e -folds measured from the initial time, N i -N , and the vertical axis is -(d 2 φ / d N 2 ) / (d φ /d N ). The magenta solid (dashed) line is for δ = 0 . 1 ( -0 . 1), and the cyan solid (dashed) line is for δ = 0 . 5 ( -0 . 5) with δ defined in Eq. (3.14). The red dashed line is the CR condition (3.9). \n<!-- image -->", '3.1 Stability of constant-roll condition in quadratic model': "For the CR inflation, there exists a critical value β = -3 / 2, which determines whether the CR condition (3.1) can be satisfied as the attractor solution [19]. Let us see how this critical value relates to the quadratic CR model. \nBy substituting Eq. (3.8) into Eq. (2.4), we have \nλ ± = 3 ±| 2 β +3 | 2 . (3.10) \nThus, the case with β = -3 / 2 gives the degenerate limit λ + = λ -= 3 / 2 [1]. If β > -3 / 2, then we have \nλ + = 3 + β , λ -= -β , (3.11) \nor if β < -3 / 2, then \nλ + = -β , λ -= 3 + β . (3.12) \nAs discussed in Ref. [1], because of λ + > λ -, the attractor solution for (2.3) is φ ( N ) ∝ C -e λ -( N -N ∗ ) which takes over at late times. In this sense, the CR condition (3.1) is stable for β > -3 / 2. Based on the formulation discussed in § 2, we should however take account of the other 'hidden' decaying solution with λ + = 3 + β to calculate δN . \nThe duality between β and -(3 + β ) is nothing but the duality in CR inflation found in Refs. [13, 19, 20]. Therefore, we expect that the logarithmic duality in the quadratic potential [1] can be generalized to the CR model. Below we check the above argument by solving numerically Eq. (3.7).", '3.1.1 β > -3 / 2': 'Based on the above discussion, in the case with β > -3 / 2, it is expected that the solution, \nφ ( N ) ∝ e -β ( N -N ∗ ) → π ( N ) = β φ ( N ) , (3.13) \nFigure 2 . Left : Background trajectories in phase space for β = -2 as an example of the CR model with β < -3 / 2 where the condition (3.1) is unstable. The red line shows the solution with Eq. (3.13) and the blue one corresponds to the solution with Eq. (3.15). Right : Attractor behaviour to the CR phase for β = -2. The horizontal axis is the number of e -folds measured from the initial time, N i -N , and the vertical axis is -(d 2 φ / d N 2 ) / (d φ /d N ). The magenta solid (dashed) line is for δ = 0 . 01 ( -0 . 01), and the cyan solid (dashed) line is for δ = 0 . 05 ( -0 . 05) with δ being defined in Eq. (3.14). The red (blue) dashed line is corresponding to the solution (3.13) (the solution (3.15)). \n<!-- image --> \nis the attractor. As can be seen in the left panel of Fig. 1, the background trajectories indeed converge into the attractor solution at a late time. In the right panel of Fig. 1, we can also check the stability with respect to the CR condition (3.9). In this plot, we change the initial condition for π as \nπ ( N i ) = β (1 + δ ) φ ( N i ) , (3.14) \nand vary the value of δ .', '3.1.2 β < -3 / 2': 'Based on the above discussion, in the case with β < -3 / 2, it is expected that the solution (3.13) is not an attractor one, while \nφ ( N ) ∝ e (3+ β )( N -N ∗ ) → π ( N ) = -(3 + β ) φ ( N ) , (3.15) \ncan be expected to be the attractor one, which corresponds to the λ --mode for the case with β < -3 / 2. As shown in the left panel of Fig. 2, the background trajectories indeed converge into the blue line which corresponds to the solution with Eq. (3.15), not the red line which corresponds to the one with Eq. (3.13). In the right panel of Fig. 2, we also investigate the stability of the background solution for the case with β = -2 < -3 / 2. From this figure, even for the model with constant Hubble and quadratic potential, one can also find the transition of the solution from -(d 2 φ / d N 2 ) / (d φ /d N ) = β (= -2) to -(3 + β ) (= -1) with the small perturbation for the initial condition, as seen in, e.g., Fig. 4 in Ref. [19]. The similar behaviour can be checked also in the case with β = -4 ( < -3) as shown in Fig. 3.', '3.2 δN in the constant-roll model': 'Let us evaluate the δ ( N -N ∗ ) in the CR model by making use of Eq. (2.7). We assume that the inflaton undergoes a sufficiently long CR regime, which already follows the attractor solution at N = N ∗ . Namely, at N = N ∗ we have \nπ ∗ ≈ -λ -φ ∗ , (3.16) \n<!-- image --> \nFigure 3 . Left : Background trajectories in phase space for β = -4 < -3. The red line shows the solution with Eq. (3.13) and the blue one corresponds to the solution with Eq. (3.15). Right : Attractor behaviour to the CR phase for β = -4. The horizontal axis is the number of e -folds measured from the initial time, N i -N , and the vertical axis is -(d 2 φ / d N 2 ) / (d φ /d N ). The magenta solid (dashed) line is for δ = 10 -8 ( -10 -8 ), and the cyan solid (dashed) line is for δ = 10 -5 ( -10 -5 ) with δ being defined in Eq. (3.14). This figure is, in fact, corresponding to Figure 2 in Ref. [1]. \n<!-- image --> \nwhere \n̸ \nλ -= { -β , ( β > -3 / 2 , β = 0) , 3 + β , ( β < -3 / 2 , β = -3) . (3.17) \n̸ \n̸ \n̸ \n̸ \nAs mentioned above, we focus on the case λ -= 0, i.e., we assume β = 0 and β = -3. Either β = 0 or 3 corresponds to λ -= 0 and λ + = 3, of which the non-attractor solution is the well-known USR inflaiton. \nAround N = N ∗ , (3.16) guarantees that π ∗ + λ -φ ∗ ≪ π ∗ + λ + φ ∗ , which turns Eq. (2.9) into [1] \n( 1 + δπ ∗ π ∗ + λ -φ ∗ ) ≈ ( 1 + δπ + λ -δφ π + λ -φ )( 1 + δπ + λ + δφ π + λ + φ ) -λ + λ -. (3.18) \nBy substituting this approximate form into Eq. (2.7) with the upper sign, we have \nδ ( N -N ∗ ) = 1 λ + ln [ 1 + δπ + λ -δφ π + λ -φ ] -1 λ + ln [ 1 + δπ ∗ π ∗ + λ -φ ∗ ] ≈ 1 λ -ln [ 1 + δπ + λ + δφ π + λ + φ ] . (3.19) \nThis resulting expression is nothing but the first (and also dominant) term on the right-hand side of Eq. (2.7) with the lower sign, which displays the logarithmic duality when the inflaton is already in the attractor at N ∗ . As mentioned in Ref. [1], in the case where the background dynamics is characterized by Eq. (3.7), due to the linearity of the equation, perturbations follow exactly the same equations for the background. Therefore, ( δπ + λ + δφ ) / ( π + λ + φ ) should be conserved on super-horizon scales, and we can evaluate this at the attractor phase where π ≈ -λ -φ and δπ ≈ -λ -δφ can be used. Finally, we can obtain the simple expression \n̸ \nδ ( N -N ∗ ) ≃ 1 λ -ln [ 1 + δφ φ ] = -1 β ln [ 1 + δφ φ ] , ( β > -3 / 2 , β = 0) , 1 3 + β ln [ 1 + δφ φ ] , ( β < -3 / 2 , β = -3) . (3.20) \n̸ \nOnce we assume that the curvature perturbations do not evolve after N = N ∗ , we can relate δ ( N -N ∗ ) with the late-time curvature perturbations denoted by R . By assuming the Gaussianity of δφ/φ , we have \n̸ \nR = δ ( N -N ∗ ) = -1 β ln [1 -β R G ] , ( β > -3 / 2 , β = 0) , 1 3 + β ln [1 + (3 + β ) R G ] , ( β < -3 / 2 , β = -3) . (3.21) \n̸ \n̸ \nR G := -δφ βφ , ( β > -3 / 2 , β = 0) , δφ (3 + β ) φ , ( β < -3 / 2 , β = -3) . (3.22) \n̸ \nand then we can discuss the non-Gaussianity of R through this expression. \nThe PDF of R can be obtained as \n̸ \nP ( R ) = ∣ ∣ ∣ ∣ d R g d R ∣ ∣ ∣ ∣ P g ( R g ( R )) = 1 √ 2 πσ 2 exp [ -1 2 β 2 σ 2 ( e -β R -1 ) 2 -β R ] , ( β > -3 / 2 , β = 0) , 1 √ 2 πσ 2 exp [ -1 2(3 + β ) 2 σ 2 ( e (3+ β ) R -1 ) 2 +(3 + β ) R ] , ( β < -3 / 2 , β = -3) , (3.23) \n̸ \nwhere σ 2 = ⟨R 2 G ⟩ . This expression is completely equivalent to Eq. (34) in Ref. [1] with replacing λ -to -β or 3 + β for the case with β > -3 / 2 or β < -3 / 2, respectively. \nFocusing on the stable CR models with β > -3 / 2, we see that there are two kinds of the tail behaviour of the PDF of the curvature perturbation R . For the case with -3 / 2 < β < 0, λ -= -β > 0 and this PDF corresponds to the Gumbel-distribution-like tail as \nP ( R ) ∼ exp [ -c 2 e -2 β R ] , (3.24) \nwith c 2 = (2 β 2 σ 2 ) -1 . On the other hand, a positive β leads to the exponential tail as \nP ( R ) ∼ exp[ -β R ] . (3.25) \nNote that the exponential tail has been known to show up from non-attractor models [9, 11, 12]. The exponential tail (3.25) from attractor models is a novel example. Recalling that the scale dependence of the power spectrum P R of the curvature perturbation is given by Eq. (3.4), i.e., d ln P R /dln k = -2 β in the CR model with β > -3 / 2, the tail behaviour can be classified by the spectral index: the blue-tilted curvature perturbations show the Gumbel tail while the red-tilted ones do the exponential tail.', '3.3 Consistency relation': "One may think of the so-called Maldacena's consistency relation [21] as our CR solution is an attractor for β > -3 / 2. The consistency relation claims that the correlation among n \nwhere \nhard modes and one soft mode is related to the scale dependence of the n -hard correlation function, known as a realisation of the cosmological soft theorems associated with the dilatation symmetry for the soft curvature perturbation or the shear transformation for the soft graviton [22, 23]. For example, the squeezed bispectrum of the curvature perturbation is related to the power spectrum by \nB R ( k 1 , k 2 , k 3 ) ∼ k 1 ≪ k 2 ∼ k 3 -dln P R ( k 2 ) dln k 2 P R ( k 1 ) P R ( k 2 ) , (3.26) \nwhere \n(2 π ) 3 δ (3) ( k 1 + k 2 ) P R ( k 1 ) = ⟨R k 1 R k 2 ⟩ , P R ( k ) = k 3 2 π 2 P R ( k ) , (2 π ) 3 δ (3) ( k 1 + k 2 + k 3 ) B R ( k 1 , k 2 , k 3 ) = ⟨R k 1 R k 2 R k 3 ⟩ , (3.27) \nwith the Fourier curvature perturbation R k . Substituting the second-order Taylor expansion of the logarithmic relation in the upper line of (3.21) into the definition of the bispectrum (3.27) and supposing the ordinary hierarchy P R G ( k 1 ) ≫ P R G ( k 2 ) for k 1 ≪ k 2 where P R G is the power spectrum for R G , one obtains the squeezed bispectrum as \nB R ( k 1 , k 2 , k 3 ) ∼ k 1 ≪ k 2 ∼ k 3 2 βP R G ( k 1 ) P R G ( k 2 ) . (3.28) \nOn the other hand, the CR is known to exhibit the scale dependence (3.4) at the leading order P R ≃ P R G , which results in -2 β for the model with β > -3 / 2 where the CR condition (3.1) can be realized as an attractor solution. The Maldacena's consistency relation (3.26) is hence confirmed to hold. In other words, the non-vanishing non-Gaussianity due to the nonlinear map (3.21) can be viewed as a consequence of the scale dependence of the power spectrum from the consistency relation as inferred at the end of the previous subsection. Note that, in contrast, the consistency relation is violated when the inflaton is in the non-attractor, such as CR inflation with β < -3 / 2 including USR inflation ( β = -3) [5, 8, 24].", '4 Discussion and conclusions': "̸ \nIn this paper, we investigated the CR inflation in light of the duality between solutions of the background equation of motion. While the logarithmic duality has recently been discussed in the quadratic potential, such duality also holds in a more general CR model. We confirmed that the critical value β := ¨ φ/ ( H ˙ φ ) = -3 / 2 determining whether the CR condition (3.1) is stable or not is understood as the point at which the dual solutions, i.e., the attractor and non-attractor solutions of the field equation, are interchanged. We have shown that the blue-tilted stable CR model, characterised by -3 / 2 < β < 0, results in the Gumbel tail in the PDF of the curvature perturbation, P ( R ) ∼ exp [ -c 2 e -2 β R ] , where the constant c in the Gumbel tail is related to the Gaussian variance σ 2 = ⟨R 2 G ⟩ by c 2 = (2 β 2 σ 2 ) -1 , while the red-tilted one, β > 0, shows the exponential tail P ( R ) ∼ exp[ -β R ]. In both cases, for β > -3 / 2 and β = 0 the curvature perturbation R is given by a (simplified) logarithmic map of a Gaussian random field R G as \nR ( x ) = -1 β ln [1 -β R G ( x )] . (4.1) \nThis is an interesting example of the realisations of the Gumbel tail, and also the exponential tail from an attractor model in contrast to ordinary non-attractor models exhibiting the exponential tail [9, 11, 12]. \nln \nFigure 4 . A schematic image of the peak of the power spectrum in a CR model. A CR phase with -3 / 2 < β < 0 yields a blue-tilted spectrum P R ∝ k -2 β and hence the curvatuer perturbations are amplified toward small scales. However, it also means that the CR phase must end for the power spectrum has a maximum dln P R /dln k = 0. \n<!-- image --> \nIn practice, the tail behaviour of the curvature perturbation has a significant effect on the abundance of rare astrophysical objects such as massive galaxy clusters [25], PBHs [11, 12, 26], etc. In particular, the PBH formation requires a substantial amplification of the curvature perturbation on a smaller scale ≲ Mpc, for which one needs to go beyond the simplest single-field SR paradigm [27] and hence to consider e.g., the USR or CR generalisations, or the multi-field extensions. The non-Gaussian tail is a crucial feature for such cases. It will also change the relation between the PBH abundance and the so-called scalar-induced gravitational waves as a byproduct of the large primordial perturbation [28-36]. While the impact of the exponential tail on them has intensively been studied (see, e.g., Ref. [37, 38]), we leave the Gumbel-tail effect for future work [39]. \nWe verified that Maldacena's consistency relation is valid in the attractor CR case ( β > -3 / 2), as the quadratic expansion of (4.1) together with (3.4) gives \nf NL = 5 6 β = -5 12 dln P R dln k . (4.2) \nHowever, one may note that the maximum of the power spectrum, which may correspond to the end of or after the CR phase (see Fig. 4 for a schematic image or Fig. 5 in Ref. [17] for a more realistic result), satisfies d ln P R /dln k = 0 by definition. Hence, whether the Gumbel (note that the power spectrum should be blue-tilted to have a peak) or another certain non-Gaussian feature is also exhibited on the peak scale is an intriguing question, as in fact the peak scale will be the most relevant to the PBH formation. We also leave it for future work.", 'Note added': 'During the preparation of our paper, Ref. [40] appeared in arXiv, which obtained a similar duality understanding of the CR inflation. The explicit illustration of the Gumbel tail is our \nnovelty.', 'Acknowledgments': "We would like to thank Qing Gao and Yungui Gong for discussions and useful comments. This work is supported in part by the National Key Research and Development Program of China Grant No. 2021YFC2203004 (SP), and by JSPS KAKENHI Grants No. JP22K03639 (HM), JP24K00624 (SP), JP21K13918 (YT), JP24K07047 (YT), JP20K03968 (SY), JP24K00627 (SY), and JP23H00108 (SY). SP is also supported by National Natural Science Foundation of China No. 12475066 and No.12047503. SP and SY are also supported by the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan. RI is supported by JST SPRING, Grant Number JPMJSP2125, and the 'Interdisciplinary Frontier NextGeneration Researcher Program of the Tokai Higher Education and Research System'.", 'References': "- [1] S. Pi and M. Sasaki, Logarithmic Duality of the Curvature Perturbation , Phys. Rev. Lett. 131 (2023) 011002 [ arXiv:2211.13932 ].\n- [2] Planck collaboration, Y. Akrami et al., Planck 2018 results. X. Constraints on inflation , Astron. Astrophys. 641 (2020) A10 [ arXiv:1807.06211 ].\n- [3] N. Tsamis and R. P. Woodard, Improved estimates of cosmological perturbations , Phys.Rev. D69 (2004) 084005 [ arXiv:astro-ph/0307463 ].\n- [4] W. H. Kinney, Horizon crossing and inflation with large eta , Phys.Rev. D72 (2005) 023515 [ arXiv:gr-qc/0503017 ].\n- [5] J. Martin, H. Motohashi and T. Suyama, Ultra Slow-Roll Inflation and the non-Gaussianity Consistency Relation , Phys. Rev. D 87 (2013) 023514 [ arXiv:1211.0083 ].\n- [6] D. Wands, Duality invariance of cosmological perturbation spectra , Phys. Rev. D 60 (1999) 023507 [ arXiv:gr-qc/9809062 ].\n- [7] K. Tzirakis and W. H. Kinney, Inflation over the hill , Phys. Rev. D 75 (2007) 123510 [ arXiv:astro-ph/0701432 ].\n- [8] M. H. Namjoo, H. Firouzjahi and M. Sasaki, Violation of non-Gaussianity consistency relation in a single field inflationary model , Europhys.Lett. 101 (2013) 39001 [ arXiv:1210.3692 ].\n- [9] Y.-F. Cai, X. Chen, M. H. Namjoo, M. Sasaki, D.-G. Wang and Z. Wang, Revisiting non-Gaussianity from non-attractor inflation models , JCAP 05 (2018) 012 [ arXiv:1712.09998 ].\n- [10] S. Passaglia, W. Hu and H. Motohashi, Primordial black holes and local non-Gaussianity in canonical inflation , Phys. Rev. D 99 (2019) 043536 [ arXiv:1812.08243 ].\n- [11] C. Pattison, V. Vennin, H. Assadullahi and D. Wands, Quantum diffusion during inflation and primordial black holes , JCAP 10 (2017) 046 [ arXiv:1707.00537 ].\n- [12] V. Atal, J. Cid, A. Escriv'a and J. Garriga, PBH in single field inflation: the effect of shape dispersion and non-Gaussianities , JCAP 05 (2020) 022 [ arXiv:1908.11357 ].\n- [13] H. Motohashi, A. A. Starobinsky and J. Yokoyama, Inflation with a constant rate of roll , JCAP 09 (2015) 018 [ arXiv:1411.5021 ].\n- [14] H. Motohashi and A. A. Starobinsky, Constant-roll inflation: confrontation with recent observational data , EPL 117 (2017) 39001 [ arXiv:1702.05847 ]. \n- [15] H. Motohashi and A. A. Starobinsky, f ( R ) constant-roll inflation , Eur. Phys. J. C 77 (2017) 538 [ arXiv:1704.08188 ].\n- [16] H. Motohashi and A. A. Starobinsky, Constant-roll inflation in scalar-tensor gravity , JCAP 11 (2019) 025 [ arXiv:1909.10883 ].\n- [17] H. Motohashi, S. Mukohyama and M. Oliosi, Constant Roll and Primordial Black Holes , JCAP 03 (2020) 002 [ arXiv:1910.13235 ].\n- [18] H. Motohashi and Y. Tada, Squeezed bispectrum and one-loop corrections in transient constant-roll inflation , JCAP 08 (2023) 069 [ arXiv:2303.16035 ].\n- [19] W.-C. Lin, M. J. P. Morse and W. H. Kinney, Dynamical Analysis of Attractor Behavior in Constant Roll Inflation , JCAP 09 (2019) 063 [ arXiv:1904.06289 ].\n- [20] M. J. P. Morse and W. H. Kinney, Largeη constant-roll inflation is never an attractor , Phys. Rev. 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Terada, Semianalytic calculation of gravitational wave spectrum nonlinearly induced from primordial curvature perturbations , Phys. Rev. D 97 (2018) 123532 [ arXiv:1804.08577 ].\n- [33] R.-g. Cai, S. Pi and M. Sasaki, Gravitational Waves Induced by non-Gaussian Scalar Perturbations , Phys. Rev. Lett. 122 (2019) 201101 [ arXiv:1810.11000 ].\n- [34] N. Bartolo, V. De Luca, G. Franciolini, A. Lewis, M. Peloso and A. Riotto, Primordial Black Hole Dark Matter: LISA Serendipity , Phys. Rev. Lett. 122 (2019) 211301 [ arXiv:1810.12218 ].\n- [35] S. Pi and M. Sasaki, Gravitational Waves Induced by Scalar Perturbations with a Lognormal Peak , JCAP 09 (2020) 037 [ arXiv:2005.12306 ].\n- [36] G. Dom'enech, Scalar Induced Gravitational Waves Review , Universe 7 (2021) 398 [ arXiv:2109.01398 ]. \n- [37] K. T. Abe, R. Inui, Y. Tada and S. Yokoyama, Primordial black holes and gravitational waves induced by exponential-tailed perturbations , arXiv:2209.13891 .\n- [38] S. Pi, Non-Gaussianities in primordial black hole formation and induced gravitational waves , arXiv:2404.06151 .\n- [39] R. Inui, C. Joana, H. Motohashi, S. Pi, Y. Tada and S. Yokoyama, in preparation.\n- [40] Y. Wang, Q. Gao, S. Gao and Y. Gong, On the duality in constant-roll inflation , arXiv:2404.18548 ."}

2024ApJ...975..160R

The characteristic timescale at which the variability of active galactic nuclei AGNs turns from red noise to white noise can probe the accretion physics around supermassive black holes SMBHs. A number of works have studied the characteristic timescale of quasars and obtained quite different scaling relations between the timescale and quasar physical properties. One possible reason for the discrepancies is that the characteristic timescale can be easily underestimated if the light curves are not long enough. In this work we construct welldefined AGN samples to observationally test the relationships between the characteristic timescale and AGN properties obtained by previous works. Our samples eliminate the effects of insufficient lightcurve lengths. We confirm that the timescale predictions of the Corona Heated Accretion disk Reprocessing model are consistent with our timescale measurements. The timescale predictions by empirical relations are systematically smaller than our measured ones. Our results provide further evidence that AGN variability is driven by thermal fluctuations in SMBH accretion disks. Future flagship timedomain surveys can critically test our conclusions and reveal the physical nature of AGN variability.

2024-11-01T00:00:00Z

['2024ApJ...975..160R', '10.3847/1538-4357/ad7b2a', 'arXiv:2409.09637', '10.48550/arXiv.2409.09637', '2024arXiv240909637R']

['Accretion', 'Light curves', 'Quasars', 'Supermassive black holes', '14', '918', '1319', '1663', 'Astrophysics - Astrophysics of Galaxies', 'Astrophysics - High Energy Astrophysical Phenomena']

How Long Will the Quasar UVOptical Flickering Be Damped II. The Observational Test

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']

https://arxiv.org/pdf/2409.09637.pdf

{'How Long Will the Quasar UV/Optical Flickering Be Damped? II. the Observational Test': '<!-- image --> \n1 \nDepartment of Astronomy, Xiamen University, Xiamen, Fujian 361005, China; msun88@xmu.edu.cn 2 Department of Astronomy, University of Science and Technology of China, Hefei 230026, China 3 School of Astronomy and Space Science, University of Science and Technology of China, Hefei 230026, China', 'ABSTRACT': 'The characteristic timescale at which the variability of active galactic nuclei (AGNs) turns from red noise to white noise can probe the accretion physics around supermassive black holes (SMBHs). A number of works have studied the characteristic timescale of quasars and obtained quite different scaling relations between the timescale and quasar physical properties. One possible reason for the discrepancies is that the characteristic timescale can be easily underestimated if the light curves are not long enough. In this work, we construct well-defined AGN samples to observationally test the relationships between the characteristic timescale and AGN properties obtained by previous works. Our samples eliminate the effects of insufficient light-curve lengths. We confirm that the timescale predictions (Zhou et al. 2024) of the Corona Heated Accretion disk Reprocessing model are consistent with our timescale measurements. The timescale predictions by empirically relations (e.g., Kelly et al. 2009) are systematically smaller than our measured ones. Our results provide further evidence that AGN variability is driven by thermal fluctuations in SMBH accretion disks. Future flagship timedomain surveys can critically test our conclusions and reveal the physical nature of AGN variability. \nKeywords: Accretion (14); Light curves (918); Quasars (1319); Supermassive black holes (1663)', '1. INTRODUCTION': "Active Galactic Nuclei (AGNs) are the brightest persistent sources in the Universe. They emit non-stellar continuum emission in a broad range of electromagnetic spectrum from radio to X-ray and γ -ray (for related reviews, see, e.g., Urry & Padovani 1995; Ulrich et al. 1997). The strong continuum emission, which varies significantly on timescales of minutes, hours, days, and months, is usually believed to originate in the central supermassive black hole (SMBH) accretion disk. The central engine of an AGN is often too compact to resolve directly because of its small physical scale and high redshift. The multi-band inherent and complex variations (i.e., light curves) play an essential role in probing the properties of the central engine, and the physical processes of AGNs (e.g., Cackett et al. 2021). \nThe study of AGN multi-band light curves provides valuable information about the AGN structure and \nmsun88@xmu.edu.cn \nphysical properties. By measuring the time lags between light curves at different wavelengths, the sizes of the accretion disk, broad line region, and dusty torus, and even the black hole mass of the SMBH can be measured using the reverberation mapping (RM) technique (e.g., Blandford & McKee 1982; Peterson 1993; Cackett et al. 2021). The black hole masses of more than 100 AGNs have been measured by the RM technique in many previous works (e.g., Kaspi et al. 2000; Peterson et al. 2004; Kaspi et al. 2007; Denney et al. 2010; Xiao et al. 2011; Grier et al. 2012; Bentz et al. 2013; Barth et al. 2015; Edelson et al. 2015; Shen et al. 2015; Fausnaugh et al. 2016; Grier et al. 2017; Du et al. 2018; Lu et al. 2022; Pandey et al. 2022; Chen et al. 2023; Malik et al. 2023; Zastrocky et al. 2024). Some AGNs are extremely variable on scales of months or years. These highly variable AGNs may be recognized as changing-look AGNs (CLAGNs) and challenge our understanding of the SMBH accretion (e.g., LaMassa et al. 2015; Yang et al. 2018; MacLeod et al. 2019; Guo et al. 2020; Wang et al. 2024; Zhu et al. 2024). \nAlthough many previous works studied AGN UV/optical variability (e.g., MacLeod et al. 2010; Gard- \nner & Done 2017; Lawrence 2018; Hagen et al. 2024; Kammoun et al. 2024), the physical mechanisms that cause AGN UV/optical variations are not yet fully clear. A number of theoretical models are proposed. For instance, Dexter & Agol (2011) proposed the strongly inhomogeneous quasar accretion disk model and assumed that the temperatures vary independently around the corresponding effective temperature of the standard thin accretion disk (hereafter SSD; Novikov & Thorne 1973; Shakura & Sunyaev 1973) in each independent disk zone. This model can explain the observed half-light radii from quasar microlensing observations and the observed variability amplitude. Cai et al. (2018) proposed that the local independent temperature fluctuations are affected by the large-scale temperature fluctuation. This model can reproduce the inter-band correlation and time lag of the UV/optical bands and the timescale-dependent bluer-when-brighter color variation. Krolik et al. (1991) proposed the X-ray reprocessing model (i.e., the lamppost model) to explain the time lags of multi-band light curves of AGNs. The X-ray reprocessing model has been adopted to explain continuum RM (e.g., Cameron et al. 2012; Shappee et al. 2014; Noda et al. 2016; Pahari et al. 2020; Fian et al. 2024). Sun et al. (2020) proposed the Corona Heated Accretion disk Reprocessing (CHAR) model to describe the AGN UV/optical variability. The CHAR model can yield the dependence of UV/optical variability on AGN luminosity, black hole mass, and wavelength. \nThe light curves of quasars can be studied by spectral techniques that include power spectra (e.g., Kelly et al. 2009). The stochastic variability (i.e., the light curves) of quasars can be well fitted by the damped random walk (DRW) model (e.g., Kelly et al. 2009; Kozglyph[suppress]lowski et al. 2010; MacLeod et al. 2010; Dexter & Agol 2011; Zu et al. 2013; Suberlak et al. 2021), albeit some optical light curves of AGNs diverge from this model (e.g., Mushotzky et al. 2011; Graham et al. 2014; Kelly et al. 2014). According to the DRW model, the power spectral density (PSD) of an optical light curve is well described as a power law function P ( f ) = f -2 at the high-frequency end, and white noise at the lowfrequency end. The transition frequency between the high-frequency end and the low-frequency end is f 0 = 1 / (2 πτ ), where τ represents the theoretical/intrinsic damping timescale that characterizes the light curve. Alternative models (e.g., the damped harmonic oscillator, a.k.a., DHO; Moreno et al. 2019) are proposed to fit quasar UV/optical light curves. Given the data quality, it is often true that the model parameters cannot be simultaneously well constrained in models that are more complicated than the DRW model. Further- \ne, the damping timescale of the DRW model is related to the timescales of the DHO model (e.g., Yu et al. 2022). Hence, there are still observational efforts desired to measure the damping timescale. \nSeveral previous works obtained the relationships between the theoretical/intrinsic damping timescale ( τ ) in the rest frame and several physical parameters of AGNs, including the wavelength ( λ ), monochromatic luminosity ( L λ ), and SMBH mass ( M BH ). Kelly et al. (2009) analyzed 100 quasars' optical light curves, and found that, \nτ days =(80 . 4 +66 . 9 -35 . 8 )( λL λ 10 45 erg s -1 ) -0 . 42 ± 0 . 28 × ( M BH 10 8 M ⊙ ) 1 . 03 ± 0 . 38 . (1) \nMacLeod et al. (2010) obtained τ for ∼ 9000 spectroscopically confirmed quasars' 10-year light curves in Sloan Digital Sky Survey (SDSS) Stripe 82 and reported the relationship between τ and λ , and M BH , i.e., τ ∝ λ 0 . 17 M 0 . 21 BH . Suberlak et al. (2021) analyzed 9248 quasars' 15-year light curves by combining the Panoramic Survey Telescope and Rapid Response System 1 Survey data and SDSS Stripe 82 light curves, and reported the correlations between τ and several AGN physical parameters, including M BH , λ , and the absolute i -band magnitude ( M i ), i.e., log 10 ( τ/ days) = 2 . 59 +0 . 02 -0 . 02 + 0 . 17 +0 . 02 -0 . 02 log 10 ( λ rest / 4000 ˚ A) + 0 . 035 +0 . 007 -0 . 007 ( M i + 23) + 0 . 14 +0 . 02 -0 . 02 log 10 ( M BH / 10 9 M ⊙ ). They stressed that the DRW parameters could be constrained better by a longer baseline. Burke et al. (2021) measured the damping timescale of 67 AGNs, and obtained the relationship between τ and M BH , i.e., τ = 107 +11 -12 days ( M BH 10 8 M ⊙ ) 0 . 38 +0 . 05 -0 . 04 . They proposed that the AGN SMBH mass can be estimated using this relation. Stone et al. (2022) studied the optical variability of 190 quasars in SSDS Stripe 82 with 20-year photometric light curves, and obtained a weak wavelength dependence of τ ∝ λ 0 . 2 . Zhang et al. (2024) obtained τ for 34 blazars and 7 microquasars from the Fermi-Large Area Telescope and the XMM-Newton X-ray telescope, respectively. They obtained the relation between τ and M BH , i.e., τ = 120 +15 -18 days ( M BH 10 8 M ⊙ ) 0 . 57 +0 . 02 -0 . 02 . \nThe CHAR model (Sun et al. 2020) can be used to predict the relationship between τ and AGN properties. Indeed, Zhou et al. (2024) investigated the damping timescale in the CHAR model and predicted a new relation between the theoretical/intrinsic damping timescale and the AGN properties, i.e., \nlog 10 ( τ/ days) =0 . 65 log 10 ( M BH /M ⊙ ) + 0 . 65 log 10 ˙ m +1 . 19 log 10 ( λ rest / ˚ A) -6 . 04 , \n(2) \nwhere ˙ m represents the Eddington ratio (i.e., the ratio of the bolometric luminosity to the Eddington luminosity). Why are the relations obtained from theory and observation inconsistent? Is the discrepancy real or caused by some systematic bias? \nThe length of a light curve (i.e., the baseline) must be much longer than the theoretical/intrinsic damping timescale when fitting the light curve with the DRW model. Otherwise, the damping timescale obtained from an insufficiently long light curve will be strongly underestimated (e.g., Kozlowski 2017; Suberlak et al. 2021; Hu et al. 2024). Zhou et al. (2024) emphasized that the measured damping timescale can still be significantly biased even if it is less than 10% of the baseline; note that this requirement is often adopted in observational studies; hence, these observational studies still obtained biased damping timescales. Instead, the correct criterion to ensure that the measured timescale is unbiased is that the intrinsic/expected (rather than measured) damping timescale is less than 10% of the baseline. \nIn this paper, we constructed unbiased samples based on the relationships between the theoretical/intrinsic damping timescale and the AGN properties obtained from Kelly et al. (2009) (i.e., Equation (1)) and Zhou et al. (2024) (i.e., Equation (2)). We stress that all sources in our samples satisfy the criterion that the theoretical/intrinsic damping timescale (rather than the measured timescale) is less than 10% of the baseline. Hence, the measured damping timescales should be unbiased if the relationship of Kelly et al. (2009) or Zhou et al. (2024) is correct. Then, we can use the measured damping timescale to test the two works critically. This manuscript is formatted as follows. In Section 2, we describe the sample construction procedures. In Section 3, we introduce the data analysis. In Section 4, we present the results. In Section 5, we discuss the possible implications of our results. Conclusions are made in Section 6.", '2. OBSERVATIONS AND SAMPLE SELECTION': "In this work, the quasar optical light curves are obtained from the Zwicky Transient Facility 1 (ZTF; Bellm et al. 2019; Graham et al. 2019; Masci et al. 2019) and several references. Quasar physical parameters (e.g., black hole mass, luminosity, and redshift) are obtained from the second data release of the Swift BAT AGN Spectroscopic Survey 2 (BASS DR2; Koss et al. 2022a,b). \n2.1. Physical parameters \nThe initial sample is obtained from the BASS DR2 (Koss et al. 2022b) which provide the 858 hard-X-rayselected AGNs in the Swift BAT 70-month sample. This sample provides the key properties for each AGN, e.g., R.A. ( α J2000 ) and Declination. ( δ J2000 ); AGN type defined by optical spectroscopy, including Sy1 (broad H β emission line), Sy1.9 (narrow H β and broad H α emission lines), and Sy2 (narrow H β and H α emission lines). For most of Sy1 AGNs, the redshifts ( z ) are measured by [O iii ] λ 5007 or broad emission lines (e.g., broad Mg ii and C iv ; see Koss et al. 2022b). The redshifts of narrow-line sources are obtained by the emission-line [O iii ] λ 5007 (e.g., Koss et al. 2022b); the majority of black hole masses are calculated by either broad H β or stellar velocity dispersion measurements. The spectra used for the above measurements were observed by the Palomar Hale 5 m telescope Double Beam Spectrograph (DBSP) or the Very Large Telescope (VLT) X-shooter spectrograph (e.g., Koss et al. 2022a). The bolometric luminosity ( L bol ) is calculated by the 14-150 keV intrinsic luminosity with the bolometric correction of 8, and assuming the photon index Γ = 1.8. In this work, we focus on Seyfert 1 AGNs ('Type' = Sy1). There are 359 Seyfert 1 AGNs out of the 858 hard-X-ray-selected AGNs.", '2.2. Light curves': "The light curves of the 359 Seyfert 1 AGNs are obtained from ZTF. ZTF is a time-domain survey using the 48-inch Samuel Oschin Telescope of Palomar Observatory. The entire northern visible sky has been scanned at the optical band (including three custom filters, zg , zr , and zi ) by the ZTF camera every 2 days since 2018, and the camera has a 47 deg 2 field with 600 megapixels. The effective wavelengths of the zg , zr , and zi bands are 4753, 6369, and 7915 ˚ A, respectively (Rodrigo et al. 2012; Rodrigo & Solano 2020). The ZTF data are released every 2 months, and the baselines of current g and r band light curves are ∼ 1800 days. Such highquality ZTF light curves, including the long baseline and the high cadence, are very suitable for time-domain science, such as studying quasar optical variability. The zi band light curves are not utilized in this work because their baselines are not long enough for our research. In this work, we retrieved the zg and zr light curves for each source in our sample from ZTF DR21 via the IRSA service 3 , and the cross-matching radius is 1 '' . The observations are obtained from all CCDs. For each light \ncurve, we removed observations with poor quality (i.e., 'catflags' > 0).", '2.3. Sample selection': 'As emphasized by several previous works (e.g., Kozlowski 2017; Suberlak et al. 2021; Kozlowski 2021; Hu et al. 2024; Zhou et al. 2024), the damping timescale is significantly biased if the light-curve baseline is less than ten times of the theoretical damping timescale. As we have stressed in Section 1, the commonly adopted criterion, which is that the baseline is at least ten times longer than the best-fitting damping timescale, cannot eliminate this bias (also see Zhou et al. 2024). Hence, in the following two sections, we aim to construct samples according to our new criterion, i.e., the theoretical damping timescale should be less than 10% of the baseline. As a result, the sample selection depends critically upon the damping timescale relation we aim to test in this work.', '2.3.1. Sample selection based on Kelly et al. (2009)': "In this section, we aim to construct a sample to test the damping timescale relation of Kelly et al. (2009), i.e., Equation 1. Given that the rest-frame baseline for ZTF light curves is about 1800 / (1 + z ) days ( where z represents the redshift), we expect that the measured damping timescale can be unbiased if the theoretical damping timescale is less than 180 / (1 + z ) days (i.e., less than 10% of the baseline). This requirement puts a strong constraint on M BH , luminosity, and redshift. The black hole mass of each source can be obtained from the BASS DR2 (see the description in Section 2.1). Sources without mass estimates ('logMBH' = 0) are removed. The monochromatic luminosity at 5100 ˚ A ( L 5100 ) of each source can be calculated as follows, i.e., L 5100 = L bol / 9 . 26 (e.g., Richards et al. 2006; Shen et al. 2011). For each source, we use M BH , L bol , and Equation 1 to calculate the theoretical damping timescale ( τ th , Kelly+2009 ). Only sources with τ th , Kelly+2009 < 180 / (1 + z ) days are selected. There are 142 sources in the zg band and 142 sources in the zr band of the 359 Seyfert 1 that merit this requirement and are selected for subsequent analysis, respectively.", '2.3.2. Sample selection based on Zhou et al. (2024)': "Zhou et al. (2024) reported the relationship between τ and AGN properties (i.e., M BH , ˙ m , and λ rest ), i.e., Equation (2). The dimensionless accretion ratio ˙ m = L bol / ((1+ k ) × L edd ), k = 1 3 represents the ratio between the magnetic fluctuations' power of the corona and the dissipation rate of the disk turbulent magnetic power (for details, see Sun et al. 2020), where L edd = 1 . 26 × \n10 38 × ( M BH /M ⊙ ) erg s -1 is the Eddington luminosity. Hence, Equation (2) can be expressed in the form of \nlog 10 ( τ/ days) = 0 . 65 log 10 ( L bol / erg s -1 ) +1 . 19 log 10 ( λ rest / ˚ A) -30 . 9 . (3) \nWe calculate the theoretical damping timescale ( τ th , Zhou+2024 ) of Zhou et al. (2024) for each of the 359 Seyfert 1 AGNs in BASS DR2. Then, we again only select sources with τ th , Zhou+2024 < 180 / (1+ z ) days. There are 70 and 38 sources of the 359 Seyfert 1 that meet our requirement for the zg band and zr band, respectively.", '2.4. Other sources obtained from the literature': "Several narrow-line Seyfert 1 galaxies' light curves have been analyzed in the literature. The Kepler light curve ( λ = 6600 ˚ A) of Zw 229-15 has been analyzed by Chen & Wang (2015). The black hole mass of Zw 229-15 is log 10 ( M BH /M ⊙ ) = 6 . 91 and the redshift is z = 0 . 0275 (Koss et al. 2022b). The Swift UVOT UVM 2 band ( λ = 2600 ˚ A) light curve of WPVS 007 has been analyzed by Li et al. (2019), whose M BH = 4 . 1 × 10 6 M ⊙ (Leighly et al. 2015), and z = 0 . 02882 (Grupe et al. 1995). The Swift UVOT UVW 2 band ( λ = 2120 ˚ A) light curve of Mrk 335 has been analyzed by Griffiths et al. (2021); the black hole mass and redshift of Mrk 335 are log 10 ( M BH /M ⊙ ) = 7 . 32 and z = 0 . 0259, respectively (Koss et al. 2022b). Burke et al. (2021) reported the characteristic optical variability timescale of 67 AGNs, and 6 sources (including NGC 4395, NGC 5548, DES J021822.51-043036.0, SDSS J025007.03+002525.3, SDSS J153425.58+040806.7, and SDSS J160531.85+174826.3) of 67 AGNs satisfy the condition τ th , Kelly+2009 / baseline < 10%, and τ th , Zhou+2024 / baseline < 10%. Five sources were added to our sample; NGC 4395 has been reduced since this target is included in our Swift BAT sample. Zhou et al. (2024) measured the damping timescale for 3 local AGNs (see Table 5 in Zhou et al. 2024), including NGC 4151, NGC 7469, and NGC 3516. These sources were added to our sample. We confirm that our main conclusions remain unchanged if these sources were excluded in our subsequent analysis.", '3. DATA ANALYSIS: LIGHT CURVE FITTING': "AGN light curves can be well fitted by the DRW model based on the taufit 4 code (Burke et al. 2021). This code is built upon the fast Gaussian process solver celerite (Foreman-Mackey et al. 2017). celerite fits the time series to the following specified kernel function using the Gaussian process regression \nFigure 1. Fitting the zg band light curve (in the observed-frame) of Z 493-2 with the DRW model via the taufit code. Top panel: the solid red curve and shaded regions are the best-fitting DRW model and the 3 σ confidence intervals. The gray and black data points correspond to the observations that lie outside or inside the shaded regions, respectively. Bottom-left panel: the posterior distributions of the DRW parameters obtained in our new DRW fit (i.e., the DRW fit with the black data points in the top panel). The parameter σ DRW is the long-term variability amplitude, τ DRW is the damping timescale, and σ n is the excess white noise amplitude. The medians of the posterior distributions are adopted as the best-fitting parameters, and the 1 σ uncertainties are obtained by the 16-th and 84-th percentiles of the posterior distributions. The red shaded regions in the bottom-left panel correspond to where τ DRW is greater than 10% of the baseline. Bottom-right panel: the gray curve is the PSD of the data (the black data points in the top panel), and the DRW PSD (the red solid curve) and its 1 σ uncertainties (the orange shaded regions). The red-shaded regions represent the regions of frequency space not sampled by the light curve, including the regions with timescales shorter than the minimum cadence and the timescales longer than 20% of the baseline. \n<!-- image --> \n= \n+0.04 \n<!-- image --> \nlog10 ( DRW) \n0.48 \n0.03 \nk ( t nm ) = 2 σ 2 DRW exp( -t nm /τ DRW ) + σ 2 n δ nm , (4) \nwhere t nm = | t n -t m | represents the time interval between two measurements ( t n , t m ) of the light curve. σ DRW represents the long-term standard deviation of variability. τ DRW defines the damping timescale. σ n is the excess white noise amplitude, which is added to the DRW kernel to account for the possible underestimation of the photometric uncertainties. σ 2 n represents the variance of the noise component. δ nm is the Kro- \nr delta function. The prior distribution of τ DRW was set to be a uniform distribution whose lower and upper bounds are the minimum cadence to the 10 times the baseline of each light curve, respectively. The bestfitting τ DRW is obtained by maximizing the posterior probability. The joint posterior probability distributions of the DRW parameters from celerite are obtained by the Markov Chain Monte Carlo (MCMC) code sampler emcee (Foreman-Mackey et al. 2013) implemented in Python. The median values of the posterior distributions are adopted as the best-fitting parameters, and \nFigure 2. The redshift distributions of the samples in this work. Left panel: the selected sources with theoretical damping timescales, which are calculated from Equation 1 (Kelly et al. 2009), smaller than 10% of the light curve durations. The redshift ranges from 0.001 to 0.823, and the median redshift is 0.030 (the black dashed line). Right panel: the selected sources with theoretical damping timescales, which are calculated from Equation 2 (Zhou et al. 2024), smaller than 10% of the light curve durations. The redshift ranges from 0.001 to 0.823, and the median redshift is 0.016 (the black dashed line). \n<!-- image --> \n16-th and 84-th percentiles of the posterior distributions are adopted as the 1 σ uncertainties for the bestfitting parameters. We compare the Akaike Information Criterion (AIC; Akaike 1998) of this fit (AIC best ) with the AICs of two alternative fits. The first alternative fit (AIC low ) is that we set τ DRW to be an extremely small value (0 . 02 days, i.e., about one-hundredth of the typical ZTF cadence); the second (AIC up ) is that we set τ DRW to be an extremely large value (18000 days, i.e., about 100 times of the typical ZTF baseline). The AIC of each fit is calculated as -ln(Likelihood) + 2 N , where N = 3 is the number of parameters. Statistically speaking, if the AIC difference between two models is larger than 10, the model with a larger AIC value has relatively little support (e.g., Burnham et al. 2011). Hence, we propose that the damping timescale is well constrained only if ∆AIC up = AIC up -AIC best > 10 and ∆AIC low = AIC low -AIC best > 100 (the larger ∆AIC corresponds to the better constraint of the damping timescale). \nIn this section, the light curve(in the observed-frame) of each source mentioned in Section 2 is fitted by the above method. A DRW fit example of Z 493-2 is shown in Fig. 1. The top panel shows the original light curve of Z 493-2, and the predicted light curve from the DRW model obtained by the best-fitting parameters. Note that, we removed the observations outside the 3 σ of the predicted light curve (the gray points in the top panel), and then repeated the fitting process until all observations were included within 3 σ of the predicted light curve obtained by the DRW model. The bottom left panel shows the best-fitting DRW parameters (i.e., \nσ DRW , τ DRW , and σ n ) obtained by the taufit . The bottom right panel shows the PSD of the original light curve (the gray curve), and the PSD of the light curve obtained from the posterior distribution of the celerite fit (the red curve). The model PSD is shown in the orange region (with 1 σ uncertainty). \nIn summary, our sample selection criteria are summarized as follows. \n- · The theoretical damping timescale obtained from Kelly et al. (2009) (i.e., τ th , Kelly+2009 calculated by Equation 1) or Zhou et al. (2024) (i.e., τ th , Zhou+2024 calculated by Equation 3) should be smaller than the 10% baseline in the rest frame, i.e., τ th , Kelly+2009 < 10% × baseline / (1 + z ), or τ th , Zhou+2024 < 10% × baseline / (1 + z );\n- · Following Burke et al. (2021), the best-fitting damping timescale obtained by the DRW model fitting ( τ obs ) is larger than 1.5 times cadence, i.e., τ obs > 1 . 5 × cadence;\n- · Following Burke et al. (2021), the source should have the high signal-to-noise ratio, i.e., σ 2 DRW > σ 2 n +d y 2 , where d y is the median magnitude err of each light curve;\n- · As described in Section 3, in order to ensure that the damping timescales of the sources in our sample are indeed constrained by the data, we require that ∆AIC low > 100, and ∆AIC up > 10. \nThe key parameters of each source in our sample include baseline, cadence (the median cadence of each \nFigure 3. The relationship between the damping timescales (log 10 ( τ obs ); rest-frame values) obtained from the DRW fittings and the theoretical predictions (log 10 ( τ th , Kelly+2009 )) of Kelly et al. (2009) (i.e., Equation 1). The cyan and orange solid dots indicate the damping timescales obtained by the zg and zr band light curves, respectively. The blue and purple dots represent the damping timescales obtained from Burke et al. (2021) and Zhou et al. (2024), respectively. The dark green star, diamond, and square indicate the damping timescales of Mrk 335, Zw 229-15, and WPVS 007, respectively. For more information on these sources, see Table A1. The gray point (ESO 424-12 zr band) was not considered in our analysis because of its unusually small measured damping timescale (the DRW fitting result of ESO 424-12 zr band light curve is shown in Figu. B1). The vertical solid gray line represents 10% of the baseline of the ZTF light curve. The solid black line indicates a one-to-one line, and the black dashed line indicates the uncertainties of 0.2 dex. The reduced χ 2 between log 10 ( τ th , Kelly+2009 ) and log 10 ( τ obs ) is 3.86, and the Kendall's correlation coefficient is ρ = 0.15 ( p -value = 1 . 68 × 10 -3 ). Most of the data points are located above the one-to-one line. That is, log 10 ( τ th , Kelly+2009 ) significantly underestimates the observed damping timescales. \n<!-- image --> \nlight curve), τ obs (i.e., obtained by the DRW fit), σ DRW , σ n , L bol , M BH , λ , and z (see Tables A1 and A2). The theoretical damping timescale ( τ th , Kelly+2009 and τ th , Zhou+2024 ) of our sample can be calculated by Equation 1 and Equation 3. Based on the damping timescale predicted by Kelly et al. (2009) (i.e., Equation 1) and the above criteria, 112 sources (including 187 light curves) were included in our final sample (hereafter sample K) to test the Kelly et al. (2009) relation. More details of these sources are listed in Table A1. Meanwhile, 49 sources (including 68 light curves) were included in our final sample (hereafter sample Z) based on the damping timescale predicted by Zhou et al. (2024); i.e., Equation 3 and the above conditions. More details of these sources \nFigure 4. The relationship between the damping timescales (log 10 ( τ obs ); rest-frame values) obtained from DRW fitting and the theoretical predictions (log 10 ( τ th , Zhou+2024 )) of Zhou et al. (2024) (i.e., Equation 3). The meaning of the data points and lines in this figure is the same as in Fig. 3. The triangles represent the damping timescales of the high and low states of zg and zr of the CL AGN NGC 5273, respectively. For more information on these sources, see Table A2. The reduced χ 2 between log 10 ( τ th , Zhou+2024 ) and log 10 ( τ obs ) is 1.81, and the Kendall's correlation coefficient is ρ = 0.41 ( p -value = 9 . 23 × 10 -7 ). The observations and the theoretical predictions of Zhou et al. (2024) are roughly consistent. \n<!-- image --> \nare listed in Table A2. The redshift distribution of these sources ranges from 0.001 to 0.823 (see Fig. 2), with medians of 0.030 and 0.016, respectively. All sources in the sample are low redshift AGNs.", '4.1. Observed damping timescales vs. theoretical predictions': "The relationship between the damping timescales (log 10 ( τ obs )) obtained from the DRW fitting and the theoretical predictions (log 10 ( τ th , Kelly+2009 )) of Kelly et al. (2009) (i.e., Equation 1) is shown in Fig. 3. There are 187 data points: 93 light curves obtained from the zg band (i.e., the cyan solid dots), 83 light curves obtained from the zr band (i.e., the orange solid dots), 5 data points obtained Burke et al. (2021) (i.e., the blue solid dots), 3 data points obtained from Zhou et al. (2024) (i.e., the purple solid dots), and 3 green dots (Mrk 335, Zw 229-15, and WPVS 007) are obtained from the literature (i.e., Griffiths et al. 2021; Chen & Wang 2015; Li et al. 2019). We found that there are unusually dense observations for the zr band light curve of ESO 424-12, which leads to the damping timescale of this light curve being unusually small, i.e., about 1 day (the \n<!-- image --> \nFigure 5. Left panel: the same as Fig. 3 but for sample C, which consists of sources that satisfy the criteria according to Kelly et al. (2009) (i.e., Equation 1) and Zhou et al. (2024) (i.e., Equation 3). Right panel: the same as Fig. 4 but for sample C. Again, observations are more consistent with the theoretical predictions of Zhou et al. (2024) than those of Kelly et al. (2009). \n<!-- image --> \nDRW fitting result of ESO 424-12 zr band light curve is shown in Fig. B1). We reject this light curve from our sample. Note that our conclusion remains unchanged if this target is included. The vertical solid gray line represents 10% of the baseline of the ZTF light curve (i.e., ZTFbaseline / 10 days). It is important to emphasize that all sources in our sample satisfy the theoretically predicted damping timescale being less than 10% of its baseline in the rest frame. As shown in Fig. 3, the vast majority of data points (184 of 187) lie to the left of the vertical dashed gray line. There are three data points located to the right of the vertical line because their light curves are not obtained from the ZTF. Their light curves have baselines much larger than 1800 days. We calculated the reduced χ 2 between log 10 ( τ th , Kelly+2009 ) and log 10 ( τ obs ) to be 3.86 and found that the Kendall's correlation coefficient to be ρ = 0 . 15 with a p -value of 1 . 68 × 10 -3 . The reduced χ 2 is χ 2 reduced = 1 K -n ∑ K i =1 ( (log 10 ( τ obs ,i ) -log 10 ( τ th , Kelly+2009 ,i )) 2 /σ 2 i ) = 3 . 86, where K is the total number of data points, n = 0 is the number of free parameters in Equation 1, log 10 ( τ obs ,i ) is the value at the i -th observed damping timescale, log 10 ( τ th , Kelly+2009 ,i ) is the value at the i -th predicted damping timescale, and σ i is the error associated with the i -th observed damping timescale. The large reduced χ 2 and small correlation coefficient suggest that the measured timescales cannot be well described by Equation 1 (Kelly et al. 2009). Indeed, most of the points are located above the one-to-one line. That is, log 10 ( τ th , Kelly+2009 ) significantly underestimates the observed damping timescales. \nSimilarly, the relationship between the damping timescales (log 10 ( τ obs )) obtained from the DRW fitting \nand the theoretical predictions (log 10 ( τ th , Zhou+2024 )) of Zhou et al. (2024) (i.e., Equation 3) is shown in Fig. 4. There are 68 data points: 35 are obtained from the zg band (i.e., the cyan solid dots), 22 are obtained from the zr band (i.e., the orange solid dots), 5 data points are obtained from Burke et al. (2021) (i.e., the blue solid dots), 4 data points are obtained from Zhou et al. (2024) (i.e., the purple solid dots), and 2 green dots (Mrk 335 and WPVS 007) are obtained from the literature (i.e., Griffiths et al. 2021; Li et al. 2019). NGC 5273 is a CL AGN, and four light curves of the high and low states of this source are studied separately. We again removed ESO 424-12 from our sample. As shown in Fig. 4, the vast majority of data points (65 of 68) lie to the left of the vertical dashed gray line. There are three data points located to the right of the vertical line because their light curves are not obtained from the ZTF. We calculate the reduced χ 2 between log 10 ( τ th , Zhou2024 ) and log 10 ( τ obs ) to be 1.81 and the Kendall's correlation coefficient to be ρ = 0 . 41 with a p -value of 9 . 23 × 10 -7 . This indicates that the measured timescales can be well described by the relation of Zhou et al. (2024). It can be noticed that over half of the data points (38 of 68) lie within 0.2 dex of the uncertainty of the one-to-one line. That is, the observations and the theoretical predictions of Zhou et al. (2024) are roughly consistent. \nWe also construct a new sample (hereafter sample C) by selecting sources whose theoretical predictions according to Kelly et al. (2009) (i.e., Equation 1) and Zhou et al. (2024) (i.e., Equation 3) satisfy our criteria (see Section 3); that is, sample C is an intersection of the sample in Fig. 3 and the sample in Fig. 4. There are 38 sources (including 54 light curves) in sample C. The \nFigure 6. The probability distributions of the parameters a and b in the fitting relation log 10 ( τ obs ) = a log 10 ( L bol ) + 1 . 19 log 10 ( λ rest ) + b . Bottom-left panel: the joint probability distributions of a and b . Top-left panel: the distribution of a . The median value of the distribution is adopted as the best-fitting a , and the gray region is the 1 σ uncertainties of a . The vertical solid orange line indicates the same parameter predicted by Zhou et al. (2024), which is 0 . 65. Bottomright panel: the distribution of b . The median value of the distribution is adopted as the best-fitting b , and the gray region is the 1 σ uncertainties of b . The vertical dashed orange line represents the same parameter predicted by Zhou et al. (2024), which is -30 . 9. Our best-fitting parameters and the theoretical predictions of Zhou et al. (2024) are statistically consistent. \n<!-- image --> \nrelationship between the theoretical/intrinsic damping timescales (log 10 ( τ obs )) obtained from the DRW fittings and the theoretical predictions (log 10 ( τ th )) of sample C are shown in Fig. 5. The reduced χ 2 between log 10 ( τ th ) of Equation 1 and log 10 ( τ obs ) is 5 . 92 (the left panel of Fig. 5), and the Kendall's correlation coefficient is ρ = 0 . 25 with a p -value of 8 . 96 × 10 -3 . There are less than half of the data points (22 of 54) lie within 0.2 dex of the uncertainty of the one-to-one line, and most of the points are located above the one-to-one line. Meanwhile, the reduced χ 2 between log 10 ( τ th ) of Equation 3 and log 10 ( τ obs ) is 1 . 79 (the right panel of Fig. 5), and the Kendall's correlation coefficient is ρ = 0 . 41 with a p -value of 1 . 69 × 10 -5 . More than half of the data points (32 of 54) lie within 0.2 dex of the uncertainty of the one-to-one line. In summary, the measured damp- \nes can be better described by the relation of Zhou et al. (2024) rather than Kelly et al. (2009).", '4.2. The dependencies of τ obs upon L bol and λ rest': 'As described in Section 1, some previous works reported the established empirical relations between the rest frame damping timescale and several physical parameters. The empirical relations between τ obs and L bol , and λ rest are also constructed in this section. The adopted sample is the same as in Fig. 4 because the observations are more consistent with the theoretical predictions of Zhou et al. (2024) than that of Kelly et al. (2009). The fitted equation is \nlog 10 ( τ obs days ) = a log 10 ( L bol erg s -1 ) +1 . 19 log 10 ( λ rest ˚ A ) + b. (5) \nThe best-fitting parameters are obtained by minimizing the χ 2 statistic. The χ 2 statistic is χ 2 = ∑ (( τ obs , i -τ model , i ) 2 / (( τ err , obs , i ) 2 + ( a × δ log 10 L bol , i ) 2 )), where τ err , obs , i is the 1 σ uncertainty of τ obs , i , and δ log 10 L bol , i represents the typical 0.2 dex uncertainty of log 10 L bol for each source. We use bootstrap with replacements to obtain the statistical distributions of a and b . The median values of the bootstrap distributions are adopted as the best-fitting parameters, and the 1 σ uncertainties are taken as 16-th to 84-th percentiles of the distributions. The results are shown in Fig. 6, with the best-fitting parameters with 1 σ uncertainties being a = 0 . 72 +0 . 18 -0 . 13 and b = -34 . 2 +5 . 7 -7 . 9 (i.e., the histograms and gray shaded areas in the upper left and lower right panels). The solid orange lines indicate the parameters (i.e., a Zhou+2024 = 0 . 65, b Zhou+2024 = -30 . 9) reported by Zhou et al. (2024). The best-fitting parameters obtained in this work and those of Zhou et al. (2024) are statistically consistent. Note that the parameters of the wavelength term are fixed to 1 . 19 according to Zhou et al. (2024). This is because most of the sources in our sample are from the ZTF and the wavelengths in the zg and the zr bands are close, i.e., ( λ r -λ g ) / ( λ r + λ g ) = 0 . 15. Hence, the current sample data are unable to robustly constrain the dependence of the damping timescale upon wavelength.', '5. DISCUSSIONS': 'The observed damping timescales ( τ obs ) in this work are more consistent with the theoretical predictions of Zhou et al. (2024) than that of Kelly et al. (2009). Compared with previous observational studies, we carefully select sources to ensure that the damping timescales are not biased. In order to obtain the unbiased damping \ntimescales, the theoretical/intrinsic damping timescale rather than the measured one should be smaller than the 10% baseline. We construct two well-defined samples in this work, and each source in our samples follows the condition that the theoretical damping timescale (i.e., τ th , Kelly+2009 or τ th , Zhou+2024 ) is smaller than the 10% baseline in the rest frame, i.e., τ obs in this work are unbiased. Our measured damping timescales are inconsistent with the relationship of Kelly et al. (2009) (as shown in Section 4.1, and Fig. 3), possibly because this relationship is biased. In addition, Burke et al. (2021) also established a similar relationship between the damping timescale and the black hole mass. Zhou et al. (2024) has shown that the sample of Burke et al. (2021) is biased. \nWe obtained the dependencies of τ obs upon L bol and λ rest by our well-defined sample, i.e., log 10 ( τ obs ) = 0 . 72 +0 . 18 -0 . 13 log 10 ( L bol ) + 1 . 19 log 10 ( λ rest ) -34 . 2 +5 . 7 -7 . 9 (as shown in Section 4.2, Fig. 4 and 6). There is a positive correlation between τ obs and L bol , and the coefficient ( a = 0 . 72 +0 . 18 -0 . 13 ) of L bol term is close to that of Zhou et al. (2024) ( a Zhou+2024 = 0 . 65), and the parameter b = -34 . 2 +5 . 7 -7 . 9 is also close to that of Zhou et al. (2024) ( b Zhou+2024 = -30 . 9). Hence, this suggests that the origin of AGN UV/optical variations can be described by the CHAR model; that is, the UV/optical variations of AGN are caused by the thermal fluctuations triggered by turbulence in the accretion disk, whose characteristic timescale depends mostly upon the mass accretion rate (or luminosity). In addition, as shown in Fig. 4, the CLAGN NGC 5273 has different damping timescales in the low and high states. Therefore, our results suggest that the damping timescale might be used to measure the luminosity rather than the black hole mass of quasars. \nThe unavoidable magnetohydrodynamic (MHD) turbulence plays an important role in the SMBH accretion process as it is responsible for removing the angular momentum of the accreted gas (e.g., Balbus & Hawley 1991; Balbus 2003). The MHD turbulence in the accretion disk can drive significant variability in the UV/optical emission of the accretion disks. Zhou et al. (2024) demonstrated that in the MHD turbulence scenario, the UV/optical luminosity should vary on the thermal timescale, and the damping timescale is the average of thermal timescales at different radii of the accretion disk. Dwarf galaxies exhibit flux variations on short timescales because of their low luminosities. That is, the damping timescales of dwarf galaxies can be well constrained even for short-duration light curves. Hence, dwarf galaxies are a promising class of targets that can be used to study the MHD turbulence-driven thermal fluctuations in accretion disks.', '6. CONCLUSION': 'We have constructed unbiased samples in which each source in our sample satisfies the criterion that the intrinsic/expected damping timescale is smaller than 10% of the baseline. We have obtained the best-fit parameters between the damping timescales ( τ obs ) and the bolometric luminosity ( L bol ) and the wavelength in the rest frame ( λ rest ), i.e., log 10 ( τ obs ) = 0 . 72 +0 . 18 -0 . 13 log 10 ( L bol ) + 1 . 19 log 10 ( λ rest ) -34 . 2 +5 . 7 -7 . 9 . Our results are roughly consistent with those of Zhou et al. (2024) which is based on the CHAR model (Sun et al. 2020). This agreement supports the idea that the observed UV/optical variability is due to thermal fluctuations in SMBH accretion disks. It is important to note that these samples are unable to robustly test the dependence of the damping timescale upon wavelength. In the future, sources with a larger wavelength range are needed to better test our conclusion and put strong constraints on the SMBH accretion physics.', '7. ACKNOWLEDGMENTS': 'G.W.R. would like to thank Dr. Q.Z. Yu for helpful discussions. We acknowledge support from the National Key R&D Program of China (No. 2023YFA1607903). G.W.R., S.Y.Z., and M.Y.S. acknowledge support from the National Natural Science Foundation of China (NSFC-12322303) and the Natural Science Foundation of Fujian Province of China (No. 2022J06002). Y.Q.X. acknowledges support from the National Natural Science Foundation of China (NSFC-12025303). \nBased on observations obtained with the Samuel Oschin Telescope 48 inch and the 60 inch Telescope at the Palomar Observatory as part of the Zwicky Transient Facility project (IRSA 2022). Z.T.F. is supported by the National Science Foundation under grants No. AST-1440341 and AST-2034437 and a collaboration including current partners Caltech, IPAC, the Weizmann Institute for Science, the Oskar Klein Center at Stockholm University, the University of Maryland, Deutsches Elektronen-Synchrotron and Humboldt University, the TANGO Consortium of Taiwan, the University of Wisconsin at Milwaukee, Trinity College Dublin, Lawrence Livermore National Laboratories, IN2P3, University of Warwick, Ruhr University Bochum, Northwestern University, and former partners the University of Washington, Los Alamos National Laboratories, and Lawrence Berkeley National Laboratories. Operations are conducted by COO, IPAC, and UW. \n(Hunter 2007), Numpy (Harris et al. 2020), Scipy (Virtanen et al. 2020), taufit (Burke et al. 2021).', 'A. RESULTS OF DRW FITTING TO THE LIGHT CURVE SAMPLES IN THIS WORK': 'Table A1 . Sample with τ th , Kelly+2009 / baseline < 10% \nContinued on next page \nTable A1 . Sample with τ th , Kelly+2009 / baseline < 10% \nContinued on next page \nTable A1 . Sample with τ th , Kelly+2009 / baseline < 10% \nContinued on next page \nTable A1 . Sample with τ th , Kelly+2009 / baseline < 10% \nContinued on next page \nTable A1 . Sample with τ th , Kelly+2009 / baseline < 10% \nColumn notes : Col. (1): Source name (sources with * represent objects that are not selected from the Swift BAT sample); Col. (2): The redshift; Col. (3): The black hole mass; Col. (4): The bolometric luminosity; Col. (5): Band Col. (6): The rest-frame wavelength; Col. (7): The rest-frame best-fitting damping timescale; Col. (8): The theoretical damping timescale; Col. (9): The baseline in the rest-frame; Col. (10): The τ th , Kelly+2009 -to-baseline ratio. The redshift, black hole mass, and the bolometric luminosity are adopt from BASS DR2 (Koss et al. 2022a,b). The samples with the same source name and the different λ rest correspond to the different band light curves of the same source. In comparison with Table A2, log 10 ( τ obs ) for the same light curve may be slightly different because log 10 ( τ obs ) in Tables A1 and A2 were obtained from the MCMC code. Similarly, the baseline for the same light curve may be slightly different due to the slightly different outlier rejections in each DRW MCMC fit. \nTable A2 . Sample with τ th , Zhou+2024 / baseline < 10% \nContinued on next page \nTable A2 . Sample with τ th , Zhou+2024 / baseline < 10% \nContinued on next page \nTable A2 . Sample with τ th , Zhou+2024 / baseline < 10% \nColumn notes : Col. (1): Source name (sources with * represent objects that are not selected from the Swift BAT sample); Col. (2): The redshift; Col. (3): The black hole mass; Col. (4): The bolometric luminosity; Col. (5): Band Col. (6): The rest-frame wavelength; Col. (7): The rest-frame best-fitting damping timescale; Col. (8): The theoretical damping timescale; Col. (9): The baseline in the rest-frame; Col. (10): The τ th , Kelly+2009 -to-baseline ratio. The redshift, black hole mass, and the bolometric luminosity are adopted from BASS DR2 (Koss et al. 2022a,b). The samples with the same source name and the different λ rest correspond to the different band light curves of the same source. In comparison with Table A1, log 10 ( τ obs ) for the same light curve may be slightly different because log 10 ( τ obs ) in Tables A1 and A2 were obtained from the MCMC code. Similarly, the baseline for the same light curve may be slightly different due to the slightly different outlier rejections in each DRW MCMC fit. \nFigure B1. Fitting the zr band light curve (in the observed-frame) of ESO 424-12 with the DRW model via the taufit code. The meaning of the data points and lines in this figure are the same as in Fig. 1. There are unusually dense observations (about MJD = 58500) for ESO 424-12, which leads to the damping timescale of this light curve being unusually small, i.e., about 1 day. \n<!-- image --> \n= \nlog10 ( DRW) \n0.39 \n+0.01 \n<!-- image --> \n0.01', 'REFERENCES': 'Akaike, H. 1998, A New Look at the Statistical Model Identification, ed. E. Parzen, K. Tanabe, & G. Kitagawa (New York, NY: Springer New York), 215-222, doi: 10.1007/978-1-4612-1694-0 16 Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33, doi: 10.1051/0004-6361/201322068 Balbus, S. A. 2003, ARA&A, 41, 555, doi: 10.1146/annurev.astro.41.081401.155207 Balbus, S. A., & Hawley, J. F. 1991, ApJ, 376, 214, doi: 10.1086/170270 \nBarth, A. J., Bennert, V. N., Canalizo, G., et al. 2015, ApJS, 217, 26, doi: 10.1088/0067-0049/217/2/26 Bellm, E. C., Kulkarni, S. R., Graham, M. J., et al. 2019, PASP, 131, 018002, doi: 10.1088/1538-3873/aaecbe Bentz, M. C., Denney, K. D., Grier, C. J., et al. 2013, ApJ, 767, 149, doi: 10.1088/0004-637X/767/2/149 Blandford, R. D., & McKee, C. F. 1982, ApJ, 255, 419, doi: 10.1086/159843 Burke, C. J., Shen, Y., Blaes, O., et al. 2021, Science, 373, 789, doi: 10.1126/science.abg9933'}

2024arXiv240902054S

The cores of stars are the cosmic furnaces where light elements are fused into heavier nuclei. The fusion of hydrogen to helium initially powers all stars. The ashes of the fusion reactions are then predicted to serve as fuel in a series of stages eventually transforming massive stars into a structure of concentric shells. These are composed of natal hydrogen on the outside and consecutively heavier compositions inside predicted to be dominated by helium carbonoxygen oxygenneonmagnesium and oxygensiliconsulphur. Silicon and sulphur are fused into inert iron leading to the collapse of the core and either a supernova explosion or the direct formation of a black hole. Stripped stars where the outer hydrogen layer has been removed and the internal Herich layer in WolfRayet WN stars or even the CO layer below it in WolfRayet WCWO stars are exposed provide evidence for this shell structure and the cosmic element production mechanism it reflects. The types of supernova explosions that arise from stripped stars embedded in shells of circumstellar material most notably Type Ibn supernovae from stars with outer He layers and Type Icn supernovae from stars with outer CO layers confirm this scenario. However direct evidence for the most interior shells which are responsible for the production of elements heavier than oxygen is lacking. Here we report the discovery of the firstofitskind supernova arising from a star peculiarly stripped all the way to the silicon and sulphurrich internal layer. Whereas the concentric shell structure of massive stars is not under debate it is the first time that such a thick massive silicon and sulphurrich shell expelled by the progenitor shortly before the SN explosion has been directly revealed.

2024-09-01T00:00:00Z

['2024arXiv240902054S', 'arXiv:2409.02054', '10.48550/arXiv.2409.02054']

['Astrophysics - High Energy Astrophysical Phenomena']

A cosmic formation site of silicon and sulphur revealed by a new type of supernova explosion

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.02054.pdf

{'A cosmic formation site of silicon and sulphur revealed by a new type of supernova explosion': "- 15 Research Center for the Early Universe, School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan.\n- 16 Nordita, Stockholm University and KTH Royal Institute of Technology, Hannes Alfv'ens vag 12, 106 91, Stockholm, Sweden.\n- 17 School of Physics and Astronomy, Tel Aviv University, Tel Aviv, 69978, Israel.\n- 18 Cahill Center for Astrophysics, California Institute of Technology, MC 249-17, 1200 East California Blvd, Pasadena, CA 91125, USA.\n- 19 Institute of Astronomy and Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, UK.\n- 20 Caltech Optical Observatories, California Institute of Technology, 1200 E California Blvd, Pasadena, CA 91125, USA.\n- 21 Astronomical Observatory, Volgina 7, 11060 Belgrade, Serbia.\n- 22 Department of Astronomy, Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11000 Belgrade, Serbia.\n- 23 Department of Astronomy, Kyoto University, Kitashirakawa-Oiwake-cho, Sakyo-ku, Kyoto, Kyoto 606-8502, Japan.\n- 24 School of Physics, Trinity College Dublin, The University of Dublin, Dublin, 2, Ireland.\n- 25 Department of Astronomy, The Ohio State University, Columbus, OH 43210, USA.\n- 26 Department of Astronomy, University of Maryland, College Park, MD 20742, USA.\n- 27 Joint Space-Science Institute, University of Maryland, College Park, MD 20742, USA.\n- 28 Astrophysics Science Division, NASA Goddard Space Flight Center, 8800 Greenbelt Rd, Greenbelt, MD 20771, USA.\n- 29 Yukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa-Oiwake-cho, Sakyo-ku, Kyoto, Kyoto 606-8502, Japan.\n- 30 Astrophysical Big Bang Laboratory, RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan.\n- 31 Miller Institute for Basic Research in Science, 468 Donner Lab, Berkeley, CA 94720, USA. 32 DIRAC Institute, Department of Astronomy, University of Washington, 3910 15th\n- Avenue NE, Seattle, WA 98195, USA.\n- 33 School of Physics and Astronomy, University of Minnesota, 116 Church Street S.E., Minneapolis, MN 55455, USA.\n- 34 IPAC, California Institute of Technology, 1200 E. California Blvd, Pasadena, CA 91125, USA.\n- 35 IP2I Lyon / IN2P3, IMR 5822, Universite Claude Bernard Lyon 1, CNRS, Enrico Fermi, Villeurbanne, 69622, France. \n*Corresponding author(s). E-mail(s): steve.schulze@northwestern.edu;", 'Abstract': "The cores of stars are the cosmic furnaces where light elements are fused into heavier nuclei [1-3]. The fusion of hydrogen to helium initially powers all stars. The ashes of the fusion reactions are then predicted to serve as fuel in a series of stages, eventually transforming massive stars into a structure of concentric shells. These are composed of natal hydrogen on the outside, and consecutively heavier compositions inside, predicted to be dominated by helium, carbon/oxygen, oxygen/neon/magnesium, and oxygen/silicon/sulphur [4, 5]. Silicon and sulphur are fused into inert iron, leading to the collapse of the core and either a supernova explosion or the direct formation of a black hole [6-9]. Stripped stars, where the outer hydrogen layer has been removed and the internal He-rich layer (in Wolf-Rayet WN stars) or even the C/O layer below it (in Wolf-Rayet WC/WO stars) are exposed [10], provide evidence for this shell structure, and the cosmic element production mechanism it reflects. The types of supernova explosions that arise from stripped stars embedded in shells of circumstellar material \n(most notably Type Ibn supernovae from stars with outer He layers, and Type Icn supernovae from stars with outer C/O layers) confirm this scenario [11-15]. However, direct evidence for the most interior shells, which are responsible for the production of elements heavier than oxygen, is lacking. Here, we report the discovery of the first-of-its-kind supernova arising from a star peculiarly stripped all the way to the silicon and sulphur-rich internal layer. Whereas the concentric shell structure of massive stars is not under debate, it is the first time that such a thick, massive silicon and sulphur-rich shell, expelled by the progenitor shortly before the SN explosion, has been directly revealed. \nOn 7 September 2021 at 09:56 (UTC dates are used throughout this paper), the public Northern Sky Survey of the Zwicky Transient Facility (ZTF) [16, 17] discovered the supernova (SN) 2021yfj at right ascension α = 01 h 37 m 46 . s 171 and declination δ = -01 · 15 ' 17 . '' 78 (J2000.0; Methods Section Discovery) [18]. A spectrum obtained with Keck/LRIS 24 hours after discovery shows a large number of narrow emission lines and P Cygni profiles from ionised silicon, sulphur, and argon (Si iii -iv , S iii -iv , and Ar iii ; Figure 1) superimposed on a hot blackbody spectrum ( T ≈ 15 , 000 K; Methods Section Bolometric Light Curve), previously unobserved in any supernova [13, 19, 20]. Lines of lighter elements, which are much more common in the Universe [3] and usually detected in spectra of infant SNe [13, 21, 22], are either very weak (carbon and helium) or even completely absent (e.g., hydrogen, nitrogen). This prompted us to monitor the photometric and spectroscopic evolution at optical and ultraviolet (UV) wavelengths for the next 120 days until SN 2021yfj faded below the brightness of the host galaxy ( Supplementary Material Observations and Data Reduction). The early spectra also reveal narrow absorption and emission lines from the interstellar medium (ISM) in the host galaxy at a redshift of z = 0 . 13865 ± 0 . 00004, placing SN 2021yfj at a luminosity distance of ≈ 676 . 4 Mpc, assuming Planck cosmology ( Methods Section Distance; [23]). Hereafter, all times are given with respect to the discovery time and in the rest-frame. \nThe Si iii -iv , S iii -iv , and Ar iii lines are visible up to day 7.7 (i.e., up to ≈ 5 . 4 days after the time of the g -band maximum). With time, these lines become weaker, and emission lines from singly-ionised silicon and sulphur emerge ( Extended Material Figure 1). Some lines exhibit P Cygni profiles ( Extended Material Figure 2), which can be produced either in the SN ejecta, an expanding shell of gas expelled by the progenitor prior to the explosion, or a stellar wind. The absorption minima are at a velocity of 1,000-1,500 km s -1 and the blue edge of the absorption component reaches merely 3 , 000 kms -1 . This is significantly slower than typical SN ejecta velocities ( ∼ 10 , 000 kms -1 ; [24]), but comparable to the wind velocities of Wolf-Rayet (WR) stars [10] and similar to the velocities of expanding shells of circumstellar material (CSM) around some SNe [e.g., 11-14]. Between day 11 and 20, the blackbody spectrum subsides, and a blue pseudocontinuum dominates the emission in the optical, a tell-tale sign of interaction with CSM [25]. Superimposed are emission lines from Si ii , S ii , Mg i -ii , He i , and O i that remain visible through our last spectrum at day 49.8 ( Extended Material Figure 1). Considering that silicon, sulphur, and argon are the ashes of the ephemeral oxygen-burning phase that takes place less than a few years before a massive star dies [e.g., 5], it is puzzling that the early and late spectra also show helium ( Methods Section Hydrogen and Helium Content). Helium should have been consumed during the earlier burning stages, and it is not a daughter product of the oxygen-burning phase [5]. \nSN2021yfj's unique properties vividly stand out when comparing its spectroscopic sequence to those of stripped-envelope supernovae (SESNe) that strongly interact with CSM rich in either He (Type Ibn) or C/O/Ne (Type Icn; Figure 2, Methods Section Comparison With Interaction-powered SESNe). Spectra of SNe Ibn and Icn obtained shortly after explosion exhibit narrow emission lines from carbon, nitrogen, oxygen, helium, or hydrogen, but no silicon and no sulphur. These differences remain well visible in the spectra obtained around maximum light. The Type Icn SNe 2019hgp and 2021csp also show P Cygni profiles with velocities similar to those of SN 2021yfj, which were interpreted as originating from a fast stellar wind [13, 14]. At late times, all objects are characterised by a blue pseudocontinuum with superimposed emission lines. Yet again, SN 2021yfj displays striking differences: conspicuous silicon and sulphur emission lines but no Ca near-infrared (NIR) triplet at ∼ 8 , 500 ˚ A in emission ( Figure 2), whereas the other objects show He i (SNe Ibn) and a conspicuous Ca NIR triplet in emission. Therefore, the discovery of silicon and sulphur in SN 2021yfj is not due to differences in the data quality, the timing of the observations, or the blackbody temperature. Instead, the presence of silicon and sulphur reflects true differences in the elemental composition of the CSM and, therefore, the SN progenitor. This lets us conclude that SN 2021yfj is embedded in a thick, extended CSM rich in silicon and sulphur, never observed \nin any supernova before. Since SN classes are defined by the absence or presence of particular chemical elements in their spectra [26, 27], SN 2021yfj is the first member of a previously unknown SN class: Type Ien ( Methods Section A New Type of Supernova; [28]). \nSN2021yfj is > 2 mag more luminous than a typical SESN ( M g = -19 . 0 mag for SN 2021yfj compared to M g = -17 . 3 mag for a typical SESN), and it has a shorter rise time from 50% peak flux (2.3 days vs. 8 days). The combination of being more luminous with a shorter rise time is inconsistent with Nipowered SNe. The modelling of the light curve corroborates this. Solely powering by radioactive 56 Ni or a central engine, such as a spindown of a rapidly spinning, highly magnetised neutron star, can be excluded ( Methods Section Light-Curve Modelling). Owing to the assumptions of the two models, they do not simultaneously capture the rise, peak luminosity, and decline. The bolometric light curve ( Figure 3), constructed from the multiband data is akin to those of interaction-powered SNe Ibn and Icn. At peak, SN 2021yfj reached ≳ 3 × 10 43 erg s -1 , and integrating over the entire light curve yields a radiated energy of 0.6-∼ 1 × 10 50 erg. Both values are strict lower limits because the true bolometric peak likely occurred before our multiwavelength campaign started ( Methods Section Bolometric Light Curve). Fitting the bolometric light curve with the model of ejecta-CSM interaction from Ref. [29, 30] reproduces the photometric evolution of SN 2021yfj ( Methods Section Light-Curve Modelling). The fit points to an ejecta mass of ∼ 5 M ⊙ , a CSM mass of > 1 M ⊙ and < M ejecta , and an explosion energy of (1.5-2) × 10 51 erg ( Extended Material Figure 5). The slight tension between the model and the observations during the rise could be mitigated by more complex CSM geometries and density profiles than the ones used here. \nThe ubiquity of silicon and sulphur lines in the blue part of the optical spectra at early times raises the question of the silicon and sulphur abundance in the spectrum formation region. Would a low abundance be sufficient, as can happen with iron-group elements? (For example, solar-metallicity iron can easily explain the strong blanketing observed in SNe II at the recombination epoch; see, e.g., observations [31] and models [32-34].) Or, would a high abundance be necessary, as can happen for helium in SNe Ib [e.g., 35]? To address these questions, we perform exploratory radiative-transfer simulations using mock ejecta with an elemental composition similar to the O/Si shell in massive helium stars, which is the outermost shell containing a significant amount of freshly nucleosynthesised silicon and sulphur. Our model consists of an ejecta with 3 . 24 M ⊙ , a steep density profile with a power-law exponent of -10 (a wind profile is excluded because of the weakness of emission lines), and a base velocity of 1000 km s -1 , yielding a total kinetic energy of a few 10 49 erg at that time. Furthermore, we deposit a power of 3 × 10 43 erg s -1 at a velocity of 1000 km s -1 ( Methods Section Spectral Modelling). This simple model (shown in red in Figure 4) can explain the observed features (shown in black in the same figure) at < 4 , 800 ˚ A in the discovery spectrum in terms of the relative and absolute strength as well as the line width. Increasing the mass fraction from a few percent to 30-50% yields slightly stronger features from silicon and sulphur but in growing tension with the observations ( Methods Section Spectral Modelling). Likewise, decreasing the mass fraction below 1% gives weaker features and is, therefore, inconsistent with the observations. Based on these simulations, we conclude that a mass fraction of a few × 0 . 01 suffice to explain the observed features. Furthermore, to have an optically thick CSM out to 10 15 cm requires a large CSM mass of ∼ 3 M ⊙ . This strongly suggests that SN 2021yfj is the product of a massive star that was stripped all the way to the silicon- and sulphur-rich internal layer prior to explosion. The presence of helium cannot be explained by this model. In fact, no O- or Si-rich material in massive-star models is rich in He. \nIn the framework of wind-driven mass loss, the stripping of a star all the way to the O/Si shell is very challenging to explain. Pulsational pair instabilities might accomplish that [9, 36-38]. This phenomenon is predicted to occur in stars with an initial mass of 70-140 M ⊙ , which experience a recurrent instability from producing e -e + pairs during the oxygen-burning phase. The interaction between CSM shells can produce luminous transients [9, 39, 40] with properties qualitatively matching those of SN 2021yfj. In this scenario, SN 2021yfj could be the product of collisions between the last shells ejected before the star collapses into a black hole. While this model has appeal, the detection of helium cannot be readily explained and may require, for example, a binary helium star companion. The presence of helium is also an issue for the other massive-star scenarios discussed in the Methods Section Progenitor Scenarios, which are motivated by the star-forming host of SN 2021yfj ( Methods Section Host Galaxy). A review of literature models involving accretion and burning of helium on the surface of a compact object does not reveal a suitable Si/S-forming scenario ( Methods Section Progenitor Scenarios). All these models merit additional theoretical studies. \nStellar evolution theory predicts that stars nearing the end of their lives should consist of concentric shells, with hydrogen in the outermost shell and iron at the core [1-5]. However, direct observations of this shell structure are rare. WR stars experience significant mass loss toward the end of their lives, which can expose their He (WN stars) and C/O (WC/WO stars) shells [10]. Interaction-powered SNe provide an independent view of the shell structure of stars. The CSM they interact with encodes information on the surface composition of the dying star just before it explodes, free from contamination by explosion products [41-43]. Type IIn, Ibn, and Icn supernovae probe the H, He, and C/O shells, respectively. These SNe not only complement observations of Wolf-Rayet stars but also reveal how dying stars can lose part of their outer layers just before the terminal explosion. The discovery of SN 2021yfj has three important implications for stellar evolution theory: (i) it reveals a formation site of argon, silicon, and sulphur; (ii) it likely directly confirms the complete sequence of concentric shells in massive stars; and (iii) it requires the operation of processes that can strip stars down to their inner shells. \nThe lack of known SNe that show any similarity to the early- or late-time spectra of SN 2021yfj suggests that Type Ien SNe are intrinsically rare. The ZTF Bright Transient Survey [BTS; 44, 45], which aims to spectroscopically classify all ZTF transients that peak brighter than m = 18 . 5 mag and is nearly 100% complete, has not identified a single SN 2021yfj-like event during its six years of operation. Thus, from the BTS, we conclude that the rate of SN 2021yfj-like events is < 30 Gpc -3 yr -1 [95% confidence], < 1 / 1 , 000 the rate of SNe Ib/c [ Methods Section Event Rate; 46]. As future facilities, such as the Vera C. Rubin Observatory, continue to expand the discovery space for the transient Universe, there is a great deal of hope that these surveys will uncover new classes of explosive events. SN 2021yfj represents one of these rare new transients, but it is important to note that it does not significantly stand out from the population of extragalactic transients based on its photometric evolution alone ( Extended Material Figure 4). Instead, it is the spectra that uniquely identify SN 2021yfj as belonging to an entirely new SN class: narrow silicon and sulphur lines at early times and a blue pseudocontinuum with silicon and sulphur lines at late times. This highlights the importance of spectroscopic observations and lays plain evidence that even sophisticated artificial-intelligence-powered anomaly-detection algorithms running on light curves from the Rubin Legacy Survey of Space and Time [LSST; 47] will not be able to recover every new type of transient in the LSST data stream. To detect additional SN 2021yfj-like events or discover the predicted but yet not discovered Type Id/Idn/Ie SNe ( Extended Material A New Type of Supernova), efforts are needed that tightly couple high-throughput, medium-resolution spectrographs with long-duration, highcadence time-domain surveys, which will naturally be provided by existing and future medium-deep surveys [16, 48-53] and the deep Rubin Legacy Survey of Space and Time [47]. The discovery of any additional SN 2021yfj-like objects will have a profound impact on our understanding of their nature. \nFig. 1 : Spectrum obtained 1.0 days after the first ZTF detection with Keck/LRIS in the range from 2,720 to 4,800 ˚ A, after subtracting the blackbody continuum. The spectrum reveals narrow emission lines of highly ionised species of silicon, sulphur, and argon, which have never been seen in any SN before, as well as doubly ionised carbon, singly ionised magnesium, and neutral helium. A number of the highly ionised silicon and sulphur lines also exhibit P Cygni profiles with a maximum velocity of ∼ 3 , 000 kms -1 ( Extended Material Figure 2), indicating that these lines are produced in a fast-moving, metal-rich CSM (e.g., a wind or a shell expelled by the progenitor shortly before the explosion). The spectrum is rebinned for illustration purposes. The lower bound of the ordinate axis was cropped for illustration purposes. \n<!-- image --> \nFig. 2 : Comparison of SN 2021yfj with other interaction-powered SESNe a few days after the explosion (top), the time of maximum light (middle), and late times (bottom). The earliest spectra of Type Ibn and Icn SNe are characterised by a hot blackbody continuum with a temperature similar to that of SN 2021yfj ( Extended Material Figure 6). They also show lines of helium, carbon and possibly hydrogen (in SN 2010al) (Ibn), or carbon, oxygen, and neon (Icn), but no silicon, sulphur, and argon. These strong helium, carbon, oxygen, and neon lines are clearly absent in SN 2021yfj. Well after peak brightness, the spectra of SNe Ibn, Icn, and SN 2021yfj are characterised by a blue pseudocontinuum with superimposed intermediate-width (a few 1,000 km s -1 ) emission lines, due to the interaction of the SN ejecta and CSM. Again, SN 2021yfj shows prominent silicon and sulphur emission lines that are absent in the other objects, and the comparison objects exhibit features that are clearly absent in SN 2021yfj. Therefore, the differences between SN 2021yfj and previously known classes of interaction-powered SESNe reflect true differences in the CSM composition and the progenitor populations, making SN 2021yfj the first member of a previously unknown supernova class. All spectra are rebinned for illustration purposes. \n<!-- image --> \nRest-frame wavelength (Å) \nFig. 3 : The bolometric light curve of SN 2021yfj (bottom panel) and the evolution of the blackbody temperature (top right) and radius (top left). The bolometric light curve shown in black covers the wavelength interval from 1,800 to 7,850 ˚ A. Correcting the bolometric flux for the missing far-UV and IR flux increases the luminosity by ∼ 0 . 2 dex. At peak brightness, SN 2021yfj reached a luminosity of > 3 × 10 43 erg s -1 . Integrating over the entire light curve yields a radiated energy of (0.6-1) × 10 50 erg. The smaller value covers the range from 1,800 to 7,850 ˚ A, and the larger value includes an estimate of the contribution from the far-UV and IR. Dotted lines indicate time intervals with incomplete wavelength coverage. The blackbody temperature and radius have typical values for infant interactionpowered SNe, although the gradual evolution of both properties is atypical for infant supernovae and indicative of an optically thick CSM ( Methods Section Bolometric Light Curve). The shaded bands indicate the statistical uncertainties at the 1 σ confidence level. The vertical dotted line in each panel indicates the date of the last non-detection. \n<!-- image --> \nFig. 4 : Comparison of the discovery spectrum of SN 2021yfj (black; Figure 1) with a spectral model (red). The model assumes a power of 3 × 10 43 erg s -1 , an ejecta mass of 3 . 24 M ⊙ , a velocity of 1000 km s -1 , and an elemental composition similar to the O/Si shell in massive helium stars: 0.786 (oxygen), 0.1 (neon), 0.05 (silicon), 0.03 (sulphur), 0.01 (argon), 0.01 (magnesium), 0.001 (calcium), and solar abundance for iron, cobalt and nickel. The spectrum is computed with steady-state, non-local thermodynamic equilibrium radiative transfer models (for more details see the Methods Sections Spectral Modelling). This model matches the strongest silicon and sulphur lines and also magnesium in terms of absolute and relative strength and line width, corroborating that SN 2021yfj is likely the explosion of a massive star stripped down to its O/Si shell. \n<!-- image -->", 'Extended material': "<!-- image --> \nWavelength (rest-frame, Å) \nExtended Material Fig. 1 : Spectral evolution from day 1 to 49.8 of SN 2021yfj in the UVoptical (upper panel) and near IR (lower panel). Up to day 11 the spectra are characterised by a blackbody shape with superimposed narrow emission and P Cygni lines from silicon, sulphur, argon, carbon, and helium. As the photosphere cools, the ionisation state of silicon, sulphur, and argon decreases. By day 20, a blue pseudocontinuum dominates the spectrum with superimposed intermediate-width emission lines from magnesium, silicon, sulphur, and helium. The most prominent features of both phases are marked. Regions of high atmospheric absorption are marked, and a near-IR spectrum of the opacity of Earth's atmosphere is shown as black vertical lines (black = high opacity). Host-galaxy emission lines are clipped. The original spectra are in grey, and rebinned versions are in black.", 'Evolution of selected S, Si, He and Mg line profiles at early times': "<!-- image --> \nVelocity (10 \nkms \n) \nExtended Material Fig. 2 : The evolution of the line profiles of selected lines from helium, magnesium, silicon, and sulphur. Top: At early times, all lines show well-developed P Cygni profiles. The absorption minima are at ∼ 1 , 500 kms -1 . The blue edge, a proxy of the maximum velocity, extends to ∼ 3 , 000 kms -1 . These velocities are comparable to velocities of stellar winds as seen in Wolf-Rayet stars [10] and winds around some SNe Icn [13, 14], and much slower than SN-ejecta velocities at similar phases ( ∼ 10 , 000 kms -1 ; [54]). The Si iii and S iv lines are blended with other lines and exhibit complex line profiles. The Mg ii line shows narrow absorption lines from the ISM in the host galaxy. Bottom: Up to day 6, the 5 , 876 ˚ A feature shows a well-developed P Cygni profile and is dominated by He i . At later phases, this feature transitions into a pure emission line with time-variable contributions from silicon, sulphur, and helium. The spectra are rebinned for illustration purposes. \nRest-frame wavelength (Å) \nExtended Material Fig. 3 : Spectrum obtained 1.6 days after the first ZTF detection with VLT/X-shooter, after subtracting the blackbody continuum. The full spectrum covers the wavelength range from 2,635 to 21,960 ˚ A. The displayed wavelength range is limited to 2,720-11,000 ˚ A where SN features are well visible. The top panel shows the same wavelength interval as the discovery spectrum in Figure 1 obtained 12 hours earlier. The evolution between both epochs is gradual at most. In addition to the SN features, the spectrum shows emission lines from star-forming regions in the host galaxy and narrow absorption lines from the host ISM. Strong telluric features are marked with ' ⊕ '. \n<!-- image --> \n11000 \nExtended Material Fig. 4 : SN2021yfj in a 4-dimensional light-curve feature space, together with 4032 extragalactic transients from the ZTF Bright Transient Survey (79% Type Ia SNe, 11% Type II SNe, and 10% other types of core-collapse SNe and other types of transients). The panels above the diagonal show all measurements in different projections of the feature space, the panels below present the diagonal 2-dimensional kernel-density estimates and the panels on the diagonal display 1-dimensional kernel-density estimates. The locations of SN 2021yfj and Type Ibn/Icn SNe are highlighted in all 2-dimensional projections. SN 2021yfj's light curve shares similarities with interactionpowered SNe Ibn and Icn: a fast rise and a high peak luminosity. However, it sustains a high luminosity for a significantly longer period of time, which is uncommon for interaction-powered SESNe but comparable to regular supernovae. The combination of short rise and long duration places SN 2021yfj in a sparsely populated area of the light-curve parameter space. \n<!-- image --> \nFit of the multi-band light curve with a magnetar and a nickel modelFit of the bolometric light curve with a CSM model \n<!-- image --> \n<!-- image --> \nExtended Material Fig. 5 : Fits of the light curve of SN 2021yfj with models of three different powering mechanisms. The panels in the upper half show the results using magnetar and nickel models in the software package Redback [55]. The bottom half shows a fit to the bolometric light curve using a CSM interaction model and the software package CHIPS [29, 30]. The CSM model can describe the observations. The mismatch between the observed and predicted rise can likely be mitigated with more complex CSM geometries and CSM density profiles than the ones considered here. The magnetar and nickel models can be excluded as the primary source of energy. The models do not simultaneously capture the rise, peak luminosity and peak time, and decline. Furthermore, the magnetar fit has an unphysically low opacity, and the nickel model requires an unphysically large nickel fraction. For illustration purposes, we only show the results in three filters. Details about the modelling are provided in the Extended Material Light-Curve Modelling. Non-detections are displayed as ' ▼ '. \nExtended Material Fig. 6 : Comparison of the light curves and blackbody properties of SN2021yfj with those of other interaction-powered SESNe and the Type Ic SN 2020oi . Compared to examples of interaction-powered SESN as well as the nickel-powered SN 2020oi, SN 2021yfj has a bright peak luminosity and a fast rise. Its blackbody radius and temperature evolve slowly in time compared to SNe Icn. This gradual evolution is reminiscent of some SNe IIn that are embedded in an optically thick CSM [e.g., 56], whereas the rapid evolution of Type Icn events suggests a CSM that is significantly less optically thick ( Methods Section Bolometric Light Curve). The vertical dotted line in each panel indicates the date of the last nondetection of SN 2021yfj. The statistical uncertainties at the 1 σ confidence level are indicated as vertical error bars in the left panel and as bands in all other panels. Non-detections are displayed as ' ▼ '. \n<!-- image --> \n<!-- image --> \nExtended Material Fig. 7 : The properties of SN 2021yfj's host galaxy. Left: The SN position, marked by the crosshair, is located ∼ 1 . 2 kpc south from the centre of its star-forming dwarf host galaxy ( M host r ≈ -18 . 1 mag). Right: The host of SN 2021yfj is a regular star-forming galaxy, demonstrated by its location with respect to the main sequence of star-forming galaxies (grey-shaded band, [57]). The properties are also consistent with hosts of core-collapse supernovae from the Palomar Transient Factory [58] (grey contours indicate the region encircling 68, 90, and 95%), including interaction-powered SESNe (Type Ibn and Icn SNe). The properties of all hosts were inferred from photometry using the software package Prospector [59]. The statistical uncertainties at the 1 σ confidence level are indicated. \n<!-- image --> \nExtended Material Table 1 : The extension of the SN classification scheme after the discovery of SN 2021yfj. \nThe SN classification scheme from Ref. [27] (last column) is a progression of the traditional system (second last column) The first number is the spectroscopic classifier of the ejecta composition: 0 = strong H features; 1 = strong He features but no H; 2 = strong C and O but no H and He; 3 = strong O, Ne, Mg, but no C, He, H; and 4 = strong Si, S, O but no Mg, Ne, He and H. The tag 'i' stands for interaction followed by the composition of the material with which the SN interacts: 0 = strong H features, etc. The value of ejecta and CSM composition can take fractional values to indicate transitional objects. The rows marked in grey are the new SN classes. SN 2021yfj belongs to the hitherto unknown class of Type Ien SNe (bold).", 'Discovery': "SN2021yfj, located at α = 01 h 37 m 46 . s 171 and δ = -01 · 15 ' 17 . '' 78 [J2000.0], was discovered by the public ZTF Northern Sky Survey as ZTF21abzbmhz at 09:56 (UTC dates are used throughout this paper) on 7 September 2021 with an apparent magnitude of r = 20 . 82 ± 0 . 30 mag, about 1.7 rest-frame days after the last nondetection [18]. The ZTF image-processing pipeline [60] generated an alert [61] based on image subtraction [62] with respect to a reference image. The alert was picked up by our custom 'infant supernovae' filter [21, 63] running on the ZTF Fritz Marshal system [64, 65]. It was identified by a duty astronomer, and follow-up observations were triggered using our standard methodology [21, 66]. The ALeRCE broker [67] team independently discovered ZTF21abzbmhz in the ZTF alert stream. They were also the first to report ZTF21abzbmhz to the IAU Transient Name Server 1 (TNS) [18]. ZTF21abzbmhz was allocated the name 2021yfj on 7 September 2021. Later detections were reported by the Asteroid Terrestrial-impact Last Alert System (ATLAS; [48]) survey on 12 September 2021 (internal name: ATLAS21bipz) and the Pan-STARRS Survey for Transients (PS; [68]) on 5 October 2021 (internal name: PS21ktg). On 3 September 2024, 2021yfj was designated the name SN 2021yfj [69, 70]. Unless stated otherwise, all times reported in this paper are with respect to the first detection and in the rest frame.", 'Distance': 'The Keck spectrum from day 1 shows emission lines from hydrogen, oxygen and sulphur, produced by the ionised gas in the star-forming regions in the host galaxy, at a common redshift of z = 0 . 1386. We refine redshift with the higher-resolution spectra obtained with X-shooter at the VLT. These spectra also show emission lines from hydrogen and oxygen. Averaging over all epochs, we measure a redshift of z = 0 . 13865 ± 0 . 00004. This is consistent with the redshift inferred from narrow absorption lines from Mg i λ 2852 and Mg ii λλ 2796, 2803 from the host ISM, detected in the X-shooter spectra up to 6.1 days after discovery, and the redshift inferred from the host emission lines detected in the Keck spectra at days 128.1 and 132.5. We use the redshift of z = 0 . 13865 as the SN redshift throughout the paper. The redshift translates to a luminosity distance of 676 . 4 Mpc and a distance modulus of 39.15 mag using a flat ΛCDM cosmology with H 0 = 67 . 7 kms -1 Mpc -1 , Ω m = 0 . 31, and Ω Λ = 0 . 69 [23], which we use throughout the paper.', 'Pre-explosion Limits': "The ZTF survey started monitoring the SN field ∼ 3 . 2 yr before its explosion. We search the archival ZTF data for pre-SN outbursts following the methods described by Ref. [71]. We download IPAC difference images and compute the forced-photometry light curve at the SN position. After quality cuts, we are left with 612 pre-SN observations (74 points discarded) on 295 different nights in the ZTF g , r , and i bands. We apply a baseline correction to ensure that the pre-SN light curve is centred around zero flux. The error bars are sufficiently large to account for the scatter of the pre-SN light curve and no upscaling of the statistical errors is required. We search for significant detections in unbinned and binned observations. Since the durations of the pre-explosion outbursts are unknown, we try seven different bin sizes between 1 and 90 days. We do not obtain 5 σ detections for any of the searches. For week-long bins, the median limiting magnitude is M > -18 . 8 mag in g and r and M > -19 . 5 mag in i . We can exclude outbursts that are brighter than those limits 50 (40) weeks before the explosion in r ( g ). These limits are ≳ 1 mag brighter than the most luminous precursors known [71], and hence do not pose meaningful limits on the outburst activity of SN 2021yfj's progenitor shortly before the explosion.", 'Light-Curve Properties': "The first detection is recorded at 09:56 on 7 September 2021, about 1.7 rest-frame days after the last nondetection. At the time of the discovery, SN 2021yfj is very faint, r = 20 . 73 ± 0 . 30 mag [Milky Way (MW)-extinction corrected], but already luminous, -18 . 3 mag, owing to its large distance ( Extended Material Distance). It reaches its peak brightness in 2.3 and 5.8 rest-frame days in the g and r band, respectively. The MW-extinction corrected peak apparent magnitudes are m g ∼ 19 . 5 ± 0 . 1 mag and \nm r ∼ 19 . 7 ± 0 . 1 mag and translate to MW-extinction corrected, K -corrected [72] absolute magnitudes of M g = -19 . 4 ± 0 . 1 mag and M r = -19 . 1 ± 0 . 1 mag. At the peak of the g -band light curve, SN 2021yfj has a blue g -r colour of ∼ -0 . 3 mag, K -corrected and corrected for MW extinction. The high absolute magnitude at the time of discovery precludes estimating the explosion time as is commonly done for infant SNe (e.g., [19, 21, 73]). \nTo put SN 2021yfj in the context of other SN classes, we also compute the rise time from 50% peak flux and how long the brightness stayed above 50% of the peak flux in the g band. We measure a rise time of 3 days and a duration of 21.7 days. Both measurements have an uncertainty on the order of a few days. For comparison, we choose an extended sample of extragalactic transients from the ongoing ZTF Bright Transient Survey. We apply the following selection criteria: (i) no active galactic nuclei (AGNs), (ii) the passing of the BTS data-quality cuts presented in Ref. [45], (iii) well-sampled g and r light curves before, around, and after maximum brightness, (iv) a peak magnitude of < 19 in g and r before MW-extinction correction, (v) a well-measured rise and decline timescale in both bands, (vi) a spectroscopic classification, and (vii) redshift information from either catalogues or the transient spectrum. In total, 4032 transients fulfil these criteria. The vast majority are Type Ia SNe (3178 objects, 79%; e.g., [74]) and Type II SNe (425 objects, 11%; e.g., [75]). The remaining 10% are other types of core-collapse supernovae (407 objects; e.g., 26, 27), tidal disruption events (16 objects; e.g., [76]), intermediate luminosity red transients (3 objects; e.g., [77]), fast blue optical transients (1 object; e.g., [78]), gap transients (1 object; e.g., [79]) and luminous red novae (1 object; e.g., [80]). Among those objects, 22 objects are interaction-powered SESNe (Ibn: 20; Icn: 2). Their light curve properties (rise-time, duration, absolute peak magnitude and g -r colour at peak) are computed following Ref. [45]. This comparison sample and SN 2021yfj are shown in a 4-dimensional corner plot in the Extended Material Figure 4. Relative to the bulk population of SNe, the fast rise of SN 2021yfj ( ∼ 3 days) represents its most extreme property. This, plus its relatively blue colour at peak ( g -r ≈ -0 . 3 mag) and moderately high luminosity ( M g ≈ -19 . 6 mag) is largely consistent with the known population of SNe Ibn/Icn. Relative to the class of interacting SESNe, SN 2021yfj, however, stands out for its long duration ( ∼ 22 days). In sum, aside from the short rise, SN 2021yfj has a light curve that is largely consistent with the general properties of Type Ia SNe and regular core-collapse supernovae. \nTo quantify the peculiarity of the photometric evolution of SN 2021yfj, we apply the Isolation Forest anomaly detection algorithm [81] to determine whether SN 2021yfj can be considered an outlier relative to other BTS sources. Briefly, an Isolation Forest builds a collection of decision trees and isolates individual sources by selecting a split point at random for a randomly selected feature within the feature space (4D as shown in the Extended Material Figure 4). Rare sources are, on average, isolated with fewer branch splittings within a tree than more common sources. Using the scikit-learn implementation of Isolation Forest [82] with the default settings, we train the forest on the 4032 BTS sources. We then apply the forest to SN 2021yfj and find that not only is it not an outlier, but also that ≈ 15% of the BTS sources are more 'rare' than SN 2021yfj in the 4D light-curve property feature space.", 'Bolometric Light Curve': "Following the procedure outlined in Refs. [83, 84], we compute the bolometric light curve over the wavelength range from 1,800 to 7,850 ˚ A (rest-frame), defined by the wavelength range of our photometric campaign from w 2 to z band. Figure 3 shows the final bolometric light curve in black. A tabulated version is provided in the Supplementary Material Table 7. \nThe solid black line in Figure 3 shows the time interval of the bolometric light curve with the best spectral coverage. The blue w 2 -r colour of 0 . 2 ± 0 . 1 mag at 1.5 days after the discovery of SN 2021yfj points to a substantial contribution from the far UV. Linearly extrapolating the observed spectral energy distribution (SED) to shorter wavelengths yields a missing far-UV fraction of 39 +10 -8 % and 22 +10 -7 % at day 1.5 and day 19.7, respectively. The missing flux contribution beyond 1 µ m is small. Fitting the observed SEDs from w 2 to z or from u to z yields a contribution of ∼ 5% between 1 and 10 µ m during the same time interval. The dark-grey curve in the Extended Material Figure 3 shows the bolometric light curve, including the two missing flux fractions. Other epochs have less well-observed SEDs, and we use the data between day 1.5 and day 19.7 to estimate bolometric corrections. The rising light curve was only observed in the g and r bands. This wavelength interval accounts for 15% of the bolometric flux at day 1.5. Since SN ejecta cool with time, a constant bolometric correction will progressively underestimate the true bolometric flux toward earlier epochs. The fading light curve between days 19.7 and 32 was monitored from the u to i bands and in gri between days 32.0 and 34.8. Similar to the data at the previous time \ninterval, we compute bolometric corrections. The bolometric light curve of these time intervals is shown as dashed lines in the Extended Material Figure 3. At > 34 . 8 days after discovery, SN 2021yfj is only detected in the g band. We omit these data in the bolometric light curve. \nIntegrating the entire light curve yields a radiated energy of 0 . 6 × 10 50 erg. Including the missing farUV and IR contributions increases the radiated energy by a factor of ∼ 2. Both values are comparable to those of other interaction-powered SESNe, such as SNe 2019hgp (Icn, E rad ≈ 10 50 erg; [13]), 2020bqj (Ibn, E rad ≈ 10 50 erg; [85]), and 2021csp (Icn, E rad ≈ 10 50 erg; [14]). Up to day 11.1, the spectra show a blackbody-like continuum with superimposed narrow emission lines but no broad metal absorption lines from the SN ejecta ( Supplementary Material Spectroscopic Evolution). Fitting the photometry from 2,000 to 10,000 ˚ A (observer frame) with the Planck function yields a temperature of ∼ 16 , 000 K at day 1.5 that gradually decreases to ∼ 12 , 000 K in ∼ 3 weeks and a radius that remains constant at ∼ 10 15 cm. The slow evolution of the bolometric light curve and the blackbody photosphere stand out compared to those of the well-observed interaction-powered SESNe 2010al, 2019hpg, and 2021csp ( Extended Material Figure 3). The bolometric flux of SN 2021yfj decreases by less than 0.2 dex during the first two weeks since discovery, whereas the bolometric flux of the two Type Icn SNe faded by 1.0-1.2 dex and SN 2010al grew even brighter. The reasons for this are that SN 2021yfj's photosphere remains at ∼ 10 15 cm at all times and merely gradually cools. Such an evolution has been observed in SNe IIn and was interpreted as the photosphere being located in the unshocked, optically thick CSM [56, 86]. In contrast to that, the photosphere of the Type Icn SNe 2019hgp and 2021csp undergoes a ballistic expansion, which points to a more optically thin CSM and, hence, a rapid decrease of the bolometric flux.", 'Light-Curve Modelling': 'We model the multiband light curve with two distinct powering mechanisms using the open-source Bayesian inference software package Redback [55]: (i) radioactive decay of 56 Ni [87] and (ii) spin-down of a rapidly spinning, highly magnetised neutron star (magnetar; [88]) utilising the generalised magnetar model by Ref. [89]. The two models include a component to account for the loss of γ -ray trapping at late times [90], which can increase the decline rate. Furthermore, we assume a Gaussian likelihood function, and we infer the model parameters with the nessai [91-93] sampler implemented in Bilby [94, 95]. The priors and marginalised posteriors of each model parameter are shown in the Supplementary Material Table 8. \nThe upper panels of the Extended Material Figure 5 show fits with the two magnetar and nickel models in three different bands. Both models are inadequate to describe the observations. The models are not able to fit the rise, peak and decline simultaneously. Furthermore, the magnetar model predicts an unphysically low opacity of κ ≈ 0 . 01 cm 2 g -1 , and the nickel requires an unphysically high nickel fraction of almost 100%. Therefore, we reject these two powering mechanisms as the primary source of energy. \nMotivated by the lines of evidence for CSM interaction [ Methods Sections Spectroscopic Evolution and Comparison With Interaction-powered SESNe], we model the bolometric light curve using the opensource code CHIPS 2 [29, 30]. The code uses hydrodynamical calculations, together with radiative transfer, to calculate bolometric light curves powered by SN ejecta colliding with a dense CSM of an arbitrary density profile. We consider homologously expanding ejecta with a density profile [96] \nρ ej ( r, t ) = { t -3 [ r/ ( gt )] -n ( r/t > υ t ) , t -3 ( υ t /g ) -n [ r/ ( tυ t )] -δ ( r/t < υ t ) , \nwhere n = 10 and δ = 1, commonly adopted for explosions for SESN progenitors. The constants g and υ t relate to the ejecta mass M ej and energy E ej as [97] \ng = { 1 4 π ( n -δ ) [2 (5 -δ ) ( n -5) E ej ] ( n -3) / 2 [(3 -δ ) ( n -3) M ej ] ( n -5) / 2 } 1 /n , υ t = [ 2 (5 -δ ) ( n -5) E ej (3 -δ ) ( n -3) M ej ] 1 / 2 . \nFor the CSM density profile, we adopt a double power law, characterised by a shallow inner core and a steep drop generally found for simulations of envelope eruption [96, 98, 99], \nρ CSM ( r ) = ˆ ρ CSM [ ( r/r CSM ) n in /y +( r/r CSM ) n out /y 2 ] -y , \nwhere r CSM and ˆ ρ CSM set the radius and density at the transition of the two power laws, respectively. The remaining parameters n in ( ≈ 0-3), n out ( ≈ 10-12), and y ( ≈ 2-4 . 5) are set by the envelope structure as well as the detailed hydrodynamics of the mass loss. However, the overall light-curve morphology is sensitive only to n in . We fix the other two as n out = 10 (as adopted in the Methods Section Spectral Modelling) and y = 2, inferred from simulations of partial envelope ejections from stripped progenitors [100]. CHIPS requires opacity tables for the CSM, which depend on its uncertain abundances. The spectra favour an O-dominant composition with enhanced Si/S/Ar and some He ( Methods Sections Spectroscopic Evolution, Hydrogen and Helium Content, and Spectral Modelling). To reproduce this, we take the abundance of the surface of a stripped helium-poor stellar model with an initial mass of 29 M ⊙ available in CHIPS [30], and enhance the Si, S, and Ar mass fractions to those inferred from the Methods Section Spectral Modelling with carbon correspondingly reduced. The mass fractions adopted are 0.0081 (helium), 0.2041 (carbon), 0.6805 (oxygen), 0.0130 (neon), 0.0027 (magnesium), 0.05 (silicon), 0.03 (sulfur), 0.01 (argon), and small contributions of < 0 . 001 for heavier metals. We use the Rosseland and Planck mean opacity tables for this composition, generated with TOPS [101]. While the adopted abundance may not exactly reflect that of the CSM, the bolometric light curves mostly depend on the dynamics of the interaction and much less on the composition details. We adopt a homologous CSM flow ( v ∝ r ) as in the Methods Section Spectral Modelling and set the constant of proportionality to the CSM velocity of 2 , 000 kms -1 at r = r CSM , as observed in the early spectra ( Methods Section Spectroscopic Evolution). \nAsuccessful fit to the bolometric light curve is shown in the bottom panel of Extended Material Figure 5, with the fitting parameters shown in Supplementary Material Table 9. We find that in order to reproduce the bolometric light curve one needs (i) a moderately large explosion energy ( ∼ 2 × 10 51 erg) and CSM mass ( M CSM ≳ 1 M ⊙ , with the lower limit being the estimated mass of the CSM swept up by the shock at day 30) for the long bright peak, (ii) a shallow ( n in ≈ 1), extended ( r CSM ≈ 5 × 10 15 cm) inner CSM profile for the slow decline (see, e.g., [97]), and (iii) a large M ej ( ≳ M CSM ) so that the interaction power does not sharply decay owing to significant ejecta deceleration by the CSM. The inferred high masses and energy for the ejecta and CSM are within the possible range for successive mass ejections in pulsational pair instability (PPI) models [9, 36-38]. A pre-SN mass eruption in a lower-mass star cannot be ruled out, although a mechanism to eject such a large mass is less clear ( Methods Section Progenitor Scenarios).', 'X-ray Emission': 'The interaction of the SN ejecta with CSM and heating of the SN ejecta by a central engine (e.g., magnetar or black hole) can produce thermal X-ray emission [102, 103]. SN 2021yfj was not detected in our Swift /XRT observations ( Supplementary Material Table 4). The inferred upper limits between 10 42 and a few 10 43 erg s -1 ( Supplementary Material Table 4) are comparable to the absorption-corrected luminosities of the X-ray brightest SNe [104-106], and thus do not place strong constraints on the lack of X-ray emission.', 'Spectroscopic Evolution': "Extended Material Figure 1 shows the spectral evolution from the rest-frame near-UV to near-IR from 1 to 50 days after the SN discovery. The spectra up to day 11 are characterised by a cooling blackbody (from ∼ 22 , 000 to ∼ 15 , 000 K; Figure 3) with superimposed narrow emission lines (width: ∼ 2 , 000 kms -1 , Extended Material Figure 2). Following Ref. [107], we use the atomic spectra database from the National Institute of Standards and Technology (NIST; [108]) for the line identifications. We included all elements up to mass number 18 (argon) and created two ranked line lists for each ion sorted by (i) the relative line intensity, and (ii) the Einstein coefficient for spontaneous emission. We identify most lines as transitions from S iii -iv , Si iii -iv , and Ar iii , and a very small minority of lines as transitions from Mg ii , C iii , and He i ( Figure 1, Extended Material Figure 3, Supplementary Material Table 1). \nThe Si iii -iv , S iii -iv , and Ar iii are visible for ≲ 6 days (i.e., ≈ 4 days after the g -band maximum). As the bolometric luminosity (a proxy of the CSM interaction strength) and the blackbody temperature decrease, the highly-ionised species vanish, and the lines from singly ionised silicon and sulphur emerge. Between days 11 and 20, the spectrum transforms from a blackbody with narrow P Cygni lines to a blue pseudocontinuum, akin to those of interaction-powered SESNe [13, 14, 109], with superimposed intermediate (width: a few 1 , 000 kms -1 ) emission lines from neutral and low-ionisation silicon, sulphur, helium, magnesium, and oxygen. \nThe emission features in the early spectra are visible as either pure emission lines or P Cygni lines (examples shown in the Extended Material Figure 2). P Cygni lines can be produced in the expanding SN ejecta, in an expanding shell of gas expelled by the progenitor before the explosion, or in a stellar wind. The absorption minima of the P Cygni profiles are at ∼ 1 , 500 kms -1 and the blue edge of the absorption component is at ≲ 3 , 000 kms -1 (best seen in the isolated Mg ii λλ 2796, 2803 doublet). These velocities are significantly lower than those of typical SN ejecta (e.g., ∼ 10 , 000 kms -1 ; [24]). Instead, they are comparable to those of Wolf-Rayet stars [10] and the CSM velocities of interaction-powered SESNe [13, 14], meaning that SN 2021yfj's ejecta interact with a fast-moving wind or shell of material that is rich in silicon and sulphur.", 'Hydrogen and Helium Content': "Hydrogen and helium constitute most of the baryonic matter in the Universe [110], and both elements play crucial roles in SN explosion and progenitor models [5]. Hydrogen has strong features at 4,861 and 6,563 ˚ A [108]. Throughout the evolution of SN 2021yfj, we detect no hydrogen, neither in absorption nor emission. The blackbody temperature of SN 2021yfj is similar to that of H-rich Type II SNe (e.g., [111]); therefore, SN 2021yfj's progenitor must have lost its hydrogen envelope well before the SN explosion. He i has its strongest optical transitions at 3,889, 4,471, 5,876, 6,678, and 7,065 ˚ A. The first high-resolution spectrum, obtained at 1.6 days after discovery, ( Extended Material Figure 3) shows well-defined emission lines at 3,889 and 5,876 ˚ A; the 7,065 ˚ A line is also detected but it is fainter than the 3,889 and 5,876 ˚ A lines. The intrinsically weak He i line at 6,678 ˚ A is not detected, though it is redshifted to the Telluric A band ( Extended Material Figure 3). At the location of the 4,471 ˚ A line there is a broad emission complex that is likely due to multiple species, meaning we cannot reliably determine whether a He i component exists at that wavelength. \nUsing the method of Ref. [107], we find that 3,889 and 5,876 ˚ A are expected to be the strongest lines within the wavelength range we consider [108]. A comparison with an early spectrum of the prototypical He-dominated Type Ibn SN 2006jc [112] indicates that this event shows only the 3,889 and 5,876 ˚ A lines at early times, further supporting the presence of He i in the spectrum of SN 2021yfj. The similarity between SN2021yfj and SN 2006jc is maintained during later phases ( Figure 2), where the broader He i lines at 5,876 ˚ A and 7,065 ˚ A are most prominent in the spectra of both SNe [12, 112]. We also detect an emission line at 10,830 ˚ A at early and late times ( Extended Material Figures 3, 1), where He i also has a strong transition. At late times, the 5,876-˚ A feature of SN 2021yfj reveals time-variable shoulders due to the contribution from singly-ionised silicon and sulphur ( Extended Material Figure 2). \nTherefore, we conclude that the spectroscopic data provide evidence for the existence of helium within the emitting material throughout the SN evolution.", 'Comparison With Interaction-powered SESNe': 'To understand the peculiarity of SN 2021yfj, we compare its spectral evolution to that of other interactionpowered SESNe. We chose SNe 2006jc and 2010al (as archetypes of the Type Ibn class; [12, 20, 113]), and SNe 2019hgp and 2021csp (as archetypes of the Type Icn class; [13, 14]). SNe 2019hgp and 2021csp were detected within < 2 days of their explosion dates, and spectra were acquired within hours after their discovery, offering an excellent opportunity to search for silicon, sulphur, and argon in their earliest spectra. \nFigure 2 shows snapshots of the spectral evolution before peak light (top), at maximum (centre), and more than one month after peak (bottom). The different SN classes evolve similarly. Up to the time of maximum light, the spectra are characterised by a thermal spectrum cooling from a few 10,000 K to 10,000-15,000 K and a series of narrow emission lines [full width at half-maximum intensity (FWHM) < 1000 kms -1 ]. Well after maximum brightness, a blue pseudocontinuum develops, produced by a forest of \niron emission lines in the CSM, and a small number of emission lines with a FWHM of several 1000 km s -1 . Depending on the elemental composition of the CSM, different emission lines are visible: Type Ibn helium (primarily), hydrogen, carbon, and oxygen; Type Icn - carbon, neon, oxygen (primarily) and helium; and SN 2021yfj- silicon, sulphur, argon (primarily) and helium. Furthermore, SNe 2010al and 2021csp show conspicuous emission from Ca ii at 8,500 ˚ A. This feature is absent in SN 2021yfj, which is puzzling considering that argon, calcium, silicon, and sulphur are the ashes of the oxygen-burning phase (e.g., [7]). \nIn conclusion, the nondetection of silicon, sulphur, and argon in Type Ibn and Icn SNe and likewise the nondetection of hydrogen, nitrogen and oxygen and the weak presence of carbon and helium in SN 2021yfj, is not due to differences in the ionising radiation field, insufficient data quality, or the wavelength coverage. Instead, it reflects differences in the elemental composition of the CSM and, therefore, in the progenitor stars (e.g., [25]).', 'A New Type of Supernova': "The classification of SNe is fundamentally based on spectroscopy [26, 27]. The type of a SN is determined by the dominant spectroscopic features in its peak-light spectra: Type II SNe are dominated by hydrogen lines, Type Ib SNe by helium, and Type Ic SNe lack both hydrogen and helium. This sequence is assumed to reflect the amount of stripping of the progenitor stars, with those of SNe II having retained much of their initial hydrogen envelope, those of SNe Ib having lost the hydrogen layer but retaining the He-rich layer below, and those of SNe Ic having lost all or most of the He-rich layer. Adding the suffix 'n' to the SN type is used to indicate the presence of relatively narrow spectral lines that arise from the SN progenitor having been surrounded by slowly-moving CSM whose composition reflects that of the outer stellar layer at the time of explosion or shortly prior. Thus, SNe IIn are surrounded by H-rich CSM, SNe Ibn have He-rich CSM, and SNe Icn have C/O-rich CSM. Following the theoretical shell structure of massive stars, one would expect that even further stripping would lead to the formation of stars whose outer layers are dominated by O/Ne/Mg and later O/Si/S, with natural designations of Type Id and Ie; events with narrow CSM lines would then be denoted by Idn and Ien ( Extended Material Table 1). Our observations, presented in the Supplementary Material Spectroscopic Evolution, Hydrogen and Helium Content, and Comparison With Interaction-powered SESNe, suggest that SN 2021yfj is indeed the first example of a Type Ien SN [28]. This discovery also implies the existence of Type Id, Idn and Ie SNe. A late-time spectrum of the Type Ic SN 2021ocs [114] may indicate that its progenitor was more rich in O/Mg than the progenitors of other Type Ic SNe. SN 2021ocs could be Id or Ic/Id transitional SN. However, the evidence comes solely from a nebular spectrum, whereas SN classifications are based on their peak-light spectra. While it is too early to claim the detection of a Type Id SN, the discovery of SN2021ocs is very intriguing.", 'Spectral Modelling': "Using the approach of Ref. [25], we simulate the early-time spectra of SN 2021yfj by adopting ejecta with a density profile of ρ ∝ r -10 , a velocity of 1000 km s -1 at 10 15 cm (together with a homologous flow, i.e., r/v = age = 116 days), and a composition with the following mass fractions: 0.786 (oxygen), 0.1 (neon), 0.05 (silicon), 0.03 (sulphur), 0.01 (argon), 0.01 (magnesium), 0.001 (calcium), and solar abundance for iron, cobalt, and nickel. This composition is representative of the O/Si shell in a massive He-star model at the time of explosion, such as the he12 model used in Ref. [115]. The density is scaled so that the total Rosseland-mean optical depth of the ejecta is 40, assuming a mean opacity of 0.1 cm 2 g -1 , roughly comparable with the results from the radiative-transfer calculation. This yields a total mass of 3 . 24 M ⊙ , which is quite substantial. Finally, a power of 3 × 10 43 erg s -1 is injected into the inner regions of these ejecta at v deposition = 1000 kms -1 over a characteristic scale of d v = 200 kms -1 . The deposition profile goes as exp[ -( v -v deposition ) 2 / d v 2 ], and the volume-integrated power is normalised to 3 × 10 43 erg s -1 . Given these initial conditions, the radiative-transfer solution computes the temperature, ionisation, etc. We then compute steady-state, non-local thermodynamic equilibrium (NLTE) radiative-transfer models [25]. \nThe results for the UV-optical spectrum are shown in Figure 4. The model spectra contain numerous lines of silicon and sulphur despite the relatively low abundance of a few 0.01, revealing that these elements have a strong absorption power, like iron, even for a modest abundance. Additional tests, in which we \nraise the silicon and sulphur abundances until they reach values of 0.3-0.5, yield somewhat stronger silicon and sulphur lines, although not so much stronger, but progressively in tension with the observations. Decreasing the mass fraction below 1% gives weaker features, again in tension with observations. Hence, it seems that a mass fraction of a few 0.01 of silicon and sulphur is sufficient to explain the optical spectra of SN 2021yfj. These explorations remain somewhat short of the true complexity of SN 2021yfj. Indeed, no O-rich or Si-rich material in a massive star is also rich in helium, which is in tension with the presence of He i lines detected in SN 2021yfj ( Methods Section Hydrogen and Helium Content). One way to accommodate this peculiarity is by invoking an asymmetric configuration in which He-rich material would be present in some 'equatorial region' and O/Si-rich material interacts with that material, producing an interacting SN, with emission coming both from that He-rich CSM and the Si/O-rich ejecta. Further work is needed to investigate this aspect thoroughly.", 'Host Galaxy': "SN2021yfj's host galaxy was detected in several broad-band filters from the rest-frame UV to near-IR ( m r ≈ 21 mag; Supplementary Material Table 5). The left panel of the Extended Material Figure 7 shows a false-colour image of the host galaxy, built with gri images from the DESI Legacy Imaging Surveys [116] and the software package STIFF [117] version 2.4.0. The SN explosion site is ∼ 1 . 2 kpc south of the galaxy centre. To infer the mass and star-formation rate of the host, we model the observed SED ( Supplementary Material Table 5) with the software package Prospector [59] version 1.1. 3 We assume a Chabrier initial-mass function (IMF; [122]) and approximate the star-formation history (SFH) by a linearly increasing SFH at early times followed by an exponential decline at late times [functional form t × exp ( -t/t 1 /e ) , where t is the age of the SFH episode and t 1 /e is the e -folding timescale]. To account for any reddening between the expected and the observed SED, we use the Calzetti attenuation model [123]. The priors of the model parameters are set identical to those used by Ref. [58]. The host galaxy has a stellar mass log M ⋆ /M ⊙ = 8 . 9 ± 0 . 2, a star-formation rate of 0 . 07 +0 . 10 -0 . 02 M ⊙ yr -1 , an age of 4 +4 -2 Gyr, and an attenuation of the stellar component of E ( B -V ) = 0 . 05 +0 . 07 -0 . 03 mag ( χ 2 / n . o . f . = 18 . 25 / 11, where n.o.f. is the number of photometric filters used in the SED modelling). The star-formation rate is comparable to typical star-forming galaxies of that stellar mass (grey band in Extended Material Figure 7; [57]). The mass and star-formation rate are also similar to the SN host galaxies from the Palomar Transient Factory (grey contours; [124-127]), including interaction-powered SESNe (colour-coded; the values of the SNe Icn were taken from Refs. [13, 14, 128]). \nThe SN spectra reveal emission lines from the ionised gas in H ii regions along the line of sight, summarised in the Supplementary Material Table 6. Their luminosities and flux ratios allow us to determine the metallicity of the gas in the star-forming regions, the metal enrichment, and the level of attenuation. The MW-extinction corrected H γ /H β and H δ /H β flux ratios are 0 . 46 ± 0 . 02 and 0 . 23 ± 0 . 02, respectively. Both values are consistent with the theoretically predicted values of 0.47 and 0.26, assuming typical conditions of star-forming regions: Case B recombination, electron temperature of 10 4 K, and electron density of 10 2 cm -3 [129]. The nominal excess in the flux ratio translates to E host ( B -V ) = 0 . 07 ± 0 . 06 mag, assuming the Calzetti attenuation model with R V = 4 . 05. Owing to the small amount of attenuation and its large statistical measurement error, we assume negligible extinction for all SN properties. The H α luminosity and the level of star formation are tightly correlated [130]. The attenuation-corrected starformation rate is 0 . 17 ± 0 . 03 M ⊙ yr -1 using Ref. [130] and Ref. [131] to convert from the Salpeter IMF (assumed in Ref. 130) to the Chabrier IMF (assumed in our galaxy SED modelling). Both the attenuation and the star-formation rate are consistent with the values derived from the SED modelling. The metallicity of the star-forming region can be determined from the ratios between H α , H β , [N ii ] λ 6584, and [O iii ] λ 5007 [132]. Using this O3N2 diagnostic together with the parameterisation from Ref. [133], we infer a gas-phase metallicity of 0 . 53 ± 0 . 01 solar, a normal value for a galaxy of that mass [134]. \nWe note that SN 2021yfj exploded close to the centre of its host galaxy, and the slit covered a large fraction of the host galaxy. The properties reported above are, hence, representative of the SN explosion site and the entire galaxy.", 'Event Rate': 'The WISeREP archive contains > 400 public spectra for > 70 SNe Ibn/Icn. We examined all spectra to determine whether any of the objects showed narrow silicon or sulphur lines. None of the objects has spectra exhibiting silicon, sulphur, and argon lines. This reveals that SN 2021yfj is the first member of a previously unknown supernova class and that Type Ien SNe are an even rarer class of objects than the already rare Type Ibn and Icn SNe (SNe Ibn, 0.1-0.5%; SNe Icn, 0.005-0.05% of the total CCSN rate; [14]). \nThe nondetection of Type Ien SNe in the ZTF Bright Transient Survey allows us to place an upper limit on their volumetric rate. Figure 9 in Ref [45] shows the relationship between the volumetric rate as a function of peak absolute magnitude derived from the ZTF Bright Transient Survey. Using the same methodology together with the 6-year BTS sample and assuming that SNe Ien reach an absolute magnitude of < -19 mag at peak, their volumetric rate is < 30 Gpc -1 yr -1 at 95% confidence (Poisson statistics), which is roughly < 1 / 1 , 000 the rate of SNe Ib/c [46] and < 1 / 3 , 000 of the total CCSN rate [45].', 'Progenitor Scenarios': 'Knowing that a moderate amount of silicon and sulphur suffices to explain the features of the early-time spectra of SN 2021yfj, we now explore possible progenitor channels.', 'A High-mass Massive Star': 'Massive stars can lose a substantial amount of their birth mass through stellar winds [135, 136], eruptions [136, 137], and interaction with a companion star [138, 139]. First, we focus on stars that have lost their entire hydrogen envelope, so-called He stars, with a pre-supernova mass between 30 and 133 M ⊙ . During the oxygen-burning phase of such a star, e + e -pairs are formed, reducing the radiation pressure that supports the star against gravitational collapse (so-called pair instability; [36-38]). As a result, implosive oxygen burning can produce enough energy to unbind a substantial amount of the stellar envelope. He cores above ≳ 64 M ⊙ experience a single violent pulse that unbinds the entire star [140-144]. Lessmassive stars can encounter pair instability a few times and eject shells of increasingly metal-rich CSM. The collisions of shells can produce luminous optical transients, so-called pulsational pair-instability SNe (PPISNe; [9, 145, 146]). The specific mass limits depend on the uncertain rate of the 12 C( α, γ ) 16 Oreaction rate [147-149], but for reasonable choices, a qualitatively similar behaviour is observed in the transition region between PPISNe and PISNe, wherever it occurs. PPISN models usually do not produce shells that contain much silicon and sulphur. Usually, the outer layers of helium, carbon, oxygen, magnesium, and neon are ejected. In the transition between PPISN and PISN, the first pulse can eject an arbitrary amount of mass and expose the oxygen convective shell. This extensive convective shell encompasses much of the star and may be mildly enhanced in newly synthesised silicon. \nAmong the available PPISN models, the He60/61 model of Ref. [9] with a helium-core mass of 6061 M ⊙ is of particular interest. It has an oxygen convective shell with a composition similar to that of the he12 model whose predicted spectral features matched the observed ones at early times ( Methods Section Spectral Modelling). In the He61 model, the first pulse ejects 19 M ⊙ , exposing the O/Ne shell. The remaining 42 M ⊙ are almost unbound, but eventually, the star contracts and reencounters pair instability. Enough nuclear fuel is left for a few final pulses that happen in rapid succession before the iron core directly collapses into a black hole. With each new pulse, a new shell (moving with a velocity of a few 1 , 000 kms -1 ) is enriched in increasingly heavier elements, and, eventually, material from the oxygen convective shell, which could be enhanced in silicon, can be expelled. The collisions between the shells (pulse 2 and later) can produce light curves with rise times as short as a few days, durations of several tens of days, and peak luminosities reaching a few 10 43 erg s -1 . Collisions between the shells can radiate up to 5 × 10 50 erg and produce spectra that are dominated by interaction [9]. However, the exact properties are more uncertain. They are highly sensitive to the properties of the first pulse, the kinematics of the shells, the mass they carry, the time between the pulses, mixing processes, and details of the simulations [9, 39, 150, 151]. \nIn the PPISN scenario, SN 2021yfj could be the product of a collision between the last shells ejected before the progenitor star collapsed to form a black hole. Qualitatively, the PPI model explains many of the observed properties, such as the wind velocity of 3 , 000 kms -1 , presence of silicon, sulphur, and argon \nin the early spectra, interaction-dominated spectra throughout the entire evolution ( Methods Section Spectroscopic Evolution), the rise time of 3 days, the peak luminosity of (3-5) × 10 43 erg s -1 , and the radiated energy of (0.5-1) × 10 50 erg ( Methods Section Light-Curve Properties). It could also provide a mechanism to strip a star to its oxygen convective shell. The host metallicity of ≲ 0 . 5 solar ( Methods Section Host Galaxy) and low event rate of SN 2021yfj-like transients ( Methods Section Event Rate) match further predictions of PPISN models. \nThe presence of helium ( Methods Section Hydrogen and Helium Content) in the spectra of SN 2021yfj is puzzling. There was helium on the surface of the 61/62M ⊙ models, but it was ejected about a thousand years before the explosion and now resides at ∼ 10 18 cm, maybe less if it collides with dense CSM. Since massive stars tend to live in binary systems, it may not be too unlikely to have a helium-star companion with a strong wind. All stars this massive have nearly the same lifetimes burning hydrogen and helium (3 × 10 6 and 3 × 10 5 yr, respectively), so a coeval star could be in a similar stage of evolution.', 'A Low-mass Massive Star': "In the regime of less-massive He stars, a few scenarios could give rise to SN 2021yfj, as follows. \nScenario A - Helium stars with pre-supernova masses in the range 2.0 to 2 . 6 M ⊙ have a complex final evolution that is strongly influenced by electron degeneracy [152, 153]. To make a low-mass helium star, a close interacting binary is a necessary starting point. Both oxygen and silicon burning ignite far off centre in these stars, and the fusion propagates inward as 'flames' bounding a molecular-weight inversion that might be unstable [152]. The stars also have unusually large radii ( ∼ 10 13 cm) that they develop after core helium depletion. Some also experience strong, degenerate silicon flashes that, while energetically incapable of exploding the entire star, can eject all or part of the matter outside the silicon core. A relevant scenario would be (i) the silicon flash plus any residual binary interaction removes most of the matter external to the oxygen-burning shell; (ii) during the following months, the inwardly propagating flame powers a strong wind that ejects more silicon-rich matter; and (iii) a terminal iron-core collapse creates an outgoing shock with ∼ 10 50 erg. The bright supernova results from the terminal explosion interacting with the wind and ejected shell, as shown in Figure 14 of Ref. [153]. This could be a more common occurrence than a PPISN and does not require low metallicity or a high star-formation rate. Helium would also be present in the recently ejected matter. The difficulty is the many uncertainties surrounding the energy and timing of the silicon flash; the propagation, in more than one dimension (1D), of the flames; and the extent to which the flash and winds uncover silicon-rich material prior to the terminal explosion. We note that based on simulations by Ref. [25], the CSM around such low-mass stars is expected to consist mostly of helium, which leads to strong helium features throughout the entire evolution, inconsistent with observations of SN 2021yfj [ Extended Material Figure 3, 1]. This channel might be ideal for Type Ibn SNe [25], but less so for SN 2021yfj. \nScenario B -Some massive stars produce jets during their terminal explosion [154, 155]. The appearance of Si/S material in the outer layer could be the result of a jetted explosion where jets lift and drag material from the stellar interior onto the outer layers. In this scenario, SN 2021yfj would have to be observed from a preferred direction close to on-axis. This might also produce a γ -ray flash as seen in long-duration GRBs, which are connected with the explosion of very massive stars. We found no γ -ray flash or an afterglow in the X-ray or the optical cospatial with SN 2021yfj within 2 days before the discovery of SN 2021yfj ( Supplementary Material High-Energy Observations). \nConsidering SN 2021yfj's moderately high redshift, an intrinsically weak γ -ray flash would likely have evaded detection with current γ -ray satellites [156, 157]. Furthermore, a baryon loading of as little as 10 -4 M ⊙ is expected to stifle the formation of a relativistic jet [158], leading instead to the formation of a nonrelativistic outflow. An outflow should not only dredge up silicon and sulphur; it should also transport carbon, oxygen, and neon from the layers between the inner Si/S shell and the other He shell. However, strong lines from carbon, oxygen, and neon are absent in the spectra of SN 2021yfj.", 'A Merger of Two Compact Objects': 'In the following, we explore whether silicon and sulphur could be formed on the surface of a white dwarf or possibly a neutron star. Simulations by Ref. [159] showed that helium burning at low densities \n( ∼ 10 9 g cm -3 ) and low temperatures ( ∼ 10 9 K) produces silicon and sulphur. Helium burning is an exothermal process, though. The temperature increase would cease the production silicon and sulphur and instead continue fusing the ashes to nickel [159]. The excess energy could be dissipated through the expansion of the gas. However, this would also skew the production toward heavier nuclei [159]. Cooling by emitting neutrinos is not possible since silicon and sulphur are stable against β decay in these conditions. Simulations of helium burning on the surface of C/O white dwarfs [160] revealed that the density on their surface is always ≪ 10 9 g cm -3 , making the production of silicon and sulphur subdominant to the more common elements, such as calcium, magnesium, and iron-group elements. We note, though, that helium burning on the surface of different types of compact objects (white dwarfs and neutron stars) is underexplored, and detailed simulations are needed.', 'Acknowledgements': "M. W. Coughlin acknowledges support from the U.S. National Science Foundation (NSF) with grants PHY-2308862 and PHY-2117997. A. V. Filippenko's group at UC Berkeley is grateful for financial assistance from the Christopher R. Redlich Fund, Gary and Cynthia Bengier, Clark and Sharon Winslow, Alan Eustace (W.Z. is a Bengier-Winslow-Eustace Specialist in Astronomy), William Draper, Timothy and Melissa Draper, Briggs and Kathleen Wood, Sanford Robertson (T.G.B. is a Draper-Wood-Robertson Specialist in Astronomy), and many other donors. A. Gal-Yam's research is supported by the ISF GW excellence centre, an IMOS space infrastructure grant and BSF/Transformative and GIF grants, as well as the Andr'e Deloro Institute for Space and Optics Research, the Center for Experimental Physics, a WISMIT Sagol grant, the Norman E Alexander Family M Foundation ULTRASAT Data Center Fund, and Yeda-Sela; A. Gal-Yam is the incumbent of the Arlyn Imberman Professorial Chair. N. Kneˇzevi'c was supported by the Ministry of Science, Technological Development and Innovation of the Republic of Serbia (MST-DIRS) through contract no. 451-03-66/2024-03/200002 made with the Astronomical Observatory (Belgrade) and contract no. 451-03-66/2024-03/200104 made with the Faculty of Mathematics at the University of Belgrade. R. Lunnan acknowledges support from the European Research Council (ERC) under the European Union's Horizon Europe research and innovation programme (grant agreement 1010422). K. Maeda acknowledges support from the JSPS KAKENHI grant JP20H00174 and JP24H01810. A. A. Miller and S. Schulze are partially supported by LBNL Subcontract 7707915. N. Sarin acknowledges support from the Knut and Alice Wallenberg foundation through the 'Gravity Meets Light' project. D. Tsuna is supported by the Sherman Fairchild Postdoctoral Fellowship at the California Institute of Technology. Y. Yang appreciates the generous financial support provided to the supernova group at U.C. Berkeley by Gary and Cynthia Bengier, Clark and Sharon Winslow, Sanford Robertson, and numerous other donors. We appreciate the excellent assistance of the staff at the various observatories where data were obtained. U.C. Berkeley undergraduate student Evelyn Liu is thanked for her effort in taking Lick/Nickel data. \nBased in part on observations obtained with the 48-inch Samuel Oschin Telescope and the 60-inch Telescope (P60) at the Palomar Observatory as part of the Zwicky Transient Facility project. ZTF is supported by the U.S. NSF under grants AST-1440341 and AST-2034437, and a collaboration including current partners Caltech, IPAC, the Weizmann Institute of Science, the Oskar Klein Centre at Stockholm University, the University of Maryland, Deutsches Elektronen-Synchrotron and Humboldt University, the TANGO Consortium of Taiwan, the University of Wisconsin at Milwaukee, Trinity College Dublin, Lawrence Livermore National Laboratories, IN2P3, University of Warwick, Ruhr University Bochum, Northwestern University, and former partners the University of Washington, Los Alamos National Laboratories, and Lawrence Berkeley National Laboratories. Operations are conducted by COO, IPAC, and UW. ZTF access was supported by Northwestern University and the Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA). The SED Machine on P60 is based upon work supported by NSF grant 1106171. Some of the data presented herein were obtained at Keck Observatory, which is a private 501(c)3 nonprofit organisation operated as a scientific partnership among the California Institute of Technology, the University of California, and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W. M. Keck Foundation. Based in part on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere under ESO programme(s) 105.20KC and 105.20PN. Data presented here were obtained in part with ALFOSC, which is provided by the Instituto de Astrof'ısica de Andaluc'a (IAA) under a joint agreement with the University of Copenhagen and NOT. KAIT and its ongoing operation at Lick Observatory were made possible by donations from Sun Microsystems, Inc., the Hewlett-Packard Company, \nAutoScope Corporation, Lick Observatory, the U.S. NSF, the University of California, the Sylvia & Jim Katzman Foundation, and the TABASGO Foundation. A major upgrade of the Kast spectrograph on the Shane 3 m telescope at Lick Observatory was made possible through generous gifts from William and Marina Kast as well as the Heising-Simons Foundation. Research at Lick Observatory is partially supported by a generous gift from Google. Based on observations made with the Liverpool Telescope operated on the island of La Palma by Liverpool John Moores University in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias with financial support from the UK Science and Technology Facilities Council. We acknowledge the use of public data from the Swift data archive.", 'Author Contributions': "Contributors are sorted alphabetically. \n- · Observations and Data Reduction - M. Bulla, R. Lunnan, S. Schulze (X-shooter), T. G. Brink, A. V. Filippenko, Y. Yang and W. Zheng (Keck, KAIT), K. Hinds and D. A. Perley (LT), S. Schulze and J. Sollerman (NOT), Y. Sharma and T. Sit (P200), R. Lunnan, D. A. Perley, Y. Sharma, and Y. Yao (Keck), S. Schulze ( Swift ), A. Gangopadhyay (bolometric light curve of SN 2020al), and K. Hinds and D. A. Perley (BTS catalogue)\n- · Discoverer of the Si, S, Ar lines - A. Gal-Yam\n- · Analysis - P. Chen, L. Dessart, A. Gal-Yam, I. Irani, N. Kneˇzevi'c, A. A. Miller, D. A. Perley, N. Sarin, S. Schulze, N. L. Strothjohannm D. Tsuna, and O. Yaron\n- · Discussion and Interpretation - All authors contributed to discussions and interpretation.\n- · Paper Writing - L. Dessart, A. V. Filippenko, A. Gal-Yam, A. A. Miller, S. Schulze, J. Sollerman, N. L. Strothjohann, D. Tsuna, and S. E. Woosley\n- · ZTF Infant SN Programme 2018-2023 -R.J. Bruch, M. Bulla, P. Chen, S. Dhawan, A. Gal-Yam, I. Irani, S. Schulze, J. Sollerman, N. L. Strothjohann, Y. Yang, O. Yaron, and E. A. Zimmerman", 'Data Availability': 'The reduced spectra and photometry of SN 2021yfj will be made available via the WISeREP archive and the journal webpage after the acceptance of the paper. Data from ESO, Keck, NOT, Swift , and ZTF can be obtained from their designated public data repositories.', 'Code Availability': 'Much analysis for this paper has been undertaken with publicly available codes. The details required to reproduce the analysis are contained within the manuscript.', 'Photometry': "Zwicky Transient Facility -The Zwicky Transient Facility (ZTF) uses the Samuel Oschin 48-inch (1.22 m) Schmidt telescope at Palomar Observatory on Mount Palomar (USA). It is equipped with a 47square-degree camera [161] and monitors the entire northern hemisphere every 2-3 days in the g and r bands to a depth of ∼ 20 . 7 mag (5 σ ; [16, 17]) as part of the public ZTF Northern Sky Survey [162]. We retrieved the host-subtracted photometry via the Infrared Processing and Analysis Center (IPAC) ZTF forced-photometry service [163]. This service uses the data-reduction techniques outlined in Ref. [60]. We cleaned and calibrated the data following Ref. [163]. \n2.56 m Nordic Optical Telescope - We obtained photometry in gri with the Alhambra Faint Object Spectrograph and Camera (ALFOSC) 4 on the 2.56 m Nordic Optical Telescope (NOT) at the Roque de los Muchachos Observatory on La Palma (Spain). To remove the host contribution, we obtained a final set of gri photometry in August/September 2022, after the SN had faded. We reduced the data with PyNOT 5 using standard techniques for CCD data processing and photometry. The world coordinate system was calibrated with the software package astrometry.net [164]. The host contribution was removed with custom image-subtraction and analysis software (K. Hinds, K. Taggart, et al., in prep.). The photometry was measured using point-spread-function (PSF) fitting techniques based on methods in Ref. [165]. \nPalomar 60-inch telescope -Weacquired additional ugri photometry using the Rainbow Camera of the Spectral Energy Distribution Machine (SEDM; [166, 167]) on the robotic Palomar 60-inch (1.52 m) telescope (P60; [168]) at Palomar Observatory. The data were reduced using the data-reduction pipeline FPipe [165]. \n0.76 m Katzman Automatic Imaging Telescope and 1 m Nickel Telescope - We obtained photometry in BVRI and in the Clear band (close to the R band; [169]) with the 0.76 m Katzman Automatic Imaging Telescope (KAIT) at Lick Observatory on Mount Hamilton (USA) as a part of the Lick Observatory Supernova Search (LOSS; [170]). One additional epoch of photometry was also obtained with the 1 m Nickel telescope at Lick Observatory. We reduced images using a custom pipeline 6 detailed in Ref. [171]. We performed PSF photometry with the package DAOPHOT [172] from the IDL Astronomy User's Library 7 . \n2 m Liverpool Telescope - We acquired photometry in ugriz using the Infrared-Optical Imager (IO:O) on the robotic Liverpool Telescope (LT; [173]) at Roque de los Muchachos Observatory. Reduced images were downloaded from the LT archive and processed with custom image-subtraction and analysis software (K. Hinds, K. Taggart, et al., in prep.). Image stacking and alignment were performed using SWarp [174] where required. Image subtraction was performed using a pre-explosion reference image in the appropriate filter from the Panoramic Survey Telescope and Rapid Response System (Pan-STARRS) Data Release (DR) 1 [175] or Sloan Digital Sky Survey (SDSS) DR9 [176]. The photometry is measured using PSF fitting techniques based on methods in Ref. [165]. \nNeil Gehrels Swift Observatory -We submitted a target of opportunity request to use the 30 cm Ultraviolet/Optical Telescope (UVOT; [177]) aboard the Neil Gehrels Swift Observatory [178] to expand the wavelength coverage to the UV. Between January and February 2022, we obtained deep images in all filters to remove the host-galaxy contamination from the transient photometry. We coadded all sky exposures for a given epoch to boost the signal-to-noise ratio (S/N) using uvotimsum in HEAsoft 8 version 6.32.2. Afterward, we measured the brightness of SN 2021yfj with the Swift tool uvotsource . The source aperture had a radius of 5 '' , while the background region had a significantly larger radius. To remove the host contribution in w 2, m 2, and w 1 from the earlier epochs, we arithmetically subtracted the host flux from the early measurements when the SN was bright. \nFinal photometry - The datasets were calibrated against stars from the Sloan Digital Sky Survey (SDSS; e.g., P60, LT, NOT observations; [176]) and Pan-STARRS (e.g., ZTF; [60, 175, 179]), and internal zeropoints ( Swift ; [180]). Observations in similar but not identical filters (e.g., SDSS vs. ZTF filters) could \nSupplementary Material Fig. 1 : The multiband light curves of SN 2021yfj from 1,800 to 7,850 ˚ A (rest frame) corrected for MW extinction. Vertical bars represent the epochs of spectroscopy. The absolute magnitude is computed with M = m -DM( z ) + 2 . 5 log (1 + z ), where DM is the distance modulus and z the redshift. Non-detections are displayed as ' ▼ '. \n<!-- image --> \nintroduce measurable, time-dependent colour terms [181]. Convolving spectra between days 1.0 and 49.8 with SDSS and ZTF filter response functions yielded differences in the filter systems between 0.01 and 0.07 mag. They were comparable to, if not smaller than, the measurement uncertainties. Owing to this, we merged the datasets without applying any colour terms. \nThe final photometric data are shown in the Supplementary Material Figure 1 and the Supplementary Material Table 2. All measurements are reported in the AB system [182]. The measurements in the Supplementary Material Table 2 are not corrected for Galactic extinction along the line of sight, but the Galactic extinction correction is applied to all photometric data shown in the figures and the derived properties. The MW extinction along the line of sight is E ( B -V ) = 0 . 03 mag [183]. We assumed the Cardelli parameterisation of the MW extinction [184] and a total to selective extinction ratio of R V = 3 . 1.", 'Spectroscopy': "We obtained 16 spectra with several 2-10 m-class telescopes. Supplementary Material Table 3 shows the observing log. Details about the observations and the data reductions are provided below. \n10 m Keck Telescope - We obtained 4 epochs with the Low-Resolution Imaging Spectrometer (LRIS; [185]) on the 10 m Keck I telescope at Maunakea (USA) between days 1.0 and 132.5. The first and the third epochs, acquired on 8 September 2021 (day 1) and 31 January 2022 (day 132.5), used the B600/4000 blue-side grism and the R400/8500 red-side grating, dichroic 5600, and a 1 . '' 0-wide slit. For the second and fourth epochs (days 23.8 and 128.1), we utilised the B400/3400 blue-side grism and the R400/8500 red-side grating, dichroic 5600, and a 1 . '' 0-wide slit. The integration times varied between 1,200 and 3,600 s depending on the epoch. To minimise slit losses caused by atmospheric dispersion [186], the spectra were acquired with the slit oriented at or near the parallactic angle. All spectra were reduced in a standard fashion with the data-reduction pipeline LPipe [187]. \n8.2 m ESO Very Large Telescope - We collected seven medium-resolution spectra with the Xshooter instrument [188] at the 8.2 m Very Large Telescope (VLT) at Paranal Observatory (Chile) between \n9 September and 11 November 2021 (days 1.6 to 49.8). All observations were performed in nodding mode and with 1 . '' 0/0 . '' 9/0 . '' 9-wide slits (UVB/VIS/NIR arm). Each spectrum covers the wavelength interval from 3,000 to 24,800 ˚ A. The integration times varied between 2,880 and 4,400 s, depending on the arm (UVB/VIS/NIR) and phase. All observations were done with an atmospheric dispersion corrector to minimise any flux losses. The data were reduced following Ref. [189]. In brief, we first removed cosmic rays with the tool astroscrappy 9 , which is based on the cosmic-ray removal algorithm by Ref. [190]. Afterward, the data were processed with the X-shooter pipeline v3.3.5 and the ESO workflow engine ESOReflex [191, 192]. The UVB- and VIS-arm data were reduced in stare mode to boost the S/N. The individual rectified, wavelength- and flux-calibrated 2D spectra files were coadded using tools developed by J. Selsing 10 . The NIR data were reduced in nodding mode to ensure good sky-line subtraction. In the third step, we extracted the 1D spectra of each arm in a statistically optimal way using tools by J. Selsing. Finally, the wavelength calibration of all spectra was corrected for barycentric motion. The spectra of the individual arms were stitched by averaging the overlap regions. \nPalomar 200-inch Telescope - We obtained one epoch with the Double Spectrograph (DBSP; [193]) on the Palomar 200 inch (5.1 m) telescope at Mount Palomar Observatory on 13 September 2021 (day 5.2). The observations were taken using the D-55 dichroic beam splitter, a blue grating with 600 lines mm -1 blazed at 4,000 ˚ A, a red grating with 316 lines mm -1 blazed at 7,500 ˚ A, and a 1 . '' 5-wide slit. To minimise slit losses caused by atmospheric dispersion, the spectrum was acquired with the slit oriented at the parallactic angle. The data were reduced using the Python package DBSP DRP 11 that is primarily based on PypeIt [194, 195] and utilises common methods in optical spectroscopy. \n2.56 m Nordic Optical Telescope - We collected 3 epochs of low-resolution spectroscopy with ALFOSC on the NOT between 7 January and 22 February 2019 (days 4.1 to 11.1). The spectra were obtained with grism #4 and either a 1 . '' 0- or 1 . '' 3-wide slit, depending on the weather conditions. To minimise slit losses caused by atmospheric dispersion, the spectra were acquired with the slit oriented at the parallactic angle. The data were reduced with PyNOT using standard techniques for CCD data processing and long-slit spectroscopy. \nShane 3 m telescope -Weobtained one spectrum with the Kast double spectrograph 12 mounted on the Shane 3 m telescope at Lick Observatory on 11 September 2021 (day 3.5). We utilised a 2 '' -wide slit, the 600/4310 grism in the blue, and the 300/7500 grating in the red. This instrument configuration has a combined wavelength range of ∼ 3 , 500-10,500 ˚ A. To minimise slit losses caused by atmospheric dispersion, the Kast spectrum was acquired with the slit oriented at or near the parallactic angle. The Kast data were reduced following standard techniques for CCD processing and spectrum extraction [196] utilising IRAF routines and custom Python and IDL codes 13 . Owing to the low quality of the Lick spectrum, it is not shown in any of the figures in the paper, but it can be downloaded from WISeREP like all other spectra. \nFlux calibration and host correction - The flux calibration of all spectra was achieved with spectrophotometric standard stars observed during the same nights. We also tied the absolute flux scale to our multiband photometry. As the SN faded, the relative contribution from the host galaxy increased. To remove the host contribution, we used the X-shooter observation obtained at day 49.8, which also covered the host galaxy. The host was detected in the LRIS spectra from January/February 2022, too. While the continua of the three spectra were identical, the spectra differed in the relative amplitude of the emission lines due to the different resolving powers.", 'High-Energy Observations': "While monitoring SN 2021yfj with UVOT between day 1.5 and day 148.7, Swift also observed the field with its onboard X-ray telescope XRT between 0.3 and 10 keV in photon-counting mode [197]. We analysed these data with the online tools of the UK Swift team 14 that use the software package HEASoft version 6.26.1 and methods described in Refs. [198, 199]. SN 2021yfj evaded detection at all epochs. The median 3 σ count-rate limit of each observing block is 6 × 10 -3 s -1 (0.3-10 keV). Coadding all data pushes the 3 σ count-rate limits to 0 . 6 × 10 -3 s -1 . To convert the count-rate limits into a flux, we assume a power-law spectrum with a photon index 15 of Γ = 2 and a Galactic neutral hydrogen \ncolumn density of 2 . 7 × 10 20 cm -2 [200]. The coadded count-rate limit translates to an unabsorbed flux of < 0 . 2 × 10 -13 erg cm -2 s -1 in the range of 0.3-10 keV and a luminosity of < 1 . 2 × 10 42 erg s -1 . Supplementary Material Table 4 shows a list of all limits. \nFurthermore, we queried the NASA High Energy Astrophysics Science Archive Research Center (HEASARC 16 ) to search for any X-ray and γ -ray transient preceding or accompanying SN 2021yfj. The source catalogues of the Fermi , MAXI , NICER , NuSTAR , AGILE , and INTEGRAL space missions returned no detection within 10 ' from SN2021yfj's position between 1 August 2021 (32.7 days before the discovery of SN 2021yfj) and 4 February 2022 (131.8 days after the discovery of SN 2021yfj). Placing a detection limit is not possible for this multitude of facilities.", 'Host-galaxy Observations': 'We retrieved science-ready coadded images from the Galaxy Evolution Explorer ( GALEX ) general release 6/7 [201], the DESI Legacy Imaging Surveys (LS; [116]) DR 10, and PS1. We measured the brightness of the host using LAMBDAR (Lambda Adaptive Multi-Band Deblending Algorithm in R; [202]) and the methods described in Ref. [58]. The field was also observed by the VISTA Hemisphere Survey [203] in the near-IR. We measured the brightness with the aperture photometry tool presented in Ref. [127] using an aperture similar to the ones employed for the other images. The GALEX , LS, and PanSTARRS photometry was calibrated against tabulated zeropoints, and the VHS photometry against stars from the 2MASS Point Source Catalogue [204]. Supplementary Material Table 5 summarises the measurements in the different bands. \nThe SN spectra also showed absorption lines from the ISM in the host and emission lines produced by the ionised gas in H ii . Supplementary Material Table 6 summarises the rest-frame equivalent widths of the absorption lines extracted from all early X-shooter spectra and the emission-line fluxes from X-shooter spectrum obtained at day 49.8. The flux measurements are not corrected for reddening.', 'Comparison Objects': 'We compare the spectra and light curves of SN 2021yfj to those of other objects. Below, we list the relevant references for each object. \n- · SN2006jc (Ibn) - bolometric light curve, not constructed; spectra, Ref. [12]\n- · SN2010al (Ibn) - bolometric light curve, built with data from Ref. [205] using the programme Superbol [206]; spectra, Ref. [205]\n- · SN2019hgp (Icn) - bolometric light curve, Ref. [13]; spectra, Ref. [13]\n- · SN2020oi (Ic) - bolometric light curve, Ref. [207]\n- · SN2021csp (Icn) - bolometric light curve, Ref. [14]; spectra, Ref. [14] \nAll spectra were retrieved from WISeREP. \nSupplementary Material Table 1 : Lines identified in the spectra shown in Figure 1 and the Extended Material Figure 3. \nNotes: These are the Ritz air wavelengths reported in the NIST atomic spectra database (details at https://physics.nist.gov/PhysRefData/ASD/Html/lineshelp.html). All wavelengths are rounded to one decimal place for convenience. \nSupplementary Material Table 2 : Log of photometric observations. \nNotes: All measurements are reported in the AB system and are not corrected for reddening. The phase is reported in the rest-frame with respect to the time of the first detection (MJD=59464.414). Non-detections are reported at 3 σ confidence. A machine-readable table will be made available via the WISeREP archive and the journal webpage after the acceptance of the paper. \nations. \nobserv \nectroscopic \nsp \nof \nLog \n: \n3 \nable \nT \nal \nMateri \ntary \nSupplemen \nSupplementary Material Table 4 : Log of X-ray observations. \nNotes: The modified Julian dates report the mid-exposure time. The phase is reported in the rest-frame with respect to the time of the first detection (MJD = 59,464.414). The time errors indicate the extent of the time bins. All limits are reported at 3 σ confidence. The measurements are corrected for MW absorption and are reported for the bandpass from 0.3 to 10 keV. \nSupplementary Material Table 5 : Photometry of the host galaxy. \nNotes: All measurements are reported in the AB system and are not corrected for reddening. \nSupplementary Material Table 6 : Properties of the ISM in the host galaxy. \nNotes: We report rest-frame equivalent widths EW r for absorption lines and fluxes for emission lines. The emission lines are measured from the X-shooter spectrum at day 49.8 and are not corrected for reddening. \nSupplementary Material Table 7 : The bolometric light curve and blackbody properties. \nNotes: The phase is reported in the rest frame with respect to the time of the first detection (MJD = 59,464.414). The last column indicates whether a bolometric correction was applied to determine the pseudobolometric luminosity in the wavelength interval 1,800-7,850 ˚ A. Including the missing FUV and IR flux required a bolometric correction for all data (see the Methods Section Bolometric Light Curve for details). The blackbody properties are only reported where UV and optical photometry are available and where the spectrum is adequately described by a blackbody. A machine-readable table will be made available via the WISeREP archive and the journal webpage after the acceptance of the paper. \nSupplementary Material Table 8 : Light-curve fits with Redback : models, priors and marginalised posteriors. \nNotes: We used uniform ( U ) and log-uniform (log U ) priors. The uncertainties of the marginalised posteriors are quoted at 1 σ confidence. The explosion date is measured with respect to the time of the first detection. All marginalised posteriors are reported in linear units. The Bayesian evidence is reported in log units. The opacities of the nickel model were taken from Refs. [208, 209]. \nSupplementary Material Table 9 : Light curve fit with CHIPS .', 'References': '- [1] Burbidge, E. M., Burbidge, G. R., Fowler, W. A. & Hoyle, F. Synthesis of the Elements in Stars. Reviews of Modern Physics 29 , 547-650 (1957).\n- [2] Kippenhahn, R., Weigert, A. & Weiss, A. Stellar Structure and Evolution (Springer Berlin, Heidelberg, 2013).\n- [3] Arcones, A. & Thielemann, F.-K. Origin of the elements. Astron. Astrophys. Rev. 31 , 1 (2023).\n- [4] Woosley, S. E. & Weaver, T. A. The Evolution and Explosion of Massive Stars. II. Explosive Hydrodynamics and Nucleosynthesis. Astrophys. J. Suppl. Ser. 101 , 181 (1995).\n- [5] Woosley, S. E., Heger, A. & Weaver, T. A. The evolution and explosion of massive stars. Reviews of Modern Physics 74 , 1015-1071 (2002).\n- [6] Heger, A., Fryer, C. L., Woosley, S. E., Langer, N. & Hartmann, D. H. How Massive Single Stars End Their Life. Astrophys. J. 591 , 288-300 (2003).\n- [7] Woosley, S. E. & Janka, T. The physics of core-collapse supernovae. Nature Physics 1 , 147-154 (2005).\n- [8] Muller, B. The Status of Multi-Dimensional Core-Collapse Supernova Models. Publ. Astron. Soc. Aust. 33 , e048 (2016).\n- [9] Woosley, S. E. Pulsational Pair-instability Supernovae. Astrophys. J. 836 , 244 (2017).\n- [10] Crowther, P. A. Physical Properties of Wolf-Rayet Stars. Annu. Rev. Astron. Astrophys. 45 , 177-219 (2007).\n- [11] Matheson, T., Filippenko, A. V., Ho, L. C., Barth, A. J. & Leonard, D. C. Detailed Analysis of Early to Late-Time Spectra of Supernova 1993J. Astron. J. 120 , 1499-1515 (2000).\n- [12] Pastorello, A. et al. A giant outburst two years before the core-collapse of a massive star. Nature 447 , 829-832 (2007).\n- [13] Gal-Yam, A. et al. A WC/WO star exploding within an expanding carbon-oxygen-neon nebula. Nature 601 , 201-204 (2022).\n- [14] Perley, D. A. et al. The Type Icn SN 2021csp: Implications for the Origins of the Fastest Supernovae and the Fates of Wolf-Rayet Stars. Astrophys. J. 927 , 180 (2022).\n- [15] Maeda, K. & Moriya, T. J. Properties of Type Ibn Supernovae: Implications for the Progenitor Evolution and the Origin of a Population of Rapid Transients. Astrophys. J. 927 , 25 (2022).\n- [16] Bellm, E. C. et al. The Zwicky Transient Facility: System Overview, Performance, and First Results. Publ. Astron. Soc. Pac. 131 , 018002 (2019).\n- [17] Graham, M. J. et al. The Zwicky Transient Facility: Science Objectives. Publ. Astron. Soc. Pac. 131 , 078001 (2019).\n- [18] Mu˜noz-Arancibia, A. et al. ALeRCE/ZTF Transient Discovery Report for 2021-09-07. Transient Name Server Discovery Report 2021-3075 , 1-3075 (2021).\n- [19] Bruch, R. J. et al. The Prevalence and Influence of Circumstellar Material around Hydrogen-rich Supernova Progenitors. Astrophys. J. 952 , 119 (2023).\n- [20] Pastorello, A. et al. Massive stars exploding in a He-rich circumstellar medium - I. Type Ibn (SN 2006jc-like) events. Mon. Not. R. Astron. Soc. 389 , 113-130 (2008).', '882 , 36 (2019).': "- [59] Johnson, B. D., Leja, J., Conroy, C. & Speagle, J. S. Stellar Population Inference with Prospector. Astrophys. J. Suppl. Ser. 254 , 22 (2021).\n- [60] Masci, F. J. et al. The Zwicky Transient Facility: Data Processing, Products, and Archive. Publ. Astron. Soc. Pac. 131 , 018003 (2019).\n- [61] Patterson, M. T. et al. The Zwicky Transient Facility Alert Distribution System. Publ. Astron. Soc. Pac. 131 , 018001 (2019).\n- [62] Zackay, B., Ofek, E. O. & Gal-Yam, A. Proper Image Subtraction-Optimal Transient Detection, Photometry, and Hypothesis Testing. Astrophys. J. 830 , 27 (2016).\n- [63] Gal-Yam, A. The Most Luminous Supernovae. Annu. Rev. Astron. Astrophys. 57 , 305-333 (2019).\n- [64] van der Walt, S., Crellin-Quick, A. & Bloom, J. SkyPortal: An Astronomical Data Platform. The Journal of Open Source Software 4 , 1247 (2019).\n- [65] Coughlin, M. W. et al. A Data Science Platform to Enable Time-domain Astronomy. Astrophys. J. Suppl. Ser. 267 , 31 (2023).\n- [66] Gal-Yam, A. et al. Real-time Detection and Rapid Multiwavelength Follow-up Observations of a Highly Subluminous Type II-P Supernova from the Palomar Transient Factory Survey. Astrophys. J. 736 , 159 (2011).\n- [67] Forster, F. et al. The Automatic Learning for the Rapid Classification of Events (ALeRCE) Alert Broker. Astron. J. 161 , 242 (2021).\n- [68] Huber, M. et al. The Pan-STARRS Survey for Transients (PSST) - first announcement and public release. The Astronomer's Telegram 7153 , 1 (2015).\n- [69] Gal-Yam, A. et al. The spectroscopic classification of SN 2021yfj. Transient Name Server Classification Report 2024-17883 (2024).\n- [70] Gal-Yam, A. et al. The spectroscopic classification of SN 2021yfj. Transient Name Server AstroNote 240 (2024).\n- [71] Strotjohann, N. L. et al. Bright, Months-long Stellar Outbursts Announce the Explosion of Interaction-powered Supernovae. Astrophys. J. 907 , 99 (2021).\n- [72] Hogg, D. W., Baldry, I. K., Blanton, M. R. & Eisenstein, D. J. The K correction. arXiv e-prints astro-ph/0210394 (2002).\n- [73] Miller, A. A. et al. ZTF Early Observations of Type Ia Supernovae. II. First Light, the Initial Rise, and Time to Reach Maximum Brightness. Astrophys. J. 902 , 47 (2020).\n- [74] Maguire, K. in Type Ia Supernovae (eds Alsabti, A. W. & Murdin, P.) Handbook of Supernovae 293 (Springer Cham, 2017).\n- [75] Arcavi, I. in Hydrogen-Rich Core-Collapse Supernovae (eds Alsabti, A. W. & Murdin, P.) Handbook of Supernovae 239 (Springer Cham, 2017).\n- [76] Gezari, S. Tidal Disruption Events. Annu. Rev. Astron. Astrophys. 59 , 21-58 (2021).\n- [77] Bond, H. E. et al. The 2008 Luminous Optical Transient in the Nearby Galaxy NGC 300. Astrophys. J. Lett. 695 , L154-L158 (2009).\n- [78] Ho, A. Y. Q. et al. A Search for Extragalactic Fast Blue Optical Transients in ZTF and the Rate of AT2018cow-like Transients. Astrophys. J. 949 , 120 (2023)."}

2024arXiv240906253H

We explore the stability and fate of gravitational triple systems comprising a central massive body and a tight binary of less massive pairs. Our present purpose is two fold 1 to improve the Hilltype stability criterion for the binary in those systems and 2 to examine the fate of the triple systems after the binary breakup with particular attention to the effects of the eccentricities of the inner and outer orbits. We perform direct Newtonian Nbody simulations over much longer integration times than previous studies which is essential to determine the eventual fate of those systems statistically in a reliable fashion. We obtain an empirical fitting formula of the binary stability boundary that incorporates effects of the inner and outer eccentricities the mutual inclination of the inner and outer orbits and the mass ratios of the three bodies. We also find that those triple systems are stable for a much longer timescale after the binary breakup and that their final fates ejection of the outer body merger to the central massive body and collision of two less massive bodies are very sensitive to the initial outer eccentricity.

2024-09-01T00:00:00Z

['arXiv:2409.06253', '2024arXiv240906253H', '10.48550/arXiv.2409.06253']

['Astrophysics - Solar and Stellar Astrophysics', 'Astrophysics - Earth and Planetary Astrophysics']

Stability and fate of hierarchical triples comprising a central massive body and a tight binary in eccentric orbits

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.06253.pdf

{'No Header': 'Draft version September 11, 2024 \nTypeset using L A T E X preprint style in AASTeX631', 'comprising a central massive body and a tight binary in eccentric orbits': 'Toshinori Hayashi , 1 Alessandro A. Trani , 2 and Yasushi Suto 3, 4, 5 \n- 1 \nYukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan 2 Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100 Copenhagen, Denmark 3 Research Institute, Kochi University of Technology, Tosa Yamada, Kochi 782-8502, Japan 4 Department of Physics, The University of Tokyo, Tokyo 113-0033, Japan \n5 Research Center for the Early Universe, School of Science, The University of Tokyo, Tokyo 113-0033, Japan \n(Received August 31, 2024; Revised; Accepted) \nSubmitted to ApJ', 'ABSTRACT': 'We explore the stability and fate of gravitational triple systems comprising a central massive body and a tight binary of less massive pairs. Our present purpose is two fold; (1) to improve the Hill-type stability criterion for the binary in those systems, and (2) to examine the fate of the triple systems after the binary break-up, with particular attention to the effects of the eccentricities of the inner and outer orbits. We perform direct Newtonian N-body simulations over much longer integration times than previous studies, which is essential to determine the eventual fate of those systems statistically in a reliable fashion. We obtain an empirical fitting formula of the binary stability boundary that incorporates effects of the inner and outer eccentricities, the mutual inclination of the inner and outer orbits, and the mass ratios of the three bodies. We also find that those triple systems are stable for a much longer timescale after the binary break-up, and that their final fates (ejection of the outer body, merger to the central massive body, and collision of two less massive bodies) are very sensitive to the initial outer eccentricity. \nKeywords: celestial mechanics - (stars:) binaries (including multiple): close - stars: black holes', '1. INTRODUCTION': "The stability and fate of gravitational triple systems is one of the long-standing and challenging questions in mathematical physics and astronomy. Many previous authors have approached the problem using a variety of different approximations and methodologies, including Eggleton & Kiseleva \nCorresponding author: Toshinori Hayashi \ntoshinori.hayashi@yukawa.kyoto-u.ac.jp \n(1995); Holman & Wiegert (1999); Mardling & Aarseth (1999, 2001); Georgakarakos (2013); Grishin et al. (2017); He & Petrovich (2018); Myllari et al. (2018); Wei et al. (2021); Lalande & Trani (2022); Tory et al. (2022); Vynatheya et al. (2022); Hayashi et al. (2022, 2023); Zhang et al. (2023), among others. \nIn particular, a lot of attention has been paid to a hierarchical triple configuration, in which two of them form a tight binary and interact with the tertiary object. For instance, Mardling & Aarseth (2001) (hereafter, MA01) considered their stability combining the chaos theory and direct numerical simulations, and obtained the following stability criterion: \na out (1 -e out ) a in > 2 . 8 ( 1 -0 . 3 i mut 180 · )[( 1 + m 3 m 1 + m 2 ) (1 + e out ) √ 1 -e out ] 2 / 5 . (1) \nIn the above expression, a out and a in are the semi-major axes of the outer and inner orbits, e out is the orbital eccentricity of the outer orbit, i mut is the mutual inclination between the outer and inner orbits, m 3 is the mass of the tertiary, and m 1 and m 2 are the masses of the inner binary. \nThe criterion (1) (implicitly) assumes that the tertiary mass m 3 is at most comparable to m 1 and m 2 , and has been frequently applied to the case of m 3 ≪ m 12 ≡ m 1 + m 2 . In what follows, we refer to such configurations as HT-P (Hierarchical Triple Planet-type) in which a tertiary orbits an inner massive binary. The classic criterion (1) has been tested and improved recently by Vynatheya et al. (2022); Hayashi et al. (2022, 2023), for instance. \nAnother hierarchical triple configuration, which we refer to as HT-S (Hierarchical Triple Satellitetype) below, is a central massive object orbited by a less massive binary. The dynamics of HTS configurations is important in understanding the fate of a variety of physically interesting triple systems. For instance, the enhanced merging rate of Binary Black-Holes (BBHs) around supermassive BH (SMBH) (e.g. Li et al. 2015; VanLandingham et al. 2016) may be a major target of future spacebased gravitational wave detectors (e.g. Xuan et al. 2023). Extrasolar binary-planet systems also belong to this configuration, which have been theoretically predicted/discussed (e.g. Ochiai et al. 2014; Lewis et al. 2015), but not yet detected. \nThe present paper focuses on the stability and fate of HT-S systems comprising a central massive body of mass m 0 orbited by an initially tightly bound binary of mass m 1 and m 2 . We examine a break-up condition of the binary and obtain its stability boundary as a function of the initial parameters of the systems. Then, we consider the subsequent fate of systems after the binary breakup, keeping in mind the application to black-hole triples. To avoid confusion, we use m 0 for the massive tertiary in HT-S systems, while we use m 3 for the less massive tertiary in the HT-P systems. Our current analysis assumes purely Newtonian gravity for simplicity and clarity, and the effect of general relativity will be studied in a forthcoming paper. \nThe Hill stability condition is particularly useful in considering the stability of HT-S systems. Historically, the Hill stability was first derived for the motion of moon, applying the conservation of the Jacobi integral in restricted three-body problems (e.g. Hill 1878). Later, it was extended to more general (non-restricted) three-body problems using Sundman's inequality, which determines a required condition among momenta for a gravitational multi-body system, and the sufficient conditions for some cases were derived (e.g. Marchal & Bozis 1982). \nWhile the Hill stability is rigorously defined in those papers, the condition is roughly understood as the competition between the gravitational tidal force of the binary due to the central object and the mutual gravitational attraction of the binary pairs. \nConsider two objects of masses m 1 and m 2 orbiting a more massive central object of mass m 0 . If both the inner and outer orbits are circular, under the test particle limit ( m 2 ≪ m 1 < m 0 ), a binary companion m 2 orbiting m 1 is stably bound to m 1 if its semi-major axis a 12 is less than the Hill radius a Hill : \na 12 < a Hill ≡ a 012 ( m 1 3 m 0 ) 1 / 3 . (2) \nwhere a 012 is the semi-major axis between m 1 -m 2 binary and m 0 . The above stability condition is rewritten in terms of the inner and outer orbital periods as \nP 012 > √ 3 P 12 . (3) \nwhere P 012 and P 12 are the periods of outer and inner orbits, respectively. In other words, the Hillstability is roughly equivalent to the condition that the outer orbital period is longer than the inner one. Note that, for clarity, we use the labels '12' and '012' for HT-S, instead of 'in' and 'out' of HT-P, unless otherwise specified. \n̸ \n̸ \nWhile the Hill stability condition (2) or (3) gives approximately the stability condition for the HT-S systems, it needs to be generalized to the cases of non-circular ( e 12 = 0 and/or e 012 = 0) orbits, the finite mass of m 2 , and non-coplanar orbits. \nFor instance, Grishin et al. (2017) (hereafter GPZM17) generalized the stability condition (2) by considering the mutual inclination dependence for initially near circular inner and outer orbits: \na 012 a 12 > 1 3 1 / 3 ( 3 m 0 m 1 + m 2 ) 1 / 3 (cos i mut + √ 3 + cos 2 i mut ) 2 / 3 × 1 (cos 2 i mut > 3 / 5) 9 -5 cos 2 i mut 6 (cos 2 i mut ≤ 3 / 5) . (4) \nGPZM17 derived the above criterion by replacing a 12 with a 12 (1 + 0 . 5 e 2 max ) under the von ZeipelKozai-Lidov (ZKL) oscillations (von Zeipel 1910; Kozai 1962; Lidov 1962), where e max is the maximum value of e 12 attained during the ZKL cycle. GPZM17 confirmed that the condition (4) well captures the strong destabilization of HT-S systems with near-polar orbits using numerical simulations. \n̸ \n̸ \nThe present paper aims to further generalize the condition (4) for the HT-S systems of initially non-circular orbits ( e 12 = 0 and e 012 = 0), as pioneered by MA01 for the HT-P systems. This is shown in Figure 1, where we plot the MA01 stability condition (1) for the coplanar HT-P systems (red curves). Because GPZM17 focused on initially circular HT-S systems, their result (4) is valid for e 012 = 0 alone. Figure 1 assumes that the initial inner orbits are circular ( e in (= e 12 ) = 0), but shows the mass dependence using different line types for comparison. \nIf the Hill stability is purely determined by the amplitude of the instantaneous tidal force due to the central massive body, one naively expects that the non-circular effect would be incorporated simply by replacing the initial semi-major axes a 12 and a 012 with their initial apocenter distance a 12 (1 + e 12 ), and pericenter distance a 012 (1 -e 012 ), respectively. In reality, however, this simple procedure completely neglects the evolution of the orbital elements. This is why we explore the non-circular effect using a series of systematic numerical simulations below. \nAs schematically illustrated in Figure 2, the main purpose of this paper is basically two-fold. The primary purpose, derivation of a generalized stability boundary for the HT-S systems that we \nFigure 1. Comparison of stability criteria for HT-P ( m 12 /m 3 = 2 , 10 2 , red curves) and HT-S ( m 12 /m 0 = 2 , 10 -2 , 10 -4 , black dots) in e out -a out (1 -e out ) /a in plane, where m 12 = m 1 + m 2 . For HT-S, a in , a out and e out are interpreted as a 12 , a 012 and e 012 , respectively. Note that the stability criterion of HT-P (GPZM17) focuses on circular systems, and horizontal gray lines are simple extrapolations assuming e 012 dependence is all absorbed in a 012 (1 -e 012 ). \n<!-- image --> \neout \ndiscussed in the above, corresponds to the classification 1. We attempt to extend the GPZM17 stability criterion by including initial eccentricity, mutual inclination, and mass ratio dependences, using long-term N-body integrations. We emphasize that the direct N-body simulations, instead of the orbit-averaged secular simulations, are essential to explore the Hill-stability boundary between (A) and (B). \nThe secondary purpose is to find the outcome of the Hill unstable HT-S systems. As indicated in the classification 2, there are four possible outcomes after the initial tight binary is broken. Their branching ratio is somewhat dependent on the integration time of the simulations. We practically perform the simulations up to 10 8 P 12 , and estimate the fraction of the four different outcomes evaluated at the epoch as a function of e 012 . We here note that Toonen et al. (2022) perform a pioneering study about the evolution and fate of HT-P type stellar systems, considering stellar evolution and various triple population models as well. We assume point-mass objects here and do not consider any astronomical evolution except the Newtonian dynamics. \nThe rest of the paper is organized as follows. In section 2, we first describe initial setup of numerical simulations, and how we evaluate the breaking condition of initial binary. Then, section 3 shows the resulting Hill-stability criterion, including initial eccentricity, mutual inclination, and mass ratio \nFigure 2. Schematic illustration of the contents in this paper. Classification 1 checks the breaking condition of SMBH-BBH triples, and Classification 2 checks the outcome fraction and timescale of Hill-unstable triples. Note that which of m 1 or m 2 becomes m in is in general time-dependent during evolution. While channel (A) can be computed using secular perturbation, channel (B) requires direct N-body simulations. \n<!-- image --> \ndependences. We also systematically check the initial phase dependence about the Hill stability, and show the result in the appendix. Next, in section 4, we show a demonstrative example of outcome fraction and timescale, focusing on the Hill-unstable systems with different mutual inclinations. Finally, section 5 summarizes the conclusion of this paper, and discusses possible future prospects.", '2. METHOD': 'We carry out a series of N-body simulations using TSUNAMI (see Trani & Spera 2023), and explore the parameter space for the initial values of e 12 and e 012 in particular. We also consider a set of different mass ratios for the three bodies, and four representative mutual inclinations ( i mut = 0 · , 5 · , 90 · , 180 · ). \nWe determine the stability boundary of those systems on e 012 -a 012 (1 -e 012 ) /a 12 plane, following Hayashi et al. (2022, 2023). For a given set of values for e 12 , e 012 , a 12 , i mut , m 1 , m 2 , and m 0 , we start a simulation with a sufficiently small value of a 012 so that the m 1 -m 2 binary quickly breaks due to the Hill instability. Then, we gradually increase the value of a 012 , therefore a 012 (1 -e 012 ) /a 12 , and record the corresponding break time T break of the m 1 -m 2 binary. The procedure is repeated up to our adopted integration time limit t int = 10 8 P 12 . The stability boundary is defined by the critical value of a 012 (1 -e 012 ) /a 12 with which the binary does not break within the integration time limit. To avoid a possible spurious result due to the fluctuation, we check a larger value of a 012 (1 -e 012 ) /a 12 , and if the binary does not break again, we define this critical value as the stability boundary. The robustness of the resulting boundary is separately examined by running realizations of different initial phases later (see the appendix). \nAll the above parameter survey is done for a given set of the initial phases (mean anomalies and pericenter arguments), but the initial-phase dependence is checked later for a sample of specific orbital configurations. Throughout this paper, we fix P 12 as 1 yr unless otherwise stated, but this does not virtually affect our result thanks to the scaling relation in Newtonian gravity. \nIn this procedure, the definition of the binary break is the most crucial, and we basically apply the disruption condition in Hayashi et al. (2023) as follows. At each numerical timestep during the simulation, we compute the osculating Kepler orbital elements for bounded pairs of particles; ( m 1 , m 2 ), ( m 0 , m 1 ) and ( m 0 , m 2 ) at most. Then, we define that the initial binary pair ( m 1 , m 2 ) is broken if \n0 < a 01 ( t ) < a 12 ( t ) or 0 < a 02 ( t ) < a 12 ( t ) or a 12 ( t ) < 0 . (5) \nThe above conditions basically reflect the configuration of (B) in Figure 2. We adopt the semi-major axes, instead of the binary binding energies, in the above definition of the binary break. This is because the amplitude of binding energies is always dominated by the central massive body in the present case ( m 0 ≫ m 1 ≈ m 2 ). \nGPZM17 adopted a binary break when either (1) the eccentricity of a binary exceeds unity or (2) the distance of binary exceeds 3 r Hill . We prefer the condition (5), rather than that adopted by GPZM17 for the following reasons. First, the osculating eccentricity may not be stable especially when the binary becomes close to a break-up configuration. Second, to use the Hill-stability radius as a measure of the break-up may lead to a circular argument, because our current purpose is to find the improved criterion for the Hill stability. In practice, however, we numerically made sure that the two different definitions change the stability boundary, i.e., the value of the y-axis in Figure 1, usually within tens of percent as shown in the next section.', '3. IMPROVED FORMULA FOR STABILITY BOUNDARIES OF THE HT-S SYSTEMS': 'In this section, we derive an empirical formula for stability boundaries of the HT-S system that improves the GPZM17 stability criterion (4). We examine the inner and outer eccentricity dependence for 4 different cases of the mutual inclination i mut = 0 · (coplanar-prograde), 5 · (near-coplanarprograde), 90 · (orthogonal), and 180 · (coplanar-retrograde). We considered i mut = 5 · in addition to 0 · because the final fate after the binary break is sensitive to the exact or near coplanarity of the orbits, but the stability boundary turned out to be almost identical. Furthermore, we confirmed the m 0 /m 12 -dependence in the GPZM17 criterion is valid for a wide range of m 0 /m 12 , and make sure that the result is statistically robust against the initial orbital phases of the three bodies (see the appendix). Adopting the i mut -dependence discovered by GPZM17, our stability criterion is written as follows: \nΥ ≡ ˜ r 012 ˜ r 12 > Υ crit ≡ ( m 0 m 1 + m 2 ) 1 / 3 (cos i mut + √ 3 + cos 2 i mut ) 2 / 3 h ( e 012 ) , (6) \nwhere \n˜ r 012 ≡ a 012 (1 -e 012 ) , (7) ˜ r 12 = a 12 (1 + 0 . 5 e 2 12 ) ( i mut = 0 · , 5 · ) a 12 [1 + 0 . 5 e 2 max ( i mut )] ( i mut = 90 · ) a 12 (1 + e 12 ) ( i mut = 180 · ) , (8) \nh ( e 012 ) =1 . 5 + 0 . 6 e 012 , \n(9) \nand e max is the maximum binary eccentricity e 12 under the ZKL oscillations. In the test-particle limit, \ne max = √ 1 -5 3 cos 2 i mut (cos 2 i mut ≤ 3 / 5) , (10) \nrecovering the part of the i mut -dependence in the GPZM17 criterion (4): \n1 + 0 . 5 e 2 max ( i mut ) = 9 -5 cos 2 i mut 6 . (11) \nHere, we note that Y is usually defined as a out (1 -e out ) /a in (1 + e in ) (e.g. Eggleton & Kiseleva 1995; Vynatheya et al. 2022), and we define Υ as the extension of this quantity. \nThe next subsections will describe how we obtain the parameter dependence in the stability boundary formula (6). We run a series of simulations for i mut = 0 · , 5 · , 90 · , and 180 · , fixing initial phases ( ω 12 , ω 012 ,M 12 , M 012 ) as (180 · , 0 · , 30 · , 45 · ), where M 12 , M 012 , ω 12 , and ω 012 are the inner mean anomaly, outer mean anomaly, inner pericenter argument, and outer pericenter argument, respectively (see Figure 3). We use the Jacobi coordinate system, and the invariant plane, therefore fixing Ω 12 and Ω 012 as 180 · and 0 · , respectively. \nSince our binary breaking timescale T break is evaluated in units of the initial value of P 12 , the result is also scalable with respect to the specific value of P 12 , and we set P 12 = 1 yr without loss of generality. \nWe examine the dependence of the stability boundary on the eccentricities e 12 and e 012 in § 3.1 and § 3.2 for m 0 = 10 6 M ⊙ and m 1 = m 2 = 10 M ⊙ , with a triple system comprising a massive BH and a stellar mass binary black hole in mind. Since we neglect the effect of general relativity and do not consider the merger condition here, the result is dependent on their mass ratio alone. The dependence on their mass ratio is examined in § 3.3, and the sensitivity to the initial phases is discussed in appendix A.', '3.1. Inner eccentricity dependence': 'For the initial configuration described in the above, we examine 25 models with e 012 = 0 , 0 . 2 , 0 . 4 , 0 . 6 , 0 . 8, and e 12 = 0 , 0 . 2 , 0 . 4 , 0 . 6 , 0 . 8. For each model, we gradually increase the initial value of a 012 (1 -e 012 ) /a 12 and find the stability boundary at which m 1 -m 2 binary does not break up before 10 8 P 12 . \nThe result is plotted in Figure 4 in which the binary break-up timescale of each system is indicated as a color-coded filled circle. If a system does not break up within the integration time limit (10 8 P 12 ), the system is indicated by a cross symbol. The horizontal axis of the figure is e 012 , and five sequences centered at the value of e 012 correspond to the results that we slightly shifted for visual clarity according to the value of e 12 (0, 0 . 2, 0 . 4, 0 . 6 and 0 . 8 from left to right). For reference, the dashed black lines in those panels indicate the GPZM17 criterion obtained for e 012 = 0. \nThe top four panels plot a 012 (1 -e 012 ) /a 12 (1 + 0 . 5 e 2 12 ), the pericenter distance between the binary and the central massive object in units of the orbit-averaged distance of the binary, for different i mut . While the effect of e 12 is reasonably absorbed in the above scaling for i mut = 0 · and 5 · , there remains a clear systematic trend for i mut = 90 · and 180 · . \nInstead, we found that the residual e 12 -dependence is well absorbed by changing the vertical axis as a 012 (1 -e 012 ) /a 12 (1 + 0 . 5 e 2 max ) for i mut = 90 · and a 012 (1 -e 012 ) /a 12 (1 + e 12 ) for i mut = 180 · ; see the bottom two panels in Figure 4. \nFigure 3. The initial configuration of the inner and outer orbits. \n<!-- image --> \nThe scaling with respect to e 12 for (near)-coplanar prograde cases implies that the binary break-up due to the Hill-type instability is not instantaneous generally, but happens in a somewhat cumulative fashion. This is reasonable especially around the stability/instability boundary where the outer orbital period is supposed to be much longer than the inner orbital period as indicated from the conventional Hill stability condition (3). \nThe scaling for i mut = 90 · was already suggested by GPZM17, and can be understood as the orbitaveraged binary distance should be computed from e max , instead of the initial value of e 12 , due to the ZKL oscillation. We note that Vynatheya et al. (2022) suggest that a similar approach works well for HT-P systems. \nIn contrast, the physical origin of the scaling for a coplanar retrograde is not clear. It seems as if the instability happened instantaneously when the temporal tidal interaction of the three bodies is strongest, but it is unlikely the case in reality. Thus, we leave the explanation to the future work, and simply show it as the empirical result at this point.', '3.2. Outer eccentricity dependence': 'Consider next the e 012 -dependence of the stability boundary. For that purpose, we replot Figure 4 in such a way that the vertical axis is now normalized by the GPZM17 criterion (4). This is equivalent to determine the functional form of h ( e 012 ) in our final expression (6). The result is shown in Figure 5, indicating that a simply linear function well captures the e 012 -dependence (red lines): \nh ( e 012 ) = 1 . 5 + 0 . 6 e 012 . (12) \nWe emphasize that the i mut -dependence of the GPZM17 result is in good agreement with our numerical results for i mut = 0 · , 5 · , 90 · and 180 · . \n/ \nFigure 4. Break-up time distribution in e 012 - distance ratio plane. Six panels show the results for i mut = 0 · , 5 · , 90 · , and 180 · . Each group of five represents the same value of e 012 , but different values of e 12 , shifted horizontally in the range of 0, 0 . 2, 0 . 4, 0 . 6 and 0 . 8 from left to right. The upper four panels use the average distances of inner orbits, although lower two panels use different distance measures. \n<!-- image --> \nWhile our result (12) is systematically larger than that derived by GPZM17 (4) corresponding to h ( e 012 ) = 1, it is partially explained by the difference of the break definition ( § 2) and our longer integration time (10 8 P 12 instead of 100 P 012 in GPZM17). We also note that the stability boundary at T break ∼ 100 P 012 ≲ 100 P 12 is fairly independent of the value of e 012 . Therefore, we conclude that our result is consistent with the GPZM17 result. \n% \nFigure 5. Same as Figure 4, but the vertical axis is now normalized by the GPZM17 result. The red dashed lines are empirical fits, h ( e 012 ), introduced here to include e 012 dependence. \n<!-- image --> \nIncidentally, Figure 5 suggests that h ( e 012 ) has an additional i mut -dependence. If we replace 1 . 5 by 1 . 6 for i mut = 90 · and by 1 . 8 for i mut = 180 · , for instance, the fit matches better our simulation result. Nevertheless, it does not lead to a significant difference, and we prefer the simpler empirical fit here. If we consider a simple linear interpolation, for instance, the above additional i mut -dependence might be expressed as \n˜ h ( e 012 ) = 1 . 5 + 0 . 15 i mut 90 · +0 . 6 e 012 , (13) \nalthough systematic simulations are required for further extension. We also note that the slope of h ( e 012 ), 0 . 6, coincides with that of MA01 criterion for HT-P (see equation (1)) when e 012 ≪ 1. \nThe break timescale distribution in Figure 4 (or Figure 5) also presents interesting features. For circular outer orbits ( e 012 = 0), especially for coplanar systems, a break occurs suddenly with a very short timescale (typically ≲ 10 P 12 ), even around the stability boundary. In contrast, highly eccentric systems ( e 012 > 0 . 4) and polar systems ( i mut = 90 · , in the strong ZKL oscillation regime) show gradually increasing break timescales towards the stability boundary at 10 8 P 12 . This behavior is similar to the result in Hayashi et al. (2023) for HT-P systems, and indicates the importance to define the stability boundary as a function of the timescale for HT-S configurations as well. \nSo far, we defined the stability boundary at t = 10 8 P 12 . As indicated in Figures 4 and 5, however, such boundaries defined at different epochs behave differently and exhibit different dependence on e 012 and i mut , in particular. While we do not attempt in the present paper, it is feasible to obtain the improved fit by generalizing the function h ( e 012 ) to h ( e 012 , i mut ; t ) from Figures 4 and 5. Indeed, this is also the case for the stability boundary for HT-P systems shown in Figures 8 - 10 of Hayashi et al. (2022).', '3.3. Mass ratio dependence': 'Figure 6 shows how the stability boundary depends on the mass ratios, m 1 /m 0 and m 2 /m 1 , in the cases of near-coplanar prograde systems ( i mut = 5 · ) with fixed initial phases. While our simulations adopt m 0 = 10 6 M ⊙ and P 12 = 1yr for definiteness, the result are scalable with respect to those values; see equation (14) in Hayashi et al. (2022). Upper-left, upper-right, lower-left and lower-right panels correspond to ( m 1 , m 2 ) = (10 M ⊙ , 10 M ⊙ ), (19 M ⊙ , 1 M ⊙ ), (100 M ⊙ , 100 M ⊙ ), and (1 M ⊙ , 1 M ⊙ ), respectively. The vertical axis of Figure 6 is normalized by the GPZM17 result as we did in Figure 5. \nFour panels in Figure 6 appear to be almost identical, indicating that the mass dependence of the stability boundary is described by the conventional Hill-radius scaling ( m 0 / ( m 1 + m 2 )) 1 / 3 alone, independent of m 2 /m 1 . \nFinally we also check the mass scalability, expected from the Newtonian gravity, with different m 0 values fixing mass ratios m 1 /m 0 and m 2 /m 1 , and confirm it statistically although chaotic behavior may change individual simulation result (see Hayashi et al. 2022).', '4.1. Classification of the fate of HT-S systems': 'Now we are in a position to examine the fate of HT-S systems after the initial binary of m 1 and m 2 breaks up. As illustrated in Figure 2, those systems should form a massive inner binary orbited by a less massive tertiary m out , which is similar to the HT-P configuration. While the stability of such HT-P systems has been discussed for instance by Mardling & Aarseth (1999, 2001); Hayashi et al. (2022, 2023), we are interested in black-hole triple systems in particular. Even though we do not take into account general relativity in the present analysis, we consider approximately the merger/collision of the m i -m j pair based on their Schwarzschild radii r s ( m ) = 2 Gm/c 2 ; see (B2) and (B3) in Figure 2. \nTo be specific, we adopt the the merger/collision criterion (see e.g. Tanikawa & Umemura 2014): \nr ij < 10[ r s ( m i ) + r s ( m j )] , (14) \nwhere r ij is the distance between m i and m j . This condition is not accurate and should be interpreted as qualitative. Nevertheless, it provides an insight in what circumstances general relativity plays an important role, which will be useful for future work including general relativistic effects in a postNewtonian fashion that we plan to study separately. We again note that all the result here is scalable under pure Newtonian gravity, except for this merger/collision criterion. \nAs in Figure 2, we consider the four fates of the systems: \n(B1) ejection of m out : This is the most common fate in conventional HT-P systems. We adopt a specific definition of the ejection following Hayashi et al. (2022), but using the semi-major axes rather than orbital energies this time (see § 2).', '𝑖 )*+ = 5 ∘ , 𝑚 -= 10 . 𝑀 ⊙ , 𝑃 01 = 1yr': 'Figure 6. Break-up time distribution on (˜ r 012 / ˜ r 12 ) norm -e 012 plane. The vertical axis is normalized by the GPZM17 stability boundary as in Figure 5. Four panels correspond to different sets of m 1 and m 2 , although fixing m 0 as 10 6 M ⊙ . \n<!-- image --> \n- (B2) collision of m 1 and m 2 : This fate appears very unlikely after the initial m 1 -m 2 binary breaks up. As will be shown below, such collisions occur fairly often in the coplanar cases ( i mut = 0 · and 180 · ). This branch is strongly suppressed when small mutual inclinations between the inner and outer orbits are introduced. This implies that the initial condition of the binary is imprinted somewhere; the system comes back again to a configuration close to the tightly bound m 1 -m 2 binary even after chaotic dynamical evolution, especially when the dynamics is confined in a lower-dimension parameter space.\n- (B3) m in -m 0 merger: This fate is expected to be fairly common because the HT-S systems first experience the Hill instability and form m in -m 0 binary, at least temporarily.\n- (B4) survived/undisrupted: We define survived systems that are undisrupted until the end of our integration time limit t int = 10 8 P 12 . Needless to say, this definition is quite dependent on t int that is determined due to the computational cost in practice. Nevertheless, we emphasize that our adopted value is significantly longer than the previous ones, which can identify the fraction of systems in (B1), (B2) and (B3) that may have been overlooked before.', '4.2. Branching ratio of fates of HT-S systems': "As will be shown below, the fate of HT-S systems after the initial binary break-up is determined over a much longer timescale than T break . Therefore, it is computationally demanding to survey the multi-dimensional parameter space faithfully. Instead, we perform a set of N-body simulations with the following specific initial conditions. \nAlthough we have systematically surveyed the parameter Υ in searching for the stability boundary (6), we fix Υ = 0 . 1Υ crit to check the evolution after the initial binary breaks. The outer eccentricity is varied as e 012 = 0 . 1 , 0 . 2 , 0 . 3 , · · · , 0 . 9, and we consider four different mutual inclinations; i mut = 0 · , 5 · , 90 · and 180 · . For each model, we generate 100 realizations with random initial phases ( ω 12 , ω 012 , M 12 , M 012 ). For simplicity, we fix e 12 = 0 . 5 assuming that e 12 does not significantly change the fate when the initial phases are statistically randomized. Again, we specifically adopt m 0 = 10 6 M ⊙ and m 1 = m 2 = 10 M ⊙ . \nSuch systems are very Hill-unstable, and the initial m 1 -m 2 binary breaks up typically within 10 P 12 . We continue to follow their orbital evolution up to t int = 10 8 P 12 , and obtain the statistics of the timescales when those systems reach the three different fates (B1), (B2) and (B3). The remaining undisrupted systems at t int are classified as (B4). \nFigures 7 to 10 plot the resulting timescales of the initial binary break (red), and the different fates (black) for i mut = 0 · , 5 · , 90 · and 180 · , respectively. In each figure, upper-left, upper-right, lower-left, and lower-right panels correspond to the fates (B1), (B2), (B3), and (B4), respectively. \nThose figures imply that the binary break timescales are much less than 10 P 12 , although increasing with e 012 . Instead of log 10 T break /P 12 , if we plot log 10 T break /P 012 , it is almost independent of e 012 , implying T break is basically determined by P 012 . On the other hand, the subsequent disruption events ( m out -ejection, m 1 -m 2 collision, m in -m 0 merger) occur with significantly longer timescales and are sensitive to e 012 . \nIncidentally, We found one exceptional case in which the original binary collides without experiencing the break; a black asterisk labeled 'unbreak' in Figure 7. Although this occurrence is quite rare under pure Newtonian gravity, the fraction may significantly increase including gravitational wave emissions, depending on the initial parameters. Therefore, this kind of fate may be important astrophysically in practice. \nThe strong dependence of the fates on the outer eccentricity e 012 is clearly shown in Figure 11, which plots the fraction of each fate against e 012 ; i mut = 0 · (upper-left), 5 · (upper-right), 90 · (lower-left), and 180 · (lower-right). \nExcept i mut = 0 · , the dependence of the fate fraction on e 012 is very similar for the other mutual inclinations. If the initial outer orbits are significantly eccentric ( e 012 > 0 . 7), m in -m 0 merger (B3) dominates. For near-circular outer orbits ( e 012 < 0 . 4), those systems are mostly undisrupted (B4) with small fraction experiencing the m 1 -m 2 collision (B2). For the intermediate value of the outer eccentricity (0 . 4 < e 012 < 0 . 7), the ejection of m out (B1) becomes sub-dominant. The trends of the corresponding timescales for those fates are consistent with the above picture. \nFigure 7. The distributions for timescales of binary break (red) and each outcome (black) for i mut = 0 · . Upper-left, upper-right, lower-left, lower right panels correspond to the results for (B1) m out -ejection, (B2) m 1 -m 2 collision, (B3) m in -m 0 merger, and (B4) survival at 10 8 P 12 , respectively. \n<!-- image --> \n∘ \nFigure 8. Same as Figure 7, but for i mut = 5 · . \n<!-- image --> \n𝑖 \nFigure 10. Same as Figure 7, but for i mut = 180 · . \n<!-- image --> \n= 90 \n∘ \nFigure 9. Same as Figure 7, but for i mut = 90 · . \n𝑖 \n= 180 \n<!-- image --> \n∘ \nFigure 11. Fraction of each outcome for i mut = 0 · (upper-left), 5 · (upper-right), 90 · (lower-left), and 180 · (lower-right), respectively. \n<!-- image --> \nThe case with exactly coplanar prograde orbits is exceptional. Our interpretation is that the initial condition of the tightly bound m 1 -m 2 binary is somehow imprinted in the phase space of the three bodies. After long-term dynamical evolution, the system sometimes reaches the configuration close to the initial condition, significantly increasing the probability of the m 1 -m 2 collision that is unlikely in more generic initial conditions. In particular, for coplanar systems, the trajectories of three bodies are bounded on a two-dimensional orbital plane, and their physical degree-of-freedom is expected to be more limited by that of inclined triples. Indeed, Ford et al. (2001) consider nearly coplanar star-two giant planet systems, where two planets initially have near orbits, and show that a fairly significant fraction of systems experience two planet collisions. We suggest that similar orbital configurations can be realized from coplanar HT-S systems after breaking an initial binary. \nNevertheless, the difference between coplanar prograde and retrograde systems is hardly understood from this interpretation. One possible physical interpretation may be the difference of energy and/or angular momentum exchanges affect the evolution (see Figure 14 and 15 in Hayashi et al. (2022), for example), and changes the initial condition after breaks. This may point to an interesting mathematical problem, but is beyond the scope of the present paper.", '5. SUMMARY AND CONCLUSION': "We have studied the long-term dynamical evolution of hierarchical triple systems comprising a central massive body and a tight binary in eccentric orbits, which we referred to as HT-S systems. The two major purposes are to find the Hill-type stability condition for the binary due to the tidal effect of the central body, and to classify the fate of the systems after the binary break-up as a function \nFigure 12. Stability boundaries for HT-P (red curves, e in = 0, m 12 /m 3 , m 2 /m 1 = 20 / 2, 10 / 10, respectively.), and HT-S (blue curves, e 12 = 0, m 12 /m 0 , m 2 /m 1 = 2000 / 10 6 , 1000 / 1000, respectively.) in e out -a out (1 -e out ) /a in plane. We use equation (16) in Hayashi et al. (2022) for HT-P stability boundaries, and equation (6) in this paper for HT-S stability boundaries, respectively. \n<!-- image --> \neout \nof the initial outer eccentricity e 012 . Using a series of direct N-body simulations, we obtained the stability criterion for the binary, inequality (6), which generalizes the formula (4) derived by Grishin et al. (2017). \nWe compare the binary stability boundary (6) for HT-S systems and the disruption criterion for HT-P systems, equation (16) in Hayashi et al. (2022) in Figure 12. Those systems above the boundary lines are stable up to t int = 10 8 P 12 (HT-S; this work) and t int = 10 9 P in (HT-P; Hayashi et al. 2022). Note that the vertical axis of the figure is a out (1 -e out ) /a in so as to plot both HT-S and HT-P systems simultaneously. Thus, the scaling with respect to e 12 , equation (8), is not incorporated in Figure 12. \nOur major findings are summarized as follows. \n- (A) Outer eccentricity: The most important parameter that determines the stability of hierarchical triple systems is the ratio of the semi-major axes of the outer and inner orbits, a 012 /a 12 . In general, eccentricities of both orbits, e 012 and e 12 , tend to destabilize the systems. The destabilizing effect of e 012 is mostly incorporated by replacing a 012 with a 012 (1 -e 012 ), but the stability boundaries after the scaling still weakly increase as a function of e 012 . We find that the residual effect for HT-S systems is well approximated by a linear function of e 012 , equation (12), for e 012 < 0 . 8, almost independent of the mutual inclination i mut . This is consistent with \nthe behavior of HT-P systems. The latter systems exhibit further stronger e out -dependence for e out > 0 . 9, but we did not check the regime for HT-S systems. \n- (B) Inner eccentricity: The effect of e 12 on the stability is weaker but more subtle than that of e 012 . We find empirical scaling expressions for the e 12 -dependence that is sensitive to i mut .\n- (C) Mutual inclination of the inner and outer orbits: Our simulations with i mut = 0 · , 5 · , 90 · and 180 · confirm the i mut -scaling of the binary stability criterion for HT-S systems that was discovered before by Grishin et al. (2017). After applying this scaling, we were able to separate and find the above scaling relations for e 12 and e 012 .\n- (D) Mass ratio: The mass dependence of the binary stability criterion is well described by the simply factor of ( m 0 / ( m 1 + m 2 )) 1 / 3 , at least for near-coplanar prograde systems. It may be a bit surprising that the criterion is determined by the total mass of the initial binary m 1 + m 2 , and insensitive to their ratio m 1 /m 2 .\n- (E) Fate of HT-S and HT-P systems: Stability criteria plotted in Figure 12 exhibit some similarity between HT-S and HT-P systems, but they have very different implications for their fates. For HT-P systems, the criterion is time-dependent and corresponds to their disruption event practically (almost always, the ejection of the tertiary body from the system); see detailed discussion in Hayashi et al. (2022). In contrast, the criterion for HT-S systems that we derived in the present paper concerns the Hill-type binary break-up, and still the triple system itself is undisrupted. In a sense, the criterion should be interpreted to indicate the transition from HT-S to HT-P systems; see Figure 2. Such timescales for the binary break-up are typically very short ( ≲ 10 P 12 ), and the final fates of those systems are determined through their orbital evolution over much longer timescales (more than several order-of-magnitudes longer than the binary breaking timescale, similar to those for the stability criterion for the HT-P systems plotted in Figure 12). Those investigations were made possible by performing long-time numerical integration of the direct N-body systems for the first time. We find that the HT-S systems reach different fates (ejection, collision, and merger) and that their corresponding timescales are decreasing functions of e 012 in general. Also the fraction of each fate is sensitive to the initial value of e 012 ; see discussion in § 4.2. In long-timescale calculations, stellar evolution should play an important role for stellar triples in practice. For instance, Toonen et al. (2022) study the evolution and fates of stellar triples, including stellar evolution. \nWhile we do not consider general relativity in the present work, the Hill stability including GR effects becomes important for some parameter ranges. For instance, Suzuki et al. (2020) derived a criterion using analytic treatments concerning Sundman's inequality and first-order post-Newtonian (1PN) approximate numerical simulations for coplanar near-circular systems. They suggested that the PN corrections tend to stabilize a system compared with a purely Newtonian case. It is also wellknown that the first-order GR correction (apsidal precession term) suppresses the ZKL oscillations if two timescales become comparable. Besides, gravitational wave emissions reduce the energy for a long timescale, and therefore shrink the semi-major axis of a binary. Close scatterings before collisions may also be significantly affected by the GR corrections. Therefore, the GR corrections affect the Hill stability of triple systems with relevant parameter values. \nFinally, we would like to emphasize that the result in this paper is basically scalable satisfying so called Kepler's third law, under the assumption that a system is completely dominated by the Newtonian gravity. Therefore, the result is applicable, not only for black hole triples, but also for planetary (or satellite) systems.", 'ACKNOWLEDGMENTS': 'T.H. gratefully acknowledges Atsushi Taruya for fruitful discussions about dynamics in three-body systems. The numerical simulations were carried out on the local computer cluster awamori , and the general calculation server from Center for Computational Astrophysics (CfCA), National Astronomical Observatory of Japan (NAOJ). T.H. gratefully acknowledges the fellowship by Japan Society for the Promotion of Science (JSPS). This work is supported partly by the JSPS KAKENHI grant No. JP23K25908 (Y.S.), JP21J11378 and JP23KJ1153 (T.H.), and JP21K13914 (A.A.T.).', 'A. EFFECT OF DIFFERENT INITIAL PHASES ON THE STABILITY BOUNDARY': 'So far, all the simulation runs have been performed for a given set of initial phases of three bodies. We vary the initial phases so as to test if our stability boundary is robust against those changes. For that purpose, we focus on coplanar prograde systems ( i mut = 0 · ) in circular ( e 12 = e 012 = 0) and highly eccentric ( e 12 = e 012 = 0 . 8) orbits, and vary the initial phases as follows. \nFor circular systems, the pericenter arguments ( ω 12 , ω 012 ) are irrelevant, and we arbitrarily fix them as 0 · that simply define the zero point of the locations of each body for coplanar systems. In addition, due to symmetry, we set M 12 = 0 · without loss of generality. Thus, the initial phases are parameterized by the mean anomaly difference ∆ M ≡ M 012 -M 12 = M 012 alone, and we vary it as 2 · i ( i = 1 , 2 , · · · , 180). \nFor eccentric systems, however, the initial phases are specified by three independent parameters; the relative location of pericenters (∆ ω ≡ ω 012 -ω 12 = ω 012 ), and mean anomalies ( M 12 and M 012 ). Therefore, we vary them as ∆ ω = ω 012 = 30 · j , M 12 = 60 · k , and M 012 = 60 · l , where the integers j , k , and l run from 1 to 6; see Figure 3. \nFigure 13 plots how T break /P 12 for circular systems is affected by the mean anomaly difference ∆ M . There is a clear systematic trend in the stability boundary, up to ∼ 20%, as a function of ∆ M . This result may seem somewhat counter-intuitive, because the initial phases are expected to be well mixed up after the orbital evolution. However, the binary break in the circular systems happens so quickly, T break ≲ 10 P 12 , before the memory of the initial location is lost. \nThe variation of the stability boundary for circular systems, however, turned out to be less than 20%, and thus not important in practice. While the origin of the above behavior of circular systems may be possibly related to the chaos theory (e.g. Mardling 1995a,b), it is beyond the scope of this paper and we do not consider further. \nFigure 14 shows the results for eccentric systems. Each panel corresponds to different values of M 12 , and the horizontal axis denotes M 012 here. Similarly to the visualization treatment for e 12 in the previous sections, we plot the results for different ∆ ω by shifting data sequences horizontally. Contrary to the circular systems, Figure 14 shows no systematic dependence on initial phases for eccentric systems. This is because eccentric systems around the stability boundary are relatively \n∘ \n( \nFigure 13. Break-up time distribution in ∆ M - ˜ r 012 / ˜ r 12 plane. \n<!-- image --> \nmore stable and the longer timescale for the orbital evolution before the instability loses the memory of their initial phases. So, we conclude that the initial phase dependence is negligible for stability boundaries of the eccentric systems. \n<!-- image --> \n∘ \n∘ \n∘ \n∘ \n∘ \n∘ \nFigure 14. Break-up time distribution in M 012 - ˜ r 012 / ˜ r 12 plane. \n<!-- image -->'}

2024MNRAS.535L..37M

New JWST observations are revealing the first galaxies to be prolific producers of ionizing photons which we argue gives rise to a tension between different probes of reionization. Over the last two decades a consensus has emerged where starforming galaxies are able to generate enough photons to drive reionization given reasonable values for their number densities ionizing efficiencies inlineformulatexmath idTM0001 notationLaTeXxi rm iontexmathinlineformula per unit ultraviolet luminosity and escape fractions inlineformulatexmath idTM0002 notationLaTeXfrm esctexmathinlineformula. However some new JWST observations infer high values of inlineformulatexmath idTM0003 notationLaTeXxi rm iontexmathinlineformula during reionization and an enhanced abundance of earlier inlineformulatexmath idTM0004 notationLaTeXzgtrsim 9texmathinlineformula galaxies dramatically increasing the number of ionizing photons produced at high z. Simultaneously recent lowz studies predict significant escape fractions for faint reionizationera galaxies. Put together we show that the galaxies we have directly observed inlineformulatexmath idTM0007 notationLaTeXMrm UV lt 15texmathinlineformula not only can drive reionization but would end it too early. That is our current galaxy observations taken at face value imply an excess of ionizing photons and thus a process of reionization in tension with the cosmic microwave background and Lymaninlineformulatexmath idTM0008 notationLaTeXalphatexmathinlineformula forest. Considering galaxies down to inlineformulatexmath idTM0009 notationLaTeXMrm UVapprox 11texmathinlineformula below current observational limits only worsens this tension. We discuss possible avenues to resolve this photon budget crisis including systematics in either theory or observations.

2024-11-01T00:00:00Z

['10.48550/arXiv.2404.07250', '2024MNRAS.tmpL..79M', '2024arXiv240407250M', 'arXiv:2404.07250', '2024MNRAS.535L..37M', '10.1093/mnrasl/slae086']

['Astrophysics - Cosmology and Nongalactic Astrophysics', 'Astrophysics - Astrophysics of Galaxies', 'High Energy Physics - Phenomenology']

Reionization after JWST a photon budget crisis

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']

https://arxiv.org/pdf/2404.07250.pdf

{'No Header': ', 1-9 (-)', 'Reionization after JWST: a photon budget crisis?': 'Julian B. Muñoz 1 ★ , Jordan Mirocha 2 , 3 , John Chisholm 1 , Steven R. Furlanetto 4 , and Charlotte Mason 5 \n- 1 Department of Astronomy, The University of Texas at Austin, 2515 Speedway, Stop C1400, Austin, TX 78712, USA\n- 2 Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA\n- 3 California Institute of Technology, 1200 E. California Boulevard, Pasadena, CA 91125, USA\n- 4 Department of Physics and Astronomy, University of California, Los Angeles, CA 90095, USA\n- 5 Niels Bohr Institute, University of Copenhagen, Jagtvej 128, 2200 København N, Denmark \n6 September 2024', 'ABSTRACT': 'New James Webb Space Telescope (JWST) observations are revealing the first galaxies to be prolific producers of ionizing photons, which we argue gives rise to a tension between different probes of reionization. Over the last two decades a consensus has emerged where star-forming galaxies are able to generate enough photons to drive reionization, given reasonable values for their number densities, ionizing efficiencies 𝜉 ion (per unit UV luminosity), and escape fractions 𝑓 esc. However, some new JWST observations infer high values of 𝜉 ion during reionization and an enhanced abundance of earlier ( 𝑧 ≳ 9) galaxies, dramatically increasing the number of ionizing photons produced at high 𝑧 . Simultaneously, recent low𝑧 studies predict significant escape fractions for faint reionization-era galaxies. Put together, we show that the galaxies we have directly observed ( 𝑀 UV < -15) not only can drive reionization, but would end it too early. That is, our current galaxy observations, taken at face value, imply an excess of ionizing photons and thus a process of reionization in tension with the cosmic microwave background (CMB) and Lyman𝛼 forest. Considering galaxies down to 𝑀 UV ≈ -11, below current observational limits, only worsens this tension. We discuss possible avenues to resolve this photon budget crisis, including systematics in either theory or observations. \nKey words: cosmology: theory - reionization - bubbles', '1 INTRODUCTION': "The epoch of reionization represents the last major phase transition of our universe. During reionization the intergalactic gas went from cold and neutral before the first cosmic structures formed (at redshift 𝑧 ∼ 30, or 100 Myrs after the Big Bang) to hot and ionized by 𝑧 ∼ 5 (roughly a billion years later). While we are certain that this process took place, we do not know how. The likely culprits for reionization are the first star-forming galaxies (Robertson et al. 2015, hereafter R15), but other suspects include supermassive black holes (Madau &Haardt 2015; Madau et al. 2024), and even dark matter (Liu et al. 2016). More broadly, the timing and topology of reionization hold a treasure trove of information on the astrophysics of the early universe, which we have yet to uncover. \nThe accounting of reionization is rather simple: there have to be enough photons to ionize all the intergalactic hydrogen atoms, including their recombinations. During the WMAP era this was a stringent requirement, as cosmic microwave background (CMB) data implied an approximate midpoint of reionization at 𝑧 = 10 -11 (Komatsu et al. 2011), earlier than expected from standard galaxy-formation models and beyond the reach of contemporaneous direct observations. With the advent of the Planck satellite this tension was eased, as newer CMB data preferred later reionization (with an effective 𝑧 ∼ 7 -8, Ade et al. 2016), and by then Hubble Space Telescope \n(HST) observations had characterized a population of star-forming galaxies at those redshifts (Madau & Dickinson 2014). Together, these observations alleviated the demand for ionizing photons and quickly led to the consensus that, under reasonable assumptions, star-forming galaxies were able to drive reionization (R15, Bouwens et al. 2015, Finkelstein et al. 2019, hereafter F19). In this Letter we examine whether this consensus holds in light of recent James Webb Space Telescope (JWST) observations of the high-redshift universe. \nThree key factors determine the average reionization history: the production rate of ionizing photons (by early galaxies and black holes), the fraction 𝑓 esc of those photons that escape to the intergalactic medium (IGM) and can ionize neutral hydrogen, and the number of recombinations per hydrogen atom. While there remain open questions about the last factor (Davies et al. 2021), the first two are particularly uncertain. \nThe production rate of ionizing photons is given by the earlygalaxy abundance, usually expressed through the UV luminosity function (UVLF, the comoving number density of galaxies per UV magnitude), times the ionizing efficiency 𝜉 ion of each galaxy. Though there is broad agreement on the bright end of the UVLF, the number density of ultra-faint (below 𝑀 UV ≈ -14) galaxies is virtually unconstrained. Theoretically, we expect the UVLF to 'turn over' at some magnitude 𝑀 turn UV due to feedback (Shapiro et al. 2004), and HST observations have constrained this turnover to be fainter than 𝑀 turn UV ≈ -15 (Atek et al. 2018). At the same time, some new JWST observations are finding early galaxies to have higher ionizing efficiencies 𝜉 ion than canonically assumed [with log 10 𝜉 ion / (Hz erg -1 ) \n≈ 25 . 5 -26 . 0 vs 25.2 Atek et al. 2024; Simmonds et al. 2024; Endsley et al. 2023; Prieto-Lyon et al. 2023; Curtis-Lake et al. 2023; Hsiao et al. 2023; Calabro et al. 2024, though see Matthee et al. 2023; Meyer et al. 2024; Pahl et al. 2024]. Moreover, JWST is also unveiling an enhanced population of both star-forming galaxies at 𝑧 ≳ 9 (Finkelstein et al. 2022, 2023; Eisenstein et al. 2023; Harikane et al. 2023; Castellano et al. 2022, with an unknown origin Mason et al. 2023; Ferrara et al. 2022; Muñoz et al. 2023; Mirocha & Furlanetto 2023) and supermassive black holes (Matthee et al. 2024, though they are likely obscured Greene et al. 2024), which would further boost the ionizing-photon budget. Such a wealth of photons will accelerate the process of reionization, if they escape their host galaxies. \nTheescapefraction 𝑓 esc of early galaxies is a contentious topic. The basic problem is that the ionizing-photon production is dominated by very massive, short-lived stars, which may live and die before their birth clouds are dispersed, minimizing photon escape. The BPASS models (Eldridge & Stanway 2009) provided a new hope for high escape fractions, as binary interactions help to lengthen effective stellar lifetime and so boost the effective 𝑓 esc. Models in which the escape fraction is set by local, cloud-scale physics, suggest that 𝑓 esc could be independent of galaxy properties like mass or luminosity (Ma et al. 2016). However, different simulations predict 𝑓 esc growing for brighter galaxies (Sharma et al. 2016), declining (Wise et al. 2014; Kimm & Cen 2014), or peaking at intermediate masses (e.g., Yoo et al. 2020; Ma et al. 2020; Rosdahl et al. 2022; Yeh et al. 2023). From a theoretical perspective, there seems to be no clear consensus on the nature of 𝑓 esc in high𝑧 galaxies. Observationally, it is extremely challenging to measure 𝑓 esc while there is neutral hydrogen in the IGM. However, recent studies of low𝑧 analogues of reionizationera galaxies have found a strong correlation between their escape fractions and UV slopes 𝛽 UV : bluer galaxies exhibit larger values of 𝑓 esc (Flury et al. 2022; Chisholm et al. 2022; Begley et al. 2022; Saldana-Lopez et al. 2023). JWST and HST data show that early galaxies have bluer slopes than their average low𝑧 counterparts (e.g., Topping et al. 2022; Cullen et al. 2023; Weibel et al. 2024), such that the few studies of reionization-era galaxies indicate modest 𝑓 esc values near 5-15% (Mascia et al. 2023; Lin et al. 2024). \nHere we argue that combining the abundance of directly observed reionization-era galaxies, the new JWST estimates of 𝜉 ion , and the low𝑧 insights on 𝑓 esc leads to too many ionizing photons at high redshifts, ending reionization too early. Such an early reionization is in contradiction with current CMB (Aghanim et al. 2020) and Lyman𝛼 forest observations (Bosman et al. 2022), and poses a tension in the photon budget during reionization . We will outline possible ways to ease this tension, including physical ingredients missing in our theoretical models, interpretation of observations, or both. \nThrough this paper we assume a flat Λ CDM cosmology with ℎ = 0 . 7 and Ω 𝑀 = 0 . 3 to match that assumed in Bouwens et al. (2021) and Donnan et al. (2024), all magnitudes are AB (Oke & Gunn 1983), and quantities are spatially averaged unless otherwise indicated.", '2 MODELING REIONIZATION': "Wewillfollow a simple model of reionization to solve for the volumeaveraged hydrogen neutral fraction 𝑥 HI ≡ 𝑛 HI / 𝑛 H , and its complement the ionized fraction 𝑥 HII ≡ 1 -𝑥 HI . This quantity evolves as (Madau et al. 1999) \n/ 𝑥 HII = / 𝑛 ion 𝑛 H -𝑥 HII 𝑡 rec , (1) \nFigure1. ThenewJWSTandlow𝑧 observations imply an earlier reionization, in tension with the CMB. Bottom: Evolution of the neutral fraction 𝑥 HI as a function of redshift 𝑧 for a pre-JWST model (black solid, with a cutoff at 𝑀 UV = -13 and 𝑓 esc = 0 . 2, following R15), for the same model but with a JWST-calibrated 𝜉 ion (purple dashed, following Simmonds et al. 2024), and a model where in addition 𝑓 esc is determined from low𝑧 analogues (blue dotdashed, using the fit in Chisholm et al. 2022). Green points show a collection of observational constraints from (McGreer et al. 2015; Greig et al. 2017, 2018; Sobacchi & Mesinger 2015; Mason et al. 2019; Whitler et al. 2019; Wang et al. 2020; Nakane et al. 2023) (see also Bruton et al. 2023) Top: CMB optical depth 𝜏 CMB, where the red band is the measurement from Aghanim et al. (2020). The new galaxy observations give rise to far more ionizing photons, and at face value are in severe tension with CMB data. \n<!-- image --> \nwhich showcases the competition between the 'sources' (first term) and 'sinks' (second) of ionizing photons. The former is given by the density of ionizing photons produced ( / 𝑛 ion ) divided by that of hydrogen 𝑛 H = 𝜌 b ( 1 -𝑌 He )/ 𝑚 H , where 𝑌 He is the Helium mass fraction, 𝑚 H the proton mass, and 𝜌 b the baryon energy density, which scales as ( 1 + 𝑧 ) 3 . The sink term captures the number of recombinations that hydrogen atoms suffer on average, characterized by a timescale (Shull et al. 2012) \n𝑡 rec = [ 𝐶 𝛼 B ( 1 + 𝑥 He ) 𝑛 H ] -1 (2) \nwhere 𝑥 He ≡ 𝑛 He / 𝑛 H ≈ 𝑌 He /[ 4 ( 1 -𝑌 He )] , 𝛼 B is the case-B recombination coefficient, and 𝐶 is the clumping factor. The clumping factor is difficult to estimate theoretically, as it depends on how ionized regions penetrate into high-density clumps. Simulations predict 𝐶 ≈ 2 -5 during reionization, growing towards lower 𝑧 (e.g., Pawlik et al. 2015). Recent work in Davies et al. (2021) instead suggests that more recombinations are needed to explain the short mean free path of ionizing photons at 𝑧 ∼ 6 (thanks to absorption by pervasive high-density clumps known as Lyman-limit systems, Becker et al. 2021; Zhu et al. 2023). For simplicity and comparison with past literature (R15), we will set 𝐶 = 3 and evaluate 𝛼 B at 𝑇 = 2 × 10 4 Kfor now (which yields nearly identical results to using the 𝐶 ( 𝑧 ) fit from Shull et al. 2012), and return to the effect of recombinations later. \nFor reionization to progress the sources have to win over the sinks. Our sources will be star-forming galaxies, which produce a background of ionizing photons at a rate of \n/ 𝑛 ion = ∫ 𝑑𝑀 UV Φ UV / 𝑁 ion 𝑓 esc , (3) \nwhere all factors inside the integral are assumed to depend on 𝑀 UV , and we integrate down to a cutoff magnitude 𝑀 ion . cutoff UV that will be a free parameter. Here, Φ UV is the UVLF, taken at 𝑧 ≤ 9 from the pre-JWST fit in Bouwens et al. (2021) and at 𝑧 > 9 from the JWST \ncalibrations of Donnan et al. (2024, see Appendix A for alternative analyses using only pre-JWST UVLFs, including that of Finkelstein & Bagley 2022), / 𝑁 ion ≡ 𝐿 UV 𝜉 ion is the production rate of ionizing photons per galaxy, given by their UV luminosity 𝐿 UV times the ionizing efficiency 𝜉 ion , of which a fraction 𝑓 esc escapes into the IGM. \nIt is apparent that the product 𝜉 ion × 𝑓 esc will determine the timing of reionization, and that these two factors are, at face value, fully degenerate. Increasing 𝜉 ion per galaxy while decreasing 𝑓 esc will yield identical effects on the IGM. Fortunately, though, direct Balmer-line observations can be used to tease out the amount of ionizations in the galaxy, and thus the amount of non-escaping ionizing photons 𝜉 ion ( 1 -𝑓 esc ) . Using the Robertson et al. 2013 inference 1 of log 10 𝜉 ion = 25 . 2 Hz erg -1 (though see Bouwens et al. 2016; Lam et al. 2019; De Barros et al. 2019 for higher reported values), R15 showed that 𝑓 esc = 20% is sufficient if galaxies down to 0.001 𝐿 ★ ( 𝑀 ion . cutoff UV ≈ -13) contribute to reionization. We illustrate what reionization would look like for this pre-JWST calibrated model in Fig. 1. It is over by 𝑧 ∼ 6, and produces an optical depth 𝜏 CMB ≈ 0 . 055, bringing galaxy observations into agreement with Planck CMB measurements. \nThe arrival of JWST is opening a new window to reionization. Observations from different teams are finding large values of 𝜉 ion , in some cases growing towards higher redshifts and fainter galaxies (though see Endsley et al. 2023 where 𝜉 ion is still high but grows towards the bright end instead, we study this case in Appendix A). In particular, Simmonds et al. (2024) find a consistent increase in 𝜉 ion up to 𝑧 = 9 and 𝑀 UV = -16 . 5 (where we will conservatively cap 𝜉 ion to avoid extrapolation), well fit by \nlog 10 h 𝜉 ion /( Hzerg -1 ) i ≈ 25 . 8 + 0 . 11 ( 𝑀 UV + 17 ) + 0 . 05 ( 𝑧 -7 ) . (4) \nSuch faint, early galaxies will produce ∼ 4 times more ionizing photons than expected pre-JWST (Atek et al. 2024, implying a very young stellar population). The purple line in Fig. 1 shows how reionization would progress assuming this JWST-calibrated 𝜉 ion , while keeping everything else the same. In this case the additional photons would kick-start reionization by 𝑧 ∼ 12 and finish it by 𝑧 ∼ 8, far overproducing the CMB optical depth ( 𝜏 CMB ≈ 0 . 08) when compared to observations. Here we have kept 𝑓 esc = 0 . 2 as in R15, so the astute reader may wonder if newer inferences of the escape fraction delay reionization. \nWe do not have a direct handle on 𝑓 esc during reionization, as escaping ionizing photons will be absorbed by the neutral IGM before reaching us. However, detailed studies of low𝑧 analogues find a strong correlation, with significant scatter, between the FUV continuum slopes 𝛽 UV of galaxies and their LyC escape fractions. This is physically explained by the dust along the line-of-sight simultaneously attenuating the FUV stellar continuum and the ionizing photons. We will use the fit from Chisholm et al. (2022, calibrated on the 𝑧 ∼ 0 LzLCS survey Flury et al. 2022, see Trebitsch et al. 2022 for an implementation on reionization), where \n𝑓 esc = 𝐴 𝑓 × 10 𝑏 𝑓 𝛽 UV (5) \nwith 𝐴 𝑓 = 1 . 3 × 10 -4 and 𝑏 𝑓 = -1 . 22. In this relation galaxies that are bluer have less dust and low-ionization gas along the line-ofsight, and thus fewer sinks of ionizing photons. This correlation is similarly observed at 𝑧 ∼ 3 in different surveys (Steidel et al. 2018; Pahl et al. 2021; Begley et al. 2022; Saldana-Lopez et al. 2023). We \ncan then employ the 𝑓 esc -𝛽 UV relation, with the 𝛽 UV -𝑀 UV measurements from Zhao & Furlanetto (2024, which incorporates both JWST and HST measurements from Bouwens et al. 2014; Topping et al. 2022; Cullen et al. 2023) to predict 2 𝑓 esc ( 𝑀 UV ) . Note that here, and throughout the text, we cap the UV slopes at 𝛽 UV = -2 . 7 when computing 𝑓 esc to avoid extrapolation in this relation (though we implicitly extrapolate in 𝑧 and 𝑀 UV , see Table A1), as that corresponds to the bluest galaxies where Eq. (5) is calibrated. We show the result of applying this calibration as the blue line in Fig. 1. The JWST-calibrated 𝜉 ion multiplied by 𝑓 esc (inferred using the high𝑧 𝛽 UV -𝑀 UV and low𝑧 𝑓 esc -𝛽 UV relations) produces an even earlier reionization, and consequently even more tension with 𝜏 CMB . \nThe curves shown in Fig. 1 are meant to illustrate the impact of the new 𝜉 ion and 𝑓 esc results for a particular reionization model. Let us now move to perform a more detailed study, where we vary different underlying assumptions and compare against current observations.", '3 OBSERVATIONAL CONSTRAINTS': 'To understand reionization we need to know the ionizing-photon budget, i.e., how many photons are produced and what fraction escape their galaxies. We model the former by taking the ionizing efficiency 𝜉 ion from Eq. (4), as measured in JWST observations, and integrating the UVLF down to a cutoff magnitude 𝑀 ion . cutoff UV (below where we will assume galaxies do not emit ionizing photons efficiently, either because 𝑓 esc, 𝜉 ion , or the UVLF itself goes to zero, see Appendix B for an example). For the latter we define the ionization-averaged escape fraction as \n⟨ 𝑓 esc ⟩ ion . ≡ / 𝑛 ion ( 𝑓 esc ) / 𝑛 ion ( 𝑓 esc = 1 ) , (6) \nwith / 𝑛 ion defined in Eq. (3). These two free parameters, 𝑀 ion . cutoff UV and ⟨ 𝑓 esc ⟩ ion . , encapsulate our uncertainty about the impact of high𝑧 galaxies on reionization. They must obey three different observational constraints. \nFirst, the cutoff 𝑀 ion . cutoff UV has to be fainter than -15 (Atek et al. 2018), given current HST and JWST observations, which we show as a green band in Fig. 2. Second, ⟨ 𝑓 esc ⟩ ion . should follow the constraints derived from low𝑧 analogues, shown as a blue band (using Eq. 5 with the best-fit amplitude and error from the LzLCS survey of 𝑧 ∼ 0 galaxies Chisholm et al. 2022, see Fig. A1 for the VANDELS 𝑧 ∼ 3 sample of Saldana-Lopez et al. 2023). Finally, the combination of 𝑀 ion . cutoff UV and ⟨ 𝑓 esc ⟩ ion . have to produce the correct reionization history, which we parametrize through the CMB optical depth 3 \n𝜏 CMB = ∫ 𝑑ℓ𝑛 𝑒 𝜎 𝑇 , (7) \nwhere ℓ is proper distance, 𝜎 𝑇 is the Thomson cross section, and 𝑛 𝑒 is the physical (not comoving) electron density, computed assuming that HeI reionization tracks HI, and that HeII reionization takes place at 𝑧 = 4. The regions of parameter space that predict the correct 𝜏 CMB within 1 𝜎 are shown as red bands in Fig. 2. In the CMB bands a brighter cutoff 𝑀 ion . cutoff UV requires higher values of ⟨ 𝑓 esc ⟩ ion . to compensate the missing star formation - and subsequent photon production - at the faint end. \nFigure 2. Tension in our models of reionization, expressed through the effective cutoff 𝑀 ion . cutoff UV on the UVLF (at which galaxies cease to emit ionizing photons), and the average escape fraction ⟨ 𝑓 esc ⟩ ion . above that cutoff. The three colored contours correspond to the regions allowed by the CMB optical depth 𝜏 CMB (red), the low𝑧 𝑓 esc studies (blue), and direct HST+JWST observations of no cutoff down to 𝑀 UV ≈ -15 (green). The left panel assumes a pre-JWST value of 𝜉 ion from Robertson et al. (2013), where the three colored regions nicely overlap for faint cutoffs and 𝑓 esc ≈ 0 . 2. The right panel instead takes the new JWST 𝜉 ion calibration from Simmonds et al. (2024), in which case the three regions do not overlap, showing a tension in reionization . In more detail, the blue region follows the results from the LzLCS survey of reionization-era analogues (Chisholm et al. 2022, evaluated at 𝑧 = 7, with solid lines corresponding to the 𝑀 UV directly observed, and dashed to extrapolation, in all cases capping UV slopes at 𝛽 UV = -2 . 7). We highlight three popular pre-JWST models from R15, F19, and M22 (the latter assumes a larger 𝜉 ion closer to the new JWST value) as colored stars. The red diamond and black circle on the right panel correspond to possible solutions to the tension (further explored in Fig. 3), which are in conflict with either the 𝑓 esc or 𝑀 ion . cutoff UV constraints. \n<!-- image --> \nFig. 2 showcases the tension between these three observations. The left panel shows the pre-JWST situation, where the lower 𝜉 ion allowed the three observational bounds (red, blue, and green regions) to overlap over a broad swath of parameter space fainter than 𝑀 ion . cutoff UV ≈ -15, with 𝑓 esc ≈ 15 -30%. The right panel, updated with the recent JWST observations, shows no overlap between the three. In this case the requirements from 𝜏 CMB (red) and the low𝑧 𝑓 esc studies (blue) only overlap for cutoffs brighter than 𝑀 ion . cutoff UV ≈ -15 (outside of the green region, and thus disfavored by direct HST+JWST observations). In other words, the new JWST observations imply an overproduction of photons during reionization, which would end this process earlier than allowed by the CMB. Note that the galaxies observed by JWST ( 𝑀 UV ≲ -15) already produce too many photons, including fainter objects would only worsen this tension. \nFig. 2 also shows the parameter space of three popular reionization models: R15 ( ⟨ 𝑓 esc ⟩ ion . = 0 . 2 and 𝑀 ion . cutoff UV ≈ -13), F19 (their best fit is at ⟨ 𝑓 esc ⟩ ion . ≈ 0 . 05 and 𝑀 ion . cutoff UV ≈ -11), and the Lyman𝛼 emitter (LAE) model of Matthee et al. (2022, hereafter M22, which we approximate as having ⟨ 𝑓 esc ⟩ ion . = 0 . 17 for galaxies down to 𝑀 ion . cutoff UV = -17). Each of these models was calibrated to give rise to the correct 𝜏 CMB , though with different 𝜉 ion assumptions. R15 assumed log 10 𝜉 ion ≈ 25 . 2 Hz erg -1 , F19 fit for a somewhat higher 𝑧 -dependent value, whereas M22 used a larger log 10 𝜉 ion ≈ 25 . 8 Hz erg -1 , calibrated to low𝑧 LAEs (Naidu et al. 2022). Each of these models is at odds with one of the three observational constraints, and thus outside one of the color bands in Fig. 2, either 𝜏 CMB (R15, outside red band), 𝑓 esc (F19, blue), or 𝑀 ion . cutoff UV (M22, green). As such, they illustrate three possible avenues to reduce the photon budget during reionization and reconcile galaxy and CMB observations.', '4 POSSIBLE OUTS': "Let us now discuss possible physical mechanisms that may resolve this apparent photon budget crisis. \n- · Perhaps some of the new 𝜉 ion calibrations are biased? It is possible that photometry alone cannot reliably recover 𝜉 ion , that dust produces a systematic shift in this quantity (Shivaei et al. 2018), or that the JWST samples used to infer 𝜉 ion are not representative of the high𝑧 galaxy population (if they are biased towards efficient ionizers or preferentially target galaxies in a burst). For example, the sample in Simmonds et al. (2024) is selected based on an emission line flux cut in photometry, which could bias the sample towards strong line emitters (and thus high 𝜉 ion ) at fixed UV magnitude. We have repeated our analysis with a lower fixed 𝜉 ion = 10 25 . 5 Hz erg -1 (in line with the lowest mean values reported in Endsley et al. 2023, which did not make an emission-line selection, as well as the 𝑧 > 4 mean in Pahl et al. 2024, though see e.g., Matthee et al. 2023 for lower values), finding that this still requires a cutoff at 𝑀 UV ≈ -14 or brighter (see Fig. A1 in Appendix A). An alternative solution involves keeping a high 𝜉 ion onaverage but cutting off photon production for faint galaxies (either smoothly or setting 𝜉 ion → 0 below a cutoff magnitude 𝑀 ion . cutoff UV ). Fig. 3 shows that a 𝜉 ion cutoff at 𝑀 ion . cutoff UV = -17 would be able to solve the tension. Such a cutoff would, however, be in conflict with the detections of ionizing photons down to 𝑀 UV ≈ -15 from Atek et al. (2024) and Prieto-Lyon et al. (2023, and down to 𝑀 UV ≈ -16 . 5 for the more statistically robust samples of Simmonds et al. 2024; Endsley et al. 2023). Further JWST observations of high𝑧 galaxies will be able to determine the 𝜉 ion distribution down to faint magnitudes and pinpoint the impact of burstiness on this quantity.\n- · Maybe 𝑓 esc is far lower than expected? From Fig. 2 it is apparent that little to no extrapolation of the LzLCS relation to bluer galaxies is required to overproduce reionization (see also Appendix B). One possibility is that the low𝑧 analogues in both LzLCS and VANDELS are biased (e.g., they may be more likely to be leakers), or that different mechanisms set 𝑓 esc at high and low redshifts (so that 𝑓 esc may not correlate well with 𝛽 UV at high 𝑧 ). Many of the LzLCS properties match those observed at high-redshift (Tang et al. 2023), but it is possible (perhaps likely) that high-redshift galaxies have larger neutral \ngas fractions and lower dust-to-gas ratios than the low-redshift benchmarks (Heintz et al. 2023). This could lead to significantly lower 𝑓 esc at fixed 𝛽 UV , or a turnaround towards fainter/bluer galaxies. While plausible, this 𝑓 esc -𝛽 UV redshift evolution is not observed in 𝑧 ∼ 3 galaxies, which in fact appear to have larger 𝑓 esc at fixed 𝛽 UV (Pahl et al. 2021; Saldana-Lopez et al. 2023). Another possibility is that there is a covariance between 𝑓 esc and 𝜉 ion , such that galaxies that produce large amounts of ionizing photons have lower 𝑓 esc. This has been predicted by simulations (Rosdahl et al. 2022), though Tang et al. (2019); Naidu et al. (2022) observe the opposite trend in line emitters. \nIf one wanted to integrate the UVLF down to the theoretically expected cutoff at 𝑀 UV ≈ -11 (Kuhlen & Faucher-Giguere 2012), the 𝑓 esc needed to fit 𝜏 CMB is ⟨ 𝑓 esc ⟩ ion . ≈ 3%, as shown in Fig. 3, slightly lower but comparable to F19. For such a low value, the LzLCSrelationship would require 𝛽 UV ≈ -1 . 93, significantly redder than JWST has observed at 𝑧 > 5 (Topping et al. 2022; Cullen et al. 2023). Moreover, even setting a modest ⟨ 𝑓 esc ⟩ = 5% still requires a cutoff at magnitudes brighter than 𝑀 UV ≈ -12 given the higher 𝜉 ion from JWST. These faint 𝑀 UV have not been statistically probed yet by JWST observations, but upcoming ultra-deep imaging of a gravitationally lensed cluster (the Glimpse program, PI: Atek, JWSTID: 3293) will measure 𝜉 ion and 𝛽 UV from lensed star-forming galaxies down to 𝑀 UV ≈ -12. Deeper measurements of analogues at moderate 𝑧 are also critical to examine how well the 𝑓 esc -𝛽 UV relation holds at fainter magnitudes (and bluer objects), as well as higher 𝑧 , pushing as close as possible to the epoch of reionization. \n· What about the faint end of the UVLF? The slope and turnover magnitude remain as the key uncertainties of this observable. For a turnover to match reionization measurements it would have to be at a bright 𝑀 UV ≈ -17, as illustrated in Fig. 3, far above the current UVLF limits. An alternative is a shallow faint-end slope. We have repeated our analysis with the Finkelstein & Bagley (2022) UVLF, which assumes a double power-law functional form with a flattening towards the faint end, and found that the tension persists (see Appendix A). This is not surprising, as the tension in Fig. 2 requires little to no extrapolation of the UVLFs during reionization (down to 𝑀 UV ≈ -15, where the faint-end uncertainties affect galaxy abundances at the 30% level). If a turnover or flattening of the UVLF was the solution it would be of paramount importance to understand its physical origin, whether it is due to feedback during reionization (Shapiro et al. 2004) or a exotic cosmology (Sabti et al. 2022). \n- · Maybe our theoretical models are wrong? The main uncertainty is how many recombinations take place, which we have modeled through a simple clumping factor 𝐶 . Past work has suggested additional recombinations can explain an extended reionization history inferred from the Lyman𝛼 forest at 𝑧 ∼ 5 -6 (Davies et al. 2021; Qin et al. 2021). Such a 'tax on the rich' (in terms of ionizing photons, Furlanetto & Oh 2005) could alleviate the budget crisis. As a test, we show in Fig. 3 how even a large 𝐶 = 20 - implying nearly an order of magnitude more recombinations throughout all of reionization - does not suffice to harmonize galaxy and CMB observations, still overproducing reionization (more than 3 𝜎 above 𝜏 CMB measurements). Of course, we expect the process of reionization to be complex and inhomogeneous, but we note that these many recombinations per hydrogen atom are not standard in Λ CDM cosmologies (even including mini-halos, Gnedin 2024), and could point to additional baryon fluctuations at very small scales, or missing ingredients in our theories. \nFig. 3 summarizes how different possible solutions would affect the timing of reionization. While these scenarios can be re-calibrated to produce the correct 𝜏 CMB (e.g., increasing 𝑀 ion . cutoff or decreasing \nx \nFigure 3. Possible solutions to the photon budget crisis, and how they would affect the timing of reionization. The blue dot-dashed line corresponds to our current understanding of reionization, as in Fig. 1. The pink dotted line assumes a higher clumping factor 𝐶 = 20, which still does not produce enough recombinations. The red dashed line has 𝑀 ion . cutoff UV = -11 and a low 𝑓 esc = 3%, whereas the black line posits 𝑀 ion . cutoff UV = -17 with a larger 𝑓 esc = 15%. These two models produce the correct 𝜏 CMB, but disagree with one of the two galaxy observations, as indicated by the red diamond and black circle in the right panel of Fig. 2. Their reionization histories 𝑥 HI ( 𝑧 ) are potentially distinguishable from one another (measurements in green, which have not been used to calibrate the models). \n<!-- image --> \n𝑓 esc), measurements of 𝑥 HI ( 𝑧 ) can potentially distinguish between them, as posing a cutoff 𝑀 ion . cutoff UV makes reionization faster (like oligarchic models, Naidu et al. 2019), whereas decreasing 𝑓 esc slows it down (appearing democratic, F19). Of course, a 𝑧 -dependent 𝑓 esc can mimic this effect, so clustering measurements of reionization bubbles, for instance with the 21-cm line (Furlanetto et al. 2004; Muñoz et al. 2022) will be required to break degeneracies. We emphasize that each of the mechanisms invoked in this section requires giving up the constraints from at least one of our galaxy measurements, be it the UVLFs, 𝜉 ion , 𝑓 esc, or a combination of them. Further observations of galaxies, 𝑥 HI , and 𝜏 CMB will sharpen our understanding of the reionization process, as current error-bars are still sizeable and may hide underlying systematics. While the list presented here is not exhaustive, we hope it encourages theoretical and observational work to resolve the JWST photon budget crisis.", '5 CONCLUSIONS': "The launch of JWST is allowing us to directly access the properties of the first galaxies with unprecedented sensitivity. Early observations are showing that early, faint galaxies are prolific producers of ionizing photons. Here we have combined new JWST measurements with determinations of the escape fractions 𝑓 esc of reionization-era analogues to show that our current galaxy observations predict a process of reionization that ends too early. That is, the situation has been reversed from the WMAP era, where the concern was producing enough photons to match 𝜏 CMB , to the postPlanck and JWST era, where there may be too many photons . \nTo match the CMB optical depth, the / 𝑛 ion of early galaxies must dramatically decrease. This is currently not observed in the high𝑧 galaxy population. For instance, the UVLFs do not show a significant turnover down to 𝑀 UV ≈ -15 (Atek et al. 2018), faint galaxies during the epoch of reionization are very blue down to 𝑀 UV ≈ -17 (Topping et al. 2022; Cullen et al. 2023, hinting at high 𝑓 esc), and on average galaxies produce significantly more ionizing photons than inferred from HST + Spitzer observations down to 𝑀 UV ≈ -15 (Simmonds et al. 2024; Atek et al. 2024; Endsley et al. 2023). \nAs such, taken at face value the galaxies JWST has observed already produce enough ionizing photons to reionize the universe. This does not include faint galaxies still unprobed by JWST observations, or a contribution from early black holes (which appear prevalent in early JWST observations, Matthee et al. 2024). There must be a missing ingredient in either our modeling or observations to harmonize the galaxy and CMB inferences of reionization. \nMoving forward, there are several avenues that can further audit the ionizing-photon budget. Future CMB surveys are expected to measure 𝜏 CMB to ≈ 0 . 002 (Allys et al. 2023), which would sharpen our understanding of reionization. Better constraints on the timing of reionization beyond 𝜏 CMB (e.g., via the kinematic SunyaevZel'dovich effect and the transmission of Lyman𝛼 photons from high-redshift sources Raghunathan et al. 2024; Chen et al. 2024; Nakane et al. 2023; Ouchi et al. 2020; Lu et al. 2024) and tomographic measurements of the distribution of neutral and ionized hydrogen through the 21-cm line (Morales & Wyithe 2010; Abdurashidova et al. 2022), will provide invaluable information on how different sources contribute to the photon budget. Further studies of both 𝑓 esc (at low 𝑧 ) and 𝜉 ion (at high 𝑧 ) are critical to account for selection biases in our samples, or for missed assumptions in their interpretation. The biggest theoretical uncertainty is quantifying the recombination rate, which has a substantial effect on the reionization history (see Fig. 3) and may alleviate the requirements on the sources. In particular, it is crucial to understand how an increase in the recombination rate at 𝑧 ∼ 6 due to dense IGM clumps would carry over to higher redshifts, during the bulk of reionization. \nIn summary, recent observations have found that early galaxies were numerous, efficient producers of ionizing photons, and likely to have non-negligible escape fractions. Together, these galaxy observations imply an excess in the ionizing-photon budget during reionization, which would end this cosmic epoch earlier than allowed by CMB data. The JWST era has just begun, and here we have examined how future observations and theoretical efforts can shed light on this tension. As of the time of writing, the different solutions are in conflict with at least one observational constraint. Resolving this tension on reionization is a key step to finally understanding the last major phase transition of our universe.", 'ACKNOWLEDGEMENTS': 'Weare grateful to V. Bromm, R. Endsley, S. Finkelstein, K. Hawkins, A. Pahl, C. Scarlata, M. Shull, E. Thelie, and the anonymous referee for insights on a previous version of this manuscript. JBM was supported by the National Science Foundation under Grants AST2307354 and AST-2408637, and thanks the Yukawa Institute for Theoretical Physics and the Kavli Institute for Theoretical Physics for their hospitality during part of this work. JM was supported by an appointment to the NASA Postdoctoral Program at the Jet Propulsion Laboratory / California Institute of Technology, administered by Oak Ridge Associated Universities under contract with NASA. SRF was supported by NASA through award 80NSSC22K0818 and by the National Science Foundation through award AST-2205900. CAMacknowledges support by the VILLUM FONDEN under grant 37459 and the Carlsberg Foundation under grant CF22-1322. The Cosmic Dawn Center (DAWN) is funded by the Danish National Research Foundation under grant DNRF140.', 'DATA AVAILABILITY': 'The data underlying this article will be shared on reasonable request to the author. Software: numpy (van der Walt et al. 2011), scipy (Jones et al. 2001), matplotlib (Hunter 2007), Zeus21 (Muñoz 2023), CLASS (Blas et al. 2011).', 'REFERENCES': "Lam D., et al., 2019, A&A, 627, A164 \nLin Y.-H., et al., 2024, MNRAS, 527, 4173 \n- Liu H., Slatyer T. R., Zavala J., 2016, Phys. Rev. D, 94, 063507\n- Lu T.-Y., Mason C. A., Hutter A., Mesinger A., Qin Y., Stark D. P., Endsley R., 2024, MNRAS, 528, 4872\n- Ma X., Hopkins P. F., Kasen D., Quataert E., Faucher-Giguère C.-A., Kereš D., Murray N., Strom A., 2016, MNRAS, 459, 3614\n- Ma X., Quataert E., Wetzel A., Hopkins P. 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A., van Dokkum P. G., 2012, Astrophys. J., 756, 14 \nSobacchi E., Mesinger A., 2015, Mon. Not. Roy. Astron. Soc., 453, 1843 \nSteidel C. C., Bogosavljevic M., Shapley A. E., Reddy N. A., Rudie G. C., \nPettini M., Trainor R. F., Strom A. L., 2018, Astrophys. J., 869, 123 \nTang M., Stark D. P., Chevallard J., Charlot S., 2019, MNRAS, 489, 2572 \nTang M., et al., 2023, MNRAS, 526, 1657 \nFigure A1. Constraints on reonization, as in Fig. 2, but with different assumptions about the UVLF and log 10 𝜉 ion . The red contour shows the 𝜏 CMB constraint using Finkelstein & Bagley (2022), whereas black (dotted) uses that of Bouwens et al. (2021). Both these pre-JWST determinations of the UVLF still show a tension in reionization, though slightly less severe. The purple (dashed) region shows the 𝜏 CMB contour taking a lower value of 𝜉 ion , which consequently predicts fewer ionizing photons. In this last case there is a region of parameter space where the constraints overlap, showing that a downward revision of 𝜉 ion plus a cutoff brighter than 𝑀 UV ≈ -14 could resolve the tension. We also show the median 𝑓 esc from the VANDELS sample of 𝑧 ∼ 3 galaxies (Saldana-Lopez et al. 2023), extrapolated to bluer galaxies as a dashed line. \n<!-- image --> \nTopping M. W., Stark D. P., Endsley R., Plat A., Whitler L., Chen Z., Charlot S., 2022, ApJ, 941, 153 Trebitsch M., et al., 2022 Wang F., et al., 2020, Astrophys. J., 896, 23 Weibel A., et al., 2024, arXiv e-prints, p. arXiv:2403.08872 Whitler L. R., Mason C. A., Ren K., Dijkstra M., Mesinger A., Pentericci L., Trenti M., Treu T., 2019, ] 10.1093/mnras/staa1178 Wise J. H., Demchenko V. G., Halicek M. T., Norman M. L., Turk M. J., Abel T., Smith B. D., 2014, MNRAS, 442, 2560 Yeh J. Y. C., et al., 2023, MNRAS, 520, 2757 Yoo T., Kimm T., Rosdahl J., 2020, MNRAS, 499, 5175 Zhao J., Furlanetto S. R., 2024, arXiv e-prints, p. arXiv:2401.07893 Zhu Y., et al., 2023, ApJ, 955, 115 van der Walt S., Colbert S. C., Varoquaux G., 2011, Computing in Science and Engineering, 13, 22", 'APPENDIX A: ALTERNATIVE ASSUMPTIONS': 'Through the text we have taken the pre-JWST UVLF from Bouwens et al. (2021) for 𝑧 ≤ 9, and the JWST-era UVLFs from Donnan et al. (2024) for higher 𝑧 . These data reach 𝑀 UV ≈ -17 at 𝑧 ∼ 7, so in order to find the abundance of galaxies at fainter magnitudes some extrapolation is required. Additionally, we have used the 𝛽 UV -𝑀 UV relation from Zhao & Furlanetto (2024), which includes JWST data, and the 𝜉 ion fit from Simmonds et al. (2024). The purpose of this Appendix is to cross check these assumptions. For that we will first repeat our analysis removing the new JWST calibrations of the 𝑧 ≳ 9 UVLF and 𝛽 UV and revert to pre-JWST estimates. Then we will use the Finkelstein & Bagley (2022) UVLF, which uses a compilation of data and has a different functional form that includes flattening towards the faint end. Finally, we will find whether there is still tension for a value of 𝜉 ion comparable to that of Endsley et al. (2023) or Pahl et al. (2024), rather than Simmonds et al. (2024). \nTable A1. Table summarizing the different mean/median assumed relations in this work, their origin, calibration region, and estimated uncertainty (not intrinsic scatter). Last column shows the optical depth derived by taking each relationship and its uncertainty, while keeping the rest of the analysis fixed. 𝑎 First relation shown for each variable corresponds to our fiducial through the paper. 𝑏 In both cases added to Donnan et al. (2024) for 𝑧 ≳ 9. 𝑐 Uncertainty in 𝛽 UV and Φ UV depends on magnitude and redshift, so we report typical values at 𝑧 ∼ 7 and 𝑀 UV ∼ -17. 𝑑 Assuming a cutoff at 𝑀 UV = -13. These ought to be compared to the Planck measurement of 𝜏 CMB = 0 . 054 ± 0 . 007 (Aghanim et al. 2020).', 'A1 How much do the new JWST UVLFs impact the tension?': 'Not significantly. We re-run our analysis returning to the pre-JWST UVLFfromBouwensetal.(2021), and using the 𝛽 UV -𝑀 UV relation from Bouwens et al. (2014, fixing its 𝑧 = 8 value for earlier times). Fig. A1 shows how the region that gives rise to the correct 𝜏 CMB is still in tension with galaxy observables, as it only overlaps the low𝑧 constraints on 𝑓 esc for cutoffs brighter than 𝑀 UV ≈ -15, which are disfavored (barring a tiny edge region around 𝑀 ion . cutoff UV = -15 and ⟨ 𝑓 esc ⟩ ion . = 15%). The tension is then largely driven by the high 𝜉 ion values inferred by JWST observations, rather than the enhancement of the UVLF at high 𝑧 . Nevertheless, the extra 𝑧 ≳ 9 galaxies can kickstart reionization earlier. Adding the recently discovered population of supermassive black holes in JWST would potentially increase the ionizing-photon production (unlike the pre-JWST expectations, e.g., Matsuoka et al. 2018), exacerbating the crisis if the accretion disks are unobscured.', 'A2 Re-analysis with Finkelstein & Bagley (2022)': 'Fig. A1 shows how the tension in the ionization-photon budget remains when changing the UVLF to the pre-JWST fit from Finkelstein &Bagley (2022, and 𝛽 UV from Bouwens et al. 2014). The three observations ( 𝜏 CMB , the 𝑓 esc measurement from low 𝑧 , and the no-cutoff down to the HST+JWST limit) still do not overlap. This is not surprising, since the different UVLFs broadly agree at the bright end, only diverging towards faint magnitudes and high 𝑧 (for instance, at 𝑧 = 7 the uncertainty in the faint-end slope 𝛼 ★ of both Finkelstein & Bagley 2022 and Bouwens et al. 2021 translates into 30% more or fewer 𝑀 UV = -15 galaxies). This is visible towards the faint side ( 𝑀 ion . cutoff UV ∼ -13 ) of Fig. A1, where the CMB-preferred region flattens at 𝑓 esc ≈ 6%, whereas in the Bouwens et al. (2021) case it does so at 𝑓 esc ≈ 3%. Part of the reason is the turnover built into the UVLF fit of Finkelstein & Bagley (2022, not included in the Bouwens et al. 2021 fit), regardless of our additional 𝑀 ion . cutoff UV . This test serves to benchmark the differences in the faint end of the UVLF.', 'A3 A lower 𝜉 ion value?': 'Through the main text we have used the fit for 𝜉 ion as a function of 𝑀 UV and 𝑧 from Simmonds et al. (2024). Other reionizationera results from Atek et al. (2024) and Endsley et al. (2023) also find enhanced ionizing-photon production, though in the latter case it decreases towards the faint end, rather than increase. We have found that the photometric results in Endsley et al. (2023) can be \napproximately fit by \nlog 10 𝜉 ion = 25 . 5 -0 . 03 × ( 𝑀 UV + 18 ) , (A1) \nfor the two faint bins in their calibration (and this relation underestimates 𝜉 ion for the brightest bin). We show in Table A1 how taking this relation still overpredicts 𝜏 CMB . A recent spectroscopic analysis in Pahl et al. (2024) finds a mean GLYPH<10> log 10 𝜉 ion /( Hzerg -1 ) GLYPH<11> = 25 . 38 for their 𝑧 > 4 sample, which translates into a mean ⟨ 𝜉 ion ⟩ = 10 25 . 57 Hzerg -1 (Pahl, Private Communication) as expected of a lognormal variable with a 0.4 dex scatter. We can, then, conservatively bracket the uncertainty in 𝜉 ion by performing a run with log 10 𝜉 ion /( Hzerg -1 ) = 25 . 5, comparable to the lowest mean values measured in Endsley et al. (2023, see third panel of Fig. B1) and the running mean of Pahl et al. (2024). We show the result of this analysis in Fig. A1, where the three observational constraints overlap over a small range of parameter space. This represents a possible compromise solution, requiring both a cutoff brighter than 𝑀 UV ∼ -14 (potentially detectable) plus a downward revision on 𝜉 ion (possibly indicating an observational bias or mismodeling).', 'A4 Summary of Assumptions': 'We summarize the relations taken in this work, their origin in either JWST or HST data, and the reported uncertainties in Table A1. We compute in each case the expected 𝜏 CMB and associated errorbars from each relationship, keeping the rest fixed and setting a fiducial cutoff at 𝑀 UV = -13 as in Robertson et al. (2015). We find that changing the UVLF calibration makes a difference of 20% on 𝜏 CMB (due to the cutoff included in Finkelstein & Bagley 2022), whereas changing the 𝛽 UV relation is at the sub-10% level. The biggest uncertainties are 𝑓 esc and 𝜉 ion , as expected, and in particular we find that going from the Simmonds et al. (2024) to the Endsley et al. (2023) 𝜉 ion calibration reduces 𝜏 CMB by 20%, though in all cases shown in Table A1 𝜏 CMB is higher than allowed by the CMB.', 'APPENDIX B: THE ORIGIN OF A FAINT CUTOFF': "Through the main text we have used the variable 𝑀 ion . cutoff UV to express a generic cutoff below where galaxies do not contribute to reionization. This cutoff can have an origin in three different mechanisms, which we illustrate in Fig. B1. \nFirst, the UVLF may have a 'turn over', so the abundance of star-forming galaxies drops below some magnitude. Current UVLF observations suggest that this cutoff has to be fainter than 𝑀 UV ≈ -15 during reionization (Atek et al. 2018). This is shown in the second panel of Fig. B1. \nFigure B1. The ionization-photon production ( first panel , at 𝑧 ∼ 7) can be cut off at the faint end from three different sources: the UVLF ( Φ UV, second panel ), the ionizing efficiency ( 𝜉 ion , third panel ), or the escape fraction ( 𝑓 esc, last panel ). We shade the regions that are not observed on each respective panel. Dotted curves show the result with no cutoffs, and thick with each cutoff. The vertical black line is at 𝑀 ion . cutoff UV = -17, as required to fit reionization in Fig. 3, which is in tension with observations. In the third panel we show not only the 𝜉 ion fit from Simmonds et al. (2024, S+24, used through the main text) but also measurements from Endsley et al. (2023, E+23), Atek et al. (2024, A+24), and Prieto-Lyon et al. (2023, PL+23, evaluated at 𝑧 ∼ 7). All the JWST-inferred 𝜉 ion values are well above the pre-JWST canonical value (black dotted). \n<!-- image --> \nSecond, the ionizing efficiency may vanish for faint galaxies. Some JWST observations suggest the opposite, in fact, with 𝜉 ion seemingly growing towards the faint end at least until 𝑀 UV ≈ -15 (Atek et al. 2024, or 𝑀 UV = -16 . 5 for the broader but photometric sample of Simmonds et al. 2024). This trend is also reported in Prieto-Lyon et al. (2023) at ⟨ 𝑧 ⟩ ≈ 4, which we show in Fig. B1 (extrapolating their results to 𝑧 = 7 by using the scaling in Eq. 4). The results in Endsley et al. (2023) instead point to 𝜉 ion growing towards the bright end, as shown in Fig. B1, though with a large variance in the distribution at each bin. This variance also makes the average 𝜉 ion slightly larger than expected from the median of log 10 𝜉 ion , as in \nfootnote 2, which we account for when plotting the Endsley et al. (2023) data (as it is the only one with measured variance). All the JWST measurements in the third panel of Fig. B1 are significantly above the pre-JWST canonical value, following pre-JWST hints in e.g., Maseda et al. (2020). \nLast, the escape fraction 𝑓 esc may stop growing towards the faint end. This is, however, the opposite behavior seen at low 𝑧 in both LzLCS( 𝑧 ∼ 0) and VANDELS( 𝑧 ∼ 3). For reference, the median 𝑓 esc in the VANDELS sample reported by Saldana-Lopez et al. (2023) can be fit using Eq. (5) with 𝐴 𝑓 = 1 . 12 × 10 -4 and 𝑏 𝑓 = -1 (from their Fig. 15), which we show in both Figs. A1 and B1. The bluest galaxies sampled in the LzLCS ( 𝛽 UV = -2 . 7) correspond to 𝑀 UV ≈ -15 . 7 at 𝑧 ∼ 7 (using the 𝛽 UV -𝑀 UV relation from Zhao & Furlanetto 2024), indicating we do not expect a cutoff on 𝑓 esc until at least that magnitude, as shown in the last panel of Fig. B1. \nTogether, different galaxy observations have probed the 𝑀 ion . cutoff UV ≈ -17 region, disallowing a cutoff at such magnitudes, unless there is an observational bias or systematic uncertainty. \nThis paper has been typeset from a T E X/L A T E X file prepared by the author."}

2024ApJ...974...92B

In this paper we describe the survey design for the Ultradeep NIRSpec and NIRCam Observations before the Epoch of Reionization UNCOVER Cycle 1 JWST Treasury program which executed its early imaging component in 2022 November. The UNCOVER survey includes ultradeep 2930AB imaging of 45 arcminSUP2SUP on and around the wellstudied A2744 galaxy cluster at z 0.308 and will follow up 500 galaxies with extremely deep lowresolution spectroscopy with the NIRSpecPRISM during the summer of 2023 with repeat visits in summer 2024. We describe the science goals survey design target selection and planned data releases. We also present and characterize the depths of the first NIRCam imaging mosaic highlighting previously unparalleled resolved and ultradeep 24 m imaging of known objects in the field. The UNCOVER primary NIRCam mosaic spans 28.8 arcminSUP2SUP in seven filters F115W F150W F200W F277W F356W F410M and F444W and 16.8 arcminSUP2SUP in our NIRISS parallel F115W F150W F200W F356W and F444W. To maximize early community use of the Treasury data set we publicly release the full reduced mosaics of public JWST imaging including 45 arcminSUP2SUP NIRCam and 17 arcminSUP2SUP NIRISS mosaics on and around the A2744 cluster including the Hubble Frontier Field primary and parallel footprints.

2024-10-01T00:00:00Z

['10.48550/arXiv.2212.04026', '2024ApJ...974...92B', '2022arXiv221204026B', 'arXiv:2212.04026', '10.3847/1538-4357/ad66cf']

['James Webb Space Telescope', 'Redshift surveys', 'Galaxy evolution', 'Galaxy formation', 'Observational astronomy', 'Abell clusters', 'High-redshift galaxies', 'Galaxies', '2291', '1378', '594', '595', '1145', '9', '734', '573', 'Astrophysics - Astrophysics of Galaxies']

The JWST UNCOVER Treasury Survey Ultradeep NIRSpec and NIRCam Observations before the Epoch of Reionization

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']

https://arxiv.org/pdf/2212.04026.pdf

{'Ultradeep NIRSpec and NIRCam ObserVations before the Epoch of Reionization': "Rachel Bezanson, 1 Ivo Labbe, 2 Katherine E. Whitaker, 3, 4 Joel Leja, 5, 6, 7 Sedona H. Price, 1 Marijn Franx, 8 Gabriel Brammer, 4 Danilo Marchesini, 9 Adi Zitrin, 10 Bingjie Wang ( 王 冰 洁 ), 5, 6, 7 John R. Weaver, 3 Lukas J. Furtak, 10 Hakim Atek, 11 Dan Coe, 12, 13, 14 Sam E. Cutler, 3 Pratika Dayal, 15 Pieter van Dokkum, 16 Robert Feldmann, 17 Natascha M. Forster Schreiber, 18 Seiji Fujimoto, 19, ∗ Marla Geha, 16 Karl Glazebrook, 2 Anna de Graaff, 20 Jenny E. Greene, 21 St'ephanie Juneau, 22 Susan Kassin, 12 Mariska Kriek, 8 Gourav Khullar, 1 Michael Maseda, 23 Lamiya A. Mowla, 24 Adam Muzzin, 25 Themiya Nanayakkara, 2 Erica J. Nelson, 26 Pascal A. Oesch, 27, 4 Camilla Pacifici, 12 Richard Pan, 9 Casey Papovich, 28, 29 David J. Setton, 1 Alice E. Shapley, 30 Renske Smit, 31 Mauro Stefanon, 32, 33 Edward N. Taylor, 2 and Christina C. Williams 34, 35 \n- 1 Department of Physics and Astronomy and PITT PACC, University of Pittsburgh, Pittsburgh, PA 15260, USA 2 Centre for Astrophysics and Supercomputing, Swinburne University of Technology, Melbourne, VIC 3122, Australia 3 Department of Astronomy, University of Massachusetts, Amherst, MA 01003, USA \n4 \nCosmic Dawn Center (DAWN), Niels Bohr Institute, University of Copenhagen, Jagtvej 128, København N, DK-2200, Denmark \n5 \nDepartment of Astronomy & Astrophysics, The Pennsylvania State University, University Park, PA 16802, USA \nInstitute for Computational & Data Sciences, The Pennsylvania State University, University Park, PA 16802, USA \n7 \nInstitute for Gravitation and the Cosmos, The Pennsylvania State University, University Park, PA 16802, USA \n8 \nLeiden Observatory, Leiden University, P.O.Box 9513, NL-2300 AA Leiden, The Netherlands \nDepartment of Physics and Astronomy, Tufts University, 574 Boston Ave., Medford, MA 02155, USA \n9 \n10 \nPhysics Department, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva 8410501, Israel \n- 11 Institut d'Astrophysique de Paris, CNRS, Sorbonne Universit'e, 98bis Boulevard Arago, 75014, Paris, France 12 Space Telescope Science Institute (STScI), 3700 San Martin Drive, Baltimore, MD 21218, USA\n- 13 Association of Universities for Research in Astronomy (AURA), Inc. for the European Space Agency (ESA) 14 Center for Astrophysical Sciences, Department of Physics and Astronomy, The Johns Hopkins University, 3400 N Charles St. Baltimore, MD 21218, USA\n- 15 Kapteyn Astronomical Institute, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands 16 Department of Astronomy, Yale University, New Haven, CT 06511, USA\n- 17 Institute for Computational Science, University of Zurich, Winterhurerstrasse 190, CH-8006 Zurich, Switzerland Max-Planck-Institut fur extraterrestrische Physik, Gießenbachstraße 1, 85748 Garching, Germany\n- 18 19 Department of Astronomy, The University of Texas at Austin, Austin, TX 78712, USA 20 Max-Planck-Institut fur Astronomie, Konigstuhl 17, D-69117, Heidelberg, Germany\n- 21 Department of Astrophysical Sciences, 4 Ivy Lane, Princeton University, Princeton, NJ 08544, USA \nNSF's National Optical-Infrared Astronomy Research Laboratory, 950 N. Cherry Avenue, Tucson, AZ 85719, USA \n- 23 Department of Astronomy, University of Wisconsin-Madison, 475 N. Charter St., Madison, WI 53706 USA \n24 \nDunlap Institute for Astronomy and Astrophysics, 50 St. George Street, Toronto, Ontario, M5S 3H4, Canada \n- 25 Department of Physics and Astronomy, York University, 4700 Keele Street, Toronto, Ontario, ON MJ3 1P3, Canada 26 Department for Astrophysical and Planetary Science, University of Colorado, Boulder, CO 80309, USA 27 Department of Astronomy, University of Geneva, Chemin Pegasi 51, 1290 Versoix, Switzerland \n28 \nDepartment of Physics and Astronomy, Texas A&M University, College Station, TX, 77843-4242 USA \n- 29 George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy, Texas A&M University, College Station, TX, 77843-4242 USA\n- 30 \nPhysics & Astronomy Department, University of California: Los Angeles, 430 Portola Plaza, Los Angeles, CA 90095, USA 31 Astrophysics Research Institute, Liverpool John Moores University, 146 Brownlow Hill, Liverpool L3 5RF, UK 32 Departament d'Astronomia i Astrofisica, Universitat de Valencia, C. Dr. Moliner 50, E-46100 Burjassot, Valencia, Spain 33 Unidad Asociada CSIC 'Grupo de Astrofisica Extragalactica y Cosmologi' (Instituto de Fisica de Cantabria - Universitat de Valencia) 34 NSF's National Optical-Infrared Astronomy Research Laboratory, 950 N. Cherry Avenue, T ucson, AZ 85719, USA 35 Steward Observatory, University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721, USA", 'ABSTRACT': 'In this paper we describe the survey design for the Ultradeep NIRSpec and NIRCam ObserVations before the Epoch of Reionization (UNCOVER) Cycle 1 JWST Treasury program, which executed its early imaging component in November 2022. The UNCOVER survey includes ultradeep ( ∼ 29 -30AB) \n6 \n22 \nimaging of ∼ 45 arcmin 2 on and around the well-studied Abell 2744 galaxy cluster at z = 0 . 308 and will follow-up ∼ 500 galaxies with extremely deep low-resolution spectroscopy with the NIRSpec/PRISM during the summer of 2023 , with repeat visits in summer 2024. We describe the science goals, survey design, target selection, and planned data releases. We also present and characterize the depths of the first NIRCam imaging mosaic, highlighting previously unparalleled resolved and ultradeep 2-4 micron imaging of known objects in the field. The UNCOVER primary NIRCam mosaic spans 28.8 arcmin 2 in seven filters (F115W, F150W, F200W, F277W, F356W, F410M, F444W) and 16.8 arcmin 2 in our NIRISS parallel (F115W, F150W, F200W, F356W, and F444W). To maximize early community use of the Treasury data set, we publicly release full reduced mosaics of public JWST imaging including 45 arcmin 2 NIRCam and 17 arcmin 2 NIRISS mosaics on and around the Abell 2744 cluster, including the Hubble Frontier Field primary and parallel footprints.', '1. INTRODUCTION': "A lasting legacy of the Hubble Space Telescope ( HST ) and Spitzer Space Telescope is the discovery and characterization of galaxies to z ∼ 11, looking back 97% of the time to the Big Bang (e.g., Coe et al. 2013; Oesch et al. 2016). Extragalactic deep fields with Hubble imaging and grism spectroscopy to ∼ 1 . 6 microns have provided a reasonably complete census of the cosmic star formation history (e.g., Madau & Dickinson 2014; Finkelstein et al. 2015; Oesch et al. 2018; Bouwens et al. 2022b), and combined with Spitzer imaging, constraints on the mass functions of galaxies since z = 3 (e.g., Leja et al. 2020; Duncan et al. 2014; Grazian et al. 2015; Song et al. 2016; Bhatawdekar et al. 2019; Kikuchihara et al. 2020; Furtak et al. 2021; Stefanon et al. 2021). However, despite 10,000 HST orbits and tens of thousands of hours of Spitzer observations, our picture of the Universe remains highly incomplete due to reliance on HST -selection (rest-frame UV at z> 4). This means we were biased towards the youngest, least dusty galaxies, and had been unable to study the first galaxies at z ≳ 12. Understanding these first galaxies is crucial since they mark the start of the process of cosmic reionization and are the first sources of metals in the Universe (e.g., Dayal & Ferrara 2018). \nThe JWST was specifically designed to address these limitations by providing high-throughput, sensitive imaging and spectroscopy at 1 -5 µ mwith the NIRCam (Rieke et al. 2005, 2003, 2023), NIRISS (Doyon et al. 2012) and NIRSpec instruments (Jakobsen et al. 2022; Rigby et al. 2023; Boker et al. 2023). These capabilities have begun to enable us to find galaxies deep into the Dark Ages ( z =10 -20) (e.g., Naidu et al. 2022a; Atek et al. 2023; Finkelstein et al. 2022; Donnan et al. 2023; Harikane et al. 2023; Adams et al. 2023), to directly study the faint galaxies responsible for reionizing the universe a few hundred million years later, and \nidentifying dusty and quiescent galaxies to low masses to redshift z ∼ 10 (e.g., Carnall et al. 2022; Labb'e et al. 2023; Naidu et al. 2022b; Zavala et al. 2023; Nelson et al. 2023). The intrinsically faintest galaxies can be reached with the aid of gravitational lensing. Observations with HST along gravitational lensing galaxy clusters (e.g., in the Hubble Frontier Fields, Lotz et al. 2017) have identified some exceptionally magnified individual objects (e.g., Zitrin et al. 2014; Kelly et al. 2015), increasing effective depths of e.g., reionization era UV luminosity functions by several magnitudes (e.g., Atek et al. 2018; Ishigaki et al. 2018; Bouwens et al. 2021; Kauffmann et al. 2022; Bouwens et al. 2022b,a). However, at the faintest ends, these studies remain besieged by systematics. \nIt is in this context that the Ultradeep NIRSpec and NIRCam ObserVations before the Epoch of Reionization (UNCOVER) Treasury survey was designed (PIs Labb'e & Bezanson, JWST -GO#2561). UNCOVER targets the powerful lensing cluster Abell 2744 and consists of two coordinated components executed in the same cycle. First, a deep NIRCam pre-imaging mosaic in 7 filters for ∼ 4 -6 hour per band, and second ultra-deep 3 -20 hour NIRSpec/PRISM low-resolution follow up of NIRCam-detected high-redshift galaxies - each with NIRISS or NIRCam imaging parallels. The imaging and spectroscopy leverages the gravitational lensing boost to push beyond depths achievable in blank fields. Perhaps the most exciting legacy is the potential to discover 'unknown unknowns', objects we have not yet imagined or anticipated. The advance in sensitivity, wavelength coverage, and spectral resolution (in comparison to Spitzer imaging) is so great that we are almost guaranteed to run into surprises. Covering ultra-deep parameter space with imaging and spectroscopy early in JWST 's mission helps to ensure ample time for follow up studies in future cycles. \nIn Figure 1 we show the full area of the UNCOVER survey (purple star) in context of other HST (blue circles) and JWST Cycle 1 extragalactic deep fields \nFigure 1. The UNCOVER combined imaging (purple star) probes a unique regime in the context of HST extragalactic ultradeep fields (blue circles) and JWST Cycle 1 imaging surveys (red squares); without lensing it probes deeper than previous HST surveys and wide field programs. Gravitational lensing (approximate lensing vectors indicated by dashed arrows) from the Abell 2744 allows UNCOVER imaging to probe the intrinsically faintest objects of any JWST project in Cycle 1. We note that lensing vectors are approximate, as the UNCOVER survey includes lensed and unlensed areas of Abell 2744. \n<!-- image --> \n(red squares). UNCOVER is more than a magnitude deeper than other wide-field surveys. Only the ultradeep NGDEEP (PIs: Finkelstein, Papovich, and Pirzkal, JWST-GO-2079, Bagley et al. 2024) survey and GTO ultradeep field (JADES, PI:Eisenstein, JWST-GTO-1180) probe beyond UNCOVER, and these depths do not account for the effects of gravitational lensing. The Abell 2744 lensing cluster boosts the intrinsic limits of UNCOVER, sacrificing area (see arrows indicating the effect of a factor of ∼ 5 magnification in cluster fields) to make it the deepest extragalactic field in Cycle 1 of JWST observations. Furthermore, UNCOVER is the only GO dataset that collected deep spectroscopic followup of JWST -selected objects with no proprietary time in Cycle 1. \nThe outline of this paper is as follows. The survey design of the UNCOVER program is presented in § 2. This paper includes the public release of early reduced mosaics of NIRCam cluster pre-imaging and NIRISS parallel imaging, described in § 3. We summarize the science cases that will be enabled by the \nUNCOVER dataset in § 4. Finally, we summarize the prospects of the UNCOVER survey in § 5. Throughout this paper we adopt a standard ΛCDM cosmology with H 0 = 70kms -1 Mpc -1 , Ω M = 0 . 3, and Ω Λ = 0 . 7.", '2.1. Observational Strategy': 'The first UNCOVER data consist of deep (4-6 hour per filter) NIRCam pre-imaging on the A2744 cluster to ∼ 29 . 5 -30AB magnitude in 7 filters, collected in November 2022. Nine months later July/August 2023 we targeted sources detected in NIRCam with ultra-deep 19 hour NIRSpec/PRISM low-resolution spectroscopy. Due to technical issues, a fraction of NIRSpec observations will be repeated in July/August 2024). Both sets of observations include deep parallel imaging (in NIRISS and NIRCam, respectively), to increase the area for deep photometric studies of high-redshift galaxies at mild (1 . 1 -1 . 3 × ) lensing magnification. The footprints of these components are shown in Figure 2 (left panel), and reproduced along with ancillary data from HST and VLT /MUSE (center panel), as well as other Cycle 1 JWST data (right panel).', '2.2. Field selection': 'A2744 is one of the most powerful known gravitational lensing clusters, with a large area of high magnification ( µ> 2 , 4 , 10 over > 17 , 7 , 2 arcmin 2 ), because the cluster itself consists of several merging subclusters, or massive subclumps (e.g., Merten et al. 2011; Richard et al. 2014; Wang et al. 2015; Jauzac et al. 2015; Diego et al. 2016; Kawamata et al. 2016). The full complex stretches towards the North-West of the Hubble Frontier Field cluster pointing. The NIRCam footprint is designed to span the µ > 2 magnification contour (see Figure 3). The cluster contains many HST-detected 6 <z< 10 galaxies, underscoring its efficacy. The field has excellent roll angle visibility for JWST and low infrared background (ideal for deep background-limited observations). The observing windows (November 2022 for the imaging and expected July 2023 for spectroscopy) benefit from the lowest possible backgrounds in Cycle 1. The field also contains some of the best complementary multi-wavelength data (including deep 29AB HST /ACS optical data) and is accessible by the Atacama Large Millimeter/submillimeter Array (ALMA); we summarize ancillary datasets in § 2.5.', '2.3. Deep NIRCam and NIRISS imaging': "The UNCOVER survey imaging includes deep primary NIRCam imaging on the extended cluster and NIRISS and NIRCam parallel imaging in the outskirts. \nFigure 2. The Abell 2744 cluster has extensive deep optical/NIR ( HST and VLT /MUSE) and JWST coverage. The JWST /UNCOVER footprints (purple) along with HST imaging (blue) and VLT /MUSE deep datacube (green, center) and JWST Cycle 1 imaging and spectroscopy (red, right) against imaging from the Legacy Imaging survey (Dey et al. 2019). An updated version with Cycle 2 JWST programs can be found in Suess et al. (2024). In the left panel, we highlight the detailed layout of the UNCOVER dataset, differentiating between the first epoch (dark purple, NIRCam primary imaging and NIRISS parallel imaging) and second epoch (planned NIRSpec spectroscopy and NIRCam parallel imaging, light purple). The NIRCam mosaic is designed to match the µ = 2 magnification curve (see Fig.3). The footprints of the NIRSpec spectroscopy are provisional and subject to target selection. \n<!-- image --> \nIn this section, we describe the NIRCam primary field cluster pre-imaging mosaic and the NIRISS parallel. \nThe NIRCam mosaic is designed to maximize the number of detected z> 10 galaxies. This was done by forward-modeling the theoretical luminosity functions of Mason et al. (2015) and from the DELPHI model (Dayal et al. 2014, 2022) using the CATS v4.1 lens model of A2744 (Jauzac et al. 2015) to predict the number of 6 <z< 16 galaxies to our detection limit. The number of z> 10 galaxies is maximized by a 4-pointing gap-filled NIRCam mosaic. Our expected 5 σ depths of ∼ 30 AB (given ∼ 4 hours of exposure in F200W and ∼ 6 hours in F115W, JWST ETC 2.0, see Table 1) correspond to M UV = -14 . 0 at z =6 -7 with < 3 magnitudes of lensing (where models are considered more robust, e.g., Livermore et al. 2017; Atek et al. 2018; Bouwens et al. 2021). The dominant uncertainty in expected numbers is the difference between theoretical models, which can differ by factors of > 10 at z > 10 (e.g. Oesch et al. 2018; Finkelstein et al. 2023), with a smaller contribution from cosmic variance ranging from 10% -40% at z = 7 -10 (e.g. Ucci et al. 2021). These depths compare favorably to those achieved in our final mosaic (see Table 1). \nNIRCam pre-imaging is taken with a 4-point mosaic and an 8 pointing INTRAMODULEX dither pattern for 3.7-6 hours, using six broadband filters (F115W, F150W, F200W, F277W, F356W, and F444W). A medium band filter F410M is added, to improve diagnostics of emission lines and improve photometric redshifts and stellar masses of high-z galaxies (Kauffmann et al. 2020; Roberts-Borsani et al. 2021; Labb'e et al. \n2023). We note that the same extended mosaic was followed up with JWST/NIRCam imaging as part of programs JWST-GO-4111 (PI Suess) and JWST-GO-3516 (PIs Matthee and Naidu), which added bluer broadband filters (F070W and F090W) and all remaining medium band filters. Exposure times and approximate imaging depths are listed in Table 1 and filter curves are shown in Figure 4. In that Figure, we show representative SEDs generated using Bagpipes (Carnall et al. 2018) with the NIRCam primary filters. We show delayed tau models (log M ⋆ /M ⊙ = 9 . 2) in the top four panels (100 Myr-old in purple for the earliest galaxies, and 300 Myr-old in blue for the z ∼ 6 -8, A V = 3 dusty galaxies in orange, and 500-Myr-old quiescent galaxies in red). NIRISS parallel imaging includes only F115W, F150W, F200W, F356W, and F444W broad bands. This field lacks some filters with respect to the NIRCam primary imaging as NIRISS lacks NIRCam's dichroic. The NIRISS parallel imaging fortuitously overlaps with 42orbit 29AB F814W Hubble Frontier Field A2744 ACS parallel field, obviating the need for additional optical data. The first NIRCam/ NIRISS imaging data were collected on November 2, 4, 7, and 15, 2022. The final visit was a repeat observation of visit 1:1, which initially failed guide star acquisition (on October 31, 2022). The new visit 3:1 was observed at a slightly different angle (V3PA = 45 . 00 degrees relative to V3PA = 41 . 3588 degrees for the other 3 visits) and higher background level to facilitate rapid rescheduling. This results in a slightly askew mosaic footprint, but image depths and footprint are minimally impacted by this change. \nFigure 3. Gravitational lensing magnification contours in Abell 2744 are extremely extended, due to the complex structure of the multiple cluster cores. The above curves are taken from the UNCOVER-based lensing model at z ∼ 10 (Furtak et al. 2023b). The UNCOVER NIRCam mosaic (dark purple) spans the µ = 2 curve, which is significantly larger than the Hubble Frontier Field (black dashed outline). By extending to the northern subclumps the UNCOVER mosaic enables a more detailed mapping of that region. \n<!-- image --> \nIn parallel to the follow-up NIRSpec observations, a second imaging parallel field will be taken with NIRCam using a more expanded set of seven broadband filters (F090W, F115W, F150W, F200W, F277W, F356W, F444W) and two medium band filters (F335M and F410M) with total exposure times of 2.6-5.3 hours. The expanded filter set mitigates the lack of optical data in that pointing. The NIRCam parallel observations are taken with the NIRSpec/PRISM in MOS mode with a 3-POINT-WITH-NIRCam-SIZE2 dither pattern and a three slitlet nod pattern.", '2.4. Scheduled NIRSpec/PRISM spectroscopic followup': "The spectroscopic component of the UNCOVER survey were completed in July-August 2023, aside from a failed visit that was impacted by an electrical short. We describe the final spectroscopic program design and observational strategy in Price et al. (2024). These spectra are designed to be ultradeep, using the lowresolution ( R ∼ 30 -300) PRISM mode to provide the deepest continuum depths and widest wavelength coverage (0.6-5.3 microns). A primary goal of UN- \nTable 1. UNCOVER Imaging \nNote -Imaging depths in the NIRISS/NIRCam mosaics are calculated using 0.08' radius apertures in the short wavelength bands, 0.16' radius apertures in the long wavelength bands, based on the noise properties inferred from the weight maps and corrected to total assuming point sources. The depths correspond the two-visit depth regions of the mosaic. ETC values correspond to S/N=5 point source depths using JWST ETC v2.0. We refer the reader to Weaver et al. (2024) for a more detailed discussion of effective depths of our photometric catalogs and Price et al. (2024) in preparation for the NIRCam parallel imaging. \nCOVER is to detect continuum flux and to measure continuum redshifts of any faint high redshift object detected securely with NIRCam (10 σ, ∼ 29AB). This requires SNR=3 per resolution element at 1.5 µ m, which can be reached in ∼ 20 hours integration. For galaxies with strong emission lines, we can measure redshifts and emission line strengths to ∼ 30AB if EW obs > 600 ˚ A corresponding to e.g., rest-frame EW 0 (H α ,0) > 100 ˚ A at z =6 or EW 0 ([OIII] 5007 ) > 60 ˚ A at z =9. Typical galaxies at these redshifts have > 5 × stronger emission lines (e.g., Stark 2012; Labb'e et al. 2013; Smit et al. 2014; Stefanon et al. 2021, 2022; Williams et al. 2023); we expect to measure redshifts for the vast majority of targets found by NIRCam. \nThe broad wavelength coverage spans critical spectral features for all potential targets. Simulated spectra for a number of targets are shown in Figure 5. From Cosmic Dawn through reionization, our PRISM spectra will \nFigure 4. Top Rows: model SEDs for four key galaxy types and Bottom Rows: Filter curves for deep HST and JWST imaging in the 3 UNCOVER imaging fields. HST imaging (ACS and WFC3) from the Hubble Frontier Field (HFF) program overlaps with the Abell 2744 cluster center (UNCOVER NIRCam primary, top row) and the HFF parallel imaging overlaps with the UNCOVER NIRISS parallel (middle panel). The NIRCam parallel lacks deep optical imaging from HST , but includes F090W and two medium band filters (F335M and F410M). \n<!-- image --> \nWavelength [microns] \nprobe UV flux and Ly α and will include rest-frame optical emission lines (including H α and [NII] below z ∼ 7). Deep exposure times probe rest-optical emission lines to measure low-level star-formation, dust attenuation, and AGN activity in dusty galaxies. Finally, the Balmer absorption features and Balmer/4000 ˚ A breaks can be clearly detected for even the most distant quiescent candidates. We include a similar figure with real spectra in Price et al. (2024). \nOur planned spectroscopic targets will be roughly prioritized according to scientific value and rarity: \n- · any z> 12 candidates\n- · z> 9 galaxies prioritized by brightness\n- · Pop III candidate sources\n- · faint highly magnified 6 <z< 7 galaxies \nFigure 5. Simulated PRISM spectra of a variety of UNCOVER targets , with similar SEDs to those presented in Figure 4, with total exposure times ranging from 2.7-17.4 hours. The wide wavelength coverage (0.6-5.3 microns) of the NIRSpec/PRISM spectra catch critical spectroscopic features, with a resolution up to R ∼ 300 at the red end. Ultradeep exposures probe the continuum flux for nearly all sources and the multiple mask designs provide spectra for ∼ 500 -1000 targets. \n<!-- image --> \nRest-frame Wavelength [ \nÅ \n] \n- · quiescent galaxies z> 4\n- · z> 6 AGN\n- · dusty galaxies z> 4\n- · low mass quiescent galaxies at 1 <z< 6\n- · any unusual or unexpected sources\n- · extreme emission line galaxies\n- · mass-selected galaxies sampled in bins of mass and redshift \nWeestimate that we can accommodate ∼ 15-20 sources to our full depth of 17.4 hours. Other sources require less exposure time. This is accomplished by switching individual sources in and out of overlapping MSA designs (described below). Approximate numbers for key target classes are included in Table 2, estimated from Williams \nTable 2. Predicted Galaxy Counts in UNCOVER \nNote -The photometric dataset inevitably yielded other extraordinary targets that have been included in the MSA designs including, but not limited to: black hole seeds, extremely lensed galaxies and stars. All numbers are rough estimates derived from the JAdes extraGalactic Ultradeep Artificial Realizations (JAGUAR) Mock Catalog (Williams et al. 2018) below z < 6 and at higher redshifts from the Mason et al. (2015) model. Spectroscopic followup prioritizes highly lensed or otherwise remarkable objects. \net al. (2018) and Mason et al. (2015) models. The full implementation of this MSA design will be presented in Price et al. (2024). \nThe total NIRSpec integration times are designed to be split up in 7 partially overlapping dithered sequences of 2.7-4 hours each. We therefore design 7 masks with exposure time ranging from 2.7-17.4 hours, repeating the high priority objects. Each of our seven micro-shutter array (MSA) configurations are designed in an iterative manner, with the possibility of keeping objects on multiple masks. Observations will follow a 3 point dither pattern (3-POINT-WITH-NIRCAM-SIZE2). We adopt the NRSIRS2 readout pattern, averaging 5 frames into groups to optimize the noise characteristics. Given the high target density of some lower priority targets (e.g., there will be 1000s of high-z emission line galaxies), we expect to fill each mask with ∼ 100 targets for a total of ∼ 500 unique sources in the spectroscopic sample. \nSlit loss through the MSA will be significant and wavelength dependent due to variation in the PSF, complicating precise flux calibration. However, because the UNCOVER program includes multiband NIRCam preimaging, our team is well-positioned to perform corrections for these wavelength-dependent aperture effects, leveraging flexible, spatially-resolved galaxy models (Leja et al. 2021).", '2.5. Ancillary Datasets in Abell 2744': "Abell 2744 has been targeted by a multitude of HST imaging, e.g., through the Hubble Frontier Fields program (Lotz et al. 2017) and the HST /BUFFALO sur- \nvey (Steinhardt et al. 2020), which we enumerate along with a variety of other ancillary datasets in Table 3. HST /ACS imaging in the cluster center was taken by Program #11689 (PI: Dupke) and # 13386 (PI: Rodney) and HST /WFC3 observations were collected through program #13495 (PI: Lotz). In the Hubble Frontier Field parallel (UNCOVER NIRISS parallel), HST /ACS imaging was taken by Programs # 13386 (PI: Rodney), # 13495 (PI: Lotz), and #13389 (PI: Siana) and HST /WFC3 imaging in that field was taken again by the Hubble Frontier Field program (#13495, PI: Lotz). Although these deep HST observations were limited to the field of view of individual ACS or WFC3 pointings, the footprints were expanded by the BUFFALO survey (Program # 15117, PI: Steinhardt; Steinhardt et al. 2020), which expands the main cluster and parallel footprints by a factor of four. Recently, the deep optical coverage was expanded by Program #17231 (PI: Treu) (Paris et al. 2023). Amongst other things, this wide area enables improved lens models. \nIn addition, the Abell 2744 cluster has been targeted by ground-based imaging and spectroscopic programs. One such rich dataset is provided by deep VLT /Multi Unit Spectroscopic Explorer (MUSE) observations. This mosaic of MUSE pointings covers 2' x 2' and yields untargeted spectroscopic redshifts for 514 galaxies (Mahler et al. 2018). These spectroscopic redshifts contribute to improved lensing models, especially for multiply imaged systems. Mahler et al. (2018) suggest that this improvement is approximately a factor of ∼ 2 . 5. Although Spitzer imaging in Abell 2744 is largely superseded by novel JWST imaging from the UNCOVER survey, the field has also been observed in the X-ray for 110 ks by XMM-Newton (#074385010, PI: Kneib), 75 ks with Suzaku (Eckert et al. 2016), and Chandra (#23700107, PI Bogdan). Deep groundbased optical imaging exist from Subaru/Suprime-Cam in B, R C , i' , and z' (Medezinski et al. 2016). ALMA observations of the Abell 2744 cluster include 1.1mm imaging of the cluster through the HFF-ALMA program (Gonz'alez-L'opez et al. 2017) and a 15GHz-wide spectral scan at 1.2mm through the ALMA lensing cluster survey (ALCS) (Kohno 2019). \nAlready in the first cycle of JWST observations, several other programs targeted Abell 2744. First, the Early Release Science (ERS) GLASS-JWST program (Treu et al. 2022, PI: Treu) targets the cluster center with deep targeted NIRSpec spectroscopy and NIRISS imaging and untargeted spectroscopy and obtains NIRCam parallel imaging in the cluster outskirts in the following broadband filters (F090W, F115W, F150W, F200W, F277W, F356W, F444W). For details about \nTable 3. Abell 2744 Ancillary Data \nthat program, the reader is referred to the survey design paper (Treu et al. 2022). All data from the GLASSJWST are available with no proprietary period. Additionally, Abell 2744 will be observed with NIRCam imaging as part of the JWST GTO PEARLS (Prime Extra-galactic Areas for Reionization and Lensing Science) program 1176 (PI: Windhorst), including ∼ 2 hour depth observations in F090W, F115W, F150W, F200W, F277W, F356W, F410M, and F444W to ∼ 28 -29AB 5 -σ limiting depths (Windhorst et al. 2023). Recently, a DDT program (#2756, PI: Chen) was approved to image and spectroscopically image a lensed, z ∼ 3 . 5 supernova. That dataset includes two epochs of NIRCam imaging (F115W, F150W, F200W, F277W, F356W, and F444W) and NIRSpec/PRISM spectra in the cluster center. A number of subsequent JWST programs have been completed in Abell 2744; we refer the reader to Suess et al. (2024) for an updated accounting and similar figures. Finally 4' x 6' of the cluster center (in the footprint of the UNCOVER NIRCam imaging) has been mapped by the Deep UNCOVER-ALMA Legacy High-z (DUALZ) Survey ALMA band 6 spectral scans (Fujimoto et al. 2023a).", '2.6. Planned Data Release Schedule': 'While there is no proprietary period associated with the UNCOVER dataset, our team is committed to the regular public release of reduced and high-level data products to maximize community use of this Treasury program. This paper represents the first of several data releases of the UNCOVER survey (DR1); see full anticipated release schedule in Table 4. We subsequently provided two incremental data releases (DR1 and DR2) prior to the JWST Cycle 2 proposal deadline, providing the community with an early photometric catalog based on archival HST imaging and NIRCam pre-imaging in the Abell 2744 cluster (Weaver et al. 2024) in addition to a description and catalog of derived physical properties, including photometric redshifts and stellar population synthesis modeling with Prospector (Johnson et al. 2021) (Wang et al. 2024). Additionally, our team has published lensing maps that have been updated with new JWST sources (Furtak et al. 2023b), magnification estimates and uncertainties from this model are included in photometric catalogs. We introduced a third data release to incorporate the additional broad and medium band NIRCam imaging and NIRISS parallel fields along with all photometric data products (DR3 Suess et al. 2024). Within a year of our second epoch of data acquisition, we plan to publicly release reduced \nNIRSpec/PRISM spectra (Price et al. 2024) and lensing maps that have been updated with new JWST spectroscopic redshifts DR4.', '3. IMAGE REDUCTION AND MOSAICS': 'This paper focuses on the UNCOVER NIRCam and NIRISS imaging (on the Hubble Frontier Field parallel). A full description and public release of photometric and spectroscopic catalogs will be included in subsequent data releases and associated publications.', '3.1. Image Reduction': "All public JWST NIRCam and NIRISS exposures in the Abell 2744 cluster field were downloaded from the Mikulski Archive for Space Telescopes The UNCOVER datasets can be accessed through MAST at 10.17909/zn4s-0243, which includes data from the UNCOVER program as well as imaging from the GLASSERS (PI: Treu) and DD-2756 (PI: Chen) programs. The data reduction pipeline Grism redshift and line analysis software for space-based spectroscopy ( Grizli , version 1.6.0.dev99) was used to process, align, and co-add the exposures. A detailed description of the pipeline is provided in G. Brammer et al. in preparation. \nWe start with the MAST rate.fits products produced by Stage 1 of the JWST calibration pipeline (v1.8.4) using calibration set jwst 0995.pmap . We derive a correction for the '1 /f noise' (Rauscher 2015) by subtracting the source-masked median image values computed along detector rows and columns. Bright cosmic rays that are not completely mitigated by the Level 1 exposure ramp fit can produce 'snowball' residuals (Rigby et al. 2023). Rather than re-running the ramp fits, we identify likely snowballs as large groups of pixels with the DQ = 4 bit set in the exposure data quality array and grow a conservative mask around them. \nBefore 2023 May, the flat-field reference files available in the JWST Calibration Reference Data System (CRDS) were determined from calibrations taken on the ground and did not accurately reflect the pixel-topixel structure in the in-flight NIRCam data. This is most important in the reddest filters where the background is brightest and therefore where any errors in the multiplicative flat-field correction are largest. We determined NIRCam 'sky flat' reference files 1 from on-sky commissioning data from program COM-1063 (PI: Sunnquist) from normalized, source-masked exposures in each of the filter/detector combinations used in the UNCOVER field. The NIRCam long-wavelength CRDS flat-field reference files were updated in 2023", 'The JWST UNCOVER Treasury Survey': "Table 4. UNCOVER Data Release Schedule \nMay ( jwst 1084.pmap ) based on in-flight data; however, we use the custom sky flats that were computed in a uniform way for all short- and long-wavelength filter/detector combinations. We apply an additional mask to the exposure data quality arrays to ignore any pixels where the sky flat is outside of the range [0.7, 1.4], essentially removing approximately one out of four images of the same region on sky due to the dither pattern. Any masked pixels will not contribute to the mosaic and the total effective exposure time at those positions will therefore decrease. On average, 3.7% of pixels are masked for one reason or another in the longest UNCOVER exposures (837 s in F277W), of which ≤ 2.8% come from the snowball masking. After applying the flat-field calibration, we fit and subtract the additive 'wisp' structure in some of the short-wavelength filter/detector combinations (Rigby et al. 2023) using wisp templates 2 derived from the GO-2561 UNCOVER data themselves. \nThe final step of the exposure-level processing is the photometric calibration, which was a rapidly-evolving topic of discussion at the time the UNCOVER data were taken in late 2022 (Boyer et al. 2022). This discussion was largely resolved with calibrations derived from in-flight data and provided by CRDS jwst 0989.pmap . However, we adopt a residual relative correction of 0.94 for the F277W filter in the NIRCam A module based on observations from the PRIMER program (GO-1837, PI: \nDunlop) where the same sources were observed in the same filter on both detectors 3 . \nAfter the JWST-specific corrections to the exposures described above, the grizli pipeline processing is essentially identical to the procedure adopted earlier for data from the Hubble Space Telescope (e.g., Kokorev et al. 2022). The most important step of this processing is the image alignment, which is performed in two steps. The first computes shift translations based on the positions of sources identified in each individual exposure; these shifts tend to be small ( ≲ 0 . 1 NIRCam pixel) as the pointing control and small offsetting of the telescope are quite precise (Rigby et al. 2023). The absolute alignment is performed by aligning (shift and rotation) the UNCOVER F444W exposures to sources in the groundbased NOAO LegacySurvey DR9 catalog (Dey et al. 2019). We have verified that with this procedure the UNCOVER mosaic is aligned to the absolute frame defined by Gaia DR3 (Gaia Collaboration et al. 2023) at a level of 12 mas (Weaver et al. 2024). We then create a source list from the aligned F444W mosaic to which the exposures in all of the other filters are aligned. The background pedestal level of each exposure are computed with the AstroDrizzle software (Gonzaga et al. 2012), which is then also used to create the final mosaics of each filter drizzled to a common pixel grid with \nFigure 6. JWST imaging mosaic of Abell 2744, centered on the UNCOVER NIRCam mosaic. We highlight the three observed cluster cores in inset panels; only the Southern primary cluster has been covered by HST imaging in the field. Our imaging of the Northern substructures reveals a multitude of lensed features, that are used to improve the lens modeling in that region (L. Furtak et al., in preparation). This color image combines all UNCOVER filters and includes other JWST NIRCam/NIRISS imaging in the field including GLASS (Treu et al. 2022) and DDT#2756. High resolution version of this color image is available on the UNCOVER website. \n<!-- image --> \ndrizzle parameters pixfrac = 0 . 75, kernel = square . 4 AstroDrizzle produces a 'weight' image that has units of inverse variance of the 'science' images and is made by propagating the total err noise model of each input exposure through the drizzle resampling. The weight map does not account for correlated noise of the finite drizzle pixel resampling (e.g., Casertano et al. 2000). We also compute mosaics of all of the historical Hubble imaging of the Abell 2744 field, drizzled to the same pixel grid and with datasets described by Kokorev et al. (2022) and references therein.", '3.2. DR1: Initial Release of NIRCam/NIRISS Mosaics': 'With this publication, we publicly release fully reduced, multiband mosaics for the first NIRCam cluster and NIRISS parallel imaging with 20 mas pixels in NIRCam short-wavelengths and 40 mas pixels in the NIR- \nCam long-wavelength filters and NIRISS imaging. Color images for each mosaic are shown in Figure 6 (NIRCam) and Figure 7 (NIRISS). We also provide reduced HST imaging on the same WCS grids. All of these data products will be hosted initially through Amazon Web Services, linked from the UNCOVER team website 5 , and upon publication uploaded to the Barbara A. Mikulski Archive for Space Telescopes (MAST). This data release is designed to be significantly in advance of the JWST Cycle 2 proposal deadline. Details of this plan can be found in Table 4. \nOur team intends to incrementally release higher-level data products associated with these early data (e.g., intracluster light (ICL)-subtracted mosaics, photometric catalogs and derived properties, updated gravitational lensing maps) on a short timescale. These data releases will be followed by final catalogs and spectroscopic data releases in the subsequent years. \nFigure 7. Color image of the UNCOVER NIRISS parallel, which overlaps with the Hubble Frontier field parallel. \n<!-- image -->', '4. DISCUSSION AND SCIENTIFIC OBJECTIVES OF THE UNCOVER SURVEY': "The JWST UNCOVER survey is designed to address a primary objective of the observatory: detecting and spectroscopically characterizing the properties of the first galaxies, while enabling a broad range of scientific explorations. Broadly, our scientific targets fall into four primary categories. \n- · The First Galaxies: The first JWST observations released in July 2022 quickly transformed the field. Early Release Observations in SMACSJ0723.37327 (ERO #2736, PI: Pontoppidan, Pontoppidan et al. 2022) and Early Release Science observations from GLASS (#1324, PI:Treu, Treu et al. 2022) and CEERS (#1345, PI:Finkelstein Bagley et al. 2023) revealed an unexpected abundance of bright galaxies at 10 < z < 17 (Naidu et al. 2022a; Donnan et al. 2023; Atek et al. 2023; Castellano et al. 2022; Harikane et al. 2023; Finkelstein et al. 2022; \nFurtak et al. 2023a) and a population of candidate massive ( > 10 10 M ⊙ ) galaxies at 7 < z < 10 (Labb'e et al. 2023), which may or may not be in tension with galaxy formation model predictions (Mason et al. 2023; Ferrara et al. 2023; BoylanKolchin 2023; Nath et al. 2023). The NIRCam imaging from UNCOVER extends these early efforts by providing deep 3.7 hour ( ∼ 30 AB F200W) imaging over a large area with gravitational lensing ( µ > 2 over > 17 arcmin 2 ), ideally suited to improve statistics of galaxies at z > 10 (see Table 2). The ultradeep 2 . 7 -20 hour R ∼ 100 PRISM spectra can provide spectroscopic continuum redshifts of any JWST -selected galaxy to 29AB, unbiased with respect to the presence of emission lines or targeting, providing the first comprehensive test of the photometric selection at these redshifts. The early UNCOVER spectra revealed a number of stunning examples of these incredibly \nFigure 8. The UNCOVER imaging reveals spectacular views of a vast array of galaxies across cosmic time, shown here as drawn from the history of the cosmic star formation rate density from Casey et al. (2018). These galaxies span foreground galaxies within the Abell 2744 cluster, to red and/or remarkably resolved lensed systems at Cosmic Noon (sub-mm galaxy A2744-ID02 from Gonz'alez-L'opez et al. (2017) and a quiescent galaxy candidate), and back into the most extreme - and therefore uncertain - epochs to the time of the first galaxies (GLASSz8-1 from Roberts-Borsani et al. (2022) and triply-lensed JD1 from Zitrin et al. (2014)). White scale bars indicate 0.5' in all postage stamps. \n<!-- image --> \ndistant galaxies (Wang et al. 2023; Fujimoto et al. 2023b). \n- · The Reionization Era: Understanding how and when galaxies irradiated their environment and reionized the surrounding neutral gas is a key outstanding question. To determine the contribution of galaxies to reionization requires a full accounting of the faintest galaxies that dominate the integrated UV light, the rate at which they produce ionizing photons, and the fraction of photons that escape. UNCOVER imaging will securely detect galaxies to M UV ∼-14 and should settle the issue of the potential turnover (e.g., Bouwens et al. 2022b), directly measuring 85% of the reionizing UV radiation (e.g., Livermore et al. 2017, and references therein). The 3 × spatial resolution improvement of NIRCam versus HST /WFC3 better constrains the size distribution of the galaxy population, further mitigating systematic sources of error. Ultra-deep PRISM spectra enables for the study of ionizing spectra of the faintest galaxies (down to M UV ∼-15) (Atek et al. 2024). Of the expected ∼ 1500 galaxies at z =6 -7, we estimate about ∼ 40 will have intrinsic UV magnitudes of M UV > -16 but still be bright enough ( < 28AB) to \nget high quality spectra. For these galaxies we will observe the full complement of strong optical lines ([OII], [OIII], H β , H α , [NII], [SII], with H α and [NII] marginally resolved at R ∼ 300 at ∼ 5 µ m), to study ISM properties, ages, metallicities, and ionization mechanisms. The combination of imaging and spectra has the potential to be a quantum leap in studies of reionization era galaxies. \n- · The Emergence of Dusty Galaxies: The fraction of star formation missed due to dust at z> 4 and at low luminosities remains unclear. ALMA has discovered dusty galaxies with relatively low masses and SFRs already at z > 5 (e.g., Yamaguchi et al. 2019; Wang et al. 2019; Williams et al. 2019; Fudamoto et al. 2021; Algera et al. 2023; Dayal et al. 2022), suggesting that dust obscured star formation may be more prominent than was expected. These highly obscured extreme starbursts known to exist may represent only the tip of the iceberg. Detection of starlight from ' HST -dark' galaxies was expected in deep JWST imaging; early imaging has indeed revealed a substantial population (Barrufet et al. 2023; Nelson et al. 2023; P'erezGonz'alez et al. 2023; Price et al. 2023) of extreme red, dusty galaxies extending to z ∼ 7 -8. \nUNCOVER will place strong constraints at the lowest masses and the highest redshifts, complementary to the JWST wide-field programs (e.g., COSMOS-Web (GO #1727, PI: Kartaltepe and Casey; Casey et al. 2023), PANORAMIC (GO #2514, PI: Williams and Oesch)) capable of finding massive obscured galaxies over larger volumes. The ability to detect any highly obscured M ⋆ > 10 9 M ⊙ galaxy to z =9 with NIRCam and test photometric stellar population inferences with NIRSpec spectroscopy will further expand our census of star formation. \n- · The Epoch of Quenching: Quenching of star formation is a complex phenomenon, with different processes dominating over a range of times, mass scales, and environments. The earliest massive quiescent galaxies have been spectroscopically confirmed to redshift z ∼ 4 (e.g., Glazebrook et al. 2017; Schreiber et al. 2018; Forrest et al. 2020). However, this is entirely limited by detection: quiescent galaxies almost certainly exist to lower masses and higher redshift beyond the capabilities of HST and ground-based spectroscopy. The first JWST imaging has already provided spectacular candidates of massive, quiescent galaxies to redshift ∼ 5 (Carnall et al. 2022). UNCOVER NIRCam imaging will extend the search for quiescent galaxies to unprecedented low mass and high redshift, i.e., < 10 9 M ⊙ to z =9. Marchesini et al. (2023) demonstrated the presence of 10 10 M ⊙ quiescent galaxies to z ∼ 2 . 5 with early NIRISS spectra. The UNCOVER NIRSpec/PRISM spectroscopy will be able to push this to even lower masses and higher redshifts due to the improved depth and wavelength coverage to ∼ 5 . 3 microns to confirm the quiescence through the absence of emission lines and strong Balmer breaks (e.g., Setton et al. 2024). Beyond identification and counting, the PRISM spectroscopy will also be deep enough to further constrain stellar populations to F 444 W ≲ 27 . 5AB from the stellar continuum.\n- · The Unknown Unknowns: Perhaps the most exciting legacy of deep fields like UNCOVER is the potential to discover objects that we have not yet imagined or identify predicted objects that we never hoped to detect in addition to offering possible constraints on the non-CDM nature of dark matter (e.g., Dayal et al. 2015). The advance in sensitivity, wavelength coverage, and spectral resolution is so large that we will certainly run into surprises. The UNCOVER treasury program en- \nres a comprehensive exploration of ultra-deep parameter space with public imaging and spectroscopy released early in JWST 's mission. An extraordinary example of this fruitful discovery space has been the abundance of actively accreting black holes that have been discovered and studied within the UNCOVER imaging (e.g., Furtak et al. 2023c; Greene et al. 2024) and spectroscopy (e.g., Furtak et al. 2024; Goulding et al. 2023; Kokorev et al. 2023). \nThe astronomical community has only begun to scratch the surface of possible JWST observations of the distant Universe. This paper aims to present a basic description of the UNCOVER Treasury program to optimize community use of early data products, prepare for Cycle 2 JWST proposals targeting objects in and behind the Abell 2744 cluster, and anticipate use of the scheduled NIRSpec/PRISM follow-up. These studies are likely to range from detailed analysis of previously known sources to exciting JWST discoveries. We conclude this paper by highlighting the broad range of UNCOVER scientific targets in Figure 8. Individual objects are plotted against the backdrop of dust-rich and dust-poor models for the evolution of the star formation rate density from (Casey et al. 2018). \nEven in Cycle 1, the community had access to a wide range of exciting JWST datasets probing the distant Universe. Some of these programs will enable detailed high-resolution studies of extremely lensed sources, as in TEMPLATES (ERS # 1355, PI: Rigby and Viera). The JWST -GLASS program (ERS # 1324, PI: Treu) provides an in-depth and multi-pronged view of reionization in the Abell 2744 cluster (Treu et al. 2022). The CEERS program provides a range of relatively shallow imaging and spectroscopy across the well-studied CANDELS/EGS field (ERS #1324, PI: Finkelstein; Finkelstein et al. 2023). Other GO programs will expand imaging footprints: PRIMER (GO #1837, PI: Dunlop) will map nearly 700 sq. arcmin with 10 NIRCam and MIRI filters (236 sq. arcmin in MIRI) and COSMOS-Web will map an incredible 0.6 square degrees in NIRCam and 0.2 square degrees in MIRI with a smaller number of filters (GO #1727, PI: Kartaltepe and Casey; Casey et al. 2023). Finally, the pure parallel program PANORAMIC (GO #2514, PI: Williams and Oesch) is planned to provide multi-band NIRCam imaging of spatially uncorrelated pointings covering up to 0.4 square degrees with a range of depths. \nThe UNCOVER program explores a unique parameter space; reaching within ∼ a magnitude of the GTO ultradeep field but boosted by gravitational lensing and with no proprietary access. Although this is just the first \nof several planned public data releases, our team looks forward to seeing what the community will uncover from this and all other JWST treasure troves. \nRB acknowledges support from the Research Corporation for Scientific Advancement (RCSA) Cottrell Scholar Award ID No: 27587. Cloud-based data processing and file storage for this work is provided by the AWS Cloud Credits for Research program. This work is based in part on observations made with the NASA/ESA/CSA James Webb Space Telescope . The data were obtained from the Mikulski Archive for Space Telescopes at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-03127 for JWST . These observations are associated with JWST Cycle 1 GO programs #2561 , JWST-ERS-1324, JWST-DD-2756. Support for program JWST-GO-2561 was provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Associations of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555. This research is based on observations made with the NASA/ESA Hubble Space Telescope obtained from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. These observations are associated with programs HST-GO-11689, HST-GO-13386, HST-GO/DD-13495, HST-GO-13389, HST-GO-15117, and HST-GO/DD-17231. The Cosmic Dawn Center is funded by the Danish National Research Foundation (DNRF) under grant #140. This research was supported in part by the University of Pittsburgh Center for Research Computing, RRID:SCR 022735, through the resources provided. Specifically, this work used the H2P cluster, which is supported by NSF award number OAC-2117681. LF and AZ acknowledge support by Grant No. 2020750 from the United States-Israel Binational Science Foundation (BSF) and Grant No. 2109066 from the United States National Science Foundation (NSF), and by the Ministry of Science & Technology, Israel. PD acknowledges support from the NWO grant 016.VIDI.189.162 ('ODIN') and from the European Commission's and University of Groningen's CO-FUND Rosalind Franklin program. HA acknowledges support from CNES (Centre National d'Etudes Spatiales). RS acknowledges an STFC Ernest Rutherford Fellowship (ST/S004831/1). MS acknowledges support from the CIDEGENT/2021/059 grant, from project PID2019109592GB-I00/AEI/10.13039/501100011033 from the Spanish Ministerio de Ciencia e Innovaci'on - Agencia Estatal de Investigaci'on, and from Proyecto ASFAE/2022/025 del Ministerio de Ciencia y Innovaci'on en el marco del Plan de Recuperaci'on, Transformaci'on y Resiliencia del Gobierno de Espa˜na. The work of CCW is supported by NOIRLab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. \nFacilities: JWST (NIRCam, NIRSpec, and NIRISS), HST (ACS and WFC3) \nBrammer & Matharu 2021, Prospector (Johnson et al. 2021), DrizzlePac (Gonzaga et al. 2012) \nSoftware: astropy (Astropy Collaboration et al. 2013, 2018, 2022), Bagpipes (Carnall et al. 2018), Grizli"}

2024arXiv240907063P

This work continues the research presented in the article 1 where we estimate the Gertsenshtein effects influence on the longwavelength part of relic gravitational wave spectrum. Here the differential equation system for the Gertsenshtein effect in FriedmanLeMaitreRobertsonWalker universe derived in 1 is simplified for gravitational waves in the underhorizon regime during radiation dominance epoch. Then the obtained system is solved analytically. As a result of the solution analysis a conclusion was made about a significant increase of relic GWs with the frequencies kgtrsim 1011 Hz for magnetic field strength about 1 nGs. In addition at the end of the article model dependency of the result is discussed

2024-09-01T00:00:00Z

['arXiv:2409.07063', '2024arXiv240907063P', '10.48550/arXiv.2409.07063']

['General Relativity and Quantum Cosmology', 'Astrophysics - Cosmology and Nongalactic Astrophysics']

Conversion of high frequency relic gravitational waves into photons in cosmological magnetic field

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.07063.pdf

{'Conversion of high frequency relic gravitational waves into photons in cosmological magnetic field': 'L. A. Panasenko a , A. O. Chetverikov a,b \nSeptember 12, 2024 \na \nDepartment of Physics, Novosibirsk State University, Pirogova 2, Novosibirsk 630090, Russia \nb Voevodsky Institute of Chemical Kinetics and Combustion, Siberian Branch of the Russian Academy of Science, 3, Institutskaya St., Novosibirsk 630090, Russia', 'Abstract': "This work continues the research presented in the article [1], where we estimate the Gertsenshtein effect's influence on the long-wavelength part of relic gravitational wave spectrum. Here, the differential equation system for the Gertsenshtein effect in Friedman- LeMaitre-Robertson-Walker universe, derived in [1], is simplified for gravitational waves in the under-horizon regime during radiation dominance epoch. Then, the obtained system is solved analytically. As a result of the solution analysis a conclusion was made about a significant increase of relic GWs with the frequencies k ≳ 10 -11 Hz for magnetic field strength B 0 ∼ 1 nGs. In addition, at the end of the article model dependency of the result is discussed.", '1 Introduction': "In the article [1] the equation system was derived for the conversion of gravitational wave (GW) into electromagnetic wave (EMW) in the presence of external magnetic field in curved space-time. After that, influence of that phenomenon on relic gravitational wave amplitude was estimated for low frequencies (10 -18 -10 -16 ) Hz and the present day cosmological magnetic field amplitude 1 nGs. \nIn order to make the study [1] complete and consistent, it is important to check whether the influence of the Gertsenshtein effect on high-frequency gravitational waves is negligible. Indeed, if there is a significant enhancement or suppression of relic gravitational waves with high frequencies, we may have to change the predictions about their detectability in the present Universe for different inflation models. \nIn the previous work we divided wave vector k into two components: k || B and k ⊥ B , where B is a cosmological magnetic field vector. Connection between metric perturbation and electromagnetic wave takes place only for perpendicular part of the wave vector. Moreover, separated systems were obtained for h × - and h + -polarizations of metric perturbation tensor. In the manuscript we will focus on the first polarization. \nA short discussion of results for h + -polarisation is presented in Appendix A. The main conclusion is that there is no influence of the Gertsenshtein effect on gravitational wave amplitude for h + -polarisation at all. But there may be generation of electromagnetic wave potential f y and generation of scalar metric perturbations Φ and Ψ. \nAll in all we conclude, that the Gertsenshtein effect changes the amplitude of gravitational wave only for the h × -polarization of the component propagating perpendicular to magnetic field B . This is a well known result from works, where the Gertsenshtein effect was considered in the flat static space-time. For example non-cosmological problems are considered in [2-9] and simple evaluations for cosmology and astrophysics are made in [10, 11]. \nTo estimate the influence of the Gertsenshtein effect on relic gravitational waves with high frequencies let us, firstly, remind the previous results. In FLRW metric in the case k ⊥ B and with taking into account interaction with a primary plasma during RD era from eq.(207) in [1] we have: \nf x : a 2 H 2 f x '' + aH 2 [ 1 + a H ' H +8 2 B 2 0 C 0 -a 4 16 B 2 0 C 0 -a 4 + aH Γ ] f x ' + + [ k 2 a 2 +2 aHH ' -8 H 2 4 B 2 0 C 0 + a 4 16 B 2 0 C 0 -a 4 +2Γ H + ω 2 pl ] f x -γβH 2 ( af x ' +2 f x ) = -ikB 0 a 4 m pl h × , h y x : a 2 H 2 h '' × + ( 4 aH 2 + a 2 HH ' ) h ' × + [ k 2 a 2 + 16 πGB 2 0 a 4 ( 1 -4 B 2 0 C 0 a 4 )] h × = = -16 πGB 0 ik a 2 ( 1 -16 B 2 0 C 0 a 4 ) m pl f x , (1.1) \nwhere prime means derivative over scale factor a . Denotations are the following: h x y = h × - the { x,y } component of the metric perturbation tensor, which is also the h × -polarization for the case when initial gravitational wave propagates along z axis; f x is an x -component \nof electromagnetic wave potential; H = ˙ a a - Hubble parameter; B 0 - magnitude of cosmological (intergalactic) magnetic field in the present Universe 1 ; G = 1 m 2 pl - gravitational constant; m pl - Planck mass; k - wave number; C 0 = α 2 90 m 4 e - constant in the HeisenbergEuler action (see [1]), where m e - electron mass, α - fine structure constant. We assume that magnetic field is homogeneous 2 and directed along x axis and B 0 = 1 nGs. \nIt is important to note, that previously we have taken into account the first loopcorrection to the Maxwell action. This correction is proportional to the product CB 2 and we accept, that it plays a significant role under the followong condition: CB 2 ≳ 10 -5 . This gives the condition for the magnetic field strength: B ≳ 10 10 Gs. The approximation of the effective Heisenberg-Euler action works for B ≪ m 2 e ∼ 10 13 Gs. Therefore we ultimately accept the following limits of applicability of the conversion effect estimates using the full system (1.1): \n10 10 Gs ≦ B ≦ 10 13 Gs . (1.2) \nFor a field strength less than 10 10 Gs the system (1.1) is still be correct, but calculating the terms from the loop-correction is redundant. \nWe solve the equations for 10 -9 ≤ a ≤ 10 -4 . In the chosen interval of the scale factor, the homogeneous magnetic field strength relative to the present value will be amplified by 10 8 -10 18 times. Thus, for the upper estimate of the effect on the amplitude of relic GWs, B 0 = 1 nG, the maximum value of the magnetic field strength is 10 9 Gs. Thus, the effect of the creation of a virtual electron-positron pair for the uniform field model can be neglected over the entire solution interval (see Eq. (1.2)). \nThus, the Eq. (1.1) can be written in terms of time in the following form: \n [ ∂ 2 t +3 H∂ t + ( k 2 a 2 -8 πGB 2 0 a 4 )] h × = -ik 16 πGB 0 a 2 f x , [ ∂ 2 t +3 H∂ t + k 2 a 2 ] f x = -ikB 0 2 a 4 h × . (1.3) \nwhere we neglect all the terms, originated from the Heisenberg-Euler Lagrangian, and for a while omit the terms responsible for the interaction with the plasma (we will consider them further). \nThe article has the following structure. In the next section the system of differential equations (1.3) is simplifyed for the high frequency limit, is made dimensionless and, finally, reduced to one differential equation of the forth order. Sec.3 is devoted to the analytical solution of the system. In Sec.4 we analyze the solution and evaluate the Gertsenshtein effect influence on relic GWs amplitudes for different frequencies and magnetic field strengths. Finally, in Sec.5, 6 we discuss the impact of the magnetic field coherence length and introduce interaction with the primary plasma respectively. In conclusion we resume the work.", '2 Simplification of the system': "We can neglect the term 8 πGB 2 0 a 4 under the following condition: \nka ≫ √ 8 π B 0 m pl ≈ 6 , 1 × 10 -23 Hz . (2.1) \nIn order evaluate the effect influence we need to specify the problem. Further we will consider radiation domimamce (RD) epoch from the so called hadron epoch: the scale factor a 1 = 10 -9 . Thus, we suggest that cosmological magnetic field was generated during the QCD phase transition 3 . \nValue of the multiplication ka is minimal at a 1 . Hence the condition of negligibility of the second term in round brackets on the left side of the first line of Eq. (1.3): \nk ≫ 6 , 1 × 10 -14 Hz . (2.2) \nFor values a > a 1 the right-hand side of the Eq.(2.2) only increases, so the condition works on the entire solution interval. \nNow let us make the replacement \n˜ h ≡ a ( η ) h × ( η ) , ˜ f ≡ a ( η ) f x ( η ) (2.3) \nand to rewrite the Eq.() in terms of conformal time η . We obtain \n˜ h '' -a '' a ˜ h + k 2 ˜ h = -ik 16 πGB 0 ˜ f, ˜ f '' -a '' a ˜ f + k 2 ˜ f = -ikπB 0 2 a ( η ) 2 ˜ h. (2.4) \nHere and below, the prime denotes the derivative with respect to η . \nIn the manuscript by high frequencies we mean frequencies satisfied the condition k ≫ a ' /a . Gravitational waves with such frequencies are under-horizon mode for the considered interval of scale factor values: we can neglect the term proportional to a '' a . As a result, the final system has a very simple form of two coupled pendulums with a variable inhomogeneous part. Indeed, we have \n˜ h '' + k 2 ˜ h = -ik 16 πB 0 m pl ℏ ˜ f, ˜ f '' + k 2 ˜ f = -ik 2 πB 0 τ 2 0 m pl ℏ η 2 ˜ h. (2.5) \nHere we replace dimentional electromagnetic potential by dimentionless ˜ ˜ f x ≡ ˜ f x /m pl , immediately omitting the second tilde. Also we use that for RD epoch from adη = dt it \nfollows that η = 2 √ τ 0 t = 2 τ 0 a , where τ 0 = 35 τ is a devider in the scale factor exprassion a = √ t τ 0 , τ - the Universe lifetime. The frequency k is substituted in hertz, therefore, for the accuracy of the dimensional matching, the Planck constant was returned. \nIn fact, for the RD epoch the term a '' a = 0 during all the solution interval. That means that the system of Eqs.(2.5) is correct for lower frequencies. Thus, the only condition we have to satisfy is the condition of Eq.(2.2) \nNote also that the presence of an imaginary unit in the right-hand sides of the equations (2.5) means that the phases of gravitational and electromagnetic waves are shifted relative to each other by π/ 2. This is expected since at the initial moment of time only the gravitational wave is present, and the amplitude of the electromagnetic wave is zero. \nLet's discuss the solution plan. First, we need to obtain a fourth-order equation for one of the functions ˜ f x ( η ) , ˜ h ( η ) from the equation system (2.5). After that, we look for a solution in the form of an integral function of a complex variable. At the final stage of solution it is important to choose an optimal integration contour, which gives the simplest form of the solution. \nThe reader can immediately see that the solution will be a sinusoid with some modulating function. \nWe will derive the unknown coefficients of the solutions from the initial conditions. For arbitrary initial values of h ( η 1 ) ≡ h 1 and h ' ( η 1 ) ≡ h ' 1 we have \n˜ h ( η 1 ) = a 1 h 1 , ˜ h ' ( η 1 ) = h 1 2 τ 0 + a 1 h ' 1 , ˜ f ( η 1 ) = 0 , ˜ f ' ( η 1 ) = 0 . (2.6) \nSo we have assumed that electromagnetic wave is absent at the initial time. It is nessessary to explaine, why there can exist non-zero first derivative h ' 1 at η 1 : different frequencies enter under horizon at different times 4 . After the entry the dependency of GW amplitude roughly changes from constant (beyond-horizon mode) to ˜ h ( η ) = ˜ h 0 a ( η entr ) a ( η ) cos ( kη + ϕ entr ) [12]. Here h 0 - relic GW amplitude at the end of inflation and ϕ entr - the phase, calculated from matching with constant mode at entry conformal time η entr : \n˜ h 0 cos ( kη entr + ϕ entr ) = ˜ h 0 . (2.7) \nAs a result, for different k , at the point η 1 we can obtain different phases ( kη 1 + ϕ entr ) in cosine in ˜ h 1 = ˜ h 0 cos ( kη 1 + ϕ entr ) and sine in ˜ h ' 1 = -˜ h 0 k sin ( kη 1 + ϕ entr ). The approximate time of entry can be calculated from the condition \nkη entr ∼ 1 . (2.8) \nAs will become clear below, the frequency dependence does not change when the Gertsenshtein effect is taken into account. So for simplicity of presentation, we do not write \nthese constant phases explicitly and present the result in terms of fraction of the amplitude obtained in the conventional theory of the tensor perturbations evolution. \nTo simplify further calculations, we introduce constants \nC 1 ≡ -ik 16 πB 0 m pl , C 2 ≡ -ik 2 πB 0 τ 2 0 m pl . (2.9) \nThen, the system of equations (2.5) takes the form \n˜ h '' + k 2 ˜ h = C 1 ˜ f, ˜ f '' + k 2 ˜ f = C 2 η 2 ˜ h. (2.10) \nThe first constant is dimensional, the second - dimensionless. So the right-hand sides of the equations, like all the terms on the left-hand side, have the dimension sec -2 . \nNow we differentiate the first equation of the system (2.10) twice and obtain \n˜ h '''' +2 k 2 ˜ h '' + k 4 ˜ h = C 1 C 2 η 2 ˜ h. (2.11) \nIt will also be convenient to non-dimensionalize the function argument. Let us introduce \nx ≡ √ C 1 C 2 η ≈ 3 i × 10 -3 k η. (2.12) \nReplacing all derivatives with respect to η with derivatives with respect to x , we obtain a fairly simple expression \nd 4 ˜ h dx 4 +2 γ d 2 ˜ h dx 2 + γ 2 ˜ h = ˜ h x 2 , (2.13) \nwhere we defined \nγ ≡ k 2 C 1 C 2 = -m 2 pl 32 π 2 B 2 0 τ 2 0 ≈ -0 . 9 × 10 5 . (2.14) \nWe emphasize that γ does not depend on frequency k and is less than zero.", '3 Analytical solution': "In this section an approach is discussed to determining the analytical solution to Eq.(2.13) by integral representation. \nConsidering Eq.(2.12) and Eq.(2.14), let us redefine the variables x = | x | and γ = | γ | . Then Eq.(2.13) becomes: \nd 4 ˜ h dx 4 +2 γ d 2 ˜ h dx 2 + ( γ 2 + 1 x 2 ) ˜ h = 0 . (3.1) \nLet us find a solution of this equation in the following form: \n˜ h ( x ) = ∫ c Z ( y ) e xy dy, (3.2) \nwhere c under the integral means an integration contour, along which the integrand is holomorphic. Z ( y ) - a function of a complex variable y . \nBy substituting Eq.(3.2) into Eq.(3.1) we obtain \n∫ c Z ( y ) [ F ( y ) x 2 +1 ] e xy dy = 0 , F ( y ) = ( y 2 + γ ) 2 . (3.3) \nIt is necessary to exclude dependence on x . To do it, we integrate by parts twice \n∫ c ( d 2 ( ZF ) dy 2 + Z ) e xy dy + xZFe xy | ∂c -d ( ZF ) dy e xy | ∂c = 0 . (3.4) \nBased on the initial assumption that neither the integration contour nor the function Z ( y ) depends on x , the only way to eliminate the last two terms in Eq.(3.4) is to set to zero each of them. One can satisfy this condition by choosing an appropriate contour. \nFinally, we have an equation for Z ( y ): \nd 2 ( ZF ) dy 2 + Z = 0 . (3.5) \nand two solutions are: \nZ 1 , 2 ( y ) = A 1 , 2 ( y + i √ γ ) -3 2 ± β 2 ( y -i √ γ ) -3 2 ∓ β 2 , (3.6) \nwhere \nβ = √ 1 + 1 γ , (3.7) \nA 1 and A 2 - arbitrary constants. This function is meromorphic on the complex plane with a branch cut between the points ± i √ γ . \nNow we need to choose an appropriate integration contour to set two integration constant in Eq.(3.4) to zero. \nThe first option is a closed contour. However, if this contour does not conclude singularity points y = ± i √ γ of Z 1 , 2 ( y ), then ˜ h ≡ 0, which is a trivial solution to Eq.(3.1). \nThe second option is a closed contour enclosing an area containing one or both of these points. In this case, the integral equals the sum of residues at these points. Figure 1 shows an example of such a contour c 1 . Then for the function ˜ h , up to an arbitrary constant, we have: \n˜ h ± ( x ) = ∮ c 1 ( y + i √ γ ) -3 2 ± β 2 ( y -i √ γ ) -3 2 ∓ β 2 e xy dy. (3.8) \nTo calculate the integral in Eq.(3.8), we expand the integrand into a Laurent series: \n( y + i √ γ ) -3 2 ± β 2 ( y -i √ γ ) -3 2 ∓ β 2 e xy = y -3 ∞ ∑ k =0 c ± k ( i √ γ y ) k ∞ ∑ m =0 ( xy ) m m ! , c ± k = Γ ( ± β 2 -1 2 ) 2 F 1 ( ± β 2 + 3 2 , -k ; ± β 2 -k -1 2 ; -1 ) Γ ( ± β 2 -k -1 2 ) k ! , (3.9) \nwhere Γ is the Gamma function and F - the Hypergeometric function. Both series are convergent, since | y | > √ γ on the integration contour. \nNext, we replace the variables by y = re iϕ , where ϕ runs from 0 to 2 π , r is a constant radius ( r > √ γ ). Then for Eq.(3.8) we obtain: \n˜ h ± ( x ) = ∫ 2 π 0 e -2 iϕ r 2 ∞ ∑ k =0 c ± k e -ikϕ ( i √ γ r ) k ∞ ∑ m =0 ( xr ) m m ! e imϕ dϕ = 2 πx 2 ∞ ∑ n =0 c ± n ( i √ γ x ) k ( n +2)! . (3.10) \nIn the resulting series, the terms can be regrouped in such a way as to isolate the exponent and obtain solutions in the form: \n˜ h 1 , 2 ( x ) = B 1 , 2 e ± i √ γx x 2 1 F 1 ( 3 2 + β 2 , 3 , ∓ 2 i √ γ x ) , (3.11) \nwhere B 1 , 2 are arbitrary constants. \nExpression (3.11) can be obtained from Eq.(3.8) directly if we shift to one of the function branch points, go to integer powers and bypass this point on all sheets of the Riemann surface. Such a procedure is much more complicated from the point of view of mathematical justification, but immediately provides an answer in the form of Eq.(3.11). \nTwo more linearly independent solutions can be obtained by taking as the integration contour in Eq.(3.2) the straight line from the point y 1 = i √ γ to y 2 = -∞ + i √ γ ( c 2 on the Figure 1). At these points, the terms Z 2 Fe xy and dZ 2 F dy e xy turn to zero, because the degree of the monomial ( y -iγ ) remains positive, even after taking the derivative, since β > 1. \nAs a result, for the desired function we have: \n˜ h ( x ) = ∫ -∞ + i √ γ i √ γ ( y + i √ γ ) -3 2 -β 2 ( y -i √ γ ) -3 2 + β 2 e xy dy = = e i √ γx ∫ -∞ 0 ( y +2 i √ γ ) -3 2 -β 2 y -3 2 + β 2 e xy dy. (3.12) \nThis integral, by differentiation by parts, reduces to the integral representation of the Tricomi function U ( 3 2 + β 2 , 3 , -2 i √ γx ) . This function is the second linearly independent solution of Kummer's equation. Thus: \n˜ h 3 , 4 ( x ) = B 3 , 4 e ± i √ γx x 2 U ( 3 2 + β 2 , 3 , ∓ 2 i √ γx ) . (3.13) \nIt is worth noting that reducing the original Eq.(3.1) to Kummer's equation, using substitution ˜ h ( x ) = e ± √ γx x 2 H ( x ), where H ( x ) - unknown function, is not a trivial task. \nAt the same time, the approach described above with searching for solutions in integral form made it easy to find all the solutions. \nFigure 1: Two contours c 1 , c 2 for solution in the integral form Eq.(3.2), which satisfy the conditions of the integration constants vanishing in Eq.(3.4) and give linearly independent solutions. Bold line shows the branch cut. \n<!-- image -->", '4 Analysis of the solution': 'Let us analyze the obtained solutions Eqs.(3.11, 3.13). The behavior of the amplitudes of these two pairs of solutions is determined by the following asymptotes: \n˜ h 1 , 2 ( x ) ∝ x 1 2 + β 2 , ˜ h 3 , 4 ( x ) ∝ x 1 2 -β 2 . (4.1) \nThe first group gives rising functions, the second - dropping down functions, because β > 1 by definition in Eq.(3.7). \nWhen β is an odd integer, the Kummer and Tricomi functions reduce to simpler forms. For example, when γ = 1 / 8 ( β = 3), the derived expressions allow the solution of the original equation to be written in the form: \n˜ h γ =1 / 8 ( x ) = x 2 e ix √ 8 ( B 1 + B 2 Γ ( -2 , ix √ 2 )) + x 2 e -ix √ 8 ( B 3 + B 4 Γ ( -2 , -ix √ 2 )) (4.2) \nThe final enhancement or attenuation on the finite interval x depends on the initial conditions, while they determine the ratio between corresponding constants B i . But it is important to note that γ depends on magnetic field strength only. Hence there is a common degree of the growth for all the frequencies. For example the quadratically increasing solution Eq.(4.2) corresponds to a magnetic field B 0 ≈ 8 . 5 · 10 -7 Gs. \nThis strength is larger, than the upper limit on the present day value, which is obtained from observational data and theoretical predictions [13]. But we will see below that in the \nrealistic models magnetic field strength decreases faster than 1 /a 2 during RD era [14]. Therefore the initial B 0 = 1 nGs leads in the models to the higher strengths for considered interval a ∈ [10 -9 , 10 -4 ]. Strictly speaking, for another model of magnetic field evolution we need to rewrite the system of Eqs.(1.1), but we can try to evaluate the qualitative behavior simply by increasing B 0 . We will conduct a more rigorous analysis of specific models in a future study.', '4.1 Solution in the limit β → 1': "The limit β → 1 is equivalent to the condition γ → ∞ or γ 2 ≫ 1 x 2 . It is clear, that for some relations between solution interval [ x 1 , x 2 ] and parameter γ we can neglect the term 1 /x 2 in the Eq.(3.1). Indeed γ does not depend on frequency and has a big absolute value (Eq.(2.14)), while η 1 = 2 τ 0 a 1 = 3 . 08 · 10 10 sec and, consequently from Eq.(2.12) x 1 ≈ 10 8 k [Hz]. As a result for the round brackets in Eq.(3.1) we have at the initial point x 1 : \n( γ 2 + 1 x 2 1 ) ≈ 0 . 8 · 10 10 + ( 10 -8 k [Hz] ) 2 . (4.3) \nThe general solution of the equation \nd 4 ˜ h dx 4 +2 γ d 2 ˜ h dx 2 + γ 2 ˜ h = 0 (4.4) \nhas the form \n˜ h = [ A 1 + A 2 x ] cos( √ γ x ) + [ A 3 + A 4 x ] sin( √ γ x ) , (4.5) \nwhere A i are integration constants, which can be found from the initial conditions. \nWe remind, that for free propagating GW the solution in expanding Universe is \n˜ h free ∼ cos( kη ) . (4.6) \nIn contrast, in Eq.(4.5), we see resonant terms, proportional to x cos( kη ). But the enhancement of the GW amplitude depends on the value of corresponding integration constants, that is, depends on the initial conditions. \nUsing initial conditions from Eq.(2.6) we obtain for the second and the third derivatives of ˜ h at the initial point \n˜ h '' ( x 1 ) = -˜ f ( x 1 ) C 2 -γ ˜ h ( x 1 ) , h ''' ( x 1 ) = -˜ f ' ( x 1 ) C 2 -γ ˜ h ' ( x 1 ) , (4.7) \nwhere \n˜ f ( x 1 ) = 0 , ˜ f ' ( x 1 ) = 0 . (4.8) \nIt is important to remind that we are working in variables x = | x | , γ = | γ | , starting with a Sec.3. \nNow we can find the constants A i explicitly, and the solution is \n˜ h = ( ˜ h ( x 1 ) + ˜ f ' ( x 1 ) 2 γC 2 ( x -x 1 ) ) cos( √ γ ( x -x 1 ))+ + ( ˜ h ' ( x 1 ) -˜ f ' ( x 1 ) 2 γC 2 -˜ f ( x 1 ) 2 γC 2 ( x -x 1 ) ) sin( √ γ ( x -x 1 )) √ γ . (4.9) \nTaking into account Eqs.(4.7, 4.8) it can be rewritten as \n˜ h = ˜ h ( x 1 ) cos( √ γ ( x -x 1 )) + ˜ h ' ( x 1 ) sin( √ γ ( x -x 1 )) √ γ (4.10) \nThere is no terms proportional to x in the equation for the considered initial conditions. Non-zero initial ˜ f x ( x 1 ) gives higher values of A 2 and A 4 . However, in the problem under consideration, the initial electromagnetic wave is absent. \nIn the next section we show what is the difference between the result in approximation β → 1 and the exact solution of Eq.(3.1). But here we stress that for very small magnetic field strengths there is no any change of the relic GW spectrum. That means, if precise measurements of intergalactic magnetic field give B 0 ≪ 1 nGs, then the effect is absent. \nAt the end of the section let us, using conditions Eq.(2.6, 4.7,4.8), to compose a system for four integration constants in the exact anlytical solution Eq.(3.11, 3.13) in a standard way: \n ˜ h ˜ h ' ˜ h '' ˜ h ''' ( x 1 ) = ˜ h 1 ( x 1 ) ˜ h 2 ( x 1 ) ˜ h 3 ( x 1 ) ˜ h 4 ( x 1 ) ˜ h ' 1 ( x 1 ) ˜ h ' 2 ( x 1 ) ˜ h ' 3 ( x 1 ) ˜ h ' 4 ( x 1 ) ˜ h '' 1 ( x 1 ) ˜ h '' 2 ( x 1 ) ˜ h '' 3 ( x 1 ) ˜ h '' 4 ( x 1 ) ˜ h ''' 1 ( x 1 ) ˜ h ''' 2 ( x 1 ) ˜ h ''' 3 ( x 1 ) ˜ h ''' 4 ( x 1 ) B 1 B 2 B 3 B 4 , (4.11) \nwhere ˜ h i are defined in Eqs.(3.11, 3.13).", '4.2 Results of the Cauchy problem solution': 'The Figure 2 shows the plots of analytical solutions of Eq.(3.1) and, for comparison, of the analytical solution for free propagating GW ˜ h free = ˜ h 0 · cos kη = ˜ h 0 cos √ γx and different parameters k . All the plots are normalized by the initial amplitude ˜ h 0 . \nFigure 2: GW amplitude dependence on x = 3 · 10 -3 kη ( a ) at the interval a ∈ [10 -9 , 10 -4 ] for B 0 = 1 nGs: blue - analytical solution of Eq.(3.1) according to Eqs.(3.11,3.13); green - analytical solution for the free propagating gravitational wave. Amplitude is normalized by the initial GW amplitude ˜ h 0 . High frequency oscillations are not distinguishable \n<!-- image --> \nAlmost the same picture as on the right panel of Figure 2 is obtained for all the frequencies k > 1 nHz: thirty percent amplification of the GWs amplitudes. \nWe, finally, have to consider B 0 as a free parameter of the task. Indeed, the exact magnetic field strength has not yet been measured. In addition, in realistic models magnetic field magnitude decreases faster then 1 /a 2 due to dissipation in the turbulent plasma. That means that maximum magnetic field magnitude and, consequently, a strength of the Gertsenshtein effect may be even higher for the considered interval a ∈ [10 -9 , 10 -4 ] 5 . \nFor example, for one of the models with the initial kinetic helicity of the plasma, the law of change of magnetic field magnitude during RD era is B ∝ 1 /a 3 [14]. This increases the magnetic field strength by five orders of magnitude on average (for the considered interval). Of course, for any model we must modify the system of Eqs.(1.1), and, where necessary, take into account the loop correction of light scattering on light. However, a naive estimate of the effect for a magnetic field, an order of magnitude greater than the initially accepted 1 nG, entails an increase in the amplitude of relic GWs by more than an order of magnitude for a frequency of 10 -10 Hz (Figure 3). \nFigure 3: GW amplitude dependence on x = 3 · 10 -3 kη ( a ) at the interval a ∈ [10 -9 , 10 -4 ] for B 0 = 0 . 1 nGs: blue - analytical solution of Eq.(3.1) according to Eqs.(3.11,3.13); green - analytical solution for the free propagating gravitational wave. Amplitude is normalized by the initial GW amplitude ˜ h 0 . High frequency oscillations are not distinguishable \n<!-- image --> \nHere we obtain an increase of more than an order of magnitude for GWs with k > 10 -12 Hz. \nIn the end of this section we can conclude that the Gertsenshtein effect increases the amplitudes of relic GWs for frequencies k ≳ 10 -11 Hz for the considered magnetic field magnitudes.', '5 Coherence length of the cosmological magnetic field': "So far we have made estimates of the GWs amplitude by the end of the RD era under the assumption of a homogeneous magnetic field. It is practically equivalent to the condition of a very large coherence length. \nIf we use the generally accepted model of a stochastic magnetic field, then, depending on its coherence length, we can get either the same effect or a much weaker one. This is due to the fact that when a gravitational wave overcomes a characteristic distance where the magnetic field strength can be considered constant, the direction of the magnetic field will randomly change. This will lead to a change in the direction of the driving force on the right side of Eqs. (2.10). As a result, the strengthening of the GW will be replaced by weakening, and on average, the effect of enhancement the amplitude over a large number of coherence lengths will become zero. \nLet us estimate the coherence length of the primordial magnetic field (PMI) by the end of the RD stage, and find out whether the gravitational wave front manages to overcome at least one coherence length before the moment a 2 = 10 -4 . It is important to understand that the coherence length in the above models grows quickly, and a wave front moving at the speed of light may not catch up with it. \nWe will make the estimates for the law of change of quantities (magnetic field strength and coherence length) only due to the expansion of the Universe, and will not take into account the energy losses due to heating of the primary plasma and its turbulence. \nThe reader can see for example the article [14], where the authors obtain exact dependencies of the coherence length and magnetic field magnitude for different models of PMI generation by the first-order phase transitions 6 . The general conclusion is that the comoving coherence length increases with time due to the magnetic field suppression at smaller scales. Hence, in the realistic models λ coh increases even faster than a ( t ). \nSo, we have λ coh ( a ) = λ coh 0 a , where λ coh 0 - coherence length of the present day intergalactic magnetic field. The wave front moves at the speed of light. Let us determine the value of the scale factor when the wave front crosses λ coh ( a ) for the RD era: \nc ( t -t 1 ) = λ coh 0 a, (5.1) \na ( t ) = √ t τ 0 , (5.2) \nwhere t 1 is the initial time, t > t 1 , τ 0 ≈ 35 τ and τ = 4 . 4 × 10 17 sec is the lifetime of the Universe. Let us rewrite the Eq.(5.1) in terms of the scale factor: \na [ 1 -( a 1 a ) 2 ] = λ coh 0 cτ 0 . (5.3) \nSubstituting the value a 1 = 10 -9 and τ 0 we get \na [ 1 -( 10 -9 a ) 2 ] ≈ 6 . 8 × 10 -6 λ coh 0 1 Mpc . (5.4) \nThe motivation to use a characteristic scale 1 Mpc is that current observations of the photon Compton cooling are not sensitive to coherence lengths λ coh 0 > 1 Mpc. \nLet us substitute the scale factor of overcoming a over = x · 10 -9 , where 1 < x ≤ 10 5 , and obtain a quadratic equation \nx 2 -Ax -1 = 0 , (5.5) \nwith the roots \nx 1 , 2 = 1 2 ( A ± √ A 2 +4 ) (5.6) \nwhere A = 6 . 8 × 10 3 λ coh 0 1 Mpc . Positive x can be obtained only for '+' sign. On the Fig.(4) the first intersection scale factor a over = x · 10 -9 for RD era is plotted as a function of λ coh 0 . \nFigure 4: The first intersection scale factor for RD era as a function of the coherence length in the modern Universe \n<!-- image --> \nIt is important to note that the fact that for the value λ coh 0 ∼ 10 Mpc the GW begins to catch up with the coherence length by the end of the RD epoch does not mean that the effect is zero. Of course, it diminishes, but on average it is not equal to zero 7 . At the same time, the smaller the coherence length of the magnetic field today, the negligibly small the change in the relic GW amplitude due to GWs conversion into electromagnetic waves. \nIt is worth to emphasize once again that the research objective is the relic GW spectrum that was not so far from the end of RD era. Nevertheless it is an interesting question: does the Gertsenshtein effect affect on relic GWs during matter dominance (MD) era? We can similarly evaluate the overcoming scale factor. Indeed, let us replace the law of change of the scale factor with time by \na = ( t τ ) 2 / 3 (5.7) \nand substitute in Eq.(5.1) and a MD 1 = 10 -4 we get \na over = 10 -4 ( 2 . 4 · 10 -2 λ coh 0 1 Mpc -1 ) 2 (5.8) \nBy definition a over > a MD 1 . Hence \n( 2 . 4 · 10 -2 λ coh 0 1 Mpc -1 ) > 1 , (5.9) \nand solution of this equation exists for λ coh 0 ≥ 80 Mpc. For smaller values of λ coh 0 the gravitational wave front will not catch up with the coherence length of cosmological magnetic field. \nFig.(5) shows the first intersection scale factor for MD era as a function of λ coh 0 . \nFigure 5: The first intersection scale factor for MD era as a function of the coherence length in the modern Universe \n<!-- image --> \nDepending on λ coh 0 , after the GWs have passed the MD epoch, the spectral change accumulated during the RD epoch may be somehow neutralized. In this case, if the effect is large enough to be observable, it will manifest itself more strongly in the GW spectrum obtained from the CMB observations then in the spectrum from observations in the modern Universe 8 . \nHowever, even during MD era, the conversion of GW into EMW occurs, but under the influence of a magnetic field of smaller amplitude than during the RD era. Therefore, there may be two counteracting effects here. \nIndeed, if the coherence length is too small, there will be no influence of the Gertsenshtein effect on relic GWs during the RD epoch. Then we can expect its manifestation during the MD epoch, due to the rapid growth of the coherence length. \nTo summarize this discussion, we can conclude that the coherence length evolution is model dependent and this model dependence affects the final strength of the Gertsenshtein effect influence on the relic GWs spectrum (whether it is a spectrum from recombination epoch or a modern spectrum). It is fair to note that for a coherence length of λ coh 0 ∼ 10 Mpc, the homogeneous magnetic field approximation is valid throughout both the RD and MD epochs. But ultimately, the crucial meaning in solving this issue has an accurate measurement of the parameter λ coh 0 , which may become possible in the future 9 .", '6 Interaction with primary plasma': "The last important effect that we must consider is the interaction of electromagnetic wave with the primary plasma during RD era. There are two parts of this effect, which are \ndiscussed in [1]: damping factor ζ and plasma frequency Ω pl . The equation for plasma frequency for relativistic particles with m<T has the form \nΩ 2 pl = 2 T 2 9 ∑ j e 2 j , (6.1) \nwhere the summation is done over all relativistic charged particles with charges e j . So, we will evaluate Ω 2 pl ∼ αT 2 . \nFor the damping term we use the following assessment \nζ = cσn ∼ α 2 T, (6.2) \nwhere c - the seed of light, n = 0 . 1 g ∗ T 3 - the density of charged particles in plasma, g ∗ = 10 -100 is the number of charged particle species and σ = α 2 /T 2 - the scattering cross-section. \nWe will substitute this terms as the functions of a ( η ): \nΩ 2 pl ( η ) = Ω 2 pl ( a 2 ) a 2 2 a ( η ) 2 , ζ ( η ) = ζ ( a 2 ) a 2 a ( η ) , (6.3) \nwhere a 2 = a ( η 2 ) = 10 -4 . Recalculating temperature of recombination T ( a = 1 / 1100) = 3000 K, we obtain \nΩ pl ( a 2 ) ≈ 5 · 10 9 Hz , ζ ( a 2 ) ≈ 3 · 10 6 Hz . (6.4) \nThe second line in Eq.(2.10) takes the form \n˜ f '' + [ k 2 +Ω 2 pl ( a 2 ) a 2 1 -ζ ( a 2 ) a 1 a ' a ] ˜ f + ζ ( a 2 ) a 1 ˜ f ' = C 2 η 2 ˜ h, (6.5) \nwhere all the derivatives are with respect to conformal time η . \nFinally, we obtain a fourth-order equation for ˜ f \n˜ f '''' + ζ ˜ f ''' + [ 2 k 2 -Ω 2 -ζ η ] ˜ f '' + ζ [ 2 η 2 + k 2 ] ˜ f ' + [ k 2 ( k 2 -Ω 2 -ζ η ) -C 1 C 2 η 2 -2 ζ η 3 ] ˜ f = 0 , (6.6) \nwhere we replaced ζ ≡ ζ ( a 2 ) a 1 and Ω ≡ Ω pl ( a 2 ) a 1 . \nThe Eq.(6.6) is much more complicated than the Eq.(2.10) and its solution is beyond the scope of the article. Apparently, the interaction with the plasma can suppress the effect of the GW amplitude amplification obtained in the previous sections. However, in addition, there may be a characteristic imprint of this effect on the relic GW spectrum. \nWe emphasize that without the Gertsenshtein effect interaction with the plasma would not occur, since the graviton itself is electrically neutral. In the problem, interaction with the plasma is due to the conversion of graviton into photon under the influence of the cosmological magnetic field. \nWe will devote our further study to the analysis of this problem.", '7 Conclusion': 'In the present work we simplified for the high frequencies the system from [1], describing conversion of relic GWs into EMWs under influence of the cosmological magnetic field in expanding Universe. We have found the exact analytical solution, analyze it and solve the Cauchy problem. In the last two sections the stochastic magnetic field model was discussed and the differential equation was obtained for taking into account interaction with primary plasma. \nThe conclusion was made, that there is a significant enhancement of relic GWs with the frequencies k ≳ 10 -11 Hz for the magnetic field strength B 0 ∼ 1 nGs. For B 0 ≪ 1 nGs we showed that there is no amplification effect, but this result is model-dependent. \nIndeed, we have shown that it is impossible to exclude the model dependence of the result on the coherence length of the cosmological magnetic field. There remains hope for the results of measurements based on observations. \nAll in all, the distortion of the spectrum of relic GWs due to the considered effect strongly depends on the model of the magnetic field generation and evolution. On the other hand, this characteristic dependency may become a potential method for verifying these models. \nThe first planned stage of the future study is to analyze and solve the Eq.(6.6), which takes into account the interaction of generated electromagnetic wave with the primary plasma during RD era. \nThe second planned stage is to estimate the effect on GW amplitudes caused by the Gertsenshtein effect during MD era. It seems, that the analytical approach used in the work is applicable to this problem too. \nLet us note also that the most interesting frequencies for MD epoch lye in the range 10 -9 -10 -3 Hz because of sensitivity of the the pulsar timing array and the sensitivity of the future space interferometers [15-17]. The considered in our work effect, along with other possible effects (see for example [18-20]), may contribute to the signal of nanohertz GWs, which PTA experiment apparently sees [21].', 'Acknowledgement': 'We thank A.D. Dolgov, A.P. Ulyanov and E.A. Lashina for the very important discussions. The work was supported by RSF Grant 23-42-00066.', 'Appendix A': 'Let us rewrite Eq.(24) and Eq.(154) from [1]: \n[ ∂ 2 t +3 H∂ t -∆ a 2 +3 ( a a -4 H 2 )] h 00 -1 2 ∂ 2 h + ( 4 H 2 -a a ) h i i = -16 πGT EM (1) 00 , 2 H [ ∂ j h 00 + ∂ x h xj + ∂ y h yj + ∂ z h zj a 2 ] = -16 πGT EM (1) 0 j , [ ∂ 2 t +3 H∂ t -∆ a 2 ] h i j + δ i j [ -∂ 2 h 2 + ( a a +2 H 2 ) h 00 + a a h l l ] = -16 πGT i EM (1) j , (7.1) \n[ f y -∆ f y a 2 + H ˙ f y + B∂ z h y y ] = 0 , (7.2) \nwhere dot means derivative with respect to time t , T µEM (1) ν - correction to electromagnetic energy-momentum tensor (EMT) in the first perturbation order, f y = -a 2 f y and f y are covariant and contravariant y -component of the electromagnetic wave potential and \nh i i ≡ h x x + h y y + h z z = -h xx + h yy + h zz a 2 . (7.3) \nEMT corrections are also calculated in [1] in Eqs.(134, 138-143). After simplification we have \nT EM (1) 0 j = B [ ˙ f y δ jz -˙ f z δ jy ] , T xEM (1) x = 1 2 [ 2 Bf y . z -B 2 ( h y y + h z z )] , T xEM (1) y = -Bf x . z + B 2 h x y , T xEM (1) z = Bf x . y + B 2 h x z , T y EM (1) y = T z EM (1) z = -Bf y . z + B 2 h y y + h z z 2 , T y EM (1) z = 0 , (7.4) \nwhere δ ij - Kronecker symbol, f i . j - Maxwell tensor for electromagnetic field. \nIt is important to remind that in this manuscript we neglect all the terms, originated from the Heisenberg-Euler action (proportional to CB 2 , where C = C 0 for low temperatures), but in [1] we do not. \nFrom Eqs.(7.1, 7.2) and EMT corrections Eq.(7.4) we obtain the system for { h + , f y , Φ , Ψ } in case k ⊥ B ( k || Oz ): \n [ ∂ 2 t +3 H∂ t + ( k 2 a 2 -8 πGB 2 0 a 4 )] h + = 16 πGB 0 a 2 ( ikf y +3 B 0 a 2 Ψ ) , [ ∂ 2 t +3 H∂ t + k 2 a 2 ] f y = ikB 0 2 a 4 [ h + -3Φ + 8Ψ] , ikH (Φ + Ψ) = -4 πGB 0 a 2 ˙ f y , Ψ = -ika 2 3 B 0 f y . (7.5) \nwhere we used expansion of the metric perturbation tensor h µ ν on scalar { Φ , Ψ } and traceless tensor { h + , h × } modes 10 [12] \nh tt = 2Φ( t, r ) , h z z = 2Ψ( t, r ) , h x x = 2Ψ( t, r ) + h + ( t , r ) , h y y = 2Ψ( t, r ) -h + ( t , r ) , h x y = h × ( t, r ) . (7.6) \nTo obtain the first equation in the system Eq.(7.5), the equation for h y y was subtracted from the equation for h x x (see the system of equations (7.1)). The third equation was obtained for the component h 0 z , and the fourth by adding the equations for h x x and h y y with simultaneous subtraction of the doubled equation for h z z . \nAfter substituting the fourth line of Eq.(7.5) into the first line of the same system, we obtain an equation for the h + GW polarization in the case of k ⊥ B : \n[ ∂ 2 t +3 H∂ t + ( k 2 a 2 -8 πGB 2 0 a 4 )] h + = 0 . (7.7) \nThis result is very interesting and requires explanation. The tensor GW propagating in the magnetic field generates EMW, and the EMW in turn generates scalar metric perturbations. But the EMW and Ψ are related in such a way that their corresponding components of the energy-momentum tensor completely compensate each other. As a result, the amplitude of the tensor GW h + remains unchanged (in the considered order of perturbation). \nFrom the above, we conclude that the Gertsenshtein effect affects the amplitude of the gravitational wave only for the h × -polarization of the GW component, propagating perpendicular to B . \nThe aim of the research is to investigate the Gertsenshtein effect influence on the amplitude of GWs, therefore we will leave the analysis of scalar and electromagnetic perturbations evolution outside the scope of the current study.', 'References': "- [1] Dolgov, A.D.; Panasenko L.A.; Bochko V.A. 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Springer, Singapore.\n- [16] Fitzsimons E.D.,et all. Elisa technology consolidation study overview. Proc. SPIE 10563, International Conference on Space Optics - ICSO 2014, 105632K (2017)\n- [17] Kawamura S., et all. Current status of space gravitational wave antenna DECIGO and B-DECIGO, PTEP , 2021 , 5\n- [18] Kitajima N., et all, Gravitational waves from domain wall collapse, and application to nanohertz signals with QCD-coupled axions, Phys. Lett. B , 2024 , 851\n- [19] Kolesova H., Laine M. Update on gravitational wave signals from post-inflationary phase transitions, Phys. Lett. B , 2024 , 851\n- [20] Alberto Salvio, Pulsar timing arrays and primordial black holes from a supercooled phase transition, Phys. Lett. B , 2024 , 852\n- [21] Agazie G., et al., NANOGrav, The NANOGrav 15 yr data set: evidence for a gravitational-wave background, Astrophys. J. Lett. , 2023 , 951 (1)"}

2024NatAs.tmp..246B

The physical processes that establish the morphological evolution and the structural diversity of galaxies are key unknowns in extragalactic astrophysics. Here we report the finding of the morphologically mature galaxy JADESGS53.1834327.79097 which existed within the first 700 million years of the Universes history. This starforming galaxy with a stellar mass of 400 million solar masses consists of three components a highly compact core with a halflight radius of less than 100 pc an actively starforming disc with a radius of about 400 pc and a starforming clump all of which show distinctive starformation histories. The central stellar mass density of this galaxy is within a factor of 2 of the most massive presentday ellipticals while being globally 1000 times less massive. The radial profile of the specific starformation rate is rising towards the outskirts. This evidence suggests a detection of the insideout growth of a galaxy as a protobulge and a starforming disc in the epoch of reionization.

2024-10-01T00:00:00Z

['arXiv:2306.02472', '2023arXiv230602472B', '10.48550/arXiv.2306.02472', '2024NatAs.tmp..246B', '10.1038/s41550-024-02384-8']

['Astrophysics - Astrophysics of Galaxies', 'Astrophysics - Cosmology and Nongalactic Astrophysics']

A core in a starforming disc as evidence of insideout growth in the early Universe

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML']

https://arxiv.org/pdf/2306.02472.pdf

{'A core in a star-forming disc as evidence of inside-out growth in the early Universe': "William M. Baker 1,2* , Sandro Tacchella 1,2* , Benjamin D. Johnson 3 , Erica Nelson 4 , Katherine A. Suess 5,6 , Francesco D'Eugenio 1,2 , Mirko Curti 1,2,7 , Anna de Graaff 8 , Zhiyuan Ji 9 , Roberto Maiolino 1,2,10 , Brant Robertson 5 , Jan Scholtz 1,2 , Stacey Alberts 9 , Santiago Arribas 11 , Kristan Boyett 12,13 , Andrew J. Bunker 14 , Stefano Carniani 15 , Stephane Charlot 16 , Zuyi Chen 9 , Jacopo Chevallard 14 , Emma Curtis-Lake 17 , A. Lola Danhaive 1,2 , Christa DeCoursey 9 , Eiichi Egami 9 , Daniel J. Eisenstein 3 , Ryan Endsley 18 , Ryan Hausen 19 , Jakob M. Helton 9 , Nimisha Kumari 20 , Tobias J. Looser 1,2 , Michael V. Maseda 21 , D'avid Pusk'as 1,2 , Marcia Rieke 9 , Lester Sandles 1,2 , Fengwu Sun 9 , Hannah Ubler 1,2 , Christina C. Williams 22 , Christopher N. A. Willmer 9 and Joris Witstok 1,2 \n- 1 Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge, CB3 OHA, UK. \n2 \nCavendish Laboratory - Astrophysics Group, University of \nCambridge, 19 JJ Thomson Avenue, Cambridge, CB3 OHE, UK. \n5 Department of Astronomy and Astrophysics, University of California, Santa Cruz, 1156 High Street, Santa Cruz, CA 96054, USA. \n6 Kavli Institute for Particle Astrophysics and Cosmology and Department of Physics, Stanford University, Stanford, CA 94305, USA. \n* Corresponding author(s). E-mail(s): wb308@cam.ac.uk; st578@cam.ac.uk;", 'Abstract': "The physical processes that establish the morphological evolution and the structural diversity of galaxies are key unknowns in extragalactic \nastrophysics. Here we report the finding of the morphologically-mature galaxy JADES-GS+53.18343-27.79097, which existed within the first 700 million years of the Universe's history. This star-forming galaxy with a stellar mass of 400 million solar masses consists of three components, a highly-compact core with a half-light radius of less than 100 pc, an actively star-forming disc with a radius of about 400 pc, and a star-forming clump, which all show distinctive starformation histories. The central stellar mass density of this galaxy is within a factor of two of the most massive present-day ellipticals, while being globally 1000 times less massive. The radial profile of the specific star-formation rate is rising toward the outskirts. This evidence suggests the first detection of inside-out growth of a galaxy as a proto-bulge and a star-forming disc in the Epoch of Reionization. \nKeywords: High-redshift galaxy, Morphology, Star-formation histories \nIn the hierarchical ΛCDM cosmological model, galaxies sustain their star formation for extended periods of time in a quasi-steady state of gas inflow, gas outflow, and gas consumption [1, 2]. To first order, the gas that cools at later cosmic epochs possesses higher angular momentum; therefore, it settles in a more extended star-forming disc, implying that galaxies grow from the inside out [3, 4]. However, the actual formation of galaxies in the cosmological context is more complex since a wide range of processes regulate star formation and the orbital distribution of stars, ranging from stellar feedback (from supernovae and stellar winds), black hole feedback, and cosmic rays, to galaxygalaxy interactions and mergers [5, 6, 7, 8, 9]. Therefore, the morphological structure and spatially resolved growth rates of galaxies are a sensitive - but also complicated - probe of galaxy formation physics [10, 11, 12, 13, 14, 15, 16]. \nGalaxies in the local Universe display a range of morphologies, from younger disc-dominated spiral galaxies to older bulge-dominated ellipticals [17, 18], and are typically classified by the Hubble sequence [19, 20, 21]. The growth of local star-forming galaxies has been observed on spatially resolved scales, confirming that generally galaxies grow inside out [22, 23, 24]. However, there is a diverse range of specific star-formation rate (sSFR) profiles in the local universe with some galaxies undergoing inside-out growth, while others grow outside-in [25, 26, 27], likely corresponding to different growth phases. Most of the mass of local galaxies is found to have formed during the redshift range 1 ≤ z ≤ 3, around the period of 'cosmic noon', the peak of the cosmic star formation rate density in the Universe [28]. Observations at these redshifts have revealed many galaxies with massive bulges and rotating discs [29, 30, 31, 25]. However, in order to probe the buildup of these 1 ≤ z ≤ 3 bulges, we need to investigate even earlier cosmic times, characterising galaxies during the Epoch of Reionization ( z ≳ 6, [32]). Observationally, \nlittle is known about how quickly these early galaxies grow, in particular on spatially resolved scales. The theoretical expectation is that the galaxy merger rate increases towards higher redshifts [33, 34, 35], which could lead to more pronounced central starbursts [36, 37]. Recent numerical models indicate that early (at z > 3), low-mass (10 7 -9 M ⊙ ) galaxies can undergo rapid size fluctuations and compaction driven by the competition between feedback-driven gas outflows and cold inflows [38, 39], leading to a flat or even inverted sizemass relation with more massive galaxies being smaller ( < 10 9 M ⊙ ). Direct observations will address questions about how galaxies grow their stellar mass and size in the early Universe, thereby providing insights into the physics that regulates star formation. \nJWST opens a new window to study the formation of the Hubble sequence and bulge-disc formation in the early Universe [40, 41, 42]. As part of the JWST Advanced Deep Extragalactic Survey (JADES, [43, 44, 45]), we report here the discovery of a core-disc galaxy with an off-centre clump (JADESGS+53.18343-27.79097) at a spectroscopic redshift of 7.430 (see Methods), when the Universe was only 700 million years old. JADES-GS+53.1834327.79097 appears to be growing inside-out, having built up a massive compact core at its centre before forming a surrounding star-forming disc. This is the first time we are able to characterise a core-disc system during the Epoch of Reionization and find the signature of early bulge formation.", 'Core-disc-clump decomposition': "We use NIRCam [46] imaging in nine filters (F090W, F115W, F150W, F200W, F277W, F335M, F356W, F410M and F444W) and NIRSpec/MSA [47, 48, 49] spectroscopy from JADES in the GOODS-S region [50]. This gives us extended coverage of the rest-frame ultra-violet and optical wavelengths (including the Balmer break), which constrains the stellar populations on spatially resolved scales. Furthermore, the medium band F410M probes the strength of the emission lines H β +[OIII] on spatially resolved scales. \nThe upper left panel of Fig. 1 shows a colour-composite red-greenblue (RGB, corresponding to F444W-F277W-F115W) image of JADESGS+53.18343-27.79097. The image shows a compact central component (core) surrounded by an extended, disc-like component. JADES-GS+53.1834327.79097 has a strong colour gradient between the central region and the outskirts (Methods Figure 6), implying an excess of H β +[OIII] in the outskirts as probed by F410M. In order to quantify the compactness and colour gradient, we employ the tool ForcePho (B. Johnson, in prep.) to perform a detailed morphological and photometric analysis of JADES-GS+53.18343-27.79097, forward modelling all individual exposures across all bands simultaneously and accounting for the point-spread functions (PSFs; see Methods, Section 1.4). We explore a range of different models and find that a three-component model, consisting of a disc S'ersic profile (fixing the S'ersic index to n = 1 by limiting the bounds to ± 0 . 01), a compact component ( n = 2 -5) consistent with a bulge or pseudobulge, and an off-centre clump (modelled as a quasi-point source), \n] \ncsec \n[ar \npos. \nSlit \nRadius [kpc] \nFig. 1 Image of the galaxy, radial profile of the light distribution and the 1D and 2D spectra. Upper left panel: the (F444W-F277W-F115W) colour composite image of the galaxy, where the central core and disc are prominent. A 1-kpc size bar (corresponding to 0.19 arcsec), the F444W PSF and the position of the NIRSpec slit are overplotted. Upper right panel: the total radial (azimuthally averaged) surface brightness profile of the core, disc and clump in the F356W band, where the pink points are the observational data and the black line is the PSF-convolved best-fit three-component model (consisting of the core, disc and clump). The errorbars correspond to the error propagated through from the error maps (i.e. the standard deviation). The intrinsic best-fit S'ersic profiles of the core and disc components are shown as red and blue lines, respectively. The off-centred clump at a distance of 1.4 kpc is indicated with a purple arrow. The grey dashed and dotted lines correspond to the empirical PSF (ePSF) and the WebbPSF, respectively, in the F356W band. Lower panel: the 2D and 1D NIRSpec R100 prism spectra, with the position of notable detected emission lines overplotted, which indicates that this galaxy is dominated by stellar emission. The errors on the spectrum correspond to the standard deviation of the mean signal. \n<!-- image --> \nFig. 2 Spectral energy distributions for the three components and their starformation histories. Spectral energy distribution (SED) fits for the disc (upper left), core (upper right), and clump (lower left) components. The yellow points show the photometry inferred from the ForcePho modelling, while the 2 σ upper limits are indicated as downward pointing arrows. The errors on the photometry are the 1 σ uncertainties (which are derived via standard error propagation through the reduction pipeline). The errors on the x-axis correspond to the widths of the filters. The open squares mark the photometry of the bestfit SED model. The solid lines and the shaded regions show the median and the 16th-84th percentile of the SED posterior from the Prospector modelling. Lower right: star-formation rate (SFR) against look-back time (from the observed redshift) for the disc (blue line), core (red line) and clump (purple line). The shading corresponds to the 16th and 84th percentiles of the star-formation history posterior. We find that the disc has been undergoing a burst of star formation for the last 5 Myr, while the core is entering a lower-star-formation phase. The clump has a lower SFR compared to the other two components and is over an order of magnitude less massive than the core. \n<!-- image --> \nreproduces the data best (with the smallest residuals and χ 2 ). Although this model is not unique, it is able to capture the complex morphology of this galaxy in all of the filters. We obtain a central core with a half-light radius of 16 mas (roughly 80 pc), while the disc has a half-light radius of 80 mas (about 400 pc; see Table 1), overall confirming the compact nature of the source. Although we refer to the extended component as a 'disc' from a purely morphological perspective, the ionised gas (as traced by F410M) of JADES-GS+53.1834327.79097 is actually consistent with a rotationally supported component with v ( r eff ) /σ 0 ≈ 1 . 3 [51]. \nFig. 1, upper right panel, shows the surface brightness profile of the galaxy in the F356W band, the best-fit model convolved with the PSF (black), the \nTable 1 Table showing (from left to right), stellar mass, half-time (the half-mass assembly time), star formation rate (SFR) averaged over 10 Myr, SFR averaged over 100Myr, stellar metallicity, extinction in the V-band A V , S'ersic index n , and half light radius r e , for the core, disc and clump components from our fiducial fit using ForcePho. The S'ersic index of the disc and clump are fixed to n = 1 (indicated by the star). \nunconvolved S'ersic model components (core in red and disc in blue), and the WebbPSF and empirical PSF (ePSF) of the mosaic (grey dotted and dashed line). This shows that the PSF-convolved three-component model reproduces the observed surface brightness profile (see also Methods, Section 1.5 for a detailed plot of the model and residual). We also fit a single-component model with and without the clump, plus just a core + disc fit, all of which we test against our fiducial three-component model. We find that all of these model variants fail to account for either the additional flux in the centre and/or the flux of the clump, resulting in higher χ 2 statistic values (0.45, 0.25, and 0.18 vs 0.14 for the fiducial three-component fit; see Methods, Section 1.5). We also calculate a reduced χ 2 which favours the fiducial three-component model. To check against PSF-approximation issues with ForcePho, we re-simulate the core, disc, and clump fits convolved with the WebbPSF, and then refit them. Using the WebbPSF model, we find that the results are consistent with the original fit within the uncertainties, confirming that the ForcePho PSF approximations are appropriate (see Methods, Section 1.6). \nFig. 1, bottom panel, shows the 2D and 1D NIRSpec R100 prism spectra of JADES-GS+53.18343-27.79097, including the positions of notable detected emission lines. This spectrum probes both the core and the disc, as indicated by the slit position in the upper left panel of Fig. 1. Using these data, we estimate a spectroscopic redshift of z = 7 . 430, consistent with the photometric redshift. The measured emission line fluxes from the NIRSpec prism spectrum indicate that this galaxy is consistent with stellar emission, and the high-resolution grating spectrum shows that all emission lines (in particular H β ) are narrow, implying that this galaxy does not appear to host a dominant Active Galactic Nucleus (AGN; see Methods, Section 1.11), but we note that it is difficult to fully rule out an AGN.", 'Stellar population properties': "We show in Fig. 2 the ForcePho-based spectral energy distributions (SEDs) of the core, disc, and clump components. To explore the stellar population properties of the three components, we fit the individual SEDs using the Bayesian SED-fitting tool Prospector [52]. We input the flux values and errors obtained for each band from ForcePho, and independently fit the SEDs with a flexible star-formation history (SFH) with the standard continuity prior [53], a variable dust attenuation law with a free dust attenuation law index and normalisation, \nand a nebular emission model. We also test using a bursty continuity prior for the SFH, finding that we obtain stellar masses consistent with the standard continuity prior (see Methods, Section 1.7). In Fig. 2 we plot the best fit for the SEDs of the three components, indicating that the SEDs are well reproduced by stellar emission in conjunction with dust attenuation and nebular emission. \nTable 1 gives the key stellar population properties of the core, disc, and clump components. The core is the most massive of the components with log(M ∗ / M ⊙ ) = 8 . 4 +0 . 2 -0 . 2 , while the disc has log(M ∗ / M ⊙ ) = 8 . 0 +0 . 3 -0 . 2 , despite the core having a radius ( ∼ 80 pc) less than a quarter of that of the disc ( ∼ 400 pc). This suggests that the core is dense (stellar mass surface density of Σ eff = M ∗ 2 π r 2 ≈ 6 × 10 9 M ⊙ kpc -2 ). The bottom right panel of Figure 2 shows the starformation rate (SFR) as a function of lookback time for the core (red), the disc (blue) and the clump (purple). These SFHs show the full posterior distributions, i.e., are taking into account the degeneracies with other parameters, such as the dust attenuation law. The core, disc and clump have varying SFHs, with the core undergoing an earlier period of star formation with a recent decline, while the disc is currently undergoing a burst of star formation. Consistently, the stellar age (lookback time when half of the stellar mass formed) for the core is rather old with t half = 68 +78 -31 Myr, while the disc is younger ( t half = 19 +108 -15 Myr). In total, we find that the combined core+disc galaxy has a stellar mass of log( M ∗ /M ⊙ ) = 8 . 5 +0 . 2 -0 . 2 , and SFR 10Myr = 5 . 8 +1 . 5 -1 . 3 M ⊙ / yr, giving it a specific SFR (sSFR) of log(sSFR/yr)= -7 . 7 +0 . 1 -0 . 1 (typical for 7 ≤ z ≤ 8 galaxies [54]). For comparison, from the NIRSpec spectrum we derive a dust-corrected H β SFR of SFR H β = 9 +30 -7 M ⊙ / yr, which is consistent with our SED-derived SFR over 10 Myr. We find a similar agreement for dust attenuation (Balmer decrement) and gas-phase metallicity between the SED-derived and NIRSpecderived values (see Methods, Section 1.10). Although the clump is not the focus of this study, we find it to be low in stellar mass (log( M ∗ /M ⊙ ) = 7 . 2 +0 . 4 -0 . 2 ) and highly star forming with a specific SFR (sSFR) of log(sSFR / yr -1 ) = -7 . 5 +0 . 3 -0 . 4 averaged over 10 Myr. The clump's age appears to be relatively unconstrained ( t half = 44 +146 -38 Myr), meaning that it could be young and have been formed through a disc instability, or - alternatively - it could be older and be an accreted satellite galaxy. \nIn summary, we find that the core and disc are similarly bright in the restUV and that the disc is dominating the most recent star formation and is slightly dust-attenuated. This is surprising in the z ≈ 2 picture of a red dusty bulge embedded in a blue disc with lower dust attenuation, but consistent with recent JWST observations showing populations of red dusty discs missed in previous rest-optically selected samples [55]. The observational data that drive this result is the higher F410M excess in the disc (see Methods, Figure 6), which implies strong nebular line contribution and is consistent with inside-out growth. This is a good showcase of the power of medium-band observations to constrain stellar populations [56]. \nFig. 3 Radial surface density profiles of stellar mass and star formation. Radial profiles of stellar mass surface density (Σ ∗ ; left panel), star-formation rate (SFR) surface density (Σ SFR ; middle panel), and specific SFR (sSFR; right panel) for the core (solid red lines), disc (solid blue lines) and the combination of the two (dashed black lines). The shaded regions show the 16th and 84th percentile uncertainties from both the SED fitting and the structural measurements. The half-mass radius at the time of observation is overplotted as a vertical dashed-dotted line. This figure shows that the SFR surface density of the galaxy is, at almost all radii, completely dominated by the star-forming disc, while the stellar mass surface density in the central regions is dominated by the core. The sSFR increases with radius, implying that this galaxy grows from the inside out. \n<!-- image -->", 'Radial profiles of stellar mass and star formation': "The compact size and the medium-band excess with increasing radius (probing the youngest stars via nebular emission) are indicative of a high central stellar mass density with a star-forming outskirt. We now use our multi-wavelength morphological decomposition and stellar population modelling to derive the intrinsic radial profiles of the stellar mass and SFR surface density. We use the unconvolved best-fit S'ersic profiles, normalised to the best-fit stellar mass and SFR for each component, as given (for example) for the stellar mass surface density profile by Σ ∗ (r) = M ∗ I(r) I tot , where I(r) is the intensity inferred from the S'ersic profile at radius r and I tot = ∫ 2 π I(r) r dr. The assumption is that each component has a negligible radial gradient in their stellar populations. However, since the normalisation of the individual components vary as a function of wavelength, we are able to account for stellar population gradients across the galaxy. \nFig. 3 shows the stellar mass surface density (Σ ∗ , left panel), the SFR surface density (Σ SFR , middle panel), and the sSFR (right panel) against the radius for the core and disc components and the combined profile. The Σ SFR profile shows how the disc completely dominates the profile compared to the core, while the Σ ∗ profile shows that the core's stellar mass surface density is prominent in the inner regions. We indicate in each diagram as a vertical dotdashed grey line the half-mass radius ( R ⋆ = 126 +37 -26 pc), which is the radius of the galaxy at which half the (core+disc) stellar mass is contained. \nBy construction, because they have the same S'ersic profile (that is, shape) for the SFR and stellar mass radial profiles, the sSFR (= SFR/M ∗ ) profiles are radially constant for the individual disc and core components, while their combined sSFR profile is rising with radius since sSFR(r) = Σ SFR , core +Σ SFR , disc Σ ∗ , core +Σ ∗ , disc . The sSFR profile shows where the galaxy grows, as the stellar mass doubling \n<!-- image --> \nFig. 4 Stellar mass density and half-mass size in the context of lower-redshift galaxies. Left: effective stellar mass density (Σ eff ) versus stellar mass for the core (red square), disc (blue square), and the combination of both components (black square). The data is presented as the median of the distribution with errors corresponding to the 16th and 84th percentiles. We compare this to measurements from z ≈ 2 [57] and z = 0 [58]. Although our z = 7 . 43 galaxy is low in stellar mass, Σ eff lies in the upper envelope of z ≈ 2 star-forming galaxies (SFGs) and in the range of quiescent galaxies (QGs). Right: half-mass size versus stellar mass for the core (red square), disc (blue square), and the combination of both components (black square). The errors correspond to the 16th and 84th percentiles of the resulting distribution. The dotted and dashed lines show the size-mass relation at redshift z ≈ 2 [57] for SFGs and QGs, respectively. The solid green line marks the predicted evolutionary track to z = 1 assuming the inferred SFR profile (Fig. 3), while the orange line marks the evolutionary track for a model galaxy where the scale length of the SFR density of the core and disc scale as 1 / (1 + z ). The half-mass radius at redshifts 5, 3, 2 and 1 is represented by the small squares, indicating that - given the current growth rate - both models predict that this galaxy will evolve onto the size-mass relation of quiescent galaxies at z = 2. \n<!-- image --> \ntimescale is approximately equal to 1/sSFR. We see that the sSFR is steeply rising at the half-mass radius (about 1.5 dex within the central 1 kpc): the central 100 pc has a stellar mass doubling time of ≈ 100 Myr, while the outskirt has a mass doubling timescale of ≈ 10 Myr. This implies that this galaxy increases its half-mass radius with time and grows inside out.", 'Discussion': "We now compare our z = 7 . 43 galaxy with galaxies and stellar systems at lower redshift, which allows us to gain a complementary view on the spatially resolved growth from the stellar mass and SFR distribution. Fig. 4 shows the effective stellar mass density (Σ eff ) and the half-mass radius as a function of the stellar mass for the core, disc, and the combination of both components. We compare our measurements to the ones from [57] of star-forming galaxies (SFGs) and quiescent galaxies (QGs) at z = 2 . 0 -2 . 5 and from [58], which contains data on local globular clusters (GCs), Ultra Compact Dwarfs (UCD) and compact Ellipticals (cE). In the right panel, we also plot the extrapolated \nFig. 5 Stellar mass surface density profile in comparison with local galaxies. Radial profiles of stellar mass surface density (Σ ∗ ) of JADES-GS+53.18343-27.79097 at z spec = 7 . 43 (black line with shaded region corresponding to 16th and 84th percentiles) compared to average z = 0 stellar mass density profiles of Ultra Compact Dwarfs (UCDs; grey line with triangles), low-mass ellipticals (grey dotted lines with points) and high-mass ellipticals (with core or cusp as a solid line with squares and a dashed line with squares respectively) [59]. We find that this galaxy is extremely compact: the stellar mass density is consistent within 0.2-0.3 dex of today's massive ellipticals in the central region, while it is a factor of a 1000 lower in total stellar mass. Also overplotted are the extrapolated profiles at z=2, with the constant star-formation rate scale length ( R s ) model in green and the R s ∝ 1 1+ z model in orange. We see that the redshift dependent scale model appears more reasonable compared with the redshift 0 ellipticals. \n<!-- image --> \ngrowth of the half-mass radius from redshift 7.43 to redshift 1, assuming the SFR profile from Fig. 3. \nAlthough our z = 7 . 43 galaxy has a lower stellar masses than the plotted SFGs and QGs at z ≈ 2, Σ eff lies in the upper envelope of z ≈ 2 SFGs and in the range of QGs. Looking at the half-mass radius, we find that both size and stellar mass are not probed by z ≈ 2 observations. In order to facilitate a comparison and gain insight into the formation of quiescent galaxies at Cosmic Noon ( z = 1 -3), we grow the observed stellar mass profile according to two simple recipes: ( i ) at a constant SFR with the observed SFR profile (green line in Fig. 4); and ( ii ) at a constant SFR but with the assumption that the scale length of the SFR surface density profile scales as R S ∝ 1 (1+ z ) (motivated by inside-out growth models [4]; orange line). We assume that the global SFR is constant because the decline of the star-forming main sequence and the higher SFR due to an increased stellar mass (on the star-forming main sequence) roughly cancel each other out. Although both scenarios are simplistic, both tracks naturally intersect with the z ≈ 2 size-mass relation of QGs by z ∼ 1 -3, highlighting that our z = 7 . 43 galaxy is a natural progenitor of the quiescent galaxy population at z ≈ 2. \nWe also consider how our z = 7 . 43 galaxy compares to local, z = 0 stellar systems. From the left panel of Fig. 4, we can see that this galaxy lies above Σ eff of local GCs and UCD, but is comparable to low-mass ellipticals. This can also be seen from Fig. 5, which shows the stellar mass surface density (Σ ∗ ) as a function of radius. For comparison, we also plot the profiles of local analogues, UCDs, cE, ellipticals with a cusp, and ellipticals with a core from the compilation gathered in [59]. The profile for JADES-GS+53.18343-27.79097 is similar to that of cEs and more extended than that of UCDs. Interestingly, the central ( R < 20 pc) density of our z = 7 . 43 galaxy is within 0 . 2 dex of those of massive elliptical galaxies seen in the Universe today, but we note that it contains just 0.1% of the total stellar mass of these galaxies. Specifically, this z = 7 . 43 galaxy has a stellar mass density at R = 1 kpc of Σ ∗ , 1kpc = 6 . 6 M ⊙ / kpc 2 , while the massive core ellipticals have on average Σ ∗ , 1kpc = 9 . 4 M ⊙ / kpc 2 , which is a difference of 2.8 orders of magnitude. If this galaxy evolves into such a massive elliptical by z = 0, we conclude that inside-out growth takes place in two phases: firstly as a star-forming galaxy, which we directly observe here, and then, secondly, as a quiescent galaxy from z ≈ 1 -3 to z = 0 via mergers that build up a stellar envelope [60, 61]. \nHow can a z = 7 . 43 galaxy build such a high central stellar mass density that is comparable to local ellipticals? Our analysis shows that the starformation activity is dominated by the disc component. However, it is not clear whether there was an episode of disc formation prior to the peak of the SFH of the core (that is, more than 100 Myr ago): earlier disc formation is still a possibility based on the posterior distribution of the SFH (see Methods Figure 12). Therefore, we speculate that the following two scenarios are possible to build up this core. The first is continuous inside-out growth, where early disc formation took place in a extremely compact disc, forming the currently observed core [4]. Indeed, such compact disc-shaped objects have been observed at a redshift of more than 10 [44]. An alternative is that the disc was formed first and suffered an infall of gas into the centre due to compaction [62, 8], which then formed the core. The disc would then re-form via newly accretion of gas. \nImportantly, all stellar systems, including our galaxy at z = 7 . 43, are well below the maximum stellar surface density of Σ max = 10 11 . 5 M ⊙ / kpc 2 . This universal maximum stellar surface density of dense stellar systems is a natural consequence of feedback-regulated star-formation physics [59, 63]. \nRegarding the two inferred evolutionary tracks, we find that the constant Σ SFR grows the galaxy efficiently within 1 kpc, leading to a profile at z = 2 that overpredicts the stellar mass density of local ellipticals in the central 300 pc and is comparable to Σ max at R < 10 pc. On the other hand, the track where we scale the scale length of the SFR density has a lower Σ ∗ (that is, it does not violate Σ max ) and is more consistent with the one of massive elliptical profiles up to R ≈ 3 kpc. We conclude that the inside-out growing model is more consistent with the low-redshift ellipticals. \nIn summary, our finding of JADES-GS+53.18343-27.79097, a core-disc galaxy with a star-forming clump during the Epoch of Reionization provides \nevidence for inside-out growth during the first 700 Myr of the Universe. This galaxy appears to be a potential candidate progenitor of a typical quiescent galaxy at redshift 2 and a present-day elliptical galaxy. This suggests that bulge formation can start at very early epochs and demonstrates the importance of understanding the nature of these earliest systems on spatially resolved scales.", '1.1 NIRCam imaging data': "We use photometric and spectroscopic data obtained by JWST as part of the JADES [43] collaboration. JADES consists of the NIRCam [46] and NIRSpec [48] Guaranteed Time Observations (GTO) instrument teams, and was established to be able to use a combination of imaging and spectroscopy to utilise the full capabilities of both instruments. We use the JADES NIRCam imaging of the Great Observatories Origins Deep Survey - South (GOODS-S) field [64]. This consists of imaging in F090W, F115W, F150W, and F200W shortwavelength (SW) bands, and F277W, F335M, F356W, F410M, and F444W long-wavelength (LW) bands. \nThe details of the data reduction of the NIRCam data will be presented as part of the JADES programme in Tacchella et al. (in prep.), and have already been described in some detail in [44, 65]. Briefly, we use the JWST Calibration Pipeline v1.9.2 with the CRDS pipeline mapping (pmap) context 1039. We run Stage 1 and Stage 2 of the pipeline with the default parameters, but provided our own sky-flat for the flat-fielding. Following Stage 2, we perform several custom corrections in order to account for several features in the NIRCam images [66], including the 1/f noise [67], scattered-light effects ('wisps') and the large-scale background. Since all of those effects are additive, we fit and subtract them. \nBefore constructing the final mosaic, we perform an astrometric alignment using a custom version of JWST TweakReg. We calculate both the relative and absolute astrometric correction for images grouped by visit and band by matching sources to a reference catalogue constructed from HST F814W and F160W mosaics in the GOODS-S field with astrometry tied to Gaia-EDR3 ([68]; G. Brammer priv. comm.). We achieve an overall good alignment with relative offsets between bands of less than 0.1 short-wavelength pixel ( < 3 mas). We then run Stage 3 of the JWST pipeline, combining all exposures of a given filter and a given visit.", '1.2 NIRSpec spectroscopic data': "We used the NIRSpec Micro Shutter Array (MSA), with two disperser/filter combinations: PRISM/CLEAR from programme 1180 (hereafter: R100; covers the entire spectral range 0 . 7 < λ < 5 . 3 µ m with spectral resolution R = 30-100) and the high-resolution grating G395H/F290LP from programme 1286 (hereafter: R2700) to cover the region 2 . 9 < λ < 4 . 2 µ m with resolution \nR = 2 , 700. The MSA was configured with the 'three-shutter slitlet', creating an effective slit of 0.2-arcsec width and approximately 1.5-arcsec length. The exposure times were 11,292 s (R100) and 8,009 s (R2700). \nFor a detailed description of the data reduction, we refer to [45, 69, 70]. Here we note that we applied wavelength-dependent path-loss corrections based on modelling JADES-GS+53.18343-27.79097 as a point source, and extracted the spectrum from a 0.5-arcsec box. The final reduced R100 1D and 2D spectra are shown in Fig. 1. \nThe redshift estimate is based on the [OIII] λ, λ 4959,5007 detection in the R2700 spectrum. We obtain z spec = 7 . 4303 ± 0 . 0002(random) ± 0 . 0005(systematic). To measure the emission-line fluxes we used the R100 data and Penalized PiXel-Fitting (pPXF) [71, 72], which models simultaneously the underlying continuum, as described in [70].", '1.3 Galaxy selection': 'In order to model the spatially resolved stellar populations, we selected JADES-GS+53.18343-27.79097 from the JADES GOODS-S imaging region with F335M and F410M medium-band coverage and a spectroscopic redshift. We used spectroscopic redshifts from both JADES and FRESCO [74], focusing on the redshift interval z = 7 . 0 -7 . 8, as we wanted to both probe the very earliest galaxies while also having the Balmer break falling within our filters. Out of these ( ≈ 20) galaxies, JADES-GS+53.18343-27.79097 appeared to be the most intriguing with evidence of a core-disc structure and colour gradient (see Fig. 6). We also note that this galaxy was included in the six sources analysed in [51] where they found evidence for rotation likely tracing the disc component. \nDue to this selection procedure, we make no claims about population statistics for this type of galaxy at these redshifts, only that it is a fascinating object in its own right. In a future work, we will explore the population statistics for a mass-complete sample of similar (bulge-disc) galaxies. Importantly, this galaxy does not seem to be peculiar given its stellar mass and redshift: it is only slightly above the extrapolated star-forming main sequence ([75, 76, 77, 78, 73]; see Fig. 7) and shows the typical emission line properties for galaxies at this redshift (refer to Sections 1.10 and 1.11). JADES-GS+53.18343-27.79097 has previously been identified in HST imaging for the GOODS-S field as a Lyman break galaxy at z ∼ 7 -8 (it is the source UDF-3244-4727 in [79] based on HSTNICMOS and ACS imaging, and was independently selected in subsequent HST/WFC3 imaging as GS.D-YD4 in [80]). \nFig. 6 shows images of the galaxy in the F277W band (upper left panel), the F356W band (upper middle panel), the F410M band (upper right panel), and the PSF-matched radial profiles of the F356W-F410M and F277W-F356W colour (bottom panel). The radial colour profile is computed from the PSFmatched images (PSF-matched to F444W) in order to remove gradient effects resulting from the wavelength-dependent PSF. Interestingly, we find opposite trends for the two colours: the F356W-F410M colour gets redder toward \nFig. 6 Colour gradients between the centre and outskirts of the galaxy. Images in the F277W band (upper left panel), F356W band (upper middle panel), F410M band (upper right panel) and the PSF-matched radial profiles of the F356W-F410M and F277WF356W colour (bottom panel). The size of the FWHM of the PSF (bottom right), a bar indicating 1 kpc at z = 7 . 43 (bottom left) and an elliptical aperture with a major axis length of 0.4 arcsec are shown for reference in each image cutout. The radial colour profile is computed from the PSF-matched images (PSF-matched to F444W) in order to remove gradient effects resulting from the wavelength-dependent PSF. The errors are obtained from the 1 σ uncertainty map and propagated forwards. The F277W-F356W is offset by 0.9 mag to highlight the colour gradients seen in the profile. The colour profiles show opposite trends: the F410M excess increases while the F277W-F356W colour (tracing the rest-4000 ˚ A spectral region) gets bluer toward the outskirts. This indicates that the outskirts - dominated by the disc - has younger stellar population than the central region. \n<!-- image --> \nthe outskirts, while the F277W-F356W colour gets bluer. Since the F356WF410M colour traces mainly the emission line excess ([OIII] and H β lines), we find that those emission lines become more prominent towards the outskirts. On the other hand, the F277W-F356W colour traces the rest-frame 4000 ˚ A spectral region, implying that the centre shows a Balmer/4000 ˚ A-break, while the outskirts show a Balmer jump. This indicates that the outskirts have a younger stellar population than the central region. Interestingly, the ionised gas of JADES-GS+53.18343-27.79097 is actually consistent with a rotationally supported component with v ( r eff ) /σ 0 ≈ 1 . 3, as recently shown in [51]. \nFig. 7 Star-formation activity relative to the star-forming main sequence. The SFR (averaged over 10 Myr) against the stellar mass for the core, disc and combined components plotted in red, blue and black. The result of fitting the core+disc+clump photometry in also plotted in orange. The data is presented as the median of the distribution with the errors corresponding to the 16th and 84th percentiles. For reference, the star-forming main sequence (SFMS) [73] extrapolated to redshift 7.43 and at 0 is plotted as black and grey dashed lines, respectively (the shaded region corresponds to a 0.3 dex uncertainty region on the SFMS at z = 7 . 43). All components show elevated SFRs compared to the SFMS (by 0.6 and 1.2 dex for the core and disc, respectively). \n<!-- image -->', '1.4 Morphology and photometry with ForcePho': "It is challenging to assess the morphology and spatially resolved stellar populations of galaxies at z > 3 because ( i ) they are compact with typical sizes of 0 . 1 -0 . 4 arcsec, i.e., of the order of size of the NIRCam/F200W PSF, and ( ii ) the NIRCam PSF FWHM varies by a factor 4 from the bluest (F115W) to the reddest (F444W) band. Given this variation in the PSF FWHM, there are two routes to analysing the data: (1) performing pixel-by-pixel SED fitting on images convolved to the F444W resolution, or (2) modelling the galaxy's deconvolved light distribution in each band independently. Convolving to F444W resolution would lose the high spatial resolution available to us in the blue bands, which provides crucial insights into the morphology of the source; therefore, we choose to forward-model the galaxy's light distribution. Importantly, the observed colour gradient is not caused by the PSF as we show in Fig. 6 after homogenising for the PSF, the colour gradient is still present. This implies that there is an intrinsic colour gradient. In order to model the intrinsic colour gradient, we have to vary the structural components of the galaxies as a function of wavelength (just changing the normalisation will not be enough because the radial component will cancel out in the colour term). This can be done via: ( i ) fitting an individual S'ersic profile with wavelength-dependent size and/or \n2 \nFig. 8 Data, model, and residual maps for various ForcePho models. For each of the four main setups we plot the data, residual and model. The four setups are our fiducial, three-component fit (top panels), a two-component fit consisting of galaxy+clump (upper middle panels), a two-component fit consisting of a core+disc (with no clump),and a onecomponent single galaxy fit (bottom panels). It shows that the three-component fit appears to reproduce the data significantly better, as is quantified through the χ 2 -value for the fit. This can also be seen in the residual figures for the other fits which all show more significant residuals (alongside their higher χ 2 values). \n<!-- image --> \n1.00 \nS'ersic index; or ( ii ) fitting a two-component S'ersic model with fixed size and S'ersic index, but wavelength dependent normalisation. We chose option ( ii ) because it has a stronger physical motivation. A given physical component (bulge, disc, or clump) has a certain morphology (size and shape) and stellar population properties (stellar mass, SFH, etc.). This means that a component has a certain SED, that is, the flux changes as a function of wavelength, but the shape remains the same. Furthermore, fixing the shape parameters will allow the use of the full signal-to-noise ratio across all bands. \nA challenge with this approach is that we have to choose a parametric model that describes the galaxy well. As outlined below (Section 1.5), we experimented with different models, finding that two central components with a clump in the outskirts are able to reproduce the light distribution of JADESGS+53.18343-27.79097 in all bands the best (both visually and quantitatively, see Fig. 8) . Our main aim here is to describe the stellar populations on spatially resolved scales (see Fig. 4), while the physical interpretation of the different components is of secondary importance, but remains of interest given the insights into bulge-disc systems at lower redshifts [18, 29, 30]. \nWe choose here to forward model the light distribution in all 9 NIRCam images using ForcePho (Johnson et al., in prep.). ForcePho fits multiple PSFconvolved S'ersic profiles simultaneously to all individual exposures and filters by sampling the joint posterior distribution via Markov Chain Monte Carlo (MCMC). This allows us to take into account and measure the covariances between all the parameters. We run ForcePho on the individual NIRCam exposures, which is a key advantage over other codes that run on mosaics, such as galfit [81], Lenstronomy [82] or ProFit [83]. Firstly, when the individual cal frame images (stage 2 products) are co-added to build the final mosaic, information is lost by construction. Working therefore on the mosaic means working on data with less information than the full set of individual exposures. This is particularly important for compact objects, such as JADES-GS+53.1834327.79097. The individual exposures capture these compact objects with several different pixelizations (thanks to different dither positions), while the mosaics are a single-pixel representation. Information is also lost about the correlation between pixel fluxes in the mosaics. Secondly, alternative methods work with empirical PSFs (ePSFs), which are based on a few stars that are not saturated, leading to significant uncertainty in the outskirts of the ePSFs due to noisy outskirts of individual stars. Furthermore, the ePSFs are only marginally oversampled, which leads to uncertainties in the convolution. The PSFs of the cal frame images can be well described with WebbPSF ([84, 85]; see also Section 1.6). Therefore, tools such as ForcePho that can work on individual exposures have a significant advantage over tools that work on mosaics with ePSFs. Furthermore, ForcePho has been successfully applied to modelling multiple components in [86, 44, 65]. Since ForcePho needs to perform many convolutions, it works with Gaussian mixture models for both the PSF and the S'ersic profile. The PSF is approximated with 4 positive Gaussians. The introduced systematics are investigated in Section 1.6 and motivate us to assume an error \nfloor for the parameters estimated by ForcePho (20% for the effective radii and 0.3 for the S'ersic index of the core). \nWe run ForcePho assuming a three-component model: two central components and one off-centred clump. Our data prefer this three-component model over a more simple model (one- or two-component model), as shown in Section 1.5 both visually and quantitatively. We assume that the structural parameters are constrained by a combination of the bands, while the flux is fit individually in each band. For the two central components, we fit for the centre, the axis ratio, the position angle, and the size. The prior on the size is uniform from 0.001' to 1.0'. Importantly, we fix the S'ersic index n of one of the central components to 1.0 (sampling it from a range of 0.99 to 1.01), while the other central component is allowed to vary between 2 < n < 5. The motivation for fixing the S'ersic index to 1 for one of the central components comes from restricting the number of degrees of freedom and from lower-redshift observations [18, 29], where so-called bulge-disc decomposition fits have been shown to describe well the light and stellar population distributions. The assumption of an exponential disc ( n = 1) is theoretically motivated as gas that cools in a halo leads to an exponential star-forming disc [4], while observed discs are well fit by an exponential profile [25, 26], hence most bulge-disc decomposition studies fix the S'ersic index to 1 [17, 18, 30]. Furthermore, specifically for this galaxy, [51] shows evidence for rotation in it. Therefore, we call the n = 1 component a 'disc', while the second smaller component with n = 2 -5 is referred to as a 'core'. We fit the off-centred clump as a quasi-point source whose radius is fixed to a maximum of 0.01 arcsec (51 pc) and a fixed S'ersic index of 1 to suppress prominent wings. In total, we fit a model with 43 free parameters and two fixed parameters (the S'ersic indices of the disc and the clump). \nWe check the success of our fits by exploring the overall data, residual, and model images (see Fig. 8, upper plot). Our fit's residuals are generally consistent with the background as can be seen from the imaging, with slight excess residuals seen in the F410M filter likely from the strong emission lines. We find that the best-fit centre of the core and disc align well, with an offset of 0.005', which is less than 0.2 SW pixel and less than the size of either central components. The core has a small effective size of 16 ± 3 mas and a S'ersic index of 2 . 0 ± 0 . 4, while the disc component has a larger size with 80 ± 16 mas. The left panel of Fig. 9 shows the posterior distributions of key parameters from the ForcePho fit (the flux in the F444W and F277W bands and the half-light radius). As can be seen from the corner plot in Fig. 9, ForcePho obtained informative posterior distributions for both the central core and the disc components. The MCMC approach of ForcePho also allows us to assess the degeneracies in the fitting, as apparent from the covariance in the core and disc fluxes of F444W and F277W. Importantly, the posterior distributions only include the statistical uncertainties and not systematic effects, and we assume an error floor for both the fitted photometry (of 5%) and for the morphological parameters (20% for the size, and 0.3 for the S'ersic index of the core). This is \nmotivated by the tests described below. In addition, the right panel of Fig. 9 shows the ForcePho spectral energy distributions (SEDs) of the core, disc and clump components, which indicate different stellar populations for the three different components. \nWe also test leaving the S'ersic index of the disc free to vary from 0.71.3 allowing for a broader range of values. We find that we obtain a disc with half-light radii of 0.12' ( ± 0 . 02) where the S'ersic index becomes 0.75 ( ± 0 . 30). Although this is broadly consistent with our fiducial run (2 σ ), we note that this run is close to the prior boundary, which cannot be extended to lower S'ersic indices due to the Gaussian mixture approach of ForcePho. We acknowledge that a lower S'ersic index might be preferred by the data. Because of the degeneracy of S'ersic index and size, this run also implies larger sizes, which would only strengthen our results, where the star-forming disc is more extended than the core of the galaxy. Finally, we check the chi-squared value of this run, finding a value of 0.17 and a reduced chi-squared value of 8.5. This is larger than our fiducial fixed S'ersic component run, which is still the preferred model and includes fewer degrees of freedom. \nAs shown in Section 1.5, both visually and quantitatively, two central components reproduce the observed light distribution of JADES-GS+53.1834327.79097. But what is the evidence for calling the two central components 'disc' and 'core'? Firstly, focusing on the structure, we find that the effective size of the disc is over 4 times larger than the core, which indicates that the core is a much more compact component than the disc. The S'ersic index of the core is consistent with a 'pseudo-bulge' component (2 . 0 ± 0 . 4), that is, we do not find any evidence for a classical bulge-like component. Importantly, we stress that our disc and core are photometric components, and we cannot say anything about the kinematics of those components from our analysis. However, as mentioned above, the ionised gas of JADES-GS+53.18343-27.79097 is actually consistent with a rotationally supported component with v ( r eff ) /σ 0 ≈ 1 . 3, as recently shown in [51], which motivates us to refer to the extended component as a 'disc'. The resulting SEDs of the core and disc components are shown in the right panel of Fig. 9. These SEDs lead to different stellar populations for the core and the disc (see Section 1.7), consistent with the idea of a slightly older core and a younger disc component. In summary, on the basis of the structure and the inferred SEDs and stellar populations, we find support for interpreting the two central components as a disc and a core. \nWe find a consistent interpretation from the direct colour analysis presented in Section 1.3 and Fig. 6. The F277W-F356W and the F356W-F410M colour profiles indicate an outskirt that is dominated by younger stellar populations than the central region. A direct comparison with the core and disc colour obtained from ForcePho should be taken with a grain of salt, because our decomposition allows for mixing of the different components at fixed radius. We perform aperture photometry on the PSF-matched mosaics using a central aperture of 0.2' and an outskirt aperture of 0.4'. We find that the colours for the centre are similar to that of the core although with \nlarger differences in the medium bands likely stemming from emission lines (F277W-F356W=0 . 15 +0 . 07 -0 . 07 mag, F356W-F410M=0 . 09 +0 . 30 -0 . 17 mag for the core and F277W-F356W= -0 . 06 ± 0 . 06 mag, F356W-F410M=1 . 03 ± 0 . 05 mag for the centre), whilst the colours for the outskirts are similar to that of the disc (F277W-F356W= -0 . 06 +0 . 06 -0 . 06 mag, F356W-F410M=1 . 41 +0 . 11 -0 . 05 mag for the disc and F277W-F356W= -0 . 19 ± 0 . 14 mag, F356W-F410M=1 . 07 ± 0 . 14 mag for the outskirts).", '1.5 Motivation for the multiple component fit': "It is crucial to check whether a multi-component fit is warranted by the NIRCam imaging data. Part of our motivation for a multiple-component fit is due to an expectation that the normalisation of different components will vary as a function of wavelength, therefore the structure will also vary with wavelength. One example of this is to fit the short-wavelength (SW) and long-wavelength (LW) bands separately. We find that the LW bands fit is broadly consistent with the fiducial run ( r half =0.028') for the core and 0.088' for the disc, whilst the SW fit inverts the radii of the core and disc components 0.042' and 0.029' respectively. We would expect the LW data to best mirror the fiducial run as we have more possible exposures in this wavelength range, while for the SW bands we are relying on only the exposures from four filters, hence the obvious degeneracies between the components. Fig. 8 upper panel shows the data, the residual, and the best-fit model for the three-component fit in all the JADES filters. We see that we can reproduce the observed light distribution in each filter for this three-component model. We then compare this with two simpler models. We incorporate a χ 2 measure for the ForcePho fits, where we measure the chi values of all pixels within a segmentation map image of the galaxy, as calculated from the F444W image. We then calculate the χ 2 value as the sum of the residuals of all stacked pixels within the segmentation map divided by the noise (estimated from the background). The reason we use a segmentation map is twofold: firstly, we do not want the metric to be dependent upon the size of the cutout, secondly, if we included all pixels in the cutout we would be dominated by the background pixels rather than pixels within the galaxy. We find that for the three-component model, this gives us a χ 2 value of 0.14. We note that the exact values depend on the exact choice of the segmentation map. \nWe run a single-component fit as a comparison to the three-component model. This means that we treat the whole galaxy as a single component and allow its S'ersic index to vary freely from 0.8-6, enabling it to be modelled as a disc or bulge-like component. As can be seen in Fig. 8 bottom, we find that the 1-component fit (bottom panel) under subtracts the central region (i.e., the core) compared to the three-component fit (top panel); it also significantly fails to fit the clump. The single component fit gives a χ 2 value of 0.45 indicating that it has struggled more than the three-component run to reproduce the observed light distribution. Our next test is to run a single component fit for the main galaxy plus the clump (to test whether the core disc fit is warranted). \nOnce again, the model fails to account for the flux in both the centre and the outskirts, as seen in the upper middle panel in Fig. 8 compared to the threecomponent fit in the top panel. This fit gives a χ 2 value of 0.18, showing an improvement over the single-component fit, but it still does not reproduce the observed light distribution as well as the three-component fit. \nWe also test the effect of ignoring the clump by fitting for just the core and disc components. The most significant change is that this extends the halflight radius of the disc component from 80 mas to 129 mas, whilst increasing the core radius to 0.039 mas; in essence, the components expand in radii to try to fit light from the clump. The S'ersic index of the core becomes 2.31. Importantly, we find consistent fluxes for the core and disc components compared to the three-component analysis: changes are well within the uncertainties. Specifically, the core fluxes change less than 10% in all bands except F410M and F444W, for which the fluxes increase by 26% and 25%, respectively. Since the disc fluxes for those bands remain unchanged (changes of less than 3%), this indicates that the core picks up clump long-wavelength light. We find a χ 2 value of 0.25 for the fit showing the importance of modelling the clump component. In summary, this test shows that our main results regarding the stellar population differences between disc and core still hold when ignoring the off-centred clump. \nIn addition to the χ 2 -values mentioned above, we also test metrics for goodness of fit that can incorporate the number of degrees of freedom. This is an important consideration, as the χ 2 value alone, while informing us which model best reproduces the data, cannot tell us whether it is worth the increased complexity of the model. However, determining a reduced chi-squared is not trivial in this case because ForcePho does not optimise the total chi-square itself, but rather fits the individual exposures. As with the aforementioned chisquared, we only consider the pixels within the segmentation map. Our reduced chi-squared is the previously measured chi-squared multiplied by the degrees of freedom. We define the degrees of freedom as the number of independent pixels within the segmentation map minus the number of parameters of the model. The number of independent pixels within the segmentation map is selected as the total number of pixels in the map minus the number of pixels within the full-width half-maximum of the PSF. The number of model parameters are the free parameters within the fitting for the different component fits. For the three-component core+disc+clump fit, we obtain a reduced chi-squared value of 6.7. For the two-component galaxy+clump fit we obtain a value of 11.4. The two-component core+disc fit gives a value of 15.7 and the singlecomponent galaxy gives 35.2. In summary, even after considering the degrees of freedom, our fiducial three-component core+disc+clump has the smallest reduced chi-square value.", '1.6 PSF approximations in ForcePho': "ForcePho approximates the JWST PSFs with a Gaussian mixture model. Four Gaussians are able to describe the key components of the JWST/NIRCam \nFig. 9 Posterior distributions for the ForcePho fit and spectral energy distribution for the resulting components. Left panel: corner plot of the posterior distributions for flux in the F444W filter, F277W filter, and half-light radius for the disc and core component from the ForcePho fit. The corner plot shows that ForcePho constrains these parameters well. We find that the core and disc have a size of 16 ± 2 mas (82 pc) and 80 ± 2 mas (412 pc), respectively. We boost the errors of the core and disc to be a minimum of 20% in order to better account for systematics. Right panel: the spectral energy distribution (SED) of the core, disc and clump components. The x errors correspond to the filter widths, and the y errors correspond to the 1 σ uncertainty propagated forwards from the error map. \n<!-- image --> \nPSFs as provided by WebbPSF. In this section, we explore the validity of these PSF approximations specifically for the data and JADES-GS+53.1834327.79097 presented in this work. \nWe simulate the best-fit 3-component model (Section 1.4) with Galsim [87]. Specifically, we produce the full set of Stage 2 products as given by our observations, assuming the best-fit 3-component model for our galaxy and PSFs directly obtained from WebbPSF. We then refit with a 3-component model with ForcePho, assuming the same setup as described above. Figure 10 shows the recovery of the core-to-total (C/T) ratio as a function of the filter wavelength (left panel) and the half-light radius versus the F277W flux for the three components (right panel), with the contours corresponding to the posteriors and the black point is the input value. We are able to recover to \nwithin 1 σ the C/T ratio, but all wavelengths are biased high, i.e. the recovered cores are overestimated relative to the discs. We find that the C/T ratios at all wavelengths are biased high, i.e. the recovered cores are overestimated relative to the discs. Although we can recover the C / T ratios within 1 σ , this clearly shows a bias on the level of 5 -10%. It is comforting that this bias is nearly wavelength independent (variations within < 5%), which implies that only the normalisation of the SED of the core and the disc are affected, but not their colours. This in turn means that we possibly overestimate the stellar mass of the core by 5% relative to the disc. The right-hand plot of Figure 10 shows that we are able to recover the morphological parameters (i.e. halflight size) well, within 0.05 dex (10 -20%). In order to account for possible systematics, we assume an error floor of 20% for the measured sizes, i.e., inflate the uncertainties if necessary. \nThis test leaves open whether WebbPSF provides accurate PSFs. We have tested this in detail in [88] by constructing and comparing empirical PSFs both from true observed stars (called empirical PSFs [ePSFs]) and from WebbPSF point sources injected at the Stage 2 level images (called model PSFs [mPSFs]). Specifically, we construct ePSFs using the empirical method proposed by [89]. This method solves the centroids and fluxes of a list of input point sources, and then stacks all the point sources together to get the ePSF. For the list of point sources, we visually identified a sample of 15 isolated point sources from JADES, which are bright but unsaturated. We obtain mPSFs by injecting WebbPSF-based PSFs into the Stage 2 images and then mosaiking them in the same way as our normal mosaics, producing 'star' images. We have then constructed mPSFs from those star images in the same way as the ePSFs. Importantly, we find excellent agreement between ePSF and mPSF, with a typical difference ≲ 1% in the radial profile of enclosed energy, from the central pixel out to 3 arcsec. This implies that the prediction of WebbPSF is accurate.", '1.7 SED fitting with Prospector': "Prospector [52] is an SED fitting code which takes in photometric fluxes and flux errors and fits model SEDs to them. It uses the Dynamic Nested Sampling package Dynesty [90] and models the stellar populations via Flexible Stellar Population Synthesis (FSPS [91, 92]), where we use MIST isochrones [93] and a Chabrier [94] initial mass function. \nWe assume a stellar population model similar to that in [95]. Briefly, we assume a flexible SFH with 6 different time bins, where the most recent bin covers the last 5 Myr, with the other bins being split between 5 Myr and 520 Myr ( z = 20) in log steps. We use the standard continuity prior [53], which weights against a bursty SFH. We use a top-hat prior on the log stellar mass, where it varies from 6 to 12. To model the effect of dust attenuation we use a flexible two-component dust model [96], which models a separate birth cloud component (attenuating emission from gas and stars formed in the last 10 Myr) and a diffuse component (attenuating all emission from the galaxy). We use a joint prior on the ratio between the two dust components, where the \nFig. 10 Component recovery test with ForcePho. We simulate the best-fit 3component model with Galsim, assuming the PSF from WebbPSF. We then refit this scene with ForcePho to determine the validity of ForcePho's PSF approximations. Left: the coreto-total (C/T) ratio from the original fit (truth) and the fit (recovered), plotted against filter wavelength. The data is presented as the median of the distribution with the errors corresponding to the 16th and 84th percentiles. Right: the half-light radius against flux in the F277W filter, the contours correspond to the posterior for the recovery tests, while the black point is the value from the original fit. These two plots show that ForcePho's PSF approximations are close to the true value and that we recover the fitted values close to the errors. The deviation seen in the contour plot is driven by the compact half-light radius of the core. \n<!-- image --> \nprior is a clipped normal between 0 and 2, with a mean of 1.0 and a standard deviation of 0.3. The prior on τ V , the optical depth of the diffuse component in the V band, is a clipped normal ranging from 0 to 4 with a mean of 0.3 and a standard deviation of 1. The slope of the dust attenuation law of the diffuse component is a free parameter and is modelled as a power law multiplication of the standard [97] law (with a top-hat prior from -1 to 0.4). We also use a top-hat prior for the log stellar metallicity with a minimum of -2.0 and a maximum of 0.19. For the nebular component, managed by FSPS, we have a freely varying ionisation parameter and gas-phase metallicity [98]. \nFigures 11, 12, and 13 show the corner plots for each component for the stellar mass, sSFR, optical depth of the diffuse component, stellar age (lookback time at which 50% of the stellar mass was formed), and stellar metallicity. The SFHs are shown on the top right. Despite not fully breaking the dust-agemetallicity degeneracy, we are able to constrain the stellar mass and overall SFH well. \nIn order to explore the dependency of our results on the SFH prior, we also tried the bursty-continuity prior ([99]), which enables the SFH to change more rapidly, enabling a more variable (i.e. bursty) star-formation history. In the bursty continuity prior case we obtain stellar masses of log(M ∗ / M ⊙ ) = 8 . 49 +0 . 22 -0 . 35 , log(M ∗ / M ⊙ ) = 7 . 75 +0 . 18 -0 . 09 and log(M ∗ / M ⊙ ) = 6 . 89 +0 . 37 -0 . 14 for the core, disc and clump components, respectively, compared to log(M ∗ / M ⊙ ) = 8 . 48 +0 . 16 -0 . 21 , log(M ∗ / M ⊙ ) = 8 . 05 +0 . 23 -0 . 29 and log(M ∗ / M ⊙ ) = 7 . 29 +0 . 34 -0 . 36 in the standard continuity prior case. Therefore, the stellar masses obtained in both cases \nFig. 11 Corner figure and star-formation history for the core component. Left: Corner plot showing stellar mass ( M ∗ ), specific star-formation rate (sSFR), optical depth ( τ ν ), half-time ( t half ) and stellar metallicity (Z) for the core component as obtained by SED fitting. Right: the SFH for the core component. The data is presented as the median of the distribution with the errors corresponding to the 16th and 84th percentiles. \n<!-- image --> \nare consistent within the errors, suggesting that the stellar masses obtained based on the standard continuity prior are not biased by failing to account for particularly bursty SFHs.", '1.8 SED fitting of the combined photometry': 'In order to assess how this galaxy relates to other galaxies, we need to infer the global stellar populations parameters from the combined photometry, i.e. treating the core, disc, and clump as a single SED. The combined SED can be seen in the top panel of Fig. 14, while the bottom panel shows the corner plot with the SFH inset. \nFig. 12 Corner figure and star-formation history for the disc component. Left: Corner plot showing stellar mass ( M ∗ ), specific star-formation rate (sSFR), optical depth ( τ ν ), half-time ( t half ) and stellar metallicity (Z) for the disc component as obtained by SED fitting. Right: the SFH for the disc component. The data is presented as the median of the distribution with the errors corresponding to the 16th and 84th percentiles. \n<!-- image --> \nWe obtain a stellar mass of log(M ∗ / M ⊙ ) = 8 . 44 +0 . 22 -0 . 26 and SFR 10Myr = 6 . 3 +1 . 6 -1 . 1 M ⊙ yr -1 and a sSFR of log(sSFR / yr) = -7 . 56 +0 . 31 -0 . 25 yr -1 . For comparison, the combined stellar masses of the individual components is log(M ∗ / M ⊙ ) = 8 . 65 +0 . 25 -0 . 30 , while the SFR amounts to SFR 10Myr = 11 . 5 +5 . 7 -4 . 3 M ⊙ yr -1 . This means that the results are consistent within the uncertainties quoted. We show the results of fitting this combined photometry in Fig. 7 as the orange marker. We see that it is consistent with the black marker (the results of adding the stellar masses and SFRs of the core and disc components). In summary, we find that this galaxy increases its SFH, as expected for galaxies at this epoch. \nFig. 13 Corner figure and star-formation history for the clump component. Left: Corner plot showing stellar mass ( M ∗ ), specific star-formation rate (sSFR), optical depth ( τ ν ), half-time ( t half ) and stellar metallicity (Z) for the clump component as obtained by SED fitting. Right: the SFH for the clump component. The data is presented as the median of the distribution with the errors corresponding to the 16th and 84th percentiles. \n<!-- image -->', '1.9 The clump component': "In this section, we explore the clump component seen in the imaging data and modelled as a point source. The left panel of Fig. 15 shows the RGB image of the galaxy with the position of the clump highlighted. The clump itself does not fall into the NIRSpec slit, so it is natural to wonder if it is actually at the same redshift as the core-disc components. Based on the RGB image, it appears to have similar colours, suggesting that this is likely. To quantify this further, we determine a photometric redshift by running Prospector on the ForcePho photometry with redshift left as a free parameter with a top-hat prior varying from 0.1-13. Figure 15 shows the resulting density distribution for the redshift of the clump component. It appears to be double-peaked, but \n<!-- image --> \n+0.22 \nFig. 14 SED fit, corner figure and SFH for the combined photometry. Upper panel: Spectral energy distribution (SED) fits for the single-component ForcePho fit. The yellow points show the photometry inferred from the ForcePho modelling, while the 2 σ upper limits are indicated as downward pointing arrows. The errors correspond to the 1 σ uncertainties from the photometric pipeline. The open squares mark the photometry from the best-fit SED model. The solid lines and the shaded regions show the median and the 16th-84th percentile of the SED posterior from the Prospector modelling, respectively. Lower panel: corner plot showing stellar mass ( M ∗ ), specific star-formation rate (sSFR), optical depth ( τ V ), half-time ( t half ) and stellar metallicity (Z) for the single component ForcePho fit as obtained by SED fitting, with the SFH inset. These two plots show that when fit as single component, the stellar mass and sSFR inferred, traces that of the combined galaxy. \n<!-- image --> \n<!-- image --> \nFig. 15 RGB image highlighting the clump and 1D posterior distribution for the photometric redshift of the clump component. Left: RGB image of JADES-GS+53.18343-27.79097 with the clump position highlighted. Right: the marginalised posterior distribution for the photometric redshift of the clump component. We see it is double peaked but consistent with the spectroscopic redshift of the core and disc galaxy. \n<!-- image --> \nFig. 16 The NIRSpec spectrum compared to the ForcePho photometry. The 1D NIRSpec spectrum (black) and synthetically derived photometry from the spectrum (orange points). The observed ForcePho photometry for the disc (blue), core (red) and combined core+disc galaxy (lilac) is overplotted. We find the synthetic photometry matches the photometry of the combined core + disc galaxy, highlighting that the combined mediumand broad-band photometry traces both the stellar continuum and nebular emission line. \n<!-- image --> \nis consistent with the core and disc component's spectroscopic redshift. This means that it is realistic to consider that the clump is either part of the galaxy (i.e. a violent disc instability) or a merging galaxy at the same redshift.", '1.10 Emission line properties': 'The NIRSpec R100 spectrum contains crucial information. However, as outlined above, we only include the inferred redshift as a constraint when performing SED modelling of the morphologically distinct components. The \nmain reason for this is that the spectrum (see Fig. 1) covers only parts of the galaxy and includes both core and disc components, but the degree of which is unknown. In principle, one could forward model the full ForcePho model through the slit, then perform a simultaneous fit of both SED components. Unfortunately, this is not yet possible with the Prospector framework. \nNevertheless, it is important to compare whether the photometry and spectroscopy are consistent with each other and to compare the SED derived quantities with the ones obtained only from the spectrum. Figure 16 shows the NIRSpec spectrum (black) with synthetic photometry (orange) produced by integrating the flux from the spectrum within each filter. The observed photometry for the core (red) component, the disc (blue), and the combined core+disc galaxy (lilac) is overplotted. The key comparison here is with the combined photometry (in lilac). We see that the synthetic photometry follows the same trends as the combined ForcePho photometry with strong jumps flux in F410M and F444W corresponding to the strong line emission in the galaxy. We use the measured flux from the [OIII] λ 5007 , 4963 doublet and the H β emission line from the spectrum to calculate their contribution to the F410M and F444W filters. We find that they account for 46% of the flux in the F410M band and 23% of the flux in F444W, hence both of these bands are being boosted by the emission line flux. We also see that the overall shape of the synthetic photometry and the actual spectrum mirror the actual photometry with a median χ 2 value of 2.5. If we compare the values of the emission lines from the spectrum to the photometry, we find that they are in good agreement (within 1 σ ) with values of [OIII] λ 5007 , 4963 = 7 . 52 +0 . 10 -0 . 12 [10 -18 erg cm -2 s] and H β = 0 . 91 +0 . 11 -0 . 08 [10 -18 erg cm -2 s] from the observed NIRSpec spectrum and [OIII] λ 5007 , 4963 = 8 . 45 +0 . 6 . 86 -3 . 33 [10 -18 erg cm -2 s] and H β = 1 . 89 +0 . 1 . 87 -0 . 74 [10 -18 erg cm -2 s] for the combined core+disc galaxy from the best-fit photometry. \nWe infer several important quantities from the emission lines obtained from the NIRSpec R100 prism spectra (see the bottom panel of Fig. 1). First, we do the fitting and measure the fluxes (Section 1.2), then we correct the fitted emission lines for extinction using the ratio between the H γ and H β Balmer lines (as in [100]). The intrinsic value of the ratio (assuming Case B recombination, electron temperature T e = 1 . 5 × 10 4 K and an electron density of N e = 300 cm -3 ) is 0.468. We measure H γ /H β =0.409 meaning we are seeing the effects of dust. We obtain a dust extinction of A V , gas = 0 . 8 +1 . 2 -0 . 8 mag in the V band. We also note that this dust extinction is also consistent with that obtained from our SED fitting (of only the photometry), where we obtain A V = 0 . 31 +0 . 20 -0 . 13 mag for the disc. \nWe calculate the gas-phase metallicity of JADES-GS+53.18343-27.79097 using the strong line method (e.g., [101, 102, 103, 104, 105]) which uses the ratios of strong emission lines, in this case [OIII] λ 5007, [OII] λ 3727, H β , and [NeIII] λ 3969. We use the strong-line metallicity diagnostics from [102]. \n<!-- image --> \nFig. 17 Emission line diagnostic diagram and examination of fits to the R1000 spectrum. Left panel: A classical BPT [109] diagram showing the ratio of the emission lines [OIII] λ 5007 to H β versus [ NII ] λ 6584 / H α . At this redshift NIRSpec can no longer detect [ NII ] λ 6584 and H α so we plot a red horizontal line for the value of [OIII] λ 5007/H β , the colour fill corresponds to the 1 σ error propagated through from the pipeline. We see that the galaxy is still consistent with a star-forming galaxy and shows similar [OIII] λ 5007/H β ratios to other star-forming galaxies at these ratios [69]. Right panel: fits to the H β emission line and the [OIII] doublet. We see that both can be with standard Gaussians and that there is no need for an underlying broad component. \n<!-- image --> \nWe obtain a gas-phase metallicity of 12+log(O/H)=7 . 86 +0 . 09 -0 . 09 , broadly consistent with that of other galaxies at these redshifts [106]. This is equivalent to a value of log(Z gas /Z ⊙ ) = -0.83, which is larger than the stellar metallicities inferred for the core and disc from Prospector, but broadly consistent with the average of the gas-phase metallicity inferred for the two with log(Z gas /Z ⊙ )= -1 . 46 +0 . 72 -0 . 37 for the disc and log(Z gas /Z ⊙ )= -0 . 955 +0 . 54 -0 . 26 for the core. \nWe can calculate an estimate of the SFR from the dust-corrected H β emission line by assuming a Balmer decrement flux ratio of F H α / F H β = 2 . 86, corresponding to case B recombination at a temperature of T ∼ 10 4 K (as in [107, 70]). This enables us to estimate H α . We then convert this H α flux into a H α luminosity, which we convert into a SFR using the conversion detailed in [108]. This gives us a SFR of SFR = 8 . 5 +34 . 7 -7 . 7 M ⊙ yr -1 . This is consistent with the combined SFR of the core and disc obtained via SED fitting with Prospector where SFR 10Myr = 11 . 5 +5 . 7 -4 . 3 M ⊙ yr -1 .', '1.11 Star formation vs AGN': "Is the central component a stellar core or an AGN? We know that the central component of this galaxy is compact, with a deconvolved half-light radius of about 80 pc and a S'ersic index of 2.0. We use dust corrected emission line diagnostics from the NIRSpec spectrum to investigate a possible AGN contribution. We use the ratio [OIII] λ 5007/H β from the classical BPT [109] diagram and find that this gives us a value of ∼ 4.5 (see Fig. 17), which, while large for the local Universe, is consistent with star-formation at high redshifts [69, 110]. Fig. 17 left panel shows the BPT diagram of [OIII] λ 5007/H β against [NII] λ 6584/H α . As JADES-GS+53.18343-27.79097 has a redshift of 7.43, the \n[NII] λ 6584 and H α emission lines are shifted out of the NIRSpec wavelength coverage, hence we plot the value of [OIII] λ 5007/H β as a straight red line. We also show the combined data stacks of [69] and [106]. We overplot contours from local SDSS galaxies with data from [104]. We see that the line ratio for JADESGS+53.18343-27.79097 appears consistent with those of the stacks for galaxies at similar redshifts, but this itself cannot rule out an AGN contribution (as shown in [111]). Similarly, we do not find indications for strong high-ionisation emission lines such as He λ 4686 that cannot be explained by stellar emission. Fig. 17 right panel shows line fits to the H β and [OIII] doublet. The key takeaway from this is that we see no broadening of the H β line compared to the [OIII]. This means JADES-GS+53.18343-27.79097 shows no evidence of being a type-1 AGN. \nOverall, this shows that JADES-GS+53.18343-27.79097 appears to be consistent with pure star formation, but we note that it is difficult to fully rule out an AGN.", 'Data availability': 'The JADES data is publicly available at https://jades-survey.github.io/ scientists/data.html or through the Mikulski Archive for Space Telescopes (MAST) https://archive.stsci.edu/hlsp/jades. Additional data derived from the raw products is available from the corresponding author upon reasonable request.', 'Code availability': 'AstroPy [112], Prospector [52], Dynesty [90], FSPS [91, 92], Galsim [87], WebbPSF and Photutils [113], are all publicly available, while ForcePho (Johnson et al. in prep) is publicly available via GitHub at https://github.com/ bd-j/forcepho.', 'Acknowledgements': "ST acknowledges support by the Royal Society Research Grant G125142. WB, TJL, FDE, RM, JW, LS and JS acknowledge support by the Science and Technology Facilities Council (STFC) and by the ERC through Advanced Grant 695671 'QUENCH'. RM also acknowledges funding from a research professorship from the Royal Society. JW further acknowledges support from the Fondation MERAC. This study made use of the Prospero high performance computing facility at Liverpool John Moores University. BDJ, EE, MR, BER and CNAW acknowledge support from the JWST/NIRCam Science Team contract to the University of Arizona, NAS5-02015. ECL acknowledges support of an STFC Webb Fellowship (ST/W001438/1). SC acknowledges support by European Union's HE ERC Starting Grant No. 101040227 - WINGS. AJB, JC acknowledge funding from the 'FirstGalaxies' Advanced Grant from \nthe European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No. 789056). SA acknowledges support from the research project PID2021-127718NB-I00 of the Spanish Ministry of Science and Innovation/State Agency of Research (MICIN/AEI). H Ugratefully acknowledges support by the Isaac Newton Trust and by the Kavli Foundation through a Newton-Kavli Junior Fellowship. DJE is supported as a Simons Investigator and by JWST/NIRCam contract to the University of Arizona, NAS5-02015. D.P. acknowledges support by the Huo Family Foundation through a P.C. Ho PhD Studentship. A.L.D. thanks the University of Cambridge Harding Distinguished Postgraduate Scholars Programme and Technology Facilities Council (STFC) Center for Doctoral Training (CDT) in Data intensive science at the University of Cambridge (STFC grant number 2742605) for a PhD studentship. The reserach of CCW is supported by NOIRLab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. The authors acknowledge use of the lux supercomputer at UC Santa Cruz, funded by NSF MRI grant AST 1828315. Funding for this research was provided by the Johns Hopkins University, Institute for Data Intensive Engineering and Science (IDIES). This research is supported in part by the Australian Research Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project number CE170100013.", 'Author Contributions Statement': 'WMB and ST led the writing of the paper. WMB performed the analysis (identification of the target, ForcePho fitting, SED modelling, and figure making) under the supervision of ST and BDJ. All authors have contributed to the interpretation of the results. DJE, BDJ, BR, ST, DP, RH, ZJ, and CNAW contributed to the NIRCam imaging reduction. EN, KS, DJE, BDJ, BR, and ST contributed to the analysis and interpretation of the NIRCam imaging data. FDE contributed to the development of tools for the spectroscopic data analysis. SC, SA, MC, and JW contributed to the reduction of NIRSpec data and the development of the NIRSpec pipeline. AB contributed to the design and optimisation of the MSA configurations. FDE, MC, ADG, RM and JS helped with the interpretation of the NIRSpec data.', 'Competing Interests Statement': 'The authors declare that they have no competing interests.', 'References': "- [1] Bouch'e, N. et al. The Impact of Cold Gas Accretion Above a Mass Floor on Galaxy Scaling Relations. Astrophys. J. 718 , 1001-1018 (2010). \n- [2] Dekel, A. & Krumholz, M. R. Steady outflows in giant clumps of high-z disc galaxies during migration and growth by accretion. Mon. Not. R. Astron. Soc. 432 , 455-467 (2013).\n- [3] Fall, S. M. & Efstathiou, G. Formation and rotation of disc galaxies with haloes. Mon. Not. R. Astron. Soc. 193 , 189-206 (1980).\n- [4] Mo, H. J., Mao, S. & White, S. D. M. The formation of galactic discs. Mon. Not. R. Astron. Soc. 295 , 319-336 (1998).\n- [5] van den Bosch, F. C. 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2024arXiv240905110P

Modifications in quantum mechanical phase space lead to changes in the Heisenberg uncertainty principle which can result in the Generalized Uncertainty Principle GUP or the Extended Uncertainty Principle EUP introducing quantum gravitational effects at small and large distances respectively. A combination of GUP and EUP the Generalized Extended Uncertainty Principle GEUP or EGUP further generalizes these modifications by incorporating noncommutativity in both coordinates and momenta. This paper examines the impact of GEUP on the Liouville theorem in statistical physics and density of states within nonrelativistic quantum mechanics framework. We find a weighted phase space volume element invariant under the infinitesimal time evolution in the cases of Snyderde Sitter and Yang models presenting how GEUP alters the density of states potentially affecting physical thermodynamical properties. Special cases obtained in certain limits from the above models are also discussed. New higher order types of GEUP and EUP are also proposed.

2024-09-01T00:00:00Z

['10.48550/arXiv.2409.05110', 'arXiv:2409.05110', '2024arXiv240905110P']

['High Energy Physics - Theory', 'General Relativity and Quantum Cosmology', 'Quantum Physics']

Generalized Extended Uncertainty Principles Liouville theorem and density of states Snyderde Sitter and Yang models

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.05110.pdf

{'Generalized Extended Uncertainty Principles, Liouville theorem and density of states: Snyder-de Sitter and Yang models': 'Anna Pacho/suppressl \nDepartment of Microsystems, University of South-Eastern Norway, Campus Vestfold, Norway \nModifications in quantum mechanical phase space lead to changes in the Heisenberg uncertainty principle, which can result in the Generalized Uncertainty Principle (GUP) or the Extended Uncertainty Principle (EUP), introducing quantum gravitational effects at small and large distances, respectively. A combination of GUP and EUP, the Generalized Extended Uncertainty Principle (GEUP or EGUP), further generalizes these modifications by incorporating noncommutativity in both coordinates and momenta. This paper examines the impact of GEUP on the Liouville theorem in statistical physics and density of states within non-relativistic quantum mechanics framework. We find a weighted phase space volume element, invariant under the infinitesimal time evolution, in the cases of Snyder-de Sitter and Yang models, presenting how GEUP alters the density of states, potentially affecting physical (thermodynamical) properties. Special cases, obtained in certain limits from the above models are also discussed. New higher order types of GEUP and EUP are also proposed.', 'I. INTRODUCTION': "The search for the fundamental theory of Quantum Gravity has been supported by the development of phenomenological models that explore possible modifications to the known quantum mechanical or gravitational phenomena. In this context the various purely phenomenological proposals have appeared aiming to capture potential signatures of quantum effects of gravity, hinting towards possible experimental set-ups which would help guide the way in the full formulation of the theory. What is known is that one needs to challenge the concepts of classical space-time. The idea that the structure of space-time should be modified is a common feature of various approaches to Quantum Gravity, such as String Theory, Loop quantum gravity, Causal Dynamical Triangulations, Asymptotically Safe Quantum Gravity, Horava-Lifshitz Gravity and Noncommutative Geometry, just to name a few [1-6]. Many of these share also the concept of minimum length [7]. Such minimal length can be implemented in quantum mechanics through introducing the modifications in Uncertainty Principles (UP) 1 : \n∆ x ∆ p ≥ 1 2 |〈 [ x, p ] 〉| (1) \nwhere ∆ denotes standard deviation and 〈 〉 the quantum expectation value on a given state. From the above relation, it is clear that any modifications in the canonical Heisenberg commutation relations between coordinates and momenta will imply changes in the UP. Various generalizations of UP have been considered in the literature. The most common type of such modifications is the Generalized Uncertainty Principle (GUP) with the momenta-dependent right hand side of (1). It was firstly proposed in [8] and then related to the specific algebraic structure of quantum phase space [9-12]. The right hand side of (1) may include quadratic terms in momenta (resulting in QGUP), see e.g. [11-13] or have linear and quadratic terms (LQGUP), see e.g. [14-18]. In momenta-dependent \n/negationslash \nGUP relations one introduces the parameter, usually denoted by β , which is related to the Planck (length) scale l P , hence linking GUPs with Planck scale physics and quantum gravitational effects. Such phenomenological models have attracted a lot of attention [9-35], see also [36] for a recent review and more references on the topic. \nOn the other hand, the symmetry between the position and momenta in the canonical (quantum mechanical) commutation relations, as well as Born reciprocity [37], suggests the possibility of introducing corrections to the Heisenberg uncertainty principle by including the modifications proportional to coordinates instead, i.e. so that the RHS of (1) is quadratic or linear in coordinates, instead of momenta. This complimentary type of uncertainty relation has been called an Extended Uncertainty Principle (EUP) [38, 39]. Here, the parameter of the model, usually denoted by α is related with the non-vanishing cosmological constant Λ and this way it can be embedded in the non-relativistic quantum mechanics. It has been shown [40] that this type of modification to the UP is related with the (Anti-)de Sitter geometric background, and the parameter α is then naturally linked with the (Anti-)de Sitter radius. While GUP, with β ∼ l P , exposes the gravitational modifications in quantum mechanics at the small distances; EUP, with α ∼ Λ introduces the idea of modifications at large distances. Relying on the analogy with GUPs, various types of such coordinate-dependent models have been introduced leading to different interesting physical effects, see e.g. [38, 39, 41, 42] \nCombination of GUP and EUP (i.e. considering both coordinate and momenta dependent terms on the RHS of (1)) appeared firstly in the construction of noncommutative quantum mechanics with quantum groups as a symmetry [43], where both nonzero minimal uncertainties have appeared for position and momentum observables, respectively. Later on the relation of the type: \n∆ x ∆ p ≥ /planckover2pi1 2 (1 + α 2 (∆ x ) 2 + β 2 (∆ p ) 2 ) (2) \nunder the name of Generalized Extended Uncertainty Principle (GEUP, or alternatively EGUP), has been studied in various contexts, see e.g. [38-40, 44], for more recent work see e.g. [45] where Born reciprocity is also promoted or see e.g. [46] for applications in statistical physics. \nEven though the origins of GUPs, EUPs and GEUPs can be tracked to the noncommutative geometry and quantum groups as the underlying mathematical frameworks [10, 43], these models are mainly treated as purely phenomenological in the majority of the literature related to this subject. They have been used (and have been quite fruitful) in providing predictions for various phenomenological effects. Such departure from the underlying mathematical framework resulting in the specific form of GUP, EUP or GEUPs together with the non-uniqueness of the defining commutation relations between coordinates and momenta has lead to many conceptual shortcomings and ambiguities (see e.g. the recent review [36] or [47] for discussion of some of the arising issues). Moreover, it is worth to point out that the minimal length may not necessarily appear if one bases only on the modifications of the quantum mechanical phase space, see e.g. [48-51]. \n/negationslash \nIn this paper, following the point of view that the noncommutative geometry is the underlying framework (mathematical language) of the possible fundamental theory, we assume that the quantization process of general relativity, includes the quantization of space-time [52, 53], i.e. requiring the space-time coordinates to become noncommutative [ˆ x µ , ˆ x ν ] = 0. The noncommutativity of spacetime introduces corrections to the canonical quantum-mechanical phase space relations. Hence modifications to the UP appear as a natural consequence. In this view, quadratic GUPs have been mainly linked with the Snyder model [10, 34, 35], with noncommuting coordinates and commuting momenta. EUPs, on the other hand can be linked with the (anti)-de Sitter geometric background and algebraic structure with commuting coordinates but noncommuting momenta [40]. The main aim of this paper is to show how the noncommutativity of both coordinates [ˆ x i , ˆ x j ] = 0 and mo- \n/negationslash \nmenta [ˆ p i , ˆ p j ] = 0 leading to GEUPs (and in some cases, the appearance of the nonzero minimal uncertainties in positions and momenta separately) affects the density of states and the analogue of the Liouville theorem in statistical mechanics. For this reason we limit ourselves to the case of non-relativistic quantum mechanics. We find the new form of weighted phase space volume element in the presence of GEUPs in the specific cases of Snyder-de Sitter (SdS) and Yang models. We show that noncommuting coordinates and noncommuting momenta with the corresponding GEUP require introducing the modification in the density states and this may impact various physical effects and thermodynamical properties (as it has been shown in the case of various GUP models). \nIn the next section, we summarise the framework generalized to the case where both coordinates and momenta do not commute and present the general formulae for the Jacobian arising from the variable change under the infinitesimal time evolution. In Sec. 3, we specify the model to the Snyderde Sitter (SdS) algebra, as only then we can identify the required weight factor (which depends on both coordinates and momenta) for the phase space volume element. The factor we obtain in this case does not depend dimension D . We also discuss the special cases obtained in the certain limits of the parameters, giving GUP or EUP relations (by reducing the starting SdS algebra to Snyder or dS algebras, respectively) and we give the expressions for the phase space volume element in these cases. In Sec. 4, we discuss the Yang model and consider specific realizations of its generators on the canonical phase space leading to another type for (higher order) GEUP relation. The weighted phase space volume element is obtained for this case in 1 dimension by adapting the result from SdS model. Special cases, obtained in certain limits are also discussed, one provides the known 'square-root' (or Maggiore) GUP and the other leads to the new higher order ('square-root') type of EUP. In Sec. 5, the Lie algebraic case with commuting momenta is briefly considered with the fuzzy sphere as an example. It is shown that in this case the phase space volume element stays invariant under the time evolution and there is no change in the density of states.", 'II. PRELIMINARIES': "In this section we set up the most general framework for investigating the effects of noncommuting coordinates and noncommuting momenta (and in principle the appearance of the nonzero length and momentum uncertainties) on the density of states in the phase space so that we can adapt the Liouville theorem in statistical physics to this new scenario. For this reason we do not fix the specific choice for the noncommutativity of space-time or momena and the resulting deformation of the quantum phase space commutation relations at this point yet. \nWe start with the following (most general) set of commutation relations describing the noncommutative quantum mechanical phase space algebra: \n[ˆ x i , ˆ x j ] = i /planckover2pi1 a ij (ˆ x, ˆ p ) , [ˆ p i , ˆ p j ] = i /planckover2pi1 b ij (ˆ x, ˆ p ) (3) \n[ˆ x i , ˆ p j ] = i /planckover2pi1 c ij (ˆ x, ˆ p ) , (4) \nwhere a ij (ˆ x, ˆ p ) , b ij (ˆ x, ˆ p ) , c ij (ˆ x, ˆ p ) are functions which may include all kinds of terms (linear, quadratic, higher order etc.) in space-time coordinates ˆ x and momenta ˆ p , such that the Jacobi identities are satisfied. We will focus on the non-relativistic general case in any dimension D with i, j = 1 , 2 , . . . , D . When none of the above commutators are zero (see e.g. [45, 54-57]), this will lead to the interesting types of the GEUPs: ∆ˆ x i ∆ˆ p j ≥ /planckover2pi1 2 |〈 c ij (ˆ x, ˆ p ) 〉| with the specific form of RHS depending on the concrete choice of the algebra (3), (4). \nIn the phenomenological approaches, one considers the right hand side of (4) as the definition of a new effective value of /planckover2pi1 which (in the most general case) may depend on both coordinates \nand momenta. This means that the size of the unit cell that each quantum state occupies in the phase space can be thought of as being also coordinates and momenta dependent. This will have an effect on the density of states and as a consequence affect physical, for example thermodynamical, properties. For this interpretation to be valid, the volume of phase space must evolve in such a way that the number of states does not change with time, in other words we are are looking for the analogue of the Liouville theorem in statistical physics. Similar investigation has been done in the case of the quadratic GUP [12, 13] or GUPs with higher order terms [58]. Note that here we assume that both coordinates and momenta do not commute. Nevertheless, later on we shall see how this more general case can be reduced to the special cases like the GUP or EUP (which include some commutative generators). \nStarting with the quantum mechanical commutation relations (3), (4), these will correspond to the Poisson brackets in classical mechanics: \n1 i /planckover2pi1 [ ˆ A, ˆ B ] classical limit ---------→ { A,B } . (5) \nTherefore, in the classical limit 2 we obtain (in our shortcut notation): \n{ x i , x j } = a ij , { p i , p j } = b ij , (6) \n{ x i , p j } = c ij . (7) \nThe time evolution of the coordinates and momenta are governed by the equations (where the summation convention is assumed) \n˙ x i = { x i , H } = { x i , p j } ∂H ∂p j + { x i , x j } ∂H ∂x j = c ij ∂H ∂p j + a ij ∂H ∂x j , (8) \n˙ p i = { p i , H } = -{ x j , p i } ∂H ∂x j + { p i , p j } ∂H ∂p j δt = -c ji ∂H ∂x j + b ij ∂H ∂p j . (9) \nThe Liouville theorem requires that the weighted phase space volume is invariant under time evolution. Hence we consider an infinitesimal time interval δt and the evolution of the coordinates and momenta during δt is: \nx ' i = x i + δx i , p ' i = p i + δp i \nwhere \nδx i = ˙ x i δt = ( c ij ∂H ∂p j + a ij ∂H ∂x j ) δt, δp i = ˙ p i δt = ( -c ji ∂H ∂x j + b ij ∂H ∂p j ) δt. (10) \nThe infinitesimal phase space volume after this infinitesimal time evolution will be: \nd D x ' d D p ' = Jd D xd D p \nwith the Jacobian \nwhere we used: \nJ = ∣ ∣ ∣ ∣ ∂ ( x ' 1 , . . . , x ' D , p ' 1 , . . . , p ' D ) ∂ ( x 1 , . . . , x D , p 1 , . . . , p D ) ∣ ∣ ∣ ∣ = 1 + ( ∂δx i ∂x i + ∂δp i ∂p i ) + . . . (11) \n∂x ' i ∂x j = δ ij + ∂δx i ∂x j , ∂x ' i ∂p j = ∂δx i ∂p j , (12) \n∂p ' i ∂x j = ∂δp i ∂x j , ∂p ' i ∂p j = δ ij + ∂δp i ∂p j . (13) \nUp to the first order in δt (based on (10)) we get: \n( ∂δx i ∂x i + ∂δp i ∂p i ) = ∂ ∂x i ( c ij ∂H ∂p j + a ij ∂H ∂x j ) δt + ∂ ∂p i ( -c ji ∂H ∂x j + b ij ∂H ∂p j ) δt = [( ∂ ∂x i a ij -∂ ∂p i c ji ) ∂H ∂x j + ( ∂ ∂x i c ij + ∂ ∂p i b ij ) ∂H ∂p j ] δt (14) \nwhere the the terms with mixed derivatives have cancelled each other and the antisymetricity of the Poisson brackets was used to cancel the remaining terms. This way we find the expression for the time evolved infinitesimal phase space volume (in the first order of δt ) as: \nd D x ' d D p ' = d D xd D p ( 1 + [( ∂ ∂x i a ij -∂ ∂p i c ji ) ∂H ∂x j + ( ∂ ∂x i c ij + ∂ ∂p i b ij ) ∂H ∂p j ] δt + O ( δt 2 ) ) (15) \nvalid for any Poisson algebra (6), (7) obtained as the classical limit of any noncommutative model (3), (4). Since in general the terms in the brackets will not cancel out, already here we see the need to introduce the weight to the phase space volume element so that the analogue of the Liouville theorem is satisfied. The specific factor has to be chosen in such a way that the weighted phase space volume is invariant under the time evolution, i.e. so that: \nd D x ' d D p ' F ( x ' , p ' ) ∼ d D xd D p F ( x, p ) . (16) \nIt is straightforward to notice from (15) that the noncanonical Poisson brackets for coordinates and for momenta (6), as well as the mixed relation between coordinates and momenta (7) are crucial in the choice of the weighted phase space volume. Hence the underlying noncommutative geometry (and the choice of (3), (4)) is the intrinsic feature of phenomenological models investigating the possible effects arising from such modified phase space volume element (and consequently affecting the thermodynamical properties of physical systems). To be able to investigate the time evolution of the weight factor F ( x, p ), we need to consider the concrete noncommutative model, which we shall do in the next section.", 'III. SNYDER-DE SITTER MODEL AND THE DENSITY OF STATES': "Snyder-de Sitter (SdS) model, which includes the noncommutative space-time coordinates and the noncommutative momenta, was proposed [54] as a generalization of the Snyder model [60] to a \nspace-time background of constant curvature and it was investigated in many contexts, see e.g. [61], [62]. In SdS model, the noncommutativity among space-time coordinates corresponds to the curved momentum space, and vice-versa noncommutative momenta lead to the curved space-time. Since we are interested in investigating how the noncommutativity of both coordinates and momenta affects the density of states we choose SdS set of commutation relations which exhibit these features: \n[ˆ x µ , ˆ x ν ] = i /planckover2pi1 β 2 ˆ M µν , [ˆ p µ , ˆ p ν ] = i /planckover2pi1 α 2 ˆ M µν , (17) \n[ ˆ M µν , ˆ x ρ ] = i /planckover2pi1 ( η µρ ˆ x ν -η νρ ˆ x µ ) , (18) \n[ ˆ M µν , ˆ p ρ ] = i /planckover2pi1 ( η µρ ˆ p ν -η νρ ˆ p µ ) , (19) \n[ ˆ M µν , ˆ M ρτ ] = i /planckover2pi1 ( η µρ ˆ M ντ -η µτ ˆ M νρ + η ντ ˆ M µρ -η νρ ˆ M µτ ) , (20) \nwith the modified quantum mechanical phase space relation: \n[ˆ x µ , ˆ p ν ] = i /planckover2pi1 ( η µν + α 2 ˆ x µ ˆ x ν + β 2 ˆ p µ ˆ p ν + αβ (ˆ x µ ˆ p ν + ˆ p µ ˆ x ν -ˆ M µν ) ) . (21) \nHere η µν is the flat metric with Lorentzian signature and µ, ν = 0 , 1 , . . . , D . Depending on the sign 3 of α 2 the Lorentz generators ˆ M µν and momenta ˆ p µ generate de Sitter (dS) or Anti-de Sitter (AdS) subalgebras. This model involves, besides the speed of light 4 , two other observer-independent constants 5 , the Planck length as well as the de Sitter radius which is related to the cosmological constant. More precisely, the parameter | α 2 | has the dimension of the inverse of the square of length and it can be identified with the (Anti-)de Sitter radius and a cosmological constant as its inverse, while the parameter | β 2 | has the dimension of the inverse square of mass and it can be is identified with 1 /M 2 P = l 2 P , where M P is the Planck mass. In general, α 2 and β 2 can take positive or negative value, which generates models with very different properties [61]. \nIn the following we focus on the non-relativistic quantum mechanical counterpart (which in D = 3 would give the Snyder model restricted to a three-dimensional sphere) however we keep unspecified dimension D , i, j = 1 , 2 , . . . , D . The GEUP corresponding to (21) has the form similar to (2) [40] 6 , but additional (mixed) terms may appear on the RHS depending on the choice of the representation of the angular momentum used, see e.g. [61]. Without any additional assumptions we can, at most, write: \n(∆ˆ x i ) (∆ˆ p i ) ≥ /planckover2pi1 2 ∣ ∣ 1 + α 2 (∆ˆ x ) 2 + β 2 (∆ˆ p ) 2 + γ ∣ ∣ (22) \n∣ ∣ where we used (∆ˆ x ) 2 = 〈 ˆ x 2 〉 -〈 ˆ x 〉 2 , (∆ˆ p ) 2 = 〈 ˆ p 2 〉 -〈 ˆ p 〉 2 and γ = α 2 〈 ˆ x 〉 2 + β 2 〈 ˆ p 〉 2 + αβ ( 〈 ˆ x ˆ p 〉 + 〈 ˆ p ˆ x 〉 ) as well as the standard notation ˆ x 2 = ˆ x i ˆ x i , ˆ p 2 = ˆ p i ˆ p i , ˆ x ˆ p = ˆ x i ˆ p i was assumed, while on the LHS in (∆ˆ x i ) (∆ˆ p i ) there is no summation. In 1-dimensional case, one can show [61] that, for both α 2 > 0 \nand β 2 > 0 the bounds on the localization in position and momentum space arise, while when α 2 < 0 and β 2 < 0 a combination of spatial and momentum coordinates becomes bounded instead. The classical (non-relativistic) Snyder-de Sitter Poisson algebra is given by 7 : \n{ x i , x j } = β 2 ( x i p j -x j p i ) ≡ a ij , { p i , p j } = α 2 ( x i p j -x j p i ) ≡ b ij , (23) { x i , p j } = δ ij + α 2 x i x j + β 2 p i p j +2 αβp i x j ≡ c ij . \nand supplemented by the relations with Lorentz generators. By using the general set up from the previous section and now the explicit form of SdS model (23) we obtain that the phase space volume element after the infinitesimal time evolution as: \nd D x ' d D p ' = d D xd D p ( 1 + 2 [ ( α 2 x j + αβp j ) ∂H ∂p j -( β 2 p j + αβx j ) ∂H ∂x j ] δt + O ( δt 2 ) ) (24) \nwhere we explicitly used (23) in (15) with the following: \n∂ ∂x i a ij = β 2 ( D -1) p j , ∂ ∂x i c ij = α 2 ( D +1) x j +2 αβp j , (25) ∂ ∂p i b ij = α 2 (1 -D ) x j = -α 2 ( D -1) x j , ∂ ∂p i c ji = β 2 ( D +1) p j +2 αβx j . (26) \nIn the remaining part of this section we will show that for the SdS Poisson algebra (23) the analogue of the Liouville theorem is satisfied when we consider the following weighted phase space volume \nd D xd D p (1 + α 2 x 2 + β 2 p 2 +2 αβx · p ) (27) \nwhich is invariant under the infinitesimal time evolution. \nTo show this, we consider the time evolution of each of the terms in the proposed factor F ( x, p ) = 1+ α 2 x 2 + β 2 p 2 +2 αβx · p during the infinitesimal time interval δt . At first we keep all the formulae as general as possible and express them in terms of a ij , b ij and c ij and only at the end we will specify to SdS case (23). We get the following expressions: \nx ' 2 = ( x i + δx i ) 2 = x 2 +2 x i δx i + O ( δt 2 ) = x 2 +2 x i ( c ij ∂H ∂p j + a ij ∂H ∂x j ) δt + O ( δt 2 ) , (28) \np ' 2 = ( p i + δp i ) 2 = p 2 +2 p i δp i + O ( δt 2 ) = p 2 +2 p i ( -c ji ∂H ∂x j + b ij ∂H ∂p j ) δt + O ( δt 2 ) , (29) \nand for the last term \nx ' · p ' = x ' p ' = ( x i + δx i ) ( p i + δp i ) = xp + p i δx i + x i δp i + O ( δt 2 ) = xp + p i ( c ij ∂H ∂p j + a ij ∂H ∂x j ) δt + x i ( -c ji ∂H ∂x j + b ij ∂H ∂p j ) δt + O ( δt 2 ) . (30) \nTherefore the time evolution of the whole factor F ( x, p ) can be first written generally as (where for simplicity we use 2 αβ = γ ): \n1 + α 2 x ' 2 + β 2 p ' 2 + γx ' p ' = 1 + α 2 x 2 + β 2 p 2 + γxp + + [ 2 α 2 x i c ij +2 β 2 p i b ij + γ ( p i c ij + x i b ij ) ] ∂H ∂p j δt + + [ 2 α 2 x i a ij -2 β 2 p i c ji + γ ( p i a ij -x i c ji ) ] ∂H ∂x j δt + O ( δt 2 ) . (31) \nNow specialising this to the case of SdS (23), after plugging in the expressions for a ij , b ij and c ij , we obtain (in the first order of δt ): \n1 + α 2 x ' 2 + β 2 p ' 2 + γx ' p ' = 1 + α 2 x 2 + β 2 p 2 + γxp + [ 2 α 2 ( x j + α 2 x 2 x j + β 2 ( xp ) p j + γ ( xp ) x j ) +2 β 2 α 2 ( xp ) p j -2 β 2 α 2 x j p 2 + γ ( p j + α 2 x 2 p j + β 2 p 2 p j + γp 2 x j )] ∂H ∂p j δt + + [ -2 β 2 ( p j + α 2 x j ( xp ) + β 2 p j p 2 + γ ( xp ) p j ) +2 α 2 β 2 x 2 p j -2 α 2 β 2 x j ( xp ) -γ ( x j + α 2 x 2 x j + β 2 p 2 x j + γx 2 p j )] ∂H ∂x j δt. (32) \nSince the aim is to factorise the whole expression (1 + α 2 x 2 + β 2 p 2 + γxp ) on the right hand side of this equality, we need to rearrange all the terms in the square brackets in such a way so that we can recognize the whole factor F ( x, p ). In this way, we obtain (once we returned to the notation γ = 2 αβ ): \n1 + α 2 x ' 2 + β 2 p ' 2 +2 αβx ' p ' = = 1 + α 2 x 2 + β 2 p 2 +2 αβ ( xp ) + [ 2 α 2 ( 1 + α 2 x 2 + β 2 p 2 +2 αβ ( xp ) ) x j +2 αβ ( 1 + α 2 x 2 + β 2 p 2 +2 αβxp ) p j ] ∂H ∂p j δt + + [ -2 β 2 ( 1 + α 2 x 2 + β 2 p 2 +2 αβ ( xp ) ) p j -2 αβ ( 1 + α 2 x 2 + β 2 p 2 +2 αβx · p ) x j ] ∂H ∂x j δt + O ( δt 2 ) = ( 1 + α 2 x 2 + β 2 p 2 +2 αβ ( xp ) ) [1 + ( 2 α 2 x j +2 αβp j ) ∂H ∂p j δt -( 2 β 2 p j +2 αβx j ) ∂H ∂x j δt ] + O ( δt 2 ) . (33) \nWe can see that the weighted volume element will stay invariant since the weight factor we introduced produces the same terms as the Jacobian in (24) under the infinitesimal time evolution (up \nto the first order in δt ): \nd D x ' d D p ' 1 + α 2 x ' 2 + β 2 p ' 2 +2 αβx ' p ' = = d D xd D p ( 1 + 2 [ ( α 2 x j + αβp j ) ∂H ∂p j -( β 2 p j + αβx j ) ∂H ∂x j ] δt + O ( δt 2 ) ) (1 + α 2 x 2 + β 2 p 2 +2 αβx · p ) ( 1 + 2 [ ( α 2 x j + αβp j ) ∂H ∂p j -( β 2 p j + αβx j ) ∂H ∂x j ] δt + O ( δ 2 ) ) = d D xd D p (1 + α 2 x 2 + β 2 p 2 +2 αβxp ) . (34) \nWe point out this result holds to any order of parameters α and β (as we have not done any approximations in the noncommutative parameters). \nWe note that from the SdS model considered above we can obtain two well known special cases. Namely: \n- · Snyder model [60] 8 is obtained when we take α → 0 in SdS algebra (17)-(21), i.e. we obtain the model with noncommutative coordinates and commutative momenta (curved momentum space): \n[ˆ x µ , ˆ x ν ] = i /planckover2pi1 β 2 ˆ M µν , [ˆ p µ , ˆ p ν ] = 0 , [ˆ x µ , ˆ p ν ] = i /planckover2pi1 ( η µν + β 2 ˆ p µ ˆ p ν ) , (35) \nsupplemented by the Lorentz covariance conditions (18) - (20). Such modification of quantum mechanical phase space relations leads, in the non-relativistic case, to the quadratic GUP (QGUP) 9 : \n∆ˆ x i ∆ˆ p i ≥ /planckover2pi1 2 (1 + β 2 (∆ˆ p ) 2 ) . (36) \nWe note that the algebraic set of commutation relations (35) is only one of the many possible realizations of the Snyder model and more general realizations may be considered, see e.g. [34, 35]. The non-relativistic classical Poisson algebra: \n{ x i , x j } = β 2 ( x i p j -x j p i ) , { p i , p j } = 0 , { x i , p j } = δ ij + β 2 p i p j (37) \nwill result in the following weighted phase space volume: \nd D xd D p 1 + β 2 p 2 (38) \n(in any dimension D ), obtained as the limit α → 0 in (27). \n- · (Anti-)de Sitter model (dual Snyder model) is obtained when we take β → 0 in SdS algebra (17)-(21), i.e. we obtain the model with commutative coordinates but noncommutative momenta (curved space-time 10 ): \n[ˆ x µ , ˆ x ν ] = 0 , [ˆ p µ , ˆ p ν ] = i /planckover2pi1 α 2 ˆ M µν , [ˆ x µ , ˆ p ν ] = i /planckover2pi1 ( η µν + α 2 ˆ x µ ˆ x ν ) . (39) \nwith the Lorentz covariance given by (18) - (20). The non-relativistic case results in the quadratic EUP (QEUP) 11 : \n∆ˆ x i ∆ˆ p i ≥ /planckover2pi1 2 (1 + α 2 (∆ˆ x ) 2 ) . (40) \nThe non-relativistic classical Poisson algebra is: \n{ x i , x j } = 0 , { p i , p j } = α 2 ( x i p j -x j p i ) , { x i , p j } = δ ij + α 2 x i x j . (41) \nAnd the weighted phase space volume (in any dimension D ), obtained as the limit β → 0 in (27), is: \nd D xd D p 1 + α 2 x 2 . (42) \nIt is worth to point out that there exists a way to transform SdS algebra (17)-(21) generated by (ˆ x, ˆ p, ˆ M ) into the Snyder algebra (35) generated by (ˆ x S , ˆ p S , ˆ M ) by the following noncanonical linear maps: \nˆ x i = ˆ x S i + β α λ ˆ p S i , ˆ p i = (1 -λ ) ˆ p S i -α β ˆ x S i (43) \nwhere λ is a free parameter and we have temporarily denoted the Snyder algebra generators (35) by the upper index S . This relation of the SdS algebra with the Snyder algebra was first noticed in [61]. Through such noncanonical change of basis one can use the already known machinery of realizations in Snyder spaces and consider various applications of SdS algebra, for example to find harmonic oscillator solutions [63] or in applications to quantum field theory [64]. \nOne can also consider various generalizations of the SdS model. For example, in [65] the following generalization of the last relation in SdS algebra (21) was proposed: \n[ˆ x µ , ˆ p ν ] = i /planckover2pi1 ( η µν ϕ 1 + ( α 2 ˆ x µ ˆ x ν + β 2 ˆ p µ ˆ p ν + αβ ˆ x µ ˆ p ν + αβ ˆ p µ ˆ x ν ) ϕ 2 -αβ ˆ M µν ) (44) \nwhere the functions ϕ 1 and ϕ 2 need to satisfy specific conditions (due to Jacobi identities). When ϕ 1 = ϕ 2 = 1 we get back the original SdS model (21). Various choices of ϕ 1 , ϕ 2 are discussed in [65] and for example for one specific choice of ϕ 1 and ϕ 2 we can obtain the following relation: \n[ˆ x µ , ˆ p ν ] = i /planckover2pi1 η µν √ 1 -( α 2 ˆ x 2 + β 2 ˆ p 2 + αβ ˆ x ˆ p + αβ ˆ p ˆ x ) -αβ ˆ M µν . (45) \nSuch generalizations would be interesting to investigate further in the context of GEUPs and their influence on the density of states. \nBefore completing this section, since the effects of modifications of UPs on the density of states have been investigated previously few comments are in order. In [12, 13], in the case of GUP (36) 12 \n/negationslash \nthe invariant weighted phase space volume element is obtained as: d D xd D p (1+ β 2 p 2 ) D , where the power D (dimension) is necessary since the Jacobian obtained is: d D x ' d D p ' = d D xd D p ( 1 -2 β 2 Dp k ∂H ∂x k δt + O ( δt 2 ) ) . \nThis is different than the case considered here, where in the limit α → 0 we obtain d D xd D p 1+ β 2 p 2 (38). The difference arises from the fact that commutation relations used in [12] for [ x i , p j ] have the term proportional to p 2 while here we have the terms with p i p j instead, cf. (37). In the case of the Snyder model with (37) the terms with D in the Jacobian (24) cancel out, hence the power D is not appearing in the weight factor of the volume form (38). \nOther important point is that often in the context of modified UP (1) also the inner product on the Hilbert space 13 becomes modified with appropriately chosen measure so that the observables (satisfying the modified commutation relations) stay symmetric on the dense domain of functions decaying faster than any power [66], see also [12, 13] for the inner product modifications in the case of GUP and for the relativistic case, see e.g. [67]. The inner product modifications in the case of EUP, i.e. in AdS and dS spaces were investigated e.g. in [41]. Subsequently the modified inner product can be used to study effects on solutions of Schrodinger equation, see e.g. [12, 13, 41]. Such modifications in the measure in the inner product have not been the purpose of this paper.", 'IV. YANG MODEL': "The Yang model introduced in [55] is a Lorentz invariant model incorporating noncommutative space-time coordinates as well as noncommutative momenta, depending on the pair of dimensionful parameters α and β related with the curvatures of quantum space-time and momentum spaces (in similarity to SdS model). The defining relations are as follows: \n[ˆ x µ , ˆ x ν ] = i /planckover2pi1 β 2 ˆ M µν , [ˆ p µ , ˆ p ν ] = i /planckover2pi1 α 2 ˆ M µν (46) \n[ ˆ M µν , ˆ x ρ ] = i /planckover2pi1 ( η µρ ˆ x ν -η νρ ˆ x µ ) , (47) \n[ ˆ M µν , ˆ p ρ ] = i /planckover2pi1 ( η µρ ˆ p ν -η νρ ˆ p µ ) , (48) \n[ ˆ M µν , ˆ M ρτ ] = i /planckover2pi1 ( η µρ ˆ M ντ -η µτ ˆ M νρ + η ντ ˆ M µρ -η νρ ˆ M µτ ) . (49) \nHowever, the quantum phase space is described by an additional generator ˆ r (central charge): \n[ˆ x µ , ˆ p ν ] = i /planckover2pi1 η µν ˆ r, (50) \nhence to obtain the full Yang algebra we need the additional relations: \n[ˆ r, ˆ x µ ] = i /planckover2pi1 β 2 ˆ p µ , [ˆ r, ˆ p µ ] = -i /planckover2pi1 α 2 ˆ x µ , [ ˆ M µν , ˆ r ] = 0 . (51) \nThe uncertainty principle corresponding to the Yang model, in the non-relativistic case, can be written in general as: \n∆ˆ x i ∆ˆ p j ≥ /planckover2pi1 δ ij 2 |〈 ˆ r 〉| . (52) \nwhere the generator ˆ r can be realized in terms of the phase space variables ˆ r = ˆ r (ˆ x, ˆ p ). It is also worth to mention that the Yang model is covariant (self-dual) under the Born reciprocity: \nB : ˆ x µ → ˆ p µ , ˆ p µ →-ˆ x µ , ˆ M µν ↔ ˆ M µν , ˆ r ↔ ˆ r, α ↔ β. (53) \nThe similarity between Yang model and SdS model considered in the previous section is not coincidental and it has been shown [68] that SdS algebra (17)-(21) can be viewed as a nonlinear realization of Yang model (46)-(51). The classical limit of the Yang model (cf. [69], see also [70]) is: \n{ x i , x j } = β 2 ( x i p j -x j p i ) , { p i , p j } = α 2 ( x i p j -x j p i ) (54) \nand one of the possible realizations for the ˆ r generator on the canonical phase space [69], for example, gives: \n{ x i , p j } = δ ij √ 1 -α 2 x 2 -β 2 p 2 -α 2 β 2 ( x 2 p 2 -( xp ) 2 ) . (55) \nThe remaining relations are obtained in a straightforward way. In D=1 this would simplify to: \nwith the corresponding GEUP 14 : \n{ x, p } = √ 1 -α 2 x 2 -β 2 p 2 . (56) \n∆ x ∆ p ≥ /planckover2pi1 2 √ 1 -α 2 (∆ x ) 2 -β 2 (∆ p ) 2 . (57) \nWe see that in such 1-dimensional case higher order terms would appear, which is not uncommon for GUPs see e.g. [71-73] but this would be the first such example considering higher order GEUPs, up to our knowledge. \nExpanding the RHS of (56) in deformation parameters, up to α 2 and β 2 , we can apply the results of the previous section (27) and obtain the invariant weighted phase space volume element as: \ndxdp 1 -1 2 ( α 2 x 2 + β 2 p 2 ) . (58) \nFrom the Yang model (46)-(51), in the realization (55) for ˆ r generator we can obtain two special cases. Namely: \n- · In the limit when α → 0 we obtain the so-called 'square-root modified' or 'Maggiore algebra' (see e.g. [74]): \n[ˆ x µ , ˆ x ν ] = i /planckover2pi1 β 2 ˆ M µν , [ˆ p µ , ˆ p ν ] = 0 , [ˆ x µ , ˆ p ν ] = i /planckover2pi1 η µν √ 1 -β 2 ˆ p 2 , (59) \nsupplemented by the Lorentz covariance conditions (47) - (49). Such modification of quantum mechanical phase space relations leads, in the non-relativistic case, to the higher order type of GUP 15 of the form: \n∆ˆ x i ∆ˆ p i ≥ /planckover2pi1 2 √ 1 -β 2 (∆ˆ p ) 2 . (60) \nThis version of GUP no longer produces a minimum observable length [48, 73]. But it would still result in the weighted phase space volume element: \ndxdp 1 -1 2 β 2 p 2 (61) \naffecting the density of states. \n- · By taking β → 0 in (46)-(51), we obtain \n[ˆ x µ , ˆ x ν ] = 0 , [ˆ p µ , ˆ p ν ] = i /planckover2pi1 α 2 ˆ M µν , [ˆ x µ , ˆ p ν ] = i /planckover2pi1 η µν √ 1 -α 2 ˆ x 2 , (62) \nleading to the higher order type of EUP 16 : \n∆ˆ x i ∆ˆ p i ≥ /planckover2pi1 2 √ 1 -α 2 (∆ˆ x ) 2 (63) \nwith the weighted phase space volume element as: \ndxdp 1 -1 2 α 2 x 2 (64) \naffecting the density of states. \nWe postpone the investigation of the full D-dimensional case of Yang model (46)-(51) in the context of density of states to the future work. It is also worth to mention that in [75] generalizations of the Snyder algebra to a curved space-time background with de Sitter symmetry were considered where the SdS model and Yang model were obtained as special cases. The realizations of these algebras were considered in terms of canonical phase space coordinates, up to the fourth order in the deformation parameters. Therefore the results of the present paper could be generalized to these type of models and realizations as well.", 'V. LIE-ALGEBRAIC CASE WITH COMMUTING MOMENTA: FUZZY SPHERE': "Many noncommutative (quantum) space-times proposed in the quantum gravity motivated literature have the Lie-algebraic form for the noncommutativity of coordinates and include commuting momenta. Hence, for the sake of completeness we discuss here one example of such model defined by the following commutation relations: \n[ˆ x i , ˆ x j ] = i /planckover2pi1 /epsilon1 ijk ˆ x k , [ˆ x i , ˆ p j ] = i /planckover2pi1 /epsilon1 ijk ˆ p k , [ ˆ p i , ˆ p j ] = 0 (65) \nwhere /epsilon1 ijk is totally skew-symmetric tensor, /epsilon1 123 = 1 . The subalgebra [ˆ x i , ˆ x j ] = i /planckover2pi1 /epsilon1 ijk ˆ x k supplemented by the relation ˆ x i ˆ x i = r corresponds to the fuzzy sphere, with r the constant radius of the sphere. \nSince we are interested in the effects on the density of states in this Lie algebraic case we follow the steps outlined in Sec.2. In the classical limit, we obtain: \n{ x i , x j } = /epsilon1 ijk x k ≡ a ij , { x i , p j } = /epsilon1 ijk p k ≡ c ij , { p i , p j } = 0 ≡ b ij , (66) \nwhere we identified RHSs with the notation used in Sec. 2. Directly plugging these relations into (15) we obtain: \nd D x ' d D p ' = d D xd D p ( 1 + [( ∂ ∂x i ( /epsilon1 ijk x k ) -∂ ∂p i ( /epsilon1 jik p k ) ) ∂H ∂x j + ( ∂ ∂x i ( /epsilon1 ijk p k ) ) ∂H ∂p j ] δt + O ( δt 2 ) ) = d D xd D p ( 1 + [ ( /epsilon1 ijk δ ik -/epsilon1 jik δ ik ) ∂H ∂x j +0 ] δt + O ( δt 2 ) ) = d D xd D p. (67) \nHence we the phase space volume element stays invariant under the time evolution without the need to introduce any additional factor and there will be no change in the density of states for the case of fuzzy sphere.", 'VI. FINAL REMARKS': 'In this paper we have focused on models exhibiting noncommutativity in both space-time coordinates and in momenta (i.e. models with curved space-time and curved momentum space), such that in the quantum phase space relations both the Planck length l P and the cosmological constant Λ appear as fundamental parameters on equal footing. The modified Heisenberg commutation relations lead to GEUPs where the symmetry between position and momentum is preserved (which is not the case in the usual GUPs or EUPs). Such symmetric GEUPs may be seen as an indication of quantum gravitational corrections to the classical space-time and standard quantum mechanics, at both very small and very large scales. We point out that, on the contrary to many works which have proposed multiple variants of the GUPs, EUPs and GEUPs by arbitrarily choosing specific forms of the commutation relations between space-time coordinates and momenta [ x, p ] with supposedly desirable properties, we have focued here on studying the consequences of known models which arise in the noncommutative geometry approach to quantum gravity. \nIn general since the canonical commutation relations are modified, one expects that thermodynamics and statistical mechanics will be affected by the introduced modifications, possibly leading to some new effects. As a consequence of the GEUP arising from the cases of Snyder-de Sitter and Yang models we have shown that the Liouville theorem in statistical physics requires considering the weighted phase space volume and introduces modification in the density of states, with the weight factor depending on both coordinates and momenta. Since such modification of the density states is required this will influence the statistical and thermodynamical properties of physical systems. Various applications can now be studied and the effects of both noncommutativity in coordinates and momenta (or the presence of the Planck length and the cosmological constant in modified UPs) on atomic physics, condensed matter physics, preheating phase of the universe and black holes etc. can now be investigated.', 'Acknowledgements': "AP thanks A. Wojnar and S. 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2024A&A...690A.274P

Context. Computational astronomy has reached the stage where running a gravitational Nbody simulation of a stellar system such as a Milky Way star cluster is computationally feasible but a major limiting factor that remains is the ability to set up physically realistic initial conditions. Aims. We aim to obtain realistic initial conditions for Nbody simulations by taking advantage of machine learning with emphasis on reproducing smallscale interstellar distance distributions. Methods. The computational bottleneck for obtaining such distance distributions is the hydrodynamics of star formation which ultimately determine the features of the stars including positions velocities and masses. To mitigate this issue we introduce a new method for sampling physically realistic initial conditions from a limited set of simulations using Gaussian processes. Results. We evaluated the resulting sets of initial conditions based on whether they meet tests for physical realism. We find that direct sampling based on the learned distribution of the star features fails to reproduce binary systems. Consequently we show that physicsinformed sampling algorithms solve this issue as they are capable of generating realisations closer to reality.

2024-10-01T00:00:00Z

['10.1051/0004-6361/202450995', '2024arXiv240910627P', '2024A&A...690A.274P', 'arXiv:2409.10627', '10.48550/arXiv.2409.10627']

['gravitation', 'hydrodynamics', 'methods: numerical', 'open clusters and associations: general', 'Astrophysics - Astrophysics of Galaxies']

A machine learning framework to generate star cluster realisations

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']

https://arxiv.org/pdf/2409.10627.pdf

{'A machine learning framework to generate star cluster realisations': "George P. Prodan , ⋆ 1 , 8 , Mario Pasquato 2 , 3 , Giuliano Iorio 1 , 4 , 5 , Alessandro Ballone 1 , 4 , 5 , Stefano Torniamenti 1 , 4 , 6 , Ugo Niccolò Di Carlo 7 , and Michela Mapelli 1 , 4 , 6 \n- 1 Dipartimento di Fisica e Astronomia, Università di Padova, Vicolo dell'Osservatorio 3, 35122, Padova, Italy\n- 2 IASF Milano, via Alfonso Corti 12, Milano, Italy\n- 3 Ciela - Montreal Institute for Astrophysical Data Analysis and Machine Learning, Montréal, Canada\n- 4 INFN-Padova, Via Marzolo 8, 35131, Padova, Italy\n- 5 INAF, Osservatorio Astronomico di Padova, Vicolo dell'Osservatorio 5, Padova, Italy\n- 6 Institut für Theoretische Astrophysik, ZAH, Universität Heidelberg, Albert-Ueberle-Str. 2, 69120, Heidelberg\n- 7 SISSA - Scuola Internazionale Superiore di Studi Avanzati, via Bonomea 365, I-34136 Trieste, Italy\n- 8 Faculty of Sciences, University of Craiova, A.I. Cuza 13, 200585 Craiova, Romania \nReceived June 05, 2024 ; accepted September 04, 2024", 'ABSTRACT': 'Context. Computational astronomy has reached the stage where running a gravitational N -body simulation of a stellar system, such as a Milky Way star cluster, is computationally feasible, but a major limiting factor that remains is the ability to set up physically realistic initial conditions. \nAims. We aim to obtain realistic initial conditions for N -body simulations by taking advantage of machine learning, with emphasis on reproducing small-scale interstellar distance distributions. \nMethods. The computational bottleneck for obtaining such distance distributions is the hydrodynamics of star formation, which ultimately determine the features of the stars, including positions, velocities, and masses. To mitigate this issue, we introduce a new method for sampling physically realistic initial conditions from a limited set of simulations using Gaussian processes. \nResults. We evaluated the resulting sets of initial conditions based on whether they meet tests for physical realism. We find that direct sampling based on the learned distribution of the star features fails to reproduce binary systems. Consequently, we show that physics-informed sampling algorithms solve this issue, as they are capable of generating realisations closer to reality. \nKey words. gravitation - hydrodynamics- (Galaxy:) open clusters and associations: general- methods: numerical', '1. Introduction': "The target of humanity's first deliberate attempt at interstellar radio communication was the star cluster M13 (Sta ff at the National Astronomy & Ionosphere Center 1975), and this act is a testament to the importance of star clusters to astronomy, which can hardly be overstated. For instance, star clusters are the likely birthplace of most stars (Lada & Lada 2003). \nAstronomers use gravitational N -body simulations with the purpose of studying the evolution of star clusters (Aarseth 2003; Spurzem & Kamlah 2023). The goals, among others, are to predict gravitational wave emission from black holes and neutron stars formed by cluster stars (Rastello et al. 2021; Dall'Amico et al. 2021; Iorio et al. 2023), constrain the place of origin of the Solar System (Pichardo et al. 2012; Pfalzner et al. 2015), determine the stability and habitability of exoplanets (Spurzem et al. 2009; Malmberg et al. 2011; Parker & Quanz 2012), and understand how star clusters contribute to the overall evolution of the Milky Way as part of the field called Galactic archaeology (Chung et al. 2019; Pang et al. 2021, 2022). \nStar clusters are understood to form within molecular clouds. Molecular clouds are high-density, low-temperature regions in the interstellar medium predominantly composed of molecular hydrogen. Star formation in molecular clouds occurs in small regions where the density becomes high enough (e.g. by tur- \nbulent colliding flows) to start a fast runaway gravitational collapse (Evans 1999). Gas in molecular clouds is often animated by turbulent motions, which are ultimately responsible for the irregular and complex spatial distribution of young cluster stars (Krumholz et al. 2019; Krause et al. 2020; Ballone et al. 2020). The scientific usefulness of gravitational N -body simulations depends on the realism of initial conditions (primordial positions, velocities, and masses of stars), which hinges on the ability to properly model star formation in molecular clouds. Realistic initial conditions may be obtained by running hydrodynamical simulations. These are computationally expensive, with one simulation taking around 10 5 CPU hours (Ballone et al. 2021). \nMachine learning can be leveraged to obtain samples of valid initial conditions at a fraction of the computational price by learning a generative model from the outputs of a limited number of simulations. The first attempt in this direction was presented by Torniamenti et al. (2022). Their work consists of a bespoke manipulation of hierarchical clustering models that is simple and e ff ective but lacks theoretical guarantees. Here we propose a different approach based on Gaussian processes (GPs).", '2. Gaussian processes': 'Gaussian processes serve as probabilistic models capable of predicting function values based on noisy observations (Wang 2022). Especially valuable when data is limited, GPs present a \nmore fitting alternative to deep neural networks given the constraints of our application (Gri ffi ths 2023). For star cluster realisations, the distribution of stars in a cluster can be represented as a function of key physical parameters, such as star masses, coordinates, and velocities. Leveraging GPs, we can learn the star distribution in clusters in the space defined by these parameters, enabling the generation of synthetic clusters. \nThe essence of GP modelling lies in Bayesian inference, where model beliefs are updated upon receiving new observations. A GP model is characterised by a mean function, µ ( x ) and a kernel function, K ( x , x \' ), with the kernel function reflecting the similarity degree between input points and influencing predictions (Rasmussen 2004). In our case, the kernel function aims to represent the spatial relations among stars in a cluster by capturing the relationships among the stars set at star formation and arising through dynamics via, for example, the e ff ects of gravitational attraction between the individual stars. \nThe kernel function hyperparameters of GP models include the signal variance, σ y ; the length scale, l ; and the noise level, σ n . The signal variance measures signal amplitudes, the length scale indicates covariance decay distance, and the noise caters to errors. All of these parameters are trainable. Usually, the radial basis function kernel (RBF) is employed to compute the amplitude of the covariance function, \nK ( x , x \' ) = σ 2 y exp " -( x -x \' ) 2 2 l 2 # . (1) \nThe hyperparameters of the mean function can be defined depending on the choice of the prior mean function. For instance, a constant mean function µ ( x ) = c leads to one additional trainable hyperparameter. \nTraining aims to minimize the log marginal likelihood (Rasmussen 2004) with respect to the above-mentioned hyperparatemers θ = { σ y , σ n , l , ... } , namely \nlog P ( y | X , θ ) = -1 2 ( y -µ ) T ( K + σ 2 n I ) -1 ( y -µ ) -1 2 log | K + σ 2 n I | -n 2 log 2 π. (2) \nThe first term ensures data fitting, the second one is responsible for regularisation, and the last one is a constant. Regularisation is an important part of model training that prevents overfitting (where the model learns irrelevant patterns in the data instead of the underlying relationships). By incorporating regularisation terms, the model generalizes better on unseen data, thus improving its predictive performance. Training with noise is an essential part of this work. By inserting noise, we obtained new cluster realisations that at the same time preserve some of the features of the training samples. Our aim for the GP fitting is to create a model that can reproduce these features. \nWe drew the predictions from the posterior GP, \ny ∼ GP ( µ, K + σ 2 n I ) , (3) \nwhere µ , K are the posterior mean and kernel function and I is the identity matrix. One can draw two kinds of predictions. The first is the mean prediction, which generates the expected values that are similar to those determined for the training samples. This prediction is deterministic and represents the noiseless prediction of the GP. The second involves sampling from the GP\'s posterior distribution, incorporating noise. Here, the predictions are more diverse because they consider the inherent uncertainty \nin the model. This can lead to a broader range of potential outcomes and possibly to the formation of di ff erent clusters. A complete mathematical overview on GP models is presented in Rasmussen & Williams (2006).', '3. Learning framework': "We have introduced a learning framework that utilizes GP modelling to learn the feature space distributions of stellar clusters, followed by sampling from these distributions to produce new clusters. This is an inverse problem, as the framework obtains new cluster realisations via simulation-based inference (Cranmer et al. 2020; Lueckmann et al. 2021). While training, the inputs of the GP model are the parameters of each star, θ i ( train ) with i = 1 , N for a cluster of N stars. The parameters are physical quantities of the stars, with each star having a specific mass, M ; location, r = ( x , y , z ); and velocity, v = ( vx , vy , vz ), with respect to the cluster's centre of mass and all of the parameters M , r , and v are not normalised. The GP learns a distribution over the probability density function of the feature space, f ( θ ). \nOnce the GP model input features were determined, our next step involved computing the target distribution, y = f ( θ ). For this purpose, k -nearest neighbours density estimators (Mack & Rosenblatt 1979) were implemented to ascertain the probability density values, yi , which are normalised accordingly, \ny ( train ) i = Nyi Σ N j = 1 yj . (4) \nFurther, we implemented Markov chain Monte Carlo (MCMC) approaches to simulate new cluster realisations based on the Metropolis-Hastings algorithm (Chib & Greenberg 1995; Eckhardt 1987; Robert & Casella 2011). The states proposed in the Markov Chain include a set of N features at a given iteration step. We considered an iteration step n corresponding to a state Sn = { θ ( n ) 1 , θ ( n ) 2 , ..., θ ( n ) N } . The next candidate state, Sn + 1, was obtained by a jump with probability T ( Sn → Sn + 1). The acceptance probability of this jump is given by \nα = min 1 , π ( Sn + 1) T ( Sn → Sn + 1) π ( Sn ) T ( Sn + 1 → Sn ) ! , (5) \nwhere π is the stationary distribution of the chain. The probability of the new state can be written as the joint probability of the candidate features, \nπ ( Sn ) = N Y i = 1 f ( θ i ) , (6) \nwhere f is a probability density function drawn from the GP model that is already trained on the features θ ( train ) and the corresponding target values y ( train ) of the density function. \nThis sequential method first crafts a GP statistical model based on the probability density function of the features space for the cluster stars. While sampling from a standard probability density function is generally straightforward, our specific problem requires meticulous adjustments due to the need for maintaining physical constraints and the presence of complex correlations. This ensures that the generated samples accurately reflect the underlying distribution and maintains the validity of the initial conditions. We employed traditional MCMC methods based on the Metropolis algorithm to directly sample in a sevendimensional space (DMCMC). Moreover, we propose a physicsinformed sampling approach based on learning the energy space distribution of nearest-neighbour star pairs (EMCMC).", '3.1. Direct sampling': "The direct approach is to define a seven-dimensional feature space that includes all the physical quantities of the stars in the clusters, meaning that for every star at the iteration of state n , we have θ ( n ) i = { M ( n ) i , x ( n ) i , y ( n ) i , z ( n ) i , v ( n ) xi , v ( n ) yi , v ( n ) zi } with i in 1 , N . Considering the large number of stars in the cluster, achieving convergence becomes challenging if all parameters are subject to change each time we propose a new candidate cluster. However, Eq. 5 remains valid also if we propose only a new candidate star θ ( n + 1) i such that the new state becomes Sn + 1 with \nθ ( n + 1) j = (1 -δ i j ) θ ( n ) i + δ i j θ ( n + 1) i (7) \nfor j in 1 , N , where δ i j is the Kronecker delta, and it is equal to one only if i = j ; otherwise it is zero. The acceptance rate will be \nα = min 1 , f GLYPH<16> θ ( n + 1) i GLYPH<17> f GLYPH<16> θ ( n ) i GLYPH<17> . (8) \nTherefore, at each iteration a new subset of seven features θ ( n + 1) i is proposed with i chosen randomly. We drew new candidates using a normal distribution such that θ ( n + 1) ik ∼ N ( θ ( n ) ik , σ ) for every feature k of θ i , where σ is called the step size, and it is the amplitude of the perturbation applied on the previous state of the Markov chain and, mathematically, the standard deviation of the normal distribution centred on the previous sampled feature from which we sampled the new candidate. \nThe new candidate is accepted or rejected based on the rate established by Eq. 8 and employing the same exact density function f drawn from the GP model, meaning that the GP noise seed must be kept unchanged during the sampling of one cluster realisation. The last condition is that it is essential not to break the detailed balance of the Metropolis-Hastings algorithm. \nOne can notice that there is no conditioning on the global properties of the cluster and the sampler 'builds' the cluster by sampling stars individually. Therefore, this direct approach works similar to a black box that relies solely on the correlations captured by the GP model.", '3.2. Physics-informed sampling': "Sampling star systems, especially binary systems in close interaction, by generating new samples in a seven-dimensional space is challenging due to the low probability of proposing candidate stars in proximity within the physical space. This may result in erroneous modelling of star clusters at small scales, even if the sampling algorithm successfully replicates the density function given by the GP. Our solution to this problem is based on incorporating physical laws to the sampling process by focusing on binary stars creating chains of pairs based on the nearest neighbours and learning the distribution over the potential and kinetic energies of these pairs. Thus, the cluster is reduced to a chain of this kind, and from this chain we can reconstruct other clusters based on similar chains. \nThis approach substitutes learning the full distribution of the mutual potential energies that results from N ( N -1) / 2 pairs of stars. This can be categorised as feature dimensionality reduction through feature selection (Jia et al. 2022) based on the heuristic that most of the binding energy is concentrated in the binary systems of the cluster (Torniamenti et al. 2021). \nWe defined for each pair a feature set based on the mutual potential energy of the stars and their kinetic energies, \nθ i = ( Ui , i + 1 , Ki , Ki + 1) with i in 1 , N -1. In this way, following the nearest neighbours chain, we get N -1 sets of features in the energy space. We considered that these sets of energy values fully define the energy state of the nearest neighbours chain. New energy states are sampled making use of a GP model trained on the probability density function of the energy space. \nThe sampling scheme therefore starts by proposing new energy states. The first step of the solution is the EMCMC algorithm that samples the energy state of the nearest neighbours chain. The energy state at the n -th iteration, Sn = { θ ( n ) 1 , θ ( n ) 2 , ..., θ ( n ) N } , implies the mutual potentials of the nearest neighbours, U ( n ) 12 , U ( n ) 23 , ..., U ( n ) N -1 , N , and all the kinetic energies, K ( n ) 1 , K ( n ) 2 , ..., K ( n ) N . As before, we sampled for a random i a new candidate θ ( n + 1) i = ( U ( n + 1) i , i + 1 , K ( n + 1) i , K ( n + 1) i + 1 ) from a normal distribution based on the values of the previous iteration. This also implies a change for θ ( n ) i -1 and θ ( n ) i + 1 . In this case, applying Eq. 5 and Eq. 6 leads to the following acceptance rate: \nα = min 1 , f GLYPH<16> θ ( n + 1) i -1 GLYPH<17> f GLYPH<16> θ ( n + 1) i GLYPH<17> f GLYPH<16> θ ( n + 1) i + 1 GLYPH<17> f GLYPH<16> θ ( n ) i -1 GLYPH<17> f GLYPH<16> θ ( n ) i GLYPH<17> f GLYPH<16> θ ( n ) i + 1 GLYPH<17> , (9) \nwhere f is the probability density function drawn from the GP model trained on the energy space. We note that at the boundaries, only two sets of features are subject to change. \nUsing the new sampled chain, one needs to 'reconstruct' the cluster by defining its stars as entities with a position, velocity, and mass. The magnitudes of the relative distances between the stars, ri , i + 1 = ri + 1 -ri , and the velocities are derived according to the physical laws of Newtonian gravity, \n| ri , i + 1 | = -GMiMi + 1 Ui , i + 1 (10) \n| vi + 1 | = r 2 Ki + 1 Mi + 1 , (11) \nwhere G is the universal gravitational constant and Mi and Mi + 1 are the masses. \nFollowing the chain sequentially implies that the properties of the star i are known when deriving those of the next star in the chain (the star i + 1). The mass of the next star, Mi + 1, is sampled independently with the help of another GP model trained only on mass distributions. One can then determine the magnitudes of ri , i + 1 and vi + 1 . The next step is to explore the full seven-dimensional feature space of the stars to determine the optimal directions of ri , i + 1 and vi + 1 . For this, we used a GP model trained on the phase-space, GP ( r , v ). The exploration consists of proposing several Nc candidate directions for each vector. The candidates were chosen randomly from a uniform distribution. Using the GP model predictions, we chose the most probable position in the phase-space to determine the next star i + 1. The procedure was repeated until the end of the chain. \nIn Algorithm 1, we provide the pseudocode of the proposed algorithm that works based on three GP models: GP ( θ ), GP ( M ), and GP ( r , v ). The function 'randomDirection' is employed to generate a random direction based on two random generated numbers, R θ and R ϕ , which are used to generate pairs of angles ( θ, ϕ ) that correspond to directions sampled uniformly on a three dimensional sphere. These directions are used to generate new candidates. \nWe define two hyper-parameters, maxJump and Nc . The first parameter is used to control the maximum distance between the \npairs of stars. In this way, the mass choice is conditioned on the spatial distribution of the cluster. Without this conditioning, the chain can be easily broken into multiple parts. This would lead to generating several smaller clusters. The other parameter, Nc , is the number of star candidates we generate randomly based on the values of | r i j | and | v j | . We use GP ( r , v ) afterwards to find the best candidate, that is, the star with the most probable position in the phase-space. As Nc is large enough, the resulting star features always point towards the densest region of the phase-space that is attainable at a given moment. \nAlgorithm 1 EMCMC algorithm in pseudocode to find and select new star candidates for the generated cluster. \nRequire: GP ( θ ), GP ( M ) , GP ( r , v ) , randomDirection , N , Nc , maxJump \ncluster ← empty array to store the parameters of N stars cluster [0] ← initialize parameters of the first star \nfor i from 0 to N-1 \nUi , i + 1 , Ki , Ki + 1 ← MCMC ( GP ( θ )) \nMi , r i , v i ← cluster [ i ] \n| r i , i + 1 | ← ∞ \nwhile | r i , i + 1 | > maxJump \nMi + 1 ← MCMC ( GP ( M )) \n| r i , i + 1 | ← Mimi + 1 Ui , i + 1 \nend while \n| v i + 1 | ← q 2 Ki + 1 mi + 1 \ncandidates ← empty array to store Nc candidates \nfor each candidate in candidates \nλ r ← randomDirection() \nλ v ← randomDirection() \nr i + 1 = r i + | r i , i + 1 | λ r \nv i + 1 = | v i + 1 | λ v \ncandidate ← Mi + 1 , r i + 1 , v i + 1 \nend for \nprobs ←GP ( r , v )( candidates ) \nbestCandidateIdx ← argmax ( probs ) \ncluster [ i + 1] ← candidates [ bestCandidateIdx ] \nend for", '4.1. Dataset': 'We utilized a dataset consisting of ten clusters extracted from hydrodynamical simulations conducted by Ballone et al. (2020). These simulations were designed to accurately reproduce not only the clumpiness but also the fractal nature observed in starforming regions. Each cluster in the dataset has approximately 2500 to 4000 stars, with a cumulative mass ranging between 4000 and 40000 M ⊙ (solar masses). For our analysis, we divided this dataset into a training subset, consisting of seven clusters, and a validation subset made up of the remaining three clusters.', '4.2. Model training': "We used the gpytorch library (Gardner et al. 2018) to implement our models. This library o ff ers a modular and e ffi cient framework for GPs and supports graphics processing unit (GPU) acceleration. For all of our experiments, we employed one NVIDIA GeForce RTX 3060 GPU. Our GP model was trained using the RBF kernel, employing the exact marginal log likelihood as the loss function and using the Gaussian likelihood to \nFig. 1. Training parameters (loss and noise level) with respect to the number of clusters used in the training. One validation cluster was used in the case of two or three training clusters; otherwise, we used two validation clusters. The training parameters correspond to the model's version at early stopping. \n<!-- image --> \ncalculate the posterior distribution. All models were initialised with a zero-prior mean. The necessary features for model training were derived from simulation data. Our training approach for each model integrates cross-validation (Hastie et al. 2001) and early stopping (Prechelt 1996), the latter having a patience threshold set at ten epochs. The properties of each trained model are summarised in Table 1. \nThe models were trained on ten clusters, among which three are for validation. The training itself can be done using more or fewer clusters, depending on the context. We observed that the loss slightly increases when decreasing the number of clusters used in training (see Fig. 1), suggesting a slight decline of the model's performance. Also, we noticed a sudden increase of the loss when using eight clusters for training (and the other two for validation). This could be caused by underfitting, as suggested by the larger noise level. Therefore, we continued our experiments using a training validation ratio of seven to three.", '4.3. Sampling': "The sampling is based on a probability density function drawn from GP ( M , r , v ) for DMCMC or GP ( Uij , Ki , Kj ) for EMCMC. The initial state, in both cases, is initialised from a normal distribution N ( µ, σ ), with µ and σ computed over the feature distributions retrieved from the simulation data. \nThe cluster size, that is, the number of stars in the cluster, N , is established when defining the initial state of MCMC proposals. The new candidate stars are proposed such that the state perturbations are uniformly distributed along the stars' features. This means that we do not define a maximum number of iterations, but several accepted states, NA . For instance, if NA = mN , new features for each star will be accepted exactly m times. In this way, one can recover m cluster realisations or use the first m -1 iterations as a burn-in strategy (Roy 2020).", '4.4. Generated clusters': 'The newly generated clusters were evaluated based on the distributions of the inter-particle distance, velocity, and mass. In addition, we computed global physical quantities such as the virial ratio, total mass, and the number of stars that are part of a bound system. The evaluation was based on comparing our generated clusters to the hydrodynamically simulated ones retrieved from Ballone et al. (2020). These clusters are our baseline, and our aim was to reproduce similar features. A two-dimensional spatial projection of such clusters obtained with DMCMC and EMCMC on the Oxy plane is shown in Fig. 2. \nTable 1. Training results. \nNotes. Columns: (1) Model; (2) best epoch; (3) training time per epoch; (4) scale length, l ; (5) noise level, σ n .', 'Hydrodynamical Simulations': "Fig. 2. Two-dimensional representation of one low-mass and one high-mass cluster generated with DMCMC and EMCMC, respectively. For comparison, the corresponding projections are illustrated also for the hydrodynamically simulated clusters m 1 . e 4 and m 9 . e 4 from the work of Ballone et al. (2020). The x and y coordinates are measured in parsecs. For EMCMC, we show the value of the hyper-parameter J (maxJump) and the total mass expressed in M 10 3 ⊙ = 1000 solar masses. \n<!-- image --> \nWe performed a set of experiments to sample a low-mass cluster (4000 M ⊙ ) of N = 3000 stars. The results are shown in Fig. 3. Comparing to the simulation data, for the inter-particle distance distribution of DMCMC, we noticed that there is a lack of stars at low inter-particle distances smaller than 10 -2 pc. This suggests a lack of close interactions between stars (i.e. binary systems). However, when analysing the total energy of the star pairs, we observed that there is a significant number of stars that are part of bound systems. For instance, the low-mass DMCMCgenerated cluster that corresponds to Figure 2 exhibits a number of 276 stars that are bound to other stars, whereas in the baseline training clusters there are between 400 and 800 binaries. According to the inter-particle distance distribution of the DMCMC clusters (Fig. 3), the orbital separations of this binary systems are large enough that gravitational encounters will likely unbind them during a cluster's evolution. This number is thus an upper limit to stable binaries, and the actual number of stable binaries could be significantly less, given the lack of close \nbinaries. This is not the case for the EMCMC clusters, as there are also star pairs with low inter-particle distance that are able to keep the bound system intact. \nOn the other hand, the physics-informed algorithm excels at sampling clusters with a balanced spectrum of inter-particle distances, which direct sampling in a seven-dimensional feature space fails to achieve. We also note that the energy spectrum of DMCMC exhibits a linear trend that di ff ers from those of the EMCMC-generated clusters or hydrodynamical simulations of clusters (see Fig. 5). \nOther relevant physical quantities are presented in Fig. 4. In the left subfigure, it is shown how the number of stars varies with respect to the cluster's radius. We observed that there is not any specific trend in the distribution derived for the reference clusters. We noticed sudden jumps, meaning that there are regions of higher density, but there is one exception for one of the DMCMC clusters, where the number of stars increases smoothly. On the right side, we have plotted the histogram illustrating the \n<!-- image --> \n<!-- image --> \nFig. 3. Distributions of inter-particle distances, velocity, and mass for low-mass clusters with 3000 stars generated using DMCMC and EMCMC methods. The mass distribution has a lower bound of 0 . 1. A simulated cluster from Ballone et al. (2020) is represented for comparison. \n<!-- image --> \n<!-- image --> \nFig. 4. The distributions of key physical paramters for the new sampled clusters. The left subfigure shows the number of stars as a function of the distance to the cluster's centre of mass for a set of simulated clusters, DMCMC and EMCMC (three clusters each). On the right subfigure is plotted a histogram of the angular momentum distribution of the stars for the same sets of clusters as in the left subfigure. The angular momentum is measured in the units of the international system. \n<!-- image --> \ndistribution of angular momenta resulting after computing the magnitude of angular momentum of each star as \n| L | = q ( ryvz -rzvy ) 2 + ( rzvx -rxvz ) 2 + ( rxvy -ryvx ) 2 . (12) \nHere, we note that DMCMC clusters lack stars with high angular momentum. Probably, this happens because of the lower sampling rate of stars with high velocity magnitudes (see Fig. 3). \nWe present a summary of additional experiments in Table 2. We show the global properties (virial ratio, estimated number of binary systems, and total mass) of the six realisations obtained via DMCMC and EMCMC. We performed the experiments in order to reproduce clusters with a low, intermediate, and high total mass. \nWhen examining virial ratios, that is, the total kinetic energy divided by half of the total binding energy, we observed that the Metropolis algorithm delivers varied dynamical states. The EMCMC approach consistently produces realistic clusters; 60% of the clusters we sampled fall within a virial ratio between one and two, which is consistent with the training samples. \nAll the experiments were aimed at generating clusters containing 3000 stars, which would allow us to provide computational performance comparisons. The number NA = 3000 was chosen arbitrarily; one can try di ff erent cluster sizes. However, the number of stars in the clusters should be the same order of magnitude as those used in the training (2000 - 5000 stars). More massive clusters could be sampled, but the framework performance has been tested primarily for lower-mass clusters, so \napplying it to significantly larger clusters might necessitate retraining or fine-tuning the model to ensure that it handles the increased star count e ff ectively. Sampling with DMCMC takes around 15 minutes for NA = 3000 accepted samples, while EMCMC samples 3000 new accepted states during the same time, but it needs additional time, around 10 minutes, to reconstruct the cluster from the chain. The running time will scale linearly with the number of stars, as the sampling is done sequentially. One would expect EMCMC sampling to take longer, as we are estimating the probability density for three stars at each iteration. However, the acceptance rate is also three times higher for EMCMC. The experiments were done using an NVIDIA GeForce RTX 3060 GPU and a 12th Gen Intel Core i7-12700H CPU at 4.70 GHz. The training, sampling, and evaluation routines are available online on GitHub, 1 and there one can find several results that we have generated and the models we trained to obtain these results.", '5. Conclusions': "We have introduced a learning framework to generate new realisations of positions, velocities, and masses of star clusters to be used as initial conditions in N -body simulations. This framework allowed us to bypass the computational bottleneck represented by hydrodynamical simulations of star formation in molecular clouds, when provided an adequate training set. Based on well \nTable 2. Monte Carlo sampling results using EMCMC and DMCMC algorithms. \nNotes. The generated clusters contain 3000 stars. Columns: (1) #, cluster index; (2) method; (3) the lower bound for the mass distribution, Mlower , in solar masses ( M ⊙ ) ; (4) the step size in the Metropolis algorithm; (5) the acceptance rate; (6) the virial ratio; (7) the number of binaries; (8) the total mass of the sampled cluster. \nFig. 5. Histogram with the energies of each pair of stars in the cluster. The energy for each pair was calculated as the sum of kinetic energies of the stars and their mutual potential energy. Negative energies correspond to gravitationally bound pairs. The resulting energy spectra for the clusters generated with EMCMC and DMCMC are compared to the clusters obtained via hydrodynamical simulations (Ballone et al. 2020). \n<!-- image --> \nunderstood GPs, our method yields predictable results and can be readily integrated into a simulation pipeline. \nThe learning framework consists of two steps. First, a statistical model of the data is constructed through the GP, and second, new clusters are generated by sampling from the target distribution. The probability density function of the star features was employed to train the GP model. We tackled the problem in two ways: by sampling directly in the seven-dimensional feature space of the stars (DMCMC) and by sampling in a specific energy space (EMCMC). That was followed by reconstructing the cluster using the proposed algorithm. \nIn the pursuit of sampling clusters with realistic spatial distributions and dynamical states, the EMCMC algorithm has proven to be particularly e ff ective, especially in addressing the crucial distribution of inter-particle distances, whereas DMCMC led to undesired distance distributions. The latter method and other conventional samplers work similar to a 'black box' in this case, relying only on the correlations learned by the GP model. The main di ff erence between the two approaches is that EMCMC incorporates physical principles. This physics-informed algorithm ensures that the generated samples are not only mathematically plausible but that they also possess physical significance. \nThe DMCMC experiments showed that knowing the probability density function of the mass, position, and velocity is not su ffi cient to fully reproduce a cluster realisation. These properties are a consequence of physical interactions, and trying to reproduce them by sampling an estimated probability density function proved to be challenging due to a lack of binary stars. In this \ncontext, the learned distribution can inform us about the possible location of a star in the feature space, but it cannot determine whether that star is part of a binary system, and it cannot reproduce its companion star. In contrast, with EMCMC, we did not change the sampling algorithm, but we sampled from a different target distribution that helps in answering both questions, namely, where the star is and whether it is a binary. The EMCMCgenerates clusters that not only match statistical properties but also reflect physical realities, such as binary star formation, making it a more robust and significant method for studies of stellar evolution and cluster dynamics. \nAcknowledgements. This work acknowledges financial support from the European Research Council for the ERC Consolidator grant DEMOBLACK, under contract no. 770017 (PI: Mapelli). MM and ST also acknowledge financial support from the German Excellence Strategy via the Heidelberg Cluster of Excellence (EXC 2181 - 390900948) STRUCTURES.", 'References': "Aarseth, S. J. 2003, Gravitational N-Body Simulations \nBallone, A., Mapelli, M., Carlo, U. N. 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2024arXiv240905519D

This paper investigates the normal modes of a probe scalar field in a fivedimensional AdSSchwarzschild black hole with the brick wall boundary condition near the horizon. We employ various techniques to compute the spectrum and analyze its properties. Our results reveal a linear dependence of the spectrum on the principal quantum number while demonstrating a nontrivial dependence on the angular momentum quantum number. We compute the Spectral Form Factor SFF and find a diprampplateau structure with the slope of the ramp approaching unity as the brick wall nears the horizon. We also observe that as the brick wall approaches the horizon the poles of the retarded Greens function condense on the real line leading to an emergent thermal behavior in the boundary theory. This work extends previous studies on lowerdimensional black holes to higher dimensions providing insights into the connection between black hole microstate models and boundary chaos. Our findings contribute to the ongoing discussions on the information paradox and the nature of black hole interiors in the context of AdSCFT correspondence.

2024-09-01T00:00:00Z

['arXiv:2409.05519', '2024arXiv240905519D', '10.48550/arXiv.2409.05519']

['High Energy Physics - Theory', 'General Relativity and Quantum Cosmology']

Brick Wall in AdSSchwarzschild Black Hole Normal Modes and Emerging Thermality

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.05519.pdf

{'Suman Das, a Somnath Porey, b Baishali Roy b': "- a Theory Division, Saha Institute of Nuclear Physics, A CI of Homi Bhabha National Institute, 1/AF, Bidhannagar, Kolkata 700064, India.\n- b Ramakrishna Mission Vivekananda Educational and Research Institute, Belur Math, Howrah711202, West Bengal, India \nE-mail: \nsuman.das[at]saha.ac.in, somnathhimu00[at]gm.rkmvu.ac.in, \nbaishali.roy025[at]gm.rkmvu.ac.in \nAbstract: This paper investigates the normal modes of a probe scalar field in a five-dimensional AdS-Schwarzschild black hole with the brick wall boundary condition near the horizon. We employ various techniques to compute the spectrum and analyze its properties. Our results reveal a linear dependence of the spectrum on the principal quantum number while demonstrating a non-trivial dependence on the angular momentum quantum number. We compute the Spectral Form Factor (SFF) and find a dip-ramp-plateau structure, with the slope of the ramp approaching unity as the brick wall nears the horizon. We also observe that as the brick wall approaches the horizon, the poles of the retarded Green's function condense on the real line, leading to an emergent thermal behavior in the boundary theory. This work extends previous studies on lower-dimensional black holes to higher dimensions, providing insights into the connection between black hole microstate models and boundary chaos. Our findings contribute to the ongoing discussions on the information paradox and the nature of black hole interiors in the context of AdS/CFT correspondence.", '1 Introduction': "Strongly interacting theories are generally chaotic. While chaos in classical mechanics is well-defined, its counterpart in the quantum world is less clear. Various attempts have been made to bridge these two realms at the semiclassical level [1-4], but a comprehensive understanding remains elusive. Over the years, several measures have been developed to quantify quantum chaos, yet ambiguity persists. One notable measure is the Out-of-Time-Ordered Correlator (OTOC) [5, 6], which measures early-time chaos, whereas the Spectral Form Factor (SFF) [7, 8] measures chaos at late times. Level spacing distribution (LSD) has also been a valuable tool for understanding quantum \nchaos since the pioneering work of Wigner and Dyson [9, 10]. Although strongly interacting systems are intriguing, they are notoriously difficult to solve. This is where Random Matrix Theory (RMT) universality provides critical insights into the statistical properties of these systems. RMT is universal in the sense that the symmetry class of the Hamiltonian determines the ensemble of the random matrix theory, offering a powerful framework for analyzing complex quantum systems. \nThermality is intrinsically linked with chaos, and it is both intriguing and challenging to understand how thermality arises in a closed, isolated system. Though there are several ideas to support this, including typicality and the concept of a bath (see [11] for details), the satisfactory mechanism is not yet well understood. One such mechanism is provided by the Eigenstate Thermalization Hypothesis (ETH) [11, 12]. Furthermore, for a pure state with very large entropy (e.g., large N ), quantum statistical mechanics shows that the variance of any local correlation function is suppressed by a factor of e -S [13], where S can be identified as the entropy. Therefore, in a strongly interacting system with a large number of degrees of freedom, it is possible that the low-point correlation function can be well approximated by its thermal expectation value. \nAt first glance, the concepts discussed above may seem disconnected, but they are closely related to black holes and quantum gravity. The AdS/CFT correspondence [14] offers a framework that links an interacting conformal field theory (CFT) with quantum gravity in anti-de Sitter (AdS) spacetime. Since its inception, AdS/CFT has provided numerous insights that have deepened our understanding of black holes and their semiclassical properties [15-22]. However, the questions of smoothness of the horizon and the existence of an interior remain unresolved [23, 24]. This smoothness is at the heart of the information paradox [23, 25, 26]. Despite notable recent progress [27, 28], the paradox remains unresolved, especially in the Lorentzian picture and for the higher-dimensional black holes. \nOn the other hand, string theory provides alternatives such as fuzzballs which can bypass this. According to the fuzzball proposal, the horizon is not a smooth place and must be replaced by a complex stringy structure [29-32]. For some particular cases, there exists a whole moduli space of such solutions which, when quantized and counted, give rise to perfect agreement with the Bekenstein-Hawking entropy of the black hole [33, 34]. Another interesting fact is that, in the low-energy supergravity limit, many of these solutions can be written as explicit metrics called microstate geometries (see [35] for a review). These are perfectly regular geometries and behave similarly to black holes from a distance, but cap off smoothly near the horizon. Although there are many criticisms (e.g., [36]) regarding the geometric picture, the main takeaway is that string theory (as a UV-complete theory) can provide mechanisms that support a structure \nnear the horizon 1 . \nBlack holes are expected to be chaotic objects as their boundary duals are strongly coupled interacting theories. This can be seen by computing the out-of-time-order correlators (OTOCs) in black hole geometries [6, 19]. Moreover, it has been shown that black holes are maximally chaotic in the sense that the Lyapunov exponent saturates the chaos bound, \nλ = 2 π β , (1.1) \nwhere β is the inverse Hawking temperature of the black hole. It is important to remember that OTOCs measure early-time chaos, and there is evidence (e.g., [39]) that early-time chaos does not always imply chaos for all times. This raises the question: How can we demonstrate that the system remains chaotic at late times? Tools such as the SFF and LSD require knowledge of the spectrum, necessitating the quantization of the black hole system-a challenging problem 2 . A simpler and more tractable question might be: Can we detect any hint of late-time chaos in the probe sector? \nA naive probe calculation results in complex-valued quasi-normal modes, leading to a vanishing SFF at late times, a manifestation of the information paradox as discussed in [15]. To circumvent this problem, [40, 41] proposed placing a brick wall in the geometry 3 and quantizing a probe scalar field to obtain normal modes as a function of the wall's position. It was shown that when the brick wall is placed very near to the horizon, the SFF constructed from these normal modes exhibits a clear Dip-RampPlateau structure with a linear ramp of slope one in the log-log plot. Although the LSD is not of the Wigner-Dyson type (despite exhibiting level repulsion), a WD-type LSD was achieved by imposing fluctuations in the stretched horizon, which may be natural in constructing a more realistic toy version of actual fuzzballs [41]. This approach was generalized for rotating BTZ geometry in [43]. It has been argued in [40, 41] that this behavior is a generic feature of any non-extremal black hole, with supporting calculations in the Rindler × S 1 framework. However, no explicit calculations have been performed for higher-dimensional black holes. This article fills this gap by computing the normal modes of a probe scalar field in a five-dimensional AdS Schwarzschild black hole. \nThis study is intriguing for several reasons. Firstly, in three dimensions, gravity is non-dynamical as the degrees of freedom of the graviton are zero, whereas in higher dimensions, gravity is fully dynamical. Secondly, the AdS/CFT correspondence is much better understood for AdS 5 -CFT 4 . For example, the CFT is a 3 + 1 dimensional N = 4 super Yang-Mills (SYM) theory, where the glueball operators Tr( F µν F µν ) and Tr( F µν ˜ F µν ) are dual to the dilaton and axions, which are minimally coupled scalar fields in the bulk AdS. Thus, we can relate the bulk results to a more realistic fourdimensional strongly interacting theory. \nThe organization of the paper is as follows: In the next section (Section [2]), we describe the setup and identify the radial equation as the Heun equation. Subsequently, we attempt to solve the Heun equation using different methods and extract the normal modes. In Section [3], we leverage the fact that the BPZ equation in Liouville CFTs also satisfies the Heun equation. By using crossing symmetries, we relate the solutions around different singular points. We first discuss the quasinormal modes in Section [3.1] and then extract the normal modes in Section [3.2]. In Section [4], we use the WKB approximation method to find the normal modes with Dirichlet boundary conditions on the probe field near the horizon. Additionally, we demonstrate that the SFF in this case exhibits a dip-ramp-plateau behavior, with the slope of the ramp approaching one as the Dirichlet wall moves closer to the horizon. In Section [5], we use another perturbative method to solve the Heun equation and compute the normal modes. In Section [6], we show the emergence of a branch cut-like structure in the Green's function of the probe field as the wall is moved closer to the horizon. Finally, in Section [7], we conclude with a discussion and outline a few possible future directions. Additionally, we have included two appendices ([A], [B]) to ensure the paper is self-contained.", '2 The Setup: Scalar Field in AdS Schwarzschild Black hole': "Let's consider a probe scalar field of mass µ in the AdS-Schwarzschild black hole geometry, \nds 2 = -f ( r ) dt 2 + dr 2 f ( r ) + r 2 d Ω 2 3 , (2.1) \nwith f ( r ) = ( 1 -r 2 H r 2 ) ( r 2 + r 2 H + 1), where r H represents the position of the horizon (we have set the AdS length l to 1). The mass of the black hole, M is related to the horizon as, \nr 2 H = 1 2 √ 1 + 32 G N M 3 π -1 . (2.2) \nAs mentioned in the introduction, our 'measuring probe' is a scalar field Φ in this black hole background which satisfies the following Klein-Gordon equation: \n□ Φ ≡ 1 √ | g | ∂ ν ( √ | g | ∂ ν Φ ) = µ 2 Φ . (2.3) \nWhere µ is the mass associated with the scalar field. The conformal dimension of the boundary operator dual to Φ is given by, \nµ = √ ∆(∆ -4) . (2.4) \nSince the metric is time-independent and spherically symmetric, we can use the ansatz \nΦ( t, r, Ω) ∼ e -iωt Y l, /vectorm (Ω) φ ωl ( r ) , (2.5) \nwhere Y l, /vectorm (Ω) represent spherical harmonics on S 3 . These harmonics satisfy the following equation: \n∇ 2 S 3 Y l, /vectorm = -l ( l +2) Y l, /vectorm , (2.6) \nwhere ∇ 2 S 3 is the Laplacian on S 3 and /vector m = ( m,m ' ) can go from -l/ 2 to l/ 2 (see [44]). With this, (2.3) can be written as (to simplify our notation, we will now write φ ωl ( r ) as φ ( r )), \n1 r 3 d dr ( r 3 f ( r ) dφ ( r ) dr ) + ( ω 2 f ( r ) -l ( l +2) r 2 -µ 2 ) φ ( r ) = 0 . (2.7) \nThis is a Heun's equation which can be written in the well-known normal form (2.10) by using the following transformations, \nz = r 2 r 2 H + r 2 +1 , (2.8) \nφ ( r ) = z -1 2 ( z -˜ t ) -1 2 (1 -z ) 1 2 χ ( z ) . (2.9) \nIn this new z coordinate, the horizon is located at z = ˜ t = r 2 H 2 r 2 H +1 and the boundary is at z = 1 . The radial equation is, \n( ∂ 2 z + 1 4 -a 2 1 ( z -1) 2 -1 2 -a 2 0 -a 2 1 -a 2 ˜ t + a 2 ∞ + u z ( z -1) + 1 4 -a 2 ˜ t ( z -˜ t ) 2 + u z ( z -˜ t ) + 1 4 -a 2 0 z 2 ) χ ( z ) = 0 . (2.10) \nThe various parameters appearing in equation (2.10) are detailed in Appendix B. \nNear horizon behaviour of χ ( z ) is the following \nχ hor ( z ) = c 1 ( ˜ t -z ) 1 2 + a ˜ t + c 2 ( ˜ t -z ) 1 2 -a ˜ t + . . . = c 1 χ ( ˜ t ) + ( z ) + c 2 χ ( ˜ t ) -( z ) + . . . (2.11) \nHere, the first term corresponds to the outgoing mode and the second term to the \ningoing mode near the horizon. The constants c 1 and c 2 are arbitrary. \nSimilarly, Near boundary behavior can be expressed as, \nχ bdry ( z ) = c 3 ( 1 -z 1 + r 2 H ) 1 2 + a 1 + c 4 ( 1 -z 1 + r 2 H ) 1 2 -a 1 + . . . = c 3 χ (1) + ( z ) + c 4 χ (1) -( z ) + . . . (2.12) \nWhere the first term corresponds to the normalizable mode and the second term is non-normalizable. \nSince the radial equation is a Heun's equation, a closed-form solution in terms of known functions is not available. However, we can still use techniques from Liouville CFT to write the connection formulas that relate the solutions around various singular points. These connection formulas will aid us in finding the normal modes, as we will discuss in the next section.", '3 Using CFT techniques': "In Liouville CFT, the BPZ equation reduces to the Heun's equation in normal form in the semi-classical limit. The solutions of this equation give conformal blocks associated with different channels. Therefore, using the crossing symmetries of the conformal blocks, we can determine the relations between solutions around different singular points [45]. Let's focus first on the computation of quasi-normal modes and Green's function in this black hole metric.", '3.1 Ingoing boundary condition and quasinormal modes': "The natural boundary condition near the horizon of a black hole is ingoing boundary condition which implies c 1 = 0. So, \nχ hor ( z ) = c 2 χ ( ˜ t ) -( z ) . (3.1) \nUsing the connection formulas, we can write the near-horizon solution in terms of the solution near the boundary. Let's write the relations between the near-horizon and near-boundary solutions in the following way 4 : \nχ ( ˜ t ) + = a 11 χ (1) + + a 22 χ (1) -, (3.2) \nχ ( ˜ t ) -= b 11 χ (1) + + b 22 χ (1) -. (3.3) \nUsing (3.2) in (3.1), we get \nχ hor ( z ) = c 2 ( b 11 χ (1) + + b 22 χ (1) -) . (3.4) \nTo find the quasi-normal modes we need to impose normalizablity at the boundary, which implies, \nb 22 = M -+ ( a ˜ t , a ; a 0 ) M --( a, a 1 ; a ∞ ) + M --( a ˜ t , a ; a 0 ) M + -( a, a 1 ; a ∞ ) ˜ t -2 a e ∂ a F = 0(3.5) \nThis can be solved to extract the quasinormal modes of the black hole (see [46, 47] for the details). Following the Son-Starinets prescription [48], the retarded Green's function can be expressed as the ratio of the normalizable to non-normalizable modes as follows, \nG BH R ( ω, λ ) = b 11 b 22 . (3.6)", '3.2 Dirichlet boundary condition and normal modes': "Instead of imposing an ingoing boundary condition near the horizon, we now consider the Dirichlet boundary condition Φ( z 0 ) = 0 at some z 0 > ˜ t . This boundary condition is motivated by the desire to connect the brick wall-type model of 't Hooft with the fuzzball (microstate geometry) picture. It is important to note that, in principle, the brick wall can be placed at any location since it is an ad hoc boundary condition within the realm of classical General Relativity. However, as noted in works by 't Hooft [37] and [38, 40, 41, 49, 50], interesting physics emerges when the brick wall is very close to the horizon. Therefore, we take z 0 → ˜ t . In this region Dirichlet boundary condition implies, \nχ hor ( z 0 ) = c 1 χ ( ˜ t ) + ( z 0 ) + c 2 χ ( ˜ t ) -( z 0 ) = 0 , (3.7) \nwhich implies, \nR c 1 c 2 = c 1 c 2 = -χ ( ˜ t ) -( z 0 ) χ ( ˜ t ) + ( z 0 ) = -( ˜ t -z 0 ) 1 2 -a ˜ t ( ˜ t -z 0 ) 1 2 + a ˜ t = -( ˜ t -z 0 ) -2 a ˜ t . (3.8) \nThen the near-horizon solution comes out to be, \nχ ( ˜ t ) ( z ) = c 2 ( R c 1 c 2 χ ( ˜ t ) + ( z ) + χ ( ˜ t ) -( z )) . (3.9) \nUsing (3.2) in (3.9), we can relate the near-horizon and near-boundary solutions as follows: \nχ ( ˜ t ) = c 2 ( ( R c 1 c 2 a 11 + b 11 ) χ (1) + +( R c 1 c 2 a 22 + b 22 ) χ (1) -) . (3.10) \nThen normalizable condition at the boundary implies, \nR c 1 c 2 a 22 + b 22 = 0 . (3.11) \nSubstituting (3.8) in (3.11), \nb 22 a 22 = ( ˜ t -z 0 ) -iω r H 1+2 r 2 H = /epsilon1 -iω r H 1+2 r 2 H 0 . (3.12) \nHere /epsilon1 0 measures the distance between the horizon and the brick wall. In Appendix B, we have shown that the absolute value of b 22 a 22 is always one, i.e., the absolute part is trivially satisfied 5 . Therefore, what we have is the argument part, which is \nArg ( b 22 a 22 ) = Arg ( /epsilon1 -iω r H 1+2 r 2 H 0 ) +Arg(1) , (3.13) \nwhich can be simplified as, \nArg( b 22 ) + ω r H 2(1 + 2 r 2 H ) = nπ, where n ∈ Z . (3.14) \nThe solution of this equation provides the required normal modes, which are real valued and labeled by two quantum numbers, n and l . We have solved this equation using Mathematica . The behaviour of the normal modes along the l and n directions is presented in Figure 1. Although the spectrum exhibits a non-trivial dependence on the l -quantum number, along the n direction it is almost linear, as observed for the BTZ black hole in [40]. The figure clearly shows that the dependence of the modes on the l -direction is much slower compared to the n -direction. This quasi-degeneracy along l -directions is the key feature that underpins the emergence of interesting non-trivial physics. \nIn Figure 2 (left), we present the ∆ dependence of the normal modes along l -direction. For smaller values of l , ω varies significantly with different ∆. However, as l increases, these differences gradually diminish, and the modes converge onto a single \n<!-- image --> \nFigure 1 : Spectrum along l (left) and n (right) directions obtained by solving (3.14). For the left figure, n is fixed at 0, while for the right figure l is set to 1. The parameters used are r H = 20 , ∆ = 2 . 1 and /epsilon1 0 = 10 -20 . Although the spectrum has a trivial linear dependence on n -quantum number, along l direction it is non-trivial. \n<!-- image --> \n<!-- image --> \nFigure 2 : Left: The ∆ dependence of the normal modes along the l direction is depicted. Right: Similar plots are presented for the BTZ black hole, demonstrating the ∆ dependence of the normal modes along the m direction. \n<!-- image --> \ncurve. A similar trend is observed for the normal modes of the BTZ black hole, as depicted on the right of Figure 2. \nDue to the complexity of the various functions appearing in (3.14), we were unable to solve for larger values of l and n . More advanced numerical techniques are required to extend the solution to larger l and n . Nonetheless, we can explore various approximate methods to solve the radial equation, as discussed in the next two sections.", '4 WKB Approximation Method': "In this section we will try to solve the radial equation using the WKB method 6 . For this, we will write (2.7) in Schrödinger form to get an effective potential, and then we will find the bound states of the potential which will essentially correspond to the normal modes of the scalar perturbation. \nIn general, an equation of the following form, \na ( r ) d 2 φ ( r ) dr 2 + b ( r ) dφ ( r ) dr + c ( r ) φ ( r ) = 0 , (4.1) \ncan be written in the form of Schrödinger equation (with φ ( r ) = g ( r ) ψ ( r )) as, \nd 2 ψ ( r ) dr 2 + V ( r ) ψ ( r ) = 0 , (4.2) \nwhere, \nV ( r ) = 1 4 a 2 ( b 2 -2 b a ' +2 a ( b ' -2 c ) ) . (4.3) \nThough the general form of the potential is complex and not very enlightening to write down explicitly, some general observations are interesting to note. For large r , V ( r ) behaves as, \nlim r →∞ V ( r ) = ( d 2 -1) + 4 µ 2 4 r 2 + O ( 1 r 4 ) ; (4.4) \nwhich is always positive. As r approaches to r H , V ( r ) goes to -∞ . Additionally, there is a turning point at some r c > r H where V ( r ) changes sign. The general structure of V ( r ) is depicted in Figure 3. The Dirichlet boundary condition φ ( r 0 ) = 0 = ψ ( r 0 ) implies the existence of an infinite potential barrier at the location of the brick wall ( r = r 0 ), represented by a red vertical line in Figure 3. The task is then to find the bound states of this potential well. \nAn interesting point to note is that as the angular momentum l increases, the height of the bump after the turning point also increases (see Figure 4) which is an artifact of the presence of the angular momentum barrier. For a given r 0 , we can tune l in such a way that no potential well forms, which corresponds to scattering states. These modes are not trapped from the perspective of a boundary observer due to their large angular momentum. We are not interested in those modes and only consider the bound states. \nThe expression for V ( r ) with µ = 0 and r H = 1 is given by \nFigure 3 : Generic structure of the effective Schrödinger like potential V ( r ) in (4.5). Here l = 3 , ω = 0 . 3 and r H = 1. \n<!-- image --> \nFigure 4 : Generic structure of the effective Schrödinger like potential V ( r ) in (4.5). Here l = 3 , ω = 0 . 3 and r H = 1. \n<!-- image --> \nV ( r ) = r 6 (4 l ( l +2) -4 ω 2 +22) + (4 l ( l +2) -57) r 4 -4(2 l ( l +2) + 3) r 2 +15 r 8 -4 4 r 2 ( r 4 + r 2 -2) 2 . (4.5) \nAs shown in the appendix (see also [43],[51]), to determine the spectrum, we need the closed-form expression of ∫ r c r 0 √ | V ( r ) | , which is not feasible for (4.5). However, this does not mark the end of the investigation. Interesting physics occurs when the position of the brick wall is very close to the horizon. We will further assume that the turning point is also very close to the horizon. For the second condition, we need ω l not to be very large, meaning we will focus on low-lying modes for a fixed l (i,e, low n ) which is also physical as long as we are in the probe limit. \nFigure 5 : Comparison of (4.5) and (4.6). This image shows that when the turning point is close to the horizon, we can approximate the WKB integration using V eff . Parameters: l = 3 , ω = 0 . 3, r H = 1. \n<!-- image --> \nThe near-horizon expansion of V ( r ) is, \nlim /epsilon1 → 0 V ( r = r H + /epsilon1 ) = V eff ( /epsilon1 ) = -A 2 /epsilon1 2 + A 1 /epsilon1 -A 0 + O ( /epsilon1 ) , (4.6) \nwhere r = r H + /epsilon1 , and, \nA 0 = 7 432 ( 12 l ( l +2) -ω 2 -9 ) , (4.7) \nA 1 = 18 l ( l +2) -5 ω 2 +117 108 , (4.8) \nA 2 = 9 + ω 2 36 . (4.9) \nUnder the above assumption, we can approximate the original V ( r ) by (4.6). A comparison of V ( /epsilon1 ) and V eff ( /epsilon1 ) is shown in Figure 5. With this approximation, the WKB integration can be performed exactly, yielding the following result: \n∫ r c r 0 | V ( r ) | 1 2 dr = -√ A 0 /epsilon1 2 0 -A 1 /epsilon1 0 + A 2 -1 2 √ A 2 log ( T 1 +1 T 1 -1 ) -A 1 4 √ A 0 log ( T 2 +1 T 2 -1 ) , (4.10) \nwhere, \nT 1 = A 1 /epsilon1 0 -2 A 2 2 √ A 2 ( A 2 -A 1 /epsilon1 0 + A 0 /epsilon1 2 0 ) , (4.11) \nT 2 = -2 A 0 /epsilon1 0 + A 1 2 √ A 0 ( A 2 -A 1 /epsilon1 0 + A 0 /epsilon1 2 0 ) . (4.12) \n<!-- image --> \nl \nFigure 6 : Spectrum along n (left) and l (right) direction using WKB method. Both n and l start form 10, as low-lying modes are difficult to obtain with WKB. Other parameters: r H = 1, /epsilon1 0 = 10 -7 . 5 . For the left figure, l is fixed to 5, whereas for the right figure, n = 1. \n<!-- image --> \nThen according to WKB method, \n∫ r c r 0 | V ( r ) | 1 2 dr = 3 π 4 + nπ, (4.13) \nwhere n is the principal quantum number and /epsilon1 0 = r 0 -r H . This equation cannot be solved analytically for ω n,l , so we have solved it numerically for different choices of /epsilon1 0 using Mathematica . The results are the following. \nIn Figure 6, we present the spectrum along the n and l -directions. Note the striking similarity with the figures in [40]. The spectrum along n direction is linear, resembling that of a simple harmonic oscillator. Consequently, we do not observe any ramp structure in the SFF (see Figure 7 (left)). In contrast, the non-trivial functional dependence along l direction leads to the presence of a ramp in the SFF along l direction (see Figure 7 (right)). Figure 8 shows the SFFs for different /epsilon1 0 . As /epsilon1 0 decreases, the slope of the ramp approaches one, consistent with our previous observation of a linear ramp in 2+1 dimensions. In Figure 9, we illustrate the spectrum for different values of /epsilon1 0 . As /epsilon1 0 decreases, the ω values become increasingly closer to each other. This quasi-degenerate property leads to the ramp observed in the Spectral Form Factor (SFF). Due to the instability of low-lying modes, as shown in Figure 9, our numerical method cannot handle values of /epsilon1 0 below 10 -9 . \nω \n<!-- image --> \nFigure 7 : SFF along n (left) and l (right) directions for the modes shown in Figure 6. Here β is fixed to zero. \n<!-- image --> \nFigure 8 : This set of figures shows how the slope of the ramp is approaching one as we move the brick wall close to the horizon. The yellow line has slope one. The noise at the end of the spectrum is caused by the removal of the first few roots from the spectrum. n is fixed to one. \n<!-- image --> \nFigure 9 : Behaviour of normal modes along the l direction with varying /epsilon1 0 . \n<!-- image -->", '5 Solving Heun equation perturbatively': "In this section, we will solve the radial Heun equation using a perturbative method, as discussed in [52]. To begin, we rewrite the radial equation (2.7) in terms of a dimensionless coordinate y = r 2 H r 2 , resulting in: \ny 3 (1 -y 2 ) d dy ( 1 -y 2 y φ ' ( y ) ) + ˆ ω 2 4 y -ˆ l 2 4 y (1 -y 2 ) -ˆ µ 2 4 (1 -y 2 ) φ ( y ) = 0 , (5.1) \nwhere we have defined ˆ ω = ω/r H , ˆ l 2 = l ( l +2) r 2 H and ˆ µ = µ . We choose the following ansatz for φ ( y ), \nφ ( y ) = y 2 (1 -y ) -i ˆ ω 4 ( 1 + y 2 ) -ˆ ω 4 F ( y ) . (5.2) \nSubstituting this into (5.1), we obtain: \nF '' ( y ) + ( 3 y + 1 -i ˆ ω/ 2 y -1 + 1 -ˆ ω/ 2 y +1 ) F ' ( y ) + (2 -(1 + i )ˆ ω/ 4) 2 y -q y ( y 2 -1) F ( y ) = 0 , (5.3) \nwhere \nq = 3( i -1) 4 ˆ ω -ˆ l 2 4 + ˆ ω 2 4 . (5.4) \nIntroducing another new variable x = y 2 , (5.3) can be rewritten as: \nH F ( x ) = x (1 -x ) F '' ( x ) + 1 4 ( (1 + i ) x ( ω +( -6 + 6 i )) -(1 -i ) √ xω +8 ) F ' ( x ) + 1 32 ( 8 q √ x -i ( ω +( -4 + 4 i )) 2 ) F ( x ) = 0 . (5.5) \nTo solve the equation perturbatively, we decompose the Hamiltonian into two terms ( H 0 + H 1 ) in such a way that H 1 can be treated as a perturbation in ˆ ω : \nH 0 = x (1 -x ) d 2 dx 2 + ( 2 -1 -i 4 ˆ ω -(3 -1 + i 4 ˆ ω ) x ) d dx -1 4 ( (2 -1 + i 4 ˆ ω ) 2 -q ) ) , H 1 = (1 -√ x ) ( 1 -i 4 ˆ ω d dx + q 4 √ x ) . (5.6) \nThe above decomposition has an additional benefit: H 0 is a hypergeometric differential equation. Furthermore, we expand F ( x ) as a perturbative series in ˆ ω .: \nF ( x ) = F 0 ( x ) + F 1 ( x ) + F 2 ( x ) . . . (5.7) \nThis expansion allows us to rewrite (5.5) formally as: \n( H 0 + H 1 )( F 0 + F 1 + . . . ) = 0 = ⇒H 0 F 0 +( H 0 F 1 + H 1 F 0 ) + ( H 0 F 2 + H 1 F 1 ) + . . . = 0 . \nAt zeroth order, we have: \nH 0 F 0 = 0 . \nH 0 F 1 + H 1 F 0 = 0 = ⇒ F = -H -1 H F = -D F . \n1 0 1 0 0 \nH 0 F 2 + H 1 F 1 = 0 (5.8) \n= ⇒ F 2 = H -1 0 H 1 H -1 0 H 1 F 0 \n(5.9) \n= D 2 F 0 . (5.10) \nSimilarly, at the n -th order, we have, F n = ( -1) n D n F 0 . Thus, once F 0 is known, we can in principle construct the full perturbative series for F ( x ). \nAs already mentioned earlier, at zeroth order, the radial equation reduces to a hypergeometric equation: \nx (1 -x ) F '' 0 ( x ) + ( c -(1 + a + b ) x ) F ' 0 ( x ) -abF 0 ( x ) = 0 , (5.11) \nAt first order: \nIn second order: \nwhere, \na, b = 1 + 1 2 ( -1 + i 4 ˆ ω ± √ q ) , c = 2 -1 -i 4 ˆ ω. (5.12) \nThe solution of (5.11) is given in terms of hypergeometric functions: \nF 0 ( x ) = c 1 2 F 1 ( a, b, c, x ) + c 2 x 1 -c 2 F 1 (1 + a -c, 1 + b -c, 2 -c, x ) . (5.13) \nThe near-boundary expansion of (5.13) is: \nF 0 ( x ) ∼ c 1 + c 2 x 1 -c , \nwhich implies: \nψ ∼ (1 -y ) -i ˆ ω 4 ( 1 + y 2 ) -ˆ ω 4 ( c 1 y 2 + c 2 y 1 -i 2 ˆ ω ) . \nThe normalizable boundary condition ( ψ → 0) implies c 2 = 0. \nThe near-horizon behavior of (5.13) is given by: \nF 0 ( x ) ∼ A + B (1 -x ) i ˆ ω 2 , (5.14) \nwhere, \nA = Γ( c )Γ( c -a -b ) Γ( c -a )Γ( c -b ) , B = Γ( c )Γ( a + b -c ) Γ( a )Γ( b ) . (5.15) \nIf we impose the Dirichlet boundary condition at the stretched horizon (close to the event horizon): \nF 0 ( x 0 ) ∼ A + B (1 -x 0 ) i ˆ ω 2 = 0 = ⇒ (1 -x 0 ) i ˆ ω 2 = -A B = ⇒ /epsilon1 i ˆ ω 2 = -A B = ⇒ Arg [ Γ( a + b -c )Γ( c -a )Γ( c -b ) Γ( c -a -b )Γ( a )Γ( b ) ] + ˆ ω 2 log /epsilon1 = (2 n +1) π . (5.16) \nwhere, n ∈ Z . Solving this equation in 'Mathematica', we obtain ω as a function of n and l . Figure [10] shows the behavior of the spectrum in the n and l directions. As we can see again, the spectrum has a linear behavior along n and a non-linear behavior along l . With the spectrum in hand, we can now compute the SFF. Figure [11] shows \n<!-- image --> \nl \nFigure 10 : Spectrum along n (left) and l (right) direction for r H = 10. \n<!-- image --> \nFigure 11 : SFF for the normal modes along l direction with fixed n = 0. Here l cut = 400 and β = β H \n<!-- image --> \nSFF along the l direction at temperature β H which is the Hawking temperature of the black hole, but it is worth mentioning that β is nothing but a parameter here. \nSo far, we have discussed the zeroth-order solution. However, one can find the higher-order corrections to the normal modes by adding order-by-order corrections to the wave function. Explaining all the technical details is beyond the scope of this paper. However, a thorough discussion of this topic can be found in [52].", '6.1 In momentum space': "In this section, we closely examine the analytic structure of the Green's function, following the approaches of [53] and [49]. For any asymptotic AdS geometry, the Son-Starinets [48] prescription provides a method to compute the Green's function of the boundary \n<!-- image --> \nFigure 12 : Plot of Green's function, G ( ω, l = 1) for two different /epsilon1 0 . The main point to note is that as /epsilon1 0 decreases the poles start to accumulate on the real line. \n<!-- image --> \ntheory as the ratio of the normalizable and non-normalizable modes. From (3.10), this implies: \nG = b 11 + R c 1 c 2 a 11 b 22 + R c 1 c 2 a 22 . (6.1) \nThe poles of this Green's function correspond to the excitations of the system when perturbed by the boundary operator dual to the scalar field in the bulk. Since the system is in a pure state (with no infalling boundary condition), we expect the poles to lie on the real line. Except that, the function is analytic in the whole complex plane. In Figure 12, we plot the Green's function G as a function of ω . As z 0 is decreased, i.e., as the brick wall approaches the horizon, the poles come closer together. In the limit /epsilon1 0 → 0, these poles are so close that we can approximate them as a branch cut. However, there is a caveat: for any given /epsilon1 0 , there is always a maximum gap ∆ ω (noting that the spectrum is not equidistant), which sets a minimum timescale of ∼ 1 ∆ ω . Beyond this timescale, a boundary observer begins to probe the discreteness of the spectrum. \nIn the position space correlator, the implication of this pole condensation is very interesting. To see this, let's first rewrite G as follows: \nG = b 11 + R c 1 c 2 a 11 b 22 + R c 1 c 2 a 22 = b 11 b 22 + R c 1 c 2 a 11 b 22 1 + R c 1 c 2 a 22 b 22 = G BH R + R c 1 c 2 a 11 b 22 1 + R c 1 c 2 a 22 b 22 , (6.2) \nwhere the retarded correlator of the black hole, G BH R , is given in (3.6). The poles of the Green's function occur when the denominator of (6.2) goes to zero, which is nothing but the quantization equation (3.12). This happens because the condition: denominator= 0 corresponds to the vanishing of the non-normalizable mode, which is exactly the quantization condition (remember that the Dirichlet condition is already imposed in writing (3.10)). \nThe residue at these simple poles (labeled by ω k ) is given by, \nRes( G,ω k ) = ( G BH R + R c 1 c 2 a 11 b 22 ) ∣ ∣ ∣ ∣ ∣ ω = ω k ∂ ω ( 1 + R c 1 c 2 a 22 b 22 ) ∣ ∣ ∣ ∣ ∣ ω = ω k . (6.3) \nNow at ω = ω k , denominator of (6.2) is zero i.e. R c 1 c 2 = -b 22 b 11 which implies numerator can be written as, \n( G BH R + R c 1 c 2 a 11 b 22 ) ∣ ∣ ∣ ∣ ∣ ω = ω k = G BH R -a 11 a 22 ∣ ∣ ∣ ∣ ∣ ω = ω k = G BH R -( G BH R ) ∗ ∣ ∣ ∣ ∣ ∣ ω = ω k = 2 i Im G BH R ( ω, l ) ∣ ∣ ∣ ∣ ∣ ω = ω k , (6.4) \nwhereas denominator becomes \n∂ ω ( 1 + R c 1 c 2 a 22 b 22 ) ∣ ∣ ∣ ∣ ∣ ω = ω k = ∂ ω e -iθ ( ω,l ) ∣ ∣ ∣ ∣ ∣ ω = ω k , (6.5) \nwhere θ ( ω, l ) is given in a footnote after (3.12). Thus, the residue is, \nRes( G,ω k ) = 2 i Im G BH R ( ω, l ) ∣ ∣ ∣ ∣ ∣ ω = ω k ∂ ω e -iθ ( ω,l ) ∣ ∣ ∣ ∣ ∣ ω = ω k . (6.6) \nIf a function has simple poles at ω 1 , ω 2 , . . . , ω k , . . . , it can be expressed as, \nG ( ω, l ) = ∑ k ( ω 2 ω 2 k ) δ Res( G,ω k ) ω -ω k . (6.7) \nThe additional factor is included to ensure the convergence of G on the contour as ω →∞ . When the poles are very closely spaced, the sum can be approximated by an integral, given by the following expression: \nG ( ω, l ) = ∫ dω k ρ ω ( ω k , l ) ω -ω k with ρ ω ( ω k , l ) = dk dω k ( ω 2 ω 2 k ) δ Res( G,ω k ) , (6.8) \nwhich implies \nG ( ω + i/epsilon1, l ) -G ( ω -i/epsilon1, l ) ≈ ∫ dω k ρ ω ( ω k , l )2 πiδ ( ω -ω k ) = 2 πiρ ω ( ω, l ) . (6.9) \nThus, when the poles are densely packed, they can be approximated by a branch cut, with the discontinuity given by (6.9). \nWeobserve a similar feature as the brick wall approaches the event horizon, allowing us to make analogous approximations for the correlator in (6.2). To compute the discontinuity, we evaluate: \ndk dω k = 1 2 π dθ ( ω k , l ) dω k = -1 2 πi e iθ ( ω k ,l ) ∂ ω k e -iθ ( ω k ,l ) = -1 2 πi ∂ ω k e -iθ ( ω k ,l ) . (6.10) \nSubstituting this into the expression in (6.8), we obtain, \nρ ω ( ω k , l ) = -1 π ( ω 2 ω 2 k ) ∆ Im G BH R ( ω, l 0 ) ∣ ∣ ∣ ∣ ∣ ω = ω k . (6.11) \nGiven the correlator in (6.2), we can define the retarded, Feynman, and Wightman correlators as follows: \nG R ( ω, l ) = G ( ω + i/epsilon1, l ) = ∑ k ( ω 2 ω 2 k ) ∆ Res( G,ω k ) ω -ω k + i/epsilon1 , (6.12) \nG F ( ω, l ) = ∑ ω k > 0 ( ω 2 ω 2 k ) ∆ Res( G,ω k ) ω -ω k + i/epsilon1 + ∑ ω k < 0 ( ω 2 ω 2 k ) ∆ Res( G,ω k ) ω -ω k -i/epsilon1 , (6.13) \nG W ( ω, l ) = -sign ω Im G R ( ω, l ) . (6.14) \nThus, the imaginary part of the retarded Green's function can be expressed as: \nIm G R ( ω, l ) = 1 2 i ( G ( ω + i/epsilon1, l ) -G ( ω -i/epsilon1, l )) = πρ ω ( ω, l ) . (6.15) \nSubstituting (6.11) into (6.15), we find, \nIm G R ( ω, l ) = -Im G BH R ( ω, l ) ∣ ∣ ∣ ∣ ∣ ω = ω k . (6.16)", '6.2 In position space': "So far, we have discussed the momentum space Green's function. In this section, we turn our attention to the position space Green's function, focusing primarily on the Feynman propagator, which is given by 7 : \nC ( u, v ) = N ∫∫ dwdpe ipv -iwu G F ( w, p ) = N ∫∫ dwdpe ipv -iwu ∑ w k ( w 2 w 2 k ) ∆ Res( G,w k ) w -w k ± i/epsilon1 . (6.17) \nHere, the ± sign corresponds to positive and negative w k , respectively. For simplicity, we will consider u > 0 without loss of generality, which allows us to select only the positive w k . After integrating over w , the expression simplifies to: \nC ( u, v ) = N ∑ w k ∫ dp e ipv -iw k u Res( G,w k ) . (6.18) \nAs mentioned earlier, when the brick wall is very close to the horizon, the sum over k can be approximated as an integral. In this approximation, the discrete w k becomes a continuous variable w . Thus, the expression for C ( u, v ) becomes 8 : \nC ( u, v ) = N ∫∫ dpdw e ipv -iwu Im G BH R ( w, l ) = N ∫∫ dpdw e ipv -iwu ( G BH R ( w, l ) -G BH R ( w, l ) ∗ ) . (6.19) \nHere, the first term has poles in the lower half-plane, while the second term has poles in the upper half-plane. Consequently, only the first term contributes, and it reproduces the exact position-space Green's function, which corresponds to that of a thermal one with a temperature equal to the Hawking temperature of the black hole. \nThis result implies that, although we start from a pure state, the two-point correlator can be well approximated by a thermal correlator when the brick wall is close to the horizon. Therefore, a boundary observer cannot distinguish this pure state from a thermal state unless they wait for an exceptionally long time or measure higher-point correlators.", '7 Discussion': "In this study, we explored a minimally coupled scalar field in the bulk five-dimensional AdS-Schwarzschild spacetime, which is dual to the gauge-invariant composite scalar operators Tr( F µν F µν ) or Tr( F µν ˜ F µν ) in the boundary four dimensional N = 4 SYM theory. The scalar field was quantized using a Dirichlet boundary condition, ensuring that the scalar field vanishes on the stretched horizon, effectively imposing a perfectly reflecting boundary condition instead of the more conventional ingoing boundary condition near the horizon. This quantization results in normal modes that are real-valued, in contrast to the complex-valued quasi-normal modes typically observed. These normal modes are labeled by two quantum numbers: the principal quantum number n and the angular momentum quantum number l . A significant degeneracy exists in the spectrum; for each l , there can be (2 l + 1) 2 states with the same ω . Since the radial equation takes the form of a Heun equation, no exact solution is available, necessitating the use of various approximate methods to determine the normal modes. We employed techniques from Liouville CFT, as previously utilized in [46], to calculate these normal modes. Although the spectrum shows linearity along the n direction, a non-trivial dependence is observed along the l direction. However, we were unable to obtain a sufficiently large number of modes to perform a SFF analysis. We anticipate that a linear ramp with a slope of one should exist in the SFF along the l direction. Improving numerical techniques to identify higher-lying modes could be valuable, not only for SFF analysis but also for extracting black hole quasi-normal modes. Insights from these modes could provide a deeper understanding of strongly coupled thermal N = 4 super Yang-Mills (SYM) theory. We aim to address this issue in future work. \nWe then applied the WKB method to solve the spectrum, which allowed us to easily compute the high-lying modes. Using this approach, we found that the corresponding single-particle spectral form factor (SFF) along l direction exhibits a clear Dip-RampPlateau (DRP) structure, with a linear ramp of slope one in the log-log plot (see Figure 8). It is important to note that the spectrum is deterministic, rather than random as seen in various Gaussian ensembles while choosing the matrix elements. Despite this determinism, the slope of the ramp is one, similar to that observed in random \nmatrix theory 9 . However, it is important to note that our numerical methods currently encounter difficulties as we approach very close to the horizon, where the numerics break down for extremely small values of the cut-off /epsilon1 0 . It is important to note that while the SFF exhibits a distinct DRP structure, the LSD does not follow the conventional Wigner-Dyson distribution. However, by generalizing this simple boundary condition, as demonstrated in [41], one can achieve a Wigner-Dyson LSD accompanied by a linear ramp. \nWe have attempted to solve the Heun equation perturbatively to determine the normal modes and the corresponding SFF. So far, we have only used the zeroth order equation to compute the normal modes. Although it is, in principle, possible to find higher-order corrections to the wave functions and normal modes, it is quite challenging in practice. Despite this, we have been able to show a linear ramp of slope one the SFF along l -direction. \nSince chaos is closely related to thermality, we expect observing thermal behavior emerging from the system. This behavior is demonstrated by computing the Green's function and showing that the Green's function of this pure state can be approximated by the thermal retarded Green's function of N = 4 SYM at a temperature equal to the black hole's Hawking temperature. This approximation holds when the brick wall is placed very close to the horizon, where the poles of the Green's function become so densely packed that they can be approximated as a branch cut. The discontinuity around this branch cut captures the thermal correlator's information. However, over long time scales, the pole structure becomes apparent instead of the branch cut. In [38], it is suggested that this transition occurs around the Page time, approximately O ( N ), and it would be valuable to examine this timescale by computing the gap in the spectrum. For a more detailed discussion, see [55]. \nThe main result of this article is the clear Dip-Ramp-Plateau (DRP) structure with a linear ramp of slope one in the SFF constructed from the normal mode spectrum along the l direction. This behavior arises from the non-trivial dependence of the spectrum on l quantum number when the brick wall is placed very close to the horizon. In this limit, the retarded correlator can be effectively approximated by a thermal correlator. \nThere are several promising directions for future work. One avenue is to compute the normal modes for charged and rotating black holes. Another direction is to generalize this framework to fermionic or gauge fields. Additionally, an interesting open question arises from the observation in [56] that the retarded Green's function of thermal SYM exhibits branch cuts in the lower half-plane in the weak coupling limit, where \nthe bulk picture is highly quantum. In contrast, in the strong coupling limit, which corresponds to a classical black hole geometry, the Green's function contains complexvalued poles that are identified as quasi-normal modes. In our study, we observed that placing a brick wall in the geometry leads to poles that manifest as a branch cut when the wall is very close to the horizon. It is important to emphasize that these poles lie on the real line since there is no dissipation in the system. If, instead of a Dirichlet boundary condition, we introduce some loss by imposing a Neumann boundary condition (a not-so-hard brick wall), it is conceivable that the poles could shift away from the real line, still manifesting as a branch cut when the brick wall is close to the horizon. We plan to address this issue in the future. \nIn a recent work [57], the author has conjectured a correspondence between the excited states of the free scalar field in this brick-wall geometry to that of the vacuum or excited states of a CFT under modular quantization[58] (quantization for the modular Hamiltonian). Additionally, in a separate study [59], it was observed that a particular class of Floquet CFT dynamics corresponds to the extended modular Hamiltonian of a subsystem between two fixed points of the dynamics and the near horizon Virasoro algebra resembles the Virasoro algebra under Modular quantization. These observations prompt an intriguing future direction to check the validity of this conjecture (or a similar one) in higher dimensions (especially in the context of AdS 5 /CFT 4 ). In particular, one might use the results of [60], which generalizes the study of floquet CFTs in d>2, to check this in higher dimensions.", '8 Acknowledgements': 'Weexpress our sincere gratitude to Alexander Zhiboedov, Arnab Kundu, Bobby Ezhuthachan, and Chethan Krishnan for their valuable discussions and insightful comments.', 'A WKB approximation method': "In this section, we briefly review the WKB approximation method. For simplicity, we will restrict the discussions to one dimension. We start with the Schrödinger equation for the stationary states of a single particle moving in a time-independent potential, \n-ℏ 2 2 m d 2 ψ ( x ) dx 2 + V ( x ) ψ ( x ) = Eψ ( x ) , (A.1) \nand consider p ( x ) = √ 2 m ( E -V ( x )). If the potential is constant, the solutions of (A.1), for E > V ( x ), are 10 \nψ ( x ) ∼ A ± e ± ipx ℏ . \nHowever, if the potential is not constant, the solution is not as straightforward. However, we assume an ansatz for the solution as follows. \nψ ( x ) = C ( x ) ± e ± iφ ( x ) ℏ . (A.2) \nSubstituting (A.2) into (A.1) and separating the real and imaginary parts, we obtain two differential equations: \nC '' ( x ) -C ( x ) φ ' ( x ) 2 + p ( x ) 2 C ( x ) = 0 , (A.3) \n( C ' ( x ) 2 φ ' ( x )) ' = 0 . (A.4) \nThe first of the two equations above cannot be solved in general. However, we assume that the amplitude C ( x ) is slowly varying, i.e. C '' ( x ) can be neglected. Under this assumption, the equation can be solved, and in the 'classically allowed region' ( E > V ( x )), the wave function takes the following form: \nψ ( x ) = A 1 √ p ( x ) e i ℏ ∫ x ∗ x p ( x ' ) dx ' + B 1 √ p ( x ) e -i ℏ ∫ x ∗ x p ( x ' ) dx ' , (A.5) \nwhereas, in the 'classically forbidden region' ( E < V ( x )), it is given by: \nψ ( x ) = A 2 √ | p ( x ) | e -1 ℏ ∫ x x ∗ | p ( x ' ) | dx ' + B 2 √ | p ( x ) | e 1 ℏ ∫ x x ∗ | p ( x ' ) | dx ' , (A.6) \nwhere, A 1 , 2 , B 1 , 2 are two sets of arbitrary constants that are yet to be fixed. \nAs we can see from above, there is a significant problem at the 'classical turning points' where E = V ( x ). At these points, both (A.5) and (A.6) diverge. Therefore, these points must be treated separately. The solutions at the turning points allow us to connect the constants of the solutions in (A.5) and (A.6), as the turning point acts as a bridge between the two regions. We assume that the potential varies slowly near the turning point such that \nV ( x ) = E + V ' ( x ∗ )( x -x ∗ ) , (A.7) \n∗ \nwhere, x is the classical turning point which is derived by solving V ( x ) = E . If we substitute (A.7) in (A.1), the Schrödinger equation becomes, \nd 2 ψ ( x ) dx 2 = ζ 3 ( x -x ∗ ) ψ ( x ) , (A.8) \nwith ζ = ( 2 m ℏ 2 V ' ( x ∗ ) ) 1 / 3 . After a change of variable, z ≡ ζ ( x -x ∗ ), the above equation takes the form of the Airy's equation: \nd 2 ψ ( z ) dz 2 = zψ ( z ) . \nTherefore, the solutions are given as a sum of the Airy functions: \nψ ( z ) = αAi ( z ) + βBi ( z ) . (A.9) \nLet us consider a scenario (Figure [13]) to illustrate how this method is used in practice. Assume that the potential is such that the region where -∞ < x < x ∗ is classically forbidden (region II), and the region where ∞ > x > x ∗ is classically allowed (region I). In this case, the wave function in region I is given by (A.5), and in region II, it is given by (A.6). Since the asymptotic forms of both Ai ( z ) and Bi ( z ) are known, by matching (A.9) with (A.5) and (A.6) in the two asymptotic regions z /greatermuch 0 and z /lessmuch 0 we can derive the WKB connection formulae relating the constants of the solutions { A 1 , B 1 } , { A 2 , B 2 } and { α, β } . \nIn particular, we can consider an example of the type Figure [3] with one turning point. Then, using the normalizability condition for solution in z /greatermuch 0 region, which implies B 2 = 0, one can show that the connection formulae are given by: \nFigure 13 : Schematic diagram of a potential, illustrating the 'classically allowed' and 'classically forbidden' regions for a particle with energy E. The point x ∗ denotes the turning point. \n<!-- image --> \nA 1 = -ie i π 4 A 2 , B 1 = ie -i π 4 A 2 , \nand correspondingly the solutions in the two regions are given by: \nψ ( x ) = 2 A 2 √ p ( x ) cos ( 1 ℏ ∫ x ∗ x p ( x ' ) dx ' -π 4 ) , for region-I A 2 √ | p ( x ) | exp ( -1 ℏ ∫ x x ∗ | p ( x ' ) | dx ' ) , for region-II . \nThis type of potential has a cut-off at ψ ( x 0 ) = 0, for some x 0 < x ∗ , then from the solution for region-II, one finds that cos ( 1 ℏ ∫ x ∗ x 0 p ( x ' ) dx ' -π 4 ) must be zero for ψ ( x 0 ) = 0, in other words, the condition \n1 ℏ ∫ x ∗ x 0 p ( x ' ) dx ' = 3 π 4 + nπ, ∀ n ∈ Z \nmust be satisfied.", 'Validity of the WKB approximation:': "We will end this section by discussing the validity of the WKB approximation method. To do that, we put ψ ( x ) = e i φ ( x ) ℏ in (A.1) and we get \ni ℏ φ '' ( x ) -( ψ ' ( x )) 2 + p ( x ) 2 = 0 (A.10) \nTherefore, in the quasi-classical limit one would expect \n| ℏ φ '' ( x ) | << ( ψ ' ( x )) 2 or, ∣ ∣ ∣ ∣ ∣ d dx ( ℏ φ ' ( x ) )∣ ∣ ∣ ∣ ∣ << 1 \nSince, from (A.3) we know that φ ' ( x ) = p ( x ) we can write the condition as \n∣ ∣ ∣ ∣ ∣ d dx ( ℏ p ( x ) )∣ ∣ ∣ ∣ ∣ << 1 (A.11) \nFrom the above equation, we can see that, at the classical turning point, the condition does not hold since p ( x ) = 0, as we mentioned earlier also.", 'B Connection formulae': "Following the connection formulas given in [45] and [61], the incoming and outgoing modes near the horizon can be written in terms of the normalizable and nonnormalizable modes near the boundary as \nχ ( ˜ t ) + = ∑ θ ' = ± ( M ++ ( a ˜ t , a ; a 0 ) M -θ ' ( a, a 1 ; a ∞ ) ˜ t a e -1 2 ∂ a F + M + -( a ˜ t , a ; a 0 ) M + θ ' ( a, a 1 ; a ∞ ) ˜ t -a e 1 2 ∂ a F ˜ t 1 2 -a 0 + a ˜ t (1 -˜ t ) a t -a 1 e 1 2 ( ∂ a ˜ t + θ ' ∂ a 1 ) F χ (1) θ ' (B.1) \n) \nand \nχ ( ˜ t ) -= ∑ ' ( M -+ ( a ˜ t , a ; a 0 ) M -θ ' ( a, a 1 ; a ∞ ) ˜ t a e -1 2 ∂ a F + M + -( a ˜ t , a ; a 0 ) M + θ ' ( a, a 1 ; a ∞ ) ˜ t -a e 1 2 ∂ a F ) \nθ = ± ˜ t 1 2 -a 0 + a ˜ t (1 -˜ t ) a ˜ t -a 1 e 1 2 ( ∂ a ˜ t + θ ' ∂ a 1 ) F χ (1) θ ' (B.2) \nwhere, \nM θθ ' ( α, β ; γ ) = Γ( -2 θ ' β )Γ(1+2 θα ) Γ( 1 2 + θα -θ ' β + γ )Γ( 1 2 + θα -θ ' β -γ ) (B.3) \nF ( ˜ t ) = ( 1 4 -a 2 -a 2 1 + a 2 ∞ )( 1 4 -a 2 -a 2 ˜ t + a 2 0 ) ˜ t 1 2 -2 a 2 (B.4) \na 2 = ( -1 4 -u -a 2 ˜ t + a 2 0 ) ) ( 1 -( -1+2 a 2 0 +2 a 2 1 -2 a 2 ∞ +2 a 2 ˜ t -2 u )( -1+4 a 2 ˜ t -2 u ) 2( -1+4 a 2 0 +4 a 2 ˜ t -2 u ) ˜ t ) (B.5) \nIn the above, we used the expansion of the conformal block F in the parameter t [45], retaining only the leading-order term. The values of the parameters depend on the \nspecific space-time that we are considering. For the AdS-Schwarzschild black hole, the parameters are [61] \n˜ t = r 2 h 2 r 2 h +1 (B.6) \na ˜ t = iω 2 r h 2 r 2 h +1 (B.7) \na 1 = ∆ -2 2 (B.8) \na 0 = 0 (B.9) \na ∞ = ω 2 √ r 2 h +1 2 r 2 h +1 (B.10) \nu = -l ( l +2) + 2(2 r 2 h +1) + r 2 h ∆(∆ -4) 4( r 2 h +1) + r 2 h 4(1 + r 2 h ) ω 2 2 r 2 h +1 (B.11) \nSubstituting (B.1) and (B.2) in (3.1), we extract the parameters a 11 , a 22 , b 11 and b 22 to be: \na 11 = [ M ++ ( a ˜ t , a ; a 0 ) M -+ ( a, a 1 ; a ∞ ) ˜ t a e -1 2 ∂ a F + M + -( a ˜ t , a ; a 0 ) M ++ ( a, a 1 ; a ∞ ) ˜ t -a e 1 2 ∂ a F ] ˜ t 1 2 -a 0 + a ˜ t (1 -˜ t ) a t -a 1 e -1 2 ( ∂ a ˜ t + ∂ a 1 ) F a 22 = [ M ++ ( a ˜ t , a ; a 0 ) M --( a, a 1 ; a ∞ ) ˜ t a e -1 2 ∂ a F + M + -( a ˜ t , a ; a 0 ) M + -( a, a 1 ; a ∞ ) ˜ t -a e 1 2 ∂ a F ] t 1 2 -a 0 + a ˜ t (1 -˜ t ) a ˜ t -a 1 e -1 2 ( ∂ a ˜ t -∂ a 1 ) F b 11 = [ M -+ ( a ˜ t , a ; a 0 ) M -+ ( a, a 1 ; a ∞ ) ˜ t a e -1 2 ∂ a F + M --( a ˜ t , a ; a 0 ) M ++ ( a, a 1 ; a ∞ ) ˜ t -a e 1 2 ∂ a F ] ˜ t 1 2 -a 0 -a t (1 -˜ t ) a ˜ t -a 1 e -1 2 ( ∂ a ˜ t + ∂ a 1 ) F b 22 = [ M -+ ( a ˜ t , a ; a 0 ) M --( a, a 1 ; a ∞ ) ˜ t a e -1 2 ∂ a F + M --( a ˜ t , a ; a 0 ) M + -( a, a 1 ; a ∞ ) ˜ t -a e 1 2 ∂ a F ] ˜ t 1 2 -a 0 -a ˜ t (1 -˜ t ) a ˜ t -a 1 e -1 2 ( ∂ a ˜ t -∂ a 1 ) F (B.12) \nSubstituting the parameters for AdS-Schwarzchild black hole (B.3)-(B.11) in (B.12), one can check that the ratio of the parameters b 22 a 22 can be written as: \nb 22 a 22 = ( ˜ t ) -iQ Λ Λ ∗ (B.13) \nwhere, ˜ t = r 2 h 2 r 2 h +1 , Q = r h ω 2 r 2 h +1 , Λ = Γ(1 -iQ ) ( e P 1 R 1 Γ ( 1 2 -a -iQ ) 2 + e P 2 R 2 Γ ( 1 2 + a -iQ ) 2 ) and P 1 = a (1 -4 a 2 ) 2 ( -4 a 2 + r 2 h ω 2 ( 2 r 2 h +1 ) 2 +1 )( -4 a 2 -(∆ -2) 2 + ( r 2 h +1 ) ω 2 ( 2 r 2 h +1 ) 2 +1 ) , \n1 = Γ(2 a )Γ(2 a +1)Γ ( 1 2 ( -2 a +∆ -√ r 2 h +1 ω 2 r 2 h +1 -1 )) Γ ( 1 2 ( -2 a +∆+ ω √ r 2 h +1 2 r 2 h +1 -1 )) \nR , P 2 = a ( 8 a 2 ( r 2 h +1 ) +∆ 2 -4∆+2 ( ∆ 2 -4∆+2 ) r 2 h -ω 2 +2 ) (4 a 2 -1) ( 2rh 2 +1 ) , \nR 2 = Γ( -2 a )Γ(1 -2 a )Γ ( 1 2 ( 2 a +∆ -√ r 2 h +1 ω 2 r 2 h +1 -1 )) Γ ( 1 2 ( 2 a +∆+ ω √ r 2 h +1 2 r 2 h +1 -1 )) . 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2024arXiv240907015J

We argue that two prominent theories of ion heating in lowbeta collisionless plasmas stochastic and quasilinear heating represent similar physical processes in turbulence with different normalized cross helicities. To capture both we propose a simple phenomenology based on the power in scales at which critically balanced fluctuations reach their smallest parallel scale. Simulations of test ions interacting with turbulence confirm our scalings across a wide range of different ion and turbulence properties including with a steep ionkinetic transition range as relevant to the solar wind.

2024-09-01T00:00:00Z

['2024arXiv240907015J', 'arXiv:2409.07015', '10.48550/arXiv.2409.07015']

['Astrophysics - Solar and Stellar Astrophysics', 'Physics - Plasma Physics', 'Physics - Space Physics']

A Unified Phenomenology of Ion Heating in Lowbeta Plasmas TestParticle Simulations

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.07015.pdf

{'A Unified Phenomenology of Ion Heating in Lowβ Plasmas: Test-Particle Simulations': "Zade Johnston, ∗ Jonathan Squire, and Romain Meyrand Physics Department, University of Otago, Dunedin 9010, New Zealand \n(Dated: September 12, 2024) \nWe argue that two prominent theories of ion heating in lowβ collisionless plasmas-stochastic and quasi-linear heating-represent similar physical processes in turbulence with different normalized cross helicities. To capture both, we propose a simple phenomenology based on the power in scales at which critically balanced fluctuations reach their smallest parallel scale. Simulations of test ions interacting with turbulence confirm our scalings across a wide range of different ion and turbulence properties, including with a steep ion-kinetic transition range as relevant to the solar wind. \nIons within the turbulent lowβ solar corona and wind are observed to be continously heated, with more energization perpendicular to the local magnetic field than parallel [1-3]. Additionally, heavy ions are heated more than protons [4-6] and protons more than electrons [7], despite the low-frequency gyrokinetic model predicting that electron heating dominates at low β [8, 9]. Many mechanisms have been proposed to explain these observations, including stochastic heating by uncorrelated turbulent fluctuations [10, 11], wave-particle interactions leading to quasi-linear heating [12-14], and others [1517]. Despite this previous work, aspects of the underlying processes remain unclear, representing an important deficiency for building understanding of coronal heating and solar-wind acceleration. \nExtending previous studies [18-23], in this Letter we present a unified description of turbulent heating of ions in both balanced turbulence (with equal amplitudes of inwards and outwards Elsasser fluctuations z ± ) and imbalanced turbulence (where one dominates: | z + | ≫ | z -| ). Noting that stochastic heating (SH) and cyclotron-resonant quasi-linear heating (QLH) can be considered as two limits of similar physical processes and motivated by the principle of 'critical balance', we propose that the ion-heating rate is controlled by the power in scales at which turbulent fluctuations reach their maximum frequency, which may not be at ion-gyroradius scales in the presence of a steep ion-kinetic 'transitionrange' drop in the spectrum. We test this proposal with high-resolution simulations of test particles interacting with lowβ magnetized turbulence. Our proposed scaling accurately captures measured proton and minor-ion heating rates in turbulence across a wide range of conditions (balanced, imbalanced, with and without a 'helicity barrier'). These results clarify the physics of collisionless turbulent heating with clear application to both in-situ observational analyses and theoretical models of astrophysical plasmas [24]. \nBefore continuing we define important symbols, with species index s either ions or protons ( s = i or p): E and B are the electric and magnetic fields; k ∥ and k ⊥ are the components of the wavevector k parallel and perpendicular to B ; ρ s = v th ,s / Ω s is the thermal gyroradius with \nv th ,s = √ 2 T s /m s and Ω s = | q s | B/m s c the ion thermal speed and gyrofrequency; q s = Ze and m s = Am p are the ion charge and mass in units of proton quantities; β s = v 2 th ,s /v 2 A is the ion beta; c is the speed of light; δu λ is the rms E × B velocity at scales k ⊥ λ ∼ 1; and v A is the Alfv'en speed. \nTurbulence Theory. -Turbulence near the Sun is imbalanced, with | z + | ≫ | z -| . Recent work [25-27] argues that such turbulence forms a 'helicity barrier', which, due to its effect on turbulent fluctuations near ρ p -scales, is a key ingredient in understanding ion heating. The effect occurs because lowβ gyrokinetics conserves energy and a 'generalized helicity' [25, 28], which must be simultaneously cascaded with constant flux to be in a statistical steady-state. At scales larger than ρ p , helicity is the cross-helicity which cascades to small perpendicular scales. At scales smaller than ρ p , helicity becomes magnetic helicity, which would inverse cascade in imbalanced turbulence. For balanced turbulence, with zero helicity, energy is able to cascade to small perpendicular scales. However, in imbalanced turbulence only the balanced (zero-helicity) portion of the energy cascade is let through to small perpendicular scales; the remainder is blocked from cascading and trapped at scales above ρ p by the helicity barrier. This causes the amplitude of the turbulence to grow in time and, through critical balance [29], reach and dissipate at small parallel scales. Hybridkinetic imbalanced turbulence simulations [26] show that fluctuations at these scales become oblique ion-cyclotron waves, heating ions through quasi-linear interactions and allowing the turbulence to saturate. The helicity barrier is well supported observationally, explaining the steep 'transition-range' drop in fluctuation spectra [30, 31] and other features [32, 33]. \nIon Heating. -Two widely studied ion-heating mechanisms are stochastic heating and cyclotron-resonant quasi-linear diffusion. Stochastic heating involves ions being heated by uncorrelated kicks from fluctuations at scales k ⊥ ρ i ∼ 1. This leads to a diffusion perpendicular to B in velocity space, generally heating the plasma [10, 34, 35]. The theory explains ion heating well in lowβ balanced turbulence simulations [19, 21, 22, 36], and has also been applied to ion heating within the imbalanced near-Sun environment [37-41]. Quasi-linear theory pre- \nthat ions and waves interact strongly if the waves' Doppler-shifted frequencies are resonant with the ion gyrofrequency [42-44]. This causes ions to diffuse ions in velocity space along contours of constant energy in the frame moving at the wave's phase velocity, heating the plasma (however, note that ions may also be heated by non-resonant interactions with waves [35, 45]). \nWhile traditionally considered as separate mechanisms, we argue that SH and QLH should instead be considered as two limits of a continuum controlled by the nonlinear broadening of the fluctuations' frequency spectrum. Unbroadened fluctuations-waves with a single frequency ω at a particular k -cause QLH-like behavior; fluctuations that are broadened to a level comparable to ω itself, which thus have significant power in low-frequency fluctuations [46], cause SH-like behavior. This reveals a deep connection to the turbulence imbalance: in imbalanced turbulence z + -fluctuations have linear frequencies that exceed their nonlinear rate (the QLH-limit); in balanced turbulence the two are comparable [29] (the SH-limit). A smooth transition between these limits occurs as the imbalance is adjusted. \nTo further illustrate the connection, we argue that SH and QLH rates scale similarly with turbulent amplitude, even though SH is traditionally discussed in the context of low-frequency turbulence, while QLH is generally considered to involve high-frequency fluctuations ( ω ∼ Ω i ). SH requires turbulent amplitudes δu ρ i comparable to v th,i (generally δu ρ i /v th,i ≳ 0 . 1 [10, 19]). To understand the amplitude at which QLH will be activated, we use the principle of 'propagation critical balance' (in which parallel scales are determined by the wandering of magnetic field lines by perpendicular perturbations [46-48]), giving ω ∼ k ∥ v A ∼ k ⊥ δu λ throughout the inertial range. Thus, in order to reach ω ∼ Ω i and activate QLH requires Ω i ∼ δu ρ i /ρ i = ⇒ δu ρ i ∼ v th,i , where we used the fact that the ion-eddy interaction is strongest at ρ i scales because ω increases with k ⊥ but interactions with k ⊥ ρ i > 1 fluctuations are suppressed [49] (there may still be contributions from such eddies, however [14, 21]). This means QLH requires similar turbulent amplitudes as SH to occur; likewise, for critically balanced turbulence, the amplitudes needed to cause SH imply turbulent frequencies that approach Ω i , although due to eddies decorrelating, there remains significant power in ω < Ω i fluctuations in balanced turbulence [46]. The argument also implies that heating is strongest at the smallest parallel scale l ∥ ∼ 1 /k ∥ ∼ v A / Ω i with λ ≲ ρ i . \nBased on the argument above, we propose to extend the usual SH formula for the ion heating rate per unit mass [10], \nQ ⊥ = Ω i v 2 th,i ˆ c 1 ˆ ξ 3 i e -ˆ c 2 / ˆ ξ i , (1) \nto also describe QLH in imbalanced turbulence. Here, ˆ c 1 and ˆ c 2 are empirical constants and the exponential \nsuppression factor accounts phenomenologically for the conservation of magnetic moment at small amplitudes and/or the exponentially small number of particles resonant with lower-frequency fluctuations (in reality, this suppression factor presumably depends on imbalance and other parameters). A key change, however, is needed to correctly describe the helicity barrier, with the definition of ξ i extended from its usual one ( ξ i = δu ρ i /v th,i ) to instead capture the scale at which the fluctuations reach their highest frequency. Thus ˆ ξ i ≡ δu λ /v th,i , where δu λ is the turbulent amplitude at perpendicular scales λ ≳ ρ i corresponding to the smallest parallel scale l ∥ , estimated from the maximum of δu λ /λ via propagation critical balance: v A /l ∥ ∼ δu λ /λ ∼ k ⊥ ˜ z + k ⊥ . Here, ˜ z + k ⊥ is the rms value at scales k ⊥ λ ∼ 1 of ˜ z + , the generalized Elsasser eigenmode of the FLR-MHD model (introduced below), which reduces to its MHD counterpart z + at k ⊥ ρ i ≪ 1. \nFor turbulence without a helicity barrier (such as balanced turbulence or imbalanced RMHD), the energy cascade is able to dissipate at perpendicular scales smaller than ρ p , forming a smooth spectrum with continuously increasing δu λ /λ . Based on the argument above, we expect ions to be heated by fluctuations at scales λ ∼ ρ p , and Eq. (1) reduces to the normal SH formula with ˆ ξ i = δu ρ i /v th,i . However, with a helicity barrier, the resulting steep spectral break causes k ⊥ ˜ z + k ⊥ to peak at an intermediate scale λ > ρ p before dropping off steeply at smaller scales. We thus use λ ∼ max( ρ i , 1 /k nl ⊥ ) where \nk nl ⊥ ≡ ∫ ρ -1 i 0 d k ⊥ k ⊥ ( k ⊥ ˜ z + k ⊥ ) ∫ ρ -1 i 0 d k ⊥ k ⊥ ˜ z + k ⊥ , (2) \nthe average value of k ⊥ weighted by k ⊥ ˜ z + k ⊥ over scales k ⊥ ρ i ≤ 1, is used as a proxy for the scale at which the heating is most efficient. This also takes into account that minor ions interact with fluctuations at scales larger than the spectral break, due to their larger gyroradii. \nTo verify this proposal, we now present numerical simulations of test protons and minor ions interacting with both balanced and imbalanced turbulence with or without a helicity barrier. \nNumerical Setup. -We use the 'finite Larmor radius MHD' (FLR-MHD) model [25, 28, 50], which can be formally derived from gyrokinetics in the limit β ≪ 1 and k ⊥ d e ≫ 1 (where d e is the electron inertial length). At k ⊥ ρ i ≪ 1, FLR-MHD reduces to the well-known RMHD model [51]; for k ⊥ ρ i ≫ 1 it reduces to 'electron RMHD' [16, 22, 52]. The FLR-MHD equations are advanced in time using a pseudospectral method [25, 53] (see Suppl. Mat.). The simulation domain is a three-dimensional periodic grid of size L ⊥ = L z = 2 π and a resolution of N 2 ⊥ × N z , where the background magnetic field is along the z -direction and ⊥ represents the x - and y -directions. To test our proposed heating model across a wide range of regimes, we simulate three cases: balanced and imbalanced FLR-MHD with ρ p,gk /L ⊥ = 0 . 02, and imbalanced \nRMHD (we assume a majority hydrogen plasma, with ρ p,gk the ρ i in the gyrokinetic model). These simulations have a resolution of N 2 ⊥ × N z = 1024 2 × 1024 for the balanced FLR-MHD case and 1024 2 × 512 for the other two cases, refined from an initial resolution of 256 2 × 512 to obtain spectra that extend to scales smaller than ρ p,gk (see Suppl. Mat.). In the imbalanced FLR-MHD simulations we run ions at three different stages of the helicity barrier's evolution as it grows, at times tv A /L z ≈ 3 , 6 , and 10. \nOnce the refined turbulence simulations have reached a quasi-steady-state, we introduce test ions (for details on implementation, see Suppl. Mat.). For a given collection of ions, we can choose the initial ρ i /L ⊥ , β i , and δu ρ i /v th,i [19]. For FLR-MHD, the choice of ρ i /L ⊥ is fixed to match ρ p,gk /L ⊥ = 0 . 02; we also choose this value for ions in RMHD simulations for comparison. This scale determines δu ρ i via δu 2 ρ i = 2 ∫ k + k -d k ⊥ E E ( k ⊥ ), where E E ( k ⊥ ) is the perpendicular electric-field spectrum (equivalent to the E × B velocity spectrum in FLR-MHD) and using k ± ≡ e ± 1 / 2 /ρ i in line with previous work [10, 19]. This value of δu ρ i with the choice of δu ρ i /v th,i determines v th,i . Due to the invariance of the quantity v A ∇ ∥ in both RMHD and FLR-MHD, we are free to choose v A based on the values of v th,i and β i so long as length-scales along the background field are scaled by the same factor [46]; this rescaling is done within the ion integrator. Once these quantities are determined, the ions are uniformly distributed throughout the simulation domain and given an isotropic Maxwellian velocity distribution. For a given choice of ρ i /L ⊥ and β i , we initialize separate cohorts of N = 10 6 particles with 0 . 01 ≲ δu ρ i /v th,i ≲ 0 . 3. \nWe introduce minor ions with mass m i = Am p and charge q i = Ze by taking v th,i to be the same of that of protons within the same simulation, so that they have similar initial δu ρ i /v th,i . This choice, along with Ω i = ( Z/A )Ω p , determines the required ion gyroradius \nTABLE I. f Simulation parameters studied in this Letter. F and R correspond to the FLR-MHD and RMHD models. Coefficients c 1 , c 2 and ˆ c 1 , ˆ c 2 are fits of simulation results to Eq. (1) using δu ρ i /v th,i and ˆ ξ i , respectively; the best fit of all data using ˆ ξ i is ˆ c 1 , ˆ c 2 = 1 . 11 , 0 . 10. \nFIG. 1. Left: Space-time Fourier transforms E tot ( k z , ω ) from balanced (top) and imbalanced (bottom) FLR-MHD turbulence. Each is normalized to the maximum value at each k z , and the dotted lines are the Alfv'en wave dispersion relation ω = ± k z v A . Right: Comparison of electric-field spectra (top) and k ⊥ ˜ z + k ⊥ (bottom) as used to estimate l ∥ from the different turbulence simulations in this Letter. \n<!-- image --> \n⊥ \nρ i = ( A/Z ) ρ p corresponding to ρ p /L ⊥ . Once this is determined, the process of initializing minor ions is identical to protons. Table I summarizes the turbulence and particle parameters studied in this Letter. \nWe note that because FLR-MHD is derived from gyrokinetics it can only simulate low-frequency dynamics, and does not contain the transition to ion-cyclotron waves. Although this still allows for QLH of protons by Alfv'enic fluctuations, it does not capture their dispersion or polarization changes at k ∥ d i ∼ 1 (where d i is the ion inertial length; see Ref. [26] for the effects of ion-cyclotron waves). This approximation is less impactful for minor ions, because their smaller gyrofrequency means they interact with lower-frequency fluctuations. \nResults. -To highlight how imbalance affects the frequency spectrum of fluctuations, which was argued above to be important for controlling the heating, the left-hand plot of Fig. 1 shows the space-time Fourier transform E tot ( k z , ω ) = E + ( k z , ω ) + E -( k z , ω ), with E ± ( k z , ω ) = 1 2 〈 | ˜ z ± ( k z , k ⊥ , ω ) | 2 〉 ⊥ ; here, ⟨·⟩ ⊥ denotes an average over all k ⊥ . In balanced turbulence, E tot ( k z , ω ) has significant power at small ω , a signature that the nonlinear and linear times are comparable [46]. In contrast, imbalanced turbulence is dominated by fluctuations at a particular frequency, as seen by the band peaked around ω ≈ -k z v A . These show that increasing imbalance reduces the nonlinear broadening of fluctuations relative to their linear frequency, causing the ion heating to transition from a SH-like to QLH-like mechanism. \nFIG. 2. Evolution of the ensemble-averaged perpendicular energy per unit mass ⟨ w 2 ⊥ ⟩ of all particle cases with 0 . 02 ≤ δu ρ p /v th,p ≤ 0 . 2 (dark to light) from F-6 β p 0 . 05, along with the linear fit for Q ⊥ in dashed lines. \n<!-- image --> \nThe electric-field spectra E E ( k ⊥ ) and k ⊥ ˜ z + k ⊥ for different turbulence simulations are compared on the right of Fig. 1, where we compute k ⊥ ˜ z + k ⊥ from its spectrum: k ⊥ ˜ z + k ⊥ ≈ k ⊥ √ k ⊥ E + ( k ⊥ ) [54]. In contrast to the wide inertial range of the imbalanced RMHD simulation, FLRMHD captures the transition from an Alfv'en-wave to kinetic-Alfv'en-wave cascade [16], with the electric-field specturm approaching a -1 / 3 scaling at k ⊥ ρ i ≳ 1. The presence of a helicity barrier is clearly seen in the imbalanced FLR-MHD simulations via the steep drop in spectra at k ⊥ ρ p ≲ 1, as observed in the solar wind [31, 55] and previous simulations [25-27]. The steep spectral slope of the helicity barrier also causes k ⊥ ˜ z + k ⊥ to reach a maximum at k ⊥ ρ p ≲ 1 scales. This maximum, and the corresponding spectral break in E E ( k ⊥ ), shifts to larger scales as the energy grows in time. \nFigure 2 illustrates how the ion heating rate Q ⊥ is calculated from the evolution of the ensemble-averaged perpendicular energy per unit mass ⟨ w 2 ⊥ ⟩ for protons from F-6 β p 0 . 05 with 0 . 02 ≤ δu ρ p /v th,p ≤ 0 . 2. Here w ⊥ is the component of w ≡ v -u E × B , the thermal velocity of the protons after subtracting the local E × B velocity, perpendicular to B . Protons experience an initial heating transient lasting around one gyroperiod as they 'pick up' the local E × B velocity (also seen in Ref. [19]). Because of this, we calculate Q ⊥ by a linear fit to ⟨ w 2 ⊥ ⟩ via Q ⊥ = 0 . 5( ⟨ w 2 ⊥ ,f ⟩ - ⟨ w 2 ⊥ ,0 ⟩ ) / ( t f -t 0 ). Here t f is either the end of the simulation or when ⟨ w 2 ⊥ ,f ⟩ = 1 . 2 ⟨ w 2 ⊥ ,0 ⟩ ; we choose this as Q ⊥ decreases as ⟨ w 2 ⊥ ⟩ increases [10, 19]. For δu ρ p /v th,p ≲ 0 . 05, the initial heating transient exhibits oscillations, so to remove the effect of these from the measurement, we choose the start time t 0 Ω i / 2 π = 5 . 25 for δu ρ i /v th,i < 0 . 02, 3 . 25 for 0 . 02 ≤ δu ρ i /v th,i ≤ 0 . 05, and 0 . 75 for δu ρ i /v th,i > 0 . 05 (see Suppl. Mat.). \nFigure 3 compares the dependence of Q ⊥ on δu ρ i /v th,i , the measure commonly used in SH theory, to our proposed definition ˆ ξ i = δu λ /v th,i . We calculate k nl ⊥ in Eq. (2) using k ⊥ ˜ z + k ⊥ from Fig. 1. It is clear that the \nstandard normalization is inadequate for describing Q ⊥ , particularly for cases with a helicity barrier. These show greater heating than expected from the power in fluctuations at these scales, indicating the true ˆ ξ i is being underestimated. Minor ions show similar heating rates in both imbalanced FLR-MHD and RMHD, due to their gyroradii lying at scales larger than ρ p and close to or above 1 /k nl ⊥ . In contrast, when plotted against our proposed ˆ ξ i , the data are well described by Eq. (1) with best fit parameters ˆ c 1 = 1 . 11 , ˆ c 2 = 0 . 10 (fits of individual simulations to Eq. (1) comparing δu ρ i /v th,i and ˆ ξ i are listed in Table I); our value of ˆ c 2 is smaller than previous test particle simulations [10, 19] but similar to recent hybridkinetic simulations [22]. That Eq. (1) holds despite the difference in simulation conditions, from turbulence that is balanced, imbalanced, with or without a helicity barrier, and for both protons and minor ions, helps verify our unified model of ion heating. We note that the worst argreement is seen for imbalanced RMHD, which is also unphysical, neglecting the effect of ρ p on the turbulence. \nDespite the success of the unified model in reproducing Q ⊥ , there remains important differences between balanced- and imbalanced-turbulent heating, as demonstrated in Fig. 4 which compares the proton velocity distribution functions (VDFs) f ( w ⊥ , w ∥ ) in these two regimes. In balanced turbulence, the VDF undergoes diffusion in perpendicular velocity, as seen in the 1D VDF f ( w ⊥ ) = ∫ d w ∥ f ( w ⊥ , w ∥ ) which has a characteristic flattopped shape [22, 38]. In contrast, in imbalanced turbulence the VDF diffuses along constant-energy contours within the frame of the waves, a hallmark of QLH [42]. For Alfv'enic fluctuations, these contours are semicircles centered on the phase speed v ph = ω/k ∥ = -v A . The 1D VDF exhibits a flattopped shape similar to the balanced case (dropping off at slightly earlier upon close inspection), presumably because the contours are nearly vertical at small w ⊥ . This picture also allows an understanding of the β p -dependent dropoff in Q ⊥ seen at large ˆ ξ p in Fig. 3 (representing the largest deviation from the phenomenological model): the constant-energy contours become increasingly curved as β p increases, where v A approaches v th,p , causing ions to gain parallel energy at the expense of perpendicular energy (see the Suppl. Mat.). \nConclusion. -In this Letter we provide a unified picture for describing ion heating in collisionless plasma turbulence across a wide range of regimes. The heating mechanism changes character depending on the frequency spectrum of turbulent fluctuations, transitioning from SH to QLH as the turbulence imbalance increases and reduces the relative nonlinear broadening of fluctuations. We propose a phenomenological model, controlled by the amplitude of fluctuations at the scale where k ⊥ ˜ z + k ⊥ ∼ v A /l ∥ peaks for k ⊥ ρ i ≲ 1, which accurately captures the ion-heating rate across both regimes. In the absence of a transition-range break or for minor ions, \n‖ \nFIG. 3. Comparison of the dependence of the measured ion heating rate Q ⊥ against the commonly used stochastic heating parameter δu ρ i /v th,i (left) and the new scale-dependent version ˆ ξ i (right). Filled and hollow markers represent simulations with and without a helicity barrier, respectively. The inset shows the ratio Q ⊥ / (ˆ c 1 ˆ ξ 3 i ), with ˆ c 1 = 1 . 11 the best fit to Eq. (1) for all data, highlighting the suppression of heating for small ξ i . The solid black lines show the best fit of Eq. (1) to F-Balβ p 0 . 05 (left) and all data (right and inset). \n<!-- image --> \nFIG. 4. Comparison of two-dimensional velocity distributions f ( w ⊥ , w ∥ ) of ˆ ξ p ≈ 0 . 3 protons from F-Balβ p 0 . 05 (top) and ˆ ξ p ≈ 0 . 7 protons from F-6 β p 0 . 1 (bottom). Dashed contours represent the initial Maxwellian distribution, and solid contours represent constant-energy contours in the frame of the wave, centered on the phase speed v ph = -v A (dotted line). Insets show the 1D perpendicular distribution function f ( w ⊥ ) in blue ( f ( w ⊥ ) at t = 0 is in gray), normalized to the maximum of the initial distribution. \n<!-- image --> \nthis scale is usually at k ⊥ ρ i ∼ 1 and our model reduces to the standard SH formula; however, with a transition range, caused by the helicity barrier, this scale can lie above at k ⊥ ρ i < 1. Using high-resolution simulations of test particles interacting with turbulence, we show that our proposal works well in describing measured heating \nrates for both protons and minor ions in turbulence that is balanced, imbalanced, with or without a helicity barrier. The fact that this holds, despite the very different 'arced' VDFs that result in imbalanced turbulence, hints at the universality of the heating mechanism. \nWe note that the FLR-MHD model, on which our results are based, is highly simplified, missing out the transition to ion-cyclotron waves when ω ∼ Ω i . Additionally, we also only measure the instantaneous ion Q ⊥ from an initial Maxwellian, rather than tracking large changes of the velocity distribution function (as occurs for strong heating). Both of these issues have been addressed in a recent hybrid-kinetic study [56], which shows similar features. Nonetheless, the results of this Letter help to disentangle the signatures of ion heating and provide a base for more complex theories. The ideas presented here can also be used to understand helicity-barrier saturation on ion heating [26, 27], which is not captured by FLR-MHD and may be important for ab-initio models of coronal and solar-wind heating. \nThe authors would like to thank T. Adkins, B. D. G. Chandran, M. W. Kunz, N. F. Loureiro, M. Zhang, and M. Zhou for helpful discussions over the course of this work. Research is supported by the University of Otago, through a University of Otago Doctoral Scholarship (ZJ), and the Royal Society Te Ap¯arangi, through Marsden-Fund grant MFP-UOO2221 (JS) and MFP-U0020 (RM), as well as through the Rutherford Discovery Fellowship RDF-U001004 (JS). High-performance computing resources were provided by the New Zealand eScience Infrastructure (NeSI) under project grant uoo02637. \n- ∗ johza721@student.otago.ac.nz\n- [1] E. Marsch, K.-H. 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Klein, Extreme heating of minor ions in imbalanced solar-wind turbulence, arXiv [astro-ph.SR] (2024), arXiv:2408.04703 [astro-ph.SR].\n- [57] G. G. Howes, S. C. Cowley, W. Dorland, G. W. Hammett, E. Quataert, and A. A. Schekochihin, Astrophysical gyrokinetics: Basic equations and linear theory, Astrophys. J. 651 , 590 (2006).\n- [58] A. Zocco and A. A. Schekochihin, Reduced fluid-kinetic equations for low-frequency dynamics, magnetic reconnection, and electron heating in low-beta plasmas, Phys. Plasmas 18 , 102309 (2011).\n- [59] R. Meyrand, A. Kanekar, W. Dorland, and A. A. Schekochihin, Fluidization of collisionless plasma turbulence, Proc. Natl. Acad. Sci. U. S. A. 116 , 1185 (2019).\n- [60] V. David, S. Galtier, and R. Meyrand, Monofractality in the solar wind at electron scales: Insights from kinetic Alfv'en waves turbulence, Phys. Rev. Lett. 132 , 085201 (2024).\n- [61] J. H. Williamson, Low-storage Runge-Kutta schemes, J. Comput. Phys. 35 , 48 (1980).\n- [62] T. Adkins, R. Meyrand, and J. Squire, The effects of finite electron inertia on helicity-barrier-mediated turbulence, arXiv [physics.plasm-ph] (2024), arXiv:2404.09380 [physics.plasm-ph].\n- [63] A. Mallet, A. A. Schekochihin, and B. D. G. Chandran, Disruption of Alfv'enic turbulence by magnetic reconnection in a collisionless plasma, J. Plasma Phys. 83 , 905830609 (2017).\n- [64] N. F. Loureiro and S. Boldyrev, Collisionless reconnection in magnetohydrodynamic and kinetic turbulence, Astrophys. J. 850 , 182 (2017).\n- [65] M. Zhou, Z. Liu, and N. F. Loureiro, Electron heating in kinetic-Alfv'en-wave turbulence, Proc. Natl. Acad. Sci. U. S. A. 120 , e2220927120 (2023).\n- [66] J. P. Boris, Relativistic plasma simulation-optimization, in Proc. Fourth Conf. Num. Sim. Plasmas (1970) pp. 367.\n- [67] C. K. Birdsall and A. B. Langdon, Plasma physics via computer simulation (CRC Press, 2018).", 'The FLR-MHD Model': "We use the finite Larmor radius MHD (FLR-MHD) model [25, 28, 50], a reduced fluid model that includes effects at scales both larger and smaller than ρ i with- \nut needing to solve kinetic equations. This theory can be derived from the Vlasov-Maxwell equations by assuming all quantities vary on timescales slower than the ioncyclotron frequency, a strong background magnetic field, and that perturbations are elongated along magnetic field lines [16, 57]. Assuming a background magnetic field B 0 = B 0 ˆ z and taking the lowβ limit [28, 58] gives the equations of FLR-MHD: \n( ∂ ∂t + u ⊥ · ∇ ⊥ ) δn e n 0 e = -c 4 πen 0 e ( ∂ ∂z + b ⊥ · ∇ ⊥ ) ∇ 2 ⊥ A ∥ + D 6 ν δn e n 0 e , (3) \n( ∂ ∂t + u ⊥ · ∇ ⊥ ) A ∥ = -c ∂φ ∂z + cT 0 e e ( ∂ ∂z + b ⊥ · ∇ ⊥ ) δn e n 0 e + D 6 η A ∥ , (4) \nδn e n 0 e = -Z τ ( 1 -ˆ Γ 0 ) eφ T 0 e , (5) \nwhere τ = T 0i /T 0e is the ratio of the ion and electron temperatures, δn e /n 0 e = δn i /n 0i is the perturbed electron density (equal to the ion density by quasi-neutrality), A ∥ is the ˆ z component of the magnetic vector potential, φ is the electrostatic potential, u ⊥ = ( c/B 0 )ˆ z × ∇ ⊥ φ is the perpendicular E × B flow, and b ⊥ = -ˆ z × ∇ ⊥ A ∥ /B 0 is the magnetic field perturbation. The gyrokinetic Poisson operator 1 -ˆ Γ 0 = 1 -I 0 ( α ) e -α , where I 0 is the modified Bessel function of the first kind and α = -ρ 2 i ∇ 2 ⊥ / 2, models the transition from an Alfv'enic to a kinetic-Alfv'enwave cascade that occurs at k ⊥ ρ i ∼ 1. In our simulations a hyperdiffusion operator D 6 ν = ν 6 ⊥ ∇ 6 ⊥ + ν 6 z ∇ 6 z (and similarly for D 6 η ) is used to dissipate energy above grid scales, but does not model any specific physical process; we set η ⊥ ,z = ν ⊥ ,z in all cases except balanced FLRMHD, discussed below. \nThis model describes the evolution of the Alfv'enic component of turbulence both above and below ρ i ; the assumption of small β leads to minimal coupling between this and the compressive component of the turbulence [16]. The FLR-MHD model is valid as long as the electron inertial length d e ≪ ρ i . This is because d e sets the length scale below which the magnetic field is no longer frozen into the electron flow; in this regime the equations are coupled to the electron distribution function and are no longer closed. Further details on the properties of FLR-MHD are given in Ref. [25].", 'Eigenmodes and Conserved Invarients': "For a given wavenumber k , the FLR-MHD model supports eigenmodes that travel parallel and anti-parallel to \nB 0 given by \nΘ ± k = -Ω i v ph ( k ⊥ ) k 2 ⊥ δn e n 0 e ∓ A ∥ √ 4 πm i n 0i , (6) \nwith frequency ω = ± k z v ph ( k ⊥ ) v A . The normalized phase speed v ph ( k ⊥ ) = k ⊥ ρ i [(1 / [1 -ˆ Γ 0 ] + Z/τ ) / 2] 1 / 2 reduces to 1 at k ⊥ ρ i ≪ 1 scales and is proportional to k ⊥ ρ i at k ⊥ ρ i ≫ 1 scales, reflecting the transition from Alfv'en to kinetic-Alfv'en waves. These eigenmodes allow the definition of generalized Elsasser variables ˜ z ± = ˆ z × ∇ ⊥ Θ ± that reduce to the standard Elsasser variables of MHD ( z ± = u ⊥ ± B ⊥ / √ 4 πm i n 0i ) in the limit k ⊥ ρ i ≪ 1. \nThis model also has two conserved invariants: the free energy E = 1 4 ∑ k ( | k ⊥ Θ + | 2 + | k ⊥ Θ -| 2 ), and the generalized helicity H = 1 4 ∑ k ( | k ⊥ Θ + | 2 -| k ⊥ Θ -| 2 ) /v ph ( k ⊥ ). The helicity barrier results from the inability of the system to support a forward cascade of both E and H simultaneously at k ⊥ ρ i ≫ 1, due to the scaling v ph ∝ k ⊥ in this range [25].", 'Additional Simulation Details': 'We use asterix [25, 59, 60], a modified verison of the pseudospectral code turbo [53], to solve the equations of RMHD and FLR-MHD, which are advanced in time with a third-order modified Williamson algorithm (a four-step, low-order Runge-Kutta method [61]). We force the imbalanced FLR-MHD and RMHD simulations with energy injection ε and σ ε = 0 . 88, where σ ε = | ε H | /ε is the ratio of the helicity and energy injection rates. In contrast, the balanced FLR-MHD simulation has no helicity injected into the system ( ε H = 0). \nFigure 5 shows the magnetic-field spectra of the highresolution simulations after refinement (discussed below), \nFigure 6 compares the time-evolution of dissipation in all balanced and imbalanced RMHD and FLR-MHD cases during the refinement process. The imbalanced cases take longer to saturate compared to the balanced case for the reason discussed above. As we increase the perpendicular resolution while refining the cascade can reach smaller corresponding parallel scales due to critical balance, leading to an increase in parallel dissipation. This effect is clearly seen in the balanced FLR-MHD case: \n<!-- image --> \n⊥ \nFIG. 5. Comparison of magnetic-field spectra from the different turbulence simulations in this Letter. \ncorresponding to the electric-field spectra in Figure 1 of the Letter. All simulations exhibit an approximate -3 / 2 scaling for k ⊥ ρ i ≲ 1. With the addition of a helicity barrier, the magnetic-field spectra show similar scalings to electric-field spectra, steeping to a -4 scaling around k ⊥ ρ i ∼ 1.', 'Choice of Parallel Viscosity': "̸ \nWe use different choices of the parallel viscosity ν 6 z in the different turbulent regimes to reflect how they arrive at a steady state. For imbalanced RMHD turbulence, the majority of the energy cascade is able to reach small perpendicular scales and dissipate there. However, due to critical balance this energy also reaches small parallel scales. To ensure correct dissipation at small scales, we use a non-zero parallel viscosity. For the helicity barrier, however, we are interested in how it affects the heating of test particles as it grows in time during the 'pseudostationary' phase, before it saturates unphysically on the parallel hyperdissipation. We thus set ν 6 z = 0 for the helicity barrier simulation, studying it well before it would saturate with ν 6 z = 0. Various tests (not shown) found that this produces results similar to using a non-zero ν 6 z .", 'Simulation Refinement': "A highly imbalanced forcing causes the turbulence to take longer to reach a steady-state, as the energy cascade is less efficient for the stronger Elsasser field [46, 47]. The addition of a helicity barrier complicates this further as the system evolves through a 'pseudo-stationary' state where the balanced portion of the cascade dissipates at small perpendicular scales, whereas the imbalanced portion grows at scales k ⊥ ρ i ≲ 1 before saturating unphysically on parallel hyperdissipation [25, 26]. To run a highresolution simulation, such as the ones used in this work, to this saturation point would be time-consuming and \nFIG. 6. Time evolution of turbulent dissipation for the imbalanced RMHD (top), and the balanced and imbalanced FLR-MHD cases (middle, bottom respectively). Contributions to the parallel (orange) and perpendicular (blue) dissipation and their total (black) are normalized to the energy input ε . The red dotted lines show when the simulations were refined, doubling the perpendicular resolution from 256 2 × 512 up to 1024 2 × 512 for the imbalanced cases or 1024 3 for the balanced case. Note the difference in scale on both axes; see text for discussion. \n<!-- image --> \ncomputationally expensive. \nInstead, we start from simulations with a resolution of N 2 ⊥ × N z = 256 2 × 512 and let these evolve to a steadystate (or pseudo-stationary state for simulations with a helicity barrier). Once this point is reached, we refine the simulation by doubling the perpendicular resolution and decreasing the perpendicular viscosity ν 6 ⊥ , allowing the energy cascade to reach smaller scales before dissipating. Once the cascade has reached a new steady-state, we repeat this refinement procedure until the desired resolution is reached, which is then used as the starting point for our test particle simulations. The final resolutions are N ⊥ = 1024 , N z = 512 for the imbalanced RMHD and FLR-MHD cases, and N ⊥ = N z = 1024 for the balanced FLR-MHD case (the reason for this is discussed below). \nFigure 7 compares two-dimensional dissipation spectra D 2D ( k ⊥ , k z ) from the high-resolution imbalanced FLRMHD and RMHD simulations. In the absence of a helicity barrier, all energy is dissipated at grid scales in the RMHD case. In contrast, the presence of a helicity barrier causes a flat dissipation spectrum around k ⊥ ρ i ∼ 1, due to the steep energy spectrum slope in this range. \n<!-- image --> \n⊥ \n⊥ \n⊥ \n⊥ \nFIG. 7. Two-dimensional dissipation spectra D 2D ( k ⊥ , k z ) from the 1024 2 × 512 imbalanced FLR-MHD (left) and RMHD (right) simulations, normalized to their maximum values. The vertical dotted lines represent the test particle gyroradius scale k ⊥ ρ i = 1. \nthe initial refinement in N ⊥ leads to an increase in ε z , and the subsequent refinement in both N ⊥ and N z resulting in a decrease (due to dissipation scales occuring at larger k z ). In the imbalanced RMHD and balanced FLR-MHD cases, all energy is dissipated near the grid at predominately small perpendicular scales with ε diss ≈ ε . In contrast, the helicity barrier in the imbalanced FLRMHDcase blocks the majority of the cascade, only letting the balanced portion of the flux ( ∼ 1 -σ ε ) through to be dissipated at grid scales; the remainder is 'trapped' at large perpendicular scales. The lower-resolution simulations allow more of the cascade to be dissipated than predicted as their dissipation scales are close to ρ i -scales; this effect decreases as the range between these scales increases with resolution, as seen by the lower ε diss /ε at each refinement level, converging to ε diss /ε = 1 -σ ε .", 'Viscosity in the Balanced FLR-MHD Case': "A notable exception to the methods described above was the balanced FLR-MHD turbulence case. Due to the transition from an Alfv'enic to a kinetic-Alfv'en-wave cascade at scales k ⊥ ρ i ≳ 1 we expect the slope of the magnetic-field spectrum to steepen from -5 / 3 to -7 / 3, and the electric-field specturm to flatten from -5 / 3 to -1 / 3 [62]. However, when refining the balanced FLRMHD simulation to a resolution of 1024 2 × 512 with the method above we found that both the magnetic- and electric-field spectra were steeper than expected (dashed \n̸ \nblue and red lines in Fig. 8, respectively), with the electric field reaching a slope of around -7 / 3. These spectra (solid lines in Fig. 8) approach their theoretical predictions after refining the simulation to a resolution of 1024 3 and switching off only the viscosity ( ν ⊥ ,z = 0 and η = 0), leaving the electric field unaffected by dissipation (a method used in hybrid-kinetic simulations [21, 22, 26]). This steepening appears to be an unexpected nonlinear effect previously unseen in the FLR-MHD model. Previous work has shown that the reconnection of current sheets by subρ i effects can lead to a steepening of the energy spectrum similar to what is seen in the transition range [63-65]; however, these require the electron inertial length d e which is ordered out of FLR-MHD. Whether what we observe here is related to this physics remains unclear, and more work is required to clarify the details of this mechanism. Because of this, we integrate particles using the ν = 0 balanced FLR-MHD simulation in the main Letter to remove any influence on heating due to this effect.", 'Additional Test Particle Details': 'In order to study particle heating, we introduced the ability to integrate test particles simultaneously with the fields solved by asterix . As in previous test particle simulations (e.g. [18, 19]), the equations of motion of the particles are integrated using the Boris push [66] with electric and magnetic fields interpolated from the grid to their positions via the triangular-shaped-cloud (TSC) method [18, 67]. The test particles are run simultaneously with the fluid code so that particles can use the current fields without needing to save the full history of the simulation. The perpendicular fields are obtained via B ⊥ = B 0 b ⊥ and E ⊥ = -u ⊥ × B /c (with u ⊥ and B ⊥ as defined above via φ and A ∥ in \nFIG. 8. Magnetic- (blue) and electric-field (red) spectra and spectral slope α (measured by fitting over a range ± log 10 ( k ⊥ ) ≈ 0 . 1 around each point), highlighting the dependence on dissipation by viscosity. Dashed lines show the spectra from a N 2 ⊥ × N z = 1024 2 × 512 balanced FLR-MHD simulation with viscosity dissipation activated, and solid lines from the 1024 3 balanced simulation with ν = 0 (used in the main Letter). \n<!-- image --> \n⊥ \nFLR-MHD); any artificial parallel electric field component due to interpolation is then removed using the method of Ref. [18]. We calculate the required timestep via ∆ t particle = min(2 π/ ( N 1 Ω i ) , ∆ t fluid /N 2 ) [18], where ∆ t fluid is the fluid timestep calculated by asterix . Setting N 1 = 50 and N 2 = 5 was found to give accurate results; these results converge so long as N 1 ≳ 10 (also seen by Ref. [18]). \nThere are some important differences between RMHD and FLR-MHD that need to be taken into account when implementing test particles. Firstly, the initial ρ i /L ⊥ of the particle distribution is no longer a free parameter due to the inclusion of kinetic scales in the FLR-MHD model (in comparison to RMHD, which assumes ρ i → 0). We fix this parameter during the initialization of particles in an FLR-MHD simulation. Secondly, the inclusion of k ⊥ ρ i ≫ 1 scales introduces a electric field component parallel to B = B 0 + B 0 b ⊥ that arises from the conservation of magnetic flux in gyrokinetics [16, 22], given by \nE ∥ = E · ˆ b = -( ∂ ∂z + b ⊥ · ∇ )( T 0 e e δn e n 0 e ) \nwhere ˆ b is the unit vector along B . We include this term when interpolating fields to the particles, although it is generally small in comparison to E ⊥ . \nFIG. 9. Measured heating rate Q ⊥ of protons in 1024 3 balanced FLR-MHD turbulence (top), demonstrating the greater-than-expected heating at low turbulent amplitudes. The solid line shows the best fit to the empirical Q ⊥ (Eq. 1 in Letter) for F-Balβ p 0 . 05 with c 1 , c 2 = 1 . 83 , 0 . 18. These heating rates are measured using a linear fit (dashed lines; bottom) to the ensemble-averaged perpendicular energy per unit mass ⟨ w 2 ⊥ ⟩ for the protons with 5 × 10 -3 ≤ δu ρ p /v th,p ≤ 2 × 10 -2 (dark to light). The oscillations are clearly seen and have mostly decayed after approximately 5 gyroperiods; Q ⊥ is measured from 5 . 25 gyroperiods onward for these smallδu ρ i /v th,i particles. For clarity, ⟨ w 2 ⊥ ⟩ is normalized to its initial values; the values used for the heating rate calculations are normalized to v 2 A . \n<!-- image -->', 'Testing of Implementation': 'After its implementation, the particle code was extensively tested to verify its correctness. Simple cases with known analytical properties (such as motion in constant E ⊥ and B 0 fields as well as conservation of energy in E = -∇ φ fields) are reproduced with a relative error of less than 10 -3 . The implementation of the TSC interpolation scheme was found to exhibit second-order convergence up to a resolution of 128 3 , before additional errors dominate (e.g., from timestepping). The accuracy of this scheme was tested by comparing the use of gridinterpolated fields to those at the particle position via more complicated analytical fields (such as a linear combination of sinusoidal electromagnetic modes); for a grid resolution of 128 3 or greater, the maximum relative error in energy evolution is less than 10 -2 . \nThe convergence of statistics with particle number was tested by running cohorts of N particles (with N ranging from 100 to 10 6 in powers of 10) in the same 128 3 balanced RMHD turbulence simulation. The relative error in the statistics of an N = 10 5 and N = 10 6 run is ≲ 1%; we use N = 10 6 particles to allow for better statistics when calculating the velocity distribution functions. \nFIG. 10. Comparison of two-dimensional velocity distributions f ( w ⊥ , w ∥ ) of protons with ˆ ξ p ≈ 0 . 7 from F-6 β p 0 . 01 (top), F-6 β p 0 . 05 (middle), and F-6 β p 0 . 1 (bottom; same as Fig. 4 of main paper). Solid contours represent constant-energy contours (Eq. 7) in the frame of the wave, centered on the phase speed v ph = -v A (dashed line). \n<!-- image --> \n‖ \nParticle Heating at Small Turbulent Amplitudes \nIons experience a greater-than-expected heating rate than the empirical exponential suppression in Q ⊥ = Ω i v 2 th,i ˆ c 1 ˆ ξ 3 i e -ˆ c 2 / ˆ ξ i (Eq. 1 of Letter) when initialized with very low δu ρ i /v th,i . In Figure 9 we compare the measured heating rates of protons with 5 × 10 -3 ≤ δu ρ p /v th,p ≤ 1 × 10 -2 interacting with the same 1024 3 balanced FLR-MHD turbulence simulation as those with \n0 . 02 ≤ δu ρ p /v th,p ≤ 0 . 3 (F-Balβ p 0 . 05 from the main Letter). Due to the low-amplitude turbulence at ρ p -scales, the evolution of ⟨ w 2 ⊥ ⟩ for these protons exhibits oscillations from the large-scale E × B flow that introduces some ambiguity in measuring the heating rate; these oscillations are reduced with increasing δu ρ p /v th,p as they experience greater heating (e.g., Figure 2 of the main Letter). We find that the oscillations decay after ∼ 5 gyroperiods for ions initialized with δu ρ i /v th,i < 0 . 02, ∼ 3 for those with 0 . 02 ≤ δu ρ i /v th,i ≤ 0 . 05, and ∼ 0 . 5 for those with δu ρ i /v th,i > 0 . 05. To remove any contributions from the oscillations when measuring the linear growth of ⟨ w 2 ⊥ ⟩ , we choose the start time of the linear fit for Q ⊥ to be 5 . 25, 3 . 25, and 0 . 75 gyroperiods respectively for the above cases (with the quarter-gyroperiod offset ensuring we are in the middle of any further oscillation cycle). Despite this, the measured heating rate is greater than that expected by exponential suppression ( Q ⊥ / Ω p v 2 th,p ≲ 10 -12 for δu ρ p /v th,p ≲ 10 -2 ).', 'Beta-dependent Perpendicular Heating in Imbalanced FLR-MHD Turbulence': "In Fig. 10, we show how the diffusion of the proton velocity distribution function in imbalanced FLR-MHD turbulence depends on β p = v 2 th,p /v 2 A . Quasi-linear theory predicts that the evolution of the distribution function f ( w ⊥ , w ∥ ) is a diffusion along constant-energy contours \nK ' = m p 2 [ w 2 ⊥ +( w ∥ -v ph ) 2 ] (7) \nwithin the frame of the dominant waves that the ions interact with [42, 43]. In Fig. 10, these are circles centred on the phase speed of Alfv'enic fluctuations parallel to the magnetic field. At smaller β p , the contours of K ' are nearly vertical for ions with small w ⊥ , with diffusion then leading to an increase in perpendicular energy and a greater measured heating rate (as seen in Figure 3 of the Letter)."}

2024ApJ...974L..13O

Postflare loops are looplike plasmas observed during the decay phase of solar flares and they are expected to exist for stellar flares. However it is unclear how postflare loops are observed in stellar flares cases. To clarify behaviors of postflare loops in spatially integrated data we performed the Sunasastar analysis of the X1.6 flare that occurred on 2023 August 5 using GOES Xray flux 10SUP7SUP K extreme ultraviolet EUV images taken by Atmospheric Imaging Assembly on board the Solar Dynamic Observatory 10SUP4.9SUP K and H data taken by Solar Dynamics Doppler Imager on board the Solar Magnetic Activity Research Telescope at Hida Observatory Kyoto University 10SUP4SUP K. As a result we found that this flare showed signatures corresponding to the important dynamics of the postflare loops even in the spatially integrated data 1 The H light curve showed two distinct peaks corresponding to the flare ribbons and the postflare loops. The plasma cooling in the postflare loops generated different peak times in soft Xrays EUV and H light curves. 2 Downflows were confirmed as simultaneous redshiftedblueshifted absorptions in the H spectra. 3 The apparent rise of postflare loops was recognized as a slowing of the decay for the H light curve. These results are the key to investigating stellar postflare loops with spatially integrated data. We also discuss the dependence of our results on flare locations and their possible applications to stellar observations.

2024-10-01T00:00:00Z

['2024arXiv240907630O', '2024ApJ...974L..13O', 'arXiv:2409.07630', '10.48550/arXiv.2409.07630', '10.3847/2041-8213/ad7a70']

['Solar flares', 'Stellar flares', '1496', '1603', 'Astrophysics - Solar and Stellar Astrophysics']

Sunasastar Analysis of the X1.6 Flare on 2023 August 5 Dynamics of Postflare Loops in Spatially Integrated Observational Data

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']

https://arxiv.org/pdf/2409.07630.pdf

{'Sun-as-a-star Analysis of the X1.6 Flare on 2023 August 5: Dynamics of Post-flare Loops in Spatially Integrated Observational Data': 'Takato Otsu, 1 Ayumi Asai, 1 Kai Ikuta, 2 and Kazunari Shibata 3, 4 \n1 Astronomical Observatory, Kyoto University, Sakyo, Kyoto, Japan \n2 Department of Multidisciplinary Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan \n3 Kwasan Observatory, Kyoto University, Yamashina, Kyoto 607-8471, Japan \n4 \nSchool of Science and Engineering, Doshisha University, Kyotanabe, Kyoto 610-0321, Japan', 'ABSTRACT': "Post-flare loops are loop-like plasmas observed during the decay phase of solar flares, and they are expected to exist for stellar flares. However, it is unclear how post-flare loops are observed in stellar flares' cases. To clarify behaviors of post-flare loops in spatially integrated data, we performed the Sun-as-a-star analysis of the X1.6 flare that occurred on 2023 August 5, using GOES X-ray flux ( ∼ 10 7 K), extreme ultraviolet (EUV) images taken by Atmospheric Imaging Assembly onboard the Solar Dynamic Observatory ( ≥ 10 4 . 9 K) and H α data taken by Solar Dynamics Doppler Imager on board the Solar Magnetic Activity Research Telescope at Hida Observatory, Kyoto University ( ∼ 10 4 K). As a result, this flare showed signatures corresponding to the important dynamics of the post-flare loops even in the spatially integrated data: (1) The H α light curve showed two distinct peaks corresponding to the flare ribbons and the post-flare loops. The plasma cooling in the post-flare loops generated different peak times in soft X-rays, EUV, and H α light curves. (2) Downflows were confirmed as simultaneous redshifted/blueshifted absorptions in the H α spectra. (3) The apparent rise of post-flare loops was recognized as a slowing of the decay for the H α light curve. These results are keys to investigating stellar post-flare loops with spatially integrated data. We also discuss the dependence of our results on flare locations and their possible applications to stellar observations. \nKeywords: Solar flares (1496); Stellar flares (1603)", '1. INTRODUCTION': "Solar flares are sudden energy release in the solar atmosphere, and their dynamics are explained by the standard flare model (Shibata & Magara 2011). The energy released in the solar corona is transported to the chromosphere along magnetic loops. As a result of this, chromospheric evaporation occurs and loops are filled with hot plasma ( ∼ 10 7 K). These hot loops are observed in soft X-rays. Then, the plasma cool to a lower temperature and the loops become visible in extreme ultraviolet (EUV; ∼ 10 5 -10 6 K) bands to chromospheric lines such as H α ( ∼ 10 4 K). Such loop-structure plasma observed in main and decay phases of solar flares is called 'post-flare' loops 1 (e.g., Kamio et al. 2003). Along post-flare loops, downflows of condensed plasma are often observed at transition-region and chromospheric temperatures. These cooled plasmas are thought to be formed by runaway radiative cooling, like the formation of quiescent coronal rains which occur in coronal loops on active regions (in some literature, downflows along post-flare loops are called 'flare-driven coronal rains'; e.g., Antolin & Froment 2022; S¸ahin & Antolin 2024). \nLike the solar case, stellar flares have been observed on various types of stars (e.g., Kowalski 2024). Some studies have suggested that stellar flares are associated with eruptive phenomena (e.g., Veronig et al. 2021; Namekata et al. 2022; Inoue et al. 2023; Namekata et al. 2024; Notsu et al. 2024). The Sun-as-a-star analyses -in which solar data are spatially integrated to be compared directly with stellar data- support these possible observations of such stellar phenomena (Namekata et al. 2022; Otsu et al. 2022). Sun-as-a-star analyses of solar activity are keys to improving \nunderstanding of stellar flares and have been actively performed (Ma et al. 2024; Pietrow et al. 2024; Otsu & Asai 2024; Leitzinger et al. 2024). \nLike the eruptive phenomena, post-flare loops are also critical components of the flare model, and they are expected to be observed for stellar flares (Heinzel & Shibata 2018; Wollmann et al. 2023). In observations of M dwarf flares, white-light curves observed by the Transiting Exoplanet Survey Satellite (TESS) sometimes exhibit secondary peaks after the main impulsive peaks (Howard & MacGregor 2022). To investigate the mechanism of the secondary peaks in TESS flares, Yang et al. (2023) performed one dimensional hydrodynamic simulations along the flare loop and found that plasma condensation can lead to a secondary peak in synthetic light curves computed with the method of Heinzel & Shibata (2018). This suggests stellar post-flare loops can explain the secondary peak in TESS flares. As other possible signatures of stellar post-flare loops, Honda et al. (2018) reported redshifted absorption in H α spectra during the decay phase of an M dwarf flare, which can be interpreted as downflows along post-flare loops. Also, stellar post-flare loops are proposed as possible causes of redshifted excess emission or red asymmetry in stellar H α spectra (Wu et al. 2022; Namizaki et al. 2023; Wollmann et al. 2023; Notsu et al. 2024). For the further detailed investigation of stellar post-flare loops, direct comparisons with solar post-flare loop data are essential as well as an approach via numerical simulations. However, how stellar post-flare loops are observed in spatially integrated data is still ambiguous. Clarifying how post-flare loops are observed in spatially integrated solar data is required for detections and investigations of stellar post-flare loops. \nIn this Letter, we present our results of a Sun-as-a-star analysis for the X1.6 flare observed on 2023 August 5. This flare showed typical post-flare loops in EUV and H α images. The observation is presented in Section 2. The methods for Sun-as-a-star analyses are described in Section 3. In Section 4, we report our results, and provide discussions and conclusions in Section 5.", '2.1. Instruments': "We used H α spectral images taken by Solar Dynamics Doppler Imager (SDDI; Ichimoto et al. 2017) attached to the Solar Magnetic Activity Research Telescope (SMART; UeNo et al. 2004) at Hida Observatory, Kyoto University. The SDDI takes full-disk solar images at 73 wavelength points between H α ± 9 . 0 ˚ A with a spectral resolution of 0.25 ˚ A, a time cadence of 12 s, and pixel size of 1 '' .23, respectively. Additionally, we used EUV images taken by Atmospheric Imaging Assembly (AIA; Lemen et al. 2012) onboard the Solar Dynamic Observatory (SDO; Pesnell et al. 2012). The AIA takes full-disk solar images in EUV channels with a time cadence of 12 s and pixel size of 0 '' .6.", '2.2. Event overview': 'The target event is the X1.6 flare with the GOES flare peak at 22:21 UT on 2023 August 5 (Figure 1 (a)), which occurred in the NOAA 13386 (Figure 1 (b)) . Figure 2 shows the time development of the flare in AIA 94, 171, and 304 ˚ A, and SDDI H α images. During the impulsive phase of the flare, two ribbons were identified (Figure 2, t = 70 minutes from 2023 August 5 21:00 UT). During the decay phase of the flare, bright post-flare loops dominantly appeared in the images of EUV and even the H α line center (Figure 2, t = 120 minutes). Additionally, the dark features can be confirmed in the images of H α ± 1 . 0 ˚ A (Figure 2 (d) and (f), t = 120 minutes), and they are located along the post-flare loop. These features are supposed to correspond to downflows along the post-flare loops. The loops are located near the solar limb and are tilted against the line-of-sight direction. As a result, the downflows could be in both blue and red wings. Later on, the upper part of the loops goes outside the solar disk due to the consecutive formation of higher and higher loops, as the result of a gradual reconnection (Figure 2, t = 210 minutes). \nWe made the time-slice diagram to describe the detailed dynamics of post-flare loops. Figure 3 (a-1)-(d-1) show the time-slice diagrams of AIA channels and H α line center, with the vertical axis along the artificial slit from the white circle to the cross mark in Figure 2. The horizontal dashed line in the time-slice diagram corresponds to the limb indicated by the white dashed lines in Figure 2. The GOES soft X-ray light curve and its peak time are also over-plotted as the dash-dot line and vertical dotted line, respectively. Figure 3 (a-2)-(d-2) are the zoomed-in diagram corresponding to the regions indicated by the dashed square in Figure 3 (a-1)-(d-1). In each time-slice diagram, flare ribbons are confirmed as two stripes at around 10-40 arcsec from the start point of the slit. Almost at the same time as the GOES peak, the post-flare loops appeared in AIA 94 ˚ A (Figure 3 (a-1)-(a-2), t ≈ 81 minutes). Subsequently, the loops appeared in cooler channels, i.e., AIA 171, 304 ˚ A (Figure 3 (b-2)-(c-2), t ≈ 90 minutes). Slightly after AIA 171 and 304, the loops appeared in H α (Figure 3 (d-2), t ≈ 91 minutes). Higher loops appear with the apparent rising \nvelocity of 4.5 km s -1 , which is consistent with white-light observations of an X-class flare reported by Jejˇciˇc et al. (2018). Finally, the upper part of the loops goes outside the solar disk at around t = 150 minutes. We note that this scenario is similar to that presented in ˇ Svestka et al. (1987).', '3. ANALYSIS': 'We performed spatial integrations to investigate how post-flare loops could be observed on distant stars.', '3.1. SDO/AIA EUV data Analysis': 'We made the light curves for the AIA channels of 94 ˚ A, 171 ˚ A, and 304 ˚ A, which have peak response temperatures of ∼ 10 6 . 8 K, ∼ 10 5 . 9 K, and ∼ 10 4 . 9 K, respectively (e.g., Peter et al. 2012). First, we integrated these data over the field of view shown in Figure 2. To focus on the signatures from the target event, we selected the local regions including the target X1.6 flare as the integral regions. This restriction of the integral region equals to assuming that dominant temporal changes occurred only inside the selected region (Figure 2). Second, we subtracted pre-event data (2023-08-05 21:00 UT) from each integrated data. Finally, we normalized the integrated data with each peak value and obtained the spatially integrated and pre-event-subtracted AIA data.', '3.2. SMART/SDDI H α Data Analysis': "We introduce the method of the Sun-as-a-star analysis for H α data (see Otsu et al. 2022, for details). First, we integrated H α spectra over the integral region A including the target phenomena. The normalizations by continuum and quiet region data were performed to suppress fluctuations in the instrument and Earth's atmospheric variations. Second, we subtracted pre-event data from each integrated spectrum to extract the change in spectra due to the flare. Finally, we normalized the pre-event subtracted spectra by the full-disk integrated continuum. The resulting normalized pre-event subtracted H α spectra ∆ ˜ S Hα ( t, λ ; A ), where t is time, λ is the wavelength, and A is the integral region, express the ratio of spectral changes coming from the target phenomena to the full-disk brightness of the Sun. In this study, the integral region for the Sun-as-a-star analysis was set as A = A 1 + A 2 + A 3 in Figure 1 (c), excluding the limb ( A limb ) region. For the SDDI observation, the shift of the solar image disrupts H α spectra at the solar limb. This effect is purely due to the imaging and would not occur in the case of stellar observation. Thus, we excluded the limb region. We will also show the results with the limb region (∆ ˜ S Hα ( t, λ ; A 1 + A 2 + A 3 + A limb )) just for the comparison (Figure 4 (b-1)-(b-2)). Hereafter, we call the ∆ S Hα ( t, λ ) = ∆ ˜ S Hα ( t, λ ; A 1 + A 2 + A 3 ) the Sun-as-a-star H α spectra. In addition to the total region A = A 1 + A 2 + A 3 , we applied the above method to three regions; Region 1 ( A 1 ), Region 2 ( A 2 ), and Region 3 ( A 3 ) (Figure 1 (c)), and obtained ∆ ˜ S Hα ( t, λ ; A 1 ), ∆ ˜ S Hα ( t, λ ; A 2 ), and ∆ ˜ S Hα ( t, λ ; A 3 ) to investigate the contributions from flare ribbons, post-flare loops on disk, and off-limb post-flare loops (Figure 5). ∆ ˜ S Hα ( t, λ ; A 1 ) mainly includes the contributions from the east ribbon ( A 1 : black region in Figure 1). ∆ ˜ S Hα ( t, λ ; A 2 ) mainly includes the contributions from the post-flare loops, although it is also affected by the west ribbon ( A 2 : red region in Figure 1). ∆ ˜ S Hα ( t, λ ; A 3 ) includes the contributions from the off-limb region ( A 3 : orange region in Figure 1). We also calculated the differenced H α equivalent width to obtain the light curve of the H α line: ∆ EW ( t ; A ) = ∫ H α +∆ λ H α -∆ λ ∆ ˜ S Hα ( t, λ ; A ) dλ where ∆ λ was set as 3 . 0 ˚ A to include the whole spectral variations.", '4.1. Result of Sun-as-a-star analysis': 'First, we show the result obtained by integration over the total region (Region 1+2+3) i.e. the Sun-as-a-star H α spectra along with the result of AIA EUV channels. Figure 4 (a-1) shows the two dimensional color map of the Sun-as-a-star H α spectra ∆ S Hα ( t, λ ) obtained by the integration over the Region 1+2+3 ( A 1 + A 2 + A 3 ). Figure 4 (a-2) shows the light curves of the SDDI H α (∆ EW ( t ; A 1 + A 2 + A 3 )) and GOES in linear scale. In Figure 4 (a-3), the light curves of AIA 94, 171 and 304 ˚ A are shown. The peak time of the light curves are indicated by vertical lines in Figure 4 (a-2) and (a-3). Figure 4 (b-1) and (b-2) are the same as (a-1) and (a-2) but for the results with the limb region. The general trends such as time evolution of the light curves are same for the two cases, although the results with limb region become much more noisy compared with those without the limb region. These justify that we select the results without the limb region (Figure 4 (a-1) and (a-2)) as the Sun-as-a-star results. \nThe Sun-as-a-star H α spectra and the multi-wavelength light cures show the following features: \n- (i) The two-step increase of ∆ EW ( t ; A 1 + A 2 + A 3 ) can be confirmed. The initial increase is similar to GOES soft X-rays flux, whereas the second peak is delayed in about 13 minutes from the GOES peak. AIA light curves \nalso showed delayed peaks compared with the GOES peak. The cooler channels and H α line show delayed peak times compared with the hotter ones. \n- (ii) The redshifted/blueshifted absorptions can be confirmed in the Sun-as-a-star H α spectra, which come from the downflows along the post-flare loops.\n- (iii) The H α ∆ EW ( t ; A 1 + A 2 + A 3 ) does not monotonically decrease as in GOES but stops at around t = 150 minutes.', '4.2. Results for sub-regions': 'Here, we show the results obtained by spatial integration of the H α spectra over Region 1-3 to investigate contributions from the flare ribbons and the on-disk/off-limb post-flare loops to the Sun-as-a-star H α spectra. \n( Region 1 ) Figure 5 (a-1) shows the two dimensional color map of the pre-event subtracted H α spectra ∆ ˜ S Hα ( t, λ ; A 1 ) obtained by the integration over the Region 1. Figure 5 (a-2) shows the light curves of ∆ EW ( t ; A 1 ) of H α and GOES soft X-rays. ∆ ˜ S Hα ( t, λ ; A 1 ) show the brightening near the line center corresponding to the east ribbon. The H α and GOES soft X-rays light curves show the similar peak time and decay. \n( Region 2 ) Figure 5 (b-1) and (b-2) are the same as Figure 5 (a-1) and (a-2) but for Region 2. The H α light curve ∆ EW ( t ; A 2 ) show the two-step increase. The first increase around t = 70 -90 minutes corresponds to the west ribbon, whereas the second increase around t = 90 minutes comes from the post-flare loops. Corresponding to the two-step increase of the light curve, the Sun-as-a-star H α spectra shows brightening near the line center. Additionally, ∆ ˜ S Hα ( t, λ ; A 2 ) shows redshifted/blueshifted absorptions coming from the downflows along the post-flare loops. The redshifted and blueshifted velocities are up to 100 km s -1 and -80 km s -1 , respectively. We note that these are Doppler velocities. Due to projection, true downflow velocities will be higher than these Doppler velocities. \n( Region 3 ) Figure 5 (c-1) and (c-2) are the same as Figure 5 (a-1) and (a-2) but for Region 3. The H α light curve ∆ EW ( t ; A 3 ) show the increase around t = 150 minutes, which comes from the off-limb post-flare loops. Correspondingly, ∆ ˜ S Hα ( t, λ ; A 3 ) also shows the brightening near the line center.', '5. DISCUSSION AND CONCLUSIONS': "5.1. Dynamics of Post-flare Loops in the Sun-as-a-star Data \n5.1.1. Cooling: Peak Time Difference in Multi-temperature Light Curves \nAs shown in Figure 5 (a-1) and (a-2), the east ribbon inside the Region 1 provides the enhancement of the H α light curve which has a peak and decay similar to the GOES flux. Corresponding to the appearance of the H α post-flare loops, the H α light curve for Region 2 exhibited the delayed peak ( t = 93 . 6 minutes) compared with the GOES peak ( t = 81 minutes). However, the west ribbon provides the initial increase even in the case of Region 2 (Figure 5 (b-1)(b-2)). Thus, the initial and secondary increase in the Sun-as-a-star result comes from the flare ribbons and post-flare loops, respectively (Figure 4 (a-1) and (a-2)). Our result showed that post-flare loops can lead to non-negligible enhancement in H α compared to flare ribbons. The EUV light curves also showed the delayed peak compared to the GOES flux. The peaks appear in the order of GOES soft X-ray ( t = 81 minutes), AIA 94 ( t = 89 . 2 minutes), 171 ( t = 91 . 9 minutes), 304 ( t = 92 . 4 minutes), and H α ( t = 93 . 6 minutes), which means the cooling of the post-flare loops can be confirmed as peak time difference even in the spatially integrated data. The peak time differences from GOES, AIA 94, 171, and 304 to H α are approximately 13, 4.4, 1.7, and 1.2 minutes, respectively (Figure 4 (a-3)) \nIn this flare, the downflows of cooled and condensed plasma were observed (Figure 2, see also Section 5.1.2), which implies the radiative loss is effective in the cooling process (initially the conductive cooling may dominate in the hot loops). To investigate whether the delayed peaks represent the radiative cooling or not, we compared the peak time differences with the radiative cooling time scale. The radiative cooling time scale is calculated using the following equation: \nτ rad = 3 n e kT/ ( n 2 e Q ( T )) , (1) \nwhere n e , T , k , and Q ( T ) [erg cm 3 s -1 ] are electron density, electron temperature, Boltzmann constant, and radiative cooling function, respectively. We used the typical electron density n e = 10 11 cm -3 for post-flare loops (Kamio et al. 2003) and calculated radiative cooling function Q ( T ) using CHIANTI 10.1 (Dere et al. 1997, 2023) with coronal abundance. The cooling times for T = 10 7 K (GOES), 10 6 . 8 K (94 ˚ A), 10 5 . 9 K (171 ˚ A), and 10 4 . 9 K (304 ˚ A) are \ncalculated to be τ rad ∼ 14 , 7 , 0 . 2 , 0 . 01 minutes, respectively 2 . We confirmed that the electron density of the post-flare loops is approximately 10 10 . 7 cm -3 using AIA/DEM analysis (Hannah & Kontar 2012). We also calculated cooling times using the method in ˇ Svestka (1987) and Schmieder et al. (1995), which provided the cooling times comparable to the estimated τ rad . The peak time differences from GOES and AIA 94 to H α ( ∼ 13 , 4 . 4 minutes) are close to τ rad ∼ 14 , 7 minutes, whereas those for AIA 171 and 304 ( ∼ 1 . 7 , 1 . 2 minutes) are much larger than τ rad ∼ 0 . 2 , 0 . 01 minutes. During the evolution of post-flare loops, electron density could change. This may be the cause of the inconsistency in AIA 171 and 304. To estimate the radiative cooling time scale and compare it with peak time difference more accurately, measuring the time development of electron density in post-flare loops is critical. In this rough estimation, the coronal abundance is used for Q ( T ) but abundance should be carefully treated because the plasma fills the post-flare loops through chromospheric evaporation. Moreover, plasma of 10 6 K affect the formation of AIA 304 (O'Dwyer et al. 2010), although the typical response temperature of AIA 304 is ∼ 10 4 . 9 K. This may also make the peak time difference from 304 to H α larger than the calculated τ rad . We note that conductive cooling may also be effective for cooling of hot loops and make the cooling time shorter than our estimation. Additionally, we should consider time variation of temperature for more accurate estimate of cooling time. We will consider these factors in further investigations using multiple flare events.", '5.1.2. Downflows: Redshifted and Blueshifted Absorption in H α Spectra': 'The H α dynamic spectrum for Region 2 shows the redshifted/blueshifted absorption from the downflows along the post-flare loops, which can be confirmed even in the Sun-as-a-star H α spectra (Figure 5 (b-1)). As described in Section 2, the post-flare loops are located near the solar limb and tilted against the line-of-sight direction. As a result, the downflows are observed as a blueshifted component and a redshifted one in the H α imaging observation, and these redshifted/blueshifted absorptions appeared even in the Sun-as-a-star spectra. Unlike the present flare, the previous Sun-as-a-star study reported only the redshifted absorption -which is probably related to downflows along post-flare loops- during the decay phase of an M1.1 flare which occurred relatively close to the disk center (Event 4 in Otsu et al. 2022). The Sun-as-a-star results of the present X1.6 and previous M1.1 flares suggest that a flare with blueshifted absorptions from post-flare loops is likely to occur near a solar/stellar limb. The dependence of shifted absorptions caused by downflows on occurrence locations should be investigated with a model of flows inside loops (e.g., Ikuta & Shibata 2024). We note that dependence of red and blue asymmetry related to post-flare loops on flare locations are also discussed in Wollmann et al. (2023) for observations on an M dwarf star. Furthermore, it will be important to compare locations of stellar post-flare loops based on signatures of downflows and those of starspots deduced from starspot mapping (e.g., Ikuta et al. 2020, 2023) for ensuring scenarios of stellar flares.', '5.1.3. Rise of the Loops: Stop of the H α Decay': 'The H α light curve for Region 3 begins to increase around t = 150 minutes, corresponding to the appearance of the off-limb post-flare loops which is a consequence of the generation of higher post-flare loops (Figure 5 (c-2)). Off-limb loops have no background intensity and they are fully in emission. Thus, they would exhibit stronger enhancement than on-disk loops compared to the pre-event state. As a result, the Sun-as-a-star H α light curve showed the stop of the decay around t = 150 minutes. Our result showed that the different contrast to the background between the cases of on-disk and off-limb loops can lead to tracing the plane-of-sky motions of plasma even in spatially integrated data. Additionally, the appearance of off-limb loops reflects that the flare occurred near the solar limb. Therefore, the stop of the decay of H α light curves can also be useful to deduce the locations of stellar flares.', '5.2. Conclusion and Implications for Stellar Observations': "In this study, we performed the Sun-as-a-star analysis of the X1.6 flare on 2023 August 5 which exhibited the post-flare loops in EUV channels and H α line spectra. We found that even the Sun-as-a-star data showed three characteristics of post-flare loops. \n- (1) The H α light curve showed two distinct peaks corresponding to the flare ribbons and the post-flare loops. The plasma cooling in the post-flare loops generated different peak times in soft X-rays, EUV, and H α light curves.\n- (2) Downflows were confirmed as simultaneous redshifted/blueshifted absorptions in Sun-as-a-star H α spectra. \n- (3) The apparent rise of post-flare loops was recognized as a slowing of the decay for the H α light curve. \nOur results are crucial to investigate stellar post-flare loops in spectroscopic and multi-wavelength observations. Additionally, we emphasize that signatures of post-flare loops would reflect the occurrence location of flares. Statistical studies on solar post-flare loops are crucial for further understanding of their dependence on flare locations. \nThe authors thank the anonymous referee for constructive comments that significantly improved the quality of this Letter. We express our sincere gratitude to the staff of Hida Observatory for developing and maintaining the instrument and daily observation. We would like to acknowledge the data use from GOES and SDO. SDO is a mission for NASA's Living With a Star program. This work was supported by JSPS KAKENHI grant Nos. JP24K07093 (PI: A. A.), JP21H01131 (PI: K. S.), JP24K00680 (PI: K. S.), JP24K17082 (PI: K. I.), and JP24H00248 (PI: D. Nogami). This work was also supported by JST SPRING, Grant Number JPMJSP2110 (T.O.). \nSoftware: sunpy (SunPy Community et al. 2020), ChiantiPy (Dere 2013) \n<!-- image --> \nFigure 1. (a) The GOES soft X-ray light curve between 20:00 UT on 2023 August 5 and 04:00 UT on 2023 August 6 is shown as black solid line. The black arrow indicates the target X1.6 flare. The vertical gray dotted line indicates the time of the panels (b) and (c) ( t = 95 minutes from 2023 August 5 21:00 UT). The vertical gray dashed lines indicate the times in Figure 2 ( t = 70 , 120, and 210 minutes). (b) The H α line center solar full-disk image taken by SMART/SDDI at 22:35 UT on 2023 August 5. Solar north and west are at the top and right. The white arrow indicates the target active region NOAA 13386. The white dashed and skyblue boxes correspond to the field of view in Figure 1 (c) and Figure 2, and the quiet region for calibration, respectively. (c) The integral and limb regions for H α analyses. The H α line center image taken by the SDDI at 22:35 UT on 2023 August 5 is shown with integral and limb regions. The black dashed, red solid, orange dash-dot, and white dotted regions correspond to the Region 1, 2, 3, and limb, respectively (see the text). \n<!-- image --> \n1100 \nFigure 2. Time development of the target event in EUV images and H α spectral images taken by SDO/AIA and SMART/SDDI, respectively. From left to right, images at 2023 August 5 22:10 UT ( t = 70 minutes from 2023 August 5 21:00 UT), 23:00 UT ( t = 120 minutes), and 2023 August 6 00:30 UT ( t = 210 minutes) are shown. In the top three rows, images of AIA 94 ˚ A (a), 171 ˚ A (b), and 304 ˚ A (c) are shown. In the bottom three rows, H α +1 . 0 ˚ A (d), H α line center (e), and H α -1 . 0 ˚ A are shown. The field of view of all the panels corresponds to the white dashed box in Figure 1 (b). The white dashed line in each panel indicates the limb. Some notable points are indicated by the white arrows. The white dotted line connecting the white circle and cross is the artificial slit for the time-slice diagram in Figure 3. \n<!-- image --> \nSolar-X [arcsec] \nSolar-X [arcsec] \nSolar-X [arcsec] \nFigure 3. The time-slice diagram. From panel (a-1) to (d-1), time-slice diagrams along the slit in Figure 2 ( t = 210 minutes) are shown for AIA 94 ˚ A, 171 ˚ A, 304 ˚ A, and H α line center, respectively. The GOES light curve is over-plotted as the white dash-dot line. The inclined white dotted lines indicate the velocity of 4.5 km s -1 . The horizontal white dashed lines indicate the solar limb, which are set as same for AIA three channels but different for the H α line center. Paneles (a-2)-(d-2) show the zoomed-in diagram corresponding to the white or black dashed regions in panels (a-1)-(d-1). The GOES peak time ( t = 81 minutes) is indicated by the vertical white dotted line in panels (a-1)-(d-1) and (a-2). The vertical white or black dash-dot lines in panels (b-2)-(d-2) indicate the time of t = 90 minutes. \n<!-- image --> \nFigure 4. (a-1)-(a-3) The results of the Sun-as-a-star analysis. (b-1) and (b-2) Integrated results with limb for the comparison. (a-1) The spatially integrated pre-event-subtracted H α spectrum normalized by the full-disk integrated continuum is shown as a two-dimensional color map for the Sun-as-a-star spectra with the total region ( A 1 + A 2 + A 3 ). Orange and purple indicate emission and absorption compared with the pre-event state, respectively. (a-2) The differenced H α equivalent widths and the GOES light curve are plotted as red circles and skyblue histogram, respectively. (a-3) The light curves of AIA 94, 171 ,and 304 ˚ A are shown as the green dashed, orange solid, and dark-red dotted lines, respectively. In panel (a-2) and (a-3), the vertical blue starred, green dashed, orange solid, dark-red dotted, and purple dash-dot lines indicate the peak time of the light curves of GOES, AIA 94, 171, 304 ˚ A, and H α . The peak times are also indicated in the legends of panels (a-2) and (a-3). (b-1) and (b-2) The same as panels (a-1) and (a-2) but for results obtained over the integration involving the limb region ( A limb ) in addition to the total region. The gray regions in panels (a-1), (a-2), (b-1), and (b-2) means that no data is available for the SDDI due to the bad weather condition. \n<!-- image --> \nFigure 5. H α dynamic spectra and their light curves. Panels (a-1), (b-1), and (c-1) are the same as Figure 4 (a-1) but for the Region 1, 2, and 3 (Figure 1), respectively. In panels (a-2), (b-2), and (c-2), the differenced H α equivalent widths are shown for the Region 1, 2, and 3, respectively. The GOES light curve is also plotted as skyblue histogram in the panels. In the gray regions in all panels, no data is available for the SDDI due to the bad weather condition. \n<!-- image -->"}

2024arXiv240911178Y

Magnetars are neutron stars with superstrong magnetic fields which can exceed 1e15 G. Some magnetars the socalled soft gammarepeaters demonstrate occasionally very powerful processes of energy release which result in exceptionally strong flares of electromagnetic radiation. It is believed that these flares are associated with the presence of superstrong magnetic fields. Despite many hypotheses the mechanism of these flares remains a mystery. In afterglows of the flares one has often observed quasiperiodic oscillations QPOs of magnetar emission. They are interpreted as stellar vibrations excited by the flares which are useful for exploring the nature of magnetar activity. The incompleteness of theories employed to interpret magnetar QPOs is discussed.

2024-09-01T00:00:00Z

['10.48550/arXiv.2409.11178', '2024arXiv240911178Y', 'arXiv:2409.11178']

['Astrophysics - High Energy Astrophysical Phenomena', 'Astrophysics - Solar and Stellar Astrophysics']

Powerful flares and magnetoelastic oscillations of magnetars

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.11178.pdf

{'D. G. Yakovlev ∗': 'Ioffe Institute, Politekhnicheskaya street 26, Saint Petersburg, 194021, Russia (Dated: September 18, 2024) \nMagnetars are neutron stars with superstrong magnetic fields which can exceed 10 15 G. Some magnetars (the so-called soft gamma-repeaters - SGRs) demonstrate occasionally very powerful processes of energy release, which result in exceptionally strong flares of electromagnetic radiation. It is believed that these flares are associated with the presence of superstrong magnetic fields. Despite many hypotheses, the mechanism of these flares remains a mystery. In afterglows of the flares, one has often observed quasi-periodic oscillations (QPOs) of magnetar emission. They are interpreted as stellar vibrations, excited by the flares, which are useful for exploring the nature of magnetar activity. The incompleteness of theories employed to interpret magnetar QPOs is discussed. Published in: Zhurnal Experimentalnoi i Teoreticheskoi Fiziki, Vol. 166 (7) (2024).', '1. INTRODUCTION': 'Pyotr Leonidovich Kapitza, to whom this issue of Journal of Experimental and Theoretical Physics (ZhETF) is dedicated, made an outstanding contribution to studies of very strong magnetic fields [1]. Perhaps he would have liked magnetars - the natural laboratories of superstrong magnetic fields. \nNeutron stars are the most compact of all stars. They are well known astrophysical objects, but are still fascinating because of extreme physical conditions in and around them. They contain superdense matter with superstrong magnetic fields in the presence of enormous gravitational forces. Many properties of neutron stars (for example, the equation of state and the composition of matter in inner layers) are still poorly understood. \nA schematic structure of a neutron star is shown in Fig. 1. One can distinguish two main internal layers (e.g., [3]): the outer shell, often called the crust, and the inner core. At a typical neutron star mass, M ∼ 1 . 4 M ⊙ ( M ⊙ is the solar mass) its radius is R ∼ 12 km. The crust consists mostly of ions (atomic nuclei), electrons, and (at densities ρ ≳ 4 × 10 11 g cm -3 ) free neutrons. It is ∼ 1 km thick, has a mass of ∼ 0 . 01 M ⊙ . The density at the crust bottom is about half the standard nuclear density ρ 0 , with ρ 0 ≈ 2 . 8 × 10 14 g cm -3 . The atomic nuclei in the crust form usually Coulomb crystals. Beneath the crust, there is a massive and bulky core, containing liquid nuclear matter; its composition and equation of state are not reliably known. The central density of the star reaches several ρ 0 . \nThis paper is devoted to magnetars (as reviewed, e.g. in Ref. [4]) which are neutron stars with extraordinary strong magnetic fields. Some of them form as a special class of sources called soft gamma-ray repeaters (SGRs). Occasionally, SGRs demonstrate huge energy releases (up to ∼ 10 46 erg), observed as powerful flares of electromagnetic radiation, which then fade. It is thought that these processes are driven by superstrong magnetic fields. \nFIG. 1. Schematic structure of a neutron star. A massive and bulky core of superdense nuclear matter is surrounded by an outer shell (crust) containing an elastic crystal of atomic nuclei. A magnetar possesses superstrong magnetic fields and is surrounded by a powerful magnetosphere. \n<!-- image --> \nThere are many models (e.g., Ref. [4]) but the nature of magnetar flares is still unknown, and it will not be discussed here. \nIt is important, that the flares are accompanied by observed quasi-periodic oscillations (QPOs) of the magnetar emission at certain frequencies. These are assumed to be the frequencies of stellar oscillations excited by the flares. In principle, a correct interpretation of the observed QPOs can provide useful information on the parameters of magnetars, on the strength and geometry of their magnetic fields, and on the mechanism of their flaring activity. This motivates studies of the QPO problem. \nThe existence of QPOs in magnetar flares was theoretically predicted by Duncan [5] in 1998. The first QPOs were discovered after observations of the giant flare of SGR 1900+14 (27 Aug. 1998) and the hyper-flare of SGR 1806-20 (27 Dec. 2004). This was done by careful pro- \ncessing observational data in the 2005-2006 [6-8], which initiated serious studies of QPOs. These observations, as well as observations of other flares of magnetars, have been processed and reprocessed many times. (e.g., [912]). The data on the SGR 1806-20 hyperflare seem most representative, apparently due to the exceptionally huge energy release in the event. \nThe observed frequencies ν of magnetar QPOs fall in a wide range from tens of Hz to several kHz. The QPOs are usually divided into low-frequency ( ν ≲ 150 Hz) and high-frequency ones (at higher ν ). The detection of QPOs in magnetar flares has given rise to a variety of calculations and interpretations of oscillation frequencies (e.g, [13-37] and references therein). \nThis paper is a logical continuation of the previous work (Refs. [20] and [37]). It provides new arguments to prove that previous calculations of magnetar QPOs have mostly dealt with an incomplete set of solutions. The paper is arranged as follows. Firstly, the formalism is briefly described in Sec. 2. Then oscillation modes and torsional oscillations of non-magnetic are are outlined (in Secs. 3 and 4, respectively). In Sec. 5 we discuss magnetoelastic oscillations assuming that magnetic field effect is sufficiently weak. In Sec. 6 this case is studied for a pure dipole magnetic field in the stellar crust, and the consideration is extrapolated to higher magnetic fields. In Sec. 7, the possibility of applying the results for interpreting the observed QPOs is discussed. In Sec. 8 the results are summarized and unsolved problems are formulated.', '2. FORMALISM': "Oscillations of magnetars are described by the standard formalism of magneto-elastic oscillations of neutron stars. The formalism is well known (e.g., [26]); it sufficient to outline the basic points. For simplicity, the equations are presented neglecting relativistic effects. These effects will be included in Sec. 5. Magneto-elastic oscillations are mediated by elastic forces of crystal lattice in the stellar crust and by elastic deformations (Alfv'en perturbations) of magnetic field lines everywhere where the field is present. \nThe star is assumed to have a stationary magnetar field B ( r ) ( ∼ 10 14 -10 16 G), which is not strong enough to cause a noticeable distortion of stellar shape from the spherical one. The oscillation equations are obtained by linearizing the equations of motion of magnetized matter assuming the field is frozen into the matter. The unperturbed configuration of the star is thought to be spherically symmetric. Under these conditions, it is sufficient to study the oscillations of incompressible matter where matter elements move only along spherical surfaces. Then the perturbations of pressure and density are absent, and the emission of gravitational waves is suppressed. The perturbations excite small velocities of matter elements v ( r , t ), small displacements of these elements u ( r , t ), and small variations of magnetic magnetic \nfield B 1 ( r , t ). All these variations oscillate in time as exp(i ωt ), where ω = 2 πν is the angular oscillation frequency, and ν is the cyclic frequency. This overall oscillating factor in the equations can be dropped, leading to the stationary wave equation for small (complex) amplitudes u ( r ) and B 1 ( r ), and for the oscillation frequency ω : \nρω 2 u = T µ + T B . (1) \nHere T µ and T B are the volume densities of forces (with minus sign) determined, respectively, by the crystal elasticity and magnetic field stresses. In the first case \nT µi = -∂σ ik ∂x k , σ ik = µ ( ∂u i ∂x k + ∂u k ∂x i ) , (2) \nwhere is σ ik is the tensor of shear deformations and µ is the shear viscosity (in the isotropic crystal approximation). In the case of magnetic forces one has \nT B = 1 4 π B × curl B 1 , B 1 = curl( u × B ) . (3) \nThese equations should be supplemented with boundary conditions. Since the crystal exists only in the stellar crust, the radial components of viscous stresses must vanish at the outer and inner boundaries of the crust. The conditions for the magnetic field depend on the formulation of the problem. Alfv'en perturbations can propagate into the core and magnetosphere of the star.", '3. GENERAL REMARKS': "Let us begin with a few remarks. It is well known that the shear modulus µ determines characteristic propagation speed v µ of elastic shear deformations in a crystal. As follows from calculations (e.g., [38]), these deformations are mainly located near the bottom of the crust, at ρ ∼ 10 14 g cm -3 . Then, under typical conditions, one comes to the estimate \nv µ ∼ √ µ/ρ ∼ 10 8 cm s -1 . (4) \nAs for magnetic perturbations, they propagate with the Alfv'en velocity v A which, for the same conditions, can be estimated as \nv A = B √ 4 πρ ∼ 3 × 10 7 B 15 cm s -1 , (5) \nwhere B 15 is the magnetic field in units of 10 15 G. The velocity v A can noticeably decrease within the star and increase toward the surface. \nThe velocities (4) and (5) become close at \nB ∼ B µ ∼ 3 × 10 15 G . (6) \nThis characteristic field strength reveals the existence of three regimes of magneto-elastic oscillations (Table I). \nTABLE I. Three regimes of magneto-elastic oscillations of magnetars. \nIn regime I, the oscillations are mainly regulated by elastic shear waves in the stellar crust; Alfv'en perturbations are driven by these elastic shear stresses and weakly affect the oscillation frequencies. Such oscillations are almost completely localized in the crust being determined by the microphysics of matter and by the properties of B ( r ) in the crust. \nThe efficiencies of shear and Alfv'en waves in regime II are comparable. Alfv'en perturbations can propagate beyond the crust (e.g. [13, 14]) and spread over the whole star. Their calculation requires knowledge of the entire microphysics of the magnetar, which contains many uncertainties, including the equation of state, superfluidity and superconductivity of the stellar core, as well as magnetic field configuration within it. \nFinally, in regime III, the oscillations are mainly regulated by Alfve'n waves. The elasticity of crystal becomes nearly or even fully negligible (e.g., [17, 19]). \nThis classification of oscillations is schematic. In particular, it does not take into account possible forbidden frequency intervals of Alfve'n oscillations in the stellar core (e.g., [26]) in which case the oscillations can be locked in the crust even at very strong magnetic fields. The effects of penetration of Alfv'en perturbations from the crust to the core and back can also be important. They can lead to frequency variations, damping, and loss of coherence of crustal oscillations. \nThe employed approximation of incompressible magneto-elastic oscillations (Sec. 2) also deserves a comment. Types of neutron star oscillations are numerous (e.g., [41] and references therein). Magneto-elastic oscillations are suitable because their frequencies are low enough to explain magnetar QPOs. The shear and Alfv'en velocities, v µ and v A , are generally much lower than the speed of ordinary sound, that is determined by the full pressure of dense stellar matter. Frequencies of oscillations of many types are higher than magneto-elastic ones.", '4. ELASTIC OSCILLATIONS OF NON-MAGNETIC CRUST': "Such oscillations are often called torsional. They are basic for studying magneto-elastic oscillations. Their theory began in the 1980s in the classical works of Hansen and Cioffi [39], Schumaker and Thorne [40], and McDermott et al. [41] long before the discovery of magnetar QPOs. After the discovery, the interest to the theory was renewed (e.g., [27, 29, 36, 42-48] and references therein). \nAny torsion oscillation mode is characterized by three quantum numbers: (1) n = 0 , 1 , 2 , . . . is the number of radial nodes of the wave function, (2) the orbital number ℓ , which in this problem runs the values ℓ = 2 , 3 , . . . , (3) the azimuthal number m which takes integer values from -ℓ to ℓ . \nIn the spherical coordinates ( r, θ, ϕ ), a stationary wave function u ( r ) has only two non-trivial components: u ϕ and u θ (since u r = 0). These can be written as (e.g., [37]) \nu ϕ ( r, θ, ϕ ) = rY ( r ) e i mϕ d P m ℓ d θ , (7) \nu θ ( r, θ, ϕ ) = rY ( r ) e i mϕ i mP m ℓ sin θ , (8) \nwhere P m ℓ (cos θ ) is an associated Legendre polynomial, and Y ( r ) = Y nℓ ( r ) is a dimensionless radial wavefunction satisfying the equation \nY '' + ( 4 r + µ ' µ ) Y ' + [ ρ µ ω 2 -( ℓ +2)( ℓ -1) r 2 ] Y = 0 . (9) \n̸ \nPrime means derivative with respect to r . These oscillations are localized in the crystalline crust, r 1 ≤ r ≤ r 2 , where r 1 is the radius of the crust-core interface, and r 2 is the outer radius of the crystallization zone which is very close to the radius of the star. At both boundaries, radial elastic stresses should vanish, Y ' ( r 1 ) = Y ' ( r 2 ) = 0. The frequencies of torsional oscillations are degenerate in m : ω = ω µnℓ (the index µ indicates the elastic shear nature of these oscillations); the functions Y ( r ) are independent of m . The value Y 0 = Y ( r 2 ) characterises the angular amplitude of oscillations (in radians) at the outer edge of the crystallization region. If m = 0, crustal matter oscillates only along parallels ( u θ = 0), but at m = 0 there appear meridional motions. The value of m strongly influences the geometry of displacements u ( r ) and the angular dependence of the energy density of oscillations. A specific stellar model affects only Y ( r ), while angular dependences of u ( r ) stay standard. \nTorsional oscillations of neutron stars are divided into fundamental ( n = 0) and ordinary ( n > 0) ones. For the fundamental oscillations, a very good approximation is the weak deformability of the crystal, in which case Y is almost independent of r (e.g., [38]). In this case, ω µ 0 ℓ ≈ 1 2 ω µ 0 √ ( ℓ +2)( ℓ -1), where ω µ 0 is the basic frequency (at ℓ = 2), the lowest for all torsion oscillations. \nThe frequencies of ordinary torsion oscillations ( n > 0) are higher and strongly increase with increasing n . At a fixed n , there is a bunch of close frequencies which grow weakly with increasing ℓ (showing 'fine splitting' with respect to ℓ ). Corresponding wave functions Y nℓ ( r ) depend on ℓ rather weakly (e.g., [49]). Since torsional oscillation frequencies do not depend on m , one usually sets m = 0 for finding the oscillation spectrum, without using the states with m = 0. \nTorsional oscillations may carry a lot of energy. For example, let us choose a neutron star model with a nucleonic core and the modern BSk21 equation of state of \n̸ \ndense matter (described, e.g., in Ref. [50]). For a 1 . 4 M ⊙ neutron star, the stellar radius is R = 12 . 6 km and the crust-core radius is r 1 = 11 . 55 km. According to the results of Ref. [49], the oscillation energy of the basic mode ( n = 0, ℓ = 2, ν µ 0 = 23 . 0 Hz) is E vib ≈ 10 49 Y 2 0 erg. At a swing angle ≈ 0 . 1 · ( Y 0 ∼ 1 . 7 × 10 -3 rad) of oscillations at the outer edge of the crystalline crust, we get E vib ∼ 3 × 10 43 erg. In this case, shear stresses in a vibrating crust are still far from the crystal-breaking limit [38].", '5. OSCILLATIONS DOMINATED BY ELASTICITY OF THE CRUST': "̸ \nThis is the simplest regime I of magneto-elastic oscillations (Table I). In this case magnetic fields are not too high ( B ≪ B µ ) and can be taken into account by perturbation theory, considering the wave functions of pure torsional oscillations (Sec. 4) as zero-order wave functions, and the quantity T B in Eq. (1) as a small perturbation. In numerous studies of magneto-elastic oscillations (e.g., [13-16, 18, 21-26, 28, 30, 31, 33, 35]), the states with m = 0 have been ignored. In this way one dealt with incomplete spectrum of magneto-elastic oscillations. \nThe exceptions were the paper by Shaisultanov and Eichler [20] and the recent paper [37]. The authors of Ref. [20] argued that the magnetic field removes the degeneracy of torsion frequencies. In a magnetic field, these frequencies should split into a series of frequencies, which can be treated as the Zeeman effect in magnetars. The effect was correctly described and evaluated, but the work did not attract much attention. Ref. [37] was devoted to developing these ideas. It proposed a simple algorithm for calculating the oscillation frequencies in the first-order perturbation theory for a wide class of B -fields in the magnetar crust. For illustration, the Zeeman splitting of fundamental oscillations ( n = 0) in a dipole crustal magnetic field at 2 ≤ ℓ ≤ 5 was calculated. \nHere we extend the consideration of magneto-elastic oscillations in the first-order perturbation theory for the fundamental modes ( n = 0). The details of the theory were presented in [37]. Here we mention them only briefly. In the formulated approach, it is sufficient to assume that the oscillations are localized in the elastic crust. As in Ref. [37], we assume that the crustal magnetic field is axially symmetric about the magnetic axis: only the field components B r ( r, θ ) and B θ ( r, θ ) are different from zero. In this case \nω 2 ℓm = ω 2 µℓ + ω 2 Bℓm , (10) \nwhere ω µℓ is the frequency of purely torsional oscillations (Sec. 4), and ω Bℓm is the small 'magnetic' correction; ℓ and m have the same meaning as in the wave functions of zero-order approximation; see Eqs. (7) and (8). \nThe expressions for ω µℓ and ω Bℓm are given in [37]. In Sec. 6 of Ref. [49] it is also described which changes \nshould be introduced to the theory to account for relativistic effects. According to [37], \nω 2 µℓ = (1 -x g ∗ ) ∫ crust d V µ ∫ crust d V ( ρ +P /c 2 ) r 2 , (11) \nω 2 Bℓm = (1 -x g ∗ ) 1 4 π ∫ crust d V I B Ξ( ℓ, m ) ∫ crust d V ( ρ +P /c 2 ) r 2 . (12) \nHere P is the pressure of dense matter; c is the speed of light; d V = r 2 d r sin θ d θ d ϕ is the volume element in the approximation of locally flat crust; integration is over crystalline matter. The factor (1 -x g ∗ ) approximates the gravitational redshift of a squared oscillation frequency for a distant observer, x g ∗ = 2 GM ∗ / ( c 2 r ∗ ), G is the gravitational constant. Furthermore, r ∗ is the radius of any point in the crust (results being almost independent of its particular choice [49]), M ∗ is the gravitational mass inside a sphere of radius r ∗ . The quantity \nΞ( ℓ, m ) = 2 ℓ ( ℓ +1)( ℓ + m )! (2 ℓ +1)( ℓ -m )! (13) \nis a convenient normalization factor, and I B is a combination of B r , B θ , P m ℓ and their derivatives (see Eq. (18) in [37]) quadratic in the magnetic field, making ω 2 Bℓm ∝ B 2 . In this case, the oscillations are located in the crust and do not depend on the B ( r ) configuration outside the crust.", '6. CASE OF THE DIPOLE FIELD': "By way of illustration, following Ref. [37], we consider a purely dipole magnetic field in the stellar crust. Then B r = B 0 cos θ ( R/r ) 3 and B θ = 1 2 B 0 sin θ ( R/r ) 3 . Here B 0 is the field strength at the magnetic pole on the stellar surface. The field removes degeneracy of frequencies ω µℓ , but only partly: according to (10) the frequency ω µℓ splits into a series of ℓ +1 components ω ℓm , where m = 0 , 1 , . . . ℓ . The frequency ω ℓ 0 appears nondegenerate, while the frequencies with m> 0 remain degenerate twice (correspond to ± m states). The Zeeman splitting is determined by ω Bℓm given by Eq. (12). For the pure dipole field \nI B = -B 2 0 4 ( R r ) 6 [ P ' 2 (1 + 3 cot 2 θ ) -3 P ' P '' cot θ + P ' × P ''' + m 2 sin 2 θ ( -P ' 2 -10 PP ' cot θ +8 P 2 cot 2 θ ) ] . (14) \nHere P = P m ℓ (cos θ ), prime denotes differentiation with respect to θ . Then \nω 2 Bℓm = B 2 0 r 3 2 [( r 2 /r 1 ) 3 -1] 12 π ∫ r 2 r 1 d r r 4 ( ρ +P /c 2 ) ζ ℓm , (15) \nFIG. 2. The frequencies of magneto-elastic oscillations of a 1 . 4 M ⊙ neutron star versus the field strength B 0 at the magnetic pole on the stellar surface. Each series of frequencies corresponds to a fixed ℓ = 2 , . . . 11 and contains a bunch of ℓ + 1 Zeeman components. The components with m = 0 are shown by dashed lines. The region which contains many quasi-crossings of Zeeman components is darkened (as detailed in the text). \n<!-- image --> \nwhere \nζ ℓm = 1 B 2 0 Ξ( ℓ, m ) ∫ π 0 sin θ d θ I B ( R,θ ) . (16) \nOur Eqs. (14) and (15) correspond to Eqs. (26) and (27) in Ref. [37]. The latter contain typos, which are corrected here. All calculations in Ref. [37] were performed using correct formulae. \nIn Ref. [37] the factors ζ ℓm were calculated and approximated by \nζ ℓm = c 0 ( ℓ ) + c 2 ( ℓ ) m 2 (17) \nat ℓ ≤ 5; the values of c 0 ( ℓ ) and c 2 ( ℓ ) were tabulated. Now the values of ζ ℓm have been calculated up to ℓ = 15. Corresponding values of c 0 ( ℓ ) and c 2 ( ℓ ) can be fitted as \nc 0 ( ℓ ) = 0 . 721 [( ℓ -2)( ℓ +1)] 0 . 954 , (18) \nc 2 ( ℓ ) = 2 3 -0 . 766 ( ℓ -2) 1 . 09 1 + 0 . 532 ( ℓ -2) 1 . 15 . (19) \nThe fit accuracy is a few per cent, which seems quite satisfactory. Equations (17)-(19) are valid for a dipole magnetic field in the crust; fields of other configurations will be studied separately. \nThe results for dipole fields are illustrated below, extending thus the consideration of Ref. [37] to a wider frequency range. \nFigure 2 shows the dependence of oscillation frequencies on B 0 at n = 0, ℓ = 2 , . . . 11 and different m . As in Ref. [37], we consider the 1 . 4 M ⊙ star with the Bsk21 equation of state, mentioned in Sec. 4. Calculations are performed using Eqs. (10), (11) and (15). Figure 2 is similar to Fig. 1b from Ref. [37], but covers wider frequency range ν ≤ 140 Hz (instead of 80 Hz in [37]). \nAccording to the results of Sec. 3, Fig. 2 shows two regimes of magneto-elastic oscillations: regime I of field strengths much smaller than B µ ∼ 3 × 10 15 G, and regime II of intermediate field strengths (Table I). The equations employed are strictly valid only in regime I. In the figure, they are extrapolated to the intermediate regime. The possibility of such extrapolation requires confirmation (see below). \nAt B 0 ≤ 4 × 10 14 G and ν < 140 Hz, Fig. 2 shows 10 frequencies of fundamental torsional oscillations (Sec. 4) which are actually unaffected by the magnetic field. However, as B 0 grows up, each of these frequencies splits noticeably into Zeeman components: 10 initial frequencies decompose into 75 branches. \nAt not too high B 0 , one can clearly see 10 separate bunches of curves corresponding to certain ℓ . The oscillation frequencies in each bunch differ by the values of m . In agreement with the results of [37], the branches of oscillations with m = 0 (dashed lines) at ℓ = 2 and 3 lie below other branches in a bunch, while at higher ℓ they become higher than the others (this inversion is possibly specific for the dipole magnetic field). The higher ℓ , the richer the splitting, and the smaller the value of B 0 at which this splitting starts to be visible. \nAt ℓ > 3, the lowest branch of oscillations in any bunch corresponds to the highest m = ℓ + 1. Interestingly, as ℓ increases, such curves become more horizontal and depend weaker on B 0 . In other words, at high m the frequencies ν ℓm ( B 0 ) approach the frequencies of torsional oscillations ν µℓ of an non-magnetic star (Sec. 4). \nStarting from B 0 ≳ 1 . 5 × 10 15 G and ℓ ∼ 11, in the upper right corner of Fig. 2, there appears a special region of frequencies and magnetic fields in which the magnetoelastic oscillations behave in a complicated way. If B 0 increases to the highest depicted values (4 × 10 15 G), this region descends to ν ∼ 90 Hz (and at higher B 0 it will descend further). In this region the two effects, neglected in the calculations, can be especially important. \nFirstly, with increasing B 0 , the oscillation modes from different bunches begin to show quasi-crossings (Fig. 2). The identity of individual bunch is lost, and the region becomes densely filled with allowed oscillation frequencies. The behavior of curves near quasi-crossing points requires further analysis. As usual, in the vicinity of these points, the oscillations of converging modes interact with each other, and their frequencies are distorted. \nSecondly, in the presence of pronounced damping and loss of coherence of crustal oscillations due to the transfer of vibrational energy by Alfve'n waves into the stellar core, the oscillation modes can acquire finite shifts and widths (this effect is expected to be amplified with in- \nFIG. 3. Same as in Fig. 2, but we left only the oscillations with m = 0, which were considered in the majority of publications. \n<!-- image --> \nreasing B 0 and ν ). The frequencies of crustal oscillations are thus capable of blurring and shifting. However, there can exist forbidden frequency intervals (e.g., [26] and references therein), which may prevent penetration of Alfve'nic waves into the core. \n̸ \nIt is clear that both effects are mutually related and require self-consistent consideration. It is impossible to correctly calculate quasi-crossings without a reliable theory of interaction between crustal oscillations and Alfv'en perturbations in the core. Great efforts have been spent on the construction of such a theory (e.g, [13-15, 17, 2126, 28, 30, 31, 33, 35]) but only for axially symmetric perturbations ( m = 0). There is no theory at m = 0, it is a difficult task for future studies. It seems that both effects are most important in the special region of high frequencies and fields, while at lower ν and B 0 they are weaker.", '7. DISCUSSION': "All the above effects can be important for interpreting observed frequencies of magnetar QPOs. Below we extend the interpretation of QPOs with complete theoretical spectrum (it was started in Ref. [37]) using the data from the hyperflare of SGR 1806-20 and the giant flare of SGR 1900+14. Now we add a few higher-frequency QPOs. The unsolved problems of quasicrossing of magneto-elastic oscillation frequencies and the interaction of crustal oscillations with Alfve'n oscillations in the stellar core are not considered here. Therefore, as in Ref. [37], our consideration is illustrative and may be particularly inaccurate at sufficiently high ν and B 0 . \n̸ \nThe available observational data on magnetar QPOs have been analyzed many times. The results are summarized, for example, in Ref. [35]. They have been widely used by many authors and will be used below. An exception is Ref. [12], whose authors expressed doubt in significance of measured low-frequency QPOs based on Bayesian model-independent extraction of noise; their conclusion requires further confirmation. \n̸ \nIt has already been noted that in many interpretations the oscillation modes with m = 0 were ignored. For illustration, Fig. 3 presents only the frequencies of m = 0 oscillation modes, instead of all of them in Fig. 2. Of the 75 modes shown in Fig. 2, only 10 are left. Clearly, they constitute an incomplete set of theoretical curves. Using such a set, an interpretation of observations can be questionable. In particular, all quasi-crossings of oscillation modes in Fig. 3 disappear. \n̸ \nThe reason for neglecting solutions with m = 0 is that the displacements u ( r ) of oscillating stellar matter for such solutions depend not only on r and θ , but also on the angle ϕ ; see, e.g., Eqs. (7) and (8). In other words, for axially symmetric magnetic fields B ( r, θ ), considered by most researchers, the perturbed quantities u and B 1 at m = 0 turn out to be axially asymmetric. However, the axial symmetry of perturbations was usually postulated leading to the loss of solutions with m = 0. \nLet us add that, as seen from Fig. 3, the frequency ν 20 (the lowest dashed line) does not depend on B at all (see also Ref. [37]). This result is valid in the first-order perturbation theory for weak dipole crustal magnetic fields (Sec. 5). In reality, it means that an expansion of ν 20 ( B 0 ) in powers of B 2 0 should not contain the B 2 0 -term. Higherorder terms can be present, although their calculation requires much effort. But according to Ref. [16], devoted to oscillations of a neutron star crust with a dipole field, the B 2 0 -term is not zero. This paradox was resolved by noting [26], that the solution in [16] had been sought by expanding u ϕ into a series of functions (7) with different ℓ at m = 0. The sum over ℓ in [16] was artificially truncated, that was actually equivalent to solving an exact non-dipole magnetic field problem. This does not prove the B 2 0 -term is nonzero. \n̸ \nThe largest number of QPOs was detected by processing observations of the hyper-flare of SGR 1806-20. The low-frequency QPOs, which are discussed below, were detected at 18, 26, 30, and 150 Hz, and (with lower confidence) at 17, 21, 36, and 59 Hz. \nCan these QPOs be interpreted as fundamental magneto-elastic oscillations of a single star (with the same mass, radius, and internal structure) possessing the same crustal dipole field? This question was raised in Ref. [37], where theoretical calculations were limited to ℓ ≤ 5 and could explain QPO frequencies ν ≤ 60 Hz. It turned out that for a M = 1 . 4 M ⊙ star, only three of seven such frequencies (17, 18, 21, 26, 30, 36, and 59 Hz) could be interpreted: 26, 30, and 59 Hz, assuming B 0 ≈ (3 . 2 -3 . 4) × 10 15 G (Fig. 2a in [37]). The lowest-frequency QPOs could not be explained in this \nFIG. 4. Same as in Fig. 2, but for a 2 . 2 M ⊙ star compared to QPO frequencies (dotted horizontal lines) observed from the hyper-flare of SGR 1806-20. The vertical green band shows a possible range of B 0 , simultaneously consistent with some observed QPOs (as detailed in the text). \n<!-- image --> \nway. However, leaving the same equation of state of the neutron star matter (BSk21), but increasing the stellar mass to 2 . 2 M ⊙ (with the 2 . 27 M ⊙ maximum mass limit), one could slightly lower all theoretical frequencies due to a stronger gravitational redshift of oscillation frequencies for a more massive (and compact) star. In this case (Fig. 2b in [37]), it was possible to explain six frequencies except for one (30 Hz), assuming B 0 ≈ (3 . 5 -3 . 7) × 10 15 G. \nBy adding new calculations of this paper (up to ℓ = 11), it is possible to explain all but one (30 Hz) of the observed low-frequency QPOs of the SGR 1806-20 hyperflare assuming the same field B 0 ≈ (3 . 5 -3 . 7) × 10 15 G as in the [37]. This is shown in Fig. 4, that is similar to Fig. 2b in [37] but extended now to 120 Hz. An explanation for the highest selected QPO frequency (150 Hz) is not shown to simplify the figure, but it is evident because of very densely spaced theoretical oscillation branches at ν > 90 Hz. We do not worry on the failure to explain the 30 Hz QPO [37]: no attempt has been made to seriously explain the observations. The M = 2 . 2 M ⊙ model was chosen as an example and was not varied. The required magnetic field B 0 corresponds to regime II (Sec. 3), where quantitative accuracy of theoretical frequencies can be questioned. In addition, a purely dipole magnetic field has been assumed, whereas possible deviations from pure dipole can change theoretical results. In any case, the proposed complete theoretical set of frequencies of magneto-elastic oscillations greatly simplifies theoretical interpretation of low-frequency magnetar QPOs. \nMoreover, in Ref. [37] we tried to interpret the QPOs observed in the giant flare of SGR 1900+14. Four low- \nFIG. 5. Same as in Figs. 2 and 4, compared with the QPO frequencies (dotted horizontal lines) observed from the giant flare of SGR 1900+14. \n<!-- image --> \nfrequency QPOs (28, 53, 84, and 155 Hz) were detected. The two lowest frequencies were easily explained by the 1 . 4 M ⊙ neutron star model (Fig. 3 in [37]). By taking the same model and now increasing the theoretical frequencies to 160 Hz, it is possible to explain (Fig. 5) all four QPOs with B 0 ≈ (2 . 42 -2 . 62) × 10 15 G. Just as for Fig. 4, a more serious interpretation seems premature. \nLet us emphasize that B ∼ B µ has often been treated as a valid estimate of magnetar fields for various reasons (e.g., [35]).", '8. CONCLUSIONS': "We have attempted to develop the theory of magnetoelastic oscillations of magnetars. We have employed the standard assumption that these oscillations are excited during flares of magnetars and are observed as quasiperiodic oscillations (QPOs) at the decay phases of the flares (Sec. 1 and references therein). Correct interpretation of observations can provide useful information about the parameters of magnetars, their magnetic fields, and the nature of their flares. \nFollowing the results of Refs. [20, 37], the completeness of theoretical QPO models has been studied, especially because many previous works neglected Zeeman splitting of magnetar oscillations. We have used a simple model of low-frequency magneto-elastic oscillations without nodes of radial wave function in the magnetar crust assuming a purely dipole crustal magnetic field. We have shown that neglecting the Zeeman effect leads to essentially incomplete set of oscillation modes. In the case of axi- \nally symmetric B ( r ), this simplification reproduces only the oscillations accompanied by axially symmetric vibrational displacements of matter elements u ( r ) and magnetic field B 1 ( r ). In this way it misses a wide range of oscillations modes in which the displacements u ( r ) and B 1 ( r ) are axially asymmetric. We have demonstrated that the full set of oscillations gives a qualitatively different oscillation spectrum and can significantly change theoretical interpretation of observed QPOs. \nTherefore, the construction of complete set of magneto-elastic oscillations has only begun. Serious efforts are needed to complete it. Here are some of the problems. \nEven in the most reliable regime I of relatively low magnetic fields, only low-frequency oscillations have been considered, without nodes of wave function along the radius. Generalization to the case of oscillations with nodes ( n > 0) can be done without difficulty. Instead of the dipole field, it is easy to study poloidal magnetic fields of other types. It is also easy to consider the case in which a toroidal magnetic field is also present. In addition, our results are obtained in the approximation of a locally flat stellar crust (e.g., [37, 49]). It would be useful to solve the problem in full General Relativity. This would be especially important for the cases in which Alfv'en perturbations propagate outside the crust. \nThe oscillation regime I, which has been studied rather reliably, is insufficient for interpreting the observations. It seems that the intermediate regime II ( B ∼ B µ ) is more important for this purpose. Such oscillations can penetrate into the core of the star, which makes the above consideration quantitatively inaccurate (although it may be qualitatively applicable, especially for lowest frequencies). A firm study of oscillations in this regime is more complicated because the calculations should include microphysics of the stellar core for many possible models (superfluidity and superconductivity in the core, different magnetic field configurations there, etc.). There is also an important problem of interaction of Alfv'en oscil- \n- [1] P. L. Kapitza, Experimental research in strong magnetic fields , Physics - Uspekhi 36 (4) , 288 (1993).\n- [2] S. L. Shapiro, A. A. Teukolsky, Black holes, white dwarfs, and neutron stars: The physics of compact objects , Wiley-Interscience, New York (1983).\n- [3] P. Haensel, A. Y. Potekhin, D. G. Yakovlev, Neutron Stars. 1. Equation of State and Structure , Springer, New York (2007).\n- [4] V. M. Kaspi, A. M. Beloborodov, Annual Rev. Astron. Astrophys. 55 , 261 (2017).\n- [5] R. C. Duncan, Astrophys. J. Lett. 498 , L45 (1998).\n- [6] G. L. Israel, T. Belloni, L. Stella, Y. Rephaeli, D. E. Gruber, P. Casella, S. Dall'Osso, N. Rea, M. Persic, R. E. Rothschild, Astrophys. J. Lett. 628 , L53 (2005).\n- [7] A. L. Watts, T. E. Strohmayer, Astrophys. J. Lett. 632 , L111 (2005). \nlations in the core with crustal oscillations. The energy of crustal vibrations can flow into the core, which can lead to damping and loss of coherence of the crustal oscillations. All these effects have been studied for oscillations induced by axially symmetric perturbations. The important case of axially asymmetric perturbations has not been explored. \nAnother area of research is to improve the microphysics of neutron star matter that affects magnetar oscillations. In particular, one can improve calculations of the shear modulus in the crust, consideration of the delicate effects of superfluidity and superconductivity of crustal matter, nuclear interactions, nuclear pasta effects at the bottom of the crust, etc. (see, e.g., [27, 29, 36, 37, 43-49, 51] ). \nFinally, it is appropriate to list the main original results of this work. The studies of low-frequency magnetoelastic oscillations in Ref. [37] are extended to higher frequencies (Sec. 6, Fig. 2). Simple approximations are derived for the coefficients c 0 ( ℓ ) and c 2 ( ℓ ), Eqs. (18) and (19). They allow one to calculate the oscillation spectrum at ν ≲ 150 Hz. Quasi-crossings of oscillation frequencies with different ℓ are pointed out in the darkened region in Fig. 2, where allowable frequencies are densely spaced, but in fact can quickly decay through interaction of crustal and Alfv'en oscillations of the core. Following Ref. [37], possible interpretations of low-frequency QPOs, observed in the hyperflare of SGR 1806-20 (Fig. 4) and the giant flare of SGR 1900+14 (Fig. 5), are further discussed. The necessity for joint consideration of quasi-crossings of modes and interactions of crustal oscillations with Alfv'en perturbations in the stellar core is stressed. It is stated that, despite considerable efforts of many authors, the theory of magneto-elastic oscillations is far from being completed. \nThis work was performed within the Work Program (number FFUG-2024-0002) of A. F. Ioffe Institute. The author is grateful to M. E. Gusakov, E. M. Kantor, and A. I. Chugunov for comments and critical remarks to the previous paper [37], which were useful for writing this one. \n- [8] A. L. Watts, T. Strohmayer, Astrophys. J. Lett. 637 , L117 (2006).\n- [9] V. Hambaryan, R. Neuhauser, K. D. Kokkotas, Astron. Astrophys. 528 , A45 (2011).\n- [10] D. Huppenkothen, L. M. Heil, A. L. Watts, E. Go˘gus , , Astrophys. J. 795 , 114 (2014).\n- [11] D. Huppenkothen, C. D'Angelo, A. L. Watts, L. Heil, M. van der Klis, A. J. van der Horst, C. Kouveliotou, M. G. Baring, E. Go˘gus , , J. Granot, Y. Kaneko, L. Lin, A. von Kienlin, G. Younes, Astrophys. J. 787 , 128 (2014).\n- [12] D. Pumpe, M. Gabler, T. Steininger, T. A. Enßlin, Astron. Astrophys. 610 , A61 (2018).\n- [13] Y. Levin, Mon. Not. R. Astron. Soc. 368 , L35 (2006).\n- [14] K. Glampedakis, L. Samuelsson, N. Andersson, Mon. Not. R. Astron. Soc. 371 , L74 (2006).\n- [15] Y. Levin, Mon. Not. R. Astron. Soc. 377 , 159 (2007).\n- [16] H. Sotani, K. D. Kokkotas, N. Stergioulas, Mon. Not. R. Astron. Soc. 375 , 261 (2007).\n- [17] H. Sotani, K. D. Kokkotas, N. Stergioulas, Mon. Not. R. Astron. Soc. 385 , L5 (2008).\n- [18] U. Lee, Mon. Not. R. Astron. Soc. 385 , 2069 (2008).\n- [19] P. Cerd'a-Dur'an, N. Stergioulas, J. A. Font, Mon. Not. R. Astron. Soc. 397 , 1607 (2009).\n- [20] R. Shaisultanov, D. Eichler, Astrophys. J. Lett. 702 , L23 (2009).\n- [21] A. Colaiuda, H. Beyer, K. D. Kokkotas, Mon. Not. R. Astron. Soc. 396 , 1441 (2009).\n- [22] A. Colaiuda, K. D. Kokkotas, Mon. Not. R. Astron. Soc. 414 , 3014 (2011).\n- [23] M. van Hoven, Y. Levin, Mon. Not. R. Astron. Soc. 410 , 1036 (2011).\n- [24] M. Gabler, P. Cerd'a-Dur'an, J. A. Font, E. Muller, N. Stergioulas, Mon. Not. R. Astron. Soc. 410 , L37 (2011).\n- [25] A. Colaiuda, K. D. Kokkotas, Mon. Not. R. Astron. Soc. 423 , 811 (2012).\n- [26] M. van Hoven, Y. Levin, Mon. Not. R. Astron. Soc. 420 , 3035 (2012).\n- [27] H. Sotani, K. Nakazato, K. Iida, K. Oyamatsu, Phys. Rev. Lett. 108 , 201101 (2012).\n- [28] M. Gabler, P. Cerd'a-Dur'an, N. Stergioulas, J. A. Font, E. Muller, Mon. Not. R. Astron. Soc. 421 , 2054 (2012).\n- [29] H. Sotani, K. Nakazato, K. Iida, K Oyamatsu, Mon. Not. R. Astron. Soc. 434 , 2060 (2013).\n- [30] M. Gabler, P. Cerd'a-Dur'an, J. A. Font, E. Muller, N. Stergioulas, Mon. Not. R. Astron. Soc. 430 , 1811 (2013).\n- [31] M. Gabler, P. Cerd'a-Dur'an, N. Stergioulas, J. A. Font, E. Muller, Phys. Rev. Lett. 111 , 211102 (2013).\n- [32] A. Passamonti, S. K. Lander, Mon. Not. R. Astron. Soc. 438 , 156 (2014).\n- [33] M. Gabler, P. Cerd'a-Dur'an, N. Stergioulas, J. A. Font, \n- E. Muller, Mon. Not. R. Astron. Soc. 460 , 4242 (2016).\n- [34] B. Link, C. A. van Eysden, Astrophys. J. Lett. 823 , L1 (2016).\n- [35] M. Gabler, P. Cerd'a-Dur'an, N., Stergioulas, J. A. Font, E. Muller, Mon. Not. R. Astron. Soc. 476 , 4199 (2018).\n- [36] H. Sotani, K. Iida, K. Oyamatsu, Mon. Not. R. Astron. Soc. 479 , 4735 (2018).\n- [37] D. G. Yakovlev, Universe 9 (12) , 504 (2023).\n- [38] A. A. Kozhberov, D. G. Yakovlev, Mon. Not. R. Astron. Soc. 498 , 5149 (2020).\n- [39] C. J. Hansen, D. F. Cioffi, Astrophys. J. 238 , 740 (1980).\n- [40] B. L. Schumaker, K. S. Thorne, Mon. Not. R. Astron. Soc. 203 , 457 (1983).\n- [41] P. N. McDermott, H. M. van Horn, C. J. Hansen, Astrophys. J. 325 , 725 (1988).\n- [42] L. Samuelsson, N. Andersson, Mon. Not. R. Astron. Soc. 374 , 256 (2007).\n- [43] N. Andersson, K. Glampedakis, L. Samuelsson, Mon. Not. R. Astron. Soc. 396 , 894 (2009).\n- [44] H. Sotani, K. Nakazato, K Iida, K. Oyamatsu, Mon. Not. R. Astron. Soc. 428 , L21 (2013).\n- [45] H. Sotani, Phys. Rev. D 93 , 044059 (2016).\n- [46] H. Sotani, K. Iida, K. Oyamatsu, Mon. Not. R. Astron. Soc. 464 , 3101 (2017).\n- [47] H. Sotani, K. Iida, K. Oyamatsu, Mon. Not. R. Astron. Soc. 470 , 4397 (2017).\n- [48] H. Sotani, K. Iida, K. Oyamatsu, Mon. Not. R. Astron. Soc. 489 , 3022 (2019).\n- [49] D. G. Yakovlev, Mon. Not. R. Astron. Soc. 518 , 1148 (2023).\n- [50] A. Y. Potekhin, A. F. Fantina, N. Chamel, J. M. Pearson, S. Goriely, Astron. Astrophys. 560 , A48 (2013).\n- [51] N. A. Zemlyakov, A. I. Chugunov, Universe 9 (5) , 220 (2023)."}

2024A&A...689L..10B

We report on the detection of unintended electromagnetic radiation UEMR from the secondgeneration of Starlink satellites. Observations with the LOFAR radio telescope between 10 to 88 MHz and 110 to 188 MHz show broadband emission covering the frequency ranges from 40 to 70 MHz and 110 to 188 MHz from the v2Mini and v2Mini DirecttoCell Starlink satellites. The spectral power flux density of this broadband UEMR varies from satellite to satellite with values ranging from 15 to 1300 Jy between 56 and 66 MHz and from 2 to 100 Jy over two distinct 8 MHz frequency ranges centered at 120 and 161 MHz. We compared the detected power flux densities of this UEMR to that emitted by the first generation v1.0 and v1.5 Starlink satellites. When correcting for the observed satellite distances we find that the secondgeneration satellites emit UEMR that is up to a factor of 32 stronger compared to the first generation. The calculated electric field strengths of the detected UEMR exceed typical electromagnetic compatibility standards used for commercial electronic devices as well as recommended emission thresholds from the Radiocommunication Sector of the International Telecommunications Union ITUR aimed at protecting the 150.05153 MHz frequency range allocated to radio astronomy. We characterize the properties of the detected UEMR with the aim of assisting the satellite operator with the identification of the cause of the UEMR.

2024-09-01T00:00:00Z

['10.1051/0004-6361/202451856', '2024arXiv240911767B', 'arXiv:2409.11767', '10.48550/arXiv.2409.11767', '2024A&A...689L..10B']

['light pollution', 'space vehicles', 'telescopes', 'surveys', 'Astrophysics - Instrumentation and Methods for Astrophysics']

Bright unintended electromagnetic radiation from secondgeneration Starlink satellites

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']

https://arxiv.org/pdf/2409.11767.pdf

{'No Header': 'L etter to the E ditor', 'Bright unintended electromagnetic radiation from second-generation Starlink satellites': 'C. G. Bassa 1, ⋆ , F. Di Vruno 2 , 3, ⋆ , B. Winkel 4 , 3, ⋆ , G. I. G. Józsa 4 , 5 , 3, ⋆ , M. A. Brentjens 1 , and X. Zhang 6 \n- 1 ASTRON, Netherlands Institute for Radio Astronomy, Oude Hoogeveensedijk 4, 7991 PD Dwingeloo, The Netherlands e-mail: bassa@astron.nl\n- 2 Square Kilometre Array Observatory, Lower Withington, Macclesfield, Cheshire, SK11 9FT, United Kingdom\n- 3 European Science Foundation, Committee on Radio Astronomy Frequencies, 1, quai Lezay Marnésia BP 90015, F-67080 Strasbourg Cedex, France\n- 4 Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, 53121 Bonn, Germany\n- 5 Department of Physics and Electronics, Rhodes University, PO Box 94, Makhanda, 6140, South Africa\n- 6 LESIA, Observatoire de Paris, Université PSL, CNRS, 5 place Jules Janssen, 92195 Meudon, France \nReceived 9 August 2024; accepted 29 August 2024', 'ABSTRACT': 'We report on the detection of unintended electromagnetic radiation (UEMR) from the second-generation of Starlink satellites. Observations with the LOFAR radio telescope between 10 to 88 MHz and 110 to 188 MHz show broadband emission covering the frequency ranges from 40 to 70 MHz and 110 to 188 MHz from the v2-Mini and v2-Mini Direct-to-Cell Starlink satellites. The spectral power flux density of this broadband UEMR varies from satellite to satellite, with values ranging from 15 Jy to 1300 Jy, between 56 and 66 MHz, and from 2 to 100 Jy over two distinct 8 MHz frequency ranges centered at 120 and 161 MHz. We compared the detected power flux densities of this UEMR to that emitted by the first generation v1.0 and v1.5 Starlink satellites. When correcting for the observed satellite distances, we find that the second-generation satellites emit UEMR that is up to a factor of 32 stronger compared to the first generation. The calculated electric field strengths of the detected UEMR exceed typical electromagnetic compatibility standards used for commercial electronic devices as well as recommended emission thresholds from the Radiocommunication Sector of the International Telecommunications Union (ITU-R) aimed at protecting the 150.05-153 MHz frequency range allocated to radio astronomy. We characterize the properties of the detected UEMR with the aim of assisting the satellite operator with the identification of the cause of the UEMR. \nKey words. light pollution - space vehicles - telescopes - surveys', '1. Introduction': 'With the miniaturization of satellites and the rapid commercialization of spaceflight, the number of satellites in orbit around the Earth has dramatically increased since around 2016 (see McDowell 2020). Since then, several commercial companies have started to mass produce and launch large numbers of satellites to o ff er various communication services, primarily broadband internet and mobile connectivity. Satellites in such constellations are typically launched into shells of specific orbital altitude and orbital inclination to optimize coverage for specific geographic latitudes. \nThis growth in the number of satellites has worrisome implications for astronomy (Walker et al. 2020a,b, 2021) as the probability that satellites are passing through the fields of view of ground-based telescopes (as well as space-based) is drastically increasing, and sunlight reflected from these satellites, as well as radio signals emitted by them, are detectable by astronomical instruments (e.g., Tyson et al. 2020; Michałowski et al. 2021; Mróz et al. 2022; Kruk et al. 2023). Assessing the impact of these satellite constellations on di ff erent astronomical observatories and science cases is an increasingly important topic of recent research (e.g., Green et al. 2022; Bassa et al. 2022; Bar-', 'entine et al. 2023; Lang et al. 2023; Kovalev et al. 2023; Hainaut &Moehler 2024).': 'For radio astronomy, the use of the radio spectrum is regulated by the Radiocommunication Sector of the International Telecommunication Union (ITU-R), which publishes the relevant international treaty in the form of the Radio Regulations. These regulations cover the intentional use of the radio spectrum for di ff erent applications (or services) such as communication, remote sensing, navigation, as well as astronomy. It considers wanted and unwanted emission, for example that due to out-ofband emission from spectral side-lobes. In particular, the ITU-R allocates several frequency ranges to the radio astronomy service. Rec. ITU-R RA.769-2 provides thresholds on the received power (or power flux densities) that must not be exceeded by other active radio services in these bands. As these bands focus on spectral lines a ff ected by Galactic Doppler shifts, the frequency ranges are relatively narrow. \nIn Di Vruno et al. (2023) we introduced the concept of unintended electromagnetic radiation (UEMR), referring to any electromagnetic radiation that is radiated by (or leaking from) electrical devices and systems on board satellites, and not necessarily related to the generation and transmission of wanted electromagnetic radiation from antennas used for communication, for example. Through simulations of the aggregate e ff ect of UEMR emitted by satellites in various large satellite constellations, we \nfound that the radiation levels required to comply with ITU-R specified interference thresholds for frequency ranges assigned to radio astronomy, would be quite constraining when compared to typical electromagnetic compatibility standards used for commercial devices on Earth. Unfortunately, the detected levels of both narrowband and broadband UEMR at frequencies between 110 and 188 MHz of dozens of satellites belonging to the SpaceX Starlink constellation showed emissions above these calculated thresholds (Di Vruno et al. 2023). This detection of UEMR from the Starlink constellation has since been independently confirmed by Grigg et al. (2023). \nIn Di Vruno et al. (2023) we argued that while narrowband UEMR detected at 143.050 MHz could be attributed to reflections from the French GRAVES Space Surveillance Radar, other narrow- and broadband emissions were intrinsically radiated by the satellites. Doppler analysis of the narrowband emissions provides further evidence (Bassa et al., in prep.) of this intrinsic origin. Triggered by the initial detections of satellite UEMR we have initiated an observing program to investigate and characterize UEMR from satellites within di ff erent satellite constellations (e.g., OneWeb, IRIDIUM Next, Swarm, Planet Labs, BlueWalker), and di ff erent hardware versions of satellites within a constellation (Starlink). As part of these observations, and those triggered by a request (C. Lonsdale, priv. comm.) to confirm circumstantial evidence for possible UEMR detected in the EDGES epoch-of-reionization experiment (Bowman et al. 2008), we report on the detection of bright and broadband UEMR from satellites of the second generation of Starlink.', '2.1. Observations': "The observations reported here closely follow the observational setup used and detailed in Di Vruno et al. (2023), where satellites are detected by letting them pass through the telescope beam pattern. Two one-hour observations were obtained using the central six stations of the LOFAR radio telescope (van Haarlem et al. 2013) in the Netherlands on July 19, 2024, one covering frequencies from 10 to 88 MHz using the low-band antennas (LBAs) in the LBA\\_OUTER configuration, the other 110 to 188 MHz with the high-band antennas (HBAs). Signals from these stations were coherently added in the COBALT beamformer (Broekema et al. 2018) to form 91 tied-array beams (TABs), tiling out the primary station beam in hexagonal rings with separations near the TAB full width at half maximum (FWHM) of 42 ' for the LBA observation, and 24 ' for the HBA observation. For each tied-array beam, dynamic spectra with total intensity (Stokes I) were recorded at 41.94 ms time resolution and 12.2 kHz frequency resolution. To minimize the distance between the telescope and satellites passing through the beam pattern, the TABs for both observations tracked equatorial positions, which culminated near zenith (maximum elevation 87 · . 5) half-way through the one-hour integration.", '2.2. Analysis': 'The dynamic spectra were analyzed and searched for the presence of satellite UEMR using an adapted version of the method outlined in Di Vruno et al. (2023). First, we retrieved orbital elements in the form of two-line element sets (TLEs), from publicly available catalogs. These orbital elements are derived \nfrom observations by the United States Space Force (USSF). 1 This approach allows searching for the UEMR of any satellite in the USSF catalog, whereas in Di Vruno et al. (2023) we used ephemerides provided by SpaceX that are only available for Starlink satellites. Using these orbital elements, we used the Skyfield software to compute the predicted trajectory of each satellite through the beam pattern of both observations and determine ingress, midpoint, and egress times for the station beam and for each TAB that the satellite passes through. \nNext, for each satellite passing through the beam pattern, we extracted a time range, centered on the predicted midpoint of the passage, from the dynamic spectra of each of the 91 TABs. The width of this time range was chosen depending on the angular velocity of the satellite on the sky and the FWHM of the station beam, and varied from 12 to 40 s. Similarly, to increase the signal-to-noise ratio, the extracted dynamic spectra are averaged to a lower time resolution by a factor n bin from the native 41 . 94 ms time resolution. We ensured that the duration of the passage through a TAB is covered by at least four averaged time samples. \nThe extracted dynamic spectra of each TAB were bandpasscalibrated by normalizing with the median of the dynamic spectra of the TABs where the satellite did not pass through. This approach has the advantage that the e ff ect of low-level terrestrial radio frequency interference (RFI), which appears similar in power in all beams, is minimized. After this normalization step, small variations in intensity ( ∼ 1%) between the di ff erent tied-array beams remain, due to di ff erences in sky temperatures and astrophysical sources. We removed these by defining the onsource spectrum as the time range during which the satellite is predicted to be located inside the primary station beam, and the o ff -source spectrum as the time range it is outside of this area, and further normalizing the spectra by the median intensity of each channel for the o ff -source time range. \nWe used the o ff -source time range to determine the frequency dependent rms of the noise, and used this as input for the radiometer equation to flux-calibrate the normalized dynamic spectra. The radiometer equation relates the rms noise to the spectral flux density of each time and frequency bin in the dynamic spectra through the system temperature and the telescope gain, as well as the time ( t samp = 41 . 94 n bin ms) and frequency resolution ( ∆ ν = 12 . 2 kHz) and the number of polarizations recorded ( n pol = 2). To determine the frequency dependent system temperature and telescope gain for the LOFAR LBA and HBA observation, we used the method by Kondratiev et al. (2016), which models the e ff ective area, the station beam model, antenna and sky temperature, together with the beamformer coherency based on the number of stations used to form the tiedarray beams, as well as the sky pointing of the beams. This approach is the same as in Di Vruno et al. (2023), except for the frequency-dependent calibration, which is strictly necessary for the LBA data. \nFor each satellite, we used the predicted mid-point times of each beam through which it passed to align the flux-density calibrated dynamic spectra in time, and computed the beamaveraged dynamic spectrum to increase the signal-to-noise ratio and be more sensitive to faint UEMR. Since the satellites do not necessarily pass through the center of each tied-array beam, the average of the flux-density calibrated spectra will underestimate the actual flux density of the satellite compared to a satellite that actually passed through the center of each TAB. Hence, we used the angular response of a TAB (using the FWHM and approxi- \nmated with a Gaussian) to compute the weighting of all beams and used this to correct the flux density scale. This correction is frequency dependent since the TAB FWHM scales with observing frequency. \nThe resulting aligned and averaged dynamic spectra were inspected for the presence of both narrowband and broadband emission whose temporal width is consistent with the expected passage duration through the TAB FWHM. In cases where this emission is detected, the temporal profile is fitted with a Gaussian to represent the beam shape, and to provide power flux density measurements and time o ff sets due to the satellite running ahead or behind predictions. For narrowband emission, we subtracted the spectral baseline of the surrounding 0.5 MHz to provide measurements relative to broadband emission. To allow comparison with our earlier measurements from Di Vruno et al. (2023), we used the same narrowband frequencies (125, 135, 143.05, 150, and 175 MHz) and broadband frequency ranges (116-124, 150.05-153, and 157-165 MHz) for the HBA observation 2 . For the LBA observation, we chose narrowband frequencies at 25, 50, and 75 MHz to check for the presence of the harmonics spaced at 25 MHz intervals that were reported earlier by Di Vruno et al. (2023). Broadband frequency ranges correspond to the 37.5-38.25 MHz and 73-74.6 MHz frequency ranges where radio astronomy has a secondary and primary ITUR allocation (see Rec. ITU-R RA.769-2 and Radio Regulations, Vol. 1, footnote 5.149), respectively. We also include the 5054 MHz frequency range, which is assigned for radio amateur usage, and 56-61 MHz and 61-66 MHz, where the strongest signals are detected. \nThe figures in Appendix A show examples of aligned and averaged dynamic spectra for several satellites in the di ff erent observing bands, while the tables in Appendix B provide lists of the detected Starlink satellites, the properties of the passage through the LOFAR beam pattern, and the power flux density measurements.', '3. Results': "A total of 141 Starlink satellites were predicted to pass through at least one tied-array beam in the LBA observation covering 10 to 88 MHz, while 97 satellites did so for the HBA observation spanning 110 to 188 MHz. Publicly available information from Jonathan McDowell 3 and Gunther Krebs 4 indicates that the observed satellites are of four di ff erent versions currently in orbit. Satellite versions v1.0 and v1.5 are from the first generation of Starlink satellites, which were also observed in the 2022 observation presented in Di Vruno et al. (2023). The other two satellite versions are from the second generation of Starlink: the normal v2-Mini satellite and the direct-to-cell (DTC) version, which offers cellular mobile phone coverage. Orbital launches of the v2Mini satellite versions started in February 2023, initially into 43 · inclined orbits, and since August 2023 into orbits with 53 · inclination. Launches with the DTC type v2-Mini's started in January 2024 and are occasionally launched together with the normal v2Mini satellites. The DTC version of the v2-Mini satellites are currently all in 53 · inclined orbits. \nIn the HBA observation, all 97 Starlink satellites are detected, either through narrowband emission, predominantly at \n125 MHz, or broadband emission over parts of or most of the 110-188 MHz frequency range. The detected signals from the v1.0 and v1.5 satellite versions are consistent in flux density and spectral properties to those of the satellites we detected in the analysis of the 2022 observation in this frequency range (Di Vruno et al. 2023). The aligned and averaged dynamic spectra of the second-generation v2-Mini satellite versions appear distinctly di ff erent, as these do not show the narrowband emission at 125, 135, and 150 MHz, but instead show significantly brighter broadband UEMR than the v1.0 and v1.5 versions. \nAt the LBA frequencies between 10 and 88 MHz, UEMR from the v2-Mini and v2-Mini DTC versions is clearly detected for 27 of the 29 observed satellites; it is exceedingly bright, reaching power flux densities of hundreds of janskys, and in a few cases even exceeding 1 kJy. The emission is predominantly constrained to a ∼ 10 MHz band centered around 61 MHz, but in some cases the broadband emission is detectable down to frequencies of around 40 MHz. The broadband UEMR varies in brightness from satellite to satellite, and tends to peak at di ff erent frequencies between 60 and 64 MHz. The v1.0 and v1.5 version satellites are not detected in the LBA data, neither through broadband emission nor at the narrowband frequencies at 25, 50, and 75 MHz. In the few cases where signals were present in the aligned and averaged spectra, they could be explained by the strong UEMR from v2-Mini or v2-Mini DTC satellites that were also in or near the station beam at that time as a result of the increasing density of satellites in the sky. As the v1.0 and v1.5 satellite versions emit narrowband UEMR at 125, 150, and 175 MHz, we suggested in Di Vruno et al. (2023) that these may be harmonics of a 25 MHz clock signal on board the satellite, and that we would expect emission at the fundamental frequency of 25 MHz and the harmonics at 50 and 75 MHz. While the 25 MHz frequency is lost due to terrestrial RFI, the LBA observations show that if present, the narrow- or broadband UEMR at 50 and 75 MHz must have power flux densities below S ν < 10 Jy (3 σ ). \nFigure 1 provides an overview of the power flux density measurements at the di ff erent narrowband frequencies and broadband frequency ranges, their distances to the telescope at the time of detection, and the satellite versions. It is clear that the power flux densities of the v2-Mini and v2-Mini DTC Starlink satellites are higher than those of the first generation. However, the v2-Mini and v2-Mini DTC versions were observed at smaller distances as these satellites operate at lower orbital altitudes. To determine whether the UEMR emitted by the second generation of Starlink satellites is intrinsically brighter, we corrected the observed power flux densities by scaling them to a fixed distance of 1000 km, as shown in Fig. 2. This approach also has the advantage that these normalized power flux densities can be directly related to the electric field strength emitted by the satellites, if received by a detector at a distance of 10 m and integrating over a 120 kHz bandwidth. This implicitly assumes that the emitted UEMR is isotropic, which likely is not the case, but allows further comparison to commercial electromagnetic compatibility (EMC) standards that we used in Di Vruno et al. (2023). From Fig. 2 we find that intrinsic levels of broadband UEMR emitted by the observed satellites from the second generation of Starlink (the v2-Mini and v2-Mini DTC versions) are higher than those observed from the first-generation satellites. \nIn the LBA band, UEMR from the observed v2-Mini and v2Mini DTC satellites shows spectral structure over the entire frequency range where UEMR is detected. This structure consists of a 'comb' of regularly spaced peaks in frequency. Power spectra of this emission at the 12.2 kHz frequency resolution between 56 and 66 MHz show significant peaks at multiple, harmonically \nFig. 1. Distances and power flux density measurements of Starlink satellites that passed through the beam pattern of the two 1 hr LOFAR observations. The horizontal axis denotes the number of satellites that were observed, ordered by their NORAD catalog identifier. As this identifier sequentially increases with each launched satellite, this axis is essentially ordered in time. All satellites were observed near zenith, and hence their distances are comparable to their orbital altitudes. Power flux density measurements at the di ff erent narrowband frequencies or broadband frequency ranges are indicated with circles, where the size of the circle corresponds to the measured flux density. Nondetections are denoted by the GLYPH<5> symbol. The horizontal gray lines and bands indicate the frequency ranges that were used for the flux density determination. There are separate legends for the LBA band from 10-88 MHz and the HBA band from 110-188 MHz. The di ff erent Starlink satellite versions are indicated with di ff erent colors. \n<!-- image --> \nObserved satellite (ordered by NORAD ID) \nrelated, peaks at frequencies of 27.5, 36.66, 55, 110, and 220 kHz for the v2-Mini satellites, and at 37.5, 50, 75, and 150 kHz for the v2-Mini DTC satellites. This shows that this spacing is distinctly di ff erent between the v2-Mini and v2-Mini DTC satellite versions that were observed. Due to variations in the power of the spectral harmonics, we cannot identify a fundamental frequency of these combs. Spectral structure is less apparent for the UEMR detections in the HBA band. Power spectra over the two frequency ranges between 116-124 and 157-165 MHz show that some of the version v1.5 satellites observed have a comb with a fundamental frequency at 50 kHz in the 157-165 MHz band, similar to what was seen in our earlier observations of satellites of this version (Di Vruno et al. 2023). The v2-Mini satellites observed predominantly show periodic signals at frequencies of 48.8, 65, and 97.5 kHz in the 157 to 165 MHz band, while most of the v2-Mini DTC satellites show a comb with 50 or 150 kHz spacing in the lower band from 116 to 124 MHz. For these combs, no fundamental frequency can be identified due to the satellite-to-satellite power variations of the harmonics.", '4. Discussion and conclusions': "We find that the second generation of Starlink satellites that we observed with LOFAR emit higher levels of unintended electromagnetic radiation (UEMR) over a broader frequency range compared to that emitted by the first generation of Starlink satellites. Our observations show that in the 150.05-153 MHz primary radio astronomy band, the broadband UEMR of the second-generation v2-Mini and v2-Mini DTC satellites is, on \naverage, 15 dB and 7 dB brighter than that of the first-generation v1.0 and v1.5 Starlink satellites. On a linear scale, this corresponds to factors of 32 and 5, respectively. As is evident from Fig. 2, this trend is also present in the 116 to 124 MHz and 157 to 165 MHz bands, and in the frequency bands from 50 to 66 MHz where the satellites from the first generation are not detected. On the other hand, the strong narrowband UEMR that is seen in the v1.0 and v1.5 satellites at 125, 135, and 150 MHz appears to be absent in the v2-Mini and v2-Mini DTC satellites. While this is an improvement, it is completely negated by the stronger broadband UEMR, which a ff ects a significantly larger part of the observed frequency range. \nThe issue of the higher levels of UEMR from the secondgeneration Starlink satellites is further exacerbated by the lower orbits in which these satellites operate. These satellites are used in the (modified) Generation 2 Starlink constellation, for which the US Federal Communications Commission (FCC) has approved operational orbits at 448 and 482 km for the v2-Mini satellites, and 360 km for the v2-Mini DTC satellites. As a result of these lower orbits and resulting smaller distances to Earthbased telescopes, the signals will be 30 to 130% brighter compared to the Generation 1 Starlink constellation, which mostly operates at orbital altitudes of 550 km. \nIn Di Vruno et al. (2023), we used the ITU-R recommended equivalent power flux density (EPFD) method (Rec. ITUR M.1583-1; Rec. ITU-R S.1586-1) to simulate the aggregate impact of large numbers of satellites in several satellite constellations and estimate their compatibility with ITU-R recommended interference thresholds. For the Generation 1 Starlink \nS \n@ 1000 km (Jy) \nFig. 2. Comparison of the intrinsic UEMR power flux densities for different broadband frequency ranges per Starlink satellite version. Each panel shows a cumulative distribution function of the power flux density measurements scaled to a distance of 1000 km (top axis) or represented as the electric field strength measured by a detector at 10 m distance using 120 kHz of bandwidth (bottom axis). \n<!-- image --> \nconstellation of 4408 satellites in orbits at 550 km altitude, we found that the intrinsic electric field strength of an individual satellite had to remain below 11 . 7 dB [ µ Vm -1 ] to satisfy the -194 dB [W m -2 ] ITU-R threshold for the radio astronomy band of 150.05-153 MHz (Rec. ITU-R RA.769-2). The intrinsic broadband UEMR from observed Starlink v1.0 and v1.5 satellites from the 2022 observation had electric field strengths of 21 to 39 dB [ µ Vm -1 ], already significantly exceeding that limit. These values also exceed typical commercial EMC standards (e.g., 30 dB [ µ Vm -1 ] from CISPR, see discussion in Di Vruno et al. 2023). Given that the Generation 2 Starlink constellation will consist of even more satellites than the Generation 1 constellation, that these satellites will be operating at lower orbital altitudes, and that this constellation will consist of the v2-Mini and v2-Mini DTC satellites that are now found to emit even stronger UEMR, we can conclude that the Rec. ITU-R RA.7692 recommended interference threshold levels are exceeded even further in this radio astronomy band. The LOFAR observations presented here do not detect UEMR in the 73-74.6 MHz band allocated to radio astronomy. However, preliminary analysis of imaging observations of second-generation Starlink satellites with the NenuFAR telescope in France (Zarka et al. 2012), indicates that some may be detectable in that band (Zhang et al., in prep.). While they are beyond the scope of this paper, EPFD simulations of the Generation 2 Starlink constellation would be required to estimate the intrinsic electric field strengths required to keep the aggregate emission of UEMR of this constellation in accordance with the ITU-R recommendations. \nCurrent wording of the ITU-R radio regulations (Radio Regulations) and recommendations (e.g., Rec. ITU-R RA.769-2, and related recommendations) discuss emissions in terms of wanted and unwanted emission related to signal transmission, where the \nunwanted emission is a byproduct of the wanted emission, for example due to out-of-band emission in the spectral domain. The UEMR as defined in our earlier paper (Di Vruno et al. 2023) appears to fall outside of these regulations. As such, UEMR is not subject to the ITU-R interference limits which protect certain parts of the spectrum for radio astronomical applications. Hence, we reiterate our earlier recommendation that UEMR from satellites should be considered in the regulatory processes. \nThe impact of the observed UEMR on radio astronomy likely varies between di ff erent science cases. The first-order e ff ect will be a loss of sensitivity of low-frequency radio telescopes since the time and frequency ranges within an observation that are affected by satellite UEMR may have to be preemptively masked. However, given that low-frequency radio telescopes are primarily built for their large fields of view, the large numbers of satellites from current and future satellite constellations may lead to the situation that one or more satellites are present in the telescope's field of view at any given time. In this case, temporal masking of data will no longer provide useful data. This is the primary reason why broadband UEMR is particularly worrisome for radio astronomy; it increases the risk that the entire observing bandwidth is a ff ected by UEMR for the entire duration of the observation. A second-order e ff ect, primarily a ff ecting interferometric telescope arrays, is that for closely spaced array elements (parabolic dishes or antenna stations), satellites will appear at the same sky location. As a result, UEMR will not decorrelate on the shortest baselines between individual array elements and may introduce artifacts on large spatial scales. \nIn contrast with this situation, astronomical radio observatories put a great deal of e ff ort in mitigating their internally generated UEMR in all frequencies covered by their telescopes, as electrical devices needed to run telescopes are also prone to producing radio noise. Observatories such as LOFAR and the SKA Observatory go to great lengths, imposing extremely tight radio emission requirements on each of the subsystems that comprise the telescopes. In the SKA-Low case in Western Australia, some examples of this are the Central Processing Facility building, which is designed to shield all the computing equipment that performs the first processing of SKA-Low data, and the power and signal distribution boxes located in close proximity to SKA-Low antennas, having requirements that are more than 100 dB (10 10 times) stricter than commercial standards for radiated emissions. \nThe UEMR from equipment close to, but not associated with radio telescopes is a day-to-day reality. Like the UEMR from satellites, their spectra are generally tens to hundreds of MHz wide, and have a comb-like structure in addition to a more diffuse wide-band spectrum. This type of UEMR is typically handled in various ways, for example i) raising awareness with local stakeholders; ii) advocating for local, regional, and / or national protected geographic radio quiet zones; iii) establishing bilateral covenants requiring EMC limits stricter than typical levels specified in industry norms such as CISPR-32 and EN 55032 (e.g., wind and solar photovoltaic installations near the LOFAR core 5 ); iv) cooperation with equipment owners to mitigate at the source (mending electric fences, replacing LED lights, switching o ff CCTV cameras); and / or v) formal regulatory complaints and follow-up by national administrations in cases of noncooperative parties. The last step is generally only viable if the sources exceed EMC norms similar to CISPR-32 / EN55032. As far as we are aware, a similar regulatory or normative framework is \nlacking for space applications. In contrast to satellites, we note that all of these sources are generally greatly attenuated by the telescope as, unlike satellites, they are never directly in its main beam. \nIn the absence of regulations that address UEMR emission from satellites, the astronomical community will have to raise and address this issue with regulatory bodies as well as satellite operators, and must continue to do so. Fortunately, SpaceX / Starlink is already actively co-operating with both optical astronomy (e.g., Tyson et al. 2020) and radio astronomy (e.g., Nhan et al. 2024) to investigate and / or test mitigation strategies. Our observations and analysis, presenting properties of the UEMR (electric field strengths, emission frequencies, comb properties) of di ff erent satellite versions, possibly combined with those of other radio telescopes, may provide information that allows SpaceX / Starlink to identify the satellite components involved in the emission of UEMR and devise mitigation strategies in already operational satellites, as well as future designs of the hardware. \nAcknowledgements. We thank Colin Lonsdale for informing us of the presence of UEMR from the second-generation Starlink satellites. We acknowledge fruitful discussions with Jess Dempsey, Michiel van Haarlem and Wim van Cappellen. This paper is based on data obtained with the International LOFAR Telescope (ILT) under project code LC20\\_009. LOFAR (van Haarlem et al. 2013) is the Low Frequency Array designed and constructed by ASTRON. It has observing, data processing, and data storage facilities in several countries, that are owned by various parties (each with their own funding sources), and that are collectively operated by the ILT foundation under a joint scientific policy. The ILT resources have benefitted from the following recent major funding sources: CNRS-INSU, Observatoire de Paris and Université d'Orléans, France; BMBF, MIWF-NRW, MPG, Germany; Science Foundation Ireland (SFI), Department of Business, Enterprise and Innovation (DBEI), Ireland; NWO, The Netherlands; The Science and Technology Facilities Council, UK; Ministry of Science and Higher Education, Poland. The project leading to this publication has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 101004719. This paper made extensive use of the Python scientific software stack, and we acknowledge the developers of numpy (van der Walt et al. 2011), matplotlib (Hunter 2007), scipy (Jones et al. 2001), astropy (Astropy Collaboration et al. 2013, 2022) and Skyfield (Rhodes 2019).", 'References': '- Astropy Collaboration, Price-Whelan, A. M., Lim, P. L., et al. 2022, ApJ, 935,\n- 167 Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33 Barentine, J. C., Venkatesan, A., Heim, J., et al. 2023, Nature Astronomy, 7, 252 Bassa, C. G., Hainaut, O. R., & Galadí-Enríquez, D. 2022, A&A, 657, A75 Bowman, J. D., Rogers, A. E. E., & Hewitt, J. N. 2008, ApJ, 676, 1 Broekema, P. C., Mol, J. J. D., Nijboer, R., et al. 2018, Astronomy and Computing, 23, 180 CENELEC. 2015, EN55032:2015: Electromagnetic compatibility of multimedia equipment - Emission requirements, Tech. rep. Di Vruno, F., Winkel, B., Bassa, C. G., et al. 2023, A&A, 676, A75 Green, R. F., Luginbuhl, C. B., Wainscoat, R. J., & Duriscoe, D. 2022, A&A Rev., 30, 1 Grigg, D., Tingay, S. J., Sokolowski, M., et al. 2023, A&A, 678, L6 Hainaut, O. R. & Moehler, S. 2024, A&A, 683, A147 Hunter, J. 2007, Computing in Science Engineering, 9, 90 ITU-R. 2003, "Protection criteria used for radio astronomical measurements", Recommendation RA.769-2, International Telecommunication Union, Geneva ITU-R. 2007a, "Calculation of unwanted emission levels produced by a nongeostationary fixed-satellite service system at radio astronomy sites", Recommendation S.1586-1, International Telecommunication Union, Geneva ITU-R. 2007b, Interference calculations between non-geostationary mobilesatellite service or radionavigation-satellite service systems and radio astronomy telescope sites, Recommendation M.1583-1, International Telecommu-\n- nication Union, Geneva \nITU-R. 2020, "Radio Regulations" (Geneva: WRC-19 / Sharm el-Sheik) \n- Jones, E., Oliphant, T., Peterson, P., et al. 2001, SciPy: Open source scientific tools for Python\n- Kondratiev, V. I., Verbiest, J. P. W., Hessels, J. W. T., et al. 2016, A&A, 585, A128\n- Kovalev, M., Hainaut, O. R., Chen, X., & Han, Z. 2023, MNRAS, 525, L60\n- Kruk, S., García-Martín, P., Popescu, M., et al. 2023, Nature Astronomy, 7, 262 Lang, T., Spencer, S. T., & Mitchell, A. M. W. 2023, A&A, 677, A141 McDowell, J. C. 2020, ApJ, 892, L36\n- Michałowski, M. J., Kami\'nski, K., Kami\'nska, M. K., & Wnuk, E. 2021, Nature Astronomy, 5, 995\n- Mróz, P., Otarola, A., Prince, T. A., et al. 2022, ApJ, 924, L30\n- Nhan, B. D., De Pree, C. G., Iverson, M., et al. 2024, arXiv e-prints, arXiv:2407.21675\n- O ff ringa, A. R., de Bruyn, A. G., Zaroubi, S., et al. 2013, A&A, 549, A11\n- Rhodes, B. 2019, Skyfield: High precision research-grade positions for planets and Earth satellites generator, Astrophysics Source Code Library, record ascl:1907.024\n- Technical committee CISPR / CIS / I Electromagnetic compatibility of information technology equipment, multimedia equipment and receivers . 2015, CISPR32:2015: Electromagnetic compatibility of multimedia equipment Emission requirements, Tech. rep.\n- Tyson, J. A., Ivezi\'c, Ž., Bradshaw, A., et al. 2020, AJ, 160, 226\n- van der Walt, S., Colbert, S., & Varoquaux, G. 2011, Computing in Science Engineering, 13, 22\n- van Haarlem, M. P., Wise, M. W., Gunst, A. W., et al. 2013, A&A, 556, A2\n- Walker, C., Di Pippo, S., Aubé, M., et al. 2021, Dark & Quiet Skies II (2021), Dark & Quiet Skies II (2021), Report of the conference held 3-7 October, 2021.\n- Walker, C., Di Pippo, S., Aubé, M., et al. 2020a, Dark & Quiet Skies I (2020), Dark & Quiet Skies I (2020), Report of the conference held 5-9 October, 2020.\n- Walker, C., Hall, J., Allen, L., et al. 2020b, in Bulletin of the American Astronomical Society, Vol. 52, 0206\n- Zarka, P., Girard, J. N., Tagger, M., & Denis, L. 2012, in SF2A-2012: Proceedings of the Annual meeting of the French Society of Astronomy and Astrophysics, ed. S. Boissier, P. de Laverny, N. Nardetto, R. Samadi, D. VallsGabaud, & H. Wozniak, 687-694', 'Appendix A: Additional figures': 'This appendix shows figures of aligned and averaged dynamic spectra for Starlink satellites in which UEMR is detected. Figures A.1 and A.2 show detections for Starlink v2-Mini and v2-Mini DTC satellites in the LBA observing band between 10 and 88 MHz. In the HBA band covering 110 to 188 MHz, Figs. A.3 and A.4 show aligned and averaged dynamic spectra for a Starlink v1.5 and a v2-Mini satellite. The color scale of the dynamic spectra is in power flux density. \nThese dynamic spectra show the UEMR from Starlink satellites, as well as interference from terrestrial sources. Satellite UEMR will stand out because of its temporal signature as the satellite passes through the telescope field-of-view. For comparison, terrestrial RFI is primarily detected through the telescope side-lobes and hence has no specific, nor predictable, variation with time. \nTerrestrial RFI in the LOFAR band is primarily due to allocated services, and an overview of these is given in O ff ringa et al. (2013). In the LBA band from 10 to 88 MHz, these services predominantly a ff ect frequencies below 30 MHz, where beyond the horizon transmissions are detectable through reflections o ff the ionosphere. In the HBA band from 110 to 188 MHz, digital audio broadcasting channels continuously occupy frequencies in several 1.6 MHz wide bands from 174 MHz and higher, while air tra ffi c control (118 to 137 MHz), satellite downlinks (137 to 138 MHz) and amateur radio (144 to 146 MHz) use smaller bandwidths (a few kHz) and transmit for limited time periods (seconds to minutes). \n) \ny \nJ \n( \nS \nFig. A.1. Spectral and temporal properties of the passage of the Starlink v2-Mini satellite Starlink-31441 [60091 / 2024-117A] (average of 11 TABs) in the LBA band from 10 to 88 MHz. Normalized, aligned, and averaged dynamic spectra (in power flux density units) are shown over the entire observed bandwidth and are centered on the predicted passage time of the satellite. Time series at specific narrowband frequencies and broadband frequency ranges are shown in the top insets. The color of each time series matches the marked frequencies and frequency ranges, in the same colors as the sides of the dynamic spectra. \n<!-- image --> \nS \n(Jy)', 'A & A proofs: manuscript no. lofar\\_v2-mini': 'Fig. A.2. Spectral and temporal properties of the passage of the Starlink v2-Mini DTC satellite Starlink-11133 [DTC] [59954 / 2024-107K] (average of 11 TABs) in the LBA band from 10 to 88 MHz. \n<!-- image --> \nFig. A.3. Spectral and temporal properties of the passage of of Starlink v1.5 satellite Starlink-3349 [50813 / 2022-001L] (average of 11 TABs) in the HBA band from 110 to 188 MHz. \n<!-- image -->', 'Appendix B: Tables': 'Article number, page 8 of 12 \nC. G. Bassa et al.: Bright unintended electromagnetic radiation from second-generation Starlink satellites \n1.0 \nFig. A.4. Spectral and temporal properties of the passage of of Starlink v2-Mini satellite Starlink-30518 [58029 / 2023-156B] (average of 11 TABs) in the HBA band from 110 to 188 MHz. \n<!-- image --> \nTable B.1. Second-generation Starlink satellites that have been detected between 10 and 88 MHz. \nNotes. Satellites are identified by their NORAD catalog numer and the COSPAR international designator, which provides the launch year, launch number, and sequential identifier, and by their name. Satellites with [DTC] in their name are of the v2-Mini direct-to-cell (DTC) version; all other satellites are of the v2-Mini version. For each satellite the distance ( d in km), passage mid-point ( t mid in s) during the 1 hr observation (which started on July 19, 2024, at 06:00 UTC) and the number of tied-array beams the satellite passed through ( n TAB) are provided. Power flux density measurements ( S ν in Jy) are given for three frequency ranges. \nC. G. Bassa et al.: Bright unintended electromagnetic radiation from second-generation Starlink satellites \nTable B.2. First-generation Starlink satellites that have been detected between 110 and 188 MHz. \nNotes. Satellite versions v1.0 and v1.5 are as indicated. The column descriptions are otherwise identical to those in Table B.1. \nTable B.3. Second-generation Starlink satellites that have been detected between 110 and 188 MHz. \nNotes. Satellites with [DTC] in their name are of the v2-Mini direct-to-cell (DTC) version, all other satellites are of the v2-Mini version. The column descriptions are otherwise identical to those of Table B.1. The passage mid-point times t mid are with respect to the observation start time of July 19, 2024, at 07:30 UTC.'}

2013A&A...558A..33A

We present the first public version v0.2 of the opensource and communitydeveloped Python package Astropy. This package provides core astronomyrelated functionality to the community including support for domainspecific file formats such as flexible image transport system FITS files Virtual Observatory VO tables and common ASCII table formats unit and physical quantity conversions physical constants specific to astronomy celestial coordinate and time transformations world coordinate system WCS support generalized containers for representing gridded as well as tabular data and a framework for cosmological transformations and conversions. Significant functionality is under activedevelopment such as a model fitting framework VO client and server tools and aperture and point spread function PSF photometry tools. The core development team is actively making additions and enhancements to the current code base and we encourage anyone interested to participate in the development of future Astropy versions.

2013-10-01T00:00:00Z

['10.48550/arXiv.1307.6212', 'arXiv:1307.6212', '10.1051/0004-6361/201322068', '2013arXiv1307.6212T', '2013A&A...558A..33A']

['methods: data analysis', 'methods: miscellaneous', 'virtual observatory tools', 'Astrophysics - Instrumentation and Methods for Astrophysics']

Astropy A community Python package for astronomy

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']

https://arxiv.org/pdf/1307.6212.pdf

{'Astropy: A Community Python Package for Astronomy': "The Astropy Collaboration, Thomas P. Robitaille 1 , Erik J. Tollerud 2 , 3 , Perry Greenfield 4 , Michael Droettboom 4 , Erik Bray 4 , Tom Aldcroft 5 , Matt Davis 4 , Adam Ginsburg 6 , Adrian M. Price-Whelan 7 , Wolfgang E. Kerzendorf 8 , Alexander Conley 6 , Neil Crighton 1 , Kyle Barbary 9 , Demitri Muna 10 , Henry Ferguson 4 , Fr'ed'eric Grollier 12 , Madhura M. Parikh 11 , Prasanth H. Nair 12 , Hans M. Gunther 5 , Christoph Deil 13 , Julien Woillez 14 , Simon Conseil 15 , Roban Kramer 16 , James E. H. Turner 17 , Leo Singer 18 , Ryan Fox 12 , Benjamin A. Weaver 19 , Victor Zabalza 13 , Zachary I. Edwards 20 , K. Azalee Bostroem 4 , D. J. Burke 5 , Andrew R. Casey 21 , Steven M. Crawford 22 , Nadia Dencheva 4 , Justin Ely 4 , Tim Jenness 23 , 24 , Kathleen Labrie 25 , Pey Lian Lim 4 , Francesco Pierfederici 4 , Andrew Pontzen 26 , 27 , Andy Ptak 28 , Brian Refsdal 5 , Mathieu Servillat 29 , 5 , and Ole Streicher 30 \n- 1 Max-Planck-Institut fur Astronomie, Konigstuhl 17, Heidelberg 69117, Germany\n- 2 Department of Astronomy, Yale University, P.O. Box 208101, New Haven, CT 06510, USA\n- 3 Hubble Fellow\n- 4 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA\n- 5 Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA, 02138, USA\n- 6 Center for Astrophysics and Space Astronomy, University of Colorado, Boulder, CO 80309, USA\n- 7 Department of Astronomy, Columbia University, Pupin Hall, 550W 120th St., New York, NY 10027, USA\n- 8 Department of Astronomy and Astrophysics, University of Toronto, 50 Saint George Street, Toronto, ON M5S3H4, Canada\n- 9 Argonne National Laboratory, High Energy Physics Division, 9700 South Cass Avenue, Argonne, IL 60439, USA\n- 10 Department of Astronomy, Ohio State University, Columbus, OH 43210, USA\n- 11 S.V.National Institute of Technology, Surat., India\n- 12 Independent developer\n- 13 Max-Planck-Institut fur Kernphysik, P.O. Box 103980, 69029 Heidelberg, Germany\n- 14 European Southern Observatory, Karl-Schwarzschild-Str. 2, 85748, Garching bei Munchen, Germany\n- 15 Laboratoire d'Astrophysique de Marseille, OAMP, Universit'e Aix-Marseille et CNRS, Marseille, France\n- 16 ETH Zurich, Institute for Astronomy, Wolfgang-Pauli-Strasse 27, Building HIT, Floor J, CH-8093 Zurich, Switzerland\n- 17 Gemini Observatory, Casilla 603, La Serena, Chile\n- 18 LIGO Laboratory, California Institute of Technology, 1200 E. California Blvd., Pasadena, CA, 91125, USA\n- 19 Center for Cosmology and Particle Physics, New York University, New York, NY 10003, USA\n- 20 Department of Physics and Astronomy, Louisiana State University, Nicholson Hall, Baton Rouge, LA 70803, USA\n- 21 Research School of Astronomy and Astrophysics, Australian National University, Mount Stromlo Observatory, via Cotter Road, Weston Creek ACT 2611, Australia\n- 22 SAAO, P.O. Box 9, Observatory 7935, Cape Town, South Africa\n- 23 Joint Astronomy Centre, 660 N. A'oh¯ok¯u Place, Hilo, HI 96720, USA\n- 24 Department of Astronomy, Cornell University, Ithaca, NY 14853, USA\n- 25 Gemini Observatory, 670 N. A'oh¯ok¯u Place, Hilo, Hawaii 96720, USA\n- 26 Oxford Astrophysics, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, UK\n- 27 Department of Physics and Astronomy, University College London, London WC1E 6BT, UK\n- 28 NASA Goddard Space Flight Center, X-ray Astrophysics Lab Code 662, Greenbelt, MD 20771, USA\n- 29 Laboratoire AIM, CEA Saclay, Bat. 709, 91191 Gif-sur-Yvette, France\n- 30 Leibniz-Institut fur Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482 Potsdam, Germany \nPreprint online version: July 25, 2013", 'ABSTRACT': 'We present the first public version (v0.2) of the open-source and community-developed Python package, Astropy. This package provides core astronomy-related functionality to the community, including support for domain-specific file formats such as Flexible Image Transport System (FITS) files, Virtual Observatory (VO) tables, and common ASCII table formats, unit and physical quantity conversions, physical constants specific to astronomy, celestial coordinate and time transformations, world coordinate system (WCS) support, generalized containers for representing gridded as well as tabular data, and a framework for cosmological transformations and conversions. Significant functionality is under active development, such as a model fitting framework, VO client and server tools, and aperture and point spread function (PSF) photometry tools. The core development team is actively making additions and enhancements to the current code base, and we encourage anyone interested to participate in the development of future Astropy versions. \nKey words. Methods: data analysis - Methods: miscellaneous - Virtual Observatory Tools', '1. Introduction': "The Python programming language 1 has become one of the fastest-growing programming languages in the astronomy community in the last decade (see e.g. Greenfield 2011 for a recent review). While there have been a number of efforts to develop Python packages for astronomy-specific functionality, these efforts have been fragmented, and several dozens of packages have been developed across the community with little or no coordination. This has led to duplication and a lack of homogeneity across packages, making it difficult for users to install all the required packages needed in an astronomer's toolkit. Because a number of these packages depend on individual or small groups of developers, packages are sometimes no longer maintained, or simply become unavailable, which is detrimental to long-term research and reproducibility. \nMotivated by these issues, the Astropy project was started in 2011 out of a desire to bring together developers across the field of astronomy in order to coordinate the development of a common set of Python tools for astronomers and simplify the landscape of available packages. The project has grown rapidly, and to date, over 200 individuals are signed up to the development mailing list for the Astropy project. 2 \nOne of the primary aims of the Astropy project is to develop a core astropy package that covers much of the astronomy-specific functionality needed by researchers, complementing more general scientific packages such as NumPy (Oliphant 2006; Van Der Walt et al. 2011) and SciPy (Jones et al. 2001), which are invaluable for numerical array-based calculations and more general scientific algorithms (e.g. interpolation, integration, clustering). In addition, the Astropy project includes work on more specialized Python packages (which we call affiliated packages ) that are not included in the core package for various reasons: for some the functionality is in early stages of development and is not robust; the license is not compatible with Astropy; the package includes large files; or the functionality is mature, but too domain-specific to be included in a core package. \nThe driving interface design philosophy behind the core package is that code using astropy should result in concise and easily readable code, even by those new to Python. Typical operations should appear in code similar to how they would appear if expressed in spoken or written language. Such an interface results in code that is less likely to contain errors and is easily understood, enabling astronomers to focus more of their effort on their science objectives rather than interpreting obscure function or variable names or otherwise spending time trying to understand the interface. \nIn this paper, we present the first public release (v0.2) of the astropy package. We provide an overview of the current capabilities ( § 2), our development workflow ( § 3), and planned functionality ( § 4). This paper is not intended to provide a detailed documentation for the package (which is available online 3 ), but is rather intended to give an overview of the functionality and design.", '2. Capabilities': 'This section provides a broad overview of the capabilities of the different astropy sub-packages, which covers units and unit conversions ( § 2.1), absolute dates and times ( § 2.2), celestial coordinates ( § 2.3), tabular and gridded data ( § 2.4), common astronomical file formats ( § 2.5), world coordinate system transformations ( § 2.6), and cosmological utilities ( § 2.7). We have illustrated each section with simple and concise code examples, but for more details and examples, we refer the reader to the online documentation. 3', '2.1. Units, Quantities, and Physical Constants': 'The astropy.units sub-package provides support for physical units. It originates from code in the pynbody package (Pontzen et al. 2013), but has been significantly enhanced in behavior and implementation (with the intent that the pynbody will eventually become interoperable with astropy.units ). This sub-package can be used to attach units to scalars and arrays, convert from one set of units to another, define custom units, define equivalencies for units that are not strictly the same (such as wavelength and frequency), and decompose units into base units. Unit definitions are included in both the International System of Units (SI) and the Centimeter-Gram-Second (CGS) systems, as well as a number of astronomy- and astrophysics-specific units.', '2.1.1. Units': "The astropy.units sub-package defines a Unit class to represent base units, which can be manipulated without attaching them to values, for example to determine the conversion factor from one set of units to another. Users can also define their own units, either as standalone base units or by composing other units together. It is also possible to decompose units into their base units, or alternatively search for higher-level units that are identical. \nThis sub-package includes the concept of 'equivalencies' in units, which is intended to be used where there exists an equation that provides a relationship between two different physical quantities. A standard astronomical example is the relationships between the frequency, wavelength and energy of a photon - it is common practice to treat such units as equivalent even though they are not strictly comparable. Such a conversion can be carried out in astropy.units by supplying an equivalency list (see Figure 1). The inclusion of these equivalencies is an important improvement over existing unit-handling software, which typically does not have this functionality. Equivalencies are also included for monochromatic flux densities, which allows users to convert between F ν and F λ , and users can easily implement their own equivalencies. \nThere are multiple string representations for units used in the astronomy community. The FITS Standard (Pence et al. 2010) defines a unit standard, as well as both the Centre de Donn'ees astronomiques de Strasbourg (CDS) (Ochsenbein 2000) and NASA/Goddard's Office of Guest Investigator Programs (OGIP) (George & Angelini 1995). In addition, the International Virtual Observatory Alliance (IVOA) has a forthcoming VOUnit standard (Derriere et al. 2012) in an attempt to resolve some of these differences. Rather than choose one of these, astropy.units supports \nFig. 1. Quantity conversion using the astropy.units subpackage. Fig. 2. Using the astropy.constants sub-package. \n```\nDefine a quantity from scalars and units: >>> from astropy import units as u >>> 15.1 * u.m / u.s <Quantity 15.1 m / (s)> Convert a distance: >>> (1.15e13 * u.km).to(u.pc) <Quantity 0.372689618289 pc> Make use of the unit equivalencies: >>> e = 130. * u.eV >>> e.to(u.Angstrom, equivalencies=u.spectral()) <Quantity 95.3724560923 Angstrom> Combine quantities: >>> x = 1.4e11 * u.km / (0.7 * u.Myr) >>> x <Quantity 2e+11 km / (Myr)> Convert to SI and CGS units: >>> x.si <Quantity 6.33761756281 m / (s)> >>> x.cgs <Quantity 633.761756281 cm / (s)> Use units with NumPy arrays >>> import numpy as np >>> d = np.array([1, 2, 3, 4]) * u.m >>> d.to(u.cm) <Quantity [ 100. 200. 300. 400.] cm> >>> d.to(u.cm) * 1. / 50. * u.s ** -1 <Quantity [ 2. 4. 6. 8.] cm / (s)>\n``` \nmost of these standards (OGIP support is planned for the next major release of astropy ), and allows the user to select the appropriate one when reading and writing unit string definitions to and from external file formats.", '2.1.2. Quantities and Physical Constants': 'While the previous section described the use of the astropy.units sub-package to manipulate the units themselves, a more common use-case is to attach the units to quantities, and use them together in expressions. The astropy.units package allows units to be attached to Python scalars, or NumPy arrays, producing Quantity objects. These objects support arithmetic with other numbers and Quantity objects while preserving their units. For multiplication and division, the resulting object will retain all units used in the expression. The final object can then be converted to a specified set of units or decomposed, effectively canceling and combining any equivalent units and returning a Quantity object in some set of base units. This is demonstrated in Figure 1. \nUsing the .to() method, Quantity objects can easily be converted to different units. The units must either be dimensionally equivalent, or users should pass equivalencies through the equivalencies argument (c.f. § 2.1.1 or Figure 1). Since Quantity objects can operate with NumPy arrays, it is very simple and efficient to convert the units on large datasets. \n```\nAccess physical constants: >>> from astropy import units as u >>> from astropy import constants as c >>> print c.G Name = Gravitational constant Value = 6.67384e-11 Error = 8e-15 Units = m3 / (kg s2) Reference = CODATA 2010 Combine quantities and constants: >>> F = (c.G * (3 * c.M\\_sun) * (2 * u.kg) / ... (1.5 * u.au) ** 2) >>> F.to(u.N) <Quantity 0.0158179542881 N>\n``` \nThe Quantity objects are used to define a number of useful astronomical constants included in astropy.constants , each with an associated unit (where applicable) and additional metadata describing their provenance and uncertainties. These can be used along with Quantity objects to provide a convenient framework for computing any quantity in astronomy. Figure 2 includes a simple example that shows how the gravitational force between two bodies can be calculated in Newtons using physical constants and user-specified quantities.', '2.2. Time': 'The astropy.time package provides functionality for manipulating times and dates. Specific emphasis is placed on supporting time scales (e.g. UTC, TAI, UT1) and time formats or representations (e.g. JD, MJD, ISO 8601) that are used in astronomy (Guinot & Seidelmann 1988; Kovalevsky 2001; Wallace 2011). Examples of using this sub-package are provided in Figure 3. \nThe most common way to use astropy.time is to create a Time object by supplying one or more input time values as well as the time format or representation and time scale of those values. The input time(s) can either be a single scalar such as "2010-01-01 00:00:00" or 2455348.5 or a sequence of such values; the format or representation specifies how to interpret the input values, such as ISO, JD, or Unix time; and the scale specifies the time standard used for the values, such as Coordinated Universal Time (UTC), Terrestial Time (TT), or International Atomic Time (TAI). The full list of available time scales is given in Table 1. Many of these formats and scales are used within astronomy, and it is especially important to treat the different time scales properly when converting between celestial coordinate systems. To facilitate this, the Time class makes the conversion to a different format such as Julian Date straightforward, as well as the conversion to a different time scale, for instance from UTC to TT. We note that the Time class includes support for leap seconds in the UTC time scale. \nThis package is based on a derived version of the Standards of Fundamental Astronomy (SOFA) time and calendar library 4 (Wallace 2011). Leveraging the robust \nTable 1. Supported time scales for astropy.timeFig. 3. Time representation and conversion using the astropy.time sub-package. \n<!-- image --> \nand well-tested SOFA routines ensures that the fundamental time scale conversions are being computed correctly. An important feature of the SOFA time library which is supported by astropy.time is that each time is represented as a pair of double-precision (64-bit) floating-point values, which enables extremely high precision time computations. Using two 64-bit floating-point values allows users to represent times with a dynamic range of 30 orders of magnitude, providing for example times accurate to better than a nanosecond over timescales of tens of Gyr. All time scale conversions are done by vectorized versions of the SOFA routines using Cython (Behnel et al. 2011), a Python package that makes it easy to use C code in Python.', '2.3. Celestial Coordinates': 'An essential element of any astronomy workflow is the manipulation, parsing, and conversion of astronomical coordinates. This functionality is provided in Astropy by the astropy.coordinates sub-package. The aim of this package is to provide a common application programming interface (API) for Python astronomy packages that use coordinates, and to relieve users from having to (re)implement extremely common utilities. To achieve this, it combines API and implementation ideas from existing Python coordinates packages. Some aspects, such as coordinate transformation \napproaches from kapteyn (Terlouw & Vogelaar 2012) and class structures resembling astropysics (Tollerud 2012), have already been implemented. Others, such as the frames of palpy (Jenness & Berry 2013) and pyast (Berry & Jenness 2012) or the ephemeris system of pyephem (Rhodes 2011), are still under design for astropy . By combining the best aspects of these other packages, as well as testing against them, astropy.coordinates seeks to provide a high-quality, flexible Python coordinates library. \nThe sub-package has been designed to present a natural Python interface for representing coordinates in computations, simplify input and output formatting, and allow straightforward transformation between coordinate systems. It also supports implementation of new or custom coordinate systems that work consistently with the built-in systems. A future design goal is also to seamlessly support arbitrarily large data sets. \nTo that end, Figure 4 shows some typical usage examples for astropy.coordinates . Coordinate objects are created using standard Python object instantiation via a Python class named after the coordinate system (e.g., ICRSCoordinates ). Astronomical coordinates may be expressed in a myriad of ways: the classes support string, numeric, and tuple value specification through a sophisticated input parser. A design goal of the input parser is to be able to determine the angle value and unit from the input alone if a person can unambiguously determine them. For example, an astronomer seeing the input string \'12h53m11.5123s\' would understand the units to be in hours, minutes, and seconds, so this value is alone sufficient to pass to the angle initializer. This functionality is built around the Angle object, which can be instantiated and used on its own. It provides additional functionality such as string formatting and mechanisms to specify the valid bounds of an angle. As a convenience, it is also possible to query the online SIMBAD 5 database to resolve the name of a source (see Figure 4 for an example showing how to find the ICRS coordinates of M32). \nThe coordinate classes represent different coordinate systems, and provide most of the user-facing functionality for astropy.coordinates . The systems provide customized initializers and appropriate formatting and representation defaults. For some classes, they also contain added functionality specific to a subset of systems, such as code to precess a coordinate to a new equinox. The implemented systems include a variety of equatorial coordinate systems (ICRS, FK4, and FK5), Galactic coordinates, and horizontal (Alt/Az) coordinates, and modern (IAU 2006/200A) precession/nutation models for the relevant systems. Coordinate objects can easily be transformed from one coordinate system to another: Figure 4 illustrates the most basic use of this functionality to convert a position on the sky from ICRS to Galactic coordinates. Transformations are provided between all coordinate systems built into version v0.2 of Astropy, with the exception of conversions from celestial to horizontal coordinates. Future versions of Astropy will include additional common systems, including ecliptic systems, supergalactic coordinates, and all necessary intermediate coordinate systems for the IAU 2000/2006 equatorial-to-horizontal mapping (e.g., Soffel et al. 2003; Kaplan 2005). \nFig. 4. Celestial coordinate representation and conversion. \n```\nParse coordinate string >>> import astropy.coordinates as coords >>> c = coords.ICRSCoordinates("00h42m44.3s +41d16m9s") Access the RA/Dec values >>> c.ra <RA 10.68458 deg> >>> c.dec <Dec 41.26917 deg> >>> c.ra.degrees 10.68458333333333 >>> c.ra.hms (0.0, 42, 44.299999999999784) Convert to Galactic coordinates >>> c.galactic.l <Angle 121.17431 deg> >>> c.galactic.b <Angle -21.57280 deg> Create a separate object in Galactic coordinates >>> from astropy import units as u >>> g = c.transform\\_to(coords.GalacticCoordinates) >>> g.l.format(u.degree, sep=":", precision=3) \'121:10:27.499\' Set the distance and view the cartesian coordinates >>> c.distance = coords.Distance(770., u.kpc) >>> c.x 568.7128882165681 >>> c.y 107.30093596881028 >>> c.z 507.8899092486349 Query SIMBAD to get coordinates from object names >>> m = coords.ICRSCoordinates.from\\_name("M32") >>> m <ICRSCoordinates RA=10.67427 deg, Dec=40.86517 deg> Two coordinates can be used to get distances >>> m.distance = coords.Distance(765., u.kpc) >>> m.separation\\_3d(c) <Distance 7.36865 kpc>\n``` \nA final significant feature of astropy.coordinates is support for line-of-sight distances. While the term \'celestial coordinates\' can be taken to refer to only on-sky angles, in astropy.coordinates a coordinate object is conceptually treated as a point in three dimensional space. Users have the option of specifying a line of sight distance to the object from the origin of the coordinate system (typically the origin is the Earth or solar system barycenter). These distances can be given in physical units or as redshifts. The astropy.coordinates sub-package will in the latter case transparently make use of the cosmological calculations in astropy.cosmology (c.f. § 2.7) for conversion to physical distances. Figure 4 illustrates an application of this information in the form of computing three-dimensional distances between two objects. \nThe astropy.coordinates sub-package was designed such that it should be easy for a user to add new coordinate systems. This flexibility is achieved in astropy.coordinates through the internal use of a transformation graph, which keeps track of a network of coordi- \nFig. 5. Table input/output and manipulation using the astropy.table sub-package. \n```\nCreate an empty table and add columns >>> from astropy.table import Table, Column >>> t = Table() >>> t.add\\_column(Column(data=["a", "b", "c"], ... name="source")) >>> t.add\\_column(Column(data=[1.2, 3.3, 5.3], ... name="flux")) >>> print t source flux ------ ----a 1.2 b 3.3 c 5.3 Read a table from a file >>> t1 = Table.read("catalog.vot") >>> t1 = Table.read("catalog.tbl", format="ipac") >>> t1 = Table.read("catalog.cds", format="cds") Select all rows from t1 where the flux column is greater than 5 >>> t2 = t1[t1["flux"] > 5.0] Manipulate columns >>> t2.remove\\_column("J\\_mag") >>> t2.rename\\_column("Source", "sources") Write a table to a file >>> t2.write("new\\_catalog.hdf5", path=\'/table\') >>> t2.write("new\\_catalog.rdb") >>> t2.write("new\\_catalog.tex")\n``` \nnate systems and the transformations between them. When a coordinate object is to be transformed from one system into another, the package determines the shortest path on the transformation graph to the new system and applies the necessary sequence of transformations. Thus, implementing a new coordinate system simply requires implementing one pair of transformations to and from a system that is already connected to the transformation graph. Once this pair is specified, astropy.coordinates can transform from that coordinate system to any other in the graph. An example of a user-defined system is provided in the documentation, 6 illustrating the definition of a coordinate system useful for a specific scientific task (Price-Whelan & Johnston 2013, in prep).', '2.4. Tables and Gridded data': 'Tables and n -dimensional data arrays are the most common forms of data encountered in astronomy. The Python community has various solutions for tables, such as NumPy structured arrays or DataFrame objects in Pandas (McKinney 2012) to name only a couple. For n-dimensional data the NumPy ndarray is the most popular. \nHowever, for use in astronomy all of these implementations lack some key features. The data that is stored in arrays and tables often contains vital metadata: the data is associated with units, and might also contain additional arrays that either mask or provide additional attributes \nto each cell. Furthermore, the data often includes a set of keyword-value pairs and comments (such as FITS headers). Finally, the data comes in a plethora of astronomy specific formats (FITS, specially formatted ASCII tables, etc.), which are not recognized by the pre-existing packages. \nThe astropy.table and astropy.nddata sub-packages contain classes ( Table and NDData ) that try to alleviate these problems. They allow users to represent astronomical data in the form of tables or n-dimensional gridded datasets, including all metadata. Examples of usage of astropy.table are shown in Figure 5. \nThe Table class provides a high-level wrapper to NumPy structured arrays, which are essentially arrays that have fields (or columns) with heterogeneous data types, and any number of rows. NumPy structured arrays are however difficult to modify, so the Table class makes it easy for users to create a table from columns, add and remove columns or rows, and mask values from the table. Furthermore, tables can be easily read from and written to common file formats using the Table.read and Table.write methods. These methods are connected to sub-packages in astropy.io such as astropy.io.ascii ( § 2.5.2) and astropy.io.votable ( § 2.5.3), which allow ASCII and VO tables to be seamlessly read or written respectively. \nIn addition to providing easy manipulation and input or output of table objects, the Table class allows units to be specified for each column using the astropy.units framework ( § 2.1), and also allows the Table object to contain arbitrary metadata (stored in Table.meta ). \nSimilarly, the NDData class provides a way to store n -dimensional array data easily and builds upon the NumPy ndarray class. The actual data is stored in an ndarray , which allows for easy compatibility with other scientific packages. In addition to keyword-value metadata, the NDData class can store a boolean mask with the same dimensions as the data, several sets of flags ( n -dimensional arrays that store attributes for each cell of the data array), uncertainties, units, and a transformation between arrayindex coordinate system and other coordinate systems (c.f. § 2.6). In addition, the NDData class intends to provide methods to arithmetically combine the data in a meaningful way. NDData is not meant for direct user interaction but more for providing a framework for higher-level subclasses that can represent for example spectra or astronomical images.', '2.5.1. FITS': "Support for reading and writing FITS files is provided by the astropy.io.fits sub-package, which at the time of writing is a direct port of the PyFITS 7 project (Barrett & Bridgman 1999). Users already familiar with PyFITS will therefore feel at home with this package. \nThe astropy.io.fits sub-package implements all features from the FITS standard (Pence et al. 2010) such as images, binary tables, and ASCII tables, and includes common compression algorithms. Header-data units (HDUs) are represented by Python classes, with the data itself stored using NumPy arrays, and with the headers stored using a Header class. Files can easily be read and written, \nFig. 6. Accessing data in FITS format. \n<!-- image --> \nand once in memory can be easily modified. This includes support for transparently reading from and writing to gzipcompressed FITS files. Figure 6 shows a simple example of how to open an existing FITS file, access and modify the header and data, and write a new file back to disk. \nCreating new FITS files is also made simple. Since the code in this sub-package has been developed over more than a decade, it has been made to work with an extensive variety of FITS files, including ones that deviate from the FITS standard. This includes support for deprecated formats such as GROUPS HDUs as well as more obscure nonstandard HDU types such as FITS HDUs which allow encapsulating multiple FITS files within FITS files. Support is also included for common but non-standard header conventions such as CONTINUE cards and ESO HIERARCH cards. Two command-line utilities for working with FITS files are packaged with Astropy: fitscheck can be used to validate FITS files against the standard. fitsdiff can be used to compare two FITS files on a number of criteria, and also includes a powerful API for programmatically comparing FITS files. \nBecause the interface is exactly the same as that of PyFITS, users may directly replace PyFITS with Astropy in existing code by changing import statements such as import pyfits to from astropy.io import fits as pyfits without any additional code changes. Although PyFITS will continue to be released as a separate package in the near term, the long term plan is to discontinue PyFITS releases in favor of Astropy. It is expected that direct support of PyFITS will end mid-2014, so users of PyFITS should plan to make suitable changes to support the eventual transition to Astropy. \nBecoming integrated with Astropy as the astropy.io.fits sub-package will greatly enhance future development on the existing PyFITS code base in \nseveral areas. First and perhaps foremost is integration with Astropy's Table interface ( § 2.4) which is much more flexible and powerful than PyFITS' current table interface. We will also be able to integrate Astropy's unit support ( § 2.1) in order to attach units to FITS table columns as well as header values that specify units in their comments in accordance with the FITS standard. Finally, as the PyWCS package has also been integrated into Astropy as astropy.wcs ( § 2.6) tighter association between data from FITS files and their world coordinate system (WCS) will be possible.", '2.5.2. ASCII table formats': 'The astropy.io.ascii sub-package (formerly the standalone project asciitable 8 ) provides the ability to read and write tabular data for a wide variety of ASCIIbased formats. In addition to generic formats such as space-delimited, tab-delimited or comma-separated values, astropy.io.ascii provides classes for specialized table formats such as CDS, 9 IPAC, 10 IRAF DAOphot (Stetson 1987), and LaTeX. Also included is a flexible class for handling a wide variety of fixed-width table formats. Finally, this sub-package is designed to be extensible, making it easy for users to define their own readers and writers for any other ASCII formats.', '2.5.3. Virtual Observatory tables': 'The astropy.io.votable sub-package (formerly the standalone project vo.table ) provides full support for reading and writing VOTable format files versions 1.1, 1.2, and the proposed 1.3 (Oschenbein et al. 2004, 2009). It efficiently stores the tables in memory as NumPy structured arrays. The file is read using streaming to avoid reading in the entire file at once and greatly reducing the memory footprint. VOTable files compressed using the gzip and bzip2 algorithms are supported transparently, as are VOTable files where the table data is stored in an external FITS file. \nIt is possible to convert any one of the tables in a VOTable file to a Table object ( § 2.4), where it can be edited and then written back to a VOTable file without any loss of data. \nThe VOTable standard is not strictly adhered to by all VOTable file writers in the wild. Therefore, astropy.io.votable provides a number of tricks and workarounds to support as many VOTable sources as possible, whenever the result would not be ambiguous. A validation tool ( volint ) is also provided that outputs recommendations to improve the standard compliance of a given file, as well as validate it against the official VOTable schema.', '2.6. World Coordinate Systems': "The astropy.wcs sub-package contains utilities for managing World Coordinate System (WCS) transformations in FITS files. These transformations map the pixel locations in an image to their real-world units, such as their position on the celestial sphere. This library is specific to WCS as it relates to FITS as described in the FITS WCS papers \n(Greisen & Calabretta 2002; Calabretta & Greisen 2002; Greisen et al. 2006) and is distinct from a planned Astropy package that will handle WCS transformations in general, regardless of their representation. \nThis sub-package is a wrapper around the wcslib library. 11 Since all of the FITS header parsing is done using wcslib , it is assured the same behavior as the many other tools that use wcslib . On top of the basic FITS WCS support, it adds support for the Simple Imaging Polynomial (SIP) convention and table lookup distortions (Calabretta et al. 2004; Shupe et al. 2005). Each of these transformations can be used independently or together in a fixed pipeline. The astropy.wcs sub-package also serves as a useful FITS WCS validation tool, as it is able to report on many common mistakes or deviations from the standard in a given FITS file. \nAs mentioned above, the long-term plan is to build a 'generalized' WCS for mapping world coordinates to image coordinates (and vice versa). While only in early planning stages, such a package would aim to not be tied to the FITS representation used for the current astropy.wcs . Such a package would also include closer connection to other parts of Astropy, for example astropy.coordinates ( § 2.3).", '2.7. Cosmology': 'The astropy.cosmology sub-package contains classes for representing widely used cosmologies, and functions for calculating quantities that depend on a cosmological model. It also contains a framework for working with less frequently employed cosmologies that may not be flat, or have a timevarying pressure to density ratio, w , for dark energy. The quantities that can be calculated are generally taken from those described by Hogg (1999). Some examples are the angular diameter distance, comoving distance, critical density, distance modulus, lookback time, luminosity distance, and Hubble parameter as a function of redshift. \nThe fundamental model for this sub-package is that any given cosmology is represented by a class. An instance of this class has attributes giving all the parameters required to specify the cosmology uniquely, such as the Hubble parameter, CMB temperature and the baryonic, cold dark matter, and dark energy densities at z = 0. One can then use methods of this class to perform calculations using these parameters. \nFigure 7 shows how the FlatLambdaCDM class can be used to create an object representing a flat ΛCDM cosmology, and how the methods of this object can be called to calculate the comoving volume, age and transverse separation at a given redshift. Further calculations can be performed using the many methods of the cosmology object as described in the Astropy documentation. For users who are more comfortable using a procedural coding style, these methods are also available as functions that take a cosmology class instance as a keyword argument. \nThe sub-package provides several pre-defined cosmology instances corresponding to commonly used cosmological parameter sets. Currently parameters from the WMAP 5-year (Komatsu et al. 2009), 7-year (Komatsu et al. 2011) and 9year results (Hinshaw et al. 2012) are included (the WMAP5 , WMAP7 , and WMAP9 classes). The parameters from the Planck results (Planck Collaboration et al. 2013) will be included \nFig. 7. Cosmology utilities. \n<!-- image --> \nin the next release of Astropy. There are several classes corresponding to non-flat cosmologies, and the most common dark energy models are supported: a cosmological constant, constant w , and w ( a ) = w 0 + w a (1 -a ) (e.g. Linder 2003, here a is the scale factor). Figure 7 gives examples showing how to use the pre-defined cosmologies, and how to define a new cosmology with a time-varying dark energy w ( a ). Any other arbitrary cosmology can be represented by subclassing one of the basic cosmology classes. \nAll of the code in the sub-package is tested against the web-based cosmology calculator by Wright (2006) and two other widely-used calculators. 12 , 13 In cases when these calculators are not precise enough to enable a meaningful comparison, the code is tested against calculations performed with Mathematica .', '3. Development Approach': "A primary guiding philosophy of Astropy is that it is developed for and (at least in part) by the astronomy user community. This ensures the interface is designed with the workflow of working astronomers in mind. At the same time, it aims to make use of the expertise of software developers to design code that encourages good software practices such as a consistent and clean API, thorough documentation, and integrated testing. It is also dedicated to remaining open source to enable wide adoption and render input from all users easier, and is thus released with a 3-clause BSD-style license (A license of this sort is 'permissive' in that it allows usage of the code for any purposes as \nlong as notice of the Astropy copyright and disclaimers of warranty are given). Achieving these aims requires code collaboration between over 30 geographically-distributed developers, and here we describe our development workflow with the hope that it may be replicated by other astronomy software projects that are likely to have similar needs. \nTo enable this collaboration, we have made use of the GitHub 14 open source code hosting and development platform. The main repository for astropy is stored in a git 15 repository on GitHub, and any non-trivial changes are made via pull requests , which are a mechanism for submitting code for review by other developers prior to merging into the main code base. This workflow aids in increasing the quality, documentation and testing of the code to be included in astropy . Not all contributions are necessarily accepted - community consensus is needed for incorporating major new functionality in astropy , and any new feature has to be justified to avoid implementing features that are only useful to a minority of users, but may cause issues in the future. \nAt the time of writing, astropy includes several thousand tests, which are small units of code that check that functions, methods, and classes in astropy are behaving as expected, both in terms of scientific correctness and from a programming interface perspective. We make use of continuous integration , which is the process of running all the tests under various configurations (such as different versions of Python or NumPy, and on different platforms) in order to ensure that the package is held to the highest standard of stability. In particular, any change made via a pull request is subject to extensive testing before being merged into the core repository. For the latter, we make use of Travis, 16 while for running more extensive tests across Linux, MacOS X, and Windows, we make use of Jenkins 17 (both are examples of continuous integration systems). \nThis development workflow has worked very well so far, allowing contributions by many developers, and blurring the line between developers and users. Indeed, users who encounter bugs and who know how to fix them can submit suggested changes. We have also implemented a feature that means that anyone reading the documentation at http://docs.astropy.org can suggest improvements to the documentation with just a few clicks in a web browser without any prior knowledge of the git version control system.", '4. Planned functionality': "Development on the Astropy package is very active, and in addition to some of the incremental improvements to existing sub-packages described in the text, we are focusing on implementing major new functionality in several areas for the next (v0.3) release (some of which have already been implemented in the publicly-available developer version): \n- -Improving interoperability between packages, which includes for example seamlessly integrating the astropy.units framework across all sub-packages \n- -Adding support for NumPy arrays in the coordinates sub-package, which will allow the efficient representation and conversions of coordinates in large datasets\n- -Supporting more file formats for reading and writing Table and NDData objects\n- -Implementing a Virtual Observatory cone search tool (Williams et al. 2011)\n- -Implementing a generalized model-fitting framework\n- -Implementing statistical functions commonly used in Astronomy \nIn the longer term, we are already planning the following major functionality: \n- -Image analysis tools, including aperture and point spread function (PSF) photometry\n- -Spectroscopic analysis tools\n- -Generalized WCS transformations beyond the FITS WCS standard\n- -ASAMPserver/client (ported from the SAMPy 18 package)\n- -Support for the Simple Image Access Protocol (SIAP) (Tody et al. 2011)\n- -Support for the Table Access Protocol (TAP) (Louys et al. 2011) is under consideration \nand undoubtedly the core functionality will grow beyond this. In fact, the astropy package will likely remain a continuously-evolving package, and will thus never be considered 'complete' in the traditional sense.", '5. Summary': 'We have presented the first public release of the Astropy package (v0.2), a core Python package for astronomers. In this paper we have described the main functionality in this release, which includes: \n- -Units and unit conversions ( § 2.1)\n- -Absolute dates and times ( § 2.2)\n- -Celestial coordinate systems ( § 2.3)\n- -Tabular and gridded data ( § 2.4)\n- -Support for common astronomical file formats ( § 2.5)\n- -World Coordinate System transformations ( § 2.6)\n- -Cosmological calculations ( § 2.7). \nWe also briefly described our development approach ( § 3), which has enabled an international collaboration of scientists and software developers to create and contribute to the package. We outlined our plans for the future ( § 4) which includes more interoperability of sub-packages, as well as new functionality. \nWe invite members of the community to join the effort by adopting the Astropy package for their own projects, reporting any issues, and whenever possible, developing new functionality.', 'Acknowledgements': 'We thank the referee, Igor Chiligarian, for suggestions that helped improve this paper. We would like to thank the NumPy, SciPy, IPython and Matplotlib communities for \nproviding their packages which are invaluable to the development of Astropy. We thank the GitHub team for providing us with an excellent free development platform. We also are grateful to Read the Docs ( https://readthedocs. org/ ), Shining Panda ( https://www.shiningpanda-ci. com/ ), and Travis ( https://www.travis-ci.org/ ) for providing free documentation hosting and testing respectively. Finally, we would like to thank all the astropy users that have provided feedback and submitted bug reports. The contribution by T. Aldcroft was funded by NASA contract NAS8-39073. The name resolution functionality shown in Figure 4 makes use of the SIMBAD database, operated at CDS, Strasbourg, France.', 'References': "Barrett, P. E., & Bridgman, W. T. 1999, in Astronomical Society of the Pacific Conference Series, Vol. 172, Astronomical Data Analysis Software and Systems VIII, 483 \n- Behnel, S., Bradshaw, R., Citro, C., Dalcin, L., Seljebotn, D., & Smith, K. 2011, Computing in Science Engineering, 13, 31\n- Berry, D. S., & Jenness, T. 2012, in Astronomical Society of the Pacific Conference Series, Vol. 461, Astronomical Data Analysis Software and Systems XXI, ed. P. Ballester, D. Egret, & N. P. F. Lorente, 825\n- Calabretta, M. R., & Greisen, E. W. 2002, Astronomy & Astrophysics, 1077\n- Calabretta, M. R., Valdes, F., Greisen, E. W., & Allen, S. L. 2004, in Astronomical Society of the Pacific Conference Series, Vol. 314, Astronomical Data Analysis Software and Systems (ADASS) XIII, ed. F. Ochsenbein, M. G. Allen, & D. Egret, 551\n- Derriere, S., Gray, N., Louys, M., McDowell, J., Ochsenbein, F., Osuna, P., Rino, B., & Salgado, J. 2012, Units in the VO, Version 1.0, IVOA Proposed Recommendation 20 August 2012 edn.\n- George, I., & Angelini, L. 1995, Specification of Physical Units within OGIP (Office of Guest Investigator Programs) FITS files \nGreenfield, P. 2011, in Astronomical Society of the Pacific Conference Series, Vol. 442, Astronomical Data Analysis Software and Systems XX, ed. I. N. Evans, A. Accomazzi, D. J. Mink, & A. H. Rots, 425 Greisen, E. W., & Calabretta, M. R. 2002, Astronomy & Astrophysics, 1061 \n- Greisen, E. W., Calabretta, M. R., Valdes, F. G., & Allen, S. L. 2006, Astronomy & Astrophysics, 747\n- Guinot, B., & Seidelmann, P. K. 1988, A&A, 194, 304\n- Hinshaw, G. et al. 2012, arXiv:1212.5226\n- Hogg, D. W. 1999, ArXiv Astrophysics e-prints, arXiv:astroph/9905116\n- Jenness, T., & Berry, D. S. 2013, in ASP Conf Ser., Vol. TBD, ADASS XXII, ed. D. Friedel, M. Freemon, & R. Plante (San Francisco: ASP), in press\n- Jones, E., Oliphant, T., & Peterson, P. 2001, http://www. scipy. org/ Kaplan, G. H. 2005, U.S. Naval Observatory Circulars, 179 \nKomatsu, E. et al. 2009, ApJS, 180, 330 \n- --. 2011, ApJS, 192, 18 \nKovalevsky, J. 2001, in Journ'ees 2000 - syst'emes de r'ef'erence spatiotemporels. J2000, a fundamental epoch for origins of reference systems and astronomical models, ed. N. Capitaine, 218-224 \nLinder, E. V. 2003, Physical Review Letters, 90, 091301 \nLouys, M. et al. 2011, arXiv:1111.1758 \n- McKinney, W. 2012, Python for Data Analysis (O'Reilly Media, Incorporated)\n- Ochsenbein, F. 2000, Astronomical Catalogues and Tables Adopted Standards, Version 2.0\n- Oliphant, T. 2006, A Guide to NumPy, Vol. 1 (Trelgol Publishing USA)\n- Oschenbein, F. et al. 2004, VOTable Format Definition, Version 1.1, International Virtual Observatory Alliance (IVOA)\n- --. 2009, VOTable Format Definition, Version 1.2, International Virtual Observatory Alliance (IVOA)\n- Pence, W. D., Chiappetti, L., Page, C. G., Shaw, R. A., & Stobie, E. 2010, A&A, 524, A42\n- Planck Collaboration et al. 2013, arXiv:1303.5076\n- Pontzen, A., Roˇskar, R., Stinson, G. S., Woods, R., Reed,\n- D. M., Coles, J., & Quinn, T. R. 2013, pynbody: Astrophysics \nSimulation Analysis for Python, Astrophysics Source Code Library, ascl:1305.002 \nRhodes, B. C. 2011, PyEphem: Astronomical Ephemeris for Python, Astrophysics Source Code Library, ascl:1112.014 \nShupe, D. L., Moshir, M., Li, J., Makovoz, D., Narron, R., & Hook, R. N. 2005, in Astronomical Society of the Pacific Conference Series, Vol. 347, Astronomical Data Analysis Software and Systems XIV, ed. P. Shopbell, M. Britton, & R. Ebert, 491 \nSoffel, M. et al. 2003, AJ, 126, 2687 \nStetson, P. B. 1987, PASP, 99, 191 \nTerlouw, J. P., & Vogelaar, M. G. R. 2012, Kapteyn Package, version 2.2, Kapteyn Astronomical Institute, Groningen \nTody, D., Plante, R., & Harrison, P. 2011, arXiv:1110.0499 \nTollerud, E. 2012, Astropysics: Astrophysics utilities for python, Astrophysics Source Code Library, ascl:1207.007 \nVan Der Walt, S., Colbert, S., & Varoquaux, G. 2011, Computing in Science & Engineering, 13, 22 \nWallace, P. T. 2011, Metrologia, 48, 200 \nWilliams, R., Hanisch, R., Szalay, A., & Plante, R. 2011, arXiv:1110.0498 \nWright, E. L. 2006, PASP, 118, 1711"}

2024arXiv240319440O

We point out that the Gaussian wavepacket formalism can serve as a concrete realization of the joint measurement of position and momentum which is an essential element in understanding Heisenbergs original philosophy of the uncertainty principle in line with the universal framework of error disturbance and their uncertainty relations developed by Lee and Tsutsui. We show that our joint measurement in the Gaussian phase space being a Positive OperatorValued Measure POVM measurement smoothly interpolates between the projective measurements of position and momentum. We for the first time have obtained the LeeTsutsui LT error and the refined Lee error for the positionmomentum measurement. We find that the LT uncertainty relation becomes trivial 00 in the limiting case of projective measurement of either position or momentum. Remarkably in contrast to the LT relation the refined Lee uncertainty relation which assesses errors for local representability provides a constant lower bound unaffected by these limits and is invariably saturated for a pure Gaussian initial state. The obtained lower bound is in agreement with Heisenbergs value.

2024-03-01T00:00:00Z

['arXiv:2403.19440', '10.48550/arXiv.2403.19440', '2024arXiv240319440O']

['High Energy Physics - Phenomenology', 'General Relativity and Quantum Cosmology', 'High Energy Physics - Theory', 'Quantum Physics']

Gaussian Formalism Concrete Realization of Joint Measurement for Heisenbergs Uncertainty Relation for Errors

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2403.19440.pdf

{"Concrete Realization of Joint Measurement for Heisenberg's Uncertainty Relation for Errors": "Kin-ya Oda ∗ and Naoya Ogawa † \n∗ Department of Information and Mathematical Sciences, Tokyo Woman's Christian University, Tokyo 167-8585, Japan † Department of Complex Systems Science, Graduate School of Informatics, Nagoya University, Nagoya 464-8601, Japan \nMarch 29, 2024", 'Abstract': "We point out that the Gaussian wave-packet formalism can serve as a concrete realization of the joint measurement of position and momentum, which is an essential element in understanding Heisenberg's original philosophy of the uncertainty principle, in line with the universal framework of error, disturbance, and their uncertainty relations developed by Lee and Tsutsui. We show that our joint measurement in the Gaussian phase space, being a Positive Operator-Valued Measure (POVM) measurement, smoothly interpolates between the projective measurements of position and momentum. We, for the first time, have obtained the Lee-Tsutsui (LT) error and the refined Lee error for the positionmomentum measurement. We find that the LT uncertainty relation becomes trivial, 0 = 0, in the limiting case of projective measurement of either position or momentum. Remarkably, in contrast to the LT relation, the refined Lee uncertainty relation, which assesses errors for local representability, provides a constant lower bound unaffected by these limits and is invariably saturated, for a pure Gaussian initial state. The obtained lower bound is in agreement with Heisenberg's value.", '1 Introduction': "Heisenberg's Uncertainty Principle, introduced in 1927, is a cornerstone of quantum mechanics, fundamentally altering our understanding of measurement precision and predictability at the quantum level [1]. Originally, this principle concerned the errors and disturbances in the measurement of physical observables. This pivotal concept was then mathematically refined by Kennard, enhancing the principle's precision [2], and later generalized by Robertson to apply to any pair of observables [3]. The result of these developments is now known as the Kennard-Robertson (KR) inequality; see also Ref. [4] for Schrodinger's subsequent contribution, now known as the Schrodinger inequality. Unlike the original Heisenberg's principle, the KR and Schrodinger inequalities exclusively concern the fluctuations of physical observables, and are completely decoupled from the errors and disturbances arising from the measurements. \nBuilding upon this foundational research, the contributions of Arthurs, Kelly, and Goodman led to the formulation of what is now known as the AKG inequality, significantly advancing our understanding of errors and the costs of measurements in quantum mechanics [5, 6]. Ozawa further expanded upon these concepts, resulting in the development of the Ozawa inequality through his studies on error and error-disturbance relations [7, 8]. Estimation theory, as highlighted in the works of Yuen-Lax [9] and further developed by Watanabe-Sagawa-Ueda, culminated in the WSU inequality [10, 11], offering alternative perspectives. These collective advancements have significantly enhanced our comprehension of quantum uncertainty. \nLee and Tsutsui have provided an operationally tangible uncertainty relation, which we call the Lee-Tsutsui (LT) inequality, marking a significant advancement in the field of quantum uncertainty. The LT inequality offers a universal and geometric perspective on quantum measurement errors and disturbances, unifying the KR, Schrodinger, AKG, Ozawa, and WSU inequalities into a comprehensive, operationally interpretable framework [12]. This framework was subsequently applied to a concrete two-state system [13]. Lee further expanded this approach by introducing the concepts of local representability and joint measurability, leading to the establishment of what we call the Lee inequality [14, 15]. However, a challenge of this elegant framework is the lack of a concrete realization for the abstractly constructed mathematical objects, beyond the simple two-state system presented in Ref. [13]. The current paper aims to fill this gap, by providing a concrete realization of the key object in Lee's inequality, the representability for a joint measurement, in the case of Heisenberg's positionmomentum uncertainty. \nRecent developments in the Gaussian wave-packet formalism have opened up new possibilities in quantum field theory [16, 17, 18, 19, 20]. Particularly notable is the rigorous proof of the emergence of time-boundary effect, initially claimed in Ref. [21], which has been recently established [22]. Gaussian wave packets are significant as they form an (over)complete set that spans the position-momentum phase space, including that of the free one-particle subspace, which underpins ordinary non-relativistic quantum mechanics [16]; see also Refs. [23, 24, 25] for a viewpoint that the Gaussian basis can be regarded as a complete set of coherent states in position-momentum space. Furthermore, the Gaussian basis can be generalized to a complete set of Lorentz-invariant wave-packet basis [23, 24, 25], and further to Lorentz-covariant wave-packet basis for spinor fields, incorporating spin degrees of freedom [26]. \nThe importance of the phase space is also emphasized in Refs. [27, 28, 29], particularly in the context of providing a rigorous examination and proof of Heisenberg-type inequalities, with a focus on the precision-disturbance trade-off in quantum measurements. \nTable 1: Reminder table for the spaces. A typical element is also given for each space. \n/owner \n/owner \n̂ \n̂ \nIn this paper, we posit that a Positive Operator-Valued Measure (POVM) measurement onto the Gaussian wave-packet basis naturally serves as the joint measurement for position and momentum, a key element in the discussion on Heisenberg's uncertainty in Ref. [15]. We provide concrete expressions for the pullback and pushforward, which remained abstract in the original construction [12, 13, 15]. We will demonstrate that the Gaussian basis naturally interpolates between the position-space basis and the momentum-space basis in the limits of infinitely narrow and wide widths, respectively. \nThe organization of this paper is as follows: In Sec. 2, we briefly review the Lee-Tsutsui (LT) formalism to clarify our notation, focusing on the parts relevant to our discussion. In Sec. 3, we review the Gaussian formalism and show that the Gaussian wave packet smoothly interpolates between the position and momentum bases. In Sec. 4, we study the separate measurement of position or momentum of a Gaussian initial state in the LT formalism. We show the concrete forms of the pullback and pushforward for the position and momentum, which were previously only abstract constructs. In Sec. 5, we present our main findings: that the POVM measurement onto the Gaussian basis can be regarded as the joint measurement of position and momentum. Again we show the concrete form of the pullback and pushforward for the position and momentum for the joint measurement of the Gaussian initial state. In Sec. 6, we summarize our results and discuss future directions.", '2 Lee-Tsutsui formalism': 'We briefly review the formalism of Lee and Tsutsui [12, 13] and of Lee [14, 15], which we collectively call the LT formalism hereafter, focusing on the part relevant to our discussion.', '2.1 Basic quantum mechanics': "A physical observable is represented by a self-adjoint operator ̂ A that satisfies ̂ A † = ̂ A , where ̂ A † is the adjoint of ̂ A , on a Hilbert space H . We write the linear space of all the self-adjoint operators as S ( H ); see Table 1. A physical state is represented by a density operator ̂ ρ that is self-adjoint, ρ † = ρ , has a trace of unity, Tr[ ̂ ρ ] = 1, and has a spectrum that is entirely nonnegative, ̂ ρ ≥ 0. We write the state space Z ( H ); see Table 1. Typically in our discussion, one may assume the state to be a pure state given as a projection operator ̂ ρ = | ψ 〉 〈 ψ | , where | ψ 〉 ∈ H (and 〈 ψ | is its dual). With this case in mind, we also call a vector | ψ 〉 ∈ H the state. \nAn expectation value of (not necessarily self-adjoint) operator A on ρ is given by \n̂ ̂ 〈 ̂ A 〉 ̂ ρ := Tr [ ̂ A ̂ ρ ] , (1) \nand a semi-inner product of any (not necessarily self-adjoint) operators A and B by \n̂ where the commutator and anticommutator are [ ̂ A, ̂ B ] := ̂ A ̂ B -̂ B ̂ A and { ̂ A, ̂ B } := ̂ A ̂ B + ̂ B ̂ A , respectively. The semi-inner product naturally induces the semi-norm and the standard deviation for any (not necessarily self-adjoint) operator A on ρ : 1 \n̂ ̂ 〈 ̂ A, ̂ B 〉 ̂ ρ := 〈 { ̂ A, ̂ B } 2 〉 ρ , (2) \n̂ ̂ ∥ ∥ ∥ ̂ A ∥ ∥ ∥ ̂ ρ := √ 〈 ̂ A † , ̂ A 〉 ̂ ρ , σ ̂ ρ [ ̂ A ] := √ ∥ ∥ ∥ ̂ A ∥ ∥ ∥ 2 ̂ ρ -〈 ̂ A 〉 2 ̂ ρ . (3) \nIf ̂ A is self-adjoint, the standard deviation reduces to σ 2 ̂ ρ [ ̂ A ] = 〈 ̂ A 2 〉 ̂ ρ - 〈 ̂ A 〉 2 ̂ ρ . Further if ̂ ρ is a pure state ̂ ρ = | ψ 〉 〈 ψ | , the semi-norm reduces to the norm of the state after the operation, ‖ ̂ A ‖ 2 | ψ 〉〈 ψ | = ‖ ̂ A | ψ 〉 ‖ 2 , and accordingly the expectation value (1) reduces to 〈 ̂ A 〉 | ψ 〉〈 ψ | = 〈 ψ | ̂ A | ψ 〉 . 2.2 Measurement process \nA measurement M is an affine map ̂ ρ ↦→ p in the sense that, for any states ̂ ρ , ̂ ρ and ∀ λ ∈ [0 , 1], \nM ( λ ρ +(1 -λ ) ρ ) = λ M ρ +(1 -λ ) M ρ , (4) \n̂ Two comments are in order: First, the affine-ness (4) is the least requirement to allow the interpretation of probability mixture but is powerful enough to derive most of the results in the LT formalism. An affine map between any pair of state spaces, in the right of Table 1, is in general called a process . The LT formalism can handle all the processes on equal foot. Especially when applied to a quantum process Θ : Z ( H 1 ) → Z ( H 2 ), its disturbance can be described in quite a parallel manner to various M 's errors given below. For notational simplicity, we do not exploit this beauty of the formalism, and will stick to a measurement M as a process (except in Sec. 2.6). Second, for most of our purpose, one may assume a projective measurement that gives, ∀ ρ ∈ Z ( H ), 2 \n̂ ̂ ̂ ̂ where p is a probability density function (PDF) that satisfies ∫ Ω d ω p ( ω ) = 1 and, ∀ ω ∈ Ω , p ( ω ) ≥ 0, in which Ω is a sample space and d ω its measure. We write the convex set of all the PDFs as W ( Ω ); see Table 1. Hereafter, we write M ̂ ρ := M ̂ ρ as a reminder to reinforce that M ρ is a PDF. As a whole, M : Z ( H ) → W ( Ω ) is defined. \n̂ [ M ρ ]( ω ) = Tr [ | ω 〉 〈 ω | ρ ] . (5) \n〈 f 〉 p := ∫ Ω d ωf ( ω ) p ( ω ) , 〈 f, g 〉 p := ∫ Ω d ωf ∗ ( ω ) g ( ω ) p ( ω ) , (6) f ‖ p := √ 〈 f, f 〉 p = √ ∫ Ω d ω | f ( ω ) | 2 p ( ω ) , σ p [ f ] := √ ‖ f ‖ 2 p -∣ ∣ ∣ 〈 f 〉 p ∣ ∣ ∣ 2 , (7) \n̂ ̂ On the sample space Ω , we define the classical expectation value, semi-inner product, semi-norm, and standard deviation in accordance with their quantum counterparts: \n‖ \nwhere f, g are arbitrary real functions on Ω (but we have retained the complex conjugate, denoted by ∗ per physics community notation, to accommodate possible generalizations). More concretely for the real function f , we have σ 2 p [ f ] = ∫ Ω d ωf 2 ( ω ) p ( ω ) -(∫ Ω d ωf ( ω ) p ( ω ) ) 2 .", '2.3 LT adjoint, pullback, and pushforward': 'Given a measurement M , we define (what we call) the Lee-Tsutsui (LT) adjoint operator ̂ M /star f of a real function f by, ∀ ̂ ρ ∈ Z ( H ), \nWealso call this the pullback operator of f . 3 As a whole, we define the LT adjoint M /star : R ( Ω ) → S ( H ), where R ( Ω ) is the space of all the real functions on Ω ; see Table 1. \n〈 ̂ M /star f 〉 ̂ ρ = 〈 f 〉 M ̂ ρ . (8) \nIt is important that the following inequality follows from the Kadison-Schwarz inequality: \n‖ f ‖ M ̂ ρ ≥ ∥ ∥ ∥ ̂ M /star f ∥ ∥ ∥ ̂ ρ . (9) \nBy definitions (3), (7), and (8), we may rewrite inequality (9) into that of the standard deviations: \nσ M ̂ ρ [ f ] ≥ σ ̂ ρ [ ̂ M /star f ] . (10) \nTo quote [15], the operational cost of acquiring the expectation value of a quantum observable through measurements can never break the quantum limit imposed by the said observable. \nOnce the LT adjoint is obtained, we define the pushforward function M /star ̂ A of a given self-adjoint operator ̂ A (over a state ρ ) by, ∀ f ∈ R ( Ω ), \nor more concretely by, ∀ f ∈ R ( Ω ), \n̂ 〈 ̂ A, ̂ M /star f 〉 ̂ ρ = 〈 M /star ̂ A, f 〉 M ̂ ρ , (11) \nTr { ̂ A, ̂ M /star f } 2 ̂ ρ = ∫ Ω d ω [ M /star ̂ A ] ( ω ) f ( ω ) [ M ̂ ρ ]( ω ) . (12) \nBy construction, the pushforward function is obtained locally , namely obtained for each given ̂ ρ , and should be more properly written as M ̂ ρ/star ̂ A . However, this dependence on ̂ ρ is trivially understood for most of our purposes, and we will use the shorthand notation to avoid clutter. As a whole, we define the pushforward M /star : S ( H ) → R ( Ω ) (for each given ̂ ρ as mentioned). \n̂ A ̂ ρ ∼ ̂ B ⇐⇒ ∥ ∥ ∥ ̂ A -̂ B ∥ ∥ ∥ ̂ ρ = 0 , f p ∼ g ⇐⇒‖ f -g ‖ p = 0 . \nFor our purpose, this step is not necessary and we neglect it, though can be important for other purposes. \nFinally, we note that for the case of projective measurement (5), its LT adjoint also becomes projective, in the sense that given any function f ∈ R ( Ω ), \n̂ M /star f = ∫ Ω d ωf ( ω ) | ω 〉 〈 ω | . (13) \nAlso, the pushforward for the projective measurement of ̂ A becomes the identity M /star ̂ A = id; see Eq. (50) below for a concrete realization.', '2.4 LT error': 'We define (what we call) the LT error [12]: \nε ̂ ρ [ ̂ A ; M ] := √ ∥ ∥ ∥ ̂ A ∥ ∥ ∥ 2 ̂ ρ -∥ ∥ ∥ M /star ̂ A ∥ ∥ ∥ 2 M ̂ ρ = √ Tr [ ρ ̂ A 2 ] -∫ Ω d ω [ M /star ̂ A ] 2 ( ω ) [ M ̂ ρ ]( ω ) , (14) \nwhere [ M /star ̂ A ] 2 ( ω ) := ([ M /star ̂ A ] ( ω ) ) 2 as above. The meaning of LT error is well explained in Ref. [12], which we will briefly review hereafter. Given an estimator function f ∈ R ( Ω ), we introduce an error with respect to f ( abbr. f -error): \nε ̂ ρ [ ̂ A ; M , f ] := √ ∥ ∥ ∥ ̂ A -̂ M /star f ∥ ∥ ∥ 2 ̂ ρ + ( ‖ f ‖ 2 M ̂ ρ -∥ ∥ ∥ ̂ M /star f ∥ ∥ ∥ 2 ̂ ρ ) . (15) \nThe first term in the square root is the goodness of the fit of ̂ A by the pullback operator of the estimator f , while the second is the increase in the variance from that of the quantum pullback operator ̂ M /star f to that of the classical estimator f : \n‖ f ‖ 2 M ̂ ρ -∥ ∥ ∥ ̂ M /star f ∥ ∥ ∥ 2 ̂ ρ = ( ‖ f ‖ 2 M ̂ ρ -〈 f 〉 2 M ̂ ρ ) -( ∥ ∥ ∥ ̂ M /star f ∥ ∥ ∥ 2 ̂ ρ -〈 ̂ M /star f 〉 2 ̂ ρ ) = σ 2 M ̂ ρ [ f ] -σ 2 ̂ ρ [ ̂ M /star f ] , (16) \nwhere we used the definition of LT adjoint (8). \nThe first term in the square root in Eq. (15) reads \n∥ ∥ ∥ ̂ A -̂ M /star f ∥ ∥ ∥ 2 ̂ ρ = 〈 ̂ A, ̂ A 〉 ̂ ρ -2 〈 ̂ A, ̂ M /star f 〉 ̂ ρ + 〈 ̂ M /star f, ̂ M /star f 〉 ̂ ρ = ∥ ∥ ∥ ̂ A ∥ ∥ ∥ 2 ̂ ρ -2 〈 M /star ̂ A, f 〉 M ̂ ρ + ∥ ∥ ∥ ̂ M /star f ∥ ∥ ∥ 2 ̂ ρ , (17) where we used the definition of pushforward (11). Putting Eq. (17) into Eq. (15), we obtain \nε 2 ̂ ρ [ ̂ A ; M , f ] = ε 2 ̂ ρ [ ̂ A ; M ] + ∥ ∥ ∥ M /star ̂ A -f ∥ ∥ ∥ 2 M ̂ ρ . (18) \nThe f -error is minimized by choosing f to be the optimal pushforward function M /star ̂ A , and the LT error (14) gives the smallest error achieved by this choice. The LT error satisfies (what we call) the LT inequality: \nε ̂ ρ [ ̂ A ; M ] ε ̂ ρ [ ̂ B ; M ] ≥ √ I 2 ̂ ρ [ ̂ A, ̂ B ; M ] + R 2 ̂ ρ [ ̂ A, ̂ B ; M ] , (19) \nwhere \nI ̂ ρ [ ̂ A, ̂ B ; M ] := 〈 [ ̂ A, ̂ B ] 2 i 〉 ρ -〈 [ ̂ M /star M /star ̂ A, ̂ B ] 2 i 〉 ρ -〈 [ ̂ A, ̂ M /star M /star ̂ B ] 2 i 〉 ρ , (20) \nThe imaginary parts I ̂ ρ [ ̂ A, ̂ B ; M ] represents the quantum contribution to the lower bound, while the real parts R ̂ ρ [ ̂ A, ̂ B ; M ] the semi-classical contribution. 2.5 Lee error \n̂ ̂ ̂ R ̂ ρ [ ̂ A, ̂ B ; M ] := 〈 { ̂ A, ̂ B } 2 〉 ̂ ρ -〈 M /star ̂ A, M /star ̂ B 〉 M ̂ ρ . (21) \nFollowing Ref. [15], we define the error for local representability , which we call the Lee error hereafter: \n˜ ε ̂ ρ [ ̂ A ; M ] := √ ∥ ∥ ∥ M /star -1 ̂ A ∥ ∥ ∥ 2 M ̂ ρ -∥ ∥ ∥ ̂ A ∥ ∥ ∥ 2 ̂ ρ , (22) \nwhere M /star -1 ̂ A := ( M /star ) -1 A . 4 More explicitly, \nThe Lee inequality reads \n̂ ˜ ε 2 ̂ ρ [ ̂ A ; M ] = ∫ Ω d ω [ M /star -1 ̂ A ] 2 ( ω ) [ M ̂ ρ ]( ω ) -Tr [ ̂ A 2 ̂ ρ ] . (23) \nwhere the contributors to the lower bound are \n˜ ε ̂ ρ [ ̂ A ; M ] ˜ ε ̂ ρ [ ̂ B ; M ] ≥ √ I 2 0 ̂ ρ [ ̂ A, ̂ B ] + ˜ R 2 ̂ ρ [ ̂ A, ̂ B ; M ] , (24) \nI 0 ̂ ρ [ ̂ A, ̂ B ] := 〈 [ ̂ A, ̂ B ] 2 i 〉 ρ , (25) \nAgain the imaginary and real parts represent quantum and semi-classical contributions, respectively. \n̂ ˜ R ̂ ρ [ ̂ A, ̂ B ; M ] := 〈 { ̂ A, ̂ B } 2 〉 ̂ ρ -〈 M /star -1 ̂ A, M /star -1 ̂ B 〉 M ̂ ρ . (26)', '2.6 Joint measurement and classical projection of marginalization': "Following Lee's construction [15], we consider a joint measurement J : Z ( H ) → W ( Ω ), whose sample space (also called the outcome space) is separated into Ω = Ω 1 × Ω 2 so that the outcome PDF is a function of ω i ∈ Ω i ( i = 1 , 2), namely, p ∈ W ( Ω ) is written as p ( ω 1 , ω 2 ). \nNow we introduce the following classical projection processes π i : W ( Ω 1 × Ω 2 ) → W ( Ω i ) ( i = 1 , 2), defined by the projection to marginal: \n[ π p ]( ω 1 ) := ∫ Ω 2 d ω 2 p ( ω 1 , ω 2 ) , [ π p ]( ω 2 ) := ∫ Ω 1 d ω 1 p ( ω 1 , ω 2 ) . (27) \nIf two measurements M i : Z ( H ) → W ( Ω i ) ( i = 1 , 2) can be written as M i = π i · J , they are said to admit the joint measurement J . \nThe LT adjoint π /star i is given in parallel to that of the measurement (8): Given a function f ∈ R ( Ω i ), its LT adjoint function π /star i f ∈ R ( Ω 1 × Ω 2 ) is defined by, ∀ p ∈ W ( Ω 1 × Ω 2 ), \n〈 π /star i f 〉 p = 〈 f 〉 π i p , ( i = 1 , 2) (28) \nor more concretely, ∀ p ∈ W ( Ω 1 × Ω 2 ), \n∫ Ω 1 d ω 1 ∫ Ω 2 d ω 2 [ π /star i f ]( ω 1 , ω 2 ) p ( ω 1 , ω 2 ) = ∫ Ω i d ω i f ( ω i ) [ π i p ]( ω i ) . ( i = 1 , 2) (29) \nWe see that, by the definition (27), the LT adjoint of f is trivially \n[ π /star i f ]( ω 1 , ω 2 ) = f ( ω i ) . ( i = 1 , 2) (30)", '3 Gaussian formalism': 'We review the Gaussian wave-packet formalism within a free one-particle subspace, decoupled from the spin degrees of freedom. In this paper, we restrict our attention to the 1D position-momentum space since the generalization to higher dimensions is straightforward. Throughout the paper, we restrict ourselves to the consideration of a fixed time slice and do not consider the time evolution/translation; see Refs. [16, 17, 18, 19] for a Gaussian formalism applied to a relativistic particle, Refs. [23, 24, 25] for insights into the free one-particle subspace of the relativistic particle and generalization to a Lorentz-invariant complete basis, and Ref. [26] for an account on the transformation of the space-like hyperplane under the Lorentz transformation and for a generalization to include the relativistic spin degrees of freedom into a Lorentz-covariant complete basis.', '3.1 Plane-wave basis': "Let { | p 〉 } p ∈ R be the momentum-space basis that spans the 1D free one-particle subspace, which we write H hereafter, without spin degrees of freedom. We employ the normalization \n〈 p ∣ ∣ p ' 〉 = δ ( p -p ' ) , (31) \nwhere the right-hand side is the Dirac delta function (distribution), resulting in the standard normalization for the completeness relation (resolution of identity): \n∫ R d p | p 〉 〈 p | = ̂ 1 , (32) \nin which ̂ 1 is the identity operator on H . Here and hereafter, the integral region of momentum (and position) is always from -∞ to ∞ unless otherwise stated. 5 The dual position-space basis { | x 〉 } x ∈ R is defined by \n〈 x | p 〉 := e ipx √ 2 π . (33) \nHere and hereafter, we employ the natural unit /planckover2pi1 = 1, unless otherwise stated, so that momentum and wavenumber have the same dimension (unit). The definition (33) leads to the completeness in the position space: \n∫ R d x | x 〉 〈 x | = ̂ 1 . (34) \nWe identify this position space as an arbitrarily chosen equal-time hyperplane. It is worth reiterating that the position-space basis can be used to span any fixed time slice, even for a relativistic particle, both in quantum mechanics (see e.g. Ref. [25]) and in quantum field theory [16, 17, 18] (this viewpoint is particularly emphasized in Refs. [25, 26]). \nOn H , we define the position and momentum operators: 6 \nIt follows that \nand \n〈 x | ̂ p = -i ∂ ∂x 〈 x | , ̂ p | x 〉 = | x 〉 i ←-∂ ∂x , 〈 p | ̂ x = i ∂ ∂p 〈 p | , ̂ x | p 〉 = | p 〉 ( -i ←-∂ ∂p ) , (36) \n[ ̂ x, ̂ p ] = i ̂ 1 , [ ̂ x, ̂ x ] = [ ̂ p, ̂ p ] = 0 . (37) The projective measurement for position M pos and that for momentum M mom are defined by, ∀ ̂ ρ ∈ Z ( H ), \nEspecially when ̂ ρ is a pure state ̂ ρ = | ψ 〉 〈 ψ | , we recover the familiar PDFs: [ M pos | ψ 〉 〈 ψ | ] ( x ) = |〈 x | ψ 〉| 2 , [ M mom | ψ 〉 〈 ψ | ] ( p ) = |〈 p | ψ 〉| 2 . (39) \n̂ x | x 〉 = x | x 〉 , ̂ p | p 〉 = p | p 〉 . (35) \n[ M pos ̂ ρ ]( x ) = Tr [ | x 〉 〈 x | ̂ ρ ] , [ M mom ̂ ρ ]( p ) = Tr [ | p 〉 〈 p | ̂ ρ ] . (38)", '3.2 Gaussian basis': "We define the Gaussian wave-packet state | X,P ; σ 〉 on the momentum-space basis [16, 17, 18] (see also Ref. [25] for a historical account): \n〈 p | X,P ; σ 〉 := ( σ π ) 1 4 exp ( -ipX -σ 2 ( p -P ) 2 ) , (40) \nwhere we have normalized such that ∥ ∥ | X,P ; σ 〉 ∥ ∥ 2 = 1. Physically, P and X give the centers of the wave packet in the position and momentum spaces, respectively, and σ ( σ -1 ) roughly gives the spatial (momentum) width-squared. The corresponding wave function becomes \n〈 x | X,P ; σ 〉 = ∫ R d p 〈 x | p 〉 〈 p | X,P ; σ 〉 = 1 ( πσ ) 1 4 exp ( iP ( x -X ) -( x -X ) 2 2 σ ) . (41) \nIt is important that the Gaussian wave-packet states form a complete basis that spans H : For any fixed σ , \n∫ R 2 d X d P 2 π | X,P ; σ 〉 〈 X,P ; σ | = ̂ 1 , (42) \nwhich can be verified by sandwiching both-hand sides by 〈 p | and | p ' 〉 (or equally by 〈 x | and | x ' 〉 ). 7 It is somewhat beautiful that the phase space measure d X d P 2 π emerges automatically. Note that the spatial width-squared σ is fixed and is not summed in the completeness (42). \nIt is also noteworthy that the Gaussian basis states are not mutually orthogonal: \n〈 X,P ; σ ∣ ∣ X ' , P ' ; σ 〉 = ∫ R d p 〈 X,P ; σ | p 〉 〈 p ∣ ∣ X ' , P ' ; σ 〉 = exp ( i P + P ' 2 ( X -X ' ) -( X -X ' ) 2 4 σ -σ 4 ( P -P ' ) 2 ) . (43) \nIn this sense, the relation (42) is sometimes called the overcompleteness. The relation (42) tells that the Gaussian basis provides a non-projective POVM: { | X,P ; σ 〉 〈 X,P ; σ | } ( X,P ) ∈ R 2 for any given fixed σ .", '4 Separate measurement of position or momentum': 'For the rest of this paper (except in Eq. (86)), we will exclusively focus on measurements of an initial Gaussian pure state: \n̂ ρ in = | X in , P in ; σ in 〉 〈 X in , P in ; σ in | . (44) In this section, we separately measure either its position by M pos or its momentum by M mom , as defined in Eq. (38). Though we list both the results of position and momentum measurements hereafter for conciseness, it should be kept in mind that either one of the measurements is performed, independently from the other. \nGiven the initial state (44), its measured PDFs in the position and momentum spaces are \n[ M pos ̂ ρ in ]( x ) = |〈 x | X in , P in ; σ in 〉| 2 = 1 √ πσ in exp ( -( x -X in ) 2 σ in ) , (45) \nThey are centered around x = X in and p = P in , with the widths-squared roughly σ in and σ -1 in , respectively. \n[ M mom ̂ ρ in ]( p ) = |〈 p | X in , P in ; σ in 〉| 2 = √ σ in π exp ( -σ in ( p -P in ) 2 ) . (46)', '4.1 Expectation and variance for separate measurements': "We obtain the expectation value, the squared semi-norm, and the variance (given in Eqs. (1), (3), (6), and (7)) for M pos of x (left) and for M mom of p (right): \n̂ ̂ ‖ x ‖ 2 ̂ ρ in = ‖ id ‖ 2 M pos ̂ ρ in = X 2 in + σ in 2 , ‖ p ‖ 2 ̂ ρ in = ‖ id ‖ 2 M mom ̂ ρ in = P 2 in + 1 2 σ in , (48) \n̂ ̂ 〈 x 〉 ̂ ρ in = 〈 id 〉 M pos ̂ ρ in = X in , 〈 p 〉 ̂ ρ in = 〈 id 〉 M mom ̂ ρ in = P in , (47) \nwhere id is the identity function \n̂ ̂ σ 2 ̂ ρ in [ ̂ x ] = σ 2 M pos ̂ ρ in [id] = σ in 2 , σ 2 ̂ ρ in [ ̂ p ] = σ 2 M mom ̂ ρ in [id] = 1 2 σ in , (49) \nid( x ) = x, id( p ) = p ; (50) \n̂ ̂ ̂ ̂ We also have the following results, which are independent of measurements: \n〈 [ ̂ x, ̂ p ] 2 i 〉 ̂ ρ in = 1 2 , 〈 { ̂ x, ̂ p } 2 〉 ̂ ρ in = X in P in . (51) \nrecall that ‖ ̂ x ‖ 2 ρ in := 〈 ̂ x 2 〉 ρ in , ‖ id ‖ 2 M pos ρ in := 〈 id 2 〉 M pos ρ in , etc. \nFor some readers' ease, we list some matrix elements as well: \n〈 p | ̂ x | X in , P in ; σ in 〉 = ( X in -iσ in ( p -P in ) ) 〈 p | X in , P in ; σ in 〉 , (52) 〈 x | ̂ p | X in , P in ; σ in 〉 = ( P in + i x -X in σ in ) 〈 x | X in , P in ; σ in 〉 , (53) \nwhich follow from the derivative representation (36) and may be useful to derive expressions below.", '4.2 Pullback and pushforward for separate measurements': "Having completed our review, we now turn to new developments from here on. R \n̂ M /star pos f = ∫ R d xf ( x ) | x 〉 〈 x | , ̂ M /star mom f = ∫ R d p f ( p ) | p 〉 〈 p | , (54) \nGiven a function f ∈ R ( ), we find its LT adjoint: \nIt is straightforward to check that they satisfy, ∀ ρ ∈ Z ( H ), \nby expanding the general state as \n̂ 〈 ̂ M /star pos f 〉 ̂ ρ = 〈 f 〉 M pos ̂ ρ , 〈 ̂ M /star mom f 〉 ̂ ρ = 〈 f 〉 M mom ̂ ρ , (55) \n̂ ρ = ∫ R d x ∫ R d x ' ρ ( x, x ' ) | x 〉 〈 x ' ∣ ∣ = ∫ R d p ∫ R d p ' ˜ ρ ( p, p ' ) | p 〉 〈 p ' ∣ ∣ , (56) \nwhere ρ ( x, x ' ) := 〈 x | ̂ ρ | x ' 〉 , with ρ ∗ ( x, x ' ) = ρ ( x ' , x ) and ∫ R d x ρ ( x, x ) = 1 (in particular, it follows that [ M pos ̂ ρ ]( x ) = ρ ( x, x )); and similarly for the momentum space. The result (54) is indeed trivial because of the generality (13), but we spelled it out anyway for concreteness. \nEquipped with the definite representation (40), yielding the matrix elements (52) and (53), we can explicitly compute the pushforward of x , \n[ (57) \nand the pushforward of p , \n̂ M pos /star ̂ x ]( x ) = x, [ M mom /star ̂ x ]( p ) = X in , \n̂ ̂ One may verify these pushforwards to satisfy the defining relation (11), or more concretely (12): As an illustration, we show the derivation of M pos /star ̂ p given in the left of Eq. (58). All other computations in this paper can be performed quite similarly. The defining relation (11) reads, ∀ f ∈ R ( R ), \n̂ [ M pos /star p ]( x ) = P in , [ M mom /star p ]( p ) = p. (58) \n〈 ̂ p, ̂ M /star pos f 〉 ̂ ρ in = 〈 M pos /star ̂ p, f 〉 M pos ̂ ρ in , (59) \nTr { ̂ p, ̂ M /star pos f } 2 ̂ ρ in = ∫ R d x [ M pos /star ̂ p ]( x ) f ( x ) [ M pos ̂ ρ in ]( x ) . (60) \nor more concretely from Eq. (12), ∀ f ∈ R ( R ), \nPutting the pullback (54) into the left-hand side, we obtain \nl.h.s. = ∫ R d xf ( x ) Tr [ ̂ p | x 〉 〈 x | + | x 〉 〈 x | ̂ p 2 | X in , P in ; σ in 〉 〈 X in , P in ; σ in | ] = ∫ R d xf ( x ) /Rfractur ( 〈 X in , P in ; σ in | x 〉 〈 x | ̂ p | X in , P in ; σ in 〉 ︸ ︷︷ ︸ ( P in + i x -X in σ in ) 〈 x | X in ,P in ; σ in 〉 ) = ∫ R d xf ( x ) P in ∣ ∣ 〈 x | X in , P in ; σ in 〉 ∣ ∣ 2 ︸ ︷︷ ︸ [ M pos ̂ ρ in ]( x ) , (61) \nwhere /Rfractur denotes the real part and we used Eq. (53) in the second last line. Comparing with the right-hand side of Eq. (60), we obtain the constant solution [ M pos /star ̂ p ]( x ) = P in . 4.3 LT error for separate measurements \nHaving obtained the pushforward, we can now compute the classical semi-norm for ̂ x , ‖ M pos /star ̂ x ‖ 2 M pos ̂ ρ in = X 2 in + σ in 2 , ‖ M mom /star ̂ x ‖ 2 M pos ̂ ρ in = X 2 in , (62) and the classical semi-norm for ̂ p , ‖ M pos /star ̂ p ‖ 2 Mmom ̂ ρ in = P 2 in , ‖ M mom /star ̂ p ‖ 2 Mmom ̂ ρ in = P 2 in + 1 2 σ in . (63) Combining with the quantum semi-norm (48), the resultant LT error (14) is, for x , \n̂ ε 2 ̂ ρ in [ ̂ x ; M pos ] = 0 , ε 2 ̂ ρ in [ ̂ x ; M mom ] = σ in 2 , (64) \nand, for ̂ p , \nWe see that for both the position and momentum measurements, the left-hand side of the LT inequality (19) vanishes: \nε 2 ̂ ρ in [ ̂ p ; M pos ] = 1 2 σ in , ε 2 ̂ ρ in [ ̂ p ; M mom ] = 0 . (65) \nLet us see if the right-hand side vanishes consistently. \nε ̂ ρ in [ ̂ x ; M pos ] ε ̂ ρ in [ ̂ p ; M pos ] = 0 , ε ̂ ρ in [ ̂ x ; M mom ] ε ̂ ρ in [ ̂ p ; M mom ] = 0 . (66)", '4.4 LT inequality for separate measurements': 'The pullback of pushforward is, for x , \nand for ̂ p , \n̂ ̂ M /star pos M pos /star ̂ x = ̂ x, ̂ M /star mom M mom /star ̂ x = X in ̂ 1 , (67) \nTherefore for the position measurement, \n̂ M /star pos M pos /star ̂ p = P in ̂ 1 , ̂ M /star mom M mom /star ̂ p = ̂ p. (68) \nand for the momentum measurement, \n[ ̂ M /star pos M pos /star ̂ x, ̂ p ] = i ̂ 1 , [ ̂ x, ̂ M /star pos M pos /star ̂ p ] = 0 , (69) \n[ ̂ M /star mom M mom /star ̂ x, ̂ p ] = 0 , [ ̂ x, ̂ M /star mom M mom /star ̂ p ] = i ̂ 1 . (70) \nCombining with the measurement-independent part (51), we see that the quantum contribution to the LT lower bound (20) vanishes: \nWe can compute \nI ̂ ρ in [ ̂ x, ̂ p ; M pos ] = 0 , I ̂ ρ in [ ̂ x, ̂ p ; M mom ] = 0 . (71) \n〈 M pos /star ̂ x, M pos /star ̂ p 〉 M pos ̂ ρ in = 〈 id 〉 M pos ̂ ρ in P in = X in P in , (72) 〈 M mom /star x, M mom /star p 〉 M mom ̂ ρ in = X in 〈 id 〉 M mom ̂ ρ in = X in P in , (73) \n̂ ̂ where we used Eq. (47). Combining with the measurement-independent part (51), we see that the semi-classical contribution to the LT lower bound (21) also vanishes: \nThe LT inequality turned out to be 0 = 0! \nR ̂ ρ in [ ̂ x, ̂ p ; M pos ] = 0 , R ̂ ρ in [ ̂ x, ̂ p ; M mom ] = 0 . (74)', '4.5 Lee error for separate measurement?': "To compute the Lee error (22), we need the inverse of the pullback (54). It is trivial for (so to say) the 'diagonal' measurement: \nM /star -1 pos ̂ x ( x ) = x, M /star -1 mom ̂ p ( p ) = p. (75) \nIt is straightforward to check that they satisfy, ∀ ρ ∈ Z ( H ), \nby expanding the general state as in Eq. (56). \n̂ 〈 ̂ x 〉 ̂ ρ = 〈 M /star -1 pos ̂ x 〉 M pos ̂ ρ , 〈 ̂ p 〉 ̂ ρ = 〈 M /star -1 mom ̂ p 〉 M mom ̂ ρ , (76) \nHowever, it is hard to obtain the inverse of pullback for the 'off-diagonal' measurements M /star -1 mom ̂ x ( p ) and M /star -1 pos ̂ p ( x ). For example for the latter, we need to find f ∈ R ( R ) that satisfies ̂ p = ∫ R d xf ( x ) | x 〉 〈 x | to match Eq. (54). However, the derivative representation (36) leads to \nFormally, the solution (if it exists) can be expressed as f ( x ) = ∫ R d x ' ( -i ∂ ∂x δ ( x -x ' ) ) . 8 This expression evidently requires regularization due to the derivative of the Dirac delta function. For example, using the regularization δ σ ( x -x ' ) = ( πσ ) -1 / 2 exp ( -( x -x ' ) 2 /σ ) leads to a vanishing result for f ( x ), indicating the need for further elaboration. Below, we will demonstrate that a joint measurement in the Gaussian phase space yields a naturally regularized expression, as will be detailed in Eq. (99). \n̂ p = ∫ R d x ∫ R d x ' | x 〉 〈 x ∣ ∣ ̂ p ∣ ∣ x ' 〉 〈 x ' ∣ ∣ = ∫ R d x ∫ R d x ' ( -i ∂ ∂x δ ( x -x ' ) ) | x 〉 〈 x ' ∣ ∣ . (77)", '5 Joint measurement in phase space': "Below, we first review known facts about the Gaussian matrix element, then restart with our main new development from Sec. 5.1. \nThe completeness (42) implies that the Gaussian basis can expand any function, including another Gaussian wave packet with a different width-squared: | X in , P in ; σ in 〉 . The expansion coefficient can be computed as [18] \n〈 X,P ; σ | X in , P in ; σ in 〉 = ∫ R d p 〈 X,P ; σ | p 〉 〈 p | X in , P in ; σ in 〉 = √ 2 ( σ red σ sum ) 1 4 exp ( iP ( X -X in ) -( X -X in ) 2 2 σ sum -σ red 2 ( P -P in ) 2 ) , (78) \nwhere 9 \n1 σσ (79) \nσ sum := σ + σ in , σ red := 1 σ + 1 σ in = in σ + σ in . \nPhysically, the inner product (78) can be understood as the transition amplitude from the initial state | X in , P in ; σ in 〉 to the final state | X,P ; σ 〉 , with the transition probability being \n|〈 X,P ; σ | X in , P in ; σ in 〉| 2 = 2 π ( 1 √ πσ sum e -1 σ sum ( X -X in ) 2 )(√ σ red π e -σ red ( P -P in ) 2 ) . (80) \nWe see that the detection probability of the particle is centered around X = X in and P = P in , with the width-squared roughly σ sum and σ -1 red , respectively. \nWhen we do not observe the momentum, we integrate the probability (80) over all the possible momenta: \n∫ R d P 2 π |〈 X,P ; σ | X in , P in ; σ in 〉| 2 = 1 √ πσ sum exp ( -( X -X in ) 2 σ sum ) . (81) \nIn the limit of detector spatial resolution finer than the initial wave packet size, σ /lessmuch σ in , we obtain σ sum → σ in , recovering the position-space PDF: [ M pos ̂ ρ in ]( X ) = |〈 X | X in , P in ; σ in 〉| 2 ; see Eq. (45). In the opposite limit σ /greatermuch σ in , the probability is governed by the detector spatial resolution, 1 √ πσ exp( -1 σ ( X -X in ) 2 ), which reduces to δ ( X -X in ) in the limit σ → 0 (while keeping σ /greatermuch σ in ), whereas in the opposite (physically more reasonable) limit σ → ∞ , the position dispersion becomes infinitely large, and the information of X in is also lost. \nWhen we do not observe the position, we integrate the probability (80) over all the possible positions: \n∫ R d X 2 π |〈 X,P ; σ | X in , P in ; σ in 〉| 2 = √ σ red π exp ( -σ red ( P -P in ) 2 ) . (82) \nIn the limit of detector spatial resolution coarser than the initial wave packet size, σ /greatermuch σ in , we obtain σ red → σ in , recovering the momentum-space PDF: [ M mom ̂ ρ in ]( P ) = |〈 P | X in , P in ; σ in 〉| 2 ; see Eq. (46). In the opposite limit σ /lessmuch σ in , the probability is governed by the detector resolution, √ σ π exp( -σ ( P -P in ) 2 ), which reduces to δ ( P -P in ) in the limit σ → ∞ (while keeping σ /lessmuch σ in ), whereas in the opposite (physically more reasonable) limit σ → 0, the momentum dispersion becomes infinitely large, and the information of P in is also lost. \nFor reader's ease, we list some matrix elements, which may be useful to derive expressions below: Using the derivative representation (36), we obtain \n〈 X,P ; σ | x | X in , P in , σ in 〉 = ( X -iσ red ( P -P in ) ) 〈 X,P ; σ | X in , P in , σ in 〉 , (83) \nX := σ red ( X σ + X in σ in ) , P := σP + σ in P in σ + σ in . (85) \n̂ 〈 X,P ; σ | ̂ p | X in , P in , σ in 〉 = ( P + i X -X in σ sum ) 〈 X,P ; σ | X in , P in ; σ in 〉 , (84) \nwhere", '5.1 Projective measurement onto phase space': "We assert that the transition probability (80) can be regarded in the LT formalism as the result of the POVM measurement onto phase space M ph defined by, ∀ ρ ∈ Z ( H ), 10 \n̂ [ M ph ρ ]( X,P ) = Tr [ | X,P ; σ 〉 〈 X,P ; σ | ρ ] . (86) \nGiven an initial state | X in , P in ; σ in 〉 , or more precisely the initial pure state (44), we recover the PDF (80): \nFor some reader's ease, we list some results of the integrals: \n[ M ph ̂ ρ in ]( X,P ) = |〈 X,P ; σ | X in , P in ; σ in 〉| 2 . (87) \n〈 X 〉 M ph ̂ ρ in = X in , 〈 P 〉 M ph ̂ ρ in = P in , 〈 XP 〉 M ph ̂ ρ in = X in P in , 〈 X 〉 M ph ̂ ρ in = X in , 〈 P 〉 M ph ̂ ρ in = P in , 〈 XP 〉 M ph ̂ ρ in = X in P in , 〈 X 2 〉 M ph ̂ ρ in = X 2 in + σ sum 2 , 〈 P 2 〉 M ph ̂ ρ in = P 2 in + 1 2 σ red , 〈 X 2 〉 M ph ̂ ρ in = X 2 in + σ 2 in 2 σ sum , 〈 P 2 〉 M ph ̂ ρ in = P 2 in + σ 2 σ in σ sum , (88) \nwhere X ( X,P ) := X and P ( X,P ) := P as well as X ( X,P ) := X and P ( X,P ) := P , with X and P being given in Eq. (85), respectively. \nNow we consider the marginalization (27). We define the classical projection (marginalization) processes π pos and π mom by, ∀ p ∈ W ( R 2 ) , \n[ π pos p ]( X ) := ∫ R d P 2 π p ( X,P ) , [ π mom p ]( P ) := ∫ R d X 2 π p ( X,P ) , (89) \nwith their trivial LT adjoints (30): ∀ f ∈ R ( R ), \nMore concretely, \n[ π /star pos f ] ( X,P ) = f ( X ) , [ π /star mom f ]( X,P ) = f ( P ) . (90) \n[ π pos M ph ̂ ρ in ]( X ) = ∫ R d P 2 π |〈 X,P ; σ | X in , P in ; σ in 〉| 2 = 1 √ πσ sum exp ( -( X -X in ) 2 σ sum ) , (91) \n[ π mom M ph ̂ ρ in ]( P ) = ∫ R d X 2 π |〈 X,P ; σ | X in , P in ; σ in 〉| 2 = √ σ red π exp ( -σ red ( P -P in ) 2 ) . (92) For the Gaussian pure state (44), the joint PDF (87) has become separable: \n[ M ph ̂ ρ in ]( X,P ) = 2 π [ π pos M ph ̂ ρ in ]( X ) [ π mom M ph ̂ ρ in ]( P ) . (93) We see that the Gaussian measurement smoothly interpolates the two limiting measurements in the momentum space and in the position space: \n- · In the limit of detector spatial resolution finer than the original wave-packet size, σ /lessmuch σ in , we obtain σ sum → σ in and σ red → σ , and hence \n[ π mom M ph ̂ ρ in ]( P ) → √ σ π exp ( -σ ( P -P in ) 2 ) . (95) \n[ π pos M ph ̂ ρ in ]( X ) → 1 √ πσ in exp ( -( X -X in ) 2 σ in ) = [ M pos ̂ ρ in ]( X ) , (94) \nWe have recovered the original PDF in the position space: |〈 X | X in , P in ; σ in 〉| 2 (see Eq. (45)), whereas the information of the original wave-packet size is lost in the momentum space. Further limit of spatially coarse detector σ →∞ (while keeping σ /lessmuch σ in ) gives [ π mom M ph ̂ ρ in ]( P ) → δ ( P -P in ), whereas in the opposite (physically more reasonable) limit of spatially fine detector σ → 0 makes the momentum dispersion infinitely large, leading to the loss of information of P in . \n- · In the limit of detector spatial resolution coarser than the original wave-packet size, σ /greatermuch σ in , we obtain σ sum → σ and σ red → σ in , and hence \n[ π mom M ph ̂ ρ in ]( P ) → √ σ in π exp ( -σ in ( P -P in ) 2 ) = [ M mom ̂ ρ in ]( P ) . (97) \n[ π pos M ph ̂ ρ in ]( X ) → 1 √ πσ exp ( -1 σ ( X -X in ) 2 ) , (96) \nWe have recovered the original PDF in the momentum space |〈 P | X in , P in ; σ in 〉| 2 (see Eq. (46)), whereas the information of the original wave-packet size is lost in the position space. Further limit of spatially fine detector σ → 0 (while keeping σ /greatermuch σ in ) gives [ π pos M ph ̂ ρ in ]( X ) → δ ( X -X in ), whereas in the opposite (physically more reasonable) limit of spatially coarse detector σ →∞ makes the position dispersion infinitely large, leading to the loss of information of X in .", '5.2 Pullback and pushforward for joint measurement': "One might find the LT adjoint M /star ph slightly less trivial because of the non-orthogonality of the basis (43); see also footnote 10. However, we find that the result is diagonal too: For a given function f ∈ R ( R 2 ) , its LT adjoint is \n̂ M /star ph f = ∫ R 2 d X d P 2 π f ( X,P ) | X,P ; σ 〉 〈 X,P ; σ | , (98) \nwhich can be verified by inserting the expansion (56) into the defining relation 〈 ̂ M /star ph f 〉 ̂ ρ = 〈 f 〉 M ph ̂ ρ . This shows that the LT adjoint maintains its diagonal nature even in the context of the POVM measurement (86). \nThe position and momentum operators have the following decomposition in the Gaussian phase space: \n̂ x = ∫ R 2 d X d P 2 π X | X,P ; σ 〉 〈 X,P ; σ | , ̂ p = ∫ R 2 d X d P 2 π P | X,P ; σ 〉 〈 X,P ; σ | , (99) \nwhich can be verified by sandwiching with 〈 x | and | x ' 〉 for ̂ x , and with 〈 p | and | p ' 〉 for ̂ p . We can also confirm this by sandwiching with 〈 X ' , P ' ; σ | and | X '' , P '' ; σ 〉 , and then using Eq. (43), (83), and (84). Comparing with Eq. (98) for the general LT adjoint/pullback, we obtain its inverse: \nFrom Eq. (88), we obtain their classical semi-norms, \n∥ ∥ ∥ M /star -1 ph ̂ x ∥ ∥ ∥ 2 M ph ̂ ρ in = X 2 in + σ sum 2 , ∥ ∥ ∥ M /star -1 ph ̂ p ∥ ∥ ∥ 2 M ph ̂ ρ in = P 2 in + 1 2 σ red , (101) \n[ M /star -1 ph ̂ x ] ( X,P ) = X, [ M /star -1 ph ̂ p ] ( X,P ) = P. (100) \nand their classical semi-inner product, \n〈 M /star -1 ph ̂ x, M /star -1 ph ̂ p 〉 M ph ̂ ρ in = X in P in . (102) \nSimilarly, we obtain the pushforward of x and p (over ρ ): \n̂ ̂ where X and P are given in Eq. (85). From Eq. (88), we obtain \nand \n̂ ̂ ̂ [ M ph /star x ]( X,P ) = X, [ M ph /star p ]( X,P ) = P, (103) \n‖ M ph /star ̂ x ‖ 2 M ph ̂ ρ in = X 2 in + σ 2 in 2 σ sum , ‖ M ph /star ̂ p ‖ 2 M ph ̂ ρ in = P 2 in + σ 2 σ in σ sum , (104) \nAccordingly, the pullback of pushforward is \n〈 M ph /star ̂ x, M ph /star ̂ p 〉 M ph ̂ ρ in = X in P in . (105) \n̂ M /star ph M ph /star x = ∫ R 2 d X d P 2 π X | X,P ; σ 〉 〈 X,P ; σ | , (106)", '5.3 LT error for joint measurement': '̂ ̂ M /star ph M ph /star ̂ p = ∫ R 2 d X d P 2 π P | X,P ; σ 〉 〈 X,P ; σ | . (107) \nSubtracting the classical semi-norm squared of pushforward (104) from the quantum one of the original operator (48), we obtain the resultant LT errors (14) for x and p : \nWe see that the left-hand side of the LT inequality (19) becomes \n̂ ̂ ε 2 ̂ ρ in [ ̂ x ; M ph ] = σ red 2 , ε 2 ̂ ρ in [ ̂ p ; M ph ] = 1 2 σ sum . (108) \nε ̂ ρ in [ ̂ x ; M ph ] ε ̂ ρ in [ ̂ p ; M ph ] = 1 2 √ σ red σ sum . (109) \nε 2 ̂ ρ in [ ̂ x ; M ph ] → { 0 ( σ → 0) , σ in 2 ( σ →∞ ) , ε 2 ̂ ρ in [ ̂ p ; M ph ] → { 1 2 σ in ( σ → 0) , 0 ( σ →∞ ) . (110) \nIn the limits of spatially fine and coarse detectors σ → 0 and ∞ , respectively, we obtain \nThe product of LT errors (109) goes to zero in both the limits σ → 0 and ∞ , while it takes the maximum value 1 / 4 when σ = σ in .', '5.4 LT inequality for joint measurement': 'Using Eqs. (106) and (107), and then Eqs. (83) and (84), we obtain \n〈 [ ̂ M /star ph M ph /star ̂ x, ̂ p ] 2 i 〉 ρ in = /Ifractur 〈 X ( P + i X -X in σ sum )〉 M ph ̂ ρ in = σ in 2 σ sum , (111) \n̂ 〈 [ ̂ x, ̂ M /star ph M ph /star ̂ p ] 2 i 〉 ̂ ρ in = /Ifractur 〈 ( X -iσ red ( P -P in ) ) ∗ P 〉 M ph ̂ ρ in = σ 2 σ sum , (112) \nwhere /Ifractur denotes the imaginary part and we used Eq. (88) in the last step of each. Subtracting the above two from the measurement-independent part in the former of Eq. (51), we see that the quantum contribution to the LT lower bound (20) vanishes: \nI ̂ ρ in [ ̂ x, ̂ p ; M ph ] = 0 . (113) For the semi-classical contribution to the LT lower bound (21), we subtract the measurementdependent part (102) from the independent part in the latter of Eq. (51). As a result, we find it to vanish: \nR ̂ ρ in [ ̂ x, ̂ p ; M ph ] = 0 . (114) It is important that the LT inequality gives no lower bound for the joint measurement of position and momentum. The LT inequality (19) now becomes \n1 2 √ σ red σ sum ≥ 0 , (115) \nwhich is saturated in the limits σ → 0 and ∞ . The left-hand side takes the maximum value 1 / 4 at σ = σ in .', '5.5 Lee error for joint measurement': 'Subtracting the quantum semi-norm (48) from the classical (101), we obtain the Lee errors (22) for ̂ x and p : \nIt is remarkable that, unlike the LT error, the Lee error is solely determined from the detector resolution, independent of the width of the measured state. \n̂ ˜ ε 2 ̂ ρ in [ ̂ x ; M ph ] = σ 2 , ˜ ε 2 ̂ ρ in [ ̂ p ; M ph ] = 1 2 σ . (116) \nTheir product gives the left-hand side of the Lee inequality: \n˜ ε ̂ ρ in [ ̂ x ; M ph ] ˜ ε ̂ ρ in [ ̂ p ; M ph ] = 1 2 . (117)', '5.6 Lee inequality for joint measurement': "The quantum contribution to the Lee lower bound (25) is identical to that of Heisenberg's, arising from the measurement-independent part, in the former of Eq. (51): \nI 0 ̂ ρ in [ ̂ x, ̂ p ] = 1 2 . (118) \nOn the other hand, subtracting the measurement-dependent part (88) from the independent (51), we find that the semi-classical contribution to the Lee lower bound (26) vanishes: \n˜ R ̂ ρ in [ ̂ x, ̂ p ; M ph ] = 0 . (119) Unlike the LT inequality, the Lee inequality provides the meaningful lower bound, which has turned out to be identical to Heisenberg's, and is invariably saturated for the pure Gaussian initial state.", '6 Summary and discussion': "We have demonstrated that the Gaussian wave-packet formalism can be a practical realization of the joint measurement of position and momentum, shedding light on Heisenberg's original philosophy of the uncertainty principle within the framework of Lee and Tsutsui's error, disturbance, and their uncertainty relations. We have successfully obtained the Lee-Tsutsui (LT) error and the refined Lee error in the context of position-momentum measurement for the first time. Our findings indicate that the LT uncertainty relation, in the limiting case of projective measurement of either position or momentum, becomes trivial: 0 = 0. \nIn contrast, the refined Lee uncertainty relation, focusing on errors for local representability, provides a constant lower bound that remains unaffected by such limits and is invariably saturated for a pure Gaussian initial state. This lower bound aligns with Heisenberg's value. Our study delves into the Gaussian measurement process in the phase space, demonstrating its capability to interpolate smoothly between the projective measurements of position and momentum. \nBuilding on the foundational research of Heisenberg; Kennard and Robertson (KR); and Schrodinger, and the advancements by Arthurs, Kelly, and Goodman (AKG); Ozawa; and Watanabe, Sagawa, and Ueda (WSU), our research deepens the understanding of quantum uncertainty. We leverage the Gaussian wave-packet formalism to provide a comprehensive and tangible approach to the abstract constructs of the LT formalism. This paper advocates for the POVM measurement onto the Gaussian wave-packet basis as a natural method for joint position and momentum measurement, offering concrete expressions for previously abstract concepts like pullback and pushforward within the LT formalism. Our work bridges the theoretical and practical aspects of quantum mechanics, setting the stage for further research and applications in quantum uncertainty. \nIt has been shown that the Gaussian basis used in our current study on a fixed time slice can be extended to the whole spacetime [16, 17, 18, 22]; see Ref. [25] for the complete basis of Lorentz invariant wave packets and Ref. [26] for that of Lorentz covariant ones including the spin degrees of freedom. It would be fascinating to explore the time-energy uncertainty relation on this ground. It is also interesting to show the LT and Lee inequalities for the standard Clauser-Horne-Shimony-Holt (CHSH) inequality as a concrete realization in the finite-dimensional space. These developments will be presented in subsequent publications.", 'Acknowledgement': 'N.O. is indebted to Sho Machiyama for assistance in understanding the LT formalism and to Shogo Tanimura for fostering a welcoming research environment in his group. We are grateful to Jaeha Lee for providing the talk slides, to Kenji Nishiwaki for helpful conversation, and to Juntaro Wada for useful comments. The work of K.O. is in part supported by the JSPS KAKENHI Grant Nos. JP19H01899 and JP21H01107.', 'References': '- [1] W. Heisenberg, Uber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik , Z. Phys. 43 (1927) 172.\n- [2] E. H. Kennard, Zur Quantenmechanik einfacher Bewegungstypen , Z. Phys. 44 (1927) 326.\n- [3] H. P. Robertson, The Uncertainty Principle , Phys. Rev. 34 (1929) 163.\n- [4] E. Schrodinger, About Heisenberg uncertainty relation , Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys. ) 19 (1930) 296 [ quant-ph/9903100 ].\n- [5] E. Arthurs and J. L. Kelly, B.S.T.J. briefs: On the simultaneous measurement of a pair of conjugate observables , Bell Syst. Tech. J. 44 (1965) 725.\n- [6] E. Arthurs and M. S. Goodman, Quantum correlations: A generalized Heisenberg uncertainty relation , Phys. Rev. Lett. 60 (1988) 2447.\n- [7] M. Ozawa, Uncertainty relations for joint measurements of noncommuting observables , Physics Letters A 320 (2004) 367.\n- [8] M. Ozawa, Universally valid reformulation of the heisenberg uncertainty principle on noise and disturbance in measurement , Phys. Rev. A 67 (2003) 042105.\n- [9] H. Yuen and M. Lax, Multiple-parameter quantum estimation and measurement of nonselfadjoint observables , IEEE Trans. Info. Theor. 19 (1973) 740.\n- [10] Y. Watanabe, T. Sagawa and M. Ueda, Uncertainty relation revisited from quantum estimation theory , Phys. Rev. A 84 (2011) 042121.\n- [11] Y. Watanabe and M. Ueda, Quantum estimation theory of error and disturbance in quantum measurement , 1106.2526 .\n- [12] J. Lee and I. Tsutsui, Geometric formulation of universally valid uncertainty relation for error , 2002.04008 .\n- [13] J. Lee and I. Tsutsui, Uncertainty Relation for Errors Focusing on General POVM Measurements with an Example of Two-State Quantum Systems , Entropy 22 (2020) 1222.\n- [14] J. Lee, A Universal Formulation of Uncertainty Relation for Errors under Local Representability , 2203.08197 .'}

2024ApJ...968....4P

We study 31 little red dots LRD detected by JADESNIRCam and covered by the SMILESMIRI survey of which 70 are detected in the two bluest MIRI bands and 40 in redder MIRI filters. The medianquartiles redshifts are inlineformula mmlmath overflowscrollmmlmizmmlmimmlmommlmommlmsubsupmmlmrowmmlmn6.9mmlmnmmlmrowmmlmrowmmlmn5.9mmlmnmmlmrowmmlmrowmmlmn7.7mmlmnmmlmrowmmlmsubsupmmlmath inlineformula 55 spectroscopic. The spectral slopes flatten in the restframe nearinfrared consistent with a 1.6 m stellar bump but bluer than direct pure emission from active galactic nuclei AGN tori. The apparent dominance of stellar emission at these wavelengths for many LRDs expedites stellar mass estimation the medianquartiles are inlineformula mmlmath overflowscrollmmlmilogmmlmimmlmsubmmlmrowmmlmiMmmlmimmlmrowmmlmrowmmlmommlmommlmrowmmlmsubmmlmrowmmlmo stretchytruemmlmommlmrowmmlmsubmmlmrowmmlmiMmmlmimmlmrowmmlmrowmmlmommlmommlmrowmmlmsubmmlmommlmommlmsubsupmmlmrowmmlmn9.4mmlmnmmlmrowmmlmrowmmlmn9.1mmlmnmmlmrowmmlmrowmmlmn9.7mmlmnmmlmrowmmlmsubsupmmlmath inlineformula. The number density of LRDs is 10SUP4.00.1SUP MpcSUP3SUP accounting for 14 3 of the global population of galaxies with similar redshifts and masses. The restframe nearmidinfrared 24 m spectral slope reveals significant amounts of warm dust bolometric attenuation 34 mag. Our spectral energy distribution modeling implies the presence of lt0.4 kpc diameter knots heated by either dustenshrouded OB stars or an AGN producing a similar radiation field obscured by AV gt 10 mag. We find a wide variety in the nature of LRDs. However the bestfitting models for many of them correspond to extremely intense and compact starburst galaxies with massweighted ages 510 Myr very efficient in producing dust with their global energy output dominated by the direct in the flat restframe ultraviolet and optical spectral range and dustrecycled emission from OB stars with some contribution from an obscured AGN in the infrared.

2024-06-01T00:00:00Z

['arXiv:2401.08782', '2024arXiv240108782P', '10.3847/1538-4357/ad38bb', '2024ApJ...968....4P', '10.48550/arXiv.2401.08782']

['Galaxy formation', 'Galaxy evolution', 'High-redshift galaxies', 'Galaxy stellar content', 'Stellar populations', 'Broad band photometry', 'Galaxy ages', 'James Webb Space Telescope', 'Active galactic nuclei', '595', '594', '734', '621', '1622', '184', '576', '2291', '16', 'Astrophysics - Astrophysics of Galaxies', 'Astrophysics - Instrumentation and Methods for Astrophysics']

What Is the Nature of Little Red Dots and what Is Not MIRI SMILES Edition

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']

https://arxiv.org/pdf/2401.08782.pdf

{'What is the nature of Little Red Dots and what is not, MIRI SMILES edition': "Pablo G. P'erez-Gonz'alez , 1 Guillermo Barro , 2 George H. Rieke , 3 Jianwei Lyu , 3 Marcia Rieke , 3 Stacey Alberts , 3 Christina C. Williams , 4 Kevin Hainline , 3 Fengwu Sun , 3 D'avid Pusk'as , 5, 6 Marianna Annunziatella , 1 William M. Baker , 5, 6 Andrew J. Bunker , 7 Eiichi Egami , 3 Zhiyuan Ji , 3 Benjamin D. Johnson , 8 Brant Robertson , 9 Bruno Rodr'ıguez Del Pino , 1 Wiphu Rujopakarn , 10, 11 Irene Shivaei , 1 Sandro Tacchella , 5, 6 Christopher N. A. Willmer , 3 and Chris Willott 12 \n1 Centro de Astrobiolog'ıa (CAB), CSIC-INTA, Ctra. de Ajalvir km 4, Torrej'on de Ardoz, E-28850, Madrid, Spain 2 Department of Physics, University of the Pacific, Stockton, CA 90340 USA \n3 Steward Observatory, University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721, USA 4 NSF's National Optical-Infrared Astronomy Research Laboratory, 950 North Cherry Avenue, Tucson, AZ 85719, USA 5 Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, UK 6 Cavendish Laboratory, University of Cambridge, 19 JJ Thomson Avenue, Cambridge, CB3 0HE, UK 7 Department of Physics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, UK 8 Center for Astrophysics | Harvard & Smithsonian, 60 Garden St., Cambridge MA 02138 USA 9 Department of Astronomy and Astrophysics, University of California, Santa Cruz, 1156 High Street, Santa Cruz, CA 95064, USA 10 National Astronomical Research Institute of Thailand, Don Kaeo, Mae Rim, Chiang Mai 50180, Thailand \n11 Department of Physics, Faculty of Science, Chulalongkorn University, 254 Phayathai Road, Pathumwan, Bangkok 10330, Thailand 12 NRC Herzberg, 5071 West Saanich Rd, Victoria, BC V9E 2E7, Canada", 'ABSTRACT': 'We study 31 little red dots (LRD) detected by JADES/NIRCam and covered by the SMILES/MIRI survey, of which ∼ 70% are detected in the two bluest MIRI bands and 40% in redder MIRI filters. The median/quartiles redshifts are z = 6 . 9 7 . 7 5 . 9 (55% spectroscopic). The spectral slopes flatten in the restframe near-infrared, consistent with a 1.6 µ m stellar bump but bluer than direct pure emission from active galactic nuclei (AGN) tori. The apparent dominance of stellar emission at these wavelengths for many LRDs expedites stellar mass estimation: the median/quartiles are log M ⋆ / M ⊙ = 9 . 4 9 . 7 9 . 1 . The number density of LRDs is 10 -4 . 0 ± 0 . 1 Mpc -3 , accounting for 14 ± 3% of the global population of galaxies with similar redshifts and masses. The rest-frame near/mid-infrared (2-4 µ m) spectral slope reveals significant amounts of warm dust (bolometric attenuation ∼ 3 -4 mag). Our spectral energy distribution modeling implies the presence of < 0 . 4 kpc diameter knots, heated by either dustenshrouded OB stars or an AGN producing a similar radiation field, obscured by A(V) > 10 mag. We find a wide variety in the nature of LRDs. However, the best-fitting models for many of them correspond to extremely intense and compact starburst galaxies with mass-weighted ages 5-10 Myr, very efficient in producing dust, with their global energy output dominated by the direct (in the flat rest-frame ultraviolet and optical spectral range) and dust-recycled emission from OB stars with some contribution from an obscured AGN (in the infrared). \nKeywords: Galaxy formation (595) - Galaxy evolution (594) - High-redshift galaxies (734) - Stellar populations (1622) - Broad band photometry (184) - Galaxy ages (576) - JWST (2291) - Active galactic nuclei(16)', '1. INTRODUCTION': "In the very first month of JWST science operations, Labb'e et al. (2023a) identified a sample of compact sources with distinct spectral energy distributions (SEDs) presenting two clear breaks (Lyman and Balmer) in the NIRCam+HST data. The SEDs were also char- \ncterized by a change of slope: they had red colors at observed wavelengths between ∼ 2 and (at least) ∼ 5 µ m, the range covered by the NIRCam long-wavelength (LW) channels, and a flat SED in the short wavelength (SW) bands. Labb'e et al. (2023a) identified these SWblue+LW-red sources as very massive, M ∗ > 10 10 M ⊙ , maybe significantly-obscured, A(V) > 1 . 5 mag, galaxies \nat z = 7 -9. They have been claimed to represent a new galaxy population revealed through the new capabilities provided by JWST, easily reaching magnitudes down to 28-29 mag from 1 to 5 µ m. \nThe existence of such early massive galaxies was quickly identified as a possible severe problem in our understanding of how galaxies form and evolve. Indeed, the large masses would be very difficult to explain with the current ΛCDM paradigm, since the amount of baryons already transformed into stars could exceed their abundance in early halos (Haslbauer et al. 2022; Boylan-Kolchin 2023; Dekel et al. 2023). \nThis high-mass interpretation was challenged by adopting models with prominent nebular emission, which could account for their red nature due to the non-negligible contribution of emission lines (and nebular continuum) to the broad-band fluxes as they enter the NIRCam passbands for different redshifts, implying significantly smaller (by a factor of 10 or more) stellar masses (see Endsley et al. 2023a,b; P'erez-Gonz'alez et al. 2023a; Desprez et al. 2023). The estimated masses of early galaxies would also be reduced significantly if they had top-heavy stellar initial mass functions (Woodrum et al. 2023; Wang et al. 2023). \nA simpler explanation in some cases is redshift errors or mis-identifications. Indeed, Kocevski et al. (2023) presented spectroscopy for one of the claimed massive galaxies in Labb'e et al. (2023a), showing an overestimation of the photometric redshift which could partly explain the high mass value for this and some other sources, as opposed to the more accurate photometric redshift and (10 times) smaller mass found in P'erezGonz'alez et al. (2023a) for this same source. However, the general agreement among different authors in the determination of photometric redshifts for this type of source implies that redshift errors are not worrisome in a statistical sense (estimations agree well between papers for ∼ 75% of the samples). In addition, some purported LRDs (at the 10-20% level) have been found to be brown dwarfs based on NIRCam colors, proper motions, and spectroscopy (Hainline et al. 2023a; Langeroodi & Hjorth 2023; Burgasser et al. 2023). \nJointly with their colors, the extreme compactness of the bona fide high-redshift objects led to the term Little Red Dots (LRDs, as first suggested by Matthee et al. 2023b). Understanding LRDs has become more complex given that there is a variety of selection techniques that arrive at very similar types of objects in terms of compact morphology and red LW-to-SW colors, with high levels of overlap but also 'contamination' from other types of sources (e.g., not so compact red galaxies or little not-so-red dots). Labb'e et al. (2023a) originally \nlooked for sources with SED breaks, which resulted in a flat or blue SED at wavelengths shorter than ∼ 2 µ mand a very red one at longer wavelengths to ≥ 5 µ m (SWblue+LW-red SEDs). Eventually arriving at similar objects, Barro et al. (2023), Akins et al. (2023), or Labb'e et al. (2023b) used colors such as F277W-F444W in their selection, adding additional constraints in the SW channels. Some LRDs also entered in the selection carried out by P'erez-Gonz'alez et al. (2023a), Barrufet et al. (2023) and Williams et al. (2023a) based on F150WF356W and F150W-F444W (see also red sources selected with F200W-F444W in Rodighiero et al. 2023 and Bisigello et al. 2023), which were in some cases included in larger investigations about the nature and cosmic relevance of 'HST-dark' sources. Finally, complementary to purely photometric techniques, spectroscopic data with less comprehensive photometry have also been able to identify LRDs, mainly based on the selection of strong H α or [OIII] emitters in blind NIRCam/grism spectroscopic surveys (Matthee et al. 2023a,b). \nThe question about the general nature of LRDs was further tangled by finding evidence for the presence of Active Galactic Nuclei (AGN). Kocevski et al. (2023) identified a broad (1000 km s -1 ) component in the H α emission of one galaxy in the Labb'e et al. (2023a) sample, which implied the presence of a 10 7 M ⊙ supermassive black hole (SMBH). This large SMBH mass is consistent with the discoveries of high-redshift AGN in other studies, all indicating that black holes grew rapidly in the early Universe (D'ıaz-Santos et al. 2021; Larson et al. 2023; Maiolino et al. 2023a; Matthee et al. 2023b; Ubler et al. 2023; Harikane et al. 2023; Natarajan et al. 2023; Bogd'an et al. 2023), many lying above the Magorrian et al. (1998) relationship (Pacucci et al. 2023; Stone et al. 2023). Further spectroscopic analyses identified AGN in some additional LRDs and stellar emission in others, even both at different wavelengths in the same LRD (Matthee et al. 2023b; Greene et al. 2023; Killi et al. 2023). ALMA non-detections have also been used to defend the AGN nature of the bulk of the LRD emission (Labb'e et al. 2023b). \nSolving the challenges outlined above requires going further into the mid-infrared where stellar emission and AGNcan be distinguished. MIRI data at observed wavelengths shorter than 8 µ m for a handful of LRDs was presented in Akins et al. (2023), Barro et al. (2023, also with a F1000W detection), indicating that they could be either dusty starbursts (see also Rodighiero et al. 2023) or obscured AGN, but no definitive conclusions could be reached with just the bluest MIRI bands. Williams et al. (2023a) used longer wavelength MIRI data from the SMILES program and found that the averaged SEDs \n(stacked in observed bands) of the LRDs flattened beyond 5 µ m, interpreting this behavior as the expected turnover of a normal stellar SED at rest ∼ 1 . 6 µ m. In addition, and at odds with the AGN interpretation of the ALMA data of LRDs in the A2744 field presented by Labb'e et al. 2023b, Williams et al. (2023a) found that even deeper ALMA observations of LRDs in GOODSS could agree with SED models mostly dominated by stars (with some implications for the global dust emission SED shape). \nAs is clear from the preceding discussion, the exact nature of the LRDs is still unclear, and there is no definitive proof that they are a single type of phenomenon, i.e., they could be a mixture of the various hypotheses advanced, their nature being also affected by selection biases. \nIn this paper, we investigate the nature of LRDs using the best multi-wavelength, large area MIRI data available to date, those coming from the SMILES survey (Lyu et al. 2023). MIRI, in fact, is essential to disentangle the nature of LRDs. If dominated by star formation, the SEDs of LRDs should show a 1.6 µ m bump, redshifted to ∼ 10 -15 µ m at z = 5 -8 (e.g. Williams et al. 2023a). On the other hand, an obscured AGN will yield a steep SED. The significant uncertainty in the relative importance of star formation and nuclear activity needs both spectroscopic (e.g., Kocevski et al. 2023; Killi et al. 2023) and photometric data to cover the widest spectral range possible, as well as a careful SED analysis of a statistically representative sample. In modeling these sources, we will also explore the uncertainties by using four different sets of modeling software with a variety of approaches and assumptions. \nThe main goals of this paper are to select and study LRDs in a comprehensive way, including examples representing a range of properties, and to see how our conclusions about them are influenced by different elaborated modeling approaches. To achieve these ends, the paper is organized as follows. Section 2 presents the NIRCam and MIRI data used to select and characterize a sample of LRDs in the GOODS-S field. Section 3 describes the average SEDs of LRDs and Section 4 presents stellar and AGN emission models that will be used to characterize the physical properties of LRDs on galaxy-by-galaxy and statistical bases in Section 5. Finally, Section 6 discusses and summarizes our conclusions. We also include in this paper three appendices that describe in detail the MIRI data reduction, the dust emission models that are the most important ingredients of the SED modeling, and provide SED fits for the whole sample. \nThroughout the paper, we assume a flat cosmology with Ω M = 0 . 3 , Ω Λ = 0 . 7, and a Hubble constant \nH 0 = 70kms -1 Mpc -1 . We use AB magnitudes (Oke & Gunn 1983). All stellar mass and SFR estimations assume a universal Chabrier (2003) IMF, which is a very relevant assumption for these parameters (but might not be accurate given the extreme conditions).", '2.1. NIRCam-based selection': "The sample of LRDs analyzed in this paper has been gathered from the JWST Advanced Deep Extragalactic Survey, JADES (Eisenstein et al. 2023a) Data Release 2 catalogs (Eisenstein et al. 2023b), which also gathered data from the First Reionization Epoch Spectroscopically Complete Observations (FRESCO, Oesch et al. 2023) and the JWST Extragalactic Mediumband Survey (JEMS, Williams et al. 2023b). We also used the Rieke et al. (2023) catalog for spectroscopic redshifts, which included values measured with data taken by NIRCam (Oesch et al. 2023) and NIRSpec (Bunker et al. 2023). Given the small (nearly pointlike) nature of our sources of interest and the fact that eventually we wanted to analyze SEDs including MIRI data, we used 0.25 '' radius PSF-matched photometry as our fiducial aperture for selection, but also checked the results using smaller radii, more suitable for NIRCam only. We searched for NIRCam SWblue+LW-red sources defined by F277W-F444W > 1 mag and F150W-F200W < 0.5 mag colors, and magnitudes F444W ≤ 28 mag, following the strategies presented in Barro et al. (2023), Labb'e et al. (2023b) and Greene et al. (2023). The two latter references further use a concentration criterion to select compact sources, but we found no need for it after applying the F150WF200W < 0.5 mag cut. In any case, the typical concentration of our original sample is indicated by the flux ratio (after applying the appropriate aperture corrections) of F(F444W) r =0 . 25 '' /F(F444W) r =0 . 10 '' = 1 . 04 1 . 10 0 . 98 (median and quartiles). That is, most of the sources are nearly point-like. \nThe selection procedure is summarized in Figure 1, where we compare our sample with others found in the literature. Overall, our color selection criterion is bluer than that used in Barro et al. (2023), who imposed a redder color cut (F277W-F444W > 1.5 mag instead of 1.0 mag), and very similar to the one used for the sample presented in Labb'e et al. (2023b), although we impose a magnitude cut ∼ 0 . 5 mag deeper in F444W. \nWe further restrict the sample to the area covered by MIRI within the Systematic Mid-infrared Instrument Legacy Extragalactic Survey (SMILES, Lyu et al. 2023), arriving at a sample of 37 galaxies. As shown by Hainline et al. (2023a), this type of color selec- \nFigure 1. The left and central panels show the F 277 W -F 444 W vs . F 444 W color-magnitude and F 277 W -F 444 W vs. F 150 W -F 200 W color-color diagrams, as well as histograms of NIRCam colors, indicating the selection thresholds for LRDs ( F 277 W -F 444 W > 1 mag, F 150 W -F 200 W < 0 . 5 mag and F 444 W < 28 mag; dashed lines) relative to the bulk of the JADES DR2 galaxy catalog. The different colors indicate the subsets of LRDs detected in different MIRI bands: up to F1280W and beyond (Golden Five galaxies), up to F1000W, up to F770W, or not detected in MIRI (at the SMILES depth). Comparison LRD samples from Barro et al. (2023) and Labb'e et al. (2023b) in the CEERS and UNCOVER fields are shown with squares and triangles, respectively. The right panel shows the color vs. redshift diagram for the LRDs and JADES galaxies and the redshift distribution of LRDs (including median and quartiles). The 55% of the LRDs in our paper that have secure NIRSpecand NIRCam-based spectroscopic redshifts are marked with a black dot. \n<!-- image --> \ntion of high redshift galaxies might be contaminated by brown dwarfs. To account for that, and following their analysis, we removed from the original sample the 4 sources with color F115W-F150W < -0.5 mag. Two other LRDs were removed since they were identified with brown dwarfs with proper motion (Hainline et al. 2023a). Our final sample contains 31 galaxies detected in the SMILES 34.9 arcmin 2 area, i.e., the surface density is 0.9 LRD arcmin -2 . Postage stamps in RGB format constructed with NIRCam data are shown in Figure 2. Table 1 provides all relevant information about the selection of our sample of LRDs. \nOur final sample includes 14 sources studied in Williams et al. (2023a), as well as 2 sources included in the sample of high redshift obscured AGN candidates presented in Lyu et al. (2023), one source identified as an AGN in Matthee et al. (2023b), and 5 galaxies selected as high-redshift candidates in Hainline et al. (2023b). Compared to the galaxies in common with Williams et al. (2023a), the sample in this paper is 0.5 mag fainter in F444W (medians and quartiles 26 . 5 27 . 5 26 . 0 mag vs. 26 . 0 26 . 9 25 . 4 mag), and bluer in F277WF444W (+1 . 3 +1 . 7 +1 . 1 mag vs. +1 . 6 +2 . 5 +1 . 5 mag) and F150WF200W (+0 . 0 +0 . 2 -0 . 1 mag vs. +0 . 2 +0 . 3 +0 . 0 mag). The sources in common with Hainline et al. (2023b) are among the faintest (median F444W 27.1 mag), slightly bluer (me- \ndian F277W-F444W and F150W-F200W 1.1 mag and 0.0 mag, respectively), and at the highest redshifts in our sample (all with z > 8 . 2, median 8.4).", '2.2. Mid-infrared properties': 'In this section, we discuss the detections of our sample of LRDs in the MIRI data gathered by SMILES (Lyu et al. 2023). We first describe briefly this dataset, and then discuss the detection fractions in MIRI as a function of wavelength.', '2.2.1. MIRI data description': "The MIRI data gathered by the SMILES survey (program ID 1207, PI: G. Rieke, Lyu et al. 2023) were reduced with JWST pipeline version 1.12.3, reference files in pmap version 1138, which includes improved absolute photometric calibration and aperture corrections based on a better characterization of the PSFs, especially at the shortest wavelengths. \nApart from the official pipeline steps, our reduction includes a special treatment of the background to deal with artifacts seen in the MIRI data (e.g., striping, tree rings, and inhomogeneities in rows and columns). This is achieved with a super-background strategy that was explained briefly in ' Alvarez-M'arquez et al. (2023) and Lyu et al. (2023). Given the importance of reaching the deepest fluxes possible in the analysis of high redshift \n<!-- image --> \nFigure 2. NIRCam F150W+F277W+F444W color composite postages of the sample of 31 LRDs in this paper. North is up, East is left, sizes are 2.5 × 2.5 arcsec 2 . Galaxies detected by MIRI at wavelengths longer than F1280W (Golden Five galaxies), F1000W, and F770W are marked in gold, red, and purple (and the sources are ordered according to this MIRI detections). Those not detected in MIRI are marked in magenta. We display redshifts, with spectroscopic values written with 4 decimals. \nsources presented in this paper, we explain and characterize in detail the bespoke MIRI reduction procedures in Appendix A. \nPhotometry in the MIRI bands was performed in circular apertures with radii of 0.2 '' , 0.3 '' , 0.4 '' , and 0.5 '' , applying the corresponding aperture corrections to each measurement. Uncertainties were calculated following the procedure explained in P'erez-Gonz'alez et al. (2023b), which gathers non-contiguous (i.e., independent) pixels to avoid the effect of correlated noise. Different measurements for our sample of LRDs agreed within 0.1 mag and we eventually chose for each band the weighted mean of all fluxes, typically the one corresponding to ∼ 70% encircled energy for each band. The photometry was revised visually to avoid contamination from cosmic ray showers, which we found to affect one source. We only kept fluxes with S/N > 5 and considered 5 σ upper limits otherwise.", '2.2.2. MIRI detections': 'Out of the 31 LRDs selected with the JADES NIRCam data down to F444W=28 mag, we detect [19,22,12,7,4,2,1,0] galaxies for MIRI bands [F560W,F770W,F1000W,F1280W,F1500W,F1800W, F2100W,F2550W] respectively. The corresponding detection fractions (in %) are [61,71,39,23,13,7,3,0] at > 5 σ levels of [26.1,25.8,24.7,24.3,24.2,23.0,22.6,20.8] mag (these magnitudes are measured in circular apertures of radius equal to the PSF FWHM, corrected to infinite extent). The detection fractions depend strongly on F444W magnitude. For the brightest galaxies, F444W < 26.5 mag, the detection fractions are (in %) [93,100,73,47,20,13,7,0] in [F560W,F770W,F1000W,F1280W,F1500W,F1800W, F2100W,F2550W], respectively. \nWe divided the sample of LRDs into 4 subsets detected up to a given MIRI band: (1) 5 galaxies detected in F1280W and longer wavelengths, hereafter the Golden Five; (2) 7 galaxies detected only up to F1000W (i.e., excluding the previous type); (3) 7 sources detected only up to F770W (excluding the previous types); (4) and, lastly, the 12 galaxies not detected in any MIRI band. The panels and histograms in Figure 1, as well as the postage stamps in Figure 2, show these subsets with different colors. The first figure illustrates that the MIRI detection fraction depends strongly on the F444W magnitude and it is nearly independent of the F150WF200W color. Interestingly, the F277W-F444W color does exhibit some differences between subsets, with the Golden Five being among the bluest, and the F1000W sample among the reddest (we will comment on this in the following sections). It is worth mentioning that only \ntwo of the Golden Five galaxies would be identified as LRDs under a more restrictive color selection of F277WF444W > 1.5 mag.', '2.3. Redshifts': "Atotal of 18 galaxies out of the 31 in the whole sample (55%) have spectroscopic redshifts provided by CANDELS (Guo et al. 2013), FRESCO (Oesch et al. 2023) data (measured by the JADES team) and JADES NIRSpec data (Bunker et al. 2023; Eisenstein et al. 2023b), all of them flagged as highly secure. \nPhotometric redshifts for all galaxies were estimated with eazy (Brammer et al. 2008), using either direct flux measurements for all bands or with a modified version of the program that penalizes solutions implying fluxes above upper limits. We used v1.3 templates, which include a dusty galaxy and a high-EW emissionline galaxy spectrum, and we added a new extreme emission-line galaxy template as well as an AGN+torus model (based on the spectrum in Killi et al. 2023). Based on the comparison with spectroscopic values, we chose the maximum χ 2 photometric redshift as our main solution. We estimated redshifts both using only NIRCam fluxes and also adding MIRI measurements, obtaining better results when using NIRCam only (a conclusion based on the comparison with spectroscopic values). \nThe quality of the photometric redshifts, when compared to spectroscopic values, is remarkably high, partly as a consequence of the strong emission lines present in many of the LRDs in our sample. The average absolute difference between photometric and spectroscopic values is 0.01, and we do not have any outliers (defined as galaxies with ∆( z ) / (1 + z ) > 0 . 1). \nA histogram of the redshift distribution of our sample is given in Figure 1. The distribution is relatively flat between z ∼ 5 and z ∼ 9, peaking at z ∼ 7. This behavior is probably related to the fact that strong emission lines entering the F444W filter affect the selection significantly (as also seen in extreme emission line galaxies included in P'erez-Gonz'alez et al. 2023a). These lines are mainly [OIII]+H β and H α +[NII]+[SII], which translate to z ∼ 8 and z ∼ 6 for the lines lying within the F444W passband. Summarizing, the typical redshift (median and quartiles) of LRDs selected down to F444W ∼ 28 mag is z = 6 . 9 7 . 7 5 . 9 .", '2.4. Additional SED constraints': 'Apart from the NIRCam and MIRI fluxes, we used Hubble ACS fluxes measured in Hubble Legacy Field images (Illingworth et al. 2016; Whitaker et al. 2019). Except for one source, not covered by the NIRCam SW \nchannels, we did not use WFC3 data. We checked that none of the LRDs in this paper are detected by Spitzer with the MIPS instrument or by Herschel with PACS and SPIRE. Taking into consideration the catalogs used in Barro et al. (2019), we imposed in our SED analysis 5 σ upper limits of 30 µ Jy for the MIPS 24 µ m band, and 4, 5, 9, 11, and 11 mJy for Herschel bands at 100, 160, 250, 350, and 500 µ m, respectively.', '3.1. Average spectral energy distributions': "Figure 3 shows the rest-frame SEDs, normalized at 0.4 µ m, and their average (more specifically, the weighted mean every 10 points calculated on a wavelength-ordered flux list) for the LRDs detected by MIRI and those with just upper limits. The data are compared qualitatively with some relevant templates to understand what the MIRI observations are telling us about the general nature of LRDs. \nOverall, the plots highlight very clearly the characteristic flat, blue UV continuum plus steep, red optical continuum of LRDs. Such peculiar SEDs are difficult to model with traditional stellar population models under conventional assumptions and therefore require more complex or unusual treatment (to avoid biases in the determination of stellar masses, for example). Recent works have put forward different solutions to try to explain the emission of the LRDs using: strong, high EW( > 1000 ˚ A) emission lines that boost even the broadband photometry (Endsley et al. 2023a,b; Matthee et al. 2023b; Furtak et al. 2023; P'erez-Gonz'alez et al. 2023a; Desprez et al. 2023), complex stellar models with a flexible treatment of the dust attenuation (Labb'e et al. 2023a; Barro et al. 2023; Akins et al. 2023; Williams et al. 2023a), or hybrid models that combine stellar and AGN-driven emission with either component dominating different spectral regions or the full SED (Kocevski et al. 2023; Barro et al. 2023; Labb'e et al. 2023b; Greene et al. 2023). \nThe fully AGN-dominated models have received more attention lately based on the spectroscopic observations of LRDs showing strong, and sometimes broad emission lines (FWHM > 2000 km/s; Kocevski et al. 2023; Harikane et al. 2023; Matthee et al. 2023b; Greene et al. 2023; Maiolino et al. 2023b). Briefly, these models combine AGN continuum emission from 1) a highly obscured accretion disk, 2) a small fraction ( ∼ 3%) of unobscured, scattered light, and 3) the dust emission of the surrounding torus (overall similar to the Polletta et al. 2007 torus template shown in Figure 3). This model also explains the presence of both narrow and broad emission lines be- \nuse there is a direct sight line (albeit obscured) to the Broad Line Region, and it predicts that the red optical colors extend toward the NIR, which is roughly consistent with results showing that the handful of LRDs observed in the short-wavelength MIRI bands (F560W and F770W) continue the steep SED trend (Barro et al. 2023). \nFigure 3 indicates that the stacked SED of the SMILES LRDs agrees well with the results from previous works based on NIRCam data: the rest-frame UV up to 0.4 µ m nicely matches the emission of an unobscured, or slightly reddened, QSO, which is consistent with the low-luminosity scattered light component of the accretion disk. Remarkably, the average SED presents a bump around the MgII emission feature, indicating that the AGN makes a non-negligible contribution to the rest-frame UV spectral range. However, the scatter around the median suggests that there might be sources other than an AGN contributing to the UV, e.g., emission from stars in the host galaxy. \nThe rest-frame optical presents a very steep slope from 0.4 to 1 µ mconsistent with the emission of a heavily obscured supermassive black hole, here indicated with the Polletta et al. (2007) circumnuclear torus template. In addition, aided by the inclusion of the JEMS mediumband photometry, the stack exhibits statistically significant peaks at the wavelengths where we would expect strong H α +[NII]+[SII] and [OIII]+H β emission. The average LRD SED might be identified with a torus up to ∼ 0 . 7 µ m; although the torus does not have optical emission lines, the lines from the central engine can masquerade as a steeper (dustier) continuum. In summary, up to the reddest bands covered by NIRCam (F444W, F460M, and F480M filters, typically probing up to H α ), the average UV-optical SED presented in this paper roughly matches an obscured QSO spectrum plus a torus. \nHowever, the interpretation of LRD SEDs being dominated by a pure and/or complex (with a gray attenuation law) AGN model is not supported when we consider the rest-frame near-infrared (NIR) spectral region probed by MIRI, the key addition in this paper, as first presented in Williams et al. (2023a). The MIRI data (colored circles in Figure 3) and average SED are consistently lower than the predictions of the Polletta et al. (2007) torus template. A more general evaluation can be made by applying different reddening levels to a standard unreddened AGN template. In the ∼ 0.5-3 µ m region where the MIRI measurements fall, there is general agreement on the shape of the intrinsic template, as determined by averaging the behavior of large samples (e.g., Elvis et al. 1994; Richards et al. 2006; Glikman et al. 2006; Lyu et al. 2017). Lyu & Rieke (2022b) show \nFigure 3. Stacked LRD SEDs for sources detected by MIRI (left panel) and not (right panel), normalized at rest-frame wavelength 0.4 µ m. Black points show NIRCam fluxes for individual sources, while rainbow color points show MIRI fluxes. Arrows depict 5 σ upper limits. The average SED is shown with a magenta line (10 point averages) and its rms with a magenta shaded region. The observed average SED for LRDs is compared to 5 different templates. The orange lines show an average QSO spectrum (see text for details), and the same template extincted by A(V) = 2 mag using a Calzetti et al. (2000) attenuation law. The red lines stand for the torus template in Polletta et al. (2007), normalized to the same wavelength as the observations (continuous line) as well as normalized at the 2 µ m average SED level (dashed line) in the case of the MIRI detections plot. The blue line shows the model for an intense starburst presented in Siebenmorgen & Krugel (2007), more specifically, the sub-LIRG model with a size of 350 pc, 90% of the total luminosity coming from OB stars, optical attenuation of A(V) = 36 mag, and 10 3 cm 3 density of dust in hot spots (gas clouds surrounding and directly heated by OB stars). \n<!-- image --> \nthat this is an appropriate average behavior including the variations such as warm dust deficient and hot dust deficient cases (Lyu et al. 2017). It is therefore the appropriate foundation for fitting the LRD photometry, but with the large attenuations required to match the SED slope. To be specific, we adopt a SED from Glikman et al. (2006), extended to the mid-infrared (MIR) with models from Siebenmorgen et al. (2015). Figure 3 shows that there is no suitable solution; if reddening is selected to match the behavior between 0.5 and 1 µ m, the SED falls substantially higher than the MIRI photometry near 2 µ m (rest). Reducing the reddening to solve this problem yields a SED too low near 1 µ m. Overall, the MIRI photometry indicates a change to a significantly bluer slope at ∼ 1 µ m and even stabilizes at a roughly flat value at 1-2 µ m, where stellar emission is expected to peak. This is a fundamentally different shape from AGN SEDs. There are also hints of another steepening at > 3 µ m, but this behavior is traced only by a small subset of LRDs detected beyond 1.6 µ m restframe (see discussion in § 5.2); deeper MIRI data at the longest wavelengths would be necessary to probe this region. \nThe comparison of photometry and templates in Figure 3 strongly suggests that the rest-NIR emission of the LRDs (around 1-2 µ m) is very different from the hot-dust (T ∼ 1500 K) dominated emission of the typical QSO templates used in recent LRD papers dominated by the fit of NIRCam data alone (e.g., Barro et al. 2023, Labb'e et al. 2023b, or Greene et al. 2023). Reproducing the observed MIRI colors would require a more flexible modeling of the torus emission that can vary its IR luminosity (i.e., the relative flux level with respect to other components such as stars or the accretion disk) and the wavelength at which the emission peaks. This can be accomplished using either radiative transfer models (e.g., Nenkova et al. 2008; Stalevski et al. 2012, 2016; Siebenmorgen et al. 2015), modified blackbody templates (Kim et al. 2015, Hern'an-Caballero et al. 2016 or Killi et al. 2023), or empirical templates that can account for separate contributions of the torus dust emission at different temperatures (e.g., warm and hot, as in Lyu et al. 2017). However, such models require significant departures from typical AGN behavior in the red and near infrared. Alternatively, the result could indicate that the SED at 0.5-2.0 µ m is not AGN-dominated but rather stellar-dominated, as recently discussed in \nWilliams et al. (2023a). We demonstrate this point by plotting in Figure 3 the torus template normalized to the average SED level of LRDs at 2 µ m (dashed red line). If the emission at rest-frame wavelengths longer than 2 µ m is dominated by a torus, then the flux at around 1 µ mmust be dominated by a different component, such as the accretion disk or, more likely, stars, whose emission peaks exactly in that spectral zone (except for very young ages). \nThe right panel of Figure 3 shows the average SED of LRDs that are not detected by MIRI. The UV range is bluer than in the MIRI-detected case, more in line with the slope of the average QSO spectrum. The change in slope at 0.4 µ m is also clear (this being one of the main selection criteria of our sample). Strong emission lines might also be present; indeed we observe a similar SED rise in the [OIII] region, and two NIRCam points may be revealing strong H α emission. However, the small number of sources does not support detecting emission lines in the average SED as well as in the case for MIRI detections. For these sources, the MIRI upper limits also imply that the SED flattens at ∼ 1 µ m, i.e., stellar emission could dominate this spectral range. However, the SED possibly has a steepening compatible with the presence of strong hot dust emission at wavelengths around 2 µ m and redward; the currently available MIRI data cannot constrain this spectral region appropriately.", '3.2. MIRI colors': 'Figure 4 illustrates further what the MIRI observations reveal about the nature of LRDs, probing further into the redshift effects. Two of the top panels show the expected NIRCam-MIRI colors for different types of templates as a function of redshift. The nature of the LRD emission in the spectral range probed by the F444W-F560W or F444W-F770W colors cannot be disentangled: both AGN and stellar dominated emission would present very similar color differences of about 0.5 mag, and the influence of emission lines in one and/or the other filter at specific redshifts complicates the problem. Only when observing at longer wavelengths with MIRI, 10-15 µ m, would the data start to differentiate between the models. For our detections at these wavelengths, the MIRI data reveal flat SEDs, F770W-F1000W, and F770W-F1280W values, bluer than 0.5 mag for 60% of the sample and consistent with stellar dominated emission presenting a 1.6 µ m bump. We note however that stellar emission alone leads to flat or even negative MIRI colors, whereas the median colors of the LRDs, including the upper limits, show small, but increasingly redder colors with wavelength which suggest that there is emission from another \nsource, such as nebular continuum or dust, but not as dominant as in the QSO templates (see also discussion in § 5.2). \nIn summary, as found in the preceding section, this more detailed analysis shows how SEDs constructed up to the reddest NIRCam bands are compatible with a dominant AGN with composite emission: blue in the UV, with emission lines coming from the broad-line region, and starting to rise as expected for dust torus emission up to 5 µ m. MIRI data at 5.6 µ mdeviates very little from this scenario, but at 7.7 µ m starts to tell a different story. Even longer wavelength data clearly point to a restricted contribution of a dust torus at rest-frame wavelengths around 1-2 µ m, where the stellar emission is expected to peak. \nBoth in Figures 3 and 4, we show one of the radiative transfer models of dust-rich compact nuclear starbursts and luminous infrared galaxies (LIRGs) presented in Siebenmorgen & Krugel (2007). In particular, we plot (in blue) the template for an extreme starburst with a 350 pc star-forming region with a sub-LIRG-like luminosity (10 10 . 1 L ⊙ ; but LIRG-like luminosity density), with 90% of its luminosity arising from OB stars embedded in very thick [A(V) = 36 mag] and dense (10 3 cm -3 hydrogen density) birth clouds. Remarkably, this stellar-only model, mainly based on the presence of highly embedded OB stars within a more general stellar population, nicely reproduces the main characteristics of LRDs across the whole spectrum: the change in spectral slope at rest-frame ∼ 0 . 4 µ m and the flattening of the SED probed by MIRI. It, however, presents a steeper slope in the FUV, but we remark the stellar emission is constant across all these models and does not take into account nebular emission. The redshift dependence of the color of this template, shown in Figure 4, is also completely consistent with the measurements. \nWith the general trends discussed with Figures 3 and 4 in mind, in the following section we investigate several different new scenarios to model the NIRCam+MIRI SEDs of each LRD in our sample.', '4. SED MODELING CODES FOR LRDS': 'In this section, we present the detailed SED fitting methods that will be applied to the analysis of individual objects in our sample to infer their physical properties. The results will be presented in the next section, concentrating on those galaxies that have spectroscopic redshifts and are detected in several MIRI bands. The SEDs of all our LRDs were fitted with 4 codes, each one with different assumptions and approaches: synthesizer-AGN , prospector-SF , prospectorAGN , and prospector-AGN +. \n<!-- image --> \n<!-- image --> \nFigure 4. NIRCam and MIRI colors vs. redshift for all LRDs detected at least in F770W. The lines illustrate the color-redshift tracks for the same templates shown in Figure 3. Galaxies from different samples are marked with different colors. The first two panels in the top row show the F444W-F560W(F770W) colors which are available for the majority of the sources. The last panel at the top and the bottom panels show the MIRI-MIRI colors relative to F770W. The median and scatter of the colors are indicated in the top left corner. \n<!-- image -->', '4.1. The synthesizer-AGN code': "For the synthesizer-AGN code (P'erez-Gonz'alez et al. 2003, 2008) we assumed a composite stellar population with both a young and a more evolved starburst. The stellar emission is described by the Bruzual & Charlot (2003) models, assuming a Chabrier (2003) IMF with stellar mass limits between 0.1 and 100 M ⊙ . Both starforming events are characterized by a Star Formation History (SFH) described by delayed exponential function with timescales between 1 Myr and 1 Gyr, and with ages from 1 Myr up to the age of the Universe at the redshift of the source. Nebular emission is also considered (see P'erez-Gonz'alez et al. 2003 for details). The attenuations of the stellar and nebular emission are described by the Calzetti et al. (2000) law, with A(V) values ranging from 0 to 4 mag for each population, considered to have completely independent attenuations. The dust emission is also modeled (beyond what the AGN might contribute) with the radiative transfer templates of nuclear starbursts presented in Siebenmorgen & Krugel (2007). We use the templates for 350 pc diameter starforming regions, with different SED shapes parametrized by the total IR luminosity (from sub-LIRG to Hyper- \nLIRG ranges), visual extinction of the starburst (from 2 to 144 mag), ratio to the total luminosity (from 4090%) of the emission arising from OB stars embedded in dense molecular clouds of a variety of hydrogen number densities (from 10 2 to 10 4 cm 3 , assuming a gas-to-dust ratio of 150). \nApart from the stellar population, we include an AGN component described by a QSO template constructed with the average spectrum of QSOs presented in Vanden Berk et al. (2001) and Glikman et al. (2006), extended to the far-IR with the dust torus template in Siebenmorgen et al. (2015). This model was shown in Figure 3. The attenuation of the AGN model is also assumed to follow the Calzetti et al. (2000) law, and we impose an energy balance criterion so the energy absorbed by dust is reradiated in the mid- and far-IR, scaling up the dust torus emission. \nThere are 9 free parameters to describe the stellar population. Apart from the 4 parameters for each stellar population, i.e., age, timescale, attenuation and metallicity, we fit the stellar mass ratio between the youngest stars and the total mass, the burst strength. We add to the 9 stellar parameters, 2 more that describe the scal- \ning of the AGN template with respect to the stars and its attenuation. \nWe remark that synthesizer-AGN assumes independent extinctions for the 2 stellar populations in the host (as well as for the AGN). This is different from what is done by the next codes, where the attenuation for the stellar emission is governed by common parameters (visual attenuation and attenuation law slope), although internally young stars are treated differently from old stars.", '4.2. The prospector-SF code': 'For prospector-SF (Leja et al. 2017, Johnson et al. 2021), we use the MIST stellar evolutionary tracks and isochrones (Choi et al. 2016), a Chabrier (2003) IMF, a range in stellar metallicity between -1.0 and 0.19, and gas phase metallicity log(Z / Z ⊙ ) between -2.0 and 0.5. For the SFH, we use a non-parametric model, following the flexible SFH prescription (Leja et al. 2019a) with 6 time bins and the bursty-continuity prior (Tacchella et al. 2022). The ionization parameter for the nebular emission ranges from log U = -4 to -1. The nebular line and continuum emission are generated using CLOUDY (Ferland et al. 1998). We base the attenuation law on a dust model that combines: (1) the Charlot & Fall (2000) two-component approach, birth-cloud vs. diffuse dust screens; and (2) the Kriek & Conroy (2013a) method that parametrizes the diffuse component as a combination of a Calzetti attenuation plus a Lorentzian Drude to model the strength of the UV bump. Both components of the diffuse dust are then modulated by a power-law factor n that varies the slope of the attenuation between -1 to 0.4 relative to Calzetti ( n = 0). \nThis model includes 5 free parameters that control the ratio of SFR in six adjacent time bins; the first two bins are spaced at 0 -5 Myr and 5 -10 Myr of lookback time, and the remaining four bins are log-spaced to a maximum age of 100 Myr. In addition, it fits (1) for the ratio of the nebular to diffuse attenuation, which ranges between 0 and 2, but follows a clipped normal prior centered on 1, and also (2) for the dust index n .', '4.3. The prospector-AGN code': "A third type of fit, prospector-AGN hereafter, makes use of a hybrid galaxy+AGN model similar to the one described in Barro et al. (2023). Briefly, we use a combination of a stellar emission component that dominates in the rest-frame UV and an AGN one that dominates in the optical to IR wavelengths. By construction, this is the only model where the AGN makes up the bulk of the luminosity and, as such, it serves as a feasibility test for AGN-dominated SEDs (forced to be \ndominant, not free as in the case of the prospectorAGN + code presented below). The stellar emission is modeled with Prospector using a parametric delayedτ SFH with no attenuation. The AGN has two distinct components that account for the emission of the accretion disk and the torus, respectively. The accretion disk is modeled after the empirical QSO templates mentioned above from 0.1 µ m up to 0.6 µ m rest-frame, followed by a power-law (f ν = ν α , see for example Hern'an-Caballero et al. 2016 or more recently in Bosman et al. 2023 and Killi et al. 2023) with variable spectral index ranging between α =0 to 0.5, with default value of α = 1 / 3, and attenuated with a Calzetti et al. (2000) law. The torus emission is modeled using the clumpy torus models from Nenkova et al. (2008) included in prospector (Leja et al. 2018). These have two free parameters: the optical depth, which ranges between τ V = 1 and 150, and the total IR luminosity. Since these parameters are independent of each other, the models allow for AGN-dominated SEDs where the emission from the torus dominates at longer wavelengths ( λ ≳ 2 µ m), as opposed to the typical QSO templates where the torus emission dominates at λ ∼ 1 µ m or blueward.", '4.4. The prospector-AGN+ code': 'Last but not least, prospector-AGN + is a modified version of the original prospector code with the Flexible Stellar Population Synthesis (FSPS; Conroy et al. 2009; Conroy & Gunn 2010) for the stellar component. We assumed a Kroupa (2001) initial mass function and delayed-tau star-formation history. The stellar nebular line and continuum emission are also turned-on, as preconfigured in FSPS (Byler et al. 2017). We adopted the Calzetti attenuation curve with a flexible slope as introduced in Kriek & Conroy (2013b). For the galaxy dust emission, since the objects of this work have z ≳ 5 . 0, we adopted the empirical IR SED model of Haro 11, a lowmetallicity star-bursting dwarf galaxy that is believed to share typical features of first-generation galaxies in the early Universe (Lyu et al. 2016; De Rossi et al. 2018). \nThis code uses a set of semi-empirical AGN SED models, which have been optimized for AGN identification and characterization (Lyu et al. 2022, 2023). For this work, we adopted a similar model configuration as in Lyu et al. (2023) for the SMILES+JADES AGN identification: the AGN component includes both the AGNpowered continuum from the UV to the far-IR and the narrow and broad emission lines from the UV to the NIR derived based on empirical observations. The continuum shape and line strengths of the AGN SED can be modified by a hybrid extinction configuration featuring the SMC-like curve for the commonly seen UV-optical \nextinction in Type-1 AGNs and an empirical attenuation law for the IR obscuration. Further details can be found in Lyu et al. (2023) and references therein. \nIn total, this code uses 7 free parameters to describe the stellar component and 2 for the AGN.', '4.5. Summary': 'These four codes let us compare models of the LRDs with a wide variety of input assumptions. This is important to let us deduce which properties are likely to be intrinsic rather than due to modeling issues. synthesizer-AGN , prospectorSF , and prospector-AGN+ use three different prescriptions for the stellar populations, with three varieties for the interstellar extinction law. synthesizerAGN and prospector-AGN+ treat the stellar-heated dust emission in two different ways. synthesizerAGN , prospector-AGN , and prospector-AGN+ use three different models for the AGN emission. This latter variety is particularly important given the diversity of possibilities for AGN SEDs and the importance of capturing this variety in modeling LRDs. The first case, synthesizer-AGN , uses an empirical fixed model SED. prospector-AGN + fits empirical templates obscured by an attenuation law derived specifically for AGNs (see e.g. Lyu et al. 2017), while prospector-AGN uses a declining power-law with α = 1 / 3 in place of the empirical templates.', '5.1. Galaxy by galaxy analysis for the Golden Five LRDs': 'In this section, we present our analysis of the SEDs of the LRDs detected in several MIRI bands. We concentrate our discussion of properties on the five galaxies detected in F1280W and redder bands, which we call the Golden Five sample. We compare the global results of the SED fitting provided by each one of our 4 different codes in different subsections, and comparatively discuss how they fit the UV, optical and near-infrared spectral regions. At the end of the section, we describe the properties for the galaxies detected up to F1000W and we briefly discuss the rest of the sample. A more detailed discussion of all the galaxies in our our sample is presented in Appendix C. The main physical properties of all the LRDs are given in Table 2.', '5.1.1. Results for the Golden Five galaxies with synthesizer-AGN': "With synthesizer-AGN , the rest-frame UV spectral range is fitted with a young stellar population for all the galaxies in the Golden Five sample except for \nJADES -57356, which is fitted with the QSO component. For this galaxy, the fact that no emission lines are observed in the optical ([OIII] or H α ), but the UV spectral range is flat, favors the contribution of a QSO to the UV emission, outshined by the stars in the optical. Very young stars would contribute strong emission lines in the optical (as in JADES -204851), and thus are disfavored by this modeling. The different UV (relatively flat) slopes observed for the Golden Five sample are reproduced with ∼ 1 Myr old (mass-weighted age) stellar population and A(V) = 0 . 6 mag dust attenuation, which is also able to reproduce the emission lines observed for four of the Golden Five galaxies. \nThe rest-frame optical range of the Golden Five sample is dominated by slightly more evolved stars in all cases, typically of 10-100 Myr mass-weighted age. They account for most of the stellar mass (typically, more then 90%) but do not contribute much, in relative terms with respect to the younger stars, to the optical emission lines. This more evolved stellar populations present high attenuations, typically A(V) = 2 -3 mag. \nIf we consider the ratio of the attenuation of the farUV with respect the optical emission arising from all the stellar populations included in the synthesizerAGN modeling, κ FUV , we typically find flatter slopes, ∼ 1 . 5 -2 . 0, than what is expected for the Calzetti et al. (2000) attenuation law ( κ FUV = 2 . 6). This effect is, in fact, common to all modeling techniques, they all typically obtain gray attenuation laws. \nFinally, the rest-frame near/mid-infrared emission of the Golden Five LRDs is fitted with dust combined with star formation in three Golden Five galaxies, and dust in an AGN torus in the other two. The heating source is indicated by the possible detection of a most probably star-formation related 3 µ m PAH feature in JADES -57356, and by the differences in slopes of the SEDs for the other sources, steeper in the case of AGNdominated fits (JADES -211388 and JADES -79803). \nThe synthesizer-AGN code presents the smallest χ 2 values for the fits of the SEDs of all the Golden Five galaxies except one. \n5.1.2. Results for the Golden Five galaxies with prospector-SF \nBy construction, prospector-SF only considers stars and dust heated by stars to fit the SEDs. In general, the χ 2 values obtained by this code and the following one are larger than those obtained with synthesizer-AGN (except for JADES -219000) and smaller than prospector-AGN +. \nThe prospector-SF results imply a ∼ 30 Myr stellar population (mass-weighted age), with large attenuations A(V) = 3 mag and a flat attenuation law. These \nFigure 5. SED fitting results for source JADES -57356, the LRD in our sample detected up to F2100W. The four upper panels show: (1) the fits for synthesizer-AGN + on the top left panel, including the individual components of the model, i.e., young, old, and all stars in blue, cyan, and gray, (un)obscured AGN in orange, and regular dust emission (in principle, linked to star formation) in red; (2) results for Prospector-AGN + on the top right panel, showing the three components, stars in gray, AGN in orange, and star-formation heated dust in red; (3) Prospector-SF on the bottom left, showing stars in gray and star-formation heated dust in red; and (4) Prospector-AGN on the bottom right, including young stars in blue, total AGN emission in orange, with the torus emission shown with a dashed line. Number of bands fitted and direct (i.e., not reduced) χ 2 values are provided, as well a stellar masses, stellar mass-weighted ages, V -band stellar attenuation, ratio between the FUV and optical stellar attenuation, and fraction of bolometric luminosity coming from the AGN. The fits include NIRCam bands, shown in black, and MIRI fluxes, in color; upper limits, depicted with triangles, are also used. Below the SEDs, we give 10 '' × 10 '' postage stamps in NIRCam (upper row), MIRI (middle row), and MIRI convolved with a 5-pixels wide tophat filter, the LRD being marked with a 0.3 '' radius circle, and the S/N provided (when it is above 5). \n<!-- image --> \nFigure 6. Same as Figure 5, but for source JADES -204851, a source detected up to F1800W. \n<!-- image --> \nFigure 7. Same as Figure 5, but for source JADES -219000, a source detected up to F1500W. \n<!-- image --> \nFigure 8. Same as Figure 5, but for source JADES -211388, a source detected up to F1280W. \n<!-- image --> \nFigure 9. Same as Figure 5, but for source JADES -79803, a source detected up to F1280W. \n<!-- image --> \nproperties translate to larger stellar masses compared to other codes (a ∼ 0.5 dex difference with respect to synthesizer-AGN , for example). We remark that the dust emission models are significantly colder than what is obtained with synthesizer-AGN (see Appendix B).", '5.1.3. Results for the Golden Five galaxies with prospector-AGN': 'The main distinct characteristic of the results achieved with prospector-AGN for the Golden Five galaxies is the smaller stellar mass. Given that, by construction, this code only fits the rest-frame UV spectral region with stars, and the optical and infrared with AGN templates (including emission from the accretion disk and the torus), the stellar masses are up to a factor of 100 smaller than those estimated with the other codes. In general, better fits in terms of χ 2 values are obtained with prospector-AGN compared to prospector-SF , except for JADES -57356, the galaxy with possible PAH emission. But better fits are obtained with the mixed models in synthesizer-AGN (favoring stellar emission in the UV and optical, a variety of results in the nearinfrared).', '5.1.4. Results for the Golden Five galaxies with prospector-AGN+': 'The prospector-AGN + code fits to the SEDs of the Golden Five galaxies are very similar to those outlined for synthesizer-AGN . The rest-frame UV and optical spectral ranges are dominated by stars, except at 0.8-1.0 µ m for two galaxies (JADES -211388 and JADES -79803) whose infrared emission was fitted with an AGN torus model. This was also the best solution for synthesizer-AGN . The only significant difference with respect to the results achieved with synthesizer-AGN is found for the infrared emission of JADES -219000, whose slope if we take the MIRI upper limits at face value is too steep for a star formation model, so prospector-AGN + (which used upper limits as regular points) prefers a torus template (while the synthesizer-AGN star-formation heated dust model lies below the upper limits). The typical mass-weighted ages for the fits with the prospector-AGN + code are around 150 Myr, with extinctions around A(V) = 1 . 5 mag with a flatter attenuation law compared to the Calzetti et al. (2000) recipe. \nA detailed analysis of the SED fits for each galaxy in the Golden Five sample is presented in Appendix C. \n5.2. Implications for the nature of the NIR emission of the LRDs \nOverall, all four models fit the characteristic bimodal SED of the Golden Five LRDs relatively well. Qual- \nitatively, the UV-optical spectral range is dominated either by 1) stars: two young populations with very different attenuations in synthesizer-AGN , or a single population with an extremely gray attenuation in prospector-SF and prospector-AGN +; or 2) emission from an obscured accretion disk combined with a young stellar population that contributes only in the UV. In the next section, we discuss the implied stellar masses and other stellar properties for the Golden Five and all other LRDs that have very similar UV-tooptical SEDs (see, e.g., Figure 11) probed primarily by the NIRCam bands. However, the Golden Five, and other MIRI-detected subsamples, allow us to probe further into the rest-frame NIR of the LRDs to characterize the origin of the emission in that spectral range. \nOur results confirm the flattening of the LRD SEDs in the rest near infrared. Previous LRD papers fitting NIRCam-only SEDs with empirical AGN templates (e.g., Kocevski et al. 2023; Barro et al. 2023; Greene et al. 2023) implied that the steep rest-optical slope would continue into the NIR. This trend seemed to be confirmed by the handful of LRDs with MIRI detections up to F770W and F1000W (Akins et al. 2023; Barro et al. 2023). However, as shown in section 3.1 and also Williams et al. (2023a), the MIRI data at longer wavelengths indicate that LRDs have a flattening in the SED between 1 and 2 µ m (rest). Interestingly, the SEDs of four out of the Golden Five LRDs at redshifts z < 7, which have direct detections of the NIR continuum at λ ≳ 2 µ m, appear to show upturns with different slopes at longer wavelengths. This suggests that, while it does not dominate, the amount of dust emission from star formation or an AGN can vary substantially from object to object. \nOur four codes fit the NIR spectral range with a combination of the dominant source of UV-optical emission (i.e., stars or an accretion disk) plus a variable contribution from dust emission, either from star-formation ( synthesizer-AGN , and prospector-SF ) or from an AGN torus ( prospector-AGN , and prospectorAGN + depending on the source). The left panel of Figure 10 illustrates this variation showing the rest-frame color-color diagram for the LRDs and some templates and models. The 0.4-to-1 µ m color probes the optical to NIR slope. This color is similar to the F277WF444W used in the sample selection but it is not affected by emission lines and, thus, is a better proxy for the amount of dust attenuation. The color is also similar to the rest-frame V -J which is a known tracer of large dust attenuation in the UVJ diagram ( V -J > 1 . 5 mag for very dusty galaxies, e.g., Brammer et al. 2011, Wuyts et al. 2011). \nFigure 10 illustrates how the modeling codes populate the color-color diagram between the flat (color ∼ 0 mag) stellar-only sequence, and the hot-dust dominated sequence of the QSO1 template with increasing contributions from dust emission (i.e., larger IRluminosities) from star-formation or an AGN, relative to the stellar or accretion disk continua. The color-color \n<!-- image --> \nFigure 10. Rest-frame optical and NIR colors for the LRDs (left, with measured and upper limit fluxes shown with circles and arrows, respectively) and different templates and best-fit models (right). The 0.4-to-1 µ m color traces the optical slope and thus it is a good proxy for the dust attenuation. The 1-to-3 µ m color tracks the amount of dust emission relative to the stellar or accretion disk emission which dominates the UV-optical SED, probed by the 0.4-to-1.0 µ m color. The LRDs exhibit values in between a stellar-only sequence with 1-to-3 µ ∼ 0 (or up to 0.5 mag with increasing nebular continuum) indicated by the solid black and magenta dashed lines on the right, and the torus-dominated sequence outlined by the colors of the Polletta et al. (2007) QSO1 template with increasing A(V), indicated with a solid grey line on the top-left, converging to the color of a Polletta et al. (2007) Torus template (black square). The dashed and dashed-dotted grey lines show a similar sequence for the HDD template of Lyu et al. (2017) and the accretion disk model with slope α = 1 / 3 used in prospector-AGN . The green and blue dashed lines and markers illustrate the sequence toward redder 1-to-3 µ m colors with increasing dust emission from star-formation (log( L IR /L ⊙ ) = 12 -13) or the Nenkova et al. (2008) clumpy torus (log( L IR /erg s -1 )=44.0-45.5) relative to the stellar or accretion disk continuum (solid black and dashed grey lines on the right). \n<!-- image --> \nThe behavior can be discussed in terms of the colorcolor behavior in Figure 10. The black horizontal line in the figure shows the flat 1-to-3 µ m color of a stellar population with zero contribution from dust emission, which peaks at 1.6 µ m. The magenta region indicates the redder colors up to 0.5 mag relative to the stellar continuum due to increasing amounts of nebular continuum (magenta dashed line in the left panel). The 3 grey lines indicate the color tracks with increasing attenuation (A(V)=1 to 4 mag) for the Polletta et al. (2007) QSO1 template (solid), the hot dust deficient (HDD) template of Lyu et al. (2017) used in prospectorAGN+ (dashed-dotted), and the accretion disk model with declining slope ( α = 1 / 3) and zero dust emission used in prospector-AGN . The right panel of Figure 10 illustrates some of same trends as they affect the SED templates. \ntracks ranging from moderate to high infrared luminosities (indicated in log( L ( L ⊙ )) for the star-forming case and log( L ( ergs s -1 ) for AGNs) are computed by scaling the dust emission (f dust /f total =5% to 60% at 2 µ m) of JADES -57356. As expected, the star-forming dust templates exhibit larger luminosities than the torus at similar 1-to-3 µ m colors because their SEDs extend to longer wavelengths with a more prominent peak (see also Figure 16 in the appendix). While a single NIR color is not able to capture all the nuances of the different dust emission templates, Figure 16 shows that the torus and star-forming dust templates in Siebenmorgen & Krugel (2007) and Nenkova et al. (2008) have similar slopes for the same normalization (i.e., same f dust /f total value). Consequently, the four modeling codes can all, in principle, reproduce the NIR continuum of the LRDs. \nThe Golden Five galaxies with direct detections beyond rest-frame ∼ 2 µ m exhibit colors that are at least 1 mag redder than a flat, stellar-only SED. In particular, two of them (JADES -57356 and JADES -79803) have very red colors, [1-to-3 µ m] ∼ 2 mag, indicative of large dust emissions and IR luminosities. JADES-219000 and JADES-211388 are not well constrained beyond 2 µ m because of their higher redshifts, but upper/lower trian- \now the range in possible colors spanned between the prospector-SF and synthesizer-AGN best-fits, which feature different amounts of dust emission within the upper limits of the redder MIRI bands. These limits fall within the overall behavior of all the sources in the figure. The broad range overall in Figure 10 highlights the need for deep, long wavelength MIRI data to constrain precisely the amount and heating nature of dust emission in LRDs. \nHowever, we can already see that fitting the colors with a purely AGN-dominated model requires an accretion disk model with declining slope (dashed grey line) and only a small contribution from dust torus emission (thin blue line) to successfully reproduce the bluest 1-to3 µ m ≲ 1.5 colors at the lower limit of the HDD template. This would differ significantly from lower redshift AGN, which tend to have strong emission from their circumnuclear tori. This issue is mitigated by prospectorAGN+ , which can reproduce those colors with a hybrid of AGN and stellar emission. \nWe now discuss the LRDs detected only at shorter wavelengths. Figures 11 and 17 show the stacked SEDs for the LRDs detected up to F1000W and F770W, which reveal that their SEDs are only well constrained up to rest-frame ∼ 1.6 µ m and ∼ 1 µ m, respectively. Consequently, the 1-to-3 µ m colors of their best-fit models span a much larger range from the synthesizer-AGN , stellar-only fits with [1-to-3 µ m] ∼ 0 mag (e.g., top-left panel of Figure 11), to the much redder best-fit models of prospector-AGN+ and prospector-AGN (bottom-left and right), which sometimes fit the MIRI upper limits with pronounced upturns at λ > 2 µ m. It is worth mentioning the F770W-only sample places more restrictive constraints against very red QSO-like colors than the F1000W-only sample. This is because, as shown in Figure 4, the F1000W-only LRDs have redder F777W-F1000W colors than the upper limits of the F770W-only LRDs and thus lead to a stronger flattening of the SED in the 1-2 µ m range. Consequently, the possible upturn after 2 µ m is not nearly as red. \nIn summary, the exact contribution from dust emission to the NIR SEDs of the LRDs is still poorly constrained by the available MIRI data at long wavelengths. Nonetheless, the constraints point to a certain diversity in the dust emission. This emission is clearly larger than stellar-only SEDs but in many cases lower than prior expectations based on QSO templates. Overall, the main conclusion is that most LRDs could harbor some (relatively large) amount of dust emission and the heating source must be intense, although not necessarily a dominant obscured AGN.', '5.3. Physical properties of the LRDs detected in the bluest MIRI bands': 'All other LRDs in our sample detected up to F1000W, apart from the Golden Five, are shown jointly in the SED plots provided in Figure 11. In the same way, the SED fits for all F770W detections not included in any previous SED plot, as well as all sources not detected by MIRI are shown and discussed in Appendix C. Physical properties derived from each code for individual galaxies as well as statistical properties obtained for the whole sample and subsamples are given in Tables 2 and 3. \nOverall, the properties of those samples with fewer MIRI points in their SEDs are similar to the Golden Five galaxies; the clearest difference is that some blue and/or very flat SED sources start to enter the selection. These are selected due to a very strong F444W emission probably linked to a high-EW emission line, but the slope of the SED is not very different in the rest of the SW and LW filters, with a quite flat slope (or even blue, as in the case of JADES -187025). \nEven though the dust emission spectral region is not fully probed for the 10 µ m sample, and to an even lesser extent the 7.7 µ m and non-MIRI samples, the upper limits imposed by the MIRI data at longer wavelengths, more specifically, at 12.8 and 15 µ m, indicate a very similar behavior of the spectral range around 2-3 µ m (rest) compared to what we showed for the Golden Five galaxies. Indeed, the SED flattens, indicating that possible dust emission powered by stellar or AGN heating is not dominant and could only start adding significant flux redward of 3 µ m. The properties we infer from the UVto-NIR SEDs are also similar for the F1000W, F770W, and no-MIRI subsamples compared to those obtained for the Golden Five galaxies. There are, however, some observational trends, which translate to differences in physical properties. \nFirst, the Golden Five galaxies are brighter than the rest of the sources in the full sample (cf. Figure 1). The median and quartiles for F444W are 25 . 4 26 . 1 25 . 1 mag, compared to 26 . 1 26 . 3 25 . 8 mag for the F1000W sample, 25 . 9 26 . 1 25 . 7 mag for the F770W sample, and 27 . 6 27 . 8 26 . 8 for the sources with no MIRI detection. Concerning colors, the F1000W sample is redder than the Golden Five, and the overall sample. As shown in Figures 1 and 4, the F1000W sample is among the reddest in both the NIRCam and MIRI colors: F277W-F444W=2.2 mag, F444W-F770W=1 mag and F770W-F1000W=0.9 mag, versus for the overall sample they are 1.4 mag, 0.7 mag and 0.2 mag respectively. Comparatively, the largest difference is in F770W-F1000W color difference, where the F1000W sample has similar colors to 2 galaxies among the Golden Five, JADES-79803 and JADES- \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 11. SED fitting results for the 7 galaxies detected at F1000W and not beyond (i.e., the plot does not include any of the Golden Five galaxies). SEDs are normalized to 0.7 µ m. In gray, we show the fits to each individual galaxy, and we provide an average in brown. \n<!-- image --> \n211388, whose MIR spectral range is fitted with dust tori. We note that for JADES-211388, at z sp = 8 . 3846, F1000W lies on top of the He I+Paγ line, which partially explains the red color. \nLooking at the rest-frame colors and stacked SEDs in Figures 10 and 11, we find that the F1000W sample has also redder 0.4-to-1.0 µ m colors than the Golden Five sources, suggesting that they are dustier (see next paragraphs). Interestingly, using a longer baseline color 0.25-to-1.0 µ m (similar to NUV-J), which probes into the relatively flat UV SED of the LRDs, we find even larger differences between the Golden Five galaxies and the F1000W samples, 0.25-to-1.0 µ m = 2.1 mag for the Golden Five vs. 3.5 mag. These colors and the multicomponent SED modeling discussed in the previous section indicate that the colors relative to rest 1 µ m are partially driven by differences in the relative luminosity \nof the component dominating the rest-UV (young, unobscured stellar population) and the component dominating the rest-optical (older stellar population or obscured accretion disk). As the flat UV component scales up in brightness, it leads to bluer 0.25-to-1.0 µ m and 0.4to-1.0 µ m colors, but perhaps not because of a change in the intrinsic properties of the component dominating the optical range. It could be that a brighter UV reveals a larger fraction of the starburst emission percolating through the compact dust cloud (possibly linked to a higher burst strength or younger age, apart from dust-star relative geometry), or perhaps it shows a more massive stellar host for the obscured AGN. This interpretation also helps to explain the larger scatter in the UV region of the stacked SEDs relative to the optical region. That is, while all LRDs have distinctive blueUV and red-optical SEDs, there is a larger diversity in \nthe UV emission for a similar optical-to-NIR slope that might reflect variations in the relative luminosity of two different components. \nThe overall colors of the LRDs range from 0.25-to1.0 µ m = 2 to 4.5 mag. Taking 0.25-to-1 µ m = 3 mag as an intermediate value, we find that all of the Golden Five galaxies exhibit bluer colors, versus only 30% (2/7) of the galaxies in the F1000W and F770W samples. This emphasizes again that the Golden Five galaxies are intrinsically bluer than the other samples in all colors. Interestingly, we also find a trend toward stronger emission lines (larger EWs) with bluer 0.25-to-1.0 µ mcolors. For example, this trend is seen among the bluest Golden Five sources (JADES -79803 and JADES -204851) and in the handful of galaxies with bluer colors in the F1000W sample (JADES -210600, JADES -214552 and JADES -217926) and F770W sample (JADES -187025 and JADES -197348). \nBased on the previous observational differences, our modeling of the SEDs provides some trends also in physical properties. The Golden Five galaxies lie at smaller redshifts than the rest of sources, median values z = 5 . 5, compared to z = 6 . 3, z = 5 . 9, and z = 7 . 4 for the F1000W, F770W, and non-MIRI subsamples (check Table 3 for more statistical information). So part of the reason for the detections of the Golden Five in many MIRI bands can be linked to redshift. \nThe stellar masses of the Golden Five galaxies are 0.20.4 dex larger than those of the other subsamples, with values around 10 9 . 6 M ⊙ for the former. Interestingly, the stellar population attenuation is lower for the Golden Five galaxies, A(V) = 3 . 0 mag, compared to F1000W sources (A(V) = 3 . 2 mag), F770W galaxies (A(V) = 3 . 3 mag), and non-MIRI sources (A(V) = 3 . 9 mag). \nWe conclude that sources detected only in the bluest MIRI filters are not just fainter (less massive) versions of the Golden Five galaxies, lying at higher redshifts. Indeed, there are also other differences in physical properties (deriving from differences in SEDs) which point to larger attenuations. If the fainter MIRI sources present larger attenuations, but still are not detected by MIRI at the longest wavelengths, the interpretation would be that the host dust emission is not enhanced compared to the Golden Five galaxies, which would lead to a dominant role of dust-enshrouded star formation rather than obscured nuclear activity for a significant fraction of these sub-populations.', '5.4. Statistical stellar and AGN properties of LRDs': 'In this section, we discuss the general properties of our sample of LRDs. Figure 12 shows the distributions of stellar mass, bolometric luminosity, mass-weighted age, \nand bolometric stellar light attenuation. All statistical information is summarized in Table 3. \nThe typical stellar mass of LRDs is log M ⋆ / M ⊙ = 9 . 4 9 . 7 9 . 1 (median and quartiles) according to synthesizer-AGN . Considering the full redshift range of our sample of LRDs, 5 ≲ z ≲ 9, and the number density of galaxies in the stellar mass range 9 . 0 < log M / M ⊙ < 10 detected by CEERS based on the v0.51 catalogs (Finkelstein et al. 2023), we calculate that the LRDs account for 14 ± 3% of the full population of galaxies (subject to uncertainties due to cosmic variance). This translates to a comoving density of LRDs of 10 -4 . 0 ± 0 . 1 Mpc -3 , which is quite constant across the 5 < z < 9 redshift range, with differences < 0 . 1 dex between 5 < z < 7 and 7 < z < 9. The estimates for both the LRDs and the other galaxies are subject to a number of potential systematic errors, but the estimate indicates that the LRDs represent a significant, but not a dominant, population over this redshift range. Given the ranges of Universe ages probed by our sample, 200 Myr for 7 < z < 9 and 400 Myr for 5 < z < 7, the ∼ 10% frequency within the global population could be interpreted in terms of a duty cycle around 20-40 Myr, which points to starburst behavior. \nLarger masses are obtained by prospector-SF , log M ⋆ / M ⊙ = 9 . 9 10 . 2 9 . 2 , mainly because the mass-weighted ages are older, between 10 and 100 Myr, with median and quartiles being t m -w = 16 27 10 Myr. synthesizerAGN fits the SEDs with significantly younger stellar populations, typically t m -w = 3 20 2 Myr. The 2 stellar populations considered by synthesizer-AGN are typically younger than 20 Myr. In fact, the average SFH of LRDs obtained by synthesizer-AGN and shown in Figure 13 indicates that LRDs are experiencing a very intense episode of star formation extending for nearly 10 Myr and with a very compact size. The burst would be in part heavily dust-enshrouded, with some younger stars having cleared the interstellar medium and being directly observable through much smaller dust optical paths. The young ages are expected for starbursts with strong emission lines (as present in some of our galaxies), with large amounts of dust (confirmed for a significant fraction of the whole sample), and with gas also feeding a SMBH. synthesizer-SF provides a similar average SFH, extended almost at a constant level up to approximately 10 Myr, and decaying afterwards. However, the first age bin considered by synthesizer-SF encompasses the 2 bursts obtained by synthesizer-AGN , the former adding more mass in ages around 10 Myr (with a large scatter, shown by the shaded region in Figure 13). \nComing back to stellar content, even smaller masses compared to synthesizer-AGN are obtained by \nFigure 12. Statistical stellar properties of LRDs, according to the 4 SED-fitting codes described in Section 4. From top to bottom, left to right, we show stellar masses, bolometric luminosities (obtained by integrating the stellar emission correcting for the effects of dust attenuation), mass-weighted ages, and bolometric stellar luminosity attenuation. Medians and quartiles are shown for each distribution. For the results provided by synthesizer-AGN , we separate statistics for the young and old stellar populations (marked as you and old ) as well as the integrated values. \n<!-- image --> \nprospector-AGN , log M ⋆ / M ⊙ = 7 . 8 8 . 0 7 . 2 , which only considers contributions of stellar light to the UV spectral region. These masses are similar to the values obtained for the youngest population in the synthesizerAGN modeling. Given that prospector-AGN fits the optical and NIR spectral regions with an AGN, the stellar mass estimates are significantly smaller than what is needed to reproduce the emission at those wavelengths with stars. \nWe conclude that the prospector-AGN stellar masses should be considered lower limits, since they assume little contribution of stellar light to the optical spectral range. synthesizer-AGN and prospectorAGN + typically obtain fits for which the optical emission is dominated by stars, hence the stellar mass estimates they obtain should be interpreted as more realistic or upper limits. The differences in stellar masses derived with synthesizer-AGN and prospector-SF exemplify the effect of the SFH, which is relatively important \n(0.4-0.6 dex) for these young galaxies whose mass-tolight ratios can change significantly as the most massive stars disappear. \nThe bolometric luminosities (including dust-absorbed energy) vary by a factor of 10 between the prospectorSF and prospector-AGN runs, while synthesizerAGN lie in between, with a typical value of log L / L ⊙ bol = 12 . 0 12 . 4 11 . 6 . Combining with the stellar masses, we conclude that LRDs present mass-to-light ratios of 1/400, typical of OB stellar associations (a B2 star would have that value, approximately), as expected based on the young ages. We remind the reader that given these very young ages and the starburst nature of LRDs, the a priori assumption of a universal IMF is quite relevant. The amount of OB stars formed, the quick and efficient formation of metals and dust, and the inferred stellar masses (or even the growth of a SMBH) are all affected by the IMF. \nFigure 13. Average SFHs of LRDs according to the fitting codes presented in Section 4. Averages and scatter are shown as lines and shaded regions. \n<!-- image --> \nFinally, the bottom-right panel of Figure 12 shows the total attenuation of the stellar light in LRDs. All codes are consistent in assigning large dust content to this type of galaxy, with attenuation around 3-4 mag (i.e., 95-99% of the light being absorbed by dust). \nThe relative importance of the AGN and stellar components in LRDs is presented in Figure 14. Here we show the fraction of the total luminosity coming from the AGN and integrated in several spectral ranges. We show bolometric, UV (integrated up to 0.4 µ m), optical (from 0.4 to 2 µ m), and IR (from 2 µ m redwards) luminosity ratios for the AGN emission (with the rest coming directly from stars or dust heated by stars). We note that we only include 3 of the 4 codes in this plot, since prospector-SF does not consider any AGN contribution (although the dust models imply intense radiation fields which could be easily identified with an AGN). \nFor synthesizer-AGN , the bolometric luminosity of most of the sample is dominated by stars, with only ∼ 20% of sources presenting AGN luminosity fractions larger than f AGN L bol > 0 . 5. In contrast, prospectorAGN + obtains a much larger AGN contribution for most galaxies, with nearly 70% of the galaxies presenting a bolometric luminosity fraction > 50%. This is a direct consequence, however, of this code fitting the MIRI nondetections assuming the 5 σ upper limit as an actual flux. This means that the prospector-AGN + results about the AGN luminosity ratio should be regarded as upper limits. In the same sense, synthesizer-AGN could be regarded as lower limits, since the MIR is not fitted for MIRI non-detections nor for F770W-only sources, since all bands with measured fluxes are well reproduced by \nstellar models alone (i.e., the dust emission is loosely constrained and it is not fitted). \nMore consistent results are found for the UV luminosity fraction identified with an AGN. All codes agree that the UV spectral region of LRDs is best-reproduced by a young stellar population with varying but low dust content and relatively young ages (1-10 Myr). We remark here that prospector-AGN forces the UV to be dominated by unobscured stars, while the other 2 codes leave complete freedom in this spectral range in terms of a possible contribution from a UV-bright AGN or the amount of dust. \nA similar behavior is observed in the optical, where both synthesizer-AGN and prospector-AGN + find that the emission is dominated by stars in most ( > 75%) of the sample. The a priori assumption of prospector-AGN is that the AGN dominates the IR spectral region and results in the optical also being dominated by the nuclear activity for most sources; consequently, as we mentioned, the estimated stellar masses are considerably smaller than the values obtained by the other codes. \nFinally, the bottom-right panel of Figure 14 shows the fraction of the IR (dust) luminosity linked to the AGN. prospector-AGN +and synthesizer-AGN show opposite distributions, but two facts must be taken into account to interpret this behavior. First, as mentioned earlier, prospector-AGN + fits upper limits as regular flux points. Those upper limits increase with a slope similar to what can be expected for an AGN torus. Second, the dust emission fits provided by synthesizerAGN imply very intense radiation fields in dense, compact hot spots, whose properties would be indistinguishable between OB associations or an AGN (see discussion on Figure 5 and Appendix B).', '5.5. Spectroscopic properties of LRDs found in the literature': 'Apart from JADES -204851, which has broad H α emission at z = 5 . 4790 ( ∼ 2000 km s -1 ) (Matthee et al. 2023b), as mentioned in Section 5, there are other sources in our sample with relevant spectroscopic information. JADES -197348 was included in the JADES NIRSpec initial data release (Bunker et al. 2023) and identified with a broad-line AGN (Maiolino et al. 2023b). Its spectrum shows a ∼ 2500 km s -1 wide component that accounts for two thirds of the H α total flux, while [OIII] shows no such component. Our fits to this source are dominated by a QSO-like spectrum in the optical and NIR in the case of synthesizer-AGN and prospector-AGN . JADES -154428 is found to present a broad-line component with FWHM ∼ 1800 km s -1 \nFigure 14. Histograms of the fraction of integrated luminosity coming from an AGN, according to the SED fits presented in Section 4. On the top-left, we show results for the bolometric luminosity. On the top-right, for the UV luminosity, integrated up to 0.4 µ m. On the bottom-left, results for the optical luminosity are provided, with L opt defined as the integral between 0.4 and 2 µ m. The bottom-right panel shows the histograms for IR wavelengths longer than 2 µ m. Medians and quartiles for each spectral range and code are displayed at the top of the panels. \n<!-- image --> \n(Sun et al. 2024, in prep.); our fits include a nonnegligible contribution from a QSO-like spectrum, dominating the SED ( prospector-AGN and prospectorAGN + or accounting for ∼ 50% of the emission at specific wavelengths ( synthesizer-AGN ). No other broad H α or H β line component has been reported in the remaining 18 galaxies with available spectroscopy, i.e., 17% of the spectroscopic sample are confirmed AGN hosts. For the rest of the spectroscopic sample, the presence of an AGN cannot be ruled out, since the broad-line region could be hidden due to geometrical effects. We also note that the typical 5 σ depth of FRESCO NIRCam 3.9-5.0µ m grism spectroscopy is ∼ 5 × 10 -18 erg s -1 cm -2 for broad emission line (FWHM ∼ 1000 kms -1 ) from a point source and, therefore, faint broad line emission (possibly from an AGN not dominating the continuum) can remain undetected.', '6. SUMMARY AND CONCLUSIONS': "We characterize the nature of Little Red Dots (LRDs) in the JADES field by analyzing their spectral energy distributions including the mid-infrared fluxes provided by the SMILES program for all MIRI broad-band filters. These data probe the rest-frame near-and mid-infrared where stellar emission and/or obscured AGN emission peak. After removing brown dwarfs, which contaminate our sample at the 15% level, we arrive at a sample of 31 LRDs, the surface density being 0.9 arcmin -2 . This translates to a number density 10 -4 . 0 ± 0 . 1 Mpc -3 , accounting for 10-15% of the global population of galaxies with similar redshifts ( z ∼ 7) and stellar mass (log M ⋆ / M ⊙ = 9 . 5). Two thirds are detected in the F560W and F770W filters (all sources brighter than F444W < 26.5 mag), two fifths in F1000W, one seventh in F1500W, one thirteenth in F1800W, and one source in F2100W, down to 5 σ limits between 26.1 and 22.6 mag. The MIRI detection fraction is largely dependent on the F444W brightness, but we find an additional trend to- \nre detections at 10 µ m or beyond with bluer F277W-F444W colors. \nWe find that the observed MIRI colors of the LRDs, in combination with the reddest NIRCam bands, are bluer than the typical obscured QSO templates, which are dominated by the torus warm/hot dust emission at λ ≳ 1 µ m. Indeed, the rest-frame NIR spectral range exhibits a much shallower slope that is consistent with the peak of stellar emission at around ∼ 1 . 6 µ m. \nWe modeled the rest-frame ultraviolet to mid-infrared spectral energy distributions with a battery of codes that include AGN and stellar emission templates. The various outputs allow us to identify the best fits to the distinctive short wavelength blue plus long wavelength red colors of LRDs under a range of assumptions. They also let us examine which of our conclusions are the most robust (e.g., are reflected by a number of the modeling techniques). \nIn general, stellar-dominated models obtain a better agreement to the near-infrared at 1-2 µ mas well as in the UV at λ < 0 . 4 µ m. The AGN-dominated models where emission from an obscured accretion disk dominates the optical and NIR and the torus takes over at longer wavelengths λ ≳ 2 µ m provide a better agreement than the typical QSO templates, but still worse than the stellar models. Furthermore, this AGN-dominated model also has conceptual problems given that many of the LRDs do not present emission lines, which should be expected for the direct detection of the accretion disk (and its broad- and narrow-line regions). Consequently, we favor the interpretation that the UV-to-optical spectral range of most LRDs is dominated ( > 50% luminosity ratio) by stars. \nIn the rest-frame near-infrared, we find that the LRDs detected in the reddest MIRI bands, beyond rest wavelengths λ ≳ 2 µ m, have color differences redder than can be expected for stars alone (even accounting for nebular emission associated with a young starburst), and consistent with some amount of emission from dust heated by star formation or an AGN. The upper limit of the MIRI colors for most LRDs also rules out that the nearinfrared emission is strongly AGN-dominated, but the loose constraints in some of them can still accommodate similar amounts of dust emission as in the LRDs detected in the rest-frame mid-infrared. \nGiven that the rest-frame UV/optical spectral range is dominated by stars, we estimate stellar masses. The modeling of the stellar emission must consider the large attenuations implied by the red NIRCam longwavelength colors and MIRI fluxes and the need for two distinct (in terms of age) stellar populations, with differential attenuation levels or an extremely gray atten- \nuation law, to account for the blue/flat NIRCam-SW emission as well the presence of emission lines. With this in mind, we obtain typical stellar masses for LRDs around log M ⋆ / M ⊙ = 9 . 4 9 . 7 9 . 1 , significantly smaller than what can be obtained with simple recipes for the Star Formation History (e.g., single exponential burst) and attenuation law (e.g., single Calzetti law, as noted by Barro et al. 2023). This mass estimate can be biased due to uncertainties in the Star Formation History at the 0.5 dex level (overestimated, if, for example, the stellar emission only contributes significantly to the ultraviolet emission), and can be affected by a possible (but less favored by most of our models) significant contribution of an AGN to the optical and near-infrared spectral range at the 1.5 dex level. \nVery young stellar ages (typically around 10 Myr or younger) are supported by high equivalent width lines seen in some LRDs (with spectroscopic and imaging data), and the presence of large amounts of dust, which is a common feature of all models for the full sample, and can be expected for gas-rich, quickly enriched dense starbursts with large amounts of OB stars (affected by the IMF). However, despite the presence of large amounts of dust, LRDs are characterized by a relatively blue emission at wavelengths < 0 . 4 µ m, which can only be reproduced if this spectral range is dominated by (1) a QSO; (2) has holes in the interstellar medium surrounding star-forming regions that allow us to see unobscured very young star formation; or (3) have a gray attenuation law typically linked to significant scattering (with implications to the relative geometry of dust and stars and the dust clumpiness). Indeed, the large ∼ 3 mag scatter in the UV-to-NIR colors within the typical blue-red SED of the LRDs might indicate different burst strengths or different fractions of percolating light through the compact dust envelope. The presence of a QSO is hinted by some medium-band colors consistent with MgII emission. Nevertheless, for most of the sources, the best spectral energy distribution fit is obtained with just stars presenting a differentiated and/or gray dust attenuation law. A significant fraction of the total stellar emission of LRDs comes from OB stellar clumps mostly embedded in dense dusty regions with large optical paths, A(V) ≳ 10 mag, with the integrated stellar emission obscured at the 90-95% level, which would explain the near- and mid-infrared emission jointly with some contribution from an obscured AGN. \nWe thank the referee for their constructive comments to our original manuscript. PGP-G acknowledges support from grant PID2022-139567NBI00 funded by Spanish Ministerio de Ciencia e Innovaci'on MCIN/AEI/10.13039/501100011033, FEDER Una manera de hacer Europa . This work was supported by NASA grants NNX13AD82G and 1255094. The work was also supported by NIRCam Development Contract NAS5-02105 from NASA Goddard Space Flight Center to the University of Arizona. This work is based on observations made with the NASA/ESA/CSA James Webb Space Telescope. The data were obtained from the Mikulski Archive for Space Telescopes at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-03127 for JWST. These observations are associated with program #1207. DP acknowledges support by the Huo Family Foundation through a P.C. Ho PhD Studentship. BER acknowledges support from the NIRCam Science Team contract to the University of Arizona, NAS5-02015, and JWST Program 3215. AJB acknowledges funding from the 'FirstGalaxies' Advanced Grant from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No. 789056). \nFacilities: JWST (NIRCam), JWST (MIRI), HST (ACS), HST (WFC3). All the JWST data used in this paper can be found in MAST: 10.17909/jmxm-1695 and 10.17909/8tdj-8n28. \nSoftware: astropy (Astropy Collaboration et al. 2013, 2018), Cloudy (Ferland et al. 2013), prospector (Leja et al. 2017, 2019b; Johnson et al. 2021), synthesizer (P'erez-Gonz'alez et al. 2003, 2008). \nTable 1. Sample of LRDs \nNote -Table with basic information (used for selection) about the sample of galaxies in this paper: IAU-format name, ID based on JADES DR2 catalog (Eisenstein et al. 2023b), magnitude and colors used in the selection (see Figure 1), and subsample according to the reddest band counting with a MIRI detection. \nTable 2. Physical properties of LRDs (complete version online) \nNote -Table with physical properties of the sample of galaxies in this paper. Apart from redshift (spectroscopic values given with 4 decimals), we provide results for each of the 4 stellar population synthesis codes plus AGN described in Section 4: synthesizer-AGN (sy-AGN), prospector-AGN + (pr-AGN+), prospector-SF (pr-SF), and prospector-AGN (pr-AGN). We quote stellar masses (surviving stars), attenuation of the bolometric, far-ultraviolet (FUV, i.e., 150 nm), and visual (V, i.e., 550 nm) emission from stars (including nebular emission, and providing results for the old and young stellar populations for synthesizer-AGN in the FUV and V cases), fraction of the luminosity associated with the AGN for the bolometric, UV, optical, and IR emission (check definition in Section 5.4), and mass-weighted ages. \nTable 3. Statistical properties of LRDs \nNote -Table with statistical physical properties of the sample of galaxies in this paper. We provide medians and quartiles for the physical properties mentioned in Table 2 for the whole sample and the subsamples defined based on the detection in different MIRI bands.", 'A. BESPOKE PROCEDURES FOR THE REDUCTION OF MIRI DATA: THE SMILES CASE': "The MIRI data used in this paper were reduced with the Rainbow JWST pipeline developed within the European Consortium MIRI GTO Team to deal with MIRI, NIRCam, and NIRISS imaging data. The pipeline relies on the jwst official pipeline and adds some offline steps to improve the results, mainly dealing with the varying, full-of-structure background observed in MIRI data, especially at the shortest wavelengths. \nThe Rainbow pipeline starts with a default execution of the 3 stages of the jwst official pipeline, which provides a first full mosaic, now also dealing (to some extent, but not completely) with cosmic ray showers (or snow balls for NIRCam). This mosaic is used to detect sources with sextractor in order to produce a mask for further refinements of the calibration. We use a relatively shallow detection limit since the mosaic presents intense background gradients and structure, indicating that a deep detection would be dominated by background, not real sources. Apart from the sextractor detection, we performed a 5-pixel dilation of the segmentation maps to account for the faint outskirts of extended objects, also including the emission from PSF spikes and the cruciform feature of short wavelength MIRI data (G'asp'ar et al. 2021). \nAfter masking sources from the previous task, stage 2 data (i.e., cal.fits files) are median-filtered in rows and columns and a smooth 4-order surface is subtracted. The new calibrated data products are again mosaicked with the official pipeline. This new mosaic presents a much more well-behaved background and allows a more aggressive source detection. The new mask is used in a final step of the Rainbow pipeline which implements a super-background strategy to obtain the best results. \nThe super-background strategy consists in homogenizing the background of a given stage 2 image using all the other images taken in the same program. For programs extending over several epochs and with enough data, PRIMER for example, we use the closest data in time. We first subtract the median of the background (after masking sources detected in the previous step) for each image. Then, for each frame to reduce, we build a stacked median background image with the rest of images. If not enough data are available for a given pixel, i.e., when very few images are available and the dithering is small compared to the size of (some) objects in the field, we replace the pixel value by a random nearby background pixel chosen from the 100 closest non-masked pixels. The stacked median super-background image is subtracted from the frame we are considering, and the filtering in rows and columns is performed, as well as the subtraction of a smooth surface. \nThe astrometry of the new background-homogenized cal.fits files is calibrated with the tweakreg external routine provided by the CEERS collaboration (Bagley et al. 2023), using an external catalog constructed with IRAF's center task in centroid mode (which was checked to perform better than photutils), using in the SMILES case the JADES catalog as the WCS reference. Then stage 3 of the pipeline is executed, switching off the tweakreg step and setting the pixel scale to 60 milliarcsec in the case of MIRI data. The WCS of the final mosaic is checked again against the reference WCS catalog and a final background subtraction is performed with sextractor . \nThe procedure is evaluated in Figure 15, where we compare the histograms of pixel signal for the final mosaic of the F1500W filter reduced just with the jwst pipeline (after subtracting the median value of the full image) and reduced with our super-background method. The histograms are fitted to a gaussian, showing that our procedure is able to reduce the noise by a factor of ∼ 1 . 5. This translates to 0.5 mag deeper detection limits in the final mosaic for this band compared to what can be achieved with the official pipeline alone, which roughly agrees with the expectations provided by the ETC version v3.0. \nTable 4 . \nNote -(1) MIRI filter. (2)-(4) Observational strategy in the SMILES survey: number of groups, number of integrations, number of dithering positions. (5) Exposure time per pixel (seconds). (6) Average background measured in the data. In parenthesis, the ETC v3.0 predictions for Dec 7, 2022 are given. Units (MJy/sr). (7) FWHM of the PSF in each filter (arcsec). (8) Depth of the data for a point-like source measure in a circular aperture with radius equal to the FWHM of the PSF, aperture corrected. In parenthesis, the ETC v3.0 predictions for Dec 7, 2022 are given. Units are AB mag. (9) Aperture correction applied to previous column results (AB mag), based on files released in pmap-. \nFigure 15. Histogram of pixel values (transformed to µ Jy) of the F1500W SMILES mosaic, after subtracting the median background. In red, we show the results for the mosaic produced by the official pipeline. In green, the results obtained with the bespoke version of the pipeline, embedded in the Rainbow database, and implementing a super-background strategy. The 2 histograms are fitted to a gaussians, whose dispersion is translated to 5 σ depths for a point-like source, calculations based on measurements in an circular of radius equal to the FWHM of the PSF, and corrected for the limited size of the aperture using the calibration available in pmap 1138. \n<!-- image --> \nTable 4 provides the observation strategy, total exposure times per pixel, average background levels, and depths of the SMILES data reduced with the Rainbow JWST pipeline, compared with ETC predictions.", 'B. DUST EMISSION MODELING': "In this Appendix, we describe in detail the dust emission models used in this paper. Different recipes and emission origins are considered by the four codes described in Section 4. The main characteristics of our dust emission templates are summarized in Figure 16. \nThe synthesizer-AGN code uses the radiative transfer models of starburst nuclei and (ultra)luminous infrared galaxies (LIRG) presented in Siebenmorgen & Krugel (2007). These models assume an intense star formation event (on top a more evolved stellar population) where a fraction of the most massive (OB) stars are embedded in compact dusty clouds (hot spots in their terminology) that dominate the mid-infrared emission. The models are parametrized in terms of the total radiated luminosity (ranging from sub-LIRG to hyper-LIRG values), the size of the star-forming region, the total amount of dust described by the total V -band attenuation, the fraction of the total luminosity linked to the OB stars in the hot spots, and the gas/dust density of the hot spot clouds. \n<!-- image --> \nFigure 16. Dust emission models used in the analysis of LRDs presented in this paper. Five different models from Nenkova et al. (2008) used by prospector-AGN are shown in red colors, showcasing the shift to shorter wavelengths (from ∼ 30 to ∼ 10 µ m) of the dust emission peak arising from an AGN torus with different dust optical depths, namely, τ V equal to 5,10, 20, 40, and 150. Gray lines show the dust torus and QSO templates (the latter, extincted by 1 and 2 mag following a Calzetti et al. 2000 law) in Polletta et al. (2007), the former being used by synthesizer-AGN . Thick lines show emission from dust, thin lines show the full models, which include stellar emission. Two black-body models for warm dust (500-1200 K) are depicted. The dust emission models for nuclear starbursts, also used by synthesizer-AGN (Siebenmorgen & Krugel 2007), are shown in blue and cyan colors, with representative values of the different parameters (total luminosity, luminosity arising from OB stars, hot spot hydrogen density, and total attenuation in the V -band). The star-forming model for Haro 11 used by prospector-AGN + (Lyu et al. 2016; De Rossi et al. 2018) is shown in purple, and the models from Draine & Li (2007) used by prospector-SF in green. \n<!-- image --> \nIn Figure 16, we show models for a 10 10 L ⊙ and 10 12 L ⊙ compact (350 pc in size) region, attenuations A(V) ∼ 10 , 20 , 100 mag, OB luminosity ratios 40 and 90%, and hydrogen number densities 10 2 and 10 4 cm -3 . The templates include a fixed stellar population, which is removed for our modeling of LRDs. Models with and without stars are shown in Figure 16. \nWe remark that these radiative transfer models nicely recover the blue+red nature of the UV-to-NIR emission of LRDs. The change in slope of the stellar UV and optical emission is governed by the OB luminosity ratio, i.e., cyan models (OB90) change in slope at longer wavelengths, around 0.4 µ m rest-frame, similarly to LRDs. Their dust emission peaks at shorter wavelengths, implying a significant amount of warm dust. The attenuation of the FUV emission in these compact starburst models is, however, steeper than what is observed for our galaxies. All Siebenmorgen & Krugel (2007) models, which assume Milky Way type dust, present more or less prominent PAH bands. The typical mid- to far-IR flux density ratio is nearly 1000, which means that some of our galaxies, which present MIRI short wavelength fluxes around 1 µ Jy, are also constrained by the non-detections in Herschel bands (with 5 σ limits around 2-3 mJy at 100 and 160 µ m and 10 mJy at 250-500 µ m). \nThe MIR slopes of the Siebenmorgen & Krugel (2007) models are very similar to the slopes of AGN torus emission presented in Nenkova et al. (2008), which are used in our prospector-AGN fits. Nenkova et al. (2008) parametrizes the emission from clumpy dust tori in terms of the number of clouds intercepting the visual, which depend on other parameters such as the torus thickness (outer to inner radii), the density and angular distribution of clumps and the viewing angle. The subset of templates included in FSPS and then Prospector adopts typical assumptions for the majority of these parameters (see e.g., Leja et al. 2018) leaving e only the scaling (overall luminosity) and the optical depth of an individual dust clump at 5500 ˚ A( τ V from 5 to 150) as free parameters. For the prospector-AGN fits we remove any contribution from the accretion disk in the dust tori templates and we instead model the disk separately following a combination of empirical QSO template plus a power-law f ν = ν α (equivalent to ν f ν = ν α +1 ) with variable slope α = -0 . 5 to 0.5, as discussed in Hern'an-Caballero et al. 2016 or more recently in Bosman et al. 2023, attenuated by a Calzetti law. For reference, the accretion disk model with A(V)=1.5 is shown in Figure 16 in magenta. The main difference between the Nenkova et al. (2008) models and the ones in Siebenmorgen & Krugel (2007) dominated by star \nformation is the absence of PAH emission (especially relevant for our observations, the one at 3 µ m) and the silicate absorption present in some dust models. Overall, a dust torus (and an AGN template) is almost featureless. \nFor reference, Figure 16 also shows the torus (also used in our synthesizer and prospector-AGN + fits) and QSO template presented in Polletta et al. (2007), the latter additionally attenuated with a Calzetti et al. (2000) law assuming A(V) = 2 mag. These templates present a similar slope as the Nenkova et al. (2008) torus models, adding the contribution from the accretion disk that dominates at wavelengths shorter than ∼ 1 µ m and a tail of colder dust peaks at around 100 µ m, as the star-forming models. \nThe AGN SED models used in prospector-AGN + are largely based on empirical observations with the AGN emission strength calibrated against various observations. For example, the AGN hot dust emission predicted from the SED template is confirmed with NIR image decomposition of HST observations of lowz quasars (Lyu et al. 2017). The AGN mid- to far-IR SED shape is checked against MIR spectral decomposition and PAH strengths (e.g., Lyu & Rieke 2017). Compared to radiation transfer models, these empirical SED models can provide more realistic descriptions of the observations across a very wide range of AGN luminosity and redshift with fewer free parameters and less model degeneracy, which make them particularly preferred for AGN identifications (see review by Lyu & Rieke 2022a). The galaxy dust emission template used in prospector-AGN + is based on an empirical SED template of Haro 11, which has been also tested against real observations of very highz galaxies (Lyu et al. 2016; De Rossi et al. 2018). In contrast, the other galaxy dust emission models used in other works typically do not capture some key features of highz galaxy ISM, such as low-metallicity and the possibly different dust compositions, as argued in De Rossi et al. (2018). \nWe have normalized all models at 2 µ m. If the emission of LRDs were dominated by the dust torus at these wavelengths, considering the average redshift of our sample < z > = 6 . 5, i.e., 15 µ m observed, the slope of the torus would translate to the emission at 1 µ m rest-frame, 8 µ m observed, being ∼ 10 times fainter. The F770W-F1500W colors of LRDS are much smaller, indicating that the 1 µ m emission cannot be dominated by the torus even if the 15 µ m flux is linked to an obscured AGN.", 'C. SED FITS FOR THE FULL LRD SAMPLE IN THE SMILES/JADES FIELD': "In this Appendix, we present a detailed discussion of the SEDs of the Golden Five galaxies (shown in Figures 5 to 9) and the rest of galaxies in our LRD sample. \nC.1. Galaxies detected up to F1800W: JADES-57356 and JADES-204851 \nFigures 5 and 6 showed the fits for JADES -57356 and JADES -204851, the 2 LRDs in our sample detected up to (at least) 18 µ m. \nJADES -57356 is a canonical LRD, presenting a change in slope in its SED at around 2 µ m. No prominent emission lines are detected in any medium- and broad-band filter, although some excess is seen in the filters that would cover the Lyα and MgII emissions for z = 5 . 5 1 . To reproduce the characteristic, bimodal SED of the LRDs, 2 of the 4 codes use two distinct components. \nThe SED fit with synthesizer-AGN uses 2 stellar populations (with independent attenuations), one with a young (60 Myr) and mildly unobscured (A(V) ∼ 1 mag) starburst, which also contributes to the faint emission lines, and an older (250 Myr) stellar population with much larger reddening, A(V) ∼ 4 mag, that dominates the stellar mass content (98% of the total). \nProspector-AGN uses a similarly young (130 Myr) stellar component for the UV emission and AGN emission from a dust-obscured accretion disk for the optical. Prospector-SF , on the other hand, uses only stars to reproduce the SED up to the optical and redder wavelengths, as also obtains Prospector-AGN +. However, Prospector-SF requires an unusual, extremely gray attenuation law (n ∼ 0.4). Indeed, Barro et al. (2023) noted that a typical Calzetti law (n=0) and a single attenuation parameter would not reproduce the SED of the LRDs. In fact, as can be extracted from the data in Table 2 and the parameters written in the plots, the ratio between the far-UV (at ∼ 150 nm, FUV) and optical ( ∼ 550 nm) attenuations is ∼ 1.5 for Prospector-SF and ∼ 1.5 for Prospector-AGN +, smaller than the ∼ 2 . 6 implied by the Calzetti law. We note that the synthesizer-AGN fits assume such as law, but the combination of the independent attenuations for the old and young stellar populations results in a FUV-to-optical attenuation ratio close to the values given above ( κ FUV = 1 . 8). \nWe also remark that synthesizer-AGN does use a Calzetti law but assumes independent extinctions for the 2 stellar populations in the host, and also for the AGN, which allows a good fit to the data for JADES -57356. ProspectorAGN + does not require a two-component fit but it allows for hybrid AGN+galaxy models, which can sometimes fit the SED with 2 distinct components, each dominating a different spectral range, but can also fit the whole SED with only one of those components. This is the case for JADES -57356 which, as noted above, exhibits a similar best-fit model to the prospector-SF model for the UV-to-NIR region. \nOverall, all 4 models provide good fits to the UV-to-NIR SED, with some differences. The three stellar-dominated models suggest a relatively large stellar mass, M ⋆ = 10 10 . 4 M ⊙ for synthesizer-AGN . In contrast, in ProspectorAGN the emission from the obscured accretion disk (A(V) = 2 . 8 mag) dominates the SED up to λ = 1 µ m (and the AGN at even bluer wavelengths), which leads to a much smaller stellar mass for the galaxy host that is only visible in the rest-UV. \nWhile all the models obtain similarly good UV-to-NIR fits, they differ substantially in the MIR emission, where the SED shows a clear flattening beyond 1 µ m (F770W-F1000W ∼ 0 mag) followed by an upturn around 2 µ m that steepens quickly towards the redder bands. This SED shape is well reproduced by the stellar-continuum-dominated models that combine a stellar peak around 1.6 µ m with dust emission presenting a prominent 3 µ m PAH line in the MIRI F2100W band. For synthesizer-AGN , the overall best-fitting code ( χ 2 values given in the plots), the dust emission model is quite extreme: the best template corresponds to a ULIRG with A(V) = 10 mag, 350 pc star-forming region size, 60% luminosity ratio of OB stars in hot spots compared to total luminosity, and dust density in hot spots 10 2 cm -3 . This dust emission model peaks at rest-frame ∼ 30 µ m (see Figure 16 in Appendix B), quite a blue wavelength compared to the more typical ∼ 70 -100 µ m for (U)LIRGs (e.g., Rieke et al. 2009), revealing the important role of warm/hot dust in LRDs, and also implying a relatively low emission at (sub-)mm wavelengths, where the flux of LRDs is faint (Labb'e et al. 2023b; Williams et al. 2023a). \nThis Siebenmorgen & Krugel (2007) model points to the existence of very hot dust bathed by a very intense radiation field, coming from dust-buried OB stars (the total attenuation of stellar light in the models ranges from 2 to 4.5 mag), as indicated by the synthesizer-AGN and prospector-AGN + fits. Such an intense and compact heating source also matches well what can be expected from an AGN, which in principle could contribute to some extent to the MIR emission. Indeed, synthesizer-AGN and prospector-AGN + do show some contribution, although faint and thus uncertain, from an AGN. \nHowever, prospector-AGN , the only code that is forced to be AGN-dominated in this spectral range, provides a worse fit to the MIRI bands because the transition from disk-dominated to torus dominated emission around λ ∼ 2 µ m leads to redder MIRI colors due to the steeper slope of the dust torus relative to the dust associated with star formation in the other models (note the bad fits to the F560W and F1000W fluxes). To some degree, prospector-SF also has problems fitting the rest-NIR region because the nebular emission partially hides the flattening of the stellar continuum and provides a worse fit to those MIRI bands. \nIn summary, for this source the fit to the dust emission presents a smaller χ 2 when including the, most probably star-formation related, 3 µ m PAH. The relatively large stellar mass and old mass-weighted age (50-700 Myr) could support the AGN nature of the dust emission (for several reasons, e.g., the stars are relatively old and that could intuitively imply that there are not so obscured, or the evolved state of the galaxy gives time for the SMBH to grow), but the fits are significantly worse. \nJADES -204851, whose SED and postage stamps are shown in Figure 6, is a source with spectroscopic redshift z sp = 5 . 4790 discussed previously in Matthee et al. (2023b, GOODS-S-13971 in that paper), where they reported on the presence of a broad 2200 km s -1 H α component (S/N ∼ 5 and with some artifacts in the spectrum -certainly, this is not one of the clearest BLR sources in Matthee et al. 2023b-) arising from an AGN with a 10 7 . 5 M ⊙ SMBH. This AGN would account for half of the H α emission, according to the spectroscopic analysis. \nIn our synthesizer-AGN fits for JADES -204851 (top left panel of Figure 6), the AGN emits around 10% of the total emission at 2 µ m, also contributing to emission lines such as H α , [OIII], and MgII. The MIR emission would also have a significant contribution from star-formation powered dust emission, with a ∼ 10% possible contribution from a QSO-like emission. For this galaxy, the best fitting dust emission model corresponds to one 350 pc ULIRG star-forming region with highly embedded stars (A(V) = 72 mag), with a 40% ratio of OB stars in hot spots compared to the total luminosity, and dust density in hot spots 10 4 cm -3 . The optical-to-NIR is fitted with a combination of two stellar populations. One of them is a very young (1 Myr) unobscured (A(V) = 0 . 5 mag) starburst, which takes care of the strong emission seen in spectroscopy and detected in the NIRCam imaging and the blue continuum. The \nother population is slightly older (10 Myr) affected by a large reddening (A(V) ∼ 2 . 5 mag) and dominating (94%) the total stellar mass (M ⋆ = 10 9 . 6 M ⊙ ). \nThe prospector-AGN + results are very similar to those obtained with synthesizer-AGN in the rest-frame optical and near-infrared: the SED is dominated by stars and dust heated by stars. However, prospector-AGN + reproduces the UV part of the SED with stars alone, while some contribution from a (unobscured) QSO is obtained with synthesizer-AGN , mainly due to the possible presence of a MgII line at observed wavelength ∼ 2 µ m. The contribution is small, and thus uncertain, but the spectroscopy hints that there is a broad-line component (at 5 σ confidence level). This difference between both codes might also be caused by the different configurations of the stellar extinction laws. As pointed out by Kriek & Conroy (2013b), the standard Calzetti law (used by synthesizer-AGN ) typically provides poor fits at UV wavelengths for highz galaxies and thus the AGN component is likely selected from the model to fit the SED in synthesizer-AGN . Meanwhile, prospector-AGN + used the updated galaxy extinction law introduced by Kriek & Conroy (2013b), which is supposed to be more realistic. On the other hand, the independent treatment of the attenuation for old and young stars in synthesizer-AGN can overcome the problems with a single extinction parameter. \nThe prospector-SF also provides a qualitatively good fit with a single stellar component and a modest A(V)=1.2 mag, but very gray attenuation, n = 0 . 4 (translating to κ FUV ∼ 1 . 5, same value obtained by synthesizerAGN ). Interestingly, the fit to the MIRI bands is worse due to the presence of strong emission lines in the model which would imply a larger flux in F1500W than is observed. The dust emission in this model contributes less than 10% of the total emission at 2 µ m and has virtually no impact in the best-fit SED. \nThe prospector-AGN fit also provides a good overall fit of the NIRCam and MIRI photometry, using stars for the UV and an AGN for the optical/IR (as in the LRD characterized in Killi et al. 2023). In this galaxy, which exhibits significantly bluer MIRI colors than JADES -57356, the obscured emission from the disk (A(V)= 2 . 3 mag) dominates the optical and NIR emission up to λ ∼ 2 µ m. The emission from the torus is mostly unconstrained but it would require low IR luminosities or a very large opacity ( τ V ≳ 100). As before, the implied stellar mass of the host is the smallest of all the models, M ⋆ = 10 8 . 1 M ⊙ . \nIn any case, the different morphology seen in F814W compared to the LW NIRCam bands, with the emission in the former being dominated by a knot located to the NW of the very concentrated (and very red) emission seen in the latter, points to star formation dominating the UV spectral range. This morphological difference between the rest-frame UV and NIR is also seen in other LRDs, for example, JADES -211388 or JADES -79803, discussed later.", 'C.2. Galaxy detected up to F1500W: JADES-219000': 'Figure 7 discussed the results for JADES -219000, the third LRD in our sample detected up to 15 µ m. This source is presented in Sun et al. (2024, in prep.), with a spectroscopic redshift of z = 6 . 8119. Similarly to JADES -57356, the best-fit with synthesizer-AGN indicates a UV-to-NIR SED dominated by stars with a relatively high mass (M ⋆ = 10 10 . 4 M ⊙ ), and dust emission revealing a very intense radiation field. The dust emission model corresponds to an intense starburst with 90% luminosity arising from OB stars embedded in a A(V) = 72 mag compact (350 pc), dense (10 4 cm -3 ) and clumpy dust cloud. The very hot dust present in this object could also be heated by an AGN, but the typical torus emission used in prospector-AGN + fails to fall within the F1800W and F2100W upper limits. This, in part, explains the larger χ 2 value compared to the other codes, jointly with discrepancies in the FUV which could be interpreted as attenuation law effects. \nThe prospector-SF fit is also relatively good with similar best-fit values as the other galaxies, i.e., a young stellar population, with a small and very gray attenuation (A(V)=1.6 mag, n=0.33, κ FUV ∼ 2). The dust emission does not contribute significantly to the MIRI fluxes ( < 1% at λ = 2 µ m), the nebular emission dominates (but the F1000W is not well-fitted). The prospector-AGN fit also agrees well with the data. As before, the best-fit model also implies a low stellar mass for the host and a moderate attenuation for the disk (A(V)=1.6 mag), but it favors a more luminous torus with large opacity, τ V = 100. This means that the torus emission starts to contribute significantly at λ ∼ 1 µ m, but it peaks at longer wavelengths ( λ = 30 µ m) than the low opacity tori (see Figure 16 of the appendix), and therefore the model does not over-estimate the F1800W and F2100W upper limits.', 'C.3. Galaxies detected up to F1280W: JADES-211388 and JADES-79803': "Figure 8 showed the SED fits for JADES -211388, a spectroscopically confirmed LRD at z = 8 . 3846. This is one of the highest redshift LRDs and, consequently, the MIRI detections probe shorter rest-frame wavelengths than in previous galaxies. \nThe synthesizer-AGN best-fit indicates that the SED is dominated by young (3 Myr), slightly extincted (A(V) ∼ 1 mag) stars in the blue, with some contribution from slightly older (5 Myr) and more obscured (A(V) = 2 . 5 mag) stars in the red, for a total stellar mass of M ⋆ = 10 9 . 1 M ⊙ . Remarkably, the NIR emission is dominated by a heavily obscured AGN as opposed to the star-formation powered dust emission of previous galaxies. \nThe obscured AGN template is very similar to the best-fit model with prospector-AGN + and resembles the AGN-dominated fits in the NIR of Barro et al. (2023) or Labb'e et al. (2023b), but with significant contribution from stars up to rest-frame wavelength ∼ 2 µ m, where the AGN would contribute with 50% of the total flux), but with a lower luminosity that is still consistent with the upper limits. \nThe prospector-SF best-fit parameters are almost identical to the previous galaxies with a young stellar population obscured with a mild but gray dust attenuation (A(V)=1.7 mag, n = 0 . 39, κ FUV ∼ 1 . 4, still failing to reproduce the FUV) and a relatively flat NIR emission dominated by the nebular continuum in the λ ∼ 1 -2 µ m range. \nprospector-AGN also provides a good fit to the optical-to-NIR SED but it has some issues fitting the sharp Lyman break indicated by the blue NIRCam bands (we remind the reader that the attenuation is fixed to 0 mag in these models). The best-fit properties are again consistent with a young, low-mass galaxy host M ⋆ = 10 8 . 7 M ⊙ and an obscured AGN, A(V) = 3 . 4 mag. The lack of MIRI constraints at long wavelengths leads to best-fit models dominated only by the accretion disk emission even up to λ =4 µ m. However, a more significant contribution from the torus emission at λ ≳ 2 µ m would still be consistent with the upper limits, as shown in the synthesizer-AGN and prospector-AGN + fits. \nFinally, Figure 9 shows the SED fits for JADES -79803, another spectroscopically confirmed LRD, this time at z = 5 . 4007. The SED exhibits a flattening around λ ∼ 1 µ m, probed by F560W and F770W, followed by a steep rise in F1000W and F1280W and then another flattening indicated by the upper limits in the reddest bands. The best-fit parameters with prospector-SF are similar to JADES -211388, but the stellar mass is a bit smaller, M ⋆ = 10 8 . 6 M ⊙ , with all stars being younger than 20 Myr and presenting different attenuations between 0.5 and 2 mag. The near-tomid IR emission is reproduced again by an obscured AGN, but in this case, the F1280W detection provides stronger evidence of an upturn. \nThe prospector-AGN + also reproduces the SED with a similarly young stellar population and an obscured AGN dominating the near-to-mid IR emission. \nThe prospector-SF best-fit continues on the same trends as before, but it provides a worse fit to the reddest MIRI bands whose steep rise can not be easily reproduced with emission from star-formation heated dust. \nprospector-AGN provides a good overall fit, the best among all codes for this source, with the same trends in previous galaxies (low-mass host and obscured disk dominating the optical emission). In this case, the additional MIRI constraints motivate a fit with a larger contribution of the torus emission to the IR emission starting around λ ∼ 2 µ m, but consistent with the upper limits.", 'C.4. Galaxies detected up to F770W and with no MIRI detections': 'Figures 17 and 18 show the SED fits for all F770W-detected sources not presented in the main text and the source with no detection in any MIRI band, respectively. Results for the 4 fitting codes described in Section 4 are provided. The F770W sample shown in Figure 17 presents more heterogeneous SEDs than the Golden Five or F1000W samples discussed in Section 4. The global slope of the SED is more constant, with the difference between the SW and LW bands being less marked than for the other LRDs. In fact, this sample includes a very blue galaxy (187025, z sp = 6 . 9076), which only entered the sample because of the enhanced F444W flux due to an emission line. The SEDs, including MIRI upper limits (remarkably, for F1000W and F1280W), are very flat in the 1-2 µ m range, completely compatible with being dominated by stars without the need of much more contribution from other emitting components. Compared to other subsamples, the F770W sources are more extincted (nearly 3 mag on average for the stars dominating the mass) and present older mass-weighted ages (nearly 10 Myr, compared to the 3-4 Myr for galaxies detected at longer MIRI wavelengths). \nFigure 18 shows a similar result for MIRI undetected sources, the available data (limited to NIRCam) is well fitted by stars only. The upper limits for MIRI would be consistent with hot dust emission from an AGN torus, but the current data is inconclusive. \nConsidering the full sample of 31 sources, the median/quartiles χ 2 for synthesizer-AGN are 342 800 207 , prospectorAGN + obtains larger relative values compared to the former by a factor of 2 . 3 4 . 8 1 . 3 , as is the case for prospector-SF with 1 . 2 2 . 5 0 . 6 times larger χ 2 values, and prospector-AGN , 1 . 3 3 . 7 0 . 6 . \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 17. SED fitting results for the 7 galaxies detected at F770W and not beyond (i.e., the plot does not include any of the Golden Five or F1000W galaxies discussed in the main text). SEDs are normalized to 0.7 µ m. In gray, we show the fits to each individual galaxy, and we provide an average in brown. \n<!-- image -->', 'REFERENCES': 'Akins, H. B., Casey, C. M., Allen, N., et al. 2023, ApJ, 956, 61, doi: 10.3847/1538-4357/acef21 \n\' Alvarez-M\'arquez, J., Crespo G\'omez, A., Colina, L., et al. 2023, A&A, 671, A105, doi: 10.1051/0004-6361/202245400 Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33, doi: 10.1051/0004-6361/201322068 Astropy Collaboration, Price-Whelan, A. M., Sip"ocz, B. M., et al. 2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f Bagley, M. B., Finkelstein, S. L., Koekemoer, A. M., et al. 2023, ApJL, 946, L12, doi: 10.3847/2041-8213/acbb08 Barro, G., P\'erez-Gonz\'alez, P. 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2024arXiv240908592S

Cosmic ray CR feedback plays a vital role in shaping the formation and evolution of galaxies through their interaction with magnetohydrodynamic waves. In the CR selfconfinement scenario the waves are generated by the CR gyroresonant instabilities via CR streaming or CR pressure anisotropy and saturate by balancing wave damping. The resulting effective particle scattering rate by the waves nueff critically sets the coupling between the CRs and background gas but the efficiency of CR feedback is yet poorly constrained. We employ 1D kinetic simulations under the MagnetohydrodynamicParticleInCell MHDPIC framework with the adaptive deltaf method to quantify nueff for the saturated state of the CR pressure anisotropy instability CRPAI with ionneutral friction. We drive CR pressure anisotropy by expandingcompressing box mimicking background evolution of magnetic field strength and the CR pressure anisotropy eventually reaches a quasisteady state by balancing quasilinear diffusion. At the saturated state we measure nueff and the CR pressure anisotropy level establishing a calibrated scaling relation with environmental parameters. The scaling relation is consistent with quasilinear theory and can be incorporated to CR fluid models in either the singlefluid or pbyp treatments. Our results serve as a basis towards accurately calibrating the subgrid physics in macroscopic studies of CR feedback and transport.

2024-09-01T00:00:00Z

['2024arXiv240908592S', '10.48550/arXiv.2409.08592', 'arXiv:2409.08592']

['Astrophysics - High Energy Astrophysical Phenomena', 'Astrophysics - Astrophysics of Galaxies']

Kinetic simulations of the cosmic ray pressure anisotropy instability cosmic ray scattering rate in the saturated state

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.08592.pdf

{'No Header': 'Draft version September 16, 2024 Typeset using L A T E X twocolumn style in AASTeX631', 'Kinetic simulations of the cosmic ray pressure anisotropy instability: cosmic ray scattering rate in the saturated state': 'Xiaochen Sun, 1, 2 Xue-Ning Bai , 1, 3 and Xihui Zhao 1 \n<!-- image --> \n1 Institute for Advanced Study, Tsinghua University, Beijing 100084, China 2 Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA 3 Department of Astronomy, Tsinghua University, Beijing 100084, China', 'ABSTRACT': 'Cosmic ray (CR) feedback plays a vital role in shaping the formation and evolution of galaxies through their interaction with magnetohydrodynamic waves. In the CR self-confinement scenario, the waves are generated by the CR gyro-resonant instabilities via CR streaming or CR pressure anisotropy, and saturate by balancing wave damping. The resulting effective particle scattering rate by the waves, ν eff , critically sets the coupling between the CRs and background gas, but the efficiency of CR feedback is yet poorly constrained. We employ 1D kinetic simulations under the Magnetohydrodynamic-ParticleIn-Cell (MHD-PIC) framework with the adaptive δf method to quantify ν eff for the saturated state of the CR pressure anisotropy instability (CRPAI) with ion-neutral friction. We drive CR pressure anisotropy by expanding/compressing box, mimicking background evolution of magnetic field strength, and the CR pressure anisotropy eventually reaches a quasi-steady state by balancing quasi-linear diffusion. At the saturated state, we measure ν eff and the CR pressure anisotropy level, establishing a calibrated scaling relation with environmental parameters. The scaling relation is consistent with quasilinear theory and can be incorporated to CR fluid models, in either the single-fluid or p -byp treatments. Our results serve as a basis towards accurately calibrating the subgrid physics in macroscopic studies of CR feedback and transport. \nKeywords: Plasma astrophysics (1261) - Alfven waves (23) - Magnetohydrodynamics (1964) - Cosmic rays(329)', '1. INTRODUCTION': "Cosmic rays (CRs) are (trans-)relativistic charged particles pervading in space. With relatively low number density ( ∼ 10 -9 cm -3 ), they possess an energy density ( ≥ 1eV cm -3 ) comparable to the internal, kinetic and magnetic energy density of gas in the Galaxy (see reviews, e.g. Blasi 2013; Zweibel 2013; Grenier et al. 2015). Consequently, CRs are expected to play a dynamically important role, known as CR feedback (see reviews, e.g., Ptuskin 1997; Ferri'ere 2001; Zweibel 2017; Naab & Ostriker 2017). At a more fundamental \nCorresponding author: Xiaochen Sun, Xue-Ning Bai \nsun.xiaochen@princeton.edu \nxbai@tsinghua.edu.cn \nlevel, the interaction between the CRs and the (ionized) gas is primarily through fluctuations (waves) in background electromagnetic field (e.g., Jokipii 1966; Voelk 1975), as opposed to direct collisions (e.g., Radin et al. 1974; Ferrando et al. 1988; Schlickeiser 1989). Based on the source of such waves/fluctuations, CR feedback/transport are classified into two categories: fluctuations generated by CRs themselves, known as 'selfconfinement' (e.g., Wentzel 1969; Skilling 1971; Holman et al. 1979), and fluctuations cascaded from extrinsic turbulence (e.g., Kolmogorov 1941; Iroshnikov 1964; Kraichnan 1965; Goldreich & Sridhar 1995). As CR scattering and transport are primarily sensitive to electromagnetic fluctuations at their gyration scale, it is expected that self-confinement governs low-energy CRs (from sub-GeV to ∼ 10 2 GeV, Blasi et al. 2012; Evoli \net al. 2018), while extrinsic turbulence likely dominates the transport of higher-energy CRs. \nAs the CR energy density typically peaks around ∼ GeV (e.g., Adriani et al. 2011, 2013; Consolandi 2014), self-confinement is expected to be the dominant process driving the force of CR feedback. At microphysical level, this is mediated by the CR gyro-resonant instabilities (e.g., Lerche 1967; Wentzel 1968; Kulsrud & Pearce 1969; Ginzburg et al. 1973; Berezinskii et al. 1990). Here, magnetohydrodynamic (MHD) waves in the background gas can be destabilized when the CRs become weakly anisotropic with respect to the bulk gas (by more than about U A /c , where U A is the Alfv'en speed) through gyro-resonant interactions between the CRs and the waves. The classic flavor is the CR streaming instability (CRSI, e.g. Wentzel 1974; Skilling 1975a,b,c), which occurs when the speed of CR bulk motion relative to background gas exceeds U A . This situation naturally arises when the CRs escape from the source. With the gyro-resonant interactions, the growth of MHD waves are fed by the free energy from CR streaming, which in turn reduce the bulk CR motion towards the Alfv'en wave speed (Adkins & Schekochihin 2018). Another flavor of the CR gyro-resonant instability, which is poorly studied, is the CR pressure anisotropy instability (CRPAI, e.g. Lazarian & Beresnyak 2006; Yan & Lazarian 2011; Zweibel 2020), which arises when the level of CR pressure anisotropy exceeds ∼ U A /c . This situation naturally occurs when the bulk gas undergoes compression/expansion and/or shear, where CR anisotropy is driven by a change in background field strength ˙ B/B under the conservation of magnetic moment. The interaction between CRs and electromagnetic fluctuations tends to isotropize the CR momentum distribution to within the ∼ U A /c level in the gas co-moving frame. \nThe coupling between the CRs and the background gas is reflected in the effective scattering rate ν eff , which is determined by the amplitude of waves. In the selfconfinement regime, the wave amplitudes are set the balance between driving (from the CR gyro-resonant instabilities) and various wave damping mechanisms, which eventually determines the efficiency of CR feedback. These include the ion-neutral collisions (Kulsrud & Pearce 1969; Soler et al. 2016), the non-linear Landau damping (Lee & Volk 1973), the linear Landau damping (Foote & Kulsrud 1979; Wiener et al. 2018), the extrinsic turbulence (Farmer & Goldreich 2004; Lazarian 2016; Xu & Lazarian 2022), and the influence of charged dust grains (Squire et al. 2021). Which of the mechanisms would dominate depends on the local environment, with ion-neutral damping dominating in the neutral medium, while non-linear Landau damping is more prevalent in \ndilute gas (Zweibel 2013; Armillotta et al. 2021, 2022, 2024). \nAt macroscopic level, galaxy simulations often model CRs as a fluid and employ approximate prescriptions for the effective scattering rate. Modeled as fluids instead of individual kinetic particles, the CR population is characterized by its energy density, momentum flux, and pressure (McKenzie & Voelk 1982; Guo & Oh 2008; Pfrommer et al. 2017; Jiang & Oh 2018; Thomas & Pfrommer 2019; Chan et al. 2019). Early simulations at galactic scales typically adopt highly simplified treatment (e.g., constant diffusion or streaming) of CR scattering, generally finding significant dynamical consequences especially in heating and in driving large-scale galactic outflows (e.g., Guo & Oh 2008; Dorfi & Breitschwerdt 2012; Uhlig et al. 2012; Hanasz et al. 2013; Girichidis et al. 2016; Wiener et al. 2017b; Jacob et al. 2018; Butsky & Quinn 2018; Chan et al. 2019; Dubois et al. 2019; Dashyan & Dubois 2020), but the results are highly sensitive to the CR prescriptions. More recent works start to incorporate more realistic prescriptions of the CR scattering rates based on the CRSI and various damping mechanisms (Armillotta et al. 2021, 2024), emphasizing the importance of resolving the multi-phase medium, while current state-of-the-art models are still unable to reproduce all observational constraints from the Galaxy (Hopkins et al. 2021). \nThe main motivation of this work is two-fold. First, we note that the scattering rates in the self-confinement regime, as calculated from balance between wave growth and damping, are usually obtained under the quasilinear theory (QLT) as in the aforementioned simulations. However, it involves a number of approximations that likely yield results that deviate from the reality, and kinetic simulations are needed to clarify and calibrate these results (e.g. Bai 2022). Second, nearly all calculations so far focus on the CRSI (e.g. Holcomb & Spitkovsky 2019; Bai et al. 2019; Haggerty et al. 2019; Lemmerz et al. 2024), which indeed likely dominates CR scattering and heating in the general situations (Zweibel 2020), but the role CRPAI have not been systematically investigated. In this work, we conduct kinetic simulations of the CRPAI, and by balancing wave growth and damping, we aim to calibrate the CR scattering rates from the CRPAI from first principles. \nOur simulations employ the magnetohydrodynamicparticle-in-cell (MHD-PIC) method (Zachary & Cohen 1986; Lucek & Bell 2000; Bai et al. 2015), which treats CRs kinetically as particles while treat the background plasma as gas described by MHD. Compared to the conventional PIC methods, it avoids resolving the microscopic scales of the background plasma, thus becomes \nhighly advantageous in simulating the CR gyro-resonant instabilities. Using the ATHENA MHD code (Stone et al. 2008; Bai et al. 2015), Bai et al. (2019) developed the framework for simulating the CRSI, which captures the linear growth over a wide range of wavelengths and quasi-linear evolution over a broad range of CR energies around the peak of the energy distribution (which can be taken to be ∼ GeV). This is facilitated by the implementation of the δf method (e.g. Parker & Lee 1993; Kunz et al. 2014) to suppress Poisson noise, allowing one to use a reasonable number of particles to properly sample a weakly anisotropic particle distribution. Subsequent works of Plotnikov et al. (2021); Bambic et al. (2021) further incorporated a simple prescription of ion-neutral damping. In Bai (2022), we introduced the streaming box framework, where CRs move along an imposed CR gradient, allowing the CRSI to be continuously driven, balanced by damping. This framework thus enables the measurement of the CR scattering rates at the saturated states from first principles. \nIn this work, we migrate the MHD-PIC simulation to the ATHENA++ MHD code (Stone et al. 2020; Sun & Bai 2023), which is more flexible and computationally efficient. We have already tested that the code well captures the linear growth of the CRPAI. In preparation for this work, we have also developed the expanding/compressing box (Grappin et al. 1993; Sironi & Narayan 2015; Bott et al. 2021) framework for MHDPIC (Sun & Bai 2023), which continuously drives the CR pressure anisotropy to trigger the CRPAI. This approach may mimic a local patch in compressible turbulence, or a mixing layer between two gas phases, or perhaps also including background shear, where a background changing field drives CR pressure anisotropy. In this work, we employ the expanding/compressing box to drive the CRPAI, balanced by ion-neutral damping to achieve a steady state. This allows us to quantitatively measure the effective CR scattering rate in the alternative scenario of CRPAI, establishing their dependence on local medium properties. This study thus fills a major gap in understanding the microphysics of CR feedback. \nThis paper is structured as follows: Section 2 outlines the formulation and numerical setup. In particular, we introduce the adaptive δf method in Section 2.2. We calculate the expected CR scattering rates under our simulation setup based on QLT in Section 3. The simulation results are presented in Section 4, which also offers quantitative calibration of the CR scattering rates. We discuss the implications of the results towards realistic systems in Section 5. In Section 6, we conclude and discuss prospects for future research.", '2. FORMULATION AND NUMERICAL METHODS': 'In this section, we briefly review the MHD-PIC formulation in the expanding box framework (Sun & Bai 2023), and the treatment of ion-neutral damping (Plotnikov et al. 2021; Bambic et al. 2021; Bai 2022). We next propose a novel adaptive δf method to reduce noise in our simulations, before describing our simulation setup.', '2.1. The governing equations': "The MHD-PIC method treats CRs as super-particles, following the standard approach of PIC, and solves MHD equations for the thermal gas (here specifically just for the thermal ion plasma, see later in this subsection). Each simulation particle carries both position information x and normalized momentum information ( p /m ). The CR kinetic equations are given as: \nd x = v , \nv ≡ ( p /m ) γ , γ ≡ √ 1 + ( p /m ) 2 / C 2 . \nd t (1) d( p /m ) d t = ( q mc ) ( c E + v × B ) , (2) (3) \nHere, q/ ( mc ), v , γ and C represent the CR charge-tomass ratio, particle velocity, Lorentz factor and the numerical speed of light, respectively. The CR population meanwhile exerts a Lorentz force on the MHD gas as CR backreaction. The MHD equations with the CR backreaction read, \n∂ t ρ i + ∇· ( ρ u i ) = 0 , ∂ t ( ρ i u i ) + ∇· ( ρ i u T i u i -B T B + P ) = -ν IN ρ i u i -( Q CR c c E + j CR c × B ) , ∂ t B = -∇× ( c E ) , c E = -u i × B , (4) \nHere, the variables include ρ i for ion gas density, P ≡ ( ρ i c 2 s + B 2 / 2) I for total pressure with I being the identity tensor, c s for the isothermal sound speed, and u i for the ion gas velocity. The unit for magnetic field is normalized such that magnetic permeability equals one, thus absorbing factors of 1 / √ 4 π . We bypass the CR backreaction on the MHD gas energy in this work by applying an isothermal equation of state (see the reason in Section 3.2). The cosmic ray charge density Q CR and the current density j CR are both functions of the phase space distribution function f ( t, x , p /m ) for the \nCR population: \nQ CR c ≡ ( q mc ) m ∫ f d 3 p = ( q mc ) ρ CR , j CR c ≡ ( q mc ) m ∫ v f d 3 p , (5) \nHere, m represents the mass of an individual CR particle, and ρ CR denotes the mass density of the CR ensemble. \nGiven that the Lorentz force is approximately proportional to ρ CR , the intensity of the CR backreaction is evaluated through the mass density ratio between CR and thermal ionized gas, ρ CR /ρ i . With the CR cyclotron frequency being Ω ≡ ( q/ ( mc )) | B | , the characteristic rate for CR feedback is ( ρ CR /ρ i )Ω. \nWe consider the ion-neutral friction as the wave damping mechanism in this work. Ion-neutral damping prevails as the primary damping mechanism in the cold dense medium (Zweibel 2013; Armillotta et al. 2021, 2022; Hopkins et al. 2022), where the ionization fraction ρ i /ρ neu ≪ 1. Under typical ISM conditions, the ion-neutral damping can be treated such that we essentially only simulate the ion gas subjecting to a frictional force -ν IN ρ i u i on top of the background static neutrals (Plotnikov et al. 2021), as included in Equation 4. Here ν IN represents the collision frequency of neutrals with ions. This approach remains valid only in the short-wavelength regime, where the Alfv'en wave frequency ω ( k ) = kU A for the relevant wavenumber k exceeds ν IN (Reville et al. 2007; Plotnikov et al. 2021). Here, U A ≡ B/ √ ρ i stands for the Alfv'en velocity. For the ∼ GeV CRs we focus on in this work,the resonant wavenumber k res ∼ Ω m/p , varies around ∼ AU -1 , much shorter than ν IN /U A in typical ISM conditions (Plotnikov et al. 2021), justifying our approach. We also note that the induction equation retains its ideal MHD form. \nIn this work, we drive CR pressure anisotropy by mimicking a local uniform patch of gas among the macroscopic system (e.g., ISM/CGM, etc.) undergoing expansion/compression. This is achieved under the expanding box framework (see derivations in Grappin et al. 1993; Sironi & Narayan 2015; Sun & Bai 2023), and CR pressure anisotropy is developed owing to the conservation of the magnetic moment of individual CR particles. This framework has been implemented in the most general manner in the ATHENA++ MHD-PIC module as described in Sun & Bai (2023), where background expansion rate can be flexible in any of the three directions. In the simulations presented here, the background magnetic field lies along the x -direction. We let the background expand or compression at the rate a ( t ) in the y - and z -directions and at a rate of a 2 ( t ) in the x -direction. This configuration enables MHD perturbations to propagate freely \nalong the background magnetic field line (see Appendix of Sun & Bai 2023). \nWe reformulate the MHD equations and the CR equations of motion in comoving coordinates x ' of the expanding box while expressing the MHD gas quantities and CR momentum in the lab frame. The equations we eventually solve in this study are as follows: \n∂ t ρ i + ∇ ' · ( ρ u i ) = -4 ρ i ˙ a a , (6) \n= -ν IN ρ i u i -( Q CR c c E + j CR c × B \n∂ t ( ρ i u i ) + ∇ ' · ( ρ i u T i u i -B T B + P ) -4 ∂ t a a ρ i u i -ρ i u i · D , \n) (7) \n∇ ' · B = 0 , (8) \n∂ t B -∇ ' × ( c E ) = -4 ∂ t a a B + D · B , (9) \nc E = -u i × B , (10) \nA · d x ' d t = v , (11) \nd( p /m ) d t + D · p m = q mc ( c E + v × B ) , (12) \nwhere, \n∇ \n' = ( ∂ a 2 ∂x ' , ∂ a∂y ' , ∂ a∂z ' ) , D = diag ( 2 ∂ t a a , ∂ t a a , ∂ t a a ) A = diag ( a 2 , a, a ) . \n, \nWe note that even we use an isothermal equation of state, the background expansion or compression brings about a thermodynamic impact on the background MHD gas, following an adiabatic polytropic process (Squire et al. 2020), c s ∝ a -4 / 3 ( t ). The expansion (compression) dynamically reduces (amplifies) the background magnetic field at a rate of ˙ B/B = -2˙ a/a , and the time evolution of the background field strength is \nB g ( t ) = B g (0) exp ( -2 ˙ a a t ) . \nConsequently, the CR cyclotron frequency Ω slowly varies over time. Both the gas density and CR density in the comoving frame changes at a rate of -4˙ a/a with time. On the other hand, the mass density ratio ρ CR /ρ i , and the Alfv'en speed U A remains constant throughout.", '2.2. Adaptive δf method': 'Inherent to a particle-based representation of the CR distribution function f , the MHD-PIC method is subject to Poisson noise, which represents a major issue \nin the simulation of the CR gyro-resonant instabilities (Bai et al. 2019). Simply increasing the particle number N becomes unfeasible as the noise level only reduces as N -1 / 2 which is very inefficient. This difficulty can be overcome by the δf method (e.g. Parker & Lee 1993; Hu & Krommes 1994; Denton & Kotschenreuther 1995; Kunz et al. 2014; Bai et al. 2019). In the δf method, the phase space distribution f ( t, x , p /m ) is divided into two components: f 0 ( t, x , p /m ) as a known analytical form, and δf ( t, x , p /m ), representing the deviation δf ≡ f -f 0 . Consequently, the CR backreaction also splits into two contributions: one associated with f 0 , which can be evaluated analytically, and the other from δf . The simulation particles, along with statistical noise, only contribute to the backreaction from the δf part. As a result, when counting the CR backreaction (Equation 5) from δf , the contribution from each simulation particle is changed by multiplying a weight factor w , \nw = 1 -f 0 ( t, x ( t ) , p ( t )) f ( t, x ( t ) , p /m ( t )) . (13) \nTraditionally, f 0 is often fixed as the initial CR distribution (e.g., Bai et al. 2019; Bai 2022) or as the CR equilibrium distribution after adiabatic evolution (e.g., Sun & Bai 2023). With the expanding box, however, the bulk anisotropy level in f keeps changing due to both the ˙ B/B driving and quasi-linear evolution. This would render the δf method less effective in mitigating statistical noise. Therefore, we adaptively adjust f 0 towards matching the evolving distribution f , so that the statistical noise remains at a relatively low level. \nBy design, the adjustment of f 0 can be arbitrary. In our case, we initialize the CR particles with an isotropic κ distribution \nf iso ( p ) = ρ CR m ( πκp 2 0 ) 1 . 5 G ( κ +1) G ( κ -0 . 5) ( 1 + p 2 κp 2 0 ) -κ -1 , (14) \nwhere G (), κ and p 0 denote the Gamma function, the power index, and the peak momentum, respectively. As the background expands/compresses, the CR distribution becomes anisotropic: \nf aniso ( t, p ) = ρ CR ξ 4 ma 4 ( πκp 2 0 ) 1 . 5 G ( κ +1) G ( κ -0 . 5) ( 1 + p 2 κp 2 0 ( ξ 4 µ 2 -ξ 2 µ 2 + ξ 2 ) ) -κ -1 , (15) \nwhere µ and ξ represent the CR pitch angle and anisotropy level. The factor a 4 in the denominator accounts for the reduction in lab frame density resulting \nfrom the background expansion (see previous subsection). The pitch angle anisotropy is significant for particles whose p ≳ p 0 but becomes negligible when p ≪ p 0 , as the κ -distribution yields f ( p ≪ p 0 , µ ) ∼ const, which changes little in response to expansion/compression. Note that initial κ distribution corresponds to a = 1 and ξ = 1. Under adiabatic evolution without MHD perturbations, the CR distribution f should precisely follow Equation 15 with ξ = a ( t ) (see Appendix of Sun & Bai 2023). However, in our simulations, the MHD waves resulting from the CRPAI tend to isotropize the CR distribution due to quasi-linear diffusion, reducing the anisotropy level at certain energy ranges. Additionally, during scattering with MHD waves, the bulk CR population can lose or gain energy, and p 0 may shift. We thus parameterize the above form of the anisotropic κ distribution (Equation 15) as the ansatz for f 0 , and dynamically estimate ξ and p 0 , by taking the generalized method of moments for Equation 15 after treating f ( t, p ) as f aniso : \nξ = 2 π ∫ d 3 x d 3 p fp √ 1 -µ 2 ∫ d 3 x d 3 p fp | µ | , \np 0 = πκ ( κ -1) G ( κ -0 . 5) 2 G ( κ +1) ∫ d 3 x d 3 p fp √ ξ 4 µ 2 -ξ 2 µ 2 + ξ 2 ∫ d 3 x d 3 p f . (17) \n(16) √ \nNote that estimating ξ only requires the information about f , but the estimation on p 0 is made convenient when ξ is already known. We also mention that while ρ CR and κ can also be adaptive free parameters, in this work, they are taken as fixed. We expect the CR distribution to be largely uniform in our simulations and thus ρ CR is known from the CR mass conservation in the comoving frame. This is also why the integration above is conducted over space which also serves to substantially reduce noise. We also find changes in the parameter κ in our simulations to be minor. \nThe evaluation of ξ and p 0 requires integration over f , which is again done by the δf method using f 0 from the previous time-step, denoted as f 0 , old . The result is \nξ new = 2 π × ( √ πp 0 , old G ( κ +1) N CR 2 ξ old √ κ ( κ -1) G ( κ -0 . 5) + ∫ d 3 x d 3 p δf old p √ 1 -µ 2 ) / ( p 0 , old G ( κ +1) N CR ξ 2 old √ πκ ( κ -1) G ( κ -0 . 5) + ∫ d 3 x d 3 p δf old p | µ | ) , (18) \nN CR ≡ ∫ d 3 x ρ CR ma 4 . \nand, we utilize ξ new to calculate p 0 , new : \np 0 , new = [ √ πκ ( κ -1) G ( κ -0 . 5) 2 G ( κ +1) × ∫ d 3 x d 3 p δf old p √ ξ 4 new µ 2 -ξ 2 new µ 2 + ξ 2 new + p 0 , old N CR 2 × { α 2 + sinh -1 √ α 2 -1 √ α 2 -1 , α ≡ ξ new ξ old > 1 α 2 + sin -1 √ 1 -α 2 √ 1 -α 2 , α ≤ 1 ] /(∫ d 3 x d 3 p δf old + N CR ) . (19) \nWe have validated the above implementation by observing the adiabatic evolution of f in an expanding box in Appendix A.1. Our implementation for the adaptive δf method attains a similar signal-to-noise level in the original δf method but with more than four times the number of simulation particles (Appendix A.2). The computational cost for adaptively fitting f 0 is not negligible. We thus fit once after a fixed time interval ∆ T adapt .', '2.3. Numerical setup': "Our simulation setup closely follows that of Bai et al. (2019) and the CRPAI simulations of Sun & Bai (2023), which we briefly describe here. We initialize an isotropic homogeneous κ distribution for CRs in a 1D expanding box (Equation (14)) along the x -direction and with a constant collision frequency for ion-neutral damping ν IN . The background gas (thermal ions) has uniform density ρ 0 and uniform magnetic field B 0 along ˆ x . The expansion rates along the three directions (ˆ x, ˆ y, ˆ z ) are set to ( a 2 ( t ) , a ( t ) , a ( t )) as mentioned earlier, with a ( t ) = exp(˙ at ). The background gas is initialized with a series of forward and background-traveling, left- and rightpolarized Alfv'en waves with random phases, with initial wave intensity spectrum as | k | I ( k ) /B 2 0 = A 2 , where k is the wavenumber, so that the wave energy is equally distributed in logarithmic k space, and the normalization is such that the total wave energy integrated over k is a fraction 2 A 2 ln 10 of the background field energy. For the CRPAI, the most unstable wavenumber is given by k 0 ∼ (Ω m ) /p 0 (Sun & Bai 2023). Our simulation box size L x is thus chosen to accommodate multiple most unstable wavelengths in the comoving frame of the expanding box. In the simulations, the numerical units include the initial mass density ρ 0 = 1, the initial intensity of the background magnetic field B g ( t = 0) = B 0 = 1, the Alfv'en velocity of thermal ions U A ≡ B 0 / √ ρ 0 = 1, the initial cyclotron frequency Ω 0 ≡ ( q/ ( mc )) B 0 = 1, and the initial ion inertial length λ i = Ω 0 /U A = 1. \nTo integrate the governing equations (Equation 6 to Equation 12), we employ the two-stage van Leer time integrator (Stone & Gardiner 2009) in ATHENA++ (Stone \net al. 2020; Sun & Bai 2023). We use the Roe solver (Roe 1981) with third-order reconstruction to solve the MHDequations, and the Boris pusher (Boris et al. 1972) to integrate CR particles. Interpolation of fluid quantities and CR backreaction deposition both follow the triangular-shaped cloud scheme (Birdsall & Langdon 2004). To enhance the numerical accuracy in CR backreaction, we enable the adaptive δf method (see Section 2.2) with ∆ T adapt = 500Ω -1 0 , and divide the CR population into eight momentum bins (e.g. Bai et al. 2019) which span the momentum range from p 0 / 500 to 500 p 0 in a logarithmic uniform manner. Periodic boundary condition is applied to all fluid quantities and CR particles, except that the gyro phase of particles are randomized upon box-crossing to facilitate quasi-linear evolution (Bai et al. 2019). \nGiven the extreme level of scale separation in realistic ISM or similar conditions 1 (e.g., U A /c ∼ 10 -4 , ρ CR /ρ 0 ≲ 10 -6 ), it is impractical to adopt such values in our simulation parameters. Instead, we choose parameters in the same regime as realistic conditions, and the results should can be rescaled for real applications. Our default simulation parameters include: C = 200 U A , p 0 = m C , κ = 1 . 25 (so that f ( p ) ∝ p -4 . 5 at large p ), ρ CR = 4 × 10 -5 ρ 0 , ν IN = 2 × 10 -5 Ω 0 , ˙ a = ± 5 × 10 -7 Ω 0 . Note that our choice of ˙ a and ν IN is such that ˙ a ≪ ν IN ≪ ω ( k 0 ) ≪ Ω 0 . This ensures that compression/expansion is adiabatic, and that ν IN is much lower than the frequency of the Alfv'en modes in our simulation box (Plotnikov et al. 2021). Moreover, the anticipated CR anisotropy level ∣ ∣ ξ 2 -1 ∣ ∣ ∼ ( ν IN / Ω 0 ) ( ρ 0 /ρ CR ) ( U A /c ) is kept low at ∼ 10 -2 (see Section 3.1). Our simulation box size is chosen to be L x = 1 . 92 × 10 5 U A / Ω 0 , corresponding to about 150 most unstable wavelengths ( ≈ 2 πp 0 /m Ω 0 ), with a resolution of ∆ x = 5 U A / Ω 0 (about 250 cells per most unstable wavelength). In each cell, we initialize 64 simulation particles per momentum bin. Similar as our earlier works, we adopt initial wave amplitudes with A = 10 -3 / 3 to seed instability growth. Our simulations typically run for a duration of 4 . 5 × 10 5 Ω -1 0 . \nWe further conduct a brief parameter survey, where we reduce each of the parameters ( C , ρ CR , ν IN , ˙ a ) by half while keeping the other physical parameters fixed to the fiducial values. Additionally, we carry out highresolution simulations with the fiducial physical parameters but a reduced MHD cell size of ∆ x = 2 . 5 U A / Ω 0 , and using 48 simulation particles per momentum bin per \ncell. For the run with C being half of the fiducial value, the most unstable wavelength is also halved. To achieve equivalent resolution of the most unstable wave, we use the same MHD cell size and number of particles per cell as in the high-resolution run.", '3. THEORETICAL FRAMEWORK': 'Under our simulation setting, the expanding/compressing box continuously drives CR anisotropy, leading to the development of the CRPAI. The wave growth is countered by the ion-neutral damping, reaching a balance in steady state. In the meantime, the waves attempt to isotropize the CRs, balancing the driving force. In this section, we estimate the steady state anisotropy level, wave amplitudes, and the resulting CR scattering rates by quasi-linear theory (QLT). The result is to be compared to and calibrated by our simulation results to be presented afterwards, which can eventually be incorporated into CR (magneto-)hydrodynamics as subgrid models for CR scattering.', '3.1. Prediction on CR anisotropy level and effective scattering rate at the saturated state': "The saturated state is characterized by two balance relations. First, the waves reach a stable amplitude through the competition between CRPAI and damping mechanisms. We extend the analytical study of CRPAI (Sun & Bai 2023) from the non-relativistic regime to the relativistic regime in the expanding box, 2 \nΓ growth ( k ) ≈-ρ CR 2 ρ 0 Ω 0 [ 1 ± ( ξ 2 -1 ) Ω 0 kU A ] Q 2 ( k ) ξ 2 , ≈∓ ρ CR 2 ρ 0 Ω 0 ( ξ 2 -1 ) Ω 0 kU A Q 2 ( k ) ξ 2 , (20) \nwhere \nQ 2 ( k ) ≡ √ π κ 1 . 5 G ( κ +1) G ( κ -0 . 5) m Ω 0 kp 0 ( 1 + m 2 Ω 2 0 κk 2 p 2 0 ) -κ . (21) \nThe linear growth rate Γ growth depends on the wavenumber k and the ± sign denotes the polarization direction. The fastest growth is achieved at k 0 ∼ Ω 0 m/p 0 regardless of the propagation direction. We have also dropped 1 compared to ( ξ 2 -1)Ω 0 /kU A in the bracket of Equation 20. This is because growth is the most prominent \nnear k ∼ k 0 , making Ω 0 /kU A ∼ C /U A ≫ 1, thus even a weak level of anisotropy would make the second term in the bracket dominate over 1. It should also be noted that this estimation on growth rate assumes that the CR distribution function has the form Equation 15, with a uniform anisotropy level at all momenta characterized by ξ , which is not necessarily true as the CR particles evolve. Therefore, this estimate should only be taken as a proxy. \nThis growth is to be balanced by the wave damping rate. With our approach of ion-neutral damping (Plotnikov et al. 2021), the damping rate is largely constant for all wavelengths we consider ( ν IN ≪ kU A ), \nΓ damp ( k ) ≈ ν IN 2 . (22) \nBalancing growth and damping Γ growth = Γ damp , we can estimate the CR anisotropy level 3 as \n∣ ∣ ξ 2 -1 ∣ ∣ ( p ) ≈ kU A Ω 0 ν IN Ω 0 ρ 0 ρ CR Q -1 2 ( Ω 0 m p ) . (23) \nwhich, interestingly, depends only on ν IN but not ˙ B/B . This is the consequence of linear damping, and is similar to the case of CR streaming instability, where the expected streaming speed does not depend on CR pressure gradient under linear damping (e.g. Wiener et al. 2013; Bai 2022). Here, we further approximate ξ to be an explicit function of p (instead of being constant by definition), by substituting k with Ω 0 m/p for the resonance condition and omitting the dependence on the particle pitch angle µ . \nFurthermore, when treating the CR population as a single fluid, we can then re-express the CR anisotropy level in terms of CR pressure P CR , \nξ 2 = P CR , ⊥ P CR , ∥ ≡ ∫ d 3 p fp 2 ( 1 -µ 2 ) 2 ∫ d 3 p fp 2 µ 2 ≈ 1 ± 1 . 78 U A c ν IN Ω 0 ρ 0 ρ CR , (24) \nwhere ⊥ and ∥ denote the components perpendicular and parallel to the background magnetic field, respectively. Here the factor '1.78' is derived from Q -1 2 (Ω 0 m/p 0 ), where the CRs at the peak of the distribution are taken to be representative of the entire population, as commonly assumed. \nThe second balance is on the CR distribution, where driving on CRs by ˙ B/B should be balanced by CR scattering, which can be expressed under the quasilinear theory. Using the Fokker-Plank equation, we can \nwrite (e.g. Jokipii 1966; Kulsrud & Pearce 1969; Skilling 1975c; Schlickeiser 2002) \n∂ t f + ˙ B 2 B [ p ( 1 + µ 2 ) ∂ p f + µ ( 1 -µ 2 ) ∂ µ f ] = ∂ µ [( D µµ ∂ µ f +sgn( µ ) D pµ ∂ p f )] + 1 p 2 ∂ p [ p 2 (sgn ( µ ) D pµ ∂ µ f + D pp ∂ p f ) ] , (25) \nwhere, \nD µµ = ν ( 1 -µ 2 ) 2 , D pµ = ν ( 1 -µ 2 ) 2 pU A v , D pp = ν ( 1 -µ 2 ) 2 ( pU A v ) 2 . (26) \nThe term proportional to ˙ B/B corresponds to the anisotropy driving, ( D · p ) · ∂f/∂ p (see Equation 12). Note that it not only affects the pitch angle, but also the total momentum. The two terms on the right-hand side represent quasi-linear diffusion (QLD) in the gas comoving frame, resulting from the pitch angle diffusion and CR momentum diffusion when CRs scatter with the Alfv'en waves. The diffusion coefficients, D µµ , D pµ , and D pp , are all characterized by a single parameter, the scattering rate ν ( p, µ ) (e.g. Jokipii 1966; Kulsrud & Pearce 1969; Skilling 1975c; Schlickeiser 2002) which, under quasi-linear theory, is given by \nν ( p, µ ) = π Ω γ Ω m pµ I ( k res = Ω m pµ ) /B 2 g . (27) \nIn the saturated state of the CR population, we anticipate ∂ t ln ( P CR , ⊥ /P CR , ∥ ) = 0. We can integrate the above over pitch angle µ . The driving term yields, \ndriving = ˙ B 2 B ( ∫ d µ [( 1 -µ 4 ) p∂ p f -( 1 -6 µ 2 +5 µ 4 ) f ] ∫ d µ (1 -µ 2 ) f -∫ d µ [( µ 2 + µ 4 ) p∂ p f -( 3 µ 2 -5 µ 4 ) f ] ∫ d µµ 2 f ) . \nAssuming the CR distribution is close to isotropy (see Section 2.3), P CR , ⊥ ≈ P CR , ∥ , we may ignore the dependence of f on µ as a high-order term and simply use the isotropic distribution function, f ( p, µ ) ≈ f iso ( p ), which leads to \ndriving ≈ -˙ B 5 B ∂ ln f iso ∂ ln p (28) \nTo simplify the right-hand side of Equation 25, we firstly eliminate the momentum diffusion terms (inside \nthe second square bracket) as they are of higher order compared to the pitch angle diffusion by a factor of U A /c . We proceed by applying a partial integration, \nQLD = -9 2 f iso ∫ d µ ν ( µ -µ 3 ) 2 ( ∂ µ f +sgn( µ ) pU A v ∂ p f ) . \nTo integrate over µ , we introduce the following assumptions: the slight anisotropic distribution takes the form of f ( p, µ ) = f ( p √ ξ 4 µ 2 -ξ 2 µ 2 + ξ 2 ) ; the anisotropy level ( ξ 2 -1 ) ≪ 1; we simplify the scattering rate as an effective rate independent of µ , \nν eff ( p ) ≡ ∫ 1 -1 ( 1 -µ 2 ) ν ( p, µ ) d µ ∫ 1 -1 (1 -µ 2 ) d µ . (29) \nUnder these assumptions, the contribution from QLD can be written to a form similar to Equation 28, \nQLD ≈ -ν eff ( 3 5 ( ξ 2 -1 ) + 9 8 U A c ) ∂ ln f iso ∂ ln p . (30) \nBy inserting the wave saturation condition (Ω 0 /k ∼ p/m in Equation 20) into the CR anisotropy balance condition (driving = QLD), we build up the relation between the effective scattering rate and environmental parameters for each CR momentum bin, \nν eff ( p ) ≈ 1 3 Q 2 (Ω 0 m/p ) ∣ ∣ ∣ ∣ ∣ ˙ B B ∣ ∣ ∣ ∣ ∣ c U A Ω 0 ν IN ρ CR ρ 0 . (31) \nThe momentum-dependent factor Q 2 peaks around 0 . 56 (for κ = 1 . 25, the exact value depends on κ ). In the conventional single-fluid treatment of the CRs, peak growth rate is often employed when calculating the balance between wave growth and damping, which yields \nν eff ≈ 0 . 187 ∣ ∣ ∣ ∣ ∣ ˙ B B ∣ ∣ ∣ ∣ ∣ c U A Ω 0 ν IN ρ CR ρ 0 . (32) \nCombining Equation 27 and Equation 31, we can further estimate the saturation level of the CRPAI as: \n( δB B g ) 2 ∼ kI ( k ) B 2 g ∼ √ 1 + Ω 2 / ( kc ) 2 3 π Q 2 ( k ) ∣ ∣ ∣ ˙ B/B ∣ ∣ ∣ Ω 0 c U A Ω 0 ν IN ρ CR ρ 0 ≪ 1 , (33) \nThe theoretical calculations above have been significantly simplified under a number of approximations, and are intended to provide order-of-magnitude estimate on how the saturated state responds to the environment. Weanticipate the results to provide the appropriate scaling relations, but the pre-factors can be subject to major uncertainties, which we aim to calibrate through our simulations (Section 4).", '3.2. Implication of effective scattering rate in CR (magneto-)hydrodynamics': 'At macroscopic scales (e.g., in the Galaxy), the cosmic rays are typically modeled as fluid, known as CR (magneto-)hydrodynamics, characterized by the CR energy density E CR and the CR energy flux density F CR , \nE CR ( t, x ) ≡ ∫ d 3 p f ( t, x , p ) √ c 2 + p 2 c ≡ ∫ dln p E CR ( t, x , p ) , F CR ( t, x ) ≡ ∫ d 3 p f ( t, x , p ) p c 2 ≡ ∫ dln p F CR ( t, x , p ) , (34) \nwhere we can either approximate CRs as a single fluid or multiple fluids distributed across different energy (momentum) bins. They are coupled with background gas through wave-particle interaction, described by the scattering rates. \nIn our simulation setting with expanding box, the outcome of the CRPAI in balance with ion-neutral damping only affects E CR , as F CR = 0 in our setups. By integrating the Fokker-Plank equation (Equation 25) over the CR population, and applying similar techniques as described in Section 3.1, the CR fluid energy equation reads \n∂ t E CR -8 3 ˙ B B E CR = U A c ν eff E CR ( 1 2 ( ξ 2 -1 ) + 4 3 U A c ) . (35) \nThe term 8 ˙ B/ (3 B ) E CR corresponds to the external driving through adiabatic compression/expansion, and the right-hand side corresponds to the quasi-linear diffusion. Note the different pre-factors compared to those in Section 3.1 due to additional weighting in the moment equations. The above equation can also be naturally re-expressed in the p by p treatment, by substituting E CR ( t, x ) with E CR ( t, x , p ). \nIn CR (magneto-)hydrodynamics, the anisotropy level ξ 2 and ν eff are expected to be user-specified parameters. Our estimates in Equations 23 & 31 can be considered as sub-grid prescriptions under the assumption that ionneutral damping dominates. Under such prescriptions, the QLD effect in CR energy density equation (the righthand side of Equation 35) can be reformulated in terms of environmental parameters as, \n∂ t E CR -8 3 ˙ B B E CR ∼ -1 6 ˙ B B U A c E CR ( p ) , (36) \nwhere we expect the ( ξ 2 -1) term to dominate over the U A /c term on the right hand side of Equation 35. We \nsee that wave scattering on CRs (which is irreversible) always reduces the adiabatic cooling/heating on CR energy density from external driving, but it is only a minor effect of the order ( U A /c ) / 16. Therefore, for the pure CRPAI case, this effect is largely negligible on CR energy density evolution in our simulation setup (Yan & Lazarian 2011; Zweibel 2020). \nThis work aims to calibrate ξ 2 and ν eff in the CR energy equation (Equation 35) using kinetic simulations. The CR fluid quantities ( P CR , ⊥ , P CR , ∥ , and E CR ) can be measured by definition. The effective scattering rate of CRSI has been previously measured through the steadystate condition in the CR flux equation (Bai 2022). However, in our simulation, due to the absence of an equilibrium state for the CR energy (Equation 36), we cannot perform a similar measurement of ν eff through Equation 35. Instead, we measure ν by definition (Section 4.2) and then numerically integrate the FokkerPlanck equation (Equation 25) for the QLD effect on CR energy density. By comparing Equation 35 with the numerical integration of Equation 25, we ultimately obtain ν eff (Section 4.3). \nNote that as CR streaming is not present in our problem setup, we do not include the equation for the CR energy flux. In the more general case with CR streaming, the scattering rate will also affect the evolution of the CR energy flux, and ν eff is likely determined by the combined outcome of the CRSI and CRPAI. This is left for our future work.', '4. SIMULATION RESULTS': 'In this section, we illustrate the instability growth and the establishment of steady CR anisotropy in our simulations. The linear growth of CRPAI has been demonstrated in Sun & Bai (2023). Here, our primary focus lies on the subsequent evolution leading towards saturation. The final outcome is the effective scattering rate ν eff at the saturated state. To compute ν eff , we assess the CR scattering rate by definition and subsequently average it over the CR pitch angle and energy (momentum magnitude). Given Equation 31 is highly approximate, we want to test its scaling on environmental parameters and calibrate the coefficients, by comparing different runs at the end of this section.', '4.1. Fiducial run': 'We start by showing in Figure 1 the time evolution of the wave energy density for all our simulation runs. The initial seed waves are first damped by ion-neutral damping. In the meantime, as the CR anisotropy gradually develops owing to ˙ B/B , the CRPAI starts to overcome damping, eventually leading to wave growth. \nFigure 1. Time evolution of the wave energy density around k 0 , ∫ 5 k 0 k 0 / 5 I ( k )d k . The solid lines represent the compressing box, while the dashed lines denote the expanding box. Line colors distinguish simulations with varying parameters and resolutions. Lines from the runs sharing similar saturated states largely overlap. \n<!-- image --> \nGiven our fiducial parameters, the turning point is expected to occur when the CR anisotropy level reaches ˙ at ∼ ( U A / C )[1 + ( ν IN / Ω 0 )( ρ 0 /ρ CR ) Q -1 2 (Ω 0 m/p )], yielding t ∼ 1 . 5 × 10 4 Ω -1 0 for the fastest growing mode in the fiducial case. The actual turn-over when summing over all waves starts later around t ∼ 4 × 10 4 Ω -1 0 as it takes longer for waves at other wavelength (which grow slower) to catch up. After a few e-folding time, the waves are sufficiently grown to efficiently scatter and isotropize the CR particles. This will reduce the CR anisotropy and hence the wave growth, eventually leading to the saturation of the wave amplitudes, where wave growth due to CRPAI balances ion-neutral damping.', '4.1.1. The wave spectrum': 'To look more closely into the processes, we decompose the waves into the Fourier space, according to polarization and propagation directions (Bai et al. 2019), and plot the wave energy spectra I ( k ) of the fiducial runs at saturated state in Figure 2. In our setup, the expansion ( ˙ B/B < 0) yields an oblate CR distribution ( P CR , ⊥ /P CR , ∥ > 1), where the CRPAI amplifies lefthanded waves propagating in opposite directions and damps the right-handed counterparts, while the compression ( ˙ B/B > 0) leads the waves to saturate at the opposite polarization direction due to the prolate CR distribution ( P CR , ⊥ /P CR , ∥ < 1). The dependence of wave intensity on k are consistent with theoretical expectations of the CRPAI. The wave growth rate peaks at \n<!-- image --> \nFigure 2. Wave intensity spectra, kI ( k ), during the saturated state of the fiducial runs at t = 4 × 10 5 Ω -1 0 . The top panel corresponds to the expanding box (˙ a = 5 × 10 -7 Ω 0 ), while the bottom panel corresponds to the compressing box (˙ a = -5 × 10 -7 Ω 0 ). Solid lines represent forward propagating waves, while dashed lines denote backward propagating ones. The left- and right-handed branches are distinguished by black and red markings. Lines in the same color (polarization) largely overlap. The grey dotted lines refer to the theoretical estimate of the saturated wave intensity (Equation 33), peaking at the most unstable wave number k 0 ∼ 5 × 10 -3 Ω 0 /U A for the fiducial runs. The accompanying animation, covering the time span from t = 0 to 4 . 5 × 10 5 Ω -1 0 , is available in the online version of the journal. \n<!-- image --> \nthe resonant wave number of particles with p = p 0 given by k 0 ∼ (Ω m ) /p 0 = 5 × 10 -3 Ω 0 /U A (Equation 20). Consequently, after an initial increase in the CR anisotropy level, the waves around k 0 start to grow, and first reach the saturated amplitude (due to QLD, to be discussed next). Waves at other wave numbers require more time to grow and are expected to reach saturation slower with lower amplitudes. \nBy the end of the simulations we observe that waves spanning from k 0 / 5 to 5 k 0 have approximately reached \n<!-- image --> \n(a) \nFigure 3. The CR distribution function ( f/ ⟨ f ⟩ µ -1 ) for the fiducial runs in the expanding box (top) and the compressing box (bottom) at various evolution snapshots. Dashed lines represent particle momenta resonating with the same wave characterized by a constant k = Ω 0 m/ ( pµ ). The animation spanning from t = 10 3 to 4 . 5 × 10 5 Ω -1 0 is accessible in the online version of the journal. \n<!-- image --> \nFigure 4. The CR pressure anisotropy level as a function of CR energy (momentum) at different snapshots of the fiducial runs. The solid lines represent the case in compressing box, while the dashed lines denote the expanding box case. Transparency indicates the simulation run time. The black dotted line corresponds to the theoretical prediction given by Equation 23. \n<!-- image --> \nsaturation in the simulations (thus in Figure 1 we actually show the evolution of wave intensity only within this wavenumber range). We also observe from Figure 2 that the wave intensity in this spectral range approximately agree with the estimation from QLT (Equation 33), with small deviation by up to a factor of ∼ 2. Note that the final wave intensity slightly differs between the expanding box and the compressing box. We attribute this to the reduction/enhancement of background magnetic field due to box expansion/compression, which leads to order unity difference in B g within simulation time.', '4.1.2. Anisotropy in the CR distribution function': 'Accompanied by the saturation of wave energy is the asymptotic saturation of the CR distribution f ( p, µ ). Its evolution for the fiducial runs is illustrated in Figure 3, which is normalized by the isotropic distribution ⟨ f ⟩ µ ( p ) averaged over pitch angle. Initially, MHD waves are too weak to scatter the CRs, and hence f ( p, µ ) evolves adiabatically due to expansion/compression approximately following the anisotropic κ distribution with ξ ∼ a ( t ) (Equation 15): CRs with p ≳ p 0 concentrate toward µ ∼ 0/ µ = ± 1 in expanding/compressing box, while CRs with p ≪ p 0 in the κ distribution do not rapidly develop significant anisotropy (see Section 2.2). Following the subsequent development of the CRPAI, waves around k 0 grow fastest and effectively scatter particles whose momentum around p 0 . More precisely, scattering is the most effective for particles satisfying Ω m/ ( pµ ) ∼ k 0 (see dashed lines in Figure 15), leading to pitch angle diffusion and hence isotropization. \nThe isotropization gradually propagates towards CRs at lower and higher energies as waves in other wavelengths grow. However, even within the momentum range where scattering is the most effective (around p = p 0 ), the particles are not fully isotropized, leaving a mild anisotropy, reflecting the balance with the box expanding/compressing. The CR anisotropy for particles with p ∈ (0 . 5 p 0 , 5 p 0 ) ultimately reaches a quasi-steady state by the end of the simulations in both the expanding and compressing boxes. Finally, we note that the CR energy distribution keeps evolving following Equations 35. \nWe quantify the CR anisotropy level through its pressure anisotropy, P CR , ⊥ /P CR , ∥ -1, based on Equation 24. In the momentum-by-momentum treatment, we present the temporal evolution of CR anisotropy in Figure 4. In the expanding box, the CR pressure perpendicular to the background magnetic field consistently exceeds the parallel component, while the reverse holds in the compressing box. Owing to the inefficient driving of anisotropy towards p ≪ p 0 (consequence of the κ distribution, see footnote 4), we primarily focus on particles with p ≳ 0 . 2 p 0 where driving is effective. Initially, anisotropy levels across all CR momenta increase with time. Subsequently, quasi-linear diffusion (QLD) induced by amplified waves reduces the anisotropy, ultimately maintaining a steady anisotropic profile. The saturated anisotropy level within the momentum range (0 . 5 p 0 , 5 p 0 ) is broadly consistent with the trend predicted by Equation 23, albeit higher than the theoretical value by order unity. We attribute this deviation to the omission of pitch angle dependence in deriving Equation 23, which is the standard approach in QLT but is prone to error, and a similar situation has been discussed in Bai (2022). Lower-energy CRs in simulations fall below the predicted anisotropy level, which is likely owing to less efficient anisotropy driving. The anisotropy level in the expanding box simulation is systematically higher than that in compressing box.', '4.2. The pitch angle scattering rate and comparison with quasi-linear theory': "This subsection introduces the methodology to quantify the pitch angle scattering rate ν ( p, µ ) and subsequently compares the obtained measurements with quasi-linear theory (QLT) as a diagnostics. \nIn our setup, two independent mechanisms contribute to the evolution of CR momenta. The expansion/compression of the box stretches the CR momentum while conserving a 2 p ∥ and ap ⊥ (See the appendix of Sun & Bai 2023). Simultaneously, MHD waves interact with CRs, inducing diffusion in both pitch angle µ \nand momentum magnitude p , characterized by the diffusion coefficients D µµ , D pµ , and D pp , connected to the scattering rate ν ( p, µ ) (Equation 26). \nWe quantify the diffusion coefficients by tracing simulation particles (Bambic et al. 2021) over some time interval ∆ t = t 2 -t 1 . Since the expanding/compressing box setup itself leads to changes in particle momentum and pitch angle, we counter such effects by considering particle evolution to effectively follow a 'stretch-diffusestretch' fashion. For an arbitrary simulation particle k , its momentum p 1 ,k at time t 1 is first stretched for ∆ t/ 2 to p ' 1 ,k due to box expansion/compression only. Subsequently, the waves scatter p ' 1 ,k to p ' 2 ,k , and finally CR undergoes another stretching for ∆ t/ 2 to p 2 ,k at time t 2 . Given the measured values of p 1 ,k and p 2 ,k , we obtain \np ' 1 ,k = ( p ' x 1 ,k , p ' y 1 ,k , p ' z 1 ,k ) = ( p x 1 ,k e -˙ a ∆ t , p y 1 ,k e -˙ a ∆ t/ 2 , p z 1 ,k e -˙ a ∆ t/ 2 ) , p ' 2 ,k = ( p ' x 2 ,k , p ' y 2 ,k , p ' z 2 ,k ) = ( p x 2 ,k e ˙ a ∆ t , p y 2 ,k e ˙ a ∆ t/ 2 , p z 2 ,k e ˙ a ∆ t/ 2 ) . \nWe then consider the particle evolution from p ' 1 ,k to p ' 2 ,k to be due to pure wave scattering, acting on the particle with momentum p ∗ = ( p ' 1 ,k + p ' 2 ,k ) / 2. We choose the measurement time interval ∆ t to be 5 × 10 3 Ω -1 0 to ensure p ' 1 ,k and p ' 2 ,k do not differ significantly. The pitch angle diffusion coefficient is thus determined by definition, \nD µµ ( µ ∗ , p ∗ ) = 〈 p ' 2 ,k · b g ∣ ∣ ∣ p ' 2 ,k ∣ ∣ ∣ -p ' 1 ,k · b g ∣ ∣ ∣ p ' 1 ,k ∣ ∣ ∣ 〉 p ∗ × 2 ∆ t , (37) \nwhere ⟨⟩ p ∗ denotes averaging over all simulation particles at the midpoint state p ∗ . The measurements for D pµ and D pp follow the same procedure. \nWith the pitch angle diffusion coefficient D µµ , the scattering rate, ν ( µ, p ), is systematically measured as a function of both pitch angle µ and momentum magnitude p . Figure 5a illustrates ν ( µ, p ) at the saturated state of the fiducial run in expanding box. The scattering rate exhibits a peak around the momentum p 0 , corresponding to particles scattering with the most unstable wave at k 0 , while it diminishes at µ = 0 due to the lack of resonant waves with wave numbers k ≫ k 0 (see Figure 2a). In the context of CR streaming instability, it causes the well-known 90 · pitch angle crossing problem, where this crossing is necessary to reduce the drift anisotropy (Felice & Kulsrud 2001; Holcomb & Spitkovsky 2019; Bai et al. 2019; Zeng et al. 2024). Given the symmetry with respect to 90 · pitch angle in \nFigure 5. 5a: The CR scattering rate ν measured by definition, as a function of CR momentum p and CR pitch angle µ , corresponding to the fiducial run in the expanding box, at time around ∼ 4 . 5 × 10 5 Ω -1 0 . 5b: The comparison between D µµ and D pp measured by definition. The quasi-linear theory predicts D pp = D µµ ( pU A /v ) 2 (see Equation 25). \n<!-- image --> \nFigure 6. The comparison between the pitch angle scattering rate ν ( p, µ ) as a function of µ for different momenta measured by definition (dashed lines) and the one calculated from the wave spectrum (Equation 27, solid lines), for the fiducial run in the expanding box at time around ∼ 4 . 5 × 10 5 Ω -1 0 . \n<!-- image --> \nthe pressure anisotropy problem considered here, this crossing does not affect the isotropization process and hence we do not discuss it further. \nThe measured value of ν remains largely consistent regardless of whether it was made through D µµ , D pµ , or D pp . For example, there exists a pre-factor difference of ( pU A /v ) 2 when calculating from D µµ and D pp (Equation 26), which is verified in Figure 5b for the fiducial run in the expanding box. For the most part, the measured ratio D µµ ( pU A /v ) 2 /D pp stays around 1 as expected. There are deviations near both µ = 0 and ± 1. This is because the diffusion coefficients theoreti- \nat both µ = 0 and ± 1 (Equation 26), thus the corresponding measurements are considered less faithful. There is also larger numerical noise in the region p < p -0 . 5 0 due to lower wave amplitudes at highk . \nAdditionally, we compare ν measured from simulation particles and evaluated through the wave intensity based on QLD (Equation 27) in Figure 6. Besides deviations near the µ = 0 (QLD fails) and ± 1 (measurement of D µµ is subject to error) regions, these two independent measurements exhibit consistency, thereby supporting the predictions of QLT. Note that due to the substantial noise in wave intensity, we will employ ν obtained from tracing particles in the subsequent calculations.", '4.3. Anisotropy level and effective scattering rate at the saturated state': 'The final outcomes of our study is to yield the CR pressure anisotropy level, P CR , ⊥ /P CR , ∥ -1, and the effective scattering rate, ν eff . They are expected to be given either as a single fluid, or on p -byp bases, as a function of environmental parameters ( ˙ B/B , ν IN , etc.). We study all our simulation results and compare them with the QLT results derived in Section 3.', '4.3.1. Momentum-by-momentum results': "The relationship between P CR , ⊥ /P CR , ∥ -1 at the saturated state and environmental parameters is depicted in Figure 7. The trends of the CR anisotropy level with respect to CR momentum (energy) remain largely similar after varying environmental parameters (see Section 4.1). The qualitative dependence on magnitude aligns with theoretical expectations (Equation 23). The expansion/compression rate does not significantly impact CRs at the saturated state (within the momentum range (0 . 5 p 0 , 5 p 0 )), as expected with QLT. Reducing C /U A or ρ CR /ρ 0 , which weakens CRPAI, indeed increases the anisotropy level again consistent with Equation 23, while we find the anisotropy level is more sensitive to C /U A than ρ CR /ρ 0 . A weak damping case is counterbalanced by a slow instability growth rate with a smaller anisotropy level, in agreement with results seen in Figure 7. Within identical environmental parameters, the expansion consistently induces a more effective anisotropy compared to compression, by a factor of less than 2. We also see that doubling the resolution leads to identical results as illustrated in Figure 7, thus validating numerical convergence in our simulations. \nThe effective scattering rate ν eff , normalized with environmental parameters and as a function of the CR momentum, is illustrated in Figure 8. The effective scattering rate encapsulates the wave scattering on CR transport, by taking moments of Equation 25. We numerically integrate the Fokker-Planck equation for ν eff , \nFigure 7. The CR pressure anisotropy level, similar to Figure 4, but for different simulation runs at time 4 . 5 × 10 5 Ω -1 0 . The solid lines represent the compressing box runs, while the dashed lines denote the expanding box cases. The simulations with varying parameters and resolutions are given in different colors. The animation covering from t = 10 3 to 4 . 5 × 10 5 Ω -1 0 for all runs in different parameters is accessible in the online version of the journal. \n<!-- image --> \nleveraging the known terms in Equation 25 from the simulations. One taking moments method is to retain the dependence of ν eff on momentum (energy) p , namely the p -byp treatment (Bai 2022). This treatment divides the CR population into distinct momentum (energy) groups, with group i representing a CR fluid composed of CR particles within the momentum range from p i, min to p i, max . Through the definition given by Equation 24 & 34, we have the energy density and the anisotropy level for the group i , \nE CR ,i = ∫ p i, max p i, min dln p E CR ( p ) , ξ 2 i = ∫ p i, max p i, min d p ∫ 1 -1 d µfp 4 ( 1 -µ 2 ) 2 ∫ p i, max p i, min d p ∫ 1 -1 d µfp 4 µ 2 . \nThe QLD effect on E CR ,i can be numerically computed from the right-hand side of Equation 25, \nQLD on group i = ∫ p i, max p i, min dln p ∫ 1 -1 d µp 3 γ C 2 × ∂ ∂ ln p [ p 2 f × ( sgn( µ ) D pµ ∂ µ ln f + D pp ∂ ln p ln f p )] , \nwhere both f and the diffusion coefficients are directly measured in the simulations. By comparing the above numerical integration with the CR energy equation \n(Equation 35), we can obtain ν eff ( p ), \nν eff ( p i ) = QLD on group i U A C E CR ,i ( 1 2 | ξ 2 i -1 | + 4 U A 3 C ) , (38) \nwhere p i should be understood as √ p i, min p i, max as the mean particle momentum in the i th bin. Here numerical differentiation are applied to the logarithms of f and p to mitigate potential truncation errors. Also, to further minimize errors, we perform one partial integration in computing 'QLD on group i '. \nAfter normalized by environmental parameters, the pre-factors of ν eff ( p ) across all simulations closely follow a same curve (Figure 8), which peak around p 0 where resonant waves are the most pronounced. Following early discussions, we consider the scattering rates measured over range of (0 . 5 p 0 , 5 p 0 ) to be reliable. The curves of ν eff in the expanding box largely overlap with the theoretical QLT prediction (Equation 31). On the other hand, in the compressing box, the peak value of ν eff exceeds the QLT prediction by a factor of ∼ 2, achieved at a CR momentum of around 2 p 0 , and the scattering rate is generally higher than QLT prediction by a factor of 2-3 for particles with p ≳ p 0 . This difference is induced by the expanding/compressing box setup which contributes to additional wave damping/growth at the rate of ˙ a (which is much less than our adopted ν IN ), and also influence the peak momentum of CRs and the CR cyclotron frequency Ω. Moreover, in theoretical calculations (Section 3.1), we have assumed that the anisotropy level well exceeds U A /c , which does not entirely hold in simulations, particularly in the compressing box runs (Figure 7). In both expanding and compressing boxes, the main 'outlier' is when we double the parameter Ω 0 /ν IN , yielding pre-factors typically smaller than those from other runs by a factor of ≲ 2. We attribute this deviation to non-negligible numerical dissipation. 4 We have also verified that the results over the range of (0 . 5 p 0 , 5 p 0 ) largely converge in high-resolution runs, while the high-resolution runs yield larger values of ν eff towards low p , where their resonant shortwavelength waves are better resolved.", '4.3.2. Single-fluid results': 'More common in the formulation of CR hydrodynamics is to treat the CRs as a single fluid, effectively integrating over the momentum (energy). We determine ν eff \nFigure 8. The effective scattering rate ν eff normalized by environmental parameters, depicted as a function of CR momentum (energy) p . The solid lines represent the compressing box runs, while the dashed lines denote the expanding box cases. The simulations with varying parameters and resolutions are distinguished by line colors, albeit with substantial overlap. The black dotted line illustrates the estimation of quasi-linear theory (QLT) on ν eff (Equation 31), in the momentum by momentum treatment. \n<!-- image --> \nfor the single fluid in the same way, by taking moments for Equation 25 and calculating the pressure anisotropy for the whole CR population (Equation 24). By default, we set p max = 10 p 0 and p min = 0 . 1 p 0 , which encompasses the majority of CR particles in the κ distribution. However, we also integrate CRs over a short range, (0 . 5 p 0 , 5 p 0 ), where CR ansiotropy reaches a quasi-steady state. We plot the CR anisotropy level and effective scattering normalized by the environmental parameters for all our simulation runs in Figure 9 & 10. \nIn Figure 9, we see that the mean CR anisotropy | P CR, ⊥ /P CR, ∥ -1 | for (0 . 1 p 0 , 10 p 0 ) particles varies around 10 ( U A /c ) ( ν IN / Ω 0 ) ( ρ 0 /ρ CR ), and their corresponding effective scattering rate in Figure 10 varies around 0 . 07 ∣ ∣ ∣ ˙ B/B ∣ ∣ ∣ ( c/U A ) (Ω 0 /ν IN ) ( ρ CR /ρ 0 ). There is a factor of up to 4 difference between the expanding and compressing box cases, but both deviate substantially from single-fluid QLT predictions. However, in the compressing box cases, when averaged over the range (0 . 5 p 0 , 5 p 0 ), ν eff largely overlaps with the single-fluid QLT prediction, and the anisotropy level is only twice that predicted by the theory. In the expanding box cases, the results still deviate from the QLT predictions when averaged over the same range, but get closer than averaging over (0 . 1 p 0 , 10 p 0 ), with \nFigure 9. The total pressure anisotropy level after normalized by environmental parameters, in the single fluid treatment. The hollow markers regards the majority of CRs, within 0 . 1 p 0 < p < 10 p 0 , as a CR single fluid, while the filled markers refer to the integration over the CRs which reaches steady pressure anisotropy (0 . 5 p 0 < p < 5 p 0 ). The red circles correspond to the results in the compressing box and the black squares indicate those in the expanding box. The grey dashed line refers to the theoretical prediction, | P CR, ⊥ /P CR, ∥ -1 | ∼ 1 . 78 ( U A /c ) ( ν IN / Ω 0 ) ( ρ 0 /ρ CR ) (Equation 23). \n<!-- image --> \n| P CR, ⊥ /P CR, ∥ -1 | ∼ 8 ( U A /c ) ( ν IN / Ω 0 ) ( ρ 0 /ρ CR ) and ν eff ∼ 0 . 09 ∣ ∣ ∣ ˙ B/B ∣ ∣ ∣ ( c/U A ) (Ω 0 /ν IN ) ( ρ CR /ρ 0 ). We can see that over any momentum range considered, the CR pressure anisotropy is primarily dominated by highenergy CRs due to the nature of second moments. However, the standard single-fluid QLT estimation relies on CRs around p 0 , by substituting Q 2 (Ω 0 m/p 0 ) for the whole CR population. The anisotropy level (and effective scattering rate) of high-energy CRs is significantly higher (and lower) than that of CRs around p 0 (Figures 4 & 7). As the average range gets closer to p 0 , both | P CR, ⊥ /P CR, ∥ -1 | and ν eff better converge to the value for p 0 in the p -byp treatment, approaching the theoretical predictions. \nIn both Figure 9 and 10, the main outlier again is from the run with ν IN reduced by half. We again attribute this to the higher fraction of numerical wave damping as \nFigure 10. The effective scattering rate ν eff after normalized by environmental parameters, in the single fluid treatment. The markers share the same meaning in Figure 9. The grey dashed line refers to the theoretical prediction, ν eff ≈ 0 . 187 ∣ ∣ ∣ ˙ B/B ∣ ∣ ∣ ( c/U A ) (Ω 0 /ν IN ) (Equation 31). \n<!-- image --> \ndiscussed earlier, and the results from this run would be brought closer to the typical values obtained from other runs should numerical dissipation be mitigated. Therefore, all our simulations generally exhibit consistent results over parameter variations, confirming the scaling from QLT. There are modest-to-strong deviations in the normalization factor, which depend on momentum range considered. A convergence in numerical resolution further validates our simulation results. \nFinally, the total cooling/heating on the CR energy density, after plugging in ξ 2 and ν eff to Equation 35, yields \n∼ -0 . 3 ˙ B B U A c E CR , \nin both expanding boxes and compressing box cases, twice than the QLT prediction (Equation 36). Therefore, its contribution to irreversible CR heating/cooling raises to about (1 / 8) U A /c (instead of (1 / 16) U A /c ) compared to adiabatic heating/cooling, again being very minor to the overall energy. We find this value is insensitive to variations in environmental parameters, nor to the range of momentum involved (within the range of \n(0 . 1 p 0 , 10 p 0 ) studied here), thanks to the cancellation of such dependencies in ξ 2 and ν eff .', '5. DISCUSSION: APPLICATION OF CRPAI': "Given the dependency of ν eff on environmental parameters, we estimate the value of ν eff in realistic systems and compare the importance of CRPAI with that of the Cosmic Ray Streaming Instability (CRSI), which has been widely considered in modeling CR feedback in the galactic context. The effective scattering rate for the CRSI with ion-neutral damping scales with the environmental parameters as ∼ 0 . 054 ( c/L gal ) ( c/U A ) (Ω 0 /ν IN ) ( ρ CR /ρ 0 ) (Bai 2022), where L gal represents the typical height of the galactic disk scale, also the typical length scale of the CR gradient. Comparing the scaling relations (Equation 32), the ratio of ν eff between the pure CRPAI (Figure 10) and pure CRSI cases is 5 ∣ ∣ ∣ ˙ B/B ∣ ∣ ∣ ( L gal /c ). \nIn a general sense, the compression/expansion timescale for the CRPAI can be estimated as the eddy turnover time in turbulence. 5 Let the largest eddy size in the ISM be L gal , and the eddy turnover timescale for Alfv'enic turbulence can be estimated as ( L gal /U A ) ∼ ∣ ∣ ∣ B/ ˙ B ∣ ∣ ∣ . Therefore, the CR scattering rate resulting from the CRPAI is smaller than that of CRSI by the order of U A /c . A similar conclusion was drawn by Zweibel (2020) who focused on CR heating. Setting typical values in the galactic context for the cold neutral medium (CNM) 6 , one finds ν eff for CRSI to be around once per year (e.g., Chan et al. 2019; Zweibel 2020; Hopkins et al. 2021), while the effective scattering rate of CRPAI is on the order of 10 -3 per year. Assuming only the CRPAI operates (no CR streaming), we find that the anisotropy level of ∼ GeV CRs to be as low as 10 -5 under such parameters. \nThe CRPAI can be triggered more effectively by smaller eddies. Given an eddy size l , one can approximately write | δB | ∼ B 0 ( l/L gal ) 1 / 4 ∼ 1 / 3 (1 / 3 for the slow magnetosonic mode and 1 / 4 for the fast magnetosonic mode, see details in Goldreich & Sridhar 1995; Cho & Lazarian 2003), where B 0 represents the mean magnetic field strength. The corresponding variation timescale can be estimated as l/U A , with magnetic field change rate experienced by CRs given by ∣ ∣ ∣ ˙ B/B ∣ ∣ ∣ ∼ \n| δB/B 0 | / ( l/U A ) ∼ ( U A /L gal ) ( l/L gal ) -3 / 4 ∼-2 / 3 , which increases as the eddy size l decreases. Consider a modest scenario taking l ∼ 1pc while L gal ∼ 1kpc, the CR scattering rate resulting from the CRPAI may become comparable to that of CRSI in such cases. The transittime damping in turbulence (e.g. Fisk 1976; Eilek 1979; Achterberg 1981; Miller et al. 1996; Schlickeiser & Miller 1998), which primarily acts on the CR momenta aligned with the background magnetic field, can further enhance CRanisotropy and thereby increase the efficiency of CRPAI (Brunetti & Lazarian 2007). \nAnother possible scenario where the CRPAI may actively manifest itself is around the mixing layer of multiphase gas (Begelman & Fabian 1990; Klein et al. 1994). Multiphase gas is ubiquitous in astrophysical environments such as the shell of bubbles from stellar explosions (e.g., El-Badry et al. 2019; Lancaster et al. 2021a,b, 2024), the circumgalactic medium (CGM) (e.g., Gronke & Oh 2018; Chen & Oh 2024), galactic winds (Fielding & Bryan 2022), and cosmic filaments (e.g., Mandelker et al. 2020). Mixing layers that separate and mix up different phases at the boundary are dynamic, turbulent and usually rapidly radiating energy away(e.g. Ji et al. 2019; Fielding et al. 2020; Tan et al. 2021), with the inflow gas experiencing cooling contraction and rapid magnetic field amplification (Zhao & Bai 2023; Das & Gronke 2024). While the dynamical effect of the CRs in the mixing layers is yet to be understood, one might approximately estimate that the magnetic field changing timescale | ˙ B/B | -1 as experienced by CRs co-moving with the gas can be as short as ∼ 0 . 01 Myr 7 . In comparison to the largest eddy turnover time in ISM, approximately 10Myr, the multiphase gas mixing may efficiently induce CR anisotropy, coupling the CRs through the CRPAI, potentially leading to a ν eff up to ∼ once per year. A comparison with the CRSI is less straight- \nd since the level of CR gradient across the mixing layer is unclear, but as significant wave damping is expected in the cold phase, the CR distribution tends to be relatively flat across the phase boundaries (Armillotta et al. 2024). \nAnother potential applicability of the CRPAI is the CR bottleneck (e.g., Skilling 1971; Wiener et al. 2017a), where CRs accumulate in front of a gas density bump (a dip in the CR streaming speed U A ), leading to a staircase-like CR pressure structure devoid of gradients on both sides of the bottleneck. Such 'meso-scale' structures could be ubiquitous and play a fundamental role in CR feedback (e.g., Bruggen & Scannapieco 2020; Bustard & Zweibel 2021; Tsung et al. 2022; Quataert et al. 2022), yet our exploration and understanding of such structures is still at very early stages. The CR streaming is expected to diminish on the two sides of the bottleneck, and they can be subject to the CRPAI in the presence of ˙ B/B . While it is expected that heating by CRPAI is not necessarily efficient (Bruggen & Scannapieco 2020), it is yet to explore how the additional coupling enabled by CRPAI affect the properties of the CR bottleneck, and towards global scales.", '6. SUMMARY AND OUTLOOK': "In this study, we employ kinetic (MHD-PIC) simulations to investigate the saturated state of CRPAI in the presence of ion-neutral damping. The CR anisotropy is continuously driven by the expanding box, which mimics gas expansion and compression in turbulence (and potentially shear motion) and is characterized by the magnetic field change rate (the expansion rate) ˙ B/B . Meanwhile, the ion-neutral damping is treated in the short-wavelength limit generally applicable in the neutral medium of the ISM, where the ionized gas motion is damped at a rate ν IN / 2. We achieve the saturated state self-consistently, where the growth rate of the CRPAI (Equation 20) is balanced by the ion-neutral damping rate, while the CR anisotropy level is maintained between driving (by expanding box) and quasi-linear diffusion (QLD, Equation 25). The kinetic simulation at the saturated state enables us to calibrate the scattering rate ν eff and the equilibrium pressure anisotropy level P CR , ⊥ /P CR , ∥ -1 from first principles, which are essential for understanding the role of CR feedback at macroscopic (e.g., galactic) scales. These properties scale with local environmental conditions (Equation 23 & 31), including the mass density ratio between CRs and thermal ionized gas ρ CR /ρ 0 , the Alfv'en speed U A , among others. We also vary simulation parameters to validate the scaling relation with environmental parameters. \nOur main findings are as follows: \n- · We detailed the formulation of the quasi-linear theory (QLT) under the aforementioned balancing relations, which yields the momentum-bymomentum estimates of the CR scattering rates (Equation 27) and anisotropy level (Equation 23) based on the CRPAI as a function of environmental parameters ( ˙ B/B , ν IN , etc.).\n- · We have verified that the CR pitch angle and momentum diffusion coefficients D µµ , D pp can be accurately reproduced from the wave intensities based on quasi-linear diffusion, lending support to the general validity of QLT.\n- · For those CRs that attain the saturated state (within the momentum range (0 . 5 p 0 , 5 p 0 )) in our simulations, the CR pressure anisotropy P CR , ⊥ /P CR , ∥ -1 largely follows the the momentum dependence predicted by QLT, with highenergy CRs exhibiting greater anisotropy (Figure 4). This pressure anisotropy also scales with the environmental parameters. However, the value of anisotropy in the single-fluid treatment exceeds the QLT prediction by a factor of 2 ∼ 4 (Figure 9).\n- · With the p -byp treatment, the measured CR scattering rates ν eff ( p ) (Figure 8) are consistent with the anticipated trend and the dependence on environmental parameters as predicted by QLT (Equation 27), except that the results with compressing box yields ν eff ( p ) that are higher by a factor of about 2 for p ≳ p 0 particles. In the single fluid treatment for CRs, the values of ν eff are consistent with QLT scaling, though with a factor of up to a few difference in the normalization depending on the range of momentum considered (Figure 10).\n- · The contribution of irreversible CR heating/cooling from the CRPAI is about a factor of 2 higher than QLT prediction (Section 3.1), ∼ -0 . 3 ( ˙ B/B ) ( U A /c ) E CR , which represents minor contributions compared to adiabatic heating/cooling.\n- · In a broader context, the scattering between CRs and MHD waves resulting from CRPAI is notably weaker compared to that of CRSI, by about a factor of U A /c on global scales. Nevertheless, we speculate that the CRPAI could play a more dominant role in smaller-scale structures, including the multiphase mixing layers and the CR bottleneck. \nOur MHD-PIC simulations on the saturation of the CRPAI, together with that of the CRSI (Bai 2022), provide valuable insights into the microphysics about CR \nfeedback from the first principles. The ultimate goal of such studies is to provide calibrated CR scattering rates from the CR gyro-resonant instabilities that serve as subgrid physics for CR (magneto-)hydrodynamic simulations at macroscopic scales. So far, we have studied the pure CRSI case driven by imposed CR pressure gradient, and the pure CRPAI case driven by ˙ B/B . The methodology laid out in these two works pave the way for us to further explore the most general case, where both a CR gradient and ˙ B/B are present, which may lead to marked differences from the cases studied so far. In the meantime, future work should explore multidimensional effects (Zeng et al. 2024), as well as the case with other wave damping mechanisms, especially the non-linear Landau damping that dominates in the more volume-filling warm medium (e.g., Armillotta et al. 2021). Finally, with the adaptive δf method, it becomes \npossible to explore 'meso-scale' phenomena such as the CR bottleneck (e.g., Skilling 1971; Wiener et al. 2017a; Tsung et al. 2022; Quataert et al. 2022) fully kinetically, which would be a major effort to directly link CR feedback from microscopic to macroscopic scales. \nWe thank Eve Ostriker, Eliot Quataert, Ellen Zweibel \n1 \n- and Alexandre Lazarian for useful discussions, particu2\n- larly during the Aspen Center for Physics program on 3\n- 'Cosmic Ray Feedback in Galaxies and Galaxy Clusters'. 4\n- This work is supported by National Science Foundation 5\n- of China under grant No. 12325304, and in part by 6\n- the National Science Foundation under Grant No. NSF 7\n- PHY-2210452. 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Accuracy test: fitting method for f 0 in the expanding box': 'We simulate the adiabatic evolution of the CR population in the expanding box to validate the fitting method outlined by Equation 18 & 19. In the absence of isotropization from waves and solely driven by box expansion/compression, each CR particle conserves pµa 2 ( t ) and pa ( t ) √ 1 -µ 2 (Equation 12, also see the Appendix of Sun & Bai 2023). Consequently, the CR distribution f , starting from an initial isotropic κ distribution (Equation 14), should precisely conform to the anisotropic κ distribution (Equation 15) with ξ = a ( t ). In this benchmark problem, we utilize Equations 18 and 19 to dynamically fit the CR anisotropy parameter ξ and the peak momentum p 0 . If the fitting method performs correctly, the fitted ξ should equal to a ( t ), while the fitted p 0 should remain constant. \nThe simulation setup largely follows the fiducial runs in the expanding/compressing boxes, albeit slightly simplified for test purposes. We initialize CRs in a uniform isotropic κ distribution (Equation 14) with p 0 = 200 U A and κ = 1 . 25. The simulation particles are segregated into eight momentum bins, with 2 . 4576 × 10 6 particles allocated to each bin, matching the total number of simulation particles in the fiducial runs (Section 2.3). To prevent isotropization, we set the initial wave amplitudes to zero and disable the CR backreaction. The background field and the expanding box setup are identical to those in Section 2.3, while the expansion/compression rate is elevated to ˙ a = ± 10 -2 Ω 0 . The simulation spans 10 2 Ω -1 0 , where the CR anisotropy level becomes significant by the end of the simulation, with f 0 updated every ∆ T adapt = 1Ω -1 0 . \nThe fitting results for ξ and p 0 are depicted in Figure 11. The fitted ξ closely tracks the expansion rate a ( t ) = exp(˙ at ), while the fitted p 0 remains constant around the initial value of 200 U A , in both the expanding box and the compressing box. We note that there is a systematic deviation in the fitted p 0 less than 1% relative to the anticipated values. This small deviation is related to the initialized CR distribution, where initial simulation particles only span over a finite range of the κ distribution ( p 0 / 500 to 500 p 0 , see Section 2.3) 8 . The fitting error can be further minimized by covering over a larger range of the κ distribution or increasing the fitting frequency (decreasing ∆ T adapt ˙ a ).', 'A.2. Performance test: signal-to-noise improvement in CPPAI': 'We compare the wave spectra triggered by CRPAI in the expanding box, both with the adaptive δf method or the traditional δf method, to assess the effectiveness. The adaptive δf method can alleviate the statistical noise in the CR backreaction, particularly in scenarios where the CR distribution f is less predictable. \nThe simulation setup closely resembles that in Section 2.3, albeit with simplifications tailored for testing purposes. In comparison to the fiducial runs, we eliminate the ion-neutral damping and augment the box expansion rate to ˙ a = +5 × 10 -5 Ω 0 , to expedite both the wave isotropization and the anisotropy driving. Additionally, for a significant wave intensity within a limited runtime, we set the initial wave amplitude as A = 10 -2 and enhance the CR mass ratio to ρ CR /ρ 0 = 10 -4 . We set the 1D simulation box size to L x = 9 . 6 × 10 4 U A / Ω 0 and the resolution to 10 U A / Ω 0 . All other parameters and setups, such as the initial CR distribution and the number of momentum bins for initial simulation particles, remain identical to those of the fiducial run in the expanding box. \nFigure 11. Fitting results for the CR anisotropy parameter ξ and the peak momentum p 0 in the test of the adaptive δf method using expanding box (blue) and compressing box (red). The box expansion rate ˙ a is ± 10 -2 Ω 0 . The markers in the top two panels represent the fitting results obtained from Equation 18 and Equation 19, while the solid lines depict the expected values. The bottom panel illustrates the relative deviation of the fitted ξ from the expected values. \n<!-- image --> \nWe carry out three parallel runs by varying the choice f 0 in the δf method and particles per cell per momentum bin: the adaptive f 0 with four particles per cell (ppc) per momentum bin, the adiabatic f 0 with four ppc per momentum bin, and the adiabatic f 0 with 16 ppc per momentum bin. The adaptive f 0 fits ξ and p 0 every ∆ T adapt = 5Ω -1 0 (see Section 2.2). The adiabatic f 0 maintains ξ = a ( t ) and a constant p 0 , representing CR adiabatic evolution without wave isotropization. We illustrate the wave spectra measured at t = 3 × 10 4 Ω -1 0 in Figure 12. Comparing the spectra from the two adiabatic f 0 runs with different particles per cell, the statistical noise from the CR backreaction primarily affects the highk regime, and the run with 16ppc has significantly smaller noise level. The adaptive δf method even \nmore effectively reduce the noise at highk . With only 4ppc, the noise level is even smaller than the run with adiabatic f 0 but using four times less particles. \nThe effectiveness of the adaptive δf method depends on the specific problem at hand. In this benchmark, it significantly reduces computational costs by over four times compared to the traditional δf method, where f 0 remains the one evolving adiabatically from the initial CR distribution. However, when f is predictable, such as when isotropization from waves dominates, or when quasi-linear diffusion is negligible and CR evolution is adiabatic, the adaptive δf method offers no signifcant improvement in signal-to-noise ratio. In our study, the saturation of CRPAI, where f deviates from both isotropic distribution and adiabatic distribution, the adaptive δf method proves essential. Note that the anisotropy level in the actual simulations varies with p , and further improvement may be considered by allowing ξ to be a parameterized function of p , but this can be left for future work. \nFigure 12. Wave intensity spectra, kI ( k ), in the benchmark problem with varying f 0 and numbers of simulation particles, at time t = 3 × 10 4 Ω -1 0 . The color and line styles correspond to those in Figure 2. Lines of the same color (representing polarization) substantially overlap. \n<!-- image -->'}

2024A&A...690A.327V

Aims. We compared stellar radii derived from asteroseismic scaling relations with those estimated using two independent surface brightnesscolour relations SBCRs combined with Gaia DR3 parallaxes. Methods. We crossmatched asteroseismic and astrometric data for over 6400 red giant branch RGB and red clump RC stars from the APOK2 catalogue with the TESS Input Catalogue v8.2 to obtain precise V band magnitudes and EB V colour excesses. We then adopted two different SBCRs from the literature to derive stellar radius estimates denoted as RSUPaSUP and RSUPbSUP respectively. We analysed the ratio of these SBCRderived radii to the asteroseismic radius estimates R provided in the APOK2 catalogue. Results. Both SBCRs exhibited good agreement with asteroseismic radius estimates. On average RSUPaSUP was overestimated by 1.2 with respect to R while RSUPbSUP was underestimated by 2.5. For stars larger than 20 RSUBSUB SBCR radii are systematically lower than asteroseismic ones. The dispersion in the radius ratio was similar for the two methods around 10. The agreement with asteroseismic radii shows a strong dependence on the parallax. The dispersion is halved for stars with a parallax greater than 2.5 mas. In this subsample RSUPbSUP showed perfect agreement with R while RSUPaSUP remained slightly overestimated by 3. A trend with FeH was found at a level of 4 to 6 per dex. Additionally a clear trend with asteroseismic mass is found. For stars less massive than about 0.95 MSUBSUB SBCR radii were significantly higher than asteroseismic ones by about 6. This overestimation correlated with the presence of extended helium cores in these stars structures relative to their envelopes. Furthermore radius ratios showed a dichotomous behaviour at higher masses mainly due to the presence of several RC stars with SBCR radii significantly lower with respect to asteroseismology. This behaviour originates from a different response of asteroseismic scaling relations and SBCR to Fe abundance ratios for massive stars both in RGB and RC phases which is reported here for the first time.

2024-10-01T00:00:00Z

['10.48550/arXiv.2409.10050', '2024A&A...690A.327V', 'arXiv:2409.10050', '2024arXiv240910050V', '10.1051/0004-6361/202451473']

['methods: statistical', 'stars: evolution', 'stars: fundamental parameters', 'stars: interiors', 'Astrophysics - Solar and Stellar Astrophysics', 'Astrophysics - Astrophysics of Galaxies']

Testing the asteroseismic estimates of stellar radii with surface brightnesscolour relations and Gaia DR3 parallaxes Red giants and red clump stars

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https://arxiv.org/pdf/2409.10050.pdf

{'Red giants and red clump stars': 'G. Valle 1 , 2 , M. Dell\'Omodarme 1 , P.G. Prada Moroni 1 , 2 , S. Degl\'Innocenti 1 , 2 \n- 1 Dipartimento di Fisica "Enrico Fermi\', Università di Pisa, Largo Pontecorvo 3, I-56127, Pisa, Italy\n- 2 INFN, Sezione di Pisa, Largo Pontecorvo 3, I-56127, Pisa, Italy \nReceived ; accepted 09 / 09 / 2024', 'ABSTRACT': "Aims. We compared stellar radii derived from asteroseismic scaling relations with those estimated using two independent surface brightness-colour relations (SBCRs) combined with Gaia DR3 parallaxes. \nMethods. We cross-matched asteroseismic and astrometric data for over 6,400 red giant branch (RGB) and red clump (RC) stars from the APO-K2 catalogue with the TESS Input Catalogue v8.2 to obtain precise V band magnitudes and E ( B -V ) colour excesses. We then adopted two di ff erent SBCRs from the literature to derive stellar radius estimates, denoted as R a and R b , respectively. We analysed the ratio of these SBCR-derived radii to the asteroseismic radius estimates, R , provided in the APO-K2 catalogue. \nResults. Both SBCRs exhibited good agreement with asteroseismic radius estimates. On average, R a was overestimated by 1.2% with respect to R , while R b was underestimated by 2.5%. For stars larger than 20 R ⊙ , SBCR radii are systematically lower than asteroseismic ones. The dispersion in the radius ratio was similar for the two methods (around 10%). \nThe agreement with asteroseismic radii shows a strong dependence on the parallax. The dispersion is halved for stars with a parallax greater than 2.5 mas. In this subsample, R b showed perfect agreement with R , while R a remained slightly overestimated, by 3%. A trend with [Fe / H] was found at a level of 4% to 6% per dex. Additionally, a clear trend with asteroseismic mass is found. For stars less massive than about 0.95 M ⊙ , SBCR radii were significantly higher than asteroseismic ones, by about 6%. This overestimation correlated with the presence of extended helium cores in these stars' structures relative to their envelopes. Furthermore, radius ratios showed a dichotomous behaviour at higher masses, mainly due to the presence of several RC stars with SBCR radii significantly lower with respect to asteroseismology. This behaviour originates from a di ff erent response of asteroseismic scaling relations and SBCR to [ α / Fe] abundance ratios for massive stars, both in RGB and RC phases, which is reported here for the first time. \nKey words. Stars: fundamental parameters - methods: statistical - stars: evolution - stars: interiors", '1. Introduction': 'Accurate measurements of stellar masses and radii are crucial to constraining stellar structure and evolution models. High-quality measurements of these parameters can be obtained from detached double-lined eclipsing binaries, so these systems are routinely adopted to test stellar evolution models and to constrain their free parameters (see among many Andersen et al. 1991; Torres et al. 2010; Valle et al. 2017; Claret & Torres 2017; Valle et al. 2023). \nObtaining precise estimates of masses and radii from single stars is more problematic. Spectroscopic determinations of the e ff ective temperature, surface gravity, and metallicity of a star, combined with a known parallax, can provide the stellar radius and luminosity. However, the strong dependence of the spectroscopic atmospheric parameters on the atmosphere models and on the method used for their derivation hamper firm predictions (see e.g. Torres et al. 2012; Coelho 2014; Ivanyuk et al. 2017; Branco et al. 2024). As a matter of fact, di ff erences of about 60 Kin e ff ective temperature and 0.15 dex in log g exist between determinations from di ff erent surveys for red giant branch (RGB) stars (Heged"us et al. 2023; Yu et al. 2023). \nA di ff erent approach to evaluating the linear radius of a star is to combine an estimate of its angular radius with its distance, which has been made particularly attractive by the recent determination of accurate distances for over a billion stars from Gaia Data Release 3 (DR3; Gaia Collaboration et al. 2021). Very accurate angular size measurements can be obtained with long-baseline interferometry (LBI; e.g. Nordgren et al. 1999; Mozurkewich et al. 2003; Baines et al. 2010; Gallenne et al. 2012; Lachaume et al. 2019; Perraut et al. 2020). However, LBI is limited to stars with angular diameters greater than about 0.5 mas (Gallenne et al. 2018), rendering it impractical for most targets. Surface brightness-colour relations (SBCRs) provide an efficient alternative for determining stellar angular diameters from photometric measurements. Essentially, SBCRs establish a link between a star\'s angular size and its de-reddened brightness across various photometric bands. SBCRs are usually calibrated using samples of stars with well-determined LBI radii (see e.g. Kervella et al. 2004; Di Benedetto 2005; Salsi et al. 2021). More than 20 SBCRs exist in the literature, and many of them focus on the V and K bands, as this set of colours provides the lowest dispersion (Kervella et al. 2004). Comparisons among recent SBCRs have revealed a limited variability for late-type stars (e.g. Pietrzy\'nski et al. 2019; Salsi et al. 2022; Nardetto et al. 2023). For instance, the comparison among 19 SBCRs performed by \nNardetto et al. (2023) showed an agreement better than about 0.008 mag between 1.5 and 2.5 mag in V -K . The disagreement increased outside this range. SBCRs play a fundamental role in the distance determination of eclipsing binaries. As an example, a precise SBCR calibrated on 41 nearby red clump (RC) giant stars (Gallenne et al. 2018) has allowed the distance to the Large Magellanic Cloud to be estimated with a precision of 1% (Pietrzy\'nski et al. 2019). \nA new method for estimating stellar radii has emerged in recent years. The growth of observational asteroseismology, thanks to satellite missions such as Kepler and Transiting Exoplanet Survey Satellite (TESS; Borucki et al. 2010; Ricker et al. 2015), has opened a new way to estimate stellar properties of stars, such as mass, radius, and age. These data allow the RGB stellar radii to be estimated with a precision of about 4% (Pinsonneault et al. 2014; Martig et al. 2015; Valle et al. 2024). Notably, neither the distance from the observer nor the reddening influence the asteroseismic estimates. In fact, this method requires global seismic parameters, namely, the frequency of maximum power, ν max, a large frequency separation, ∆ ν , and a determination of the stellar e ff ective temperature. Given these ingredients, scaling relations yield the asteroseismic estimates of mass and radius (Ulrich 1986; Kjeldsen & Bedding 1995). While the validity of these scaling relations in the RGB phase has been questioned (e.g. Epstein et al. 2014; Gaulme et al. 2016; Viani et al. 2017; Brogaard et al. 2018; Buldgen et al. 2019), corrections accounting for the temperature and metallicity of the star have been proposed in the literature (e.g. Zinn et al. 2022; Stello & Sharma 2022). For stars with a metallicity [Fe / H] > -1 . 0, scaling relations are considered reliable (Epstein et al. 2014; Valentini et al. 2019; Schonhut-Stasik et al. 2024). In the following, we adopt asteroseismic radii as a reference. It is, however, important to clarify that we do not assume them to be the gold standard for radius determination. Disagreement in the measurement of radii from asteroseismic-corrected scaling relations and interferometric measurements have been reported by di ff erent authors, while other researchers have not found significant di ff erences (see e.g. Hekker 2020; Thomsen et al. 2022, and references therein). \nGiven the relevant di ff erences between asteroseismic and SBCR methods, a comparison of their results obtained when applied to a homogeneous sample is of particular interest. The recently released APO-K2 catalogue (Schonhut-Stasik et al. 2024) o ff ers a great opportunity to perform this test, as this catalogue contains high-precision data of more than 7,500 RGB and RC stars, combining spectroscopic (APOGEE DR17; Abdurro\'uf et al. 2022), asteroseismic (K2-GAP; Stello et al. 2015), and astrometric ( Gaia DR3; Gaia Collaboration et al. 2021) data.', '2. Adopted SBCRs and data selection': "The surface brightness, S λ , of a star is linked to its limbdarkened angular diameter, θ, and its apparent magnitude corrected from the extinction, m λ 0. In the V band, SV is defined as \nSV = V 0 + 5 log θ, (1) \nwhere V 0 is the V band magnitude corrected for extinction. From Eq. (1) it follows that \nθ = 10 0 . 2 ( SV -V 0) . (2) \nTherefore, an estimate of the stellar linear radius is \nr = 0 . 5 d θ, \n(3) \nArticle number, page 2 of 6 \nwhere d is the heliocentric distance of the star. \nWe adopted two di ff erent SBCRs. The first, proposed by Pietrzy'nski et al. (2019), is \nS a V = 1 . 330[( V -K )0 -2 . 405] + 5 . 869 mag , (4) \nwhere ( V -K )0 is the colour corrected for reddening. This relation was fitted in a colour range ( V -K )0 from 2.0 to 2.8 mag. The second SBCR we adopted for testing was proposed by Salsi et al. (2021). They used a slightly di ff erent formalism in their paper, but it can be written as \nS b V = 1 . 22( V -K )0 + 2 . 864 mag . (5) \nThis relation (from Table 5 in Salsi et al. 2021 for F5 / K7-II / III spectral class stars) is valid in the range ( V -K )0 from 1.8 to 3.9 mag. \nThe APO-K2 catalogue provides the Ks band magnitude from Two Micron All-Sky Survey and the parallax from Gaia DR3, corrected according to the Gaia zero-point (Lindegren et al. 2021). However, the V band magnitudes it provides are computed starting from magnitudes in the J and Ks bands (Schonhut-Stasik et al. 2024) and have low precision. The APOK2 catalogue does not provide information about colour excess E ( B -V ) or extinction in the V band AV either. To obtain both more precise V band magnitudes and E ( B -V ), we cross-matched the APO-K2 catalogue with the TESS Input Catalogue (TIC) v8.2. The TIC adopts the three-dimensional empirical dust maps from Panoramic Survey Telescope and Rapid Response System (Green et al. 2018), with a re-calibration coe ffi cient of 0.884 applied to obtain E ( B -V ) values, as prescribed by Schlafly & Finkbeiner (2011). Nevertheless, it is known that the extinction has a negligible e ff ect on the SBCR because the colour and surface brightness are sensitive to it in almost the same way. The only (tiny) di ff erence comes from the extinction in the K band (Nardetto et al. 2020). We adopted the following extinction relations from Cardelli et al. (1989): AV = 3 . 1 E ( B -V ) and AK = 0 . 114 AV . We also verified that the adoption of the relation AK = 0 . 089 AV from Nishiyama et al. (2009) does not modify the results; it leads to an average variation in the estimated radii of only 0.2%. \nData in the APO-K2 catalogue were subjected to a selection procedure, both to reject apparent outliers and to restrict the data to a metallicity range where asteroseismic scaling relations are the most reliable. Stars satisfying all the following constraints were retained in the sample: [Fe / H] > -1 . 0 dex, 4000 K < T e ff < 5300 K, -0 . 1 dex < [ α / Fe] < 0.4 dex, and log g < 3.25. Stars in the RGB with a mass lower than 0.75 M ⊙ were rejected as artefacts because single stars with such a low mass cannot be in the RGB given their long evolutionary timescale. Stars outside the range of colour for the Salsi et al. (2021) SBCR were excluded and so were stars with a binary flag set in the APO-K2 catalogue. Finally, stars with a relative error in the parallax greater than 0.1 were rejected. This allowed us to rely on distances obtained using the inverse parallax (Bailer-Jones et al. 2021; Fouesneau et al. 2023). The final sample comprises 6,420 stars, with 4,202 of them in the RGB and 2,218 in the RC phase.", '3. SBCRs to asteroseismic radii comparisons': "For all stars in the final sample, linear radii were obtained from Eq. (3), with θ (Eq. (2)) computed using the SBCR from Pietrzy'nski et al. (2019) and Salsi et al. (2021). This resulted in estimates denoted as R a and R b , respectively. The comparison of \nFigure 2 explores the dependence of radius ratios on various stellar parameters, focusing on the results obtained using the SBCR from Pietrzy'nski et al. (2019). Similar trends were obtained for the Salsi et al. (2021) SBCR. The impact of the metallicity is investigated in the bottom panel of Figure 2. There is a weak positive trend of R a / R with metallicity [Fe / H]. This trend is approximately 6% per dex for R a / R , and it is about 4% per dex for R b / R . \n<!-- image --> \nFig. 1. Ratio of the radii from SBCRs and from asteroseismology. Top : Radius ratio using the Pietrzy'nski et al. (2019) SBCR. Red and blue points correspond to RC and RGB stars, respectively. The dashed black line is a smoother of data, while the dotted one serves as a visual aid. Bottom : Same as in the top panel but using the Salsi et al. (2021) SBCR. \n<!-- image --> \nthese radii with R from asteroseismic scaling relations (Fig. 1) revealed both similarities and di ff erences. \nOverall, there is a notable agreement between the two SBCR methods and the predictions from the scaling relations. The agreement between the SBCR methods and the asteroseismic radius, R , is better for R a , with an overestimation of the radii by 1.2% over the full sample, while R b leads to an underestimation by 2.5%. Both SBCRs underestimate radii at higher values, a trend more pronounced for R b compared to R a . Interestingly, the analysed SBCRs show the same feature for RC stars because they strongly underestimate radii larger than about 12 R ⊙ for RC stars, while the trend for RGB stars is much less pronounced. This di ff erence, which we discuss in detail later, is likely due to di ff erences in the mass and metallicity distribution between the samples of RC and RGB stars. Finally, the dispersion is similar for the two SBCRs, with a standard deviation of 10%. \nA significant dependence on asteroseismic mass can be observed in the top panel of Fig. 2. At lower masses, the SBCR method clearly provides a larger radius. However, in the mass range 1.0 to 2.0 M ⊙ , the average discrepancy is less than 0.1%. The di ff erence at the lower mass end is partially due to the presence of 189 RC stars with masses below 0.75 M ⊙ , whose SBCR radii are larger than asteroseismic ones by a median of about 6%. Excluding RC stars less massive than 0.75 M ⊙ , as done for the RGB sample, is however not theoretically justified. In fact, the mass loss, occurring primarily during the later stages \n<!-- image --> \nFig. 2. Ratio of the radii from the Pietrzy'nski et al. (2019) SBCR and from asteroseismology as a function of di ff erent stellar parameters. Top : Radius ratio as a function of the stellar mass as estimated by scaling relations. Bottom : Same as in the top panel but as a function of the metallicity [Fe / H]. Colour codes and line styles are as in Fig. 1. \n<!-- image --> \nof the RGB phase, may significantly impact the stellar mass at the clump phase. For instance, studies of globular clusters suggest that progenitor stars with initial masses of 0.80 M ⊙ can lose around 0.17 M ⊙ during their evolution (Howell et al. 2022). The internal structure of these objects is dominated by a helium core that is roughly 0.5 M ⊙ in size, with minimal variation depending on the star's initial chemical composition or mass (see e.g. Sweigart & Gross 1978; Cassisi et al. 2016). Asteroseismic observations, which are sensitive to a star's internal structure, can readily identify these stars, as shown in the top panel of Fig. 4. These low-mass RC stars exhibit a higher ∆ ν value at fixed ν max with respect to more massive objects. However, the presence of these low-mass RC stars alone does not fully explain the discrepancy between SBCRs and asteroseismic radii. This is because even for RGB stars below 0.95 M ⊙ , R a / R is overestimated by a median of 4%. This overestimation is likely due to the significant helium core that grows steadily as the stars ascend the RGB. The bottom panel of Fig. 4 demonstrates a clear dependence of the R a / R ratio on the stellar log g . Stars with lower log g values have a larger helium core, which in turn leads to a greater bias in the R a / R measurement. A possible reason for the detected discrepancies is the di ff erence in mass range used for calibration. For example, the Gallenne et al. (2018) sample used to calibrate the Pietrzy'nski et al. (2019) SBCR relation did not include any stars with an estimated mass below 0.9 M ⊙ . It is, however, interesting that the theoretical investigation performed by Salsi et al. (2022) reports negligible di ff erences between log g = 3 and log g = 0 in the predicted SBCR radii. Therefore the trend detected here might also be due to an inherent bias in the asteroseismic relations. \nFor higher masses, dominated by RC stars, a more complex and dichotomous trend emerges. Some SBCR radii are estimated close to the asteroseismic ones, while others have significantly lower values. The latter group corresponds to the lower end of \nTable 1. Robust regression fits of R a / R as a function of [ α / Fe], according to the evolutionary phase and the stellar mass. \nthe arc-shaped distribution for RC stars in Figure 1. The top panel in Figure 4 shows that objects with R a / R < 0 . 85 correspond to the lower end of the ∆ ν -ν max distribution. \nThe observed dichotomous trend, most prominent for the RC population, arises from the di ff ering responses of SBCRestimated and asteroseismic radii to variations in [ α / Fe] across di ff erent mass ranges. Figure 3 shows the R a / R ratio as a function of [ α / Fe] for RGB and RC stars, further subdivided by mass. Acuto ff value of 1.7 M ⊙ was chosen, but any value up to 2.0 M ⊙ yields similar results. Using a higher cuto ff would significantly reduce the number of stars in the massive RGB bin. While radius ratios are almost constant in the lower-mass bin, significant trends with [ α / Fe] occur for massive stars. To further explore the trends, robust linear models 1 were fitted to data. The results are presented in Table 1 and over-plotted in Fig. 3. The trends are similar between RGB and RC stars, with a steeper dependence on [ α / Fe] for the more massive RC population. These behaviours suggest a theoretical problem in our understanding of SBCR and asteroseismic radius determinations for α -enhanced massive stars. However, this question cannot be solved with the available data, and further theoretical research is needed to assess the origin of the discrepancy. The evident di ff erence between SBCRs and asteroseismic radii for the RC population in Fig. 1 and the dichotomous trend in Fig. 2 arise from the superposition of the trends in the di ff erent mass bins. \nAs expected, the agreement between the radii from SBCRs and asteroseismology is significantly better for nearby stars, where parallax measurements are most accurate. Figure 5 demonstrates the noticeable decrease in the dispersion of radius ratios as the parallax increases. The standard deviation of the radius ratios for both SBCRs is halved for stars with a parallax greater than 2.5 mas, reaching 6%. Interestingly, the Salsi et al. (2021) SBCR perfectly agrees with the asteroseismic radii in this range, while the Pietrzy'nski et al. (2019) method has a positive median overestimation of about 3.0%.", '4. Conclusions': 'Leveraging the recently released APO-K2 catalogue (SchonhutStasik et al. 2024), which contains astrometric, spectroscopic, and asteroseismic information of more than 7,500 stars in the RGB and RC evolutionary phases, we performed a comparison of radii derived from asteroseismic scaling relations and those from SBCRs combined with Gaia DR3 parallaxes. We specifically adopted SBCRs from Pietrzy\'nski et al. (2019) and Salsi et al. (2021). Information about colour excess and magnitude in the V band was obtained by cross-matching the APO-K2 catalogue with TIC v8.2. The adopted stellar sample is a couple \nof orders of magnitude larger than those adopted to investigate the agreement between asteroseismic estimates of stellar radii in binary systems (e.g. Gaulme et al. 2016; Brogaard et al. 2018; Themeßl et al. 2018). The sample encompasses di ff erent evolutionary stages, from the early RGB to the RC, and covers a wide metallicity range ([Fe / H] from -1.0 to 0.5 dex). It therefore allowed us to test the agreement between the di ff erent radius estimates as a function of various stellar evolutionary characteristics. \nOverall, both SBCRs demonstrated good agreement with the asteroseismic estimates. Since asteroseismic scaling relations and SBCRs rely on very di ff erent information to obtain their estimates, this agreement is particularly relevant. Comparisons were made using asteroseismic values as a reference, though this choice does not assume them to be unbiased estimators (see Hekker 2020, and references therein for a discussion of the accuracy and precision of asteroseismic radii). The SBCR from Pietrzy\'nski et al. (2019) was found to predict a larger radius with respect to scaling relations by about 1.2% over the whole explored range, while the SBCR from Salsi et al. (2021) was found to be 2.5% smaller than asteroseismic values. Both SBCRs led to radii significantly smaller than the asteroseismic ones when the latter are larger than about 20 R ⊙ . The two SBCR methods show a similar dispersion of about 10% in the ratio of radii with asteroseismic estimates. The agreement with asteroseismic radii is found to depend strongly on the parallax because the dispersion is halved for stars with a parallax greater than 2.5 mas, where Salsi et al. (2021) estimates proved to be in perfect agreement with asteroseismic ones, while Pietrzy\'nski et al. (2019) radii were overestimated by about 3% with respect to asteroseismic values. \nA trend with metallicity [Fe / H] is found, at a level between 4% and 6% per dex. An impact of the metallicity on SBCRs has already been suggested by Kervella et al. (2004) and Boyajian et al. (2012) for main-sequence dwarf stars. For giant stars, a recent theoretical investigation by Salsi et al. (2022), adopting MARCS model atmospheres (Gustafsson et al. 2008) to compute spectra and obtain the surface brightness of stars, found a slight impact of the metallicity, which was almost negligible at the e ff ective temperature of about 5000 K and increased to 10% per dex at lower temperatures, corresponding to V -K ∼ 4. The detected trend of the radius ratios with metallicity is therefore higher than the theoretical prediction of the SBCRs, suggesting a possible bias in the asteroseismic radii. \nWe found a clear trend with the asteroseismic mass. Stellar radii from SBCRs are significantly larger, by about 6%, than the asteroseismic ones for stars less massive than 0.95 M ⊙ , while no di ff erence was detected in the 1.0 to 2.0 M ⊙ mass range. This overestimation is correlated with the presence of a significant extension of the helium core with respect to the envelope in the stellar structures. The most interesting result of the present investigation is that the radius ratio trends with asteroseismic mass show a dichotomous behaviour moving to high mass values, mainly due to the presence of several RC stars with SBCR radii significantly lower with respect to asteroseismology. Analysis of this behaviour allowed us to highlight a di ff erent response of the asteroseismic scaling relations and SBCRs to [ α / Fe] for massive stars, in both the RGB and RC phases, which is reported here for the first time. In fact, theoretical investigation by Salsi et al. (2022) did not detect any impact of the mass on their synthetic SBCR, but the tested mass range did not comprise low-mass stars, extending from 2.0 to 5.0 M ⊙ . Moreover no investigation was performed to assess the influence, if any, of α enhancement. Further investigations are encouraged in or- \nFig. 3. Ratio of the radii from the Pietrzy\'nski et al. (2019) SBCR and asteroseismology as a function of [ α / Fe] in di ff erent evolutionary phases and mass bins. Top row : Radius ratio for RGB stars. The dashed green lines show the robust regression trends. Bottom row : Same as in the top row but for RC stars. \n<!-- image --> \n<!-- image --> \nFig. 4. Seismic parameters distribution and dependence on log g for the R a / R ratio. Top row : Global seismic parameters for the investigated sample. Green symbols identify stars with R a / R < 0 . 85. Orange symbols identify RC stars with a mass lower than 0.75 M ⊙ . Bottom row : Dependence on log g for RGB stars in di ff erent mass bins. The dotted and dashed lines are a smoother of the sample with M > 0 . 95 M ⊙ and M < 0 . 95 M ⊙ , respectively. \n<!-- image --> \nnderstand these trends because they suggest a theoretical problem in the SBCR and / or asteroseismic radius estimates. \nThe strong agreement between radii determined by SBCRs and asteroseismology, particularly for RGB stars, is highly encouraging. These two methods di ff er fundamentally, not only in \n<!-- image --> \nFig. 5. Ratio of the radii from SBCRs and from asteroseismology as a function of the Gaia DR3 parallax. Top : Radius ratio for the Pietrzy\'nski et al. (2019) SBCR. Bottom : Same as in the top panel but for the Salsi et al. (2021) SBCR. Colour codes are as in Fig. 1. \n<!-- image --> \nthe observables they rely on but also in terms of observation time, instrumentation requirements, and budget constraints. In all of these aspects, SBCRs o ff er significant advantages, opening the possibility for obtaining large samples of asteroseismiccompatible radius estimates with minimal financial investment. \nAcknowledgements. G.V., P.G.P.M. and S.D. acknowledge INFN (Iniziativa specifica TAsP) and support from PRIN MIUR2022 Progetto "CHRONOS" (PI: S. Cassisi) finanziato dall\'Unione Europea - Next Generation EU.'}

2024arXiv240910398B

We focus on the inflationary predictions of betaexponential potential models in which the inflaton is a representation of the field delineating the size of extradimension. Since it offers a wellmotivated starting point for the study of physics at very high energies we incorporate an R2 term in the Palatini gravity. In addition afterward the inflation the inflaton oscillates about the minimum of the inflation potential and reheats the universe. This occurs during the reheating phase at which the inflaton decays into the standard model particles which fill the universe. We extend our examination by considering the reheating effects on inflationary observables by employing the different scenarios of the reheat temperature. Supposing the standard thermal history after inflation we display the inflationary predictions ns r mathrmdnsmathrmdln k of betaexponential potential with minimal coupling in Palatini R2 gravity. Also different kinds of constraints from a variety of observations such as BICEPKeck Planck 2018 as well as future possible detectable sensitivities that might be reached by CMB experiments CMBS4 and LiteBIRD are taken into account in this work. We indicate that our results are consistent with both the latest data and the future sensitivity forecasts of LiteBIRDPlanck and CMBS4. Finally the results in this study highlight the viability of our model even in the case of the existence of more stringent constraints expected from future achievable confidence level limits.

2024-09-01T00:00:00Z

['2024arXiv240910398B', 'arXiv:2409.10398', '10.48550/arXiv.2409.10398']

['Astrophysics - Cosmology and Nongalactic Astrophysics', 'General Relativity and Quantum Cosmology', 'High Energy Physics - Phenomenology']

Minimally coupled betaexponential inflation with an R2 term in the Palatini formulation

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.10398.pdf

{'Nilay Bostan a,b and Rafid H. Dejrah c': '- Department of Physics and Astronomy, University of Iowa, 52242 Iowa City, IA, USA\n- a b Proton Accelerator Facility, Nuclear Energy Research Institute,\n- Turkish Energy Nuclear and Mineral Research Agency, 06980, Ankara, Turkiye\n- c Department of Physics, Ankara University, Faculty of Sciences, 06100, Ankara, Turkiye \nE-mail: nilay.bostan@tenmak.gov.tr, rafid.dejrah@gmail.com \nAbstract. We focus on the inflationary predictions of β -exponential potential models, in which the inflaton is a representation of the field delineating the size of extra-dimension. Since it offers a well-motivated starting point for the study of physics at very high energies, we incorporate an R 2 term in the Palatini gravity. In addition, afterward the inflation, the inflaton oscillates about the minimum of the inflation potential, and reheats the universe. This occurs during the reheating phase, at which the inflaton decays into the standard model particles, which fill the universe. We extend our examination by considering the reheating effects on inflationary observables by employing the different scenarios of the reheat temperature. Supposing the standard thermal history after inflation, we display the inflationary predictions, n s , r, d n s / dln k of β -exponential potential with minimal coupling in Palatini R 2 gravity. Also, different kinds of constraints from a variety of observations, such as BICEP/Keck, Planck 2018, as well as future possible detectable sensitivities that might be reached by CMB experiments: CMB-S4 and LiteBIRD are taken into account in this work. We indicate that our results are consistent with both the latest data and the future sensitivity forecasts of LiteBIRD/Planck and CMB-S4. Finally, the results in this study highlight the viability of our model even in the case of the existence of more stringent constraints expected from future achievable confidence level limits. \nKeywords: inflation, physics of the early universe, modified gravity, Palatini formulation', '1 Introduction': "In the field of modern cosmology, there are a plethora of investigations for many concepts that assist us in gaining a deeper understanding of the universe. Amongst these nomenclatures comes 'inflation'; a period of exponential cosmic expansion of the universe's size and scale that occurred right after the big bang singularity. It is essential to advance our studies of the notion of inflation because the scientific community has focused on comprehending and characterizing some properties of the origin of the universe for many years. An excellent comprehension of the inflationary epoch is necessary to accomplish this. For the sake of example, the concept is taken into account in ref. [1] with details. Furthermore, the idea of inflation is very pivotal to explicate a legion of what were thought to be issues for cosmologists, for instance; the structure problem, smoothness problem, flatness problem, large-scale structure problem, and other issues. It is worth mentioning that the small-scale structure problem still exists, and is not solved by inflation. For more details on how the cosmic concept of inflation participated in solving such issues, see the following studies [2, 3]. Hence, inflation is considered by the majority to be a milestone for modern cosmology, especially for its merit that it can be extended to explain a wide range of other concepts; see ref. [4] for more detailed cases. Not only does it achieve that, but it also advances it to a more acceptable status by being supported by measurements from the cosmic microwave background (CMB) anisotropies, notwithstanding the ongoing discussion about related frameworks. \nOne can relate to the fact that exponential models are used to depict cosmic inflation since it is explained as the universe expanding quasi-exponentially in the early era. An exponential model is a mathematical description in which the universe's scale factor increases exponentially with time, and it is employed to represent the rapid expansion of the universe during the inflationary period, where the scale factor a ( t ) grows approximately as a ( t ) ∝ e Ht , and H is being the Hubble parameter. This exponential expansion aids in elucidating the observed large-scale homogeneity and isotropy of the universe, see the refs. [1, 5, 6]. In the literature, a variety of distinct models with different special properties, potentials, and inflation fields are taken into consideration in very detailed procedures; see refs. [7-11]. In this work, we consider the R 2 term in the Einstein-Hilbert action, which was first introduced by A. Starobinsky (ref. [6]). The R 2 term gained acceptance because it can be derived for the inflationary expansion by introducing a scalar degree of freedom known as the scalaron, which \nacts a role of the inflaton; hence, the model does not require a separate, ad-hoc inflation field to achieve inflation, making it one of the earliest and most accomplished inflationary models. Moreover, amongst the variations and principles, (it can be dissected in the refs. [12, 13]) that one can apply to the Einstein-Hilbert action to derive Einstein's equations, we are going to work in this paper with the Palatini formalism [on the contrary to the name, it was Einstein who introduced it, see ref. [14]]. Palatini formalism is defined as an independent variation with respect to the metric, an independent connection, and reduced standard deviation. It is remarkable to mention that theories based on this formalism satisfy the metric postulates [15]. The importance of the Palatini formalism is that it has been demonstrated to supply intriguing phenomenological implications. As one of the most pivotal examples, the differences between the metric and the Palatini formulations for inflationary predictions, can be given [16, 17]. According to studies in the literature, the Palatini formalism predicts the inflationary observables, especially for the tensor-to-scalar ratio ( r ) which makes it more prevailing than the metric formulation [16, 18]. Also, see refs. [17, 19-24] that delve into the inflation in Palatini formalism with details. These features can potentially offer better alignment with the measurements from CMB anisotropies and large-scale structure surveys, see the refs. [25-27]. Thus, by considering this type of formalism principle, our work can leverage the mentioned advantages and, in return, provide us with a robust and comprehensive analysis of the inflationary dynamics in the context in which our work is set. \nIn the literature, β -exponential inflation has already been considered in many studies so far. For this point of view, the pivotal study is the ref. [28], which investigates the inflationary predictions of β -exponential potential in a minimally coupled case. They depict the trajectories for disparate values of β parameters in the n s -r and d n s / dln k -n s planes for some chosen values of β and compare their findings with the cosmological data. In addition to ref. [28], there are two important studies in the literature that inspect the β -exponential inflation in detail; see the following refs. [29, 30]. Ref. [29] shows the n s -r plane for different values of β , as well as some selected values of λ , considering two different number of e-folds N ∗ : 50 and 60, and comparing the predictions of this model with the Planck data. Additionally, ref. [30] has calculated the inflationary predictions for the nonminimally coupled β -exponential inflation and compared their results with the CMB data. They present the cosmological consequences of the non-minimally coupled β -exponential inflation with details in the metric formulation. [Recent studies that also include the β -exponential potential model are as follows: [31, 32]]. The potential model of β -exponential inflation is very pivotal to take into account because the model can appear in the framework of brane cosmology at which the inflaton, see ref. [7] for the models that have been examined so far based on inflaton, is regarded as the field representing the size of extra-dimension. It is suitable to mention here that this type of potential model is derived using the braneworld scenario framework. For more details about the concept of inflation in brane cosmology, see the following refs. [29, 33]. Here, we refer to some insightful studies for a detailed analysis, regarding: \n- i) Examining the Inflating Branes Concept, see ref. [34].\n- ii) Phenomenological potentials in the 3-brane scenario, see the refs. [35, 36]. \nOn the other hand, the warm inflation scenario is investigated by the class of β -exponential potentials with details in ref. [31]. \nFurthermore, the couplings between the inflaton and the standard model (SM) particles are essential in indicating the dynamics of the reheating phase; a phase that transitions the universe from inflationary expansion to a hot, radiation-dominated era; see the following \nrefs. [37-45] for more details about the notion itself. In addition, these couplings result in the production of SM particles leading to the impact on the thermalization process and the subsequent evolution of the universe. The inflaton couples to other fields throughout the reheating phase, converting the remaining energy into new particles that make up the radiation energy density, see refs. [46-49]. Across this manuscript, we thoroughly analyze the reheating effects on the inflationary predictions within the context of our model in order to provide a comprehensive understanding of the reheating dynamics by calculating the inflationary observables for different reheat temperatures. [For details about the reheating dynamics, see the following refs. [48, 50, 51]]. Moreover, in this work, we depict inflationary predictions with reheating impacts that can have a consistency of our model with current observational data. In literature, refs. [52-54] can be examined for more details about the interactions and their effects between different models and the inflaton. Additionally, these references [41, 55, 56] provide the results for further analysis related to the reheating process and the subsequent cosmological observables. \nIn this manuscript, we study the inflationary predictions for the potential model that generalizes the well-known power law inflation, see the refs. [7, 57-60], through a general exponential function [7, 28], which is the β -exponential potential. The framework of braneworld scenarios can be used to generate this potential, which can be accurately compared with the observational data [28-30]. In this work, we specifically study this potential in the minimally coupled case with an R 2 term in Palatini formalism [61]. The specific case of the Palatini formalism lies in its ability to significantly reduce the tensor-to-scalar ratio r in regions of the parameter space [20]. Unlike the metric formulation which predicts large values of the tensor-to-scalar ratio r , which are not favored by observational data. The feature of the Palatini formulation makes it a privileged choice for inflationary models since it aligns better with current measurements from the CMB anisotropies. When comparing these two variation methods, one can see how the inflationary models are more viable with the Palatini formalism for the merits mentioned above. For more results that support this point, please see tables 2 and 3 in refs. [24, 62] which studied this case in detail. Additionally, one of the motivations to work with Palatini formulation is that the symmetries of the fundamental action are more manifest in this type of approach as discussed in ref. [63]. For this reason, we can get a more insightful framework for constructing and analyzing the inflationary models. \nWe analyze the inflationary observables of this potential and compare our results for the inflationary predictions with the current data from Planck and BICEP/Keck [64], as well as the future CMB-S4 [65] and LiteBIRD [66] sensitivity forecasts. On the other hand, it is good to mention here ref. [67], which recently indicates that the fundamental theory of gravity is a Palatini as opposed to being a metric when the gravity sector is extended by an αR 2 term (where α is a dimensionless parameter). It is worth mentioning that the inclusion of the αR 2 term is not primarily motivated by renormalizability, but rather by its ability to drive inflation and improve the ultraviolet (UV) behavior of the theory. Therefore, adding an αR 2 term assures a well-motivated starting point for the physics analysis at very high energies [68]. In literature, the studies that discuss the Palatini R 2 inflation can be listed as follows: [62, 68-75]. In addition, ref. [24] has considered the post-inflationary leptogenesis and the production of dark matter in the Palatini formalism with all details. \nThe paper is mapped as follows. In section 2, we introduce the framework we are going to work on, such as introducing the Einstein-Hilbert action that includes the R 2 term in the Palatini formalism for different frames. Moreover, we introduce the β -exponential inflation model, and its mathematical properties to provide a better understanding for our analytical \nand numerical analysis later on. We also introduce the Einstein frame potential form in the section alongside an illustrated study for a variety of choices of β to get a better image. The slow-roll parameters are provided in both the canonical scalar field ζ and in the terms of the original scalar field ϕ . The number of e-folds N ∗ is also given in two different forms for both analytical and numerical calculations, and to get better results for the latter one the reheat temperature concept T reh is also introduced, for three different scenarios. We have also introduced the brane inflation with detailed analysis in section 2, since it is connected to the context of our paper. In section 3, we show and discuss our analytical and numerical results by illustrating a thorough figure that helps visualize our analytical results and then a detailed table by which one can get a clearer image of our discussions. We have considered a variety of observational data choices as well. Finally, we summarize and conclude the paper in section 4. In the appendix section A, we provide related analytical approximations of section 3. We adopt M P to unity for our calculations 1 that we depict with details in section 3.", '2 Palatini β -exponential inflation with an R 2 term': 'We commence by acquainting the action that is taken into consideration in this work [67, 68] \nS J = ∫ d 4 x √ -g ( 1 2 M 2 P R + α 4 R 2 -1 2 ∇ µ ϕ ∇ µ ϕ -V ( ϕ ) ) , (2.1) \nwhere g is the determinant of the metric tensor g µν , J indicates that the action is given in the Jordan frame. R is the Ricci scalar, which is defined by R = g µν R µν (Γ), where R µν is the Ricci tensor derived by using the Christoffel symbols, Γ λ µν . Also, ϕ is the scalar field, called the inflaton, and V ( ϕ ) is the potential given in the Jordan frame. M P is the reduced Planck mass, and α is the dimensionless parameter. The Jordan frame action given in eq. (2.1) can be defined in terms of the auxiliary scalar field χ dynamically, as follows [67, 68]: \nS J = ∫ d 4 x √ -g ( 1 2 M 2 P ( 1 + αχ 2 ) R -α 4 χ 4 -1 2 ∇ µ ϕ ∇ µ ϕ -V ( ϕ ) ) . (2.2) \nIn addition, one can switch from the Jordan frame ( J ) to the Einstein ( E ) frame by applying a Weyl rescaling [76]. By performing the Weyl transformation of the metric with \n˜ g µν → Ω g µν , Ω ≡ 1 + αχ 2 M 2 P , (2.3) \none can write the action within the Einstein frame, resulting in [67]: \nS E = ∫ d 4 x √ -˜ g ( 1 2 M 2 P ˜ R -1 2Ω ∇ µ ϕ ∇ µ ϕ -( M P Ω ) 2 V ( ϕ, χ ) ) , (2.4) \nhere V ( ϕ, χ ) = V ( ϕ ) + α 4 χ 4 . Variation of this action given in eq. (2.4) with respect to the χ is obtained for the constraint equation [67, 68, 70, 71] \nδS E δχ = 0 → χ 2 M 2 P = 4 V ( ϕ ) + ∇ µ ϕ ∇ µ ϕ M 4 P -α ∇ µ ϕ ∇ µ ϕ . (2.5) \nSubstituting eq. (2.5) into eq. (2.4), one can obtain the form [61]: \nS E ≃ ∫ d 4 x √ -˜ g 1 2 M 2 P ˜ R -1 2 ∇ µ ϕ ∇ µ ϕ ( 1 + 4 α M 4 P V ( ϕ ) ) -V ( ϕ ) ( 1 + 4 α M 4 P V ( ϕ ) ) . (2.6) \nIt is important to mention here, by making the following field redefinition, \nd ζ = d ϕ √ 1 + 4 α M 4 P V ( ϕ ) = d ϕ √ Z ( ϕ ) , (2.7) \nthe action for a minimally coupled scalar field ζ with a canonical kinetic term can be obtained. Here, Z ( ϕ ) = 1 + 4 α M 4 P V ( ϕ ) is known as the field space metric. Eq. (2.7) can be computed with respect to the form of the specific potential models that are considered. In addition, we can indicate the Einstein frame potential from the action described in eq. (2.6) as follows: \nV E ( ϕ ) = V ( ϕ ) ( 1 + 4 α M 4 P V ( ϕ ) ) . (2.8) \nAs one can notice from eq. (2.8), the potential V ( ϕ ) is scaled by a factor as written in the denominator of the equation, which is a consequence of the existence of the term R 2 . \nIn this work, we focus on the inflationary predictions of the β -exponential potential, which we describe in the following section. We display the inflationary parameters for this potential, the spectral index n s , the tensor-to-scalar ratio r , and the running of the spectral index d n s / dln k , by supposing the standard thermal history afterward inflation, and for this potential, we present the compatible regions for the spectral index n s and the tensor-toscalar ratio r within the recent Planck + BICEP/Keck data and the future CMB-S4 and LiteBIRD/Planck achievable sensitivity forecasts. We show the cosmological consequences of β -exponential inflation with an R 2 term in Palatini formalism, presenting the results of inflationary predictions of this potential.', '2.1 β -exponential inflation': 'In this work, we study the β -exponential potential model, which was first introduced and studied in ref. [28] as a generalization of the power law inflation phenomenological [7, 57, 59, 60], which is already mentioned in the introduction. We start constructing our model with the usual exponential function [7] which is defined by \nV ( ϕ ) = M 4 exp( -λϕ/M P ) . (2.9) \nIn this work , we discuss a possible generalization for the inflation potential, which is given in eq. (2.9), with the following form: \nV ( ϕ ) = M 4 exp 1 -β ( -λϕ/M P ) , (2.10) \nwhere the definition of the generalized exponential function exp 1 -β is as follows [28, 77, 78] \nexp 1 -β ( f ) = [1 + βf ] 1 /β , (2.11) \nfor 1 + βf > 0 exp 1 -β ( f ) = 0, otherwise. \nFor f > 0 and g > 0, this function satisfies the following identities (as it is already delineated and discussed with details in [7, 28]): \nexp 1 -β [ln 1 -β ( f )] = f , \nand \nln 1 -β ( f ) + ln 1 -β ( g ) = ln 1 -β ( fg ) -β [ln 1 -β ( f ) ln 1 -β ( g )] , \nwhere ln 1 -β ( f ) = ( f β -1) /β is the generalized logarithmic function. The β -exponential potential can fulfill the disruption of the slow-roll regime with the end of inflation [31], thus it makes the tiny values for the tensor-to-scalar ratio, r ([28, 31]). \nThe Jordan frame potential, V ( ϕ ), for the β -exponential inflation can be written in the following form 2 , where the constraints provided by eq. (2.10) are taken into consideration [79]: \nV ( ϕ ) = V 0 ( 1 -λβ ϕ M P ) 1 /β , (2.12) \nwhere the deviation from the pure exponential function is controlled by constant β , while λ is a dimensionless constant. \nIn this work, with the form of eq. (2.8), we delve into the Einstein frame minimally coupled potential for the β -exponential inflation with an R 2 term in the Palatini formalism. By using eq. (2.12), we can define the potential model in the Einstein frame as follows: \nV E ( ϕ ) = V 0 ( 1 -λβ ϕ M P ) 1 /β ( 1 + 4 α V 0 ( 1 -λβ ϕ M P ) 1 /β M 4 P ) . (2.13) \nIn figure 1, we illustrate how the Einstein frame β -exponential potential, which is given in eq. (2.13) changes according to the values of the β parameter, which we select and how this parameter controls the potential model, as well as deviation from the usual exponential function. \nS 4 = ∫ d 4 x √ -g 4 ( 1 2 σ ˙ L 2 -V eff ( L ) ) , \nwhere σ is the brane tension, L is the position of the brane with respect to r = 0, here r is the fifth coordinate of the 5D bulk. Also, the effective potential is defined as V eff ( L ) = V 0 (1 + c 1 L ) 1 λc 1 + 1 2 σ . In addition, with some changes in the definitions of the effective potential, V eff ( L ), the equivalent form of the β -exponential potential, which is presented in eq. (2.12) can be found, see ref. [29]. According to the studies refs. [29, 30], it can be inferred that the brane tension σ is related to the ratio β/λ . Both β and λ are constrained by this connection; that is, β must be greater than λ , with β ≥ 1 / 2. Thus, considering the β -exponential potential models is highly motivated for the context of braneworld scenarios and brane dynamics [30]. For more details thoroughly, please see [29, 30]. \n8 \nFigure 1 . The Einstein frame β -exponential potential with minimal coupling in Palatini R 2 gravity as a function of ϕ . The colors show different values of the β parameter. We fixed λ = 0 . 1, α = 10 8 , V 0 = 10 -9 , as well as taken M P unity. \n<!-- image -->', '2.2 Inflationary observables': "As long as the Einstein frame potential can be obtained in terms of the canonical scalar field ζ , inflationary predictions can be calculated by exploiting the slow-roll parameters as follows [80]: \nϵ = M 2 P 2 ( V ζ V ) 2 , η = M 2 P V ζζ V , κ 2 = M 4 P V ζ V ζζζ V 2 , (2.14) \nwhere ζ in the subscript denotes the derivatives with respect to the canonical scalar field ζ . Here, ϵ is the parameter that measures the steepness of the inflationary potential, and it is an indicator of how quickly the scalar field ζ is rolling down potential. The lower values of ϵ results in slower roll and more sustained inflation. η , on the other hand, is the parameter that describes the slope of the potential changes that affect the stability and duration of inflation. Moreover, κ 2 investigates finer details about the shape of potential and the dynamics of the inflation, in instances refs. [81-83] have studied these parameters in detail. Inflationary observables, for example, the spectral index n s , the tensor-to-scalar ratio r , and the running of the spectral index d n s / dln k are given in the following forms for the slow-roll approximation: \nn s = 1 -6 ϵ +2 η , r = 16 ϵ, d n s dln k = 16 ϵη -24 ϵ 2 -2 κ 2 . (2.15) \nRecently, more precise constraints on the inflationary predictions have been provided by BICEP/Keck [64], especially for the tensor-to-scalar ratio r , which tightens to r < 0 . 035 at 95% CL. This strong constraint elucidates the amplitude of the primordial gravitational waves, as well as the inflationary scale. Moreover, recent BICEP/Keck results also constrain the spectral index n s to the range [0 . 957 , 0 . 976] at 2 σ of confidence level. These constraints are with the pivot scale, which is selected at k ∗ = 0 . 002 Mpc -1 . In addition, the next generation \nof CMB surveys, such as CMB-S4 [65], which aims for r ≃ O (10 -3 ). Also, the future measurement from the LiteBIRD experiment [66] will be able to test the inflationary models precisely. Moreover, another pivotal constraint arises from the Planck 2018 measurements along with the results from the baryon acoustic oscillations (BAO). They provide the constraint on d n s / dln k = -0 . 0041 ± 0 . 0067 to base ΛCDM in 68%, TT,TE,EE +lowE+lensing+BAO [84]. In the future, some improvements are expected from the observations of the 21-cm line, [8587]. In addition, for the case of the tensor-to-scalar ratio r > 0 . 003, at larger than 5 σ of confidence level, primordial gravitational waves can be detectable in the future by CMBS4 [65]. The highest limit of the tensor-to-scalar ratio r < 0 . 001 at 95% CL may be reached through future observations done by CMB-S4, even in the absence of a detection, this limit would still provide important new insights for the inflation [65]. \nThe number of e-folds N ∗ in the slow-roll approximation is given by \nN ∗ = 1 M 2 P ∫ ζ ∗ ζ e V d ζ V ζ , (2.16) \nwhere the subscript ' ∗ ' denotes the quantities when the pivot scale exits the horizon, and ζ e is the value of the inflaton at which the inflation ends, we can compute ζ e via ϵ ( ζ e ) = 1. \nFurthermore, the amplitude of the curvature perturbation can be calculated by using the following relation: \n∆ R = 1 2 √ 3 πM 3 P V 3 / 2 | V ζ | , (2.17) \nthe best fit value for the pivot scale k ∗ = 0 . 002 Mpc -1 is ∆ 2 R ≈ 2 . 1 × 10 -9 [84], which is obtained from the Planck measurements. \nOn the other hand, it may not always be easy or possible to obtain the analytical expression of an inflation potential defined as V J ( ϕ ) in the Jordan frame as V E ( ζ ) in the Einstein frame. This depends on the form of the inflation potential which is taken into account. In this case, the analysis of its predictions for the considered potential can be made in terms of the original scalar field ϕ numerically instead of the canonical scalar field ζ . Further analysis related to ζ is mentioned in detail in section 3. We use such equations to perform numerical calculations in terms of the scalar field ϕ when necessary. Furthermore, for the numerical calculations, one needs to have the slow-roll parameters in terms of the field ϕ to be able to calculate the inflationary predictions of the potential model in terms of the general values of free parameters. Thus, the slow-roll parameters should be acquired in terms of the scalar field ϕ , and these parameters can be written as follows [88]: \nϵ = Zϵ ϕ , η = Zη ϕ +sgn( V ' ) Z ' √ ϵ ϕ 2 , κ 2 = Z ( Zκ 2 ϕ +3sgn( V ' ) Z ' η ϕ √ ϵ ϕ 2 + Z '' ϵ ϕ ) , (2.18) \nwhere the slow-roll parameters are defined in terms of ϕ as the following: \nϵ ϕ = 1 2 ( V ' V ) 2 , η ϕ = V '' V , κ 2 ϕ = V ' V ''' V 2 . (2.19) \nHere ' ' ' represents the derivatives with respect to the original scalar field ϕ . Furthermore, eqs. (2.16) and (2.17) can be written with regard to ϕ resulting in the following forms: \nN ∗ = sgn(V ' ) ∫ ϕ ∗ ϕ e d ϕ Z( ϕ ) √ 2 ϵ ϕ , (2.20) \n∆ R = 1 2 √ 3 π V 3 / 2 √ Z | V ' | . (2.21) \nTo calculate the numerical values of observables; the spectral index n s , the tensor-to-scalar ratio r , and the running of the spectral index d n s / dln k , the numerical value of the number of e-folds N ∗ is also required. Supposing a standard thermal history afterward inflation, one can express the number of e-folds N ∗ for the pivot scale k ∗ = 0 . 002 Mpc -1 in the following form [89]: \nN ∗ ≈ 64 . 7 + 1 2 ln ρ ∗ M 4 P -1 3(1 + ω r ) ln ρ e M 4 P + ( 1 3(1 + ω r ) -1 4 ) ln ρ r M 4 P , (2.22) \nhere, ρ e = (3 / 2) V ( ϕ e ) is the energy density at the end of inflation, ρ r is the energy density at the end of reheating, and ρ ∗ ≈ V ( ϕ ∗ ) is the energy density when the scale corresponding to k ∗ exits the horizon. ω r is the equation of the state parameter during reheating. The definitions of ρ r and ρ ∗ are given in the following forms: \nρ r = ( π 2 30 g ∗ ) T 4 reh , ρ ∗ = 3 π 2 ∆ 2 R r 2 , (2.23) \nwhere the standard model value g ∗ = 106 . 75, which gives the number of relativistic degrees of freedom, can be employed to compute ρ r . Also, T reh indicates the reheat temperature; the temperature at which the universe is in thermal equilibrium and radiation dominates. At the end of the reheating phase, thermal equilibrium is reached and the universe is fully filled with radiation [90]. The inflation potential represents the majority of the universe's energy density during inflation. When the potential steepens and the inflation field (inflaton) gains kinetic energy, inflation terminates. The SM particles then need to receive the energy from the inflaton sector. In addition, the hot big bang is initiated by reheating. The inflaton's decay produce a particle soup that eventually approach thermal equilibrium, with the radiation and particle fields present at the time, ensuring that energy is uniformly distributed across the universe's constituents [91], at a certain temperature, reheat temperature, as a result of particle interactions. The energy density ρ r at the end of the reheating period gives this reheat temperature. For much more details and highlights on the thermal history of the universe and reheating, please see the following refs. [90, 92, 93]. Moreover, here are the detailed studies related to the reheating concept and constraints on inflationary predictions [94, 95]. \nIn this work, we consider two different cases for the number of e-folds N ∗ which is defined in eq. (2.22): \n- · First case: We take w r = 1 / 3, it is the instant reheating assumption. With this assumption, the number of e-folds N ∗ in eq. (2.22) reduces to the following form: \nN ∗ ≈ 64 . 7 + 1 2 ln ρ ∗ M 4 P -1 4 ln ρ e M 4 P . (2.24) \nEquation (2.24) demonstrates that the inflationary predictions should not depend on T reh for the instant reheating assumption. \n- · Second case: We take w r = 0, and with this selection, the number of e-folds N ∗ should depend on the reheat temperature since the potential does not have a minimum, then the reheating cannot happen in a standard way. In our numerical calculations, we take T reh = 10 8 GeV and T reh = 10 14 GeV in the following section. Ref. [74] has studied the reheating mechanisms for the quintessential inflation in Palatini R 2 gravity, they have found the maximum reheating temperature to be T reh = 10 14 GeV, accordingly in this study we select 10 14 GeV as the highest reheat temperature, as well as we show the predictions for the T reh = 10 8 GeV because, at this specific limit, our model works and consistent with the cosmological data in a good manner. It is also important to note that here, in the second case, the value of the e-folds number is less than the ones for the case when w r = 1 / 3 (instant reheating). By taking w r = 0, eq. (2.22) becomes: \nN ∗ ≈ 64 . 7 + 1 2 ln ρ ∗ M 4 P -1 3 ln ρ e M 4 P + 1 12 ln ρ r M 4 P . (2.25) \nIt is clear that the second case depends on the reheat temperature due to the existence of the term ρ r in the expression itself. It is concluded that equation (2.25) can be calculated by taking different values of T reh so that one can see the relation between reheat temperature and the number of e-folds N ∗ , as well as its effects on inflationary predictions. Throughout the next section, we will present the inflationary predictions of the Einstein frame β -exponential potential with minimal coupling in Palatini R 2 gravity supposing the standard thermal history afterward inflation. We first present our results analytically with rough approximations as an example, as well as we show the inflationary predictions numerically for both first (instant reheating) and second ( T reh = 10 8 GeV and T reh = 10 14 GeV) cases for the number of e-folds, N ∗ that we describe above.", '3 Results and Discussion': "We inaugurate this section by expressing the canonically normalized field ζ with respect to the original scalar field ϕ . By using eq. (2.7), the canonical scalar field ζ ( ϕ ) can be found for the β -exponential inflation, which is given in eq. (2.12) as follows: \nd ζ = d ϕ √ 1 + 4 αV 0 M 4 P ( 1 -λβ ϕ M P ) 1 /β , (3.1) \nto able to compute and solve this differential equation, we integrate both sides: \nζ ( ϕ ) = ∫ d ϕ √ 1 + 4 αV 0 M 4 P ( 1 -λβ ϕ M P ) 1 /β . (3.2) \nLet γ ≡ 1 -λβ ϕ M P , then the integral becomes: \nζ ( γ ) = -M P λβ ∫ d γ √ 1 + 4 αV 0 M 4 P γ 1 /β (3.3) \nFigure 2 . Plot of ζ ( ϕ ) vs. ϕ . We set the values as follows: V 0 = 10 -9 , α = 10 8 , λ = 0 . 1, β = 0 . 5 ( M P is set to 1). \n<!-- image --> \nThis integral can be evaluated in terms of a hypergeometric function. We obtain the final result by inserting γ ≡ 1 -λβ ϕ M P , as follows: \nζ ( ϕ ) = ( βλϕ -M P ) × 2 F 1 ( 1 2 , β ; β +1; -4 V 0 α ( 1 -ϕβλ M P ) 1 /β M 4 P ) βλ , (3.4) \nwhere 2 F 1 ( a, b ; c ; z ) is the hypergeometric function. In order to illustrate the behavior of the function of ζ ( ϕ ), we compute it numerically as a function of ϕ . The resultant plot is given in figure 2, by setting V 0 = 10 -9 , α = 10 8 , λ = 0 . 1, β = 0 . 5 and M P = 1. \nIt is important to emphasize that the expression of our potential model in terms of ϕ ( ζ ) is not straightforward. As in our case, some inflationary potentials are difficult to express in terms of the canonical scalar field ζ with the exact form due to the complex structures of the potentials, for this case, the analytical approximations for the inflationary potentials are required. For further analytical approximations for our model in this study, please see the appendix section A. In addition, because of this situation, for the general values of free parameters in the potential, we compute the inflationary predictions of our model through the numerical techniques with respect to the original scalar field ϕ to find the inflationary observables more precisely. \nNext, we analyze the inflationary parameters in the slow-roll approximation. Throughout the subsequent analysis in this section, M P will be set to unity. By using eq. (2.18), the slow-roll parameters, ϵ ( ϕ ∗ ) and η ( ϕ ∗ ), can be found for the Einstein frame minimally coupled \nβ -exponential potential in Palatini R 2 gravity, which is defined in eq. (2.13) as follows: \nϵ ( ϕ ∗ ) ≃ λ 2 8 αV 0 x 1 β +2 +2 x 2 , η ( ϕ ∗ ) ≃ λ 2 ( -2 β + 3 4 αV 0 x 1 /β +1 -1 ) 2 x 2 , (3.5) \nwhere 1 -βλϕ ∗ ≡ x . Also, by utilizing eq. (2.15), the main observational parameters, the spectral index n s and the tensor-to-scalar ratio r , can be acquired analytically for the β -exponential potential as follows: \nn s ( ϕ ∗ ) ≃ 1 -(2 β +1) λ 2 x 2 , r ( ϕ ∗ ) ≃ 8 λ 2 x 2 ( 4 αV 0 x 1 /β +1 ) . (3.6) \nAlso, by using eq. (2.21), the amplitude of the curvature perturbation can be found with the form: \n∆ 2 R ( ϕ ∗ ) ≃ V 3 0 x 3 /β 12 π 2 ( 4 αV 0 x 1 /β +1 ) 4 ∣ ∣ ∣ ∣ x 1 β -1 λV 0 ( 4 αV 0 x 1 /β +1 ) 2 ∣ ∣ ∣ ∣ 2 . (3.7) \nIn our numerical analysis, we take into account the slow-roll conditions to calculate ϕ e by using ϵ ( ϕ e ) = 1 in eq. (2.18), and to compute ϕ ∗ we use the CMB constraint by using eq. (2.21) with ∆ 2 R ( ϕ ∗ ) ≈ 2 . 1 × 10 -9 , as well as with these field values, then we set that 50 ≲ N ∗ ≲ 60 should be satisfied for both the eq. (2.20) and (2.22). It is also important to note that in the case of instant reheating, N ∗ ∼ 55 -60. On the other hand, since the reheat temperature is included in the e-fold expression for the w r = 0 case, the number of e-folds varies depending on the reheating temperature. We can highlight that as the reheat temperature decreases, the number of e-folds decreases accordingly. For instance, for T reh = 10 8 GeV, N ∗ ∼ 50 -55. \nFurthermore, regarding eq. (3.6), it is important to mention that for the β -exponential potential in minimal coupling with an R 2 term in Palatini formalism, even though this is not the case for the spectral index n s predictions, tensor-to-scalar ratio r depends on the α parameter significantly, which is confirmed by our numerical results shown in table 1, and related more analysis can be acquired through studying figure 7 in a specific range of α values, we will elaborate on this point during the analysis of the related figures. From the analytical results that are given by eq. (3.6), it can be mentioned that the predictions of the tensor-to-scalar ratio r should decrease as the α parameter increases. In addition, the number of e-folds N ∗ is obtained by using eq. (2.20) for our inflationary model in the following form: \nN ∗ ≃ ϕ ∗ ( βλϕ ∗ -2) 2 λ . (3.8) \nOne can find the spectral index n s , the tensor-to-scalar ratio r , and the amplitude of the curvature perturbation expressions given in eqs. (3.6) and (3.7) in terms of the number of e-folds N ∗ for the β -exponential potential by considering different kinds of approximations. For instance, let us assume βϕ 2 ∗ / 2 ≫ ϕ ∗ /λ in eq. (3.8), then we can obtain x ≈ 1 ∓ ( λ √ 2 βN ∗ ). If one inserts this into the equations (3.6) and (3.7), the predictions can be acquired in terms \nFigure 3 . Predictions for the n s -r parameter space for the selected parameters of minimally coupled β -exponential inflation with an R 2 term in Palatini formalism, where α varies in the range of [10 7 -10 15 ]. Purple (dot-dashed)(dashed) contours indicate the recent 95%(68%) CL given by BICEP/Keck [64], while the magenta (dot-dashed)(dashed) lines correspond to the prospect of future CMB-S4 constraints [65]. Blue (dot-dashed)(dashed) lines represent the 95%(68%) CL upper limits achievable with LiteBIRD/Planck [66] in the future. \n<!-- image --> \nof the number of e-folds N ∗ as follows: \nn s ≃ 1 -(2 β +1) λ 2 ( 1 ∓ ( λ √ 2 βN ∗ ) ) 2 \n, r ≃ 8 λ 2 ( 1 ∓ ( λ √ 2 βN ∗ ) ) 2 ( 4 αV 0 ( 1 ∓ ( λ √ 2 βN ∗ ) ) 1 /β +1 ) , (3.9) \n∆ 2 R ≃ V 3 0 (1 ∓ ( λ √ 2 βN ∗ )) 3 /β 12 π 2 ( 4 αV 0 (1 ∓ ( λ √ 2 βN ∗ )) 1 /β +1 ) 4 ∣ ∣ ∣ ∣ (1 ∓ ( λ √ 2 βN ∗ )) 1 β -1 λV 0 ( 4 αV 0 (1 ∓ ( λ √ 2 βN ∗ )) 1 /β +1 ) 2 ∣ ∣ ∣ ∣ 2 . (3.10) \nIn addition, for the case of βλϕ ∗ ≪ 1, one can find x ≈ 1 + λ 2 βN ∗ . With this approximation, the predictions can be obtained in terms of the number of e-folds N ∗ analytically as follows: \n2 \nn s ≃ 1 -(2 β +1) λ (1 + λ 2 βN ∗ ) 2 , r ≃ 8 λ 2 (1 + λ 2 βN ∗ ) 2 ( 4 αV 0 (1 + λ 2 βN ∗ ) 1 /β +1 ) . (3.11) \n∆ 2 R ≃ V 3 0 (1 + λ 2 βN ∗ ) 3 /β 12 π 2 ( 4 αV 0 (1 + λ 2 βN ∗ ) 1 /β +1 ) 4 ∣ ∣ ∣ ∣ (1+ λ 2 βN ∗ ) 1 β -1 λV 0 ( 4 αV 0 (1+ λ 2 βN ∗ ) 1 /β +1 ) 2 ∣ ∣ ∣ ∣ 2 . (3.12) \nIt is also worth emphasizing that these analytical approaches are rough approximations, these are considered for our model in this work for the sake of presenting how the expressions can be written approximately in terms of the number of e-folds in general case. As for the next step, we will begin by discussing our numerical results. It is important to mention that with the assumption of the standard thermal history after inflation, we use eq. (2.22) for our numerical calculations. Throughout our numerical computations for the inflationary predictions, for each steps, it was also taken into consideration that the values calculated from equation (2.22) should be very close, almost similar values to the values computed from equation (2.20). The consistency of the results of these two equations ensures that the calculated inflationary parameters are more reliable with high precision. \nOur examination in figure 3 takes into consideration different options of the reheating scenarios in order to have a better image of our model's consistency, which leads to a both better and more accurate analysis. We take the highest T reh = 10 14 GeV, as well as take the lower value, T reh = 10 8 GeV to show the differences of reheating effects on inflationary predictions, the instant reheating scenario is included as well. Moreover, one can relate the consistency of the plot to the equations derived and presented in this work. As ϵ decreases, one can see that the predicted values of the tensor-to-scalar ratio r decrease as well, and this leads to lines with different slopes. The positions of the lines in this plot imply the sensitivity of the tensor-to-scalar ratio r and the spectral index n s to parameters as β, λ, and the reheating scenarios. Comparing the constrained predictions obtained from our model to the observational data, one can spot how our model fits the data right and well, especially given that the solid lines approach the best-fit regions of the contours, which makes our model viable even for the sensitivity forecasts for the future CMB-S4 and LiteBIRD/Planck. Additionally, this figure shows the essential importance of taking into consideration multiple missions, such as CMB-S4 and LiteBIRD/Planck, since both can provide enhanced future sensitivity on the inflationary parameters. As one can observe from this figure, the magenta, and blue data contours represent the 95% and 68% confidence levels for the achievable upper limits in the future; hence, from these missions, we will be able to get further robustness that is in alignment with our model. It is worth mentioning that the consistency with the CMB-S4 and LiteBIRD/Planck predictions highlights the our potential model to be highly viable with the upcoming observational data. \nFurthermore, ref. [30] has examined the β -exponential inflationary model for both minimally and non-minimally coupled scalar fields with gravity. They have shown a n s -r plane for both cases. In particular, for the minimally coupled case, they have found their model predictions of n s -r are in good agreement at 2 σ CL when using Planck 2015 data but for the most recent data from Planck 2018 + BAO measurements, the agreement between their results and the recent data is lost. On the contrary, our results in this work show that the inflationary predictions for the β -exponential inflation with minimally coupled in Palatini R 2 gravity can have a good agreement within the recent Planck + BICEP/Keck data even in 1 σ CL region. Also, for the non-minimal coupling case, ref. [30] has indicated the agreement between their results and recent cosmological data for some of the selected non-minimal coupling parameters. \nIn addition, figure 4 depicts the relation between the tensor-to-scalar ratio r and the parameter α for different values of β, λ , and the reheat temperature T reh cases. It can be noticed from this figure that the tensor-to-scalar ratio r decreases as α values increase for all sets of β, λ, and T reh . This kind of behavior is consistent with the expectation that for larger α values, we can spot more suppressed tensor modes, resulting in lower tensor-to-scalar \nFigure 4 . α -r plane for the selected parameters of minimally coupled β -exponential inflation with an R 2 term in Palatini formalism. \n<!-- image --> \nratio r values. On the other hand, for lower values of β , in a specific choice of α , we can notice higher values of the tensor-to-scalar ratio r as well, reflecting in the sensitivity of the inflationary dynamics to this mentioned parameter. Additionally, we cannot ignore the impact of the reheat temperature T reh choice or case as well, for that in some given β and λ values; instant reheating scenario leads to slightly different tensor-to-scalar ratio r values compared to the scenario of T reh = 10 8 GeV. This difference can be interpreted due to the interplay between the reheating phase and the dynamics of the scalar field ϕ during inflation. \nAlso, figure 4 shows that for lower values of β , the tensor-to-scalar ratio r tends to be higher an inverse relation ; however, it has a positive correlation with the given values of λ . This type of behavior of the curves aligns with what one can expect that larger deviations from the standard inflationary potential reflect in a higher tensor-to-scalar ratio r . Furthermore, the curves also illustrate the sensitivity of the tensor-to-scalar ratio r to the post-inflationary reheating phase, which is captured by different values of the number of e-folds N ∗ , and this is very important since it points out to the fact that the inflationary predictions depend on the dynamics of the scalar field ϕ , the slow-roll parameters, and the reheating phase. For the considered scenario of T reh = 10 14 GeV, the tensor-to-scalar ratio r exhibits slightly higher values when we compare it to the instant reheating case, particularly at lower α values. This behavior further highlights the sensitivity of the inflationary predictions to the reheating phase, where a higher T reh can lead to a more pronounced tensor mode contribution, thus slightly elevating the tensor-to-scalar ratio r . \nFor figure 5, we can comment with the following. The top panel presents the relation between the number of e-folds N ∗ and the parameter α for selected values of β and λ . The figure shows that as the parameter α increases, the number of e-folds N ∗ decreases for each set of parameters, indicating that a stronger R 2 term reduces the duration of inflation. This reduction is more pronounced for lower β values, reflecting the dependence of the inflationary dynamics on the shape of the potential, as β controls the steepness of the potential. The effect of the dimensionless parameter α in the top plot is subtle and does not play a major role when one compares its relation with the slow-roll parameters, which is stronger as we can notice from the other figures (e.g., figure 4), and this explains why there is no direct and obvious dependence of the number of e-folds on the parameter α in eq. (3.8). Hence this shows the alignment between our analytical and numerical results, which makes our \nFigure 5 . For the minimally coupled β -exponential inflation with an R 2 term in Palatini formalism, the top panel depicts α -N ∗ plane for the selected parameters, as well as bottom panel shows how the N ∗ -n s predictions change depending on the α parameter with β = 0 . 5 and λ = 0 . 1 for the instant reheating assumption. \n<!-- image --> \npaper more viable. The bottom panel examines how the spectral index n s varies with the number of e-folds N ∗ for the fixed values of β = 0 . 5 and λ = 0 . 1, under the assumption of instant reheating. The color gradient represents different values of the logarithmic scale of α . Additionally, the figure illustrates that as the number of e-folds N ∗ increases, the spectral index n s also increases slightly. In the case of T reh = 10 14 GeV, the spectral index n s exhibits slightly higher values for the same number of e-folds N ∗ with comparison to the instant reheating scenario, indicating that a higher reheating temperature can lead to a less tilted curve; hence, affecting the inflationary predictions. \nMoreover, figure 6 represents the tensor-to-scalar ratio r as a function of the number of e-folds N ∗ for various sets of β and λ . Different curves correspond to the different reheating scenarios. The curves generally show a decreasing trend of the tensor-to-scalar ratio r as the number of e-folds N ∗ increases. For the higher number of e-folds N ∗ values, the tensor-to-scalar ratio r tends to stabilize, showing less variation, which is consistent with the expectation that as the inflationary phase progresses, the contributions of the number of e-folds N ∗ to the tensor-to-scalar ratio r diminish. The comparison between instant reheating \nFigure 6 . N ∗ -r plane for the selected parameters of minimally coupled β -exponential inflation with an R 2 term in Palatini formalism. \n<!-- image --> \nFigure 7 . The plot depicts how the n s -r predictions change depending on the α parameter for the β -exponential inflation with an R 2 term in Palatini formalism with β = 0 . 5 and λ = 0 . 1 for the instant reheating assumption. \n<!-- image --> \nand reheating at T reh = 10 8 GeV illustrates that lower reheating temperatures generally lead to slightly lower values of the tensor-to-scalar ratio r for the same number of e-folds N ∗ . This is consistent with the expectation that a longer reheating phase (lower T reh ) allows for additional red-shifting of the tensor modes, reducing the tensor-to-scalar ratio r . As shown from the figure, the scenario at which T reh = 10 14 GeV results in slightly higher values of the tensor-to-scalar ratio r compared to the lower reheating temperature scenarios, for the same number of e-folds N ∗ . This indicates that a higher reheating temperature reduces the duration of the reheating phase, leading to less red-shifting of tensor modes and thus a higher tensor-to-scalar ratio r . \nIn figure 7, we take advantage of the color gradient technique to indicate the continuous dependence of the tensor-to-scalar ratio r on both the spectral index n s and the parameter \nTable 1 . The inflationary parameter sets of approximate values for the minimally coupled β -exponential inflation with an R 2 term in Palatini formalism. The table presents essential inflationary observables, the scalar spectral index n s , the tensor-to-scalar ratio r , and the running of the spectral index d n s / dln k evaluated at the horizon exit of the pivot scale, ϕ ∗ . The values of ϕ ∗ and ϕ e correspond to the field values at horizon exit and at the end of inflation, respectively. Two distinct cases for the parameter α are considered: α = 10 8 and α = 10 15 , with λ = 0 . 1 and β = 0 . 5 fixed for both cases. The results for different values of N ∗ , the number of e-folds, are shown to remark the dependence of the inflationary observables (note that the results of predictions in the table were calculated by taking M P set to unity). \nα , which provides us with a good alignment with what we have already expected analytically earlier in this section. From this figure, a positive correlation between the tensor-to-scalar ratio r and the spectral index n s can be noticed, which is something typical of many inflationary models. However, as the spectral index n s gets larger, the curve starts to behave in a more flat manner, and this can indicate that the sensitivity of the change of the tensor-to-scalar r with respect to the spectral index n s becomes less sensitive to changes in α . One can refer to this last sentence as output to the slow-roll parameters ϵ and η stabilizing in this regime, which is something that can be expected. Looking at the ranges of the tensor-to-scalar ratio log 10 r , which is produced by our model, we can see their agreement with the observational data. \nLastly, we also show our results in table 1, which is a comprehensive table regarding three different parameters of α, λ, and β , and the results presented here have been checked by running the calculations through a numerical technique to obtain a very valid set of results. It is worth mentioning that these choices of the parameter values presented in the table are the ones at which the numerical results are significantly important. From this table, we can mention that for the selected values of e-folds, 45 , 55, and 61, the spectral index n s predictions remain similar for the larger and smaller values of α parameter, but the change in the r values is quite large which is something we expect after observing our both results from analytical approximations and figure 7. Thus, it can be concluded that the β -exponential inflation with an R 2 term in Palatini formalism can be well aligned with the recent cosmological data for the larger α values, which makes the inflationary model compatible with the data. We can see that as the parameter α value increases, the tensor-to-scalar ratio r value becomes very tiny. In addition, from our results, we can mention that the spectral index n s values generally show a change depending on reheating scenarios. It is pivotal to note that here for the larger \nnumber of e-folds N ∗ , larger spectral index n s values are obtained. Also, the predictions of the running of the spectral index d n s / dln k do not alter so much with the α values. Lastly, we find the running of the spectral index d n s / dln k predictions are very tiny to be observed at least in the near future.", '4 Summary and conclusions': "In this work, we have studied the minimally coupled β -exponential inflation with an R 2 term in Palatini formalism. For this model, we calculate the inflationary predictions thoroughly, as well as compare them with the recent cosmological data and sensitivity prospects of future CMB experiments: CMB-S4 and LiteBIRD. \nThe analysis provided in this paper explores the impact of the reheating dynamics, particularly on the spectral index n s , and the tensor-to-scalar ratio r , across a variety of reheat temperature scenarios. It is important to remark that higher values of reheat temperature are specifically favorable for non-thermal dark matter production and leptogenesis since higher temperatures increase the chances of interactions and processes that are essential for the occurrence of these phenomena. In addition, we have shown that the inclusion of the reheating impacts gives us clearer information about the effects of the reheating on the inflationary predictions, which not only refines our model's predictions for the inflationary models but also aligns our results closely with the observational constraints from current data, and makes it sensitive for future achievable upper limits forecasts by CMB-S4 and LiteBIRD/Planck, as one can note from figure 3. \nWe have found that our considered potential model can be aligned and in good agreement with the recent cosmological data for larger α values, which makes our model consistent with the data, taking very tiny r values, by reaching r ∼ 10 -9 . This numerical result is in good agreement with the results we have found analytically for this case. In addition, this result is consistent with the studies in the literature as we discussed in the previous section. We have also discussed our results by considering different reheat temperature scenarios. We show that depending on the values of reheat temperature, the difference appears in the spectral index n s predictions for the selected parameters in our inflationary model. In addition, we display the predictions of the running of the spectral index d n s / dln k are too small for our model. \nIn addition, it is important to note that the studies in literature for the β -exponential inflation models that are taking into consideration the different scenarios and gravity models are still ongoing, thus it is considered that our results in this work will be very important to be depicted to build a bridge between different inflationary scenarios, gravity theories and braneworld cosmological frameworks regarding this type of potential model. The β -exponential potential models can be obtained through the braneworld scenario framework, and considering these kinds of potential models within the context of the braneworld scenarios is strongly motivated for the primordial inflation. Additionally, it is essential to notice that the potential described in this work is an effective one, which becomes truncated when the arguments of the β -exponential function turn negative. This kind of truncation reflects the limit of the effective description, as the model does not describe a physical relevance at this point. Therefore, this limitation should be highly considered when interpreting the behavior of the potential, specifically when analyzing the implications for the early universe dynamics. \nLast but not least, while the analysis followed in this paper is based around the β -exponential model, the methodology and the analysis we employed for the predictions of \nCMB observables and the reheating dynamics are generic and can be applied to a broad range of classes for the inflationary models. For this and the good alignment of our model with the observables, one can agree on the good viability of the model in this paper.", 'A Analytical approximations of ζ ( ϕ )': "In this section, we provide a more detailed analysis for finding the inverse relation of the expression ζ ( ϕ ), which is mentioned with further text in eq. (3.4). The main challenge comes from the involvement of the hypergeometric function 2 F 1 , which complicates the computation of finding the inversion of ζ ( ϕ ). Hence, we provide an approximation to find this inversion by employing different approaches to illustrate this issue mathematically, then we depict the results with figures that help us in getting a grasp of our analytical results. \nMoreover, this appendix contains a detailed examination of how the field ϕ evolves with respect to ζ , which is essential for understanding the slow-roll parameters and the overall inflationary phase. Through obtaining ϕ ( ζ ) with the related approximations, we can better analyze the behavior of the potential for the slow-roll regime, which is important for evaluating the number of e-folds N ∗ as mentioned in details in section 3, and the circumstances under which the inflation ends. \nTo find the inverse ϕ ( ζ ) analytically from the expression for ζ ( ϕ ), we employ equation (3.4): \nζ ( ϕ ) = ( βλϕ -M P ) × 2 F 1 ( 1 2 , β ; β +1; -4 V 0 α ( 1 -ϕβλ M P ) 1 /β M 4 P ) βλ . (A.1) \nIt is important to mention here finding the inverse ϕ ( ζ ) analytically is challenging because this expression involves the hypergeometric function 2 F 1 ( a, b ; c ; z ), which is not easily invertible in the closed form. However, we can proceed as follows: as the initial step, we express ϕ as a function of ζ . \nFor clarity, let us define: \nν ≡ βλϕ M P , (A.2) \nthen, the equation (A.1) becomes: \nζ ( ν ) = M P ( ν -1) × 2 F 1 ( 1 2 , β ; β +1; -4 αV 0 (1 -ν ) 1 /β M 4 P ) βλ . (A.3) \nSince the equation for ζ ( ν ) is not easily invertible in closed form due to the presence of the hypergeometric function [96], which can be defined by power series for | z | < 1 as follows: \n2 F 1 ( a, b ; c ; z ) = 1 + ab c z + a ( a +1) b ( b +1) 2 c ( c +1) z 2 + · · · , (A.4) \nand using this series expansion, we can approximate: \n2 F 1 ( 1 2 , β ; β +1; -4 αV 0 (1 -ν ) 1 /β M 4 P ) ≈ 1 -2 αV 0 β (1 -ν ) 1 /β M 4 P ( β +1) + O ( (1 -ν ) 2 /β M 8 P ) . (A.5) \nFigure 8 . Plot of ϕ ( ζ ) vs. ϕ . We set the values as follows: V 0 = 10 -9 , α = 10 8 , λ = 0 . 1, β = 0 . 5 ( M P is set to 1). \n<!-- image --> \nSubstituting this approximation back into the expression of ζ ( ν ) in eq. (A.3), we can get the form for the leading-order term as follows \nζ ( ν ) ≈ M P ( ν -1) [ 1 -2 αV 0 β (1 -ν ) 1 /β M 4 P ( β +1) ] βλ . (A.6) \nNow, we aim to invert this equation for ν ( ζ ), which corresponds to finding ϕ ( ζ ). Let us first isolate the leading-order term in ν by neglecting higher-order terms in the series expansion: \nζ ( ν ) ≈ M P ( ν -1) βλ . (A.7) \nRecall that ν ≡ βλϕ M P from eq. (A.2), so we can now solve for ζ ( ν ) in terms of ϕ : \nζ ( ϕ ) ≈ M P ( βλϕ M P -1 ) βλ . (A.8) \nNow after we used the previous analytical approximations; starting by eq. (A.5), we can easily invert ζ ( ϕ ) in eq. (A.8), and then we get the following expression: \nϕ ( ζ ) ≈ M P βλ + ζ ( ϕ ) . (A.9) \nTo improve this approximation, we can include the next term from the expansion of the hypergeometric function as expressed in eq. (A.6). \nExpanding ν ( ζ ) to account for this term requires solving a more complicated equation, which may need numerical inversion. However, the leading-order approximation given above \nprovides a reasonable analytical form for ϕ ( ζ ) resulting in the same expression as eq. (A.9). Higher-order corrections can be computed by including terms from the hypergeometric series expansion, but these generally require numerical inversion methods. \nFurthermore, we use eq. (A.9) to obtain the figure 8 by illustrating the field ϕ ( ζ ) as a function of the parameter ζ approximately within the context of β -exponential potential. The figure points out a characteristic slow-roll inflationary behavior. At lower ζ values, ϕ ( ζ ) remains nearly constant, indicating a stable and prolonged inflationary phase, which is essential for generating a sufficient number of e-folds N ∗ , and this is in favor of the analysis presented throughout the paper. This flat region corresponds to the slow-roll approximation, where the potential is sufficiently flat to satisfy inflation. As ζ increases beyond a certain point, the field begins to grow rapidly, showing the larger field value at the beginning of inflation then the field values will get lower by the slow rolling scenario as mentioned in details and calculations in subsection 2.2 and section 3. We highlight that we are just plotting the expression ϕ ( ζ ) which is computed through rough approximations, so the resultant figure's main purpose is to give us a small hint about the dynamics of the investigated expression of ϕ ( ζ ). \nWe can now write the potential in the Einstein frame by substituting eq. (A.9) into eq. (2.13) resulting into getting the potential in terms of ζ instead of ϕ , and this can be achieved by following steps: \nV E ( ζ ) = V 0 ( 1 -λβ M P βλ + ζ M P ) 1 /β 1 + 4 α V 0 ( 1 -λβ M P βλ + ζ M P ) 1 /β M 4 P . (A.10) \nSimplifying the argument of the potential through taking: \n1 -λβ ϕ M P = 1 -λβ ( M P βλ + ζ M P ) = -λβ ζ M P , (A.11) \nthus, the expression for the potential V E ( ζ ) can be written as follows \nV E ( ζ ) = V 0 ( -λβ ζ M P ) 1 /β ( 1 + 4 α V 0 ( -λβ ζ M P ) 1 /β M 4 P ) , (A.12) \nwhich is depicted in figure 9 3 in order to illustrate the behavior of the potential in the Einstein frame in terms of ζ for a variety of choices of β . It is worth keeping in mind that this plot represents only an approximation of computing V E ( ζ ) analytically and it does not represent the full behavior of our potential, so this plot is solely an attempt to visualize and give a general image of our potential in terms of ζ . \nWe try to visualize that regardless of the different representations of our potential, the main behavior of the potential itself is retained. We express our potential in terms of ϕ \nFigure 9 . The Einstein frame β -exponential potential with minimal coupling in Palatini R 2 gravity as a function of ζ . The colors show different values of the β parameter. We fixed λ = 0 . 1, α = 10 8 , V 0 = 10 -9 , as well as taken M P unity. \n<!-- image --> \nas in eq. (2.13). Additionally, in this section, after following an analytical approach and approximations, we are able to represent our potential itself in terms of ζ as can be seen in eq. (A.12), so having these two representations we consider analyzing their behavior in both the frequency and the field domains closely. Along this paper, we expressed our potential V E in terms of ϕ and ζ , and to explore the consistency and the behavior of our model within these different representations we illustrate in figure 10 the contour V E ( ϕ, ζ ); which is defined by taking the average of the sum of our potentials' different representations, and this step helps us to illustrate how both V E ( ϕ ) and V E ( ζ ) are behaving similarly. From this figure, one can notice that regardless of the transformation between ϕ and ζ our potential maintains its expected structure and dynamics ensuring that our model behaves consistently under these different descriptions. This is a solid step to confirm the robustness of the inflationary dynamics through the potential in this paper, which is essential for ensuring accurate cosmological predictions. \nMoreover, we harness the Fourier transform which gives us the chance to break down these potentials into their frequency components, providing insights into their structure and the impact of different frequency modes. In the same plot, we can observe the two linear lines that represent a very smooth varying near the x axis, and they represent the low-frequency elements that correspond to the overall trend of the two different representations of our potential resulting in no sharp oscillations or rapid changes. From the top panel of figure 11 4 , one can notice that the overall frequency content of both potentials is similar; however, due to the approximations that are used to get the expression of ϕ ( ζ ) we can spot subtle", 'Contour Plots for V E ( , ) for Different Values': 'Figure 10 . Contour plots of V E ( ϕ, ζ ); which are calculated as the average of both our expressed potential in terms of ϕ as in eq. (2.13), and in terms of ζ as shown in eq. (A.12). The different panels of the plot show the potential behavior for various values of β : -0 . 8 , -0 . 5 , 0 . 5 , 1 . 5 , and 3. We fixed λ = 0 . 1, α = 10 8 , V 0 = 10 -9 , as well as taken M P unity. \n<!-- image --> \ndifferences, particularly at lower frequencies. For the bottom panel of the same figure, we plot the inverse Fourier transform, which depicts the original form of the potentials. Regardless of the approximations used in this section, both V E ( ϕ ) and V E ( ζ ) follow the same general trends; both of them grow over a range of the field but with different profiles. This shows us that, despite the used approximations we retain the key similarities to V E ( ϕ ), which indicates that the approximations that are used in calculating ϕ ( ζ ) affect finer details without drastically altering the overall structure of our potential. \nWe can also evaluate ζ ( ϕ ) in eq. (3.4) for the case of β = 1 / 2 by utilizing the special \n<!-- image --> \nInverse Fourier Transform (Reconstructed \nVE \n( \n) \nand \nVE \n( ) \n) \n1e 9 \nFigure 11 . The top panel represents the Fourier Transform (FFT) of the potentials V E ( ϕ ) (solid blue line) and V E ( ζ ) (dashed red line), highlighting their magnitude spectra in the low-frequency domain. Additionally, both the horizontal lines represent the corresponding average value of each potential. The plot at the bottom panel shows the reconstructed potentials, with the same color definitions of the top plot, obtained via the inverse Fourier Transform (iFFT). We fixed β = 0 . 5, λ = 0 . 1, α = 10 8 , V 0 = 10 -9 , as well as taken M P unity. \n<!-- image --> \ncase for the hypergeometric function 2 F 1 ( 1 2 , 1 2 ; 3 2 ; -Λ 2 ) 5 : \n2 F 1 ( 1 2 , 1 2 ; 3 2 ; -Λ 2 ) = sinh -1 (Λ) Λ , (A.13) \nusing this identity, where Λ = √ 4 αV 0 ( 1 -ϕλ 2 M P ) 2 M 4 P , the expression for ζ ( ϕ ) defined by eq. (3.4) becomes: \nFigure 12 . Plot of ϕ ( ζ ) vs. ϕ through using the hypergeometric hyperbolic sine formula for the special case of β = 1 2 . We set the values as follows: V 0 = 10 -9 , α = 10 8 , λ = 0 . 1 ( M P is set to 1). \n<!-- image --> \nζ ( ϕ ) = ( λϕ -2 M P 2 ) M 2 P sinh -1 ( 2 √ αV 0 ( 1 -ϕλ 2 M P ) M 2 P ) λ √ αV 0 ( 1 -ϕλ 2 M P ) . (A.14) \nTo find ϕ ( ζ ), we start by isolating the term involving ϕ . Multiply both sides by λ √ αV 0 ( 1 -ϕλ 2 M P ) : \nζ ( ϕ ) · λ √ αV 0 ( 1 -ϕλ 2 M P ) = ( λϕ -2 M P ) 2 M 2 P sinh -1 2 √ αV 0 ( 1 -ϕλ 2 M P ) M 2 P , (A.15) \nthen dividing by λϕ -2 M P 2 , it gives: \nζ ( ϕ ) · λ √ αV 0 ( 1 -ϕλ 2 M P ) λϕ -2 M P 2 = M 2 P sinh -1 2 √ αV 0 ( 1 -ϕλ 2 M P ) M 2 P , (A.16) \ndividing both sides by M 2 P , the expression results in: \n2 ζ ( ϕ ) · λ √ αV 0 ( 1 -ϕλ 2 M P ) ( λϕ -2 M P ) M 2 P = sinh -1 2 √ αV 0 ( 1 -ϕλ 2 M P ) M 2 P . (A.17) \nBy applying the hyperbolic sine function (sinh) to both sides, we get: \n2 √ αV 0 ( 1 -ϕλ 2 M P ) M 2 P = sinh 2 ζ ( ϕ ) · λ √ αV 0 ( 1 -ϕλ 2 M P ) ( λϕ -2 M P ) M 2 P , (A.18) \nthen solving this result for 1 -ϕλ 2 M P , it gives us the following result \n1 -ϕλ 2 M P = M 2 P 2 √ αV 0 sinh 2 ζ ( ϕ ) · λ √ αV 0 ( 1 -ϕλ 2 M P ) ( λϕ -2 M P ) M 2 P . (A.19) \nFinally, solving for ϕ ( ζ ) provides us with the following expression: \nϕ ( ζ ) = 2 M P λ 1 -M 2 P 2 √ αV 0 sinh 2 ζ ( ϕ ) · λ √ αV 0 ( 1 -ϕλ 2 M P ) ( λϕ -2 M P ) M 2 P . (A.20) \nThus, the inverse ϕ ( ζ ) involves solving a transcendental equation, which in general cannot be expressed in a simple closed form, which highlights the intricacy of the underlying dynamics in this inflationary model; one of the points that support to this latter point is that we used eq. (A.9) in plotting ϕ ( ζ ) vs. ζ rather than using eq. (A.20). The equation for ϕ ( ζ ) must be solved numerically or through iterative methods, and this behavior is essential for our understanding of the period of inflation and the conditions under which inflation ends, making the numerical solution very essential for analyzing our scenario for β -exponential inflation. In conclusion, through different approaches and approximations we formulated two expressions of the invert ϕ ( ζ ); however, it is worth mentioning that both resultant expressions describe the very same dynamics within our model framework. The expression provided by eq. (A.20) is more intricacy; however, for the sake of showing the viability and consistency of this expression with the first one in eq. (A.9), and plotted in figure 8, we solve eq. (A.20) numerically through the Root-finding method [97] and then plot it as shown in figure 12. Consequently, both of the figures for the two expressions show the consistency of the behavior of the two given expressions of ϕ ( ζ ).', 'Acknowledgements': 'The authors would like to thank Alberto Salvio, Antonio Racioppi and Anish Ghoshal for the useful comments and suggestions to improve the manuscript. NB also thanks Ozan Sargın for the fruitful suggestions on the hypergeometric function. \nNote added. 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2024arXiv240905533P

We construct an explicit model of inhomogeneous gravitational collapse leading to a naked singularity in which gravitational absorption is both efficient and observable. We propose that the infeasibility of graviton detection is simply a consequence of Natures conspiracy to hide regions of strong curvature behind event horizons.

2024-09-01T00:00:00Z

['10.48550/arXiv.2409.05533', '2024arXiv240905533P', 'arXiv:2409.05533']

['General Relativity and Quantum Cosmology', 'High Energy Physics - Phenomenology', 'High Energy Physics - Theory']

Infeasibility of Graviton Detection as Cosmic Censorship

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.05533.pdf

{'Infeasibility of Graviton Detection as Cosmic Censorship': 'Andrea Palessandro ∗ \nDeloitte AI Institute', 'Abstract': "We construct an explicit model of inhomogeneous gravitational collapse leading to a naked singularity in which gravitational absorption is both efficient and observable. We propose that the infeasibility of graviton detection is simply a consequence of Nature's conspiracy to hide regions of strong curvature behind event horizons.", '1 Introduction': "In the theory of General Relativity, gravitational singularities are both inevitable and ubiquitous [1, 2]. The Cosmic Censorship Conjecture (CCC) posits that these singularities are always hidden from view behind an event horizon 1 [3, 4]. Given that quantum gravitational effects are expected to become visible in regions of large space-time curvature, the CCC is sometimes taken to imply the unobservability of quantum gravity [5]. If this view is correct, an observer could probe the singularity and establish the quantization of gravity, but would not be able to communicate their results with the outside world due to the existence of an uncrossable event horizon. \nConceptually, the simplest experiment that could be performed to prove the quantum nature of gravity is the detection of single gravitons [6-10]. This can be done, for example, by sending gravitational radiation through a cloud of atoms. If the gravitational field is quantized, gravitons of a certain wavelength will be absorbed, resulting in absorption lines in the gravitational spectrum [11], a telltale sign of the field's granularity. \nGiven a cloud of atoms of constant density ρ and total mass M , the optical depth of a graviton traveling through the cloud is [6] \nτ = nσR, (1.1) \nwhere n ≡ ρ/µ is the number density of atoms, µ their mass, σ ∼ G the gravitational absorption cross section, and R the extension of the cloud. Assuming the Compton wavelength of a single \natom is contained within the cloud, µR > 1, imposing τ > 1 gives R < GM , which is the condition for gravitational collapse. This means that the parameter space that allows for graviton detection via visible absorption lines ( τ > 1) corresponds to a collapsed atomic cloud and is thus usually considered to be hidden from view. \nThe conclusion rests crucially on the assumption that gravitational collapse generically results in black holes, i.e. singularities hidden behind event horizons. Were this not true, for example in the case of a naked singularity, one could imagine an experiment in which graviton absorption is both efficient ( τ > 1) and visible from the outside. The aim of this paper is to demonstrate this by explicit construction. \nThe experiment's setup is described in Figure 1. The detector is a gas cloud of extension r 0 and total mass M with an inhomogeneous mass distribution that slowly collapses under its own gravity. A gravitational wave source is placed at distance r = ϵr 0 ≪ r 0 from the central singularity, emitting radiation to infinity. An observer placed outside the cloud analyzes the incoming gravitational radiation and looks for absorption lines to establish the quantization of gravity. Clearly, two conditions have to be satisfied in order for this experiment to be successful: \n- · The detector has to be efficient, i.e. τ > 1 for a graviton traveling through the gas cloud.\n- · The gravitational radiation ought to escape the gas cloud, meaning that the central singularity has to be (globally) visible. \nIn the rest of the paper we will construct an explicit example of inhomogeneous spherically symmetric gravitational collapse in which both conditions above are satisfied. In particular, in § 2 we study the evolution of the detector (gas cloud) in the special case of a spherically symmetric mass distribution. In § 3 we work out the necessary conditions for the local visibility of the central singularity, while in § 4 the conditions for its global visibility. In § 5 we demonstrate the efficiency of the detector if the gravitational source is placed sufficiently close to the (naked) singularity. Finally, we present our concluding remarks in § 6.", '2 Spherically symmetric gravitational collapse': "In this section we study a specific model of spherically symmetric gravitational collapse which can give rise to a naked singularity [12,13]. In the two sections that follow we will specify the conditions for this to happen both locally and globally. We work in natural units ℏ = c = 1 and with metric signature ( -, + , + , +). Dots and primes indicate differentiation with respect to time and space, respectively. \nThe general class of solutions describing the evolution of a spherically symmetric inhomogeneous dust cloud is given by the Lemaitre-Tolman-Bondi metric [14-16] \nds 2 = -dt 2 + R ' ( t, r ) 2 1 -k ( r ) dr 2 + R ( t, r ) 2 ( dθ 2 +sin 2 θdϕ 2 ) , (2.1) \nwhere R ( t, r ) is the proper radius of a matter shell at comoving coordinates ( t, r ), and k ( r ) < 1 controls the curvature of the spatial slices at constant t . \nFigure 1: The experimental setup. An observer looks for absorption lines in the gravitational spectrum of radiation emitted by a source close to the center of an inhomogeneously distributed atomic gas cloud. \n<!-- image --> \nThe metric is sourced by the energy-momentum tensor of a pressureless fluid: \nT µν = ρ ( t, r ) δ 0 µ δ 0 ν , (2.2) \nwhere ρ ( t, r ) is the matter density of the dust cloud. We assume the cloud is made up of atoms, with total mass M . \nThe Einstein field equations give [17] \n˙ R 2 + k R 2 = 2 Gm R 3 . (2.3) \nwhere m ( r ) is the mass enclosed in a sphere of radius R ( r ): \nm ' = 4 πR 2 R ' ρ. (2.4) \nThe model is called bound, marginally bound or unbound depending on whether k > 0, k = 0 or k < 0. For pedagogical clarity, we analyze here the marginally bound case. Integration of (2.3) with k = 0 gives \nR ( t, r ) = ( r 3 / 2 -3 2 √ 2 Gmt ) 2 / 3 . (2.5) \nNote that, since we are interested in gravitational collapse, we have taken the solution with ˙ R < 0. Moreover, we have used the remaining coordinate freedom to equate proper and coordinate distance on the initial hypersurface, i.e. R (0 , r ) = r . \nClearly, the mass function is fixed once the initial density distribution ρ (0 , r ) ≡ ρ ( r ) is given: \nm ( r ) = 4 π ∫ ρ ( r ) r 2 dr. (2.6) \nWe further assume that ρ ( r ) is of the form \nρ ( r ) = ρ 0 [ 1 -( r r 0 ) n ] for 0 ≤ r ≤ r 0 , ρ ( r ) = 0 for r > r 0 , (2.7) \nwhere r 0 is the initial extension of the gas cloud, and ρ 0 the initial matter density at r = 0. By Birkhoff's theorem the spacetime at r > r 0 is described by the Schwarzschild metric with total mass M . If n > 0, the density of the cloud decreases monotonically as one moves out from the center. Given (2.7), the mass function is \nm ( r ) = M [ 1 + 3 n -3 n ( r r 0 ) n ]( r r 0 ) 3 , (2.8) \nwhere m ( r 0 ) = M ≡ 4 πnρ 0 r 3 0 / (3( n +3)) is the total mass of the cloud. \nGravitational singularities are defined as points at the boundary of spacetime where the energy density or the curvature scalars diverge. One such example is the Kretschmann scalar K = R abcd R abcd , which for the metric (2.1) is given by \nK = 4 ( k + ˙ R 2 ) 2 R 4 +8 R 2 R 2 +2 ( k ' +2 ˙ R ˙ R ' ) 2 R 2 R ' 2 +4 R ' 2 R ' 2 = 48 G 2 m 2 R 6 -32 G 2 mm ' R 5 R ' +12 G 2 m ' 2 R 4 R ' 2 , (2.9) \nwhere the second equality follows from (2.3). Clearly, the Kretschmann scalar diverges for both R = 0 (with R ' , m ' = 0) and R ' = 0 (with R,m ' = 0). \n̸ \n̸ \nThe former is called a shell-focusing singularity and takes place when the physical radius of a matter shell shrinks to zero. According to (2.5), this happens at the time \nt c ( r ) = 2 3 r 3 / 2 √ 2 Gm . (2.10) \nThis is the time of collapse for a matter shell at comoving distance r from the center. In general, for an inhomogeneous mass distribution different shells will meet the singularity at different times depending on the value of r . \nThe latter is called a shell-crossing singularity, and generically occurs whenever t c ( r ) is not a monotonically increasing function, i.e. when matter shells cross [18,19]. At a crossing event the matter density and certain components of the Riemann curvature tensor blow up, but the causal structure of spacetime can be extended through it [20]. Unlike shell-crossing singularities, spacetime admits no extension through a shell-focusing singularity, which is therefore the only type of 'genuine' singularity in a causal sense [21]. Given the mass function (2.8), the \ntime of collapse (2.10) is a monotonically increasing function of r , therefore no shell-crossing singularities occur in our model. We focus then on the shell-focusing singularities. \nIn order to determine the nature of the singularity (hidden or naked), one must study the behavior of outgoing non-spacelike geodesics in the spacetime (2.1). For simplicity we will focus on outgoing radial null geodesics, which, for k = 0, are described by \ndt dr = R ' . (2.11) \nAlong an outgoing radial null geodesics during the collapsing phase we have [22] \ndR dt = ˙ R + R ' dr dt = 1 -√ 2 Gm R . (2.12) \nTherefore, dR/dt < 0 whenever R < 2 Gm and the corresponding region is trapped, meaning that all light rays converge towards the singularity. The outer boundary of the trapped region is called the apparent horizon, and lies at R = 2 Gm . Given (2.5), the apparent horizon forms at time \nt ah ( r ) = t c ( r ) -4 3 Gm. (2.13) \n̸ \nClearly then, t ah ( r ) ≤ t c ( r ) in any neighborhood of r = 0, therefore a non-central shell-focusing singularity is always hidden. A central shell-focusing singularity can be locally naked if in a neighborhood of r = 0, t ah ( r ) > t c (0), so that an outgoing radial null geodesics can probe the singularity without encountering any trapped surface. Similarly, the singularity can be globally naked (visible to observers at infinity) if the validity of the condition t ah ( r ) > t c (0) extends to the outer edge of the gas cloud. This condition, however, only provides a necessary (but not sufficient) criterion for visibility, as we will explain in the next section.", '3 Local Visibility': "As we discussed in the previous section, only the central shell-focusing singularity can be naked. This forms at the time \nt c (0) ≡ t 0 = 2 3 √ n n +3 r 0 2 GM r 0 = 1 √ 6 πGρ 0 . (3.1) \nNear r = 0, we can write (2.10) as \nt c ( r ) = t 0 [ 1 + 3 2( n +3) ( r r 0 ) n ] + O ( r n +1 ) . (3.2) \nSince t c ( r ) ≥ t 0 , the central singularity at r = 0 forms first, followed by the outer shells, in order of distance from the center. The limiting case n → ∞ corresponds to homogeneous collapse (the Oppenheimer-Snyder model [23]) with ρ ( r ) = ρ 0 , in which all matter shells collapse simultaneously at t c ( r ) = t 0 regardless of r . \nIn order for the central singularity to be naked, at least locally, it is necessary for the apparent horizon to form after collapse, i.e. t ah ( r ) > t 0 in a neighborhood of r = 0. Near r = 0, the apparent horizon forms at the time \nt ah ( r ) = t c ( r ) -4 3 Gm = t 0 -4( n +3) 3 n GM ( r r 0 ) 3 + 3 t 0 2( n +3) ( r r 0 ) n + O ( r n +1 ) . (3.3) \nFor n →∞ (homogeneous collapse) t ah ( r ) ≤ t 0 and the singularity is hidden. For finite values of n , if n = 1 , 2, t ah ( r ) ≥ t 0 around r = 0 and the singularity is potentially naked. If n = 3 the singularity is potentially naked only when t 0 > 32 / 3 GM , or r 0 ≳ 10 GM . For n ≥ 4 the singularity is again hidden behind an event horizon, and the collapse always results in a black hole. \nAs mentioned in the previous section, the condition t ah ( r ) > t 0 is necessary, but not sufficient: local visibility requires the existence of an outgoing light-like geodesic emitting from the singularity with no trapped surfaces in its path. This can only happen if the apparent horizon forms sufficiently late. In general, whenever the formation of the apparent horizon is sufficiently delayed, for example due to strong shearing effects [24], the singularity is exposed to external observers, at least locally, and becomes naked. In black hole formation, instead, the apparent horizon forms before gravitational collapse and the singularity is hidden behind a global event horizon. \nGiven that only the cases n = 1 , 2 , 3 allow for naked singularities, we can check for local visibility by explicit construction. We assume an outgoing radial null geodesic starting at the singularity of the form [25] \nt = t 0 + a ( r r 0 ) α , (3.4) \nto leading order in r , with a, α > 0. In order for the geodesic to lie in the ambient spacetime, we require t ≤ t c ( r ), which by (3.2) is satisfied for all α > n , and for a < 3 t 0 / 2( n +3) if α = n . To leading order in r , (2.5) is \nR = [ 1 -( 1 -3 2( n +3) ( r r 0 ) n ) t t 0 ] 2 / 3 r. (3.5) \nDifferentiating with respect to r we get \nR ' = [ 1 -( 1 -3 2( n +3) ( r r 0 ) n ) t t 0 ] -1 / 3 [ 1 -( 1 -2 n +3 2( n +3) ( r r 0 ) n ) t t 0 ] . (3.6) \nGiven that by (3.4) dt/dr = α ( a/r 0 )( r/r 0 ) α -1 , (2.11) evaluated on the assumed geodesic gives \nα a r 0 ( r r 0 ) α -1 = 1 -( 1 -2 n +3 2( n +3) ( r r 0 ) n )( 1 + a t 0 ( r r 0 ) α ) [ 1 -( 1 -3 2( n +3) ( r r 0 ) n )( 1 + a t 0 ( r r 0 ) α )] 1 / 3 (3.7) \nIf the equation above admits self-consistent solutions, the singularity is locally naked, meaning that there exists at least one outgoing null geodesics which terminates arbitrarily close to the singularity. \nLet's consider first the case α > n . At leading order, (3.7) gives \nα a r 0 ( r r 0 ) α -1 = ( 1 + 2 n 3 )( 3 2( n +3) ) 2 / 3 ( r r 0 ) 2 n/ 3 , (3.8) \nwhich implies α = 1 + 2 n/ 3 and a/r 0 = (3 / 2( n + 3)) 2 / 3 . The condition α > n translates to n < 3, meaning that the singularity is locally naked for n = 1 and n = 2, confirming our previous analysis. \nIn the case α = n , (3.7) gives \nn a r 0 ( r r 0 ) n -1 = 2 n +3 2( n +3) -a t 0 ( 3 2( n +3) -a t 0 ) 1 / 3 ( r r 0 ) 2 n/ 3 , (3.9) \nwhich requires n = 3. With this choice of n , the expression above reduces to \n3 y = 3 4 -y ( 1 4 -y ) 1 / 3 r 0 t 0 , (3.10) \nsubject to the constraint y < 1 / 4, where y ≡ a/t 0 . The equation above admits real solutions only when t 0 /r 0 > (4 + 2 √ 3) / 3, or, equivalently, when r 0 > (28 + 16 √ 3) GM ≈ 56 GM , a stronger constraint than the one we deduced by just requiring t ah > t 0 .", '4 Global Visibility': "In order for the singularity to be visible to observers at infinity, the geodesic is prohibited from crossing any apparent horizon throughout the collapsing cloud. \nFirst, then, we need to check that there are no trapped surfaces on the initial hypersurface, i.e. we must require r > 2 Gm for all r ≤ r 0 at t = 0. Given (2.8), the condition translates to \n2 GM r 0 < n ( n +3 -3 x n ) x 2 for all 0 ≤ x ≤ 1 , (4.1) \nwith x ≡ r/r 0 . The function on the right-hand side of the equation has a minimum at x n = (2 n +6) / (3 n +6), therefore (4.1) entails \n2 GM r 0 < 2 + n 3 + n ( 3 n +6 2 n +6 ) 2 /n . (4.2) \nThe function on the right hand side is ≈ 1 for n = 1 , 2 , 3, so the constraint is merely the statement that the cloud is not a black hole initially. \nThe sufficient condition for global visibility was found in [26] and is given by \nt ' c ( r ) > G 3 (26 + 15 √ 3) m ' ( r ) for all 0 ≤ r ≤ r 0 . (4.3) \nBy (2.8) and (2.10), this yields \nn √ 2(3 + n ) x n -3 (1 -x n ) ( 3+ n n -3 n x n ) 3 / 2 ( r 0 GM ) 3 / 2 > 26 + 15 √ 3 for all 0 ≤ x ≤ 1 . (4.4) \nNow, one has to distinguish the cases n = 1 , 2 and n = 3. In the former case, the function on the left hand side of (4.4) has a minimum at \nx = ( 12 + 3 n -√ 25 n 2 +8 n 3 6(2 + n ) ) 1 /n , (4.5) \ntherefore condition (4.4) is satisfied if \n( r 0 GM ) 3 / 2 > (26 + 15 √ 3) (3 + n )(3 + √ 25 + 8 n )(7 + 2 n + √ 25 + 8 n ) 3 / 2 2 × 6 3 n (2 + n ) 3( n +2) 2 n (12 -n ( √ 25 + 8 n -3)) n -3 n . (4.6) \nThis gives r 0 ≳ 28 GM for n = 1, and r 0 ≳ 36 GM for n = 2. In the latter case, the function on the left hand side of (4.4) has a minimum at x = 0, therefore the condition becomes r 0 > (28+16 √ 3) GM ≈ 56 GM . This is the same condition we obtained in the previous section where we studied local visibility. This means that in the special case n = 3, if the singularity is locally naked, it is also globally naked. \nTo summarize, in marginally bound spherically symmetric collapse models, the central singularity is \n- · locally naked for n = 1 , 2 and for n = 3 if r 0 ≳ 56 GM ,\n- · globally naked for n = 1 if r 0 ≳ 28 GM , n = 2 if r 0 ≳ 36 GM , and n = 3 if r 0 ≳ 56 GM . \nIn all other cases, the singularity is hidden. \nIt is worth pointing out that, as shown in [27,28], naked singularities in LTB spacetimes are generic, in the sense that given an initial density profile for the cloud, there is a non-zero measure set of configurations leading to the formation of a naked singularity. Moreover, the choice of considering only radial geodesics to characterize the naked singularity is not overly restrictive, as it can be shown that the existence of future-directed non-radial null geodesics emanating from the singularity is guaranteed by the existence of the corresponding future-directed radial null geodesics [29].", '5 Absorption Efficiency': "Having established that, given certain generic initial conditions, the singularity can be globally naked and the radiation emitted visible to an outside observer, we now turn to proving explicitly the detector's efficiency. \nAssuming the source is kept stationary at a distance ϵr 0 from the singularity 2 , we can fire gravitons to infinity by suitably choosing the initial mass density. Will those gravitons also be absorbed with high probability? To determine that, we need to compute the optical depth of the graviton through the dust cloud, which is defined as \nτ = ∫ σn ( t, r ( t )) dt = G µ ∫ r 0 ϵr 0 ρ ( t ( r ) , r ) R ' dr, (5.1) \nwhere σ ∼ G is the graviton absorption cross section and n ≡ ρ/µ the number density of atoms, with ρ = m ' / 4 πR 2 R ' and µ the mass of a single atom. The atoms have to be contained inside the detector, so their Compton wavelength should at least be smaller than the detector's radius, i.e. µr 0 > 1. \nIn terms of x ≡ r/r 0 , and using (2.5) and (2.8), the integral is \nτ = G µ ∫ r 0 ϵr 0 m ' 4 πR 2 dr = Gρ 0 r 0 µ ∫ 1 ϵ 1 -x n ( 1 -(1 + ax α ) √ 1 -3 3+ n x n ) 4 / 3 dx. (5.2) \nSince ax α > 0, the optical depth is bounded from below by \nτ > Gρ 0 r 0 µ ∫ 1 ϵ 1 -x n ( 1 -√ 1 -3 3+ n x n ) 4 / 3 dx. (5.3) \nThe integral above is dominated by values around x = 0, therefore it is well approximated by \nτ > Gρ 0 r 0 µ ∫ 1 ϵ 1 -x n ( 1 -√ 1 -3 3+ n x n ) 4 / 3 dx ≈ Gρ 0 r 0 µ ( 2(3 + n ) 3 ) 4 / 3 ∫ 1 ϵ (1 -x n ) x -4 n 3 dx = Gρ 0 r 0 µ ( 2(3 + n ) 3 ) 4 / 3 3 ( ϵ 1 -4 n 3 -1 ) 4 n -3 + 3 ( ϵ 1 -n 3 -1 ) 3 -n . (5.4) \nFor all values n ≥ 1, the optical depth can be made arbitrarily large in the limit ϵ → 0. This means that in models of spherically symmetric gravitational collapse with a mass density of the form (2.7) and n = 1 , 2 , 3, if the initial size of the cloud is large enough graviton detection is both efficient and observable. \nA particular caveat of this construction is that by taking the limit ϵ → 0 one gets arbitrarily close to the singularity where unknown quantum gravity effects can spoil the experiment. It's worth pointing out then that the limit ϵ → 0 is not strictly necessary, as all we want for efficient graviton detection is for the optical depth to be greater than one. This can be achieved for reasonable values of ϵ by tuning the initial parameters r 0 , M and µ . For example, if one takes n = 2, µr 0 ∼ 10 2 , and r 0 /GM ∼ 10 2 , numerical integration of (5.3) shows that the optical \ndepth becomes order one for ϵ ≈ 5 × 10 -3 , meaning that the gravitational source is sitting at a distance ϵr 0 = 0 . 5 GM from the singularity. Clearly, if M is large enough, the source is undisturbed by putative quantum gravity effects. \nIn a realistic scenario, one could consider a cloud of hydrogen atoms with µ ∼ 5 × 10 -20 m p , and total mass M = 10 M ⊙ ∼ 10 39 m p . Then, assuming n = 3 and r 0 /GM ∼ 10 2 , µr 0 ∼ 10 22 and ϵ ∼ 10 -8 , meaning that the source is at a distance of ϵr 0 ∼ 1 cm from the singularity. The Kretschmann scalar (2.9) at that distance is of order K = 48 G 2 m 2 /R 6 ∼ G 2 M 2 /ϵ 4 n r 6 0 ∼ 10 -72 m 4 p , well below the Planck density.", '6 Conclusions': "We have shown, in a specific model of inhomogeneous gravitational collapse, that graviton detection can be both visible and efficient in the absence of an event horizon. This demonstrates that, if the CCC is violated, gravitons can be detected. The converse implication, namely that if gravitons can be detected then the CCC is necessarily violated, if true, is presumably much more difficult to prove. Thus, even though it is tempting to conclude that the CCC is the fundamental reason for the infeasibility of graviton detection, more work is needed to establish this connection. \nHowever, there is some circumstantial evidence: \n- · Gravitational wave detectors based on laser interferometry, such as LIGO, need a sensitivity of less than a Planck length to be able to detect single gravitons [6]. Resolving such distances is deemed impossible due to black hole formation [31].\n- · Many thought experiments trying to establish the quantization of gravity [32-35] in the spirit of Bohr and Rosenfeld [36] fail to do so due to the Planck length acting as a fundamental limit on spatial resolution [37-39], as above.\n- · The Gertsenshtein effect could in principle be used to detect single gravitons [6,7]. However, it can be shown that such a detector is always inefficient at sub-horizon scales due to nonlinear electromagnetic effects that break quantum coherence [10]. The cosmological event horizon effectively hides graviton-photon oscillations [40], preventing single graviton detection.\n- · Measurement of tensor modes in the CMB could be used to establish the quantization of gravity [41]. However, it can be shown that tensor modes are unobservable in models that hinder the growth of trans-Planckian fluctuations [42, 43]. In these models, transPlanckian fluctuations are hidden behind the cosmological horizon and thus unable to turn classical and affect macroscopic observations. \nNote that in the last two experiments listed, it is the cosmic event horizon that prevents their successful completion, and as such the CCC, as commonly understood, does not apply. However, one can strenghten the formulation of the CCC to include these cosmological cases as well. The CCC posits that any singularity in spacetime must lie behind an event horizon. This \ncan be taken as a particular instance of a more general conjecture on the unobservability of quantum gravity effects in the universe, namely that Nature conspires to hide the quantization of the gravitational field behind event horizons, either astrophysical or cosmic. In this more general sense, the CCC prevents the successful completion of all graviton detection experiments listed above. \nFinally, one could establish the quantization of the gravitational field by detecting its quantum-induced noise in the lengths of the arms of a LIGO-like gravitational wave detector [44, 45]. This noise is negligible for coherent states but is greatly enhanced in thermal and squeezed states. However, as shown in [8], the simple observation of enhanced noise in the measuring apparatus cannot be used to distinguish a quantum model of the gravitational field from a classical one. In order to demonstrate quantization one would need to observe sub-vacuum levels of noise. Crucially, any deviation from noise at the standard quantum limit is proportional to the detector's efficiency [46], which means that, even in the case of highly squeezed graviton states, the quantum signature of the gravitational field is unobservable unless the detector is already highly efficient, and the evidence so far shows that efficient graviton detection always lies beyond an event horizon.", 'References': "- [1] Penrose, R. (1965). Gravitational collapse and space-time singularities. Physical Review Letters, 14(3), 57.\n- [2] Hawking, S. W. (1972). Black holes in general relativity. Communications in Mathematical Physics, 25, 152-166.\n- [3] Penrose, R. (1969). Gravitational collapse: The role of general relativity.\n- [4] Penrose, R. (1999). The question of cosmic censorship. Journal of Astrophysics and Astronomy, 20(3), 233-248.\n- [5] Joshi, P. S., and Malafarina, D. (2011). Recent developments in gravitational collapse and spacetime singularities. International Journal of Modern Physics D, 20(14), 2641-2729.\n- [6] Dyson, F. (2013). Is a graviton detectable?. International Journal of Modern Physics A, 28(25), 1330041.\n- [7] Rothman, T., and Boughn, S. (2006). Can gravitons be detected?. Foundations of Physics, 36, 1801-1825.\n- [8] Carney, D., Domcke, V., and Rodd, N. L. (2024). Graviton detection and the quantization of gravity. Physical Review D, 109(4), 044009.\n- [9] Tobar, G., Manikandan, S. K., Beitel, T., and Pikovski, I. (2024). Detecting single gravitons with quantum sensing. Nature Communications, 15(1), 7229.\n- [10] Palessandro, A. (2024). Graviton-photon oscillations as a probe of quantum gravity. arXiv preprint arXiv:2405.01407.\n- [11] Palessandro, A., and Sloth, M. S. (2020). Gravitational absorption lines. Physical Review D, 101(4), 043504.\n- [12] Christodoulou, D. (1984). Violation of cosmic censorship in the gravitational collapse of a dust cloud. Communications in Mathematical Physics, 93, 171-195.\n- [13] Joshi, P. S., and Dwivedi, I. H. (1993). Naked singularities in spherically symmetric inhomogeneous Tolman-Bondi dust cloud collapse. Physical Review D, 47(12), 5357.\n- [14] Lemaˆıtre, G. (1933). L'univers en expansion. In Annales de la Soci'et'e scientifique de Bruxelles (Vol. 53, p. 51).\n- [15] Tolman, R. C. (1934). Effect of inhomogeneity on cosmological models. Proceedings of the National Academy of Sciences, 20(3), 169-176.\n- [16] Bondi, H. (1947). Spherically symmetrical models in general relativity. Monthly Notices of the Royal Astronomical Society, 107(5-6), 410-425."}

2024arXiv240913556V

We present constraints on the fR gravity model using a sample of 1005 galaxy clusters in the redshift range 0.25 1.78 that have been selected through the thermal SunyaevZeldovich effect tSZE from South Pole Telescope SPT data and subjected to optical and nearinfrared confirmation with the Multicomponent Matched Filter MCMF algorithm. We employ weak gravitational lensing mass calibration from the Dark Energy Survey DES Year 3 data for 688 clusters at z lt 0.95 and from the Hubble Space Telescope HST for 39 clusters with 0.6 lt z lt 1.7. Our cluster sample is a powerful probe of fR gravity because this model predicts a scaledependent enhancement in the growth of structure which impacts the halo mass function HMF at cluster mass scales. To account for these modified gravity effects on the HMF our analysis employs a semianalytical approach calibrated with numerical simulations. Combining calibrated cluster counts with primary cosmic microwave background CMB temperature and polarization anisotropy measurements from the Planck2018 release we derive robust constraints on the fR parameter fR0. Our results log10 fR0 lt 5.32 at the 95 credible level are the tightest current constraints on fR gravity from cosmological scales. This upper limit rules out fRlike deviations from general relativity that result in more than a sim20 enhancement of the cluster population on mass scales Mmathrm200cgt3times1014Modot.

2024-09-01T00:00:00Z

['2024arXiv240913556V', 'arXiv:2409.13556', '10.48550/arXiv.2409.13556']

['Astrophysics - Cosmology and Nongalactic Astrophysics']

Constraints on fR gravity from tSZEselected SPT galaxy clusters and weak lensing mass calibration from DES and HST

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.13556.pdf

{'Constraints on f ( R ) gravity from tSZE-selected SPT galaxy clusters and weak lensing mass calibration from DES and HST': "S. M. L. Vogt, 1, 2, 3, ∗ S. Bocquet, 1 C. T. Davies, 1 J. J. Mohr, 1, 4 F. Schmidt, 3 C.-Z. Ruan, 5 B. Li, 6 C. Hern'andez-Aguayo, 3 S. Grandis, 7, 1 L. E. Bleem, 8, 9 M. Klein, 1, 4 T. Schrabback, 7 M. Aguena, 10 D. Brooks, 11 D. L. Burke, 12, 13 A. Campos, 14, 15 A. Carnero Rosell, 16, 10 J. Carretero, 17 M. Costanzi, 18, 19, 20 L. N. da Costa, 10 M. E. S. Pereira, 21 J. De Vicente, 22 P. Doel, 11 S. Everett, 23 I. Ferrero, 24 J. Frieman, 25, 26 J. Garc'ıa-Bellido, 27 M. Gatti, 28 G. Giannini, 17, 26 D. Gruen, 1 R. A. Gruendl, 29, 30 S. R. Hinton, 31 D. L. Hollowood, 32 S. Lee, 33 M. Lima, 34, 10 J. L. Marshall, 35 J. Mena-Fern'andez, 36 R. Miquel, 37, 17 J. Myles, 38 M. Paterno, 25 A. Pieres, 10, 39 A. A. Plazas Malag'on, 12, 13 C. L. Reichardt, 40 A. K. Romer, 41 S. Samuroff, 42 A. Sarkar, 43 E. Sanchez, 22 I. Sevilla-Noarbe, 22 M. Smith, 44 E. Suchyta, 45 M. E. C. Swanson, 29 G. Tarle, 46 V. Vikram, 47 N. Weaverdyck, 48, 49 and J. Weller 1, 4 \n(the SPT and DES Collaborations) 1 University Observatory, Faculty of Physics, Ludwig-Maximilians-Universitat, Scheinerstr. 1, 81679 Munich, Germany 2 Excellence Cluster Origins, Boltzmannstr. 2, 85748 Garching, Germany 3 Max Planck Institute for Astrophysics, Karl-Schwarzschild-Str. 1, 85748 Garching, Germany 4 Max Planck Institute for Extraterrestrial Physics, Giessenbachstr. 2, 85748 Garching, Germany 5 Institute of Theoretical Astrophysics, University of Oslo, 0315 Oslo, Norway 6 Institute for Computational Cosmology, Department of Physics, Durham University, South Road, Durham DH1 3LE, UK 7 Universitat Innsbruck, Institut fur Astro- und Teilchenphysik, Technikerstr. 25/8, 6020 Innsbruck, Austria 8 High-Energy Physics Division, Argonne National Laboratory, 9700 South Cass Avenue, Lemont, IL 60439, USA 9 Kavli Institute for Cosmological Physics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA 10 Laborat'orio Interinstitucional de e-Astronomia - LIneA, Rua Gal. Jos'e Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil 11 Department of Physics & Astronomy, University College London, Gower Street, London, WC1E 6BT, UK 12 Kavli Institute for Particle Astrophysics & Cosmology, P. O. Box 2450, Stanford University, Stanford, CA 94305, USA 13 SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA 14 Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15312, USA 15 NSF AI Planning Institute for Physics of the Future, Carnegie Mellon University, Pittsburgh, PA 15213, USA 16 Instituto de Astrofisica de Canarias, E-38205 La Laguna, Tenerife, Spain 17 Institut de F'ısica d'Altes Energies (IFAE), The Barcelona Institute of Science and Technology, Campus UAB, 08193 Bellaterra (Barcelona) Spain 18 Astronomy Unit, Department of Physics, University of Trieste, via Tiepolo 11, I-34131 Trieste, Italy 19 INAF-Osservatorio Astronomico di Trieste, via G. B. Tiepolo 11, I-34143 Trieste, Italy 20 Institute for Fundamental Physics of the Universe, Via Beirut 2, 34014 Trieste, Italy 21 Hamburger Sternwarte, Universitat Hamburg, Gojenbergsweg 112, 21029 Hamburg, Germany 22 Centro de Investigaciones Energ'eticas, Medioambientales y Tecnol'ogicas (CIEMAT), Madrid, Spain 23 California Institute of Technology, 1200 East California Blvd, MC 249-17, Pasadena, CA 91125, USA 24 Institute of Theoretical Astrophysics, University of Oslo. P.O. Box 1029 Blindern, NO-0315 Oslo, Norway 25 Fermi National Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510, USA 26 Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA 27 Instituto de Fisica Teorica UAM/CSIC, Universidad Autonoma de Madrid, 28049 Madrid, Spain 28 Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA 29 Center for Astrophysical Surveys, National Center for Supercomputing Applications, 1205 West Clark St., Urbana, IL 61801, USA 30 Department of Astronomy, University of Illinois at Urbana-Champaign, 1002 W. Green Street, Urbana, IL 61801, USA 31 School of Mathematics and Physics, University of Queensland, Brisbane, QLD 4072, Australia 32 Santa Cruz Institute for Particle Physics, Santa Cruz, CA 95064, USA 33 Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Dr., Pasadena, CA 91109, USA 34 Departamento de F'ısica Matem'atica, Instituto de F'ısica, Universidade de S˜ao Paulo, CP 66318, S˜ao Paulo, SP, 05314-970, Brazil 35 George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy, and Department of Physics and Astronomy, Texas A&M University, College Station, TX 77843, USA 36 LPSC Grenoble - 53, Avenue des Martyrs 38026 Grenoble, France 37 \nInstituci'o Catalana de Recerca i Estudis Avan¸cats, E-08010 Barcelona, Spain \n38 Department of Astrophysical Sciences, Princeton University, Peyton Hall, Princeton, NJ 08544, USA 39 Observat'orio Nacional, Rua Gal. Jos'e Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil 40 School of Physics, University of Melbourne, Parkville, VIC 3010, Australia 41 Department of Physics and Astronomy, Pevensey Building, University of Sussex, Brighton, BN1 9QH, UK 42 Department of Physics, Northeastern University, Boston, MA 02115, USA 43 Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, 70 Vassar St, Cambridge, MA 02139 44 Physics Department, Lancaster University, Lancaster, LA1 4YB, UK 45 Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831 46 Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA 47 Argone National Laboratory, 9700 S. Cass Avenue, Lemont, IL 60439, USA 48 Department of Astronomy, University of California, Berkeley, 501 Campbell Hall, Berkeley, CA 94720, USA 49 Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA \nWe present constraints on the f ( R ) gravity model using a sample of 1,005 galaxy clusters in the redshift range 0 . 25 -1 . 78 that have been selected through the thermal Sunyaev-Zel'dovich effect (tSZE) from South Pole Telescope (SPT) data and subjected to optical and near-infrared confirmation with the Multi-component Matched Filter (MCMF) algorithm. We employ weak gravitational lensing mass calibration from the Dark Energy Survey (DES) Year 3 data for 688 clusters at z < 0 . 95 and from the Hubble Space Telescope (HST) for 39 clusters with 0 . 6 < z < 1 . 7. Our cluster sample is a powerful probe of f ( R ) gravity, because this model predicts a scale-dependent enhancement in the growth of structure, which impacts the halo mass function (HMF) at cluster mass scales. To account for these modified gravity effects on the HMF, our analysis employs a semianalytical approach calibrated with numerical simulations. Combining calibrated cluster counts with primary cosmic microwave background (CMB) temperature and polarization anisotropy measurements from the Planck 2018 release, we derive robust constraints on the f ( R ) parameter f R 0 . Our results, log 10 | f R 0 | < -5 . 32 at the 95 % credible level, are the tightest current constraints on f ( R ) gravity from cosmological scales. This upper limit rules out f ( R )-like deviations from general relativity that result in more than a ∼ 20% enhancement of the cluster population on mass scales M 200c > 3 × 10 14 M ⊙ .", 'I. INTRODUCTION': "One of the most challenging questions in modern cosmology is understanding the nature of the accelerating expansion of the Universe [1, 2]. Various cosmological theories have been proposed to explain this phenomenon. Within the framework of general relativity (GR), this acceleration can be explained by introducing a cosmological constant Λ to the Einstein-Hilbert action, leading to the well-known Λ cold dark matter (ΛCDM) model. However, adding a cosmological constant to the EinsteinHilbert action offers little physical insight into the nature of Dark Energy. Therefore, there is strong motivation to consider modifications to the Einstein-Hilbert action that give rise to modified gravity models (see e.g., reviews [35]). \nThese modifications impact the growth of cosmic structures. Consequently, the abundance of massive galaxy clusters, as the end products of the hierarchical growth of cosmic structures, is sensitive to the different matter clustering and therefore, serves as an excellent probe for constraining modified gravity models and offering an independent test of GR. \nIn this work, we focus on a specific modified gravity model that introduces a non-linear function f ( R ) of the \nscalar curvature R into the Einstein-Hilbert action [6]. We employ the widely studied Hu & Sawicki model for the function f ( R ) [7]. Physically, this model introduces an additional gravitational-strength fifth force, altering structure formation in a scale-dependent manner and enhancing structure formation on galaxy cluster scales. The extent to which the f ( R ) model deviates from GR is encoded in the single parameter f R 0 , which has been constrained using various observations on cosmological scales. \nBecause the effects of f ( R ) gravity persist to very small scales, constraints on galactic and solar-system scales are very stringent for this model. Studies using galaxy rotation curves and morphology report log 10 | f R 0 | < -6 . 1 and log 10 | f R 0 | < -7 . 55 at 95% credible level, respectively [8, 9]. However, because these studies probe small scales, systematics such as uncertainties in galaxy formation play a key role. This makes constraints from larger scales that are less sensitive to galaxy formation a complementary test. \nThe current tightest constraint from clusters comes from a combination of ROSAT clusters, primary cosmic microwave background (CMB) anisotropy data, Supernovae (SNe), and baryonic acoustic oscillations (BAO), with an upper bound of log 10 | f R 0 | < -4 . 79 at the 95% credible level [10]. The recent analysis of eROSITA clusters reports log 10 | f R 0 | < -4 . 12 at the 95 % credible level using clusters alone and marginalizing over the neutrino \nmass [11]. \nStronger constraints from large scales probes are obtained from a weak-lensing peak analysis, which used data from the Canada-France-Hawaii-Telescope Lensing Survey (CFHTLenS), and are given by log 10 | f R 0 | < -5 . 16 [12]. Similar constraints come from the cross-correlation of galaxies with CMB lensing and galaxy weak-lensing, CMB, SNe, and BAO, reporting log 10 | f R 0 | < -4 . 61 and log 10 | f R 0 | < -4 . 5 at the 95 % credible level respectively [13, 14]. \nIn the analysis presented here, we focus on the weak lensing informed galaxy cluster abundance to constrain f ( R ) gravity. We are motivated to pursue this study partly because it has long been recognized that galaxy cluster surveys would be powerful probes of cosmic growth and therefore the action of gravity [15, 16] and also because in recent years cluster surveys have been successfully employed to study the standard ΛCDM model [17-25] as well as modified gravity models [11, 26-30]. \nOne of the most promising cluster samples currently available has been constructed using South Pole Telescope (SPT) [31] survey data and the thermal SunyaevZel'dovich effect (tSZE) [32]. The tSZE is caused by high-energy electrons in the intra-cluster medium (ICM) scattering off CMB photons, resulting in a spectral distortion of the CMB at the cluster position. Because the tSZE is a direct tracer of the hot ICM, it enables the detection of massive galaxy clusters. Moreover, the cluster tSZE signature is strongly mass dependent and approximately redshift-independent and thus can be employed to identify galaxy clusters up to the highest redshifts where clusters of sufficient mass exist. To constrain cosmological parameters with galaxy clusters, one has to relate observables such as the tSZE detection significance to the underlying halo mass at all relevant redshifts. These observable-mass relations can be empirically calibrated using weak gravitational lensing data and are typically modeled as power laws in cluster mass and redshift. \nIn this study, we employ the sample of 1,005 galaxy clusters detected using SPT data and confirmed using the MCMF algorithm [33-35] with optical and near-infrared data from the Dark Energy Survey (DES) [36-38] and the Wide-field Infrared Survey Explorer (WISE) [39]. To obtain mass estimates for the cluster sample, we use weak-lensing measurements from DES and targeted observations from the Hubble Space Telescope (HST). The f ( R ) analysis framework employed for the SPT clusters with mass calibration from DES and HST is based on the state-of-the-art method developed for the recent ΛCDM analyses of this same sample [24, 40] (hereafter SB24a and SB24b). In a recent paper, this same framework was modified and employed to carry out validation tests and forecasts for f ( R ) gravity constraints from upcoming Stage-III and Stage-IV surveys [41] (hereafter SV24a). \nFollowing SV24a, we incorporate f ( R ) gravity into our analysis by modifying the halo mass function (HMF), which is enhanced relative to GR, in a mass- and redshiftdependent way. We implement this modification by \nintroducing a multiplicative factor to the GR HMF [28, 42], dependent on the spherical collapse threshold for halo collapse in f ( R ) gravity for which we use a semianalytical model [43, 44]. In the present analysis we calibrate this semi-analytical HMF model against the f ( R ) FORGE numerical simulations to obtain a more accurate halo mass function [45]. The simulations are not used directly to predict the HMF, because the available mass range from the simulations is limited, whereas the HMF from the semi-analytical model can be calculated for the wide mass range needed in this analysis. \nThis paper is organized as follows. Section II presents a summary of the SPT cluster dataset and the DES and HST weak-lensing data. We review in Sec. III f ( R ) gravity and the f ( R ) HMF model used in this work as well as the calibration to the FORGE simulations. In Sec. IV, we discuss our analysis method, including the observablemass relations, weak-lensing model, likelihood approach, and priors. The results are presented in Sec. V, and we conclude with a summary in Sec. VI. \nThroughout this paper, U ( a, b ) denotes a uniform distribution between limits a and b , and N ( µ, σ 2 ) is a Gaussian distribution with mean µ and variance σ 2 . We adopt the halo mass definition M 200c , which is the mass within a radius where the mean density is 200 times the critical density.", 'II. DATA': 'This section gives a brief summary of the cluster and weak-lensing data we use in this work. A detailed description of the data products is presented in SB24a.', 'A. SPT cluster catalog': 'The tSZE selected cluster catalogs from the SPT-SZ, SPTpol ECS and SPTpol 500d surveys employed here cover a total solid angle of 5 , 270 deg 2 of the southern sky [34, 35, 46, 47]. Note that the whole SPTpol 500d survey lies within the SPT-SZ footprint, and we use only the data from the deeper SPTpol 500d survey in the overlapping region. Cluster candidates of these surveys are selected in tSZE detection significance ˆ ζ and confirmed using optical and infrared data, which also add redshift information. Over the whole SPT survey region, only clusters with z > 0 . 25 are included in the sample, because the impact of atmospheric and primary CMB noise on the selection function at low redshift becomes dominant. \nIn the SPT survey region that is not covered by DES (1 , 327 deg 2 , ca. 27 % of the total solid angle), the cluster candidates are confirmed by targeted observations that also provide redshift measurements. Candidates in this \nregion are selected by \nˆ ζ > 5 z > 0 . 25 , (1) \nwhich results in a sample with 110 clusters and a purity ≳ 95 % [46, 47]. \nFor the region covered by DES (3 , 567 deg 2 , 75 % of the SPT area) cluster candidates are optically confirmed using the Multi-Component Matched Filter cluster confirmation tool [MCMF; 33, 34], and we follow the work of Refs. [34, 35]. Moreover, measurements for cluster redshift z , optical richness ˆ λ and optical center position are obtained using MCMF. Because DES data are only reliable for redshifts z ≤ 1 . 1, WISE data are used to compute richnesses and redshifts for clusters with z > 1 . 1. There is the chance that an overdensity of galaxies is a random superposition along the line of sight of a tSZE noise fluctuation. To exclude chance associations, we apply a redshift dependent richness cut, ˆ λ min ( z ), which ensures a > 98 % purity over the entire redshift range probed and a cluster candidate is then confirmed if the measured richness is larger than this threshold. Using MCMF for cluster confirmation allows us to validate clusters with lower tSZE detection significance while achieving a high sample purity, and results in a cluster sample, which is 30 % larger using the MCMF tool. \nDue to the different depths of the individual tSZE surveys, different selection thresholds in ˆ ζ are applied to obtain an approximately constant purity for the combined SPT sample. The selection criteria are \nˆ ζ > 4 . 25 / 4 . 5 / 5 (500d / SZ / ECS) , ˆ λ > ˆ λ min ( z ) , z > 0 . 25 . (2) \nThese selections result in a sample of 895 confirmed clusters over this region. \nTo summarize, the total cluster sample consists of 1,005 clusters, each characterized by the observables tSZE detection significance ˆ ζ and redshift z . In the SPT area covered by DES, these clusters also have additional measurements of richness ˆ λ and cluster center position from either DES ( z ≤ 1 . 1) or WISE ( z > 1 . 1). \nFigure 1 shows the 1,005 confirmed cluster sample in the space of tSZE detection significance ˆ ζ and redshift (left) as well as in optical richness ˆ λ and redshift (middle). The clusters are color coded according to which of the three SPT surveys they originate from. In addition, the corresponding selection thresholds are shown as lines in each observable. One can see in the ˆ ζ -z distribution that while the sample extends to z ∼ 1 . 8 the bulk of the clusters lie at z < 1. In addition, this distribution makes clear that the majority of the cluster lies close to the tSZE detection thresholds of the surveys. The ˆ λ -z distribution displayed in the middle panel shows that the bulk of the clusters have relatively high richnesses \nˆ λ > 30, and that the optical selection thresholds are not significantly impacting the completeness of the confirmed cluster sample. Indeed, the MCMF selection thresholds largely impact the high purity of the sample, because the noise fluctuations present in the tSZE selected candidate clusters are efficiently removed due to their low optical richnesses ( ˆ λ < 10).', 'B. DES Y3 weak-lensing data': 'The DES was conducted in the griz Y bands and covers a sky area of 5 , 000 deg 2 . The DES Y3 weak lensing shape catalog [48] utilized data from the first three years of observations and covers approximately 4 , 143 deg 2 of the sky after masking. 3 , 567 deg 2 of the DES region overlaps with the SPT surveys, corresponding to 75 % of the whole SPT survey area. The weak-lensing shape catalog of DES Y3 is built with the Metacalibration pipeline from the r , i and z bands [49, 50]. Lensing source galaxies are selected in four tomographic redshift bins as employed in the 3 × 2pt analysis of DES [51]. \nFor each cluster in the overlapping region of SPT and DES, we use the weak-lensing shear profiles within the radial range 0 . 5 < r/ ( h -1 Mpc) < 3 . 2 (1 + z cluster ) -11 around the optical cluster center. The lower limit on the radial range excludes the inner region of the cluster, which is largely affected by feedback from active galactic nuclei, miscentering, blending, cluster member contamination and non-linear shear. The upper limit on the radial range guarantees that only the one-halo term region is used for mass calibration [52]. We only use weaklensing data from DES for clusters with z < 0 . 95, corresponding to the median redshift of the highest redshift tomographic bin [SB24b, SB24a]. \nIn our analysis, we account for systematic and statistical uncertainties such as cluster member contamination, miscentering of the shear profile, shear and photo-z calibration, halo mass modeling and the impact of largescale structure. A detailed description of the modeling of these uncertainties can be found in detail in SB24a, Section V. Note that the calibration of these uncertainties was performed within the ΛCDM paradigm, but we expect these to not change significantly in f ( R ) gravity [53]. In total, our analysis includes 688 cluster shear profiles from 555 , 912 source galaxies with an average of 808 shear source galaxies per cluster [SB24a]. We show for illustration the averaged matter density profile of the DES Y3 data in the right panel of Fig. 1. The profiles are broken into four redshift ranges of comparable signalto-noise. The combined dataset corresponds to a 31 . 2 σ detection of the matter profiles of these clusters. \nFIG. 1. Left : tSZE detection significance ˆ ζ and redshift distribution for the three SPT surveys. Dashed-colored lines show the detection threshold for the corresponding survey in the region that overlaps with DES. The black dashed line shows the ˆ ζ threshold for the clusters outside of the DES region (same as the cut for the SPTpol ECS survey). Middle : optical richness ˆ λ and redshift distribution of the cluster sample color coded by the three surveys. Colored lines correspond to the ˆ λ detection threshold of the given survey. The SPTpol 500d is significantly deeper than the other two surveys (yellow-green dashed line) and thus a lower ˆ λ ( z ) threshold is applied (solid blue line). Right : averaged weak-lensing inferred projected matter profiles for DES Y3 data shown for four redshift bins in purple to yellow. The redshift bins are chosen such that the signal-to-noise is approximately equal in each bin. The HST data are shown in blue. Over the radial range used for weak-lensing mass calibration, the DES Y3 data have a signal-to-noise of 31 . 2 and the HST data have 9 . 7. \n<!-- image -->', 'C. HST weak-lensing data': 'DES lensing data are only reliable for z ≲ 0 . 95 and therefore, we complement the weak-lensing dataset with HST data to obtain weak-lensing information for the mass calibration at high redshift. We use the HST39 dataset [54-56] to obtain weak-lensing shear profiles. This dataset contains 39 clusters of our tSZE selected sample in the redshift range 0 . 6 -1 . 7. More details about the dataset and the analysis can be found in Refs. [54-59]. The averaged matter density profile from the 39 clusters from HST is shown in Fig. 1 in the right panel in blue. These data correspond to a 9 . 7 σ detection of the matter profiles of these halos.', 'III. f ( R ) MODIFIED GRAVITY': 'f ( R ) gravity modifies GR by introducing an arbitrary function f ( R ) of the Ricci scalar R into the EinsteinHilbert action [6] \nS = ∫ d 4 x √ -g [ R + f ( R ) 16 πG + L m ] , (3) \nwhere g denotes the determinant of the metric tensor, G is the gravitational constant and L m is the matter Lagrangian density. Note that we use natural units where c = ℏ = 1 and if f ( R ) = -2Λ we recover GR plus a cosmological constant, hence a ΛCDM cosmology. The \nfield equation obtained from this modified action takes the form \nG µν + f R R µν -( f 2 -□ f R ) g µν -∇ µ ∇ ν f R = 8 πGT µν . (4) \nHere G µν is the Einstein tensor, R µν is the Ricci tensor, T µν represents the energy-momentum tensor and f R = d f ( R ) / d R is an additional scalar degree of freedom, which indicates the strength of the modifications to GR. \nThe equation of motion for f R is derived from the trace of the field equation in the quasistatic and weak-field limit \n∇ 2 δf R = 1 3 ( δR -8 πGδρ ) , (5) \nwhere δx = x -¯ x is the perturbation of the quantity x with respect to the cosmic mean. Additionally, from the time-time component of the field equation, Eq (4), we derive the modified Poisson equation in f ( R ) gravity: \n∇ 2 Φ = 16 πG 3 δρ -1 6 δR, (6) \nwith Φ the Newtonian potential which is defined via 2Φ = δg 00 /g 00 . Combining Eqs. (5) and (6) shows how the Poisson equation and thus the structure growth depends on the strength of the f ( R ) gravity model \n∇ 2 Φ = 4 πGδρ -1 2 ∇ 2 δf R . (7) \nCompared to GR the Poisson equation includes a term proportional to the Laplacian of δf R and thus depends directly on the strength of the model. \nThe modified Poisson equation, Eq. (6), shows that the strength of the gravitational force depends on the environment. In low curvature regions, δR ≪ 8 πGδρ , and consequently the f ( R ) Poisson equation reduces to a modified Poisson equation with gravitational forces enhanced by a factor of 4 / 3 compared to GR. In high curvature environments, the Ricci scalar R is approximately 8 πGδρ and thus Eq. (6) corresponds to the unmodified GR Poisson equation, i. e. in high-density regions f ( R ) gravity falls back to GR. This property of driving f R → 0 in high-density regions is the so-called chameleon screening mechanism [60], which makes f ( R ) gravity consistent with Solar systems tests [7, 61]. The two limits show that structure growth in f ( R ) gravity is environmentdependent in contrast to GR. \nIn this paper, we adopt the widely used and studied Hu & Sawicki model [7] \nf ( R ) = -m 2 c 1 ( R m 2 ) n c 2 ( R m 2 ) n +1 , (8) \nwith m 2 = Ω m H 2 0 , H 0 the Hubble constant and the free parameters n , c 1 , c 2 . In the high curvature regime we have c 1 /n 2 R/m 2 ≫ 1 and Eq. (8) is approximately given by \nf ( R ) ≈ -m 2 c 1 c 2 -f R 0 R n +1 0 nR n . (9) \nHere R 0 is the present background curvature and f R 0 := f R ( R 0 ), which is the parameter that quantifies the strength of this f ( R ) gravity model. \nTo have a modified gravity scenario that is active at late times and large scales, when the acceleration of the Universe happens, and that does not spoil the successful description of early Universe observables such as BBN and CMB the function f ( R ) has to satisfy f R < 0, i. e. it is a decreasing function of R . Therefore, the approximation above is correct up to order ∼ ( f R 0 ) 2 , and since the constraints from our cluster sample are better than | f R 0 | ∼ 10 -4 , the approximation in Eq. (9) is entirely sufficient. We work with log 10 | f R 0 | for numerical convenience, and, given that the theory gives no strong prior on the scale of f R 0 , we impose a uniform prior on log 10 | f R 0 | . \nTo obtain an expansion history consistent with ΛCDM in the limit | f R 0 | → 0, the parameters c 1 and c 2 are given by \nc 1 c 2 = 6 Ω Λ Ω m . (10) \nWe further adopt n = 1 for the Hu & Sawicki model; see Ref. [62] for an approach to approximately rescale constraints from n = 1 to other values of n .', 'A. The halo mass function in f ( R ) gravity': 'As we constrain f ( R ) gravity with the help of cluster abundance datasets we need a model for the distribution of halo mass and redshift within this theory, in other words the (differential) halo mass function. In this work we adopt a two-component model [28, 42] for which the first component is the HMF in GR, and the second factor models the enhancement of the f ( R ) gravity, which accounts for the scale-dependent clustering due to the scale-dependent structure growth described in Sec. III: \nd n dln M = d n dln M ∣ ∣ ∣ ∣ GR ×R . (11) \nFor the GR HMF, we use the halo mass function from Ref. [63] \nd n dln M ∣ ∣ ∣ ∣ GR = -¯ ρ m 2 M f ( σ ) T dln σ 2 dln M , (12) \nwith the multiplicity function of the form \nf ( σ ) T = ˜ A [ ( σ ˜ b ) -˜ a +1 ] e -˜ c σ 2 , (13) \nwhere ˜ A, ˜ a, ˜ b and ˜ c are parameters calibrated using Nbody simulations [see 63, table 2] and σ = σ ( M ) is the variance of the overdensity field on a mass scale M in the corresponding GR cosmology. \nThe enhancement factor R is calculated from the ratio of the Sheth-Tormen HMF [64] in f ( R ) gravity to GR, i. e. \nR = d n dln M ∣ ∣ ST , f ( R ) d n dln M ∣ ∣ ST , GR . (14) \nThis HMF can account for the scale-dependent clustering through the spherical collapse threshold δ crit . In this work, we calculate the spherical collapse threshold δ crit from the spherical collapse model of Refs. [43, 44]. With this, the Sheth-Tormen HMF is given by \nd n dln M ∣ ∣ ∣ ∣ ST = ¯ ρ m M f ( ν ) ST [ dln δ crit dln M -1 2 dln σ 2 dln M ] . (15) \nHere ν = δ crit /σ is the peak height and f ( ν ) ST refers to the Sheth-Tormen multiplicity function [64], \nf ( ν ) ST = A √ aν 2 2 π [ 1 + ( aν 2 ) -p ] e -aν 2 2 , (16) \nwith A, a , p are free parameters for which we adopt the parameters of Ref. [65]. \nThe change in the clustering and thus in the HMF is completely captured by the scale-dependent spherical collapse threshold δ crit and therefore, we use an effective variance σ derived from the corresponding GR cosmology [44, 66]. Moreover, when including massive neutrinos, the HMF shape is closer to universal if the halo \nmass and variance are calculated neglecting the neutrino component [67, 68], i. e. using the baryon and cold dark matter power spectrum instead of the total matter power spectrum for the variance. Since there is a degeneracy between massive neutrinos and log 10 | f R 0 | (at least for high values of Σ m ν and log 10 | f R 0 | ) [see e. g. 69-71] we use the approach of Ref. [67, 68] to account for the non-zero total neutrino mass. \nThe derivation of δ crit with the spherical collapse model requires solving a computationally expensive system of coupled differential equations for each value of mass, redshift and cosmology [43, 44]. To speed up the calculations we use emulators to predict the values of δ crit and d δ crit / dln M . The emulators adopted in this analysis are presented in SV24a. \nFigure 2 shows the f ( R ) HMF on top and the ratio R on the bottom for different values of log 10 | f R 0 | and redshift. In f ( R ) gravity, the HMF is larger compared to GR as expected from the enhanced structure growth. The shape and strength of the enhancement depend on the value of log 10 | f R 0 | with a larger enhancement for stronger f ( R ) models. Furthermore, the enhancement becomes smaller at higher redshifts, because modified gravity falls back to GR in the early universe. A large enhancement of the most massive halos is only seen for the strongest model with log 10 | f R 0 | = -4 . 5 and at lower redshifts, because even the most massive halos are only partially screened in this scenario. In the other cases, the potential Ψ of the high mass halos becomes compatible with f R and the fifth force is screened in these massive halos, reducing the modified gravity effect on the halo abundance.', 'B. HMF Calibration using FORGE simulations': 'The model for the HMF presented above follows a semi-analytical approach where the f ( R ) gravity is incorporated via the spherical collapse threshold δ crit calculated from spherical collapse theory in f ( R ) gravity. Besides semi-analytical models, the HMF can also be derived from simulations. Therefore, we make a comparison with f ( R ) simulations to validate the semi-analytical model and to check for any kind of discrepancy. \nIn this work, we use the state-of-the-art FORGE N -body simulations which encompass 49 f ( R ) gravity cosmologies (nodes) [45]. The simulations sample the cosmological parameters Ω m , h , S GR 8 = σ GR 8 √ Ω m / 0 . 3, and log 10 | f R 0 | (hereafter FORGE parameters) with a latin hypercube, while the other cosmological parameters are fixed to n s = 0 . 9652 Ω b = 0 . 049199, Ω ν = 0 and Ω Λ = 1 -Ω m [45]. Each FORGE f ( R ) gravity node has a ΛCDM counterpart. The parameter ranges explored in \nFIG. 2. The f ( R ) HMF, Eq. (11), (top) and the ratio R of the f ( R ) and the GR Sheth-Tormen HMF, Eq. (15), (bottom) for different values of log 10 | f R 0 | (different colors) and three redshifts (different lifestyles) corresponding to the minimum, mean and maximum redshift of the SPT cluster sample. The deviation from GR depends on the strength of the f ( R ) gravity and redshift, whereby weaker f ( R ) models and higher redshifts show less enhancement in the HMF relative to GR. \n<!-- image --> \n/circledot \nthe FORGE simulations are \n0 . 11 < Ω m < 0 . 54 , 0 . 61 < h < 0 . 81 , 0 . 6 < S GR 8 < 0 . 9 , (17) -6 . 17 < log 10 | f R 0 | < -4 . 51 . \nThe ranges in Ω m and S GR 8 translate to a range in σ GR 8 of 0 . 49 < σ GR 8 < 1 . 31. \nRef. [53] presents an emulator for the f ( R ) HMF that is based upon the FORGE simulations. In this work, the authors trained a neural network to directly predict the enhancement factor of the HMF due to f ( R ) gravity for each mass, redshift and set of FORGE parameters. However, by design, the resulting halo mass function is only valid within the mass, redshift and cosmological parameter range used in the neural network training set. To make predictions outside the mass range available within the FORGE simulations, we calibrate the semi-analytical HMF model, which is valid up to halo masses of 10 16 h -1 M ⊙ , with the HMF retrieved from the FORGE simulations. \nTo calibrate the semi-analytical HMF model with the \nFORGE simulations we use the high-resolution simulations with 1024 3 dark matter particles in a box with length L = 500 h -1 Mpc and a mass resolution of 9 . 5 × 10 9 (Ω m / 0 . 3) h -1 M ⊙ . We extract halo catalogs from all f ( R ) simulations as well as from their corresponding ΛCDM nodes at redshifts z = 0 . 00 , 0 . 25 , 0 . 5 , 0 . 75 , 1 . 00 , 1 . 25 , 1 . 50 , 1 . 75, and 2 . 00, using a bin width of 0.1 in log 10 M within the mass range 10 13 h -1 M ⊙ ≤ M 200c ≤ 5 × 10 15 h -1 M ⊙ . To ensure no empty bins at the high-mass end, we combine the last high-mass bins into one (large) bin such that it contains at least 20 halos 2 . We compute covariance matrices for each halo catalog using 5 3 jackknife samples to account for noise due to sample variance and shot noise. \nWe characterize the difference between the semianalytical model and the simulation by comparing the enhancement in the HMF, R , from the simulations and the semi-analytical HMF model. This has the advantage that the cosmic variance in the simulations partially cancels in this ratio. With the halo catalogs from f ( R ) gravity and ΛCDM FORGE simulations, we can calculate the enhancement in the cluster counts, R FORGE , for each node and redshift. The enhancement from the semi-analytical model, R SAM , is also calculated for each FORGE cosmology and redshift. \nThe top panels of Fig. 3 show the ratios between the enhancements in each mass bin, i. e. R FORGE / R SAM , for four of the nine different redshifts with error bars derived from the jackknife covariance of the simulations. There is a bias between the semi-analytical HMF and the FORGE simulations, which varies with mass, redshift and the FORGE parameters. In Fig. 3 the ratio is color-coded based on the value of log 10 | f R 0 | and the discrepancy is larger with higher values of log 10 | f R 0 | . Note that also the other cosmological parameters can drive the difference between the semi-analytical HMF and the simulations. Therefore, we assume that the bias depends on all FORGE parameters. Overall the semi-analytical HMFmodel predicts more clusters than the FORGE simulations, and the agreement is better for weaker f ( R ) models, i. e. smaller values of log 10 | f R 0 | . \nBased on the top panels of Fig. 3 we model the ratio and thus the correction to the semi-analytical HMF with a broken linear function in log 10 M with a pivot scale log 10 M piv and smooth transition with strength k between the two linear functions at given logarithmic mass log 10 M 1 . To be precise, for log 10 M smaller than log 10 M 1 , the ratio is modeled with a linear function in log 10 M with slope a and intercept b . For log 10 M larger than log 10 M 1 , the same ratio is modeled by a different linear function with slope a 2 = a + ∆ a and intercept fixed by continuity at the transition mass M 1 . We then smooth the transition between the two linear relations by a power-law interpolation controlled by a parameter k . For fixed pivot mass, transition mass and smoothing \nstrength, the fitting function for the correction has three free parameters and is given by \nc ( M, p , z ) = a ( p , z )(log 10 M -log 10 M piv ) + b ( p , z ) +∆ a ( p , z ) ln(1 + e -k (log 10 M -log 10 M 1 ) ) k +∆ a ( p , z )(log 10 M -log 10 M 1 ) . (18) \nHere p represents the vector containing the FORGE parameters, a ( p , z ), ∆ a ( p , z ) , b ( p , z ) are the fitting parameters of the correction for which we assume a dependence on the FORGE parameters and redshift. We decided to fit for the parameter ∆ a ( p , z ) instead of a 2 ( p , z ) = a ( p , z ) + ∆ a ( p , z ) because it showed a better behavior during fitting. \nThe parameters of the broken linear function are obtained by fitting the correction function, Eq. (18), to the ratio between the enhancements, R FORGE /R SAM , where we fix the parameters k = 2, log 10 M 1 = 14 . 5 and log 10 M piv = 13. With this correction, we achieve good agreement with the simulations within the error bars as shown in the bottom panels of Figure 3. \nBecause we assume that the fitting parameters depend on cosmology and redshift we have to predict these parameters for an arbitrary cosmology and redshift to use the calibrated HMF in our analysis. To do so we build emulators based on Gaussian process regression to predict the three fitting parameters as a function of the FORGE parameters and redshift. \nWith the emulators, we can calculate the correction to the HMF for an arbitrary cosmology (within the parameter ranges Eq. (17)), and our calibrated semi-analytical HMF is given by \nd n dln M = d n dln M ∣ ∣ ∣ ∣ GR × c ×R , (19) \nwhere R is the enhancement factor in the semi-analytical halo mass function from Eq. (14) and c is the correction function, Eq. (18), calibrated with the FORGE simulations. As we model the correction function c ( M, p , z ) with a broken linear function, the enhancement of the HMF, c R , may be reduced to values below one, i. e. the f ( R ) HMF is suppressed rather than enhanced. Because the semi-analytical model always predicts an enhancement and also the FORGE simulations show no sign of suppression apart from noise at the high-mass end, we enforce c R ≥ 1 for the mass range considered in this work. Specifically, we set c R = 1 for all masses above the mass where the corrected HMF factor drops below unity. \nUsing the 49 FORGE simulations at 9 different redshifts we calibrate the semi-analytical HMF model presented in Sec. III A; however, there are limitations to this approach that we have to address. First, due to the relatively small box size of 500 h -1 Mpc the halo catalogs contain only halos with M ≲ 4 × 10 15 h -1 M ⊙ depending on cosmology and redshift. Thus the correction function is only calibrated in the mass range \nFIG. 3. Comparison of the enhancement in the HMF from the FORGE simulations, R FORGE , and the semi-analytical HMF model, R SAM for different redshifts, color-coded by the log 10 | f R 0 | values. Grey dashed lines are plotted at 10 % deviation to guide the eye. The upper four plots show the comparison of the FORGE simulations to the semi-analytical model, Eq. (11), without a correction. The four plots at the bottom show the comparison to the corrected semi-analytical HMF, Eq. (19), leading to a better agreement between the two HMF enhancements. Error bars are derived from the jackknife covariance of the FORGE simulations. \n<!-- image --> \n/circledot \n/circledot \n/circledot \n/circledot \nthat is available from the simulations. However, the semi-analytical prediction gives an overall shape of the f ( R ) HMF. Moreover, the halo mass function is exponentially suppressed at high masses and therefore a correction in this regime has negligible impact on our cosmological analysis. The second limitation is the available range for log 10 | f R 0 | values. FORGE samples log 10 | f R 0 | in the range -6 . 17 < log 10 f R 0 < -4 . 51, so we can only predict the parameters of the correction function in this range. Therefore, we adopt a hard upper limit prior of log 10 | f R 0 | = -4 . 51 in our analysis. We extend our analysis to log 10 | f R 0 | = -7 based on the fact that the f ( R ) HMF approaches the GR HMF for log 10 | f R 0 | → -∞ and thus a, ∆ a → 0 and b → 1. Thus, we interpolate the parameters between log 10 f R 0 = -6 . 17 and log 10 f R 0 = -7 under the assumption that a model with log 10 f R 0 = -7 is indistinguishable from GR with the SPT dataset. Moreover, the correction from the simulation is smaller as we approach GR. We will show in our analysis that the upper bound has no impact on the f R 0 posterior, as the data strongly disfavor values of log 10 f R 0 > -4 . 51. Given these limitations on the calibration, we present both the results derived when using the semi-analytical HMF model and when using the FORGE-informed calibrated semi-analytical HMF model. \nIn the analysis presented below, we account for the impact of remaining uncertainties in the HMF by following the approach of Ref. [72] and introducing uncertainties \nin the amplitude and the logarithmic mass trend of the HMF, i. e. the HMF with uncertainties given by \nd n dln M = d n dln M ( q + s ln ( M 200c 10 14 h -1 M ⊙ )) , (20) \nwhere q is the uncertainty in the amplitude and s is the uncertainty in the trend with logarithmic mass and we marginalize over the q and s in our analysis (see Sec. IV D)', 'IV. ANALYSIS METHOD': 'The method we employ in this work is based on the state-of-the-art weak lensing informed cluster cosmological analysis of the SPT sample [SB24a]. The method was also used and validated in the recent f ( R ) gravity forecast [SV24a] for SPT-3G [73] and CMB-S4 [74] cluster samples with next-generation weak-lensing data like those expected to come from the Euclid mission [75, 76] or the Vera C. Rubin Observatory [77, 78].', 'A. Observable-mass relations': 'In tSZE cluster surveys, galaxy clusters are identified and selected by observables such as the tSZE detection significance and richness and observable-mass relations link these observables to the halo mass [e.g., 79, 80]. \nThrough gravitational weak lensing calibration of these relations, we can relate the observed cluster sample to the HMF, which describes the abundance of halos depending on cosmology, mass, and redshift. In this analysis, we employ observable-mass relations that are empirically calibrated with weak-lensing data [e.g. 24, 81-86]. This section outlines the observable-mass relation for tSZE detection significance and optical/NIR richness.', '1. tSZE ζ -mass relation': 'As in previous SPT studies, we first relate the observed tSZE detection significance ˆ ζ to the intrinsic detection significance ζ to account for noise in the data. The relation between ˆ ζ and ζ is given by [87] \nP ( ˆ ζ | ζ ) = N ( √ ζ 2 +3 , 1 ) . (21) \nThe distribution accounts for the Gaussian noise present in the survey maps, with a correction factor of 3 due to the noise resulting from the matched-filter search for peaks in three dimensions. The mean intrinsic tSZE detection significance ζ is then modeled by \n⟨ ln ζ ⟩ = ln ζ 0 + ζ M ln ( M 200c 3 × 10 14 h -1 M ⊙ ) + ζ z ln ( E ( z ) E (0 . 6) ) , (22) \nwhere ζ 0 , ζ M and ζ z are the parameters corresponding to the normalization, mass and redshift trend of the scaling relation and E ( z ) = H ( z ) /H 0 . Adiitionally, we assume a lognormal intrinsic scatter in ζ with width σ ln ζ . \nBecause the SPT surveys vary in depth, and we want to employ one ζ -mass relation for all surveys, the normalization ζ 0 and the redshift trend ζ z are rescaled for each field [35, 46, 47], i. e. ζ 0 , field = γ field ζ 0 and ζ z, field = ζ z +constant. In the case of the SPTpol ECS survey fields the normalization is difficult to calibrate and thus the parameter γ ECS is allowed to vary in the analysis [SB24a]. For the redshift trend ζ z the variation of the rescaling parameter c across fields is negligible in the SPT-SZ and SPTpol surveys. Therefore, we rescale ζ z for each survey where the SPT-SZ survey is taken as the reference [35]: \nζ z, SPT-SZ = ζ z , ζ z, SPTpol ECS = ζ z -0 . 09 , ζ z, SPTpol 500d = ζ z +0 . 26 . (23)', '2. Cluster richness λ -mass relation': 'As for the tSZE detection significance, the observed richness ˆ λ is related to the intrinsic richness λ by a Gaussian distribution of the form \nP ( ˆ λ | λ ) = N ( λ, √ λ ) . (24) \nThis relationship accounts for Poisson sampling noise. Note that the Gaussian approximation of a Poisson distribution approximately holds for λ ≳ 10, which is below the richness cut we apply to our sample. Similar to the ζ -mass scaling relation we assume for the mean intrinsic richness a power law, in mass and (1 + redshift): \n⟨ ln λ ⟩ = ln λ 0 + λ M ln ( M 200c 3 × 10 14 h -1 M ⊙ ) + λ z ln ( 1 + z 1 . 6 ) . (25) \nThe parameters λ 0 , λ M and λ z govern the normalization, mass and redshift trend, respectively. The cluster intrinsic richness varies around this relation by a log-normal distribution with a width σ ln λ . \nAs mentioned in Sec. II A we use richness measurements from DES for clusters with z ≤ 1 . 1 and data from WISE for high-redshift clusters. As matching two distinct types of richness measurements is challenging, we use two separate λ -mass relations for the DES and WISE data [SB24b], hereafter denoted by subscripts DES and WISE respectively.', 'B. Weak-lensing model in f ( R ) gravity': 'With the above described observable-mass relations, we can model the cluster sample in the ζ -λ -z space by transforming the halo mass function into the halo observable function and using it to predict cosmological parameters. However, there are no informative priors on the parameters of the ζ and λ scaling relations and their scatters. To empirically calibrate these relations and the corresponding scatters we rely on weak-lensing data. It has been shown that weak-lensing measurements are a robust way to measure halo masses with well-characterized and controllable biases [SB24b, SB24a]. \nIn this analysis, we assume that any f ( R )-gravity modification of the mapping from cluster potential to lensing signal can be neglected, as in previous works [11, 41]. First, the f ( R ) effect on the lensing signal for a given fixed mass distribution is given by a rescaling of the GR signal by a factor of (1 + | f R 0 | ) -1 [88, 89], which is negligible for the values of | f R 0 | we consider in this work. Second, while the cluster observables and halo profiles do undergo modifications in f ( R ) gravity, these effects are small [53, 90, 91]. Any changes to the cluster observables ˆ ζ and ˆ λ will be accounted by the empirical calibration of the observable mass relations. Changes to the halo profiles are more concerning, because they could impact the weak lensing inferred cluster masses. Accounting for these effects self-consistently within the weak lensing model described below would require a study of the halo shapes using f ( R ) numerical simulations and measurement of any changes in the inferred weak-lensing masses. Because we know these effects are smaller than the current uncertainties on the weak lensing model, which are \ndominated by uncertainties in the hydrodynamical effects and on photometric redshift systematics, we adopt the GR-based calibration of the weak lensing model presented in SB24a and described below.', '1. DES weak-lensing model': 'The model we adopt for DES weak-lensing data was studied and described in detail in SB24a and works referenced therein. Here we provide a summary of the method. The weak-lensing observable is the reduced tangential shear profile, which is related to the underlying projected halo mass distribution Σ by \ng t ( r, M WL ) = ∆Σ( r, M WL ) Σ -1 crit 1 -Σ( r, M WL ) Σ -1 crit . (26) \nHere ∆Σ( r ) ≡ ⟨ Σ( < r ) ⟩-Σ( r ) is the surface density contrast and Σ -1 crit is the lensing efficiency or inverse critical surface mass density, given by \nΣ -1 crit = 4 πG c 2 D l D s × max[0 , D ls ] , (27) \nwhere c is the speed of light and D s , D l , D ls are the angular diameter distances between the observer and the source, the observer and the lens, and the source and the lens, respectively. We model Σ by the line of sight integral of a Navarro-Frenk-White profile (NFW) [92, 93] and we refer to the associated mass as the weak-lensing mass, M WL . We account for possible miscentering of the selected cluster center by assuming a constant density within the cluster miscentering radius R min , i. e. Σ( R ) = Σ( R min ) for R ≤ R min (see SB24a section IVC for more details). Cluster member contamination f cl ( r ) is corrected by a factor (1 -f cl ( r )) to the reduced tangential shear profile [SB24a]. \nGiven that the cluster profiles are not perfectly described by an NFW profile, the computed weak-lensing mass M WL is a biased and noisy estimator of the true cluster mass M 200c [94, 95]. To account for the bias we use a scaling relation between M WL and M 200c with a mean relation of [52] \n〈 ln ( M WL M 0 )〉 = ln M WL 0 ( z ) + M WL M ln ( M 200c M 0 ) . (28) \nHere ln M WL 0 is the logarithmic mass bias normalization and M WL M is the mass trend in this bias at a pivot mass M 0 = 2 × 10 14 h -1 M ⊙ . We assume a log-normal scatter of the true relations with a width described by \nln σ ln WL 2 = ln σ 2 ln WL 0 ( z ) + σ 2 ln WL M ln ( M 200c M 0 ) , (29) \nwhere ln σ 2 ln WL 0 is the normalization and σ 2 ln WL M is the mass trend of the scatter. \nThe parameters of the above mean scaling relation and scatter are calibrated from simulations by extracting the \nTABLE I. Normalization and uncertainties of the amplitude and scatter of the weak-lensing-mass-to-halo-mass relation derived from the simulations at redshifts z ∈ { 0 . 252 , 0 . 470 , 0 . 783 , 0 . 963 } . \nweak-lensing inferred mass from hydrodynamical simulations and calculating the corresponding cluster mass from the matched N -body simulation at different redshifts [52]. The calibration results in a mean value and uncertainty obtained from the posterior for each of the above parameters. \nIn this model, the logarithmic mass bias normalization, ln M WL 0 , and the normalization of the scatter, ln σ 2 ln WL 0 , are functions of redshift and calibrated from the simulations at four redshift values: z ∈ { 0 . 252 , 0 . 470 , 0 . 783 , 0 . 963 } . Therefore, we model the two parameters in the analysis as \np = N (¯ p, (∆ p ) 2 ) = ¯ p ( z ) + ∆ p ( z ) N (0 , 1) , (30) \nwhere ¯ p ( z ) is the mean value and ∆ p ( z ) is the uncertainty of the corresponding parameter p ( z ) at redshift z . We interpolate linearly to obtain the values for these parameters at any intermediate redshift. To accurately describe the uncertainty of the logarithmic mass bias ln M WL 0 ( z ), the uncertainty in this parameter, ∆ln M WL 0 ( z ), is modeled as a linear combination of two redshift-dependent functions [SB24a]: \n∆ln M WL 0 ( z ) = ∆ 1 ln M WL 0 ( z ) + ∆ 2 ln M WL 0 ( z ) . (31) \nThe values of the bias and scatter normalization parameters, Eqs. (28) and (29), as well as their uncertainties at the simulation redshifts used in this work are summarized in Tab. I. The uncertainties of these parameters include various elements such as uncertainties from baryonic effects, photoz calibration, miscentering and shear calibration [52]. The total uncertainty is primarily influenced by uncertainties in baryonic effects at low redshifts, while at high redshifts, the uncertainty in photoz calibration becomes dominant. Overall uncertainty from the weak-lensing model remains small across the calibrated redshift range, contributing to approximately 1 % of the total uncertainty (see Figure 10 in SB24a).', '2. HST weak-lensing model': 'A similar model is applied to the HST-39 dataset. The shear profiles from HST are modeled by the line of sight integral of an NFW profile with a concentration from \nRef. [96]. From the NFW a weak-lensing mass M WL is calculated and related to the true halo mass with a mean relation [54]. \n⟨ ln M WL ⟩ = ln M WL 0 +ln M 200c . (32) \nThe true relation scatters around the mean by a Gaussian distribution with width σ ln WL . The scatter σ ln WL accounts for all sources of uncertainties in the M WL -M 200c relation. Here each cluster has its own bias and scatter and associated uncertainties by calibrating Eq. (32) for each cluster individually. We refer the reader to the original works for a more detailed explanation of the cluster lensing model employed in the HST dataset [54-56, 59].', 'C. Multivariate observable-mass relation': 'To account for possible correlation among the three observables, unbiased tSZE detection significance ζ , intrinsic richness λ and weak-lensing mass M WL we employ the multivariant observable-mass relation from the work of SB24a. For this, the lognormal scatters of the observables, σ ln ζ , σ ln λ and σ ln WL are combined into a covariance matrix of the form \nΣ = σ 2 ln ζ ρ SZ , WL σ ln ζ σ ln WL ρ SZ ,λ σ ln ζ σ ln λ ρ SZ , WL σ ln ζ σ ln WL σ ln WL 2 ρ WL ,λ σ ln WL σ ln λ ρ SZ ,λ σ ln ζ σ ln λ ρ WL ,λ σ ln WL σ ln λ σ 2 ln λ (33) \nwhere ρ SZ ,λ , ρ SZ , WL and ρ WL ,λ are the correlation coefficients between ζ and M WL , ζ and λ , and λ and M WL respectively. The joint multi-observable-mass relation is then given by a multivariate Gaussian with correlation matrix Σ \nP ( ln ζ ln M WL ln λ ∣ ∣ M,z, p ) = N ( ⟨ ln ζ ⟩ ( M,z, p ) ⟨ ln M WL ⟩ ( M,z, p ) ⟨ ln λ ⟩ ( M,z, p ) , Σ ) . (34)', 'D. Likelihood and priors': 'The analysis relies on Bayesian statistics and we obtain cosmological and scaling relation parameters p using a cluster population model. The likelihood model employed in this analysis is based on the recent ΛCDM SPT × DES+HST analysis of SB24b, SB24a and was verified for an f ( R ) cosmology in our forecast work [SV24a]. Following SB24a and SB24b, we approximate the multiobservable cluster abundance likelihood with a Poisso- \n, \nnian distribution: \nln L ( p ) = ∑ i ln ∫ ∞ ˆ λ min d ˆ λ d 3 N ( p ) d ˆ ζ d ˆ λ d z ∣ ∣ ∣ ˆ ζ i ,z i -∫ z max z min d z ∫ ∞ ˆ ζ min d ˆ ζ ∫ ∞ ˆ λ min d ˆ λ d 3 N ( p ) d ˆ ζ d ˆ λ d z + ∑ i ln d 4 N ( p ) d ˆ ζ d ˆ λ d g t d z ∣ ∣ ∣ ˆ ζ i , ˆ λ i , g t ,i ,z i ∫ ∞ ˆ λ min d ˆ λ d 3 N ( p ) d ˆ ζ d ˆ λ d z ∣ ∣ ∣ ˆ ζ i ,z i +const . , (35) \nwhere both sums run over all clusters i . The differential cluster numbers d 3 N dobs in the above likelihood are the differential halo observable function (HOF) and is given in the ˆ ζ -ˆ λ -z space by \nd 3 N ( p ) d ˆ ζ d ˆ λ d z = ∫ dΩ s ∫∫∫ d M d λ d ζ P ( ˆ ζ | ζ ) P ( ˆ λ | λ ) P ( ζ, λ | M,z, p ) d 2 N ( M,z, p ) d M d V d 2 V ( z, p ) d z dΩ s , (36) \nand in the ˆ ζ -ˆ λ -g t -z by \nd 4 N ( p ) d ˆ ζ d ˆ λ d g t d z = ∫ dΩ s ∫∫∫∫ d M d ζ d λ d M WL P ( g t | M WL , p ) P ( ˆ ζ | ζ ) P ( ˆ λ | λ ) P ( ζ, λ, M WL | M,z, p ) d 2 N ( M,z, p ) d M d V d 2 V ( z, p ) d z dΩ s . (37) \nHere Ω s is the survey solid angle, the factors d 2 N ( M,z, p ) d M d z and d 2 V ( z, p ) d z dΩ s are the HMF and the differential volume element for the corresponding cosmology whereas P ( ˆ ζ | ζ ) and P ( ˆ λ | λ ) relate the observed quantity to the intrinsic one given by Eqs. (21) and (24) respectively. Moreover, P ( ζ, λ | M,z, p ) and P ( ζ, λ, M WL | M,z, p ) are obtained from the multivariant observable-mass relation Eq. (34), and P ( g t | M WL , p ) is given by the product of Gaussian probabilities in each radial bin i of the tangential shear profiles g t with the shape noise ∆ g t ,i \nP ( g t | M WL , p ) = ∏ i ( √ 2 π ∆ g t ,i ) -1 exp [ -1 2 ( g t ,i -g t ,i ( M WL , p ) ∆ g t ,i ) 2 ] . (38) \nNote that the first two terms of the total Poisson likelihood, Eq. (35), are independent of the weak-lensing data and thus are associated with the Poisson likelihood of the cluster sample in the ˆ ζ -z space with the condition ˆ λ > ˆ λ min ( z ). The last term in Eq. (35) includes the information of the mass calibration from the weak-lensing data and is therefore often called the mass-calibration likelihood. The likelihood is implemented in the CosmoSIS framework as a Python module [97] and we use the \nnested sampling algorithm nautilus to run the MCMC chains [98]. \nGiven the large number of cosmological and nuisance parameters considered in the analysis, we combine the SPT cluster data set with the primary CMB data from Planck (Planck 2018 TT,TE,EE+lowE) [99] to break parameter degeneracies and recover meaningful constraints on f ( R ) gravity. This combination of data is sound, because the standard cosmological analysis from SB24b showed no statistically significant tension between the SPT-clusters × WL dataset and the Planck data. \nWe emphasize that primary CMB data like those from Planck 2018 place only weak constraints on f ( R ) gravity of the order of log 10 | f R 0 | ≲ -3 [13, 100], and thus the constraints in log 10 | f R 0 | are primarily coming from the SPT cluster sample. However, CMB data are essential to constrain the remaining cosmological parameters such as Ω m h 2 . The Planck 2018 likelihood is implemented in CosmoSIS , and because one can assume that the cluster likelihood and the Planck 2018 likelihood are independent, we multiply the two likelihoods in a joint analysis. We account for the effect of f ( R ) gravity in the Planck 2018 likelihood by using the f ( R ) power spectrum computed with MGCAMB [101-104]. \nIn this analysis, we vary 23 nuisance parameters and eight cosmological parameters. All parameters with their priors are listed in Table II. For the standard cosmological parameters, we adopt uniform priors with ranges that are based on the Planck 2018 posteriors, since these parameters will be best constrained by the Planck 2018 dataset. For the f ( R ) gravity parameter log 10 | f R 0 | we apply a uniform prior U ( -7 , -3). Note that the GR limit at | f R 0 | = 0 cannot be reached when using a logarithmic prior and thus would introduce an infinitely large parameter volume below our lower bound when we calculate the upper limit of the f ( R ) parameter. To avoid the sensitivity of the upper limit of the f ( R ) parameter on the choice of the lower prior boundary, we transform the parameter space from logarithmic to linear. In this linear space, the parameter volume between 0 and 10 -7 is negligible. We account for the uncertainty in the weak-lensing model as described in Sec. IV B 1 by Gaussian priors on these parameters. To ensure a positive-definite covariance matrix of the multivariate observable-mass relation we assume priors on the correlation coefficients of U ( -0 . 5 , 0 . 5). No informative priors are applied for the tSZE and richness observable-mass relation parameters, and we adopt sufficiently wide uniform priors for these parameters. To account for systematic uncertainties in the HMF, we use Gaussian priors on the amplitude and slope of the HMF as defined in Eq. (20).', 'V. RESULTS': 'In this section, we present our constraints on the f ( R ) gravity model derived from the analysis of DES and HST weak lensing calibrated SPT clusters combined with \nTABLE II. Parameters and priors of our f ( R ) gravity analysis of the DES and HST weak-lensing informed SPT cluster sample and Planck 2018. The prior on Ω ν h 2 corresponds to a prior on the sum of neutrino masses Σ m ν ∼ U (0 , 0 . 6) eV. \nPlanck 2018 data. For our baseline analysis, we employ the FORGE-calibrated HMF model, because it is considered to be more accurate due to its empirical calibration with the FORGE numerical simulations. For comparison, we also include results obtained using the uncorrected semi-analytical HMF model. Note that all reported upper limits are provided at the 95 % credible level and uncertainties at 68 % credibility.', 'A. Comparison to Λ CDM results': 'We first compare our results to those from the ΛCDM analysis of SB24b. Because we report a tight upper limit on log 10 | f R 0 | (see next section) and thus the deviation from a ΛCDM cosmology is relatively small, we expect consistent results between the two analyses for all standard cosmological parameters. Figure 4 shows the posterior distribution of the cosmological parameters from both analyses. Overall, the standard cosmological parameters are in good agreement with the ΛCDM results. The only discernible parameter shifts are observed in Ω b h 2 and n s , which are both primarily constrained by the Planck 2018 data. However, the shift of the two parameters is within the 68 % credible contours and within the statistical uncertainties. An explanation for the shift can be given by the fact that Ω b h 2 and n s as well as log 10 | f R 0 | change the power spectrum on small scales, and in opposite directions. \nA complete comparison of all parameters from the f ( R ) and ΛCDM analysis can be found in Appendix A, Fig. 6. We summarize in Tab. III the constraints on the ζ -mass and λ -mass relation parameters, as well as on the correlation coefficients for the f ( R ) and ΛCDM analyses. The results are in good agreement with ΛCDM, indicating no significant deviations in the observable-mass relations under the f ( R ) gravity model. Compared to the ΛCDM constraints, we obtain slightly weaker constraints for most of the observable-mass relation parameters (see the third column of Tab. III). The largest increase in the uncertainties is observed in the ζ -mass relation parameters, the amplitude mass trend of the λ -mass relation for the DES richness, and the scatter of the λ -mass relation for the WISE richness. An overall increase in the uncertainties is expected due to the additional degrees of freedom introduced by the f ( R ) gravity model. \nWe observe a 1 . 5 σ shift in the correlation parameter between the tSZE detection significance and richness, ρ SZ,λ , compared to the ΛCDM results, which is attributed to different models for the scatter in richness assumed in the two analyses: in the ΛCDM analysis, a lognormal richness scatter with width λ -1 / 2 was used, which resulted in a negative correlation between the tSZE detection significance and richness. In this f ( R ) analysis, we use a Gaussian approximation of Poisson noise (see Eq (24)) and report a vanishing correlation between the two quantities. When using the same richness scatter model as in the ΛCDM analysis, we obtain the same results for this parameter. The same is true for the scatter parameter of the WISE richness σ ln λ, WISE .', 'B. f ( R ) gravity constraints': "Our f ( R ) analysis of the combination of weak-lensing mass calibrated SPT clusters and the Planck 2018 dataset results in the current tightest constraints available from clusters and CMB data and other probes of the large- \nTABLE III. Constraints on the observable-mass relation parameters, the scatter correlation coefficients and cosmological parameters for the f ( R ) and ΛCDM analyses in the second and third column respectively. The last column shows the relative increase of the parameter uncertainties in the f ( R ) analysis with respect to ΛCDM. Note that all reported upper limits are provided at the 95 % credible level and uncertainties at 68 % credibility. \nscale structure. With our baseline analysis using the FORGE-calibrated HMF, we obtain an upper bound on the f ( R ) parameter: \nlog 10 | f R 0 | < -5 . 32 (95 % credible level) . (39) \nThis result is consistent with a ΛCDM cosmology and excludes all f ( R ) parameter space that does not lead to substantial screening of halos. When applying the semianalytical HMF model, we achieve a 3 % tighter upper bound on log 10 | f R 0 | : \nlog 10 | f R 0 | < -5 . 46 (95 % credible level) . (40) \nThe slightly stronger constraint from the semi-analytical HMF model can be explained by the larger enhancement in the halo mass function compared to the FORGE simulations, see Sec. III B. Therefore, one expects a greater sensitivity to log 10 | f R 0 | , which leads to tighter constraints on the f ( R ) parameter. The remaining cosmological parameter posteriors are found to be consistent \nFIG. 4. Posterior distribution of the cosmological parameters for our f ( R ) analysis (gray and red) and the ΛCDM analysis from SB24b (blue) of the DES and HST weak-lensing mass calibrated SPT clusters combined with Planck 2018 CMB data. Because the two f ( R ) HMF models give similar results, the gray contours from the semi-analytical HMF (see Sec. III A) are hidden by the red ones from the FORGE calibrated HMF (see Sec. III B). As expected, the results from the analysis show consistent results in the nonf ( R ) cosmological parameters. \n<!-- image --> \nΩ \nb \nh \nwith those in the ΛCDM analysis. This is because we use Planck 2018 data in our analysis, which tightly constrains the ΛCDM cosmological parameters and thus eliminates potential degeneracies. Moreover, degeneracies between log 10 | f R 0 | and other cosmological parameters that have previously been found are more pronounced for higher \nvalues of log 10 | f R 0 | , which are excluded by our dataset [30, 70, 105]. \nFor comparison, the best previous constraints on f ( R ) gravity from clusters are presented in Ref. [28]. The authors obtained log 10 | f R 0 | < -4 . 79 using clusters from ROSAT and the Massive Cluster Survey, combined with \nprimary CMB, SN, and BAO data. Our analysis improves upon this result by a factor of 3.4 in f R 0 without using any information from SNe or BAO. The improved constraints are due to the large cluster sample and the weak lensing mass calibration dataset that we use in this analysis, with 1 , 005 clusters compared to 224 clusters in the analysis of Ref. [28]. Recent results from the eROSITA cluster analysis reported an upper limit of log 10 | f R 0 | < -4 . 12, considering the neutrino mass as a free parameter [11]. Although this constraint is significantly weaker than ours, it is important to note that their analysis was based on clusters alone. eROSITA can place meaningful constraints from the clusters alone, because of the larger weak-lensing calibrated cluster sample of 5 , 259 clusters and lower redshift range for which the effect of f ( R ) gravity is larger. On the other hand, they employed a different HMF based on the model of Ref. [30], which predicts a slightly smaller enhancement of the f ( R ) HMF. For a fairer comparison with the eROSITA results, we also perform our analysis using the HMF of Ref. [30] but still including Planck 2018 data. We obtain log 10 | f R 0 | < -5 . 11 (95 % credible level), which is an order of magnitude better than the results from Ref. [11] and 62 % weaker in f R 0 than our baseline result. These different results show that a reliable f ( R ) HMF model is needed to obtain accurate constraints on this modified gravity model. Note that different work apply different prior on log 10 | f R 0 | which affects this parameter's upper limit due to the infinite volume below the lower prior. Therefore, a comparison is always affected by the prior choice when using a uniform prior in log 10 | f R 0 | . \nThe strongest constraints from large-scale probes of f ( R ) gravity are derived from weak-lensing peak abundance using weak-lensing data from CFHTLenS [12]. In this study, the authors found an upper bound of log 10 | f R 0 | < -5 . 16 (95 % credible level) with priors from the Planck 15 analysis on Ω m and A s . This result is comparable, but slightly weaker than the constraints presented here. \nOther recent cosmological constraints on f ( R ) gravity have been derived from the cross-correlation of galaxies from BOSS combined with primary CMB and lensing data, yielding log 10 | f R 0 | ≤ -4 . 61 [13] and a combination of galaxy weak-lensing shear from the Canada-France-Hawaii Telescope Lensing Survey, CMB, SN and BAO was used to obtain log 10 | f R 0 | ≤ -4 . 50 [12]. Figure 5 shows the comparison of the constraints from the different cosmological probes discussed in this section. \nModified gravity models such as f ( R ) gravity generally enhance structure formation and thus lead to larger values of σ 8 . Therefore, they do not provide a solution to the S 8 tension. Additionally, galaxy clusters are not sensitive to H 0 , so our analysis does not offer any insights into the H 0 tension. \nFIG. 5. Comparison of log 10 | f R 0 | constraints from SPTclusters + CMB with the FORGE calibrated HMF (top bar) and the semi-analytical HMF model (second bar) with other recent results on cosmological scales. All limits are given at the 95 % credible level. \n<!-- image --> \n| \n|", 'VI. SUMMARY': 'This work presents constraints on f ( R ) gravity derived from the DES and HST weak lensing informed SPT cluster abundance combined with primary CMB data from Planck 2018. We use a sample of 1 , 005 galaxy clusters selected from the SPT-SZ, SPTpolECS, and SPT500d [34, 35, 46, 47] surveys with redshifts z > 0 . 25. 688 of these clusters have weak-lensing information from DES [SB24b, SB24a] and 39 from HST [54, 55, 59]. Our analysis framework is based on the methodology established by SB24a and in the recent f ( R ) gravity forecast for upcoming Stage-III and -IV surveys [SV24a]. \nf ( R ) gravity alters gravity and leads, compared to GR, to a scale-dependent enhancement of structure formation and thus modifies the HMF. This enhancement in the abundance of massive galaxy clusters makes cluster samples, such as those from the SPT surveys, powerful probes for testing modified gravitational models like f ( R ) gravity. \nTo capture the effects of f ( R ) gravity on the HMF, we employ a semi-analytical approach for calculating the mass-dependent spherical collapse threshold δ crit [44]. The f ( R ) HMF is then given by the GR HMF scaled by an enhancement factor, which includes the massdependent spherical collapse threshold. We emphasize that this model is designed to also capture the nontrivial screening effects in this modified gravity model, which play a major role at small modified gravity amplitudes that our cluster sample is able to probe. We compare the predictions of the HMF from this semi-analytical approach with those from the FORGE simulations [45], and we find a discrepancy between the two that depends on cosmology, redshift and cluster mass. We use the simulations to calibrate the semi-analytical model to obtain a more robust HMF, while still allowing for an analysis within a broader parameter range than the simulations allow. \nIn this analysis, we neglect the f ( R ) gravity effects on the gravitational lensing potential, which are subdomi- \nnant compared to the weak-lensing mass calibration uncertainties derived from GR simulations [52]. Furthermore, modifications to the observable-mass relations and the halo profiles are minimal, keeping the weak-lensing parameters similar to those in GR within their uncertainties [53, 90, 91]. \nWe achieve consistent results in all parameters compared to the ΛCDM analysis presented in SB24b. This is reassuring and expected, since our constraints on log 10 | f R 0 | do not indicate a preference for a deviation from the ΛCDM cosmology. \nWe report an upper bound of log 10 | f R 0 | < -5 . 32 at the 95% credible level with the HMF calibrated by the FORGE simulations. A slightly tighter constraint of log 10 | f R 0 | < -5 . 46 is obtained when using the semianalytical HMF. The difference in the constraints shows the necessity for a reliable f ( R ) HMF to place accurate constraints on f ( R ) gravity. The constraints reported here are the tightest constraints on f ( R ) gravity from clusters and on cosmological scales published to date. \nUpcoming Stage-III and Stage-IV surveys, such as SPT-3G [73] and CMB-S4 [74] or the Simons Observatory [106], will provide significantly larger cluster samples, which cover a broader redshift range [107]. In combination with next-gravitational lensing data like from the Euclid [75, 76] satellite or the Vera C. Rubin observatory [77, 78], these cluster data sets will lead to improved constraints on f ( R ) gravity [SV24a]', 'ACKNOWLEDGMENTS': "This research was supported by 1) the Excellence Cluster ORIGINS, which is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy - EXC2094-390783311, by 2) the Max Planck Society Faculty Fellowship program at MPE, and by 3) the LudwigMaximilians-Universitat in Munich. \nThe FORGE simulations and analyses of this project made use of the DiRAC@Durham facility managed by the Institute for Computational Cosmology (ICC) on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). The equipment was funded by BEIS capital funding via STFC capital grants ST/K00042X/1, ST/P002293/1, ST/R002371/1 and ST/S002502/1, Durham University and STFC operations grant ST/R000832/1. DiRAC is part of the National eInfrastructure. \nBL is supported by STFC via Consolidated Grant ST/X001075/1 \nThe Innsbruck authors acknowledge support provided by the Austrian Research Promotion Agency (FFG) and the Federal Ministry of the Republic of Austria for Climate Action, Environment, Mobility, Innovation and Technology (BMK) via the Austrian Space Applications Programme with grant numbers 899537, 900565, and 911971. \nThe South Pole Telescope program is supported by the National Science Foundation (NSF) through the Grant No. OPP-1852617 and 2332483. Partial support is also provided by the Kavli Institute of Cosmological Physics at the University of Chicago. Work at Argonne National Lab is supported by UChicago Argonne LLC, Operator of Argonne National Laboratory (Argonne). Argonne, a U.S. Department of Energy Office of Science Laboratory, is operated under contract no. DE-AC02-06CH11357 \nFunding for the DES Projects has been provided by the U.S. Department of Energy, the U.S. National Science Foundation, the Ministry of Science and Education of Spain, the Science and Technology Facilities Council of the United Kingdom, the Higher Education Funding Council for England, the National Center for Supercomputing Applications at the University of Illinois at Urbana-Champaign, the Kavli Institute of Cosmological Physics at the University of Chicago, the Center for Cosmology and Astro-Particle Physics at the Ohio State University, the Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M University, Financiadora de Estudos e Projetos, Funda¸c˜ao Carlos Chagas Filho de Amparo 'a Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de Desenvolvimento Cient'ıfico e Tecnol'ogico and the Minist'erio da Ciˆencia, Tecnologia e Inova¸c˜ao, the Deutsche Forschungsgemeinschaft and the Collaborating Institutions in the Dark Energy Survey. \nThe Collaborating Institutions are Argonne National Laboratory, the University of California at Santa Cruz, the University of Cambridge, Centro de Investigaciones Energ'eticas, Medioambientales y Tecnol'ogicas-Madrid, the University of Chicago, University College London, the DES-Brazil Consortium, the University of Edinburgh, the Eidgenossische Technische Hochschule (ETH) Zurich, Fermi National Accelerator Laboratory, the University of Illinois at Urbana-Champaign, the Institut de Ci'encies de l'Espai (IEEC/CSIC), the Institut de F'ısica d'Altes Energies, Lawrence Berkeley National Laboratory, the Ludwig-Maximilians-Universitat Munchen and the associated Excellence Cluster Origins, the University of Michigan, NSF's NOIRLab, the University of Nottingham, The Ohio State University, the University of Pennsylvania, the University of Portsmouth, SLAC National Accelerator Laboratory, Stanford University, the University of Sussex, Texas A&M University, and the OzDES Membership Consortium. \nBased in part on observations at Cerro Tololo InterAmerican Observatory at NSF's NOIRLab (NOIRLab Prop. ID 2012B-0001; PI: J. Frieman), which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. \nThe DES data management system is supported by the National Science Foundation under Grant Numbers AST-1138766 and AST-1536171. The DES participants from Spanish institutions are partially supported by MICINN under grants ESP2017-89838, PGC2018094773, PGC2018-102021, SEV-2016-0588, SEV-2016- \n0597, and MDM-2015-0509, some of which include ERDF funds from the European Union. IFAE is partially funded by the CERCA program of the Generalitat de Catalunya. Research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Program (FP7/20072013) including ERC grant agreements 240672, 291329, and 306478. 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2022arXiv221115742K

We construct examples of black hole formation from regular oneended asymptotically flat Cauchy data for the EinsteinMaxwellcharged scalar field system in spherical symmetry which are exactly isometric to extremal ReissnerNordstrm after a finite advanced time along the event horizon. Moreover in each of these examples the apparent horizon of the black hole coincides with that of a Schwarzschild solution at earlier advanced times. In particular our result can be viewed as a definitive disproof of the third law of black hole thermodynamics. The main step in the construction is a novel Ck characteristic gluing procedure which interpolates between a light cone in Minkowski space and a ReissnerNordstrm event horizon with specified charge to mass ratio eM. Our setup is inspired by the recent work of AretakisCzimekRodnianski on perturbative characteristic gluing for the Einstein vacuum equations. However our construction is fundamentally nonperturbative and is based on a finite collection of scalar field pulses which are modulated by the BorsukUlam theorem.

2022-11-01T00:00:00Z

['10.48550/arXiv.2211.15742', '2022arXiv221115742K', 'arXiv:2211.15742']

['General Relativity and Quantum Cosmology', 'Mathematical Physics', 'Mathematics - Analysis of PDEs', 'Mathematics - Differential Geometry']

Gravitational collapse to extremal black holes and the third law of black hole thermodynamics

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2211.15742.pdf

{'Gravitational collapse to extremal black holes and the third law of black hole thermodynamics': 'Christoph Kehle ∗1 and Ryan Unger †2 \n1 Institute for Theoretical Studies & Department of Mathematics, ETH Zürich, Clausiusstrasse 47, 8092 Zürich, Switzerland \n2 Department of Mathematics, Princeton University, Washington Road, Princeton NJ 08544, United States of America \nFebruary 15, 2024', 'Abstract': "We construct examples of black hole formation from regular, one-ended asymptotically flat Cauchy data for the Einstein-Maxwell-charged scalar field system in spherical symmetry which are exactly isometric to extremal Reissner-Nordström after a finite advanced time along the event horizon. Moreover, in each of these examples the apparent horizon of the black hole coincides with that of a Schwarzschild solution at earlier advanced times. In particular, our result can be viewed as a definitive disproof of the 'third law of black hole thermodynamics.' \nThe main step in the construction is a novel C k characteristic gluing procedure, which interpolates between a light cone in Minkowski space and a Reissner-Nordström event horizon with specified charge to mass ratio e/M . Our setup is inspired by the recent work of Aretakis-Czimek-Rodnianski on perturbative characteristic gluing for the Einstein vacuum equations. However, our construction is fundamentally nonperturbative and is based on a finite collection of scalar field pulses which are modulated by the Borsuk-Ulam theorem.", '1 Introduction': "Following pioneering work of Christodoulou [Chr70] and Hawking [Haw71] on energy extraction from rotating black holes, Bardeen, Carter, and Hawking [BCH73] proposed-via analogy to classical thermodynamicsthe celebrated four laws of black hole thermodynamics . In particular, letting the surface gravity κ of the black hole take the role of its temperature, an identification later vindicated by the discovery of Hawking radiation [Haw75], they proposed a third law in analogy to 'Nernst's theorem' in classical thermodynamics. \nConjecture (The third law of black hole thermodynamics) . A subextremal black hole cannot become extremal in finite time by any continuous process, no matter how idealized, in which the spacetime and matter fields remain regular and obey the weak energy condition. \nThis version is distilled from the literature, particularly from the work of Israel [Isr86; Isr92] who added explicit mention of regularity and the weak energy condition to avoid previously known examples [DI67; Kuc68; Bou73; FH79; SI80; Pró83] which would otherwise violate the third law. In this paper, we show that the third law is fundamentally flawed in a manner that does not appear to be salvageable by further reformulation. Indeed, we construct counterexamples in the Einstein-Maxwell-charged scalar field model in spherical symmetry, a model which satisfies the dominant energy condition, arising from arbitrarily regular initial data on a one-ended asymptotically flat hypersurface. \nTheorem 1. Subextremal black holes can become extremal in finite time, evolving from regular initial data. In fact, there exist regular one-ended Cauchy data for the Einstein-Maxwell-charged scalar field system which undergo gravitational collapse and form an exactly Schwarzschild apparent horizon, only for the spacetime to form an exactly extremal Reissner-Nordström event horizon at a later advanced time. In particular, the 'third law of black hole thermodynamics' is false. \nFigure 1: Penrose diagram of our counterexample to the third law arising from regular initial data on Σ . The northwest edge of the Schwarzschild region is exactly isometric to a section of the r = 2 M hypersurface in Schwarzschild. The outermost apparent horizon A ' is initially indistinguishable from Schwarzschild and then jumps out in finite time to be exactly isometric to the event horizon of extremal Reissner-Nordström. For speculations about the future boundary of the interior, see already Section 1.5.1. The behavior of our solutions can be modified to be subextremal near i 0 , see already Remark 1.1. \n<!-- image --> \nOur result also clarifies some issues raised by Israel in [Isr86; Isr92] who seemingly associated a disconnected outermost apparent horizon with a severe lack of regularity of the spacetime metric and/or matter fields. We stress that our examples are regular despite the disconnectedness of the apparent horizon. We note moreover that Israel seemed to associate extremization with the black hole 'losing its trapped surfaces.' This confusion appears to be related to his implicit assumption that the apparent horizon is connected. Since the Einstein-Maxwell-charged scalar field matter manifestly obeys the dominant energy condition, trapped surfaces are not lost in any sense, nonetheless, the black hole becomes extremal in finite time. In the examples we construct, there exists an open set of trapped spheres inside the black hole region, which persist for all advanced time until they encounter the Cauchy horizon or a curvature singularity inside the black hole. However, there is a neighborhood of the event horizon which does not contain any (strictly) trapped surfaces. For an extended discussion of these issues, see already Section 1.4. \nRemark 1.1 . Note that in discussions of the third law, the focus is typically on dynamics near the event horizon and apparent horizon, in late advanced time. Our counterexamples depicted in Fig. 1 are isometric to extremal Reissner-Nordström for all sufficiently late advanced times and all retarded times to the past of the event horizon, in particular near spatial infinity i 0 . However, by using a scattering argument as in [Keh22], one can easily modify our examples so as to be subextremal in a neighborhood of i 0 , if desired. \nOur falsification of the third law (Theorem 1) is preceded by our following more general result. We construct regular one-ended Cauchy data for the Einstein-Maxwell-charged scalar field system in spherical symmetry whose black hole exterior evolves (in fact is eventually isometric) to a Schwarzschild black hole with prescribed mass M > 0 or to a subextremal or extremal Reissner-Nordström black hole with prescribed mass M > 0 and prescribed charge to mass ratio q . = e/M ∈ [ -1 , 1] . The Einstein-Maxwell-charged scalar field (EMCSF) system reads \nR µν ( g ) -1 2 R ( g ) g µν = 2 ( T EM µν + T CSF µν ) , (1.1) \n∇ µ F µν = 2 e Im( ϕD ν ϕ ) , (1.2) \ng µν D µ D ν ϕ = 0 , (1.3) \nfor a quintuplet ( M , g, F, A, ϕ ) , where ( M , g ) is a (3+1)-dimensional Lorentzian manifold, ϕ is a complexvalued scalar field, A is a real-valued 1-form, F = dA is a real-valued 2-form, D = d + i e A is the gauge covariant derivative, e ∈ R \\ { 0 } is a fixed coupling constant representing the charge of the scalar field, and the energy momentum tensors are defined by \nT EM µν . = g αβ F αν F βµ -1 4 F αβ F αβ g µν , (1.4) \nT CSF µν . = Re( D µ ϕD ν ϕ ) -1 2 g µν g αβ D α ϕD β ϕ. (1.5) \nWe refer to Section 2 for the form of the EMCSF system in spherical symmetry. \nRemark 1.2 . All of the results in this paper also hold for the Einstein-Maxwell-charged Klein-Gordon system in which the wave equation (1.3) is replaced by the Klein-Gordon equation \ng µν D µ D ν ϕ = m 2 ϕ, \nwhere m ∈ R > 0 represents the mass of the scalar field and satisfies m M ≪ e M . Here M denotes the mass of the black holes, see already Theorem 2 below. \nWe emphasize that not only are our data in the above examples regular, but the spacetimes arise from gravitational collapse, i.e., the initial data surface is one-ended, has a regular center, lies entirely in the domain of outer communication, and the black hole forms strictly to the future of initial data. In particular, in contrast to what has been suggested numerically [TA14; CIP21], there is no upper bound (strictly less than unity) on the charge to mass ratio of a black hole which can be achieved in gravitational collapse for this model. \nThe key step toward the construction of one-ended Cauchy data evolving to black holes with prescribed mass and charge is a novel characteristic / null gluing result. The study of the characteristic gluing problem for the Einstein vacuum equations (outside of spherical symmetry) was recently initiated by Aretakis, Czimek, and Rodnianski [ACR21a; ACR21b; ACR21c] in the perturbative regime around Minkowski space. Our setup is directly inspired by their work. In contrast, however, our null gluing construction (while in spherical symmetry) necessarily exploits the large data regime in order to glue a cone of Minkowski space to a black hole event horizon along a null hypersurface within the EMCSF model. The construction of Cauchy data on Σ ∼ = R 3 collapsing to an extremal or subextremal event horizon will then follow from Theorem 2 as Corollary 1 presented in Section 1.2. \nOn the basis of our spherically symmetric horizon gluing construction in Theorem 2, the results and framework introduced in [ACR21a; ACR21b; ACR21c; CR22; DHR], and Remark 1.5 below, we formulate the following \nConjecture. There exist regular one-ended Cauchy data for the Einstein vacuum equations \nRic( g ) = 0 \nwhich undergo gravitational collapse and form an exactly Schwarzschild apparent horizon, only for the spacetime to form an exactly extremal Kerr event horizon at a later advanced time. In particular, already in vacuum, the 'third law of black hole thermodynamics' is false.", '1.1 Event horizon gluing': "We will now state the rough version of our main null gluing theorem, which concerns gluing a null cone in Minkowski space to a Reissner-Nordström event horizon. \nTheorem 2 (Rough version) . Let k ∈ N be a regularity index, q ∈ [ -1 , 1] a charge to mass ratio, and e ∈ R \\ { 0 } a fixed coupling constant. For any M sufficiently large depending on k , q , and e , there exist spherically symmetric characteristic data for the Einstein-Maxwell-charged scalar field system with coupling constant e gluing a Minkowski null cone of radius 1 2 M to a Reissner-Nordström event horizon with mass M and charge e = q M up to order k . \nWe also refer to Fig. 2 for an illustration of our construction. \nFigure 2: Setup of Theorem 2. \n<!-- image --> \nFor the precise version of Theorem 2 we refer to Theorem 2A and Theorem 2B in Section 3.4. In fact, more generally, we can replace the Minkowski sphere with certain Schwarzschild exterior spheres at v = 0 , which is important for constructing counterexamples to the third law of black hole thermodynamics (see already Section 1.4). Furthermore, when q = 0 we may take the scalar field to be real-valued, in which case the EMCSF system collapses to the Einstein-scalar field system. \nRemark 1.3 . For the proofs of Corollary 2 and Corollary 3 below, we will use versions of Theorem 2 where the top sphere is not located on a horizon. See Theorem 2C and Theorem 2C ' in Section 3.4 below. \nRemark 1.4 . With our methods one can also construct characteristic data which are exactly Minkowski initially and then settle down, but only asymptotically, to a Schwarzschild or (sub-)extremal ReissnerNordström event horizon of prescribed mass and charge. The rate of decay can be chosen to be | ∂ v ϕ | ≈ v -p , p > 1 2 , in a standard Eddington-Finkelstein gauge for Schwarzschild or subextremal Reissner-Nordström black holes. This provides examples of 'global' characteristic data settling down at certain prescribed rates as assumed in [Van18; GL19; KV21].", '1.2 Gravitational collapse from event horizon gluing': "For appropriate matter models, the Einstein equations \nRic( g ) -1 2 R ( g ) g = 2 T \nare well-posed (see [Fou52; CG69] for the vacuum case) as a Cauchy problem for suitable initial data posed on a 3 -manifold Σ , which will then be isometrically embedded as a spacelike hypersurface in a Lorentzian manifold ( M , g ) . The textbook explicit black hole solutions such as the Schwarzschild spacetime do not contain one-ended Cauchy surfaces Σ ∼ = R 3 but are instead foliated by two-ended hypersurfaces Σ ∼ = R × S 2 . Thus, a natural and physically relevant problem is to construct regular asymptotically flat data on Σ ∼ = R 3 which evolve to a black hole spacetime. The first example of gravitational collapse , that is a black hole \nspacetime containing a one-ended Cauchy surface which lies outside of the black hole region, was constructed by Oppenheimer and Snyder [OS39] for the Einstein-massive dust model in spherical symmetry. \nUsing Theorem 2, by solving the Einstein equations backwards , we construct examples of gravitational collapse where the domain of outer communication is eventually exactly isometric to Reissner-Nordström with prescribed mass and charge. The proof of the following Corollary 1 is given in Section 5.2. \nCorollary 1 (Exact Reissner-Nordström arising from gravitational collapse) . For any regularity index k ∈ N and charge to mass ratio q ∈ [ -1 , 1] , there exist spherically symmetric, asymptotically flat Cauchy data for the Einstein-Maxwell-charged scalar field system, with Σ ∼ = R 3 and a regular center, such that the maximal future globally hyperbolic development ( M 4 , g ) has the following properties: \n- · All dynamical quantities are at least C k -regular.\n- · Null infinity I + is complete. \n̸ \n- · The black hole region is non-empty, BH . = M\\ J -( I + ) = ∅ .\n- · The Cauchy surface Σ lies in the causal past of future null infinity, Σ ⊂ J -( I + ) . In particular, Σ does not intersect the event horizon H + . = ∂ ( BH ) . Furthermore, Σ contains no trapped or antitrapped surfaces.\n- · For sufficiently late advanced times v ≥ v 0 , the domain of outer communication, including the event horizon, is isometric to that of a Reissner-Nordström solution with charge to mass ratio q . For v ≥ v 0 , the event horizon of the spacetime can be identified with the event horizon of Reissner-Nordström. \nFigure 3: Penrose diagram for Corollary 1. The textured line segment is where the data constructed in Theorem 2 live. \n<!-- image --> \nNote that in the case | q | = 1 , this does not yet furnish a counterexample to the third law of black hole thermodynamics, as the spacetime does not necessarily contain a subextremal apparent horizon. For the counterexample we must defer to Theorem 1 in Section 1.4.4 below. \nHowever, in our proof of Corollary 1, forming an extremal black hole with | q | = 1 is no different from any subextremal charge to mass ratio | q | < 1 (see already Section 1.4.5). In particular, in contrast with what has been suggested by numerical simulations [TA14; CIP21], there is no universal upper bound (strictly less than unity) for | q | . Given that we have now proved that extremal Reissner-Nordström can arise in gravitational collapse, it would be interesting to rethink the numerical approach to this problem and develop a scheme to construct such solutions numerically. Because our construction is fundamentally teleological (see already Section 5.1), it might be challenging to directly find suitable data on Σ by trial and error. \nThe formation of black holes is a very well studied problem in spherical symmetry. We mention here only the Einstein-scalar field model, for which Christodoulou [Chr91] first showed that concentration of the scalar field can lead to formation of a black hole. This result played a decisive role in Christodoulou's proof of weak cosmic censorship in spherical symmetry [Chr99]. Dafermos constructed solutions of the Einstein-scalar field \nsystem which collapse to the future but are complete and regular to the past [Daf09]. For work on other matter models, see for example [And14; AL22]. \nOutside of spherical symmetry (for the Einstein vacuum equations), formation of black holes was studied by Christodoulou in the seminal monograph [Chr09]. Christodoulou constructed characteristic data for the Einstein vacuum equations containing no trapped surfaces, but whose evolution contains trapped surfaces in the future. Li and Yu [LY15] showed how to combine Christodoulou's construction with the spacelike gluing technique of Corvino and Schoen [CS06] to construct asymptotically flat Cauchy data containing no trapped surfaces, but whose evolution contains trapped surfaces in the future. Later, Li and Mei [LM20] observed that the Corvino-Schoen gluing can be done 'behind the event horizon,' which yields a genuine construction of gravitational collapse in vacuum arising from one-ended asymptotically flat Cauchy data. \nHowever, the constructions of the above type rely on the observation that if an additional restriction is imposed on the seed data in [Chr09], then the resulting spacetime has a region of controlled size which is close to Schwarzschild. The Corvino-Schoen gluing then selects very slowly rotating Kerr parameters for the exterior region. \nWe emphasize that our gluing approach yields collapsing spacetimes with exactly specified (in particular, extremal, if desired) parameters, but is so far limited to spherical symmetry. \nRemark 1.5 . Our derivation of Corollary 1 from Theorem 2 is completely soft and does not make use of spherical symmetry. Therefore, if versions of the main gluing theorems were known for the Einstein vacuum equations (for example, gluing a Minkowski cone to an extremal Kerr event horizon, or more generally a Schwarzschild exterior sphere to an extremal Kerr event horizon), then our procedure would yield vacuum spacetimes arising from gravitational collapse which are eventually isometric to extremal Kerr. Furthermore, such a construction would also yield a disproof of the third law in vacuum. \nRemark 1.6 . By the very nature of our gluing procedure, the constructions in this paper have finite regularity ( C k for arbitrarily large k ). It would be mathematically interesting to create such examples with C ∞ regularity. See already Remark 1.8. \nRemark 1.7 . The existence of dynamical spacetimes satisfying the dominant energy condition which are extremal at spacelike infinity i 0 does not contradict the positive mass theorem 'with charge' [GHHP83; CRT06] because the matter itself carries charge. Concretely, condition (27) in [GHHP83] is false for various charged matter models, in particular the Einstein-Maxwell-charged scalar field model with small (or zero) mass.", '1.3.1 Previous work on characteristic gluing': "The gluing problem along characteristic hypersurfaces for hyperbolic equations and associated null constraints already appears for the linear wave equation on Minkowski space. On R 3+1 , let u = 1 2 ( t -r ) , v = 1 2 ( t + r ) , and let ϕ be a spherically symmetric solution to the wave equation, i.e., \n∂ u ∂ v ( rϕ ) = 0 . (1.6) \nLet C ∪ C be a spherically symmetric bifurcate null hypersurface, that is, C = { u = u 0 } ∩ { v ≥ v 0 } and C = { u 1 ≥ u ≥ u 0 } ∩ { v = v 0 } . The wave equation (1.6) implies that ∂ u ( rϕ ) is conserved along the outgoing cone C . This implies that ∂ u ϕ cannot be freely prescribed along C , but is in fact determined by ∂ u ϕ on the bifurcation sphere C ∩ C . Indeed, the characteristic initial value problem is well posed with just ϕ itself prescribed along C ∪ C -the full 1-jet of ϕ can then be recovered from (1.6). For general spacetimes, the question of null gluing for the linear wave equation was studied by Aretakis [Are17]. \nFor a general wave equation, ingoing derivatives satisfy transport equations along outgoing null cones. The general C k characteristic gluing problem is to be given two spheres S 1 and S 2 along an outgoing null cone C , and k ingoing and outgoing derivatives of ϕ at S 1 and S 2 . One then seeks to prescribe ϕ along the part of C between S 1 and S 2 so that the outgoing derivatives agree with the given ones and the solutions of the transport equations for the ingoing derivatives have the specified initial and final values. In general, the linear characteristic gluing problem is obstructed due to the presence of conserved charges stemming from conservation laws along C . \nRemark 1.8 . Even in the absence of conservation laws at any order, C ∞ gluing of transverse derivatives may be obstructed in linear theory. This can be seen already for the (1 + 1) -dimensional wave equation ∂ u ∂ v ϕ = f ( u, v ) ϕ for generic f ∈ C ∞ ( R 1+1 ) . For such an f , there are no conservation laws at any order and by imposing trivial data at S 1 and very rapidly growing (in k ) ∂ k u -derivatives at S 2 , one can show that C ∞ gluing cannot be achieved. Note that in 1 + 1 dimensions, S 1 and S 2 are points. \nThe null gluing problem for the Einstein vacuum equations was recently initiated by Aretakis, Czimek, and Rodnianski in a fundamental series of papers [ACR21a; ACR21b; ACR21c]. Their proof uses the inverse function theorem to reduce the nonlinear problem to a linear characteristic gluing problem for the linearized Einstein equations in double null gauge around Minkowski space, in the formalism of Dafermos-HolzegelRodnianski [DHR]. This linearized problem is carefully analyzed and the authors identify infinitely many conserved charges which are obstructions to the linear gluing problem. All but ten of the charges turn out to be related to linearized gauge transformations (cf. the 'pure gauge solutions' of [DHR]). An inverse function theorem argument then gives nonlinear gluing close to Minkowski space, provided that, a posteriori , the 10 transported gauge-invariant charges at S 2 agree with the prescribed charges on S 2 (which is in fact also perturbed to deal with the gauge-dependent charges). Very recently, Czimek and Rodnianski [CR22] carefully exploited the nonlinear structure of the null constraint equations to nonlinearly compensate for failure of matching of the linearly conserved charges. In this way, the authors prove obstruction-free gluing for characteristic and Cauchy data near Minkowski space. \nIn the present paper, we are however interested in a different regime of gluing. We wish to glue two specific null cones: a light cone in Minkowski space and a Reissner-Nordström event horizon, as a solution of the EMCSF null constraint system. On the one hand, this is a genuine 'large data' gluing problem, as these cones are very dissimilar in a gauge invariant sense and there is no known spacetime around which one could reasonably linearize the equations. On the other hand, we study our problem in spherical symmetry, which makes it considerably more tractable. We refer to Section 3.1 below for a precise definition of characteristic gluing in spherical symmetry.", '1.3.2 Outline of the proof of Theorem 2': "In the Einstein-Maxwell-charged scalar field model in spherical symmetry, the spacetime metric is written in double null gauge as \ng = -Ω 2 dudv + r 2 g S 2 , \nwhere Ω 2 is the lapse and r the area-radius. We also have a complex-valued scalar field ϕ and a real-valued charge Q , which is related to the only nonzero component of the electromagnetic tensor F . We choose an electromagnetic gauge in which A = A u du , where A is a gauge potential for F . The dynamical variables to be glued along an outgoing cone (which we will call C -1 . = { u = -1 } ) are ( r, Ω 2 , ϕ, Q, A u ) . The charge Q solves first order equations in u and v , A u is computed from Q via F = dA , and the variables r , Ω 2 , and ϕ solve coupled nonlinear wave equations involving also Q and A u . See already equations (2.4)-(2.9). Since the value of Ω 2 along any given null cone (or bifurcate null hypersurface) can be adjusted by reparametrizing the double null gauge, we impose that Ω 2 ≡ 1 . \nWe first consider Raychaudhuri's equation (see already (2.11)), which reads in the gauge Ω 2 ≡ 1 \n∂ 2 v r = -r | ∂ v ϕ | 2 . (1.7) \nThis equation gives a nonlinear constraint on C -1 and completely determines r on C -1 given r and ∂ v r at one point of C -1 and ϕ along C -1 . Thus, in the gauge Ω 2 ≡ 1 along C -1 , up to specifying the dynamical quantities at a sphere, the free data in this problem is exactly ϕ on C -1 : All other dynamical quantities and their derivatives (both in the u and v coordinates) along C -1 can be obtained from ϕ and the equations (2.4)-(2.11). \nWe will choose ϕ to be compactly supported on the textured segment in Fig. 2 and set \n∂ u ϕ (0) = · · · = ∂ k u ϕ (0) = 0 , \nwhere k is the order at which we wish to glue. A first attempt to solve the gluing problem would be to set ( r, Ω 2 , Q, A u ) and derivatives to have their 'Minkowski values' at the sphere v = 0 and then prescribe ϕ ( v ) so that the dynamical variables reach their 'Reissner-Nordström values' at v = 1 . However, specifying \na 'Minkowski value' for ∂ v r is essentially another gauge choice, and the gauge invariance of the equations enables a much more convenient strategy. \nGiven that ϕ vanishes to order k at v = 0 , to know that the sphere v = 0 is a sphere in Minkowski space to order k , we merely need to know that r (0) > 0 and that the charge Q and the Hawking mass (see already (2.1)) both vanish. See already Lemma 4.1. This reduces to the statement that in the gauge Ω 2 ≡ 1 , \n∂ u r (0) ∂ v r (0) = -1 4 . \nSince r solves a wave equation (see already (2.5)), ∂ u r solves a first order equation in v , so it is determined on C -1 by ∂ u r (0) alone. Given ϕ , we solve Raychaudhuri's equation (1.7) backwards , i.e., we teleologically normalize r at the final sphere by setting \nr (1) = r + . = ( 1 + √ 1 -q 2 ) M ∂ v r (1) = 0 \nand then set \n∂ u r (0) . = -1 4 ∂ v r (0) . \nTherefore the only 'constraint' is that ∂ v r (0) > 0 , which will be automatically satisfied by the monotonicity property of Raychaudhuri's equation as long as r > 0 . \nThe charge Q is determined by Maxwell's equation (see already (2.8)) \n∂ v Q = e r 2 Im( ϕ∂ v ϕ ) . \nIntegrating this forwards in v yields the charge condition \n∫ 1 0 e r 2 Im( ϕ∂ v ϕ ) dv = q M. (1.8) \nAt this point we note that if r (0) ≥ 1 2 M , then the left-hand side of this equation is ≈ M 2 ∫ Im( ϕ∂ v ϕ ) . So by modulating ∫ Im( ϕ∂ v ϕ ) , we can hope to satisfy this equation just on the basis of scaling ϕ itself. \nFigure 4: Schematic illustration of the pulses. \n<!-- image --> \nOur ansatz for the scalar field will be \nϕ α = ∑ 1 ≤ j ≤ 2 k +1 α j ϕ j , \n̸ \nwhere α = ( α 1 , . . . , α 2 k +1 ) ∈ R 2 k +1 and the ϕ j 's are smooth compactly supported complex-valued functions with disjoint supports. We assume q = 0 now, the q = 0 case being in fact much easier. The charge condition (1.8) is examined on every ray R + ˆ α ∈ R 2 k +1 , ˆ α ∈ S 2 k . We show that for a given choice of baseline profiles ϕ j , there is a smooth starshaped hypersurface Q 2 k ⊂ R 2 k +1 which is isotopic to the unit sphere S 2 k and invariant under the antipodal map α ↦→ -α such that (1.8) holds for every α ∈ Q 2 k . \nThe condition that M is large depending on k , q , and e in Theorem 2 comes from natural conditions that arise when attempting to construct the hypersurface Q 2 k . The charge condition (1.8) implies | e | M 2 | α | 2 ≈ \n| q | M on Q 2 k . However, to keep r ≥ 1 2 M on C -1 , we find the condition | α | ≲ 1 , see already Lemma 4.4. These conditions are consistent only if | e | M ≳ | q | . Furthermore, this condition is crucially used to propagate the condition ∂ u r < 0 , see already Lemma 4.8. \nThe remaining equations ( 2 k real equations since the scalar field is complex) \n∂ i u ϕ α (1) = 0 1 ≤ i ≤ k (1.9) \ncan naturally be viewed as odd equations as a function of α . So when restricted to α ∈ Q 2 k , we can use the classical Borsuk-Ulam theorem to find a simultaneous solution. Once we have an α ∈ Q 2 k such that (1.9) is satisfied, ϕ α will glue all relevant quantities to k -th order, as desired. \nTheorem 3 (Borsuk-Ulam [Bor33]) . If f : S k → R k is a continuous odd function, i.e., f ( -x ) = -f ( x ) for every x ∈ S k , then f has a root. \nFor a nice proof using only basic degree theory and transversality arguments, see Nirenberg's lecture notes [Nir01].", '1.4 Retiring the third law of black hole thermodynamics': "In this section we give more details on the background and history of the third law of black hole thermodynamics put forth by Bardeen, Carter, and Hawking in [BCH73], and how our present work fits into the picture. \nWhile the zeroth , first , and second laws of black hole thermodynamics are by now well understood in the literature (see e.g. [Wal01]), the validity of the third law has been a source of debate up until today. In the original form of Bardeen-Carter-Hawking (BCH), in analogy to Nernst's version of the third law of classical thermodynamics [Ner26] 1 , it reads: \nIt is impossible by any procedure, no matter how idealized, to reduce κ to zero by a finite sequence of operations. \nA number of arguably pathological (e.g. singular or energy condition violating) examples of extremal black hole formation were put forth in [Kuc68; DI67; Pró83; Bou73; FH79; SI80], which Israel [Isr86; Isr92] took into account to make the third law more precise: \nA nonextremal black hole cannot become extremal (i.e., lose its trapped surfaces) at a finite advanced time in any continuous process in which the stress-energy tensor of accreted matter stays bounded and satisfies the weak energy condition in a neighborhood of the outer apparent horizon. \nThe parenthetical comment '(i.e., lose its trapped surfaces)' is an extra source of confusion which will be specifically addressed in Section 1.4.3. We will now discuss the papers [Kuc68; DI67; Pró83; Bou73; FH79; SI80; Isr86; Isr92] and where the issues lie.", '1.4.1 The singular massive dust shell model': "It has been known since the 60's that an extremal black hole can be formed instantly by collapsing an infinitesimally thin shell of charged massive dust [Kuc68; DI67; Pró83; Bou73]. Later, Farrugia and Hajicek [FH79] showed how to 'turn a subextremal Reissner-Nordström spacetime into an extremal one' by firing an appropriately charged singular massive shell into the black hole. The resulting spacetime metric is not C 2 -regular. The Penrose diagram of the spacetime they construct is similar to our Fig. 1 (see [FH79, p. 296 Fig. 2]). In particular, we note the presence of a disconnected outermost apparent horizon in their example. Israel seemed to associate the disconnectedness of the apparent horizon with a singularity of the matter and/or spacetime: 'Violations can also be produced by any process that induces discontinuous behavior of the apparent horizon-for example, absorption of an infinitely thin massive shell, which will force this horizon to jump outward.' See already Section 1.4.3. On the basis of this, he dismissed this example in his formulation of the third law by explicitly requiring regularity. We note, however, that Farrugia and Hajicek suggest that their construction can in principle be desingularized-we do not know if this point was ever addressed again, because if true, it would seem to provide an alternative route to constructing a counterexample apart from our own.", '1.4.2 The charged null dust model': 'An interesting example motivating explicit mention of the weak energy condition in the third law was provided by Sullivan and Israel [SI80] in spherical symmetry, with the charged null dust matter model. This matter model allows for dynamical violations of the weak energy condition-even if the initial data satisfies the weak energy condition, the solution might violate it in the future. Sullivan and Israel showed that extremization is impossible in this model without such a violation, which can also be seen from Penrose diagrams. They interpreted this result as further evidence that the third law holds as long as the weak energy condition is demanded near the apparent horizon. We note, however, that Ori has proposed a different interpretation of the model studied by Sullivan and Israel which does not violate the weak energy condition [Ori91].', "1.4.3 'Losing trapped surfaces' and connectedness of the outermost apparent horizon": "We will now clarify the issue of 'losing trapped surfaces' appearing prominently in [Isr86; Isr92] and the implicit assumption of connectedness of the outermost apparent horizon. \nThe black hole region in a subextremal Reissner-Nordström or Kerr spacetime is foliated by trapped spheres. Conversely, extremal Reissner-Nordström and Kerr black holes have no trapped surfaces, but the event horizon is a marginally trapped tube in both cases. As | q | → 1 (where we take q . = e/M for ReissnerNordström and q . = a/M for Kerr), r -→ r + , and one might be inclined to think that extremizing involves 'squeezing' away the trapped region inside the black hole. However, it is an immediate consequence of Raychaudhuri's equation [HE73; Wal84] that trapped surfaces persist in evolution as long as the spacetime satisfies the weak energy condition. Since the typical explicit extremal black holes have no trapped surfaces (in particular none near the event horizon), one might wonder if Raychaudhuri's equation alone could be used to 'prove' the third law. \nThis is what Israel attempted to do in [Isr86; Isr92]. We will formalize his observation in Definition 1.1 and Proposition 1.1 below. However, as should be clear from our main theorem, this does not in fact capture the intended meaning of the third law. \nIn order to reconstruct Israel's argument mathematically, let us formulate the following definition. For precise definitions relating to spherical symmetry, see already Section 2. \nDefinition 1.1. Let H be a connected dynamical apparent horizon, i.e., a connected, achronal curve in the (1+1) -dimensional reduction ( Q , g Q ) of a spherically symmetric spacetime ( M , g ) , along which ∂ v r vanishes identically. We say that H becomes extremal in finite time in the sense of Israel if \n- 1. H is not completely contained in a null cone.\n- 2. Let τ ↦→ H ( τ ) be a parametrization of H . Then there exists a τ 0 ∈ R so that for all τ ≥ τ 0 , τ ↦→ H ( τ ) is a future-directed constant u curve.\n- 3. There exists a τ 1 > τ 0 and a neighborhood N of H τ ≥ τ 1 such that N \\ H τ ≥ τ 0 contains only strictly untrapped spheres ( ∂ v r > 0 ). \nRemark 1.9 . The outermost apparent horizon A ' (see already Section 5.1), if connected, is an example of a connected dynamical apparent horizon. \nAs a simple consequence of Raychaudhuri's equation in a spacetime satisfying the weak energy condition [HE73; Wal84], we have \nProposition 1.1 (Israel's observation) . Let ( M , g ) be a spherically symmetric black hole spacetime. If the spacetime satisfies the weak energy condition, has a nonempty trapped region, and a connected outermost apparent horizon A ' as defined in [Kom13], then the outermost apparent horizon A ' does not become extremal in finite time in the sense of Israel. \nHowever, it is clear that in view of our main theorem, the correct reading of this proposition is the contrapositive , namely that violations of the third law necessarily have a disconnected apparent horizon. This effect has nothing to do with singularities of spacetime or the matter model (and there was never actually any a priori reason to believe that the outermost apparent horizon was connected). This situation is depicted in Fig. 5. \nFigure 5: Illustration of the contrapositive of Proposition 1.1. The outermost apparent horizon A ' = A ' 1 ∪A ' 2 becomes disconnected when a black hole with trapped surfaces 'becomes extremal,' while the spacetime and matter fields remain regular. The trapped region begins to the north of A ' 1 and persists for all advanced time. \n<!-- image -->", '1.4.4 Disproving the third law': "With this discussion out of the way, we present now a detailed version of our counterexample to the third law. It is essentially a corollary of the more general version of our main gluing result Theorem 2 with a Schwarzschild exterior sphere in place of a Minkowski sphere (see already Section 3.4) and will be given in Section 5.3. For an illustration of the spacetime, we refer the reader back to Fig. 1. \nTheorem 1 (Gravitational collapse to ERN with a Schwarzschild piece) . For any regularity index k ∈ N , there exist spherically symmetric, asymptotically flat Cauchy data for the Einstein-Maxwell-charged scalar field system, with Σ ∼ = R 3 and a regular center, such that the maximal future globally hyperbolic development ( M 4 , g ) has the following properties: \n- · The spacetime satisfies all the conclusions of Corollary 1 with q = 1 , including C k -regularity of all dynamical quantities.\n- · The black hole region contains an isometrically embedded portion of a Schwarzschild exterior horizon neighborhood. In particular, there is a portion of a null cone behind the event horizon of ( M , g ) which can be identified with a portion of the apparent horizon of Schwarzschild.\n- · The 'Schwarzschild horizon' piece is a part of the outermost apparent horizon A ' of the spacetime. The set A ' is disconnected and agrees with the event horizon H + to the future of the first marginally trapped sphere on the event horizon.\n- · There is a neighborhood of the event horizon that contains no trapped surfaces. Nonetheless, the black hole region contains trapped surfaces. In fact, there are trapped surfaces at arbitrarily late advanced time in the interior of the black hole. \nTo reiterate, the scalar field collapses to form an exact Schwarzschild spacetime, including the horizon, only to collapse further to form an exact extremal Reissner-Norström for all late advanced time. The spacetime is regular (for any fixed k ≥ 1 , one can construct an example which is C k ) and the matter model satisfies the dominant energy condition.", '1.4.5 Exceptionality and stability of third law violating solutions': "The third law is manifestly concerned with exceptional behavior, which is why the phrases 'no matter how idealized' [BCH73] or 'in any continuous process' [Isr86] are specifically included in formulations of the third law. Indeed, keeping a horizon at exactly constant temperature (or equivalently constant surface gravity), any temperature, is of course exceptional. (Exactly stationary behavior on the horizon for all late \nadvanced times is itself an infinite codimension phenomenon in the moduli space of solutions.) In view of our construction, the case of gravitational collapse to zero temperature in finite time is no more exceptional than any other fixed temperature. \nWe would also like to address the interesting question of whether creating asymptotically extremal black holes should be viewed any differently from the subextremal case. Indeed, any mechanism which forms a black hole with exactly specified parameters is inherently unstable, because a small perturbation can just change the parameters. As an example of this, we note the codimension-3 nonlinear stability of the Schwarzschild family by Dafermos-Holzegel-Rodnianski-Taylor [DHRT]. In order to preserve the final black hole parameters, only a codimension-3 submanifold of the moduli space of data is admissible in their theorem. \nThe stability problem for extremal black holes is exceptional because they suffer from a linear instability known as the Aretakis instability [Are11a; Are11b; Are15; Ape22]. This instability is weak, and a restricted form of nonlinear stability is nevertheless conjectured to hold with the same codimensionality as in the subextremal case. See [DHRT, Section IV.2] for conjectures about stability of extremal black holes, [Ang16; AAG20] for stability results on a nonlinear model problem, and numerical work [MRT13; LMRT13] which is consistent with the above conjecture. The Aretakis instability should not be thought of as a manifestation of the third law and understanding its ramifications in the full nonlinear theory is a fundamental open problem in general relativity. \nTherefore, asymptotic stability for any fixed parameter ratio (up to and including extremality) should be formulated as a positive codimension statement. In our spherically symmetric setting, we are led to conjecture that for every solution constructed in Corollary 1, there exists a codimension-1 family of perturbations which asymptote to a Reissner-Nordström black hole with the same final parameter ratio. Since the conjectured codimension is the same for every ratio, we are then led to conclude that asymptotically extremal black holes are not qualitatively rarer than any fixed positive temperature. \nWe hope that our construction demystifies the scenario of matter collapsing to exactly extremal black holes. More generally, the considerations in this paper open up a new window to studying critical behavior in gravitational collapse, which is fundamentally different from the regime studied in [Cho93; GM07].", '1.4.6 Aside: Extremal horizons with nearby trapped surfaces': 'Though not directly relevant for the considerations of the present paper, we would like to point out that there is another issue with the attempt to characterize extremality by the lack of trapped surfaces near the horizon, i.e., by the third property of Definition 1.1. In fact, it would appear that the property of having no trapped surfaces in the interior near the horizon is actually stronger than being extremal. \nFor a spacetime ( M , g ) with Killing field K , a Killing horizon H is said to be extremal if the surface gravity κ , defined by ∇ K K = κK on H , vanishes identically. Equivalently, extremality means that g ( K,K ) vanishes to at least second order along null geodesics crossing H transversely. If K is timelike to the past of H and g ( K,K ) vanishes to an even order on H , then K passes from timelike, to null, then back to timelike across H , and there are no strictly trapped surfaces near the horizon. This is precisely the situation for extremal Reissner-Nordström and Kerr black holes, where g ( K,K ) vanishes to second order on the event horizon. \nHowever, there exist spacetimes for which g ( K,K ) vanishes to an odd order (at least three), in which case there may be trapped surfaces just behind the horizon. Indeed, in Proposition A.1 of Appendix A we construct an example of a stationary spacetime containing an extremal Killing horizon, with trapped surfaces just behind the horizon, and satisfying the dominant energy condition. In this case g ( K,K ) is exactly cubic in an ingoing null coordinate system. It would be interesting to construct such a spacetime with a specific matter model, or an extremal black hole with this behavior. \nWhile extremal Kerr, Reissner-Nordström, and other known examples are extremal in the sense of Definition 1.1, it is far from obvious that all hairy (i.e., carrying non-EM matter fields) extremal black holes should be free of trapped surfaces. In view of our example in Appendix A, any mechanism which enforces this must necessarily be global in nature and/or depend on particular properties of the matter model in question. \nOne could define the notion of a nondegenerate extremal Killing horizon, i.e., the Killing field K has the property that g ( K,K ) vanishes only to second order, which would then be compatible with Definition 1.1. See already Remark A.1. \nFor more discussion about possible definitions of extremality, see for instance [BF08; Boo16; MRT13].', '1.5 Future boundary of the interior and Cauchy horizon gluing': "The future boundary of the black hole region of dynamical black holes formed from gravitational collapse in the EMCSF system is known to be intricate (see e.g. [Daf03; Kom13; Van18]). We refer to [Kom13] for a detailed description of the most general possible structure of the interior, but see already Fig. 10 for a summary of the most salient features. In this subsection we will first discuss the future boundary of the black hole interior in Theorem 1. Further, we will present additional corollaries of our characteristic gluing method which provide examples of gravitational collapse to black holes with a piece of null boundary (a 'Cauchy horizon') and a construction of spacetimes for which a Cauchy horizon closes off the interior region.", '1.5.1 Future boundary of the interior in Theorem 1': 'For our main counterexample to the third law in Theorem 1, we obtain that the regular center Γ extends into the black hole region. Regarding the future boundary of the spacetime, we do not know whether there exists a piece of possibly singular null boundary emanating from i + as in the subextremal case [Daf03; Van18] or whether a spacelike singularity emanates from i + . Note that the result of [GL19], which shows the existence of a Cauchy horizon emanating from i + , does not apply directly since their analysis requires | e | M ≤ 0 . 1 , whereas our construction requires | e | M large. Nevertheless, one may speculate that a piece of Cauchy horizon occurs (for which the linear analysis of [Gaj17a; Gaj17b] would be relevant), which could eventually turn into a spacelike singularity. (Note that one can readily set up the data such that the future boundary of the interior in Theorem 1 has a piece of spacelike singularity. See however already Section 1.5.3.)', '1.5.2 Gravitational collapse with a piece of smooth Cauchy horizon': 'Another corollary of our method is the construction of regular one-ended Cauchy data which evolve to a subextremal or extremal black hole for which there exists a piece of Cauchy horizon emanating from i + . We refer to Fig. 6 for the Penrose diagram of the spacetime constructed in Corollary 2. The proof of Corollary 2 is given in Section 5.4. \nFigure 6: Penrose diagram depicting Corollary 2: Gravitational collapse to Reissner-Nordström with nonempty piece of Cauchy horizon CH + . \n<!-- image --> \nCorollary 2 (Gravitational collapse to RN with a smooth Cauchy horizon) . For any regularity index k ∈ N and nonzero charge to mass ratio q ∈ [ -1 , 1] \\ { 0 } , there exist spherically symmetric, asymptotically flat Cauchy data for the Einstein-Maxwell-charged scalar field system in spherical symmetry, with Σ ∼ = R 3 and a regular center, such that the maximal future globally hyperbolic development ( M 4 , g ) has the following properties: \n̸ \n- · The spacetime satisfies all the conclusions of Corollary 1 with q = 0 , including C k -regularity of all dynamical quantities.\n- · The black hole region contains an isometrically embedded portion of a Reissner-Nordström Cauchy horizon neighborhood with charge to mass ratio q . \nRemark 1.10 . When | q | = 1 , the spacetime constructed in Corollary 2 does not contain trapped symmetry spheres in the dark shaded region in Fig. 6. By a slight modification of the argument in Proposition B.1 below, this implies no trapped surfaces intersect the dark shaded region. In particular, the trapped region (in the sense of [HE73, p. 319]) of the spacetime (if nonempty) avoids a whole double null neighborhood of the event horizon. Nevertheless, the event horizon agrees with the outermost apparent horizon for late advanced times.', '1.5.3 Black hole interiors for which the Cauchy horizon closes off spacetime': "Our horizon gluing method can also be extended to glue Reissner-Nordström interior spheres to a regular center along an ingoing cone, see already Theorem 2C ' . Using this, we construct asymptotically flat Cauchy data for which the future boundary of the black hole region BH is a Cauchy horizon CH + which closes off spacetime. We refer to Fig. 7 for the Penrose diagram of the spacetime constructed in Corollary 3. The proof of Corollary 3 is given in Section 5.5. \nFigure 7: Penrose diagram depicting Corollary 3: The Cauchy horizon is regular and closes off the spacetime in a regular fashion. \n<!-- image --> \nCorollary 3 (Cauchy horizon that closes off the spacetime) . For any regularity index k ∈ N , and nonzero charge to mass ratio q ∈ [ -1 , 1] \\ { 0 } , there exist spherically symmetric, asymptotically flat Cauchy data for the Einstein-Maxwell-charged scalar field system, with Σ ∼ = R 3 and a regular center, such that the maximal future globally hyperbolic development ( M 4 , g ) has the following properties: \n- · All dynamical quantities are at least C k -regular. \n̸ \n- · The black hole region is non-empty, BH . = M\\ J -( I + ) = ∅ .\n- · The future boundary of BH is a C k -regular Cauchy horizon CH + which closes off spacetime.\n- · The black hole exterior is isometric to a Reissner-Nordström exterior with charge to mass ratio q . In particular, null infinity I + is complete.\n- · The spacetime does not contain antitrapped surfaces.\n- · When | q | = 1 , the spacetime does not contain trapped surfaces. \nRemark 1.11 . In contrast to our previous constructions, the Cauchy surface Σ in Corollary 3 could contain trapped surfaces and Σ intersects the black hole region. It would be interesting to construct a spacetime as in Corollary 3 which depicts genuine gravitational collapse, i.e., for which Σ ⊂ J -( I + ) . \nIn the subextremal case, the behavior exhibited by our construction can be seen as exceptional as one generically expects a Cauchy horizon which forms in gravitational collapse to be a weak null singularity [Daf03; Van18; LO19]. In particular, in the case where the Cauchy horizon CH + is weakly singular, Van de Moortel [Van19] showed that the Cauchy horizon CH + cannot close off spacetime in the sense of Fig. 7. Thus, our construction in Corollary 3 makes [Van19] sharp in the sense that the singularity assumption of CH + in [Van19] is needed. Restricted to the extremal case, however, on the basis of a more regular Cauchy horizon as in [GL19], one may speculate that there exists a set of data (open as a subset of the positive codimension set of data settling down to ERN) for which the Cauchy horizon closes off spacetime as depicted in Fig. 7.", 'Acknowledgments': 'The authors wish to express their gratitude to Mihalis Dafermos for suggesting the problem and many helpful discussions. We also thank Jonathan Luk, Hamed Masaood, Thomas Massoni, Georgios Moschidis, Harvey Reall, Igor Rodnianski, Jaydeep Singh, and Nina Zubrilina for helpful conversations. C.K. acknowledges support by a grant from the Institute for Advanced Study, by Dr. Max Rössler, the Walter Haefner Foundation, and the ETH Zürich Foundation. R.U. acknowledges partial support from the grant NSF-1759835 and the hospitality of the Institute for Advanced Study, the Centro de Ciencias de Benasque, and the University of Cambridge.', '2 The characteristic initial value problem for the Einstein-Maxwellcharged scalar field system in spherical symmetry': 'In this section, we give a detailed explanation of the setup and characteristic initial value problem for the Einstein equations with charged scalar fields in spherical symmetry, with a view towards the characteristic gluing problem. See [Kom13] for more details on the EMCSF system.', '2.1.1 Spherically symmetric spacetimes': "We say that a smooth, connected, time-oriented, four-dimensional Lorentzian manifold ( M , g ) is a spherically symmetric spacetime with (possibly empty) center of symmetry Γ ⊂ M if M\\ Γ splits diffeomorphically as M\\ Γ ∼ = ˚ Q× S 2 with metric \ng = g Q + r 2 g S 2 , \nwhere ( Q , g Q ) for Q = ˚ Q∪ Γ is a (1+1)-dimensional Lorentzian spacetime with (possibly empty) boundary Γ , g S 2 is the round metric on the sphere, and r : Q → R ≥ 0 is a function which can be geometrically interpreted as the area radius of the orbits of the isometric SO(3) action on ( M , g ) . In mild abuse of notation, we denote with Γ both the center of symmetry in M and its projection to Q . Moreover, if Γ is non-empty, we assume that the SO(3) action fixes Γ and that Γ consists of one timelike geodesic along which r = 0 . We further assume that ( Q , g Q ) admits a global double-null foliation (locally a double-null foliation always exists) with null coordinates ( u, v ) such that the metric g takes the form \ng = -Ω 2 dudv + r 2 g S 2 \nfor a nowhere vanishing function Ω 2 = -2 g Q ( ∂ u , ∂ v ) on Q and such that ∂ u and ∂ v are future-directed. We further assume that along the center Γ , the coordinate v is outgoing and u is ingoing, i.e., ∂ v r | Γ > 0 , ∂ u r | Γ < 0 . While we introduced the above notions in the smooth category, we will also consider spacetimes which are less regular ( C k ≥ 1 ) . We note that all notions introduced above also apply in this less regular case. We will also make use of the Hawking mass m : M→ R defined as \nm . = r 2 (1 -g ( ∇ r, ∇ r )) \nwhich also can be viewed as a function on Q as \nm = r 2 ( 1 + 4 ∂ u r∂ v r Ω 2 ) . (2.1) \nFinally, we note that the double null coordinates ( u, v ) are not unique and for any smooth functions U, V : R → R with U ' , V ' > 0 , we obtain new global double null coordinates (˜ u, ˜ v ) = ( U ( u ) , V ( v )) such that the metric g = -˜ Ω 2 d ˜ ud ˜ v + r 2 g S 2 , where ˜ Ω 2 (˜ u, ˜ v ) = ( U ' V ' ) -1 Ω 2 ( U -1 (˜ u ) , V -1 (˜ v )) and r (˜ u, ˜ v ) = r ( U -1 (˜ u ) , V -1 (˜ v )) .", '2.1.2 The Einstein-Maxwell-charged scalar field system': 'For the Einstein-Maxwell-scalar field system (1.1)-(1.5) in spherical symmetry, additionally to the spherically symmetric spacetime ( M 4 , g ) , we assume that the field ϕ is complex-valued and spherically symmetric, so that ϕ descends to a function Q → C , and that F and A are spherically symmetric such that F can be written as \nF = Q 2 r 2 Ω 2 du ∧ dv \nfor charge Q : Q → R . The potential 1 -form reads \nA = A u du + A v dv. \nWe also define the gauge covariant derivative operator D = d + i e A . The Einstein-Maxwell-scalar field system is invariant with respect to the following gauge transformations \nϕ ↦→ e -i e χ ϕ, A ↦→ A + dχ (2.2) \nfor real-valued functions χ = χ ( u, v ) , where e is a dimensionful coupling constant representing the charge of the scalar field. More abstractly, the Einstein-Maxwell-scalar field system is a U (1) -gauge theory and we refer to [Kom13] for more details. In order to break the symmetry we will use the global electromagnetic gauge \nA v = 0 (2.3) \nthroughout the paper. In this gauge, the Einstein-Maxwell-scalar field system (1.1)-(1.3) in spherical symmetry reduces to the following set of equations.', 'Wave equations for scalar field and metric components:': "∂ u ∂ v ϕ = -∂ u ϕ∂ v r r -∂ u r∂ v ϕ r + i e Ω 2 Q 4 r 2 ϕ -i e A u ∂ v r r ϕ -i e A u ∂ v ϕ (2.4) \n∂ u ∂ v r = -Ω 2 4 r -∂ u r∂ v r r + Ω 2 4 r 3 Q 2 (2.5) \n∂ u ∂ v log(Ω 2 ) = Ω 2 2 r 2 +2 ∂ u r∂ v r r 2 -Ω 2 r 4 Q 2 -2Re( D u ϕ∂ v ϕ ) (2.6) \nMaxwell's equations: \n∂ u Q = -e r 2 Im( ϕD u ϕ ) (2.7) \n∂ v Q = e r 2 Im( ϕ∂ v ϕ ) (2.8) \n∂ v A u = -Q Ω 2 2 r 2 (2.9) \n∂ u ( ∂ u r Ω 2 ) = -r Ω 2 | D u ϕ | 2 (2.10) \n∂ v ( ∂ v r Ω 2 ) = -r Ω 2 | ∂ v ϕ | 2 (2.11) \nRaychaudhuri's equations: \nFrom these equations we easily derive \n∂ v ( r∂ u r ) = -Ω 2 4 ( 1 -Q 2 r 2 ) (2.12) \nand \n∂ v ∂ u ( rϕ ) = -Ω 2 m 2 r 2 ϕ + i e Ω 2 Q 4 r ϕ + Ω 2 Q 2 4 r 3 ϕ -i e A u ∂ v ( rϕ ) , (2.13) \nas well as \n∂ v m = 2Ω -2 r 2 ( -∂ u r ) | ∂ v ϕ | 2 + 1 2 Q 2 r 2 ∂ v r (2.14) \nwhich will be useful later.", '2.2 The characteristic initial value problem': 'With the equations of the EMCSF system at hand, we can precisely define what we mean by a C k solution. We may for now restrict attention to solutions away from the center.', '2.2.1 Bifurcate characteristic data': 'Definition 2.1. Let k ∈ N . A C k solution for the Einstein-Maxwell-charged scalar field system in the EM gauge (2.3) consists of a domain Q ⊂ R 1+1 u,v and functions r ∈ C k +1 ( Q ) and Ω 2 , ϕ, Q, A u ∈ C k ( Q ) , such that r > 0 , Ω 2 > 0 , ϕ is complex-valued, ∂ k +1 v A u ∈ C 0 ( Q ) , and the functions satisfy 2 equations (2.4)-(2.11). \nNext, we formulate the characteristic initial value problem for this class of solutions. Let R 1+1 u,v denote the standard (1 + 1) -dimensional Minkowski space. We introduce the bifurcate null hypersurface C ∪ C ⊂ R 1+1 u,v , where \nC . = C -1 . = { u = -1 } ∩ { v ≥ 0 } C . = C 0 . = { v = 0 } ∩ { u ≥ -1 } . \nThe special point ( -1 , 0) is called the bifurcation sphere . We pose data for ϕ , Q , r , Ω 2 and A u for the Einstein-Maxwell-charged-scalar field system on C ∪ C . \nDefinition 2.2. Let k ∈ N . A C k bifurcate characteristic initial data set on C ∪ C for the EinsteinMaxwell-charged scalar field system in the EM gauge (2.3) consists of continuous functions r > 0 , Ω 2 > 0 , ϕ (complex-valued), Q , and A u on C ∪ C . It is required that r ∈ C k +1 , Ω 2 ∈ C k , ϕ ∈ C k , Q ∈ C k , and A u ∈ C k on C ∪ C . 3 Finally, the data are required to satisfy equations (2.7)-(2.11), which implies also ∂ k +1 v A u ∈ C 0 ( C ) . \nGiven characteristic initial data on a portion of C ∪ C containing the bifurcation sphere, we can solve in a full double null neighborhood to the future. The proof is a standard iteration argument. \nProposition 2.1. Given a C k bifurcate characteristic initial data set for the EMCSF system on \n( { u = -1 } × { 0 ≤ v ≤ v 0 } ) ∪ ( {-1 ≤ u ≤ u 0 } × { v = 0 } ) ⊂ C ∪ C, \nwhere u 0 > -1 and v 0 > 0 , there exists a number δ > 0 and a unique spherically symmetric C k solution of the EMCSF system on \n( {-1 ≤ u ≤ -1 + δ } × { 0 ≤ v ≤ v 0 } ) ∪ ( {-1 ≤ u ≤ u 0 } × { 0 ≤ v ≤ δ } ) \nwhich extends the initial data on C ∪ C .', '2.2.2 Determining transversal derivatives from tangential data': 'Now that we know that the data on C ∪ C extends to a solution of the system (2.4)-(2.11), we can use the equations to compute all the partial derivatives of the solution along C ∪ C . We describe a procedure for determining all u -derivatives on C just in terms of r, Ω 2 , ϕ, Q , and A u (as functions of v ) and their u derivatives at the bifurcation sphere. \nProposition 2.2. Let ( r, Ω 2 , ϕ, Q, A u ) be a C k bifurcate characteristic initial data set as in Definition 2.2. Then the EMCSF system can be used to determine as many u -derivatives of r, Ω 2 , ϕ, Q , and A u on C as is consistent with Definition 2.1, explicitly from the data on C ∪ C . \nProof. Since ( r, Ω 2 , ϕ, Q, A u ) are all given on C , we can compute as many u -derivatives of these quantities at the bifurcation sphere ( -1 , 0) as the regularity k allows. We describe an inductive procedure for computing u -derivatives of ( r, Ω 2 , ϕ, Q, A u ) on C , starting with ∂ u r . Since ∂ u r ( -1 , 0) is known, and the wave equation (2.5) can be written as \n( ∂ v + ∂ v r r ) ∂ u r = -Ω 2 4 r + Ω 2 4 r 3 Q 2 , \nwhere everything on the right-hand side is already known, ∂ u r ( -1 , v ) can be found by solving this ODE. In the same manner, ∂ u ϕ ( -1 , v ) and then ∂ u log(Ω 2 )( -1 , v ) can be found. To find ∂ u Q ( -1 , v ) , differentiate (2.7) in v and then integrate. (Alternatively, differentiate (2.8) in u .) Finally, ∂ u A u ( -1 , v ) is found by differentiating (2.9) in v and then integrating. \nProceeding in this way, by commuting all the equations with ∂ i u , every partial derivative of ( r, Ω 2 , ϕ, Q, A u ) which is consistent with the initial C k regularity can be found. We finally note that ∂ k +1 u r ( -1 , v ) is found from differentiating (2.10) an appropriate number of times, since the wave equation it satisfies is not consistent with the level of regularity of the rest of the dynamical variables. \nRemark 2.1 . Both Proposition 2.1 and Proposition 2.2 exploit the null condition satisfied by the EMCSF system in double null gauge. For a general nonlinear wave equation, the solution may not exist in a full double null neighborhood of the initial bifurcate null hypersurface as in Proposition 2.1. Indeed, the null condition means the transport equations for transversal derivatives in Proposition 2.2 are linear and hence do not blow up in finite time.', '2.3.1 Sphere data': "In order to define a notion of characteristic gluing later, we introduce a notion of sphere data inspired by [ACR21a; ACR21b]. Given a C k solution of the EMCSF system in spherical symmetry, for every ( u 0 , v 0 ) ∈ Q one can extract a list of numbers corresponding to r ( u 0 , v 0 ) , Ω 2 ( u 0 , v 0 ) , ϕ ( u 0 , v 0 ) , Q ( u 0 , v 0 ) , ∂ u r ( u 0 , v 0 ) etc. Our definition of sphere data formalizes this (long) list of numbers and incorporates the constraints (2.7)(2.11), so we may refer to the data induced by a C k solution on a sphere without reference to an actual solution of the equations themselves. \nDefinition 2.3. Let k ≥ 1 . A sphere data set with regularity index k for the Einstein-Maxwell-charged scalar field in the EM gauge (2.3) is the following list of numbers 4 : \n1. ϱ > 0 , ϱ 1 u , . . . , ϱ k +1 u , ϱ 1 v , . . . , ϱ k +1 v ∈ R \n2. ω > 0 , ω 1 u , . . . , ω k u , ω 1 v , . . . , ω k v ∈ R \n- 3. φ, φ 1 u , . . . , φ k u , φ 1 v , . . . , φ k v ∈ C\n- 4. q, q 1 u , . . . , q k u , q 1 v , . . . , q k v ∈ R\n- 5. a, a 1 u , . . . , a k u , a 1 v , . . . , a k v , a k +1 v ∈ R \nsubject to the following conditions: \n- (i) ϱ i +2 u can be expressed as a rational function of ϱ j +1 u , ω j +1 u , φ j +1 u , and a j u for 0 ≤ j ≤ i by formally differentiating (2.10),\n- (ii) ϱ i +2 v can be expressed as a rational function of ϱ j +1 v , ω j +1 v , and φ j +1 v for 0 ≤ j ≤ i by formally differentiating (2.11),\n- (iii) q i +1 u can be expressed as a polynomial of ϱ j u , φ j u , and a j u for 0 ≤ j ≤ i by formally differentiating (2.7),\n- (iv) q i +1 v can be expressed as a polynomial of ϱ j u , and φ j u for 0 ≤ j ≤ i by formally differentiating (2.8), and\n- (v) a i +1 v can be expressed as a rational function of ϱ j v , ω j v , and q j v for 0 ≤ j ≤ i by formally differentiating (2.9), \nwhere we have adopted the convention that ϱ 0 u = ϱ , etc. We denote by D k the set of such sphere data sets with regularity index k . \nGauge freedom is a very important aspect of the study of the EMCSF system. Our next definition records the gauge freedom present in sphere data. We need to consider both double null gauge transformations \nu = f ( U ) , v = g ( V ) , \nwhere f and g are increasing functions on R and EM gauge transformations (2.2) \nϕ ↦→ e -i e χ ϕ, A ↦→ A + dχ, \nwhere χ is a function of u alone, i.e. ∂ v χ = 0 , in order to satisfy (2.3). \nDefinition 2.4. We define the full gauge group of the Einstein-Maxwell-charged scalar field system in spherically symmetric double null gauge with the EM gauge condition (2.3) as \nG . = { ( f, g ) : f, g ∈ Diff + ( R ) , f (0) = g (0) = 0 } × C ∞ ( R ) , \nwith the group multiplication given by 5 \n(( f 2 , g 2 ) , χ 2 ) · (( f 1 , g 1 ) , χ 1 ) = (( f 2 · f 1 , g 2 · g 1 ) , χ 2 · f -1 1 + χ 1 ) . \nThe gauge group defines an action on sphere data as follows. Given sphere data D ∈ D k , assign functions r ( u, v ) , Ω 2 ( u, v ) , ϕ ( u, v ) , Q ( u, v ) , and A u ( u, v ) whose jets agree with the sphere data D . For τ = (( f, g ) , χ ) ∈ G , let \n˜ r ( u, v ) = r ( f ( u ) , g ( v )) (2.15) \n˜ Ω 2 ( u, v ) = f ' ( u ) g ' ( v )Ω 2 ( f ( u ) , g ( v )) (2.16) \n˜ ϕ ( u, v ) = e -i e χ ( f ( u )) ϕ ( f ( u ) , g ( v )) (2.17) \n˜ Q ( u, v ) = Q ( f ( u ) , g ( v )) (2.18) \n˜ A u ( u, v ) = f ' ( u ) A u ( f ( u )) + f ' ( u ) χ ' ( f ( u )) . (2.19) \nThe components of τD are then defined by formally differentiating equations (2.15)-(2.19) and evaluating at u = v = 0 . For example, τ ( ϱ ) = ϱ , τ ( ϱ 1 v ) = g ' (0) ϱ 1 v , and τ ( φ 1 u ) = (1 -i e χ ' (0)) e -i e χ (0) φ . \nIf one is given a bifurcate characteristic initial data set ( r, Ω 2 , ϕ, Q, A u ) , the lapse Ω 2 can be set to unity on C ∪ C by reparametrizing u and v . In the sphere data setting, we have an analogous notion: \nDefinition 2.5. A sphere data set D ∈ D k is said to be lapse normalized if ω = 1 and ω i u = ω i v = 0 for 1 ≤ i ≤ k . Every sphere data set is gauge equivalent to a lapse normalized sphere data set.", '2.3.2 Cone data and seed data': "In the previous subsection, we saw how a C k solution ( r, Ω 2 , ϕ, Q, A u ) on Q gives rise to a continuous map Q → D k . For the purpose of characteristic gluing, it is convenient to consider one-parameter families of sphere data which are to be thought of as being induced by constant u cones in Q . \nMore precisely, if we consider a null cone C ⊂ Q , parametrized by v ∈ [ v 1 , v 2 ] , then a solution of the EMCSF system induces a continuous map D : [ v 1 , v 2 ] →D k by sending each v to its associated sphere data D ( v ) . In fact, this map can be produced by knowing only D ( v 1 ) and the values of ( r, Ω 2 , ϕ, Q, A u ) on C . Arguing as in Proposition 2.2 with D ( v 1 ) taking the role of the bifurcation sphere gives: \nProposition 2.3. Let k ∈ N , v 1 < v 2 ∈ R , r, A u ∈ C k +1 ([ v 1 , v 2 ]) , and Ω 2 , ϕ, Q ∈ C k ([ v 1 , v 2 ]) which satisfy the constraints (2.8) , (2.9) , and (2.11) on [ v 1 , v 2 ] . Let D 1 ∈ D k such that all v -components of D 1 agree with the corresponding v -derivatives of ( r, Ω 2 , ϕ, Q, A u ) at v 1 . Then there exists a unique continuous function D : [ v 1 , v 2 ] → D k such that D ( v 1 ) = D 1 and upon identification of the formal symbols ϱ ( D ( v )) , ϱ 1 u ( D ( v )) , etc., with the dynamical variables ( r, Ω 2 , ϕ, Q, A u ) and their u - and v -derivatives, satisfies the EMCSF system and agrees with ( r, Ω 2 , ϕ, Q, A u ) in the v -components for every v ∈ [ v 1 , v 2 ] . \nDefinition 2.6. Let k ∈ N and v 1 < v 2 ∈ R . A C k cone data set for the Einstein-Maxwell-charged scalar field in spherical symmetry is a continuous function D : [ v 1 , v 2 ] → D k satisfying the conclusion of Proposition 2.3, i.e., formally satisfying the EMCSF system. \nWe now discuss a procedure for generating solutions of the 'tangential' constraint equations, (2.8), (2.9), and (2.11), which were required to be satisfied in the previous proposition. \nProposition 2.4 (Seed data) . Let k ∈ N , v 1 < v 2 ∈ R , and D 1 ∈ D k be lapse normalized. For any ϕ ∈ C k ([ v 1 , v 2 ]) such that ∂ i v ϕ ( v 1 ) = φ i v ( D 1 ) for 0 ≤ i ≤ k , there exist unique functions r, A u ∈ C k +1 ([ v 1 , v 2 ]) and Q ∈ C k ([ v 1 , v 2 ]) such that ( r, Ω 2 , ϕ, Q, A u ) satisfies the hypotheses of Proposition 2.3 with Ω 2 ( v ) = 1 for every v ∈ [ v 1 , v 2 ] . \nProof. When Ω 2 ≡ 1 , Raychaudhuri's equation (2.11) reduces to \n∂ 2 v r = -r | ∂ v ϕ | 2 , \nwhich is a second order ODE for r ( v ) . Setting r ( v 1 ) = ϱ ( D 1 ) and ∂ v r ( v 1 ) = ϱ 1 v ( D 1 ) , we obtain a unique solution r ∈ C k +1 ([ v 1 , v 2 ]) . The charge is obtained by integrating Maxwell's equation (2.8): \nQ ( v ) = q ( D 1 ) + ∫ v 0 e r 2 ( v ' )Im( ϕ ( v ' ) ∂ v ϕ ( v ' )) dv ' . \nFinally, the gauge potential is obtained by integrating (2.9): \nA u ( v ) = a ( D 1 ) -∫ v 0 Q ( v ' ) 2 r 2 ( v ' ) dv ' . \nThe v -derivatives of ( r, Ω 2 , ϕ, Q, A u ) agree with the v -components of D 1 by virtue of the definitions.", '3 The main gluing theorems': 'In this section we give precise statements of our main theorems. In order to do this, we carefully define the notion of characteristic gluing .', '3.1 Characteristic gluing in spherical symmetry': 'Definition 3.1 (Characteristic gluing) . Let k ∈ N . Let D 1 , D 2 ∈ D k be sphere data sets. We say that D 1 can be characteristically glued to D 2 to order k in the Einstein-Maxwell-charged scalar field system in spherical symmetry if there exist v 1 < v 2 and a C k cone data set D : [ v 1 , v 2 ] →D k such that D ( v 1 ) is gauge equivalent to D 1 and D ( v 2 ) is gauge equivalent to D 2 . \nRemark 3.1 . It is clear that if D 1 and D 2 can be characteristically glued and τ 1 , τ 2 ∈ G , then τ 1 D 1 and τ 2 D 2 can be characteristically glued. \nRemark 3.2 . Definition 3.1 on characteristic gluing along an outgoing cone has a natural analog defining characteristic gluing along an ingoing cone by parametrizing the cone data with u and letting v denote the transverse null coordinate, but keeping the definition of sphere data unchanged. \nBy Proposition 2.4, characteristic gluing is equivalent to choosing an appropriate seed ϕ in the following sense. By applying a gauge transformation to D 1 , we may assume it to be lapse normalized. Then cone data sets with Ω 2 ≡ 1 agreeing with D 1 at v 1 are parametrized precisely by functions ϕ ∈ C k ([ v 1 , v 2 ]; C ) with the correct v -jet at v 1 . Therefore, characteristic gluing reduces to finding ϕ so that the final data set D ( v 2 ) produced by Proposition 2.3 is gauge equivalent to D 2 .', '3.2 Spacetime gluing from characteristic gluing': 'If the two sphere data sets in Definition 3.1 come from spheres in two spherically symmetric EMCSF spacetimes, we can use local well posedness for the EMCSF characteristic initial value problem, Proposition 2.1, to glue parts of the spacetimes themselves. This principle underlies all of our constructions in Section 5. \nFigure 8: Spacetime gluing obtained from characteristic gluing. The two spacetimes (dark gray) are glued along the cone u = -1 . Note that the dark gray regions are causally disconnected except for the cone u = -1 . Such a spacetime exists if and only if D 1 and D 2 can be characteristically glued. \n<!-- image --> \nProposition 3.1. Let ( Q 1 , r 1 , Ω 2 1 , ϕ 1 , Q 1 , A u 1 ) and ( Q 2 , r 2 , Ω 2 2 , ϕ 2 , Q 2 , A u 2 ) be two C k solutions of the EMCSF system in spherical symmetry, where each Q i is a double null rectangle, i.e., \n. \nQ 1 = [ u 0 , 1 , u 1 , 1 ] × [ v 0 , 1 , v 1 , 1 ] Q 2 = [ u 0 , 2 , u 1 , 2 ] × [ v 0 , 2 , v 1 , 2 ] \nLet D 1 be the sphere data induced by the first solution on ( u 0 , 1 , v 1 , 1 ) and D 2 be the sphere data induced by the second solution on ( u 1 , 2 , v 0 , 2 ) . If D 1 can be characteristically glued to D 2 to order k , then there exists a spherically symmetric C k solution ( Q , r, Ω 2 , ϕ, Q, A u ) of the EMCSF system with the following property: There exists a global double null gauge ( u, v ) on Q containing double null rectangles \nR 1 = [ -1 , u 2 ] × [ v 0 , v 1 ] , R 2 = [ u 0 , -1] × [ v 2 , v 3 ] , \nsuch that the restricted solutions ( R i , r, Ω 2 , ϕ, Q, A u ) are isometric to the solutions ( Q i , r i , Ω 2 i , ϕ i , Q i , A ui ) for i = 1 , 2 , the sphere data induced on ( -1 , v 1 ) is equal to D 1 to k -th order, and the sphere data induced on ( -1 , v 2 ) is gauge equivalent to D 2 to k -th order. \nProof. In this proof, we will refer to spherically symmetric solutions of the EMCSF system by their domains alone. \nBy Definition 3.1, since D 1 and D 2 can be characteristically glued, we obtain v 1 < v 2 , functions r, Ω 2 , ϕ, Q, and A u on [ v 1 , v 2 ] , and a gauge transformation τ ∈ D k which acts on D 2 . We now build the spacetime out of two pieces which will then be pasted along u = -1 and match to order C k . See Fig. 9. \nFirst, we prepare the given spacetimes. We relabel the double null gauge on Q 1 by changing the southeast edge to be u = -1 and the northeast edge to be v = v 1 . This also determines u 2 and v 0 and we apply no further gauge transformation to Q 1 . We denote this region by R 1 . \nNext, the gauge transformation τ is extended and applied to Q 2 . We relabel the double null gauge to have u = -1 on the northwest edge and v = v 2 on the southwest edge. We denote this region by R 2 . \nWe now construct the left half of Fig. 9 as follows. Extend the cone u = -1 in R 1 until v = v 3 , and extend the functions ( r, Ω 2 , ϕ, Q, A u ) on u = -1 by taking them from the definition of characteristic gluing for v ∈ [ v 1 , v 2 ] , and then from the induced data on u = -1 in R 2 for v ∈ [ v 2 , v 3 ] . We now appeal to local existence, Proposition 2.1, the EMCSF system in spherical symmetry to construct the solution in a thin slab S 1 to the future of \n( { u = -1 } × [ v 1 , v 3 ]) ∪ ([ -1 , u 2 ] ×{ v = v 1 } ) . \nThis completes the construction of R 1 ∪ S 1 . \nThe region R 2 ∪ S 2 is constructed similarly, with the cone u = -1 now being extended backwards, first using the characteristic gluing data and then using the tangential data induced by R 1 on u = -1 . Again, Proposition 2.1 is used to construct the thin strip S 2 . \nFinally, the spacetime is constructed by taking Q . = ( R 1 ∪S 1 ) ∪ ( R 2 ∪S 1 ) and pasting r, Ω 2 , ϕ, Q, and A u . From the construction, it is clear that the dynamical variables, together with all v -derivatives consistent with C k regularity are continuous on Q . To show that all u -derivatives are continuous across u = -1 , we observe that all transverse quantities are initialized consistently to k -th order at ( -1 , v 1 ) and that the tangential data agrees by construction. Now Proposition 2.2 implies that the transverse derivatives through order k are equal on u = -1 in both R 1 ∪ S 1 and R 2 ∪ S 2 . This completes the proof. \nFigure 9: Proof of Proposition 3.1. \n<!-- image --> \nRemark 3.3 . If the characteristic gluing hypothesis is C k but no better and the original solutions Q 1 and Q 2 are more regular than C k , then one expects ( k +1) -th derivatives of dynamical quantities to jump across any of the null hypersurfaces bordering the light gray regions in Fig. 8.', '3.3 Sphere data in Minkowski, Schwarzschild, and Reissner-Nordström': 'Before stating our main gluing results, we need to precisely define the terms Minkowski sphere , Schwarzschild event horizon sphere , and Reissner-Nordström event horizon sphere . \nDefinition 3.2 (Minkowski sphere data) . Let k ∈ N and R > 0 . The unique lapse normalized sphere data set satisfying \n- · ϱ = R ,\n- · ϱ 1 u = -1 2 ,\n- · ϱ 1 v = 1 2 , and\n- · all other components zero, \nis called the Minkowski sphere data of radius R and is denoted by D M R,k . \nDefinition 3.3 (Schwarzschild sphere data) . Let k ∈ N , R > 0 , and 0 ≤ 2 M ≤ R . The unique lapse normalized sphere data set satisfying \n- · ϱ = R ,\n- · ϱ 1 u = -1 2\n- · ϱ 1 v = 1 2 (1 -2 M/R ) , and\n- · all other components zero, \nis called the Schwarzschild sphere data of mass M and radius R and is denoted by D S M,R,k . Note that D S 0 ,R,k = D M R,k . \nDefinition 3.4 (Reissner-Nordström horizon sphere data) . Let k ∈ N , M > 0 , and 0 ≤ | e | ≤ M . The unique lapse normalized sphere data set satisfying \n- · ϱ = r + . = M + √ M 2 -e 2 ,\n- · ϱ 1 u = -1 2 ,\n- · ϱ 1 v = 0 ,\n- · q = e , and\n- · all other components zero, \nis called the Reissner-Nordström horizon sphere data with parameters M and e and is denoted by D RN H M,e,k . Note that D RN H M, 0 ,k = D S M, 2 M,k . \nWe will also define sphere data for general Reissner-Nordström spheres. To do so, we extend the Hawking mass (2.1) to a function on sphere data sets D ∈ D k by setting \nm ( D ) . = ϱ 2 ( 1 + 4 ϱ 1 u ϱ 1 v ω ) . \nWe also define the modified Hawking mass of a spherically symmetric spacetime with charge by \nϖ . = m + Q 2 2 r \nand extend it to sphere data sets by \nϖ ( D ) . = m ( D ) + q 2 2 ϱ . \nIn a Reissner-Nordström spacetime of mass M and charge e , any sphere data set D associated to a symmetry sphere has ϖ ( D ) = M . Note that given ϱ > 0 , ϱ 1 u < 0 , ω , and q , ϱ 1 v is determined uniquely by ϖ ( D ) . Recall that the horizons of Reisser-Nordström with parameters | e | ≤ M are located at \nr ± = M ± √ M 2 -e 2 . \nDefinition 3.5 (Reissner-Nordström sphere data) . Let k ∈ N , e ∈ R , and R > 0 satisfy M > e 2 / (2 R ) . A lapse normalized sphere data set satisfying \n- · ϱ = R ,\n- · q = e ,\n- · ϖ = M ,\n- · | ϱ 1 v | = 1 2 , or | ϱ 1 u | = 1 2 , or ϱ 1 v = ϱ 1 u = 0 ,\n- · all other components zero \nis called a Reissner-Nordström sphere data set of modified Hawking mass M , charge e , and radius R and is denoted by D RN M,e,R,k . \nRemark 3.4 . A Reissner-Nordström sphere data set of modified Hawking mass M , charge e , and radius R , D RN M,e,R,k , gives rise to unique sphere data if either ϱ 1 v = ϱ 1 u = 0 , or one additionally specifies sgn( ϱ 1 v ) ∈ { + , -} or sgn( ϱ 1 u ) ∈ { + , -} .', '3.4 Main gluing theorems': "With the previous definitions of Section 3.1 and Section 3.3 at hand, we are now in a position to state our main gluing results. \nOur first gluing theorem concerns gluing a sphere in Minkowski space to a Schwarzschild event horizon with a real scalar field. When the scalar field ϕ in the EMCSF system is real-valued, Maxwell's equation decouples from the rest of the system and the charge Q is constant throughout the spacetime. Since Q must vanish on any sphere in Minkowski space, it vanishes everywhere and the EMCSF system reduces to the Einstein-scalar field system. \nTheorem 2A. For any k ∈ N and 0 < R i < 2 M f , the Minkowski sphere of radius R i , D M R i ,k , can be characteristically glued to the Schwarzschild event horizon sphere with mass M f , D S M f ,k , to order C k within the Einstein-scalar field model in spherical symmetry. \nThe proof of Theorem 2A is given in Section 4.1. We have separated out Minkowski to Schwarzschild gluing as a special case because it is simpler and highlights our topological argument. We will actually use this special case as the first step to produce our counterexample to the third law in Section 5.3. \nOur second gluing theorem concerns gluing a sphere in the domain of outer communication of a Schwarzschild spacetime to a Reissner-Nordström event horizon with specified mass and charge to mass ratio. \nTheorem 2B. For any k ∈ N , q ∈ [ -1 , 1] , and e ∈ R \\ { 0 } , there exists a number M 0 ( k, q , e ) ≥ 0 such that if M f > M 0 , 0 ≤ M i ≤ 1 8 M f , and 2 M i < R i ≤ 1 2 M f , then the Schwarzschild sphere of mass M i and radius R i , D S M i ,R i ,k , can be characteristically glued to the Reissner-Nordström event horizon with mass M f and charge to mass ratio q , D RN H M f , q M f ,k , to order C k within the Einstein-Maxwell-charged scalar field model with coupling constant e . The associated characteristic data can be chosen to have no spherically symmetric antitrapped surfaces, i.e. ∂ u r < 0 everywhere. \nThe proof of Theorem 2B is given in Section 4.2. \nRemark 3.5 . The data constructed in the proof of Theorem 2A will automatically not contain spherically symmetric antitrapped surfaces because of a special monotonicity property in the absence of charge. Namely, \n∂ v ( r∂ u r ) = -Ω 2 4 , (3.1) \nso r∂ u r is decreasing. In particular, since r∂ u r is negative in Minkowski space, the sign will propagate in view of (3.1) for the Einstein-scalar field model. \nOur next gluing theorem supersedes Theorem 2A and Theorem 2B by relaxing the requirement that the final sphere lie on the event horizon. The proof is slightly more involved than Theorem 2B but has the same basic structure and is given in Section 4.3 below. \nTheorem 2C. For any k ∈ N , q ∈ R , e ∈ R \\ { 0 } and r > 0 , there exists a number M 0 ( k, q , e , r ) > 0 such that if M f > M 0 and \nR f ≥ M f 2 (1 + r ) q 2 , (3.2) \nthen there exists R i ∈ (0 , R f ) such that the Minkowski sphere of radius R i , D M R i ,k , can be characteristically glued to the Reissner-Nordström sphere with modified Hawking mass M f , charge q M f , and radius R f , D RN M f , q M f ,R f ,k with ϱ 1 u < 0 , to order C k within the Einstein-Maxwell-charged scalar field system with coupling constant e . The associated characteristic data can be chosen to have no spherically symmetric antitrapped surfaces, i.e., ∂ u r < 0 everywhere. \nRemark 3.6 . Reissner-Nordström spheres with modified Hawking mass M , charge q M and radius R ≤ M 2 q 2 have non-positive Hawking mass, m ≤ 0 . In this sense, the assumption r > 0 in Theorem 2C is necessary. Indeed, one immediately sees that (3.2) implies \nm ≥ r 1 + r M f , \nso that r > 0 ensures m> 0 . \nRemark 3.7 . Theorem 2C also allows for gluing of Minkowski space to Reissner-Nordström Cauchy horizons located at r = r -. This is achieved by setting r = q 2 / 4 in Theorem 2C, see already the proof of Corollary 3. \nWhile all the above theorems are stated as gluing results along outgoing cones, by mapping u ↦→ -v and v ↦→ -u , they also hold true for gluing along ingoing cones, recall Remark 3.2. In particular, restating Theorem 2C for gluing along ingoing cones gives \nTheorem 2C ' . For any k ∈ N , q ∈ R , e ∈ R \\ { 0 } and r > 0 , there exists a number M 0 ( k, q , e , r ) > 0 such that if M f > M 0 and \nR f ≥ M f 2 (1 + r ) q 2 , (3.3) \nthen there exists R i ∈ (0 , R f ) such that the Reissner-Nordström sphere with modified Hawking mass M f , charge q M f , and radius R f , D RN M f , q M f ,R f ,k with ϱ 1 v > 0 , can be characteristically glued along an ingoing cone to the Minkowski sphere of radius R i , D M R i ,k , to order C k within the Einstein-Maxwell-charged scalar field system with coupling constant e . The associated characteristic data can be chosen to have no spherically symmetric trapped surfaces, i.e., ∂ v r > 0 everywhere.", '4 Proofs of the main gluing theorems': "We begin with two lemmas which identify the orbits of Schwarzschild and Reissner-Nordström sphere data under the action of the full gauge group. This essentially amounts to a version of Birkhoff's theorem for sphere data. \nLemma 4.1 (Schwarzschild exterior sphere identification) . If D ∈ D k satisfies \n- · ϱ = R > 0 ,\n- · ϱ 1 u < 0 ,\n- · ϱ 1 v > 0 ,\n- · 1 2 ϱ (1 + 4 ϱ 1 u ϱ 1 v ) = M,\n- · q = 0 , and\n- · φ i u = φ i v = 0 for 0 ≤ i ≤ k , \nthen R > 2 M and D is equivalent to D S M,R,k up to a gauge transformation. \nProof. First, we observe that by the relations obtained from Maxwell's equations, q i u = q i v = 0 for 1 ≤ i ≤ k . Since φ i u = φ i v = 0 , we can perform an EM gauge transformation to make a i u = 0 for 0 ≤ i ≤ k . Also, a i v = 0 for 1 ≤ i ≤ k from F = d ( A u du ) . Next, we can normalize the lapse. Finally, R > 2 M follows from the definitions and ϱ 1 u ϱ 1 v < 0 . \nLemma 4.2 (Reissner-Nordström horizon sphere identification) . If D ∈ D k satisfies \n- · ϱ = (1 + √ 1 -q 2 ) M for q ∈ [ -1 , 1] and M > 0 ,\n- · ϱ 1 u < 0 ,\n- · ϱ 1 v = 0 ,\n- · q = q M , and\n- · φ i u = φ i v = 0 for 0 ≤ i ≤ k , \nthen D is equivalent to D RN M, q M,k up to a gauge transformation. \nProof. As before, the charge vanishes to all orders and we normalize the gauge potential and lapse. We then use the additional double null gauge freedom u ↦→ λu , v ↦→ λ -1 v to make ϱ 1 u = -1 2 . \nRemark 4.1 . Without the condition ϱ 1 u < 0 in the previous lemma, the sphere data in the extremal case could also arise from the Bertotti-Robinson universe. \nWith these lemmas and Remark 3.1 in mind, we follow the strategy discussed in Section 3.1. We fix the interval [0 , 1] , set Ω 2 ≡ 1 , and solve Raychaudhuri's equation, Maxwell's equation, and the transport equation for transverse derivatives of ϕ with appropriate initial and final values. We do not have to track transverse derivatives of ∂ u r , Ω 2 , Q , or A u , because these will be 'gauged away' at the end of the proof.", '4.1 Proof of Theorem 2A': "In this subsection we prove Theorem 2A. We first note that if the scalar field is chosen to be real-valued, the Einstein-Maxwell-charged scalar field system collapses to the Einstein-scalar field system. If the initial data has no charge ( Q (0) = 0 ), then this is equivalent to setting e = 0 and A u and all its derivatives to be identically zero. \nWe will first set up our scalar field ansatz as a collection of pulses. To do so, let \n0 = v 0 < v 1 < · · · < v k < v k +1 = 1 \nbe an arbitrary partition of [0 , 1] . For each 1 ≤ j ≤ k +1 , fix a nontrivial bump function \nχ j ∈ C ∞ c (( v j -1 , v j ); R ) . \nIn the rest of this section, the functions χ 1 , . . . , χ k +1 are fixed and our constructions depend on these choices. Let α = ( α 1 , . . . , α k +1 ) ∈ R k +1 and set \nϕ α ( v ) . = ϕ ( v ; α ) . = ∑ 1 ≤ j ≤ k +1 α j χ j ( v ) . (4.1) \nWe set Ω 2 ( v ; α ) ≡ 1 along [0 , 1] and define r ( v ; α ) as the unique solution of Raychauduri's equation (2.11) with this scalar field ansatz, \n∂ 2 v r ( v ; α ) = -r ( v ; α )( ∂ v ϕ α ( v )) 2 , (4.2) \nr (1; α ) = 2 M \nf \n∂ v r (1; α ) = 0 . \nwith prescribed 'final values' \nLet 0 < ε < 2 M f -R i . By Cauchy stability and monotonicity properties of Raychaudhuri's equation (4.2), there exists a δ > 0 such that for every 0 < | α | ≤ δ , \nsup [0 , 1] | r ( · ; α ) -2 M f | ≤ ε, inf [0 , 1] ∂ v r ( · ; α ) ≥ 0 , ∂ v r (0; α ) > 0 . \n̸ \nThe final inequality follows from the fact that α = 0 . We now consider the sphere S k δ . = { α ∈ R k +1 : | α | = δ } . For each α ∈ S k δ , define D α (0) ∈ D k by setting \n- · ϱ = r (0; α ) > 0 ,\n- · ϱ 1 v = ∂ v r (0; α ) > 0 ,\n- · ϱ 1 u = -1 4 ( ϱ 1 v ) -1 ,\n- · ω = 1 , and\n- · all other components to zero. \nBy Lemma 4.1, D α (0) is equivalent to D M up to a gauge transformation. \nr (0; α ) ,k For each α ∈ S k δ , we now apply Proposition 2.3 and Proposition 2.4 to uniquely determine cone data \nD α : [0 , 1] →D k , \nwith initialization D α (0) above and seed data ϕ α given by (4.1). By standard ODE theory, D α ( v ) is jointly continuous in v and α . Note that ϱ ( D α ( v )) = r ( v ; α ) and φ ( D α ( v )) = ϕ ( v ; α ) by definition. We now use the notation \n∂ i u ϕ ( v ; α ) . = φ i u ( D α ( v )) \nfor i = 1 , . . . , k to denote the transverse derivatives of the scalar field obtained by Proposition 2.3. By construction, the data set D α (1) satisfies \n- · ϱ = 2 M f ,\n- · ϱ 1 u < 0 ,\n- · ϱ 1 v = 0 ,\n- · ω = 1 , and\n- · φ i v = 0 for 0 ≤ i ≤ k . \nThe second property follows from the initialization of ϱ 1 u in D α (0) and the monotonicity of \n( r∂ u r )( v ; α ) . = ϱ ( D α ( v )) ϱ 1 u ( D α ( v )) \nin the Einstein-scalar field system discussed in Remark 3.5. \nIn order to glue to Schwarzschild at v = 1 , by Lemma 4.2, it suffices to find an α ∗ ∈ S k δ for which additionally \n∂ u ϕ (1; α ∗ ) = · · · = ∂ k u ϕ (1; α ∗ ) = 0 . \nThe following discrete symmetry of the Einstein-scalar field system plays a decisive role in finding α ∗ . A function f ( v ; α ) is even in α if f ( v ; -α ) = f ( v ; α ) and odd in α if f ( v ; -α ) = -f ( v ; α ) . \nLemma 4.3. As functions on [0 , 1] × S k δ , the metric coefficients r ( v ; α ) , Ω 2 ( v ; α ) and all their ingoing and outgoing derivatives are even functions of α . The scalar field ϕ ( v ; α ) and all its ingoing and outgoing derivatives are odd functions of α . In particular, the map \nF : S k δ → R k (4.3) α ↦→ ( ∂ u ϕ (1; α ) , . . . , ∂ k u ϕ (1; α ) ) \nis continuous and odd. \nProof. The scalar field itself is odd by the definition (4.1). Since Raychaudhuri's equation (4.2) involves the square of ∂ v ϕ ( v ; α ) , r ( v ; α ) will be automatically even. Next, ∂ u r ( v ; α ) is found by integrating the wave equation for the radius (2.5), forwards in v with initial value determined by D α (0) . Since ϕ enters into this equation with an even power (namely zero), ∂ u r ( v ; α ) will also be even. The wave equation for rϕ in the Einstein-scalar field model can be derived from (2.13) and reads \n∂ u ∂ v ( rϕ ) = -Ω 2 m 2 r 3 rϕ, \nand the right-hand side is odd in α (the Hawking mass is constructed from metric coefficients so is also even). Recall from Proposition 2.3 that this wave equation is used to compute φ i u ( D α ( v )) . By inspection ∂ u ( rϕ ) is odd, whence ∂ u ϕ ( v ; α ) is also odd. The proof now follows by inductively following the procedure of Proposition 2.2, taking note of the fact that the transport equations for ingoing derivatives of r and Ω 2 only involve even powers of ϕ and its derivatives, whereas the transport equations for ingoing derivatives of ϕ only involve odd powers. \nThe claim about the map F follows from the oddness of ingoing derivatives of ϕ and the continuity of all dynamical quantities in α , per standard ODE theory. \nWe now complete the proof of Theorem 2A. By the Borsuk-Ulam theorem stated as Theorem 3, F ( α ∗ ) = 0 for some α ∗ ∈ S k δ , where F is as in (4.3). By Lemma 4.2, D α ∗ (1) is gauge equivalent to D S M f ,k . \nSo far we have glued D M r (0; α ) ,k to D S M f ,k , and since r (0; α ) > R i , we extend the data trivially in order to glue D M R i ,k to D S M f ,k , which concludes the proof of Theorem 2A.", '4.2 Proof of Theorem 2B': "̸ \nIn this subsection we prove Theorem 2B. We assume that q = 0 , the q = 0 version of this result being essentially a repeat of the arguments in the previous section combined with the new initialization of ∂ u r (0; α ) in (4.17) below. \nIn this subsection we adopt the notational convention that A ≲ B means A ≤ CB , where C is a constant that depends only on k and the baseline scalar field profile, but not on q , e , M i , M f , or α . The notation A ≈ B means A ≲ B and B ≲ A . \nLet \n0 = v 0 < v 1 < · · · < v 2 k < v 2 k +1 = 1 \nbe an arbitrary partition of [0 , 1] . For each 1 ≤ j ≤ 2 k +1 , fix a nontrivial bump function \nχ j ∈ C ∞ c (( v j -1 , v j ); R ) . \nIn the rest of this section, the functions χ 1 , . . . , χ 2 k +1 are fixed and our constructions depend on these choices. \nFor α = ( α 1 , . . . , α 2 k +1 ) ∈ R 2 k +1 , set \nϕ α ( v ) . = ϕ ( v ; α ) . = ∑ 1 ≤ j ≤ 2 k +1 α j χ j ( v ) e -iv . (4.4) \nRemark 4.2 . If e > 0 , this choice of ϕ will make Q ≥ 0 , which is consistent with q > 0 . If e > 0 and q < 0 , then we replace -iv in the exponential with + iv . Similarly, the cases e < 0 , q > 0 and e < 0 , q < 0 can be handled. Therefore, we assume without loss of generality that e > 0 , q > 0 . \nFor ˆ α ∈ S 2 k (the unit sphere in R 2 k +1 ) and β ≥ 0 , it is convenient to define r ( v ; β, ˆ α ) = r ( v ; β ˆ α ) , etc. We again set Ω 2 ( v ; α ) ≡ 1 and study the equations (2.11) and (2.8) for v ∈ [0 , 1] with the ϕ α ansatz: \n∂ 2 v r ( v ; α ) = -| α | 2 r ( v ; α ) | ∂ v ϕ ˆ α ( v ) | 2 , (4.5) \n∂ v Q ( v ; α ) = e | α | 2 r ( v ; α ) 2 Im( ϕ ˆ α ( v ) ∂ v ϕ ˆ α ( v )) . (4.6) \nIn addition, we again define r at v = 1 by \nand Q at v = 0 by \nfor v ∈ [0 , 1] , where \nFurthermore, \n∂ v r (0; β ˆ α ) > 0 . (4.10) \nProof. This is a simple bootstrap argument in v . Assume that on [ v 0 , 1] ⊂ [0 , 1] , we have \n[ \ninf v 0 , 1] r ≥ 0 \ninf [ v 0 , 1] ∂ v r ≥ 0 . \nThis is clear for v 0 close to 1 by Cauchy stability. From Raychaudhuri's equation (4.5), r ≥ 0 implies ∂ v r is monotone decreasing, hence is bounded above by ∂ v r ( v 0 ) , which can be estimated by \n∂ v r ( v 0 ) = ∫ 1 v 0 β 2 r | ∂ v ϕ ˆ α | 2 dv ≲ β 2 r + , (4.11) \nsince r ≤ r + on [ v 0 , 1] . It follows that \nr ( v 0 ) = r + -∫ 1 v 0 ∂ v r dv ≥ r + -Cβ 2 r + (4.12) \nfor some C ≲ 1 . Choosing β > 0 sufficiently small shows r ( v 0 ) ≥ 1 2 r + which improves the bootstrap assumptions and proves the desired estimate (4.8). Finally, note that (4.10) holds true as ∂ v r is monotone decreasing and r is not constant ( β > 0 and the scalar field is not identically zero). \nr (1; α ) = r + , ∂ v r (1; α ) = 0 , \nQ (0; α ) = 0 , \n(4.7) \nwhich together with (4.5) and (4.6) uniquely determine r and Q on [0 , 1] . Note that we will initialize ∂ u r only later in (4.17). \nWe first note that basic calculations yield \n| ∂ v ϕ ˆ α | 2 = ∑ 1 ≤ j ≤ 2 k +1 ˆ α 2 j ( χ 2 j + χ ' 2 j ) \nand \nTherefore, \nIm( ϕ ˆ α ∂ v ϕ ˆ α ) = ∑ 1 ≤ j ≤ 2 k +1 ˆ α 2 j χ 2 j . \n∫ 1 0 | ∂ v ϕ ˆ α | 2 dv ≈ ∫ 1 0 Im( ϕ ˆ α ∂ v ϕ ˆ α ) dv ≈ 1 \nfor any ˆ α ∈ S 2 k . \nLemma 4.4. There exists a constant 0 < c ≲ 1 such that if 0 < β ≤ c , then for any ˆ α ∈ S 2 k , r ( · ; β ˆ α ) satisfies \nr \n( \nv \n; \nβ \nˆ α \n) \n≥ \n1 \n2 \nr \n+ \n(4.8) \n∂ v r ( v ; β ˆ α ) ≥ 0 (4.9) \nr + . = ( 1 + √ 1 -q 2 ) M f . \nLemma 4.5. By potentially making the constant c from Lemma 4.4 smaller, we have that for any 0 < β ≤ c and ˆ α ∈ S 2 k , the following estimate holds \n∂ ∂β Q (1; β, ˆ α ) > 0 . \nProof. Integrating Maxwell's equation (4.6) and using (4.7), we find \nQ (1; β, ˆ α ) = ∫ 1 0 e β 2 r 2 Im( ϕ ˆ α ∂ v ϕ ˆ α ) dv. \nA direct computation yields \n∂ β Q (1; β, ˆ α ) = 2 e β ∫ 1 0 ( r 2 + βr∂ β r )Im( ϕ ˆ α ∂ v ϕ ˆ α ) dv. \nNote that Im( ϕ ˆ α ∂ v ϕ ˆ α ) ≥ 0 pointwise and is not identically zero. Since 0 < β ≤ c , we use Lemma 4.4 to estimate \nr 2 + βr∂ β r ≥ 1 4 r 2 + -Cβr 2 + = r 2 + ( 1 4 -Cβ ) , \nwhere we also used | ∂ β r | ≲ r + which follows directly from differentiating (4.5) with respect to β = | α | . Therefore, by choosing c even smaller, we obtain ∂ β Q (1; β, ˆ α ) > 0 . \nLemma 4.6. If e M f / q is sufficiently large depending only on k and the choice of profiles, then there is a smooth function β Q : S 2 k → (0 , ∞ ) so that Q (1; β Q (ˆ α ) , ˆ α ) = q M f for every ˆ α ∈ S 2 k , which also satisfies \nβ Q (ˆ α ) ≈ √ q M f √ e r + (4.13) \nβ Q ( -ˆ α ) = β Q (ˆ α ) (4.14) \nfor every ˆ α ∈ S 2 k . \nProof. As in the proof of Lemma 4.5 we have \nQ (1; β, ˆ α ) = e β 2 ∫ 1 0 r 2 Im( ϕ ˆ α ∂ v ϕ ˆ α ) . \nIf β is sufficiently small so that Lemma 4.4 and Lemma 4.5 apply, we estimate \nQ (1; β, ˆ α ) ≈ e β 2 r 2 + . \nFor e M f / q sufficiently large as in the assumption, we apply now the intermediate value theorem, to obtain a β Q (ˆ α ) satisfying 0 < β Q (ˆ α ) ≤ c such that \nQ (1; β Q , ˆ α ) = q M f . (4.15) \n̸ \nNote that β Q (ˆ α ) is unique since Q (1; · , ˆ α ) is strictly increasing as shown in Lemma 4.5. Moreover, since Q (1; · , · ) is smooth (note that ˆ α ∈ S 2 k and β > 0 enter as smooth parameters in (4.6) which defines Q ), a direct application of the implicit function theorem using that ∂ β Q (1; · , ˆ α ) = 0 shows that β Q : S 2 k → (0 , ∞ ) is smooth. \nMoreover, by (4.2) and (4.15), β Q satisfies \ne β 2 Q r 2 + ≈ q M f \nwhich shows (4.13). Finally, note that Q (1; β, -ˆ α ) = Q (1; β, ˆ α ) , from which (4.14) follows. \nLemma 4.7. Let e M f / q be sufficiently large (depending only on k and the choice of profiles) so that Lemma 4.6 applies. Then \np Q : S 2 k → Q 2 k ˆ α ↦→ β Q (ˆ α )ˆ α \nis a diffeomorphism, where \nQ 2 k . = { β Q (ˆ α )ˆ α : ˆ α ∈ S 2 k } ⊂ R 2 k +1 \nis the radial graph of β Q . Moreover, Q 2 k is invariant under the antipodal map A ( α ) = -α and p Q commutes with the antipodal map. \nProof. By definition of Q 2 k and the facts that β Q is smooth, positive, and invariant under the antipodal map as proved in Lemma 4.6, the stated properties of Q 2 k and p Q follow readily. \nHaving identified the set Q 2 k which guarantees gluing of the charge Q , for the rest of the section we will always take α ∈ Q 2 k . Recall from (4.13) that for every α ∈ Q 2 k : \n| α | ≈ √ q M f √ e r + . (4.16) \nBefore proceeding to choose sphere data, we will need to examine the equation for ∂ u r because this will place a further restriction on α which must be taken into account before setting up the topological argument. We continue by using the definition of the Hawking mass m in (2.1), to impose the condition \nm (0; α ) = M i \nby initializing \nso \n∂ u r (0; α ) = -( 1 -2 M i r (0; α ) ) 1 4 ∂ v r (0; α ) . (4.17) \nThe transverse derivative ∂ u r ( v ; α ) is now determined by solving (2.5), \n∂ v ∂ u r ( v ; α ) = -1 4 r ( v ; α ) 2 -∂ u r ( v ; α ) ∂ v r ( v ; α ) r ( v ; α ) 2 + Q ( v ; α ) 2 4 r ( v ; α ) 3 , (4.18) \nwith initialization (4.17). \nNote that (4.17) is well-defined by (4.10) and (4.8) from Lemma 4.4. Furthermore, \n1 -2 M i r (0; α ) ≥ 1 -4 M i M f > 0 , \n∂ u r (0; α ) < 0 . (4.19) \nHaving initialized ∂ u r at v = 0 , we determine ∂ u r ( v ; α ) using (4.18), and we will now show that for e M f / q sufficiently large, ∂ u r ( v ; α ) < 0 for all v ∈ [0 , 1] . \nLemma 4.8. If e M f / q is sufficiently large depending only on k and the choice of profiles and if 0 ≤ M i ≤ 1 8 M f , then \nsup v ∈ [0 , 1] ∂ u r ( v ; α ) < 0 (4.20) \nfor every α ∈ Q 2 k . \nProof. Since r > 0 on [0 , 1] , it suffices to show that \nsup [0 , 1] r∂ u r < 0 . \nFirst, by (2.12), \nas \nQ ( v ; α ) ≤ Q (1; α ) = q M f ≲ r ( v ; α ) , \nwhere we used (4.8). Integrating (4.21), we have \nsup v ∈ [0 , 1] r ( v ) ∂ u r ( v ) ≤ r (0) ∂ u r (0) + C 1 , (4.22) \nwhere C 1 ≲ 1 is a constant. Analogously to (4.11), we estimate \n∂ v r (0; α ) ≲ | α | 2 r + ≲ q e , \nwhere we used (4.16). Now, using (4.17), \n-r (0) ∂ u r (0) = r (0) -2 M i 4 ∂ v r (0) ≳ e q ( 1 2 M f -2 M i ) ≳ e q M f . \nTherefore, we improve (4.22) to \nsup v ∈ [0 , 1] r ( v ) ∂ u r ( v ) ≤ -C 2 e q M f + C 1 \nfor some C 2 ≲ 1 . Thus, if e M f / q is sufficiently large we obtain (4.20). \nTo continue the proof of Theorem 2B, we now put our construction into the framework of the sphere data in Section 3.3. For each α ∈ Q 2 k , define D α (0) ∈ D k by setting \n- · ϱ = r (0; α ) ≥ 1 2 r + (see (4.8)),\n- · ϱ 1 v = ∂ v r (0; α ) > 0 (see (4.10)),\n- · ϱ 1 u = ∂ u r (0; α ) < 0 (see (4.17) and (4.19)),\n- · ω = 1 , and\n- · all other components to zero. \nBy Lemma 4.1, D α (0) is equivalent to D S M i ,r (0; α ) ,k up to a gauge transformation. For each α ∈ Q 2 k , we now apply Proposition 2.3 and Proposition 2.4 to uniquely determine cone data \nD α : [0 , 1] →D k , \nwith initialization D α (0) above and seed data ϕ α given by (4.4). By standard ODE theory, D α ( v ) is jointly continuous in v and α . Note that ϱ ( D α ( v )) = r ( v ; α ) , φ ( D α ( v )) = ϕ ( v ; α ) , and q ( D α ( v )) = Q ( v ; α ) by definition. As in the proof of Theorem 2A, we use the notation \n∂ i u ϕ ( v ; α ) . = φ i u ( D α ( v )) \nfor i = 1 , . . . , k to denote the transverse derivatives of the scalar field obtained by Proposition 2.3. Note also that \n∂ u r ( v ; α ) = ϱ 1 u ( D α ( v )) , \nwhere ∂ u r ( v ; α ) is as in (2.5) above. \nBy construction, the data set D α (1) satisfies \n| ∂ v ( r∂ u r ) | = ∣ ∣ ∣ ∣ 1 4 ( 1 -Q 2 r 2 )∣ ∣ ∣ ∣ ≲ 1 , (4.21) \n- · ϱ = 2 M f ,\n- · ϱ 1 u < 0 (see Lemma 4.8),\n- · ϱ 1 v = 0 ,\n- · ω = 1 ,\n- · q = q M f (definition of Q 2 k ), and\n- · φ i v = 0 for 0 ≤ i ≤ k . \nIn order to glue to the appropriate Reissner-Nordström event horizon sphere, by Lemma 4.2, it suffices to find an α ∗ ∈ Q 2 k for which additionally \n∂ u ϕ (1; α ∗ ) = · · · = ∂ k u ϕ (1; α ∗ ) = 0 . \nAnalogously to Lemma 4.3 we first establish \nLemma 4.9. The metric coefficients r ( v ; α ) , Ω 2 ( v ; α ) , the electromagnetic quantities Q ( v ; α ) , A u ( v ; α ) , and all their ingoing and outgoing derivatives are even functions of α . The scalar field ϕ ( v ; α ) and all its ingoing and outgoing derivatives are odd functions of α . \nProof. The proof is essentially the same as Lemma 4.3, noting that equations (2.7), (2.8), and (2.9) are also even in ϕ . \nWe now complete the proof of Theorem 2B. Recall from Lemma 4.7 that p Q : S 2 k → Q 2 k is a diffeomorphism which commutes with the antipodal map. We now argue similarly to Section 4.1. By Lemma 4.9, the function \nF : Q 2 k → C k α ↦→ ( ∂ u ϕ (1; α ) , . . . , ∂ k u ϕ (1; α ) ) \nis continuous and odd. Therefore, the Borsuk-Ulam theorem, stated as Theorem 3, applied to \n(Re F 1 , Im F 1 , . . . , Re F k , Im F k ) · p Q : S 2 k → R 2 k , \nwhere F i is the i th component of F , shows that there is an α ∗ ∈ Q 2 k such that F ( α ∗ ) = 0 . By Lemma 4.2, D α ∗ (1) is gauge equivalent to D RN H M f , q M f ,k which concludes the gluing construction. Since we have already established that ∂ u r < 0 for all v ∈ [0 , 1] in Lemma 4.8, this concludes the proof of Theorem 2B.", '4.3 Proof of Theorem 2C': "In this section we extend our characteristic gluing result Theorem 2B to allow for sphere data at the final sphere which is not necessarily located on a horizon. Recall Definition 3.5 for the definition of general Reissner-Nordström sphere data. As the steps in the proof below are direct generalizations of the proof of Theorem 2B, our presentation here will have fewer details. \n̸ \nProof of Theorem 2C. We only consider the case q = 0 , the case q = 0 being strictly easier and requiring only 'gluing 3' below. Without loss of generality, we may also assume R f ≤ 3 M f as for r ≥ 3 M f we can extend trivially with Reissner-Nordström data satisfying ∂ v r > 0 and ∂ u r < 0 . In the following proof, we use the convention that all constants appearing in ≲ , ≳ and ≈ to also depend on q , r and e . The theorem is proved as a consequence of the following three intermediate gluings: \n- 1. D M R i ,k is glued to D RN M ' ,Q f ,R 1 ,k with a complex scalar field,\n- 2. D RN M ' ,Q f ,R 1 ,k is glued to D RN M ' ,Q f ,R 2 ,k trivially (i.e., with identically vanishing scalar field), and\n- 3. D RN M ' ,Q f ,R 2 ,k is glued to D RN M f ,Q f ,R f ,k with a real scalar field, \nwhere R i . = R f -M 3 / 4 f , 0 < M ' < M f is an intermediate modified Hawking mass, Q f . = q M f , R 1 , R 2 are intermediate radii which satisfy R i < R 1 < R 2 . \nGluing 1. In the interval v ∈ [0 , 1] we impose the ansatz (4.4). At v = 0 , we set \nr (0) = R i , m (0) = Q (0) = 0 , ∂ u r (0) = -1 2 M 1 / 2 f , ∂ v r (0) = M 1 / 2 f 2 . (4.23) \nThe pulse parameters α ∗ which achieve gluing of transverse derivatives of ϕ are determined by the procedure of Section 4.2, with charge condition Q (1; α ) = Q f . As in Section 4.2 we find that the gluing can be performed with parameters satisfying | α ∗ | 2 ≲ M -1 f . Using this estimate on α ∗ , we obtain from Raychaudhuri's equation (2.11) and (4.23) that 1 2 M 1 / 2 f ≥ ∂ v r ≥ 1 4 M 1 / 2 f for every v ∈ [0 , 1] by choosing M 0 ( k, q , e , r ) sufficiently large. This also implies R i ≤ r ≤ R i + 1 2 M 1 / 2 f . Using r ≥ R f -M 3 / 4 f and the estimate analogous to (4.21) we infer | r∂ u r -r (0) ∂ u r (0) | ≲ 1 for every v ∈ [0 , 1] , i.e., 0 < -∂ u r ≤ M -1 / 2 f . We now estimate the Hawking mass at v = 1 by integrating (2.14), \nm (1) = ∫ 1 0 2 r 2 ( -∂ u r ) | ∂ v ϕ | 2 dv + ∫ 1 0 Q 2 2 r 2 ∂ v r dv ≲ M 2 f M -1 / 2 f M -1 f + M 1 / 2 f ≲ M 1 / 2 f . (4.24) \nSetting R 1 = r (1) and M ' = m (1) + Q 2 f / (2 R 1 ) , we have shown that R i < R 1 ≤ R i + 1 2 M 1 / 2 f . The condition (3.2) shows that Q 2 f / (2 R f ) ≤ M f / (1 + r ) . In particular, since R 1 ≥ R f -M 3 / 4 f we estimate \nM ' = m (1) + Q 2 f 2 R 1 ≤ 1 2 ( 1 + 1 1 + r ) M f = 2 + r 2 + 2 r M f (4.25) \nby possibly taking M 0 ( k, q , e , r ) larger. This completes the first gluing step. \nGluing 3. It is more convenient to now carry out the third gluing step and simply ensure that R 2 > R 1 . We use a collection of k +1 real-valued pulses as in (4.1) on v ∈ [0 , 1] . We impose \nr (1) = R f , ∂ u r (1) = -M f , Q (1) = Q f , ϖ (1) = M f . (4.26) \nThis uniquely determines ∂ v r (1) which can have either sign but satisfies | ∂ v r (1) | ≲ M -1 f . We also note that as long as | α | 2 ≤ M -3 / 2 f , we have | ∂ v r | ≲ M -1 / 2 f and thus | r -R f | ≲ M -1 / 2 f on [0 , 1] . This also gives -∂ u r ≈ M f . Using \nϖ (1) -ϖ (0) = ∫ 1 0 2 r 2 ( -∂ u r ) | ∂ v ϕ | 2 dv \nand (4.25), we write the mass condition ϖ (1) = M f and ϖ (0) = M ' as a sphere of α 's ( | α | 2 ≈ M -2 f ) for which we will apply the Borsuk-Ulam argument. We use here that M f -M ' = M f r / (2 + 2 r ) . With | α ∗ | 2 ≈ M -2 f we have the improved estimate | ∂ v r | ≲ M -1 f for v ∈ [0 , 1] and thus, | r (0) -R f | ≲ M -1 f . Taking now M 0 ( k, q , e , r ) sufficiently large makes R 2 . = r (0) > R 1 . \nGluing 2. By the previous constructions, we have R 1 < R 2 , ϖ (0) = ϖ (1) = M ' , Q (0) = Q (1) = Q f , ∂ v r (0) > 0 , and ∂ u r (0) < 0 . Now D RN M ' ,Q f ,R k can be trivially glued to D RN M ' ,Q f ,R 2 ,k by choosing ϕ ≡ 0 , and we must merely ensure that ∂ u r < 0 along the way. Since ∂ v r > 0 by Raychauduri's equation, this amounts to proving 2 m r < 1 . Indeed, \nm ( v ) ≤ m (0) + ∫ 1 0 Q 2 f 2 r 2 ∂ v r dv = m (0) + ∫ R 2 R 1 Q 2 f 2 r 2 dr ≲ m (0) + ( R 2 -R 1 ) ≲ M 1 / 2 f + M 3 / 4 f , \nwhere we used (4.24). In particular, by choosing M 0 ( k, q , e , r ) larger, we can make m ( v ) /M f arbitrarily small and thus ∂ u r < 0 throughout gluing 2.", '5 Constructing the spacetimes and Cauchy data': 'In this final section we will prove our main result Theorem 1 as well as Corollary 1, Corollary 2, and Corollary 3.', '5.1 Maximal future developments of asymptotically flat data for EMCSF': "Our theorems and corollaries in this paper are stated in the framework of the Cauchy problem for the Einstein-Maxwell-charged scalar field system. We recall that Cauchy data for the EMCSF system consist of the usual Cauchy data (Σ , g 0 , k 0 ) for the Einstein equations, where Σ is a 3-manifold, g 0 a Riemannian metric on Σ , and k 0 a symmetric 2 -tensor field, together with initial data for the matter fields, namely initial electric and magnetic fields, E 0 and B 0 , and finally the scalar field ϕ 0 and its 'time derivative' ϕ 1 . (See e.g. [Cho09, Section VI.10] for a treatment of the Einstein-Maxwell Cauchy problem.) Associated to a Cauchy data set is a unique maximal future globally hyperbolic development ( M 4 , g, F, A, ϕ ) [Fou52; CG69]. If the Cauchy data are moreover spherically symmetric, then the maximal development will be spherically symmetric by uniqueness. \nWe will not, however, actually construct our spacetimes by directly evolving Cauchy data. Rather, we construct the spacetimes teleologically by gluing together explicit spacetimes with the help of our characteristic gluing results and Proposition 3.1. In each case, a Cauchy hypersurface Σ is then found, within the spacetime, whose future domain of dependence contains the physically relevant region, and contains no antitrapped spheres. At this point, all attention is restricted to this future domain of dependence. A posteriori , by the existence and uniqueness theory for the maximal globally hyperbolic development, the spacetime will then be contained in the maximal development of the induced data on the Cauchy hypersurface Σ . \nFigure 10: General structure of the MFGHD of asymptotically flat Cauchy data Σ in the EMCSF system in spherical symmetry [Kom13]. What is depicted is the quotient manifold Q as a bounded subset of R 1+1 u,v with boundary suitably labeled. Note that various components of the diagram can be empty. \n<!-- image --> \nSince our examples are maximal globally hyperbolic developments of asymptotically flat, spherically symmetric Cauchy data for the EMCSF system with no antitrapped spheres of symmetry, we can make use of a general characterization of the boundary of spacetime in this context appearing in [Kom13]. In particular one can rigorously associate a global Penrose diagram, and unambiguously identify a nonempty null boundary component future null infinity I + , domain of outer communication J -( I + ) , (possibly empty) black hole region BH . = M\\ J -( I + ) , (possibly empty) event horizon H + . = ∂ ( BH ) , (possibly empty) Cauchy horizon CH + , (possibly empty) r = 0 singularity S , and (possibly empty) null boundary component N emanating from a (possibly absent) 'locally naked' singularity at the center. The Penrose diagram Q ⊂ R 1+1 u,v can be viewed as a global double null chart for the spacetime, with v the 'outgoing' null coordinate and u the 'ingoing' coordinate. See Fig. 10. 6 \nFor use in the statement and proof of Theorem 1 below, we recall that in the EMCSF system in spherical \nsymmetry, the apparent horizon is defined by \nA . = { ∂ v r = 0 } ⊂ BH . \nSince A might have a complicated structure (in particular, it might have nonempty interior), we define an appropriate notion of boundary as follows. The outermost apparent horizon A ' consists of those points p ∈ A whose past-directed ingoing null segment lies in the strictly untrapped region { ∂ v r > 0 } and eventually exits the black hole region, i.e., enters J -( I + ) . A ' is a possibly disconnected achronal curve in the (1 + 1) -dimensional reduction Q of M . Note, as depicted in Fig. 10, that A ' does not necessarily asymptote to future timelike infinity i + . \nFor definiteness, we will make extensive use of these notions in our theorems and corollaries. However, our notation and usage should be sufficiently familiar to readers acquainted with standard concepts in general relativity so that they may read our diagrams and understand our theorems without specific reference to [Kom13]. \nWe also note that when referring to spherically symmetric subsets of ( M , g ) , such as the event horizon H + , we may view them as objects in M or in the reduced space Q . The context will make it clear which point of view we are taking. \nRemark 5.1 . In Appendix B, we show by a barrier argument that since ∂ u r < 0 in a spacetime satisfying the hypotheses of [Kom13], there are also no nonspherically symmetric antitrapped surfaces.", '5.2 Construction of gravitational collapse to Reissner-Nordström': "We now state a more precise version of Corollary 1 as follows. \nCorollary 1. For any k ∈ N , q ∈ [ -1 , 1] \\ { 0 } , and e ∈ R \\ { 0 } , let M 0 ( k, q , e ) be as in Theorem 2B. Then for any M ≥ M 0 there exist asymptotically flat, spherically symmetric Cauchy data (Σ , g 0 , k 0 , E 0 , B 0 , ϕ 0 , ϕ 1 ) for the EMCSF system, with Σ ∼ = R 3 and a regular center, such that the maximal future globally hyperbolic development ( M 4 , g, F, A, ϕ ) has the following properties: \n- · All dynamical quantities are at least C k -regular.\n- · Null infinity I + is complete. \n̸ \n- · The black hole region is nonempty, BH . = M\\ J -( I + ) = ∅ .\n- · The Cauchy surface Σ lies in the domain of outer communication J -( I + ) . In particular, it does not intersect the event horizon H + . = ∂ ( BH ) .\n- · The initial data hypersurface does not contain trapped surfaces.\n- · The spacetime does not contain antitrapped surfaces.\n- · For sufficiently late advanced times v ≥ v 0 , the domain of outer communication, including the event horizon, is isometric to that of a Reissner-Nordström solution with mass M charge to mass ratio q . For v ≥ v 0 , the event horizon of the spacetime can be identified with the event horizon of ReissnerNordström. \nRemark 5.2 . A similar statement can be made with q = 0 for the Einstein-scalar field model, using instead Theorem 2A. In that case, there will also be no assumption made on the mass. \nProof. We refer the reader to Fig. 11 for a visual guide to the proof. Using Theorem 2B with regularity index k +1 (see footnote below) and Proposition 3.1, a portion of Minkowski space \nt + r ≤ 1 2 M, t -r ≥ -1 2 M, \ncan be glued to a Reissner-Nordström solution with parameters M and q M . Note that as depicted, one can solve for a complete future neighborhood of the event horizon, which might not be a complete double null neighborhood. \nFigure 11: Penrose diagram for the proof of Corollary 1. \n<!-- image --> \nSince we are in spherical symmetry, standard techniques (see [Chr93, Section 5] or [LOY18, Section 3]) allow the 'local existence' region emanating from the Reissner-Nordström portion of the spacetime to be extended all the way up to the center. 7 (In this figure, this region is denoted 'Cauchy stability' for reasons that will become clear below.) \nWe now identify a spacelike curve Σ connecting spacelike infinity i 0 in the exactly Reissner-Nordström region to the center, to the past of the cone u = -1 . The curve Σ can be chosen so the induced data on it is asymptotically flat near i 0 . For example, it may be taken to be a constant t curve near i 0 in standard coordinates. Furthermore, by having Σ hug the gluing region closely enough, we are guaranteed to have no spherically symmetric antitrapped surfaces on Σ . \nCompleteness of null infinity I + is inherited from the exact Reissner-Nordström solution. By inspecting Fig. 11, we see that the null hypersurface C -1 is the event horizon H + = ∂J -( I + ) of the spacetime and that Σ can be arranged to lie in the domain of outer communication J -( I + ) . The statement about trapped surfaces follows from Proposition B.2 below. \nWe now consider the (unique) maximal future globally hyperbolic development ( M 4 , g, F, A, ϕ ) of the induced data (Σ , g 0 , k 0 , E 0 , B 0 , ϕ 0 , ϕ 1 ) on Σ . By uniqueness of the MFGHD, it contains the domain of dependence of Σ in the gluing spacetime (and thus all shaded regions to the future of Σ in Fig. 11). Therefore, by construction, ( M 4 , g, F, A, ϕ ) has all the properties listed in the statement of Corollary 1. Note that the property of having no antitrapped symmetry spheres is propagated to the whole development by Raychaudhuri's equation (2.10). By Proposition B.2, the spacetime does not contain any nonspherically symmetric antitrapped surfaces either. This concludes the proof. \nThe above proof made use of spherical symmetry in the local existence region and the region up to the center. In view of potentially extending our work to the Einstein vacuum equations in the future, we give a second construction of these regions which does not invoke spherical symmetry. First, the 'local existence region' can be constructed outside of spherical symmetry by the well-known theorem of Luk [Luk12]. Once such a region has been constructed, we can use the fact that it lies 'outside' of a Minkowski region to construct the rest of the spacetime, up to the center, by Cauchy stability: \nLemma 5.1. Let B r 0 and B r 1 denote the (open) balls of radii r 0 > 0 and r 1 > r 0 in R 3 , respectively. Consider on B r 1 data for the Einstein-Maxwell-charged scalar field system corresponding to Minkowski space, \n( δ, 0 , 0 , 0 , 0 , 0) . Let D . = ( g 0 , k 0 , E 0 , B 0 , ϕ 0 , ϕ 1 ) be a C k (for k ∈ N sufficiently large and not assumed to be spherically symmetric) initial data set for the Einstein-Maxwell-charged scalar field system defined on B r 1 which agrees with the Minkowski data set on B r 0 . Then the maximal globally hyperbolic development of D contains the Minkowski cone over B r 0 'in its interior' in the following sense: \nThere exists an ε > 0 and a development ( g, F, A, ϕ ) of the data D on K r 0 + ε . = { t + r < r 0 + ε }∩{ t ≥ 0 } ⊂ R 3+1 so that the development of the Minkowski portion of the data is defined on K r 0 . = { t + r < r 0 }∩{ t ≥ 0 } and is the Minkowski metric in those coordinates. \nProof. Since this is a standard Cauchy stability argument we merely sketch the proof. For 0 < ε < r 1 -r 0 2 , let θ ε be a cutoff function which is equal to one on B r 0 + ε and vanishes outside B r 0 +2 ε . On B r 1 , we consider the 'initial data set' \nD ε . = ( θ ε g 0 +(1 -θ ε ) δ, θ ε k 0 , θ ε E 0 , θ ε B 0 , θ ε ϕ 0 , θ ε ϕ 1 ) . \nThis does not solve the constraints everywhere, but it does solve them on B r 0 + ε , where it equals D . We assume that k ≥ 5 and show that D ε is O ( ε ) -close to the Minkowski data set in H 4 . Then Cauchy stability for the reduced Einstein equations (in harmonic coordinates) will show that a solution to the reduced equations with data D ε exists on K r 0 +2 ε for ε sufficiently small. By domain of dependence arguments, a genuine solution will then exist on a smaller domain which still contains the entirety of K r 0 in its interior. \nTo show that D ε is close to Minkowski data we must check it componentwise. For brevity, we only check θ ε k 0 . Note first that \n∥ θ ε k 0 ∥ H 4 ≲ ∥ θ ε k 0 ∥ C 4 . \nNow since k 0 vanishes on B r 0 and is at least C 5 , Taylor's theorem implies \n| ∂ i r / ∇ j k 0 | ≲ max { 0 , r -r 0 } 5 -i -j , \nif 0 ≤ i + j ≤ 5 . In the region where either θ ε or ∂ r θ i are nonvanishing, max { 0 , r -r 0 } ≲ ε . It follows that \n∥ θ ε k 0 ∥ H 4 ≲ ∑ 0 ≤ i + j ≤ 4 sup B r 1 | ∂ i r / ∇ j ( θ ε k 0 ) | ≲ ε, \nwhich proves the claim and hence the lemma.", '5.3 Construction of counterexample to the third law': "In this section we prove Theorem 1 with an analogous approach as in the proof of Corollary 1. We first restate the result in more detail. \nTheorem 1. For any k ∈ N and e ∈ R \\ { 0 } , there exist asymptotically flat, spherically symmetric Cauchy data (Σ , g 0 , k 0 , E 0 , B 0 , ϕ 0 , ϕ 1 ) , with Σ ∼ = R 3 and a regular center, for the EMCSF system such that the maximal future globally hyperbolic development ( M 4 , g, F, A, ϕ ) has the following properties: \n- · All dynamical quantities are at least C k -regular.\n- · The spacetime and Cauchy data satisfy all the conclusions of Corollary 1 with q = 1 and final mass M f ≥ M 0 (1 , e , k ) + 8 .\n- · The spacetime contains a double null rectangle of the form R . = {-2 ≤ u ≤ -1 } ∩ { 1 ≤ v ≤ 2 } which is isometric to a double null rectangle in a Schwarzschild spacetime of mass 1 .\n- · The cone { u = -1 } ∩ R lies in the outermost apparent horizon A ' of the spacetime and is isometric to an appropriate portion of the r = 2 hypersurface in the Schwarzschild spacetime of mass 1 .\n- · The outermost apparent horizon A ' is disconnected.\n- · The spacetime contains trapped surfaces in the black hole region, for all arbitrarily late advanced time. More precisely, for every symmetry sphere S u,v ⊂ H + , J + ( S u,v ) contains a trapped sphere.\n- · There exists a neighborhood U of H + in M such that there are no trapped surfaces S ⊂ U . \nFigure 12: Penrose diagram for the proof of Theorem 1. \n<!-- image --> \nProof. We refer to Fig. 12 for a Penrose diagram illustrating the proof. The proof begins as the proof of Corollary 1 (recall also Proposition 3.1), by gluing a Minkowski cone to a Schwarzschild event horizon of unit mass along { u = -1 } . Then, attach a double null rectangle R of Schwarzschild along the hypersurface r = 2 , as in Corollary 1, but stop after a finite advanced time v = 2 . Now place u = -2 so that \nsup { u = -2 }∩ R r = 2 + ε ≤ 3 . \nFor ε sufficiently small, the first strip down to the center can be constructed as in the proof of Corollary 1. Now let M f ≥ M 0 +8 and extend the cone u = -2 to the future with trivial scalar field until r = 1 2 ( M 0 +8) ≫ 3 . Then using Theorem 2B, extremal Reissner-Nordström of mass M f can be attached. We again solve backward up to the center as in Corollary 1 and have now constructed the spacetime depicted in Fig. 12. \nAs in the proof of Corollary 1, we again find an asymptotically flat spacelike curve Σ connecting i 0 with the center and lying entirely in J -( I + ) . The maximal future globally hyperbolic development ( M , g, F, A, ϕ ) of the induced data on Σ contains the domain of dependence of Σ in the spacetime constructed above (and thus all shaded regions to the future of Σ in Fig. 12) and satisfies all the conclusions of Corollary 1 with q = 1 and final mass M f ≥ M 0 (1 , e , k ) + 8 . By construction, M contains the double null rectangle R which satisfies the stated properties. Further, the cone { u = -1 } ∩ R lies in the apparent horizon A of ( M , g ) and { u = -1 } ∩ R is isometric to an appropriate portion of the r = 2 hypersurface in the Schwarzschild spacetime of mass 1 . \nWe readily see that ( M , g ) contains trapped surfaces in any (future) neighborhood of { u = -1 } ∩ R as ∂ v r = 0 along { u = -1 } ∩ R and (2.12) evaluated on { u = -1 } ∩ R gives \n∂ u ( r∂ v r ) = -Ω 2 4 . \nTo prove that trapped surfaces exist for arbitrarily late advanced time, we invoke the general boundary \ncharacterization of [Kom13]. If the r = 0 singularity S is empty, then the outgoing cone starting from one of these trapped spheres terminates on the Cauchy horizon CH + and the claim is clearly true by Raychaudhuri's equation (2.11). If S is nonempty, then every outgoing null cone which terminates on S is eventually trapped since r extends continuously by zero on S . Furthermore, S terminates at the Cauchy horizon CH + or future timelike infinity i + , so the claim is also true in this case. \nWe now show that there exists a neighborhood U of H + in M which does not contain spherically symmetric trapped surfaces. It suffices to show that there is a neighborhood V of H + in Q such that ∂ v r > 0 on V \\ H + , where we use the same symbol for the event horizon in M and Q . Let p ∈ H + be any sphere after the final gluing sphere, see Fig. 12. Then r ( p ) = Q ( p ) = M f , ∂ v r ( p ) = 0 , and ϕ ( p ) = 0 . Reparametrize the double null gauge so that Ω ≡ 1 on the ingoing cone C passing through p . By the wave equation for the radius (2.5), \n∂ u ∂ v r ( p ) = -1 4 M f + M 2 f 4 M 3 f = 0 . \nDifferentiating (2.5) in u , we find \n∂ 2 u ∂ v r = ∂ u r 4 r 2 -∂ u ( ∂ u log r ) ∂ v r -( ∂ u log r ) ∂ u ∂ v r -3 Q 2 ∂ u r 4 r 4 + Q∂ u Q 2 r 3 . \nEvaluating at p , we find ∂ u Q ( p ) = 0 by Maxwell's equation (2.7), so we have \n∂ 2 u ∂ v r ( p ) = ∂ u r ( p ) 4 M 2 f -3 M 2 f ∂ u r ( p ) 4 M 4 f = -2 ∂ u r ( p ) M 2 f > 0 . \nTherefore, ∂ v r becomes immediately positive for all points along C sufficiently close to the event horizon but not on it (see also Fig. 5). 8 \nBy the monotonicity of Raychaudhuri's equation (2.11) and since p ∈ H + after the final gluing sphere was arbitrary, this shows that there exists a neighborhood V of H + contained in Q that does not contain trapped symmetry spheres except for H + itself. That there are also no nonspherically symmetric trapped surfaces in U . = V × S 2 now follows immediately from Proposition B.1 below. \nThe claim about the disconnectedness of the outermost apparent horizon A ' now follows from the fact that A ' ∩H + is one connected component of A ' which does not contain { u = -1 } ∩ R ⊂ A ' . This concludes the proof.", '5.4 Construction of collapse to Reissner-Nordström with piece of Cauchy horizon': 'In this section, we show that a mild modification of the proof of Corollary 1 allows us to construct examples of gravitational collapse such that the black hole region admits a piece of future boundary which is a Cauchy horizon which is isometric to a subextremal or extremal Reissner-Nordström Cauchy horizon. \nCorollary 2. For any k ∈ N , q ∈ [ -1 , 1] \\{ 0 } , and e ∈ R \\{ 0 } , let M 0 ( k, q , e , 1 / 2) be as in Theorem 2C. Then for any M ≥ M 0 there exist asymptotically flat, spherically symmetric Cauchy data (Σ , g 0 , k 0 , E 0 , B 0 , ϕ 0 , ϕ 1 ) , with Σ ∼ = R 3 and a regular center, for the EMCSF system such that the maximal future globally hyperbolic development ( M 4 , g, F, A, ϕ ) has the following properties: \n- · All dynamical quantities are at least C k -regular.\n- · The spacetime and Cauchy data satisfy all the conclusions of Corollary 1.\n- · The black hole region contains an isometrically embedded portion of a Reissner-Nordström Cauchy horizon neighborhood with parameters M and q M , in particular CH + = ∅ . \n̸ \nFigure 13: Penrose diagram depicting the proof of Corollary 2. \n<!-- image --> \nProof. The proof is completely analogous to the proof of Corollary 1. We apply the gluing construction of Theorem 2C to glue a sphere in Minkowski space to a Reissner-Nordström interior sphere with radius R f < r + and r + -R f small. Indeed, this can be achieved by setting r = 1 2 in Theorem 2C as then 1 2 M f (1 + r ) q 2 ≤ 3 4 M f < M f ≤ r + . We then apply the local existence and Cauchy stability argument as in the proof of Corollary 1. We note that the u -width of the local existence and Cauchy stability argument remains uniform as R f → r + so by choosing R f sufficiently close to r + , we guarantee that we find a Cauchy hypersurface Σ which does not intersect the event horizon. We refer to Fig. 13 for the Penrose diagram explaining the proof. \nRemark 5.3 . As in Remark 5.2, we note that a similar statement with a piece of Schwarzschild interior including the { r = 0 } singularity can be made with q = 0 .', '5.5 Construction of black hole interior for which the Cauchy horizon closes off spacetime': "We now give our construction of a spacetime for which the Cauchy horizon closes off the black hole region. \nCorollary 3. For any k ∈ N , q ∈ [ -1 , 1] \\{ 0 } , e ∈ R \\{ 0 } , let ˜ M 0 ( k, q , e , q 2 / 4) be as in Theorem 2C ' . Then for any M ≥ ˜ M 0 there exist asymptotically flat, spherically symmetric Cauchy data (Σ , g 0 , k 0 , E 0 , B 0 , ϕ 0 , ϕ 1 ) , with Σ ∼ = R 3 and a regular center, for the EMCSF system such that the maximal future globally hyperbolic development ( M 4 , g, F, A, ϕ ) has the following properties: \n- · All dynamical quantities are at least C k -regular.\n- · The spacetime does not contain antitrapped surfaces. \n̸ \n- · The black hole region is nonempty, BH . = M\\ J -( I + ) = ∅ .\n- · The future boundary of the black hole region is a C k -regular Cauchy horizon CH + which closes off spacetime, i.e., N ∪ S = ∅ in Fig. 10.\n- · The exterior region is isometric to a Reissner-Nordström exterior with mass M and charge q M . In particular, future null infinity I + is complete. \nProof. Analogous to the proof of Corollary 2 we glue a Reissner-Nordström interior sphere with R f < r -and r --R f small to a sphere in Minkowski space along an ingoing cone using Theorem 2C ' . We can choose R f arbitrarily close to r -in Theorem 2C ' by setting r = q 2 / 4 . Indeed, in this case \nr --M f 2 ( 1 + q 2 4 ) q 2 = M f ( 1 -√ 1 -q 2 ) -M f 2 ( 1 + q 2 4 ) q 2 = M f ( 1 -√ 1 -q 2 -q 2 2 -q 4 8 ) ≥ M f q 6 16 , \nwhere \nFigure 14: Penrose diagram depicting the proof of Corollary 3. \n<!-- image --> \nwhere in the last step we used the Taylor expansion of √ 1 -q 2 around q = 0 . The rest of the proof is now analogous to Corollary 2 and can be read off from Fig. 14. We note that an isometric copy of the Reissner-Nordström exterior can be attached to the past of H + in Fig. 14.", 'A An isolated extremal horizon with nearby trapped surfaces': "In this appendix we show that, in the context of the dominant energy condition, there is no local mechanism forcing a stationary extremal Killing horizon to have no trapped surfaces 'just inside' of the horizon. We also refer back to Section 1.4.6. \nProposition A.1. There exists a C ∞ spherically symmetric spacetime ( M 4 , g ) with a complete null hypersurface H ⊂ M and a Killing vector field T with the following properties. The Killing field T is spherically symmetric, timelike in I -( H ) , spacelike in I + ( H ) , null and tangent along H , where it also satisfies ∇ T T = 0 , i.e., its integral curves are affinely parametrized null generators of H . Furthermore, ( M , g ) contains no antitrapped symmetry spheres, i.e., ∂ u r < 0 , and satisfies the dominant energy condition. Therefore, H is an extremal Killing horizon and I + ( H ) is foliated by trapped symmetry spheres. \nWe recall that a spacetime ( M , g ) satisfies the dominant energy condition if for all future directed causal vectors X ∈ T M , -G ( · , X ) ♯ is future directed causal or zero. Here G denotes the Einstein tensor of g , \nG ( g ) . = Ric( g ) -1 2 R ( g ) g. \nProof. The spacetime is given by the spherically symmetric ansatz \n2 \nM = Q× S g = g Q + r 2 g S 2 , \nQ = { ( t, u ) ∈ R 2 : t ∈ R , -ε < u < ε } \nfor ε to be chosen later, and r = r ( u ) . Let f = f ( u ) and set \ng Q = fdt 2 -2 dtdu. \nThe vector field L = ∂ u is geodesic and null and we declare it to be future directed. The Killing vector field T = ∂ t satisfies g ( T, T ) = f . Letting f ( u ) = u 2 F ( u ) for a smooth function F ( u ) makes H = { u = 0 } an extremal Killing horizon and ∂ t is future directed where it is causal. The conjugate null vector to L is L = ∂ t + 1 2 f∂ u such that g ( L, L ) = -1 . The symmetry spheres S t 0 ,u 0 = { t = t 0 } ∩ { u = u 0 } are trapped if \nLr < 0 \nLr < 0 , \nwhich can be more simply written as \nf ( u ) r ' ( u ) < 0 r ' ( u ) < 0 . \nFrom this we see that r ' ( u ) < 0 implies no antitrapped spheres of symmetry and f ( u ) < 0 for u < 0 and f ( u ) > 0 for u > 0 implies the symmetry spheres to the past (respectively, future) of H are untrapped (respectively, trapped). This also makes T timelike to the past of H . Since we require f ( u ) = u 2 F ( u ) but also that f changes sign, we in fact have f ( u ) = u 3 ˜ F ( u ) . \nWe will now see which restrictions on f , r , and ε enforce the dominant energy condition. The Einstein tensor of g is given by \nwhere \nG = -θg Q -2 r '' r du 2 + ζr 2 g S 2 , (A.1) \nθ . = 1 + ( r ' ) 2 f + rf ' r ' +2 frr '' r 2 , ζ . = -1 2 f '' -r ' r f ' -r '' r f. \nFor f ( u ) = u 3 ˜ F ( u ) and r ( u ) fixed and ε > 0 sufficiently small, we have θ ( u ) > 0 and | ζ ( u ) | ≪ θ ( u ) for | u | < ε . \nLet X be a future causal vector, that is \ng Q ( X,X ) + r 2 g S 2 ( X,X ) ≤ 0 , g Q ( L + L, X ) < 0 . (A.2) \nTo show that -G ( · , X ) ♯ is causal or zero, it suffices to show that \ng µν G µ ρ G ν σ X ρ X σ ≤ 0 . (A.3) \nTo simplify the calculation, we assume r '' vanishes identically and then the left-hand side of (A.3), using (A.1) and (A.2), can be estimated as \ng µν G µ ρ G ν σ X ρ X σ = θ 2 g Q ( X,X ) + ζ 2 r 2 g S 2 ( X,X ) ≤ ( ζ 2 -θ 2 ) r 2 g S 2 ( X,X ) . \nSince ζ 2 -θ 2 ≤ 0 , this proves that -G ( · , X ) ♯ is causal. To show that -G ( · , X ) ♯ is future directed we compute using (A.2) \ng ( L + L, -G ( · , X ) ♯ ) = -G ( L + L, X ) = θg Q ( L + L, X ) < 0 . \nFinally, an explicit example of a metric satisfying all of our conditions is \ng = u 3 dt 2 -2 dtdu +(1 -u ) 2 g S 2 . \nRemark A.1 . Extremal Reissner-Nordström has f ( u ) ∼ -u 2 . One might say that an extremal horizon constructed in the above manner with f ( u ) vanishing faster than u 2 is a degenerate extremal horizon .", 'B General trapped and antitrapped surfaces in spherically symmetric spacetimes': "In this appendix we infer the absence of nonspherically symmetric trapped or antitrapped surfaces from the absence of spherically symmetric trapped or antitrapped surfaces. \nOur definition of trapped surface is completely standard, see Definition B.1 below. (Note that we assume trapped surfaces to be closed and strictly trapped.) Our definition of antitrapped is as in [Chr93; Kom13], i.e., an antitrapped surface is closed and past weakly outer trapped, see Definition B.2 below. \nProposition B.1. Let ( M 4 , g ) be a spherically symmetric spacetime as defined in Section 2.1. Then there are no trapped surfaces contained in the sets \nA . = { p ∈ M : ∂ u r ≥ 0 } , (B.1) \nB . = { p ∈ M : ∂ v r ≥ 0 } . (B.2) \nRemark B.1 . Note that there could be trapped surfaces contained in A ∪ B . There might also be trapped surfaces which merely intersect A or B . \nProposition B.2. Let ( M 4 , g, F, A, ϕ ) be a spherically symmetric spacetime arising as the maximal future globally hyperbolic development from one-ended asymptotically flat Cauchy data for the EMCSF system with no antitrapped spheres of symmetry as in [Kom13]. Then: \n- 1. If S is a trapped surface in M , then S ∩ J -( I + ) = ∅ .\n- 2. M does not contain any antitrapped surfaces. \nRemark B.2 . Under stronger assumptions on I + , the first part of the previous proposition would follow from a classical result of Hawking [Haw72; HE73, Proposition 9.2.1]. \nFor the proofs, we recall some facts from Lorentzian geometry [Gal00]. Let H be a null hypersurface in a spacetime ( M 4 , g ) , i.e., H is a 3-dimensional submanifold of M and admits a future-directed normal vector field L which is null and whose integral curves can be reparametrized to be null geodesics. We say that L is a (future-directed) null generator of H . \nThe second fundamental form of H with respect to L is given by \nB L ( X,Y ) = g ( ∇ X L, Y ) (B.3) \nfor X,Y ∈ TH . If e 1 and e 2 are an orthonormal pair of spacelike vectors at p ∈ H , we define the null expansion of H with respect to L by \nθ L = B L ( e 1 , e 1 ) + B L ( e 2 , e 2 ) (B.4) \nat p , and this definition is independent of the pair e 1 and e 2 . If ˜ L is another future-directed null generator of H , then there is a positive function f on H such that ˜ L = fL . In this case, we have \nθ ˜ L = fθ L . (B.5) \nLemma B.1 (Comparison principle for null hypersurfaces) . Let H 1 and H 2 be null hypersurfaces in ( M 4 , g ) , with H 1 to the future of H 2 and generated by L 1 and L 2 , respectively. If H 1 and H 2 are tangent at a point p , and L 1 ( p ) = L 2 ( p ) , then \nθ L 1 H 1 ( p ) ≥ θ L 2 H 2 ( p ) . (B.6) \nProof. By (B.5), it suffices to prove (B.6) with respect to some choice of null generators of H 1 and H 2 which agree at p . Let ( t, x, y, z ) be normal coordinates for g based at p so that ∂ t is future-directed and { 1 2 ( ∂ t + ∂ x ) , ∂ y , ∂ z } spans T p H 1 = T p H 2 . We introduce approximate null coordinates u = t -x and v = t + x , so that \n∂ u = 1 2 ( ∂ t -∂ x ) , ∂ v = 1 2 ( ∂ t + ∂ x ) . \nNote that ∂ u and ∂ v are only guaranteed to be null at p . \nBy the implicit function theorem, there exist functions f 1 ( v, y, z ) and f 2 ( v, y, z ) defined near p , so that, upon defining \nζ 1 ( u, v, y, z ) . = f 1 ( v, y, z ) -u, ζ 2 ( u, v, y, z ) . = f 2 ( v, y, z ) -u, \nwe have H i = { ζ i = 0 } for i = 1 , 2 . Note that f 1 ( p ) = f 2 ( p ) = 0 and that p is a critical point for f 1 and f 2 . The vector fields Z i = grad ζ i are null on H i and define there future-directed null generators. In particular, we have Z 1 ( p ) = Z 2 ( p ) = ∂ v | p . \nWe first show that f 1 ≥ f 2 near p . If a point q = ( u, v, y, z ) lies to the past of H 1 , then ζ 1 ( q ) ≥ 0 . If q ∈ H 2 , then ζ 2 ( q ) = 0 , so combining these inequalities yields \nf 1 ( v, y, z ) = ζ 1 ( q ) + u ≥ ζ 2 ( q ) + u = f 2 ( v, y, z ) , \nas claimed. \nWe now show that \nB Z 1 H 1 ( ∂ y , ∂ y )( p ) ≥ B Z 2 H 2 ( ∂ y , ∂ y )( p ) , (B.7) \nthe corresponding statement and proof for ∂ z being the same. By (B.4) this will complete the proof. Since f 1 ≥ f 2 near p , p is a local minimum for f 1 -f 2 . It follows that \n∂ 2 y ( f 1 -f 2 )( p ) ≥ 0 (B.8) \nby the second derivative test. Since we are working in a normal coordinate system, \nB Z i H i ( ∂ y , ∂ y )( p ) = g ( ∇ ∂ y ∇ ζ i , ∂ y )( p ) = ∂ 2 y f i ( p ) , \nwhence (B.8) proves (B.7), which completes the proof. \nDefinition B.1. A closed spacelike 2-surface S in a spacetime ( M 4 , g ) is always the intersection of two locally defined null hypersurfaces. We say that S is trapped if both of these hypersurfaces have negative future null expansion along S . \nProof of Proposition B.1. We show that there is no trapped surface S ⊂ B . The argument for S ⊂ A is analogous after noting that A ∩ Γ = ∅ by our definition of spherical symmetry and convention for u . \nLet S ⊂ { ∂ v r ≥ 0 } be a closed 2-surface. Let π : M→Q be the projection of the spherically symmetric spacetime to its Penrose diagram. Then π ( S ) is a compact subset of Q and hence u attains a minimum u 0 on π ( S ) . \nTherefore, there exists a symmetry sphere S u 0 ,v 0 on which ∂ v r ≥ 0 such that S lies to the future of C u 0 and is tangent to this cone at a point p ∈ S u 0 ,v 0 . Note that p / ∈ Γ because C u 0 is not regular there. The condition ∂ v r ≥ 0 means C u 0 has nonnegative future expansion. By Lemma B.1, one of the two null hypersurfaces emanating from S also has nonnegative future expansion, so S is not trapped. \nDefinition B.2. Let ( M 4 , g ) be a spacetime satisfying the hypotheses of Proposition B.2. A closed spacelike 2-surface S which bounds a compact spacelike hypersurface Ω is said to be antitrapped if its future-directed inward null expansion is nonnegative. Here the (locally defined) inward null hypersurface H in emanating from S is chosen to be the one which smoothly extends the boundary of the causal past of Ω . \n̸ \nProof of Proposition B.2. 1. Since r →∞ at I + [Kom13], Raychaudhuri's equation (2.11) implies ∂ v r > 0 in J -( I + ) . Let S be a closed 2-surface such that S ∩ J -( I + ) = ∅ . Let π : M→Q be the projection to the Penrose diagram. Then u attains a minimum u 0 on π ( S ) . By the causal properties of J -( I + ) , there exists a symmetry sphere S u 0 ,v 0 ⊂ J -( I + ) such that S lies to the future of C u 0 and is tangent to the cone at p ∈ S u 0 ,v 0 . Arguing as in the proof of Proposition B.1, we see that one of the null hypersurfaces emanating from S has positive future expansion, so S is not trapped. \n- 2. Let π : M→Q be again the projection. Then v attains a maximum v 0 on π ( S ) and again there exists a non-central symmetry sphere S u 0 ,v 0 such that ∂ u r ( u 0 , v 0 ) < 0 , S lies to the past of C v 0 , and is tangent to the cone at a point p ∈ S u 0 ,v 0 . Now C v 0 is tangent to H in at p and lies to the future, so by Lemma B.1, H in has negative null expansion at p . Therefore, S is not antitrapped."}

2017arXiv170800006B

Tensor network methods are taking a central role in modern quantum physics and beyond. They can provide an efficient approximation to certain classes of quantum states and the associated graphical language makes it easy to describe and pictorially reason about quantum circuits channels protocols open systems and more. Our goal is to explain tensor networks and some associated methods as quickly and as painlessly as possible. Beginning with the key definitions the graphical tensor network language is presented through examples. We then provide an introduction to matrix product states. We conclude the tutorial with tensor contractions evaluating combinatorial counting problems. The first one counts the number of solutions for Boolean formulae whereas the second is Penroses tensor contraction algorithm returning the number of 3edgecolorings of 3regular planar graphs.

2017-07-01T00:00:00Z

['arXiv:1708.00006', '10.48550/arXiv.1708.00006', '2017arXiv170800006B']

['Quantum Physics', 'Condensed Matter - Disordered Systems and Neural Networks', 'General Relativity and Quantum Cosmology', 'High Energy Physics - Theory', 'Mathematical Physics']

Tensor Networks in a Nutshell

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/1708.00006.pdf

{'Quantum Tensor Networks in a Nutshell': "Jacob Biamonte 1, 2, ∗ and Ville Bergholm 1, † 1 Quantum Software Initiative Skolkovo Institute of Science and Technology, Skoltech Building 3, Moscow 143026, Russia 2 \nInstitute for Quantum Computing \nUniversity of Waterloo, Waterloo, N2L 3G1 Ontario, Canada \nTensor network methods are taking a central role in modern quantum physics and beyond. They can provide an efficient approximation to certain classes of quantum states, and the associated graphical language makes it easy to describe and pictorially reason about quantum circuits, channels, protocols, open systems and more. Our goal is to explain tensor networks and some associated methods as quickly and as painlessly as possible. Beginning with the key definitions, the graphical tensor network language is presented through examples. We then provide an introduction to matrix product states. We conclude the tutorial with tensor contractions evaluating combinatorial counting problems. The first one counts the number of solutions for Boolean formulae, whereas the second is Penrose's tensor contraction algorithm, returning the number of 3 -edge-colorings of 3 -regular planar graphs. \n<!-- image -->", '1. QUANTUM LEGOS': "Tensors are a mathematical concept that encapsulates and generalizes the idea of multilinear maps, i.e. functions of multiple parameters that are linear with respect to every parameter. A tensor network is simply a countable collection of tensors connected by contractions. 'Tensor network methods' is the term given to the entire collection of associated tools, which are regularly employed in modern quantum information science, condensed matter physics, mathematics and computer science. \nTensor networks come with an intuitive graphical language that can be used to reason about them. This diagrammatic language dates back to at least the early 1970s by Roger Penrose [1]. These methods have seen many advancements and adaptations to different domains of physics, mathematics and computer science. An important milestone was David Deutsch's use of the diagrammatic notation in quantum computing, developing the quantum circuit (a.k.a. quantum computational network) model [2]. Quantum circuits are a special class of tensor networks, in which the arrangement of the tensors and their types are restricted. A related diagrammatic language slightly before that is due to Richard Feynman [3]. The quantum circuit model-now well over two decades old-is widely used to describe quantum algorithms and their experimental implementations, to quantify the resources they use (by e.g. counting the quantum gates required), to classify the entangling properties and computational power of specific gate families, and more. \nThere is now a lot of excitement about tensor network algorithms-for reviews see [416]. Some of the best known applications of tensor networks are 1D Matrix Product States (MPS), Tensor Trains (TT), Tree Tensor Networks (TTN), the Multi-scale Entanglement Renormalization Ansatz (MERA), Projected Entangled Pair States (PEPS)-which generalize matrix product states to higher dimensions-and various other renormalization methods [5-8, 12, 15, 17]. The excitement is based on the fact that certain classes of quantum systems can now be simulated more efficiently, studied in greater detail, and this has opened new avenues for a greater understanding of certain physical systems. \nThese methods approximate a complicated quantum state using a tensor network with a simplistic, regular structure-essentially applying lossy data compression that preserves the most important properties of the quantum state. To give the reader a rough idea how these methods work, below we conceptually depict how the quantum state ψ could be represented (or approximated) using tensor networks in various ways. \n<!-- image --> \nWe assume that most readers will have a basic understanding of some quantum theory, linear algebra and tensors. In Appendix A, we provide a short mathematical definition of tensors and tensor products. However, readers may wish to skip these definitions and for now proceed with a more informal or intuitive understanding of the idea. \nThere are several notational approaches to tensors. We begin by using abstract index notation, and then explain how it connects to the Dirac notation used in quantum computing. The connection between tensor networks and quantum circuits is elucidated at the end of Section 2.", '2. FROM TENSORS TO NETWORKS': "- 1. Drawing tensors. In the tensor diagram notation, a tensor is a labelled shape such as a box, oval or triangle, with zero or more open output legs (or arms ) pointing up, and zero or more open input legs pointing down. Individual arms and legs each correspond to upper and lower indices, respectively. 1 The arm and leg wires may be labelled with the indices they represent, or with the vector spaces they correspond to, if necessary. An order-(0 , 0) tensor without any open arms or legs is simply a complex number. \ni k j (a) (b) A k j i (c) T (2) \nFor example, diagram (a) above represents the tensor ψ i with a single upper index (a vector), diagram (b) the tensor A j k (a matrix), and diagram (c) the tensor T i jk . \n- 2. Tensor juxtaposition. When two or more disconnected tensors appear in the same diagram they are multiplied together using the tensor product. In quantum physics notation, they would have a tensor product sign ⊗ between them. In the abstract index notation the tensor product sign is omitted. \nTensors can be freely moved past each other. This is sometimes called planar deformation. \n<!-- image --> \nFrom the diagram above, using equations we have \n( 1 1 ⊗ B )( A ⊗ 1 1) = A ⊗ B = ( A ⊗ 1 1)( 1 1 ⊗ B ) , (4) \nwhere we make use of the wire also playing the role of the identity tensor 1 1-detailed in Section 3. As we shall soon see, wires are allowed to cross tensor symbols and other wires, as long as the wire endpoints are not changed. This is one reason why tensor diagrams are often simpler to deal with than their algebraic counterparts. \n<!-- image --> \nIn the diagram above we did not label the wires, since it is an arbitrary assignment. If we did, we could for example denote it as Q deg b R f ac . \n- 3. Connecting wires. Connecting two tensor legs with a wire means that the corresponding indices are contracted (summed over). \n(a) \n(b) \n<!-- image --> \nIn diagram (a) above we find a matrix multiplying a vector which results in another vector. \nDiagram (a) is equivalent to the expression \nA j i ψ i = φ j , (7) \nwhere we notice that the wire labeled i is fully connected, and hence the corresponding index is summed over. We used the Einstein summation convention , in which any index that appears exactly twice in a term is summed over. \n- In (b) we face a slightly more complicated case, departing from familiar vectors and matrices, contracting two indices between two order-3 tensors. Diagram (b) is equivalent to \nΓ i jk ∆ jk l = B i l . (8) \nTogether, two or more tensors in a diagram form a tensor network . If none of the tensors have any open arms or legs the network is said to be fully contracted: it evaluates to some complex number, a scalar. \nYou have probably noticed that we are being strict and are leaving spaces to identify the order of indices as we read them left to right across the page (e.g. in Γ i jk we have i , followed by jk ). In Section 3 we are going to talk about this in more detail, and justify why it is not necessary: we are doing it here for ease of illustration. \nSome authors like to put arrows on the tensor diagram wires [18], to denote the difference between vector spaces and their duals. 2 In this work we mostly deal with finite-dimensional vector spaces over real or complex numbers, and for each vector space V we pick a preferred basis (called the computational basis in quantum computing). The basis can be used to define an inner product, turn V into a Hilbert space, and establish a bijective mapping between V and its dual V ∗ . We then use this mapping to equate dual vectors in V ∗ with their counterparts in V . In this case arrows on the wires add nothing essential, so we omit them. \n- 4. Connection to quantum computing notation. As mentioned, tensors are multilinear maps. They can be expanded in any given basis, and expressed in terms of their components. In quantum information science one often introduces a computational basis {| k 〉} k for each Hilbert space and expands the tensors in it, using kets ( | 〉 ) for vectors and bras ( 〈 | ) for dual vectors: \nT = ∑ ijk T i jk | i 〉〈 jk | . (9) \nHere T i jk is understood not as abstract index notation but as the actual components of the tensor in the computational basis. In practice there is little room for confusion. The Einstein summation convention is rarely used in quantum information science, hence we write the sum sign explicitly. \nSo far we have explained how tensors are represented in tensor diagrams, and what happens when wires are connected. The ideas are concluded by four examples; we urge the reader to work through the examples and check the results for themselves. \nThe first example introduces a familiar structure from linear algebra in tensor form. The next two examples come from quantum entanglement theory-see connecting tensor networks with invariants [19, 20]. The fourth one showcases quantum circuits, a subclass of tensor networks widely used in the field of quantum information. The examples are chosen to illustrate properties of tensor networks and should be self-contained. \nExample 1 (The glyph[epsilon1] tensor) . A tensor is said to be fully antisymmetric if swapping any pair of indices will change its sign: A ij = -A ji . The glyph[epsilon1] tensor is used to represent the fully antisymmetric Levi-Civita symbol, which in two dimensions can be expressed as \nglyph[epsilon1] 00 = glyph[epsilon1] 11 = 0 , glyph[epsilon1] 01 = -glyph[epsilon1] 10 = 1 . (10) \nThe glyph[epsilon1] tensor can be used to compute the determinant of a matrix. In two dimensions we have \ndet( S ) = glyph[epsilon1] ij S i 0 S j 1 . (11) \nUsing this we obtain \n= S S · det S (12) \nas can be seen by labeling the wires in the diagram. In equational form this is \nglyph[epsilon1] ij S i m S j n = det( S ) glyph[epsilon1] mn . (13) \nIn terms of quantum mechanics, glyph[epsilon1] corresponds to the two-qubit singlet state: \n1 √ 2 | glyph[epsilon1] 〉 = 1 √ 2 ( | 01 〉 - | 10 〉 ) . (14) \nThis quantum state is invariant under any transformation of the form U ⊗ U , where U is a 2 × 2 unitary, as it only gains an unphysical global phase factor det( U ). \nExample 2 (Concurrence and entanglement) . Given a two-qubit pure quantum state | ψ 〉 , its concurrence C ( ψ ) = | C ' ( ψ ) | is the absolute value of the following tensor network expression [21]: \nC'( ) = (15) \nHere ψ is the complex conjugate of ψ in the computational basis. The concurrence is an entanglement monotone, a function from states to nonnegative real numbers that measures how entangled the state is. | ψ 〉 is entangled if and only if the concurrence is greater than zero. \nConsider now what happens when we act on | ψ 〉 by an arbitrary local unitary operation, i.e. | ψ ' 〉 = ( U 1 ⊗ U 2 ) | ψ 〉 . Using the result of Example 1 we obtain \nC (( U 1 ⊗ U 2 ) | ψ 〉 ) = C ( ψ ) | det( U 1 ) det( U 2 ) | . (16) \nDue to the unitarity | det U 1 | = | det U 2 | = 1, which means that the value of the concurrence is invariant (i.e. does not change) under local unitary transformations. This is to be expected, as local unitaries cannot change the amount of entanglement in a quantum state. We will revisit concurrence in Example 11. \nMore complicated invariants can also be expressed as tensor networks [19]. We will leave it to the reader to write the following network as an algebraic expression: \n<!-- image --> \nIf | ψ 〉 is a 3-qubit quantum state, τ ( ψ ) = 2 | τ ' ( ψ ) | represents the entanglement invariant known as the 3-tangle [22]. It is possible to form invariants also without using the epsilon tensor. For example, the following expression represents the 3-qubit entanglement invariant known as the Kempe invariant [23]: \nK ( ψ ) = ψ ijk ψ ilm ψ nlo ψ pjo ψ pqm ψ nqk . (18) \nThe studious reader would draw the equivalent tensor network. \nExample 3 (Quantum circuits) . Quantum circuits are a restricted subclass of tensor networks that is widely used in the field of quantum information. In a quantum circuit diagram each horizontal wire represents the Hilbert space associated with a quantum subsystem, typically a single qubit. The tensors attached to the wires represent unitary propagators acting on those subsystems, and are called quantum gates . Additional symbols may be used to denote measurements. The standard notation is described in [24]. \nHere we will consider a simple quantum circuit that can generate entangled Bell states. It consists of two tensors, a Hadamard gate ( H ) and a controlled NOT gate ( CNOT , denoted by the symbol inside the dashed region): \ni j k l m /uniF048 (19) \nThe CNOT and Hadamard gates are defined as \nCNOT = ∑ | a, a ⊕ b 〉〈 a, b | and \nH = √ 2 ∑ ( -1) ab | a 〉〈 b | , \nab (20) 1 ab (21) \nwhere the addition in the CNOT is modulo 2. 3 The reader should verify that acting on the quantum state | 00 〉 the above circuit yields the Bell state 1 √ 2 ( | 00 〉 + | 11 〉 ), and acting on | 11 〉 it yields the singlet state 1 √ 2 ( | 01 〉 - | 10 〉 ). \nExample 4 ( COPY and XOR tensors) . One can view the CNOT gate itself as a contraction of two order-three tensors [25]: \n<!-- image --> \nThe top tensor ( · with three legs) is called the COPY tensor. It equals unity when all the indices are assigned the same value (0 or 1), and vanishes otherwise: \n<!-- image --> \nHence, COPY acts to copy the binary inputs 0 and 1: \n<!-- image --> \nThe bottom tensor ( ⊕ with three legs) is called the parity or XOR tensor. It equals unity when the index assignment contains an even number of 1s, and vanishes otherwise: \n<!-- image --> \nThe XOR and COPY tensors are related via the Hadamard gate as \n= 1 √ 2 H H H (26) \nThus one can think of XOR as being a (scaled) copy operation in another basis: \n1 √ 2 XOR | + 〉 = | + 〉| + 〉 , (27a) 1 √ 2 XOR |-〉 = |-〉|-〉 , (27b) \nwhere | + 〉 := H | 0 〉 and |-〉 := H | 1 〉 . In terms of components, \nCOPY ij = (1 -i )(1 -j )(1 -k ) + ijk, \nk (28a) XOR qr s = 1 -( q + r + s ) + 2( qr + qs + sr ) -4 qrs. (28b) \nThe CNOT gate is now obtained as the tensor contraction \n∑ m COPY qm i XOR r mj = CNOT qr ij . (29) \nThe COPY and XOR tensors will be explored further in later examples and have many convenient properties [26, 27, 39].", '3. BENDING AND CROSSING WIRES': "' It now ceases to be important to maintain a distinction between upper and lower indices. ' - Roger Penrose, 1971 [1] \n- 1. Cups and caps. As explained in the previous section, wires are used to denote the contraction of pairs of tensor indices. However, it is often useful to interpret certain wire structures as independent tensors of their own. We start with three of these special wire tensors that allow one to rearrange the arms and legs of another tensor: \nij (a) (b) (c) ij = = = (30) \nThe identity tensor (a) is used for index contraction by connecting the corresponding legs. The cup (b) and the cap (c) raise and lower tensor indices by bending the corresponding tensor legs. 4 Expanding them in the computational basis we obtain \n1 1 = ∑ ij δ i j | i 〉〈 j | = ∑ k | k 〉〈 k | , (31) \n|∪〉 = ∑ ij δ ij | ij 〉 = ∑ k | kk 〉 , (32) \n〈∩| = ∑ ij δ ij 〈 ij | = ∑ k 〈 kk | . (33) \nIn a quantum information context, the cup also corresponds to an (unnormalized) Bell state, generalized so that it is not defined just for qubits. \n- 2. Snake equation. One can raise and then lower an index or vice versa, which amounts to doing nothing at all. This idea is captured diagrammatically by the so called snake or zig-zag equation [1]. \n= = (34) \nIn abstract index notation it is expressed succinctly as δ ij δ jk = δ i k = δ kj δ ji . \n- 3. SWAP gate. Crossing two wires (as in diagram (a) below) can be thought of as swapping the relative order of two vector spaces. It corresponds to the SWAP gate used in quantum computing. If both wires represent the same vector space, it can be alternatively understood as swapping the states of the two subsystems. \n(a) (b) = (35) \nEquation (b) illustrates that the SWAP operation is self inverse. It may be written as SWAP ij kl = δ j k δ i l , or expanded in the computational basis as SWAP = ∑ ij | ij 〉〈 ji | . It also has a well-known implementation in terms of three CNOT gates as \n= (36) \nSWAP is the simplest nontrivial example of a permutation tensor. More complicated permutations may be built out of the δ i j tensors in an obvious way. We will return to this idea in Example 7. \n- 4. Transpose. Given A i j , we may reverse the positions of its indices using a cup and a cap. This is equivalent to transposing the corresponding linear map in the computational basis: \nA = A (37) \n- 5. Trace. In the tensor diagram notation, trace is given by appropriately joining all the output wires of a tensor to corresponding input wires. Diagram (a) below represents the trace A i i . Diagram (b) represents the trace B iq iq . \n(a) A (b) B (c) C i i q p i j k (38) \nPartial trace means contracting only some of the outputs with their corresponding inputs, such as with the tensor C ijk pk shown in diagram (c). \nExample 5 (Partial trace) . The following is an early rewrite representing entangled pairs due to Penrose [28]. \n<!-- image --> \nThe diagram on the left represents the partial trace of | ψ 〉〈 ψ | over the second subsystem. Readers can prove that this equality follows by interpreting the bent wires as cups and caps, and the crossing wires as SWAP s. \nExample 6 (Partial trace of Bell states) . Continuing on from Example 5, if we choose | ψ 〉 = |∪〉 , i.e. | ψ 〉 is an unnormalized Bell state, we obtain \n<!-- image --> \nExample 7 (Relation between glyph[epsilon1] and SWAP ) . For any order-2 tensor T ij we can define its antisymmetrization as T [ ij ] = 1 2 ( T ij -T ji ). Here we used the notation of putting brackets around a group of indices-[ ij ]-to denote their antisymmetrization. Only indices of the same dimension may be antisymmetrized (otherwise the expression would be undefined for some index values). \nThe fully antisymmetric glyph[epsilon1] tensor from Example 1 has an interesting relation to the SWAP gate: \n= -(41) \nor alternatively \nglyph[epsilon1] kl glyph[epsilon1] ij = δ k i δ l j -δ k j δ l i . (42) \nIt is now easy to show that for any tensor T ij (for which both indices are two-dimensional) we can write T [ kl ] = 1 2 glyph[epsilon1] kl glyph[epsilon1] ij T ij . \nBoth the concept of antisymmetrization and the epsilon tensor can be extended to more than two indices. The antisymmetrizer of n d -dimensional vector spaces is an order-( n, n ) tensor A i 1 ··· i n j 1 ··· j n . It can be expressed as the sum of all n -element permutations multiplied by their signatures. 5 It antisymmetrizes n d -dimensional indices by contraction: \nT [ i 1 ··· i n ] = A i 1 ··· i n j 1 ··· j n T j 1 ··· j n . (43) \nWhen d < n the only possible antisymmetric combination is a zero tensor, and the corresponding A vanishes identically. \nFor the general order-(0 , n ) epsilon tensor, all the n vector spaces need to be n -dimensional. We then define glyph[epsilon1] 012 ... ( n -1) = 1, and all the other components are fixed by requiring complete antisymmetry, i.e. change of sign under the interchange of any two indices. In particular, if any index value is repeated the corresponding component is zero. Now 1 n ! glyph[epsilon1] i 1 ··· i n glyph[epsilon1] j 1 ··· j n is an antisymmetrizer. \nWe shall use order-three epsilon tensors to count graph edge colorings by tensor contraction in Section 6 2. \nExample 8 (Quantum circuits for cups and epsilon states) . The quantum circuit from Example 3 is typically used to generate entangled qubit pairs. For instance, acting on the state | 00 〉 yields the familiar Bell state-as a tensor network, this is equal to a normalized cup. Here we also show the mathematical relationship the XOR and COPY tensors have with the cup (here | + 〉 := | 0 〉 + | 1 〉 ): \n<!-- image --> \nSimilarly, one can use the circuit (19) to generate the epsilon state. Let us denote the Pauli matrices by X := | 0 〉〈 1 | + | 1 〉〈 0 | , Y := -i | 0 〉〈 1 | + i | 1 〉〈 0 | and Z := | 0 〉〈 0 | - | 1 〉〈 1 | . The \nZ gate commutes with the COPY tensor, and the X or NOT gate (which we denote with ⊕ ) commutes with XOR . Commuting those tensors to the right hand side, allows us to apply Eq. (44). Making use of the Pauli algebra identity ZX = iY , one recovers the epsilon state: \n<!-- image --> \nExample 9 (Map-state duality) . The index raising and lowering using cups and caps can be interpreted as a linear map between bipartite vectors and linear maps leading to a relationship between quantum states and operators acting on them. We will start with the linear map A . Raising the second index using a cup (a) yields a tensor with two output legs (b), i.e. something that can be interpreted as a bipartite vector | A 〉 . \n(a) (b) A (c) A = = A (46) \nFinally, using the snake equation (34) together with the equality (37) we can see that this is equal to (c) the transposed map A glyph[latticetop] with a cup raising the input index to the other side. Indeed, a tensor may be moved around a cup or a cap by transposing it. The relationship (46) arises in practice in the following scenario from quantum information science: \n<!-- image --> \nHere an entangled state | ψ 〉 acted on by a map A can instead be viewed as a map ψ acting on a state | A glyph[latticetop] 〉 . 6 This is a diagrammatic form of map-state duality underlying bipartite entanglement evolution [29, 30]. See e.g. the survey [15] which includes a detailed discussion on reshaping tensors. \n- 6. Dagger and complex conjugation. If the order-( p, q ) tensor T is a map between Hilbert spaces, we may define its (Hermitian) adjoint T † , an order-( q, p ) tensor, using the inner product: 〈 x, T y 〉 = 〈 T † x, y 〉 for all x, y . Diagrammatically the adjoint is obtained by mirroring the tensor network such that input and output wires switch places, the relative order of the tensors in the diagram is reversed, and each tensor symbol is decorated with a dagger (with T †† = T ). \nSimilarly, the dagger operation maps Hilbert space ket vectors one-to-one to their dual bra vectors (and vice versa) in the sense of the Riesz representation theorem. This is denoted | a 〉 † = 〈 a | . Note that we have been using this notation implicitly, as it should be familiar to many readers from basic quantum mechanics. \nThe dagger is an antilinear (or conjugate-linear) operation, ( cT + dU ) † = c T † + dU † , since we have to take the complex conjugate of the scalars c and d . This means that it cannot be represented by the linear cups and caps alone, unlike the transpose. However, it can be represented as transpose together with complex conjugation in the same basis. This is summarized in the following adjoint square . \n<!-- image --> \nNote that some authors mark one of the corners of each tensor box, and use the convention that mirroring a diagram across the horizontal plane corresponds to the † operation. We do not adhere to this convention and instead place the dagger on the symbol such as A → A † . \nFor kets and bras, bending a wire on a ket yields the complex conjugate of the corresponding bra, and vice versa. \n= (49) \n7. Index position. Now we will consider the set of operations formed by bending tensor wires forwards and backwards using cups and caps, as well as exchanging the order of wires using SWAP . If one conceptualizes a tensor as an array of numbers, these transforms correspond to array reshapes and reorderings. As the snake equation (34) shows, the action of cups and caps can be inverted, and SWAP is self inverse (35). This means that all possible configurations of a tensor's wires obtained using these operations are isomorphic. \nAs an example, given a tensor T i j one can use cups and caps to naively rearrange the index elevations and positions, arriving at \nT i j , T ij , T ij , T j i , T j i , T ji , T ji and T i j , (50) \nfor a total of eight possible reshapes. If the tensor has more than two indices, one can additionally use SWAP s to arrange the indices in any relative order. Thus one might think that for a general n -index tensor there are n ! · 2 n different ways of arranging the indices ( n ! different permutations of the indices, with each index being either up or down). However, this way one overcounts the number of index configurations that are truly different. In our \nexample, in fact T i j = T i j and T j i = T j i , as can be seen from the diagram below: \nT j i = = (51) \nConsequently, the tensor T i j in actuality only has six unique reshapes: \n(a) (d) (b) (e) (c) (f) T i j T ij Tij T j i = = = = = = (52) \nMore generally, one finds that the number of unique reshapes generated by cups, caps and SWAP s for an order-( p, q ) tensor T i 1 ··· i p j 1 ··· j q is ( p + q +1)!.", '4. DIAGRAMMATIC SVD': "In this section, we explain the diagrammatic version of the singular value decomposition (SVD). This method is at the heart of many numerical simulation algorithms in wide use today-we are explaining it as a prelude leading to Section 5. The SVD factors an arbitrary order-(1 , 1) tensor into well defined building blocks with simple properties: (i) an order-(1 , 1) diagonal tensor storing the singular values, and (ii) two order-(1 , 1) unitary tensors. As several tensor legs can always be grouped together to form a single leg, the method works for any tensor of order two or higher. \nInterpreting order-(1 , 1) tensors as linear maps (or simply as matrices), we may use the SVD to factor any tensor T : A → B (for vector spaces A and B ) as \nT b a = U b j Σ j i V i a , (53) \nwhere U and V are unitary, and Σ is real, non-negative, and diagonal in the computational basis. Σ has the singular values { σ k } k of T on its diagonal, typically arranged in a nonincreasing order: σ 1 ≥ σ 2 ≥ . . . ≥ σ min(dim A, dim B ) ≥ 0. It can be expanded as \nΣ = ∑ k σ k | k 〉 B 〈 k | A . (54) \nDiagrammatically this is represented as \n= U A A V /uni03A3 A B B B T . (55) \nAs will be seen in the next section, the SVD is centrally employed in the efficient representation of certain quantum states by tensor networks. The idea is to represent a small but physically relevant portion of the Hilbert space-such as low entanglement states-by repeated application of the SVD paired with low-rank approximations. \nThe rank of a matrix T is the number of non-zero singular values it has. To determine its optimal rankr approximation (with r < rank( T )), we can turn to a classic theorem by Eckart and Young which was generalized by Mirsky. Given the SVD T = U Σ V † , we will discard rank( T ) -r smallest singular values in Σ by setting them to zero, obtaining Σ ' . This process is often called trimming. \nThis gives rise to T ' = U Σ ' V † , an approximation of T . The Eckart-Young-Mirsky theorem states that \n‖ T -T ' ‖ = min rank( ˆ T ) ≤ r ‖ T -ˆ T ‖ (56) \nfor any unitarily invariant matrix norm. 7 Here ˆ T is any approximation to T of the same or lesser rank as T ' . This implies that truncating or trimming Σ in this way yields as good of an approximation as one can expect. In the following section, we will specifically consider the induced error for such an approximation. \nUsing the wire bending techniques from Section 3, we immediately obtain the Schmidt decomposition as a corollary to the SVD: \n<!-- image --> \nGiven a vector | ψ 〉 ∈ A ⊗ B (for example a ket vector describing a pure state of a bipartite quantum system), we may use the snake equation to convert it into a linear map ψ : A → B (inside the dashed region). Now we apply the SVD on ψ as above. Diagram reorganization leads to the diagrammatic Schmidt decomposition \n| ψ 〉 A ⊗ B = ∑ i σ i | ϕ i 〉 A | φ i 〉 B . (58) \nThe singular values { σ k } k now correspond to the Schmidt coefficients. If | ψ 〉 is normalized, we have ∑ k σ 2 k = 1. \nExample 10 (Entanglement topology) . The topology of the bipartite quantum state | ψ 〉 depends solely on the Schmidt coefficients { σ k } k . \n<!-- image --> \nIf the coefficients are all equal, we may replace the diamond in (a) with a cup tensor times a scaling factor as in diagram (b). This corresponds to a maximally entangled state. In the other extreme, illustrated in diagram (c), we have just one nonzero Schmidt coefficient σ 1 = 1. In this case the diagram breaks into two pieces and thus corresponds to a factorizable state. The number of nonzero Schmidt coefficients is called the Schmidt rank of the decomposition-see Def. 17 for the relation with R'enyi entropy of order zero. \nExample 11 (Concurrence-part II) . Continuing on from Example 2, one can apply the Schmidt decomposition to the tensor network defining the concurrence of a two-qubit state | ψ 〉 . We obtain \n<!-- image --> \nand thus \n= = C'( ) = ·det( )·det( ) (60) \nC ( ψ ) = | C ' ( ψ ) | = ∣ ∣ Tr(( glyph[epsilon1] Σ) 2 ) ∣ ∣ = 2 σ 1 σ 2 = 2 | det( ψ ) | . (61) \nThe unitaries glyph[squaresolid] and glyph[square] vanish from the expression since | det( glyph[squaresolid] ) | = | det( glyph[square] ) | = 1. Given the basis expansion of the state, \n| ψ 〉 = a | 00 〉 + b | 01 〉 + c | 10 〉 + d | 11 〉 , (62) \nits two Schmidt coefficients are given as \nσ 2 k = 1 2 ( 1 + ( -1) k +1 √ 1 -4 | ad -bc | 2 ) . (63) \nExample 12 (Purification backwards) . For any bipartite ket vector | ψ 〉 , the partial trace of | ψ 〉〈 ψ | yields a positive semidefinite operator. We can see this by using the Schmidt decomposition, and then reorganizing the diagram so that two of the unitaries cancel: \n<!-- image --> \nOn the right hand side we have an eigendecomposition of a square matrix with strictly nonnegative eigenvalues { σ 2 k } k , which can be interpreted as a density matrix if | ψ 〉 is normalized. Conversely, any density matrix representing a mixed quantum state can be purified , or expressed as the partial trace of a bipartite pure state.", '5. MATRIX PRODUCT STATES': "Matrix product states (MPSs) are quantum states presented as a linear chain or ring of tensors. Any quantum state can be exactly represented in this form and the representation is known to approximate a class of 1D gapped systems efficiently [31]. We will explain the basic ideas of the MPS representation here and point the reader to [6, 7] and the references therein for additional information. \nGiven an n -party quantum state | ψ 〉 , fully describing this state generally requires an amount of information (or computer memory) that grows exponentially with n . If | ψ 〉 represents the state of n qubits, \n| ψ 〉 = ∑ ij ··· k ψ ij ··· k | ij · · · k 〉 , (65) \nthe number of independent coefficients ψ ij ··· k in the basis expansion in general would be 2 n which quickly grows into a computationally unmanageable number as n increases. The goal is to find an alternative representation of | ψ 〉 which is less data-intensive. We wish to write | ψ 〉 as \n| ψ 〉 = ∑ ij ··· k Tr( A [1] i A [2] j · · · A [ n ] k ) | ij · · · k 〉 , (66) \nwhere A [1] i , A [2] j , . . . , A [ n ] k are indexed sets of matrices. Calculating the components of | ψ 〉 then becomes a matter of calculating the products of matrices, hence the name matrix product state . \nIf the matrices are bounded in size, the representation becomes efficient in the sense that the amount of information required to describe them is only linear in n . The point of the method is to choose these matrices such that they provide a good (and compact) approximation to | ψ 〉 . For instance, if the matrices are at most χ by χ , the size of the representation scales as ndχ 2 , where d is the dimension of each subsystem. \nWithout loss of generality, we will now show how to obtain an MPS representation of an arbitrary four-party state | ψ 〉 . The key ingredient is the recursive application of the singular value decomposition (SVD) presented in Section 4. We start by considering the tensor which represents the state vector. We select a bipartition that separates one leg from the rest (starting at either end), and apply the SVD. This process is then repeated, traversing the entire tensor. This results in a 1D tensor network representation of the state, as shown below. \n<!-- image --> \nFinally the tensors are grouped-though the grouping has some ambiguity-resulting in the typical form shown below. \n<!-- image --> \nWe note that one can recover the form in Eq. (66) as \n| ψ 〉 = ∑ ijkm A [1] i A [2] j A [3] k A [4] m | ijkm 〉 . (69) \nIn this case the first and last sets of matrices A [1] i and A [4] m have just a single row and a single column, respectively, so the product always yields a scalar and the trace is not required. The tensor network in Eq. (68) is called an MPS with open boundary conditions. MPSs also come with periodic boundary conditions, in which case the tensors form a ring instead of a chain, and the trace represents the contraction that closes the ring. \nYou might have noticed that we made a choice to perform the factorization starting from the left of the tensor and applying the SVD successively on tensors as we moved to the right. This apparent ambiguity has been characterized in detail [32]. For open boundary conditions as have been considered here, there is a canonical choice unique up to degeneracies in the spectrum of local reduced density operators [32]. \nFor a general n -qubit quantum state it can be shown that the MPS matrix size can be bounded by χ = 2 glyph[floorleft] n/ 2 glyph[floorright] , which still grows exponentially as expected. A compact approximate representation is obtained by choosing a cutoff value ξ for the singular values across each partition, or a maximum number χ of singular values to be kept-see Example 13. This allows one to compress data by truncating the Hilbert space and is at the heart of the MPS algorithms. If an MPS is cut into two parts, the Schmidt rank of the decomposition, describing the degree of entanglement between the parts, is always ≤ χ , the dimension of the internal wire that was cut-see Example 10 for Schmidt rank and set q = 0 in Eq. (86) for the connection to entropy. \nAbove we have illustrated how to obtain the MPS representation of any pure quantum state, but it is normally not practical to factor states in this way for computational reasons. Instead, efficient MPS-generating algorithms are given an indirect, compact description of a state e.g. in the form of a nearest-neighbor Hamiltonian whose ground state we are interested in, and they then iteratively produce an MPS that closely approximates that state. The seminal algorithm of this type is the Density Matrix Renormalization Group (DMRG), which essentially works as a variational method in MPS space. Another class of algorithms can efficiently time-evolve MPSs under a nearest-neighbor Hamiltonian. One of the most used methods of this type is Time-Evolving Block Decimation (TEBD). \nExample 13 (MPS approximation error) . As explained, matrix product state algorithms employ repeated application of the singular value decomposition. The size of the representation can be reduced by lossy truncation in one of two ways. Each of these rely on truncation of singular values. For a fixed rank, the Eckart-Young-Mirsky theorem-from the last Section-tells us that truncation of the singular values is the best approximation that one can expect. \nIn the first approximation, one can simply discard some fixed number of lowest singular values and their corresponding vectors-in other words, we will fix the dimension χ of all internal wires. In another approximation-which we will consider here-one will pick a cutoff value ξ , and truncate all singular values which are less than this. Such a cutoff value is not guaranteed to provide a useful partition of the singular values-e.g. it could be smaller than the smallest singular value. Here we will analyze the errors of this truncation assuming this cutoff partitions the singular values-which in practice is very often the case. \nGiven a bipartite state | ψ 〉 we write \n| ψ 〉 = k ∑ i =1 σ i | u i 〉| v i 〉 = k ∑ i =1 σ i | i 〉 . (70) \nwhere to simplify the notation we write | u i 〉| v i 〉 as just | i 〉 . We order the singular values in \na non-decreasing sequence and introduce a cutoff ξ > 0: \n0 < σ 1 ≤ σ 2 ≤ σ 3 ≤ · · · ≤ σ n ≤ ξ ≤ σ n +1 ≤ · · · ≤ σ k . (71) \nAs a heuristic, the small cutoff ξ can be chosen such that ξ < 1 √ dim H where H is the vector space acted on by | ψ 〉 -you could also write ξ < 1 √ d q where d is the dimension of q constituent spaces. \nAnd then we will partition our space in terms of the n singular values less than ξ and the k -n ones that are greater than ξ . \n| ψ 〉 = n ∑ i =1 σ i | i 〉 + k ∑ j = n +1 σ j | j 〉 = n ∑ i =1 σ i | i 〉 + | ψ ' 〉 , (72) \nwhere | ψ ' 〉 in (72) represents a (so far, non-normalized) approximation to | ψ 〉 . To understand the limits of validity of this approximation, we consider the inequality \nn ∑ i =1 σ i ≤ n · σ n ≤ n · ξ, (73) \nand hence n · σ 2 n ≤ n · ξ 2 . \nProvided ξ is small, and for constant n we can return to (72) and consider another normalized but otherwise arbitrary vector | φ 〉 (element of the same space as | ψ 〉 and | ψ ' 〉 ) \n|〈 φ | ψ -ψ ' 〉| = ∣ ∣ ∣ ∣ ∣ n ∑ i =1 σ i 〈 φ | i 〉 ∣ ∣ ∣ ∣ ∣ ≤ n ∑ i =1 σ i |〈 φ | i 〉| ≤ n ∑ i =1 σ i ≤ n · ξ, (74) \nwhich scales linearly in the upper-bound of the error independent of | φ 〉 . \nIt's common to introduce a function O ( ξ ) to collect terms up to a constant that approach zero no slower than O 's argument. So (74) simply says that the absolute value of the difference between | ψ ' 〉 and | ψ 〉 projected onto 〈 φ | is bounded by a function proportional to ξ . From this we can readily conclude that the error in the approximation considered here is linear in ξ -or in other words, the error scales at most as O ( ξ ). \nHowever, as | ψ 〉 is normalized, we note that \n〈 ψ | ψ 〉 = n ∑ i =1 σ 2 i + k ∑ j = n +1 σ 2 j = 1 , (75) \nand then calculate the inner product \n〈 ψ | ψ ' 〉 = k ∑ j = n +1 σ 2 j = 1 -n ∑ i =1 σ 2 i ≥ 1 -nξ 2 = 1 + O ( ξ 2 ) . (76) \nWe hence conclude that the error in the inner product is only quadratic. In fact, if we normalize | ψ ' 〉 → | ψ '' 〉 and calculate \n|〈 ψ | ψ '' 〉| 2 = 1 -n ∑ i =1 σ 2 i ≥ 1 -nξ 2 = 1 + O ( ξ 2 ) , (77) \nwe recover the same thing. \nExample 14 (MPS for the GHZ state) . The standard MPS representation of the GreenbergerHorne-Zeilinger (GHZ) state is given as \n| GHZ 〉 = 1 √ 2 Tr ( | 0 〉 0 0 | 1 〉 ) n = 1 √ 2 ( | 00 . . . 0 〉 + | 11 . . . 1 〉 ) . (78) \nAlternatively, we may use a quantum circuit made of CNOT gates to construct the GHZ state, and then use the rewrite rules employed in Examples 4 and 8 to recover the familiar MPS comb-like structure consisting of COPY tensors: \n<!-- image --> \nDiagrammatically, any tensor network formed from connected COPY -tensors reduces to a single dot with the appropriate number of input and output legs. Hence one might write the n-party GHZ-state as \n| GHZ 〉 = 1 √ 2 ∑ ijk...l COPY ijk...l | ijk . . . l 〉 . (80) \nExample 15 (MPS for the W state) . Like the GHZ state from Example 14, the n -qubit Wstate ( n ≥ 3) has the following MPS representation: \n| W 〉 = 1 √ n ( | 1 〉 | 0 〉 ) ( | 0 〉 0 | 1 〉 | 0 〉 ) n -2 ( | 0 〉 | 1 〉 ) = 1 √ n ( | 10 . . . 0 〉 + | 010 . . . 0 〉 + . . . + | 0 . . . 01 〉 ) . (81) \nExample 16 (The AKLT model) . The AKLT model [33] (named after the authors Affleck, Kennedy, Lieb and Tasaki) is a theoretically important exactly solvable model of a spin-1 Heisenberg chain with an extra quadratic interaction term: \nH = ∑ j glyph[vector] S j · glyph[vector] S j +1 + 1 3 ( glyph[vector] S j · glyph[vector] S j +1 ) 2 , (82) \nwhere glyph[vector] S is a three-vector of the familiar spin-1 operators. The exact ground state of this Hamiltonian has an elegant expression as a matrix product state. Here we will carry on from Examples 1 and 7 which define and use the glyph[epsilon1] tensor-see also Example 8 which provides a quantum circuit realization for the corresponding singlet state. We start with a tensor product of these states, \n... ... (83) \nand then project each neighboring qubit pair onto a three-dimensional (spin-1) Hilbert space using the projectors P : \n<!-- image --> \nThe open wires on the P 's have three degrees of freedom (spin-1). The projectors are defined as \nP = | +1 〉〈 11 | + 1 √ 2 | 0 〉 ( 〈 01 | + 〈 10 | ) + | -1 〉〈 00 | . (85) \nThe kets | +1 〉 , | 0 〉 and | -1 〉 are the standard spin-1 basis states and clearly the bond dimension χ = 2. The AKLT state obtained after projection is rotationally symmetric but has a non-trivial entanglement structure and a host of other interesting properties-see e.g. [4] and the references therein. \nThe singular values found from the MPS factorization can be used to form a complete polynomial basis to express invariant quantities related to an MPS. This has a close connection to Schmidt rank and other concepts-see Examples 10 and 11. \nDefinition 17 (R'enyi and von Neumann entropies) . Given a density operator ρ , its R'enyi entropy of order q is defined to be \nH q ( ρ ) = 1 1 -q log Tr( ρ q ) = 1 1 -q log ∑ i λ q i , (86) \nwhere λ i are the eigenvalues of ρ . The case q → 0 gives the rank of ρ (here we define 0 0 = 0), and the limit q → 1 recovers the familiar von Neumann entropy \nH q → 1 ( ρ ) = -Tr( ρ log ρ ) . (87) \nFor a pure bipartite state | ψ 〉 with subsystems A and B , the entropies of the reduced density operator ρ A = Tr B ( | ψ 〉〈 ψ | ) can be used to quantify bipartite entanglement in the state. In this context they are called entanglement entropies. The eigenvalues of ρ A (and ρ B ) are the squares of the Schmidt coefficients of | ψ 〉 , and H 0 ( ρ A ) is equal to the Schmidt rank of the bipartition-see Example 10. \nArea laws are quantified in terms of the scaling of entropy across tensor network partitions [34]. Whenever you bipartition a tensor network representing a pure quantum state, the total dimension χ of wires 'cut' by this partition gives an upper bound to the entanglement entropy: H 0 ( ρ A ) ≤ χ and H 1 ( ρ A ) ≤ log χ .", '6. COUNTING BY TENSOR CONTRACTION': "In the tensor network language, counting problems can be expressed by evaluating fully contracted diagrams [1, 28, 35, 36]. \nTo understand counting problems, imagine a phone book. But this phone book has a problem. The names are arranged randomly, without the standard alphabetical order we'd \nall expect. If you're task is to determine Stephen Clark's phone number, the average number of names you'll need to examine before finding 'Stephen Clark' is exactly half the number in the phone book-assuming the name is unique. This is an example of a search problem. If we tell you the page number and location on the page where Stephen Clark's name appears, you can easily check and see if we're correct or not. Often times in computer science, problems are classified not in terms of how hard they are to solve-which is often unknown-but in terms of how hard it is to check if a given solution is correct or not. \nCounting problems arise from search problems in the following way. Imagine you want to determine all the entries in this random phone book with the last name 'Jaksch'. This is then the counting version of the above search problem. \nIn these examples, both problems can be solved efficiently in the number of entries in the phone book. 'Efficiently' in the language of computer science means that the computational memory and runtime required is less than a polynomial in the problem size-in this case, the number of phone book entries. \nIn the language of computer science, more generally search problems are complete for the complexity class NP . Counting versions of NP -complete problems are #P -complete [37] in the language of computational complexity theory (pronounced sharp-P). \nConnections between counting problems and tensor networks arose from the very early days of tensor networks. Penrose showed [1] that certain graph coloring problems can be solved by tensor contraction. We're going to outline Penrose's algorithm in Section 6 2. Before doing that, we will describe in Section 6 1 how tensor contractions can count the number of inputs that cause a given Boolean function to output 1.", '1. Counting Boolean Formula Solutions': "A Boolean function (a.k.a. switching function) takes an n -bit string of binary numberse.g. 00110101011-and maps this to a single binary digit (either 0 or 1). Several problems in physics can be mapped to Boolean functions [38]. In what follows, we will often denote this n -bit string as x . We will show how to associate the Boolean function f ( x ) with a non-normalized quantum state, written as a tensor network. We will establish the following (Remark 18) by a few examples. \nDefinition 18 (The class of Boolean tensors [35, 36]) . Every Boolean function f ( x ) gives rise to a Boolean tensor \nf = ∑ x | f ( x ) 〉〈 x | , (88) \nwith binary coefficients in { 0 , 1 } . Moreover, a tensor network representing this state is determined from the classical logic gate network description of f ( x ). \nAs an example, consider the logical AND operation which takes Boolean input variables x 1 and x 2 and outputs the Boolean product x 1 ∧ x 2 -as a tensor, we write \nAND := ∑ x 1 ,x 2 | x 1 ∧ x 2 〉〈 x 1 , x 2 | = | 0 〉〈 00 | + | 0 〉〈 01 | + | 0 〉〈 10 | + | 1 〉〈 11 | . (89) \nWe will use the standard AND gate symbol also for the corresponding tensor: \n(90) \nUsing (88) we can map any Boolean tensor f to an (unnormalized) Boolean state | f 〉 by contracting the output with | 1 〉 : \n| f 〉 = f glyph[latticetop] | 1 〉 = ∑ x 〈 f ( x ) | 1 〉| x 〉 = ∑ x f ( x ) | x 〉 . (91) \nIn addition to AND we have already seen other examples from this class of states, the GHZ and W states from Examples 14 and 15. The corresponding Boolean functions (for three variables/qubits) are \nf GHZ ( x ) = x 1 x 2 x 3 +(1 -x 1 )(1 -x 2 )(1 -x 3 ) (92) \nand \nf W ( x ) = x 1 x 2 (1 -x 3 ) + x 1 (1 -x 2 ) x 3 +(1 -x 1 ) x 2 x 3 . (93) \nInserting these functions in Eq. (91), we obtain the corresponding (unnormalized) quantum states: \n| GHZ 〉 = | 000 〉 + | 111 〉 , (94) \n| W 〉 = | 001 〉 + | 010 〉 + | 100 〉 . (95) \nNow, given a Boolean function f ( x ), each input for which the function returns 1 is said to satisfy the function. The number of satisfying inputs is then equal to the size of the support of f ( x ). \n<!-- image --> \nWe may count the number of inputs that satisfy f ( x ) by contracting the tensor f with | 1 〉 to obtain the corresponding Boolean state | f 〉 (diagram (b)). The contraction post-selects the tensor network such that its support now consists solely of inputs that satisfy f ( x ). \nTo explicitly translate this into a counting problem, we compute the squared norm, \n‖| f 〉‖ 2 = 〈 f | f 〉 = ∑ x , y f ( x ) f ( y ) 〈 x | y 〉 = ∑ x f ( x ) 2 = ∑ x f ( x ) , (97) \nwhich clearly gives the number of satisfying inputs. In general for Boolean states the square of the two-norm always equals the one-norm since f ( x ) ∈ { 0 , 1 } . In diagram form this is depicted as \n<!-- image --> \nwhere | + 〉 := | 0 〉 + | 1 〉 and thus | + 〉 ⊗ | + 〉 ⊗ . . . ⊗| + 〉 = ∑ x | x 〉 . More formally, \n1 \nTheorem 19 (Counting SAT solutions [35, 36]) . Let f be a SAT instance. Then the standard two-norm length squared of the corresponding Boolean state | f 〉 gives the number of satisfying assignments of the problem instance. \nSolving the counting problem for general formula is known to be #P -complete [37]. Just to remind you what we talked about in the introduction to this section, computational complexity jargon for the set of the counting problems associated with the decision problemsdecision problems seek to determine if a state is satisfiable at all, whereas counting problems seek to determine the total number of satisfying solutions. Indeed, the condition 〈 f | f 〉 > 0 implies that the SAT instance f has a satisfying assignment. Determining whether this condition holds for general Boolean states is a NP -complete decision problem-as described in the introduction to this section. \nFormulating these problems as tensor contractions allows the adaptation of tools developed to simulate quantum systems and circuits such that they now apply to an areas traditionally considered in computer science [35, 36]. We adapted these tools and discovered efficient tensor network descriptions of finite Abelian lattice gauge theories [27]. These tools also lead to the discovery of a wide class of efficiently contractable tensor networks, representing counting problems [36]. \nExample 20 (AND from Toffoli) . The Toffoli gate has long been studied in reversible computing, and also in quantum computing. One can view this operation as being formed internally by an AND tensor, two copy tensors (black dots) and one XOR tensor ( ⊕ )-see Examples 3 and 14. One can create the | AND 〉 state in an experiment by preparing the state \n| + 〉| + 〉| 0 〉 , (99) \nwhere | + 〉 := | 0 〉 + | 1 〉 , and then applying the Toffoli gate. This is illustrated with the following tensor diagram. \n<!-- image --> \nExample 21 (Hadamard from AND) . By considering the contraction formed with the state |-〉 := 1 √ 2 ( | 0 〉 - | 1 〉 ) and the output of the AND tensor, one recovers the Hadamard gate defined in Eq. (21). Note that we've been using | + 〉 without normalization and here we normalize |-〉 . \n<!-- image --> \nExample 22 (De Morgan's laws) . A common identity used in Boolean algebra is \n¬ ( a ∧ b ) = ( ¬ a ) ∨ ( ¬ b ) , (102) \nwhere negation denoted as ¬ , logical OR as ∨ , and logical AND as ∧ . When expressed as a tensor network, \n<!-- image --> \nthis relation has the same structure as the relationship between the COPY and XOR tensors and the Hadamard gate in Eq. (26) Such diagrammatic rewrite rules can be used to formalize a system of graphical reasoning, and reduce calculations through easily employed graphical rewrite identities. \nThe synthesis problem seeks to determine how to build a logical circuit from basic logic gates (such as AND) that realizes a given Boolean function. Given this logical circuit, we can obtain the corresponding tensor network simply by replacing the gates with their tensor counterparts. Using tensors that represent classical logical gates provides an alternative means to determine tensor networks representing for instance the GHZ and AND states [25], compared to the MPS representations given in Examples 14 and 15.", '2. Counting Graph Colorings': 'Given a 3-regular planar graph 8 , how many possible edge colorings using three colors exist, such that all edges connected to each node have distinct colors? This counting problem can be solved in an interesting (if not computationally efficient) way using the order-3 glyph[epsilon1] tensor, which is defined in terms of components as \nglyph[epsilon1] 012 = glyph[epsilon1] 120 = glyph[epsilon1] 201 = 1 , glyph[epsilon1] 021 = glyph[epsilon1] 210 = glyph[epsilon1] 102 = -1 , (104) \notherwise zero. The counting algorithm is stated as \nTheorem 23 (Planar graph 3-colorings, Penrose 1971 [1]) . The number K of proper 3 -edge-colorings of a planar 3 -regular graph is obtained by replacing each node with an order-3 epsilon tensor, replacing each edge with a wire, and then contracting the resulting tensor network. \nWe will first consider the simplest case, a graph with just two nodes. In this case we obtain \n2 \n<!-- image --> \nThere are indeed 6 distinct edge colorings for this graph, given as \n<!-- image --> \nTo understand Theorem 23, note first that the contraction K of the epsilon tensor network is the sum of all possible individual assignments of the index values to the epsilon tensors comprising the network. Each of the three possible index values can be understood as a color choice for the corresponding edge. Whenever the index values for a given epsilon tensor are not all different, the corresponding term in K is zero. Hence only allowed color assignments result in nonzero contributions to K , and for a graph that does not admit a proper 3-edge-coloring we will have K = 0. For instance, for the non-3-colorable Petersen graph we obtain \n<!-- image --> \nHowever, for K to actually equal the number of allowed colorings, each nonzero term must have the value 1 (and not -1). This is only guaranteed if the graph is planar, as can \nbe seen by considering the non-planar graph K 3 , 3 : \n<!-- image --> \nThe edges can be colored with three colors-in 12 different ways-yet the contraction vanishes. \nThe computational complexity of this problem has been studied in [40]. Interesting, by a well known result (Heawood 1897), the 3-colorings as stated above, are one quarter of the ways of coloring the faces of the graph with four colors, so that no two like-colored faces have an edge in common. \nExample 24 (Physical implementation of glyph[epsilon1] abc in quantum computing) . In quantum computing, typically one works with qubits (two level quantum systems) but implementations using qutrits exist (three level quantum systems, available in e.g. nitrogen vacancy centers in diamond-see for instance [41]). The epsilon tensor glyph[epsilon1] abc could be realized directly as a locally invariant 3-party state using qutrits, and can also be embedded into a qubit system. We leave it to the reader to show that by pairing qubits, glyph[epsilon1] abc can be represented with six qubits, where each leg now represents a qubit pair. (Note that a basis of 3 states can be isometrically embedded in 4-dimensional space in any number of ways.) Show further that the construction can be done such that the two qubit pairs (together representing one leg) are symmetric under exchange. Note the similarities with Eq. (85) in Example 16.', '7. FRONTIERS IN TENSOR NETWORKS': "Tensor network methods represent a vibrant area of active research, with new results and ideas appearing regularly. Here we have covered the elementary aspects of the tensor network language and three applications. The first was the matrix product state representation, the next was tensor contractions to count Boolean formula solutions and the final application focused on tensor contractions to evaluate 3-edge-colorings of 3-regular planar graphs. These three sample applications should provide a good base to move forward into active research. \nThe most common tensor network structures and algorithms used in quantum mechanics include Matrix Product States (MPS) [42-44] (see Section 5) and the related Density Matrix Renormalization Group (DMRG) [8], Matrix Product Operators (MPO) [45], Tensor \nTrains [15], Tree Tensor Networks (TTN) [46], the Multiscale Entanglement Renormalization Ansatz (MERA) [5, 47-49], Projected Entangled Pair States (PEPS) [50], Correlator Product States (CPS) [51] and Time-Evolving Block Decimation (TEBD) [52]-see also time-evolution with MERA [53, 54]. \nOur reference list above is admittedly very much incomplete but should provide a solid starting place for further study. Going further, there are several reviews that we encourage readers to consult. These proceed largely towards approaches that map lattice problems to tensor networks as tools to solve models of strongly correlated systems [4-8, 10, 11, 14]see also the viewpoint [9]. There is a growing and active community exploring the use of tensor network algorithms as a means to discover and understand new properties of quantum systems. \nIn terms of the graphical language, category theory is a branch of mathematics well suited to describe a wide range of networks [18]. Quantum circuits were first given a 'categorical model' in pioneering work by Lafont in 2003 [55] and dagger compact closed categories [56], also called Baez-Dolan † -categories, were first derived to describe both standard quantum theory as well as classes of topological quantum field theories in seminal work published in 1995 [56]. (See [18] for a well written review of categorical quantum mechanics.) For practical purposes, the graphical language turns out to be mathematically equivalent to the categorical formulation. \nThere has been some recent excitement surrounding MERA [5, 47-49]-which is capable of representing the ground state of certain many-body models at their critical pointsand its connection to quantum gravity research. Several interesting discoveries [57] have recently been made around the so called tensor network incarnation of the AdS/MERA correspondence; networks which realize a discrete anti-de Sitter space have a corresponding MERA network which represents the ground state of a critical system [57, 58]. This has generated significant recent interest and excitement. If this sounds exciting to you, you might want to take a look at [11, 13, 59].", 'ACKNOWLEDGMENTS': "We've had a number of excellent collaborators over the years, who certainly influenced our understanding of the topic. The least we can do is thank them here. In alphabetical order we thank John Baez, Stephen Clark, Sam Denny, Dieter Jaksch, Tomi Johnson, Marco Lanzagorta, Jason Morton, Lea Trenkwalder, Jacob Turner, Chris Wood and Zolt'an Zimbor'as, as well as others we're probably forgetting. J.B. acknowledges AFOSR grant FA9550-16-10300, Models and Protocols for Quantum Distributed Computation, for financial support. Diagrams and cover courtesy of Lusa Zheglova (illustrator). \n- [4] R. Or'us, 'A practical introduction to tensor networks: Matrix product states and projected entangled pair states,' Annals of Physics 349 , 117-158 (2014), arXiv:1306.2164.\n- [5] G. Vidal, 'Entanglement renormalization: an introduction,' in Understanding Quantum Phase Transitions , edited by Lincoln D. Carr (Taylor & Francis, Boca Raton, 2010).\n- [6] F. Verstraete, V. Murg, and J. I. Cirac, 'Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems,' Advances in Physics 57 , 143-224 (2008), arXiv:0907.2796.\n- [7] J. I. Cirac and F. Verstraete, 'Renormalization and tensor product states in spin chains and lattices,' J. Phys. A Math. Theor. 42 , 504004 (2009), arXiv:0910.1130.\n- [8] U. Schollwock, 'The density-matrix renormalization group in the age of matrix product states,' Annals of Physics 326 , 96-192 (2011), arXiv:1008.3477.\n- [9] S. Sachdev, 'Viewpoint: Tensor networks-a new tool for old problems,' Physics 2 , 90 (2009), arXiv:1006.0675.\n- [10] Ulrich Schollwock, 'The density-matrix renormalization group: a short introduction,' Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 369 , 2643-2661 (2011).\n- [11] R. Or'us, 'Advances on tensor network theory: symmetries, fermions, entanglement, and holography,' European Physical Journal B 87 , 280 (2014), arXiv:1407.6552.\n- [12] J. Eisert, 'Entanglement and tensor network states,' Modeling and Simulation 3 , 520 (2013), arXiv:1308.3318.\n- [13] G. Evenbly and G. Vidal, 'Tensor Network States and Geometry,' Journal of Statistical Physics 145 , 891-918 (2011), arXiv:1106.1082.\n- [14] J. C. Bridgeman and C. T. Chubb, 'Hand-waving and Interpretive Dance: An Introductory Course on Tensor Networks,' ArXiv e-prints (2016), arXiv:1603.03039 [quant-ph].\n- [15] Andrzej Cichocki, Namgil Lee, Ivan Oseledets, Anh-Huy Phan, Qibin Zhao, and Danilo P. Mandic, 'Tensor networks for dimensionality reduction and large-scale optimization: Part 1 low-rank tensor decompositions,' Foundations and Trends in Machine Learning 9 , 249-429 (2016), 1609.00893.\n- [16] Andrzej Cichocki, Anh-Huy Phan, Qibin Zhao, Namgil Lee, Ivan Oseledets, Masashi Sugiyama, and Danilo P. Mandic, 'Tensor networks for dimensionality reduction and largescale optimization: Part 2 applications and future perspectives,' Foundations and Trends in Machine Learning 9 , 431-673 (2017).\n- [17] M. J. Hartmann, J. Prior, S. R. Clark, and M. B. Plenio, 'Density matrix renormalization group in the Heisenberg picture,' Phys. Rev. Lett. 102 , 057202 (2009), arXiv:0808.0666.\n- [18] J. C. Baez and A. Lauda, 'A Prehistory of n-Categorical Physics,' in Deep Beauty (Cambridge University Press, 2011) pp. 13-128, arXiv:0908.2469.\n- [19] Jacob D. Biamonte, Ville Bergholm, and Marco Lanzagorta, 'Tensor network methods for invariant theory,' J. Phys. A: Math. Theor. 46 , 475301 (2013), arXiv:1209.0631.\n- [20] A. Critch and J. Morton, 'Algebraic Geometry of Matrix Product States,' SIGMA 10 , 095 (2014), arXiv:1210.2812 [quant-ph].\n- [21] William K. Wootters, 'Entanglement of formation of an arbitrary state of two qubits,' Phys. Rev. Lett. 80 , 2245-2248 (1998), arXiv:quant-ph/9709029.\n- [22] V. Coffman, J. Kundu, and W. K. Wootters, 'Distributed entanglement,' Phys. Rev. A 61 , 052306 (2000), arXiv:quant-ph/9907047.\n- [23] Julia Kempe, 'Multiparticle entanglement and its applications to cryptography,' Phys. Rev. A 60 , 910-916 (1999), arXiv:quant-ph/9902036. \n- [24] Adriano Barenco, Charles H. Bennett, Richard Cleve, David P. DiVincenzo, Norman Margolus, Peter Shor, Tycho Sleator, John A. Smolin, and Harald Weinfurter, 'Elementary gates for quantum computation,' Phys. Rev. A 52 , 3457-3467 (1995), arXiv:quant-ph/9503016.\n- [25] J. D. Biamonte, S. R. Clark, and D. Jaksch, 'Categorical tensor network states,' AIP Advances 1 , 042172 (2011).\n- [26] Ville Bergholm and Jacob D. Biamonte, 'Categorical quantum circuits,' J. Phys. A: Math. Theor. 44 , 245304 (2011), arXiv:1010.4840.\n- [27] S. J. Denny, J. D. Biamonte, D. Jaksch, and S. R. Clark, 'Algebraically contractible topological tensor network states,' Journal of Physics A Mathematical General 45 , 015309 (2012), arXiv:1108.0888.\n- [28] Roger Penrose, 'The theory of quantized directions,' in Collected Works , Vol. 1 (October 1953-67) (Oxford University Press, 2010) Chap. 31, pp. 769-800.\n- [29] Christopher J. Wood, Jacob D. Biamonte, and David G. Cory, 'Tensor networks and graphical calculus for open quantum systems,' Quant. Inf. Comp. 15 , 759-811 (2015), arXiv:1111.6950.\n- [30] S. Meznaric and J. Biamonte, 'Tensor networks for entanglement evolution,' in Quantum Information and Computation for Chemistry: Advances in Chemical Physics , Vol. 154, edited by Sabre Kais (John Wiley & Sons, 2014) pp. 561-574, arXiv:1204.3599.\n- [31] M. B. Hastings, 'An area law for one-dimensional quantum systems,' Journal of Statistical Mechanics: Theory and Experiment 8 , 08024 (2007), arXiv:0705.2024.\n- [32] D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac, 'Matrix product state representations,' Quant. Inf. and Comp. 7 , 401-430 (2007), arXiv:quant-ph/0608197.\n- [33] I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, 'Rigorous results on valence-bond ground states in antiferromagnets,' Phys. Rev. Lett. 59 , 799-802 (1987).\n- [34] J. Eisert, M. Cramer, and M. B. Plenio, 'Colloquium: Area laws for the entanglement entropy,' Reviews of Modern Physics 82 , 277-306 (2010), arXiv:0808.3773.\n- [35] T. H. Johnson, J. D. Biamonte, S. R. Clark, and D. Jaksch, 'Solving search problems by strongly simulating quantum circuits,' Scientific Reports 3 , 1235 (2013), arXiv:1209.6010.\n- [36] Jacob Biamonte, Jason Morton, and Jacob Turner, 'Tensor network contractions for #SAT,' Journal of Statistical Physics 160 , 1389-1404 (2015), arXiv:1405.7375.\n- [37] Leslie G Valiant, 'The complexity of computing the permanent,' Theoretical computer science 8 , 189-201 (1979).\n- [38] J. D. Whitfield, M. Faccin, and J. D. Biamonte, 'Ground-state spin logic,' EPL (Europhysics Letters) 99 , 57004 (2012), arXiv:1205.1742 [quant-ph].\n- [39] Jacob Biamonte, 'Charged string tensor networks,' Proceedings of the National Academy of Sciences 114 , 2447 (2017).\n- [40] Mingji Xia, Peng Zhang, and Wenbo Zhao, 'Computational complexity of counting problems on 3-regular planar graphs,' Theoretical Computer Science 384 , 111-125 (2007).\n- [41] F. Dolde et al. , 'High-fidelity spin entanglement using optimal control,' Nature Communications 5 , 3371 (2014), arXiv:1309.4430.\n- [42] M. Fannes, B. Nachtergaele, and R. F. Werner, 'Finitely correlated states on quantum spin chains,' Communications in Mathematical Physics 144 , 443-490 (1992).\n- [43] Stellan Ostlund and Stefan Rommer, 'Thermodynamic limit of density matrix renormalization,' Phys. Rev. Lett. 75 , 3537-3540 (1995), arXiv:cond-mat/9503107.\n- [44] Stefan Rommer and Stellan Ostlund, 'Class of ansatz wave functions for one-dimensional spin systems and their relation to the density matrix renormalization group,' Phys. Rev. B 55 , 2164-2181 (1997), arXiv:cond-mat/9606213. \n- [45] F. Verstraete, J. J. Garc'ıa-Ripoll, and J. I. Cirac, 'Matrix product density operators: Simulation of finite-temperature and dissipative systems,' Phys. Rev. Lett. 93 , 207204 (2004), arXiv:cond-mat/0406426.\n- [46] Y.-Y. Shi, L.-M. Duan, and G. Vidal, 'Classical simulation of quantum many-body systems with a tree tensor network,' Phys. Rev. A 74 , 022320 (2006), arXiv:quant-ph/0511070.\n- [47] G. Vidal, 'Entanglement renormalization,' Phys. Rev. Lett. 99 , 220405 (2007), arXiv:condmat/0512165.\n- [48] V. Giovannetti, S. Montangero, and R. Fazio, 'Quantum Multiscale Entanglement Renormalization Ansatz Channels,' Physical Review Letters 101 , 180503 (2008), arXiv:0804.0520.\n- [49] G. Vidal, 'Class of quantum many-body states that can be efficiently simulated,' Phys. Rev. Lett. 101 , 110501 (2008), arXiv:quant-ph/0610099.\n- [50] F. Verstraete, M. M. Wolf, D. Perez-Garcia, and J. I. Cirac, 'Criticality, the area law, and the computational power of projected entangled pair states,' Physical Review Letters 96 , 220601 (2006), arXiv:quant-ph/0601075.\n- [51] S. Al-Assam, S. R. Clark, C. J. Foot, and D. Jaksch, 'Capturing long range correlations in two-dimensional quantum lattice systems using correlator product states,' Phys. Rev. B 84 , 205108 (2011), arXiv:1107.0936.\n- [52] G. Vidal, 'Efficient classical simulation of slightly entangled quantum computations,' Physical Review Letters 91 , 147902 (2003), arXiv:quant-ph/0301063.\n- [53] M. Rizzi, S. Montangero, and G. Vidal, 'Simulation of time evolution with multiscale entanglement renormalization ansatz,' Phys. Rev. A 77 , 052328 (2008), arXiv:0706.0868.\n- [54] J. Molina-Vilaplana and J. Prior, 'Entanglement, tensor networks and black hole horizons,' General Relativity and Gravitation 46 , 1823 (2014), arXiv:1403.5395.\n- [55] Yves Lafont, 'Towards an algebraic theory of boolean circuits,' Journal of Pure and Applied Algebra 184 , 2003 (2003).\n- [56] John C. Baez and James Dolan, 'Higher-dimensional algebra and topological quantum field theory,' Journal of Mathematical Physics 36 , 6073-6105 (1995).\n- [57] B. Swingle, 'Entanglement renormalization and holography,' Phys. Rev. D 86 , 065007 (2012), arXiv:0905.1317.\n- [58] N. Bao, C. Cao, S. M. Carroll, A. Chatwin-Davies, N. Hunter-Jones, J. Pollack, and G. N. Remmen, 'Consistency conditions for an AdS multiscale entanglement renormalization ansatz correspondence,' Phys. Rev. D 91 , 125036 (2015), arXiv:1504.06632.\n- [59] M. Van Raamsdonk, 'Lectures on gravity and entanglement,' (2016), arXiv:1609.00026 [hepth].", 'Appendix A: Tensors and Tensor Products': "The definition of a tensor starts with the tensor product ⊗ . There are many equivalent ways to define it, but perhaps the simplest one is through basis vectors. Let V and W be finite-dimensional vector spaces over the same field of scalars K . In physics-related applications K is typically either the real numbers R or the complex numbers C . Now V ⊗ W is also a vector space over K . If V and W have the bases { e j } j and { f k } k , respectively, the symbols { e j ⊗ f k } jk form a basis for V ⊗ W . Thus, for finite-dimensional spaces dim( V ⊗ W ) = dim V dim W . \nThe tensor product of two individual vectors v ∈ V and w ∈ W is denoted as v ⊗ w . For vectors the tensor product is a bilinear map V × W → V ⊗ W , i.e. one that is linear in both input variables. For finite-dimensional spaces one can obtain the standard basis coordinates of the tensor product of two vectors as the Kronecker product of the standard basis coordinates of the individual vectors: \n( v ⊗ w ) jk = v j w k . (A1) \nIt is important to notice that due to the bilinearity ⊗ maps many different pairs of vectors ( v, w ) to the same product vector: v ⊗ ( sw ) = ( sv ) ⊗ w = s ( v ⊗ w ), where s ∈ K . For inner product spaces (such as the Hilbert spaces encountered in quantum mechanics) the tensor product space inherits the inner product from its constituent spaces: \n〈 v 1 ⊗ w 1 , v 2 ⊗ w 2 〉 V ⊗ W = 〈 v 1 , v 2 〉 V 〈 w 1 , w 2 〉 W . (A2) \nA tensor T is an element of the tensor product of a finite number of vector spaces over a common field of scalars K . The dual space V ∗ of a vector space V is defined as the space of linear maps from V to K . It is not hard to show that V ∗ is a vector space over K on its own. This leads us to define the concept of an order-( p, q ) tensor, an element of the tensor product of p primal spaces and q dual spaces: \nT ∈ W 1 ⊗ W 2 ⊗ . . . ⊗ W p ⊗ V ∗ 1 ⊗ V ∗ 2 ⊗ . . . ⊗ V ∗ q . (A3) \nGiven a basis { e ( i ) k } k for each vector space W i and a dual basis { η ( i ) k } k for each dual space V ∗ i , we may expand T in the tensor products of these basis vectors: \nT = T i 1 ...i p j 1 ...j q e (1) i 1 ⊗ . . . ⊗ e ( p ) i p ⊗ η (1) j 1 ⊗ . . . ⊗ η ( q ) j q . (A4) \nT i 1 ...i p j 1 ...j q is simply an array of scalars containing the basis expansion coefficients. Here we have introduced the Einstein summation convention , in which any index that is repeated exactly twice in a term, once up, once down, is summed over. This allows us to save a considerable number of sum signs, without compromising on the readability of the formulas. Traditionally basis vectors carry a lower (covariant) index and dual basis vectors an upper (contravariant) index. \nA tensor is said to be simple if it can be written as the tensor product of some elements of the underlying vector spaces: T = v (1) ⊗ . . . ⊗ v ( q ) ⊗ ϕ (1) ⊗ . . . ⊗ ϕ ( p ) . This is not true for most tensors; indeed, in addition to the bilinearity, this is one of the properties that separates tensors from mere Cartesian products of vectors. However, any tensor can be written as a linear combination of simple tensors, e.g. as in Eq. (A4). \nFor every vector space W there is a unique bilinear map W ⊗ W ∗ → K , w ⊗ φ ↦→ φ ( w ) called a natural pairing, where the dual vector maps the primal vector to a scalar. One can apply this map to any pair of matching primal and dual spaces in a tensor. It is called a contraction of the corresponding upper and lower indices. For example, if we happen to have W 1 = V 1 we may contract the corresponding indices on T : \nC 1 , 1 ( T ) = T i 1 ...i p j 1 ...j q η (1) j 1 ( e (1) i 1 ) e (2) i 2 ⊗ . . . ⊗ e ( p ) i p ⊗ η (2) j 2 ⊗ . . . ⊗ η ( q ) j q = T k i 2 ...i p k i 2 ...j q e (2) i 2 ⊗ . . . ⊗ e ( p ) i p ⊗ η (2) j 2 ⊗ . . . ⊗ η ( q ) j q , (A5) \nsince the defining property of a dual basis is η (1) j 1 ( e (1) i 1 ) = δ j 1 i 1 . Hence the contraction eliminates the affected indices ( k is summed over), lowering the tensor order by (1 , 1). \nWe can see that an order-(1 , 0) tensor is simply a vector, an order-(0 , 1) tensor is a dual vector, and can define an order-(0 , 0) tensor to correspond to a plain scalar. But what about general, order-( p, q ) tensors? How should they be understood? Using contraction, they can be immediately reinterpreted as multilinear maps from vectors to vectors: \nT ' : V 1 ⊗ . . . ⊗ V q → W 1 ⊗ . . . ⊗ W p , T ' ( v (1) ⊗ . . . ⊗ v ( q ) ) = T i 1 ...i p j 1 ...j q e (1) i 1 ⊗ . . . ⊗ e ( p ) i p × η (1) j 1 ( v (1) ) × . . . × η ( q ) j q ( v ( q ) ) , (A6) \nwhere we tensor-multiply T and the vectors to be mapped together, and then contract the corresponding indices. However, this is not the only possible interpretation. We could just as easily see them as mapping dual vectors to dual vectors: \nT '' : W ∗ 1 ⊗ . . . ⊗ W ∗ p → V ∗ 1 ⊗ . . . ⊗ V ∗ q , T '' ( ϕ (1) ⊗ . . . ⊗ ϕ ( p ) ) = T i 1 ...i p j 1 ...j q ϕ (1) ( e (1) i 1 ) × . . . × ϕ ( p ) ( e ( p ) i p ) × η (1) j 1 ⊗ . . . ⊗ η ( q ) j q . (A7) \nEssentially we may move any of the vector spaces to the other side of the arrow by taking their dual: \nW ⊗ V ∗ ∼ = K → W ⊗ V ∗ ∼ = V → W ∼ = V ⊗ W ∗ → K ∼ = W ∗ → V ∗ , (A8) \nwhere all the arrows denote linear maps. Any and all input vectors are mapped to scalars by the corresponding dual basis vectors in expansion (A4), whereas all input dual vectors map the corresponding primal basis vectors to scalars. \nIf we expand the input vectors v ( k ) in Eq. (A6) using the same bases as when expanding the tensor T, we obtain the following equation for the expansion coefficients: \nT ' ( v (1) ⊗ . . . ⊗ v ( q ) ) i 1 ...i p = T i 1 ...i p j 1 ...j q v (1) j 1 · · · v ( q ) j q . (A9) \nThis is much less cumbersome than Eq. (A6), and contains the same information. This leads us to adopt the abstract index notation for tensors, in which the indices no longer denote the components of the tensor in a particular basis, but instead signify the tensor's order. Tensor products are denoted by simply placing the tensor symbols next to each other. Within each term, any repeated index symbol must appear once up and once down, and denotes contraction over those indices. Hence, x a denotes a vector (with one contravariant index), ω a a dual vector (with one covariant index), and T ab c an order-(2 , 1) tensor with \ntwo contravariant and one covariant indices. S ab cde x c y d P e a denotes the contraction of an order-(2 , 3) tensor S , an order-(1 , 1) tensor P , and two vectors, x and y , resulting in an order-(1 , 0) tensor with one uncontracted index, b . \nIn many applications, for example in differential geometry, the vector spaces associated with a tensor are often copies of the same vector space V or its dual V ∗ , which means that any pair of upper and lower indices can be contracted, and leads to the tensor components transforming in a very specific way under basis changes. This specific type of a tensor is called an order-( p, q ) tensor on the vector space V . However, here we adopt a more general definition, allowing { V k } k and { W k } k to be all different vector spaces."}

2023A&A...677A..88B

We present JADES JWSTNIRSpec spectroscopy of GNz11 the most luminous candidate z gt 10 Lyman break galaxy in the GOODSNorth field with MSUBUVSUB 21.5. We derive a redshift of z 10.603 lower than previous determinations based on multiple emission lines in our low and medium resolution spectra over 0.7 5.3 m. We significantly detect the continuum and measure a blue restUV spectral slope of 2.4. Remarkably we see spatially extended Lyman in emission despite the highly neutral intergalactic medium expected at this early epoch offset 555 km sSUP1SUP redwards of the systemic redshift. From our measurements of collisionally excited lines of both low and high ionisation including O II 3727 Ne III 3869 and C III 1909 we infer a high ionisation parameter log U 2. We detect the rarely seen N IV 1486 and N III 1748 lines in both our low and medium resolution spectra with other high ionisation lines seen in the low resolution spectrum such as He II blended with O III and C IV with a possible PCygni profile. Based on the observed restUV line ratios we cannot conclusively rule out photoionisation from an active galactic nucleus AGN although the high C IIIHe II and N IIIHe II ratios are compatible with a star formation explanation. If the observed emission lines are powered by star formation then the strong N III 1748 observed may imply an unusually high NO abundance. Balmer emission lines H H are also detected and if powered by star formation rather than an AGN we infer a star formation rate of 20 30 MSUBSUB yrSUP1SUP depending on the initial mass function and low dust attenuation. Our NIRSpec spectroscopy confirms that GNz11 is a remarkable galaxy with extreme properties seen 430 Myr after the Big Bang.

2023-09-01T00:00:00Z

['2023arXiv230207256B', '10.1051/0004-6361/202346159', 'arXiv:2302.07256', '10.48550/arXiv.2302.07256', '2023A&A...677A..88B']

['galaxies: high-redshift', 'galaxies: evolution', 'galaxies: groups: individual: GN-z11', 'galaxies: abundances', 'Astrophysics - Astrophysics of Galaxies', 'Astrophysics - Cosmology and Nongalactic Astrophysics']

JADES NIRSpec Spectroscopy of GNz11 Lyman emission and possible enhanced nitrogen abundance in a z 10.60 luminous galaxy

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']

https://arxiv.org/pdf/2302.07256.pdf

{'JADES NIRSpec Spectroscopy of GN-z11: LymanGLYPH<11> emission and possible enhanced nitrogen abundance in a z = 10 : 60 luminous galaxy': "Andrew J. Bunker 1, ? , Aayush Saxena 1,2 , Alex J. Cameron 1 , Chris J. Willott 3 , Emma Curtis-Lake 4 , Peter Jakobsen 5,6 , Stefano Carniani 7 , Renske Smit 8 , Roberto Maiolino 9,10,2 , Joris Witstok 9,10 , Mirko Curti 9,10,11 , Francesco D'Eugenio 9,10 , Gareth C. Jones 1 , Pierre Ferruit 12 , Santiago Arribas 13 , Stephane Charlot 14 , Jacopo Chevallard 1 , Giovanna Giardino 15 , Anna de Graa GLYPH<11> 16 , Tobias J. Looser 9,10 , Nora Lützgendorf 17 , Michael V. Maseda 18 , Tim Rawle 17 , Hans-Walter Rix 16 , Bruno Rodríguez Del Pino 13 , Stacey Alberts 19 , Eiichi Egami 19 , Daniel J. Eisenstein 20 , Ryan Endsley 21 , Kevin Hainline 19 , Ryan Hausen 22 , Benjamin D. Johnson 20 , George Rieke 19 , Marcia Rieke 19 , Brant E. Robertson 23 , Irene Shivaei 19 , Daniel P. Stark 19 , Fengwu Sun 19 , Sandro Tacchella 9,10 , Mengtao Tang 19 , Christina C. Williams 24,19 , Christopher N. A. Willmer 19 , William M. Baker 9,10 , Stefi Baum 25 , Rachana Bhatawdekar 12,26 , Rebecca Bowler 27 , Kristan Boyett 28,29 , Zuyi Chen 19 , Chiara Circosta 12 , Jakob M. Helton 19 , Zhiyuan Ji 19 , Jianwei Lyu 19 , Erica Nelson 30 , Eleonora Parlanti 7 , Michele Perna 13 , Lester Sandles 9,10 , Jan Scholtz 9,10 , Katherine A. Suess 23,31 , Michael W. Topping 19 , Hannah Übler 9,10 , Imaan E. B. Wallace 1 , and Lily Whitler 19 \n(A GLYPH<14> liations can be found after the references)", 'ABSTRACT': 'We present JADES JWST / NIRSpec spectroscopy of GN-z11, the most luminous candidate z > 10 Lyman break galaxy in the GOODS-North field with MUV = GLYPH<0> 21 : 5. We derive a redshift of z = 10 : 603 (lower than previous determinations) based on multiple emission lines in our low and medium resolution spectra over 0 : 8 GLYPH<0> 5 : 3 GLYPH<22> m. We significantly detect the continuum and measure a blue rest-UV spectral slope of GLYPH<12> = GLYPH<0> 2 : 4. Remarkably, we see spatially-extended LymanGLYPH<11> in emission (despite the highly-neutral IGM expected at this early epoch), o GLYPH<11> set 555 km s GLYPH<0> 1 redward of the systemic redshift. From our measurements of collisionally-excited lines of both low- and high-ionization (including [O ii ] GLYPH<21> 3727, [Ne iii ] GLYPH<21> 3869 and C iii ] GLYPH<21> 1909) we infer a high ionization parameter (log U GLYPH<24> GLYPH<0> 2). We detect the rarely-seen N iv ] GLYPH<21> 1486 and N iii ] GLYPH<21> 1748 lines in both our low and medium resolution spectra, with other high ionization lines seen in the low resolution spectrum such as He ii (blended with O iii ]) and C iv (with a possible P-Cygni profile). Based on the observed rest-UV line ratios, we cannot conclusively rule out photoionization from AGN, although the high C iii ] / He ii and N iii ] / He ii ratios are compatible with a star-formation explanation. If the observed emission lines are powered by star formation, then the strong N iii ] GLYPH<21> 1748 observed may imply an unusually high N = O abundance. Balmer emission lines (H GLYPH<13> , H GLYPH<14> ) are also detected, and if powered by star formation rather than an AGN we infer a star formation rate of GLYPH<24> 20 GLYPH<0> 30 M GLYPH<12> yr GLYPH<0> 1 (depending on the IMF) and low dust attenuation. Our NIRSpec spectroscopy confirms that GN-z11 is a remarkable galaxy with extreme properties seen 430 Myr after the Big Bang. \nKey words. Galaxies: high-redshift - Galaxies: formation - Galaxies: active - (Cosmology:) dark ages, reionization, first stars', '1. Introduction': 'Spectroscopically confirming galaxies formed within the first few hundred million years after the Big Bang, and understanding their nature and evolution, represents one of the biggest challenges of modern astrophysics, and one of the main drivers behind the James Webb Space Telescope (JWST) . Probing the formation of some of the very first galaxies helps establish the epoch of first light in the Universe, i.e. the timescales of the formation of the first stars, bringing an end to the so-called cosmic Dark Ages. Deep NIRSpec follow-up of some of the highest redshift galaxy candidates has already yielded spectroscopic confirmations via clear detection of the Lyman break in four galaxies at z > 10 (Curtis-Lake et al. 2023; Robertson et al. 2023). \nThe onset of the first star formation began the process of reionizing the intergalactic medium (IGM), although the exact details of this reionization are still uncertain. Observations of LymanGLYPH<11> emission and absorption provide strong constraints on \nhow and when the di GLYPH<11> use gas in the IGM transitions from neutral to ionized (see Robertson 2022 for a review). An important observation is the decrease in LymanGLYPH<11> emission line equivalent width with increasing redshift above z = 6 (Stark et al. 2010; Schenker et al. 2014; Caruana et al. 2014; Jung et al. 2020), consistent with stronger absorption from an increasingly neutral IGM. However, this picture was largely based on moderateluminosity galaxies. Luminous galaxies at 7 : 5 < z < 9 often show LymanGLYPH<11> emission (Zitrin et al. 2015; Oesch et al. 2015; Stark et al. 2017; Larson et al. 2022; Finkelstein et al. 2013; Roberts-Borsani et al. 2016; Jung et al. 2019; Song et al. 2016), at a redshift where quasar damping wing studies suggest the IGM is significantly neutral ( x hi GLYPH<24> 0 : 5; Greig et al. 2017; Davies et al. 2018; Wang et al. 2020). This indicates that around luminous (and potentially massive) galaxies, LymanGLYPH<11> escapes more easily, perhaps as a result of ionized bubbles that grew early in overdense regions (Endsley & Stark 2022; Jung et al. 2022; cf. Saxena et al. 2023). Another possible LymanGLYPH<11> escape mechanism is through resonant scattering, with high velocity neutral \ngas (perhaps associated with outflows) redshifting the photons to lower frequencies at which they are no longer absorbed by the intervening neutral IGM (Dijkstra 2014; Mason et al. 2018a). Deep spectroscopy with JWST is vastly increasing our knowledge in this area, both by detecting rest-frame optical lines of known LymanGLYPH<11> emitters to derive systemic redshifts and details of nebular physical conditions, and by detecting LymanGLYPH<11> further into the reionization epoch than has been possible from the ground (e.g. Tang et al. 2023). \nBefore the launch of JWST , the most distant galaxy with a tentative but plausible spectroscopic redshift was GN-z11 (Oesch et al. 2016). This was first selected as a likely high redshift Lyman-break candidate through multi-colour imaging with HST (Bouwens et al. 2010; Oesch et al. 2015), and subsequent HST / WFC3 slitless grism spectroscopy revealed a possible Lyman break in the continuum (Oesch et al. 2016), yielding a redshift of z grism = 11 : 09. With an apparent H 160 magnitude of 26 : 0 GLYPH<6> 0 : 1, GN-z11 is remarkably bright, up to 3 times more luminous than the characteristic rest-UV luminosity ( L ? ) measured from luminosity functions at z GLYPH<24> 6 GLYPH<0> 8 (e.g. Finkelstein et al. 2015; Bouwens et al. 2015). Using Spitzer / IRAC fluxes, Oesch et al. (2016) estimated its stellar mass to be M ? = 10 9 M GLYPH<12> , indicating a rapid build up of stellar mass in the very early Universe. \nThrough ground-based near-infrared spectroscopy using MOSFIRE on the Keck Telescope, the redshift of GN-z11 was further refined by Jiang et al. (2021) via the possible detection of the [C iii ] GLYPH<21> 1907 + C iii ] GLYPH<21> 1909 doublet, yielding a redshift of z = 10 : 957. If real, the intense C iii ] emission line might originate partly due to an active galactic nucleus (AGN) hosted by the galaxy, or due to rapid carbon enrichment (Jiang et al. 2021). \nGiven the unique nature of this source and the low signal-tonoise ratio ( S = N ) of existing continuum break and emission line detections of this galaxy, NIRSpec on JWST now o GLYPH<11> ers a chance to confirm its true distance and nature through high S = N detection of multiple rest-UV and optical emission lines as well as its bright continuum. Several diagnostics relying on the ratios and strengths of rest-UV and optical emission lines can help di GLYPH<11> erentiate between photoionization due to AGN or star-formation alone, and further help characterize the ionization conditions in the interstellar medium (ISM) of this remarkably luminous distant galaxy. \nIn this paper, we report an unambiguous spectroscopic redshift of z = 10 : 6034 for GN-z11 using deep NIRSpec observations in the GOODS-North field via the robust detection of several emission lines including N iv ] GLYPH<21> 1486, N iii ] GLYPH<21>GLYPH<21> 1747 ; 1749, [C iii ] GLYPH<21> 1907 + C iii ] GLYPH<21> 1909, [O ii ] GLYPH<21>GLYPH<21> 3726 ; 3729, [Ne iii ] GLYPH<21>GLYPH<21> 3869 ; 3967, H GLYPH<14> and H GLYPH<13> . Although the measured redshift is lower than previously reported in other work, this still places GN-z11 as comfortably the most luminous source currently confirmed at z > 10. This galaxy is given the designation JADES-GN-z10-0 in the JADES spectroscopic database, but for the remainder of this paper we use the more familiar name GN-z11. The photometric properties from our JWST / NIRCam imaging are reported in a companion paper (Tacchella et al. 2023). \nThe layout of this paper is as follows. In Section 2 we describe our JWST / NIRSpec observations of GN-z11 and the data reduction strategies adopted. In Section 3 we present the 1D and 2D spectra of GN-z11, emission line measurements and discuss the inferred physical properties. In Section 4 we conclude our findings. Throughout this work, we assume the Planck 2018 cosmology (Planck Collaboration et al. 2020) and the AB magnitude system (Oke & Gunn 1983).', '2. Observations': 'The NIRSpec observations of GN-z11 were taken as part of the JWST Advanced Deep Extragalactic Survey (JADES), a collaboration between the Instrument Science Teams of NIRSpec and NIRCam to study galaxy evolution out to high redshift through imaging and spectroscopy in the two GOODS fields. The Guaranteed Time Observations presented here are part of program ID 1181 (P.I.: D. Eisenstein), with spectroscopic targets based on preJWST data, largely HST imaging. \nOur observations of GN-z11 were taken on UT 5 & 7 February 2023, using NIRSpec (Jakobsen et al. 2022) in its microshutter array (MSA) mode (Ferruit et al. 2022). The MSA comprises four arrays of 365 GLYPH<2> 171 independently-operable shutters, each covering 98 00 GLYPH<2> 91 00 on the sky. GN-z11 was targeted in four independent MSA configurations. Each configuration acquired 3100 s of integration in each of the medium-resolution G140M / F070LP, G235M / F170LP, and G395M / F290LP grating / filter combinations (with resolving power R GLYPH<25> 1000 and combined spectral coverage over 0 : 7 GLYPH<0> 5 : 3 GLYPH<22> m) and 6200 s in the low-resolution PRISM / CLEAR mode (with R GLYPH<24> 100 and continuous coverage 0 : 7 GLYPH<0> 5 : 3 GLYPH<22> m). Targets were assigned three shutter slitlets, with the targets nodded into each of the three shutters during the observing sequence to facilitate background subtraction. As GN-z11 was one of our highest priority targets, we ensured that its spectra did not overlap with other targets, even for the gratings (where the spectra are more extended on the detector than the low-dispersion prism). Individual integrations used the NRSIRS2 readout mode with 14 groups (1035 s each) to limit correlated readout noise. In total our integration time was 3.45 hours in each of the three gratings, and 6.9 hours in the prism. \nThese observations were processed with algorithms developed by the ESA NIRSpec Science Operations Team and the NIRSpec GTO Team. We refer the reader to Cameron et al. (2023) for more details of the processing. We note that for the G140M / F070LP grating / filter combination we extended the calibration of the spectrum up to 1.84 GLYPH<22> m, taking into account the transmission filter throughput beyond the nominal wavelength range of this configuration (0.70 GLYPH<22> m-1.27 GLYPH<22> m). Since GN-z11 is at z > 10 with no flux at wavelengths below LymanGLYPH<11> , there is no second order light to overlap with the extended wavelength range of 1.27 GLYPH<22> m-1.84 GLYPH<22> m. Wavelength-dependent path-loss corrections were made based on the object position within the shutter and modelling the galaxy as a point-like source. GN-z11 is very compact, so this is a good approximation. In all four of the configurations, GN-z11 was located not more than 40% of the illuminated slit width or slit height from the centre along either axis of the 0 00 : 20 GLYPH<2> 0 00 : 46 slitlet, and Figure 1 shows the locations of the open areas of the microshutters overlaid on the NIRCam F200W image of GN-z11. Individual calibrated one-dimensional (1D) and 2D spectra were combined excluding bad pixels by using an an iterative sigma clipping algorithm at the 3 GLYPH<27> level. The wavelength calibration takes into account the position of the galaxy within the slit at each pointing. Before the combination process a spatial median of each exposure is subtracted to remove any residual background. 1D spectral extractions were made from the rectified 2D spectra using box extractions of height 3 and 5 pixels (0 00 : 3 and 0 00 : 5, respectively).', '3. Results': 'Our NIRSpec spectra of GN-z11 show well-detected continuum in the prism (Figure 2) where we have S = N > 20 per \nFig. 1. The NIRCam F200W image of GN-z11 (see Tacchella et al. 2023) with the NIRSpec microshutters overlaid for the four di GLYPH<11> erent pointings. The green rectangles denote the illuminated region of each microshutter (0 00 : 2 GLYPH<2> 0 00 : 46). North is up and East to the left, and the image is 3 arcsec on the side. The flux density units on the colour bar are MJy / sr. \n<!-- image --> \nspectral resolution element at wavelength above LymanGLYPH<11> and out to GLYPH<24> 3 GLYPH<22> m. We also see the continuum at lower S = N in the medium-resolution gratings (Figure 3, Figure B.1). A strong spectral break at LymanGLYPH<11> is observed, with no significant flux at shorter wavelengths. We have robust detections of several emission lines, most of which are seen in both the prism (Figure 2) and grating (Figure 3) spectra. We also see evidence of interstellar absorption lines, but in this paper we focus on the emission lines properties. \nIn the subsections below, we use the spectrum of GN-z11 to infer physical properties. As well as using empirical diagnostics from the emission line fluxes and ratios, we also use the beagle Bayesian SED fitting code (Chevallard & Charlot 2016) on our full prism spectrum, the exact details and results are presented in Table 2 and in Appendix A.', '3.1. Emission lines and Redshift Determination': 'The full list of detected lines is given in Table 1. Line wavelengths are measured from the grating spectra because they have higher resolution resulting in less blending and more accurate line centroids. Line fluxes are measured from both the prism and gratings, and we discuss the relative flux calibration between the two in Appendix C. We perform a cubic-spline fit to the continuum in the prism (excluding the emission lines from the fit), and use this to subtract the continuum level in both the prism and the grating spectra (since the continua in the grating spectra have low signal-to-noise ratio). We then fit each emission line with a single Gaussian model, except for blended lines (noted in Table 1) where we measure the total flux of the complex. The uncertainty is computed taking into account the Poisson \ncounting statistics and the readout noise. This is preferred to taking the intra-pixel standard deviation in counts in the spectrum, which would underestimate the true noise since the pixels are sub-sampled and interpolated from their native size by the data reduction pipleline and hence the noise is correlated. One emission line, N iv ] GLYPH<21> 1486, falls in the spectral coverage of both the G140M and G235M gratings, and we report both measurements in Table 1. The wavelength and flux of this line are consistent between the gratings within the uncertainties. \nIn determining the redshift from the vacuum rest-frame wavelengths, we exclude LymanGLYPH<11> (which has a velocity o GLYPH<11> -set, see Section 3.3), Mg ii (which is only significantly detected in the low-resolution prism) and He i GLYPH<21> 3889 (which is blended with Balmer-8). A weighted fit of the remaining 8 well-detected emission lines ( S = N > 5) from the grating spectra give a redshift z = 10 : 6034 GLYPH<6> 0 : 0013, where we assume a [C iii ] GLYPH<21> 1907 / C iii ] GLYPH<21> 1909 doublet ratio of 1 : 5 for these spectrally-unresolved lines. \nThe redshift we measure is considerably lower than the previously reported redshift values of z = 11 : 09 + 0 : 08 GLYPH<0> 0 : 012 from HST grism (Oesch et al. 2016) and z = 10 : 957 GLYPH<6> 0 : 001 from Keck MOSFIRE (Jiang et al. 2021). The 2D HST grism observation shows flux down to the wavelength we measure for the Lyman break (1 : 41 GLYPH<22> m), but due to noise fluctuations their fitted model break was at a longer wavelength of 1 : 47 GLYPH<22> m. The Keck MOSFIRE redshift was based on possibles detections of the [C iii ] GLYPH<21> 1907 and C iii ] GLYPH<21> 1909 lines at 2 : 2797 GLYPH<22> m and 2 : 282 GLYPH<22> m respectively, at 2 : 6 GLYPH<27> and 5 : 3 GLYPH<27> . We do not find any significant emission lines at these observed wavelengths in our data, where they would have been detected at 20 GLYPH<27> and 40 GLYPH<27> for the line fluxes quoted in Jiang et al. (2021). Instead, we do detect C iii ] but at a shorter wavelength consistent with our measured z = 10 : 603.', '3.2. Is GN-z11 an AGN?': 'GN-z11 has a compact morphology and the continuum spatial extent in our NIRSpec 2D spectroscopy is barely resolved. In a companion paper, Tacchella et al. (2023) analyze JADES NIRCam imaging data and derive the best size constraint so far, finding an intrinsic half-light radius of only 0 : 016 GLYPH<6> 0 : 005 00 (64 GLYPH<6> 20 pc). The possibility of a significant point source contribution to the total flux leaves open the question of whether some of the light originates from an AGN. Our data do contain several high ionization lines and we wish to explore the excitation mechanism. \nWe have detected a large number of emission lines of varying ionization potential in GN-z11. In particular the N iv ] GLYPH<21> 1486 line (ionization potential E > 47 : 5 eV) is often a signature of an AGN, although it has been seen in some high-redshift star forming galaxies (e.g., Fosbury et al. 2003 and McGreer et al. 2018) and it is detected in 6 of 44 galaxies in the low-redshift CLASSY survey (Mingozzi et al. 2022). N iv ] GLYPH<21> 1486 or N iii ] GLYPH<21> 1748 is seen in 1% of SDSS quasars at 1 : 7 < z < 4 : 0 (Jiang et al. 2008). Vanzella et al. (2010) detect N iv ] GLYPH<21> 1486 in an object at z = 5 : 6 which may be an AGN, although Raiter et al. (2010) are able to model this spectrum with stars. For GN-z11, the higher ionization Nitrogen line N v ( E > 77 : 5 eV), which is a clear signature of AGN activity, is undetected in the G140M grating (where the 3 GLYPH<27> sensitivity is 4 GLYPH<2> 10 GLYPH<0> 19 erg cm GLYPH<0> 2 s GLYPH<0> 1 ), and is blended with LymanGLYPH<11> continuum break at the prism resolution. \nWe note that in the prism ( R GLYPH<24> 100) spectrum we see prominent emission features arising from blended He ii and [O iii ] GLYPH<21>GLYPH<21> 1660 ; 1666 lines, as well as a P-Cygni type feature from \nFig. 2. 2D (top) and 1D (bottom) spectra of GN-z11 using PRISM / CLEAR configuration of NIRSpec. The 1D spectrum has been extracted using a 3 pixel wide aperture that leads to improved S / N in this highly compact object. Prominent emission lines present in the spectra are marked. The signal to noise ratio (SNR) of the continuum is high and the emission lines are clearly seen in both the 1D and 2D spectra. \n<!-- image --> \nC iv (see Figure 3), with redshifted emission and blueshifted absorption relative to the systemic redshift. civ emission with a PCygni-like profile is also detected in the grating, along with a low S = N tentative detection (3 GLYPH<27> ) of of He ii and [O iii ] GLYPH<21>GLYPH<21> 1660 ; 1666. \nThe reliable detection of C iii ] and N iii ] lines in the grating spectra, however, enables us to investigate rest-UV line ratios that can be compared with predictions from photoionization models to di GLYPH<11> erentiate between an AGN or a star-formation origin (e.g. Feltre et al. 2016). In Figure 4 we plot the line ratios C iii ] GLYPH<21> 1909 / He ii GLYPH<21> 1640 versus C iii ] GLYPH<21> 1909 / C iv GLYPH<21>GLYPH<21> 1548 ; 1550, along with predictions from photoionization models of Feltre et al. (2016) for type 2 AGN, and Gutkin et al. (2016) for star formation. We consider a density range of log( nH = cm GLYPH<0> 3 ) range of 2 GLYPH<0> 4, and metallicities in the range Z = 0 : 001 GLYPH<0> 0 : 002, corresponding to Z = Z GLYPH<12> = 0 : 066 GLYPH<0> 0 : 131 (based on Z GLYPH<12> = 0 : 0152 assumed by Feltre et al. 2016), which is consistent with the gasphase metallicity inferred from SED fitting using beagle (Section A), and the estimates from emission line ratios in Section 3.5. \nWe find that neither AGN nor SFG model predictions are able to conclusively explain the observed line ratios in GNz11. As previously noted, there is C iv absorption visible in the spectrum, blueshifted from the systemic redshift. However, it is unclear how much of the nebular component of the C iv line flux is being attenuated by this blueshifted absorption as there is not enough S / N in the grating to disentangle the nebular and stellar components of C iv . If a significant amount of nebular C iv emission is also being absorbed, then the data point would move downwards on the y -axis in Figure 4. \nAdditionally, using the C iii ] and He ii based diagnostics from Nakajima et al. (2018), we find that the observed strength of C iii ] emission and the C iii ] / He ii ratio once again lie between photoionization model predictions due to type-2 AGN and starformation. Nakajima et al. (2018) found a parameter space in their diagnostic plots where both AGN and star-forming models could overlap due to low metallicities and high C / O ratios, which is where the measurements from GN-z11 suggest it could lie. We note that when considering the photoionization models of Nakajima & Maiolino (2022), the limit on C iii ] / C iv we derive is compatible with the envelope of expectations from AGN over a range of metallicities. \nAratio of N iii ] / He ii GLYPH<25> 3 : 3 is consistent with photoionization due to star-formation (e.g. Hirschmann et al. 2019), and interestingly, no type 2 AGN scenario in the models of Hirschmann et al. (2019) predicts a N iii ] / He ii ratio greater than 1. Composite models containing contribution from both AGN and star-formation can achieve N iii ] / He ii ratios GLYPH<24> 1, but only star-formation is favoured at ratios > 1. \nOverall, we find that the C iii ] and N iii ] emission and their ratios with respect to He ii and C iv do not obviously favour photoionization due to AGN. However, the presence of other rare lines (e.g. N iv ]) that have previously been observed in the spectra of AGN makes ruling out the presence of an AGN less obvious (see Übler et al. 2023, for example). Given the expected extreme nature of GN-z11, together with a lack of any observational insights into the expected spectroscopic properties of AGN at z > 10, we are unable to draw definitive conclusions about the dominant source of photoionization in GN-z11. \nFinally, we note that the grating spectra do not show obvious evidence for the presence of a broad component of permitted lines (see Figure 3), which would be ascribed to the Broad Line Region (BLR) of an AGN. This is not necessarily conclusive proof against the AGN scenario, as the BLR is often obscured along our line of sight in most AGN, however it is another element consistent with the lack of dominant contribution from an AGN.', '3.3. LymanGLYPH<11> Emission': 'The prism spectrum shows a near-total Gunn-Peterson trough at wavelengths below LymanGLYPH<11> , consistent with a highly neutral intervening IGM (Gunn & Peterson 1965). Although the spectral break is fairly sharp in wavelength at the low dispersion of the prism, we do see some evidence of a damping wing absorption. \nIn spectra from the bluest G140M grating, an emission line is seen at 14132 Å, close to the sharp Lyman break observed with the prism. Taking the systemic redshift of GN-z11 to be z = 10 : 6034 (see Section 3.1), the rest-frame wavelength is 1217.92 Å, consistent with being LymanGLYPH<11> in emission, but with the line centroid redshifted by 555 GLYPH<6> 32 km s GLYPH<0> 1 (see Figure 5). \nFig. 3. Gallery of the most prominent emission lines seen in the spectrum GN-z11 from the Medium resolution (R1000) gratings using a 3 pixel 1D spectral extraction. \n<!-- image --> \nFig. 4. Measured C iii ] / He ii vs C iii ] / C iv ratios for GN-z11 shown along with predictions from photoionization due to AGN (circles) and starformation (stars) from Feltre et al. (2016) and Gutkin et al. (2016) considering in the range of Z = Z GLYPH<12> = 0 : 066 GLYPH<0> 0 : 131 and gas densities in the range log( nH ) = cm GLYPH<0> 3 = 2 GLYPH<0> 4. Based on the observed line ratios neither photoionization from AGN or star-formation alone can conclusively explain the observations, placing GNz11 right between the model predictions from the two. \n<!-- image --> \nFig. 5. Velocity o GLYPH<11> set of the Ly GLYPH<11> emission line (blue solid line) compared with the H GLYPH<13> line (green dashed line). The Ly GLYPH<11> line is redshifted by 555 km s GLYPH<0> 1 compared to the redshift derived from other emission lines in the spectrum. \n<!-- image --> \nThis emission line is seen in all 12 of the individual G140M \nTable 1. Emission line fluxes (in units of GLYPH<2> 10 GLYPH<0> 19 erg s GLYPH<0> 1 cm GLYPH<0> 2 ) detected in the prism ( R GLYPH<24> 100) and grating ( R GLYPH<24> 1000) spectra measured from both the 3 pixel and the 5 pixel extraction. \ngrating exposures (in each of the 3 nod positions in the 4 MSA configurations), and is > 10 GLYPH<27> in the combined grating spectrum. \nFigure 6 shows a close-up of the G140M spectral region around LymanGLYPH<11> . There is zero transmitted flux shortward of the systemic LymanGLYPH<11> wavelength. Any such flux would require an ionized bubble around the galaxy, but the lack of such flux rules out an optically-thin H II region around GN-z11 (e.g. Mason & Gronke 2020). The redshifted line is well approximated by a Gaussian, without significant asymmetry. We measure a FWHM of 566 GLYPH<6> 61 km s GLYPH<0> 1 , which is extended beyond the instrumental line spread function of GLYPH<25> 200 km s GLYPH<0> 1 (de Graa GLYPH<11> et al. in prep) for a compact source in the G140M grating at this wavelength. Removing the line spread function in quadrature suggests an intrinsic velocity spread of GLYPH<14> v FWHM = 530 GLYPH<6> 65 km s GLYPH<0> 1 . \nIn the 2D spectrum of Figure 6 it is apparent that the LymanGLYPH<11> emission is more spatially extended than the continuum. Whilst the continuum flux is largely contained within 2 pixels (0.2 00 ), as expected based on the small size measured in our NIRCam imaging presented in Tacchella et al. (2023), the LymanGLYPH<11> emission extends further to the south-west. The LymanGLYPH<11> extension beyond the continuum is at least 2 pixels, corresponding to an extra 0.8 kpc. We note the LymanGLYPH<11> could extend further \nsince the MSA shutters are only 5 pixels high, so beyond this region there is self-subtraction, but a visual check of the 2D spectrum without background subtraction did not show LymanGLYPH<11> in neighbouring shutters. A similar check on the extent of other well-detected lines in the grating spectra ([O ii ] and H GLYPH<13> ) shows that these lines have the same spatial profile as their nearby continuum. \nThe fact that LymanGLYPH<11> is spatially extended is a remarkable result, which may be suggestive of a LymanGLYPH<11> halo. The presence of such haloes around individual star-forming galaxies has been reported at lower redshifts (e.g. Rauch et al. 2008; Wisotzki et al. 2016; Leclercq et al. 2017; Kusakabe et al. 2022) and we may be seeing the gas in the circum-galactic medium (CGM), from LymanGLYPH<11> fluorescence or shock heating. \nUsing a 3 pixel (0 : 3 00 ) extraction aperture, the measured LymanGLYPH<11> flux is (1 : 51 GLYPH<6> 0 : 15) GLYPH<2> 10 GLYPH<0> 18 erg s GLYPH<0> 1 cm GLYPH<0> 2 , with a restframe equivalent width (with respect to the continuum longward of the LymanGLYPH<11> break) of EW0 = 12 Å. Using a larger \'fullshutter" extraction aperture of height 5 pixels (0 : 5 00 ) gives a significantly higher flux of (2 : 30 GLYPH<6> 0 : 19) GLYPH<2> 10 GLYPH<0> 18 erg s GLYPH<0> 1 cm GLYPH<0> 2 , and the rest-frame equivalent width rises to EW0 = 18 Å. The emission \nFig. 6. Zoom in on the Ly GLYPH<11> emission line in the G140M 1D (lower) and 2D (upper) spectra. The grey dashed line shows the systemic wavelength of the Ly GLYPH<11> transition. The histogram (top-left) of the Ly GLYPH<11> spatial profile (yellow) and that of the continuum (blue), shows the Ly GLYPH<11> emission from GN-z11 is more extended towards the south-west (up in the MSA shutter in this view). \n<!-- image --> \nFig. 7. LymanGLYPH<11> velocity o GLYPH<11> set ( GLYPH<1> v Ly GLYPH<11> ) versus MUV for GN-z11 (star) and other high-redshift galaxies, color coded by LymanGLYPH<11> EW. We overplot data of z > 6 galaxies with ground-based observations from the literature (Cuby et al. 2003; Pentericci et al. 2011, 2016, 2018; Vanzella et al. 2011; Willott et al. 2013, 2015; Maiolino et al. 2015; Oesch et al. 2015; Stark et al. 2015, 2017; Furusawa et al. 2016; Knudsen et al. 2016; Carniani et al. 2017; Laporte et al. 2017; Mainali et al. 2017; Hashimoto et al. 2019; see Endsley et al. 2022 and the Table 4 therein) in circles, and Ly GLYPH<11> emitting galaxies at z GLYPH<24> 7 GLYPH<0> 9 from CEERS NIRSpec observations (Tang et al. 2023) in squares. Prediction of the correlation between Ly GLYPH<11> velocity o GLYPH<11> set and MUV at z = 7 from Mason et al. (2018b) is shown by the grey dashed line. GN-z11 has properties similar to the Ly GLYPH<11> emitting galaxies at z GLYPH<24> 7 GLYPH<0> 9. \n<!-- image --> \nline flux of LymanGLYPH<11> is about twice that of HGLYPH<13> (the strongest Balmer line we detect). From Case B recombination and assuming no dust as found from the Balmer line ratio, LymanGLYPH<11> would have about 50 GLYPH<2> the line flux of HGLYPH<13> , so it appears to be suppressed by about a factor of 26 GLYPH<2> (i.e., f esc,Ly GLYPH<11> = 0 : 038 GLYPH<6> 0 : 004), presumably through resonant scattering e GLYPH<11> ects. \nThe discovery of LymanGLYPH<11> emission at such high redshift is remarkable, given the expected highly neutral IGM at this epoch so much earlier than the end of reionization at z GLYPH<25> 6 (Fan et al. 2001). However, perhaps this result is not so surprising when one considers the high rate of LymanGLYPH<11> detection in luminous 7 : 5 < z < 9 galaxies (Zitrin et al. 2015; Oesch et al. 2015; Stark et al. 2017; Larson et al. 2022; Finkelstein et al. 2013; Roberts-Borsani et al. 2016; Jung et al. 2019; Song et al. 2016) at redshifts prior to complete reionization. GN-z11 is a similarly luminous galaxy with M UV = GLYPH<0> 21 : 5 so the e GLYPH<11> ects that make LymanGLYPH<11> detectable in such galaxies (see Mason et al. 2018a) may be at play for GN-z11. \nThere are two aspects of our LymanGLYPH<11> observations that may explain the significant transmission of f esc,Ly GLYPH<11> = 0 : 04. Firstly, the large velocity o GLYPH<11> set of 555 GLYPH<6> 32 km s GLYPH<0> 1 and rest-frame equivalent width are similar to those measured in luminous 7 : 5 < z < 9 galaxies with LymanGLYPH<11> (Figure 7; see also Tang et al. 2023). Large velocity o GLYPH<11> sets are key to the escape of LymanGLYPH<11> photons from galaxies in a highly-neutral IGM. Since the damping wing of the IGM and proximate HI will absorb photons close to the resonant frequency, photons that escape must resonantly scatter in the wings. Those that scatter far enough to the red may then be able to escape the system without being absorbed (Dijkstra 2014). We note that the infall motion of the neutral gas around galaxies may also play a role (Santos 2004; Sadoun et al. 2017; Weinberger et al. 2019; Park et al. 2021; Smith et al. 2022). If the peculiar velocity of neutral gas near a Lyman-a source is infalling, then even photons redward of the LymanGLYPH<11> will be resonantly absorbed, necessitating large velocity o GLYPH<11> sets of LymanGLYPH<11> to facilitate its escape. \nAdditionally, the intense star formation in luminous galaxies will be driving powerful and fast-moving outflows. Outflows on the far side of the galaxy may provide a redshifted medium from which the photons can backscatter to our line-of-sight with the required velocity o GLYPH<11> set. In this context, our observation of spatially-extended LymanGLYPH<11> emission to the south-west suggests that if an outflow is present, it would extend in the north-east direction.', '3.4. Rest-frame UV properties of GN-z11': "From our low-dispersion prism spectra, where we have high S / N detection of the continuum, we measure a UV spectral slope of GLYPH<12> = GLYPH<0> 2 : 36 GLYPH<6> 0 : 10 (over the range GLYPH<21> rest = 1500 GLYPH<0> 2600 Å), consistent with that of GLYPH<12> = GLYPH<0> 2 : 4 reported from our NIRCam imaging in Tacchella et al. (2023). We measure a luminosity of M UV AB = GLYPH<0> 21 : 50 GLYPH<6> 0 : 02 over the range GLYPH<21> rest = 1400 GLYPH<0> 1600 Å (adopting a luminosity distance of 113,148.8 Mpc from our chosen cosmology). This corresponds to a luminosity density of L UV = 1 : 7 GLYPH<2> 10 29 erg s GLYPH<0> 1 cm GLYPH<0> 2 Hz GLYPH<0> 1 around GLYPH<21> rest = 1500 Å. \nGLYPH<23> \nThe rest-UV will be potentially a GLYPH<11> ected by dust reddening. However, from the prism spectrum, we measure a Balmer line ratio using H GLYPH<14> / H GLYPH<13> of 0 : 53 GLYPH<6> 0 : 06, which is very close and within the errors of the intrinsic ratio of 0 : 55 expected in an H ii region with electron density ne = 300 cm GLYPH<0> 3 and temperature Te = 15 ; 000 K, conditions expected in very high redshift galaxies (e.g. Curti et al. 2023; Katz et al. 2023; Isobe et al. 2023). We note that the wavelength baseline between H GLYPH<14> and H GLYPH<13> is short, so does not provide a large lever arm to quantify the dust attenuation accurately, and also that the H GLYPH<14> / H GLYPH<13> ratio of the fluxes from our 3-pixel medium grating spectrum is 0 : 68 GLYPH<6> 0 : 20, above the case B value of 0 : 55 but consistent within the errors. Our full-SED beagle fits to the prism spectrum, assuming that \nthis is a star-forming galaxy rather than an AGN, also suggest low attenuation ( AV GLYPH<25> 0 : 17, only 70% of which is from dust outside H ii regions a GLYPH<11> ecting nebular emission). Therefore, from the observed Balmer decrement and SED fitting we do not measure any considerable dust attenuation or reddening in the spectrum of GN-z11. \nWe now turn to the production and potential escape of ionizing photons in GN-z11. We can potentially constrain the escape fraction of ionizing photons ( f esc) by looking at how many produce local recombinations (as tracked by the hydrogen Balmer emission lines), compared with the total number produced. Zackrisson et al. (2017) provide tracks of the rest-UV spectral slope ( GLYPH<12> , which is related to the hardness of the spectrum and hence the production of ionizing photons) and the equivalent width of H GLYPH<12> . Although we do not measure H GLYPH<12> in GN-z11, as it is beyond our spectral range, we can use the continuum slope we measure ( GLYPH<12> GLYPH<25> GLYPH<0> 2 : 4) along with the flux in H GLYPH<13> and the case B ratio (appropriate if f esc is low) of H GLYPH<13> / H GLYPH<12> = 0.468 to estimate log(EW(H GLYPH<12> ) = Å) = 1 : 8. From the plots of Zackrisson et al. (2017), this high H GLYPH<12> equivalent width suggests a low escape fraction of f esc . 0 : 1. Our beagle SED fits also yield a low f esc = 0 : 03 + 0 : 05 GLYPH<0> 0 : 02 . We do not see strong evidence for a 'reverse Balmer break' in our NIRSpec spectra, which might be expected from nebular continuum if the escape fraction is indeed low, although in the best-fit beagle SED the reverse Balmer break is small. Conversely, we also see no evidence for a Balmer break that would be indicative of a moderately evolved stellar population. This is consistent with the very young age determined by the beagle SED fitting ( t GLYPH<24> 19 Myr), with the light in the rest-frame UV and blue wavelengths we probe being completely dominated by a young stellar population. \nAdditional constraints on f esc could potentially be derived from the presence of Mg ii GLYPH<21>GLYPH<21> 2795 ; 2802 emission in the prism spectrum (e.g. Chisholm et al. 2020). The considerable restframe EW of 12 Å for Mg ii might suggest the presence of ionized channels in the galaxy, potentially facilitating the escape of ionizing radiation and LymanGLYPH<11> . Similar to LymanGLYPH<11> , the resonant nature of Mg ii routinely causes strong absorption by low-ionization gas, while pure Mg ii emission is thought to indicate a porous ISM (Feltre et al. 2018; Henry et al. 2018; Witstok et al. 2021). Following the predicted relationship in Witstok et al. (2021) between the strength of Mg ii and the [Ne iii ] and [O ii ] lines, we estimate an intrinsic Mg ii GLYPH<21> 2795 flux of 3 : 1 GLYPH<2> 10 GLYPH<0> 19 erg s GLYPH<0> 1 cm GLYPH<0> 2 . This results in a Mg ii escape fraction of f esc, Mg ii GLYPH<24> 60% under the assumption of a typical doublet ratio of F 2795 = F 2802 GLYPH<25> 1 : 7 (which itself depends on f esc, Mg ii and the dust content; Chisholm et al. 2020). This is much larger than the estimated LymanGLYPH<11> escape fraction, and also larger than the f esc inferred above from the equivalent width of the Balmer lines and the BEAGLE SED fitting. However, it has been suggested that Mg ii escape could be more sensitive to dust rather than f esc particularly for galaxies in the optically thick regime, which may explain the discrepancy between the escape fractions measured from Balmer emission and Mg ii (e.g. Katz et al. 2022). \nWe now consider the production of ionizing photons, under the assumption of a low escape fraction fesc as discussed above, and compare these with the non-ionizing UV continuum detected. In case B, f ( H GLYPH<11> ) = f ( H GLYPH<13> ) = 6 : 11, and 45% of recombinations result in an H GLYPH<11> photon being emitted (Osterbrock & Ferland 2006). We take the observed H GLYPH<13> line flux to be 1 : 2 GLYPH<2> 10 GLYPH<0> 18 erg s GLYPH<0> 1 cm GLYPH<0> 2 from the 5-pixel extraction of the prism spectrum (since the agreement of the flux calibration with the NIRCam imaging is better than for the grating, Appendix C). We \nmake the assumption of no dust attenuation to obtain a hydrogen ionizing photon production rate of N ion = 8 : 8 GLYPH<2> 10 54 photons s GLYPH<0> 1 . This gives an ionizing photon production e GLYPH<14> ciency of GLYPH<24> ion = N ion = L UV GLYPH<23> = 5 : 2 GLYPH<2> 10 25 erg GLYPH<0> 1 Hz. This value log GLYPH<24> ion = 25 : 7 agrees with that from the beagle SED fitting log GLYPH<24> ion = 25 : 67 GLYPH<6> 0 : 02, and has higher ionizing e GLYPH<14> ciency than galaxies at much lower redshift (e.g., Chevallard et al. 2018 find log GLYPH<24> ion = 25 : 2 GLYPH<0> 25 : 8 in extreme galaxies at z GLYPH<24> 0, whereas Bouwens et al. 2016 find log GLYPH<24> ion = 25 : 3 in subL GLYPH<3> galaxies at z = 4 GLYPH<0> 5) but is comparable with that seen in z GLYPH<24> 7 GLYPH<0> 8 galaxies (e.g., Tang et al. 2023 who find log GLYPH<24> ion = 25 : 7 GLYPH<0> 26 : 0). \nAlthough we have not ruled out an AGN component of GN-z11, we can place upper limits on the star formation rate based on the assumption that the observed line emission is powered solely by star formation. From the rate of ionizing photons, we can estimate the star formation rate subject to assumptions about the star formation history and initial mass function (IMF) of stars. Using the Kennicutt (1998) relation, SFR = 1 : 08 GLYPH<2> 10 GLYPH<0> 53 ( N ion = s GLYPH<0> 1 ) M GLYPH<12> yr GLYPH<0> 1 assuming a Salpeter (1955) IMF, gives a star formation rate of 90 M GLYPH<12> yr GLYPH<0> 1 . For a Chabrier (2003) IMF the star formation rate is 54 M GLYPH<12> yr GLYPH<0> 1 from the Kennicutt (1998) relation. Using the more recent H GLYPH<11> -based relation from Reddy et al. (2018), which is more representative of the conditions found in galaxies at high redshifts, we obtain a star formation rate of 35 M GLYPH<12> yr GLYPH<0> 1 again assuming Chabrier (2003) IMF with an upper mass cut-o GLYPH<11> of 100 M GLYPH<12> . Out beagle fit has a star formation rate of GLYPH<24> 19 M GLYPH<12> yr GLYPH<0> 1 for a Chabrier IMF with a higher upper-mass cut-o GLYPH<11> of 300 M GLYPH<12> . Using an upper-mass cut-o GLYPH<11> of 100 M GLYPH<12> reduces the number of ionizing photons per unit SFR to 62%, bringing the star formation rate to 31 M GLYPH<12> yr GLYPH<0> 1 , in agreement with the SFR derived using the Reddy et al. (2018) conversion. \nWe can also potentially use the rest-frame UV continuum to infer the star formation rate (or an upper limit on this, if there is an AGN contribution to the rest-frame UV). Kennicutt (1998) give a relation SFR = 1 : 4 GLYPH<2> 10 28 GLYPH<2> ( L UV GLYPH<23> = erg s GLYPH<0> 1 Hz GLYPH<0> 1 ) M GLYPH<12> yr GLYPH<0> 1 for a Salpeter IMF, which would translate to a star formation rate of 24 M GLYPH<12> yr GLYPH<0> 1 for GN-z11. However, this relation is probably inappropriate since it assumes constant star formation for 100 Myr, and GN-z11 is likely much younger, so the UV luminosity will still be increasing even if star formation is constant, causing the star formation rate to be underestimated.", '3.5. ISM ionization and enrichment': "In this section we use line ratio diagnostics to explore the ionization state and metal enrichment of the ISM, again under the assumption that the emission line fluxes are not dominated by an AGN contribution. We detect a number of collisionally-excited metal lines, both of low ionization ([O ii ]) and high ionization (including N iii ], [Ne iii ], C iii ]), as well as Balmer lines from hydrogen recombination. Our wavelength coverage does not extend to the widely-used [O iii ] GLYPH<21> 5007, however we do have a robust detection of [Ne iii ] GLYPH<21> 3869 which has a similar ionization potential. Hence, we consider the line flux ratio [Ne iii ] GLYPH<21> 3869 / [O ii ] GLYPH<21>GLYPH<21> 3726, 3729 as a probe of ionization parameter ( U ) - [Ne iii ] / [O ii ] has been shown to track [O iii ] / [O ii ] well (e.g. Levesque & Richardson 2014; Witstok et al. 2021), which is the most widely-used indicator of U . We measure [Ne iii ] / [O ii ] = 1 : 12 GLYPH<6> 0 : 13 from the 3-pixel grating extraction ([Ne iii ] GLYPH<21> 3869 is blended in the prism), which is comparable to the redshift z GLYPH<24> 5 : 5 GLYPH<0> 9 : 5 NIRSpec sample presented in Cameron et al. (2023) from our JADES survey, and also z & 7 galaxies observed in the CEERS survey (Tang et al. 2023). Following the calibration set \nTable 2. Estimates of GN-z11 physical parameters derived from beagle SED fitting of the prism spectrum of Figure 2 \n: \nbeagle SED fitting of the prism spectrum with the uncertainties giving the extent of the 1 GLYPH<27> credible regions: stellar mass ( M , accounting for mass returned to the ISM through stellar winds and supernova explosions), star formation rate ( ), maximum age of the stars ( t ), the mass-weighted age of stars ( t m ), nebular metallicity ( Z neb), ionization parameter (log U S), V -band dust attenuation (A V ), ionizing photon production e GLYPH<14> ciency ( GLYPH<24> ion) and escape fraction of H-ionizing photons ( f esc; see Appendix A for details). \nout in Witstok et al. (2021), this corresponds to an ionization parameter of log U = GLYPH<0> 2 : 03 GLYPH<6> 0 : 04. We find a similar value of log U = GLYPH<0> 2 : 25 GLYPH<6> 0 : 97 from our beagle SED fitting. \nWereport a marginal detection of the [O iii ] GLYPH<21> 4363 line in our prism spectrum (partially blended with H GLYPH<13> ; Figure 3), which has already been observed in a number of z > 7 galaxies (e.g. Curti et al. 2023; Katz et al. 2023). Although this line can in theory be used to derive a Te -based ('direct method') metallicity, the absence of [O iii ] GLYPH<21> 5007 from our data means we cannot measure the temperature with the standard approach. The O iii ] GLYPH<21>GLYPH<21> 1660,1666 / [O iii ] GLYPH<21> 4363 ratio can also be used as a temperature diagnostic, but the low significance of the [O iii ] GLYPH<21> 4363 coupled with the marginal detection of O iii ] GLYPH<21>GLYPH<21> 1660, 1666 in our grating spectrum means that any derived temperature would be highly uncertain. Thus, we instead consider using strongline ratios to constrain the metallicity of GN-z11. A widely-used metallicity indicator is R23 (the log of the ratio of [O ii ] + [O iii ] to H GLYPH<12> ), but since [O iii ] GLYPH<21> 5007 and H GLYPH<12> fall beyond our spectral coverage, we cannot measure this ratio. We instead consider an analogous ratio of ([Ne iii ] GLYPH<21> 3869 + [O ii ] GLYPH<21>GLYPH<21> 3727) / H GLYPH<14> . All three of these emission lines are well detected in our grating spectra, and conveniently lie at very similar wavelengths which minimizes any uncertainties arising due to wavelength-dependent attenuation. We measure a ratio of log 10 (([Ne iii ] + [O ii ]) = H GLYPH<14> ) = 0 : 50 GLYPH<6> 0 : 07 from the grating (3-pixel spectral extraction). Following the calibrations from Witstok et al. (2021) (which provides [O iii ] / [Ne iii ] GLYPH<25> 15 at the derived ionization parameter) and assuming H GLYPH<14>= H GLYPH<12> = 0 : 268, this would be equivalent to R 23 GLYPH<25> 0 : 85. These values place GN-z11 in fairly close alignment with the median values presented in the Cameron et al. (2023) sample; their stacked spectra at z GLYPH<24> 6 ( z GLYPH<24> 8) show R 23 = 0 : 88 (0 : 86) and log([Ne iii ] / [O ii ]) = 0 : 05 (0 : 04). According to the binned average relationships presented in Nakajima et al. (2022), this suggests a metallicity in the range 7 : 59 < 12 + log(O / H) < 7 : 76, which corresponds to 0 : 08 GLYPH<0> 0 : 12 Z GLYPH<12> assuming a solar abundance of 12 + log( O = H ) GLYPH<12> = 8 : 69. Our beagle SED fitting yields a consistent value of Z n eb = 0 : 12 GLYPH<6> 0 : 02 Z GLYPH<12> . \nIn Figure 8 we compare our [Ne iii ] / [O ii ] and ([Ne iii ] + [O ii ]) / H GLYPH<14> measurements from GN-z11 (plotted separately from the prism and the grating data) with measurements from z > 5 : 5 \ngalaxies from Cameron et al. (2023), z GLYPH<24> 0 galaxies from SDSS MPA-JHU catalogs 1 (Aihara et al. 2011) and photoionization model grids from Gutkin et al. (2016). This line-ratio diagram is analogous to the widely used R23-O32 'ionization vs. excitation' diagram since, as described above, [Ne iii ] / [O ii ] traces ionization and ([Ne iii ] + [O ii ]) / H GLYPH<14> traces excitation of both the high- and low-ionization metal ions. \nThe Gutkin et al. (2016) models in Figure 8 demonstrate the two-valued nature of ([Ne iii ] GLYPH<21> 3869 + [O ii ] GLYPH<21>GLYPH<21> 3727) / H GLYPH<14> with metallicity. Although the signal-to-noise ratio requirements significantly cut down the available SDSS sample, one can still see clear evidence of this two-valued relation. The z > 5 : 5 sample from Cameron et al. (2023) appears to follow an extrapolation of the low-metallicity (high-ionization) branch of this two-valued sequence. We see that GN-z11 (diamond symbol) lies in good agreement with the sequence formed by these z > 5 : 5 galaxies. It falls between the Gutkin et al. (2016) Z = Z GLYPH<12> = 0.07 and Z = Z GLYPH<12> = 0.15 model lines, suggesting 12 + l og ( O = H ) GLYPH<25> 7 : 7, and lies proximal to the model values with log U = GLYPH<0> 2 : 0, consistent with the empirical values derived above. \nWe now consider where GN-z11 might fall on the massmetallicity relation (see Maiolino & Mannucci 2019 for a review). The stellar mass estimated from beagle of log( M GLYPH<3> = M GLYPH<12> ) = 8 : 73 + 0 : 06 GLYPH<0> 0 : 06 is consistent with that derived from our NIRCam photometry of log( M GLYPH<3> = M GLYPH<12> ) = 9 : 1 GLYPH<6> 0 : 3 0 : 4 presented in Tacchella et al. (2023), again assuming that the light is dominated by the stellar population rather than an AGN. Our observed spectrum shows no evidence of a Balmer Break, and if the continuum is purely stellar, is dominated by a young stellar population. It is possible a more stochastic star formation history would fit a higher stellar mass. Comparing our metallicity and mass estimates for GN-z11 with the average reported for 8 < z < 10 galaxies in Nakajima et al. (2023) we find GN-z11 is o GLYPH<11> set to somewhat lower metallicity, albeit within the uncertainty quoted there. We note that the sample presented in that paper is still small and our understanding of the metallicities of galaxies at z > 8 will no doubt continue to evolve significantly over the coming years. We also note that the uncertainties on our derived metallicity are large. In particular, we caution that the set of emission lines used to determine the metallicity presented here has not been robustly calibrated. The systematic uncertainties associated with this quoted metallicity are likely very high, so robust conclusions cannot be drawn from this about the evolution of the mass-metallicity relation. Further work is needed to robustly calibrate shorter-wavelength metallicity diagnostics suitable for the study of z > 10 galaxies with NIRSpec. \nWhat is more puzzling is the strong N iii ] and N iv ] emission observed in the rest-frame UV, especially given the absence of a convincing detection of O iii ] GLYPH<21>GLYPH<21> 1660,1666 in our grating spectrum (although the blend with He ii is detected in the lowdispersion prism spectrum). The N iii ] GLYPH<21> 1748 emission line complex is not often seen in the spectra of star-forming galaxies, although it is detected in 2 of 44 galaxies in the low-redshift CLASSY survey (Mingozzi et al. 2022), including Mk996 (James et al. 2009). At intermediate redshifts, N iii ] GLYPH<21> 1748 has been observed in stacks of rest-UV galaxy spectra at z GLYPH<24> 3 (e.g. Saxena et al. 2022) and is weakly detected in SL2SJ021737051329, a lensed arc at z = 1 : 84 with low metallicity and high ionization (Berg et al. 2018). However, N iii ] GLYPH<21> 1748 is typically observed to be weaker than the nearby [O iii ] GLYPH<21> 1660, 1666 lines. \nThis ratio of N iii ] GLYPH<21> 1748 / O iii ] GLYPH<21>GLYPH<21> 1660, 1666 can be used to place constraints on nebular N / O abundance ratios (Garnett \nFig. 8. Line ratio diagram, showing ([Ne iii ] + [O ii ]) / H GLYPH<14> vs. [Ne iii ] / [O ii ], featuring GN-z11 - the yellow diamond denotes the line ratios derived from the medium-dispersion G395M grating, and the cyan diamond uses the low-dispersion prism spectrum (where we have corrected for the blending of [Ne iii ] GLYPH<21> 3869 with He i GLYPH<21> 3889 using the flux ratio from the grating). The background grey 2D PDF shows the subset of SDSS galaxies with 0 : 03 < z < 0 : 1 for which [Ne iii ] GLYPH<21> 3869, [O ii ] GLYPH<21>GLYPH<21> 3726, 3729 and H GLYPH<14> are all detected with S = N > 5. Purple squares show z > 5 : 5 galaxies from Cameron et al. (2023) after adjusting the reported ratios to be in terms of H GLYPH<14> by assuming a fixed value of H GLYPH<14>= H GLYPH<12> = 0 : 268. Solid lines show model grids from Gutkin et al. (2016), plotted for nine di GLYPH<11> erent values of metallicity ( Z = Z GLYPH<12> = 0.04, 0.07, 0.15, 0.30, 0.45, 0.60, 0.75, 1.0, 1.5) indicated by the di GLYPH<11> erent colours, and seven values of ionization parameter, indicated by marker sizes, in steps of 0 : 5 from log U = GLYPH<0> 4 : 0 (smallest) to log U = GLYPH<0> 1 : 0 (largest). \n<!-- image --> \net al. 1995). The conversion of N iii ] / O iii ] to N ++ / O ++ depends on the electron temperature- and density-sensitive emissivities of these emission lines. However, the temperature and density dependence of these emission lines are remarkably similar. Adopting N iii ] / O iii ] & 2 : 6 (see Table 1), any adopted values of temperature 1 GLYPH<20> Te = 10 4 K GLYPH<20> 55 and density 2 GLYPH<20> log( ne = cm GLYPH<0> 3 ) GLYPH<20> 10 results in emissivities that imply 0 : 72 GLYPH<20> N ++ = O ++ GLYPH<20> 1 : 09, significantly higher than the solar N / O value of 0.14 2 \nThe measured N ++ = O ++ may over-estimate the total N / O if there is a significant fraction of oxygen in other ionisation states. The second and third ionisation energies of nitrogen (29.6 and 47.4 eV) are milder than those of oxygen (35.1 and 54.9 eV). Assuming a simply multi-zone model of the ionisation structure of the ISM, this implies that N ++ = (O + + O ++ ) should be a lower limit on the total N / O. However, from detections of [O ii ] GLYPH<21>GLYPH<21> 3727 and [O iii ] GLYPH<21> 4363 emission for GN-z11, if we assume the [O ii ] emission arises from gas with density below the critical density of [O ii ] GLYPH<21>GLYPH<21> 3727 (i.e. ne . 10 4 K ), we get an approximate lower limit of O ++ = O + & 1 : 5, consistent with the expectation of a highly ionised ISM. Thus, even if we conservatively assume O ++ = O + = 1 : 5, this only lowers the inferred N / O by a factor of 0.6, implying a lower limit on the total nitrogen to oxygen abundance ratio of N / O > 0 : 43, or log 10(N = O) > GLYPH<0> 0 : 36, more than two times higher than the solar abundance ratio. This would appear quite unusual with respect to z GLYPH<24> 0 GLYPH<0> 2 galaxies (e.g. Pérez-Montero & Contini 2009; Hayden-Pawson et al. \n2022), and strongly inconsistent with canonical chemical evolution models (see Maiolino & Mannucci 2019 or Kobayashi 2022 for reviews). \nWe note that this simple calculation is independent of the excitation source (i.e. stellar photoionisation or AGN). However, we cannot rule out the scenario in which only a small fraction of the gas in GN-z11 is highly nitrogen enriched, but that this gas is extremely luminous in emission and dominates the global spectrum. More detailed modelling of the ionisation states of nitrogen and oxygen throughout the ISM of GN-z11 would be required to derive a more precise value of N / O, which is beyond the scope of this paper. \nWe also detect the [C iii ] GLYPH<21> 1907 + C iii ] GLYPH<21> 1909 line in our G235M spectrum. The C iii ] line has been much more widely observed in star-forming galaxies at high redshift (e.g. Saxena et al. 2022; Arellano-Córdova et al. 2022; Jones et al. 2023), and its presence does not necessarily point to unusual C / Oabundance ratios (Arellano-Córdova et al. 2022; Jones et al. 2023). \nIn summary, the emission line ratios measured for GN-z11 suggest a very high ionization parameter and low oxygen abundance in the vicinity of 10 % solar, broadly in line with findings from galaxies at z GLYPH<24> 6 GLYPH<0> 10 (Cameron et al. 2023; Sanders et al. 2023; Mascia et al. 2023; Nakajima et al. 2023; Tang et al. 2023). However, the detection of strong N iii ] emission suggests unexpected abundance patterns, which may have deeper implications for chemical enrichment histories.", '4. Conclusions': 'We present JWST / NIRSpec spectroscopy of one of the most luminous galaxies at z > 10. GN-z11 is in the GOODS-North field and had previously been identified as a Lyman break galaxy candidate by Oesch et al. (2015), with a tentative redshift of z = 11 : 1 from a continuum break in slitless HST / WFC3 spectroscopy (Oesch et al. 2016). We see numerous emission lines and a strong LymanGLYPH<11> break in our NIRSpec spectroscopy, and we unambiguously measure the redshift to be z = 10 : 603. Our grating spectrum reveals LymanGLYPH<11> in emission, making it the first object at z > 9 with confirmed LymanGLYPH<11> emission. The rest-frame equivalent width is W 0 = 18 Å. The emission is o GLYPH<11> -set 555 km s GLYPH<0> 1 redward of the systemic redshift and spatially extended. These properties are consistent with models of LymanGLYPH<11> backscattering o GLYPH<11> the far side of galactic scale outflows. \nThe NIRSpec spectrum of GN-z11 is remarkably rich with emission lines, enabling us to study the ISM properties at z > 10. Based on the high [Ne iii ] / [O ii ] ratio we infer a high ionization parameter (log( U ) > GLYPH<0> 2 : 0). We report a significant detection of the very rarely-seen N iii ] GLYPH<21> 1748 line, which could suggest unusually high N / O ratios. While some high ionization lines are detected, the He ii GLYPH<21> 1640 and C iv GLYPH<21> 1550 lines, which are typically associated with photoionization due to AGN, are weak. Although we cannot conclusively rule our the contribution of an AGN, if this galaxy is indeed powered by star formation then the Balmer emission lines and UV continuum suggest a current star formation rate of GLYPH<24> 30 M GLYPH<12> yr GLYPH<0> 1 and low dust attenuation. \nWe have presented a very high signal-to-noise spectrum of a galaxy at z > 10, showing continuum and line emission, highlighting the power of our JADES observations to not only measure redshifts but to do detailed studies of the physical and chemical properties of galaxies formed within the first few hundred million years of the Big Bang. \nAcknowledgements. AJB, AS, AJC, GCJ, JC, and IEBW acknowledge funding from the "FirstGalaxies" Advanced Grant from the European Research \nCouncil (ERC) under the European Union\'s Horizon 2020 research and innovation programme (Grant agreement No. 789056). ECL acknowledges support of an STFC Webb Fellowship (ST / W001438 / 1). The Cosmic Dawn Center (DAWN) is funded by the Danish National Research Foundation under grant no.140. RS acknowledges support from a STFC Ernest Rutherford Fellowship (ST / S004831 / 1). RM, JW, MC, FDE, JS, TJL, LS, and WMB acknowledge support by the Science and Technology Facilities Council (STFC) and by the ERC through Advanced Grant 695671 "QUENCH". RM also acknowledges funding from a research professorship from the Royal Society. JW also acknowledges funding from the Fondation MERAC. This research is supported in part by the Australian Research Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project number CE170100013. FS, EE, DJE, BDJ, MR, BER, IS, and CNAW acknowledge a JWST / NIRCam contract to the University of Arizona NAS5-02015. DJE is also supported as a Simons Investigator. SC acknowledges support by European Union\'s HE ERC Starting Grant No. 101040227 - WINGS. SA, BRDP, and MP acknowledges support from the research project PID2021-127718NB-I00 of the Spanish Ministry of Science and Innovation / State Agency of Research (MICIN / AEI). HÜ gratefully acknowledges support by the Isaac Newton Trust and by the Kavli Foundation through a Newton-Kavli Junior Fellowship. Funding for this research was provided by the Johns Hopkins University, Institute for Data Intensive Engineering and Science (IDIES). RB acknowledges support from an STFC Ernest Rutherford Fellowship [grant number ST / T003596 / 1]. MP also acknowledges support from the Programa Atracción de Talento de la Comunidad de Madrid via grant 2018-T2 / TIC11715. LW acknowledges support from the National Science Foundation Graduate Research Fellowship under Grant No. DGE-2137419. DP acknowledges support by the Huo Family Foundation through a P.C. Ho PhD Studentship. This work is based [in part] on observations made with the NASA / ESA / CSA James Webb Space Telescope. 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We fit varying all stellar and nebular parameters (metallicity, ionization parameter, dust-to-gas mass ratio), employ a delayed exponential star formation history with recent 10 Myr of constant star formation that can vary independently. Finally, we model attenuation by dust using the Charlot & Fall (2000) two-component dust law. We also allow for a damping wing for the neutral intergalactic medium following the prescription described in Curtis-Lake et al. (2023), though we do not report the values here as they will be a GLYPH<11> ected by the Ly GLYPH<11> flux which is unresolved in the PRISM spectrum. The resulting spectral fit and derived parameters are shown in Fig. A.1 and reported in Table 2. The Gutkin et al. (2016) nebular models employ a relation between N / O and O / H abundances, which gives low N / O abundance at low metallicity. As such, our fit does not reproduce the rest-frame UV Nitrogen lines. We therefore mask them to prevent their presence a GLYPH<11> ecting the fit to the rest-frame UV continuum. We also mask the region around C iv , which shows an o GLYPH<11> set in the emission from the expected wavelength.', 'Appendix B: R GLYPH<24> 1000 grating 1D and 2D spectra': 'Figure B.1 shows the full NIRSpec medium-resolution grating spectra with detected emission lines marked.', 'Appendix C: Comparison of NIRSpec flux calibration to NIRCam imaging': "As a check on our spectroscopic flux calibration, we compare the GNz-11 fluxes derived from the NIRSpec spectroscopy with the NIRCam fluxes across several filters reported in Tacchella et al. (2023). Given the small spatial size of the source, and the narrow (0 00 : 2-wide) NIRSpec microshutters, we use the 0 00 : 2-diameter NIRCam aperture photometry (corrected to total magnitudes assuming a point source) reported in Tacchella et al. (2023) and our 3-pixel (0 00 : 3-wide) NIRSpec spectral extraction (again corrected for slit losses to approximate total flux assuming a point source at the location of GN-z11 with the microshutter). Each extracted spectrum was converted to photons per unit wavelength (since NIRCam detects photons, not energy), and the flux density integrated over the bandpass of each filter, weighting by the filter transmission curve. We then computed the brightness (in nJy) of a source with a spectrum uniform in f GLYPH<23> (i.e. flat in AB magnitudes) which would produce the same integrated flux, so as to compare with the quoted fluxes for NIRCam on the AB system (see Table C.1). \nAs can be seen from Figure C.1, the agreement in flux between the low-dispersion prism spectrum and the NIRCam photometry is excellent, with most filters agreeing with the spectral fluxes within the nominal error bars (the agreement is generally within 5%). The flux calibration of the grating spectra is less accurate. The grating spectra show significantly higher fluxes than NIRCam (or the prism spectrum) at 2 : 5 < GLYPH<21> < 4 GLYPH<22> m at the GLYPH<25> 15 GLYPH<0> 20% level, greater than the nominal uncertainties in the photon statistics, although at other wavelengths the agreement is \nTable C.1. Comparison of the NIRCam photometry of GN-z11 with the NIRSpec spectroscopy. \nThe NIRCam photometry of GN-z11 in a 0 00 : 2-diameter aperture (column 2) from Tacchella et al. (2023) is compared with that inferred from the spectroscopy in the low-dispersion prism (column 3) and medium-dispersion gratings (flux in column 4, and grating which overlaps that filter in column 5). The spectroscopic measurements use the 3-pixel extraction. \nbetter. The exact origin of this is unclear, but the flux calibration of the gratings may be less good, or there may be background subtraction issues a GLYPH<11> ecting the measured flux. \nFig. A.1. Triangle plot showing the 2D (1D on diagonal) posterior probability distributions for the derived stellar mass ( M , accounting for mass returned to the ISM through stellar winds and supernova explosions), star formation rate ( ), maximum age of the stars ( t ), e GLYPH<11> ective V -band attenuation optical depth ( ˆ GLYPH<28> v = AV = 1 : 086), ionization parameter (log U S), nebular metallicity ( Z neb), escape fraction of H-ionizing photons ( f esc) and ionizing photon production e GLYPH<14> ciency ( GLYPH<24> ion). The contours in the 2D posterior plots show the 1, 2 and 3 GLYPH<27> credible regions in light, medium and dark blue, respectively. The inset shows the resulting fit to the prism spectrum, with the spectrum and 1 GLYPH<27> standard errors shown as red line and shaded region respectively, and 1 GLYPH<27> range of fitted model spectra in blue. Regions that are masked in the spectrum are shown as fainter red (data) and fainter blue (model) shaded regions. \n<!-- image --> \nbrew install ilmbase \nFig. B.1. Full coverage of the 2D and 1D spectra from the medium resolution G140M (top), G235M (middle) and G395M (bottom) gratings. The main emission lines observed in the spectra have been marked. \n<!-- image --> \nFig. C.1. The NIRSpec low-dispersion prism spectrum of GN-z11 (3-pixel extraction), compared with the NIRCam photometry in di GLYPH<11> erent filters (red error bars, denoting the flux uncertainty and the wavelength span of the filter bandpass). The large diamond symbols denote the flux from the NIRSpec spectrum integrated over the filter response curve, with the small blue error bars within these diamond symbols. \n<!-- image --> \nFig. C.2. The NIRSpec medium-dispersion grating spectrum of GN-z11 (3-pixel extraction, smoothed in the spectral direction with a 11-pixel boxcar), compared with the NIRCam photometry in di GLYPH<11> erent filters (red error bars, denoting the flux uncertainty and the wavelength span of the filter bandpass). The light grey spectrum is the G235M, with the G140M and G395M grating spectra in black. The ' + ' symbols and blue error bars denote the flux from the NIRSpec spectrum integrated over the filer response curve. \n<!-- image -->"}

2024A&A...690A.386B

Context. Stellar activity is comprised of various phenomena mainly spots and faculae. It is one of the main sources of noise in exoplanetary observations because it affects both spectroscopic and photometric observations. In studying young active planetary systems we need to model the activity of the host stars to remove astrophysical noise from our observational data. Aims. We model the contribution of stellar spots in photometric observations. Through the use of multiband photometry we aim to extract the geometric properties of the spots and constrain their temperatures. Methods. We analysed multiband photometric observations acquired with the 80 cm Marcon telescope of the Osservatorio Polifunzionale del Chianti of V1298 Tau assuming the photometric modulation observed in different bands is attributed to cold spots. Results We constrained the effective temperature of the active regions present on the surface of V1298 Tau resulting from a combination of spots and faculae. We tested our hypothesis on solar data verifying that we successfully measured the size of the dominant active region and its averaged effective temperature.

2024-10-01T00:00:00Z

['10.48550/arXiv.2409.11034', '2024A&A...690A.386B', '10.1051/0004-6361/202450036', 'arXiv:2409.11034', '2024arXiv240911034B']

['techniques: photometric', 'Sun: activity', 'stars: activity', 'Astrophysics - Solar and Stellar Astrophysics', 'Astrophysics - Earth and Planetary Astrophysics']

Spot modelling through multiband photometry Analysis of V1298 Tau

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']

https://arxiv.org/pdf/2409.11034.pdf

{'Analysis of V1298 Tau': "Alfredo Biagini 12 , Antonino Petralia 1 , Claudia Di Maio 1 , Lorenzo Betti 34 , Emanuele Pace 34 and Giuseppina Micela 1 \n- 1 INAF - Osservatorio Astronomico di Palermo, Piazza del Parlamento 1, 90134 Palermo, Italy\n- 2 Università degli Studi di Palermo, Dipartimento di Fisica e Chimica, Via Archirafi 36, Palermo, Italy \ne-mail: \nalfredo.biagini@inaf.it \n- 3 Dipartimento di Fisica ed Astronomia, Università degli Studi di Firenze, Via Sansone, 150019 Sesto Fiorentino (FI), Italy\n- 4 Osservatorio Polifunzionale del Chianti, Barberino Val d'Elsa, Florence, Italy", 'ABSTRACT': 'Context. Stellar activity consists of di ff erent phenomena, mainly spots and faculae, and it is one of the main sources of noise in exoplanetary observations because it a ff ects both spectroscopic and photometric observations. If we want to study young active planetary systems we need to model the activity of the host stars in order to remove astrophysical noise from our observational data. Aims. We modelled the contribution of stellar spots in photometric observations. Through the use of multiband photometry, we aim to extract the geometric properties of the spots and constrain their temperature. \nMethods. We analyzed multiband photometric observations acquired with the 80 cm Marcon telescope of the Osservatorio Polifunzionale del Chianti of V1298 Tau, assuming that the photometric modulation observed in di ff erent bands should be due to cold spots. Results. We constrained the e ff ective temperature of the active regions present on the surface of V1298 Tau, which is composed by the contemporary presence of spots and faculae. We tested our hypothesis on solar data, verifying that we measure the size of the dominant active region and its averaged e ff ective temperature. \nKey words. Stars: activity - Sun: activity - Techniques: photometric', '1. Introduction': 'Stellar variability involves a variety of inhomogeneities on the stellar surface, produced by the magnetic activity of the interior of the star (Berdyugina 2005) that leads to a large variety of phenomena like stellar spots (Solanki 2003), faculae and flares (Ballerini et al. 2012) and their intrinsic evolution with a large variety of timescales: minutes for flares, days for the evolution of the active regions (faculae and spots) on the stellar surface, years for active regions cycles, billions of years for the evolution of the magnetic activity of the star (Berdyugina 2005). \nThe analysis of the stellar activity is crucial for understanding the stellar interior processes. Moreover, it is also fundamental for exoplanetary observations, since it is the main source of noise both for photometry and spectroscopy (e.g. Pont et al. 2008; Czesla et al. 2009; Sing et al. 2009, 2011b,a; Agol et al. 2010; Désert et al. 2011; Berta et al. 2011; Ballerini et al. 2012; Micela 2015; Scandariato & Micela 2015). \nStellar spots are regions with a lower temperature with respect to the quiet stellar surface because the local magnetic field configuration halts the convection process leading to a local cooling of the stellar plasma together with a change in local gravity and magnetic field (Berdyugina 2005; Ballerini et al. 2012). The presence of stellar spots, for instance, can influence both spectroscopic and photometric observations: it can distort spectral lines, thus hampering accurate radial velocity estimation (Pont et al. 2008; Carleo et al. 2020; Damasso et al. 2023) or a ff ect the photometric curves analysis by hiding or altering the signal of a transit in stellar light curves (Tsiaras et al. 2018). Furthermore, stellar activity could impact the extraction of the exoplanetary \ntransmission spectra(Changeat et al. 2020), hampering the detection of atmospheric molecules and atmospheric features like clouds (Pont et al. 2008; Ballerini et al. 2012; Micela 2015). \nThe modelling of the stellar inhomogeneities on the stellar surface could play a crucial role in the search and characterization of exoplanetary atmosphere for mission such as JWST (Greene et al. 2016) and for the incoming ARIEL mission (Atmospheric Remote-Sensing Infrared Exoplanet Large-survey, Tinetti et al. 2018), the M4-ESA mission dedicated to study the exoplanetary atmospheres. \nStellar activity is a very common phenomenon among solar-type stars and it is due to the e ff ects of the magnetic dynamo inside the star (Berdyugina 2005). Because stellar magnetic dynamos increase along with the stellar rotation (Pallavicini et al. 1981; Walter & Bowyer 1981; Pizzolato et al. 2003; Berdyugina 2005) and due to the typical age-rotation relation (Baliunas et al. 1995), we find that stellar activity is particularly strong for young stars (Baliunas et al. 1995; Berdyugina 2005). Despite this, in recent years, young stars have become targets of significant interest because they can provide valuable insights into the early stages of formation and evolution of exoplanetary systems (David et al. 2016; Carleo et al. 2020; Rizzuto et al. 2020; Plavchan et al. 2020; Damasso et al. 2023; Mantovan et al. 2024). \nIn recent years, ground- and space-based facilities have enabled us to discover thousands of exoplanets orbiting various types of stars, revealing a variety of these bodies with characteristics different from those of our own Solar System. \nIn order to determine the properties of young planetary systems and analyze their planetary atmospheres, it is fundamental to understand the stellar activity of the host stars to remove its e ff ects \nfrom the observational data or to study its impact on planets. In this paper, we focus our work on spots analysis. In particular, we study only the change in temperature in the spots, with the hypothesis that local gravity and magnetic field changes of the spot have much more little e ff ects on stellar observations. \nDue to their lower temperature, the spot emission curve di ff ers from that of the quiet photosphere, thereby altering the photometric signal, with an e ff ect that decreases along the wavelength observed, or distorting the spectral lines of the star and leading to incorrect radial velocity measurements (Berdyugina 2005). For instance, during a photometric observation, the presence of a spot occulted by a transiting planet can produce bumps in the observed light-curve. This occurs because the planet obscures a region with a lower flux compared to the unspotted surface, thereby increasing the observed total intensity coming from the star (Czesla et al. 2009; Sing et al. 2011b). \nOn the other hand, also an unocculted spot can a ff ect the measurement of the transit depth (Czesla et al. 2009) since it can affect both photometric and spectroscopic in-transit observations without clear signs of its presence, leading to incorrect estimations of the exoplanets parameters. In particular, the chromaticity of these e ff ects leads to inaccurate measures of the planetary radius at di ff erent wavelengths. The presence of a spot can also mimic a transit and, as a consequence, more than one observations are needed to confirm the presence of a transiting planet. \nIn order to remove the e ff ects of the spots on our data, we need to know their properties, positions and temperature. For this reason, in the following, we will develop a method to model the spot configuration in active stars based on a multiband photometry analysis. In this work, we used multiband photometric groundbased observations to model the spots of a very interesting young stellar object (YSO), V1298 Tau, that we will describe in Section 2. In Section 3 we will describe the methodology of our observations and in Section 4 our retrieval method, while in Section 5 we will discuss the validation of our model using solar data. Our final results will be discussed in Section 6.', '2. Stellar target: V1298 Tau': 'V1298 Tau is a K0-K1.5 young star ( ∼ 20 Myr) with a mass about 1 . 1M ⊙ located at ∼ 108 pc from Earth. It is an ultra-fast rotator with a rotational period of less than 3 days (David et al. 2019a). In Table 1 we show the properties of V1298 Tau from the literature. \nV1298 Tau is one of the youngest stars known to host more than one transiting planet (Suárez Mascareño et al. 2021), not a very common occurrence (Latham et al. 2011).The star and its planets have already been studied both through photometry (for example by TESS, Feinstein et al. 2022) and spectroscopic observations (through radial velocities, for example by HARPS-N, Suárez Mascareño et al. 2021). \nWe know also from previous studies, such as the spectral crosscorrelation (CCF) function analysis from Di Maio et al. (2023), that the star shows large spots at high latitudes ( > 60°) and smaller spots at low latitudes ( < 40°), but the relation between the spectral distortion due to stellar activity and the photometric flux variations of the star is not clear and the spectral analysis of the spots has been based on the hypothesis of not emitting spots (i.e. black spots, surface regions with temperature of 0 K) that is not physically correct. In this paper, we are interested in modelling the activity of the star due to spots on the stellar surface using multiband photometric data to better constrain their temperature and geometric parameters. \nTable 1. V1298 Tau properties (David et al. 2019a,b).Table 2. Scheme of our observing campaign of V1298 Tau. OPC is "Osservatorio Polifunzionale del Chianti".', '3. Observations and data analysis': "Using the photometric light curve to model the spots leads to the so-called inverse light curve inversion problem (Luger et al. 2021): we can't see directly the spots, so we have many geometrical degeneracies in our models. For example, it is di ffi cult to distinguish spots on each specific hemisphere of the star and we have a degeneracy between temperature, radius and latitude of the spots. For isolated observations, we have also a degeneracy between the latitude and longitude of the spots that can be broken by observing the star multiple times during its rotation. We aim to break these degeneracies by monitoring the star for a few days through di ff erent photometric bands at the same time. We chose to use multiband photometric observations because we want to take advantage of the chromaticity of the spot contribution to the stellar observation: we expect in fact to have a di ff erent ratio between spots flux and stellar flux at di ff erent wavelengths because they correspond in first approximation to two blackbodies with di ff erent temperatures. The di ff erence between the spots and stellar flux should decrease at longer wavelengths, as far as in IR band: for example, a star with T ⊙ and a spot with a filling factor of 1% and a ∆ T of 1250 K will present variations of ∆ FU FU = 9 × 10 -3 and ∆ FK FK = 3 × 10 -3 , respectively (Ballerini et al. 2012). \nThrough simultaneous multi-band photometric observations, we aim to estimate the temperature di ff erence between the stellar surface (whose temperature is known from literature) and the", '3.1. Observations campaigns': "We have organized multiband photometric observing campaigns from Osservatorio Polifunzionale del Chianti (OPC) with a Ritchey-Chretien telescope with the following characteristics: \n- -80 cm diameter\n- -f / 8\n- -field of view (FoV) ∼ 20'x20'\n- -Johnson filters B-V-R-I \nEach observing campaign spanned between 3 and 5 days, i.e. between 1 and slightly less than 2 rotational periods (about 2.8 days) of the star (see Table 2). The star's location near the ecliptic during the period from October to March poses challenges in coordinating observations for this target. Thus, in addition to the usual constraints arising from potential bad weather conditions, we also had to account for the lunar interference by avoiding days when the Moon passed near the target within a 60° range. Finally, to model the stellar photosphere, we avoided all the days with planetary transits for this target. In conclusion, we observed the star in three di ff erent periods: in February 2021, December 2021 and February 2022. We evaluated the best observing runs selecting only observed sets with SNR > 20. Finally, to avoid problems due to possible spot evolution we limited our analysis to maximum 3 days (see Section 4), obtaining these final databases: \n- -21-22-23, 22-23-24 and 23-24-25 (February 2021) with B, V and R bands\n- -12-13-14 and 13-14-15 (December 2021) with B, V and R bands\n- -11-12-13 (December 2021) with V and R bands\n- -21-22-23 (February 2022) with V, R and I bands. \nWe calibrated and analyzed the images using AstroImageJ (a software developed by TESS collaboration, Collins et al. 2017) and we applied the so-called di ff erential photometry method: we did not measure the absolute photometry of the target star, but its variability with respect to some carefully chosen stable check stars in the same field of view of V1298 Tau, used as reference. The choice of these check stars required a delicate compromise between the necessity of the highest number of them and the requirement of stability and comparable magnitude with respect to the target. In the FoV of the telescope (20' x 20') around the star there are only a few stars and most of them were saturated, very faint or variable, so in the end we found only 6 stable reference stars. It should be noted that we used the same reference stars for all the datasets. \nUsing this procedure we obtained the light-curve of V1298 Tau in the selected photometric bands shown in Figure 1. To retrieve the rotational period of the star, we employed a sinusoidal function and an o ff set to fit the data. This function feeds the nested sampling algorithm Multinest v3.10 (Feroz et al. 2009) to derive parameters best-fit values and errors throughout the python package PyMultiNest v2.12 (Buchner et al. 2014) and with the following likelihood: \nln p ( yn , tn , σ, θ ) = -1 / 2 X n h ( yn -F ( tn , θ )) 2 /σ 2 + ln 2 πσ 2 i \nTable 4. Priors on spot parameters of MultiNest for our retrieval procedure to model V1298 activity. Spot temperature's prior is set with a maximum temperature exceeding the unspotted surface one to include the possibility of retrieving a facula instead of a spot. \nTable 3. Flux variations amplitudes measured analyzing V1298 Tau data in di ff erent photometric bands, obtained fitting the data with a sinusoidal function and an o ff set. \n(1) \nwhere yn and tn are, respectively, data fluxes and times, σ = q σ 2 n + σ 2 j with σ n as data errors and σ j as white noise \njitter term, introduced to account for unknown source of errors. The algorithm was set to use 1000 live points, leaving all other parameters to their default values. \nThis analysis allowed to estimate the flux variations (see Table 3) of the star and verify the coherence in each observation. \nTherefore, from Figures 1 we observed variability in the amplitude of the sinusoidal fit, which on average tends to decrease at longer wavelengths, consistent with the assumption of a spotdominated star. The results of this analysis are summarized in Table 3.", '4. Forward Model and Retrieval': "We modelled the spot properties (latitude, longitude, radius and the temperature of the spots) following the approach of Cracchiolo et al. (2021a,b) and we imposed some assumptions: \nFig. 1. Data observed from OPC in B, V, R bands in February 2021 and December 2021 and in V, R and I bands in February 2022. The plotted line is an estimation of the flux variation of the star obtained by fitting the data by using a sinusoidal function with an o ff set. For the observations of February 2022 and for the first day of observation of December 2021, B band data were discarded because they had a signal-to-noise ratio (SNR) < 20. \n<!-- image --> \n- -Spots corotating with the stellar surface\n- -Not evolving spots during a rotational period of the star\n- -Approximation of circular spots projected on the stellar surface\n- -Same temperature for all the spots \nWith these hypotheses, we aimed to analyze flux variations of the star as generated by the luminosity contrast between the spots and the photosphere. Since the star has an inclination of nearly 90°, as we can see by combining the v sin i value of the star with its rotational period (see table 1), we had a degeneration in latitude because we could not distinguish the hemisphere in which the specific spot was located. We removed instead the degeneracy in longitude for the spots thanks to the rotation of the star, so we could retrieve their longitude from the time variations of the stellar flux. We chose to analyze datasets of 3 consecutive days (about a rotational period of the star) in order to avoid relevant contributions from spots evolution. In order to build our forward model for the retrieval procedure, we divided the star into concentric rings, with a finer division near the limb of the stellar \ndisk to better account for the e ff ect of limb darkening. We also took into account the limb-darkening e ff ect of the star with a linear approximation. We derived the limb darkening coe ffi cients for each possible temperature of the stellar surface (and spots) and di ff erent wavelengths using ExoTETHyS package (Morello et al. 2020). \nTo retrieve the spots properties from our data we used the nested sampling algorithm MultiNest whose priors are shown in Table 4, setting 3000 live points to achieve high precision in the retrieval. It should be noted that spot temperature's prior is set with a maximum temperature exceeding that estimated for the unspotted surface to include the possibility of retrieving a facula instead of a spot. \nIn order to fit the data, our forward model simulates the light curve of the star for each spot configuration tested, calculating at each temporal step the portion of the spotted and unspotted stellar surface visible from Earth. Using these informations it estimates the contributions to the total stellar flux of the spotted and of the unspotted visible surface of the star using the method \nFig. 2. Comparison of the retrieval posteriors of spots for V1298 Tau, obtained by OPC observations on 23-24-25 / 02 / 2021 through a separate analysis for data in B (top) and R (bottom) photometric bands. Blue lines (for right image) and red lines (for left image) mark median values, while black lines mark maximum probability (MAP) values for both images. \n<!-- image --> \ndeveloped in Cracchiolo et al. (2021b), then it rotates the star of a given time step and uses this loop to calculate the lightcurve of the star. To estimate the fluxes of spots and stellar surface in \nthe chosen range of temperatures (see Table 4) we used Phoenix models (Claret et al. 2012; Husser et al. 2013) and we applied the limb darkening coe ffi cients derived by ExoTETHyS (Morello \nFig. 3. Corner plot obtained by the retrieval procedure for the multiband photometric data acquired by OPC in the first run of observation, specifically on February 23-24-25th, 2021. σ i are the jitter parameters and γ i are the o ff sets of the forward model used in the retrieval for each band observed (B, V and R). Red lines mark median values, while green lines mark MAP values. \n<!-- image --> \net al. 2020). Finally, MultiNest calculates the fit and Bayesian evidence to estimate the goodness of the fit and in the end the parameters of our model.", '5.1. One band vs Multiband approach': 'Figure 2 shows some examples of results obtained from the analysis of single photometric band. It is clear that a degeneracy exists among the latitude of the spots, their radius, and their temperature, as we observe only the emission contrast between the spots and the stellar surface. These degeneracies lead to high uncertainties in spots parameters: in particular, we found a temperature error bar of several hundreds of K, while the latitude spans between 13° and 64° in the R band. By assuming that the stel- \nr spots and the photosphere emit as star surfaces at di ff erent temperatures, the emission contrast should change at di ff erent wavelengths, producing a di ff erent flux variation across di ff erent spectral bands. Consequently, through the analysis of simultaneous multiband photometric observations, we could be able to estimate the spots\' temperature, thereby breaking or at least reducing the degeneracy between the radius, temperature and latitude of the spots. Therefore, we performed a new retrieval procedure imposing for each 3 days sequence that all the bands share the same spot parameters, but di ff erent flux scales. We applied our model assuming the simultaneous presence of one, two, three, or four spots on the entire stellar surface. However, due to our temporal coverage and sampling, each model with more than one spot resulted in overfitting of our data, so we chose to model only one spot per dataset. Examples of the resulting corner plots are shown in Figure 3, 4 and 5 and the retrieved spot temperatures \nFig. 4. Same as for Figure 3, but in the case of B, V and R bands on December 13-14-15th, 2021. \n<!-- image --> \nare shown in Table 5. In comparison to the single band analysis, we found a more limited interval for spots temperature, with a decrease of the spot temperature uncertainty by a factor of two and a strong reduction of latitude uncertainties. The constraints on the spots parameters are typically wider when B filter data are not available, like in the observations of December 11-12-13rd, 2021 and February 21-22-23rd, 2022. Finally, we checked for the consistency of the spot positions and geometry along each observing run, plotting all the configurations of the spots with weights (of MultiNest algorithm) over the 0.85 quantiles. The above selection guarantees that only high likelihood values are selected and, consequently, shown. \nFor each observing run, we found a good compatibility of the radius and position of the retrieved spot, even if with some hints of evolution on a timescale similar to the rotational period of the star as we can see comparing the retrieved spots in di ff erent 3 days sequences: in each one we had imposed the absence of evolution for the spot, but we can see how the same \nspot changes between di ff erent sequences of days in the same observing run. An example is shown in Figure 6 for the best observing run (February 2021) results. The spot configurations here shown correspond to the same time of the star rotation as obtained using data by each 3 days-sequence of this observing run. Comparing our results with those of Di Maio et al. (2023), obtained through a CCF analysis of V1298 Tau\'s spectra with the hypothesis of non-emitting black spots, our method seems to be sensible only to bigger spots. Our retrieved spots\' projected filling factors range at their peak between 0.015 (February 2021) and 0.20 (December 2021 and February 2022), with respect to the distribution of filling factors between 0 and 0.15 found showed in Figure 13 from Di Maio et al. (2023). Our retrieved projected filling factors are higher than those retrieved through spot CCF, but still compatible. Moreover, our big spots are in the same range of latitude found for big spots using CCF. Our higher retrieved projected filling factors could be explained by our not-optimal time coverage that doesn\'t allow us to \nFig. 5. Same as for Figure 3, but in the case of V, R and I bands on February 21-22-23rd, 2022. \n<!-- image --> \ndistinguish more than one active areas on the star. The retrieved filling factor is also compatible with results from (Morris 2020), even if it is larger for the last set of data (February 2022). \nThe retrieved spot temperature intervals for each observing run are shown in Table 6. It is quite evident a decrease of the retrieved spots temperature between the first period of observations and the other two. A possible scenario is that in our approach we are measuring an e ff ective temperature of the inhomogeneities of the stellar surface, which are more complex than simple spots and what we retrieve is in fact the whole active region of the star with properties that result from the average e ff ect of little surface features (like spots and faculae). Then our e ff ective temperature should be sensitive, for example, to a change in the ratio between the filling factor of faculae and spots on the star. \n5.2. Interpretation of "spot temperature" \nIn order to verify the hypothesis that our analysis retrieves the properties of the complex dominant active region, we analyzed the spectroscopic activity indicators of V1298 Tau to get some clues about a possible change of the stellar activity. In particular, we analyzed HARPS-N (Cosentino et al. 2012) data acquired in a span of 4 years (2018-2023) using the method described in Di Maio et al. (2020), looking for temporal variability of activity indicators such as CaII H and CaII K (Figure 7) and H α (Figure 8). Vertical lines in the plots mark the periods of our observations. In Figure 8 we can see a linear trend for the measured intensity that leads this line from absorption to emission, while in Figure 7 we can see an increase in the scatter of CaII H and K line intensities along the time. These e ff ects point towards a change in activity between the first and the second period of observations, with an increase of the ratio between H α and CaII H + CaII K line intensities (Figure 9). Our photometric observations support \nTable 5. Spot temperature range estimates for each independent set of observations in the hypothesis of one spot. For the set marked with "*" we used only V and R data because B data had a low SNR, while for the same reason we used V, R and I data for the last set (marked with "**").Table 6. Spot temperature range estimates for each observing run in the 1 spot hypothesis. For observing runs longer than 3 days, we took the common range of the various 3 days-sequences of that run. \nthat the reason for this evolution could be a change in the ratio of stellar surface covered by spots and faculae.', '5.3. Validation of the model: solar case': "For further validation, we tested the model using Sun observations, where photometric data can be directly compared with spatially resolved active regions present on its surface. We used the multiband photometric data from VIRGO-SPM (Chaplin & Appourchaux 1999), the three channels sun photometer (SPM) of the experiment VIRGO (Fröhlich et al. 1997, Variability of solar IRradiance and Gravity Oscillations), an experiment of the space mission SOHO (Domingo et al. 1995, Solar and Heliospheric Observatory). VIRGO-SPM observed the Sun from April 11th 1996 until March 30th 2014 with an image every 60 seconds in three narrow bands at 402, 500, and 862 nm with a bandwidth of about 5 nm. \nIn particular, we focused on the rotational period starting from 12 / 30 / 2013, during which the Sun exhibited a higher level of activity. This was evident through the presence of two large active regions separated by approximately 180° in longitude. We analyzed the most active one fitting half period. Subsequently, we rebinned the data to obtain two data points per day (which is reasonable for observations of a long-period star like our Sun), and then applied our method to all three bands simultaneously, assuming one dominant spot. As prior for the retrieval model we imposed a spot radius between 0 and 0.25 and a spot temperature between 3800 and 5900 K to include also the possibility of \nFig. 6. V1298 Tau spots positions of the 3 observing sets of February 2021, given by the distribution of solutions of MultiNest retrieval. Each image shows the star as seen on the same day, February 23th 2021. The spots are shown both from a polar view (on the right) and from the equator, i.e. from Earth (on the left). \n<!-- image --> \nFig. 7. CaIIH + CaIIK values obtained by HARPS-N for V1298 Tau. The coloured lines correspond to our periods of observations: February 2021, December 2021 and February 2022. \n<!-- image --> \nretrieval of faculae instead of spots. The fit results for the first half-period analyzed are shown in Figure 10. \nFig. 8. H alpha values obtained by HARPS-N for V1298 Tau. The coloured lines correspond to our periods of observations: February 2021, December 2021 and February 2022. \n<!-- image --> \nFig. 9. Ratio of H α/ (CaIIH + CaIIK) values obtained by HARPS-N spectrometer for V1298 Tau. The coloured lines correspond to our periods of observations: February 2021, December 2021 and February 2022. \n<!-- image --> \nFigure 11 shows the comparison between the simulation of the retrieved solar active region (left panel) and the actual contemporary image of the Sun (right panel), with its real spots, taken from Helioseismic and Magnetic Imager (Schou et al. 2012, HMI), while the bottom panel of the figure shows how our model fit the observed solar data in the three photometric bands. The comparison between the two upper panels of Figure 11 shows that we were able to constrain with great precision the position of the active regions of the Sun breaking the degeneracy between their latitude and radius, while Figure 10 shows that the degeneracy between active region's latitude and temperature is also small and confined to a narrow parameters range. Moreover, we found that the retrieved radius encloses not only the main big spot but also all the smaller spots, penumbra areas, faculae and some of the photosphere inside and around them, leading to very little contrast between the retrieved active region's temperature and the photospheric temperature, confirming our initial hypothesis that our procedure retrieves the average temperature of the entire active region of the star.", '6. Summary and Conclusions': 'In conclusion, modelling multiband photometry and assuming a single spot-dominated surface, we measured an e ff ective temperature of the active regions of V1298 Tau which is probably a weighted average of the temperature of the spots, faculae and photosphere around them. Multiband photometry allows us to estimate this property, with the bluest photometric bands that provide the strongest constraint. We may reduce the degeneracy between radius, latitude and temperature of the active regions of the star, thanks to simultaneous multiband data, especially retrieving correctly its geometric properties, as verified on the solar case. More bands, especially bluer bands, and a denser temporal coverage, likely, would help in reducing the temperature uncertainties and in distinguishing spots and other features of the surface and better constraining their positions. Finally, we developed a method to characterize surface inhomogeneities of quite active and young stars that has the advantage of using little and medium-size telescopes for few days. Our results strongly suggest that the evolution of the e ff ective temperature of the active regions may reflect the evolution of stellar activity, indicating the needs for a monitoring of the star on various timescales. In conclusion, we suggest that this method, together with ground-based high-resolution spectroscopy and photometric observations from space, could improve our comprehension of active stars. \nAcknowledgements. The authors thank the anonymous reviewer for his / her very useful comments and suggestions. The authors acknowledge the support of Osservatorio Polifunzionale del Chianti for the acquisition of the data. \nThe authors acknowledge the support of the ASI-INAF agreement 2021-5-HH.0. Part of the research activities described in this paper were carried out with contribution of the Next Generation EU funds within the National Recovery and Resilience Plan (PNRR), Mission 4 - Education and Research, Component 2 - From Research to Business (M4C2), Investment Line 3.1 - Strengthening and creation of Research Infrastructures, Project IR0000034 - \'STILES - Strengthening the Italian Leadership in ELT and SKA\'. 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M., Lecavelier Des Etangs, A., et al. 2009, A&A, 505, 891 Sing, D. K., Pont, F., Aigrain, S., et al. 2011b, MNRAS, 416, 1443 Solanki, S. K. 2003, A&A Rev., 11, 153 Suárez Mascareño, A., Damasso, M., Lodieu, N., et al. 2021, Nature Astronomy, 6, 232 Tinetti, G., Drossart, P., Eccleston, P., et al. 2018, Experimental Astronomy, 46, 135\n- Tsiaras, A., Waldmann, I. P., Zingales, T., et al. 2018, AJ, 155, 156 \nWalter, F. M. & Bowyer, S. 1981, ApJ, 245, 671', 'Retrieved Sunspot': 'Fig. 11. Upper-left: simulated sunspot according to our model results, shown in Figure 10. Upper-right: a real image of the Sun during the period studied, from HMI. Bottom: best fit of our model compared to the observed data for the Sun in all the three bands, from left to right, B, V and R filter, shifted in time. \n<!-- image -->'}

2024PASA...41...88M

We present the Pilot Survey Phase 2 data release for the Widefield ASKAP Lband Legacy Allsky Blind surveY WALLABY carriedout using the Australian SKA Pathfinder ASKAP. We present 1760 H I detections with a default spatial resolution of 30SUPSUP from three pilot fields including the NGC 5044 and NGC 4808 groups as well as the Vela field covering a total of inlineformula texmath sim 180 texmath inlineformula deginlineformula texmath 2 texmath inlineformula of the sky and spanning a redshift up to inlineformula texmath z simeq 0.09 texmath inlineformula. This release also includes kinematic models for over 126 spatially resolved galaxies. The observed median rms noise in the image cubes is 1.7 mJy per 30SUPSUP beam and 18.5 kHz channel. This corresponds to a 5inlineformula texmath sigma texmath inlineformula H I column density sensitivity of inlineformula texmath sim 9.1times10191 z4 texmath inlineformula cminlineformula texmath 2 texmath inlineformula per 30SUPSUP beam and inlineformula texmath sim 20 texmath inlineformula km sinlineformula texmath 1 texmath inlineformula channel and a 5inlineformula texmath sigma texmath inlineformula H I mass sensitivity of inlineformula texmath sim 5.5times108 D100 texmath inlineformula Mpcinlineformula texmath 2 texmath inlineformula Minlineformula texmath odot texmath inlineformula for point sources. Furthermore we also present for the first time 12SUPSUP highresolution images cutouts and catalogues for a subsample of 80 sources from the Pilot Survey Phase 2 fields. While we are able to recover sources with lower signaltonoise ratio compared to sources in the Public Data Release 1 we do note that some data quality issues still persist notably flux discrepancies that are linked to the impact of side lobes associated with the dirty beams due to inadequate deconvolution. However in spite of these limitations the WALLABY Pilot Survey Phase 2 has already produced roughly a third of the number of HIPASS sources making this the largest spatially resolved H I sample from a single survey to date.

2024-11-01T00:00:00Z

['10.1017/pasa.2024.91', '10.48550/arXiv.2409.13130', '2024PASA...41...88M', 'arXiv:2409.13130', '2024arXiv240913130M']

['zgalaxies: evolution', 'galaxies: fundamental parameters', 'galaxies: ISM', 'galaxies: kinematics and dynamics', 'Astrophysics - Astrophysics of Galaxies']

WALLABY Pilot Survey Public data release of 1800 H I sources and highresolution cutouts from Pilot Survey Phase 2

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML']

https://arxiv.org/pdf/2409.13130.pdf

{'No Header': '<!-- image -->', 'WALLABY Pilot Survey: Public data release of ∼ 1800 H /I.sc sources and high-resolution cut-outs from Pilot Survey Phase 2': "C. Murugeshan, 1,2 N. Deg, 3 T. Westmeier, 4,5 A. X. Shen, 1 B. -Q. For, 4,5 K. Spekkens, 6,3 O. I. Wong, 1,4,5 L. Staveley-Smith, 4,5 B. Catinella, 4,5 K. Lee-Waddell, 4,1,7 H.Dénes, 8 J. Rhee, 4 L. Cortese, 4,5 S. Goliath, 9 R. Halloran, 10 J. M. van der Hulst, 11 P. Kamphuis, 22 B. S. Koribalski, 33,44 R. C. Kraan-Korteweg, 55 F. Lelli, 66 P. Venkataraman, 77 L. Verdes-Montenegro, 88 and N. Yu 99,101010 \n1 ATNF, CSIRO, Space and Astronomy, PO Box 1130, Bentley, WA 6102, Australia \n- 2 ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), Australia\n- 3 Department of Physics, Engineering Physics, and Astronomy, Queen's University, Kingston, ON, K7L 3N6, Canada\n- 4 International Centre for Radio Astronomy Research (ICRAR), The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia\n- 5 ARC Centre of Excellence for All Sky Astrophysics (ASTRO 3D), Australia\n- 6 Department of Physics and Space Science, Royal Military College of Canada, PO Box 17000, Station Forces, Kingston, Ontario, Canada, K7K 7B4 7 International Centre for Radio Astronomy Research (ICRAR), Curtin University, Bentley, WA 6102, Australia\n- 8 School of Physical Sciences and Nanotechnology, Yachay Tech University, Hacienda San José S/N, 100119, Urcuquí, Ecuador\n- 9 Canadian Astronomy Data Centre, NRC Herzberg, 5071 West Saanich Road, Victoria, British Columbia, Canada, V9E 2E7.\n- 10 Queens University, 99 University Ave, Kingston, ON, K7L3N6, Canada\n- 11 Kapteyn Astronomical Institute, P.O. Box 800, 9700AV Groningen, The Netherlands\n- 22 Ruhr University Bochum, Faculty of Physics and Astronomy, Astronomical Institute (AIRUB), 44780 Bochum, Germany\n- 33 Australia Telescope National Facility, CSIRO, Space and Astronomy, P.O. Box 76, Epping, NSW 1710, Australia\n- 44 School of Science, Western Sydney University, Locked Bag 1797, Penrith, NSW 2751, Australia\n- 55 Department of Astronomy, University of Cape Town, Private Bag X3, Rondebosch 7701, South Africa\n- 66 INAF, Arcetri Astrophysical Observatory, Largo E. Fermi 5, Florence 50125, Italy\n- 77 CIRADA, Dunlap Institute for Astronomy and Astrophysics, University of Toronto, Toronto, ON M5S 3H4, Canada\n- 88 Instituto de Astrofísica de Andalucía (CSIC), Spain\n- 99 National Astronomical Observatories, Chinese Academy of Sciences, 20A Datun Rd, Chaoyang District, Beijing 100101, China\n- 101010 Key Laboratory of Radio Astronomy and Technology, Chinese Academy of Sciences, 20A Datun Rd, Chaoyang District, Beijing 100101, China Author for correspondence: T. Westmeier, Email: tobias.westmeier@uwa.edu.au. \n(Received dd Mmm YYYY; revised dd Mmm YYYY; accepted dd Mmm YYYY; first published online 22 September 2020)", 'Abstract': 'We present the Pilot Survey Phase 2 data release for the Wide-field ASKAP L-band Legacy All-sky Blind surveY (WALLABY), carried-out using the Australian SKA Pathfinder (ASKAP). We present 1760 H /I.sc detections (with a default spatial resolution of 30 \'\' ) from three pilot fields including the NGC 5044 and NGC 4808 groups as well as the Vela field, covering a total of ∼ 180 deg 2 of the sky and spanning a redshift up to z ≃ 0.09. This release also includes kinematic models for over 126 spatially resolved galaxies. The observed median rms noise in the image cubes is 1.7 mJy per 30 \'\' beam and 18.5 kHz channel. This corresponds to a 5 σ H/I.sc column density sensitivity of ∼ 9.1 × 10 19 (1 + z ) 4 cm -2 per 30 \'\' beam and ∼ 20 km s -1 channel, and a 5 σ H/I.sc mass sensitivity of ∼ 5.5 × 10 8 ( D /100 Mpc) 2 M ⊙ for point sources. Furthermore, we also present for the first time 12 \'\' high-resolution images (\'cut-outs") and catalogues for a sub-sample of 80 sources from the Pilot Survey Phase 2 fields. While we are able to recover sources with lower signal-to-noise ratio compared to sources in the Public Data Release 1, we do note that some data quality issues still persist, notably, flux discrepancies that are linked to the impact of side lobes associated with the dirty beams due to inadequate deconvolution. However, in spite of these limitations, the WALLABY Pilot Survey Phase 2 has already produced roughly a third of the number of HIPASS sources, making this the largest spatially resolved H /I.sc sample from a single survey to date. \nKeywords: galaxies: evolution- galaxies: fundamental parameters- galaxies: ISM- galaxies: kinematics and dynamics', '1. Introduction': "The role of neutral hydrogen (H /I.sc) gas as the primary fuel for star formation in galaxies is now well-established. Several surveys utilizing single-dish (e.g., Meyer et al. 2004; Koribalski et al. 2004; Wong et al. 2006; Giovanelli et al. 2005; Catinella et al. 2010) and interferometric observations (e.g., van der Hulst et al. 2001; Verdes-Montenegro et al. 2005; Walter et al. 2008; Begum et al. 2008; Heald et al. 2011; Cappellari et al. 2011; Hunter et al. 2012; Ott et al. 2012; Koribalski et al. 2018), have shown the significance of the H /I.sc gas in understanding galaxy evolution. While significant progress has been made in \nstudying galaxy evolution through resolved H /I.sc observations, a thorough perspective of the H /I.sc gas distribution in galaxies, its statistical properties and its relation to star formation necessitates more resolved observations of tens of thousands of galaxies from unbiased surveys. \nThe Wide-field ASKAP L-band Legacy All-sky Blind surveY (WALLABY; Koribalski et al. 2020) is already contributing on this front and is expected to detect over ∼ 200, 000 sources out to a redshift of z ∼ 0.1 covering the majority of the southern sky using the Australian SKA Pathfinder (ASKAP; Hotan et al. 2021) telescope. This is almost a factor \nof 10 better than the number of sources detected in ALFALFA (Giovanelli et al. 2005; Haynes et al. 2018). In addition, WALLABY will be able to resolve tens of thousands of galaxies with a default resolution of 30 '' , while also producing higherresolution 12 '' 'cut-outs' for a select sub-sample of galaxies. The 12 '' data products will become part of regular full WALLABY survey data releases. The aim is to image all HIPASS sources ( N ∼ 5000) in high resolution, in addition, the WALLABY team is compiling a catalogue of galaxies selected based on their optical properties which we also intend to image at 12 '' resolution. As such, WALLABY will deliver 12 '' data products for thousands of galaxies in its first 5-year survey period. Some of the main goals that can be achieved with the higher resolution data include but are not limited to: \n- · Studying the H /I.sc morphology of galaxies at higher resolution and detailed kinematic studies of local galaxies by accurately modelling the H /I.sc distribution. In addition, the higher resolution also allows for complementary studies with IFU observations. This will also enable us to look for kinematical misalignment between the H /I.sc gas and/or the ionised gas and stars in galaxies (e.g., Wong et al. 2015; Bryant et al. 2019).\n- · Dynamical scaling laws of rotation supported galaxies and resolved angular momentum studies (e.g., McGaugh et al. 2000; Verheijen & Sancisi 2001; Lelli et al. 2017; Murugeshan et al. 2020; Kurapati et al. 2018; Mancera Piña et al. 2021; Sorgho et al. 2024) and tracing the effects of nonaxisymmetric potentials such as bars and bulges on the H/I.sc gas in galaxies (e.g., Masters et al. 2012; Murugeshan et al. 2023). With the higher resolution, we may also be able to trace warps in the discs of galaxies more accurately (Garcia-Ruiz et al., 2002).\n- · Probing the dynamics of galaxies using reliable and robust rotation curves derived from the higher resolution data (de Blok et al. 2008; Lelli et al. 2012). This will also enable us to probe the dark matter distribution in local galaxies and additionally address the core-cusp problem relating to dwarf galaxies (Katz et al., 2017).\n- · Studying the star formation properties and star formation laws pertaining to the high column density H /I.sc gas (N H/I.sc ≥ 10 20 cm -2 ). In addition, we may also be able to probe the H/I.sc gas and star formation properties of well resolved local dwarf galaxies (e.g., Roychowdhury et al. 2014; Bacchini et al. 2020). \nThese science cases highlight the need for high-resolution H/I.sc imaging of targeted (and potentially interesting) galaxies. As such, WALLABY will truly pave the way for highresolution H /I.sc studies of local galaxies to an unprecedented scale by imaging thousands of galaxies at 12 '' resolution. \nFor more specific details on the WALLABY survey we refer the reader to the original WALLABY paper (Koribalski et al., 2020). We summarise some important updated WALLABY survey parameters for the next 5-year period in Table 1. Prepilot and pilot surveys were conducted to assess ASKAP data quality and to plan full survey strategies. The targeted fields \nof the pre-pilot surveys are listed in the following WALLABY pre-pilot survey papers by For et al. (2021), Wong et al. (2021) and Murugeshan et al. (2021), while the details of the public data release of the Pilot Survey Phase 1 (hereafter Phase 1 or PDR1) observations are described in Westmeier et al. (2022) and Deg et al. (2022). \nTable 1: Important updated WALLABY survey parameters \nIn this paper we present the public data release of the H /I.sc catalogues and associated data products from the WALLABY Pilot Survey Phase 2 (hereafter also Phase 2 or PDR2) observations. Section 2 gives details of the targeted fields, observations, data reduction and briefly introduces the methods employed for the validation of the observations. In Section 3 we highlight the source finding strategy and provide specific notes for each target field. In Section 4, we present the general properties of the detected 30 '' sample. Section 5 introduces the high-resolution 12 '' data, the data reduction pipeline and characteristics of the sources. In Section 6 we describe an observed flux discrepancy in the WALLABY data and give details on the simulation studies undertaken to uncover the origins of this flux discrepancy. Section 7 describes the kinematic modelling pipeline and presents the kinematic models along with some comparisons between the 30 '' and 12 '' models. Finally, Section 8 provides details on how to access the data, while in Section 9 we provide a summary and the future goals of the WALLABY survey.", '2. Observations and data reduction': 'The data used in this work has been acquired via ASKAP observations of the WALLABY Pilot Survey Phase 2 fields NGC 4808, NGC 5044 and the Vela group. Located at the Inyarrimanha Ilgari Bundara, the Murchison Radio-astronomy Observatory (MRO), ASKAP (Hotan et al., 2021) is a stateof-the-art radio interferometer comprising of 36 12-meter antennas, equipped with Mk II phased array feeds (PAFs; DeBoer et al. 2009; Chippendale et al. 2010; Hotan et al. 2014). ASKAP is able to form 36 beams simultaneously on the sky using the advantage of the PAF, thus covering a very large area on the sky in a single pointing. For WALLABY, the 36 \nbeams are typically arranged in the form of 6 × 6 square footprints (see Figure 1). The simultaneous field of view (FOV) of ASKAP is ∼ 30 deg 2 at 1.4 GHz. For the WALLABY survey observations, two 6 × 6 square footprints (footprint A and B) are interleaved to attain the required uniform sensitivity across the field. A combination of both footprints A and B is referred to as a tile. \nThe observations of the various Phase 2 fields were carried out between April 2021 and May 2022 (for exact observing dates refer to Table 2) with an integration time of ∼ 8 h for each footprint and thus a total on-source time of ∼ 16 h per tile. During the observations, most of the 36 antennas were used to correlate the data, although a few antennas were flagged as bad during the data reduction process (for details, refer to Table. 2). \nWe note that the observations were carried out in the frequency range of 1152 - 1440 MHz, with a total bandwidth of 288 MHz, consisting of 15,552 channels corresponding to a spectral resolution of 18.5 kHz. As with Phase 1 observations, we note that only the upper half of the band above ∼ 1300 MHz has been processed as the observations below this frequency are severely affected by Radio Frequency Interference (RFI) due to Global Positioning System/Global Navigation Satellite System (GPS/GNSS).', '2.1 Field selection': 'For the Phase 2 observations, each ASKAP Science Survey Project (SSP) was allocated a total of 100 h of observing time. In Figure 1, we show the targeted pilot Phase 2 fields. The field selection was decided based on the following criteria: \nScientific merit - The Phase 2 fields were chosen on their merit, ensuring that multi-wavelength data is readily available, and in addition, have the potential to maximise the science goals, which include probing large scale structures in the zone of avoidance (ZOA) and investigating environmental effects on galaxy groups.', 'Commensality with other ASKAP Science Survey Teams': "- WALLABY is commensal with other ASKAP surveys such as the Evolutionary Map of the Universe (EMU; Norris et al. 2011) survey, Polarisation Sky Survey of the Universe's Magnetism (POSSUM; Gaensler et al. 2010), the Galactic ASKAP Survey (GASKAP; Dickey et al. 2013) and the Commensal Real-time ASKAP Fast Transients Survey (CRAFT; Macquart et al. 2010). The NGC 5044 tile 3 field was chosen to be the EMU-POSSUM-WALLABY three-way commensal field. While the Vela field was chosen to be commensal with GASKAP, wherein observations in the Galactic range ( Vsys < 500 km s -1 ) were reduced in 'zoom mode' with a full spectral resolution of 2 km s -1 . \nSource finding strategy - The NGC 5044 fields were targeted as they cover a contiguous region on the sky (see Figure 1). Observing overlapping fields/tiles was necessary so as to test our source finding strategy in preparation for the full \nsurvey, which will involve running the source finding pipeline on contiguous adjacent fields. Refer to Section 3.1 for more details on the source finding strategy.", "2.2 Default 30 '' WALLABY data reduction pipeline": "For a detailed description of the data reduction process of the default 30 '' WALLABY observations, we refer the reader to Westmeier et al. (2022). We describe very briefly the different stages of the default pipeline. We note that each of the steps below are performed independently for each ASKAP beam before they are mosaicked to form the final image cubes. First, the pipeline runs an automated flagging procedure which identifies bad antennas and flags the bad data for each beam. After the flagging procedure the pipeline proceeds to perform the bandpass calibration, followed by imaging the continuum. Then using the component and sky models derived from the continuum imaging, continuum subtraction is performed in the UV-domain. The next steps involve imaging each ASKAP beam separately, which also includes the deconvolution step, where the data is cleaned to a peak residual flux density of 3.5 mJy, followed by a deeper cleaning (within the pixels corresponding to the identified clean components) to a residual peak flux density threshold of 0.5 mJy. This is then followed by restoring the clean components convolved with a 30 '' Gaussian beam and adding back the residuals to form the image cubes. After the restoring phase, an image-based continuum subtraction routine is performed. A primary beam correction is then performed after which all the beams are mosaicked together to form two footprint (A and B) image cubes, which are then mosaicked to form the final full sensitivity image cube. We note that the main change in the data reduction pipeline for Phase 2 is the use of holographic measurements of the actual ASKAP primary beams used for the observations (Hotan, 2016) for the primary beam correction, as opposed to the static Gaussian primary beam correction that was used for the Phase 1 data reduction. The introduction of the holography models for the correction provides more accurate primary beam model weights leading to more accurate flux recovery from detections across the entire FOV compared to the flux based on the static Gaussian primary beam model.", '2.3 Data quality assessment and validation': "RFI and antenna flagging are performed on a beam-bybeam basis. The overall flagged visibility fraction ranges from 5 to 30% across all beams, and typically all 36 antennas were utilised for all beams. \nWe evaluate the data quality of each footprint image cube based on a set of metrics. These metrics were established based on the data in the WALLABY early science field of M83 and pre-pilot field of Eridanus (see For et al. 2019; For et al. 2021), which include RMS, minimum and maximum flux densities, 1 percentile noise level and median absolute deviation of median flux (MADMF). Each set consists of values for three types of image cubes, i.e., before and after continuum subtraction image cubes as well as a residual cube. \nThebroadband RFI/artefacts are evaluated with the MADMF \n4 \n<!-- image --> \n<!-- image --> \nFigure 1: The ASKAP footprints covering the Pilot Phase 2 fields overlaid on top of their PanSTARRS composite optical images. The green points show the location of the HIPASS sources imaged with a 12 '' resolution for the high-resolution cut-outs. \n<!-- image --> \nTable 2: Details of the observations. Col (1): Name of the field; Col (2): tile/footprint; Col (3): ASKAP Scheduling block identifier (SBID) used to tag the data in CASDA; Col (4): Date of observation; Col (5) - (6): RA and Dec of the centre of the footprint, respectively, in J2000; Col (7): Phase rotation of the footprint on the sky in deg; Col (8): Number of antennas used; Col (9): Flagged fraction. a EMU-POSSUM-WALLABY commensal field; b GASKAP-WALLABY commensal field. \nstatistic. This metric is sensitive to strong artefacts. The distribution of flux density values for all voxels in each beam at the 1 percentile level indicates any bandpass calibration and/or sidelobe issues. All these metrics and observation information are presented in a HTML style summary report for each footprint. The report of each footprint and description of each metric is available at the CSIRO ASKAP Science Data Archive (CASDA; Huynh et al. 2020). \nFollowing this, a quality checking pipeline verifies that the footprints in CASDA meet the data quality requirements as mentioned above. The pipeline is executed when a new observation is available on CASDA. We run the Source Finding Application (SoFiA; Serra et al. 2015; Westmeier et al. 2021) described in detail in Section 3 on the mosaicked image cubes to generate moment 0 images of the field. Then, we verify by eye that there are no significant artefacts in the source finding output. Footprints that show significant artefacts from the source finding run are rejected by the team, and marked to be re-observed. Accepted footprints are recorded in a database (for more details see Appendix 3), which is used by the main source finding pipeline.", '3. Source Finding and parametrisation': "Source finding on the final image cubes was performed using the Source Finding Application (SoFiA; Serra et al. 2015; Westmeier et al. 2021) version 2.3.1. For this purpose, each tile was split into sub-regions of approximately 1500 × 1500 spatial pixels and 1400 spectral channels for parallel processing on multiple nodes of the Nimbus computing cluster at the Pawsey Supercomputing Centre in Perth. In total, the frequency range of 1305 - 1418 MHz, corresponding to a recession velocity range of 500 ≲ c z ≲ 26, 500 km s -1 , was searched for H /I.sc emission. \nEach sub-region was first multiplied by the square root of the associated weights cube to normalise the noise across the data cube. This was followed by automatic flagging of artefacts and the positions of radio continuum sources with flux densities > 150 mJy in the Rapid ASKAP Continuum Survey (RACS; McConnell et al. 2020) catalogue. A circle with a radius of 5 pixels (or 30 '' ) was flagged around the position of each such continuum source, flagging the entire frequency range (including any H /I.sc emission) within those pixels, creating a circular hole in the affected area. If an H /I.sc detection is affected by flagging, the flag parameter in the catalogue is set accordingly to alert users of the fact that the detection is adjacent to flagged pixels. Additional noise normalisation in a running window of size 51 × 51 spatial pixels and 51 spectral channels was carried out to normalise any remaining noise variation that was not accurately reflected by the weights cube. In addition, a robust first-order polynomial was fitted to each spectrum to remove any remaining low-level continuum residuals, and SoFiA's ripple filter was employed to remove any low-level bandpass ripples due to RFI. \nAfter these preconditioning steps, SoFiA's 'smooth and clip' (S+C) algorithm was used to detect emission above a threshold of 3.8 times the noise level in each smoothing iteration. \n5 \nFigure 2: Strategy for source finding in the NGC 5044 field which has overlapping regions. Tiles are shown as blue shaded regions while each orange box corresponds to a central ∼ 4 · × 4 · area, where the source finding is performed. For the NGC 5044 field central regions are processed when both footprints have been observed, and overlapping regions are processed when adjacent tiles are completed. The light green boxes represent ∼ 4 · × 4 · areas where source finding is run when appropriate adjacent tiles are available (or become available in the future). \n<!-- image --> \nSmoothing kernels of 0, 5 and 10 spatial pixels and 0, 3, 7, 15 and 31 spectral channels were employed to boost the signal-tonoise ratio of faint, extended H /I.sc emission on spatial scales of up to 1 arcmin and velocity widths of up to about 120 km s -1 (at z ≈ 0). All detected pixels were then merged into coherent detections across a merging length of 2 spatial pixels and 3 spectral channels, with detections smaller than 5 pixels or channels discarded. Next, SoFiA's reliability module was used to discard all detections with a reliability of less than 0.7 or an integrated signal-to-noise ratio of less than 3. In addition, detections with a total of less than 300 spatial and spectral pixels were also discarded as unreliable. \nThe remaining detections were then parameterised before SoFiA generated the final source catalogue and output products, including cubelets, moment maps and integrated spectra for all detections. Table 5 in Appendix 2 lists some important SoFiA parameter values used for the 30 '' source finding runs.", '3.1 Source finding strategy': 'A pipeline has been developed through the Australian SKA Regional Centre (AusSRC) to run the source finding for the WALLABY survey. The pipeline communicates with external databases such as CASDA and the WALLABY database to automatically check for new footprints (and tiles) that have been uploaded on to CASDA. When a new observing tile has been deposited in CASDA, the pipeline mosaics overlapping regions of adjacent tiles outside the central 4 · × 4 · (orange boxes in Figure 2) and executes the source finding application. \nThe Vela and NGC 4808 fields were covered by only a sin- \ngle ASKAP tile each, and hence source finding was performed on the full tile at once. For the four-tile NGC 5044 mosaic we instead employed a staged source finding approach to account for the gradual completion and release of observations for this field. Figure 2 gives a visual representation of the source finding strategy adopted for the NGC 5044 field. First, SoFiA was run on the central 4 · × 4 · region (orange boxes) of each individual NGC 5044 tile. This central region corresponds to the area across which the noise level is roughly constant in an individual tile. Beyond the central 4 · × 4 · region, the noise in the outskirts of the tile typically tend to increase by a factor of two or more (see Appendix A in Westmeier et al. 2022). Once adjacent tiles became available, SoFiA was then additionally run on the overlapping regions between those tiles in steps of 4 · × 4 · regions (green boxes), to gradually build up a source catalogue of the entire NGC 5044 field. This staged source finding approach will also be applied to the full WALLABY survey in the future. The NGC 5044 mosaic provided us with the opportunity to develop and test this approach in anticipation of the full survey observations. \nDetections from the source finding pipeline are uploaded into a database and WALLABY sources are then manually accepted following visual inspection. For more details on the manual workflow refer to Appendix 3.', '3.2 Notes on individual fields': "In this section we present some pertinent notes on the individual fields. We note that while due care has been taken to avoid artefacts and false positives in the final source catalogues through visual inspection of all raw detections from SoFiA, we caution that some false positives may still remain in the final source catalogue as well as the issue of blended sources and/or sources broken into separate detections. This is true for all three Phase 2 fields. Where possible, comments are made in the source catalogue highlighting such issues. In addition, we also added 'multiplet' and 'component' tags to mark such cases.", '3.2.1 NGC 5044': 'The data quality for the NGC 5044 tiles is good overall with only a few artefacts still present in the final mosaicked image cube. This release consists of the source finding detections from the four NGC 5044 tiles covering 4 × 30 ∼ 120 deg 2 and spanning a velocity range of cz ∼ 500 - 26, 500 km s -1 ( z < 0.089) using the full RFI-free higher frequency band available for WALLABY. Some artefacts still remain in the data cube particularly related to faint continuum residuals and sidelobes that have affected the northern edge of tile 1, the southern edge of tile 2 and a small region of tile 4 of the NGC 5044 mosaic. We note that this may have reduced the completeness of the source finding runs in the affected regions. \nAfter the source finding run, all detections were visually inspected and obvious artefacts were removed following which 1326 detections remain. We note that the NGC 5044 tile 4 was the only Phase 2 tile for which a Gaussian primary beam model was used for primary beam correction instead of using a holography model, due to which we anticipate minor \nflux-related issues such a potential increase in flux by about ∼ 15 - 20% for sources that lie further away from the beam centre and/or close to the edge of the tile/footprint. For tiles 1, 2 and 3 of the NGC 5044 field, the holography-based primary beam correction was performed.', '3.2.2 NGC 4808': 'The data release for the NGC 4808 field covers 30 deg 2 of the sky with a velocity range of cz ∼ 500 - 26, 500 km s -1 ( z < 0.089). There were no major issues identified with the NGC 4808 field and the data quality is overall good, with very few artefacts in the image cube. The source finding run resulted in the retention of 231 detections following removal of few faint artefacts.', '3.2.3 Vela': 'The Vela field covers 30 deg 2 with a redshift range of cz ∼ 500 - 25, 400 km s -1 ( z < 0.085). As mentioned in Section 2.1, this field was observed commensally with the GASKAP-H/I.sc project in spectral zoom mode and processed at the full spectral channel width of 9.26 kHz. After this, the extragalactic frequency range of the data was re-binned to the default WALLABY spectral resolution of 18.5 kHz prior to spectral imaging. However, due to flagging preceding binning, some faint RFI from global navigational satellite systems was not fully flagged in the higher spectral resolution data, which has resulted in a significant number of false detections at frequencies of ν ≃ 1380 MHz and ν ≃ 1310 MHz. This therefore has resulted in the reliability of detections at those frequencies to be reduced which may have resulted in some genuine H/I.sc sources being omitted by SoFiA. Overall, 203 detections are retained after visual inspection and removing artefacts and false positives.', '4. Source characterization': "In this section we highlight some characteristics of the source properties from the Phase 2 source finding runs, such as the distribution of the signal-to-noise-ratio (SNR) of the detected sources, size distribution, H /I.sc mass distribution as well as their H /I.sc mass - distance plot. We also compare the Phase 2 source properties with the Phase 1 detections in order to highlight the significant improvement in the data quality. \nPanel a) in Figure 3 shows the distribution of the barycentric redshift for the Phase 2 detections (in blue) compared to the redshift distribution of sources in Phase 1. We find that the median redshift of the sources in Phase 2 is ∼ 0.027 ( cz ∼ 8094 km s -1 ). The median redshift of sources in the NGC 5044 field is ∼ 0.025 ( cz ∼ 7495 km s -1 ), the NGC 4808 field is ∼ 0.039 ( cz ∼ 11692 km s -1 ) and the Vela field is ∼ 0.04 ( cz ∼ 11992 km s -1 ). We see the clumping in redshifts in two distinct peaks in Figure 3. In comparison, the Phase 1 sources were mainly from nearby groups and clusters and as such, show a median barycentric redshift of ∼ 0.014 ( cz ∼ 4197 km s -1 ). \nIn panel b) of Figure 3 we show the SNR (defined as the ratio of the integrated flux to the uncertainty in the integrated \nPublications of the Astronomical Society of Australia \n7 \nFigure 3: a) Distribution of the barycentric redshifts of the Phase 2 sources (blue) compared to the Phase 1 detections (orange). b) Histogram of the Signal-to-noise (SNR) for both the Phase 2 and Phase 1 detections. c) Local noise distribution in the images cubes for the Phase 2 and Phase 1 detections. d) Distribution of the w 20 H/I.sc line-width distribution. e) Histogram of the major axis size (in units of 30 '' beams) for the two samples. f ) The H/I.sc mass distribution for the Phase 2 and Phase 1 samples. In all plots, the dashed and dotted black lines represents the median value of the distribution for the Phase 2 and Phase 1 detections, respectively. \n<!-- image --> \n/circledot \nflux measured by SoFiA) of the detected sources for both the Phase 1 and Phase 2 fields. As reported in Westmeier et al. (2022), the peak of the SNR for the Phase 1 data is ∼ 9 (with median ∼ 11), while the peak of the SNR distribution for the Phase 2 detections is ∼ 6 (with median ∼ 7). This significant improvement in detecting low SNR sources in Phase 2 from the source finding runs can mainly be attributed to the following reasons - a) the overall data quality of the Phase 2 observations has improved significantly compared to Phase 1 data mainly because the fields targeted in Phase 2 were chosen specifically to avoid continuum sources brighter than 2 Jy. This leads to better data quality with fewer continuumrelated artefacts, leading to the source finding runs being more complete out to low SNR; b) the on-dish calibrators were switched off for Phase 2, as they had caused a lot of RFI in the Phase 1 data, particularly in the corner beams; c) the SoFiA settings were fine-tuned based on the experience participating in the SKA Science Data Challenge 2 (Hartley et al., 2023), which has also contributed to a higher completeness of the catalogue in Phase 2. \nPanel c) in Figure 3 shows the distribution of the local rms noise in the image cubes for both Phase 1 and 2 sources. The median rms in the image cubes for Phase 2 is ∼ 1.7 mJy per 30 '' beam and 18.5 kHz ( ∼ 4 km s -1 ) channel width, which is close to the expected theoretical rms noise in the image cube for WALLABY (Koribalski et al., 2020). This translates to a 5 σ H/I.sc column density (N H/I.sc ) sensitivity of ∼ 9.1 × 10 19 (1 + z ) 4 \ncm -2 per 30 '' beam and ∼ 20 km s -1 channel, and a 5 σ H/I.sc mass sensitivity of ∼ 5.5 × 10 8 ( D /100 Mpc) 2 M ⊙ for point sources, where D is the Hubble distance to the source. \nIn terms of the line width of the detections in Phase 2, we show the distribution of the w 20 line-widths (defined as the spectral width corresponding to 20% of the peak flux in the integrated spectrum) for both the Phase 1 and Phase 2 samples in panel d) in Figure 3. The median w 20 value for both the samples is ∼ 170 km s -1 . \nPanel e) in Figure 3 shows the distribution of the major axis size of the ellipse fit to the moment 0 map of the detections. It can be seen that the median size of sources detected in Phase 2 is ∼ 1.3 beams, at the nominal 30 '' resolution, compared to a median value of ∼ 1.6 for the Phase 1 detections. This means that WALLABY has managed to detect a larger number of marginally resolved galaxies in Phase 2, primarily because the median redshift of Phase 2 detections is a factor of two higher than the median value for Phase 1 observations. \nPanel f ) in Figure 3 shows the distribution of the H /I.sc mass for all pilot Phase 1 and 2 detections. The H /I.sc mass is computed using equation 7 in the PDR1 paper (Westmeier et al., 2022). We observe that the Phase 2 detections have a median H /I.sc mass of log 10 ( M H/I.sc /M ⊙ ) ∼ 9.6 which is consistent with the median H/I.sc mass value of log 10 ( M H/I.sc /M ⊙ ) ∼ 9.5 for the pilot Phase 1 detections. The phase 2 median H /I.sc mass is slightly higher than the Phase 1 median H /I.sc mass, which is expected from the higher median redshift of the Phase 2 sample. We note that we make \nuse of the Hubble distance, D = vH 0 , of the sources to estimate their H /I.sc mass. Where v is the measured barycentric velocity and H 0 = 70 km s -1 Mpc -1 is the Hubble constant. We caution that this distance is only an approximation and will be prone to large errors of up to ∼ 20% due to effects of peculiar velocities in the local Universe, as well as systematic errors from using barycentric redshifts (Strauss & Willick 1995; Willick et al. 1997). We have used the Hubble distances for this release to remain consistent with the distance estimates used in Phase 1. However, going forward, for the full survey, the WALLABY team plans to apply more sophisticated flow models and correct the redshifts appropriately before measuring derived quantities such as distances and H /I.sc masses. \nFigure 4 shows the distribution of the H /I.sc mass of the detections from both pilot Phase 1 (grey points) and Phase 2 (color-coded by the different fields) as a function of their measured Hubble distance ( D = vH 0 ). Also plotted is the 5 σ H/I.sc mass detection threshold (dashed black line) measured across a 1 MHz frequency bandwidth and assuming the median local RMS noise level of ∼ 1.71 mJy in the image cubes derived from the SoFiA runs. The 5 σ H/I.sc mass detection threshold is computed as follows: \nM H/I.sc (5 σ ) M ⊙ = 5 × 49.7 × ( σ Jy Hz ) √ ∆ν / d ν ( D Mpc ) 2 (1) \nWhere, σ = 1.71 × 10 3 Jy Hz and ∆ν = 1000 kHz is the 1 MHz channel width and d ν = 18.5 kHz is the default spectral resolution. We see that our completeness at 5 σ is close to zero in accordance with Figure 5 in Section 4.1. As with the Phase 1 detections, we find large-scale clustering at various distances corresponding to the different groups detected in the Phase 2 fields. For example, for the NGC 4808 field, we find galaxies clustered at ∼ 30 Mpc, ∼ 100 Mpc and another over-density close to ∼ 200 Mpc. Similarly, for the Vela field, we find an over-density of galaxies corresponding to a distance of ∼ 50 Mpc, at ∼ 180 Mpc and another at ∼ 260 Mpc. The overdensity at ∼ 260 Mpc in Vela field is particularly interesting as it lies in the Zone of Avoidance (ZOA) and as such there are limited redshifts. However, a few previous optical studies (e.g., Hudson et al. 2004; Hoffman et al. 2015) have hinted at the existence of a large over-density corresponding to a systemic velocity of ∼ 18000 km s -1 (roughly a distance of 260 Mpc). This was later confirmed by Kraan-Korteweg et al. (2017), who measured the spectra from ∼ 4500 galaxies to map the composition and structure of the over-density. Studying and understanding this large-scale structure will add immensely to our knowledge of modelling bulk flows in the local Universe, as well as mapping the large-scale structures in the ZOA.", '4.1 Completeness': 'In order to estimate the completeness of the source catalogue, we plot in Figure 5 the number of sources, N , as a function of integrated signal-to-noise ratio (SNR) in doublelogarithmic space. As before the SNR is defined here as the \nFigure 4: The H /I.sc mass plotted against the estimated Hubble distance for the combined Pilot Phase 2 sample. The orange circles represent the NGC 5044 field, green triangles the NGC 4808 field and the purple squares the Vela field. The grey circles in the background represent the Phase 1 detections. The dashed black line represents the 5 σ H/I.sc mass threshold as a function of distance, assuming a 1 MHz frequency band width. \n<!-- image --> \nFigure 5: Histogram of the number of detected sources, N , as a function of integrated signal-to-noise ratio, SNR, in doublelogarithmic space in bins of ∆ log 10 (SNR) = 0.025 (black data points). The error bars correspond to √ N . The red, dashed line shows the result of a linear fit in the range of 0.9 < log 10 (SNR) < 1.4. The resulting completeness, defined as the observed source count divided by the fit, is shown as the green, solid curve at SNR ≲ 7 where incompleteness effects are evident. \n<!-- image --> \nratio of the integrated flux and the statistical uncertainty of the integrated flux measurement within the source mask produced by SoFiA. As expected from an untargeted survey, the source count follows an almost perfect power-law with a turnover at SNR ≲ 7. Under the assumption that the intrinsic population continues to follow a power law at low SNR and that the turnover therefore is entirely caused by incompleteness, we can estimate completeness as a function of SNR. We do this by fitting a straight line to the data points in the range of 0.9 < log 10 (SNR) < 1.4 (red, dashed line) which yields a slope, and hence power-law exponent, of -2.54. The completeness of our source catalogue as a function of SNR can then be estimated by dividing the number of detected sources in each bin by the expected number of sources predicted by the power-law fit. \nThe resulting completeness curve is shown as the green, solid line in Figure 5. We reach 100% completeness at SNR ≈ 7 beyond which we do not plot the actual completeness curve any more, as it would eventually show a large scatter around 1 due to stochastic errors as a result of low source counts at high SNR. 50% completeness is reached at SNR ≈ 5.5 below which our completeness rapidly declines to near zero at SNR ≈ 4.', "5. High-resolution 12 '' cut-outs": "One of the objectives of the WALLABY survey is to generate high-resolution (12 '' ) cut-outs for a sub-sample of galaxies. We use the calibrated visibility data derived from the default ASKAP spectral-line processing pipeline (Guzman et al. 2019; Whiting 2020) to image a sub-sample of galaxies at high angular resolution. As mentioned earlier the default spatial resolution of the WALLABY survey is 30 '' , which was determined to be the optimal resolution that gives a good compromise between resolution, sensitivity and computational resources required to process large volumes of data. In contrast, the computational resources required to image the data in the full 12 '' resolution will be significantly higher due to the additional baselines and increasing image sizes. However, it is still possible to image a sub-sample of the WALLABY detections in high-resolution by limiting the bandwidth to be imaged to a few hundred channels and only encompassing the velocity range of the target galaxies. This way, we drastically reduce the computing and storage requirements to process the data. We tested this functionality in preparation for the full WALLABY survey in Phase 2. \nFor Phase 2, we selected all HIPASS sources from the three fields. We targeted HIPASS sources, as these are likely to be detected in the WALLABY data and also as they are well resolved (tens of 12 '' beams across the major axis). We note that for the full WALLABY survey, apart from the HIPASS targets, some optically-selected target galaxies are also expected to be included. We emphasise here that since the target galaxies for the high-resolution cut-outs are HIPASS galaxies and therefore H/I.sc-selected, this will naturally introduce biases in the sample, which the users need to consider and account for while using the data for their analysis. \nTo perform the high-resolution imaging making use of \nthe full visibility including the longest baselines, we split out individual ASKAP primary beams containing (and surrounding) our target sources. We split out 250 channels ( ∼ 4.6 MHz) encompassing the velocity range of the source. For the WALLABY channel width of ∼ 4 km s -1 , this translates to a total velocity range of ∼ 1000 km s -1 , which is sufficient to contain the emission from even the most massive and rotationallydominated galaxies. We split out only 250 channels mainly to bring down the storage and processing costs required for each source. We split out up to 3 PAF beams from each footprint for each source, i.e., up to a total of 6 beams for a single source from both footprints. Each calibrated visibility data set of 250 channels for each beam is ∼ 15GB in size, therefore the total storage cost for each source for 6 beams is ∼ 90GB. The splitout visibilities are then uploaded on to CASDA. The splitting of the visibilities described above is performed automatically whenever a new field has been observed and processed. \nThe relevant visibilities for each source are then downloaded from CASDA and used to make the high-resolution image cubes using the 'high-resolution' imaging pipeline (hereafter high-res pipeline). All data have been reduced on Pawsey Supercomputing Facility's dedicated High Performance Computing clusters. We make use of ASKAPSoft to process and image the cut-outs. We now describe the various stages of the high-res pipeline. The pipeline is a Python script that reads in a catalogue of sources that need to be imaged, and a user-defined configuration file containing essential information such as the location of the split-out calibrated visibility, holography and footprint data. The main Python pipeline job then creates all the necessary bash scripts such as the parsets and the corresponding slurm job submission scripts for each task (e.g., imager, imcontsub etc). These jobs for the various tasks are then submitted as dependencies for each beam for each individual source in a parallel framework. \nThe imaging is first carried out beam-by-beam and then all beams are mosaicked to produce the final image cube for the individual sources. The first step is to image the visibilities for each beam using the cimager task in ASKAPSoft. We make an image of size 384 × 384 pixels, with a pixel size of 2 '' . We use a Wiener filter with a robust parameter value set to 0.5 and apply a Gaussian taper of 12 '' to achieve a synthesized beam close to 12 '' . The spectral resolution is kept at 18.5 kHz ( ∼ 4 km s -1 ). In addition, the deconvolution process is also performed within the task cimager . For more details on the ASKAPSoft parameters used for the imaging, refer to Table 4 in Appendix 1. The imaging step is then followed by the image-based continuum subtraction using the task imcontsub . The pipeline then performs the primary beam correction using the holography model with the task linmos . These steps are performed for each of the 6 beams that encompass the target HIPASS source. As the final step, all 6 beams are mosaicked to form the final 'mosaicked' cube for the source. This is again performed using the mosaicking task linmos . The above workflow is adopted for all sources and a number of jobs are submitted on the cluster to simultaneously image the data for multiple sources at any given time. We now present an overview of the cut-outs sample, and give \ndetails of the quality of the data, including the typical SNR of the detections, size distribution and their H /I.sc mass range. In addition, we also compare the properties of the 12 '' detections with their corresponding 30 '' counterparts.", "5.1 12 '' imaging results": "A total of 73 HIPASS target galaxies were imaged in highresolution as part of the Pilot Survey Phase 2. We note that in the majority of cases the target HIPASS galaxy is the only genuine detection in the image cube. However, in a few cases, source finding on some target HIPASS image cubes resulted in the detection of genuine smaller sources surrounding the target HIPASS galaxy. Once the source finding is complete, each tentative detection is visually examined to verify if it is a genuine source and is then added to the final source catalogue. A total of 80 sources are detected from the source finding runs from all three Phase 2 fields combined. Most detections in the high-resolution image cubes are also detected in the default 30 '' data cubes, however, in some cases it is observed that a 30 '' source in the default WALLABY catalogue is split-up into multiple components, with each component being a genuine nearby galaxy in the vicinity of a large galaxy. In such cases, each 12 '' component is assigned a unique WALLABY name. \nIn Figure 6, we plot some of the source characteristics of the 12 '' sample. Given that most targeted 12 '' sources are HIPASS detections, the redshift distribution of the sample ranges from 0.002 < z < 0.04, with a median z ∼ 0.01. We find that the median SNR of the 12 '' detections is ∼ 20, while the rms in the local image cubes of the 12 '' detections is found have a median value of ∼ 1.8 mJy, which is close to the expected theoretical rms of 1.75 mJy (using robust=0.5 and all baselines including 6 km). This translates to a 5 σ H/I.sc column density (N H/I.sc ) sensitivity limit of ∼ 6 × 10 20 (1 + z ) 4 cm -2 assuming a 12 '' beam and a 20 km s -1 channel width, which is a factor of 6.6 higher compared to the 30 '' data. This is a natural compromise between sensitivity and spatial resolution that is associated with higher-resolution observations and we advice the user to be cognisant of this compromise in sensitivity when dealing with the high-resolution data. \nWe also note that the distribution of the w 20 H/I.sc linewidth for the 12 '' detections ranges from 46 < w 20 (km s -1 ) < 597, with a median w 20 ∼ 172 km s -1 , which indicates that the majority of the high-resolution sources are likely to be rotationally-supported late-type galaxies. We examined the moment maps and the corresponding optical image for the obvious outlier (with w 20 ≈ 597 km s -1 ) and find that the H /I.sc emission is much more extended compared to the optical disk, along with kinematic warps and other signatures indicating that this galaxy is likely undergoing an interaction and may have accreted H /I.sc gas from a gas-rich low-mass companion. As the SoFiA mask encompasses all the H /I.sc emission, it results in considerably broadening the velocity width of this detection. Most of the 12 '' detections are well resolved with their major axis size typically spanning ∼ 7 (12 '' ) beams across. Finally, we note that the H /I.sc mass distribution of the high-resolution sample is 8.0 ≤ log 10 ( M H/I.sc M ⊙ ) ≤ 10.2, with a sample median of \nFigure 7 shows moment 0 (intensity) and 1 (velocity) maps of two interacting system of galaxies in the default 30 '' and 12 '' -resolution. From the images it is very clear that finer details in the H /I.sc morphology begin to show-up in the high-resolution images. The high-resolution moment maps highlight the distribution of the high-column density H /I.sc gas in the galaxies, which are otherwise washed-out in the 30 '' images. In addition, in Figure 8 we show the 30 '' and 12 '' resolution H /I.sc contours overlaid on top of a composite (g,z,i-band) DESI Legacy Survey image for the galaxy NGC 5054. The two contours show H /I.sc column densities of 2.4 × 10 20 cm -2 (light orange) and 7.2 × 10 20 cm -2 (dark orange), respectively. The contours correspond to a SNR of 4 and 10 in the 30 '' image respectively, while corresponding to a SNR of 2 and 6 in the 12 '' image. Compared to the 30 '' resolution H /I.sc contours, the 12 '' resolution contours clearly trace the high-column density H /I.sc gas along the spiral arms in NGC 5044, allowing us to study both the H /I.sc gas and star formation properties at a much higher resolution. A factor of ∼ 3 improvement in resolution will significantly aid in studies directed towards understanding the distribution of the high-column density gas in galaxies and also enable us to more accurately probe the connection between H/I.sc gas, star formation and star formation laws. In addition, the higher resolution enables us to model the kinematics of the H /I.sc gas more accurately. \nlog 10 ( M H/I.sc M ⊙ ) ∼ 9.42.", '5.2 Data quality and known issues with the high-resolution data': "We do note that while the overall quality of the 12 '' data is good, there were some issues identified with the imaging pipeline as well as the data products. We list below some of the known issues with the cut-outs in this data release. \nFlux discrepancy: Wenote that the 12 '' sources show a higher integrated flux compared to their 30 '' counterparts. The flux of the 12 '' sources is on average ∼ 15% higher compared to their 30 '' counterparts. We present a more thorough discussion on this flux discrepancy in Section 6 and also highlight the likely origins of the discrepancy. \nDifferent synthesized beam size: Some sources from the NGC 5044 field (tile4) have a different angular resolution. These data sets have a synthesized beam of ∼ 17 '' instead of 12 '' . There are 14 such sources. This is because a slightly different tapering was applied during the imaging stage. The visibilities for these sources were not stored as the observations for the NGC 5044 tile 4 were carried-out before the scheme of storing visibilities on to CASDA was introduced. As such, the visibilities for these sources were unfortunately unavailable to be re-imaged to a 12 '' resolution. We have included a comment in the source catalogue for all relevant affected sources to highlight this. \nUnreliable spectra: 7 sources in the 12 '' data show bad spectra. These are typically edge-on galaxies with large spectral \nFigure 6: Plots show the source properties of the 12 '' detections in the Phase 2 sample. Top left: Distribution of the barycentric redshifts of the 12 '' detections. Top centre: Histogram of the Signal-to-noise (SNR) of the 12 '' detections. Top right: Local rms noise distribution in the images cubes. Bottom left: Distribution of the w 20 H/I.sc line-width distribution. Bottom centre: Histogram of the major axis size (in units of 12 '' beams). Bottom right: The H/I.sc mass distribution. In all plots, the dashed black line represents the median value of the distribution. \n<!-- image --> \n/circledot \nwidths. Given that only 250 channels are split-out for the highresolution imaging, we suspect that there were not enough line-free channels for the image-based continuum subtraction routine in ASKAPSoft to properly perform the continuum subtraction, leading to over-subtraction. Sources affected by this issue have a qflag = 128 in the source catalogue. \nNo default 30 '' WALLABY cross-match: We note that 6 sources in the cut-outs source catalogue do not have a corresponding default 30 '' WALLABY detection. Upon further examination, it was found that the missing sources in the 30 '' WALLABY catalogue are due to one of the following reasons. \n- · Source lies in the Galactic velocity range. The default 30 '' source finding runs are only performed on the extragalactic velocity range ( cz ∼ 500 - 26500 km s -1 ) and as a consequence all sources below a velocity of cz < 500 km s -1 are excluded from the source finding runs. Two sources are missing due to this limitation.\n- · Source is in the corner of a footprint. The SoFiA source finding runs are only performed on the inner 4 · × 4 · area of the mosaicked footprint as the outer edges of the footprint suffer from lower SNR and sensitivity as the noise increases by a factor of two. For this reason, some sources in the outer parts of the specific footprint may have been omitted in the current default 30 '' source finding run. These sources will however be added to the catalogue whenever overlapping footprints are subsequently processed and\n- available for source finding. Three sources are missed due to this.\n- · Very faint sources near the detection threshold may be missed in the global 30 '' source finding, as the completeness curve is known to gradually decrease below an SNR of ∼ 7 - 8 (see Figure 5). Since the high-resolution source finding involves checking and verifying each individual detection, in some cases it is possible to detect sources close to the detection threshold of the source finding runs. One source is missed due to this issue.", '6. Flux discrepancy': "As mentioned in the previous sections, the 30 '' detections are observed to have a flux deficiency of ∼ 15% compared to single-dish observations (see upper panel in Figure 9). The flux discrepancy between the 30 '' detections and their singledish observations was also observed in the Phase 1 release. A number of independent effects may be contributing to this issue including flux that is genuinely missed in the 30 '' data from extended diffuse emission; inadequate deconvolution threshold, which leads to contributions from uncleaned flux and negative side lobes associated with the 30 '' dirty beam (see Westmeier et al. 2022); as well as systematic flux offsets due to the different procedures implemented for the primary beam correction between Phase 1 and Phase 2. While it is difficult to estimate the typical fraction of flux that is missed from diffuse emission, it is easier to resort to simulations of ASKAP \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 7: The comparison of moment 0 and moment 1 maps for two galaxies (top: HIPASS J0949-047b, bottom: HIPASS J1005-44b) with a resolution of 30 '' and 12 '' . In each row, panels (a) and (c) show the moment 0 and 1 maps with a resolution of 30 '' while panels (b) and (d) show the corresponding 12 '' maps. At the bottom of each figure, we show the respective beam size as black circles and a scale bar set to 10 kpc. \n<!-- image --> \nobservations in order to specifically understand the impact of inadequate deconvolution thresholds on the measured flux. We discuss this in more detail in Section 6.1. \nIn addition, we observe a flux discrepancy between the 12 '' and the 30 '' sources, wherein the 12 '' detections typically show ∼ 15% higher flux compared to their corresponding 30 '' detections (see lower panel in Figure 10), while being consistent (albeit coincidentally) with their single-dish detections (see upper panel in Figure 10). We examined the ASKAP dirty beams for the 12 '' and 30 '' observations and find that there are significant positive sidelobes associated with the 12 '' beam, whereas there are significant negative sidelobes observed in the 30 '' dirty beams (see Figs. 15 and 16 in Appendix 5). This observation is true for all declination ranges (-47 · < δ < +8 · ) covered by Phase 2 and could potentially lead to flux offsets, as any uncleaned flux in the residual maps impacted by the sidelobes of the dirty beams will contribute to the final image cube. In order to study the impact of the side lobes of the dirty beams for the two resolutions we resorted to simulations of ASKAP observations. In the subsection below we describe the details of the simulations and their main outcomes.", '6.1 Simulating ASKAP observations': "In order to generate mock ASKAP observations, we make use of the MIRIAD software package (Sault et al., 1995). We use the MIRIAD task UVGEN to generate the mock visibilities. \nUVGEN takes in details of the mock observations, such as the positions of the ASKAP antennas, the correlator setup, the frequency of the observations, the RA and Dec of the pointing, hours of observation, integration time, the latitude of the observatory and the system temperature of ASKAP. This then generates the expected visibilities. We note that we generated mock visibilities at six different declinations (+8, +2, -11, -19, -24 and -45 · ) to represent the declination range observed in the Phase 2 fields. We generated 4 × 8 h mock ASKAP observations to approximately emulate the ASKAP beams which are processed and imaged independently before mosaicking them to form the final image cube with an rms noise close to ∼ 1.6 mJy per beam per channel, representative of the noise in typical WALLABY image cubes. \nWe Fourier-transform the mock visibilities to generate the dirty beam and dirty images at the 12 '' and 30 '' resolutions, using the MIRIAD task INVERT . We note that we were unable to yield a synthesised beam ∼ 12 '' using a robust value of 0.5 and thus set the robust weighting to 0 and applied appropriate Gaussian tapering to the mock visibility data in order to achieve nearly 12 '' and 30 '' dirty beams. Next, we generate ∼ 60 mock galaxies per declination range, of varying size, surface densities, integrated flux, position angle and inclination angles using BBarolo 's (Di Teodoro & Fraternali, 2015) GALMOD task. The size and the flux range of the model galaxies is set appropriately to reflect the respective range observed in the real 30 '' and 12 '' WALLABYdetections. The model galaxy \n<!-- image --> \nFigure 8: Left: 30 '' resolution H /I.sc contours overlaid on top of a composite (g,z,i) DESI Legacy Survey image of the galaxy NGC 5054. Right: Corresponding 12 '' resolution H /I.sc contours. In both cases the contours levels are set at column densities of 2.4 × 10 20 cm -2 (light orange) and 7.2 × 10 20 cm -2 (dark orange). \n<!-- image --> \nis then convolved with the 12 '' and 30 '' dirty beams, respectively, using the MIRIAD task CONVOL . The convolved galaxy models are then injected into the ASKAP observations dirty image cube (noise cube) using the task MATHS . This is then followed by the deconvolution step using MIRIAD's CLEAN task. We set the clean thresholds to match the settings used in the default ASKAPSoft imaging pipeline. This corresponds to a clean threshold in the minor cycle of 3.5 mJy ( ∼ 2 σ ) and an additional second deep clean threshold of 0.5 mJy ( ∼ 0.3 σ ). We set the maximum number of iterations ( niters ) for the minor clean cycles to be 800, again replicating the ASKAPSoft pipeline settings. \nAfter the cleaning stage, we restore the images using the task RESTOR . This takes in the dirty beam, the dirty image (with the injected model galaxy) and the clean components from the deconvolution step and generates the residual map as well as the final restored image cube by convolving the clean components with 12 '' and 30 '' Gaussian beams, respectively, and adding them to the residual map. We then mosaic all four simulated image cubes to generate the final image cube on which source finding is performed using SoFiA. \nIn addition, we also convolved the model galaxies with a Gaussian of full width at half maximum (fwhm) of 12 '' and 30 '' which will represent perfectly 'cleaned' data. The fluxes derived from these Gaussian beam convolved data sets will not have the influence of the sidelobes that is typically observed in interferometric data sets where some uncleaned flux may \nremain, impacting the final measured flux as well as the quality of the final image cubes. \nWe run SoFiA on the simulated 30 '' image cubes by using the default parameter settings currently used for the WALLABY source finding. We appropriately change some SoFiA parameters for the high-resolution data sets as the default 30 '' parameters are not optimal for source finding in the 12 '' image cubes. For a list of important SoFiA parameters used for source finding on the 30 '' and 12 '' data-sets we refer the reader to Table 5 in Appendix 2.", '6.2 Origin of the flux discrepancy': "We now report the main observations from our simulation experiment and delve into the details of the origin of the flux discrepancy that is observed between the single-dish, 30 '' and 12 '' detections. \nPanel a) in Figure 11 shows a plot of the ratio of the 30 '' integrated flux to the total flux of the injected model galaxy ( F 30 F model ) against the injected model galaxy flux ( F model ). The detections are color-coded based on their SNR values, which are computed as the total flux divided by the uncertainty in the flux measurement, both of which are provided by SoFiA in the source catalogue. As mentioned before, the CLEAN thresholds are set according to the default ASKAPSoft pipeline which is cleaning to a residual peak flux density threshold of 3.5 mJy. We find that indeed the measured flux of the mock \nFigure 9: Top: The ratio of the WALLABY 30 '' integrated flux to the single-dish integrated flux plotted against the WALLABY integrated flux for those galaxies which have a corresponding single-dish cross-match, either in ALFALA and/or HIPASS. For the NGC 5044 and Vela fields, we use the HIPASS data and for the NGC 4808 field, we use the ALFALFA data for the flux comparison. Bottom: Similar plot as above, but now the WALLABY fluxes have been corrected using a polynomial fit to the data. The horizontal black line represents a flux ratio of one in both cases. \n<!-- image --> \nFigure 10: Plot shows the ratio of the integrated flux of the 12 '' ( F 12 ) to the 30 '' flux ( F 30 ) for the overlapping sample. The black solid line represents the expected one-to-one line, and the dashed black line represents the median value of the F 12 F 30 ratio. \n<!-- image --> \ngalaxies in the 30 '' resolution is consistently lower than the flux of the injected model galaxies. The mean flux discrepancy is ∼ 4%, while the discrepancy becomes more and more prominent towards the low-flux and low-SNR regime, where the offset can be as much as 15%, corresponding closely to the discrepancy observed in the real data. \nFollowing this, we explored the observed flux discrepancy between the 12 '' and 30 '' data. Panel c) in Figure 11 shows the ratio of the 12 '' to 30 '' fluxes ( F 12 F 30 ) plotted against the 30 '' fluxes of injected model galaxies. We find that the 12 '' fluxes are typically ∼ 10% higher compared to the 30 '' fluxes, which is comparable to what is observed in real ASKAP observations (about ∼ 15%). From these simulations, it is evident that the impact of the positive sidelobes on the uncleaned flux in the 12 '' resolution is quite significant. A similar observation has also been reported in Radcliffe et al. (2024), who undertook a simulation study to investigate the impact of asymmetric PSFs from interferometers on the recovered flux from sources. Their study points to the fact that non Gaussian dirty beams (PSFs) lead to consistent flux offsets compared to the flux of the injected model source. We note that our simulations are in agreement with their observations, wherein the flux is more discrepant for the marginally resolved, low SNR sources, while being somewhat more consistent for extended sources. \nAs noted previously, this flux discrepancy may be an effect of incomplete cleaning, which results in the uncleaned flux being included into the final image cubes, impacting the final \n<!-- image --> \nR \nN \nS \nR \nN \nS \n<!-- image --> \n<!-- image --> \nFigure 11: a): Circles show the ratio of the integrated flux of the injected model source convolved with the 30 '' PSF ( F 30 ) to the total flux of the injected model galaxy ( F model ) for over 350 simulated galaxies in the declination range -47 · ≤ δ ≤ +8 · . The data was cleaned to a residual flux threshold of 3.5 mJy in the minor CLEAN cycles. The inverted yellow triangles represent the flux ratio of the model sources convolved with a perfect 30 '' Gaussian beam to that of the total flux of the injected source into the image cubes. b): Same as panel a), but now the sources were cleaned deeper to a residual flux threshold of 0.9 mJy. c): Shows the ratio of the integrated flux from the 12 '' and 30 '' model sources injected into to the image cubes and cleaned to a residual flux threshold of 3.5 mJy. d): Same as panel c), but now cleaned to a deeper residual flux threshold of 0.9 mJy. The points are color-coded based on the SNR of the 30 '' detections. The black solid line represents the expected one-to-one ratio, while the dashed red line shows the mean flux discrepancy of the distribution. \n<!-- image --> \nmeasured integrated flux. In order to investigate the impact of deeper cleaning, we cleaned the data to a peak residual flux threshold of ∼ 0.5 σ = 0.9 mJy in the minor cycles with an additional deep cleaning threshold set to 0.1 mJy. The deeper deconvolution thresholds lead to better flux recovery for both the 30 '' and 12 '' data sets, with the measured and injected fluxes in better agreement. This is shown in panels b) and d) in Figure 11. \nTo summarise, we note the following observations from the simulation study. \n- · We find that the 30 '' integrated flux is consistently lower by about 4% compared to the integrated flux of the input model galaxy. However, at the low flux (or SNR) end, the flux discrepancy can be as high as 15%, consistent with observations. This effect is observed in panel a) in Figure 11 (also Figure 9), were we observe that sources with an SNR < 20 show a higher flux discrepancy.\n- · We find that the 12 '' fluxes are on average consistently higher than the 30 '' fluxes by ∼ 4 - 10%, depending on a number of factors, including the SNR of the data, as well as how extended and bright the source is.\n- · The 12 '' and 30 '' fluxes for the Gaussian beam convolved data sets are consistent with each other for the high SNR (SNR > 20), while not surprisingly the 12 '' fluxes for the marginally resolved and/or low SNR sources is typically lower than the 30 '' fluxes.\n- · We note that by cleaning deeper in both the 30 '' and 12 '' data sets, we recover most of the flux, almost completely resolving the flux discrepancy. This suggests that deeper CLEANing thresholds are essential to fully recover the flux from WALLABY observations. \nWedoacknowledge that while care was taken to carry-out the simulations to reflect as closely as possible the ASKAPSoft pipeline, there are many subtle differences that might still impact the way the data is processed and hence the recovered fluxes. Such effects are likely to impact sources in the lowflux (-SNR) regime more than well resolved and higher flux sources. However, despite these caveats, the simulations do highlight the importance of deeper cleaning of the ASKAP observations in order to recover fluxes properly.", '6.3 Correcting the fluxes': "We showed in the previous section through simulations that in order to properly recover the total flux from a source, we need to clean much deeper (potentially 0.5 σ or deeper) than the current deconvolution thresholds set in ASKAPSoft, in addition to using source masks generated from shallower CLEAN runs for further cleaning. Another important point is the implementation of joint deconvolution routines that enable the visibility data from all ASKAP primary beams be jointly imaged and deconvolved, so that the data can be cleaned to the appropriate deeper CLEAN thresholds. However, such a system is still not in place in the current ASKAPSoft pipeline and furthermore will require significantly more computational resources to process the data. This means that one cannot fully \nclean the data and therefore the impact of the sidelobes on the residual flux will remain an issue for both the 12 '' and 30 '' data-sets. However, the WALLABY team is currently testing newly implemented parameters in ASKAPSoft's deconvolution algorithm and optimal CLEAN thresholds will be implemented for the full WALLABY survey accordingly. \nIn the interim, we statistically correct the 30 '' fluxes of Phase 2 sources by appropriately scaling their integrated fluxes to their corresponding single dish values from ALFALFA and HIPASS. We find that the data is best fit by a second order polynomial of the form \nlog 10 ( F W F SD ) = -0.006448 ϕ 2 + 0.103635 ϕ - 0.439071 (2) \nwhere ϕ = log 10 ( F W Jy Hz ) . The red line in the upper panel in Figure 9 shows the fit to the data, while the bottom panel shows the corrected fluxes. We see that this seems to systematically bring the flux level up, making it more consistent with the single-dish data. We note that we resorted to the new polynomial fit for the Phase 2 data, as the third order polynomial fit from PDR1 (dashed grey line in the upper panel in Figure 9) does not fit the data very well and seems to over-correct the fluxes in the low-flux end. The reason for the very different flux offsets observed between Phase 1 and Phase 2 data is likely stemming from the fact that Phase 2 observations utilised the holography-based primary beam correction as opposed to the use of a Gaussian primary beam correction in Phase 1, which will lead to systematic offsets in the flux. \nWe note that these corrections have not been applied to the data products for each source included as part of this public data release, however, we have included the corrected fluxes in the catalogue and advise the users to be aware of this issue and apply the necessary correction to the fluxes when using the image cubes and moment maps for any analysis. The keywords f\\_sum\\_corr and err\\_f\\_sum\\_corr in the source catalogue represent the corrected flux and error on the corrected flux, respectively. Similarly, the keyword log\\_m\\_hi\\_corr represents the H /I.sc mass derived from the corrected flux and using the Hubble distance to the source.", '7. Kinematic Modelling': "One of the goals of WALLABY is to generate kinematic models for as many galaxies as possible. For Phase 1 Deg et al. (2022) developed the WALLABY Kinematic Analysis ProtoPipeline (WKAPP a ) that is optimized for the low resolution and signal-to-noise (S/N) of the standard 30 '' data. It uses a combination of two different tilted ring (TR) modelling algorithms to generate reliable kinematic models from observed source cubelets. It was used to generate the 109 kinematic models of WALLABY Phase 1 and we use it here on both the 30 '' and 12 '' source cubelets. \nTilted-ring modelling treats a galaxy as a series of nested rings described by a number of observational parameters (center, systemic velocity, position angle, and inclination angle) and intrinsic ones (surface density, disk thickness, rotation velocity, and velocity dispersion). This technique, introduced by (Rogstad et al., 1974), was first developed for 2D images and has been adapted to work with 3D data cubes. There are a number of advantages to working in 3D including the ability to apply more complex models to well resolved, high S/N data (see for instance Józsa et al. 2009; Khoperskov et al. 2014; Di Teodoro & Peek 2021; Józsa et al. 2021). More relevant to the WALLABY context, 3D TR algorithms are also able to model galaxies at lower spatial resolutions across a wider range of disk geometries than equivalent 2D algorithms (e.g. Kamphuis et al., 2015; Di Teodoro & Fraternali, 2015; Lewis, 2019; Jones et al., 2021). \nWhile a full description of WKAPP is found in Deg et al. (2022) we will briefly describe the key points here. WKAPP combines fits from two different 3D TR algorithms to generate its models - Fully Automated T/I.scR/I.scF/I.scC (FAT, Kamphuis et al. 2015), which itself is built on the Tilted Ring Fitting Code (TiRiFiC; Józsa et al. 2007); and the 3D-Based Analysis of Rotating Objects From Line Observations (BBAROLO; Di Teodoro & Fraternali 2015). Both codes are run in a 'flat-disk' mode, where the observed geometry is constant across all rings. Deg et al. (2022) found that the differences between the two codes tended to be larger than the reported uncertainties of either algorithm. As such, WKAPP uses half the difference between the models as the better estimate of the model uncertainty, which is applied to all galaxies with either a S/O.scF/I.scA ell\\_maj ≥ 2 beams or an integrated log( S / N ) ≥ 1.25. The fits for each code are compared, and if both fits are reasonable, the two are averaged to create the final kinematic model. \nIn this section we describe the results of applying WKAPP to both the 30 '' and 12 '' data. Section 7.1 focuses on the 30 '' Phase 2 data and how the models compare to Phase 1. Section 7.2 focuses on the 12 '' data and how those models compare to the 30 '' models.", '7.1 Normal Resolution Modelling': "While Phase 2 contains many more detections than Phase 1 ( ∼ 1800 unique detections compared to the ∼ 600 unique detections of Phase 1), these galaxies tend to be further away and smaller in size than the Phase 1 observations. Of these Phase 2 galaxies, only 275 have the requisite size and S/N to attempt kinematic modelling. For comparison, Phase 1 contains 209 unique galaxies that satisfy the ell\\_maj ≥ 2 or log( S / N ) ≥ 1.25 criteria. Table 3 lists the sources, the galaxies that satisfied the modelling attempt criteria, and the total number of kinematic models for Phase 2. Considering only those galaxies where kinematic modelling is attempted, the 30 '' Phase 2 sources have a success rate of 45%, which is comparable to Phase 1. \nUsing WKAPP, the final catalogue of Phase 2 kinematic models contains 127 unique galaxies. The left-hand panels of Figure 12 shows the kinematic models for these 127 models. These models span a wide range of rotation velocities \nTable 3: The number of sources, attempts, and successful models in each release (where TR refers to Team Release). Note that there are no double sources in the 12 '' data so a 'Unique' 12 '' row is the same as the 'Total' 12 '' row. \nand extents, including a number of low-mass dwarfs. It is worth noting that the surface densities are calculated through ellipse-fitting on the moment 0 map using the averaged model geometry. As such the deprojected profiles shown in Figure 12 have not been corrected for beam smearing effects, which may be important for some applications.", '7.2 High Resolution Kinematic Modelling': "WKAPPwasdeveloped with an eye towards the 30 '' WALLABY data, and it is not clear that this approach is appropriate for the higher resolution 12 '' data. Nonetheless, we have applied the proto-pipeline data to the high resolution data and obtained 27 kinematic models from the 80 detections. The rotation curves and deprojected surface density profiles for these models are shown in the right-hand panels of Figure 12. \nComparing the left and right panels of Figure 12, it is clear that the distribution of 12 '' and 30 '' models are not the same. The 12 '' models are biased towards higher rotation velocities and H/I.sc masses compared to the 30 '' models (which show a wider range of velocities and H /I.sc masses). This is evidenced by the fact that very few modelled galaxies in the 12 '' resolution have rotation velocities lower than 80 km s -1 in the outer/flat parts of their rotation curves. This is likely due to HIPASS preferentially finding relatively nearby gas-rich galaxies, which are typically observed to have higher rotation velocities. \nThe middle panel of Figure 12 shows the 30 '' models for galaxies that also have a 12 '' model. Some galaxies in the 12 '' sample that are successfully modelled do not have an equivalent 30 '' model. Thus, while there are 27 12 '' models, there are only 18 cross-matched 30 '' models. Comparing the middle and right columns of Figure 12 reveals that the rotation curves are broadly equivalent for the cross-matched models. However the 12 '' models tend to be truncated relative to the 30 '' data. \nTo gain a better understanding of the Phase 2 models and the 12 '' data, Figure 13 shows the S/O.scF/I.scA ell\\_maj parameter and integrated S/N for the Phase 2 sources. As noted in Deg et al. (2022), there is a clear relationship between the size \nFigure 12: The rotation curves (top row) and deprojected surface density profiles (bottom row) for Phase 2. The left hand panels shows the models for all 30 '' data while the right hand panels show the models for the 12 '' data. The middle column shows the 30 '' models for galaxies that also have a model from their 12 '' data. The dashed horizontal line in the surface density panels is at 1 M ⊙ pc -2 , which is the standard value used to define R H/I.sc . \n<!-- image --> \nand the integrated S/N. In the left-hand panel that shows all Phase 2 sources, it is clear that there is a diagonal limit above which no successful models are generated. For a given size, a higher S/N leads to a higher chance of kinematically modelling a galaxy. Conversely, larger galaxies with the same S/N as smaller galaxies are more difficult to model with WKAPP. \nFocusing on the 12 '' detections and the cross-matched 30 '' sources reveals a number of interesting behaviours. Firstly, the majority of the cross-matched sources have approximately the same log( S / N ). Secondly, the approximate size of the modelled disc has not increased by a factor of 2.5. This is expected as the smaller beam size results in a worse column density sensitivity, which means that the most extended gas will be below the noise limit. This, combined with the fact that the beam smearning effects are minimised in the 12 '' resolution also explains the decreased radial extent of the 12 '' kinematic models seen in Figure 12. \nA third, and perhaps more important result is apparent in Figure 13. Only 8 galaxies do not have a 30 '' kinematic model and a successful 12 '' model (indicated by blue lines in Figure 13). By contrast there are 18 sources that were successfully modelled using their 30 '' data that were not modelled with the 12 '' data. These results show that, for WKAPP, the increased noise of the 12 '' data leads to poorer results in terms of kinematic modelling despite the increased resolution. It is important to note here that WKAPP is being run in precisely the same way for the 12 '' as for the 30 '' data. If a more tailored \napproach were adopted it is possible that the kinematic modelling would be significantly more successful. Additionally, the increased resolution brings many of the galaxies into the regime where various 2D algorithms are applicable. Collapsing the data to moment maps effectively increases the S/N and may lead to greater success than the 3D approach of WKAPP. These ideas will be explored in a future work.", '8. Data access': "The WALLABY Pilot Survey Phase 2 data and associated catalogues are available to the public through the CSIRO ASKAP Science Data Archive (CASDA) and the Canadian Astronomy Data Centre (CADC). The data release is similar to Public Data Release 1 and includes all the 30 '' source data products, kinematic models and respective catalogues. In addition, in this release we are also including the high-resolution 12 '' data products, kinematic models and catalogue. We also provide descriptions and details on the various data products, data quality issues and list details of the various column names in the catalogues. \nWe note that the source catalogue in this release also includes all detections from the Public Data Release 1 from Westmeier et al. (2022) for easy accessibility to both DR1 and DR2 detections. Furthermore, the new catalogue will include all the relevant updated columns such as the corrected fluxes ( f\\_sum\\_corr and err\\_f\\_sum\\_corr ) as well as the corrected H /I.sc masses ( log\\_mi\\_hi\\_corr ) for both the DR1 and DR2 samples \nFigure 13: The size and integrated S/N of the Phase 2 sources. The circles show the 30 '' detections, while the stars and triangles shows the 12 '' detections. The different 12 '' symbols indicate whether there is a cross-matched 30 '' source for the 12 '' source (stars) or not (triangles). The black, red, and blue points indicate galaxies where kinematic modelling was not attempted, attempted and failed, or successfully modelled respectively. The left hand panel shows all Phase 2 detections, while the right hand panel only shows the 12 '' sources and their crossmatched 30 '' counterpart (if a crossmatched source exists). In the right hand panel the lines connect the cross-matched sources. Occasionally a 30 '' source is broken into two different sources and will have two lines originate from the source. If the kinematic modelling result has not changed (failed for both or successful for both), the line is black. If the 30 '' source is kinematically modelled while the 12 '' source is not the line is red, and when the situation is reversed the line is blue. \n<!-- image --> \nmaking it convenient for the user to use the corrected values. \nThe combined footprint A and B mosaics are available on CASDA via https://doi.org/10.25919/hg66-4v60 . These are very large (typically ∼ 600 GB) and we recommend that users interact with these via the CASDA cutout service. These cutouts can be made either through the CASDA Data Access Portal (DAP) or by using the Simple Image Access Protocol (SIAP) coupled with the Serverside Operations for Data Access (SODA) protocol. In the second case, the user interacts using a Python script or Jupyter notebook to select the region and channel range of interest. Additionally, the CASDA module of the Astropy Astroquery package b can also generate cutouts. \nUsers can access the 30 '' data via CASDA using the following links for the various types of data-sets. a) Source data products (including moment maps, cubelet, channel map, source mask and spectra) and complete source catalogue: https://doi. org/10.25919/qw7w-tn96 ; b) 30 '' kinematic modelling data products and catalogue: https://doi.org/10.25919/7w8n-9h19 . \nThe 12 '' source data products, which includes all SoFiA source data products (moment maps, cubelet, channel map, source mask and spectra), kinematics models and catalogue (including kinematic modelling parameter values) can be accessed via CASDA using the following link: https://doi.org/ 10.25919/47tr-k441 . \nAll the above data products for both the 30 '' and 12 '' data can be accessed via CADC through a TAP service using ADQL queries. For more details on how to access the data through CADCwe refer the reader to the Public Data Release 1 papers (Westmeier et al. 2022; Deg et al. 2022). Furthermore, users can also get detailed instructions and links to the data releases through WALLABY's data access page c .", '9. Summary and Future': 'In this data release paper, we present the catalogue, data products including moment maps and spectra for over 1800 galaxies from the WALLABY Pilot Survey Phase 2. The observations were carried out on three selected fields which include the NGC 5044, NGC 4808 and Vela groups. The total observed sky area is ∼ 180 deg 2 and the redshift limit corresponding to z ∼ 0.09. The median rms noise levels in the data cubes is ∼ 1.7 mJy, which is close to the expected theoretical noise for the WALLABY observations. This translates to a 5 σ column density sensitivity of ∼ 9.1 × 10 19 (1 + z ) 4 cm -2 assuming a 30 \'\' beam and a 20 km s -1 channel width. \nIn addition to the default 30 \'\' data products, in Phase 2 we have also presented the high-resolution 12 \'\' cut-outs of select HIPASS galaxies demonstrating the true potential of WALLABY to produce high spatial and spectral resolution H /I.sc observations of several thousand galaxies (including all HIPASS galaxies) in the 5-year survey period, thereby forming the largest sample of high spatial resolution H /I.sc maps of galaxies until the SKA-mid begins observations. As such, these highresolution cut-outs carry immense legacy value. \nWe highlighted the significant improvement in the quality of the data compared to Phase 1 which is mainly attributed to the fact that the Pilot Phase 2 fields were selected in a way as to avoid bright continuum sources, but also due to the introduction of the holography-based primary beam correction for the ASKAP observations, which results in more accurate fluxes for the sources. It is to be noted that there is ongoing work to implement appropriate \'peeling" techniques into the ASKAPSoft data reduction pipeline in order to properly subtract residual continuum that is associated with bright continuum sources, likely improving the quality of the data significantly. While the data quality in general is very good, we note the observed flux discrepancy in the ASKAP observations. The issue was first highlighted in the Phase 1 paper (Westmeier et al., 2022), wherein the integrated flux of the 30 \'\' WALLABY detections were observed to be ∼ 15% lower than the corresponding single-dish flux. This was alluded to improper deconvolution and the impact of residual sidelobes still present in the image cubes. In order to fully understand this issue, we undertook simulations of ASKAP observations and injected model galaxies by varying their properties such as flux and size and find that up on performing the source finding using SoFiA, the simulated galaxies in the 30 \'\' resolution are indeed observed to show consistently lower flux compared to the flux of the injected model galaxies. We attribute this to the contribution of the uncleaned flux in the data, which is impacted by the severe negative sidelobes that systematically brings down the integrated flux. We also note that marginally-resolved and/or low-SNR sources are more severely impacted by this. \nFurthermore, we also note that the integrated flux of the 12 \'\' sources is observed to be consistently higher than their 30 \'\' counterparts, which again is attributed to the impact of uncleaned flux in the data and which has the imprint of the highly non-Gaussian 12 \'\' ASKAP dirty beam with strong positive sidelobes. This uncleaned flux therefore artificially boosts the flux of the 12 \'\' detections to about ∼ 15% depending on a number of factors including the SNR and spatial extent of the source. In order to minimise the impact of the uncleaned flux on the data, going forward for the full WALLABY survey, it is necessary to set appropriate cleaning thresholds and making sure that the thresholds are reached during the clean cycles. In addition, a two stage cleaning approach involving a shallow clean followed by a deeper cleaning using a source mask might result in better flux recovery. This will of course considerably increase the time and resources required to process the data, however, such a scheme may be implemented in the ASKAPSoft spectral-line imaging pipeline given the ASKAP observations are now being processed in the new upgraded Pawsey HPC Setonix , which is capable of handling large data volumes.', 'Acknowledgement': "We would like to thank the anonymous referee for their useful comments which improved the clarity of this paper. We would also like to sincerely thank Minh Huynh (CSIRO) for all the efforts towards releasing the data onto CASDA. \nThis scientific work uses data obtained from Inyarrimanha Ilgari Bundara / the Murchison Radio-astronomy Observatory. We acknowledge the Wajarri Yamaji People as the Traditional Owners and native title holders of the Observatory site. CSIRO's ASKAP radio telescope is part of the Australia Telescope National Facility (https://ror.org/05qajvd42). Operation of ASKAP is funded by the Australian Government with support from the National Collaborative Research Infrastructure Strategy. ASKAP uses the resources of the Pawsey Supercomputing Research Centre. Establishment of ASKAP, Inyarrimanha Ilgari Bundara, the CSIRO Murchison Radioastronomy Observatory and the Pawsey Supercomputing Research Centre are initiatives of the Australian Government, with support from the Government of Western Australia and the Science and Industry Endowment Fund. \nThis research used the facilities of the Canadian Astronomy Data Centre operated by the National Research Council of Canada with the support of the Canadian Space Agency. \nThe Canadian Initiative for Radio Astronomy Data Analysis (CIRADA) is funded by a grant from the Canada Foundation for Innovation 2017 Innovation Fund (Project 35999) and by the Provinces of Ontario, British Columbia, Alberta, Manitoba and Quebec, in collaboration with the National Research Council of Canada, the US National Radio Astronomy Observatory and Australia's Commonwealth Scientific and Industrial Research Organisation. \nThis paper includes archived data obtained through the CSIROASKAPScienceDataArchive, CASDA(http://data.csiro.au). WALLABY acknowledges technical support from the Australian SKA Regional Centre (AusSRC) and Astronomy Data \nAnd Computing Services (ADACS). \nThis research has made use of the NASA/IPAC Extragalactic Database (NED), which is funded by the National Aeronautics and Space Administration and operated by the California Institute of Technology. \nParts of this research were supported by the Australian Research Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project number CE170100013. \nKS acknowledges funding from the Natural Sciences and Enginneeing Research Council of Canada. \nLC acknowledges support from the Australian Research \nCouncil via the Discovery Project funding scheme (DP210100337) NYacknowledges the fellowship of the China Postdoctoral Science Foundation (grant: 2022M723175, GZB20230766). 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G., et al. 2015, MNRAS, 447, 3311 Wong, O. I., Meurer, G. R., Bekki, K., et al. 2006, MNRAS, 370, 1607 Wong, O. I., Stevens, A. R. H., For, B. Q., et al. 2021, MNRAS, 507, 2905", "Appendix 1. ASKAPSo/f\\_t imaging parameters for the 30 '' and 12 '' data reduction": "In Table 4 we list some relevant ASKAPSoft imaging parameters used to process both the default 30 '' and 12 '' data. For more details on the definition of each of the parameters we refer the reader to the ASKAPSoft User Documentation. d", "Appendix 2. SoFiA parameters for the 30 '' and 12 '' source finding runs": "In Table 5, we list some important SoFiA parameter values used for the source finding runs for the 30 '' and 12 '' data sets.", 'Appendix 3. Manual Inspection Workflow': 'A source finding pipeline run generates detections and associated data products, which are then added to a database. The database is populated through a manual process whereby all detections are visually examined by the WALLABY team to ensure that artefacts, false detections, and duplicates are \nremoved. A web portal has been developed for conveniently executing the various stages of this inspection workflow. \nTo determine whether a detection is a real source, a WALLABYteammemberispresented with key detection properties (such as flux, RA, Dec) in a table, and a visual summary of data products (moment 0 and 1 maps, spectra, and an overlay of the moment 0 H/I.sc contours on to a Digitized Sky Survey (DSS) optical image. An example of the summary figure is shown in Figure 14. Detections that pass this first check are selected as potential genuine sources.', 'WALLABY J133541-240427': "Figure 14: Summary figure presenting the moment 0, moment 1 map, spectra, and optical DSS image of a source. These summary figures, along with properties of the detection from the source finding application are used by the WALLABY team to identify and remove false detections. \n<!-- image --> \nIn the case where there is a staggered approach to selecting sources, for example, overlapping regions of the sky subsequently processed by the source finding pipeline may give rise to duplicate detections of already accepted sources. This is the case for the NGC 5044 field, where overlapping regions are shown in Figure 2 in the darker shade of green in the corners of the central 4 · × 4 · processing regions. In such cases, an external cross matching routine is performed by the inspection workflow allowing the WALLABY team to identify and handle potential duplicate detections. The spatial and spectral locations of the detections from each new source finding run are compared against accepted source entries in the catalogue. If they are within a tight spatial or spectral threshold ( ∆ spat ± 5 '' , ∆ spec ± 0.05 MHz), they are automatically marked as duplicates and are removed from the database. If the candidate is within a lenient spatial and spectral threshold ( ∆ spat ± 90 '' , ∆ spec ± 2 MHz), they are marked for an additional visual inspection step, before being accepted as a genuine detection and assigned a WALLABY source name. Once this workflow is completed, \nTable 4: Important ASKAPSoft imaging, pre-conditioning, deconvolution and tapering parameters for the 30 '' and 12 '' data processing \nthe accepted sources are ready for release.", 'Appendix 4. Output source catalogue': "Table 6 provides details of all the parameters that are included in the source catalogue for all PDR2 detections. Two additional parameters are listed in the 12 '' source catalogue that represent the 12 '' integrated flux corrected to the original 30 '' integrated flux and the associated statistical uncertainty.", 'Appendix 5. Dirty beams (PSFs) from simulations': "In Figure. 15 we show the simulated PSFs of the 12 '' beam for various declinations using MIRIAD tasks. We used a robust parameter of 0 and applied appropriate tapering in order to generate a dirty beam that is approximately 12 '' . The positive sidelobes associated with the 12 '' dirty beam is very significant, making the beam highly non-Gaussian-like. In the case of the 30 '' PSFs (see Figure 16), while the central part of the beam is more Gaussian-like, there are significant negative sidelobes associated with the dirty beams. \nTable 5: SoFiA parameter values for the 30 '' and 12 '' source finding runs. \nTable 6: List of parameters in the source catalogue. \nFigure 15: The 12 '' dirty beams for various declinations from the simulations. \n<!-- image --> \nFigure 16: The 30 '' dirty beams for various declinations from the simulations. \n<!-- image -->"}

2024ApJ...975L..10G

Detecting atmospheres around planets with a radius below 1.6 R SUBSUB commonly referred to as rocky planets has proven to be challenging. However rocky planets orbiting M dwarfs are ideal candidates due to their favorable planettostar radius ratio. Here we present one transit observation of the SuperEarth L9859 d 1.58 R SUBSUB and 2.31 M SUBSUB at the limit of rockygasrich using the JWST NIRSpec G395H mode covering the 2.85.1 m wavelength range. The extracted transit spectrum from a single transit observation deviates from a flat line by 2.65.6 depending on the data reduction and retrieval setup. The hints of an atmospheric detection are driven by a large absorption feature between 3.3 and 4.8 m. A stellar contamination retrieval analysis rejected the source of this feature as being due to stellar inhomogeneities making the best fit an atmospheric model including sulfurbearing species suggesting that the atmosphere of L9859 d may not be at equilibrium. This result will need to be confirmed by the analysis of the second NIRSpec G395H visit in addition to the NIRISS SOSS transit observation.

2024-11-01T00:00:00Z

['2024arXiv240815855G', '10.3847/2041-8213/ad73d1', '10.48550/arXiv.2408.15855', 'arXiv:2408.15855', '2024ApJ...975L..10G']

['Exoplanet atmospheres', 'Exoplanet atmospheric composition', 'Transmission spectroscopy', 'Astronomy data reduction', 'Planetary atmospheres', 'Stellar atmospheres', 'Infrared spectroscopy', 'Super Earths', 'Extrasolar rocky planets', '487', '2021', '2133', '1861', '1244', '1584', '2285', '1655', '511', 'Astrophysics - Earth and Planetary Astrophysics']

Hints of a Sulfurrich Atmosphere around the 1.6 R SUBSUB SuperEarth L9859 d from JWST NIRspec G395H Transmission Spectroscopy

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']

https://arxiv.org/pdf/2408.15855.pdf

{'Hints of a sulfur-rich atmosphere around the 1.6 R ⊕ Super-Earth L98-59 d from JWST NIRSpec G395H transmission spectroscopy': "Am'elie Gressier , 1 N'estor Espinoza , 1, 2 Natalie H. Allen , 2 David K. Sing , 3, 4 Agnibha Banerjee , 5 Joanna K. Barstow , 5 Jeff A. Valenti , 1 Nikole K. Lewis , 6 Stephan M. Birkmann , 7 Ryan C. Challener , 8 Elena Manjavacas , 9, 3 Catarina Alves de Oliveira, , 7 Nicolas Crouzet , 10 and 1 \nTracy. L Beck \n1 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA \n- 2 William H. Miller III Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA\n- 3 Department of Physics and Astronomy, Johns Hopkins University, 3400 N. Charles Street, Baltimore, MD 21218, USA \nDepartment of Earth and Planetary Sciences, Johns Hopkins University, 3400 N. Charles Street, Baltimore, MD 21218, USA \n5 School of Physical Sciences, The Open University, Milton Keynes, MK7 6AA, UK \n- 6 Department of Astronomy and Carl Sagan Institute, Cornell University, 122 Sciences Drive, Ithaca, NY 14853, USA \n7 European Space Agency, European Space Astronomy Centre, Camino Bajo del Castillo s/n, E-28692 Villanueva de la Ca˜nada, Madrid, Spain \n8 Department of Astronomy, Cornell University, 122 Sciences Drive, Ithaca, NY 14853, USA \n9 AURA for the European Space Agency (ESA), ESA Office, Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA \n10 Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA Leiden, The Netherlands \n(Accepted for publication in the Astrophysical Journal Letters, August 25, 2024.)", 'ABSTRACT': 'Detecting atmospheres around planets with a radius below 1.6 R ⊕ , commonly referred to as rocky planets (Rogers 2015; Rogers et al. 2021), has proven to be challenging. However, rocky planets orbiting M-dwarfs are ideal candidates due to their favorable planet-to-star radius ratio. Here, we present one transit observation of the Super-Earth L98-59 d (1.58 R ⊕ , 2.31 M ⊕ ), at the limit of rocky/gas-rich, using the JWST NIRSpec G395H mode covering the 2.8 to 5.1 µ m wavelength range. The extracted transit spectrum from a single transit observation deviates from a flat line by 2.6 to 5.6 σ , depending on the data reduction and retrieval setup. The hints of an atmospheric detection are driven by a large absorption feature between 3.3 to 4.8 µ m. A stellar contamination retrieval analysis rejected the source of this feature as being due to stellar inhomogeneities, making the best fit an atmospheric model including sulfur-bearing species, suggesting that the atmosphere of L98-59 d may not be at equilibrium. This result will need to be confirmed by the analysis of the second NIRSpec G395H visit in addition to the NIRISS SOSS transit observation.', '1. INTRODUCTION': "Pushing the instrumental limit to possibly detect an atmosphere around an Earth-like planet is the first step in understanding whether any of these planets could support life and to what extent we could detect it. The large planet-to-star radius ratio, coupled with the high frequency of the transit event, creates favorable conditions for searching for an atmosphere using transmission spectroscopy around planets orbiting M-dwarfs. However, in the search for an atmospheric signal around a rocky planet, one must balance the M-dwarf opportunity with the possibility that long exposure to X-ray and EUV radiations can strip away the atmosphere (Owen & Wu 2013; Owen 2019). Besides, M-dwarf stars can mimic \nplanetary atmospheric features in the transmission spectrum from photospheric inhomogeneities, known as stellar contamination (Rackham et al. 2018). \nSpace- and ground-based observations of rocky planets (R P < 1.6R ⊕ ) have not yet provided strong and compelling evidence of atmospheric signals. The Hubble Space Telescope Wide Field Camera 3 Grism 141 (HST WFC3 G141), widely used in transmission spectroscopy for its water band at 1.4 µ m, did not yield conclusive and/or significant evidence due to its precision and limited spectral range (de Wit et al. 2016, 2018; Wakeford et al. 2019; Mugnai et al. 2021; Edwards et al. 2021; Gressier et al. 2021; Garcia et al. 2022). Similarly, no strong constraints were found using ground- \n4 \nbased (Diamond-Lowe et al. 2018) and Spitzer observations (Demory et al. 2011, 2012, 2015; Demory et al. 2016; Kreidberg et al. 2019; Zieba et al. 2022; Crossfield et al. 2022). \nThe atmospheric features in transmission for warm rocky planets are expected to reach 20 to 100 ppm, which approaches the expected noise floor of the James Webb Space Telescope instruments(Greene et al. 2016). Even for the JWST, detecting an atmosphere around a terrestrial world is a challenge, and this has been confirmed within the first years of observations. To date, no strong evidence for an atmosphere around a rocky exoplanet has been found (Lustig-Yaeger et al. 2023; Moran et al. 2018; May et al. 2023; Kirk et al. 2024). TRAPPIST-1b and c photometric data points and transmission spectra are so far consistent with no or small atmospheric signals, with the interpretation of the transmission spectrum complicated by the presence of significant stellar contamination (Greene et al. 2023; Zieba et al. 2023; Lim et al. 2023). \nThe L98-59 system is composed of 3 rocky transiting planets (Cloutier et al. 2019; Kostov et al. 2019), with radii below 1.58 R ⊕ , and a non-transiting planet (Demangeon et al. 2021) orbiting a bright M3-dwarf (K=7.01 mag). Spanning the HST and JWST cycles 1 and 2, the L98-59 system stands out as one of the most extensively studied multi-planetary systems in emission and transmission spectroscopy (cycle 1 GO and GTO programs 1201, 1224, 2512 and cycle 2 GO 3942, 3730, 4098). So far, the HST WFC3 G141 observations in the 1.1 to 1.7 µ m range have not provided strong evidence of an atmosphere for any planets in the system. Both Damiano et al. (2022) for planet b and Zhou et al. (2023) for planets c and d rejected cloud-free hydrogen and helium atmospheres. However, they could not rule out primary cloudy/hazy or water-rich atmospheres. \nHere we present the result of the JWST-GTO-1224 program observations of the third planet of the system, L98-59 d, consisting of one transit covering the 2.87 to 5.17 µ m wavelength range. With a radius of 1.58 ± 0.08R ⊕ and a mass of 2.31 +0 . 46 -0 . 45 M ⊕ (Luque & Pall'e 2022), L98-59 d lies on the boundary defined by Rogers (2015) between rocky and gas-rich planets, making it a highly intriguing target. In this letter, we first present the observations in Section 2. In Section 3, we detail the two independent data analyses used to extract the transmission spectrum. Section 4 describes the interpretations we carried out, including atmospheric and stellar contamination forward and retrieval modeling. We discuss the results in the final Section 5.", '2. JWST OBSERVATIONS OF L98-59d': 'We observed one transit of L98-59 d using the NIRSpec/G395H mode as part of the JWST GTO cycle 1 Program 1224, led by PI Stephan Birkmann. The G395H instrument mode covers the wavelength range from 2.87 to 5.17 µ m, at a native resolution of R ∼ 2700. The grating is split over two detectors NRS1 and NRS2, with a gap between 3.72 to 3.82 µ m. The observations started on June 25 nd 2023, covered the full transit and sufficient baseline for a total of 5.34 hours, resulting in 2121 integrations, each composed of 6 groups up-theramp. The observation used the NIRSpec Bright Object Time Series mode with the NRSRAPID readout pattern, S1600A1 slit, and the SUB2048 subarray.', '3. DATA REDUCTION': 'The reduction of L98-59 d NIRSpec G395H observations was obtained using two independent pipelines, transitspectroscopy (Espinoza 2022) and FIREFLy (Rustamkulov et al. 2022, 2023) which have already been benchmarked for the JWST Early Release Science (ERS) program (Alderson et al. 2023; Rustamkulov et al. 2023). FIREFLy have provided data reductions for smaller planets observations with the JWST NIRSpec G395H (Lustig-Yaeger et al. 2023; Moran et al. 2023; May et al. 2023) which makes it a good comparison.', '3.1. Data reduction with transitspectroscopy': "We used the jwst pipeline version 1.12.5 and the transitspectroscopy pipeline (Espinoza 2022) 1 to reduce the NIRSpec G395H transit of L98-59d. The reduction process followed the jwst pipeline from uncal.fits to rateints.fits, excluding the jump step detection, which we customized using the transitspectroscopy algorithm. This algorithm identifies outliers at the pixel level within each group by calculating group differences, applying a median filter, and detecting jumps. Spectral tracing was performed using the trace spectrum function, smoothed with a spline function. We determined the trace position using a cross-correlation method with a Gaussian input, covering specified pixel ranges for NRS1 and NRS2. Background estimation involved masking pixels around the trace and calculating the median of the remaining pixels. To address 1/f noise (Jakobsen et al. 2022; Birkmann et al. 2022) - noise arising from the electronics of the readout pattern, which appears as column striping in the subarray image - we subtracted the median outof-transit frame and iteratively removed the median of \nFigure 1. Top: Raw white light curves for NRS1 (left) and NRS2 (right) from transitspectroscopy and spectral light curves map. Middle : Corrected white light curves (colored data points) overplotted with the best-fit transit model (black line) from juliet . Bottom : residuals between raw light curves and transit model. \n<!-- image --> \nnon-omitted pixels near the trace position. The stellar spectrum was extracted using the getSimpleSpectrum routine with a 3-pixel radius aperture, minimizing outof-transit flux. We employed a box-extraction method to sum flux within the aperture and replaced outliers exceeding a 5 σ threshold with a 1D median filter. White light curves and pixel-level light curves were generated from the time series of 1D stellar spectra for both detectors. \nWe independently fitted white and spectral light curves from each detector using the juliet Python package (Espinoza et al. 2019), employing nested sampling via dynesty (Speagle 2020). We restricted the time series observations to +1.0 hours from the start, as the transit begins in the first half-hour (see top panel Figure 1). The period ( P = 7 . 4507245 days), eccentricity ( e = 0), and argument of periastron ( ω = 90), are fixed based on values from (Luque & Pall'e 2022; Demangeon et al. 2021). We fitted the planet-to-star radius ratio, mid-transit time, impact parameter, and semi-major axis ratio, including limb-darkening coefficients determined with a square-root law and a uniform prior. Additionally, we included a mean-out-oftransit offset ( mflux ) and a jitter parameter( sigma w ) \nfor white noise. To account for correlated noise we used Gaussian Processes (GP) using the george package with a Mat'ern 3/2 kernel (Ambikasaran et al. 2015; ForemanMackey et al. 2017). This method captures complex systematics and has been validated in ERS WASP-39b studies (Feinstein et al. 2023; Alderson et al. 2023). A log-uniform prior was set for the GP amplitude between 0.001 and 1000 ppm, and an exponential prior was set for the GP length-scale, using time and the full-width half maximum (FWHM) as GP regressors. The white light curve fitting results are shown in Figure 1, and the detailed best-fit parameters and confidence intervals are provided in Appendix Table 2. We determined the optimal systematics detrending model based on Bayesian log-evidence, finding that the GP model with both time and FWHM as regressors offers the best fit. We evaluated several detrending approaches, including linear models with time and FWHM, and a GP with time only using the Mat'ern 3/2 kernel via the celerite package (Foreman-Mackey et al. 2017). For NRS1, the logevidence values were 6700 for the transit-only model, 6778 for the linear time model, 6780 for the linear time plus FWHM model, 6782 for the GP with time only, and 6784 for the GP with both time and FWHM. Similarly, for NRS2, the GP model including both time and \nFigure 2. Transmission spectra of L98-59 d obtained using transitspectroscopy (pink) and FIREFLy (blue) pipelines. The spectral light curves from the two indepedent reductions were fitted using juliet with a similar parametrization at pixel resolution and then binned to R ∼ 100. The residuals between the two reductions show a global offset between the two reductions for the NRS2 detector. \n<!-- image --> \nFWHM achieved the highest log-evidence value of 6405, compared to 6302 for the transit-only model, 6212 for the linear time model, 6219 for the linear time plus FWHM model, and 6282 for the GP with time only. \nFor the spectral light curves, we used a similar setup but fixed the mid-transit time, impact parameter, and semi-major axis ratio based on the best-fit results from the white light curves. The fitted values are T 0 (BJD TDB ) = 2460121 . 112518755 ± 7 × 10 -5 , b = 0 . 925 ± 0 . 004, and a/R ⋆ = 37 . 11 ± 0 . 55. Limbdarkening coefficients were determined using a truncated normal distribution centered on values estimated by the exotic (Grant & Wakeford 2022) package and PHOENIX models (Husser et al. 2013). Given the high impact parameter, constraining these coefficients was challenging, as discussed in Appendix B. The transmission spectrum was obtained by binning pixel-level transit depths to a resolution of R ∼ 100, resulting in 56 data points. Uncertainties in the final spectrum were calculated by determining the weighted mean and variance for \neach bin, with errors derived from the weighted contributions of the high-resolution spectra's errors and their associated variance. This spectrum can be found in Appendix 3.", '3.2. Data reduction with FIREFLy': 'We reduced the L98-59 d data using the Fast InfraRed Exoplanet Fitting Lyghtcurve ( FIREFLy ; Rustamkulov et al. 2022, 2023) reduction suite. FIREFLy starts with a customised reduction using the STScI pipeline and the uncalibrated images and includes 1/ f destriping at the group level before the ramp is fit. We applied a custom superbias step, where the STScI pipeline superbias was scaled separately for NRS1 and NRS2 to match the trace-masked count level median value of each integration. This procedure was introduced in Moran et al. (2023) to help reduce offsets between the NRS1 and NRS2 detectors. The jump-step and dark-current stages of the STScI pipeline are skipped. We then use the custom-run 2D images after the gain scale step, and \nperform customised cleaning of bad pixels, cosmic rays and hot pixels. \nWe fit the transit light curves using a quadratic limbdarkening model ( q 1 , q 2 ; Kipping 2013, along with a polynomial function of time to remove overall baseline trends. In addition, we tested detrending the lightcurves with the X and Y positions on the detector and the scaling factor used in the superbias step. We found the NRS1 detector to have a step trend which evolved over the first 0.04 days requiring a polynomial in time with orders 1, 2 , 5 and 6. The higher-order trends modelled the trends seen at the beginning of the observation well. Conversely, NRS2 only required a second-order polynomial though a trend in Y-position was found to be needed. In addition a trend for both NRS1 and NRS2 with the superbias scaling factor was found to be needed. We first fit for both q 1 and q 2 , but found q 2 to be weakly constrained and consistent with 0, so q 2 was fixed to zero throughout the rest of the analysis. a/R ⋆ , T 0 , and the impact parameter b were fit for both NRS1 and NRS2, then fixed to the weighted-average value of both detectors. The spectroscopic fits followed the same procedure. However, q 1 was found not to have a strong wavelength dependence when fit spectroscopically and was weakly constrained, so it was fixed to the weighted average value for all wavelength channels ( q 1 =0.117 ± 0.017). The lightcurves were binned to produce a final transmission spectra with a resolution near R =60 with a total of 38 points.', '3.3. Data reductions comparison': "Wefit the pixel-light curves from transitspectroscopy and FIREFLy reductions with the parametrization described in Section 3.1. The results are in Figure 2. While both spectra exhibit similar shapes, a systematic offset is observed for NRS2 detector transit depths between the two reductions, with FIREFLy 's transmission spectrum exhibiting higher transit depth values than transitspectroscopy . Specifically, there is a higher median of residuals for NRS2 (+46ppm) compared to NRS1 ( -8ppm), potentially attributable to a different background treatment for NRS2 detector. The root mean square of the residuals is found to be 56 ppm for NRS2 and 20 ppm for NRS1. Similarly, the light curve fitting setup described in Section 3.2 is applied to transitspectroscopy 's light curves (see AppendixB.6).", '4. INTERPRETATION': 'We interpret the results using forward and retrieval modelling focusing on the transmission spectrum obtained with transitspectroscopy at R ∼ 100.', '4.1. Evidence of an atmosphere from 1D forward modelling': "We used Exo-REM (Baudino et al. 2015; Charnay et al. 2018; Blain et al. 2021) to create a grid of transmission spectra for L98-59 d. Exo-REM is a 1D selfconsistent radiative-convective code for exoplanets and brown dwarfs' atmospheres. The atmosphere is modelled using 80 layers between 10 -3 and 10 7 Pa, with an Eddy Diffusion coefficient of 10 -8 cm 2 /s -1 . We simulated a 300 × solar atmosphere, including 13 absorbing species in a hydrogen-rich atmosphere, considering both equilibrium and out-of-equilibrium chemistry. The metallicity is the factor by which all the elemental abundances except H are multiplied compared to their solar abundances (Lodders 2010). For out-of-equilibrium chemistry, Exo-REM incorporates disequilibrium processes for CH 4 -CO, CO-CO 2 , and N 2 -NH 3 -HCN reactions based on Zahnle & Marley (2014). The quenching level is determined by comparing the reaction timescale to the mixing time H 2 /K zz , where H is the atmospheric scale height and K zz the Eddy diffusion coefficient. The abundance of the species is governed by the thermochemical equilibrium below the quenching level, where the temperature and pressure are high enough so that the kinetics dominates. The internal temperature was set to 70 K, controlling the position of the convective layer. We also tested a pure H 2 O, CH 4 , CO 2 and H 2 S atmosphere. \nFigure 3 shows the transmission spectrum obtained with transitspectroscopy and grid models from Exo-REM , along with corresponding χ 2 statistics. The pure H 2 S model provides the best fit to the spectrum with a chi-square difference of around 19 with the flatline. This represents a statistically significant improvement in fit, exceeding 3 σ (dof = 56). However, the observed variations in the transmission spectrum for a pure H 2 S model, which show differences of approximately 40 ppm, exceed the simple scale height prediction of ∼ 5 ppm. This is estimated using a 416 K equilibrium temperature, 34 g/cm 3 mean molecular weight. This discrepancy may arise from several factors including the modeling of the atmosphere with ExoREM, which simulates a pure H 2 S atmosphere within a hydrogendominated framework, potentially resulting in an inflated effective scale height. Additionally, the use of cross-section opacities made for hydrogen dominated atmosphere and detailed atmospheric physics in our models may explain the larger signal compared to the simplified scale height calculation. Further investigation into the support for sulfur-bearing species opacities in the atmospheric model will be conducted using a Bayesian framework in Section 4.2. Cloud-free models with 300 \nFigure 3. Transmission spectrum of L98-59 d obtained using transitspectroscopy and binned to R ∼ 100 (black points) compared to forward models from Exo-REM (coloured dashed and filled lines). The pure H 2 S model provides the best-fit according to χ 2 statistics. No offset is added between NRS1 and NRS2. \n<!-- image --> \ntimes solar abundance exhibit large CH 4 features at 3.3 µ m and CO/CO 2 features around 4.5 µ m, which are not observed in the L98-59 d transmission spectrum. Despite the pure H 2 S model being a likely non-physical atmospheric model, it is the best-fit according to χ 2 statistics. The modeled flat-line presented in Figure 3 is computed using the results of the retrieval analysis with TauREx for consistency (see Section 4.2). Using a weighted mean of spectrum points yields a chi-squared of 80.47, instead of 81.62. This results might indicate an atmosphere with the presence of sulfur-bearing species or stellar contamination.", '4.2.1. Retrieval configurations': "We conduct an atmospheric and a stellar contamination retrievals of L98-59 d's properties based on its observed spectra obtained through transitspectroscopy using TauREx 3 (Al-Refaie et al. 2021) 2 and exoretrievals (Espinoza et al. 2019). \nFor the atmospheric retrieval analysis, we used the Multinest algorithm (Feroz et al. 2009; Buchner et al. \n2014) with 100 layers in the atmosphere, spanning from 10 -3 to 10 6 Pa. The model included an evidence tolerance of 0.5 and 1500 live points. Stellar parameters were fixed (radius: 0.303 R ⊙ , T star : 3415 K, metallicity: -0.46). The planetary radius was fitted within the range of 0.79 to 2.37 R ⊕ to explore the full parameter space, account for uncertainties including limb-darkening effects, and ensure robustness in the model. The temperature-pressure profile ranged from 200 to 600 K. Clouds were modeled as grey clouds with a top pressure between 10 -3 and 10 6 Pa. The helium-to-hydrogen ratio was set to 0.17, and an offset between NRS1 and NRS2 transmission spectra was fitted uniformly between -100 and +100 ppm using the taurex-offset plugin 3 . \nEquilibrium chemistry : We used ggchem 4 (Woitke et al. 2018) for equilibrium chemistry, including molecular line lists and continuum from the ExoMol project (Tennyson et al. 2016; Chubb et al. 2021), HITEMP (Tennyson & Yurchenko 2018), and HITRAN (Rothman et al. 1987; Rothman et al. 2010). The active molecules included are : H 2 O (Polyansky et al. 2018), CO (Li et al. 2015), CO 2 (Rothman et al. \n2010), CH 4 (Yurchenko et al. 2017), NH 3 (Yurchenko et al. 2011), H 2 S (Azzam et al. 2016), HCN(Barber et al. 2014), C 2 H 2 (Chubb et al. 2020), C 2 H 4 (Mant et al. 2018), SiO(Yurchenko et al. 2021), SO 2 (Underwood et al. 2016), TiO(McKemmish et al. 2019), and VO(McKemmish et al. 2016). Collision-induced absorption (CIA) and Rayleigh scattering were also considered. We fitted for C/O and S/O ratios (0.1 to 2) and atmospheric metallicity (1 to 1000 times solar). \nFree chemistry : The active molecules included are H 2 O, CO, CO 2 , CH 4 , SiO, H 2 S, HCN, NH 3 , and SO 2 . CIA and Rayleigh opacities were included. We fitted the abundance of each molecule (log(X VMR )) between 10 -12 and 0.3 (between -12 and -0.5 in log space), ensuring the total molecular abundances never exceed one. \nFree chemistry with N 2 : The setup is similar to the free chemistry retrieval but included N 2 as an inactive gas. The N 2 /H 2 ratio was fitted between 0.001 and 2. \nFlat-line : The flat-line is a retrieval with no opacity sources, fitting for planetary radius and temperature. This was used to assess the significance of the atmospheric detection using Bayesian evidence, evaluated using the Bayes's theorem. The Bayes factor, positively designed is computed between an atmospheric model and the flat-line and then converted in σ significance using the formalism of Trotta (2008) and Benneke & Seager (2013). This retrieval is performed for each atmospheric retrieval on the transmission spectrum with the retrieved offset applied and fixed. \n4.2.2. Retrieval results : an atmosphere around L98-59d ? \nThe retrieval results using transitspectroscopy reduction are summarised in Table 1. We detected an atmosphere around L98-59 d at 5.6 σ with the free chemistry retrieval and at 2.7 σ with the equilibrium chemistry retrieval. Both retrievals report a high abundance of H 2 S: log(H 2 S)= -0 . 74 +0 . 14 -0 . 49 (free chemistry) and log(H 2 S)= -0 . 21 +0 . 07 -0 . 09 (equilibrium), with Bayesian evidences log Z = 464 . 99 and log Z = 460 . 31, respectively. The flat-line fits yield log Z = 451 . 19 and log Z = 458 . 00. \nThe free chemistry retrieval best-fit model is presented in Figure 4 and the corresponding posterior distributions are in Appendix C.7 while the equilibrium chemistry retrieval is presented in Appendix C.8. The results align with Paper II using NEMESISPY(Irwin et al. 2008; Yang et al. 2023). An offset of 37 . 6 +15 . 0 -16 . 4 ppm \nwas found between NRS2 and NRS1 with transit depths higher for NRS2. Both models suggest significant H 2 S opacity, with CO 2 (equilibrium) and SO 2 (free chemistry) as additional opacity contributers. The free chemistry model fits significantly better than the equilibrium model by 3.8 σ suggesting that L98-59 d's atmosphere is not at equilibrium. The SO 2 VMR in the free retrieval shows a bi-modal distribution, with a constraint either above -1 . 0 or an upper limit below -1 . 5. However, we can put upper limits on every other molecules of the fit. In particular, H 2 O's VMR is constrained below 10 -5 suggesting a water-poor atmosphere for this planet that has density compatible with a water-rich composition (Luque & Pall'e 2022). Excluding both H 2 S and SO 2 from the free chemistry retrieval indicates that the model with these molecules is favored at 4.5 σ using model comparison with the Bayes factor. \nAdding N 2 as an inactive gas in the free chemistry retrieval did not significantly alter results, favoring the atmospheric model over a flat-line at 4.5 σ . The only strongly constrained parameters are the planetary radius at 10 bars, the log VMR of H 2 S, toward the edge of the prior (0.3), and the top pressure of the grey clouds. The log VMR of SO 2 is tentatively constrained. For this reason, we run a fit with H 2 S and SO 2 as the only active molecular opacity sources, increasing the prior range to 1, and fit for their molecular abundances while fixing the temperature to the equilibrium temperature at 416K. This atmospheric model is preferred at 5.3 σ to a flat-line. \n4.2.3. Stellar contamination : is there an atmosphere or stellar inhomogeneities ? \nOne of the early results from observations of small planets with JWST is how important it is to consider heterogeneities on the surface of the host star when looking for atmospheric signals in transmission. Multiple small planet observations have shown contamination from the presence of hot and cold spots on the photosphere of the host star during transit, which imprint false slopes and can potentially even inject false molecular features into the transmission spectrum (see e.g. Moran et al. 2023; Lim et al. 2023). This is especially important for planets orbiting small M stars like L9859, which are known to show a high level of magnetic activity and associated surface features (Rackham et al. 2018). \nTo test if the spectrum could be explained by stellar contamination caused by inhomogeneities (hot and/or cold spots) on the stellar surface, we carry out retrievals using exoretrievals (Espinoza et al. 2019). Based on the framework presented in Rackham et al. (2018), this allows us to model the spectrum created by occulted \nFigure 4. Top panel: Transmission spectrum of L98-59 d obtained using transitspectroscopy and binned to R ∼ 100 (black points) compared to a free chemistry retrieval model from TauREx (purple) and NEMESISPY (dashed blue) from Banerjee et al. 2024, hereafter Paper II. Middle : Opacity contributions from the best-fit model. The model suggests a contribution from H 2 S (purple) and SO 2 (green) opacities. Bottom : Transmission spectrum compared to stellar (yellow) and atmospheric (purple) retrievals models. \n<!-- image --> \nTable 1. Summary of Atmospheric retrieval results using TauREx on L98-59d transmission spectrum obtained with transitspectroscopy 's reduction at R ∼ 100. \nand/or unocculted spots on the stellar surface during the transit, by replacing a portion of the surface with a stellar model of another temperature. Since we do not see any evidence of spot crossing events in the light curves, we only consider unocculted spots in our retrieval. We consider the cases of only hot spots, only cold spots, or both being present on the stellar surface, and allow these spot temperatures to vary over a large parameter space, from 1800 K to 3250 K and from 3550 K to 5000 K, respectively, for a photospheric temperature of 3415 ± 135. We also place a wide prior on the spot covering fraction, allowing the spots to together cover up to the entirety of the unocculted photosphere. We also tested using both default PHOENIX models and BT-SETTL models considering the relatively low photospheric temperature of L98-59, but saw no appreciable difference in the results. Even with this very large explored parameter space, none of the stellar contamination models are able to explain the features seen in the spectrum, especially those at the longest wavelengths. Figure 4 bottom panel shows the best-fit model from the stellar contamination retrieval results. The corresponding posterior distributions are in Appendix C.10. We also show the best-fit model from an atmospheric retrieval analysis with TauREx . In these fits, no offset between NRS1 and NRS2 is applied. The H 2 S/SO 2 rich atmosphere is favored compared to the stellar contamination suggesting the large feature between 3.3 to 4.8 µ m, if confirmed by follow-up observations is originated from the atmosphere.", '4.2.4. Comparison between data reductions': "To confirm the atmospheric detection, we applied the free chemistry retrieval setup to the FIREFLy transmission spectrum at R ∼ 100 and to the spectrum obtained at R ∼ 60 from an independent fitting method (see Figure 2). The results of these fits are shown in Appendix Figure C.11. The atmospheric detection was less significant with FIREFLy , yielding 3.0 σ for R ∼ 100 and 2.6 σ for R ∼ 60 spectra. \nThe discrepancy at R ∼ 100 primarily arises from differences at longer wavelengths in the NRS2 detectors. transitspectroscopy data showed a decrease in transit depth between 4.5 and 5 µ m, fitted with H 2 S opacity (see the middle panel in Figure 4) while FIREFLy 's transmission spectrum remains flat across the entire wavelength range of the NRS2 detector. However, sulfurbearing species consistently appeared as the primary source of opacity in all retrievals. The NRS1 and NRS2 offset varied by reduction method: +37 ppm for transitspectroscopy and -14 ppm for FIREFLy , both consistent with zero at 1 σ . Removing this offset is un- \nly to significantly change the FIREFLy result given the small offset within 1 σ uncertainty. \nThe R ∼ 60 spectrum from FIREFLy with an independent light curve fitting showed a tentative atmospheric detection (2.6 σ ). This finding raises concerns, suggesting that the detection made with other binning and light curve fitting may be attributed to random noise. Specifically, the discrepancy observed at shorter wavelengths, between 3 and 3.3 µ m, between the two reductions could account for this result (see Figure B.6). The variability observed in FIREFLy 's transmission spectrum in this wavelength range resulted in a seemingly flatter spectrum and a statistically non-detection, whereas transitspectroscopy 's data indicated a pronounced decrease in transit depths, well-fitted by H 2 S's opacity (see the middle panel in Figure 4). We note that employing the same light curve fitting and binning, the decrease is observed in FIREFLy 's reduction as well (see Figure 2). \nIn AppendixC, we assess the impact of introducing an offset between NRS1 and NRS2 transit depths on atmospheric detection for L98-59 d (see Appendix C.5) and evaluate the influence of spectral resolution (see Appendix C.6).", '5. CONCLUSIONS': "We presented one transit observation of the SuperEarth L98-59 d with the JWST NIRSpec G395H mode. Our study highlights several challenges encountered in the data reduction process when analyzing transmission spectra of small planets. First, the limb-darkening coefficients pose a challenge for grazing planets due to their impact on the light curve fitting process. Even though our two data reductions agree when using the same limb-darkening coefficients, remaining discrepancies at specific wavelength led to different interpretation in the retrieval analysis. \nFrom this single transit, our analysis suggests no stellar contamination but hints at potential atmospheric detection, though significance varies with data reduction methods. In the best case scenario, we detected an atmosphere with a 5.6 σ significance. An independent analysis led to a 2.6 σ tentative detection. Atmospheric retrieval models suggest the presence of sulfur-bearing species with hydrogen and helium as background gases, although the inferred high abundance levels are not yet well understood. The discovery of sulfur dioxide in the atmospheres of hot Jupiters (Rustamkulov et al. 2023; Alderson et al. 2023; Powell et al. 2024) and Neptunelike planets(Dyrek et al. 2023; Benneke et al. 2024; Holmberg & Madhusudhan 2024) are key-result in early JWST analysis being evidence of photo-chemistry. Sul- \nr species and clouds are expected in giant planets and brown-dwarf atmosphere (Morley et al. 2012; Tsai et al. 2023; Polman et al. 2023). Crossfield (2023) showed that the ability to detect SO 2 in exoplanet atmospheres provides a crucial test for different planet formation models, revealing that sulfur's volatility and abundance can distinguish between planetesimal and pebble-accretion scenarios. Photochemistry could also produce H 2 S and SO 2 in terrestrial atmosphere as it was predicted by Hu et al. (2013). The presence of sulfur-bearing species in rocky planets could also be explained by out-gassing, volcanism, interaction between the atmosphere and the rocky surface. The recent study of Janssen et al. (2023) investigated sulfur's presence and detectability in rocky exoplanet atmospheres using thermochemical equilibrium models at the crust-atmosphere interface, considering surface temperatures of 500-5000 K and pressures of 1-100 bar, with various element abundances based on common rock compositions. They showed that at temperatures between 1000 and 2000 K, gaseous sulfur concentrations can reach up to 25%. They conclude that the most abundant sulfur molecules are SO, SO 2 , H 2 S, and S 2 with potentially detectable features in transmission spectra around 4 µ m, 7-8 µ m, and beyond 15 µ m. \nThe difference between the data reductions and the uncertainties on the data points are magnified when dealing with observations from a single visit. (May et al. 2023) highlighted for GJ 1132 b the importance of multiple visits when claiming atmospheric detection for rocky planets. Our transmission spectrum might have random noise fluctuations that suggested the detection of an atmosphere which emphasizes the need for multiple transits observations. This result will have to be compared to the second G395H visit from Cycle 2 GO program 4098 (PI: Benneke) but also to the NIRISS SOSS transit from Cycle 1 GTO program 1201 (PI: Lafreni'ere). \nOnly hints of atmospheric detections have been found so far for rocky planets with radii below 1.6 R ⊕ around M-dwarfs. Notably, planets from the GO program 1981 (PIs: Stevenson and Lustig-Yaeger), in order of increasing radius, include GJ 341 b with a radius of 0.92 R ⊕ (Kirk et al. 2024), LHS 475 b with 0.99 R ⊕ (LustigYaeger et al. 2023), GJ 1132 b with 1.1 R ⊕ (May et al. 2023), and GJ 486 b with 1.3 R ⊕ (Moran et al. 2023), yet they have not provided compelling evidence of an atmosphere. However, similar JWST observations on planets with radii above 1.6R ⊕ , such as K2-18 b (2.61 R ⊕ ) (Madhusudhan et al. 2023) and TOI-270d (2.31R ⊕ ), 55 Cancri e (1.97 R ⊕ )(Hu et al. 2024) have provided strong molecular detections. If confirmed, the detection of sulfur-bearing species in an hydrogen-dominated atmosphere around L98-59 d, a planet with a radius of 1.58 Earth radii, would be a significant result, as it lies right at the cutoff predicted by Rogers (2015); Rogers et al. (2021) for planets to have retained their primary hydrogen-helium atmosphere. Planets with a radius above 1.6R ⊕ have a density too low to be compatible with a silicate and iron-only core composition. A second study Paper II, from the NIRSpec GTO collaboration will explore in more details the possible atmospheric scenarios for L98-59 d. \nN.H.A. acknowledges support by the National Science Foundation Graduate Research Fellowship under Grant No. DGE1746891. \nThe JWST data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST) at the Space Telescope Science Institute. The specific observations analyzed can be accessed via 10.17909/nrxs-cx46. \nFacilities: JWST NIRSPEC G395H", 'REFERENCES': "Al-Refaie, A. F., Changeat, Q., Waldmann, I. P., & Tinetti, G. 2021, The Astrophysical Journal, 917, 37, \ndoi: 10.3847/1538-4357/ac0252 Alderson, L., Wakeford, H. R., Alam, M. K., et al. 2023, Nature, 614, 664-669, doi: 10.1038/s41586-022-05591-3 Ambikasaran, S., Foreman-Mackey, D., Greengard, L., Hogg, D. W., & O'Neil, M. 2015, IEEE Transactions on Pattern Analysis and Machine Intelligence, 38, 252, doi: 10.1109/TPAMI.2015.2448083 \nAzzam, A. A. A., Tennyson, J., Yurchenko, S. N., & Naumenko, O. V. 2016, Monthly Notices of the Royal Astronomical Society, 460, 4063-4074, \ndoi: 10.1093/mnras/stw1133 \nBarber, R. J., Strange, J. K., Hill, C., et al. 2014, Monthly Notices of the Royal Astronomical Society, 437, 1828, \ndoi: 10.1093/mnras/stt2011 Baudino, J.-L., B'ezard, B., Boccaletti, A., et al. 2015, Astronomy & Astrophysics, 582, A83, doi: 10.1051/0004-6361/201526332 Benneke, B., & Seager, S. 2013, The Astrophysical Journal, 778, 153, doi: 10.1088/0004-637x/778/2/153 Benneke, B., Roy, P.-A., Coulombe, L.-P., et al. 2024, JWST Reveals CH 4 , CO 2 , and H 2 O in a Metal-rich Miscible Atmosphere on a Two-Earth-Radius Exoplanet. https://arxiv.org/abs/2403.03325 \nUnderwood, D. S., Tennyson, J., Yurchenko, S. N., et al. \n2016, Monthly Notices of the Royal Astronomical \nSociety, 459, 3890-3899, doi: 10.1093/mnras/stw849 \nWakeford, H. R., Lewis, N. K., Fowler, J., et al. 2019, The Astronomical Journal, 157, 11, \ndoi: 10.3847/1538-3881/aaf04d \n- Woitke, P., Helling, C., Hunter, G. H., et al. 2018, Astronomy & Astrophysics, 614, A1, \ndoi: 10.1051/0004-6361/201732193 \n- Yang, J., Irwin, P. G. J., & Barstow, J. K. 2023, Testing 2D temperature models in Bayesian retrievals of atmospheric properties from hot Jupiter phase curves. https://arxiv.org/abs/2305.10249 \nYurchenko, S. N., Amundsen, D. S., Tennyson, J., & Waldmann, I. P. 2017, Astronomy & Astrophysics, 605, A95, doi: 10.1051/0004-6361/201731026 \n- Yurchenko, S. N., Barber, R. J., & Tennyson, J. 2011, Monthly Notices of the Royal Astronomical Society, 413, 1828-1834, doi: 10.1111/j.1365-2966.2011.18261.x \nYurchenko, S. N., Tennyson, J., Syme, A.-M., et al. 2021, Monthly Notices of the Royal Astronomical Society, 510, 903-919, doi: 10.1093/mnras/stab3267 \nZahnle, K. J., & Marley, M. S. 2014, The Astrophysical Journal, 797, 41, doi: 10.1088/0004-637x/797/1/41 Research in Astronomy and Astrophysics, 23, 025011, \n- Zhou, L., Ma, B., Wang, Y.-H., & Zhu, Y.-N. 2023, doi: 10.1088/1674-4527/acaceb \nZieba, S., Zilinskas, M., Kreidberg, L., et al. 2022, Astronomy & Astrophysics, 664, A79, doi: 10.1051/0004-6361/202142912 \nZieba, S., Kreidberg, L., Ducrot, E., et al. 2023, Nature, 620, 746-749, doi: 10.1038/s41586-023-06232-z", 'A. DATA REDUCTION : WHITE LIGHT CURVE BEST-FIT RESULTS': 'Table 2. L98-59 d White Light Curve fitting Results for the transitspectroscopy reduction using juliet . \nDescription of parameters:Mid-transit time (BJD TDB ). Ratio of the semi-major axis to the stellar radius. Planet to star radius ratio. \nFirst limb-darkening coefficient of the squareroot law with the Kipping (2013) parametrization. Second limb-darkening coefficient. Out-of-transit flux offset. White noise jitter parameter. GP amplitude. GP length scales.', 'B. DATA REDUCTION: TREATMENT OF THE LIMB-DARKENING COEFFICIENTS': "L98-59 d had a high impact parameter b > 0 . 9, which makes it challenging to constrain the limb-darkening coefficient in the light curve fitting. The choice of the limb-darkening coefficient treatment impact the extracted transit spectrum particularly the absolute transit depth value. First, we ran tests: one involving fitting the limb-darkening coefficients uniformly between 0 and 1 using a quadratic law, and another using a square-root law. For this test, we fit the light curves from the transitspectroscopy 's reduction at pixel resolution and bin the spectrum after. Results of this test are in Appendix Figure B.5. We note that there is a 400ppm offset between the two resulting transit spectra. The fit involving the quadratic law has a higher mean transit depth and lower precision. The shape of the spectrum is the same but the transit depth values and precision are sensitive to the choice of the limb-darkening treatment. \nAdditionally, we perform a fit with fixed limb-darkening coefficients ( q 1 = 0 . 13 and q 2 = 0 . 0), derived from the combined best-fit of NRS1 and NRS2 white light curves, with q 2 = 0 . 0 being unconstrained. This parametrization is applied to pixel-level transitspectroscopy light curves, while FIREFLy light curves are binned to a R ∼ 60 and fitted with these fixed coefficients and an independent detrending linear model. This fitting is described in Section 3.2. The results of this additionnal test is in Appendix B.6. \nFigure B.5. Top: Transmission spectra of L98-59 d obtained using the reductions from transitspectroscopy and binned to a resolution of R ∼ 60. The two spectra are derived by fitting the limb-darkening coefficients uniformly between 0 and 1 using a quadratic law (orange) and employing a square-root law (blue). The light orange points are obtained by applying an offset corresponding to the median differences between the two spectra. Bottom: Residuals between the two transmission spectra after subtraction of the median transit depth. Each detector is treated independently. The shape of the two spectra is similar but there is a 400ppm offset between the two reductions. \n<!-- image -->", 'C. ADDITIONAL RETRIEVAL ANALYSIS AND FIGURES': "C.1. Free chemistry retrieval analysis \n- C.2. Equilibrium chemistry retrieval analysis\n- C.3. Stellar contamination retrieval analysis \nC.4. Atmospheric retrieval analysis on the FIREFLy data reduction \nC.5. Does the offset between NRS1 and NRS2 impact the atmospheric detection ? \nTo assess the influence of introducing an offset between the NRS1 and NRS2 transit depth values, we conducted an atmospheric retrieval fit on the transmission spectrum of L98-59 d at R ∼ 100 obtained from transitspectroscopy . The best-fit model is shown in Figure 4 bottom panel along with the stellar retrieval model. The atmospheric model is preferred at 3.4 σ compared to a flat-line with SO 2 and H 2 S as the main opacities. Moreover, incorporating these two molecules is favored at a significance of 3.2 σ compared to a model without them. Introducing the offset led to higher NRS2 transit depths in transitspectroscopy 's data, thereby reinforcing the atmospheric detection, particularly of SO 2 at 4.2 µ m. However, removing the offset did not impact the detection of an atmosphere in transitspectroscopy 's reduction. \nFigure B.6. Transmission spectra of L98-59 d obtained using transitspectroscopy (pink) and FIREFLy (blue) pipelines. The spectral light curves from the two independent reductions were fitted independently. transitspectroscopy are fitted at pixel-level resolution using Gaussian Processes to model the noise and then binned to the native resolution of the FIREFLy 's reduction to approximately a R ∼ 60. FIREFLy light curves are binned to R ∼ 60 and then fitted using a linear de-trending model. For both fitting the coefficients of the limb-darkening coefficients are fixed to q 1 = 0 . 13 and q 2 = 0 . 0. \n<!-- image -->", 'C.6. Does the spectral resolution impact the atmospheric detection ?': "To examine the impact of spectral resolution on our results, we applied the atmospheric free chemistry retrieval setup to transitspectroscopy 's data binned at different resolutions, following the fit at pixel-level resolution. We then evaluated the atmospheric detection by comparing Bayesian evidences with a flat-line, representing a fit without opacity sources. We did not fit for an offset between NRS1 and NRS2 in this analysis. The choice of spectral resolution did not significantly affect the atmospheric detection obtained with transitspectroscopy 's reduction. \nFigure C.7. Posterior distributions from the free retrieval analysis. Not all parameters in the fit are plotted for clarity. \n<!-- image --> \nTable 3. L98-59 d extracted transit depths from transitspectroscopy at R ∼ 100 \nFigure C.8. Top panel: Transmission spectrum of L98-59 d obtained using transitspectroscopy and binned to R ∼ 100 (black points) compared to an equilibrium chemistry retrieval model from TauREx (light green). Bottom : Opacity contributions from the best-fit model. The model suggests a contribution from H 2 S (purple) and CO 2 (red) opacities. \n<!-- image --> \nFigure C.9. Posterior distributions from the equilibrium chemistry retrieval analysis. Not all parameters in the fit are plotted for clarity. \n<!-- image --> \nFigure C.10. Posterior distributions from the stellar retrieval analysis. \n<!-- image --> \nFigure C.11. Top Left:Transmission spectrum of L98-59 d obtained using transitspectroscopy (pink data points) and FIREFLy (blue data points) and binned to R ∼ 100 compared to atmospheric retrievals models with 1 σ uncertainty. An Offset is fitted between NRS1 and NRS2. Top Right : Posterior distributions from the atmospheric retrieval analyses. We do not represent all parameters for clarity. The values are printed for the FIREFLy atmospheric fit. Bottom Left :Transmission spectrum of L98-59 d obtained using FIREFLy (blue data points) binned to R ∼ 60 compared to an atmospheric retrieval models with 1 σ uncertainty. An Offset is fitted between NRS1 and NRS2. Bottom Right : Posterior distributions from the atmospheric retrieval analysis. \n<!-- image -->"}